From 9fa229f1d16f523cbfb457e9e0a2f2e4ba76206e Mon Sep 17 00:00:00 2001 From: David Doret Date: Wed, 23 Aug 2023 13:48:25 +0200 Subject: [PATCH 01/21] reviewing tests + complete method signatures for free-variable declarations --- punctilious/core.py | 63 ++++++++++++++++++----------- tests/test_variable_substitution.py | 38 +++++++++-------- 2 files changed, 61 insertions(+), 40 deletions(-) diff --git a/punctilious/core.py b/punctilious/core.py index fa01e2ea..245f26ef 100644 --- a/punctilious/core.py +++ b/punctilious/core.py @@ -2522,12 +2522,17 @@ class Status(repm.ValueName): scope_initialization_status = Status('scope_initialization_status') closed_scope_status = Status('closed_scope_status') - def __init__(self, nameset=None, universe_of_discourse=None, status=None, scope=None, - echo=None): + def __init__(self, u: UniverseOfDiscourse, status: (None, FreeVariable.Status) = None, + scope: (None, Formula, typing.FrozenSet[Formula]) = None, + symbol: (None, str, StyledText) = None, index: (None, int) = None, + auto_index: (None, bool) = None, dashed_name: (None, str, StyledText) = None, + acronym: (None, str, StyledText) = None, abridged_name: (None, str, StyledText) = None, + name: (None, str, StyledText) = None, explicit_name: (None, str, StyledText) = None, + nameset: (None, str, NameSet) = None, echo: (None, bool) = None) -> None: echo = prioritize_value(echo, configuration.echo_free_variable_declaration, configuration.echo_default, False) - status = FreeVariable.scope_initialization_status if status is None else status - scope = frozenset() if scope is None else scope + status = prioritize_value(status, FreeVariable.scope_initialization_status) + scope = prioritize_value(scope, frozenset()) scope = {scope} if isinstance(scope, Formula) else scope verify(isinstance(scope, frozenset), 'The scope of a FreeVariable must be of python type frozenset.') @@ -2535,19 +2540,20 @@ def __init__(self, nameset=None, universe_of_discourse=None, status=None, scope= 'The status of a FreeVariable must be of the FreeVariable.Status type.') self._status = status self._scope = scope - assert isinstance(universe_of_discourse, UniverseOfDiscourse) - if nameset is None: + assert isinstance(u, UniverseOfDiscourse) + if symbol is None: symbol = configuration.default_free_variable_symbol - index = universe_of_discourse.index_symbol(symbol=symbol) - nameset = NameSet(symbol=symbol, index=index) - if isinstance(nameset, str): + index = u.index_symbol(symbol=symbol) + if isinstance(symbol, str): # If symbol was passed as a string, # assume the base was passed without index. # TODO: Analyse the string if it ends with index in subscript characters. - symbol = StyledText(plaintext=nameset, text_style=text_styles.serif_bold) - index = universe_of_discourse.index_symbol(symbol=symbol) - nameset = NameSet(symbol=symbol, index=index) - super().__init__(nameset=nameset, universe_of_discourse=universe_of_discourse, echo=False) + symbol = StyledText(plaintext=symbol, text_style=text_styles.serif_bold) + if index is None and auto_index: + index = u.index_symbol(symbol=symbol) + super().__init__(universe_of_discourse=u, symbol=symbol, index=index, auto_index=auto_index, + dashed_name=dashed_name, acronym=acronym, abridged_name=abridged_name, name=name, + explicit_name=explicit_name, nameset=nameset, echo=False) # self.universe_of_discourse.cross_reference_variable(x=self) super()._declare_class_membership(declarative_class_list.free_variable) if echo: @@ -7066,7 +7072,7 @@ def variable_substitution(self) -> VariableSubstitutionDeclaration: return self._variable_substitution @property - def vs(self) -> InferenceRuleDeclaration: + def vs(self) -> VariableSubstitutionDeclaration: """An inference-rule: P โŠข P' Abridged property: u.i.vs @@ -8478,7 +8484,7 @@ def variable_substitution(self) -> VariableSubstitutionInclusion: return self._variable_substitution @property - def vs(self) -> InferenceRuleInclusion: + def vs(self) -> VariableSubstitutionInclusion: return self.variable_substitution @@ -8629,18 +8635,18 @@ def cross_reference_simple_objct(self, o: SimpleObjct): if o not in self.simple_objcts: self.simple_objcts[o.nameset] = o - def cross_reference_symbolic_objct(self, o: SymbolicObject): + def cross_reference_symbolic_objct(self, o: TheoreticalObject): """Cross-references a symbolic-objct in this universe-of-discourse. :param o: a symbolic-objct. """ - verify(is_in_class(o, classes.symbolic_objct), + verify(is_in_class(o=o, c=classes.symbolic_objct), 'Cross-referencing a symbolic-objct in a universe-of-discourse requires ' 'an object of type SymbolicObjct.', o=o, slf=self) duplicate = self.symbolic_objcts.get(o.nameset) - verify(duplicate is None, - 'A symbolic-object already exists in the current universe-of-discourse with a ' - 'duplicate (symbol, index) pair.', o=o, duplicate=duplicate, slf=self) + verify(severity=verification_severities.warning, assertion=duplicate is None, + msg='A symbolic-object already exists in the current universe-of-discourse with a ' + 'duplicate (symbol, index) pair.', o=o, duplicate=duplicate, slf=self) self.symbolic_objcts[o.nameset] = o def cross_reference_theory(self, t: TheoryElaborationSequence): @@ -8670,7 +8676,10 @@ def declare_formula(self, relation: Relation, *parameters, nameset: (None, str, nameset=nameset, lock_variable_scope=lock_variable_scope, echo=echo) return phi - def declare_free_variable(self, symbol=None, echo=None): + def declare_free_variable(self, symbol: (None, str, StyledText) = None, + dashed_name: (None, str, StyledText) = None, acronym: (None, str, StyledText) = None, + abridged_name: (None, str, StyledText) = None, name: (None, str, StyledText) = None, + explicit_name: (None, str, StyledText) = None, echo: (None, bool) = None): """Declare a free-variable in this universe-of-discourse. A shortcut function for FreeVariable(universe_of_discourse=u, ...) @@ -8867,7 +8876,11 @@ def take_note(self, t: TheoryElaborationSequence, content: str, # @FreeVariableContext() @contextlib.contextmanager - def v(self, symbol=None, echo=None): + def v(self, symbol: (None, str, StyledText) = None, index: (None, int) = None, + auto_index: (None, bool) = None, dashed_name: (None, str, StyledText) = None, + acronym: (None, str, StyledText) = None, abridged_name: (None, str, StyledText) = None, + name: (None, str, StyledText) = None, explicit_name: (None, str, StyledText) = None, + echo: (None, bool) = None): """Declare a free-variable in this universe-of-discourse. This method is expected to be as in a with statement, @@ -8881,8 +8894,10 @@ def v(self, symbol=None, echo=None): use declare_free_variable() instead. """ # return self.declare_free_variable(symbol=symbol) - x = FreeVariable(universe_of_discourse=self, nameset=symbol, - status=FreeVariable.scope_initialization_status, echo=echo) + status = FreeVariable.scope_initialization_status + x = FreeVariable(u=self, status=status, symbol=symbol, index=index, auto_index=auto_index, + dashed_name=dashed_name, acronym=acronym, abridged_name=abridged_name, name=name, + explicit_name=explicit_name, echo=echo) yield x x.lock_scope() diff --git a/tests/test_variable_substitution.py b/tests/test_variable_substitution.py index 40062d70..073b64e0 100644 --- a/tests/test_variable_substitution.py +++ b/tests/test_variable_substitution.py @@ -10,15 +10,17 @@ def test_variable_substitution_without_variable(self): u = pu.UniverseOfDiscourse() o1 = u.o.declare() o2 = u.o.declare() - r1 = u.r.declare(1, signal_proposition=True) - r2 = u.r.declare(2, signal_proposition=True) + r1 = u.r.declare(arity=1, signal_proposition=True) + r2 = u.r.declare(arity=2, signal_proposition=True) t = u.t() - a = u.declare_axiom(random_data.random_sentence()) - ap = t.include_axiom(a) - p_formula = u.f(r1, u.f(r2, o1, o2)) - p_statement = t.i.axiom_interpretation.infer_statement(ap, p_formula, echo=True) + a = u.declare_axiom(natural_language=random_data.random_sentence()) + ap = t.include_axiom(a=a) + p_formula = r1(r2(o1, o2)) + p_statement = t.i.axiom_interpretation.infer_statement(axiom=ap, formula=p_formula, + echo=True) # y_sequence = tuple() - p_prime = t.i.vs.infer_statement(p_statement, echo=True) + p_prime = t.i.vs.infer_statement(p=p_statement, phi=(), echo=True) + self.assertTrue(p_prime.is_formula_syntactically_equivalent_to(p_statement)) self.assertEqual('๐‘Ÿโ‚(๐‘Ÿโ‚‚(๐‘œโ‚, ๐‘œโ‚‚))', p_prime.rep_formula(encoding=pu.encodings.unicode)) def test_variable_substitution_with_free_variables(self): @@ -30,14 +32,18 @@ def test_variable_substitution_with_free_variables(self): o4 = u.o.declare() o5 = u.o.declare() o6 = u.o.declare() - r1 = u.r.declare(1, signal_proposition=True) - r2 = u.r.declare(2, signal_proposition=True) - t = u.t('test_variable_substitution_with_free_variables') + f = u.r.declare(arity=1, signal_proposition=True, symbol='f', auto_index=False) + g = u.r.declare(arity=2, signal_proposition=True, symbol='g', auto_index=False) + t = u.t() a = u.declare_axiom(random_data.random_sentence()) ap = t.include_axiom(a) - with u.v() as x, u.v() as y, u.v() as z: - p_formula = u.f(r1, u.f(r2, u.f(r2, z, u.f(r2, u.f(r1, x), y)), u.f(r2, x, y))) - p_statement = t.i.axiom_interpretation.infer_statement(ap, p_formula, echo=True) - y_sequence = (o4, o6, o5) # sequence: (z, x, y) - p_prime = t.i.vs.infer_statement(p_statement, *y_sequence, echo=True) - self.assertEqual('โ—†โ‚(โ—†โ‚‚(โ—†โ‚‚(โ„ดโ‚„, โ—†โ‚‚(โ—†โ‚(โ„ดโ‚†), โ„ดโ‚…)), โ—†โ‚‚(โ„ดโ‚†, โ„ดโ‚…)))', p_prime.rep_formula()) + with u.v(symbol='x', auto_index=False) as x, u.v(symbol='y', auto_index=False) as y, u.v( + symbol='z', auto_index=False) as z: + p_statement = t.i.axiom_interpretation.infer_statement(axiom=ap, + formula=f(g(g(z, g(f(x), y)), g(x, y))), echo=True) + self.assertEqual('๐‘“(๐‘”(๐‘”(๐ณ, ๐‘”(๐‘“(๐ฑ), ๐ฒ)), ๐‘”(๐ฑ, ๐ฒ)))', + p_statement.rep_formula(encoding=pu.encodings.unicode)) + p_prime = t.i.vs.infer_statement(p=p_statement, phi=(o4, o6, o5), echo=True) + self.assertEqual('๐‘“(๐‘”(๐‘”(๐‘œโ‚„, ๐‘”(๐‘“(๐‘œโ‚†), ๐‘œโ‚…)), ๐‘”(๐‘œโ‚†, ๐‘œโ‚…)))', + p_prime.rep_formula(encoding=pu.encodings.unicode)) + p_prime.is_formula_syntactically_equivalent_to(o2=f(g(g(o4, g(f(o6), o5)), g(o6, o5)))) From 20cda451948c45cb24df0e4fe8fbf743358fad7b Mon Sep 17 00:00:00 2001 From: David Doret Date: Wed, 23 Aug 2023 23:14:58 +0200 Subject: [PATCH 02/21] code cleanup --- punctilious/__init__.py | 15 ++- punctilious/core.py | 162 -------------------------------- tests/test_title.py | 37 -------- tests/test_two_columns_proof.py | 11 --- 4 files changed, 7 insertions(+), 218 deletions(-) delete mode 100644 tests/test_title.py delete mode 100644 tests/test_two_columns_proof.py diff --git a/punctilious/__init__.py b/punctilious/__init__.py index 848c4d15..3fb40e8d 100644 --- a/punctilious/__init__.py +++ b/punctilious/__init__.py @@ -6,14 +6,13 @@ from repm import monospace, prnt, serif_bold from core import Article, AxiomDeclaration, AxiomInclusion, ComposableBlockSequence, classes, \ configuration, consistency_values, DashedName, create_universe_of_discourse, \ - DefinitionInclusion, DirectDefinitionInference, Encoding, encodings, \ - FailedVerificationException, Formula, FreeVariable, Header, InconsistencyWarning, \ - interpret_formula, interpret_statement_formula, is_in_class, InferredStatement, NoteInclusion, \ - Paragraph, prioritize_value, QuasiQuotation, Relation, rep_two_columns_proof_item, ScriptNormal, \ - SansSerifBold, SansSerifNormal, SerifNormal, SimpleObjct, Statement, Subscript, subscriptify, \ - paragraph_headers, ComposableText, NameSet, SymbolicObject, TextStyle, text_styles, \ - TheoreticalObject, TheoryElaborationSequence, TheoryPackage, paragraph_headers, \ - UniverseOfDiscourse + DefinitionInclusion, Encoding, encodings, FailedVerificationException, Formula, FreeVariable, \ + Header, InconsistencyWarning, interpret_formula, interpret_statement_formula, is_in_class, \ + InferredStatement, NoteInclusion, Paragraph, prioritize_value, QuasiQuotation, Relation, \ + rep_two_columns_proof_item, ScriptNormal, SansSerifBold, SansSerifNormal, SerifNormal, \ + SimpleObjct, Statement, Subscript, subscriptify, paragraph_headers, ComposableText, NameSet, \ + SymbolicObject, TextStyle, text_styles, TheoreticalObject, TheoryElaborationSequence, \ + TheoryPackage, paragraph_headers, UniverseOfDiscourse # from foundation_system_1 import foundation_system_1, ft, u diff --git a/punctilious/core.py b/punctilious/core.py index 245f26ef..22d67430 100644 --- a/punctilious/core.py +++ b/punctilious/core.py @@ -1623,111 +1623,6 @@ def to_dict(self): 'explicit_name': self._explicit_name} -class TitleOBSOLETE: - """A title to introduce some symbolic-objcts in reports. - - TODO: QUESTION: rename Title to Name, Category to Nature, Reference to Abridged and - Unabridged Name. - - - Features: - - long-names have a short (reference) and long (long-name) version. - - long-names do not have an index. - - long-names comprise a category that must be consistent with - the declarative class of the symbolic-objct. - """ - - def __init__(self, nameset: (None, NameSet) = None, ref: (None, str, ComposableText) = None, - paragrapha_header: (None, ParagraphHeader) = None, - subtitle: (None, str, ComposableText) = None, abr: (None, str, ComposableText) = None): - if isinstance(ref, str): - ref = StyledText(s=ref, text_style=text_styles.sans_serif_bold) - self._ref = ref - if isinstance(abr, str): - abr = StyledText(s=abr, text_style=text_styles.sans_serif_bold) - self._abr = abr - self._paragraph_header = paragraph_headers.uncategorized if paragrapha_header is None else paragrapha_header - if isinstance(subtitle, str): - subtitle = StyledText(s=subtitle, text_style=text_styles.sans_serif_normal) - self._nameset = nameset - self._subtitle = subtitle - self._styled_title = None - self._styled_ref = None - - def __eq__(self, other): - return hash(self) == hash(other) - - def __hash__(self): - """Note that the title attribute is not hashed, - because title is considered purely decorative. - """ - return hash((self.cat, self.ref)) - - def __repr__(self) -> str: - return self.rep(encoding=encodings.plaintext) - - def __str__(self) -> str: - return self.rep(encoding=encodings.plaintext) - - @property - def cat(self) -> ParagraphHeader: - """The category of this statement.""" - return self._paragraph_header - - @property - def subtitle(self) -> ComposableText: - """A conditional complement to the automatically structured title.""" - return self._subtitle - - @property - def ref(self) -> ComposableText: - """Unabridged name.""" - return self._ref - - @property - def abr(self) -> ComposableText: - """Abridged name.""" - return self._abr - - @property - def nameset(self) -> NameSet: - return self._nameset - - @nameset.setter - def nameset(self, nameset: NameSet): - self._nameset = nameset - - def rep(self, encoding: (None, Encoding) = None, cap: bool = False) -> str: - """Return the default representation for this long-name. - - :param cap: Whether the representation must be capitalized (default: False). - :return: str - """ - return self.rep_ref(encoding=encoding, cap=cap) - - def rep_title(self, encoding: (None, Encoding) = None, cap: (None, bool) = None) -> str: - """ - - :param cap: - :return: - """ - return f'{StyledText(s=self.cat.natural_name, text_style=text_styles.sans_serif_bold).rep(encoding=encoding, cap=cap)}' \ - f'{"" if self.ref is None else " " + self.ref.rep(encoding=encoding)}' \ - f'{"" if self.subtitle is None else " - " + self.subtitle.rep(encoding=encoding)}' - - def rep_ref(self, encoding: (None, Encoding) = None, cap: (None, bool) = None) -> str: - return StyledText(s=self._paragraph_header.abridged_name, - text_style=text_styles.sans_serif_bold).rep(encoding=encoding, - cap=cap) + '' if self._ref is None else str( - ' ' + self._ref.rep(encoding=encoding)) + '' if self._nameset is None else str( - ' (' + self._nameset.rep_symbol(encoding=encoding) + ')') - - def rep_mention(self, encoding: (None, Encoding) = None, cap: (None, bool) = None) -> str: - return f'{"" if self.ref is None else self.ref.rep(encoding=encoding) + " "}' + StyledText( - s=self.cat.natural_name, text_style=text_styles.sans_serif_normal).rep( - encoding=encoding, cap=cap) - - class DashedName: """A dashed-name to provide more semantically meaningful names to symbolic-objcts in reports than symbols. @@ -3733,61 +3628,6 @@ def __init__(self, theory, position, phi, proof): self.proof = proof -class DirectDefinitionInference(FormulaStatement): - """ - - Definition: - A theoretical-statement that states that x = some other theoretical-object. - When an object is defined like this, it means that for every formula - where x is present, the same formula with the substitution of x by x' can be substituted in - all package. - TODO: QUESTION: Should we create a base "Alias" object that is distinct from simple-objct??? - XXXXXXX - """ - - def __init__(self, p: Formula, d: DefinitionInclusion, t: TheoryElaborationSequence, - nameset: (None, str, NameSet) = None, title: (None, str, TitleOBSOLETE) = None, - dashed_name: (None, str, DashedName) = None, echo: (None, bool) = None): - echo = prioritize_value(echo, configuration.echo_definition_direct_inference, - configuration.echo_default, False) - verify(t.contains_theoretical_objct(d), 'The definition-endorsement โŒœdโŒ must be contained ' - 'in the hierarchy of theory-elaboration โŒœtโŒ.', d=d, - t=t) - verify(p.universe_of_discourse is t.universe_of_discourse, - 'The universe-of-discourse of the valid-proposition โŒœpโŒ must be ' - 'consistent with the universe-of-discourse of theory-elaboration โŒœtโŒ.', p=p, t=t) - verify(p.relation is t.universe_of_discourse.r.equality, - 'The root relation of the valid-proposition โŒœpโŒ must be ' - 'the well-known equality-relation โŒœ=โŒ in the universe-of-discourse.', p=p, - p_relation=p.relation) - self.definition = d - super().__init__(theory=t, valid_proposition=p, nameset=nameset, - paragraphe_header=paragraph_headers.formal_definition, title=title, - dashed_name=dashed_name, echo=False) - assert d.statement_index < self.statement_index - super()._declare_class_membership(declarative_class_list.direct_definition_inference) - if echo: - repm.prnt(self.rep_report()) - - def compose_class(self) -> collections.abc.Generator[Composable, None, None]: - # TODO: Instead of hard-coding the class name, use a meta-theory. - yield SerifItalic(plaintext='definition-interpretation') - - def rep_report(self, proof: (None, bool) = None): - """Return a representation that expresses and justifies the statement. - - The representation is in two parts: - - The formula that is being stated, - - The justification for the formula.""" - output = f'{self.rep_title(cap=True)}: {self.valid_proposition.rep_formula(expand=True)}' - if proof: - output = output + f'\n\t' \ - f'{repm.serif_bold("Derivation from natural language definition")}' - output = output + f'\n\t{self.valid_proposition.rep_formula(expand=True):<70} โ”‚ ' \ - f'Follows from {self.definition.rep_ref()}.' - return output + f'\n' - - universe_of_discourse_symbol_indexes = dict() @@ -5392,8 +5232,6 @@ def __init__(self, section_title: str, t: TheoryElaborationSequence, self._max_subsection_number = 0 self.category = section_category symbol = NameSet(symbol=self.category.symbol_base, index=self.statement_index) - title = TitleOBSOLETE(ref=self._section_reference, paragrapha_header=section_category, - subtitle=section_title) super().__init__(nameset=symbol, theory=t, echo=False) super()._declare_class_membership(declarative_class_list.note) if echo: diff --git a/tests/test_title.py b/tests/test_title.py deleted file mode 100644 index 6d25a4df..00000000 --- a/tests/test_title.py +++ /dev/null @@ -1,37 +0,0 @@ -from unittest import TestCase -import punctilious as pu -import random_data - - -class TestTitle(TestCase): - def test_title(self): - pu.configuration.echo_default = False - pu.configuration.encoding = pu.encodings.plaintext - - title1 = pu.TitleOBSOLETE('1.1.1') - self.assertEqual('uncategorized 1.1.1', title1.rep_title()) - self.assertEqual('Uncategorized 1.1.1', title1.rep_title(cap=True)) - self.assertEqual('uncat. 1.1.1', title1.rep_ref()) - - title2 = pu.TitleOBSOLETE('1.1.2', - pu.paragraph_headers._hypothesis_statement_in_child_theory) - self.assertEqual('proposition 1.1.2', title2.rep_title()) - self.assertEqual('Proposition 1.1.2', title2.rep_title(cap=True)) - self.assertEqual('prop. 1.1.2', title2.rep_ref()) - self.assertEqual('๐—ฝ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป ๐Ÿญ.๐Ÿญ.๐Ÿฎ', title2.rep_title(encoding=pu.encodings.unicode)) - self.assertEqual('๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป ๐Ÿญ.๐Ÿญ.๐Ÿฎ', - title2.rep_title(encoding=pu.encodings.unicode, cap=True)) - self.assertEqual('๐—ฝ๐—ฟ๐—ผ๐—ฝ. ๐Ÿญ.๐Ÿญ.๐Ÿฎ', title2.rep_ref(encoding=pu.encodings.unicode)) - self.assertEqual('\\boldsymbol\\mathsf{proposition}} \\boldsymbol\\mathsf{1.1.2}}', - title2.rep_title(encoding=pu.encodings.latex)) - self.assertEqual('\\boldsymbol\\mathsf{Proposition}} \\boldsymbol\\mathsf{1.1.2}}', - title2.rep_title(encoding=pu.encodings.latex, cap=True)) - self.assertEqual('\\boldsymbol\\mathsf{prop.}} \\boldsymbol\\mathsf{1.1.2}}', - title2.rep_ref(encoding=pu.encodings.latex)) - - complement3 = random_data.random_sentence() - title3 = pu.TitleOBSOLETE(f'1.1.3', pu.paragraph_headers.lemma, subtitle=complement3) - self.assertEqual(f'lemma 1.1.3 - {complement3}', title3.rep_title()) - self.assertEqual(f'Lemma 1.1.3 - {complement3}', title3.rep_title(cap=True)) - self.assertEqual(f'lem. 1.1.3', title3.rep_ref()) - self.assertEqual(f'๐—น๐—ฒ๐—บ. ๐Ÿญ.๐Ÿญ.๐Ÿฏ', title3.rep_ref(encoding=pu.encodings.unicode)) diff --git a/tests/test_two_columns_proof.py b/tests/test_two_columns_proof.py deleted file mode 100644 index 70333bec..00000000 --- a/tests/test_two_columns_proof.py +++ /dev/null @@ -1,11 +0,0 @@ -from unittest import TestCase -import punctilious as pu -import random_data - - -class TestTwoColumnsProof(TestCase): - def test_rep_two_columns_proof_item(self): - left = random_data.random_sentence(min_words=50) - right = random_data.random_sentence(min_words=20) - report = pu.rep_two_columns_proof_item(left, right) - print(report) From 070aea060243fe9e0d07d2b121aa3c9e8322e023 Mon Sep 17 00:00:00 2001 From: David Doret Date: Wed, 23 Aug 2023 23:47:40 +0200 Subject: [PATCH 03/21] code cleanup --- punctilious/__init__.py | 8 +- punctilious/core.py | 22 ++- punctilious/locale_en_us.py | 32 +++- punctilious/unicode_utilities.py | 149 ++++++------------ tests/test_hypothesis.py | 10 +- ..._inconsistency_by_negation_introduction.py | 2 +- tests/test_note.py | 2 +- 7 files changed, 102 insertions(+), 123 deletions(-) diff --git a/punctilious/__init__.py b/punctilious/__init__.py index 3fb40e8d..a43d656c 100644 --- a/punctilious/__init__.py +++ b/punctilious/__init__.py @@ -9,10 +9,10 @@ DefinitionInclusion, Encoding, encodings, FailedVerificationException, Formula, FreeVariable, \ Header, InconsistencyWarning, interpret_formula, interpret_statement_formula, is_in_class, \ InferredStatement, NoteInclusion, Paragraph, prioritize_value, QuasiQuotation, Relation, \ - rep_two_columns_proof_item, ScriptNormal, SansSerifBold, SansSerifNormal, SerifNormal, \ - SimpleObjct, Statement, Subscript, subscriptify, paragraph_headers, ComposableText, NameSet, \ - SymbolicObject, TextStyle, text_styles, TheoreticalObject, TheoryElaborationSequence, \ - TheoryPackage, paragraph_headers, UniverseOfDiscourse + rep_two_columns_proof_item, ScriptNormal, SansSerifBold, SansSerifNormal, SerifBoldItalic, \ + SerifItalic, SerifNormal, SimpleObjct, Statement, Subscript, subscriptify, paragraph_headers, \ + ComposableText, NameSet, SymbolicObject, TextStyle, text_styles, TheoreticalObject, \ + TheoryElaborationSequence, TheoryPackage, paragraph_headers, UniverseOfDiscourse # from foundation_system_1 import foundation_system_1, ft, u diff --git a/punctilious/core.py b/punctilious/core.py index 22d67430..b05902e9 100644 --- a/punctilious/core.py +++ b/punctilious/core.py @@ -306,6 +306,10 @@ def __init__(self): unicode_table_index=unicode_utilities.unicode_serif_bold_index, start_tag=ComposableText(plaintext='', unicode='', latex='\\mathbf{'), end_tag=ComposableText(plaintext='', unicode='', latex='}')) + self.serif_bold_italic = TextStyle(name='serif-bold-italic', + unicode_table_index=unicode_utilities.unicode_serif_bold_italic_index, + start_tag=ComposableText(plaintext='', unicode='', latex='\\mathbold{'), + end_tag=ComposableText(plaintext='', unicode='', latex='}')) self.serif_italic = TextStyle(name='serif-italic', unicode_table_index=unicode_utilities.unicode_serif_italic_index, start_tag=ComposableText(plaintext='', unicode='', latex='\\mathit{'), @@ -735,6 +739,13 @@ def __init__(self, plaintext: str, unicode: (None, str) = None, latex=latex) +class SerifBoldItalic(StyledText): + def __init__(self, s: (None, str) = None, plaintext: (None, str) = None, + unicode: (None, str) = None, latex: (None, str) = None) -> None: + super().__init__(s=s, text_style=text_styles.serif_bold_italic, plaintext=plaintext, + unicode=unicode, latex=latex) + + class SerifItalic(StyledText): def __init__(self, s: (None, str) = None, plaintext: (None, str) = None, unicode: (None, str) = None, latex: (None, str) = None) -> None: @@ -4631,14 +4642,15 @@ def __init__(self, universe_of_discourse: UniverseOfDiscourse, echo: (None, bool dashed_name = 'inconsistency-by-negation-introduction' explicit_name = 'inconsistency by negation introduction inference rule' name = 'inconsistency by negation introduction' + definition = '๐“ฃ.๐‘ท, ๐“ฃ.ยฌ(๐‘ท) โŠข ๐ผ๐‘›๐‘(๐“ฃ)' # Assure backward-compatibility with the parent class, # which received these methods as __init__ arguments. infer_formula = InconsistencyByNegationIntroductionDeclaration.infer_formula verify_args = InconsistencyByNegationIntroductionDeclaration.verify_args - super().__init__(infer_formula=infer_formula, verify_args=verify_args, - universe_of_discourse=universe_of_discourse, symbol=symbol, auto_index=auto_index, - dashed_name=dashed_name, acronym=acronym, abridged_name=abridged_name, name=name, - explicit_name=explicit_name, echo=echo) + super().__init__(definition=definition, infer_formula=infer_formula, + verify_args=verify_args, universe_of_discourse=universe_of_discourse, symbol=symbol, + auto_index=auto_index, dashed_name=dashed_name, acronym=acronym, + abridged_name=abridged_name, name=name, explicit_name=explicit_name, echo=echo) def infer_formula(self, p: FormulaStatement = None, not_p: FormulaStatement = None, inconsistent_theory: TheoryElaborationSequence = None, @@ -4649,7 +4661,7 @@ def infer_formula(self, p: FormulaStatement = None, not_p: FormulaStatement = No def compose_paragraph_proof(self, o: InferredStatement) -> collections.abc.Generator[ Composable, Composable, bool]: - output = yield from configuration.locale.compose_inconsistency_introduction_paragraph_proof( + output = yield from configuration.locale.compose_inconsistency_introduction_by_negation_introduction_paragraph_proof( o=o) return output diff --git a/punctilious/locale_en_us.py b/punctilious/locale_en_us.py index 931e8b08..daea8a10 100644 --- a/punctilious/locale_en_us.py +++ b/punctilious/locale_en_us.py @@ -300,16 +300,38 @@ def compose_inconsistency_by_inequality_introduction_paragraph_proof(self, # Retrieve the parameters from the statement p_eq_q: FormulaStatement = o.parameters[0] p_neq_q: FormulaStatement = o.parameters[1] - yield from p_eq_q.compose_formula() + yield from p_eq_q.valid_proposition.compose_formula() yield SansSerifNormal(' follows from ') yield from p_eq_q.compose_ref_link() yield SansSerifNormal('. ') - yield from p_neq_q.compose_formula() + yield from p_neq_q.valid_proposition.compose_formula() yield SansSerifNormal(' follows from ') yield from p_neq_q.compose_ref_link() yield SansSerifNormal('. ') return True + def compose_inconsistency_introduction_by_negation_introduction_paragraph_proof(self, + o: InferredStatement) -> collections.abc.Generator[Composable, Composable, bool]: + global text_dict + # Retrieve the parameters from the statement + p: FormulaStatement = o.parameters[0] + not_p: FormulaStatement = o.parameters[1] + yield SansSerifNormal('Let ') + yield SerifBoldItalic('๐‘ท') + yield SerifItalic(' := ') + yield from p.valid_proposition.compose_formula() + yield SansSerifNormal(', which follows from ') + yield from p.compose_ref_link() + yield SansSerifNormal('. ') + yield SansSerifNormal('Let ') + yield SerifBoldItalic('ยฌ(๐‘ท)') + yield SerifItalic(' := ') + yield from not_p.valid_proposition.compose_formula() + yield SansSerifNormal(', which follows from ') + yield from not_p.compose_ref_link() + yield SansSerifNormal('. ') + return True + def compose_inference_rule_declaration(self, i: InferenceRuleDeclaration) -> \ collections.abc.Generator[Composable, Composable, bool]: global text_dict @@ -381,9 +403,11 @@ def compose_inferred_statement_report(self, o: InferredStatement, proof: (None, yield SansSerifNormal(': ') yield from o.inference_rule.compose_paragraph_proof(o=o) # Proof conclusion - yield SansSerifNormal(' Therefore, by the ') + yield SansSerifNormal('Therefore, by the ') yield from o.inference_rule.compose_dashed_name() - yield SansSerifNormal(' inference rule, it follows that ') + yield SansSerifNormal(' inference rule: ') + yield o.inference_rule.definition + yield SansSerifNormal(', it follows that ') yield from o.valid_proposition.compose_formula() yield SansSerifNormal('. ') yield self.qed diff --git a/punctilious/unicode_utilities.py b/punctilious/unicode_utilities.py index a9d9bd06..42a715d4 100644 --- a/punctilious/unicode_utilities.py +++ b/punctilious/unicode_utilities.py @@ -17,70 +17,37 @@ unicode_monospace_index = 12 unicode_double_struck_index = 13 -unicode_styled_characters = { - 'a': 'a๐š๐‘Ž๐’‚๐–บ๐—ฎ๐˜ข๐™–๐’ถ๐“ช๐”ž๐–†๐šŠ๐•’', - 'b': 'b๐›๐‘๐’ƒ๐–ป๐—ฏ๐˜ฃ๐™—๐’ท๐“ซ๐”Ÿ๐–‡๐š‹๐•“', - 'c': 'c๐œ๐‘๐’„๐–ผ๐—ฐ๐˜ค๐™˜๐’ธ๐“ฌ๐” ๐–ˆ๐šŒ๐•”', - 'd': 'd๐๐‘‘๐’…๐–ฝ๐—ฑ๐˜ฅ๐™™๐’น๐“ญ๐”ก๐–‰๐š๐••', - 'e': 'e๐ž๐‘’๐’†๐–พ๐—ฒ๐˜ฆ๐™šโ„ฏ๐“ฎ๐”ข๐–Š๐šŽ๐•–', - 'f': 'f๐Ÿ๐‘“๐’‡๐–ฟ๐—ณ๐˜ง๐™›๐’ป๐“ฏ๐”ฃ๐–‹๐š๐•—', - 'g': 'g๐ ๐‘”๐’ˆ๐—€๐—ด๐˜จ๐™œโ„Š๐“ฐ๐”ค๐–Œ๐š๐•˜', - 'h': 'h๐กโ„Ž๐’‰๐—๐—ต๐˜ฉ๐™๐’ฝ๐“ฑ๐”ฅ๐–๐š‘๐•™', - 'i': 'i๐ข๐‘–๐’Š๐—‚๐—ถ๐˜ช๐™ž๐’พ๐“ฒ๐”ฆ๐–Ž๐š’๐•š', - 'j': 'j๐ฃ๐‘—๐’‹๐—ƒ๐—ท๐˜ซ๐™Ÿ๐’ฟ๐“ณ๐”ง๐–๐š“๐•›', - 'k': 'k๐ค๐‘˜๐’Œ๐—„๐—ธ๐˜ฌ๐™ ๐“€๐“ด๐”จ๐–๐š”๐•œ', - 'l': 'l๐ฅ๐‘™๐’๐—…๐—น๐˜ญ๐™ก๐“๐“ต๐”ฉ๐–‘๐š•๐•', - 'm': 'm๐ฆ๐‘š๐’Ž๐—†๐—บ๐˜ฎ๐™ข๐“‚๐“ถ๐”ช๐–’๐š–๐•ž', - 'n': 'n๐ง๐‘›๐’๐—‡๐—ป๐˜ฏ๐™ฃ๐“ƒ๐“ท๐”ซ๐–“๐š—๐•Ÿ', - 'o': 'o๐จ๐‘œ๐’๐—ˆ๐—ผ๐˜ฐ๐™คโ„ด๐“ธ๐”ฌ๐–”๐š˜๐• ', - 'p': 'p๐ฉ๐‘๐’‘๐—‰๐—ฝ๐˜ฑ๐™ฅ๐“…๐“น๐”ญ๐–•๐š™๐•ก', - 'q': 'q๐ช๐‘ž๐’’๐—Š๐—พ๐˜ฒ๐™ฆ๐“†๐“บ๐”ฎ๐––๐šš๐•ข', - 'r': 'r๐ซ๐‘Ÿ๐’“๐—‹๐—ฟ๐˜ณ๐™ง๐“‡๐“ป๐”ฏ๐–—๐š›๐•ฃ', - 's': 's๐ฌ๐‘ ๐’”๐—Œ๐˜€๐˜ด๐™จ๐“ˆ๐“ผ๐”ฐ๐–˜๐šœ๐•ค', - 't': 't๐ญ๐‘ก๐’•๐—๐˜๐˜ต๐™ฉ๐“‰๐“ฝ๐”ฑ๐–™๐š๐•ฅ', - 'u': 'u๐ฎ๐‘ข๐’–๐—Ž๐˜‚๐˜ถ๐™ช๐“Š๐“พ๐”ฒ๐–š๐šž๐•ฆ', - 'v': 'v๐ฏ๐‘ฃ๐’—๐—๐˜ƒ๐˜ท๐™ซ๐“‹๐“ฟ๐”ณ๐–›๐šŸ๐•ง', - 'w': 'w๐ฐ๐‘ค๐’˜๐—๐˜„๐˜ธ๐™ฌ๐“Œ๐”€๐”ด๐–œ๐š ๐•จ', - 'x': 'x๐ฑ๐‘ฅ๐’™๐—‘๐˜…๐˜น๐™ญ๐“๐”๐”ต๐–๐šก๐•ฉ', - 'y': 'y๐ฒ๐‘ฆ๐’š๐—’๐˜†๐˜บ๐™ฎ๐“Ž๐”‚๐”ถ๐–ž๐šข๐•ช', - 'z': 'z๐ณ๐‘ง๐’›๐—“๐˜‡๐˜ป๐™ฏ๐“๐”ƒ๐”ท๐–Ÿ๐šฃ๐•ซ', - 'A': 'A๐€๐ด๐‘จ๐– ๐—”๐˜ˆ๐˜ผ๐’œ๐“๐”„๐•ฌ๐™ฐ๐”ธ', - 'B': 'B๐๐ต๐‘ฉ๐–ก๐—•๐˜‰๐˜ฝโ„ฌ๐“‘๐”…๐•ญ๐™ฑ๐”น', - 'C': 'C๐‚๐ถ๐‘ช๐–ข๐—–๐˜Š๐˜พ๐’ž๐“’โ„ญ๐•ฎ๐™ฒโ„‚', - 'D': 'D๐ƒ๐ท๐‘ซ๐–ฃ๐——๐˜‹๐˜ฟ๐’Ÿ๐““๐”‡๐•ฏ๐™ณ๐”ป', - 'E': 'E๐„๐ธ๐‘ฌ๐–ค๐—˜๐˜Œ๐™€โ„ฐ๐“”๐”ˆ๐•ฐ๐™ด๐”ผ', - 'F': 'F๐…๐น๐‘ญ๐–ฅ๐—™๐˜๐™โ„ฑ๐“•๐”‰๐•ฑ๐™ต๐”ฝ', - 'G': 'G๐†๐บ๐‘ฎ๐–ฆ๐—š๐˜Ž๐™‚๐’ข๐“–๐”Š๐•ฒ๐™ถ๐”พ', - 'H': 'H๐‡๐ป๐‘ฏ๐–ง๐—›๐˜๐™ƒโ„‹๐“—โ„Œ๐•ณ๐™ทโ„', - 'I': 'I๐ˆ๐ผ๐‘ฐ๐–จ๐—œ๐˜๐™„โ„๐“˜โ„‘๐•ด๐™ธ๐•€', - 'J': 'J๐‰๐ฝ๐‘ฑ๐–ฉ๐—๐˜‘๐™…๐’ฅ๐“™๐”๐•ต๐™น๐•', - 'K': 'K๐Š๐พ๐‘ฒ๐–ช๐—ž๐˜’๐™†๐’ฆ๐“š๐”Ž๐•ถ๐™บ๐•‚', - 'L': 'L๐‹๐ฟ๐‘ณ๐–ซ๐—Ÿ๐˜“๐™‡โ„’๐“›๐”๐•ท๐™ป๐•ƒ', - 'M': 'M๐Œ๐‘€๐‘ด๐–ฌ๐— ๐˜”๐™ˆโ„ณ๐“œ๐”๐•ธ๐™ผ๐•„', - 'N': 'N๐๐‘๐‘ต๐–ญ๐—ก๐˜•๐™‰๐’ฉ๐“๐”‘๐•น๐™ฝโ„•', - 'O': 'O๐Ž๐‘‚๐‘ถ๐–ฎ๐—ข๐˜–๐™Š๐’ช๐“ž๐”’๐•บ๐™พ๐•†', - 'P': 'P๐๐‘ƒ๐‘ท๐–ฏ๐—ฃ๐˜—๐™‹๐’ซ๐“Ÿ๐”“๐•ป๐™ฟโ„™', - 'Q': 'Q๐๐‘„๐‘ธ๐–ฐ๐—ค๐˜˜๐™Œ๐’ฌ๐“ ๐””๐•ผ๐š€โ„š', - 'R': 'R๐‘๐‘…๐‘น๐–ฑ๐—ฅ๐˜™๐™โ„›๐“กโ„œ๐•ฝ๐šโ„', - 'S': 'S๐’๐‘†๐‘บ๐–ฒ๐—ฆ๐˜š๐™Ž๐’ฎ๐“ข๐”–๐•พ๐š‚๐•Š', - 'T': 'T๐“๐‘‡๐‘ป๐–ณ๐—ง๐˜›๐™๐’ฏ๐“ฃ๐”—๐•ฟ๐šƒ๐•‹', - 'U': 'U๐”๐‘ˆ๐‘ผ๐–ด๐—จ๐˜œ๐™๐’ฐ๐“ค๐”˜๐–€๐š„๐•Œ', - 'V': 'V๐•๐‘‰๐‘ฝ๐–ต๐—ฉ๐˜๐™‘๐’ฑ๐“ฅ๐”™๐–๐š…๐•', - 'W': 'W๐–๐‘Š๐‘พ๐–ถ๐—ช๐˜ž๐™’๐’ฒ๐“ฆ๐”š๐–‚๐š†๐•Ž', - 'X': 'X๐—๐‘‹๐‘ฟ๐–ท๐—ซ๐˜Ÿ๐™“๐’ณ๐“ง๐”›๐–ƒ๐š‡๐•', - 'Y': 'Y๐˜๐‘Œ๐’€๐–ธ๐—ฌ๐˜ ๐™”๐’ด๐“จ๐”œ๐–„๐šˆ๐•', - 'Z': 'Z๐™๐‘๐’๐–น๐—ญ๐˜ก๐™•๐’ต๐“ฉโ„จ๐–…๐š‰โ„ค', - '0': '0๐ŸŽ0๐ŸŽ๐Ÿข๐Ÿฌ๐Ÿข๐Ÿฌ๐Ÿข๐Ÿฌ๐Ÿข๐Ÿฌ๐Ÿถ๐Ÿ˜', - '1': '1๐Ÿ1๐Ÿ๐Ÿฃ๐Ÿญ๐Ÿฃ๐Ÿญ๐Ÿฃ๐Ÿญ๐Ÿฃ๐Ÿญ๐Ÿท๐Ÿ™', - '2': '2๐Ÿ2๐Ÿ๐Ÿค๐Ÿฎ๐Ÿค๐Ÿฎ๐Ÿค๐Ÿฎ๐Ÿค๐Ÿฎ๐Ÿธ๐Ÿš', - '3': '3๐Ÿ‘3๐Ÿ‘๐Ÿฅ๐Ÿฏ๐Ÿฅ๐Ÿฏ๐Ÿฅ๐Ÿฏ๐Ÿฅ๐Ÿฏ๐Ÿน๐Ÿ›', - '4': '4๐Ÿ’4๐Ÿ’๐Ÿฆ๐Ÿฐ๐Ÿฆ๐Ÿฐ๐Ÿฆ๐Ÿฐ๐Ÿฆ๐Ÿฐ๐Ÿบ๐Ÿœ', - '5': '5๐Ÿ“5๐Ÿ“๐Ÿง๐Ÿฑ๐Ÿง๐Ÿฑ๐Ÿง๐Ÿฑ๐Ÿง๐Ÿฑ๐Ÿป๐Ÿ', - '6': '6๐Ÿ”6๐Ÿ”๐Ÿจ๐Ÿฒ๐Ÿจ๐Ÿฒ๐Ÿจ๐Ÿฒ๐Ÿจ๐Ÿฒ๐Ÿผ๐Ÿž', - '7': '7๐Ÿ•7๐Ÿ•๐Ÿฉ๐Ÿณ๐Ÿฉ๐Ÿณ๐Ÿฉ๐Ÿณ๐Ÿฉ๐Ÿณ๐Ÿฝ๐ŸŸ', - '8': '8๐Ÿ–8๐Ÿ–๐Ÿช๐Ÿด๐Ÿช๐Ÿด๐Ÿช๐Ÿด๐Ÿช๐Ÿด๐Ÿพ๐Ÿ ', - '9': '9๐Ÿ—9๐Ÿ—๐Ÿซ๐Ÿต๐Ÿซ๐Ÿต๐Ÿซ๐Ÿต๐Ÿซ๐Ÿต๐Ÿฟ๐Ÿก' -} +unicode_styled_characters = {'a': 'a๐š๐‘Ž๐’‚๐–บ๐—ฎ๐˜ข๐™–๐’ถ๐“ช๐”ž๐–†๐šŠ๐•’', 'b': 'b๐›๐‘๐’ƒ๐–ป๐—ฏ๐˜ฃ๐™—๐’ท๐“ซ๐”Ÿ๐–‡๐š‹๐•“', + 'c': 'c๐œ๐‘๐’„๐–ผ๐—ฐ๐˜ค๐™˜๐’ธ๐“ฌ๐” ๐–ˆ๐šŒ๐•”', 'd': 'd๐๐‘‘๐’…๐–ฝ๐—ฑ๐˜ฅ๐™™๐’น๐“ญ๐”ก๐–‰๐š๐••', + 'e': 'e๐ž๐‘’๐’†๐–พ๐—ฒ๐˜ฆ๐™šโ„ฏ๐“ฎ๐”ข๐–Š๐šŽ๐•–', 'f': 'f๐Ÿ๐‘“๐’‡๐–ฟ๐—ณ๐˜ง๐™›๐’ป๐“ฏ๐”ฃ๐–‹๐š๐•—', + 'g': 'g๐ ๐‘”๐’ˆ๐—€๐—ด๐˜จ๐™œโ„Š๐“ฐ๐”ค๐–Œ๐š๐•˜', 'h': 'h๐กโ„Ž๐’‰๐—๐—ต๐˜ฉ๐™๐’ฝ๐“ฑ๐”ฅ๐–๐š‘๐•™', + 'i': 'i๐ข๐‘–๐’Š๐—‚๐—ถ๐˜ช๐™ž๐’พ๐“ฒ๐”ฆ๐–Ž๐š’๐•š', 'j': 'j๐ฃ๐‘—๐’‹๐—ƒ๐—ท๐˜ซ๐™Ÿ๐’ฟ๐“ณ๐”ง๐–๐š“๐•›', + 'k': 'k๐ค๐‘˜๐’Œ๐—„๐—ธ๐˜ฌ๐™ ๐“€๐“ด๐”จ๐–๐š”๐•œ', 'l': 'l๐ฅ๐‘™๐’๐—…๐—น๐˜ญ๐™ก๐“๐“ต๐”ฉ๐–‘๐š•๐•', + 'm': 'm๐ฆ๐‘š๐’Ž๐—†๐—บ๐˜ฎ๐™ข๐“‚๐“ถ๐”ช๐–’๐š–๐•ž', 'n': 'n๐ง๐‘›๐’๐—‡๐—ป๐˜ฏ๐™ฃ๐“ƒ๐“ท๐”ซ๐–“๐š—๐•Ÿ', + 'o': 'o๐จ๐‘œ๐’๐—ˆ๐—ผ๐˜ฐ๐™คโ„ด๐“ธ๐”ฌ๐–”๐š˜๐• ', 'p': 'p๐ฉ๐‘๐’‘๐—‰๐—ฝ๐˜ฑ๐™ฅ๐“…๐“น๐”ญ๐–•๐š™๐•ก', + 'q': 'q๐ช๐‘ž๐’’๐—Š๐—พ๐˜ฒ๐™ฆ๐“†๐“บ๐”ฎ๐––๐šš๐•ข', 'r': 'r๐ซ๐‘Ÿ๐’“๐—‹๐—ฟ๐˜ณ๐™ง๐“‡๐“ป๐”ฏ๐–—๐š›๐•ฃ', + 's': 's๐ฌ๐‘ ๐’”๐—Œ๐˜€๐˜ด๐™จ๐“ˆ๐“ผ๐”ฐ๐–˜๐šœ๐•ค', 't': 't๐ญ๐‘ก๐’•๐—๐˜๐˜ต๐™ฉ๐“‰๐“ฝ๐”ฑ๐–™๐š๐•ฅ', + 'u': 'u๐ฎ๐‘ข๐’–๐—Ž๐˜‚๐˜ถ๐™ช๐“Š๐“พ๐”ฒ๐–š๐šž๐•ฆ', 'v': 'v๐ฏ๐‘ฃ๐’—๐—๐˜ƒ๐˜ท๐™ซ๐“‹๐“ฟ๐”ณ๐–›๐šŸ๐•ง', + 'w': 'w๐ฐ๐‘ค๐’˜๐—๐˜„๐˜ธ๐™ฌ๐“Œ๐”€๐”ด๐–œ๐š ๐•จ', 'x': 'x๐ฑ๐‘ฅ๐’™๐—‘๐˜…๐˜น๐™ญ๐“๐”๐”ต๐–๐šก๐•ฉ', + 'y': 'y๐ฒ๐‘ฆ๐’š๐—’๐˜†๐˜บ๐™ฎ๐“Ž๐”‚๐”ถ๐–ž๐šข๐•ช', 'z': 'z๐ณ๐‘ง๐’›๐—“๐˜‡๐˜ป๐™ฏ๐“๐”ƒ๐”ท๐–Ÿ๐šฃ๐•ซ', + 'A': 'A๐€๐ด๐‘จ๐– ๐—”๐˜ˆ๐˜ผ๐’œ๐“๐”„๐•ฌ๐™ฐ๐”ธ', 'B': 'B๐๐ต๐‘ฉ๐–ก๐—•๐˜‰๐˜ฝโ„ฌ๐“‘๐”…๐•ญ๐™ฑ๐”น', + 'C': 'C๐‚๐ถ๐‘ช๐–ข๐—–๐˜Š๐˜พ๐’ž๐“’โ„ญ๐•ฎ๐™ฒโ„‚', 'D': 'D๐ƒ๐ท๐‘ซ๐–ฃ๐——๐˜‹๐˜ฟ๐’Ÿ๐““๐”‡๐•ฏ๐™ณ๐”ป', + 'E': 'E๐„๐ธ๐‘ฌ๐–ค๐—˜๐˜Œ๐™€โ„ฐ๐“”๐”ˆ๐•ฐ๐™ด๐”ผ', 'F': 'F๐…๐น๐‘ญ๐–ฅ๐—™๐˜๐™โ„ฑ๐“•๐”‰๐•ฑ๐™ต๐”ฝ', + 'G': 'G๐†๐บ๐‘ฎ๐–ฆ๐—š๐˜Ž๐™‚๐’ข๐“–๐”Š๐•ฒ๐™ถ๐”พ', 'H': 'H๐‡๐ป๐‘ฏ๐–ง๐—›๐˜๐™ƒโ„‹๐“—โ„Œ๐•ณ๐™ทโ„', + 'I': 'I๐ˆ๐ผ๐‘ฐ๐–จ๐—œ๐˜๐™„โ„๐“˜โ„‘๐•ด๐™ธ๐•€', 'J': 'J๐‰๐ฝ๐‘ฑ๐–ฉ๐—๐˜‘๐™…๐’ฅ๐“™๐”๐•ต๐™น๐•', + 'K': 'K๐Š๐พ๐‘ฒ๐–ช๐—ž๐˜’๐™†๐’ฆ๐“š๐”Ž๐•ถ๐™บ๐•‚', 'L': 'L๐‹๐ฟ๐‘ณ๐–ซ๐—Ÿ๐˜“๐™‡โ„’๐“›๐”๐•ท๐™ป๐•ƒ', + 'M': 'M๐Œ๐‘€๐‘ด๐–ฌ๐— ๐˜”๐™ˆโ„ณ๐“œ๐”๐•ธ๐™ผ๐•„', 'N': 'N๐๐‘๐‘ต๐–ญ๐—ก๐˜•๐™‰๐’ฉ๐“๐”‘๐•น๐™ฝโ„•', + 'O': 'O๐Ž๐‘‚๐‘ถ๐–ฎ๐—ข๐˜–๐™Š๐’ช๐“ž๐”’๐•บ๐™พ๐•†', 'P': 'P๐๐‘ƒ๐‘ท๐–ฏ๐—ฃ๐˜—๐™‹๐’ซ๐“Ÿ๐”“๐•ป๐™ฟโ„™', + 'Q': 'Q๐๐‘„๐‘ธ๐–ฐ๐—ค๐˜˜๐™Œ๐’ฌ๐“ ๐””๐•ผ๐š€โ„š', 'R': 'R๐‘๐‘…๐‘น๐–ฑ๐—ฅ๐˜™๐™โ„›๐“กโ„œ๐•ฝ๐šโ„', + 'S': 'S๐’๐‘†๐‘บ๐–ฒ๐—ฆ๐˜š๐™Ž๐’ฎ๐“ข๐”–๐•พ๐š‚๐•Š', 'T': 'T๐“๐‘‡๐‘ป๐–ณ๐—ง๐˜›๐™๐’ฏ๐“ฃ๐”—๐•ฟ๐šƒ๐•‹', + 'U': 'U๐”๐‘ˆ๐‘ผ๐–ด๐—จ๐˜œ๐™๐’ฐ๐“ค๐”˜๐–€๐š„๐•Œ', 'V': 'V๐•๐‘‰๐‘ฝ๐–ต๐—ฉ๐˜๐™‘๐’ฑ๐“ฅ๐”™๐–๐š…๐•', + 'W': 'W๐–๐‘Š๐‘พ๐–ถ๐—ช๐˜ž๐™’๐’ฒ๐“ฆ๐”š๐–‚๐š†๐•Ž', 'X': 'X๐—๐‘‹๐‘ฟ๐–ท๐—ซ๐˜Ÿ๐™“๐’ณ๐“ง๐”›๐–ƒ๐š‡๐•', + 'Y': 'Y๐˜๐‘Œ๐’€๐–ธ๐—ฌ๐˜ ๐™”๐’ด๐“จ๐”œ๐–„๐šˆ๐•', 'Z': 'Z๐™๐‘๐’๐–น๐—ญ๐˜ก๐™•๐’ต๐“ฉโ„จ๐–…๐š‰โ„ค', + '0': '0๐ŸŽ0๐ŸŽ๐Ÿข๐Ÿฌ๐Ÿข๐Ÿฌ๐Ÿข๐Ÿฌ๐Ÿข๐Ÿฌ๐Ÿถ๐Ÿ˜', '1': '1๐Ÿ1๐Ÿ๐Ÿฃ๐Ÿญ๐Ÿฃ๐Ÿญ๐Ÿฃ๐Ÿญ๐Ÿฃ๐Ÿญ๐Ÿท๐Ÿ™', + '2': '2๐Ÿ2๐Ÿ๐Ÿค๐Ÿฎ๐Ÿค๐Ÿฎ๐Ÿค๐Ÿฎ๐Ÿค๐Ÿฎ๐Ÿธ๐Ÿš', '3': '3๐Ÿ‘3๐Ÿ‘๐Ÿฅ๐Ÿฏ๐Ÿฅ๐Ÿฏ๐Ÿฅ๐Ÿฏ๐Ÿฅ๐Ÿฏ๐Ÿน๐Ÿ›', + '4': '4๐Ÿ’4๐Ÿ’๐Ÿฆ๐Ÿฐ๐Ÿฆ๐Ÿฐ๐Ÿฆ๐Ÿฐ๐Ÿฆ๐Ÿฐ๐Ÿบ๐Ÿœ', '5': '5๐Ÿ“5๐Ÿ“๐Ÿง๐Ÿฑ๐Ÿง๐Ÿฑ๐Ÿง๐Ÿฑ๐Ÿง๐Ÿฑ๐Ÿป๐Ÿ', + '6': '6๐Ÿ”6๐Ÿ”๐Ÿจ๐Ÿฒ๐Ÿจ๐Ÿฒ๐Ÿจ๐Ÿฒ๐Ÿจ๐Ÿฒ๐Ÿผ๐Ÿž', '7': '7๐Ÿ•7๐Ÿ•๐Ÿฉ๐Ÿณ๐Ÿฉ๐Ÿณ๐Ÿฉ๐Ÿณ๐Ÿฉ๐Ÿณ๐Ÿฝ๐ŸŸ', + '8': '8๐Ÿ–8๐Ÿ–๐Ÿช๐Ÿด๐Ÿช๐Ÿด๐Ÿช๐Ÿด๐Ÿช๐Ÿด๐Ÿพ๐Ÿ ', '9': '9๐Ÿ—9๐Ÿ—๐Ÿซ๐Ÿต๐Ÿซ๐Ÿต๐Ÿซ๐Ÿต๐Ÿซ๐Ÿต๐Ÿฟ๐Ÿก'} def prioritize_value(*args) -> typing.Any: @@ -135,7 +102,7 @@ def unicode_serif_italic(s: str): def unicode_serif_bold_italic(s: str): - return unicode_format(s=s, index=unicode_serif_italic_bold_index) + return unicode_format(s=s, index=unicode_serif_bold_italic_index) def unicode_script_normal(s: str): @@ -162,45 +129,21 @@ def unicode_double_struck(s: str): return unicode_format(s=s, index=unicode_double_struck_index) -unicode_subscript_dictionary = { - '0': u'โ‚€', - '1': u'โ‚', - '2': u'โ‚‚', - '3': u'โ‚ƒ', - '4': u'โ‚„', - '5': u'โ‚…', - '6': u'โ‚†', - '7': u'โ‚‡', - '8': u'โ‚ˆ', - '9': u'โ‚‰', - 'a': u'โ‚', - 'e': u'โ‚‘', - 'o': u'โ‚’', - 'x': u'โ‚“', - # '???': u'โ‚”', - 'h': u'โ‚•', - 'k': u'โ‚–', - 'l': u'โ‚—', - 'm': u'โ‚˜', - 'n': u'โ‚™', - 'p': u'โ‚š', - 's': u'โ‚›', - 't': u'โ‚œ', - '+': u'โ‚Š', - '-': u'โ‚‹', - '=': u'โ‚Œ', - '(': u'โ‚', - ')': u'โ‚Ž', - 'j': u'โฑผ', - 'i': u'แตข', # Alternative from the Unicode Phonetic Extensions block: แตข - 'r': u'แตฃ', # Source: Unicode Phonetic Extensions block. - 'u': u'แตค', # Source: Unicode Phonetic Extensions block. - 'v': u'แตฅ', # Source: Unicode Phonetic Extensions block. - 'ฮฒ': u'แตฆ', # Source: Unicode Phonetic Extensions block. - 'ฮณ': u'แตง', # Source: Unicode Phonetic Extensions block. +unicode_subscript_dictionary = {'0': u'โ‚€', '1': u'โ‚', '2': u'โ‚‚', '3': u'โ‚ƒ', '4': u'โ‚„', '5': u'โ‚…', + '6': u'โ‚†', '7': u'โ‚‡', '8': u'โ‚ˆ', '9': u'โ‚‰', 'a': u'โ‚', 'e': u'โ‚‘', + 'o': u'โ‚’', 'x': u'โ‚“', # '???': u'โ‚”', + 'h': u'โ‚•', 'k': u'โ‚–', 'l': u'โ‚—', 'm': u'โ‚˜', 'n': u'โ‚™', 'p': u'โ‚š', + 's': u'โ‚›', 't': u'โ‚œ', '+': u'โ‚Š', '-': u'โ‚‹', '=': u'โ‚Œ', '(': u'โ‚', + ')': u'โ‚Ž', 'j': u'โฑผ', 'i': u'แตข', + # Alternative from the Unicode Phonetic Extensions block: แตข + 'r': u'แตฃ', # Source: Unicode Phonetic Extensions block. + 'u': u'แตค', # Source: Unicode Phonetic Extensions block. + 'v': u'แตฅ', # Source: Unicode Phonetic Extensions block. + 'ฮฒ': u'แตฆ', # Source: Unicode Phonetic Extensions block. + 'ฮณ': u'แตง', # Source: Unicode Phonetic Extensions block. # '???': u'แตจ', # Source: Unicode Phonetic Extensions block. - 'ฯ†': u'แตฉ', # Source: Unicode Phonetic Extensions block. - 'ฯ‡': u'แตช' # Source: Unicode Phonetic Extensions block. + 'ฯ†': u'แตฉ', # Source: Unicode Phonetic Extensions block. + 'ฯ‡': u'แตช' # Source: Unicode Phonetic Extensions block. } diff --git a/tests/test_hypothesis.py b/tests/test_hypothesis.py index 0c419c45..f55a2a21 100644 --- a/tests/test_hypothesis.py +++ b/tests/test_hypothesis.py @@ -30,10 +30,10 @@ def test_hypothesis(self): hypothetical_theory = hypothesis.hypothesis_child_theory hypothetical_conjunction = hypothetical_theory.i.ci.infer_statement((o1 | r1 | o2), hypothetical_proposition) - proposition_1 = hypothetical_theory.i.vs.infer_statement(conditional, o1, # x - o2, # y - o3) # z - conclusion_1 = hypothetical_theory.i.mp.infer_statement(proposition_1, - hypothetical_conjunction) + proposition_1 = hypothetical_theory.i.vs.infer_statement(p=conditional, phi=(o1, # x + o2, # y + o3)) # z + conclusion_1 = hypothetical_theory.i.mp.infer_statement(p_implies_q=proposition_1, + p=hypothetical_conjunction) self.assertEqual('๐‘Ÿโ‚(๐‘œโ‚, ๐‘œโ‚ƒ)', conclusion_1.valid_proposition.rep_formula(pu.encodings.unicode)) diff --git a/tests/test_inconsistency_by_negation_introduction.py b/tests/test_inconsistency_by_negation_introduction.py index be4bcb0e..0e2b7270 100644 --- a/tests/test_inconsistency_by_negation_introduction.py +++ b/tests/test_inconsistency_by_negation_introduction.py @@ -48,7 +48,7 @@ def test_inconsistency_introduction_2(self): t2 = t1_h1.hypothesis_child_theory t2_p5 = t1_h1.hypothesis_statement_in_child_theory t2_p6 = t2.i.conjunction_introduction.infer_statement(p=t1_p1, q=t1_p2) - t2_p7 = t2.i.variable_substitution.infer_statement(t1_p3_implication, o1, o2, o3) + t2_p7 = t2.i.variable_substitution.infer_statement(p=t1_p3_implication, phi=(o1, o2, o3)) # t2_p8: ๐‘Ÿโ‚(๐‘œโ‚, ๐‘œโ‚ƒ) by modus ponens t2_p8 = t2.i.modus_ponens.infer_statement(p_implies_q=t2_p7, p=t2_p6) # p5 is the negation of p8, which is a contradiction in t2 diff --git a/tests/test_note.py b/tests/test_note.py index 6232d50b..b44c10f5 100644 --- a/tests/test_note.py +++ b/tests/test_note.py @@ -15,7 +15,7 @@ def test_note_introduction(self): ap = t.include_axiom(a) t.i.axiom_interpretation.infer_statement(ap, u.f(r1, o1)) note = t.take_note('Hello world!', ref='1.1.1') - self.assertEqual('๐—ก๐—ผ๐˜๐—ฒ ๐Ÿญ.๐Ÿญ.๐Ÿญ (๐Ÿ—…โ‚‚): Hello world!', note.rep_report()) + self.assertEqual('๐—ก๐—ผ๐˜๐—ฒ ๐Ÿญ.๐Ÿญ.๐Ÿญ (๐Ÿ—…โ‚): ๐–ง๐–พ๐—…๐—…๐—ˆ ๐—๐—ˆ๐—‹๐—…๐–ฝ!', note.rep_report()) comment = t.take_note('Foo', ref='1.1.2', paragraph_header=pu.paragraph_headers.comment) self.assertEqual('๐—–๐—ผ๐—บ๐—บ๐—ฒ๐—ป๐˜ ๐Ÿญ.๐Ÿญ.๐Ÿฎ (๐Ÿ—…โ‚ƒ): Foo', comment.rep_report()) remark = t.take_note('Bar', ref='1.1.3', paragraph_header=pu.paragraph_headers.remark) From b3ec5a6c5c7606168acbb435fb0f1beee0c24592 Mon Sep 17 00:00:00 2001 From: David Doret Date: Wed, 23 Aug 2023 23:48:41 +0200 Subject: [PATCH 04/21] theory build --- ...no_axioms_interactive_graph_plaintext.html | 2 +- ...eano_axioms_interactive_graph_unicode.html | 2 +- ...peano_axioms_report_noproof_enus_latex.txt | 2 +- ...o_axioms_report_noproof_enus_plaintext.txt | 2 +- ...ano_axioms_report_noproof_enus_unicode.txt | 2 +- ...e_peano_axioms_report_proof_enus_latex.txt | 2 +- ...ano_axioms_report_proof_enus_plaintext.txt | 126 +++++++++--------- ...peano_axioms_report_proof_enus_unicode.txt | 126 +++++++++--------- 8 files changed, 132 insertions(+), 132 deletions(-) diff --git a/theory/build/tao_2006_2_1_the_peano_axioms_interactive_graph_plaintext.html b/theory/build/tao_2006_2_1_the_peano_axioms_interactive_graph_plaintext.html index a0cec182..aba3fe3b 100644 --- a/theory/build/tao_2006_2_1_the_peano_axioms_interactive_graph_plaintext.html +++ b/theory/build/tao_2006_2_1_the_peano_axioms_interactive_graph_plaintext.html @@ -88,7 +88,7 @@

// parsing and collecting nodes and edges from the python - nodes = new vis.DataSet([{"color": "#81C784", "id": "A1", "label": "A1 (2.1) : \"0 is a\nnatural number.\"", "shape": "box"}, {"color": "#FFF59D", "id": "P1", "label": "P1 : (0 is-a\nnatural-number)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P1): (0 is-a\nnatural-number). Proof: \"0 is a\nnatural number.\" is postulated\nby axiom 2.1 (A1). (0 is-a\nnatural-number) is a valid\nformula statement interpreted\nfrom that axiom. Therefore, by\nthe axiom-interpretation\ninference rule, it follows that\n(0 is-a natural-number). QED"}, {"color": "#81C784", "id": "A2", "label": "A2 (2.2) : \"If n is\na natural number,\nthen n++ is a\nnatural number.\"", "shape": "box"}, {"color": "#FFF59D", "id": "P2", "label": "P2 : ((n1 is-a\nnatural-number) ==\u003e\n((n1)++ is-a\nnatural-number))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P2): ((n1 is-a\nnatural-number) ==\u003e ((n1)++ is-a\nnatural-number)). Proof: \"If n\nis a natural number, then n++ is\na natural number.\" is postulated\nby axiom 2.2 (A2). ((n1 is-a\nnatural-number) ==\u003e ((n1)++ is-a\nnatural-number)) is a valid\nformula statement interpreted\nfrom that axiom. Therefore, by\nthe axiom-interpretation\ninference rule, it follows that\n((n1 is-a natural-number) ==\u003e\n((n1)++ is-a natural-number)).\nQED"}, {"color": "#FFF59D", "id": "P3", "label": "P3 : ((0 is-a\nnatural-number) ==\u003e\n((0)++ is-a natural-\nnumber))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P3): ((0 is-a\nnatural-number) ==\u003e ((0)++ is-a\nnatural-number)). Proof: ((n1\nis-a natural-number) ==\u003e ((n1)++\nis-a natural-number)) follows\nfrom prop. (P2). Let n1 = 0.\nTherefore, by the variable-\nsubstitution inference rule, it\nfollows that ((0 is-a natural-\nnumber) ==\u003e ((0)++ is-a natural-\nnumber)). QED"}, {"color": "#FFF59D", "id": "P4", "label": "P4 (2.2.3) : ((0)++\nis-a natural-number)", "labelHighlightBold": true, "shape": "box", "title": "Proposition 2.2.3 (T1.P4):\n((0)++ is-a natural-number).\nProof: ((0 is-a natural-number)\n==\u003e ((0)++ is-a natural-number))\nfollows from prop. (P3).(0 is-a\nnatural-number) follows from\nprop. (P1). Therefore, by the\nmodus-ponens inference rule, it\nfollows that ((0)++ is-a\nnatural-number). QED"}, {"color": "#90CAF9", "id": "D1", "label": "D1 : \"We define 1 to\nbe the number 0++, 2\nto be the number\n(0++)++, 3 to be the\nnumber\n((0++)++)++,etc. (In\nother words, 1 :=\n0++, 2 := 1++, 3 :=\n2++, etc. In this\ntext I use \"x := y\"\nto denote the\nstatement that x is\ndefined to equal\ny.)\"", "shape": "box"}, {"color": "#FFF59D", "id": "P5", "label": "P5 : (1 = (0)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P5): (1 =\n(0)++). Proof: \"We define 1 to\nbe the number 0++, 2 to be the\nnumber (0++)++, 3 to be the\nnumber ((0++)++)++,etc. (In\nother words, 1 := 0++, 2 := 1++,\n3 := 2++, etc. In this text I\nuse \"x := y\" to denote the\nstatement that x is defined to\nequal y.)\" is postulated by def.\n(D1). (1 = (0)++) is an\ninterpretation of that\ndefinition. Therefore, by the\ndefinition-interpretation\ninference rule, it follows that\n(1 = (0)++). QED"}, {"color": "#FFF59D", "id": "P6", "label": "P6 : (2 = ((0)++)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P6): (2 =\n((0)++)++). Proof: \"We define 1\nto be the number 0++, 2 to be\nthe number (0++)++, 3 to be the\nnumber ((0++)++)++,etc. (In\nother words, 1 := 0++, 2 := 1++,\n3 := 2++, etc. In this text I\nuse \"x := y\" to denote the\nstatement that x is defined to\nequal y.)\" is postulated by def.\n(D1). (2 = ((0)++)++) is an\ninterpretation of that\ndefinition. Therefore, by the\ndefinition-interpretation\ninference rule, it follows that\n(2 = ((0)++)++). QED"}, {"color": "#FFF59D", "id": "P7", "label": "P7 : (3 =\n(((0)++)++)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P7): (3 =\n(((0)++)++)++). Proof: \"We\ndefine 1 to be the number 0++, 2\nto be the number (0++)++, 3 to\nbe the number ((0++)++)++,etc.\n(In other words, 1 := 0++, 2 :=\n1++, 3 := 2++, etc. In this text\nI use \"x := y\" to denote the\nstatement that x is defined to\nequal y.)\" is postulated by def.\n(D1). (3 = (((0)++)++)++) is an\ninterpretation of that\ndefinition. Therefore, by the\ndefinition-interpretation\ninference rule, it follows that\n(3 = (((0)++)++)++). QED"}, {"color": "#FFF59D", "id": "P8", "label": "P8 : (4 =\n((((0)++)++)++)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P8): (4 =\n((((0)++)++)++)++). Proof: \"We\ndefine 1 to be the number 0++, 2\nto be the number (0++)++, 3 to\nbe the number ((0++)++)++,etc.\n(In other words, 1 := 0++, 2 :=\n1++, 3 := 2++, etc. In this text\nI use \"x := y\" to denote the\nstatement that x is defined to\nequal y.)\" is postulated by def.\n(D1). (4 = ((((0)++)++)++)++) is\nan interpretation of that\ndefinition. Therefore, by the\ndefinition-interpretation\ninference rule, it follows that\n(4 = ((((0)++)++)++)++). QED"}, {"color": "#FFF59D", "id": "P9", "label": "P9 : (((0)++ is-a\nnatural-number) ==\u003e\n(((0)++)++ is-a\nnatural-number))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P9): (((0)++\nis-a natural-number) ==\u003e\n(((0)++)++ is-a natural-\nnumber)). Proof: ((n1 is-a\nnatural-number) ==\u003e ((n1)++ is-a\nnatural-number)) follows from\nprop. (P2). Let n1 = (0)++.\nTherefore, by the variable-\nsubstitution inference rule, it\nfollows that (((0)++ is-a\nnatural-number) ==\u003e (((0)++)++\nis-a natural-number)). QED"}, {"color": "#FFF59D", "id": "P10", "label": "P10 : (((0)++)++\nis-a natural-number)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P10): (((0)++)++\nis-a natural-number). Proof:\n(((0)++ is-a natural-number) ==\u003e\n(((0)++)++ is-a natural-number))\nfollows from prop. (P9).((0)++\nis-a natural-number) follows\nfrom prop. 2.2.3 (P4).\nTherefore, by the modus-ponens\ninference rule, it follows that\n(((0)++)++ is-a natural-number).\nQED"}, {"color": "#FFF59D", "id": "P11", "label": "P11 : ((((0)++)++\nis-a natural-number)\n==\u003e ((((0)++)++)++\nis-a natural-\nnumber))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P11):\n((((0)++)++ is-a natural-number)\n==\u003e ((((0)++)++)++ is-a natural-\nnumber)). Proof: ((n1 is-a\nnatural-number) ==\u003e ((n1)++ is-a\nnatural-number)) follows from\nprop. (P2). Let n1 = ((0)++)++.\nTherefore, by the variable-\nsubstitution inference rule, it\nfollows that ((((0)++)++ is-a\nnatural-number) ==\u003e\n((((0)++)++)++ is-a natural-\nnumber)). QED"}, {"color": "#FFF59D", "id": "P12", "label": "P12 : ((((0)++)++)++\nis-a natural-number)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P12):\n((((0)++)++)++ is-a natural-\nnumber). Proof: ((((0)++)++ is-a\nnatural-number) ==\u003e\n((((0)++)++)++ is-a natural-\nnumber)) follows from prop.\n(P11).(((0)++)++ is-a natural-\nnumber) follows from prop.\n(P10). Therefore, by the modus-\nponens inference rule, it\nfollows that ((((0)++)++)++ is-a\nnatural-number). QED"}, {"color": "#FFF59D", "id": "P13", "label": "P13 :\n(((((0)++)++)++ is-a\nnatural-number) ==\u003e\n(((((0)++)++)++)++\nis-a natural-\nnumber))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P13):\n(((((0)++)++)++ is-a natural-\nnumber) ==\u003e (((((0)++)++)++)++\nis-a natural-number)). Proof:\n((n1 is-a natural-number) ==\u003e\n((n1)++ is-a natural-number))\nfollows from prop. (P2). Let n1\n= (((0)++)++)++. Therefore, by\nthe variable-substitution\ninference rule, it follows that\n(((((0)++)++)++ is-a natural-\nnumber) ==\u003e (((((0)++)++)++)++\nis-a natural-number)). QED"}, {"color": "#FFF59D", "id": "P14", "label": "P14 :\n(((((0)++)++)++)++\nis-a natural-number)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P14):\n(((((0)++)++)++)++ is-a natural-\nnumber). Proof: (((((0)++)++)++\nis-a natural-number) ==\u003e\n(((((0)++)++)++)++ is-a natural-\nnumber)) follows from prop.\n(P13).((((0)++)++)++ is-a\nnatural-number) follows from\nprop. (P12). Therefore, by the\nmodus-ponens inference rule, it\nfollows that (((((0)++)++)++)++\nis-a natural-number). QED"}, {"color": "#FFF59D", "id": "P15", "label": "P15 : ((0)++ = 1)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P15): ((0)++ =\n1). Proof: (1 = (0)++) follows\nfrom prop. (P5). Therefore, by\nthe equality-commutativity\ninference rule, it follows that\n((0)++ = 1). QED"}, {"color": "#FFF59D", "id": "P16", "label": "P16 : (2 = (1)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P16): (2 =\n(1)++). Proof: (2 = ((0)++)++)\nfollows from prop. (P6). ((0)++\n= 1) follows from prop. (P15).\nTherefore, by the equal-terms-\nsubstitution inference rule, it\nfollows that (2 = (1)++). QED"}, {"color": "#FFF59D", "id": "P17", "label": "P17 : (((0)++)++ =\n2)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P17): (((0)++)++\n= 2). Proof: (2 = ((0)++)++)\nfollows from prop. (P6).\nTherefore, by the equality-\ncommutativity inference rule, it\nfollows that (((0)++)++ = 2).\nQED"}, {"color": "#FFF59D", "id": "P18", "label": "P18 : (3 = (2)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P18): (3 =\n(2)++). Proof: (3 =\n(((0)++)++)++) follows from\nprop. (P7). (((0)++)++ = 2)\nfollows from prop. (P17).\nTherefore, by the equal-terms-\nsubstitution inference rule, it\nfollows that (3 = (2)++). QED"}, {"color": "#FFF59D", "id": "P19", "label": "P19 : ((((0)++)++)++\n= 3)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P19):\n((((0)++)++)++ = 3). Proof: (3 =\n(((0)++)++)++) follows from\nprop. (P7). Therefore, by the\nequality-commutativity inference\nrule, it follows that\n((((0)++)++)++ = 3). QED"}, {"color": "#FFF59D", "id": "P20", "label": "P20 : ((2)++ = 3)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P20): ((2)++ =\n3). Proof: ((((0)++)++)++ = 3)\nfollows from prop. (P19).\n(((0)++)++ = 2) follows from\nprop. (P17). Therefore, by the\nequal-terms-substitution\ninference rule, it follows that\n((2)++ = 3). QED"}, {"color": "#FFF59D", "id": "P21", "label": "P21 (2.1.4) : (3\nis-a natural-number)", "labelHighlightBold": true, "shape": "box", "title": "Proposition 2.1.4 (T1.P21): (3\nis-a natural-number). Proof:\n((((0)++)++)++ is-a natural-\nnumber) follows from prop.\n(P12). ((((0)++)++)++ = 3)\nfollows from prop. (P19).\nTherefore, by the equal-terms-\nsubstitution inference rule, it\nfollows that (3 is-a natural-\nnumber). QED"}, {"color": "#FFF59D", "id": "P22", "label": "P22 : (4 =\n((((0)++)++)++)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P22): (4 =\n((((0)++)++)++)++). Proof: \"We\ndefine 1 to be the number 0++, 2\nto be the number (0++)++, 3 to\nbe the number ((0++)++)++,etc.\n(In other words, 1 := 0++, 2 :=\n1++, 3 := 2++, etc. In this text\nI use \"x := y\" to denote the\nstatement that x is defined to\nequal y.)\" is postulated by def.\n(D1). (4 = ((((0)++)++)++)++) is\nan interpretation of that\ndefinition. Therefore, by the\ndefinition-interpretation\ninference rule, it follows that\n(4 = ((((0)++)++)++)++). QED"}, {"color": "#FFF59D", "id": "P23", "label": "P23 :\n(((((0)++)++)++)++ =\n4)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P23):\n(((((0)++)++)++)++ = 4). Proof:\n(4 = ((((0)++)++)++)++) follows\nfrom prop. (P8). Therefore, by\nthe equality-commutativity\ninference rule, it follows that\n(((((0)++)++)++)++ = 4). QED"}, {"color": "#FFF59D", "id": "P24", "label": "P24 : ((3)++ = 4)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P24): ((3)++ =\n4). Proof: (((((0)++)++)++)++ =\n4) follows from prop. (P23).\n((((0)++)++)++ = 3) follows from\nprop. (P19). Therefore, by the\nequal-terms-substitution\ninference rule, it follows that\n((3)++ = 4). QED"}, {"color": "#FFF59D", "id": "P25", "label": "P25 :\n(((((0)++)++)++ is-a\nnatural-number) ==\u003e\n(((((0)++)++)++)++\nis-a natural-\nnumber))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P25):\n(((((0)++)++)++ is-a natural-\nnumber) ==\u003e (((((0)++)++)++)++\nis-a natural-number)). Proof:\n(((((0)++)++)++ is-a natural-\nnumber) ==\u003e (((((0)++)++)++)++\nis-a natural-number)) follows\nfrom prop. (P13). ((3)++ = 4)\nfollows from prop. (P24).\nTherefore, by the equal-terms-\nsubstitution inference rule, it\nfollows that (((((0)++)++)++\nis-a natural-number) ==\u003e\n(((((0)++)++)++)++ is-a natural-\nnumber)). QED"}, {"color": "#FFF59D", "id": "P26", "label": "P26 : (4 is-a\nnatural-number)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P26): (4 is-a\nnatural-number). Proof:\n(((((0)++)++)++)++ is-a natural-\nnumber) follows from prop.\n(P14). (((((0)++)++)++)++ = 4)\nfollows from prop. (P23).\nTherefore, by the equal-terms-\nsubstitution inference rule, it\nfollows that (4 is-a natural-\nnumber). QED"}, {"color": "#81C784", "id": "A3", "label": "A3 (2.3) : \"0 is not\nthe successor of any\nnatural number;\ni.e., we have n++ 0\nfor every natural\nnumber n.\"", "shape": "box"}, {"color": "#FFF59D", "id": "P27", "label": "P27 : ((n2 is-a\nnatural-number) ==\u003e\n((n2)++ neq 0))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P27): ((n2 is-a\nnatural-number) ==\u003e ((n2)++ neq\n0)). Proof: \"0 is not the\nsuccessor of any natural number;\ni.e., we have n++ 0 for every\nnatural number n.\" is postulated\nby axiom 2.3 (A3). ((n2 is-a\nnatural-number) ==\u003e ((n2)++ neq\n0)) is a valid formula statement\ninterpreted from that axiom.\nTherefore, by the axiom-\ninterpretation inference rule,\nit follows that ((n2 is-a\nnatural-number) ==\u003e ((n2)++ neq\n0)). QED"}, {"color": "#FFF59D", "id": "P28", "label": "P28 : ((3 is-a\nnatural-number) ==\u003e\n((3)++ neq 0))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P28): ((3 is-a\nnatural-number) ==\u003e ((3)++ neq\n0)). Proof: ((n2 is-a natural-\nnumber) ==\u003e ((n2)++ neq 0))\nfollows from prop. (P27). Let n2\n= 3. Therefore, by the variable-\nsubstitution inference rule, it\nfollows that ((3 is-a natural-\nnumber) ==\u003e ((3)++ neq 0)). QED"}, {"color": "#FFF59D", "id": "P29", "label": "P29 : ((3)++ neq 0)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P29): ((3)++ neq\n0). Proof: ((3 is-a natural-\nnumber) ==\u003e ((3)++ neq 0))\nfollows from prop. (P28).(3 is-a\nnatural-number) follows from\nprop. 2.1.4 (P21). Therefore, by\nthe modus-ponens inference rule,\nit follows that ((3)++ neq 0).\nQED"}, {"color": "#FFF59D", "id": "P30", "label": "P30 (2.1.6) : (4 neq\n0)", "labelHighlightBold": true, "shape": "box", "title": "Proposition 2.1.6 (T1.P30): (4\nneq 0). Proof: ((3)++ neq 0)\nfollows from prop. (P29). ((3)++\n= 4) follows from prop. (P24).\nTherefore, by the equal-terms-\nsubstitution inference rule, it\nfollows that (4 neq 0). QED"}, {"color": "#81C784", "id": "A4", "label": "A4 (2.4) :\n\"Different natural\nnumbers must have\ndifferent\nsuccessors; i.e., if\nn, m are natural\nnumbers and n m,\nthen n++ m++.\nEquivalently, if n++\n= m++, then we must\nhave n = m.\"", "shape": "box"}, {"color": "#FFF59D", "id": "P31", "label": "P31 : ((((n3 is-a\nnatural-number) and\n(m1 is-a natural-\nnumber)) and (n3 neq\nm1)) ==\u003e ((n3)++ neq\n(m1)++))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P31): ((((n3\nis-a natural-number) and (m1\nis-a natural-number)) and (n3\nneq m1)) ==\u003e ((n3)++ neq\n(m1)++)). Proof: \"Different\nnatural numbers must have\ndifferent successors; i.e., if\nn, m are natural numbers and n\nm, then n++ m++. Equivalently,\nif n++ = m++, then we must have\nn = m.\" is postulated by axiom\n2.4 (A4). ((((n3 is-a natural-\nnumber) and (m1 is-a natural-\nnumber)) and (n3 neq m1)) ==\u003e\n((n3)++ neq (m1)++)) is a valid\nformula statement interpreted\nfrom that axiom. Therefore, by\nthe axiom-interpretation\ninference rule, it follows that\n((((n3 is-a natural-number) and\n(m1 is-a natural-number)) and\n(n3 neq m1)) ==\u003e ((n3)++ neq\n(m1)++)). QED"}, {"color": "#FFF59D", "id": "P32", "label": "P32 : ((((n4 is-a\nnatural-number) and\n(m2 is-a natural-\nnumber)) and ((n4)++\n= (m2)++)) ==\u003e (n4 =\nm2))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P32): ((((n4\nis-a natural-number) and (m2\nis-a natural-number)) and\n((n4)++ = (m2)++)) ==\u003e (n4 =\nm2)). Proof: \"Different natural\nnumbers must have different\nsuccessors; i.e., if n, m are\nnatural numbers and n m, then\nn++ m++. Equivalently, if n++ =\nm++, then we must have n = m.\"\nis postulated by axiom 2.4 (A4).\n((((n4 is-a natural-number) and\n(m2 is-a natural-number)) and\n((n4)++ = (m2)++)) ==\u003e (n4 =\nm2)) is a valid formula\nstatement interpreted from that\naxiom. Therefore, by the axiom-\ninterpretation inference rule,\nit follows that ((((n4 is-a\nnatural-number) and (m2 is-a\nnatural-number)) and ((n4)++ =\n(m2)++)) ==\u003e (n4 = m2)). QED"}, {"color": "#FFF59D", "id": "P33", "label": "P33 : ((((4 is-a\nnatural-number) and\n(0 is-a natural-\nnumber)) and (4 neq\n0)) ==\u003e ((4)++ neq\n(0)++))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P33): ((((4 is-a\nnatural-number) and (0 is-a\nnatural-number)) and (4 neq 0))\n==\u003e ((4)++ neq (0)++)). Proof:\n((((n3 is-a natural-number) and\n(m1 is-a natural-number)) and\n(n3 neq m1)) ==\u003e ((n3)++ neq\n(m1)++)) follows from prop.\n(P31). Let n3 = 4, m1 = 0.\nTherefore, by the variable-\nsubstitution inference rule, it\nfollows that ((((4 is-a natural-\nnumber) and (0 is-a natural-\nnumber)) and (4 neq 0)) ==\u003e\n((4)++ neq (0)++)). QED"}, {"color": "#FFF59D", "id": "P34", "label": "P34 : ((4 is-a\nnatural-number) and\n(0 is-a natural-\nnumber))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P34): ((4 is-a\nnatural-number) and (0 is-a\nnatural-number)). Proof: (4 is-a\nnatural-number) follows from\nprop. (P26). (0 is-a natural-\nnumber) follows from prop. (P1).\nTherefore, by the conjunction-\nintroduction inference rule, it\nfollows that ((4 is-a natural-\nnumber) and (0 is-a natural-\nnumber)). QED"}, {"color": "#FFF59D", "id": "P35", "label": "P35 : (((4 is-a\nnatural-number) and\n(0 is-a natural-\nnumber)) and (4 neq\n0))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P35): (((4 is-a\nnatural-number) and (0 is-a\nnatural-number)) and (4 neq 0)).\nProof: ((4 is-a natural-number)\nand (0 is-a natural-number))\nfollows from prop. (P34). (4 neq\n0) follows from prop. 2.1.6\n(P30). Therefore, by the\nconjunction-introduction\ninference rule, it follows that\n(((4 is-a natural-number) and (0\nis-a natural-number)) and (4 neq\n0)). QED"}, {"color": "#FFF59D", "id": "P36", "label": "P36 : ((4)++ neq\n(0)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P36): ((4)++ neq\n(0)++). Proof: ((((4 is-a\nnatural-number) and (0 is-a\nnatural-number)) and (4 neq 0))\n==\u003e ((4)++ neq (0)++)) follows\nfrom prop. (P33).(((4 is-a\nnatural-number) and (0 is-a\nnatural-number)) and (4 neq 0))\nfollows from prop. (P35).\nTherefore, by the modus-ponens\ninference rule, it follows that\n((4)++ neq (0)++). QED"}, {"color": "#FFF59D", "id": "P37", "label": "P37 : (5 = (((((0)++\n)++)++)++)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P37): (5 =\n(((((0)++)++)++)++)++). Proof:\n\"We define 1 to be the number\n0++, 2 to be the number (0++)++,\n3 to be the number\n((0++)++)++,etc. (In other\nwords, 1 := 0++, 2 := 1++, 3 :=\n2++, etc. In this text I use \"x\n:= y\" to denote the statement\nthat x is defined to equal y.)\"\nis postulated by def. (D1). (5 =\n(((((0)++)++)++)++)++) is an\ninterpretation of that\ndefinition. Therefore, by the\ndefinition-interpretation\ninference rule, it follows that\n(5 = (((((0)++)++)++)++)++). QED"}, {"color": "#FFF59D", "id": "P38", "label": "P38 : ((((((0)++)++)\n++)++)++ = 5)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P38):\n((((((0)++)++)++)++)++ = 5).\nProof: (5 =\n(((((0)++)++)++)++)++) follows\nfrom prop. (P37). Therefore, by\nthe equality-commutativity\ninference rule, it follows that\n((((((0)++)++)++)++)++ = 5). QED"}, {"color": "#FFF59D", "id": "P39", "label": "P39 : ((4)++ = 5)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P39): ((4)++ =\n5). Proof:\n((((((0)++)++)++)++)++ = 5)\nfollows from prop. (P38).\n(((((0)++)++)++)++ = 4) follows\nfrom prop. (P23). Therefore, by\nthe equal-terms-substitution\ninference rule, it follows that\n((4)++ = 5). QED"}, {"color": "#FFF59D", "id": "P40", "label": "P40 : (5 = (4)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P40): (5 =\n(4)++). Proof: ((4)++ = 5)\nfollows from prop. (P39).\nTherefore, by the equality-\ncommutativity inference rule, it\nfollows that (5 = (4)++). QED"}, {"color": "#FFF59D", "id": "P41", "label": "P41 : ((((5 is-a\nnatural-number) and\n(1 is-a natural-\nnumber)) and (5 neq\n1)) ==\u003e ((5)++ neq\n(1)++))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P41): ((((5 is-a\nnatural-number) and (1 is-a\nnatural-number)) and (5 neq 1))\n==\u003e ((5)++ neq (1)++)). Proof:\n((((n3 is-a natural-number) and\n(m1 is-a natural-number)) and\n(n3 neq m1)) ==\u003e ((n3)++ neq\n(m1)++)) follows from prop.\n(P31). Let n3 = 5, m1 = 1.\nTherefore, by the variable-\nsubstitution inference rule, it\nfollows that ((((5 is-a natural-\nnumber) and (1 is-a natural-\nnumber)) and (5 neq 1)) ==\u003e\n((5)++ neq (1)++)). QED"}, {"color": "#FFF59D", "id": "P42", "label": "P42 : ((4 is-a\nnatural-number) ==\u003e\n((4)++ is-a natural-\nnumber))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P42): ((4 is-a\nnatural-number) ==\u003e ((4)++ is-a\nnatural-number)). Proof: ((n1\nis-a natural-number) ==\u003e ((n1)++\nis-a natural-number)) follows\nfrom prop. (P2). Let n1 = 4.\nTherefore, by the variable-\nsubstitution inference rule, it\nfollows that ((4 is-a natural-\nnumber) ==\u003e ((4)++ is-a natural-\nnumber)). QED"}, {"color": "#FFF59D", "id": "P43", "label": "P43 : ((4 is-a\nnatural-number) ==\u003e\n(5 is-a natural-\nnumber))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P43): ((4 is-a\nnatural-number) ==\u003e (5 is-a\nnatural-number)). Proof: ((4\nis-a natural-number) ==\u003e ((4)++\nis-a natural-number)) follows\nfrom prop. (P42). ((4)++ = 5)\nfollows from prop. (P39).\nTherefore, by the equal-terms-\nsubstitution inference rule, it\nfollows that ((4 is-a natural-\nnumber) ==\u003e (5 is-a natural-\nnumber)). QED"}, {"color": "#FFF59D", "id": "P44", "label": "P44 : (5 is-a\nnatural-number)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P44): (5 is-a\nnatural-number). Proof: ((4 is-a\nnatural-number) ==\u003e (5 is-a\nnatural-number)) follows from\nprop. (P43).(4 is-a natural-\nnumber) follows from prop.\n(P26). Therefore, by the modus-\nponens inference rule, it\nfollows that (5 is-a natural-\nnumber). QED"}, {"color": "#FFF59D", "id": "P45", "label": "P45 : ((((5 is-a\nnatural-number) and\n(1 is-a natural-\nnumber)) and (5 neq\n1)) ==\u003e ((5)++ neq\n(1)++))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P45): ((((5 is-a\nnatural-number) and (1 is-a\nnatural-number)) and (5 neq 1))\n==\u003e ((5)++ neq (1)++)). Proof:\n((((n3 is-a natural-number) and\n(m1 is-a natural-number)) and\n(n3 neq m1)) ==\u003e ((n3)++ neq\n(m1)++)) follows from prop.\n(P31). Let n3 = 5, m1 = 1.\nTherefore, by the variable-\nsubstitution inference rule, it\nfollows that ((((5 is-a natural-\nnumber) and (1 is-a natural-\nnumber)) and (5 neq 1)) ==\u003e\n((5)++ neq (1)++)). QED"}, {"color": "#FFF59D", "id": "P46", "label": "P46 : ((4)++ neq\n(0)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P46): ((4)++ neq\n(0)++). Proof: ((((4 is-a\nnatural-number) and (0 is-a\nnatural-number)) and (4 neq 0))\n==\u003e ((4)++ neq (0)++)) follows\nfrom prop. (P33).(((4 is-a\nnatural-number) and (0 is-a\nnatural-number)) and (4 neq 0))\nfollows from prop. (P35).\nTherefore, by the modus-ponens\ninference rule, it follows that\n((4)++ neq (0)++). QED"}, {"color": "#FFF59D", "id": "P47", "label": "P47 : (5 neq (0)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P47): (5 neq\n(0)++). Proof: ((4)++ neq (0)++)\nfollows from prop. (P46). ((4)++\n= 5) follows from prop. (P39).\nTherefore, by the equal-terms-\nsubstitution inference rule, it\nfollows that (5 neq (0)++). QED"}, {"color": "#FFF59D", "id": "P48", "label": "P48 : (6 = ((((((0)+\n+)++)++)++)++)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P48): (6 =\n((((((0)++)++)++)++)++)++).\nProof: \"We define 1 to be the\nnumber 0++, 2 to be the number\n(0++)++, 3 to be the number\n((0++)++)++,etc. (In other\nwords, 1 := 0++, 2 := 1++, 3 :=\n2++, etc. In this text I use \"x\n:= y\" to denote the statement\nthat x is defined to equal y.)\"\nis postulated by def. (D1). (6 =\n((((((0)++)++)++)++)++)++) is an\ninterpretation of that\ndefinition. Therefore, by the\ndefinition-interpretation\ninference rule, it follows that\n(6 = ((((((0)++)++)++)++)++)++).\nQED"}, {"color": "#FFF59D", "id": "P49", "label": "P49 : (((((((0)++)++\n)++)++)++)++ = 6)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P49):\n(((((((0)++)++)++)++)++)++ = 6).\nProof: (6 =\n((((((0)++)++)++)++)++)++)\nfollows from prop. (P48).\nTherefore, by the equality-\ncommutativity inference rule, it\nfollows that\n(((((((0)++)++)++)++)++)++ = 6).\nQED"}, {"color": "#FFF59D", "id": "P50", "label": "P50 : (1 is-a\nnatural-number)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P50): (1 is-a\nnatural-number). Proof: ((0)++\nis-a natural-number) follows\nfrom prop. 2.2.3 (P4). ((0)++ =\n1) follows from prop. (P15).\nTherefore, by the equal-terms-\nsubstitution inference rule, it\nfollows that (1 is-a natural-\nnumber). QED"}, {"color": "#FFF59D", "id": "P51", "label": "P51 : ((5)++ = 6)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P51): ((5)++ =\n6). Proof:\n(((((((0)++)++)++)++)++)++ = 6)\nfollows from prop. (P49).\n((((((0)++)++)++)++)++ = 5)\nfollows from prop. (P38).\nTherefore, by the equal-terms-\nsubstitution inference rule, it\nfollows that ((5)++ = 6). QED"}, {"color": "#FFF59D", "id": "P52", "label": "P52 : (6 = (5)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P52): (6 =\n(5)++). Proof: ((5)++ = 6)\nfollows from prop. (P51).\nTherefore, by the equality-\ncommutativity inference rule, it\nfollows that (6 = (5)++). QED"}, {"color": "#FFF59D", "id": "P66", "label": "P66 : Inc(H1)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P66): Inc(H1).\nProof: P65 follows from prop.\n(P65). P30 follows from prop.\n2.1.6 (P30). Therefore, by the\ninconsistency-by-inequality-\nintroduction inference rule, it\nfollows that Inc(H1). QED"}, {"color": "#FFF59D", "id": "P65", "label": "P65 : (4 = 0)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (H1.P65): (4 = 0).\nProof: ((((4 is-a natural-\nnumber) and (0 is-a natural-\nnumber)) and ((4)++ = (0)++))\n==\u003e (4 = 0)) follows from prop.\n(P64).(((4 is-a natural-number)\nand (0 is-a natural-number)) and\n((4)++ = (0)++)) follows from\nprop. (P63). Therefore, by the\nmodus-ponens inference rule, it\nfollows that (4 = 0). QED"}, {"color": "#FFF59D", "id": "P64", "label": "P64 : ((((4 is-a\nnatural-number) and\n(0 is-a natural-\nnumber)) and ((4)++\n= (0)++)) ==\u003e (4 =\n0))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (H1.P64): ((((4 is-a\nnatural-number) and (0 is-a\nnatural-number)) and ((4)++ =\n(0)++)) ==\u003e (4 = 0)). Proof:\n((((n4 is-a natural-number) and\n(m2 is-a natural-number)) and\n((n4)++ = (m2)++)) ==\u003e (n4 =\nm2)) follows from prop. (P32).\nLet n4 = 4, m2 = 0. Therefore,\nby the variable-substitution\ninference rule, it follows that\n((((4 is-a natural-number) and\n(0 is-a natural-number)) and\n((4)++ = (0)++)) ==\u003e (4 = 0)).\nQED"}, {"color": "#FFF59D", "id": "P63", "label": "P63 : (((4 is-a\nnatural-number) and\n(0 is-a natural-\nnumber)) and ((4)++\n= (0)++))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (H1.P63): (((4 is-a\nnatural-number) and (0 is-a\nnatural-number)) and ((4)++ =\n(0)++)). Proof: ((4 is-a\nnatural-number) and (0 is-a\nnatural-number)) follows from\nprop. (P62). ((4)++ = (0)++)\nfollows from prop. (P61).\nTherefore, by the conjunction-\nintroduction inference rule, it\nfollows that (((4 is-a natural-\nnumber) and (0 is-a natural-\nnumber)) and ((4)++ = (0)++)).\nQED"}, {"color": "#FFF59D", "id": "P62", "label": "P62 : ((4 is-a\nnatural-number) and\n(0 is-a natural-\nnumber))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (H1.P62): ((4 is-a\nnatural-number) and (0 is-a\nnatural-number)). Proof: (4 is-a\nnatural-number) follows from\nprop. (P26). (0 is-a natural-\nnumber) follows from prop. (P1).\nTherefore, by the conjunction-\nintroduction inference rule, it\nfollows that ((4 is-a natural-\nnumber) and (0 is-a natural-\nnumber)). QED"}, {"color": "#FFF59D", "id": "P61", "label": "P61 : ((4)++ =\n(0)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (H1.P61): ((4)++ =\n(0)++). Proof: ((4)++ = 1)\nfollows from prop. (P60). (1 =\n(0)++) follows from prop. (P5).\nTherefore, by the equal-terms-\nsubstitution inference rule, it\nfollows that ((4)++ = (0)++).\nQED"}, {"color": "#FFF59D", "id": "P60", "label": "P60 : ((4)++ = 1)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (H1.P60): ((4)++ =\n1). Proof: (5 = 1) follows from\nprop. (P59). (5 = (4)++) follows\nfrom prop. (P40). Therefore, by\nthe equal-terms-substitution\ninference rule, it follows that\n((4)++ = 1). QED"}, {"color": "#FFF59D", "id": "P59", "label": "P59 : (5 = 1)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (H1.P59): (5 = 1).\nProof: ((((5 is-a natural-\nnumber) and (1 is-a natural-\nnumber)) and ((5)++ = (1)++))\n==\u003e (5 = 1)) follows from prop.\n(P58).(((5 is-a natural-number)\nand (1 is-a natural-number)) and\n((5)++ = (1)++)) follows from\nprop. (P57). Therefore, by the\nmodus-ponens inference rule, it\nfollows that (5 = 1). QED"}, {"color": "#FFF59D", "id": "P58", "label": "P58 : ((((5 is-a\nnatural-number) and\n(1 is-a natural-\nnumber)) and ((5)++\n= (1)++)) ==\u003e (5 =\n1))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (H1.P58): ((((5 is-a\nnatural-number) and (1 is-a\nnatural-number)) and ((5)++ =\n(1)++)) ==\u003e (5 = 1)). Proof:\n((((n4 is-a natural-number) and\n(m2 is-a natural-number)) and\n((n4)++ = (m2)++)) ==\u003e (n4 =\nm2)) follows from prop. (P32).\nLet n4 = 5, m2 = 1. Therefore,\nby the variable-substitution\ninference rule, it follows that\n((((5 is-a natural-number) and\n(1 is-a natural-number)) and\n((5)++ = (1)++)) ==\u003e (5 = 1)).\nQED"}, {"color": "#FFF59D", "id": "P57", "label": "P57 : (((5 is-a\nnatural-number) and\n(1 is-a natural-\nnumber)) and ((5)++\n= (1)++))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (H1.P57): (((5 is-a\nnatural-number) and (1 is-a\nnatural-number)) and ((5)++ =\n(1)++)). Proof: ((5 is-a\nnatural-number) and (1 is-a\nnatural-number)) follows from\nprop. (P56). ((5)++ = (1)++)\nfollows from prop. (P55).\nTherefore, by the conjunction-\nintroduction inference rule, it\nfollows that (((5 is-a natural-\nnumber) and (1 is-a natural-\nnumber)) and ((5)++ = (1)++)).\nQED"}, {"color": "#FFF59D", "id": "P56", "label": "P56 : ((5 is-a\nnatural-number) and\n(1 is-a natural-\nnumber))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (H1.P56): ((5 is-a\nnatural-number) and (1 is-a\nnatural-number)). Proof: (5 is-a\nnatural-number) follows from\nprop. (P44). (1 is-a natural-\nnumber) follows from prop.\n(P50). Therefore, by the\nconjunction-introduction\ninference rule, it follows that\n((5 is-a natural-number) and (1\nis-a natural-number)). QED"}, {"color": "#FFF59D", "id": "P55", "label": "P55 : ((5)++ =\n(1)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (H1.P55): ((5)++ =\n(1)++). Proof: ((5)++ = 2)\nfollows from prop. (P54). (2 =\n(1)++) follows from prop. (P16).\nTherefore, by the equal-terms-\nsubstitution inference rule, it\nfollows that ((5)++ = (1)++).\nQED"}, {"color": "#FFF59D", "id": "P54", "label": "P54 : ((5)++ = 2)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (H1.P54): ((5)++ =\n2). Proof: (6 = 2) follows from\nprop. (P53). (6 = (5)++) follows\nfrom prop. (P52). Therefore, by\nthe equal-terms-substitution\ninference rule, it follows that\n((5)++ = 2). QED"}, {"color": "#FFF59D", "id": "P53", "label": "P53 : (6 = 2)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (H1.P53): (6 = 2).\nProof: \"By hypothesis, assume (6\n= 2) is true.\" is postulated by\naxiom (A5). (6 = 2) is a valid\nformula statement interpreted\nfrom that axiom. Therefore, by\nthe axiom-interpretation\ninference rule, it follows that\n(6 = 2). QED"}, {"color": "#81C784", "id": "A5", "label": "A5 : \"By hypothesis,\nassume (6 = 2) is\ntrue.\"", "shape": "box"}, {"color": "#FFF59D", "id": "P67", "label": "P67 (2.1.8) : (6 neq\n2)", "labelHighlightBold": true, "shape": "box", "title": "Proposition 2.1.8 (T1.P67): (6\nneq 2). Proof: Let hyp. (H1) be\nthe hypothesis (6 = 2). Inc(H1)\nfollows from prop. (P66).\nTherefore, by the proof-by-\nrefutation-of-equality inference\nrule, it follows that (6 neq 2).\nQED"}, {"color": "#FFF59D", "id": "P68", "label": "P68 : ((1)++ = 2)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P68): ((1)++ =\n2). Proof: (((0)++)++ = 2)\nfollows from prop. (P17). ((0)++\n= 1) follows from prop. (P15).\nTherefore, by the equal-terms-\nsubstitution inference rule, it\nfollows that ((1)++ = 2). QED"}, {"color": "#FFF59D", "id": "P69", "label": "P69 : (5 neq 1)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P69): (5 neq 1).\nProof: (5 neq (0)++) follows\nfrom prop. (P47). ((0)++ = 1)\nfollows from prop. (P15).\nTherefore, by the equal-terms-\nsubstitution inference rule, it\nfollows that (5 neq 1). QED"}, {"color": "#FFF59D", "id": "P70", "label": "P70 : ((((5 is-a\nnatural-number) and\n(1 is-a natural-\nnumber)) and (5 neq\n1)) ==\u003e (6 neq\n(1)++))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P70): ((((5 is-a\nnatural-number) and (1 is-a\nnatural-number)) and (5 neq 1))\n==\u003e (6 neq (1)++)). Proof: ((((5\nis-a natural-number) and (1 is-a\nnatural-number)) and (5 neq 1))\n==\u003e ((5)++ neq (1)++)) follows\nfrom prop. (P45). ((5)++ = 6)\nfollows from prop. (P51).\nTherefore, by the equal-terms-\nsubstitution inference rule, it\nfollows that ((((5 is-a natural-\nnumber) and (1 is-a natural-\nnumber)) and (5 neq 1)) ==\u003e (6\nneq (1)++)). QED"}, {"color": "#FFF59D", "id": "P71", "label": "P71 : ((((5 is-a\nnatural-number) and\n(1 is-a natural-\nnumber)) and (5 neq\n1)) ==\u003e (6 neq 2))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P71): ((((5 is-a\nnatural-number) and (1 is-a\nnatural-number)) and (5 neq 1))\n==\u003e (6 neq 2)). Proof: ((((5\nis-a natural-number) and (1 is-a\nnatural-number)) and (5 neq 1))\n==\u003e (6 neq (1)++)) follows from\nprop. (P70). ((1)++ = 2) follows\nfrom prop. (P68). Therefore, by\nthe equal-terms-substitution\ninference rule, it follows that\n((((5 is-a natural-number) and\n(1 is-a natural-number)) and (5\nneq 1)) ==\u003e (6 neq 2)). QED"}, {"color": "#FFF59D", "id": "P72", "label": "P72 : ((5 is-a\nnatural-number) and\n(1 is-a natural-\nnumber))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P72): ((5 is-a\nnatural-number) and (1 is-a\nnatural-number)). Proof: (5 is-a\nnatural-number) follows from\nprop. (P44). (1 is-a natural-\nnumber) follows from prop.\n(P50). Therefore, by the\nconjunction-introduction\ninference rule, it follows that\n((5 is-a natural-number) and (1\nis-a natural-number)). QED"}, {"color": "#FFF59D", "id": "P73", "label": "P73 : (((5 is-a\nnatural-number) and\n(1 is-a natural-\nnumber)) and (5 neq\n1))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P73): (((5 is-a\nnatural-number) and (1 is-a\nnatural-number)) and (5 neq 1)).\nProof: ((5 is-a natural-number)\nand (1 is-a natural-number))\nfollows from prop. (P72). (5 neq\n1) follows from prop. (P69).\nTherefore, by the conjunction-\nintroduction inference rule, it\nfollows that (((5 is-a natural-\nnumber) and (1 is-a natural-\nnumber)) and (5 neq 1)). QED"}, {"color": "#FFF59D", "id": "P74", "label": "P74 : (6 neq 2)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P74): (6 neq 2).\nProof: ((((5 is-a natural-\nnumber) and (1 is-a natural-\nnumber)) and (5 neq 1)) ==\u003e (6\nneq 2)) follows from prop.\n(P71).(((5 is-a natural-number)\nand (1 is-a natural-number)) and\n(5 neq 1)) follows from prop.\n(P73). Therefore, by the modus-\nponens inference rule, it\nfollows that (6 neq 2). QED"}, {"color": "#81C784", "id": "A6", "label": "A6 : \"Let P(n) be\nany property\npertaining to a\nnatural number n.\nSuppose that P(O) is\ntrue, and suppose\nthat whenever P(n)\nis true, P(n++) is\nalso true. Then P(n)\nis true for every\nnatural number n.\"", "shape": "box"}, {"color": "#FFF59D", "id": "P75", "label": "P75 : (((n5 is-a\nnatural-number) and\n(P1(0) and (P1(n5)\n==\u003e P1((n5)++))))\n==\u003e ((m3 is-a\nnatural-number) ==\u003e\nP1(m3)))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P75): (((n5 is-a\nnatural-number) and (P1(0) and\n(P1(n5) ==\u003e P1((n5)++)))) ==\u003e\n((m3 is-a natural-number) ==\u003e\nP1(m3))). Proof: \"Let P(n) be\nany property pertaining to a\nnatural number n. Suppose that\nP(O) is true, and suppose that\nwhenever P(n) is true, P(n++) is\nalso true. Then P(n) is true for\nevery natural number n.\" is\npostulated by axiom schema (A6).\n(((n5 is-a natural-number) and\n(P1(0) and (P1(n5) ==\u003e\nP1((n5)++)))) ==\u003e ((m3 is-a\nnatural-number) ==\u003e P1(m3))) is\na valid formula statement\ninterpreted from that axiom.\nTherefore, by the axiom-\ninterpretation inference rule,\nit follows that (((n5 is-a\nnatural-number) and (P1(0) and\n(P1(n5) ==\u003e P1((n5)++)))) ==\u003e\n((m3 is-a natural-number) ==\u003e\nP1(m3))). QED"}]); + nodes = new vis.DataSet([{"color": "#81C784", "id": "A1", "label": "A1 (2.1) : \"0 is a\nnatural number.\"", "shape": "box"}, {"color": "#FFF59D", "id": "P1", "label": "P1 : (0 is-a\nnatural-number)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P1): (0 is-a\nnatural-number). Proof: \"0 is a\nnatural number.\" is postulated\nby axiom 2.1 (A1). (0 is-a\nnatural-number) is a valid\nformula statement interpreted\nfrom that axiom.Therefore, by\nthe axiom-interpretation\ninference rule: \ud835\udc9c \u22a2 P, it\nfollows that (0 is-a natural-\nnumber). QED"}, {"color": "#81C784", "id": "A2", "label": "A2 (2.2) : \"If n is\na natural number,\nthen n++ is a\nnatural number.\"", "shape": "box"}, {"color": "#FFF59D", "id": "P2", "label": "P2 : ((n1 is-a\nnatural-number) ==\u003e\n((n1)++ is-a\nnatural-number))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P2): ((n1 is-a\nnatural-number) ==\u003e ((n1)++ is-a\nnatural-number)). Proof: \"If n\nis a natural number, then n++ is\na natural number.\" is postulated\nby axiom 2.2 (A2). ((n1 is-a\nnatural-number) ==\u003e ((n1)++ is-a\nnatural-number)) is a valid\nformula statement interpreted\nfrom that axiom.Therefore, by\nthe axiom-interpretation\ninference rule: \ud835\udc9c \u22a2 P, it\nfollows that ((n1 is-a natural-\nnumber) ==\u003e ((n1)++ is-a\nnatural-number)). QED"}, {"color": "#FFF59D", "id": "P3", "label": "P3 : ((0 is-a\nnatural-number) ==\u003e\n((0)++ is-a natural-\nnumber))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P3): ((0 is-a\nnatural-number) ==\u003e ((0)++ is-a\nnatural-number)). Proof: ((n1\nis-a natural-number) ==\u003e ((n1)++\nis-a natural-number)) follows\nfrom prop. (P2). Let n1 =\n0.Therefore, by the variable-\nsubstitution inference rule: (P,\n\ud835\udef7) \u22a2 P\u0027, it follows that ((0\nis-a natural-number) ==\u003e ((0)++\nis-a natural-number)). QED"}, {"color": "#FFF59D", "id": "P4", "label": "P4 (2.2.3) : ((0)++\nis-a natural-number)", "labelHighlightBold": true, "shape": "box", "title": "Proposition 2.2.3 (T1.P4):\n((0)++ is-a natural-number).\nProof: ((0 is-a natural-number)\n==\u003e ((0)++ is-a natural-number))\nfollows from prop. (P3).(0 is-a\nnatural-number) follows from\nprop. (P1).Therefore, by the\nmodus-ponens inference rule: ((P\n\u27f9 Q), P) \u22a2 Q, it follows that\n((0)++ is-a natural-number). QED"}, {"color": "#90CAF9", "id": "D1", "label": "D1 : \"We define 1 to\nbe the number 0++, 2\nto be the number\n(0++)++, 3 to be the\nnumber\n((0++)++)++,etc. (In\nother words, 1 :=\n0++, 2 := 1++, 3 :=\n2++, etc. In this\ntext I use \"x := y\"\nto denote the\nstatement that x is\ndefined to equal\ny.)\"", "shape": "box"}, {"color": "#FFF59D", "id": "P5", "label": "P5 : (1 = (0)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P5): (1 =\n(0)++). Proof: \"We define 1 to\nbe the number 0++, 2 to be the\nnumber (0++)++, 3 to be the\nnumber ((0++)++)++,etc. (In\nother words, 1 := 0++, 2 := 1++,\n3 := 2++, etc. In this text I\nuse \"x := y\" to denote the\nstatement that x is defined to\nequal y.)\" is postulated by def.\n(D1). (1 = (0)++) is an\ninterpretation of that\ndefinition.Therefore, by the\ndefinition-interpretation\ninference rule: \ud835\udc9f \u22a2 P, it\nfollows that (1 = (0)++). QED"}, {"color": "#FFF59D", "id": "P6", "label": "P6 : (2 = ((0)++)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P6): (2 =\n((0)++)++). Proof: \"We define 1\nto be the number 0++, 2 to be\nthe number (0++)++, 3 to be the\nnumber ((0++)++)++,etc. (In\nother words, 1 := 0++, 2 := 1++,\n3 := 2++, etc. In this text I\nuse \"x := y\" to denote the\nstatement that x is defined to\nequal y.)\" is postulated by def.\n(D1). (2 = ((0)++)++) is an\ninterpretation of that\ndefinition.Therefore, by the\ndefinition-interpretation\ninference rule: \ud835\udc9f \u22a2 P, it\nfollows that (2 = ((0)++)++).\nQED"}, {"color": "#FFF59D", "id": "P7", "label": "P7 : (3 =\n(((0)++)++)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P7): (3 =\n(((0)++)++)++). Proof: \"We\ndefine 1 to be the number 0++, 2\nto be the number (0++)++, 3 to\nbe the number ((0++)++)++,etc.\n(In other words, 1 := 0++, 2 :=\n1++, 3 := 2++, etc. In this text\nI use \"x := y\" to denote the\nstatement that x is defined to\nequal y.)\" is postulated by def.\n(D1). (3 = (((0)++)++)++) is an\ninterpretation of that\ndefinition.Therefore, by the\ndefinition-interpretation\ninference rule: \ud835\udc9f \u22a2 P, it\nfollows that (3 =\n(((0)++)++)++). QED"}, {"color": "#FFF59D", "id": "P8", "label": "P8 : (4 =\n((((0)++)++)++)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P8): (4 =\n((((0)++)++)++)++). Proof: \"We\ndefine 1 to be the number 0++, 2\nto be the number (0++)++, 3 to\nbe the number ((0++)++)++,etc.\n(In other words, 1 := 0++, 2 :=\n1++, 3 := 2++, etc. In this text\nI use \"x := y\" to denote the\nstatement that x is defined to\nequal y.)\" is postulated by def.\n(D1). (4 = ((((0)++)++)++)++) is\nan interpretation of that\ndefinition.Therefore, by the\ndefinition-interpretation\ninference rule: \ud835\udc9f \u22a2 P, it\nfollows that (4 =\n((((0)++)++)++)++). QED"}, {"color": "#FFF59D", "id": "P9", "label": "P9 : (((0)++ is-a\nnatural-number) ==\u003e\n(((0)++)++ is-a\nnatural-number))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P9): (((0)++\nis-a natural-number) ==\u003e\n(((0)++)++ is-a natural-\nnumber)). Proof: ((n1 is-a\nnatural-number) ==\u003e ((n1)++ is-a\nnatural-number)) follows from\nprop. (P2). Let n1 =\n(0)++.Therefore, by the\nvariable-substitution inference\nrule: (P, \ud835\udef7) \u22a2 P\u0027, it follows\nthat (((0)++ is-a natural-\nnumber) ==\u003e (((0)++)++ is-a\nnatural-number)). QED"}, {"color": "#FFF59D", "id": "P10", "label": "P10 : (((0)++)++\nis-a natural-number)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P10): (((0)++)++\nis-a natural-number). Proof:\n(((0)++ is-a natural-number) ==\u003e\n(((0)++)++ is-a natural-number))\nfollows from prop. (P9).((0)++\nis-a natural-number) follows\nfrom prop. 2.2.3 (P4).Therefore,\nby the modus-ponens inference\nrule: ((P \u27f9 Q), P) \u22a2 Q, it\nfollows that (((0)++)++ is-a\nnatural-number). QED"}, {"color": "#FFF59D", "id": "P11", "label": "P11 : ((((0)++)++\nis-a natural-number)\n==\u003e ((((0)++)++)++\nis-a natural-\nnumber))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P11):\n((((0)++)++ is-a natural-number)\n==\u003e ((((0)++)++)++ is-a natural-\nnumber)). Proof: ((n1 is-a\nnatural-number) ==\u003e ((n1)++ is-a\nnatural-number)) follows from\nprop. (P2). Let n1 =\n((0)++)++.Therefore, by the\nvariable-substitution inference\nrule: (P, \ud835\udef7) \u22a2 P\u0027, it follows\nthat ((((0)++)++ is-a natural-\nnumber) ==\u003e ((((0)++)++)++ is-a\nnatural-number)). QED"}, {"color": "#FFF59D", "id": "P12", "label": "P12 : ((((0)++)++)++\nis-a natural-number)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P12):\n((((0)++)++)++ is-a natural-\nnumber). Proof: ((((0)++)++ is-a\nnatural-number) ==\u003e\n((((0)++)++)++ is-a natural-\nnumber)) follows from prop.\n(P11).(((0)++)++ is-a natural-\nnumber) follows from prop.\n(P10).Therefore, by the modus-\nponens inference rule: ((P \u27f9 Q),\nP) \u22a2 Q, it follows that\n((((0)++)++)++ is-a natural-\nnumber). QED"}, {"color": "#FFF59D", "id": "P13", "label": "P13 :\n(((((0)++)++)++ is-a\nnatural-number) ==\u003e\n(((((0)++)++)++)++\nis-a natural-\nnumber))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P13):\n(((((0)++)++)++ is-a natural-\nnumber) ==\u003e (((((0)++)++)++)++\nis-a natural-number)). Proof:\n((n1 is-a natural-number) ==\u003e\n((n1)++ is-a natural-number))\nfollows from prop. (P2). Let n1\n= (((0)++)++)++.Therefore, by\nthe variable-substitution\ninference rule: (P, \ud835\udef7) \u22a2 P\u0027, it\nfollows that (((((0)++)++)++\nis-a natural-number) ==\u003e\n(((((0)++)++)++)++ is-a natural-\nnumber)). QED"}, {"color": "#FFF59D", "id": "P14", "label": "P14 :\n(((((0)++)++)++)++\nis-a natural-number)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P14):\n(((((0)++)++)++)++ is-a natural-\nnumber). Proof: (((((0)++)++)++\nis-a natural-number) ==\u003e\n(((((0)++)++)++)++ is-a natural-\nnumber)) follows from prop.\n(P13).((((0)++)++)++ is-a\nnatural-number) follows from\nprop. (P12).Therefore, by the\nmodus-ponens inference rule: ((P\n\u27f9 Q), P) \u22a2 Q, it follows that\n(((((0)++)++)++)++ is-a natural-\nnumber). QED"}, {"color": "#FFF59D", "id": "P15", "label": "P15 : ((0)++ = 1)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P15): ((0)++ =\n1). Proof: (1 = (0)++) follows\nfrom prop. (P5). Therefore, by\nthe equality-commutativity\ninference rule: (x = y) \u22a2 (y =\nx), it follows that ((0)++ = 1).\nQED"}, {"color": "#FFF59D", "id": "P16", "label": "P16 : (2 = (1)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P16): (2 =\n(1)++). Proof: (2 = ((0)++)++)\nfollows from prop. (P6). ((0)++\n= 1) follows from prop.\n(P15).Therefore, by the equal-\nterms-substitution inference\nrule: (P, (Q = R)) \u22a2 P\u0027, it\nfollows that (2 = (1)++). QED"}, {"color": "#FFF59D", "id": "P17", "label": "P17 : (((0)++)++ =\n2)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P17): (((0)++)++\n= 2). Proof: (2 = ((0)++)++)\nfollows from prop. (P6).\nTherefore, by the equality-\ncommutativity inference rule: (x\n= y) \u22a2 (y = x), it follows that\n(((0)++)++ = 2). QED"}, {"color": "#FFF59D", "id": "P18", "label": "P18 : (3 = (2)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P18): (3 =\n(2)++). Proof: (3 =\n(((0)++)++)++) follows from\nprop. (P7). (((0)++)++ = 2)\nfollows from prop.\n(P17).Therefore, by the equal-\nterms-substitution inference\nrule: (P, (Q = R)) \u22a2 P\u0027, it\nfollows that (3 = (2)++). QED"}, {"color": "#FFF59D", "id": "P19", "label": "P19 : ((((0)++)++)++\n= 3)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P19):\n((((0)++)++)++ = 3). Proof: (3 =\n(((0)++)++)++) follows from\nprop. (P7). Therefore, by the\nequality-commutativity inference\nrule: (x = y) \u22a2 (y = x), it\nfollows that ((((0)++)++)++ =\n3). QED"}, {"color": "#FFF59D", "id": "P20", "label": "P20 : ((2)++ = 3)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P20): ((2)++ =\n3). Proof: ((((0)++)++)++ = 3)\nfollows from prop. (P19).\n(((0)++)++ = 2) follows from\nprop. (P17).Therefore, by the\nequal-terms-substitution\ninference rule: (P, (Q = R)) \u22a2\nP\u0027, it follows that ((2)++ = 3).\nQED"}, {"color": "#FFF59D", "id": "P21", "label": "P21 (2.1.4) : (3\nis-a natural-number)", "labelHighlightBold": true, "shape": "box", "title": "Proposition 2.1.4 (T1.P21): (3\nis-a natural-number). Proof:\n((((0)++)++)++ is-a natural-\nnumber) follows from prop.\n(P12). ((((0)++)++)++ = 3)\nfollows from prop.\n(P19).Therefore, by the equal-\nterms-substitution inference\nrule: (P, (Q = R)) \u22a2 P\u0027, it\nfollows that (3 is-a natural-\nnumber). QED"}, {"color": "#FFF59D", "id": "P22", "label": "P22 : (4 =\n((((0)++)++)++)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P22): (4 =\n((((0)++)++)++)++). Proof: \"We\ndefine 1 to be the number 0++, 2\nto be the number (0++)++, 3 to\nbe the number ((0++)++)++,etc.\n(In other words, 1 := 0++, 2 :=\n1++, 3 := 2++, etc. In this text\nI use \"x := y\" to denote the\nstatement that x is defined to\nequal y.)\" is postulated by def.\n(D1). (4 = ((((0)++)++)++)++) is\nan interpretation of that\ndefinition.Therefore, by the\ndefinition-interpretation\ninference rule: \ud835\udc9f \u22a2 P, it\nfollows that (4 =\n((((0)++)++)++)++). QED"}, {"color": "#FFF59D", "id": "P23", "label": "P23 :\n(((((0)++)++)++)++ =\n4)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P23):\n(((((0)++)++)++)++ = 4). Proof:\n(4 = ((((0)++)++)++)++) follows\nfrom prop. (P8). Therefore, by\nthe equality-commutativity\ninference rule: (x = y) \u22a2 (y =\nx), it follows that\n(((((0)++)++)++)++ = 4). QED"}, {"color": "#FFF59D", "id": "P24", "label": "P24 : ((3)++ = 4)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P24): ((3)++ =\n4). Proof: (((((0)++)++)++)++ =\n4) follows from prop. (P23).\n((((0)++)++)++ = 3) follows from\nprop. (P19).Therefore, by the\nequal-terms-substitution\ninference rule: (P, (Q = R)) \u22a2\nP\u0027, it follows that ((3)++ = 4).\nQED"}, {"color": "#FFF59D", "id": "P25", "label": "P25 :\n(((((0)++)++)++ is-a\nnatural-number) ==\u003e\n(((((0)++)++)++)++\nis-a natural-\nnumber))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P25):\n(((((0)++)++)++ is-a natural-\nnumber) ==\u003e (((((0)++)++)++)++\nis-a natural-number)). Proof:\n(((((0)++)++)++ is-a natural-\nnumber) ==\u003e (((((0)++)++)++)++\nis-a natural-number)) follows\nfrom prop. (P13). ((3)++ = 4)\nfollows from prop.\n(P24).Therefore, by the equal-\nterms-substitution inference\nrule: (P, (Q = R)) \u22a2 P\u0027, it\nfollows that (((((0)++)++)++\nis-a natural-number) ==\u003e\n(((((0)++)++)++)++ is-a natural-\nnumber)). QED"}, {"color": "#FFF59D", "id": "P26", "label": "P26 : (4 is-a\nnatural-number)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P26): (4 is-a\nnatural-number). Proof:\n(((((0)++)++)++)++ is-a natural-\nnumber) follows from prop.\n(P14). (((((0)++)++)++)++ = 4)\nfollows from prop.\n(P23).Therefore, by the equal-\nterms-substitution inference\nrule: (P, (Q = R)) \u22a2 P\u0027, it\nfollows that (4 is-a natural-\nnumber). QED"}, {"color": "#81C784", "id": "A3", "label": "A3 (2.3) : \"0 is not\nthe successor of any\nnatural number;\ni.e., we have n++ 0\nfor every natural\nnumber n.\"", "shape": "box"}, {"color": "#FFF59D", "id": "P27", "label": "P27 : ((n2 is-a\nnatural-number) ==\u003e\n((n2)++ neq 0))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P27): ((n2 is-a\nnatural-number) ==\u003e ((n2)++ neq\n0)). Proof: \"0 is not the\nsuccessor of any natural number;\ni.e., we have n++ 0 for every\nnatural number n.\" is postulated\nby axiom 2.3 (A3). ((n2 is-a\nnatural-number) ==\u003e ((n2)++ neq\n0)) is a valid formula statement\ninterpreted from that\naxiom.Therefore, by the axiom-\ninterpretation inference rule: \ud835\udc9c\n\u22a2 P, it follows that ((n2 is-a\nnatural-number) ==\u003e ((n2)++ neq\n0)). QED"}, {"color": "#FFF59D", "id": "P28", "label": "P28 : ((3 is-a\nnatural-number) ==\u003e\n((3)++ neq 0))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P28): ((3 is-a\nnatural-number) ==\u003e ((3)++ neq\n0)). Proof: ((n2 is-a natural-\nnumber) ==\u003e ((n2)++ neq 0))\nfollows from prop. (P27). Let n2\n= 3.Therefore, by the variable-\nsubstitution inference rule: (P,\n\ud835\udef7) \u22a2 P\u0027, it follows that ((3\nis-a natural-number) ==\u003e ((3)++\nneq 0)). QED"}, {"color": "#FFF59D", "id": "P29", "label": "P29 : ((3)++ neq 0)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P29): ((3)++ neq\n0). Proof: ((3 is-a natural-\nnumber) ==\u003e ((3)++ neq 0))\nfollows from prop. (P28).(3 is-a\nnatural-number) follows from\nprop. 2.1.4 (P21).Therefore, by\nthe modus-ponens inference rule:\n((P \u27f9 Q), P) \u22a2 Q, it follows\nthat ((3)++ neq 0). QED"}, {"color": "#FFF59D", "id": "P30", "label": "P30 (2.1.6) : (4 neq\n0)", "labelHighlightBold": true, "shape": "box", "title": "Proposition 2.1.6 (T1.P30): (4\nneq 0). Proof: ((3)++ neq 0)\nfollows from prop. (P29). ((3)++\n= 4) follows from prop.\n(P24).Therefore, by the equal-\nterms-substitution inference\nrule: (P, (Q = R)) \u22a2 P\u0027, it\nfollows that (4 neq 0). QED"}, {"color": "#81C784", "id": "A4", "label": "A4 (2.4) :\n\"Different natural\nnumbers must have\ndifferent\nsuccessors; i.e., if\nn, m are natural\nnumbers and n m,\nthen n++ m++.\nEquivalently, if n++\n= m++, then we must\nhave n = m.\"", "shape": "box"}, {"color": "#FFF59D", "id": "P31", "label": "P31 : ((((n3 is-a\nnatural-number) and\n(m1 is-a natural-\nnumber)) and (n3 neq\nm1)) ==\u003e ((n3)++ neq\n(m1)++))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P31): ((((n3\nis-a natural-number) and (m1\nis-a natural-number)) and (n3\nneq m1)) ==\u003e ((n3)++ neq\n(m1)++)). Proof: \"Different\nnatural numbers must have\ndifferent successors; i.e., if\nn, m are natural numbers and n\nm, then n++ m++. Equivalently,\nif n++ = m++, then we must have\nn = m.\" is postulated by axiom\n2.4 (A4). ((((n3 is-a natural-\nnumber) and (m1 is-a natural-\nnumber)) and (n3 neq m1)) ==\u003e\n((n3)++ neq (m1)++)) is a valid\nformula statement interpreted\nfrom that axiom.Therefore, by\nthe axiom-interpretation\ninference rule: \ud835\udc9c \u22a2 P, it\nfollows that ((((n3 is-a\nnatural-number) and (m1 is-a\nnatural-number)) and (n3 neq\nm1)) ==\u003e ((n3)++ neq (m1)++)).\nQED"}, {"color": "#FFF59D", "id": "P32", "label": "P32 : ((((n4 is-a\nnatural-number) and\n(m2 is-a natural-\nnumber)) and ((n4)++\n= (m2)++)) ==\u003e (n4 =\nm2))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P32): ((((n4\nis-a natural-number) and (m2\nis-a natural-number)) and\n((n4)++ = (m2)++)) ==\u003e (n4 =\nm2)). Proof: \"Different natural\nnumbers must have different\nsuccessors; i.e., if n, m are\nnatural numbers and n m, then\nn++ m++. Equivalently, if n++ =\nm++, then we must have n = m.\"\nis postulated by axiom 2.4 (A4).\n((((n4 is-a natural-number) and\n(m2 is-a natural-number)) and\n((n4)++ = (m2)++)) ==\u003e (n4 =\nm2)) is a valid formula\nstatement interpreted from that\naxiom.Therefore, by the axiom-\ninterpretation inference rule: \ud835\udc9c\n\u22a2 P, it follows that ((((n4 is-a\nnatural-number) and (m2 is-a\nnatural-number)) and ((n4)++ =\n(m2)++)) ==\u003e (n4 = m2)). QED"}, {"color": "#FFF59D", "id": "P33", "label": "P33 : ((((4 is-a\nnatural-number) and\n(0 is-a natural-\nnumber)) and (4 neq\n0)) ==\u003e ((4)++ neq\n(0)++))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P33): ((((4 is-a\nnatural-number) and (0 is-a\nnatural-number)) and (4 neq 0))\n==\u003e ((4)++ neq (0)++)). Proof:\n((((n3 is-a natural-number) and\n(m1 is-a natural-number)) and\n(n3 neq m1)) ==\u003e ((n3)++ neq\n(m1)++)) follows from prop.\n(P31). Let n3 = 4, m1 =\n0.Therefore, by the variable-\nsubstitution inference rule: (P,\n\ud835\udef7) \u22a2 P\u0027, it follows that ((((4\nis-a natural-number) and (0 is-a\nnatural-number)) and (4 neq 0))\n==\u003e ((4)++ neq (0)++)). QED"}, {"color": "#FFF59D", "id": "P34", "label": "P34 : ((4 is-a\nnatural-number) and\n(0 is-a natural-\nnumber))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P34): ((4 is-a\nnatural-number) and (0 is-a\nnatural-number)). Proof: (4 is-a\nnatural-number) (P) follows from\nprop. (P26). (0 is-a natural-\nnumber) (Q) follows from prop.\n(P1).Therefore, by the\nconjunction-introduction\ninference rule: (P, Q) \u22a2 (P \u22c0\nQ), it follows that ((4 is-a\nnatural-number) and (0 is-a\nnatural-number)). QED"}, {"color": "#FFF59D", "id": "P35", "label": "P35 : (((4 is-a\nnatural-number) and\n(0 is-a natural-\nnumber)) and (4 neq\n0))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P35): (((4 is-a\nnatural-number) and (0 is-a\nnatural-number)) and (4 neq 0)).\nProof: ((4 is-a natural-number)\nand (0 is-a natural-number)) (P)\nfollows from prop. (P34). (4 neq\n0) (Q) follows from prop. 2.1.6\n(P30).Therefore, by the\nconjunction-introduction\ninference rule: (P, Q) \u22a2 (P \u22c0\nQ), it follows that (((4 is-a\nnatural-number) and (0 is-a\nnatural-number)) and (4 neq 0)).\nQED"}, {"color": "#FFF59D", "id": "P36", "label": "P36 : ((4)++ neq\n(0)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P36): ((4)++ neq\n(0)++). Proof: ((((4 is-a\nnatural-number) and (0 is-a\nnatural-number)) and (4 neq 0))\n==\u003e ((4)++ neq (0)++)) follows\nfrom prop. (P33).(((4 is-a\nnatural-number) and (0 is-a\nnatural-number)) and (4 neq 0))\nfollows from prop.\n(P35).Therefore, by the modus-\nponens inference rule: ((P \u27f9 Q),\nP) \u22a2 Q, it follows that ((4)++\nneq (0)++). QED"}, {"color": "#FFF59D", "id": "P37", "label": "P37 : (5 = (((((0)++\n)++)++)++)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P37): (5 =\n(((((0)++)++)++)++)++). Proof:\n\"We define 1 to be the number\n0++, 2 to be the number (0++)++,\n3 to be the number\n((0++)++)++,etc. (In other\nwords, 1 := 0++, 2 := 1++, 3 :=\n2++, etc. In this text I use \"x\n:= y\" to denote the statement\nthat x is defined to equal y.)\"\nis postulated by def. (D1). (5 =\n(((((0)++)++)++)++)++) is an\ninterpretation of that\ndefinition.Therefore, by the\ndefinition-interpretation\ninference rule: \ud835\udc9f \u22a2 P, it\nfollows that (5 =\n(((((0)++)++)++)++)++). QED"}, {"color": "#FFF59D", "id": "P38", "label": "P38 : ((((((0)++)++)\n++)++)++ = 5)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P38):\n((((((0)++)++)++)++)++ = 5).\nProof: (5 =\n(((((0)++)++)++)++)++) follows\nfrom prop. (P37). Therefore, by\nthe equality-commutativity\ninference rule: (x = y) \u22a2 (y =\nx), it follows that\n((((((0)++)++)++)++)++ = 5). QED"}, {"color": "#FFF59D", "id": "P39", "label": "P39 : ((4)++ = 5)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P39): ((4)++ =\n5). Proof:\n((((((0)++)++)++)++)++ = 5)\nfollows from prop. (P38).\n(((((0)++)++)++)++ = 4) follows\nfrom prop. (P23).Therefore, by\nthe equal-terms-substitution\ninference rule: (P, (Q = R)) \u22a2\nP\u0027, it follows that ((4)++ = 5).\nQED"}, {"color": "#FFF59D", "id": "P40", "label": "P40 : (5 = (4)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P40): (5 =\n(4)++). Proof: ((4)++ = 5)\nfollows from prop. (P39).\nTherefore, by the equality-\ncommutativity inference rule: (x\n= y) \u22a2 (y = x), it follows that\n(5 = (4)++). QED"}, {"color": "#FFF59D", "id": "P41", "label": "P41 : ((((5 is-a\nnatural-number) and\n(1 is-a natural-\nnumber)) and (5 neq\n1)) ==\u003e ((5)++ neq\n(1)++))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P41): ((((5 is-a\nnatural-number) and (1 is-a\nnatural-number)) and (5 neq 1))\n==\u003e ((5)++ neq (1)++)). Proof:\n((((n3 is-a natural-number) and\n(m1 is-a natural-number)) and\n(n3 neq m1)) ==\u003e ((n3)++ neq\n(m1)++)) follows from prop.\n(P31). Let n3 = 5, m1 =\n1.Therefore, by the variable-\nsubstitution inference rule: (P,\n\ud835\udef7) \u22a2 P\u0027, it follows that ((((5\nis-a natural-number) and (1 is-a\nnatural-number)) and (5 neq 1))\n==\u003e ((5)++ neq (1)++)). QED"}, {"color": "#FFF59D", "id": "P42", "label": "P42 : ((4 is-a\nnatural-number) ==\u003e\n((4)++ is-a natural-\nnumber))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P42): ((4 is-a\nnatural-number) ==\u003e ((4)++ is-a\nnatural-number)). Proof: ((n1\nis-a natural-number) ==\u003e ((n1)++\nis-a natural-number)) follows\nfrom prop. (P2). Let n1 =\n4.Therefore, by the variable-\nsubstitution inference rule: (P,\n\ud835\udef7) \u22a2 P\u0027, it follows that ((4\nis-a natural-number) ==\u003e ((4)++\nis-a natural-number)). QED"}, {"color": "#FFF59D", "id": "P43", "label": "P43 : ((4 is-a\nnatural-number) ==\u003e\n(5 is-a natural-\nnumber))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P43): ((4 is-a\nnatural-number) ==\u003e (5 is-a\nnatural-number)). Proof: ((4\nis-a natural-number) ==\u003e ((4)++\nis-a natural-number)) follows\nfrom prop. (P42). ((4)++ = 5)\nfollows from prop.\n(P39).Therefore, by the equal-\nterms-substitution inference\nrule: (P, (Q = R)) \u22a2 P\u0027, it\nfollows that ((4 is-a natural-\nnumber) ==\u003e (5 is-a natural-\nnumber)). QED"}, {"color": "#FFF59D", "id": "P44", "label": "P44 : (5 is-a\nnatural-number)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P44): (5 is-a\nnatural-number). Proof: ((4 is-a\nnatural-number) ==\u003e (5 is-a\nnatural-number)) follows from\nprop. (P43).(4 is-a natural-\nnumber) follows from prop.\n(P26).Therefore, by the modus-\nponens inference rule: ((P \u27f9 Q),\nP) \u22a2 Q, it follows that (5 is-a\nnatural-number). QED"}, {"color": "#FFF59D", "id": "P45", "label": "P45 : ((((5 is-a\nnatural-number) and\n(1 is-a natural-\nnumber)) and (5 neq\n1)) ==\u003e ((5)++ neq\n(1)++))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P45): ((((5 is-a\nnatural-number) and (1 is-a\nnatural-number)) and (5 neq 1))\n==\u003e ((5)++ neq (1)++)). Proof:\n((((n3 is-a natural-number) and\n(m1 is-a natural-number)) and\n(n3 neq m1)) ==\u003e ((n3)++ neq\n(m1)++)) follows from prop.\n(P31). Let n3 = 5, m1 =\n1.Therefore, by the variable-\nsubstitution inference rule: (P,\n\ud835\udef7) \u22a2 P\u0027, it follows that ((((5\nis-a natural-number) and (1 is-a\nnatural-number)) and (5 neq 1))\n==\u003e ((5)++ neq (1)++)). QED"}, {"color": "#FFF59D", "id": "P46", "label": "P46 : ((4)++ neq\n(0)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P46): ((4)++ neq\n(0)++). Proof: ((((4 is-a\nnatural-number) and (0 is-a\nnatural-number)) and (4 neq 0))\n==\u003e ((4)++ neq (0)++)) follows\nfrom prop. (P33).(((4 is-a\nnatural-number) and (0 is-a\nnatural-number)) and (4 neq 0))\nfollows from prop.\n(P35).Therefore, by the modus-\nponens inference rule: ((P \u27f9 Q),\nP) \u22a2 Q, it follows that ((4)++\nneq (0)++). QED"}, {"color": "#FFF59D", "id": "P47", "label": "P47 : (5 neq (0)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P47): (5 neq\n(0)++). Proof: ((4)++ neq (0)++)\nfollows from prop. (P46). ((4)++\n= 5) follows from prop.\n(P39).Therefore, by the equal-\nterms-substitution inference\nrule: (P, (Q = R)) \u22a2 P\u0027, it\nfollows that (5 neq (0)++). QED"}, {"color": "#FFF59D", "id": "P48", "label": "P48 : (6 = ((((((0)+\n+)++)++)++)++)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P48): (6 =\n((((((0)++)++)++)++)++)++).\nProof: \"We define 1 to be the\nnumber 0++, 2 to be the number\n(0++)++, 3 to be the number\n((0++)++)++,etc. (In other\nwords, 1 := 0++, 2 := 1++, 3 :=\n2++, etc. In this text I use \"x\n:= y\" to denote the statement\nthat x is defined to equal y.)\"\nis postulated by def. (D1). (6 =\n((((((0)++)++)++)++)++)++) is an\ninterpretation of that\ndefinition.Therefore, by the\ndefinition-interpretation\ninference rule: \ud835\udc9f \u22a2 P, it\nfollows that (6 =\n((((((0)++)++)++)++)++)++). QED"}, {"color": "#FFF59D", "id": "P49", "label": "P49 : (((((((0)++)++\n)++)++)++)++ = 6)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P49):\n(((((((0)++)++)++)++)++)++ = 6).\nProof: (6 =\n((((((0)++)++)++)++)++)++)\nfollows from prop. (P48).\nTherefore, by the equality-\ncommutativity inference rule: (x\n= y) \u22a2 (y = x), it follows that\n(((((((0)++)++)++)++)++)++ = 6).\nQED"}, {"color": "#FFF59D", "id": "P50", "label": "P50 : (1 is-a\nnatural-number)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P50): (1 is-a\nnatural-number). Proof: ((0)++\nis-a natural-number) follows\nfrom prop. 2.2.3 (P4). ((0)++ =\n1) follows from prop.\n(P15).Therefore, by the equal-\nterms-substitution inference\nrule: (P, (Q = R)) \u22a2 P\u0027, it\nfollows that (1 is-a natural-\nnumber). QED"}, {"color": "#FFF59D", "id": "P51", "label": "P51 : ((5)++ = 6)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P51): ((5)++ =\n6). Proof:\n(((((((0)++)++)++)++)++)++ = 6)\nfollows from prop. (P49).\n((((((0)++)++)++)++)++ = 5)\nfollows from prop.\n(P38).Therefore, by the equal-\nterms-substitution inference\nrule: (P, (Q = R)) \u22a2 P\u0027, it\nfollows that ((5)++ = 6). QED"}, {"color": "#FFF59D", "id": "P52", "label": "P52 : (6 = (5)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P52): (6 =\n(5)++). Proof: ((5)++ = 6)\nfollows from prop. (P51).\nTherefore, by the equality-\ncommutativity inference rule: (x\n= y) \u22a2 (y = x), it follows that\n(6 = (5)++). QED"}, {"color": "#FFF59D", "id": "P66", "label": "P66 : Inc(H1)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P66): Inc(H1).\nProof: (4 = 0) follows from\nprop. (P65). (4 neq 0) follows\nfrom prop. 2.1.6 (P30).\nTherefore, by the inconsistency-\nby-inequality-introduction\ninference rule: ((P = Q), (P \u2260\nQ)) \u22a2 Inc(\ud835\udcaf), it follows that\nInc(H1). QED"}, {"color": "#FFF59D", "id": "P65", "label": "P65 : (4 = 0)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (H1.P65): (4 = 0).\nProof: ((((4 is-a natural-\nnumber) and (0 is-a natural-\nnumber)) and ((4)++ = (0)++))\n==\u003e (4 = 0)) follows from prop.\n(P64).(((4 is-a natural-number)\nand (0 is-a natural-number)) and\n((4)++ = (0)++)) follows from\nprop. (P63).Therefore, by the\nmodus-ponens inference rule: ((P\n\u27f9 Q), P) \u22a2 Q, it follows that (4\n= 0). QED"}, {"color": "#FFF59D", "id": "P64", "label": "P64 : ((((4 is-a\nnatural-number) and\n(0 is-a natural-\nnumber)) and ((4)++\n= (0)++)) ==\u003e (4 =\n0))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (H1.P64): ((((4 is-a\nnatural-number) and (0 is-a\nnatural-number)) and ((4)++ =\n(0)++)) ==\u003e (4 = 0)). Proof:\n((((n4 is-a natural-number) and\n(m2 is-a natural-number)) and\n((n4)++ = (m2)++)) ==\u003e (n4 =\nm2)) follows from prop. (P32).\nLet n4 = 4, m2 = 0.Therefore, by\nthe variable-substitution\ninference rule: (P, \ud835\udef7) \u22a2 P\u0027, it\nfollows that ((((4 is-a natural-\nnumber) and (0 is-a natural-\nnumber)) and ((4)++ = (0)++))\n==\u003e (4 = 0)). QED"}, {"color": "#FFF59D", "id": "P63", "label": "P63 : (((4 is-a\nnatural-number) and\n(0 is-a natural-\nnumber)) and ((4)++\n= (0)++))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (H1.P63): (((4 is-a\nnatural-number) and (0 is-a\nnatural-number)) and ((4)++ =\n(0)++)). Proof: ((4 is-a\nnatural-number) and (0 is-a\nnatural-number)) (P) follows\nfrom prop. (P62). ((4)++ =\n(0)++) (Q) follows from prop.\n(P61).Therefore, by the\nconjunction-introduction\ninference rule: (P, Q) \u22a2 (P \u22c0\nQ), it follows that (((4 is-a\nnatural-number) and (0 is-a\nnatural-number)) and ((4)++ =\n(0)++)). QED"}, {"color": "#FFF59D", "id": "P62", "label": "P62 : ((4 is-a\nnatural-number) and\n(0 is-a natural-\nnumber))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (H1.P62): ((4 is-a\nnatural-number) and (0 is-a\nnatural-number)). Proof: (4 is-a\nnatural-number) (P) follows from\nprop. (P26). (0 is-a natural-\nnumber) (Q) follows from prop.\n(P1).Therefore, by the\nconjunction-introduction\ninference rule: (P, Q) \u22a2 (P \u22c0\nQ), it follows that ((4 is-a\nnatural-number) and (0 is-a\nnatural-number)). QED"}, {"color": "#FFF59D", "id": "P61", "label": "P61 : ((4)++ =\n(0)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (H1.P61): ((4)++ =\n(0)++). Proof: ((4)++ = 1)\nfollows from prop. (P60). (1 =\n(0)++) follows from prop.\n(P5).Therefore, by the equal-\nterms-substitution inference\nrule: (P, (Q = R)) \u22a2 P\u0027, it\nfollows that ((4)++ = (0)++).\nQED"}, {"color": "#FFF59D", "id": "P60", "label": "P60 : ((4)++ = 1)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (H1.P60): ((4)++ =\n1). Proof: (5 = 1) follows from\nprop. (P59). (5 = (4)++) follows\nfrom prop. (P40).Therefore, by\nthe equal-terms-substitution\ninference rule: (P, (Q = R)) \u22a2\nP\u0027, it follows that ((4)++ = 1).\nQED"}, {"color": "#FFF59D", "id": "P59", "label": "P59 : (5 = 1)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (H1.P59): (5 = 1).\nProof: ((((5 is-a natural-\nnumber) and (1 is-a natural-\nnumber)) and ((5)++ = (1)++))\n==\u003e (5 = 1)) follows from prop.\n(P58).(((5 is-a natural-number)\nand (1 is-a natural-number)) and\n((5)++ = (1)++)) follows from\nprop. (P57).Therefore, by the\nmodus-ponens inference rule: ((P\n\u27f9 Q), P) \u22a2 Q, it follows that (5\n= 1). QED"}, {"color": "#FFF59D", "id": "P58", "label": "P58 : ((((5 is-a\nnatural-number) and\n(1 is-a natural-\nnumber)) and ((5)++\n= (1)++)) ==\u003e (5 =\n1))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (H1.P58): ((((5 is-a\nnatural-number) and (1 is-a\nnatural-number)) and ((5)++ =\n(1)++)) ==\u003e (5 = 1)). Proof:\n((((n4 is-a natural-number) and\n(m2 is-a natural-number)) and\n((n4)++ = (m2)++)) ==\u003e (n4 =\nm2)) follows from prop. (P32).\nLet n4 = 5, m2 = 1.Therefore, by\nthe variable-substitution\ninference rule: (P, \ud835\udef7) \u22a2 P\u0027, it\nfollows that ((((5 is-a natural-\nnumber) and (1 is-a natural-\nnumber)) and ((5)++ = (1)++))\n==\u003e (5 = 1)). QED"}, {"color": "#FFF59D", "id": "P57", "label": "P57 : (((5 is-a\nnatural-number) and\n(1 is-a natural-\nnumber)) and ((5)++\n= (1)++))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (H1.P57): (((5 is-a\nnatural-number) and (1 is-a\nnatural-number)) and ((5)++ =\n(1)++)). Proof: ((5 is-a\nnatural-number) and (1 is-a\nnatural-number)) (P) follows\nfrom prop. (P56). ((5)++ =\n(1)++) (Q) follows from prop.\n(P55).Therefore, by the\nconjunction-introduction\ninference rule: (P, Q) \u22a2 (P \u22c0\nQ), it follows that (((5 is-a\nnatural-number) and (1 is-a\nnatural-number)) and ((5)++ =\n(1)++)). QED"}, {"color": "#FFF59D", "id": "P56", "label": "P56 : ((5 is-a\nnatural-number) and\n(1 is-a natural-\nnumber))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (H1.P56): ((5 is-a\nnatural-number) and (1 is-a\nnatural-number)). Proof: (5 is-a\nnatural-number) (P) follows from\nprop. (P44). (1 is-a natural-\nnumber) (Q) follows from prop.\n(P50).Therefore, by the\nconjunction-introduction\ninference rule: (P, Q) \u22a2 (P \u22c0\nQ), it follows that ((5 is-a\nnatural-number) and (1 is-a\nnatural-number)). QED"}, {"color": "#FFF59D", "id": "P55", "label": "P55 : ((5)++ =\n(1)++)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (H1.P55): ((5)++ =\n(1)++). Proof: ((5)++ = 2)\nfollows from prop. (P54). (2 =\n(1)++) follows from prop.\n(P16).Therefore, by the equal-\nterms-substitution inference\nrule: (P, (Q = R)) \u22a2 P\u0027, it\nfollows that ((5)++ = (1)++).\nQED"}, {"color": "#FFF59D", "id": "P54", "label": "P54 : ((5)++ = 2)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (H1.P54): ((5)++ =\n2). Proof: (6 = 2) follows from\nprop. (P53). (6 = (5)++) follows\nfrom prop. (P52).Therefore, by\nthe equal-terms-substitution\ninference rule: (P, (Q = R)) \u22a2\nP\u0027, it follows that ((5)++ = 2).\nQED"}, {"color": "#FFF59D", "id": "P53", "label": "P53 : (6 = 2)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (H1.P53): (6 = 2).\nProof: \"By hypothesis, assume (6\n= 2) is true.\" is postulated by\naxiom (A5). (6 = 2) is a valid\nformula statement interpreted\nfrom that axiom.Therefore, by\nthe axiom-interpretation\ninference rule: \ud835\udc9c \u22a2 P, it\nfollows that (6 = 2). QED"}, {"color": "#81C784", "id": "A5", "label": "A5 : \"By hypothesis,\nassume (6 = 2) is\ntrue.\"", "shape": "box"}, {"color": "#FFF59D", "id": "P67", "label": "P67 (2.1.8) : (6 neq\n2)", "labelHighlightBold": true, "shape": "box", "title": "Proposition 2.1.8 (T1.P67): (6\nneq 2). Proof: Let hyp. (H1) be\nthe hypothesis (6 = 2). Inc(H1)\nfollows from prop.\n(P66).Therefore, by the proof-\nby-refutation-of-equality\ninference rule: (\u210b(P=Q), Inc(\u210b))\n\u22a2 (P \u2260 Q), it follows that (6\nneq 2). QED"}, {"color": "#FFF59D", "id": "P68", "label": "P68 : ((1)++ = 2)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P68): ((1)++ =\n2). Proof: (((0)++)++ = 2)\nfollows from prop. (P17). ((0)++\n= 1) follows from prop.\n(P15).Therefore, by the equal-\nterms-substitution inference\nrule: (P, (Q = R)) \u22a2 P\u0027, it\nfollows that ((1)++ = 2). QED"}, {"color": "#FFF59D", "id": "P69", "label": "P69 : (5 neq 1)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P69): (5 neq 1).\nProof: (5 neq (0)++) follows\nfrom prop. (P47). ((0)++ = 1)\nfollows from prop.\n(P15).Therefore, by the equal-\nterms-substitution inference\nrule: (P, (Q = R)) \u22a2 P\u0027, it\nfollows that (5 neq 1). QED"}, {"color": "#FFF59D", "id": "P70", "label": "P70 : ((((5 is-a\nnatural-number) and\n(1 is-a natural-\nnumber)) and (5 neq\n1)) ==\u003e (6 neq\n(1)++))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P70): ((((5 is-a\nnatural-number) and (1 is-a\nnatural-number)) and (5 neq 1))\n==\u003e (6 neq (1)++)). Proof: ((((5\nis-a natural-number) and (1 is-a\nnatural-number)) and (5 neq 1))\n==\u003e ((5)++ neq (1)++)) follows\nfrom prop. (P45). ((5)++ = 6)\nfollows from prop.\n(P51).Therefore, by the equal-\nterms-substitution inference\nrule: (P, (Q = R)) \u22a2 P\u0027, it\nfollows that ((((5 is-a natural-\nnumber) and (1 is-a natural-\nnumber)) and (5 neq 1)) ==\u003e (6\nneq (1)++)). QED"}, {"color": "#FFF59D", "id": "P71", "label": "P71 : ((((5 is-a\nnatural-number) and\n(1 is-a natural-\nnumber)) and (5 neq\n1)) ==\u003e (6 neq 2))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P71): ((((5 is-a\nnatural-number) and (1 is-a\nnatural-number)) and (5 neq 1))\n==\u003e (6 neq 2)). Proof: ((((5\nis-a natural-number) and (1 is-a\nnatural-number)) and (5 neq 1))\n==\u003e (6 neq (1)++)) follows from\nprop. (P70). ((1)++ = 2) follows\nfrom prop. (P68).Therefore, by\nthe equal-terms-substitution\ninference rule: (P, (Q = R)) \u22a2\nP\u0027, it follows that ((((5 is-a\nnatural-number) and (1 is-a\nnatural-number)) and (5 neq 1))\n==\u003e (6 neq 2)). QED"}, {"color": "#FFF59D", "id": "P72", "label": "P72 : ((5 is-a\nnatural-number) and\n(1 is-a natural-\nnumber))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P72): ((5 is-a\nnatural-number) and (1 is-a\nnatural-number)). Proof: (5 is-a\nnatural-number) (P) follows from\nprop. (P44). (1 is-a natural-\nnumber) (Q) follows from prop.\n(P50).Therefore, by the\nconjunction-introduction\ninference rule: (P, Q) \u22a2 (P \u22c0\nQ), it follows that ((5 is-a\nnatural-number) and (1 is-a\nnatural-number)). QED"}, {"color": "#FFF59D", "id": "P73", "label": "P73 : (((5 is-a\nnatural-number) and\n(1 is-a natural-\nnumber)) and (5 neq\n1))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P73): (((5 is-a\nnatural-number) and (1 is-a\nnatural-number)) and (5 neq 1)).\nProof: ((5 is-a natural-number)\nand (1 is-a natural-number)) (P)\nfollows from prop. (P72). (5 neq\n1) (Q) follows from prop.\n(P69).Therefore, by the\nconjunction-introduction\ninference rule: (P, Q) \u22a2 (P \u22c0\nQ), it follows that (((5 is-a\nnatural-number) and (1 is-a\nnatural-number)) and (5 neq 1)).\nQED"}, {"color": "#FFF59D", "id": "P74", "label": "P74 : (6 neq 2)", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P74): (6 neq 2).\nProof: ((((5 is-a natural-\nnumber) and (1 is-a natural-\nnumber)) and (5 neq 1)) ==\u003e (6\nneq 2)) follows from prop.\n(P71).(((5 is-a natural-number)\nand (1 is-a natural-number)) and\n(5 neq 1)) follows from prop.\n(P73).Therefore, by the modus-\nponens inference rule: ((P \u27f9 Q),\nP) \u22a2 Q, it follows that (6 neq\n2). QED"}, {"color": "#81C784", "id": "A6", "label": "A6 : \"Let P(n) be\nany property\npertaining to a\nnatural number n.\nSuppose that P(O) is\ntrue, and suppose\nthat whenever P(n)\nis true, P(n++) is\nalso true. Then P(n)\nis true for every\nnatural number n.\"", "shape": "box"}, {"color": "#FFF59D", "id": "P75", "label": "P75 : (((n5 is-a\nnatural-number) and\n(P1(0) and (P1(n5)\n==\u003e P1((n5)++))))\n==\u003e ((m3 is-a\nnatural-number) ==\u003e\nP1(m3)))", "labelHighlightBold": false, "shape": "box", "title": "Proposition (T1.P75): (((n5 is-a\nnatural-number) and (P1(0) and\n(P1(n5) ==\u003e P1((n5)++)))) ==\u003e\n((m3 is-a natural-number) ==\u003e\nP1(m3))). Proof: \"Let P(n) be\nany property pertaining to a\nnatural number n. Suppose that\nP(O) is true, and suppose that\nwhenever P(n) is true, P(n++) is\nalso true. Then P(n) is true for\nevery natural number n.\" is\npostulated by axiom schema (A6).\n(((n5 is-a natural-number) and\n(P1(0) and (P1(n5) ==\u003e\nP1((n5)++)))) ==\u003e ((m3 is-a\nnatural-number) ==\u003e P1(m3))) is\na valid formula statement\ninterpreted from that\naxiom.Therefore, by the axiom-\ninterpretation inference rule: \ud835\udc9c\n\u22a2 P, it follows that (((n5 is-a\nnatural-number) and (P1(0) and\n(P1(n5) ==\u003e P1((n5)++)))) ==\u003e\n((m3 is-a natural-number) ==\u003e\nP1(m3))). QED"}]); edges = new vis.DataSet([{"arrows": "to", "from": "A1", "to": "P1"}, {"arrows": "to", "from": "A2", "to": "P2"}, {"arrows": "to", "from": "P2", "to": "P3"}, {"arrows": "to", "from": "P3", "to": "P4"}, {"arrows": "to", "from": "P1", "to": "P4"}, {"arrows": "to", "from": "D1", "to": "P5"}, {"arrows": "to", "from": "D1", "to": "P6"}, {"arrows": "to", "from": "D1", "to": "P7"}, {"arrows": "to", "from": "D1", "to": "P8"}, {"arrows": "to", "from": "P2", "to": "P9"}, {"arrows": "to", "from": "P9", "to": "P10"}, {"arrows": "to", "from": "P4", "to": "P10"}, {"arrows": "to", "from": "P2", "to": "P11"}, {"arrows": "to", "from": "P11", "to": "P12"}, {"arrows": "to", "from": "P10", "to": "P12"}, {"arrows": "to", "from": "P2", "to": "P13"}, {"arrows": "to", "from": "P13", "to": "P14"}, {"arrows": "to", "from": "P12", "to": "P14"}, {"arrows": "to", "from": "P5", "to": "P15"}, {"arrows": "to", "from": "P6", "to": "P16"}, {"arrows": "to", "from": "P15", "to": "P16"}, {"arrows": "to", "from": "P6", "to": "P17"}, {"arrows": "to", "from": "P7", "to": "P18"}, {"arrows": "to", "from": "P17", "to": "P18"}, {"arrows": "to", "from": "P7", "to": "P19"}, {"arrows": "to", "from": "P19", "to": "P20"}, {"arrows": "to", "from": "P17", "to": "P20"}, {"arrows": "to", "from": "P12", "to": "P21"}, {"arrows": "to", "from": "P19", "to": "P21"}, {"arrows": "to", "from": "D1", "to": "P22"}, {"arrows": "to", "from": "P8", "to": "P23"}, {"arrows": "to", "from": "P23", "to": "P24"}, {"arrows": "to", "from": "P19", "to": "P24"}, {"arrows": "to", "from": "P13", "to": "P25"}, {"arrows": "to", "from": "P24", "to": "P25"}, {"arrows": "to", "from": "P14", "to": "P26"}, {"arrows": "to", "from": "P23", "to": "P26"}, {"arrows": "to", "from": "A3", "to": "P27"}, {"arrows": "to", "from": "P27", "to": "P28"}, {"arrows": "to", "from": "P28", "to": "P29"}, {"arrows": "to", "from": "P21", "to": "P29"}, {"arrows": "to", "from": "P29", "to": "P30"}, {"arrows": "to", "from": "P24", "to": "P30"}, {"arrows": "to", "from": "A4", "to": "P31"}, {"arrows": "to", "from": "A4", "to": "P32"}, {"arrows": "to", "from": "P31", "to": "P33"}, {"arrows": "to", "from": "P26", "to": "P34"}, {"arrows": "to", "from": "P1", "to": "P34"}, {"arrows": "to", "from": "P34", "to": "P35"}, {"arrows": "to", "from": "P30", "to": "P35"}, {"arrows": "to", "from": "P33", "to": "P36"}, {"arrows": "to", "from": "P35", "to": "P36"}, {"arrows": "to", "from": "D1", "to": "P37"}, {"arrows": "to", "from": "P37", "to": "P38"}, {"arrows": "to", "from": "P38", "to": "P39"}, {"arrows": "to", "from": "P23", "to": "P39"}, {"arrows": "to", "from": "P39", "to": "P40"}, {"arrows": "to", "from": "P31", "to": "P41"}, {"arrows": "to", "from": "P2", "to": "P42"}, {"arrows": "to", "from": "P42", "to": "P43"}, {"arrows": "to", "from": "P39", "to": "P43"}, {"arrows": "to", "from": "P43", "to": "P44"}, {"arrows": "to", "from": "P26", "to": "P44"}, {"arrows": "to", "from": "P31", "to": "P45"}, {"arrows": "to", "from": "P33", "to": "P46"}, {"arrows": "to", "from": "P35", "to": "P46"}, {"arrows": "to", "from": "P46", "to": "P47"}, {"arrows": "to", "from": "P39", "to": "P47"}, {"arrows": "to", "from": "D1", "to": "P48"}, {"arrows": "to", "from": "P48", "to": "P49"}, {"arrows": "to", "from": "P4", "to": "P50"}, {"arrows": "to", "from": "P15", "to": "P50"}, {"arrows": "to", "from": "P49", "to": "P51"}, {"arrows": "to", "from": "P38", "to": "P51"}, {"arrows": "to", "from": "P51", "to": "P52"}, {"arrows": "to", "from": "P32", "to": "P64"}, {"arrows": "to", "from": "P64", "to": "P65"}, {"arrows": "to", "from": "P26", "to": "P62"}, {"arrows": "to", "from": "P1", "to": "P62"}, {"arrows": "to", "from": "P62", "to": "P63"}, {"arrows": "to", "from": "P32", "to": "P58"}, {"arrows": "to", "from": "P58", "to": "P59"}, {"arrows": "to", "from": "P44", "to": "P56"}, {"arrows": "to", "from": "P50", "to": "P56"}, {"arrows": "to", "from": "P56", "to": "P57"}, {"arrows": "to", "from": "A5", "to": "P53"}, {"arrows": "to", "from": "P53", "to": "P54"}, {"arrows": "to", "from": "P52", "to": "P54"}, {"arrows": "to", "from": "P54", "to": "P55"}, {"arrows": "to", "from": "P16", "to": "P55"}, {"arrows": "to", "from": "P55", "to": "P57"}, {"arrows": "to", "from": "P57", "to": "P59"}, {"arrows": "to", "from": "P59", "to": "P60"}, {"arrows": "to", "from": "P40", "to": "P60"}, {"arrows": "to", "from": "P60", "to": "P61"}, {"arrows": "to", "from": "P5", "to": "P61"}, {"arrows": "to", "from": "P61", "to": "P63"}, {"arrows": "to", "from": "P63", "to": "P65"}, {"arrows": "to", "from": "P65", "to": "P66"}, {"arrows": "to", "from": "P30", "to": "P66"}, {"arrows": "to", "from": "P66", "to": "P67"}, {"arrows": "to", "from": "P17", "to": "P68"}, {"arrows": "to", "from": "P15", "to": "P68"}, {"arrows": "to", "from": "P47", "to": "P69"}, {"arrows": "to", "from": "P15", "to": "P69"}, {"arrows": "to", "from": "P45", "to": "P70"}, {"arrows": "to", "from": "P51", "to": "P70"}, {"arrows": "to", "from": "P70", "to": "P71"}, {"arrows": "to", "from": "P68", "to": "P71"}, {"arrows": "to", "from": "P44", "to": "P72"}, {"arrows": "to", "from": "P50", "to": "P72"}, {"arrows": "to", "from": "P72", "to": "P73"}, {"arrows": "to", "from": "P69", "to": "P73"}, {"arrows": "to", "from": "P71", "to": "P74"}, {"arrows": "to", "from": "P73", "to": "P74"}, {"arrows": "to", "from": "A6", "to": "P75"}]); nodeColors = {}; diff --git a/theory/build/tao_2006_2_1_the_peano_axioms_interactive_graph_unicode.html b/theory/build/tao_2006_2_1_the_peano_axioms_interactive_graph_unicode.html index 7a3d0808..a4a233c8 100644 --- a/theory/build/tao_2006_2_1_the_peano_axioms_interactive_graph_unicode.html +++ b/theory/build/tao_2006_2_1_the_peano_axioms_interactive_graph_unicode.html @@ -88,7 +88,7 @@

// parsing and collecting nodes and edges from the python - nodes = new vis.DataSet([{"color": "#81C784", "id": "A1", "label": "\ud835\udc34\u2081 (\ud835\udfee.\ud835\udfed) : \u231c\ud835\udfe2 \ud835\ude2a\ud835\ude34 \ud835\ude22\n\ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33.\u231d", "shape": "box"}, {"color": "#FFF59D", "id": "P1", "label": "\ud835\udc43\u2081 : (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2081): (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: \u231c\ud835\udfe2 \ud835\ude2a\ud835\ude34 \ud835\ude22\n\ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33.\u231d \ud835\uddc2\ud835\uddcc \ud835\uddc9\ud835\uddc8\ud835\uddcc\ud835\uddcd\ud835\uddce\ud835\uddc5\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddbd\n\ud835\uddbb\ud835\uddd2 \ud835\uddee\ud835\ude05\ud835\uddf6\ud835\uddfc\ud835\uddfa \ud835\udfee.\ud835\udfed (\ud835\udc34\u2081). (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \ud835\uddc2\ud835\uddcc \ud835\uddba \ud835\uddcf\ud835\uddba\ud835\uddc5\ud835\uddc2\ud835\uddbd\n\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddc6\ud835\uddce\ud835\uddc5\ud835\uddba \ud835\uddcc\ud835\uddcd\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddc6\ud835\uddbe\ud835\uddc7\ud835\uddcd \ud835\uddc2\ud835\uddc7\ud835\uddcd\ud835\uddbe\ud835\uddcb\ud835\uddc9\ud835\uddcb\ud835\uddbe\ud835\uddcd\ud835\uddbe\ud835\uddbd\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd \ud835\uddba\ud835\uddd1\ud835\uddc2\ud835\uddc8\ud835\uddc6. \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2\n\ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc4e\ud835\udc65\ud835\udc56\ud835\udc5c\ud835\udc5a-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5d\ud835\udc5f\ud835\udc52\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n(0 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \u220e"}, {"color": "#81C784", "id": "A2", "label": "\ud835\udc34\u2082 (\ud835\udfee.\ud835\udfee) : \u231c\ud835\ude10\ud835\ude27 \ud835\ude2f \ud835\ude2a\ud835\ude34\n\ud835\ude22 \ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33,\n\ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude2f \ud835\ude2f++ \ud835\ude2a\ud835\ude34 \ud835\ude22\n\ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33.\u231d", "shape": "box"}, {"color": "#FFF59D", "id": "P2", "label": "\ud835\udc43\u2082 : ((\ud835\udc27\u2081 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9\n((\ud835\udc27\u2081)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2082): ((\ud835\udc27\u2081 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((\ud835\udc27\u2081)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: \u231c\ud835\ude10\ud835\ude27 \ud835\ude2f\n\ud835\ude2a\ud835\ude34 \ud835\ude22 \ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33, \ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude2f \ud835\ude2f++ \ud835\ude2a\ud835\ude34\n\ud835\ude22 \ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33.\u231d \ud835\uddc2\ud835\uddcc \ud835\uddc9\ud835\uddc8\ud835\uddcc\ud835\uddcd\ud835\uddce\ud835\uddc5\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddbd\n\ud835\uddbb\ud835\uddd2 \ud835\uddee\ud835\ude05\ud835\uddf6\ud835\uddfc\ud835\uddfa \ud835\udfee.\ud835\udfee (\ud835\udc34\u2082). ((\ud835\udc27\u2081 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((\ud835\udc27\u2081)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \ud835\uddc2\ud835\uddcc \ud835\uddba \ud835\uddcf\ud835\uddba\ud835\uddc5\ud835\uddc2\ud835\uddbd\n\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddc6\ud835\uddce\ud835\uddc5\ud835\uddba \ud835\uddcc\ud835\uddcd\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddc6\ud835\uddbe\ud835\uddc7\ud835\uddcd \ud835\uddc2\ud835\uddc7\ud835\uddcd\ud835\uddbe\ud835\uddcb\ud835\uddc9\ud835\uddcb\ud835\uddbe\ud835\uddcd\ud835\uddbe\ud835\uddbd\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd \ud835\uddba\ud835\uddd1\ud835\uddc2\ud835\uddc8\ud835\uddc6. \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2\n\ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc4e\ud835\udc65\ud835\udc56\ud835\udc5c\ud835\udc5a-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5d\ud835\udc5f\ud835\udc52\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n((\ud835\udc27\u2081 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9\n((\ud835\udc27\u2081)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \u220e"}, {"color": "#FFF59D", "id": "P3", "label": "\ud835\udc43\u2083 : ((0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9\n((0)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2083): ((0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((0)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((\ud835\udc27\u2081\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((\ud835\udc27\u2081)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2082). \ud835\uddab\ud835\uddbe\ud835\uddcd \ud835\udc27\u2081 = 0.\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc63\ud835\udc4e\ud835\udc5f\ud835\udc56\ud835\udc4e\ud835\udc4f\ud835\udc59\ud835\udc52-\n\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((0 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((0)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \u220e"}, {"color": "#FFF59D", "id": "P4", "label": "\ud835\udc43\u2084 (\ud835\udfee.\ud835\udfee.\ud835\udfef) : ((0)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)", "labelHighlightBold": true, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb \ud835\udfee.\ud835\udfee.\ud835\udfef (\ud835\udcaf\u2081.\ud835\udc43\u2084):\n((0)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((0 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)\n\u27f9 ((0)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083).(0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2081). \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc5a\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc60-\ud835\udc5d\ud835\udc5c\ud835\udc5b\ud835\udc52\ud835\udc5b\ud835\udc60 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((0)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \u220e"}, {"color": "#90CAF9", "id": "D1", "label": "\ud835\udc37\u2081 : \u231c\ud835\ude1e\ud835\ude26 \ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26 \ud835\udfe3 \ud835\ude35\ud835\ude30\n\ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 \ud835\udfe2++, \ud835\udfe4\n\ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33\n(\ud835\udfe2++)++, \ud835\udfe5 \ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26\n\ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33\n((\ud835\udfe2++)++)++,\ud835\ude26\ud835\ude35\ud835\ude24. (\ud835\ude10\ud835\ude2f\n\ud835\ude30\ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude33 \ud835\ude38\ud835\ude30\ud835\ude33\ud835\ude25\ud835\ude34, \ud835\udfe3 :=\n\ud835\udfe2++, \ud835\udfe4 := \ud835\udfe3++, \ud835\udfe5 :=\n\ud835\udfe4++, \ud835\ude26\ud835\ude35\ud835\ude24. \ud835\ude10\ud835\ude2f \ud835\ude35\ud835\ude29\ud835\ude2a\ud835\ude34\n\ud835\ude35\ud835\ude26\ud835\ude39\ud835\ude35 \ud835\ude10 \ud835\ude36\ud835\ude34\ud835\ude26 \"\ud835\ude39 := \ud835\ude3a\"\n\ud835\ude35\ud835\ude30 \ud835\ude25\ud835\ude26\ud835\ude2f\ud835\ude30\ud835\ude35\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26\n\ud835\ude34\ud835\ude35\ud835\ude22\ud835\ude35\ud835\ude26\ud835\ude2e\ud835\ude26\ud835\ude2f\ud835\ude35 \ud835\ude35\ud835\ude29\ud835\ude22\ud835\ude35 \ud835\ude39 \ud835\ude2a\ud835\ude34\n\ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26\ud835\ude25 \ud835\ude35\ud835\ude30 \ud835\ude26\ud835\ude32\ud835\ude36\ud835\ude22\ud835\ude2d\n\ud835\ude3a.)\u231d", "shape": "box"}, {"color": "#FFF59D", "id": "P5", "label": "\ud835\udc43\u2085 : (1 = (0)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2085): (1 =\n(0)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: \u231c\ud835\ude1e\ud835\ude26 \ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26 \ud835\udfe3 \ud835\ude35\ud835\ude30\n\ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 \ud835\udfe2++, \ud835\udfe4 \ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26\n\ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 (\ud835\udfe2++)++, \ud835\udfe5 \ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26\n\ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 ((\ud835\udfe2++)++)++,\ud835\ude26\ud835\ude35\ud835\ude24. (\ud835\ude10\ud835\ude2f\n\ud835\ude30\ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude33 \ud835\ude38\ud835\ude30\ud835\ude33\ud835\ude25\ud835\ude34, \ud835\udfe3 := \ud835\udfe2++, \ud835\udfe4 := \ud835\udfe3++,\n\ud835\udfe5 := \ud835\udfe4++, \ud835\ude26\ud835\ude35\ud835\ude24. \ud835\ude10\ud835\ude2f \ud835\ude35\ud835\ude29\ud835\ude2a\ud835\ude34 \ud835\ude35\ud835\ude26\ud835\ude39\ud835\ude35 \ud835\ude10\n\ud835\ude36\ud835\ude34\ud835\ude26 \"\ud835\ude39 := \ud835\ude3a\" \ud835\ude35\ud835\ude30 \ud835\ude25\ud835\ude26\ud835\ude2f\ud835\ude30\ud835\ude35\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26\n\ud835\ude34\ud835\ude35\ud835\ude22\ud835\ude35\ud835\ude26\ud835\ude2e\ud835\ude26\ud835\ude2f\ud835\ude35 \ud835\ude35\ud835\ude29\ud835\ude22\ud835\ude35 \ud835\ude39 \ud835\ude2a\ud835\ude34 \ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26\ud835\ude25 \ud835\ude35\ud835\ude30\n\ud835\ude26\ud835\ude32\ud835\ude36\ud835\ude22\ud835\ude2d \ud835\ude3a.)\u231d \ud835\uddc2\ud835\uddcc \ud835\uddc9\ud835\uddc8\ud835\uddcc\ud835\uddcd\ud835\uddce\ud835\uddc5\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbb\ud835\uddd2 \ud835\uddf1\ud835\uddf2\ud835\uddf3.\n(\ud835\udc37\u2081). (1 = (0)++) \ud835\uddc2\ud835\uddcc \ud835\uddba\ud835\uddc7\n\ud835\uddc2\ud835\uddc7\ud835\uddcd\ud835\uddbe\ud835\uddcb\ud835\uddc9\ud835\uddcb\ud835\uddbe\ud835\uddcd\ud835\uddba\ud835\uddcd\ud835\uddc2\ud835\uddc8\ud835\uddc7 \ud835\uddc8\ud835\uddbf \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n\ud835\uddbd\ud835\uddbe\ud835\uddbf\ud835\uddc2\ud835\uddc7\ud835\uddc2\ud835\uddcd\ud835\uddc2\ud835\uddc8\ud835\uddc7. \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc51\ud835\udc52\ud835\udc53\ud835\udc56\ud835\udc5b\ud835\udc56\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5d\ud835\udc5f\ud835\udc52\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n(1 = (0)++). \u220e"}, {"color": "#FFF59D", "id": "P6", "label": "\ud835\udc43\u2086 : (2 = ((0)++)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2086): (2 =\n((0)++)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: \u231c\ud835\ude1e\ud835\ude26 \ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26 \ud835\udfe3\n\ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 \ud835\udfe2++, \ud835\udfe4 \ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26\n\ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 (\ud835\udfe2++)++, \ud835\udfe5 \ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26\n\ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 ((\ud835\udfe2++)++)++,\ud835\ude26\ud835\ude35\ud835\ude24. (\ud835\ude10\ud835\ude2f\n\ud835\ude30\ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude33 \ud835\ude38\ud835\ude30\ud835\ude33\ud835\ude25\ud835\ude34, \ud835\udfe3 := \ud835\udfe2++, \ud835\udfe4 := \ud835\udfe3++,\n\ud835\udfe5 := \ud835\udfe4++, \ud835\ude26\ud835\ude35\ud835\ude24. \ud835\ude10\ud835\ude2f \ud835\ude35\ud835\ude29\ud835\ude2a\ud835\ude34 \ud835\ude35\ud835\ude26\ud835\ude39\ud835\ude35 \ud835\ude10\n\ud835\ude36\ud835\ude34\ud835\ude26 \"\ud835\ude39 := \ud835\ude3a\" \ud835\ude35\ud835\ude30 \ud835\ude25\ud835\ude26\ud835\ude2f\ud835\ude30\ud835\ude35\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26\n\ud835\ude34\ud835\ude35\ud835\ude22\ud835\ude35\ud835\ude26\ud835\ude2e\ud835\ude26\ud835\ude2f\ud835\ude35 \ud835\ude35\ud835\ude29\ud835\ude22\ud835\ude35 \ud835\ude39 \ud835\ude2a\ud835\ude34 \ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26\ud835\ude25 \ud835\ude35\ud835\ude30\n\ud835\ude26\ud835\ude32\ud835\ude36\ud835\ude22\ud835\ude2d \ud835\ude3a.)\u231d \ud835\uddc2\ud835\uddcc \ud835\uddc9\ud835\uddc8\ud835\uddcc\ud835\uddcd\ud835\uddce\ud835\uddc5\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbb\ud835\uddd2 \ud835\uddf1\ud835\uddf2\ud835\uddf3.\n(\ud835\udc37\u2081). (2 = ((0)++)++) \ud835\uddc2\ud835\uddcc \ud835\uddba\ud835\uddc7\n\ud835\uddc2\ud835\uddc7\ud835\uddcd\ud835\uddbe\ud835\uddcb\ud835\uddc9\ud835\uddcb\ud835\uddbe\ud835\uddcd\ud835\uddba\ud835\uddcd\ud835\uddc2\ud835\uddc8\ud835\uddc7 \ud835\uddc8\ud835\uddbf \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n\ud835\uddbd\ud835\uddbe\ud835\uddbf\ud835\uddc2\ud835\uddc7\ud835\uddc2\ud835\uddcd\ud835\uddc2\ud835\uddc8\ud835\uddc7. \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc51\ud835\udc52\ud835\udc53\ud835\udc56\ud835\udc5b\ud835\udc56\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5d\ud835\udc5f\ud835\udc52\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n(2 = ((0)++)++). \u220e"}, {"color": "#FFF59D", "id": "P7", "label": "\ud835\udc43\u2087 : (3 =\n(((0)++)++)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2087): (3 =\n(((0)++)++)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: \u231c\ud835\ude1e\ud835\ude26\n\ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26 \ud835\udfe3 \ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 \ud835\udfe2++, \ud835\udfe4\n\ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 (\ud835\udfe2++)++, \ud835\udfe5 \ud835\ude35\ud835\ude30\n\ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 ((\ud835\udfe2++)++)++,\ud835\ude26\ud835\ude35\ud835\ude24.\n(\ud835\ude10\ud835\ude2f \ud835\ude30\ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude33 \ud835\ude38\ud835\ude30\ud835\ude33\ud835\ude25\ud835\ude34, \ud835\udfe3 := \ud835\udfe2++, \ud835\udfe4 :=\n\ud835\udfe3++, \ud835\udfe5 := \ud835\udfe4++, \ud835\ude26\ud835\ude35\ud835\ude24. \ud835\ude10\ud835\ude2f \ud835\ude35\ud835\ude29\ud835\ude2a\ud835\ude34 \ud835\ude35\ud835\ude26\ud835\ude39\ud835\ude35\n\ud835\ude10 \ud835\ude36\ud835\ude34\ud835\ude26 \"\ud835\ude39 := \ud835\ude3a\" \ud835\ude35\ud835\ude30 \ud835\ude25\ud835\ude26\ud835\ude2f\ud835\ude30\ud835\ude35\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26\n\ud835\ude34\ud835\ude35\ud835\ude22\ud835\ude35\ud835\ude26\ud835\ude2e\ud835\ude26\ud835\ude2f\ud835\ude35 \ud835\ude35\ud835\ude29\ud835\ude22\ud835\ude35 \ud835\ude39 \ud835\ude2a\ud835\ude34 \ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26\ud835\ude25 \ud835\ude35\ud835\ude30\n\ud835\ude26\ud835\ude32\ud835\ude36\ud835\ude22\ud835\ude2d \ud835\ude3a.)\u231d \ud835\uddc2\ud835\uddcc \ud835\uddc9\ud835\uddc8\ud835\uddcc\ud835\uddcd\ud835\uddce\ud835\uddc5\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbb\ud835\uddd2 \ud835\uddf1\ud835\uddf2\ud835\uddf3.\n(\ud835\udc37\u2081). (3 = (((0)++)++)++) \ud835\uddc2\ud835\uddcc \ud835\uddba\ud835\uddc7\n\ud835\uddc2\ud835\uddc7\ud835\uddcd\ud835\uddbe\ud835\uddcb\ud835\uddc9\ud835\uddcb\ud835\uddbe\ud835\uddcd\ud835\uddba\ud835\uddcd\ud835\uddc2\ud835\uddc8\ud835\uddc7 \ud835\uddc8\ud835\uddbf \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n\ud835\uddbd\ud835\uddbe\ud835\uddbf\ud835\uddc2\ud835\uddc7\ud835\uddc2\ud835\uddcd\ud835\uddc2\ud835\uddc8\ud835\uddc7. \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc51\ud835\udc52\ud835\udc53\ud835\udc56\ud835\udc5b\ud835\udc56\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5d\ud835\udc5f\ud835\udc52\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n(3 = (((0)++)++)++). \u220e"}, {"color": "#FFF59D", "id": "P8", "label": "\ud835\udc43\u2088 : (4 =\n((((0)++)++)++)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2088): (4 =\n((((0)++)++)++)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: \u231c\ud835\ude1e\ud835\ude26\n\ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26 \ud835\udfe3 \ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 \ud835\udfe2++, \ud835\udfe4\n\ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 (\ud835\udfe2++)++, \ud835\udfe5 \ud835\ude35\ud835\ude30\n\ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 ((\ud835\udfe2++)++)++,\ud835\ude26\ud835\ude35\ud835\ude24.\n(\ud835\ude10\ud835\ude2f \ud835\ude30\ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude33 \ud835\ude38\ud835\ude30\ud835\ude33\ud835\ude25\ud835\ude34, \ud835\udfe3 := \ud835\udfe2++, \ud835\udfe4 :=\n\ud835\udfe3++, \ud835\udfe5 := \ud835\udfe4++, \ud835\ude26\ud835\ude35\ud835\ude24. \ud835\ude10\ud835\ude2f \ud835\ude35\ud835\ude29\ud835\ude2a\ud835\ude34 \ud835\ude35\ud835\ude26\ud835\ude39\ud835\ude35\n\ud835\ude10 \ud835\ude36\ud835\ude34\ud835\ude26 \"\ud835\ude39 := \ud835\ude3a\" \ud835\ude35\ud835\ude30 \ud835\ude25\ud835\ude26\ud835\ude2f\ud835\ude30\ud835\ude35\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26\n\ud835\ude34\ud835\ude35\ud835\ude22\ud835\ude35\ud835\ude26\ud835\ude2e\ud835\ude26\ud835\ude2f\ud835\ude35 \ud835\ude35\ud835\ude29\ud835\ude22\ud835\ude35 \ud835\ude39 \ud835\ude2a\ud835\ude34 \ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26\ud835\ude25 \ud835\ude35\ud835\ude30\n\ud835\ude26\ud835\ude32\ud835\ude36\ud835\ude22\ud835\ude2d \ud835\ude3a.)\u231d \ud835\uddc2\ud835\uddcc \ud835\uddc9\ud835\uddc8\ud835\uddcc\ud835\uddcd\ud835\uddce\ud835\uddc5\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbb\ud835\uddd2 \ud835\uddf1\ud835\uddf2\ud835\uddf3.\n(\ud835\udc37\u2081). (4 = ((((0)++)++)++)++) \ud835\uddc2\ud835\uddcc\n\ud835\uddba\ud835\uddc7 \ud835\uddc2\ud835\uddc7\ud835\uddcd\ud835\uddbe\ud835\uddcb\ud835\uddc9\ud835\uddcb\ud835\uddbe\ud835\uddcd\ud835\uddba\ud835\uddcd\ud835\uddc2\ud835\uddc8\ud835\uddc7 \ud835\uddc8\ud835\uddbf \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n\ud835\uddbd\ud835\uddbe\ud835\uddbf\ud835\uddc2\ud835\uddc7\ud835\uddc2\ud835\uddcd\ud835\uddc2\ud835\uddc8\ud835\uddc7. \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc51\ud835\udc52\ud835\udc53\ud835\udc56\ud835\udc5b\ud835\udc56\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5d\ud835\udc5f\ud835\udc52\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n(4 = ((((0)++)++)++)++). \u220e"}, {"color": "#FFF59D", "id": "P9", "label": "\ud835\udc43\u2089 : (((0)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9\n(((0)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2089): (((0)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9\n(((0)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((\ud835\udc27\u2081 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((\ud835\udc27\u2081)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2082). \ud835\uddab\ud835\uddbe\ud835\uddcd \ud835\udc27\u2081 = (0)++.\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc63\ud835\udc4e\ud835\udc5f\ud835\udc56\ud835\udc4e\ud835\udc4f\ud835\udc59\ud835\udc52-\n\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (((0)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 (((0)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \u220e"}, {"color": "#FFF59D", "id": "P10", "label": "\ud835\udc43\u2081\u2080 : (((0)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2081\u2080): (((0)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3:\n(((0)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9\n(((0)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2089).((0)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. \ud835\udfee.\ud835\udfee.\ud835\udfef (\ud835\udc43\u2084).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc5a\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc60-\ud835\udc5d\ud835\udc5c\ud835\udc5b\ud835\udc52\ud835\udc5b\ud835\udc60\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n(((0)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f).\n\u220e"}, {"color": "#FFF59D", "id": "P11", "label": "\ud835\udc43\u2081\u2081 : ((((0)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)\n\u27f9 ((((0)++)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2081\u2081):\n((((0)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)\n\u27f9 ((((0)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((\ud835\udc27\u2081 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((\ud835\udc27\u2081)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2082). \ud835\uddab\ud835\uddbe\ud835\uddcd \ud835\udc27\u2081 = ((0)++)++.\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc63\ud835\udc4e\ud835\udc5f\ud835\udc56\ud835\udc4e\ud835\udc4f\ud835\udc59\ud835\udc52-\n\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((((0)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((((0)++)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \u220e"}, {"color": "#FFF59D", "id": "P12", "label": "\ud835\udc43\u2081\u2082 : ((((0)++)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2081\u2082):\n((((0)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((((0)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((((0)++)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2081\u2081).(((0)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2081\u2080). \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc5a\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc60-\ud835\udc5d\ud835\udc5c\ud835\udc5b\ud835\udc52\ud835\udc5b\ud835\udc60 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((((0)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \u220e"}, {"color": "#FFF59D", "id": "P13", "label": "\ud835\udc43\u2081\u2083 :\n(((((0)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9\n(((((0)++)++)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2081\u2083):\n(((((0)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 (((((0)++)++)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3:\n((\ud835\udc27\u2081 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9\n((\ud835\udc27\u2081)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2082). \ud835\uddab\ud835\uddbe\ud835\uddcd \ud835\udc27\u2081\n= (((0)++)++)++. \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2\n\ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc63\ud835\udc4e\ud835\udc5f\ud835\udc56\ud835\udc4e\ud835\udc4f\ud835\udc59\ud835\udc52-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n(((((0)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 (((((0)++)++)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \u220e"}, {"color": "#FFF59D", "id": "P14", "label": "\ud835\udc43\u2081\u2084 :\n(((((0)++)++)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2081\u2084):\n(((((0)++)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (((((0)++)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9\n(((((0)++)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2081\u2083).((((0)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2081\u2082). \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc5a\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc60-\ud835\udc5d\ud835\udc5c\ud835\udc5b\ud835\udc52\ud835\udc5b\ud835\udc60 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (((((0)++)++)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \u220e"}, {"color": "#FFF59D", "id": "P15", "label": "\ud835\udc43\u2081\u2085 : ((0)++ = 1)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2081\u2085): ((0)++ =\n1). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (1 = (0)++) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2085). \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2\n\ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59\ud835\udc56\ud835\udc61\ud835\udc66-\ud835\udc50\ud835\udc5c\ud835\udc5a\ud835\udc5a\ud835\udc62\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc63\ud835\udc56\ud835\udc61\ud835\udc66\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n((0)++ = 1). \u220e"}, {"color": "#FFF59D", "id": "P16", "label": "\ud835\udc43\u2081\u2086 : (2 = (1)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2081\u2086): (2 =\n(1)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (2 = ((0)++)++)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2086). ((0)++\n= 1) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2081\u2085).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\n\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (2 = (1)++). \u220e"}, {"color": "#FFF59D", "id": "P17", "label": "\ud835\udc43\u2081\u2087 : (((0)++)++ =\n2)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2081\u2087): (((0)++)++\n= 2). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (2 = ((0)++)++)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2086).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59\ud835\udc56\ud835\udc61\ud835\udc66-\n\ud835\udc50\ud835\udc5c\ud835\udc5a\ud835\udc5a\ud835\udc62\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc63\ud835\udc56\ud835\udc61\ud835\udc66 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (((0)++)++ = 2). \u220e"}, {"color": "#FFF59D", "id": "P18", "label": "\ud835\udc43\u2081\u2088 : (3 = (2)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2081\u2088): (3 =\n(2)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (3 =\n(((0)++)++)++) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2087). (((0)++)++ = 2)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2081\u2087).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\n\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (3 = (2)++). \u220e"}, {"color": "#FFF59D", "id": "P19", "label": "\ud835\udc43\u2081\u2089 : ((((0)++)++)++\n= 3)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2081\u2089):\n((((0)++)++)++ = 3). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (3 =\n(((0)++)++)++) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2087). \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59\ud835\udc56\ud835\udc61\ud835\udc66-\ud835\udc50\ud835\udc5c\ud835\udc5a\ud835\udc5a\ud835\udc62\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc63\ud835\udc56\ud835\udc61\ud835\udc66 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe\n\ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n((((0)++)++)++ = 3). \u220e"}, {"color": "#FFF59D", "id": "P20", "label": "\ud835\udc43\u2082\u2080 : ((2)++ = 3)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2082\u2080): ((2)++ =\n3). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((((0)++)++)++ = 3)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2081\u2089).\n(((0)++)++ = 2) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2081\u2087). \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n((2)++ = 3). \u220e"}, {"color": "#FFF59D", "id": "P21", "label": "\ud835\udc43\u2082\u2081 (\ud835\udfee.\ud835\udfed.\ud835\udff0) : (3\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)", "labelHighlightBold": true, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb \ud835\udfee.\ud835\udfed.\ud835\udff0 (\ud835\udcaf\u2081.\ud835\udc43\u2082\u2081): (3\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3:\n((((0)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2081\u2082). ((((0)++)++)++ = 3)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2081\u2089).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\n\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (3 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \u220e"}, {"color": "#FFF59D", "id": "P22", "label": "\ud835\udc43\u2082\u2082 : (4 =\n((((0)++)++)++)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2082\u2082): (4 =\n((((0)++)++)++)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: \u231c\ud835\ude1e\ud835\ude26\n\ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26 \ud835\udfe3 \ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 \ud835\udfe2++, \ud835\udfe4\n\ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 (\ud835\udfe2++)++, \ud835\udfe5 \ud835\ude35\ud835\ude30\n\ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 ((\ud835\udfe2++)++)++,\ud835\ude26\ud835\ude35\ud835\ude24.\n(\ud835\ude10\ud835\ude2f \ud835\ude30\ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude33 \ud835\ude38\ud835\ude30\ud835\ude33\ud835\ude25\ud835\ude34, \ud835\udfe3 := \ud835\udfe2++, \ud835\udfe4 :=\n\ud835\udfe3++, \ud835\udfe5 := \ud835\udfe4++, \ud835\ude26\ud835\ude35\ud835\ude24. \ud835\ude10\ud835\ude2f \ud835\ude35\ud835\ude29\ud835\ude2a\ud835\ude34 \ud835\ude35\ud835\ude26\ud835\ude39\ud835\ude35\n\ud835\ude10 \ud835\ude36\ud835\ude34\ud835\ude26 \"\ud835\ude39 := \ud835\ude3a\" \ud835\ude35\ud835\ude30 \ud835\ude25\ud835\ude26\ud835\ude2f\ud835\ude30\ud835\ude35\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26\n\ud835\ude34\ud835\ude35\ud835\ude22\ud835\ude35\ud835\ude26\ud835\ude2e\ud835\ude26\ud835\ude2f\ud835\ude35 \ud835\ude35\ud835\ude29\ud835\ude22\ud835\ude35 \ud835\ude39 \ud835\ude2a\ud835\ude34 \ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26\ud835\ude25 \ud835\ude35\ud835\ude30\n\ud835\ude26\ud835\ude32\ud835\ude36\ud835\ude22\ud835\ude2d \ud835\ude3a.)\u231d \ud835\uddc2\ud835\uddcc \ud835\uddc9\ud835\uddc8\ud835\uddcc\ud835\uddcd\ud835\uddce\ud835\uddc5\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbb\ud835\uddd2 \ud835\uddf1\ud835\uddf2\ud835\uddf3.\n(\ud835\udc37\u2081). (4 = ((((0)++)++)++)++) \ud835\uddc2\ud835\uddcc\n\ud835\uddba\ud835\uddc7 \ud835\uddc2\ud835\uddc7\ud835\uddcd\ud835\uddbe\ud835\uddcb\ud835\uddc9\ud835\uddcb\ud835\uddbe\ud835\uddcd\ud835\uddba\ud835\uddcd\ud835\uddc2\ud835\uddc8\ud835\uddc7 \ud835\uddc8\ud835\uddbf \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n\ud835\uddbd\ud835\uddbe\ud835\uddbf\ud835\uddc2\ud835\uddc7\ud835\uddc2\ud835\uddcd\ud835\uddc2\ud835\uddc8\ud835\uddc7. \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc51\ud835\udc52\ud835\udc53\ud835\udc56\ud835\udc5b\ud835\udc56\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5d\ud835\udc5f\ud835\udc52\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n(4 = ((((0)++)++)++)++). \u220e"}, {"color": "#FFF59D", "id": "P23", "label": "\ud835\udc43\u2082\u2083 :\n(((((0)++)++)++)++ =\n4)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2082\u2083):\n(((((0)++)++)++)++ = 4). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3:\n(4 = ((((0)++)++)++)++) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2088). \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2\n\ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59\ud835\udc56\ud835\udc61\ud835\udc66-\ud835\udc50\ud835\udc5c\ud835\udc5a\ud835\udc5a\ud835\udc62\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc63\ud835\udc56\ud835\udc61\ud835\udc66\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n(((((0)++)++)++)++ = 4). \u220e"}, {"color": "#FFF59D", "id": "P24", "label": "\ud835\udc43\u2082\u2084 : ((3)++ = 4)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2082\u2084): ((3)++ =\n4). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (((((0)++)++)++)++ =\n4) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2082\u2083).\n((((0)++)++)++ = 3) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2081\u2089). \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n((3)++ = 4). \u220e"}, {"color": "#FFF59D", "id": "P25", "label": "\ud835\udc43\u2082\u2085 :\n(((((0)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9\n(((((0)++)++)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2082\u2085):\n(((((0)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 (((((0)++)++)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3:\n(((((0)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 (((((0)++)++)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2081\u2083). ((3)++ = 4)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2082\u2084).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\n\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (((((0)++)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9\n(((((0)++)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \u220e"}, {"color": "#FFF59D", "id": "P26", "label": "\ud835\udc43\u2082\u2086 : (4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2082\u2086): (4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3:\n(((((0)++)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2081\u2084). (((((0)++)++)++)++ = 4)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2082\u2083).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\n\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (4 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \u220e"}, {"color": "#81C784", "id": "A3", "label": "\ud835\udc34\u2083 (\ud835\udfee.\ud835\udfef) : \u231c\ud835\udfe2 \ud835\ude2a\ud835\ude34 \ud835\ude2f\ud835\ude30\ud835\ude35\n\ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude34\ud835\ude36\ud835\ude24\ud835\ude24\ud835\ude26\ud835\ude34\ud835\ude34\ud835\ude30\ud835\ude33 \ud835\ude30\ud835\ude27 \ud835\ude22\ud835\ude2f\ud835\ude3a\n\ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33;\n\ud835\ude2a.\ud835\ude26., \ud835\ude38\ud835\ude26 \ud835\ude29\ud835\ude22\ud835\ude37\ud835\ude26 \ud835\ude2f++ \u2260\n\ud835\udfe2 \ud835\ude27\ud835\ude30\ud835\ude33 \ud835\ude26\ud835\ude37\ud835\ude26\ud835\ude33\ud835\ude3a \ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d\n\ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 \ud835\ude2f.\u231d", "shape": "box"}, {"color": "#FFF59D", "id": "P27", "label": "\ud835\udc43\u2082\u2087 : ((\ud835\udc27\u2082 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9\n((\ud835\udc27\u2082)++ \u2260 0))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2082\u2087): ((\ud835\udc27\u2082 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((\ud835\udc27\u2082)++ \u2260 0)).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: \u231c\ud835\udfe2 \ud835\ude2a\ud835\ude34 \ud835\ude2f\ud835\ude30\ud835\ude35 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude34\ud835\ude36\ud835\ude24\ud835\ude24\ud835\ude26\ud835\ude34\ud835\ude34\ud835\ude30\ud835\ude33\n\ud835\ude30\ud835\ude27 \ud835\ude22\ud835\ude2f\ud835\ude3a \ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33; \ud835\ude2a.\ud835\ude26., \ud835\ude38\ud835\ude26\n\ud835\ude29\ud835\ude22\ud835\ude37\ud835\ude26 \ud835\ude2f++ \u2260 \ud835\udfe2 \ud835\ude27\ud835\ude30\ud835\ude33 \ud835\ude26\ud835\ude37\ud835\ude26\ud835\ude33\ud835\ude3a \ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d\n\ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 \ud835\ude2f.\u231d \ud835\uddc2\ud835\uddcc \ud835\uddc9\ud835\uddc8\ud835\uddcc\ud835\uddcd\ud835\uddce\ud835\uddc5\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbb\ud835\uddd2\n\ud835\uddee\ud835\ude05\ud835\uddf6\ud835\uddfc\ud835\uddfa \ud835\udfee.\ud835\udfef (\ud835\udc34\u2083). ((\ud835\udc27\u2082 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((\ud835\udc27\u2082)++ \u2260 0))\n\ud835\uddc2\ud835\uddcc \ud835\uddba \ud835\uddcf\ud835\uddba\ud835\uddc5\ud835\uddc2\ud835\uddbd \ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddc6\ud835\uddce\ud835\uddc5\ud835\uddba \ud835\uddcc\ud835\uddcd\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddc6\ud835\uddbe\ud835\uddc7\ud835\uddcd\n\ud835\uddc2\ud835\uddc7\ud835\uddcd\ud835\uddbe\ud835\uddcb\ud835\uddc9\ud835\uddcb\ud835\uddbe\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd \ud835\uddba\ud835\uddd1\ud835\uddc2\ud835\uddc8\ud835\uddc6.\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc4e\ud835\udc65\ud835\udc56\ud835\udc5c\ud835\udc5a-\n\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5d\ud835\udc5f\ud835\udc52\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe,\n\ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((\ud835\udc27\u2082 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((\ud835\udc27\u2082)++ \u2260 0)).\n\u220e"}, {"color": "#FFF59D", "id": "P28", "label": "\ud835\udc43\u2082\u2088 : ((3 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9\n((3)++ \u2260 0))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2082\u2088): ((3 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((3)++ \u2260 0)).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((\ud835\udc27\u2082 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)\n\u27f9 ((\ud835\udc27\u2082)++ \u2260 0)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2082\u2087). \ud835\uddab\ud835\uddbe\ud835\uddcd \ud835\udc27\u2082 = 3.\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc63\ud835\udc4e\ud835\udc5f\ud835\udc56\ud835\udc4e\ud835\udc4f\ud835\udc59\ud835\udc52-\n\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((3 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((3)++ \u2260 0)). \u220e"}, {"color": "#FFF59D", "id": "P29", "label": "\ud835\udc43\u2082\u2089 : ((3)++ \u2260 0)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2082\u2089): ((3)++ \u2260\n0). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((3 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((3)++ \u2260 0)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2082\u2088).(3 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. \ud835\udfee.\ud835\udfed.\ud835\udff0 (\ud835\udc43\u2082\u2081). \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2\n\ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc5a\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc60-\ud835\udc5d\ud835\udc5c\ud835\udc5b\ud835\udc52\ud835\udc5b\ud835\udc60 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe,\n\ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((3)++ \u2260 0). \u220e"}, {"color": "#FFF59D", "id": "P30", "label": "\ud835\udc43\u2083\u2080 (\ud835\udfee.\ud835\udfed.\ud835\udff2) : (4 \u2260\n0)", "labelHighlightBold": true, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb \ud835\udfee.\ud835\udfed.\ud835\udff2 (\ud835\udcaf\u2081.\ud835\udc43\u2083\u2080): (4 \u2260\n0). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((3)++ \u2260 0) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2082\u2089). ((3)++ = 4)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2082\u2084).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\n\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (4 \u2260 0). \u220e"}, {"color": "#81C784", "id": "A4", "label": "\ud835\udc34\u2084 (\ud835\udfee.\ud835\udff0) :\n\u231c\ud835\ude0b\ud835\ude2a\ud835\ude27\ud835\ude27\ud835\ude26\ud835\ude33\ud835\ude26\ud835\ude2f\ud835\ude35 \ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d\n\ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33\ud835\ude34 \ud835\ude2e\ud835\ude36\ud835\ude34\ud835\ude35 \ud835\ude29\ud835\ude22\ud835\ude37\ud835\ude26\n\ud835\ude25\ud835\ude2a\ud835\ude27\ud835\ude27\ud835\ude26\ud835\ude33\ud835\ude26\ud835\ude2f\ud835\ude35\n\ud835\ude34\ud835\ude36\ud835\ude24\ud835\ude24\ud835\ude26\ud835\ude34\ud835\ude34\ud835\ude30\ud835\ude33\ud835\ude34; \ud835\ude2a.\ud835\ude26., \ud835\ude2a\ud835\ude27\n\ud835\ude2f, \ud835\ude2e \ud835\ude22\ud835\ude33\ud835\ude26 \ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d\n\ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33\ud835\ude34 \ud835\ude22\ud835\ude2f\ud835\ude25 \ud835\ude2f \u2260 \ud835\ude2e,\n\ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude2f \ud835\ude2f++ \u2260 \ud835\ude2e++.\n\ud835\ude0c\ud835\ude32\ud835\ude36\ud835\ude2a\ud835\ude37\ud835\ude22\ud835\ude2d\ud835\ude26\ud835\ude2f\ud835\ude35\ud835\ude2d\ud835\ude3a, \ud835\ude2a\ud835\ude27 \ud835\ude2f++\n= \ud835\ude2e++, \ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude2f \ud835\ude38\ud835\ude26 \ud835\ude2e\ud835\ude36\ud835\ude34\ud835\ude35\n\ud835\ude29\ud835\ude22\ud835\ude37\ud835\ude26 \ud835\ude2f = \ud835\ude2e.\u231d", "shape": "box"}, {"color": "#FFF59D", "id": "P31", "label": "\ud835\udc43\u2083\u2081 : ((((\ud835\udc27\u2083 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227\n(\ud835\udc26\u2081 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (\ud835\udc27\u2083 \u2260\n\ud835\udc26\u2081)) \u27f9 ((\ud835\udc27\u2083)++ \u2260\n(\ud835\udc26\u2081)++))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2083\u2081): ((((\ud835\udc27\u2083\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (\ud835\udc26\u2081 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (\ud835\udc27\u2083 \u2260 \ud835\udc26\u2081)) \u27f9\n((\ud835\udc27\u2083)++ \u2260 (\ud835\udc26\u2081)++)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3:\n\u231c\ud835\ude0b\ud835\ude2a\ud835\ude27\ud835\ude27\ud835\ude26\ud835\ude33\ud835\ude26\ud835\ude2f\ud835\ude35 \ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33\ud835\ude34 \ud835\ude2e\ud835\ude36\ud835\ude34\ud835\ude35\n\ud835\ude29\ud835\ude22\ud835\ude37\ud835\ude26 \ud835\ude25\ud835\ude2a\ud835\ude27\ud835\ude27\ud835\ude26\ud835\ude33\ud835\ude26\ud835\ude2f\ud835\ude35 \ud835\ude34\ud835\ude36\ud835\ude24\ud835\ude24\ud835\ude26\ud835\ude34\ud835\ude34\ud835\ude30\ud835\ude33\ud835\ude34; \ud835\ude2a.\ud835\ude26.,\n\ud835\ude2a\ud835\ude27 \ud835\ude2f, \ud835\ude2e \ud835\ude22\ud835\ude33\ud835\ude26 \ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33\ud835\ude34 \ud835\ude22\ud835\ude2f\ud835\ude25\n\ud835\ude2f \u2260 \ud835\ude2e, \ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude2f \ud835\ude2f++ \u2260 \ud835\ude2e++.\n\ud835\ude0c\ud835\ude32\ud835\ude36\ud835\ude2a\ud835\ude37\ud835\ude22\ud835\ude2d\ud835\ude26\ud835\ude2f\ud835\ude35\ud835\ude2d\ud835\ude3a, \ud835\ude2a\ud835\ude27 \ud835\ude2f++ = \ud835\ude2e++, \ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude2f\n\ud835\ude38\ud835\ude26 \ud835\ude2e\ud835\ude36\ud835\ude34\ud835\ude35 \ud835\ude29\ud835\ude22\ud835\ude37\ud835\ude26 \ud835\ude2f = \ud835\ude2e.\u231d \ud835\uddc2\ud835\uddcc\n\ud835\uddc9\ud835\uddc8\ud835\uddcc\ud835\uddcd\ud835\uddce\ud835\uddc5\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbb\ud835\uddd2 \ud835\uddee\ud835\ude05\ud835\uddf6\ud835\uddfc\ud835\uddfa \ud835\udfee.\ud835\udff0 (\ud835\udc34\u2084).\n((((\ud835\udc27\u2083 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227\n(\ud835\udc26\u2081 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (\ud835\udc27\u2083\n\u2260 \ud835\udc26\u2081)) \u27f9 ((\ud835\udc27\u2083)++ \u2260 (\ud835\udc26\u2081)++)) \ud835\uddc2\ud835\uddcc \ud835\uddba\n\ud835\uddcf\ud835\uddba\ud835\uddc5\ud835\uddc2\ud835\uddbd \ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddc6\ud835\uddce\ud835\uddc5\ud835\uddba \ud835\uddcc\ud835\uddcd\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddc6\ud835\uddbe\ud835\uddc7\ud835\uddcd\n\ud835\uddc2\ud835\uddc7\ud835\uddcd\ud835\uddbe\ud835\uddcb\ud835\uddc9\ud835\uddcb\ud835\uddbe\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd \ud835\uddba\ud835\uddd1\ud835\uddc2\ud835\uddc8\ud835\uddc6.\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc4e\ud835\udc65\ud835\udc56\ud835\udc5c\ud835\udc5a-\n\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5d\ud835\udc5f\ud835\udc52\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe,\n\ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((((\ud835\udc27\u2083 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (\ud835\udc26\u2081 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (\ud835\udc27\u2083 \u2260 \ud835\udc26\u2081)) \u27f9\n((\ud835\udc27\u2083)++ \u2260 (\ud835\udc26\u2081)++)). \u220e"}, {"color": "#FFF59D", "id": "P32", "label": "\ud835\udc43\u2083\u2082 : ((((\ud835\udc27\u2084 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227\n(\ud835\udc26\u2082 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((\ud835\udc27\u2084)++ =\n(\ud835\udc26\u2082)++)) \u27f9 (\ud835\udc27\u2084 =\n\ud835\udc26\u2082))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2083\u2082): ((((\ud835\udc27\u2084\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (\ud835\udc26\u2082 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((\ud835\udc27\u2084)++ =\n(\ud835\udc26\u2082)++)) \u27f9 (\ud835\udc27\u2084 = \ud835\udc26\u2082)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3:\n\u231c\ud835\ude0b\ud835\ude2a\ud835\ude27\ud835\ude27\ud835\ude26\ud835\ude33\ud835\ude26\ud835\ude2f\ud835\ude35 \ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33\ud835\ude34 \ud835\ude2e\ud835\ude36\ud835\ude34\ud835\ude35\n\ud835\ude29\ud835\ude22\ud835\ude37\ud835\ude26 \ud835\ude25\ud835\ude2a\ud835\ude27\ud835\ude27\ud835\ude26\ud835\ude33\ud835\ude26\ud835\ude2f\ud835\ude35 \ud835\ude34\ud835\ude36\ud835\ude24\ud835\ude24\ud835\ude26\ud835\ude34\ud835\ude34\ud835\ude30\ud835\ude33\ud835\ude34; \ud835\ude2a.\ud835\ude26.,\n\ud835\ude2a\ud835\ude27 \ud835\ude2f, \ud835\ude2e \ud835\ude22\ud835\ude33\ud835\ude26 \ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33\ud835\ude34 \ud835\ude22\ud835\ude2f\ud835\ude25\n\ud835\ude2f \u2260 \ud835\ude2e, \ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude2f \ud835\ude2f++ \u2260 \ud835\ude2e++.\n\ud835\ude0c\ud835\ude32\ud835\ude36\ud835\ude2a\ud835\ude37\ud835\ude22\ud835\ude2d\ud835\ude26\ud835\ude2f\ud835\ude35\ud835\ude2d\ud835\ude3a, \ud835\ude2a\ud835\ude27 \ud835\ude2f++ = \ud835\ude2e++, \ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude2f\n\ud835\ude38\ud835\ude26 \ud835\ude2e\ud835\ude36\ud835\ude34\ud835\ude35 \ud835\ude29\ud835\ude22\ud835\ude37\ud835\ude26 \ud835\ude2f = \ud835\ude2e.\u231d \ud835\uddc2\ud835\uddcc\n\ud835\uddc9\ud835\uddc8\ud835\uddcc\ud835\uddcd\ud835\uddce\ud835\uddc5\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbb\ud835\uddd2 \ud835\uddee\ud835\ude05\ud835\uddf6\ud835\uddfc\ud835\uddfa \ud835\udfee.\ud835\udff0 (\ud835\udc34\u2084).\n((((\ud835\udc27\u2084 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227\n(\ud835\udc26\u2082 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227\n((\ud835\udc27\u2084)++ = (\ud835\udc26\u2082)++)) \u27f9 (\ud835\udc27\u2084 = \ud835\udc26\u2082))\n\ud835\uddc2\ud835\uddcc \ud835\uddba \ud835\uddcf\ud835\uddba\ud835\uddc5\ud835\uddc2\ud835\uddbd \ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddc6\ud835\uddce\ud835\uddc5\ud835\uddba \ud835\uddcc\ud835\uddcd\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddc6\ud835\uddbe\ud835\uddc7\ud835\uddcd\n\ud835\uddc2\ud835\uddc7\ud835\uddcd\ud835\uddbe\ud835\uddcb\ud835\uddc9\ud835\uddcb\ud835\uddbe\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd \ud835\uddba\ud835\uddd1\ud835\uddc2\ud835\uddc8\ud835\uddc6.\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc4e\ud835\udc65\ud835\udc56\ud835\udc5c\ud835\udc5a-\n\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5d\ud835\udc5f\ud835\udc52\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe,\n\ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((((\ud835\udc27\u2084 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (\ud835\udc26\u2082 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((\ud835\udc27\u2084)++ =\n(\ud835\udc26\u2082)++)) \u27f9 (\ud835\udc27\u2084 = \ud835\udc26\u2082)). \u220e"}, {"color": "#FFF59D", "id": "P33", "label": "\ud835\udc43\u2083\u2083 : ((((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (4 \u2260 0))\n\u27f9 ((4)++ \u2260 (0)++))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2083\u2083): ((((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (4 \u2260 0)) \u27f9\n((4)++ \u2260 (0)++)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((((\ud835\udc27\u2083\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (\ud835\udc26\u2081 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (\ud835\udc27\u2083 \u2260 \ud835\udc26\u2081)) \u27f9\n((\ud835\udc27\u2083)++ \u2260 (\ud835\udc26\u2081)++)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083\u2081). \ud835\uddab\ud835\uddbe\ud835\uddcd \ud835\udc27\u2083 = 4, \ud835\udc26\u2081 = 0.\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc63\ud835\udc4e\ud835\udc5f\ud835\udc56\ud835\udc4e\ud835\udc4f\ud835\udc59\ud835\udc52-\n\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((((4 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (4 \u2260 0)) \u27f9 ((4)++ \u2260\n(0)++)). \u220e"}, {"color": "#FFF59D", "id": "P34", "label": "\ud835\udc43\u2083\u2084 : ((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2083\u2084): ((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2082\u2086). (0 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2081).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc50\ud835\udc5c\ud835\udc5b\ud835\udc57\ud835\udc62\ud835\udc5b\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\n\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc5f\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((4 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \u220e"}, {"color": "#FFF59D", "id": "P35", "label": "\ud835\udc43\u2083\u2085 : (((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (4 \u2260 0))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2083\u2085): (((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (4 \u2260 0)).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((4 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)\n\u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083\u2084). (4 \u2260\n0) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. \ud835\udfee.\ud835\udfed.\ud835\udff2\n(\ud835\udc43\u2083\u2080). \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc50\ud835\udc5c\ud835\udc5b\ud835\udc57\ud835\udc62\ud835\udc5b\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc5f\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n(((4 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (4 \u2260\n0)). \u220e"}, {"color": "#FFF59D", "id": "P36", "label": "\ud835\udc43\u2083\u2086 : ((4)++ \u2260\n(0)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2083\u2086): ((4)++ \u2260\n(0)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (4 \u2260 0)) \u27f9\n((4)++ \u2260 (0)++)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083\u2083).(((4 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (4 \u2260 0)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083\u2085). \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc5a\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc60-\ud835\udc5d\ud835\udc5c\ud835\udc5b\ud835\udc52\ud835\udc5b\ud835\udc60 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((4)++ \u2260 (0)++). \u220e"}, {"color": "#FFF59D", "id": "P37", "label": "\ud835\udc43\u2083\u2087 : (5 = (((((0)++\n)++)++)++)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2083\u2087): (5 =\n(((((0)++)++)++)++)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3:\n\u231c\ud835\ude1e\ud835\ude26 \ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26 \ud835\udfe3 \ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33\n\ud835\udfe2++, \ud835\udfe4 \ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 (\ud835\udfe2++)++,\n\ud835\udfe5 \ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33\n((\ud835\udfe2++)++)++,\ud835\ude26\ud835\ude35\ud835\ude24. (\ud835\ude10\ud835\ude2f \ud835\ude30\ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude33\n\ud835\ude38\ud835\ude30\ud835\ude33\ud835\ude25\ud835\ude34, \ud835\udfe3 := \ud835\udfe2++, \ud835\udfe4 := \ud835\udfe3++, \ud835\udfe5 :=\n\ud835\udfe4++, \ud835\ude26\ud835\ude35\ud835\ude24. \ud835\ude10\ud835\ude2f \ud835\ude35\ud835\ude29\ud835\ude2a\ud835\ude34 \ud835\ude35\ud835\ude26\ud835\ude39\ud835\ude35 \ud835\ude10 \ud835\ude36\ud835\ude34\ud835\ude26 \"\ud835\ude39\n:= \ud835\ude3a\" \ud835\ude35\ud835\ude30 \ud835\ude25\ud835\ude26\ud835\ude2f\ud835\ude30\ud835\ude35\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude34\ud835\ude35\ud835\ude22\ud835\ude35\ud835\ude26\ud835\ude2e\ud835\ude26\ud835\ude2f\ud835\ude35\n\ud835\ude35\ud835\ude29\ud835\ude22\ud835\ude35 \ud835\ude39 \ud835\ude2a\ud835\ude34 \ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26\ud835\ude25 \ud835\ude35\ud835\ude30 \ud835\ude26\ud835\ude32\ud835\ude36\ud835\ude22\ud835\ude2d \ud835\ude3a.)\u231d\n\ud835\uddc2\ud835\uddcc \ud835\uddc9\ud835\uddc8\ud835\uddcc\ud835\uddcd\ud835\uddce\ud835\uddc5\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbb\ud835\uddd2 \ud835\uddf1\ud835\uddf2\ud835\uddf3. (\ud835\udc37\u2081). (5 =\n(((((0)++)++)++)++)++) \ud835\uddc2\ud835\uddcc \ud835\uddba\ud835\uddc7\n\ud835\uddc2\ud835\uddc7\ud835\uddcd\ud835\uddbe\ud835\uddcb\ud835\uddc9\ud835\uddcb\ud835\uddbe\ud835\uddcd\ud835\uddba\ud835\uddcd\ud835\uddc2\ud835\uddc8\ud835\uddc7 \ud835\uddc8\ud835\uddbf \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n\ud835\uddbd\ud835\uddbe\ud835\uddbf\ud835\uddc2\ud835\uddc7\ud835\uddc2\ud835\uddcd\ud835\uddc2\ud835\uddc8\ud835\uddc7. \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc51\ud835\udc52\ud835\udc53\ud835\udc56\ud835\udc5b\ud835\udc56\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5d\ud835\udc5f\ud835\udc52\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n(5 = (((((0)++)++)++)++)++). \u220e"}, {"color": "#FFF59D", "id": "P38", "label": "\ud835\udc43\u2083\u2088 : ((((((0)++)++)\n++)++)++ = 5)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2083\u2088):\n((((((0)++)++)++)++)++ = 5).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (5 =\n(((((0)++)++)++)++)++) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083\u2087). \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2\n\ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59\ud835\udc56\ud835\udc61\ud835\udc66-\ud835\udc50\ud835\udc5c\ud835\udc5a\ud835\udc5a\ud835\udc62\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc63\ud835\udc56\ud835\udc61\ud835\udc66\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n((((((0)++)++)++)++)++ = 5). \u220e"}, {"color": "#FFF59D", "id": "P39", "label": "\ud835\udc43\u2083\u2089 : ((4)++ = 5)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2083\u2089): ((4)++ =\n5). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3:\n((((((0)++)++)++)++)++ = 5)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083\u2088).\n(((((0)++)++)++)++ = 4) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2082\u2083). \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2\n\ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n((4)++ = 5). \u220e"}, {"color": "#FFF59D", "id": "P40", "label": "\ud835\udc43\u2084\u2080 : (5 = (4)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2084\u2080): (5 =\n(4)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((4)++ = 5)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083\u2089).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59\ud835\udc56\ud835\udc61\ud835\udc66-\n\ud835\udc50\ud835\udc5c\ud835\udc5a\ud835\udc5a\ud835\udc62\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc63\ud835\udc56\ud835\udc61\ud835\udc66 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (5 = (4)++). \u220e"}, {"color": "#FFF59D", "id": "P41", "label": "\ud835\udc43\u2084\u2081 : ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1))\n\u27f9 ((5)++ \u2260 (1)++))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2084\u2081): ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1)) \u27f9\n((5)++ \u2260 (1)++)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((((\ud835\udc27\u2083\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (\ud835\udc26\u2081 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (\ud835\udc27\u2083 \u2260 \ud835\udc26\u2081)) \u27f9\n((\ud835\udc27\u2083)++ \u2260 (\ud835\udc26\u2081)++)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083\u2081). \ud835\uddab\ud835\uddbe\ud835\uddcd \ud835\udc27\u2083 = 5, \ud835\udc26\u2081 = 1.\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc63\ud835\udc4e\ud835\udc5f\ud835\udc56\ud835\udc4e\ud835\udc4f\ud835\udc59\ud835\udc52-\n\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1)) \u27f9 ((5)++ \u2260\n(1)++)). \u220e"}, {"color": "#FFF59D", "id": "P42", "label": "\ud835\udc43\u2084\u2082 : ((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9\n((4)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2084\u2082): ((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((4)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((\ud835\udc27\u2081\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((\ud835\udc27\u2081)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2082). \ud835\uddab\ud835\uddbe\ud835\uddcd \ud835\udc27\u2081 = 4.\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc63\ud835\udc4e\ud835\udc5f\ud835\udc56\ud835\udc4e\ud835\udc4f\ud835\udc59\ud835\udc52-\n\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((4 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((4)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \u220e"}, {"color": "#FFF59D", "id": "P43", "label": "\ud835\udc43\u2084\u2083 : ((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 (5\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2084\u2083): ((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 (5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((4\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((4)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2084\u2082). ((4)++ = 5)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083\u2089).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\n\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((4 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 (5 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \u220e"}, {"color": "#FFF59D", "id": "P44", "label": "\ud835\udc43\u2084\u2084 : (5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2084\u2084): (5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 (5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2084\u2083).(4 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2082\u2086). \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc5a\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc60-\n\ud835\udc5d\ud835\udc5c\ud835\udc5b\ud835\udc52\ud835\udc5b\ud835\udc60 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (5 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \u220e"}, {"color": "#FFF59D", "id": "P45", "label": "\ud835\udc43\u2084\u2085 : ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1))\n\u27f9 ((5)++ \u2260 (1)++))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2084\u2085): ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1)) \u27f9\n((5)++ \u2260 (1)++)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((((\ud835\udc27\u2083\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (\ud835\udc26\u2081 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (\ud835\udc27\u2083 \u2260 \ud835\udc26\u2081)) \u27f9\n((\ud835\udc27\u2083)++ \u2260 (\ud835\udc26\u2081)++)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083\u2081). \ud835\uddab\ud835\uddbe\ud835\uddcd \ud835\udc27\u2083 = 5, \ud835\udc26\u2081 = 1.\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc63\ud835\udc4e\ud835\udc5f\ud835\udc56\ud835\udc4e\ud835\udc4f\ud835\udc59\ud835\udc52-\n\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1)) \u27f9 ((5)++ \u2260\n(1)++)). \u220e"}, {"color": "#FFF59D", "id": "P46", "label": "\ud835\udc43\u2084\u2086 : ((4)++ \u2260\n(0)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2084\u2086): ((4)++ \u2260\n(0)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (4 \u2260 0)) \u27f9\n((4)++ \u2260 (0)++)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083\u2083).(((4 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (4 \u2260 0)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083\u2085). \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc5a\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc60-\ud835\udc5d\ud835\udc5c\ud835\udc5b\ud835\udc52\ud835\udc5b\ud835\udc60 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((4)++ \u2260 (0)++). \u220e"}, {"color": "#FFF59D", "id": "P47", "label": "\ud835\udc43\u2084\u2087 : (5 \u2260 (0)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2084\u2087): (5 \u2260\n(0)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((4)++ \u2260 (0)++)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2084\u2086). ((4)++\n= 5) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083\u2089).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\n\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (5 \u2260 (0)++). \u220e"}, {"color": "#FFF59D", "id": "P48", "label": "\ud835\udc43\u2084\u2088 : (6 = ((((((0)+\n+)++)++)++)++)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2084\u2088): (6 =\n((((((0)++)++)++)++)++)++).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: \u231c\ud835\ude1e\ud835\ude26 \ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26 \ud835\udfe3 \ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26\n\ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 \ud835\udfe2++, \ud835\udfe4 \ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33\n(\ud835\udfe2++)++, \ud835\udfe5 \ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33\n((\ud835\udfe2++)++)++,\ud835\ude26\ud835\ude35\ud835\ude24. (\ud835\ude10\ud835\ude2f \ud835\ude30\ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude33\n\ud835\ude38\ud835\ude30\ud835\ude33\ud835\ude25\ud835\ude34, \ud835\udfe3 := \ud835\udfe2++, \ud835\udfe4 := \ud835\udfe3++, \ud835\udfe5 :=\n\ud835\udfe4++, \ud835\ude26\ud835\ude35\ud835\ude24. \ud835\ude10\ud835\ude2f \ud835\ude35\ud835\ude29\ud835\ude2a\ud835\ude34 \ud835\ude35\ud835\ude26\ud835\ude39\ud835\ude35 \ud835\ude10 \ud835\ude36\ud835\ude34\ud835\ude26 \"\ud835\ude39\n:= \ud835\ude3a\" \ud835\ude35\ud835\ude30 \ud835\ude25\ud835\ude26\ud835\ude2f\ud835\ude30\ud835\ude35\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude34\ud835\ude35\ud835\ude22\ud835\ude35\ud835\ude26\ud835\ude2e\ud835\ude26\ud835\ude2f\ud835\ude35\n\ud835\ude35\ud835\ude29\ud835\ude22\ud835\ude35 \ud835\ude39 \ud835\ude2a\ud835\ude34 \ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26\ud835\ude25 \ud835\ude35\ud835\ude30 \ud835\ude26\ud835\ude32\ud835\ude36\ud835\ude22\ud835\ude2d \ud835\ude3a.)\u231d\n\ud835\uddc2\ud835\uddcc \ud835\uddc9\ud835\uddc8\ud835\uddcc\ud835\uddcd\ud835\uddce\ud835\uddc5\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbb\ud835\uddd2 \ud835\uddf1\ud835\uddf2\ud835\uddf3. (\ud835\udc37\u2081). (6 =\n((((((0)++)++)++)++)++)++) \ud835\uddc2\ud835\uddcc \ud835\uddba\ud835\uddc7\n\ud835\uddc2\ud835\uddc7\ud835\uddcd\ud835\uddbe\ud835\uddcb\ud835\uddc9\ud835\uddcb\ud835\uddbe\ud835\uddcd\ud835\uddba\ud835\uddcd\ud835\uddc2\ud835\uddc8\ud835\uddc7 \ud835\uddc8\ud835\uddbf \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n\ud835\uddbd\ud835\uddbe\ud835\uddbf\ud835\uddc2\ud835\uddc7\ud835\uddc2\ud835\uddcd\ud835\uddc2\ud835\uddc8\ud835\uddc7. \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc51\ud835\udc52\ud835\udc53\ud835\udc56\ud835\udc5b\ud835\udc56\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5d\ud835\udc5f\ud835\udc52\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n(6 = ((((((0)++)++)++)++)++)++).\n\u220e"}, {"color": "#FFF59D", "id": "P49", "label": "\ud835\udc43\u2084\u2089 : (((((((0)++)++\n)++)++)++)++ = 6)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2084\u2089):\n(((((((0)++)++)++)++)++)++ = 6).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (6 =\n((((((0)++)++)++)++)++)++)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2084\u2088).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59\ud835\udc56\ud835\udc61\ud835\udc66-\n\ud835\udc50\ud835\udc5c\ud835\udc5a\ud835\udc5a\ud835\udc62\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc63\ud835\udc56\ud835\udc61\ud835\udc66 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n(((((((0)++)++)++)++)++)++ = 6).\n\u220e"}, {"color": "#FFF59D", "id": "P50", "label": "\ud835\udc43\u2085\u2080 : (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2085\u2080): (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((0)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. \ud835\udfee.\ud835\udfee.\ud835\udfef (\ud835\udc43\u2084). ((0)++ =\n1) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2081\u2085).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\n\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (1 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \u220e"}, {"color": "#FFF59D", "id": "P51", "label": "\ud835\udc43\u2085\u2081 : ((5)++ = 6)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2085\u2081): ((5)++ =\n6). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3:\n(((((((0)++)++)++)++)++)++ = 6)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2084\u2089).\n((((((0)++)++)++)++)++ = 5)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083\u2088).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\n\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((5)++ = 6). \u220e"}, {"color": "#FFF59D", "id": "P52", "label": "\ud835\udc43\u2085\u2082 : (6 = (5)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2085\u2082): (6 =\n(5)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((5)++ = 6)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2085\u2081).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59\ud835\udc56\ud835\udc61\ud835\udc66-\n\ud835\udc50\ud835\udc5c\ud835\udc5a\ud835\udc5a\ud835\udc62\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc63\ud835\udc56\ud835\udc61\ud835\udc66 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (6 = (5)++). \u220e"}, {"color": "#FFF59D", "id": "P66", "label": "\ud835\udc43\u2086\u2086 : \ud835\udc3c\ud835\udc5b\ud835\udc50(\u210b\u2081)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2086\u2086): \ud835\udc3c\ud835\udc5b\ud835\udc50(\u210b\u2081).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: \ud835\udc43\u2086\u2085 \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2086\u2085). \ud835\udc43\u2083\u2080 \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n\ud835\udfee.\ud835\udfed.\ud835\udff2 (\ud835\udc43\u2083\u2080). \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc56\ud835\udc5b\ud835\udc50\ud835\udc5c\ud835\udc5b\ud835\udc60\ud835\udc56\ud835\udc60\ud835\udc61\ud835\udc52\ud835\udc5b\ud835\udc50\ud835\udc66-\ud835\udc4f\ud835\udc66-\ud835\udc56\ud835\udc5b\ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59\ud835\udc56\ud835\udc61\ud835\udc66-\n\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc5f\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd \ud835\udc3c\ud835\udc5b\ud835\udc50(\u210b\u2081). \u220e"}, {"color": "#FFF59D", "id": "P65", "label": "\ud835\udc43\u2086\u2085 : (4 = 0)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\u210b\u2081.\ud835\udc43\u2086\u2085): (4 = 0).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((((4 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((4)++ = (0)++)) \u27f9 (4\n= 0)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2086\u2084).(((4 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)\n\u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227\n((4)++ = (0)++)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2086\u2083). \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc5a\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc60-\ud835\udc5d\ud835\udc5c\ud835\udc5b\ud835\udc52\ud835\udc5b\ud835\udc60 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (4 = 0). \u220e"}, {"color": "#FFF59D", "id": "P64", "label": "\ud835\udc43\u2086\u2084 : ((((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((4)++ =\n(0)++)) \u27f9 (4 = 0))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\u210b\u2081.\ud835\udc43\u2086\u2084): ((((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((4)++ =\n(0)++)) \u27f9 (4 = 0)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3:\n((((\ud835\udc27\u2084 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227\n(\ud835\udc26\u2082 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227\n((\ud835\udc27\u2084)++ = (\ud835\udc26\u2082)++)) \u27f9 (\ud835\udc27\u2084 = \ud835\udc26\u2082))\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083\u2082). \ud835\uddab\ud835\uddbe\ud835\uddcd \ud835\udc27\u2084\n= 4, \ud835\udc26\u2082 = 0. \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc63\ud835\udc4e\ud835\udc5f\ud835\udc56\ud835\udc4e\ud835\udc4f\ud835\udc59\ud835\udc52-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe\n\ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((4)++ =\n(0)++)) \u27f9 (4 = 0)). \u220e"}, {"color": "#FFF59D", "id": "P63", "label": "\ud835\udc43\u2086\u2083 : (((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((4)++ =\n(0)++))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\u210b\u2081.\ud835\udc43\u2086\u2083): (((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((4)++ =\n(0)++)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2086\u2082). ((4)++ = (0)++)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2086\u2081).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc50\ud835\udc5c\ud835\udc5b\ud835\udc57\ud835\udc62\ud835\udc5b\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\n\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc5f\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (((4 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((4)++ = (0)++)). \u220e"}, {"color": "#FFF59D", "id": "P62", "label": "\ud835\udc43\u2086\u2082 : ((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\u210b\u2081.\ud835\udc43\u2086\u2082): ((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2082\u2086). (0 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2081).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc50\ud835\udc5c\ud835\udc5b\ud835\udc57\ud835\udc62\ud835\udc5b\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\n\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc5f\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((4 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \u220e"}, {"color": "#FFF59D", "id": "P61", "label": "\ud835\udc43\u2086\u2081 : ((4)++ =\n(0)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\u210b\u2081.\ud835\udc43\u2086\u2081): ((4)++ =\n(0)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((4)++ = 1)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2086\u2080). (1 =\n(0)++) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2085).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\n\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((4)++ = (0)++). \u220e"}, {"color": "#FFF59D", "id": "P60", "label": "\ud835\udc43\u2086\u2080 : ((4)++ = 1)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\u210b\u2081.\ud835\udc43\u2086\u2080): ((4)++ =\n1). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (5 = 1) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2085\u2089). (5 = (4)++) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2084\u2080). \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2\n\ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n((4)++ = 1). \u220e"}, {"color": "#FFF59D", "id": "P59", "label": "\ud835\udc43\u2085\u2089 : (5 = 1)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\u210b\u2081.\ud835\udc43\u2085\u2089): (5 = 1).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((5)++ = (1)++)) \u27f9 (5\n= 1)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2085\u2088).(((5 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)\n\u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227\n((5)++ = (1)++)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2085\u2087). \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc5a\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc60-\ud835\udc5d\ud835\udc5c\ud835\udc5b\ud835\udc52\ud835\udc5b\ud835\udc60 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (5 = 1). \u220e"}, {"color": "#FFF59D", "id": "P58", "label": "\ud835\udc43\u2085\u2088 : ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((5)++ =\n(1)++)) \u27f9 (5 = 1))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\u210b\u2081.\ud835\udc43\u2085\u2088): ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((5)++ =\n(1)++)) \u27f9 (5 = 1)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3:\n((((\ud835\udc27\u2084 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227\n(\ud835\udc26\u2082 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227\n((\ud835\udc27\u2084)++ = (\ud835\udc26\u2082)++)) \u27f9 (\ud835\udc27\u2084 = \ud835\udc26\u2082))\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083\u2082). \ud835\uddab\ud835\uddbe\ud835\uddcd \ud835\udc27\u2084\n= 5, \ud835\udc26\u2082 = 1. \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc63\ud835\udc4e\ud835\udc5f\ud835\udc56\ud835\udc4e\ud835\udc4f\ud835\udc59\ud835\udc52-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe\n\ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((5)++ =\n(1)++)) \u27f9 (5 = 1)). \u220e"}, {"color": "#FFF59D", "id": "P57", "label": "\ud835\udc43\u2085\u2087 : (((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((5)++ =\n(1)++))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\u210b\u2081.\ud835\udc43\u2085\u2087): (((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((5)++ =\n(1)++)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2085\u2086). ((5)++ = (1)++)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2085\u2085).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc50\ud835\udc5c\ud835\udc5b\ud835\udc57\ud835\udc62\ud835\udc5b\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\n\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc5f\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (((5 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((5)++ = (1)++)). \u220e"}, {"color": "#FFF59D", "id": "P56", "label": "\ud835\udc43\u2085\u2086 : ((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\u210b\u2081.\ud835\udc43\u2085\u2086): ((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2084\u2084). (1 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2085\u2080). \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc50\ud835\udc5c\ud835\udc5b\ud835\udc57\ud835\udc62\ud835\udc5b\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc5f\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n((5 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \u220e"}, {"color": "#FFF59D", "id": "P55", "label": "\ud835\udc43\u2085\u2085 : ((5)++ =\n(1)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\u210b\u2081.\ud835\udc43\u2085\u2085): ((5)++ =\n(1)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((5)++ = 2)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2085\u2084). (2 =\n(1)++) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2081\u2086).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\n\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((5)++ = (1)++). \u220e"}, {"color": "#FFF59D", "id": "P54", "label": "\ud835\udc43\u2085\u2084 : ((5)++ = 2)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\u210b\u2081.\ud835\udc43\u2085\u2084): ((5)++ =\n2). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (6 = 2) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2085\u2083). (6 = (5)++) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2085\u2082). \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2\n\ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n((5)++ = 2). \u220e"}, {"color": "#FFF59D", "id": "P53", "label": "\ud835\udc43\u2085\u2083 : (6 = 2)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\u210b\u2081.\ud835\udc43\u2085\u2083): (6 = 2).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: \u231c\ud835\ude09\ud835\ude3a \ud835\ude29\ud835\ude3a\ud835\ude31\ud835\ude30\ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude34\ud835\ude2a\ud835\ude34, \ud835\ude22\ud835\ude34\ud835\ude34\ud835\ude36\ud835\ude2e\ud835\ude26 (\ud835\udfe8\n= \ud835\udfe4) \ud835\ude2a\ud835\ude34 \ud835\ude35\ud835\ude33\ud835\ude36\ud835\ude26.\u231d \ud835\uddc2\ud835\uddcc \ud835\uddc9\ud835\uddc8\ud835\uddcc\ud835\uddcd\ud835\uddce\ud835\uddc5\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbb\ud835\uddd2\n\ud835\uddee\ud835\ude05\ud835\uddf6\ud835\uddfc\ud835\uddfa (\ud835\udc34\u2085). (6 = 2) \ud835\uddc2\ud835\uddcc \ud835\uddba \ud835\uddcf\ud835\uddba\ud835\uddc5\ud835\uddc2\ud835\uddbd\n\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddc6\ud835\uddce\ud835\uddc5\ud835\uddba \ud835\uddcc\ud835\uddcd\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddc6\ud835\uddbe\ud835\uddc7\ud835\uddcd \ud835\uddc2\ud835\uddc7\ud835\uddcd\ud835\uddbe\ud835\uddcb\ud835\uddc9\ud835\uddcb\ud835\uddbe\ud835\uddcd\ud835\uddbe\ud835\uddbd\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd \ud835\uddba\ud835\uddd1\ud835\uddc2\ud835\uddc8\ud835\uddc6. \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2\n\ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc4e\ud835\udc65\ud835\udc56\ud835\udc5c\ud835\udc5a-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5d\ud835\udc5f\ud835\udc52\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n(6 = 2). \u220e"}, {"color": "#81C784", "id": "A5", "label": "\ud835\udc34\u2085 : \u231c\ud835\ude09\ud835\ude3a \ud835\ude29\ud835\ude3a\ud835\ude31\ud835\ude30\ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude34\ud835\ude2a\ud835\ude34,\n\ud835\ude22\ud835\ude34\ud835\ude34\ud835\ude36\ud835\ude2e\ud835\ude26 (\ud835\udfe8 = \ud835\udfe4) \ud835\ude2a\ud835\ude34\n\ud835\ude35\ud835\ude33\ud835\ude36\ud835\ude26.\u231d", "shape": "box"}, {"color": "#FFF59D", "id": "P67", "label": "\ud835\udc43\u2086\u2087 (\ud835\udfee.\ud835\udfed.\ud835\udff4) : (6 \u2260\n2)", "labelHighlightBold": true, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb \ud835\udfee.\ud835\udfed.\ud835\udff4 (\ud835\udcaf\u2081.\ud835\udc43\u2086\u2087): (6 \u2260\n2). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: \ud835\uddab\ud835\uddbe\ud835\uddcd \ud835\uddf5\ud835\ude06\ud835\uddfd. (\ud835\udc3b\u2081) \ud835\uddbb\ud835\uddbe \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\uddc1\ud835\uddd2\ud835\uddc9\ud835\uddc8\ud835\uddcd\ud835\uddc1\ud835\uddbe\ud835\uddcc\ud835\uddc2\ud835\uddcc (6 = 2). \ud835\udc3c\ud835\udc5b\ud835\udc50(\u210b\u2081)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2086\u2086).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc5d\ud835\udc5f\ud835\udc5c\ud835\udc5c\ud835\udc53-\ud835\udc4f\ud835\udc66-\n\ud835\udc5f\ud835\udc52\ud835\udc53\ud835\udc62\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\ud835\udc5c\ud835\udc53-\ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59\ud835\udc56\ud835\udc61\ud835\udc66 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe\n\ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (6 \u2260 2). \u220e"}, {"color": "#FFF59D", "id": "P68", "label": "\ud835\udc43\u2086\u2088 : ((1)++ = 2)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2086\u2088): ((1)++ =\n2). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (((0)++)++ = 2)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2081\u2087). ((0)++\n= 1) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2081\u2085).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\n\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((1)++ = 2). \u220e"}, {"color": "#FFF59D", "id": "P69", "label": "\ud835\udc43\u2086\u2089 : (5 \u2260 1)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2086\u2089): (5 \u2260 1).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (5 \u2260 (0)++) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2084\u2087). ((0)++ = 1) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2081\u2085). \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2\n\ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n(5 \u2260 1). \u220e"}, {"color": "#FFF59D", "id": "P70", "label": "\ud835\udc43\u2087\u2080 : ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1))\n\u27f9 (6 \u2260 (1)++))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2087\u2080): ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1)) \u27f9 (6\n\u2260 (1)++)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1)) \u27f9\n((5)++ \u2260 (1)++)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2084\u2085). ((5)++ = 6) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2085\u2081). \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2\n\ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1))\n\u27f9 (6 \u2260 (1)++)). \u220e"}, {"color": "#FFF59D", "id": "P71", "label": "\ud835\udc43\u2087\u2081 : ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1))\n\u27f9 (6 \u2260 2))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2087\u2081): ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1)) \u27f9 (6\n\u2260 2)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1)) \u27f9 (6\n\u2260 (1)++)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2087\u2080). ((1)++ = 2) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2086\u2088). \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1))\n\u27f9 (6 \u2260 2)). \u220e"}, {"color": "#FFF59D", "id": "P72", "label": "\ud835\udc43\u2087\u2082 : ((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2087\u2082): ((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2084\u2084). (1 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2085\u2080). \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc50\ud835\udc5c\ud835\udc5b\ud835\udc57\ud835\udc62\ud835\udc5b\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc5f\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n((5 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \u220e"}, {"color": "#FFF59D", "id": "P73", "label": "\ud835\udc43\u2087\u2083 : (((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2087\u2083): (((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1)).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((5 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)\n\u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2087\u2082). (5 \u2260\n1) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2086\u2089).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc50\ud835\udc5c\ud835\udc5b\ud835\udc57\ud835\udc62\ud835\udc5b\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\n\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc5f\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (((5 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1)). \u220e"}, {"color": "#FFF59D", "id": "P74", "label": "\ud835\udc43\u2087\u2084 : (6 \u2260 2)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2087\u2084): (6 \u2260 2).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1)) \u27f9 (6 \u2260 2))\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2087\u2081).(((5\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1))\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2087\u2083).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc5a\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc60-\ud835\udc5d\ud835\udc5c\ud835\udc5b\ud835\udc52\ud835\udc5b\ud835\udc60\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n(6 \u2260 2). \u220e"}, {"color": "#81C784", "id": "A6", "label": "\ud835\udc34\u2086 : \u231c\ud835\ude13\ud835\ude26\ud835\ude35 \ud835\ude17(\ud835\ude2f) \ud835\ude23\ud835\ude26\n\ud835\ude22\ud835\ude2f\ud835\ude3a \ud835\ude31\ud835\ude33\ud835\ude30\ud835\ude31\ud835\ude26\ud835\ude33\ud835\ude35\ud835\ude3a\n\ud835\ude31\ud835\ude26\ud835\ude33\ud835\ude35\ud835\ude22\ud835\ude2a\ud835\ude2f\ud835\ude2a\ud835\ude2f\ud835\ude28 \ud835\ude35\ud835\ude30 \ud835\ude22\n\ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 \ud835\ude2f.\n\ud835\ude1a\ud835\ude36\ud835\ude31\ud835\ude31\ud835\ude30\ud835\ude34\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude22\ud835\ude35 \ud835\ude17(\ud835\ude16) \ud835\ude2a\ud835\ude34\n\ud835\ude35\ud835\ude33\ud835\ude36\ud835\ude26, \ud835\ude22\ud835\ude2f\ud835\ude25 \ud835\ude34\ud835\ude36\ud835\ude31\ud835\ude31\ud835\ude30\ud835\ude34\ud835\ude26\n\ud835\ude35\ud835\ude29\ud835\ude22\ud835\ude35 \ud835\ude38\ud835\ude29\ud835\ude26\ud835\ude2f\ud835\ude26\ud835\ude37\ud835\ude26\ud835\ude33 \ud835\ude17(\ud835\ude2f)\n\ud835\ude2a\ud835\ude34 \ud835\ude35\ud835\ude33\ud835\ude36\ud835\ude26, \ud835\ude17(\ud835\ude2f++) \ud835\ude2a\ud835\ude34\n\ud835\ude22\ud835\ude2d\ud835\ude34\ud835\ude30 \ud835\ude35\ud835\ude33\ud835\ude36\ud835\ude26. \ud835\ude1b\ud835\ude29\ud835\ude26\ud835\ude2f \ud835\ude17(\ud835\ude2f)\n\ud835\ude2a\ud835\ude34 \ud835\ude35\ud835\ude33\ud835\ude36\ud835\ude26 \ud835\ude27\ud835\ude30\ud835\ude33 \ud835\ude26\ud835\ude37\ud835\ude26\ud835\ude33\ud835\ude3a\n\ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 \ud835\ude2f.\u231d", "shape": "box"}, {"color": "#FFF59D", "id": "P75", "label": "\ud835\udc43\u2087\u2085 : (((\ud835\udc27\u2085 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227\n(\ud835\udc0f\u2081(0) \u2227 (\ud835\udc0f\u2081(\ud835\udc27\u2085) \u27f9\n\ud835\udc0f\u2081((\ud835\udc27\u2085)++)))) \u27f9 ((\ud835\udc26\u2083\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)\n\u27f9 \ud835\udc0f\u2081(\ud835\udc26\u2083)))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2087\u2085): (((\ud835\udc27\u2085 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (\ud835\udc0f\u2081(0) \u2227\n(\ud835\udc0f\u2081(\ud835\udc27\u2085) \u27f9 \ud835\udc0f\u2081((\ud835\udc27\u2085)++)))) \u27f9 ((\ud835\udc26\u2083\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 \ud835\udc0f\u2081(\ud835\udc26\u2083))).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: \u231c\ud835\ude13\ud835\ude26\ud835\ude35 \ud835\ude17(\ud835\ude2f) \ud835\ude23\ud835\ude26 \ud835\ude22\ud835\ude2f\ud835\ude3a \ud835\ude31\ud835\ude33\ud835\ude30\ud835\ude31\ud835\ude26\ud835\ude33\ud835\ude35\ud835\ude3a\n\ud835\ude31\ud835\ude26\ud835\ude33\ud835\ude35\ud835\ude22\ud835\ude2a\ud835\ude2f\ud835\ude2a\ud835\ude2f\ud835\ude28 \ud835\ude35\ud835\ude30 \ud835\ude22 \ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33\n\ud835\ude2f. \ud835\ude1a\ud835\ude36\ud835\ude31\ud835\ude31\ud835\ude30\ud835\ude34\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude22\ud835\ude35 \ud835\ude17(\ud835\ude16) \ud835\ude2a\ud835\ude34 \ud835\ude35\ud835\ude33\ud835\ude36\ud835\ude26,\n\ud835\ude22\ud835\ude2f\ud835\ude25 \ud835\ude34\ud835\ude36\ud835\ude31\ud835\ude31\ud835\ude30\ud835\ude34\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude22\ud835\ude35 \ud835\ude38\ud835\ude29\ud835\ude26\ud835\ude2f\ud835\ude26\ud835\ude37\ud835\ude26\ud835\ude33 \ud835\ude17(\ud835\ude2f)\n\ud835\ude2a\ud835\ude34 \ud835\ude35\ud835\ude33\ud835\ude36\ud835\ude26, \ud835\ude17(\ud835\ude2f++) \ud835\ude2a\ud835\ude34 \ud835\ude22\ud835\ude2d\ud835\ude34\ud835\ude30 \ud835\ude35\ud835\ude33\ud835\ude36\ud835\ude26.\n\ud835\ude1b\ud835\ude29\ud835\ude26\ud835\ude2f \ud835\ude17(\ud835\ude2f) \ud835\ude2a\ud835\ude34 \ud835\ude35\ud835\ude33\ud835\ude36\ud835\ude26 \ud835\ude27\ud835\ude30\ud835\ude33 \ud835\ude26\ud835\ude37\ud835\ude26\ud835\ude33\ud835\ude3a\n\ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 \ud835\ude2f.\u231d \ud835\uddc2\ud835\uddcc \ud835\uddc9\ud835\uddc8\ud835\uddcc\ud835\uddcd\ud835\uddce\ud835\uddc5\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddbd\n\ud835\uddbb\ud835\uddd2 \ud835\uddee\ud835\ude05\ud835\uddf6\ud835\uddfc\ud835\uddfa \ud835\ude00\ud835\uddf0\ud835\uddf5\ud835\uddf2\ud835\uddfa\ud835\uddee (\ud835\udc34\u2086). (((\ud835\udc27\u2085 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (\ud835\udc0f\u2081(0) \u2227\n(\ud835\udc0f\u2081(\ud835\udc27\u2085) \u27f9 \ud835\udc0f\u2081((\ud835\udc27\u2085)++)))) \u27f9 ((\ud835\udc26\u2083\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 \ud835\udc0f\u2081(\ud835\udc26\u2083)))\n\ud835\uddc2\ud835\uddcc \ud835\uddba \ud835\uddcf\ud835\uddba\ud835\uddc5\ud835\uddc2\ud835\uddbd \ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddc6\ud835\uddce\ud835\uddc5\ud835\uddba \ud835\uddcc\ud835\uddcd\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddc6\ud835\uddbe\ud835\uddc7\ud835\uddcd\n\ud835\uddc2\ud835\uddc7\ud835\uddcd\ud835\uddbe\ud835\uddcb\ud835\uddc9\ud835\uddcb\ud835\uddbe\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd \ud835\uddba\ud835\uddd1\ud835\uddc2\ud835\uddc8\ud835\uddc6.\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc4e\ud835\udc65\ud835\udc56\ud835\udc5c\ud835\udc5a-\n\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5d\ud835\udc5f\ud835\udc52\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe,\n\ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (((\ud835\udc27\u2085 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (\ud835\udc0f\u2081(0) \u2227\n(\ud835\udc0f\u2081(\ud835\udc27\u2085) \u27f9 \ud835\udc0f\u2081((\ud835\udc27\u2085)++)))) \u27f9 ((\ud835\udc26\u2083\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 \ud835\udc0f\u2081(\ud835\udc26\u2083))).\n\u220e"}]); + nodes = new vis.DataSet([{"color": "#81C784", "id": "A1", "label": "\ud835\udc34\u2081 (\ud835\udfee.\ud835\udfed) : \u231c\ud835\udfe2 \ud835\ude2a\ud835\ude34 \ud835\ude22\n\ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33.\u231d", "shape": "box"}, {"color": "#FFF59D", "id": "P1", "label": "\ud835\udc43\u2081 : (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2081): (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: \u231c\ud835\udfe2 \ud835\ude2a\ud835\ude34 \ud835\ude22\n\ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33.\u231d \ud835\uddc2\ud835\uddcc \ud835\uddc9\ud835\uddc8\ud835\uddcc\ud835\uddcd\ud835\uddce\ud835\uddc5\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddbd\n\ud835\uddbb\ud835\uddd2 \ud835\uddee\ud835\ude05\ud835\uddf6\ud835\uddfc\ud835\uddfa \ud835\udfee.\ud835\udfed (\ud835\udc34\u2081). (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \ud835\uddc2\ud835\uddcc \ud835\uddba \ud835\uddcf\ud835\uddba\ud835\uddc5\ud835\uddc2\ud835\uddbd\n\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddc6\ud835\uddce\ud835\uddc5\ud835\uddba \ud835\uddcc\ud835\uddcd\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddc6\ud835\uddbe\ud835\uddc7\ud835\uddcd \ud835\uddc2\ud835\uddc7\ud835\uddcd\ud835\uddbe\ud835\uddcb\ud835\uddc9\ud835\uddcb\ud835\uddbe\ud835\uddcd\ud835\uddbe\ud835\uddbd\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd \ud835\uddba\ud835\uddd1\ud835\uddc2\ud835\uddc8\ud835\uddc6.\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2\n\ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc4e\ud835\udc65\ud835\udc56\ud835\udc5c\ud835\udc5a-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5d\ud835\udc5f\ud835\udc52\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: \ud835\udc9c \u22a2 P, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (0 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \u220e"}, {"color": "#81C784", "id": "A2", "label": "\ud835\udc34\u2082 (\ud835\udfee.\ud835\udfee) : \u231c\ud835\ude10\ud835\ude27 \ud835\ude2f \ud835\ude2a\ud835\ude34\n\ud835\ude22 \ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33,\n\ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude2f \ud835\ude2f++ \ud835\ude2a\ud835\ude34 \ud835\ude22\n\ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33.\u231d", "shape": "box"}, {"color": "#FFF59D", "id": "P2", "label": "\ud835\udc43\u2082 : ((\ud835\udc27\u2081 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9\n((\ud835\udc27\u2081)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2082): ((\ud835\udc27\u2081 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((\ud835\udc27\u2081)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: \u231c\ud835\ude10\ud835\ude27 \ud835\ude2f\n\ud835\ude2a\ud835\ude34 \ud835\ude22 \ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33, \ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude2f \ud835\ude2f++ \ud835\ude2a\ud835\ude34\n\ud835\ude22 \ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33.\u231d \ud835\uddc2\ud835\uddcc \ud835\uddc9\ud835\uddc8\ud835\uddcc\ud835\uddcd\ud835\uddce\ud835\uddc5\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddbd\n\ud835\uddbb\ud835\uddd2 \ud835\uddee\ud835\ude05\ud835\uddf6\ud835\uddfc\ud835\uddfa \ud835\udfee.\ud835\udfee (\ud835\udc34\u2082). ((\ud835\udc27\u2081 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((\ud835\udc27\u2081)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \ud835\uddc2\ud835\uddcc \ud835\uddba \ud835\uddcf\ud835\uddba\ud835\uddc5\ud835\uddc2\ud835\uddbd\n\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddc6\ud835\uddce\ud835\uddc5\ud835\uddba \ud835\uddcc\ud835\uddcd\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddc6\ud835\uddbe\ud835\uddc7\ud835\uddcd \ud835\uddc2\ud835\uddc7\ud835\uddcd\ud835\uddbe\ud835\uddcb\ud835\uddc9\ud835\uddcb\ud835\uddbe\ud835\uddcd\ud835\uddbe\ud835\uddbd\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd \ud835\uddba\ud835\uddd1\ud835\uddc2\ud835\uddc8\ud835\uddc6.\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2\n\ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc4e\ud835\udc65\ud835\udc56\ud835\udc5c\ud835\udc5a-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5d\ud835\udc5f\ud835\udc52\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: \ud835\udc9c \u22a2 P, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((\ud835\udc27\u2081 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((\ud835\udc27\u2081)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \u220e"}, {"color": "#FFF59D", "id": "P3", "label": "\ud835\udc43\u2083 : ((0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9\n((0)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2083): ((0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((0)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((\ud835\udc27\u2081\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((\ud835\udc27\u2081)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2082). \ud835\uddab\ud835\uddbe\ud835\uddcd \ud835\udc27\u2081 =\n0.\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc63\ud835\udc4e\ud835\udc5f\ud835\udc56\ud835\udc4e\ud835\udc4f\ud835\udc59\ud835\udc52-\n\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P,\n\ud835\udef7) \u22a2 P\u0027, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((0\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((0)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \u220e"}, {"color": "#FFF59D", "id": "P4", "label": "\ud835\udc43\u2084 (\ud835\udfee.\ud835\udfee.\ud835\udfef) : ((0)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)", "labelHighlightBold": true, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb \ud835\udfee.\ud835\udfee.\ud835\udfef (\ud835\udcaf\u2081.\ud835\udc43\u2084):\n((0)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((0 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)\n\u27f9 ((0)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083).(0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2081).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc5a\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc60-\ud835\udc5d\ud835\udc5c\ud835\udc5b\ud835\udc52\ud835\udc5b\ud835\udc60 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: ((P\n\u27f9 Q), P) \u22a2 Q, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n((0)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \u220e"}, {"color": "#90CAF9", "id": "D1", "label": "\ud835\udc37\u2081 : \u231c\ud835\ude1e\ud835\ude26 \ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26 \ud835\udfe3 \ud835\ude35\ud835\ude30\n\ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 \ud835\udfe2++, \ud835\udfe4\n\ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33\n(\ud835\udfe2++)++, \ud835\udfe5 \ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26\n\ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33\n((\ud835\udfe2++)++)++,\ud835\ude26\ud835\ude35\ud835\ude24. (\ud835\ude10\ud835\ude2f\n\ud835\ude30\ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude33 \ud835\ude38\ud835\ude30\ud835\ude33\ud835\ude25\ud835\ude34, \ud835\udfe3 :=\n\ud835\udfe2++, \ud835\udfe4 := \ud835\udfe3++, \ud835\udfe5 :=\n\ud835\udfe4++, \ud835\ude26\ud835\ude35\ud835\ude24. \ud835\ude10\ud835\ude2f \ud835\ude35\ud835\ude29\ud835\ude2a\ud835\ude34\n\ud835\ude35\ud835\ude26\ud835\ude39\ud835\ude35 \ud835\ude10 \ud835\ude36\ud835\ude34\ud835\ude26 \"\ud835\ude39 := \ud835\ude3a\"\n\ud835\ude35\ud835\ude30 \ud835\ude25\ud835\ude26\ud835\ude2f\ud835\ude30\ud835\ude35\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26\n\ud835\ude34\ud835\ude35\ud835\ude22\ud835\ude35\ud835\ude26\ud835\ude2e\ud835\ude26\ud835\ude2f\ud835\ude35 \ud835\ude35\ud835\ude29\ud835\ude22\ud835\ude35 \ud835\ude39 \ud835\ude2a\ud835\ude34\n\ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26\ud835\ude25 \ud835\ude35\ud835\ude30 \ud835\ude26\ud835\ude32\ud835\ude36\ud835\ude22\ud835\ude2d\n\ud835\ude3a.)\u231d", "shape": "box"}, {"color": "#FFF59D", "id": "P5", "label": "\ud835\udc43\u2085 : (1 = (0)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2085): (1 =\n(0)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: \u231c\ud835\ude1e\ud835\ude26 \ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26 \ud835\udfe3 \ud835\ude35\ud835\ude30\n\ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 \ud835\udfe2++, \ud835\udfe4 \ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26\n\ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 (\ud835\udfe2++)++, \ud835\udfe5 \ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26\n\ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 ((\ud835\udfe2++)++)++,\ud835\ude26\ud835\ude35\ud835\ude24. (\ud835\ude10\ud835\ude2f\n\ud835\ude30\ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude33 \ud835\ude38\ud835\ude30\ud835\ude33\ud835\ude25\ud835\ude34, \ud835\udfe3 := \ud835\udfe2++, \ud835\udfe4 := \ud835\udfe3++,\n\ud835\udfe5 := \ud835\udfe4++, \ud835\ude26\ud835\ude35\ud835\ude24. \ud835\ude10\ud835\ude2f \ud835\ude35\ud835\ude29\ud835\ude2a\ud835\ude34 \ud835\ude35\ud835\ude26\ud835\ude39\ud835\ude35 \ud835\ude10\n\ud835\ude36\ud835\ude34\ud835\ude26 \"\ud835\ude39 := \ud835\ude3a\" \ud835\ude35\ud835\ude30 \ud835\ude25\ud835\ude26\ud835\ude2f\ud835\ude30\ud835\ude35\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26\n\ud835\ude34\ud835\ude35\ud835\ude22\ud835\ude35\ud835\ude26\ud835\ude2e\ud835\ude26\ud835\ude2f\ud835\ude35 \ud835\ude35\ud835\ude29\ud835\ude22\ud835\ude35 \ud835\ude39 \ud835\ude2a\ud835\ude34 \ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26\ud835\ude25 \ud835\ude35\ud835\ude30\n\ud835\ude26\ud835\ude32\ud835\ude36\ud835\ude22\ud835\ude2d \ud835\ude3a.)\u231d \ud835\uddc2\ud835\uddcc \ud835\uddc9\ud835\uddc8\ud835\uddcc\ud835\uddcd\ud835\uddce\ud835\uddc5\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbb\ud835\uddd2 \ud835\uddf1\ud835\uddf2\ud835\uddf3.\n(\ud835\udc37\u2081). (1 = (0)++) \ud835\uddc2\ud835\uddcc \ud835\uddba\ud835\uddc7\n\ud835\uddc2\ud835\uddc7\ud835\uddcd\ud835\uddbe\ud835\uddcb\ud835\uddc9\ud835\uddcb\ud835\uddbe\ud835\uddcd\ud835\uddba\ud835\uddcd\ud835\uddc2\ud835\uddc8\ud835\uddc7 \ud835\uddc8\ud835\uddbf \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n\ud835\uddbd\ud835\uddbe\ud835\uddbf\ud835\uddc2\ud835\uddc7\ud835\uddc2\ud835\uddcd\ud835\uddc2\ud835\uddc8\ud835\uddc7.\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc51\ud835\udc52\ud835\udc53\ud835\udc56\ud835\udc5b\ud835\udc56\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5d\ud835\udc5f\ud835\udc52\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: \ud835\udc9f \u22a2 P, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (1 = (0)++). \u220e"}, {"color": "#FFF59D", "id": "P6", "label": "\ud835\udc43\u2086 : (2 = ((0)++)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2086): (2 =\n((0)++)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: \u231c\ud835\ude1e\ud835\ude26 \ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26 \ud835\udfe3\n\ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 \ud835\udfe2++, \ud835\udfe4 \ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26\n\ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 (\ud835\udfe2++)++, \ud835\udfe5 \ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26\n\ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 ((\ud835\udfe2++)++)++,\ud835\ude26\ud835\ude35\ud835\ude24. (\ud835\ude10\ud835\ude2f\n\ud835\ude30\ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude33 \ud835\ude38\ud835\ude30\ud835\ude33\ud835\ude25\ud835\ude34, \ud835\udfe3 := \ud835\udfe2++, \ud835\udfe4 := \ud835\udfe3++,\n\ud835\udfe5 := \ud835\udfe4++, \ud835\ude26\ud835\ude35\ud835\ude24. \ud835\ude10\ud835\ude2f \ud835\ude35\ud835\ude29\ud835\ude2a\ud835\ude34 \ud835\ude35\ud835\ude26\ud835\ude39\ud835\ude35 \ud835\ude10\n\ud835\ude36\ud835\ude34\ud835\ude26 \"\ud835\ude39 := \ud835\ude3a\" \ud835\ude35\ud835\ude30 \ud835\ude25\ud835\ude26\ud835\ude2f\ud835\ude30\ud835\ude35\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26\n\ud835\ude34\ud835\ude35\ud835\ude22\ud835\ude35\ud835\ude26\ud835\ude2e\ud835\ude26\ud835\ude2f\ud835\ude35 \ud835\ude35\ud835\ude29\ud835\ude22\ud835\ude35 \ud835\ude39 \ud835\ude2a\ud835\ude34 \ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26\ud835\ude25 \ud835\ude35\ud835\ude30\n\ud835\ude26\ud835\ude32\ud835\ude36\ud835\ude22\ud835\ude2d \ud835\ude3a.)\u231d \ud835\uddc2\ud835\uddcc \ud835\uddc9\ud835\uddc8\ud835\uddcc\ud835\uddcd\ud835\uddce\ud835\uddc5\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbb\ud835\uddd2 \ud835\uddf1\ud835\uddf2\ud835\uddf3.\n(\ud835\udc37\u2081). (2 = ((0)++)++) \ud835\uddc2\ud835\uddcc \ud835\uddba\ud835\uddc7\n\ud835\uddc2\ud835\uddc7\ud835\uddcd\ud835\uddbe\ud835\uddcb\ud835\uddc9\ud835\uddcb\ud835\uddbe\ud835\uddcd\ud835\uddba\ud835\uddcd\ud835\uddc2\ud835\uddc8\ud835\uddc7 \ud835\uddc8\ud835\uddbf \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n\ud835\uddbd\ud835\uddbe\ud835\uddbf\ud835\uddc2\ud835\uddc7\ud835\uddc2\ud835\uddcd\ud835\uddc2\ud835\uddc8\ud835\uddc7.\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc51\ud835\udc52\ud835\udc53\ud835\udc56\ud835\udc5b\ud835\udc56\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5d\ud835\udc5f\ud835\udc52\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: \ud835\udc9f \u22a2 P, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (2 = ((0)++)++). \u220e"}, {"color": "#FFF59D", "id": "P7", "label": "\ud835\udc43\u2087 : (3 =\n(((0)++)++)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2087): (3 =\n(((0)++)++)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: \u231c\ud835\ude1e\ud835\ude26\n\ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26 \ud835\udfe3 \ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 \ud835\udfe2++, \ud835\udfe4\n\ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 (\ud835\udfe2++)++, \ud835\udfe5 \ud835\ude35\ud835\ude30\n\ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 ((\ud835\udfe2++)++)++,\ud835\ude26\ud835\ude35\ud835\ude24.\n(\ud835\ude10\ud835\ude2f \ud835\ude30\ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude33 \ud835\ude38\ud835\ude30\ud835\ude33\ud835\ude25\ud835\ude34, \ud835\udfe3 := \ud835\udfe2++, \ud835\udfe4 :=\n\ud835\udfe3++, \ud835\udfe5 := \ud835\udfe4++, \ud835\ude26\ud835\ude35\ud835\ude24. \ud835\ude10\ud835\ude2f \ud835\ude35\ud835\ude29\ud835\ude2a\ud835\ude34 \ud835\ude35\ud835\ude26\ud835\ude39\ud835\ude35\n\ud835\ude10 \ud835\ude36\ud835\ude34\ud835\ude26 \"\ud835\ude39 := \ud835\ude3a\" \ud835\ude35\ud835\ude30 \ud835\ude25\ud835\ude26\ud835\ude2f\ud835\ude30\ud835\ude35\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26\n\ud835\ude34\ud835\ude35\ud835\ude22\ud835\ude35\ud835\ude26\ud835\ude2e\ud835\ude26\ud835\ude2f\ud835\ude35 \ud835\ude35\ud835\ude29\ud835\ude22\ud835\ude35 \ud835\ude39 \ud835\ude2a\ud835\ude34 \ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26\ud835\ude25 \ud835\ude35\ud835\ude30\n\ud835\ude26\ud835\ude32\ud835\ude36\ud835\ude22\ud835\ude2d \ud835\ude3a.)\u231d \ud835\uddc2\ud835\uddcc \ud835\uddc9\ud835\uddc8\ud835\uddcc\ud835\uddcd\ud835\uddce\ud835\uddc5\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbb\ud835\uddd2 \ud835\uddf1\ud835\uddf2\ud835\uddf3.\n(\ud835\udc37\u2081). (3 = (((0)++)++)++) \ud835\uddc2\ud835\uddcc \ud835\uddba\ud835\uddc7\n\ud835\uddc2\ud835\uddc7\ud835\uddcd\ud835\uddbe\ud835\uddcb\ud835\uddc9\ud835\uddcb\ud835\uddbe\ud835\uddcd\ud835\uddba\ud835\uddcd\ud835\uddc2\ud835\uddc8\ud835\uddc7 \ud835\uddc8\ud835\uddbf \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n\ud835\uddbd\ud835\uddbe\ud835\uddbf\ud835\uddc2\ud835\uddc7\ud835\uddc2\ud835\uddcd\ud835\uddc2\ud835\uddc8\ud835\uddc7.\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc51\ud835\udc52\ud835\udc53\ud835\udc56\ud835\udc5b\ud835\udc56\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5d\ud835\udc5f\ud835\udc52\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: \ud835\udc9f \u22a2 P, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (3 =\n(((0)++)++)++). \u220e"}, {"color": "#FFF59D", "id": "P8", "label": "\ud835\udc43\u2088 : (4 =\n((((0)++)++)++)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2088): (4 =\n((((0)++)++)++)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: \u231c\ud835\ude1e\ud835\ude26\n\ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26 \ud835\udfe3 \ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 \ud835\udfe2++, \ud835\udfe4\n\ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 (\ud835\udfe2++)++, \ud835\udfe5 \ud835\ude35\ud835\ude30\n\ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 ((\ud835\udfe2++)++)++,\ud835\ude26\ud835\ude35\ud835\ude24.\n(\ud835\ude10\ud835\ude2f \ud835\ude30\ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude33 \ud835\ude38\ud835\ude30\ud835\ude33\ud835\ude25\ud835\ude34, \ud835\udfe3 := \ud835\udfe2++, \ud835\udfe4 :=\n\ud835\udfe3++, \ud835\udfe5 := \ud835\udfe4++, \ud835\ude26\ud835\ude35\ud835\ude24. \ud835\ude10\ud835\ude2f \ud835\ude35\ud835\ude29\ud835\ude2a\ud835\ude34 \ud835\ude35\ud835\ude26\ud835\ude39\ud835\ude35\n\ud835\ude10 \ud835\ude36\ud835\ude34\ud835\ude26 \"\ud835\ude39 := \ud835\ude3a\" \ud835\ude35\ud835\ude30 \ud835\ude25\ud835\ude26\ud835\ude2f\ud835\ude30\ud835\ude35\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26\n\ud835\ude34\ud835\ude35\ud835\ude22\ud835\ude35\ud835\ude26\ud835\ude2e\ud835\ude26\ud835\ude2f\ud835\ude35 \ud835\ude35\ud835\ude29\ud835\ude22\ud835\ude35 \ud835\ude39 \ud835\ude2a\ud835\ude34 \ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26\ud835\ude25 \ud835\ude35\ud835\ude30\n\ud835\ude26\ud835\ude32\ud835\ude36\ud835\ude22\ud835\ude2d \ud835\ude3a.)\u231d \ud835\uddc2\ud835\uddcc \ud835\uddc9\ud835\uddc8\ud835\uddcc\ud835\uddcd\ud835\uddce\ud835\uddc5\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbb\ud835\uddd2 \ud835\uddf1\ud835\uddf2\ud835\uddf3.\n(\ud835\udc37\u2081). (4 = ((((0)++)++)++)++) \ud835\uddc2\ud835\uddcc\n\ud835\uddba\ud835\uddc7 \ud835\uddc2\ud835\uddc7\ud835\uddcd\ud835\uddbe\ud835\uddcb\ud835\uddc9\ud835\uddcb\ud835\uddbe\ud835\uddcd\ud835\uddba\ud835\uddcd\ud835\uddc2\ud835\uddc8\ud835\uddc7 \ud835\uddc8\ud835\uddbf \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n\ud835\uddbd\ud835\uddbe\ud835\uddbf\ud835\uddc2\ud835\uddc7\ud835\uddc2\ud835\uddcd\ud835\uddc2\ud835\uddc8\ud835\uddc7.\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc51\ud835\udc52\ud835\udc53\ud835\udc56\ud835\udc5b\ud835\udc56\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5d\ud835\udc5f\ud835\udc52\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: \ud835\udc9f \u22a2 P, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (4 =\n((((0)++)++)++)++). \u220e"}, {"color": "#FFF59D", "id": "P9", "label": "\ud835\udc43\u2089 : (((0)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9\n(((0)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2089): (((0)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9\n(((0)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((\ud835\udc27\u2081 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((\ud835\udc27\u2081)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2082). \ud835\uddab\ud835\uddbe\ud835\uddcd \ud835\udc27\u2081 =\n(0)++.\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc63\ud835\udc4e\ud835\udc5f\ud835\udc56\ud835\udc4e\ud835\udc4f\ud835\udc59\ud835\udc52-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe\n\ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, \ud835\udef7) \u22a2 P\u0027, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (((0)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 (((0)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \u220e"}, {"color": "#FFF59D", "id": "P10", "label": "\ud835\udc43\u2081\u2080 : (((0)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2081\u2080): (((0)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3:\n(((0)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9\n(((0)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2089).((0)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. \ud835\udfee.\ud835\udfee.\ud835\udfef (\ud835\udc43\u2084).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe,\n\ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc5a\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc60-\ud835\udc5d\ud835\udc5c\ud835\udc5b\ud835\udc52\ud835\udc5b\ud835\udc60 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe\n\ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: ((P \u27f9 Q), P) \u22a2 Q, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (((0)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \u220e"}, {"color": "#FFF59D", "id": "P11", "label": "\ud835\udc43\u2081\u2081 : ((((0)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)\n\u27f9 ((((0)++)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2081\u2081):\n((((0)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)\n\u27f9 ((((0)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((\ud835\udc27\u2081 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((\ud835\udc27\u2081)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2082). \ud835\uddab\ud835\uddbe\ud835\uddcd \ud835\udc27\u2081 =\n((0)++)++.\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc63\ud835\udc4e\ud835\udc5f\ud835\udc56\ud835\udc4e\ud835\udc4f\ud835\udc59\ud835\udc52-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe\n\ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, \ud835\udef7) \u22a2 P\u0027, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((((0)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((((0)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \u220e"}, {"color": "#FFF59D", "id": "P12", "label": "\ud835\udc43\u2081\u2082 : ((((0)++)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2081\u2082):\n((((0)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((((0)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((((0)++)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2081\u2081).(((0)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2081\u2080).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc5a\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc60-\ud835\udc5d\ud835\udc5c\ud835\udc5b\ud835\udc52\ud835\udc5b\ud835\udc60 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: ((P\n\u27f9 Q), P) \u22a2 Q, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n((((0)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \u220e"}, {"color": "#FFF59D", "id": "P13", "label": "\ud835\udc43\u2081\u2083 :\n(((((0)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9\n(((((0)++)++)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2081\u2083):\n(((((0)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 (((((0)++)++)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3:\n((\ud835\udc27\u2081 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9\n((\ud835\udc27\u2081)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2082). \ud835\uddab\ud835\uddbe\ud835\uddcd \ud835\udc27\u2081\n= (((0)++)++)++.\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2\n\ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc63\ud835\udc4e\ud835\udc5f\ud835\udc56\ud835\udc4e\ud835\udc4f\ud835\udc59\ud835\udc52-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, \ud835\udef7) \u22a2 P\u0027, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (((((0)++)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9\n(((((0)++)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \u220e"}, {"color": "#FFF59D", "id": "P14", "label": "\ud835\udc43\u2081\u2084 :\n(((((0)++)++)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2081\u2084):\n(((((0)++)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (((((0)++)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9\n(((((0)++)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2081\u2083).((((0)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2081\u2082).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc5a\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc60-\ud835\udc5d\ud835\udc5c\ud835\udc5b\ud835\udc52\ud835\udc5b\ud835\udc60 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: ((P\n\u27f9 Q), P) \u22a2 Q, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n(((((0)++)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \u220e"}, {"color": "#FFF59D", "id": "P15", "label": "\ud835\udc43\u2081\u2085 : ((0)++ = 1)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2081\u2085): ((0)++ =\n1). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (1 = (0)++) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2085). \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2\n\ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59\ud835\udc56\ud835\udc61\ud835\udc66-\ud835\udc50\ud835\udc5c\ud835\udc5a\ud835\udc5a\ud835\udc62\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc63\ud835\udc56\ud835\udc61\ud835\udc66\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (x = y) \u22a2 (y =\nx), \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((0)++ = 1).\n\u220e"}, {"color": "#FFF59D", "id": "P16", "label": "\ud835\udc43\u2081\u2086 : (2 = (1)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2081\u2086): (2 =\n(1)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (2 = ((0)++)++)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2086). ((0)++\n= 1) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2081\u2085).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\n\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe\n\ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, (Q = R)) \u22a2 P\u0027, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (2 = (1)++). \u220e"}, {"color": "#FFF59D", "id": "P17", "label": "\ud835\udc43\u2081\u2087 : (((0)++)++ =\n2)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2081\u2087): (((0)++)++\n= 2). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (2 = ((0)++)++)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2086).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59\ud835\udc56\ud835\udc61\ud835\udc66-\n\ud835\udc50\ud835\udc5c\ud835\udc5a\ud835\udc5a\ud835\udc62\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc63\ud835\udc56\ud835\udc61\ud835\udc66 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (x\n= y) \u22a2 (y = x), \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n(((0)++)++ = 2). \u220e"}, {"color": "#FFF59D", "id": "P18", "label": "\ud835\udc43\u2081\u2088 : (3 = (2)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2081\u2088): (3 =\n(2)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (3 =\n(((0)++)++)++) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2087). (((0)++)++ = 2)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2081\u2087).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\n\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe\n\ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, (Q = R)) \u22a2 P\u0027, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (3 = (2)++). \u220e"}, {"color": "#FFF59D", "id": "P19", "label": "\ud835\udc43\u2081\u2089 : ((((0)++)++)++\n= 3)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2081\u2089):\n((((0)++)++)++ = 3). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (3 =\n(((0)++)++)++) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2087). \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59\ud835\udc56\ud835\udc61\ud835\udc66-\ud835\udc50\ud835\udc5c\ud835\udc5a\ud835\udc5a\ud835\udc62\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc63\ud835\udc56\ud835\udc61\ud835\udc66 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe\n\ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (x = y) \u22a2 (y = x), \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((((0)++)++)++ =\n3). \u220e"}, {"color": "#FFF59D", "id": "P20", "label": "\ud835\udc43\u2082\u2080 : ((2)++ = 3)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2082\u2080): ((2)++ =\n3). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((((0)++)++)++ = 3)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2081\u2089).\n(((0)++)++ = 2) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2081\u2087).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, (Q = R)) \u22a2\nP\u0027, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((2)++ = 3).\n\u220e"}, {"color": "#FFF59D", "id": "P21", "label": "\ud835\udc43\u2082\u2081 (\ud835\udfee.\ud835\udfed.\ud835\udff0) : (3\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)", "labelHighlightBold": true, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb \ud835\udfee.\ud835\udfed.\ud835\udff0 (\ud835\udcaf\u2081.\ud835\udc43\u2082\u2081): (3\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3:\n((((0)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2081\u2082). ((((0)++)++)++ = 3)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2081\u2089).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\n\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe\n\ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, (Q = R)) \u22a2 P\u0027, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (3 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \u220e"}, {"color": "#FFF59D", "id": "P22", "label": "\ud835\udc43\u2082\u2082 : (4 =\n((((0)++)++)++)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2082\u2082): (4 =\n((((0)++)++)++)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: \u231c\ud835\ude1e\ud835\ude26\n\ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26 \ud835\udfe3 \ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 \ud835\udfe2++, \ud835\udfe4\n\ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 (\ud835\udfe2++)++, \ud835\udfe5 \ud835\ude35\ud835\ude30\n\ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 ((\ud835\udfe2++)++)++,\ud835\ude26\ud835\ude35\ud835\ude24.\n(\ud835\ude10\ud835\ude2f \ud835\ude30\ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude33 \ud835\ude38\ud835\ude30\ud835\ude33\ud835\ude25\ud835\ude34, \ud835\udfe3 := \ud835\udfe2++, \ud835\udfe4 :=\n\ud835\udfe3++, \ud835\udfe5 := \ud835\udfe4++, \ud835\ude26\ud835\ude35\ud835\ude24. \ud835\ude10\ud835\ude2f \ud835\ude35\ud835\ude29\ud835\ude2a\ud835\ude34 \ud835\ude35\ud835\ude26\ud835\ude39\ud835\ude35\n\ud835\ude10 \ud835\ude36\ud835\ude34\ud835\ude26 \"\ud835\ude39 := \ud835\ude3a\" \ud835\ude35\ud835\ude30 \ud835\ude25\ud835\ude26\ud835\ude2f\ud835\ude30\ud835\ude35\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26\n\ud835\ude34\ud835\ude35\ud835\ude22\ud835\ude35\ud835\ude26\ud835\ude2e\ud835\ude26\ud835\ude2f\ud835\ude35 \ud835\ude35\ud835\ude29\ud835\ude22\ud835\ude35 \ud835\ude39 \ud835\ude2a\ud835\ude34 \ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26\ud835\ude25 \ud835\ude35\ud835\ude30\n\ud835\ude26\ud835\ude32\ud835\ude36\ud835\ude22\ud835\ude2d \ud835\ude3a.)\u231d \ud835\uddc2\ud835\uddcc \ud835\uddc9\ud835\uddc8\ud835\uddcc\ud835\uddcd\ud835\uddce\ud835\uddc5\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbb\ud835\uddd2 \ud835\uddf1\ud835\uddf2\ud835\uddf3.\n(\ud835\udc37\u2081). (4 = ((((0)++)++)++)++) \ud835\uddc2\ud835\uddcc\n\ud835\uddba\ud835\uddc7 \ud835\uddc2\ud835\uddc7\ud835\uddcd\ud835\uddbe\ud835\uddcb\ud835\uddc9\ud835\uddcb\ud835\uddbe\ud835\uddcd\ud835\uddba\ud835\uddcd\ud835\uddc2\ud835\uddc8\ud835\uddc7 \ud835\uddc8\ud835\uddbf \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n\ud835\uddbd\ud835\uddbe\ud835\uddbf\ud835\uddc2\ud835\uddc7\ud835\uddc2\ud835\uddcd\ud835\uddc2\ud835\uddc8\ud835\uddc7.\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc51\ud835\udc52\ud835\udc53\ud835\udc56\ud835\udc5b\ud835\udc56\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5d\ud835\udc5f\ud835\udc52\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: \ud835\udc9f \u22a2 P, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (4 =\n((((0)++)++)++)++). \u220e"}, {"color": "#FFF59D", "id": "P23", "label": "\ud835\udc43\u2082\u2083 :\n(((((0)++)++)++)++ =\n4)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2082\u2083):\n(((((0)++)++)++)++ = 4). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3:\n(4 = ((((0)++)++)++)++) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2088). \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2\n\ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59\ud835\udc56\ud835\udc61\ud835\udc66-\ud835\udc50\ud835\udc5c\ud835\udc5a\ud835\udc5a\ud835\udc62\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc63\ud835\udc56\ud835\udc61\ud835\udc66\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (x = y) \u22a2 (y =\nx), \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n(((((0)++)++)++)++ = 4). \u220e"}, {"color": "#FFF59D", "id": "P24", "label": "\ud835\udc43\u2082\u2084 : ((3)++ = 4)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2082\u2084): ((3)++ =\n4). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (((((0)++)++)++)++ =\n4) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2082\u2083).\n((((0)++)++)++ = 3) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2081\u2089).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, (Q = R)) \u22a2\nP\u0027, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((3)++ = 4).\n\u220e"}, {"color": "#FFF59D", "id": "P25", "label": "\ud835\udc43\u2082\u2085 :\n(((((0)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9\n(((((0)++)++)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2082\u2085):\n(((((0)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 (((((0)++)++)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3:\n(((((0)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 (((((0)++)++)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2081\u2083). ((3)++ = 4)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2082\u2084).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\n\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe\n\ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, (Q = R)) \u22a2 P\u0027, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (((((0)++)++)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9\n(((((0)++)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \u220e"}, {"color": "#FFF59D", "id": "P26", "label": "\ud835\udc43\u2082\u2086 : (4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2082\u2086): (4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3:\n(((((0)++)++)++)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2081\u2084). (((((0)++)++)++)++ = 4)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2082\u2083).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\n\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe\n\ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, (Q = R)) \u22a2 P\u0027, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (4 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \u220e"}, {"color": "#81C784", "id": "A3", "label": "\ud835\udc34\u2083 (\ud835\udfee.\ud835\udfef) : \u231c\ud835\udfe2 \ud835\ude2a\ud835\ude34 \ud835\ude2f\ud835\ude30\ud835\ude35\n\ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude34\ud835\ude36\ud835\ude24\ud835\ude24\ud835\ude26\ud835\ude34\ud835\ude34\ud835\ude30\ud835\ude33 \ud835\ude30\ud835\ude27 \ud835\ude22\ud835\ude2f\ud835\ude3a\n\ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33;\n\ud835\ude2a.\ud835\ude26., \ud835\ude38\ud835\ude26 \ud835\ude29\ud835\ude22\ud835\ude37\ud835\ude26 \ud835\ude2f++ \u2260\n\ud835\udfe2 \ud835\ude27\ud835\ude30\ud835\ude33 \ud835\ude26\ud835\ude37\ud835\ude26\ud835\ude33\ud835\ude3a \ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d\n\ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 \ud835\ude2f.\u231d", "shape": "box"}, {"color": "#FFF59D", "id": "P27", "label": "\ud835\udc43\u2082\u2087 : ((\ud835\udc27\u2082 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9\n((\ud835\udc27\u2082)++ \u2260 0))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2082\u2087): ((\ud835\udc27\u2082 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((\ud835\udc27\u2082)++ \u2260 0)).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: \u231c\ud835\udfe2 \ud835\ude2a\ud835\ude34 \ud835\ude2f\ud835\ude30\ud835\ude35 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude34\ud835\ude36\ud835\ude24\ud835\ude24\ud835\ude26\ud835\ude34\ud835\ude34\ud835\ude30\ud835\ude33\n\ud835\ude30\ud835\ude27 \ud835\ude22\ud835\ude2f\ud835\ude3a \ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33; \ud835\ude2a.\ud835\ude26., \ud835\ude38\ud835\ude26\n\ud835\ude29\ud835\ude22\ud835\ude37\ud835\ude26 \ud835\ude2f++ \u2260 \ud835\udfe2 \ud835\ude27\ud835\ude30\ud835\ude33 \ud835\ude26\ud835\ude37\ud835\ude26\ud835\ude33\ud835\ude3a \ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d\n\ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 \ud835\ude2f.\u231d \ud835\uddc2\ud835\uddcc \ud835\uddc9\ud835\uddc8\ud835\uddcc\ud835\uddcd\ud835\uddce\ud835\uddc5\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbb\ud835\uddd2\n\ud835\uddee\ud835\ude05\ud835\uddf6\ud835\uddfc\ud835\uddfa \ud835\udfee.\ud835\udfef (\ud835\udc34\u2083). ((\ud835\udc27\u2082 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((\ud835\udc27\u2082)++ \u2260 0))\n\ud835\uddc2\ud835\uddcc \ud835\uddba \ud835\uddcf\ud835\uddba\ud835\uddc5\ud835\uddc2\ud835\uddbd \ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddc6\ud835\uddce\ud835\uddc5\ud835\uddba \ud835\uddcc\ud835\uddcd\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddc6\ud835\uddbe\ud835\uddc7\ud835\uddcd\n\ud835\uddc2\ud835\uddc7\ud835\uddcd\ud835\uddbe\ud835\uddcb\ud835\uddc9\ud835\uddcb\ud835\uddbe\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n\ud835\uddba\ud835\uddd1\ud835\uddc2\ud835\uddc8\ud835\uddc6.\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc4e\ud835\udc65\ud835\udc56\ud835\udc5c\ud835\udc5a-\n\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5d\ud835\udc5f\ud835\udc52\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: \ud835\udc9c\n\u22a2 P, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((\ud835\udc27\u2082 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((\ud835\udc27\u2082)++ \u2260 0)).\n\u220e"}, {"color": "#FFF59D", "id": "P28", "label": "\ud835\udc43\u2082\u2088 : ((3 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9\n((3)++ \u2260 0))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2082\u2088): ((3 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((3)++ \u2260 0)).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((\ud835\udc27\u2082 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)\n\u27f9 ((\ud835\udc27\u2082)++ \u2260 0)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2082\u2087). \ud835\uddab\ud835\uddbe\ud835\uddcd \ud835\udc27\u2082 =\n3.\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc63\ud835\udc4e\ud835\udc5f\ud835\udc56\ud835\udc4e\ud835\udc4f\ud835\udc59\ud835\udc52-\n\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P,\n\ud835\udef7) \u22a2 P\u0027, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((3\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((3)++ \u2260\n0)). \u220e"}, {"color": "#FFF59D", "id": "P29", "label": "\ud835\udc43\u2082\u2089 : ((3)++ \u2260 0)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2082\u2089): ((3)++ \u2260\n0). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((3 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((3)++ \u2260 0)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2082\u2088).(3 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. \ud835\udfee.\ud835\udfed.\ud835\udff0 (\ud835\udc43\u2082\u2081).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2\n\ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc5a\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc60-\ud835\udc5d\ud835\udc5c\ud835\udc5b\ud835\udc52\ud835\udc5b\ud835\udc60 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe:\n((P \u27f9 Q), P) \u22a2 Q, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((3)++ \u2260 0). \u220e"}, {"color": "#FFF59D", "id": "P30", "label": "\ud835\udc43\u2083\u2080 (\ud835\udfee.\ud835\udfed.\ud835\udff2) : (4 \u2260\n0)", "labelHighlightBold": true, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb \ud835\udfee.\ud835\udfed.\ud835\udff2 (\ud835\udcaf\u2081.\ud835\udc43\u2083\u2080): (4 \u2260\n0). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((3)++ \u2260 0) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2082\u2089). ((3)++ = 4)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2082\u2084).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\n\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe\n\ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, (Q = R)) \u22a2 P\u0027, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (4 \u2260 0). \u220e"}, {"color": "#81C784", "id": "A4", "label": "\ud835\udc34\u2084 (\ud835\udfee.\ud835\udff0) :\n\u231c\ud835\ude0b\ud835\ude2a\ud835\ude27\ud835\ude27\ud835\ude26\ud835\ude33\ud835\ude26\ud835\ude2f\ud835\ude35 \ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d\n\ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33\ud835\ude34 \ud835\ude2e\ud835\ude36\ud835\ude34\ud835\ude35 \ud835\ude29\ud835\ude22\ud835\ude37\ud835\ude26\n\ud835\ude25\ud835\ude2a\ud835\ude27\ud835\ude27\ud835\ude26\ud835\ude33\ud835\ude26\ud835\ude2f\ud835\ude35\n\ud835\ude34\ud835\ude36\ud835\ude24\ud835\ude24\ud835\ude26\ud835\ude34\ud835\ude34\ud835\ude30\ud835\ude33\ud835\ude34; \ud835\ude2a.\ud835\ude26., \ud835\ude2a\ud835\ude27\n\ud835\ude2f, \ud835\ude2e \ud835\ude22\ud835\ude33\ud835\ude26 \ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d\n\ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33\ud835\ude34 \ud835\ude22\ud835\ude2f\ud835\ude25 \ud835\ude2f \u2260 \ud835\ude2e,\n\ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude2f \ud835\ude2f++ \u2260 \ud835\ude2e++.\n\ud835\ude0c\ud835\ude32\ud835\ude36\ud835\ude2a\ud835\ude37\ud835\ude22\ud835\ude2d\ud835\ude26\ud835\ude2f\ud835\ude35\ud835\ude2d\ud835\ude3a, \ud835\ude2a\ud835\ude27 \ud835\ude2f++\n= \ud835\ude2e++, \ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude2f \ud835\ude38\ud835\ude26 \ud835\ude2e\ud835\ude36\ud835\ude34\ud835\ude35\n\ud835\ude29\ud835\ude22\ud835\ude37\ud835\ude26 \ud835\ude2f = \ud835\ude2e.\u231d", "shape": "box"}, {"color": "#FFF59D", "id": "P31", "label": "\ud835\udc43\u2083\u2081 : ((((\ud835\udc27\u2083 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227\n(\ud835\udc26\u2081 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (\ud835\udc27\u2083 \u2260\n\ud835\udc26\u2081)) \u27f9 ((\ud835\udc27\u2083)++ \u2260\n(\ud835\udc26\u2081)++))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2083\u2081): ((((\ud835\udc27\u2083\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (\ud835\udc26\u2081 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (\ud835\udc27\u2083 \u2260 \ud835\udc26\u2081)) \u27f9\n((\ud835\udc27\u2083)++ \u2260 (\ud835\udc26\u2081)++)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3:\n\u231c\ud835\ude0b\ud835\ude2a\ud835\ude27\ud835\ude27\ud835\ude26\ud835\ude33\ud835\ude26\ud835\ude2f\ud835\ude35 \ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33\ud835\ude34 \ud835\ude2e\ud835\ude36\ud835\ude34\ud835\ude35\n\ud835\ude29\ud835\ude22\ud835\ude37\ud835\ude26 \ud835\ude25\ud835\ude2a\ud835\ude27\ud835\ude27\ud835\ude26\ud835\ude33\ud835\ude26\ud835\ude2f\ud835\ude35 \ud835\ude34\ud835\ude36\ud835\ude24\ud835\ude24\ud835\ude26\ud835\ude34\ud835\ude34\ud835\ude30\ud835\ude33\ud835\ude34; \ud835\ude2a.\ud835\ude26.,\n\ud835\ude2a\ud835\ude27 \ud835\ude2f, \ud835\ude2e \ud835\ude22\ud835\ude33\ud835\ude26 \ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33\ud835\ude34 \ud835\ude22\ud835\ude2f\ud835\ude25\n\ud835\ude2f \u2260 \ud835\ude2e, \ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude2f \ud835\ude2f++ \u2260 \ud835\ude2e++.\n\ud835\ude0c\ud835\ude32\ud835\ude36\ud835\ude2a\ud835\ude37\ud835\ude22\ud835\ude2d\ud835\ude26\ud835\ude2f\ud835\ude35\ud835\ude2d\ud835\ude3a, \ud835\ude2a\ud835\ude27 \ud835\ude2f++ = \ud835\ude2e++, \ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude2f\n\ud835\ude38\ud835\ude26 \ud835\ude2e\ud835\ude36\ud835\ude34\ud835\ude35 \ud835\ude29\ud835\ude22\ud835\ude37\ud835\ude26 \ud835\ude2f = \ud835\ude2e.\u231d \ud835\uddc2\ud835\uddcc\n\ud835\uddc9\ud835\uddc8\ud835\uddcc\ud835\uddcd\ud835\uddce\ud835\uddc5\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbb\ud835\uddd2 \ud835\uddee\ud835\ude05\ud835\uddf6\ud835\uddfc\ud835\uddfa \ud835\udfee.\ud835\udff0 (\ud835\udc34\u2084).\n((((\ud835\udc27\u2083 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227\n(\ud835\udc26\u2081 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (\ud835\udc27\u2083\n\u2260 \ud835\udc26\u2081)) \u27f9 ((\ud835\udc27\u2083)++ \u2260 (\ud835\udc26\u2081)++)) \ud835\uddc2\ud835\uddcc \ud835\uddba\n\ud835\uddcf\ud835\uddba\ud835\uddc5\ud835\uddc2\ud835\uddbd \ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddc6\ud835\uddce\ud835\uddc5\ud835\uddba \ud835\uddcc\ud835\uddcd\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddc6\ud835\uddbe\ud835\uddc7\ud835\uddcd\n\ud835\uddc2\ud835\uddc7\ud835\uddcd\ud835\uddbe\ud835\uddcb\ud835\uddc9\ud835\uddcb\ud835\uddbe\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n\ud835\uddba\ud835\uddd1\ud835\uddc2\ud835\uddc8\ud835\uddc6.\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc4e\ud835\udc65\ud835\udc56\ud835\udc5c\ud835\udc5a-\n\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5d\ud835\udc5f\ud835\udc52\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: \ud835\udc9c\n\u22a2 P, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((((\ud835\udc27\u2083 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (\ud835\udc26\u2081 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (\ud835\udc27\u2083 \u2260 \ud835\udc26\u2081)) \u27f9\n((\ud835\udc27\u2083)++ \u2260 (\ud835\udc26\u2081)++)). \u220e"}, {"color": "#FFF59D", "id": "P32", "label": "\ud835\udc43\u2083\u2082 : ((((\ud835\udc27\u2084 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227\n(\ud835\udc26\u2082 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((\ud835\udc27\u2084)++ =\n(\ud835\udc26\u2082)++)) \u27f9 (\ud835\udc27\u2084 =\n\ud835\udc26\u2082))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2083\u2082): ((((\ud835\udc27\u2084\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (\ud835\udc26\u2082 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((\ud835\udc27\u2084)++ =\n(\ud835\udc26\u2082)++)) \u27f9 (\ud835\udc27\u2084 = \ud835\udc26\u2082)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3:\n\u231c\ud835\ude0b\ud835\ude2a\ud835\ude27\ud835\ude27\ud835\ude26\ud835\ude33\ud835\ude26\ud835\ude2f\ud835\ude35 \ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33\ud835\ude34 \ud835\ude2e\ud835\ude36\ud835\ude34\ud835\ude35\n\ud835\ude29\ud835\ude22\ud835\ude37\ud835\ude26 \ud835\ude25\ud835\ude2a\ud835\ude27\ud835\ude27\ud835\ude26\ud835\ude33\ud835\ude26\ud835\ude2f\ud835\ude35 \ud835\ude34\ud835\ude36\ud835\ude24\ud835\ude24\ud835\ude26\ud835\ude34\ud835\ude34\ud835\ude30\ud835\ude33\ud835\ude34; \ud835\ude2a.\ud835\ude26.,\n\ud835\ude2a\ud835\ude27 \ud835\ude2f, \ud835\ude2e \ud835\ude22\ud835\ude33\ud835\ude26 \ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33\ud835\ude34 \ud835\ude22\ud835\ude2f\ud835\ude25\n\ud835\ude2f \u2260 \ud835\ude2e, \ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude2f \ud835\ude2f++ \u2260 \ud835\ude2e++.\n\ud835\ude0c\ud835\ude32\ud835\ude36\ud835\ude2a\ud835\ude37\ud835\ude22\ud835\ude2d\ud835\ude26\ud835\ude2f\ud835\ude35\ud835\ude2d\ud835\ude3a, \ud835\ude2a\ud835\ude27 \ud835\ude2f++ = \ud835\ude2e++, \ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude2f\n\ud835\ude38\ud835\ude26 \ud835\ude2e\ud835\ude36\ud835\ude34\ud835\ude35 \ud835\ude29\ud835\ude22\ud835\ude37\ud835\ude26 \ud835\ude2f = \ud835\ude2e.\u231d \ud835\uddc2\ud835\uddcc\n\ud835\uddc9\ud835\uddc8\ud835\uddcc\ud835\uddcd\ud835\uddce\ud835\uddc5\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbb\ud835\uddd2 \ud835\uddee\ud835\ude05\ud835\uddf6\ud835\uddfc\ud835\uddfa \ud835\udfee.\ud835\udff0 (\ud835\udc34\u2084).\n((((\ud835\udc27\u2084 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227\n(\ud835\udc26\u2082 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227\n((\ud835\udc27\u2084)++ = (\ud835\udc26\u2082)++)) \u27f9 (\ud835\udc27\u2084 = \ud835\udc26\u2082))\n\ud835\uddc2\ud835\uddcc \ud835\uddba \ud835\uddcf\ud835\uddba\ud835\uddc5\ud835\uddc2\ud835\uddbd \ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddc6\ud835\uddce\ud835\uddc5\ud835\uddba \ud835\uddcc\ud835\uddcd\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddc6\ud835\uddbe\ud835\uddc7\ud835\uddcd\n\ud835\uddc2\ud835\uddc7\ud835\uddcd\ud835\uddbe\ud835\uddcb\ud835\uddc9\ud835\uddcb\ud835\uddbe\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n\ud835\uddba\ud835\uddd1\ud835\uddc2\ud835\uddc8\ud835\uddc6.\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc4e\ud835\udc65\ud835\udc56\ud835\udc5c\ud835\udc5a-\n\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5d\ud835\udc5f\ud835\udc52\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: \ud835\udc9c\n\u22a2 P, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((((\ud835\udc27\u2084 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (\ud835\udc26\u2082 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((\ud835\udc27\u2084)++ =\n(\ud835\udc26\u2082)++)) \u27f9 (\ud835\udc27\u2084 = \ud835\udc26\u2082)). \u220e"}, {"color": "#FFF59D", "id": "P33", "label": "\ud835\udc43\u2083\u2083 : ((((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (4 \u2260 0))\n\u27f9 ((4)++ \u2260 (0)++))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2083\u2083): ((((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (4 \u2260 0)) \u27f9\n((4)++ \u2260 (0)++)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((((\ud835\udc27\u2083\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (\ud835\udc26\u2081 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (\ud835\udc27\u2083 \u2260 \ud835\udc26\u2081)) \u27f9\n((\ud835\udc27\u2083)++ \u2260 (\ud835\udc26\u2081)++)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083\u2081). \ud835\uddab\ud835\uddbe\ud835\uddcd \ud835\udc27\u2083 = 4, \ud835\udc26\u2081 =\n0.\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc63\ud835\udc4e\ud835\udc5f\ud835\udc56\ud835\udc4e\ud835\udc4f\ud835\udc59\ud835\udc52-\n\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P,\n\ud835\udef7) \u22a2 P\u0027, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((((4\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (4 \u2260 0)) \u27f9\n((4)++ \u2260 (0)++)). \u220e"}, {"color": "#FFF59D", "id": "P34", "label": "\ud835\udc43\u2083\u2084 : ((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2083\u2084): ((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) (\ud835\udc43) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2082\u2086). (0 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) (\ud835\udc44) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2081).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc50\ud835\udc5c\ud835\udc5b\ud835\udc57\ud835\udc62\ud835\udc5b\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc5f\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, Q) \u22a2 (P \u22c0\nQ), \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \u220e"}, {"color": "#FFF59D", "id": "P35", "label": "\ud835\udc43\u2083\u2085 : (((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (4 \u2260 0))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2083\u2085): (((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (4 \u2260 0)).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((4 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)\n\u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) (\ud835\udc43)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083\u2084). (4 \u2260\n0) (\ud835\udc44) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. \ud835\udfee.\ud835\udfed.\ud835\udff2\n(\ud835\udc43\u2083\u2080).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc50\ud835\udc5c\ud835\udc5b\ud835\udc57\ud835\udc62\ud835\udc5b\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc5f\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, Q) \u22a2 (P \u22c0\nQ), \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (4 \u2260 0)). \u220e"}, {"color": "#FFF59D", "id": "P36", "label": "\ud835\udc43\u2083\u2086 : ((4)++ \u2260\n(0)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2083\u2086): ((4)++ \u2260\n(0)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (4 \u2260 0)) \u27f9\n((4)++ \u2260 (0)++)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083\u2083).(((4 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (4 \u2260 0)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083\u2085).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc5a\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc60-\ud835\udc5d\ud835\udc5c\ud835\udc5b\ud835\udc52\ud835\udc5b\ud835\udc60 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: ((P\n\u27f9 Q), P) \u22a2 Q, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n((4)++ \u2260 (0)++). \u220e"}, {"color": "#FFF59D", "id": "P37", "label": "\ud835\udc43\u2083\u2087 : (5 = (((((0)++\n)++)++)++)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2083\u2087): (5 =\n(((((0)++)++)++)++)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3:\n\u231c\ud835\ude1e\ud835\ude26 \ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26 \ud835\udfe3 \ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33\n\ud835\udfe2++, \ud835\udfe4 \ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 (\ud835\udfe2++)++,\n\ud835\udfe5 \ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33\n((\ud835\udfe2++)++)++,\ud835\ude26\ud835\ude35\ud835\ude24. (\ud835\ude10\ud835\ude2f \ud835\ude30\ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude33\n\ud835\ude38\ud835\ude30\ud835\ude33\ud835\ude25\ud835\ude34, \ud835\udfe3 := \ud835\udfe2++, \ud835\udfe4 := \ud835\udfe3++, \ud835\udfe5 :=\n\ud835\udfe4++, \ud835\ude26\ud835\ude35\ud835\ude24. \ud835\ude10\ud835\ude2f \ud835\ude35\ud835\ude29\ud835\ude2a\ud835\ude34 \ud835\ude35\ud835\ude26\ud835\ude39\ud835\ude35 \ud835\ude10 \ud835\ude36\ud835\ude34\ud835\ude26 \"\ud835\ude39\n:= \ud835\ude3a\" \ud835\ude35\ud835\ude30 \ud835\ude25\ud835\ude26\ud835\ude2f\ud835\ude30\ud835\ude35\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude34\ud835\ude35\ud835\ude22\ud835\ude35\ud835\ude26\ud835\ude2e\ud835\ude26\ud835\ude2f\ud835\ude35\n\ud835\ude35\ud835\ude29\ud835\ude22\ud835\ude35 \ud835\ude39 \ud835\ude2a\ud835\ude34 \ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26\ud835\ude25 \ud835\ude35\ud835\ude30 \ud835\ude26\ud835\ude32\ud835\ude36\ud835\ude22\ud835\ude2d \ud835\ude3a.)\u231d\n\ud835\uddc2\ud835\uddcc \ud835\uddc9\ud835\uddc8\ud835\uddcc\ud835\uddcd\ud835\uddce\ud835\uddc5\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbb\ud835\uddd2 \ud835\uddf1\ud835\uddf2\ud835\uddf3. (\ud835\udc37\u2081). (5 =\n(((((0)++)++)++)++)++) \ud835\uddc2\ud835\uddcc \ud835\uddba\ud835\uddc7\n\ud835\uddc2\ud835\uddc7\ud835\uddcd\ud835\uddbe\ud835\uddcb\ud835\uddc9\ud835\uddcb\ud835\uddbe\ud835\uddcd\ud835\uddba\ud835\uddcd\ud835\uddc2\ud835\uddc8\ud835\uddc7 \ud835\uddc8\ud835\uddbf \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n\ud835\uddbd\ud835\uddbe\ud835\uddbf\ud835\uddc2\ud835\uddc7\ud835\uddc2\ud835\uddcd\ud835\uddc2\ud835\uddc8\ud835\uddc7.\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc51\ud835\udc52\ud835\udc53\ud835\udc56\ud835\udc5b\ud835\udc56\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5d\ud835\udc5f\ud835\udc52\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: \ud835\udc9f \u22a2 P, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (5 =\n(((((0)++)++)++)++)++). \u220e"}, {"color": "#FFF59D", "id": "P38", "label": "\ud835\udc43\u2083\u2088 : ((((((0)++)++)\n++)++)++ = 5)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2083\u2088):\n((((((0)++)++)++)++)++ = 5).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (5 =\n(((((0)++)++)++)++)++) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083\u2087). \ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2\n\ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59\ud835\udc56\ud835\udc61\ud835\udc66-\ud835\udc50\ud835\udc5c\ud835\udc5a\ud835\udc5a\ud835\udc62\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc63\ud835\udc56\ud835\udc61\ud835\udc66\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (x = y) \u22a2 (y =\nx), \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n((((((0)++)++)++)++)++ = 5). \u220e"}, {"color": "#FFF59D", "id": "P39", "label": "\ud835\udc43\u2083\u2089 : ((4)++ = 5)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2083\u2089): ((4)++ =\n5). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3:\n((((((0)++)++)++)++)++ = 5)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083\u2088).\n(((((0)++)++)++)++ = 4) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2082\u2083).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2\n\ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, (Q = R)) \u22a2\nP\u0027, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((4)++ = 5).\n\u220e"}, {"color": "#FFF59D", "id": "P40", "label": "\ud835\udc43\u2084\u2080 : (5 = (4)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2084\u2080): (5 =\n(4)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((4)++ = 5)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083\u2089).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59\ud835\udc56\ud835\udc61\ud835\udc66-\n\ud835\udc50\ud835\udc5c\ud835\udc5a\ud835\udc5a\ud835\udc62\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc63\ud835\udc56\ud835\udc61\ud835\udc66 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (x\n= y) \u22a2 (y = x), \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n(5 = (4)++). \u220e"}, {"color": "#FFF59D", "id": "P41", "label": "\ud835\udc43\u2084\u2081 : ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1))\n\u27f9 ((5)++ \u2260 (1)++))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2084\u2081): ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1)) \u27f9\n((5)++ \u2260 (1)++)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((((\ud835\udc27\u2083\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (\ud835\udc26\u2081 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (\ud835\udc27\u2083 \u2260 \ud835\udc26\u2081)) \u27f9\n((\ud835\udc27\u2083)++ \u2260 (\ud835\udc26\u2081)++)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083\u2081). \ud835\uddab\ud835\uddbe\ud835\uddcd \ud835\udc27\u2083 = 5, \ud835\udc26\u2081 =\n1.\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc63\ud835\udc4e\ud835\udc5f\ud835\udc56\ud835\udc4e\ud835\udc4f\ud835\udc59\ud835\udc52-\n\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P,\n\ud835\udef7) \u22a2 P\u0027, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((((5\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1)) \u27f9\n((5)++ \u2260 (1)++)). \u220e"}, {"color": "#FFF59D", "id": "P42", "label": "\ud835\udc43\u2084\u2082 : ((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9\n((4)++ \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2084\u2082): ((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((4)++ \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((\ud835\udc27\u2081\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((\ud835\udc27\u2081)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2082). \ud835\uddab\ud835\uddbe\ud835\uddcd \ud835\udc27\u2081 =\n4.\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc63\ud835\udc4e\ud835\udc5f\ud835\udc56\ud835\udc4e\ud835\udc4f\ud835\udc59\ud835\udc52-\n\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P,\n\ud835\udef7) \u22a2 P\u0027, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((4\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((4)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \u220e"}, {"color": "#FFF59D", "id": "P43", "label": "\ud835\udc43\u2084\u2083 : ((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 (5\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2084\u2083): ((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 (5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((4\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 ((4)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2084\u2082). ((4)++ = 5)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2083\u2089).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\n\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe\n\ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, (Q = R)) \u22a2 P\u0027, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((4 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 (5 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \u220e"}, {"color": "#FFF59D", "id": "P44", "label": "\ud835\udc43\u2084\u2084 : (5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2084\u2084): (5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 (5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2084\u2083).(4 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2082\u2086).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc5a\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc60-\n\ud835\udc5d\ud835\udc5c\ud835\udc5b\ud835\udc52\ud835\udc5b\ud835\udc60 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: ((P \u27f9 Q),\nP) \u22a2 Q, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \u220e"}, {"color": "#FFF59D", "id": "P45", "label": "\ud835\udc43\u2084\u2085 : ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1))\n\u27f9 ((5)++ \u2260 (1)++))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2084\u2085): ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1)) \u27f9\n((5)++ \u2260 (1)++)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((((\ud835\udc27\u2083\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (\ud835\udc26\u2081 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (\ud835\udc27\u2083 \u2260 \ud835\udc26\u2081)) \u27f9\n((\ud835\udc27\u2083)++ \u2260 (\ud835\udc26\u2081)++)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083\u2081). \ud835\uddab\ud835\uddbe\ud835\uddcd \ud835\udc27\u2083 = 5, \ud835\udc26\u2081 =\n1.\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc63\ud835\udc4e\ud835\udc5f\ud835\udc56\ud835\udc4e\ud835\udc4f\ud835\udc59\ud835\udc52-\n\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P,\n\ud835\udef7) \u22a2 P\u0027, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((((5\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1)) \u27f9\n((5)++ \u2260 (1)++)). \u220e"}, {"color": "#FFF59D", "id": "P46", "label": "\ud835\udc43\u2084\u2086 : ((4)++ \u2260\n(0)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2084\u2086): ((4)++ \u2260\n(0)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (4 \u2260 0)) \u27f9\n((4)++ \u2260 (0)++)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083\u2083).(((4 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (4 \u2260 0)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083\u2085).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc5a\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc60-\ud835\udc5d\ud835\udc5c\ud835\udc5b\ud835\udc52\ud835\udc5b\ud835\udc60 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: ((P\n\u27f9 Q), P) \u22a2 Q, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n((4)++ \u2260 (0)++). \u220e"}, {"color": "#FFF59D", "id": "P47", "label": "\ud835\udc43\u2084\u2087 : (5 \u2260 (0)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2084\u2087): (5 \u2260\n(0)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((4)++ \u2260 (0)++)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2084\u2086). ((4)++\n= 5) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2083\u2089).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\n\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe\n\ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, (Q = R)) \u22a2 P\u0027, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (5 \u2260 (0)++). \u220e"}, {"color": "#FFF59D", "id": "P48", "label": "\ud835\udc43\u2084\u2088 : (6 = ((((((0)+\n+)++)++)++)++)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2084\u2088): (6 =\n((((((0)++)++)++)++)++)++).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: \u231c\ud835\ude1e\ud835\ude26 \ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26 \ud835\udfe3 \ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26\n\ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 \ud835\udfe2++, \ud835\udfe4 \ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33\n(\ud835\udfe2++)++, \ud835\udfe5 \ud835\ude35\ud835\ude30 \ud835\ude23\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33\n((\ud835\udfe2++)++)++,\ud835\ude26\ud835\ude35\ud835\ude24. (\ud835\ude10\ud835\ude2f \ud835\ude30\ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude33\n\ud835\ude38\ud835\ude30\ud835\ude33\ud835\ude25\ud835\ude34, \ud835\udfe3 := \ud835\udfe2++, \ud835\udfe4 := \ud835\udfe3++, \ud835\udfe5 :=\n\ud835\udfe4++, \ud835\ude26\ud835\ude35\ud835\ude24. \ud835\ude10\ud835\ude2f \ud835\ude35\ud835\ude29\ud835\ude2a\ud835\ude34 \ud835\ude35\ud835\ude26\ud835\ude39\ud835\ude35 \ud835\ude10 \ud835\ude36\ud835\ude34\ud835\ude26 \"\ud835\ude39\n:= \ud835\ude3a\" \ud835\ude35\ud835\ude30 \ud835\ude25\ud835\ude26\ud835\ude2f\ud835\ude30\ud835\ude35\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude26 \ud835\ude34\ud835\ude35\ud835\ude22\ud835\ude35\ud835\ude26\ud835\ude2e\ud835\ude26\ud835\ude2f\ud835\ude35\n\ud835\ude35\ud835\ude29\ud835\ude22\ud835\ude35 \ud835\ude39 \ud835\ude2a\ud835\ude34 \ud835\ude25\ud835\ude26\ud835\ude27\ud835\ude2a\ud835\ude2f\ud835\ude26\ud835\ude25 \ud835\ude35\ud835\ude30 \ud835\ude26\ud835\ude32\ud835\ude36\ud835\ude22\ud835\ude2d \ud835\ude3a.)\u231d\n\ud835\uddc2\ud835\uddcc \ud835\uddc9\ud835\uddc8\ud835\uddcc\ud835\uddcd\ud835\uddce\ud835\uddc5\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbb\ud835\uddd2 \ud835\uddf1\ud835\uddf2\ud835\uddf3. (\ud835\udc37\u2081). (6 =\n((((((0)++)++)++)++)++)++) \ud835\uddc2\ud835\uddcc \ud835\uddba\ud835\uddc7\n\ud835\uddc2\ud835\uddc7\ud835\uddcd\ud835\uddbe\ud835\uddcb\ud835\uddc9\ud835\uddcb\ud835\uddbe\ud835\uddcd\ud835\uddba\ud835\uddcd\ud835\uddc2\ud835\uddc8\ud835\uddc7 \ud835\uddc8\ud835\uddbf \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n\ud835\uddbd\ud835\uddbe\ud835\uddbf\ud835\uddc2\ud835\uddc7\ud835\uddc2\ud835\uddcd\ud835\uddc2\ud835\uddc8\ud835\uddc7.\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc51\ud835\udc52\ud835\udc53\ud835\udc56\ud835\udc5b\ud835\udc56\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5d\ud835\udc5f\ud835\udc52\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: \ud835\udc9f \u22a2 P, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (6 =\n((((((0)++)++)++)++)++)++). \u220e"}, {"color": "#FFF59D", "id": "P49", "label": "\ud835\udc43\u2084\u2089 : (((((((0)++)++\n)++)++)++)++ = 6)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2084\u2089):\n(((((((0)++)++)++)++)++)++ = 6).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (6 =\n((((((0)++)++)++)++)++)++)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2084\u2088).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59\ud835\udc56\ud835\udc61\ud835\udc66-\n\ud835\udc50\ud835\udc5c\ud835\udc5a\ud835\udc5a\ud835\udc62\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc63\ud835\udc56\ud835\udc61\ud835\udc66 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (x\n= y) \u22a2 (y = x), \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n(((((((0)++)++)++)++)++)++ = 6).\n\u220e"}, {"color": "#FFF59D", "id": "P50", "label": "\ud835\udc43\u2085\u2080 : (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2085\u2080): (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((0)++\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. \ud835\udfee.\ud835\udfee.\ud835\udfef (\ud835\udc43\u2084). ((0)++ =\n1) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2081\u2085).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\n\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe\n\ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, (Q = R)) \u22a2 P\u0027, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (1 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f). \u220e"}, {"color": "#FFF59D", "id": "P51", "label": "\ud835\udc43\u2085\u2081 : ((5)++ = 6)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2085\u2081): ((5)++ =\n6). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3:\n(((((((0)++)++)++)++)++)++ = 6)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2084\u2089).\n((((((0)++)++)++)++)++ = 5)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2083\u2088).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\n\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe\n\ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, (Q = R)) \u22a2 P\u0027, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((5)++ = 6). \u220e"}, {"color": "#FFF59D", "id": "P52", "label": "\ud835\udc43\u2085\u2082 : (6 = (5)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2085\u2082): (6 =\n(5)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((5)++ = 6)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2085\u2081).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59\ud835\udc56\ud835\udc61\ud835\udc66-\n\ud835\udc50\ud835\udc5c\ud835\udc5a\ud835\udc5a\ud835\udc62\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc63\ud835\udc56\ud835\udc61\ud835\udc66 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (x\n= y) \u22a2 (y = x), \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n(6 = (5)++). \u220e"}, {"color": "#FFF59D", "id": "P66", "label": "\ud835\udc43\u2086\u2086 : \ud835\udc3c\ud835\udc5b\ud835\udc50(\u210b\u2081)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2086\u2086): \ud835\udc3c\ud835\udc5b\ud835\udc50(\u210b\u2081).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (4 = 0) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2086\u2085). (4 \u2260 0) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. \ud835\udfee.\ud835\udfed.\ud835\udff2 (\ud835\udc43\u2083\u2080).\n\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc56\ud835\udc5b\ud835\udc50\ud835\udc5c\ud835\udc5b\ud835\udc60\ud835\udc56\ud835\udc60\ud835\udc61\ud835\udc52\ud835\udc5b\ud835\udc50\ud835\udc66-\n\ud835\udc4f\ud835\udc66-\ud835\udc56\ud835\udc5b\ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59\ud835\udc56\ud835\udc61\ud835\udc66-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc5f\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: ((P = Q), (P \u2260\nQ)) \u22a2 Inc(\ud835\udcaf), \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n\ud835\udc3c\ud835\udc5b\ud835\udc50(\u210b\u2081). \u220e"}, {"color": "#FFF59D", "id": "P65", "label": "\ud835\udc43\u2086\u2085 : (4 = 0)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\u210b\u2081.\ud835\udc43\u2086\u2085): (4 = 0).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((((4 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((4)++ = (0)++)) \u27f9 (4\n= 0)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2086\u2084).(((4 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)\n\u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227\n((4)++ = (0)++)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2086\u2083).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc5a\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc60-\ud835\udc5d\ud835\udc5c\ud835\udc5b\ud835\udc52\ud835\udc5b\ud835\udc60 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: ((P\n\u27f9 Q), P) \u22a2 Q, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (4\n= 0). \u220e"}, {"color": "#FFF59D", "id": "P64", "label": "\ud835\udc43\u2086\u2084 : ((((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((4)++ =\n(0)++)) \u27f9 (4 = 0))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\u210b\u2081.\ud835\udc43\u2086\u2084): ((((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((4)++ =\n(0)++)) \u27f9 (4 = 0)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3:\n((((\ud835\udc27\u2084 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227\n(\ud835\udc26\u2082 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227\n((\ud835\udc27\u2084)++ = (\ud835\udc26\u2082)++)) \u27f9 (\ud835\udc27\u2084 = \ud835\udc26\u2082))\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083\u2082). \ud835\uddab\ud835\uddbe\ud835\uddcd \ud835\udc27\u2084\n= 4, \ud835\udc26\u2082 = 0.\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc63\ud835\udc4e\ud835\udc5f\ud835\udc56\ud835\udc4e\ud835\udc4f\ud835\udc59\ud835\udc52-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe\n\ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, \ud835\udef7) \u22a2 P\u0027, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((((4 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)\n\u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227\n((4)++ = (0)++)) \u27f9 (4 = 0)). \u220e"}, {"color": "#FFF59D", "id": "P63", "label": "\ud835\udc43\u2086\u2083 : (((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((4)++ =\n(0)++))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\u210b\u2081.\ud835\udc43\u2086\u2083): (((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((4)++ =\n(0)++)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) (\ud835\udc43) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2086\u2082). ((4)++ =\n(0)++) (\ud835\udc44) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2086\u2081).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc50\ud835\udc5c\ud835\udc5b\ud835\udc57\ud835\udc62\ud835\udc5b\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc5f\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, Q) \u22a2 (P \u22c0\nQ), \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((4)++ =\n(0)++)). \u220e"}, {"color": "#FFF59D", "id": "P62", "label": "\ud835\udc43\u2086\u2082 : ((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\u210b\u2081.\ud835\udc43\u2086\u2082): ((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) (\ud835\udc43) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2082\u2086). (0 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) (\ud835\udc44) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2081).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc50\ud835\udc5c\ud835\udc5b\ud835\udc57\ud835\udc62\ud835\udc5b\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc5f\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, Q) \u22a2 (P \u22c0\nQ), \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((4 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (0 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \u220e"}, {"color": "#FFF59D", "id": "P61", "label": "\ud835\udc43\u2086\u2081 : ((4)++ =\n(0)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\u210b\u2081.\ud835\udc43\u2086\u2081): ((4)++ =\n(0)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((4)++ = 1)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2086\u2080). (1 =\n(0)++) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2085).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\n\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe\n\ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, (Q = R)) \u22a2 P\u0027, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((4)++ = (0)++). \u220e"}, {"color": "#FFF59D", "id": "P60", "label": "\ud835\udc43\u2086\u2080 : ((4)++ = 1)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\u210b\u2081.\ud835\udc43\u2086\u2080): ((4)++ =\n1). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (5 = 1) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2085\u2089). (5 = (4)++) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2084\u2080).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2\n\ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, (Q = R)) \u22a2\nP\u0027, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((4)++ = 1).\n\u220e"}, {"color": "#FFF59D", "id": "P59", "label": "\ud835\udc43\u2085\u2089 : (5 = 1)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\u210b\u2081.\ud835\udc43\u2085\u2089): (5 = 1).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((5)++ = (1)++)) \u27f9 (5\n= 1)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2085\u2088).(((5 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)\n\u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227\n((5)++ = (1)++)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2085\u2087).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc5a\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc60-\ud835\udc5d\ud835\udc5c\ud835\udc5b\ud835\udc52\ud835\udc5b\ud835\udc60 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: ((P\n\u27f9 Q), P) \u22a2 Q, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (5\n= 1). \u220e"}, {"color": "#FFF59D", "id": "P58", "label": "\ud835\udc43\u2085\u2088 : ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((5)++ =\n(1)++)) \u27f9 (5 = 1))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\u210b\u2081.\ud835\udc43\u2085\u2088): ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((5)++ =\n(1)++)) \u27f9 (5 = 1)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3:\n((((\ud835\udc27\u2084 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227\n(\ud835\udc26\u2082 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227\n((\ud835\udc27\u2084)++ = (\ud835\udc26\u2082)++)) \u27f9 (\ud835\udc27\u2084 = \ud835\udc26\u2082))\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2083\u2082). \ud835\uddab\ud835\uddbe\ud835\uddcd \ud835\udc27\u2084\n= 5, \ud835\udc26\u2082 = 1.\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc63\ud835\udc4e\ud835\udc5f\ud835\udc56\ud835\udc4e\ud835\udc4f\ud835\udc59\ud835\udc52-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe\n\ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, \ud835\udef7) \u22a2 P\u0027, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)\n\u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227\n((5)++ = (1)++)) \u27f9 (5 = 1)). \u220e"}, {"color": "#FFF59D", "id": "P57", "label": "\ud835\udc43\u2085\u2087 : (((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((5)++ =\n(1)++))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\u210b\u2081.\ud835\udc43\u2085\u2087): (((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((5)++ =\n(1)++)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) (\ud835\udc43) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2085\u2086). ((5)++ =\n(1)++) (\ud835\udc44) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2085\u2085).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc50\ud835\udc5c\ud835\udc5b\ud835\udc57\ud835\udc62\ud835\udc5b\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc5f\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, Q) \u22a2 (P \u22c0\nQ), \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 ((5)++ =\n(1)++)). \u220e"}, {"color": "#FFF59D", "id": "P56", "label": "\ud835\udc43\u2085\u2086 : ((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\u210b\u2081.\ud835\udc43\u2085\u2086): ((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) (\ud835\udc43) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2084\u2084). (1 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) (\ud835\udc44) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2085\u2080).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc50\ud835\udc5c\ud835\udc5b\ud835\udc57\ud835\udc62\ud835\udc5b\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc5f\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, Q) \u22a2 (P \u22c0\nQ), \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \u220e"}, {"color": "#FFF59D", "id": "P55", "label": "\ud835\udc43\u2085\u2085 : ((5)++ =\n(1)++)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\u210b\u2081.\ud835\udc43\u2085\u2085): ((5)++ =\n(1)++). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((5)++ = 2)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2085\u2084). (2 =\n(1)++) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2081\u2086).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\n\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe\n\ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, (Q = R)) \u22a2 P\u0027, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((5)++ = (1)++). \u220e"}, {"color": "#FFF59D", "id": "P54", "label": "\ud835\udc43\u2085\u2084 : ((5)++ = 2)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\u210b\u2081.\ud835\udc43\u2085\u2084): ((5)++ =\n2). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (6 = 2) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2085\u2083). (6 = (5)++) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2085\u2082).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2\n\ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, (Q = R)) \u22a2\nP\u0027, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((5)++ = 2).\n\u220e"}, {"color": "#FFF59D", "id": "P53", "label": "\ud835\udc43\u2085\u2083 : (6 = 2)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\u210b\u2081.\ud835\udc43\u2085\u2083): (6 = 2).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: \u231c\ud835\ude09\ud835\ude3a \ud835\ude29\ud835\ude3a\ud835\ude31\ud835\ude30\ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude34\ud835\ude2a\ud835\ude34, \ud835\ude22\ud835\ude34\ud835\ude34\ud835\ude36\ud835\ude2e\ud835\ude26 (\ud835\udfe8\n= \ud835\udfe4) \ud835\ude2a\ud835\ude34 \ud835\ude35\ud835\ude33\ud835\ude36\ud835\ude26.\u231d \ud835\uddc2\ud835\uddcc \ud835\uddc9\ud835\uddc8\ud835\uddcc\ud835\uddcd\ud835\uddce\ud835\uddc5\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbb\ud835\uddd2\n\ud835\uddee\ud835\ude05\ud835\uddf6\ud835\uddfc\ud835\uddfa (\ud835\udc34\u2085). (6 = 2) \ud835\uddc2\ud835\uddcc \ud835\uddba \ud835\uddcf\ud835\uddba\ud835\uddc5\ud835\uddc2\ud835\uddbd\n\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddc6\ud835\uddce\ud835\uddc5\ud835\uddba \ud835\uddcc\ud835\uddcd\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddc6\ud835\uddbe\ud835\uddc7\ud835\uddcd \ud835\uddc2\ud835\uddc7\ud835\uddcd\ud835\uddbe\ud835\uddcb\ud835\uddc9\ud835\uddcb\ud835\uddbe\ud835\uddcd\ud835\uddbe\ud835\uddbd\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd \ud835\uddba\ud835\uddd1\ud835\uddc2\ud835\uddc8\ud835\uddc6.\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2\n\ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc4e\ud835\udc65\ud835\udc56\ud835\udc5c\ud835\udc5a-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5d\ud835\udc5f\ud835\udc52\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: \ud835\udc9c \u22a2 P, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (6 = 2). \u220e"}, {"color": "#81C784", "id": "A5", "label": "\ud835\udc34\u2085 : \u231c\ud835\ude09\ud835\ude3a \ud835\ude29\ud835\ude3a\ud835\ude31\ud835\ude30\ud835\ude35\ud835\ude29\ud835\ude26\ud835\ude34\ud835\ude2a\ud835\ude34,\n\ud835\ude22\ud835\ude34\ud835\ude34\ud835\ude36\ud835\ude2e\ud835\ude26 (\ud835\udfe8 = \ud835\udfe4) \ud835\ude2a\ud835\ude34\n\ud835\ude35\ud835\ude33\ud835\ude36\ud835\ude26.\u231d", "shape": "box"}, {"color": "#FFF59D", "id": "P67", "label": "\ud835\udc43\u2086\u2087 (\ud835\udfee.\ud835\udfed.\ud835\udff4) : (6 \u2260\n2)", "labelHighlightBold": true, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb \ud835\udfee.\ud835\udfed.\ud835\udff4 (\ud835\udcaf\u2081.\ud835\udc43\u2086\u2087): (6 \u2260\n2). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: \ud835\uddab\ud835\uddbe\ud835\uddcd \ud835\uddf5\ud835\ude06\ud835\uddfd. (\ud835\udc3b\u2081) \ud835\uddbb\ud835\uddbe \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\uddc1\ud835\uddd2\ud835\uddc9\ud835\uddc8\ud835\uddcd\ud835\uddc1\ud835\uddbe\ud835\uddcc\ud835\uddc2\ud835\uddcc (6 = 2). \ud835\udc3c\ud835\udc5b\ud835\udc50(\u210b\u2081)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2086\u2086).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc5d\ud835\udc5f\ud835\udc5c\ud835\udc5c\ud835\udc53-\n\ud835\udc4f\ud835\udc66-\ud835\udc5f\ud835\udc52\ud835\udc53\ud835\udc62\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\ud835\udc5c\ud835\udc53-\ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59\ud835\udc56\ud835\udc61\ud835\udc66\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (\u210b(P=Q), Inc(\u210b))\n\u22a2 (P \u2260 Q), \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (6 \u2260\n2). \u220e"}, {"color": "#FFF59D", "id": "P68", "label": "\ud835\udc43\u2086\u2088 : ((1)++ = 2)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2086\u2088): ((1)++ =\n2). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (((0)++)++ = 2)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2081\u2087). ((0)++\n= 1) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2081\u2085).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\n\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe\n\ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, (Q = R)) \u22a2 P\u0027, \ud835\uddc2\ud835\uddcd\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((1)++ = 2). \u220e"}, {"color": "#FFF59D", "id": "P69", "label": "\ud835\udc43\u2086\u2089 : (5 \u2260 1)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2086\u2089): (5 \u2260 1).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (5 \u2260 (0)++) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2084\u2087). ((0)++ = 1) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2081\u2085).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2\n\ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, (Q = R)) \u22a2\nP\u0027, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (5 \u2260 1). \u220e"}, {"color": "#FFF59D", "id": "P70", "label": "\ud835\udc43\u2087\u2080 : ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1))\n\u27f9 (6 \u2260 (1)++))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2087\u2080): ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1)) \u27f9 (6\n\u2260 (1)++)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1)) \u27f9\n((5)++ \u2260 (1)++)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2084\u2085). ((5)++ = 6) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc\n\ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2085\u2081).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2\n\ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, (Q = R)) \u22a2\nP\u0027, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1)) \u27f9 (6\n\u2260 (1)++)). \u220e"}, {"color": "#FFF59D", "id": "P71", "label": "\ud835\udc43\u2087\u2081 : ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1))\n\u27f9 (6 \u2260 2))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2087\u2081): ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1)) \u27f9 (6\n\u2260 2)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1)) \u27f9 (6\n\u2260 (1)++)) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2087\u2080). ((1)++ = 2) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2086\u2088).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc52\ud835\udc5e\ud835\udc62\ud835\udc4e\ud835\udc59-\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5a\ud835\udc60-\ud835\udc60\ud835\udc62\ud835\udc4f\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc61\ud835\udc62\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, (Q = R)) \u22a2\nP\u0027, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1)) \u27f9 (6\n\u2260 2)). \u220e"}, {"color": "#FFF59D", "id": "P72", "label": "\ud835\udc43\u2087\u2082 : ((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2087\u2082): ((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: (5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) (\ud835\udc43) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6\n\ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2084\u2084). (1 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) (\ud835\udc44) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2085\u2080).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc50\ud835\udc5c\ud835\udc5b\ud835\udc57\ud835\udc62\ud835\udc5b\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc5f\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, Q) \u22a2 (P \u22c0\nQ), \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd ((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)). \u220e"}, {"color": "#FFF59D", "id": "P73", "label": "\ud835\udc43\u2087\u2083 : (((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2087\u2083): (((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1)).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((5 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)\n\u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) (\ud835\udc43)\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2087\u2082). (5 \u2260\n1) (\ud835\udc44) \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2086\u2089).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe\n\ud835\udc50\ud835\udc5c\ud835\udc5b\ud835\udc57\ud835\udc62\ud835\udc5b\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b-\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc5f\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc50\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\n\ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: (P, Q) \u22a2 (P \u22c0\nQ), \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (((5 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1)). \u220e"}, {"color": "#FFF59D", "id": "P74", "label": "\ud835\udc43\u2087\u2084 : (6 \u2260 2)", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2087\u2084): (6 \u2260 2).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: ((((5 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\n\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1)) \u27f9 (6 \u2260 2))\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd. (\ud835\udc43\u2087\u2081).(((5\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (1 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)) \u2227 (5 \u2260 1))\n\ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddfd\ud835\uddff\ud835\uddfc\ud835\uddfd.\n(\ud835\udc43\u2087\u2083).\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc5a\ud835\udc5c\ud835\udc51\ud835\udc62\ud835\udc60-\n\ud835\udc5d\ud835\udc5c\ud835\udc5b\ud835\udc52\ud835\udc5b\ud835\udc60 \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: ((P \u27f9 Q),\nP) \u22a2 Q, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (6 \u2260 2).\n\u220e"}, {"color": "#81C784", "id": "A6", "label": "\ud835\udc34\u2086 : \u231c\ud835\ude13\ud835\ude26\ud835\ude35 \ud835\ude17(\ud835\ude2f) \ud835\ude23\ud835\ude26\n\ud835\ude22\ud835\ude2f\ud835\ude3a \ud835\ude31\ud835\ude33\ud835\ude30\ud835\ude31\ud835\ude26\ud835\ude33\ud835\ude35\ud835\ude3a\n\ud835\ude31\ud835\ude26\ud835\ude33\ud835\ude35\ud835\ude22\ud835\ude2a\ud835\ude2f\ud835\ude2a\ud835\ude2f\ud835\ude28 \ud835\ude35\ud835\ude30 \ud835\ude22\n\ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 \ud835\ude2f.\n\ud835\ude1a\ud835\ude36\ud835\ude31\ud835\ude31\ud835\ude30\ud835\ude34\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude22\ud835\ude35 \ud835\ude17(\ud835\ude16) \ud835\ude2a\ud835\ude34\n\ud835\ude35\ud835\ude33\ud835\ude36\ud835\ude26, \ud835\ude22\ud835\ude2f\ud835\ude25 \ud835\ude34\ud835\ude36\ud835\ude31\ud835\ude31\ud835\ude30\ud835\ude34\ud835\ude26\n\ud835\ude35\ud835\ude29\ud835\ude22\ud835\ude35 \ud835\ude38\ud835\ude29\ud835\ude26\ud835\ude2f\ud835\ude26\ud835\ude37\ud835\ude26\ud835\ude33 \ud835\ude17(\ud835\ude2f)\n\ud835\ude2a\ud835\ude34 \ud835\ude35\ud835\ude33\ud835\ude36\ud835\ude26, \ud835\ude17(\ud835\ude2f++) \ud835\ude2a\ud835\ude34\n\ud835\ude22\ud835\ude2d\ud835\ude34\ud835\ude30 \ud835\ude35\ud835\ude33\ud835\ude36\ud835\ude26. \ud835\ude1b\ud835\ude29\ud835\ude26\ud835\ude2f \ud835\ude17(\ud835\ude2f)\n\ud835\ude2a\ud835\ude34 \ud835\ude35\ud835\ude33\ud835\ude36\ud835\ude26 \ud835\ude27\ud835\ude30\ud835\ude33 \ud835\ude26\ud835\ude37\ud835\ude26\ud835\ude33\ud835\ude3a\n\ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 \ud835\ude2f.\u231d", "shape": "box"}, {"color": "#FFF59D", "id": "P75", "label": "\ud835\udc43\u2087\u2085 : (((\ud835\udc27\u2085 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227\n(\ud835\udc0f\u2081(0) \u2227 (\ud835\udc0f\u2081(\ud835\udc27\u2085) \u27f9\n\ud835\udc0f\u2081((\ud835\udc27\u2085)++)))) \u27f9 ((\ud835\udc26\u2083\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f)\n\u27f9 \ud835\udc0f\u2081(\ud835\udc26\u2083)))", "labelHighlightBold": false, "shape": "box", "title": "\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfd\ud835\uddfc\ud835\ude00\ud835\uddf6\ud835\ude01\ud835\uddf6\ud835\uddfc\ud835\uddfb (\ud835\udcaf\u2081.\ud835\udc43\u2087\u2085): (((\ud835\udc27\u2085 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (\ud835\udc0f\u2081(0) \u2227\n(\ud835\udc0f\u2081(\ud835\udc27\u2085) \u27f9 \ud835\udc0f\u2081((\ud835\udc27\u2085)++)))) \u27f9 ((\ud835\udc26\u2083\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 \ud835\udc0f\u2081(\ud835\udc26\u2083))).\n\ud835\udde3\ud835\uddff\ud835\uddfc\ud835\uddfc\ud835\uddf3: \u231c\ud835\ude13\ud835\ude26\ud835\ude35 \ud835\ude17(\ud835\ude2f) \ud835\ude23\ud835\ude26 \ud835\ude22\ud835\ude2f\ud835\ude3a \ud835\ude31\ud835\ude33\ud835\ude30\ud835\ude31\ud835\ude26\ud835\ude33\ud835\ude35\ud835\ude3a\n\ud835\ude31\ud835\ude26\ud835\ude33\ud835\ude35\ud835\ude22\ud835\ude2a\ud835\ude2f\ud835\ude2a\ud835\ude2f\ud835\ude28 \ud835\ude35\ud835\ude30 \ud835\ude22 \ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33\n\ud835\ude2f. \ud835\ude1a\ud835\ude36\ud835\ude31\ud835\ude31\ud835\ude30\ud835\ude34\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude22\ud835\ude35 \ud835\ude17(\ud835\ude16) \ud835\ude2a\ud835\ude34 \ud835\ude35\ud835\ude33\ud835\ude36\ud835\ude26,\n\ud835\ude22\ud835\ude2f\ud835\ude25 \ud835\ude34\ud835\ude36\ud835\ude31\ud835\ude31\ud835\ude30\ud835\ude34\ud835\ude26 \ud835\ude35\ud835\ude29\ud835\ude22\ud835\ude35 \ud835\ude38\ud835\ude29\ud835\ude26\ud835\ude2f\ud835\ude26\ud835\ude37\ud835\ude26\ud835\ude33 \ud835\ude17(\ud835\ude2f)\n\ud835\ude2a\ud835\ude34 \ud835\ude35\ud835\ude33\ud835\ude36\ud835\ude26, \ud835\ude17(\ud835\ude2f++) \ud835\ude2a\ud835\ude34 \ud835\ude22\ud835\ude2d\ud835\ude34\ud835\ude30 \ud835\ude35\ud835\ude33\ud835\ude36\ud835\ude26.\n\ud835\ude1b\ud835\ude29\ud835\ude26\ud835\ude2f \ud835\ude17(\ud835\ude2f) \ud835\ude2a\ud835\ude34 \ud835\ude35\ud835\ude33\ud835\ude36\ud835\ude26 \ud835\ude27\ud835\ude30\ud835\ude33 \ud835\ude26\ud835\ude37\ud835\ude26\ud835\ude33\ud835\ude3a\n\ud835\ude2f\ud835\ude22\ud835\ude35\ud835\ude36\ud835\ude33\ud835\ude22\ud835\ude2d \ud835\ude2f\ud835\ude36\ud835\ude2e\ud835\ude23\ud835\ude26\ud835\ude33 \ud835\ude2f.\u231d \ud835\uddc2\ud835\uddcc \ud835\uddc9\ud835\uddc8\ud835\uddcc\ud835\uddcd\ud835\uddce\ud835\uddc5\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddbd\n\ud835\uddbb\ud835\uddd2 \ud835\uddee\ud835\ude05\ud835\uddf6\ud835\uddfc\ud835\uddfa \ud835\ude00\ud835\uddf0\ud835\uddf5\ud835\uddf2\ud835\uddfa\ud835\uddee (\ud835\udc34\u2086). (((\ud835\udc27\u2085 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (\ud835\udc0f\u2081(0) \u2227\n(\ud835\udc0f\u2081(\ud835\udc27\u2085) \u27f9 \ud835\udc0f\u2081((\ud835\udc27\u2085)++)))) \u27f9 ((\ud835\udc26\u2083\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 \ud835\udc0f\u2081(\ud835\udc26\u2083)))\n\ud835\uddc2\ud835\uddcc \ud835\uddba \ud835\uddcf\ud835\uddba\ud835\uddc5\ud835\uddc2\ud835\uddbd \ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddc6\ud835\uddce\ud835\uddc5\ud835\uddba \ud835\uddcc\ud835\uddcd\ud835\uddba\ud835\uddcd\ud835\uddbe\ud835\uddc6\ud835\uddbe\ud835\uddc7\ud835\uddcd\n\ud835\uddc2\ud835\uddc7\ud835\uddcd\ud835\uddbe\ud835\uddcb\ud835\uddc9\ud835\uddcb\ud835\uddbe\ud835\uddcd\ud835\uddbe\ud835\uddbd \ud835\uddbf\ud835\uddcb\ud835\uddc8\ud835\uddc6 \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd\n\ud835\uddba\ud835\uddd1\ud835\uddc2\ud835\uddc8\ud835\uddc6.\ud835\uddb3\ud835\uddc1\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddbf\ud835\uddc8\ud835\uddcb\ud835\uddbe, \ud835\uddbb\ud835\uddd2 \ud835\uddcd\ud835\uddc1\ud835\uddbe \ud835\udc4e\ud835\udc65\ud835\udc56\ud835\udc5c\ud835\udc5a-\n\ud835\udc56\ud835\udc5b\ud835\udc61\ud835\udc52\ud835\udc5f\ud835\udc5d\ud835\udc5f\ud835\udc52\ud835\udc61\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\uddc2\ud835\uddc7\ud835\uddbf\ud835\uddbe\ud835\uddcb\ud835\uddbe\ud835\uddc7\ud835\uddbc\ud835\uddbe \ud835\uddcb\ud835\uddce\ud835\uddc5\ud835\uddbe: \ud835\udc9c\n\u22a2 P, \ud835\uddc2\ud835\uddcd \ud835\uddbf\ud835\uddc8\ud835\uddc5\ud835\uddc5\ud835\uddc8\ud835\uddd0\ud835\uddcc \ud835\uddcd\ud835\uddc1\ud835\uddba\ud835\uddcd (((\ud835\udc27\u2085 \ud835\udc56\ud835\udc60-\ud835\udc4e\n\ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u2227 (\ud835\udc0f\u2081(0) \u2227\n(\ud835\udc0f\u2081(\ud835\udc27\u2085) \u27f9 \ud835\udc0f\u2081((\ud835\udc27\u2085)++)))) \u27f9 ((\ud835\udc26\u2083\n\ud835\udc56\ud835\udc60-\ud835\udc4e \ud835\udc5b\ud835\udc4e\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc4e\ud835\udc59-\ud835\udc5b\ud835\udc62\ud835\udc5a\ud835\udc4f\ud835\udc52\ud835\udc5f) \u27f9 \ud835\udc0f\u2081(\ud835\udc26\u2083))).\n\u220e"}]); edges = new vis.DataSet([{"arrows": "to", "from": "A1", "to": "P1"}, {"arrows": "to", "from": "A2", "to": "P2"}, {"arrows": "to", "from": "P2", "to": "P3"}, {"arrows": "to", "from": "P3", "to": "P4"}, {"arrows": "to", "from": "P1", "to": "P4"}, {"arrows": "to", "from": "D1", "to": "P5"}, {"arrows": "to", "from": "D1", "to": "P6"}, {"arrows": "to", "from": "D1", "to": "P7"}, {"arrows": "to", "from": "D1", "to": "P8"}, {"arrows": "to", "from": "P2", "to": "P9"}, {"arrows": "to", "from": "P9", "to": "P10"}, {"arrows": "to", "from": "P4", "to": "P10"}, {"arrows": "to", "from": "P2", "to": "P11"}, {"arrows": "to", "from": "P11", "to": "P12"}, {"arrows": "to", "from": "P10", "to": "P12"}, {"arrows": "to", "from": "P2", "to": "P13"}, {"arrows": "to", "from": "P13", "to": "P14"}, {"arrows": "to", "from": "P12", "to": "P14"}, {"arrows": "to", "from": "P5", "to": "P15"}, {"arrows": "to", "from": "P6", "to": "P16"}, {"arrows": "to", "from": "P15", "to": "P16"}, {"arrows": "to", "from": "P6", "to": "P17"}, {"arrows": "to", "from": "P7", "to": "P18"}, {"arrows": "to", "from": "P17", "to": "P18"}, {"arrows": "to", "from": "P7", "to": "P19"}, {"arrows": "to", "from": "P19", "to": "P20"}, {"arrows": "to", "from": "P17", "to": "P20"}, {"arrows": "to", "from": "P12", "to": "P21"}, {"arrows": "to", "from": "P19", "to": "P21"}, {"arrows": "to", "from": "D1", "to": "P22"}, {"arrows": "to", "from": "P8", "to": "P23"}, {"arrows": "to", "from": "P23", "to": "P24"}, {"arrows": "to", "from": "P19", "to": "P24"}, {"arrows": "to", "from": "P13", "to": "P25"}, {"arrows": "to", "from": "P24", "to": "P25"}, {"arrows": "to", "from": "P14", "to": "P26"}, {"arrows": "to", "from": "P23", "to": "P26"}, {"arrows": "to", "from": "A3", "to": "P27"}, {"arrows": "to", "from": "P27", "to": "P28"}, {"arrows": "to", "from": "P28", "to": "P29"}, {"arrows": "to", "from": "P21", "to": "P29"}, {"arrows": "to", "from": "P29", "to": "P30"}, {"arrows": "to", "from": "P24", "to": "P30"}, {"arrows": "to", "from": "A4", "to": "P31"}, {"arrows": "to", "from": "A4", "to": "P32"}, {"arrows": "to", "from": "P31", "to": "P33"}, {"arrows": "to", "from": "P26", "to": "P34"}, {"arrows": "to", "from": "P1", "to": "P34"}, {"arrows": "to", "from": "P34", "to": "P35"}, {"arrows": "to", "from": "P30", "to": "P35"}, {"arrows": "to", "from": "P33", "to": "P36"}, {"arrows": "to", "from": "P35", "to": "P36"}, {"arrows": "to", "from": "D1", "to": "P37"}, {"arrows": "to", "from": "P37", "to": "P38"}, {"arrows": "to", "from": "P38", "to": "P39"}, {"arrows": "to", "from": "P23", "to": "P39"}, {"arrows": "to", "from": "P39", "to": "P40"}, {"arrows": "to", "from": "P31", "to": "P41"}, {"arrows": "to", "from": "P2", "to": "P42"}, {"arrows": "to", "from": "P42", "to": "P43"}, {"arrows": "to", "from": "P39", "to": "P43"}, {"arrows": "to", "from": "P43", "to": "P44"}, {"arrows": "to", "from": "P26", "to": "P44"}, {"arrows": "to", "from": "P31", "to": "P45"}, {"arrows": "to", "from": "P33", "to": "P46"}, {"arrows": "to", "from": "P35", "to": "P46"}, {"arrows": "to", "from": "P46", "to": "P47"}, {"arrows": "to", "from": "P39", "to": "P47"}, {"arrows": "to", "from": "D1", "to": "P48"}, {"arrows": "to", "from": "P48", "to": "P49"}, {"arrows": "to", "from": "P4", "to": "P50"}, {"arrows": "to", "from": "P15", "to": "P50"}, {"arrows": "to", "from": "P49", "to": "P51"}, {"arrows": "to", "from": "P38", "to": "P51"}, {"arrows": "to", "from": "P51", "to": "P52"}, {"arrows": "to", "from": "P32", "to": "P64"}, {"arrows": "to", "from": "P64", "to": "P65"}, {"arrows": "to", "from": "P26", "to": "P62"}, {"arrows": "to", "from": "P1", "to": "P62"}, {"arrows": "to", "from": "P62", "to": "P63"}, {"arrows": "to", "from": "P32", "to": "P58"}, {"arrows": "to", "from": "P58", "to": "P59"}, {"arrows": "to", "from": "P44", "to": "P56"}, {"arrows": "to", "from": "P50", "to": "P56"}, {"arrows": "to", "from": "P56", "to": "P57"}, {"arrows": "to", "from": "A5", "to": "P53"}, {"arrows": "to", "from": "P53", "to": "P54"}, {"arrows": "to", "from": "P52", "to": "P54"}, {"arrows": "to", "from": "P54", "to": "P55"}, {"arrows": "to", "from": "P16", "to": "P55"}, {"arrows": "to", "from": "P55", "to": "P57"}, {"arrows": "to", 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{"arrows": "to", "from": "P71", "to": "P74"}, {"arrows": "to", "from": "P73", "to": "P74"}, {"arrows": "to", "from": "A6", "to": "P75"}]); nodeColors = {}; diff --git a/theory/build/tao_2006_2_1_the_peano_axioms_report_noproof_enus_latex.txt b/theory/build/tao_2006_2_1_the_peano_axioms_report_noproof_enus_latex.txt index 1e2517f4..57af3e54 100644 --- a/theory/build/tao_2006_2_1_the_peano_axioms_report_noproof_enus_latex.txt +++ b/theory/build/tao_2006_2_1_the_peano_axioms_report_noproof_enus_latex.txt @@ -1 +1 @@ -\mathsf{}\mathsf{}\section{\boldsymbol\mathsf{Theory properties}}}\mathsf{}\boldsymbol\mathsf{Consistency: }}undetermined\mathsf{}\mathsf{}\boldsymbol\mathsf{Stabilized: }}False\mathsf{}\mathsf{}\boldsymbol\mathsf{Extended theory: }}N/A\mathsf{}\section{\boldsymbol\mathsf{Simple-objects declarations}}}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{0}\right\ulcorner, \left\ulcorner\mathit{natural-number}\right\ulcorner, \left\ulcorner\mathit{1}\right\ulcorner, \left\ulcorner\mathit{2}\right\ulcorner, \left\ulcorner\mathit{3}\right\ulcorner, \left\ulcorner\mathit{4}\right\ulcorner, \left\ulcorner\mathit{5}\right\ulcorner, \left\ulcorner\mathit{6}\right\ulcorner\mathsf{ be }\mathit{simple-objects}\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\section{\boldsymbol\mathsf{Relations}}}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{++}\right\ulcorner\mathsf{, }\left\ulcorner\mathit{Inc}\right\ulcorner\mathsf{ be }\mathit{unary}\mathit{-relations}\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{=}\right\ulcorner\mathsf{, }\left\ulcorner\mathit{\land}\right\ulcorner\mathsf{, }\left\ulcorner\mathit{is-a}\right\ulcorner\mathsf{, }\left\ulcorner\mathit{\neq}\right\ulcorner\mathsf{, }\left\ulcorner\mathit{\implies}\right\ulcorner\mathsf{ be }\mathit{binary}\mathit{-relations}\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\section{\boldsymbol\mathsf{Inference rules}}}\mathsf{}\mathsf{The following inference rules are considered valid under this theory:}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{axiom-interpretation}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner๐’œ โŠข P\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{conjunction-introduction}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner(P, Q) โŠข (P โ‹€ Q)\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{definition-interpretation}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner๐’Ÿ โŠข P\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{equal-terms-substitution}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner(P, (Q = R)) โŠข P'\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{equality-commutativity}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner(x = y) โŠข (y = x)\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{inconsistency-by-inequality-introduction}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner((P = Q), (P โ‰  Q)) โŠข Inc(๐’ฏ)\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{modus-ponens}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner((P โŸน Q), P) โŠข Q\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{proof-by-refutation-of-equality}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner(โ„‹(P=Q), Inc(โ„‹)) โŠข (P โ‰  Q)\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{variable-substitution}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner(P, ๐›ท) โŠข P'\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\section{\boldsymbol\mathsf{Theory elaboration sequence}}}\mathsf{}# \boldsymbol\mathsf{2}}: \boldsymbol\mathsf{The natural numbers}}\mathsf{}\mathsf{}## \boldsymbol\mathsf{2.1}}: \boldsymbol\mathsf{The peano axioms}}\mathsf{}\mathsf{}### \boldsymbol\mathsf{Informal definition of natural number}}\mathsf{}\mathsf{}### \boldsymbol\mathsf{Axiom 2.1}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Axiom}} \boldsymbol\mathsf{2.1}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{A}_{1}\mathsf{)}\mathsf{: Let }\mathit{axiom}\mathsf{ }\mathcal{A}_{1}\mathsf{ }\left\ulcorner\text{\sffamily{\itshape{0 is a natural number.}}}\right\ulcorner\mathsf{ be included (postulated) in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{axiom-interpretation}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{axiom-interpretation}\mathsf{ defined as }\left\ulcorner๐’œ โŠข P\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{1}\mathsf{)}\mathsf{: }\left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{}\mathsf{}### \boldsymbol\mathsf{Axiom 2.2}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Axiom}} \boldsymbol\mathsf{2.2}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{A}_{2}\mathsf{)}\mathsf{: Let }\mathit{axiom}\mathsf{ }\mathcal{A}_{2}\mathsf{ }\left\ulcorner\text{\sffamily{\itshape{If n is a natural number, then n++ is a natural number.}}}\right\ulcorner\mathsf{ be included (postulated) in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{2}\mathsf{)}\mathsf{: }\left(\left(\mathbf{n}_{1} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathbf{n}_{1}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{variable-substitution}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{variable-substitution}\mathsf{ defined as }\left\ulcorner(P, ๐›ท) โŠข P'\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{3}\mathsf{)}\mathsf{: }\left(\left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathit{0}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{modus-ponens}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{modus-ponens}\mathsf{ defined as }\left\ulcorner((P โŸน Q), P) โŠข Q\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}} \boldsymbol\mathsf{2.2.3}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{4}\mathsf{)}\mathsf{: }\left(\left(\mathit{0}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Definition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{D}_{1}\mathsf{)}\mathsf{: Let }\mathit{definition}\mathsf{ }\mathcal{D}_{1}\mathsf{ }\left\ulcorner\text{\sffamily{\itshape{We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)}}}\right\ulcorner\mathsf{ be included (postulated) in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{definition-interpretation}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{definition-interpretation}\mathsf{ defined as }\left\ulcorner๐’Ÿ โŠข P\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{5}\mathsf{)}\mathsf{: }\left(\mathit{1} \mathit{=} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{6}\mathsf{)}\mathsf{: }\left(\mathit{2} \mathit{=} \left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{7}\mathsf{)}\mathsf{: }\left(\mathit{3} \mathit{=} \left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{8}\mathsf{)}\mathsf{: }\left(\mathit{4} \mathit{=} \left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{9}\mathsf{)}\mathsf{: }\left(\left(\left(\mathit{0}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{10}\mathsf{)}\mathsf{: }\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{11}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{12}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{13}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{14}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{equality-commutativity}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{equality-commutativity}\mathsf{ defined as }\left\ulcorner(x = y) โŠข (y = x)\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{15}\mathsf{)}\mathsf{: }\left(\left(\mathit{0}\right)\mathit{++} \mathit{=} \mathit{1}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{equal-terms-substitution}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{equal-terms-substitution}\mathsf{ defined as }\left\ulcorner(P, (Q = R)) โŠข P'\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{16}\mathsf{)}\mathsf{: }\left(\mathit{2} \mathit{=} \left(\mathit{1}\right)\mathit{++}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{17}\mathsf{)}\mathsf{: }\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{2}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{18}\mathsf{)}\mathsf{: }\left(\mathit{3} \mathit{=} \left(\mathit{2}\right)\mathit{++}\right)\mathsf{.}\mathsf{}\mathsf{}### \boldsymbol\mathsf{3 is a natural number}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{19}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{3}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{20}\mathsf{)}\mathsf{: }\left(\left(\mathit{2}\right)\mathit{++} \mathit{=} \mathit{3}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}} \boldsymbol\mathsf{2.1.4}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{21}\mathsf{)}\mathsf{: }\left(\mathit{3} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{22}\mathsf{)}\mathsf{: }\left(\mathit{4} \mathit{=} \left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{23}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{4}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{24}\mathsf{)}\mathsf{: }\left(\left(\mathit{3}\right)\mathit{++} \mathit{=} \mathit{4}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{25}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{26}\mathsf{)}\mathsf{: }\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{}\mathsf{}### \boldsymbol\mathsf{Axiom 2.3}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Axiom}} \boldsymbol\mathsf{2.3}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{A}_{3}\mathsf{)}\mathsf{: Let }\mathit{axiom}\mathsf{ }\mathcal{A}_{3}\mathsf{ }\left\ulcorner\text{\sffamily{\itshape{0 is not the successor of any natural number; i.e., we have n++ 0 for every natural number n.}}}\right\ulcorner\mathsf{ be included (postulated) in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{27}\mathsf{)}\mathsf{: }\left(\left(\mathbf{n}_{2} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathbf{n}_{2}\right)\mathit{++} \mathit{\neq} \mathit{0}\right)\right)\mathsf{.}\mathsf{}\mathsf{}### \boldsymbol\mathsf{4 is not equal to 0.}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{28}\mathsf{)}\mathsf{: }\left(\left(\mathit{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathit{3}\right)\mathit{++} \mathit{\neq} \mathit{0}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{29}\mathsf{)}\mathsf{: }\left(\left(\mathit{3}\right)\mathit{++} \mathit{\neq} \mathit{0}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}} \boldsymbol\mathsf{2.1.6}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{30}\mathsf{)}\mathsf{: }\left(\mathit{4} \mathit{\neq} \mathit{0}\right)\mathsf{.}\mathsf{}\mathsf{}### \boldsymbol\mathsf{Axiom 2.4}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Axiom}} \boldsymbol\mathsf{2.4}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{A}_{4}\mathsf{)}\mathsf{: Let }\mathit{axiom}\mathsf{ }\mathcal{A}_{4}\mathsf{ }\left\ulcorner\text{\sffamily{\itshape{Different natural numbers must have different successors; i.e., if n, m are natural numbers and n m, then n++ m++. Equivalently, if n++ = m++, then we must have n = m.}}}\right\ulcorner\mathsf{ be included (postulated) in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{31}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathbf{n}_{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{m}_{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathbf{n}_{3} \mathit{\neq} \mathbf{m}_{1}\right)\right) \mathit{\implies} \left(\left(\mathbf{n}_{3}\right)\mathit{++} \mathit{\neq} \left(\mathbf{m}_{1}\right)\mathit{++}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{32}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathbf{n}_{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{m}_{2} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\left(\mathbf{n}_{4}\right)\mathit{++} \mathit{=} \left(\mathbf{m}_{2}\right)\mathit{++}\right)\right) \mathit{\implies} \left(\mathbf{n}_{4} \mathit{=} \mathbf{m}_{2}\right)\right)\mathsf{.}\mathsf{}\mathsf{}### \boldsymbol\mathsf{6 is not equal to 2.}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{33}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{4} \mathit{\neq} \mathit{0}\right)\right) \mathit{\implies} \left(\left(\mathit{4}\right)\mathit{++} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{conjunction-introduction}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{conjunction-introduction}\mathsf{ defined as }\left\ulcorner(P, Q) โŠข (P โ‹€ Q)\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{34}\mathsf{)}\mathsf{: }\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{35}\mathsf{)}\mathsf{: }\left(\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{4} \mathit{\neq} \mathit{0}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{36}\mathsf{)}\mathsf{: }\left(\left(\mathit{4}\right)\mathit{++} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{37}\mathsf{)}\mathsf{: }\left(\mathit{5} \mathit{=} \left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{38}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{5}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{39}\mathsf{)}\mathsf{: }\left(\left(\mathit{4}\right)\mathit{++} \mathit{=} \mathit{5}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{40}\mathsf{)}\mathsf{: }\left(\mathit{5} \mathit{=} \left(\mathit{4}\right)\mathit{++}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{41}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\left(\mathit{5}\right)\mathit{++} \mathit{\neq} \left(\mathit{1}\right)\mathit{++}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{42}\mathsf{)}\mathsf{: }\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathit{4}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{43}\mathsf{)}\mathsf{: }\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{44}\mathsf{)}\mathsf{: }\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{45}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\left(\mathit{5}\right)\mathit{++} \mathit{\neq} \left(\mathit{1}\right)\mathit{++}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{46}\mathsf{)}\mathsf{: }\left(\left(\mathit{4}\right)\mathit{++} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{47}\mathsf{)}\mathsf{: }\left(\mathit{5} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{48}\mathsf{)}\mathsf{: }\left(\mathit{6} \mathit{=} \left(\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{49}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{6}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{50}\mathsf{)}\mathsf{: }\left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{51}\mathsf{)}\mathsf{: }\left(\left(\mathit{5}\right)\mathit{++} \mathit{=} \mathit{6}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{52}\mathsf{)}\mathsf{: }\left(\mathit{6} \mathit{=} \left(\mathit{5}\right)\mathit{++}\right)\mathsf{.}\mathsf{}\mathsf{}#### \boldsymbol\mathsf{Proof by contradiction}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Hypothesis}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{H}_{1}\mathsf{)}\mathsf{: }\left(\mathit{6} \mathit{=} \mathit{2}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{inconsistency-by-inequality-introduction}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{inconsistency-by-inequality-introduction}\mathsf{ defined as }\left\ulcorner((P = Q), (P โ‰  Q)) โŠข Inc(๐’ฏ)\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{66}\mathsf{)}\mathsf{: }\mathit{Inc}\left(\mathcal{H}_{1}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{proof-by-refutation-of-equality}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{proof-by-refutation-of-equality}\mathsf{ defined as }\left\ulcorner(โ„‹(P=Q), Inc(โ„‹)) โŠข (P โ‰  Q)\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}} \boldsymbol\mathsf{2.1.8}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{67}\mathsf{)}\mathsf{: }\left(\mathit{6} \mathit{\neq} \mathit{2}\right)\mathsf{.}\mathsf{}\mathsf{}#### \boldsymbol\mathsf{Direct proof}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{68}\mathsf{)}\mathsf{: }\left(\left(\mathit{1}\right)\mathit{++} \mathit{=} \mathit{2}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{69}\mathsf{)}\mathsf{: }\left(\mathit{5} \mathit{\neq} \mathit{1}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{70}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\mathit{6} \mathit{\neq} \left(\mathit{1}\right)\mathit{++}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{71}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\mathit{6} \mathit{\neq} \mathit{2}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{72}\mathsf{)}\mathsf{: }\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{73}\mathsf{)}\mathsf{: }\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{74}\mathsf{)}\mathsf{: }\left(\mathit{6} \mathit{\neq} \mathit{2}\right)\mathsf{.}\mathsf{}\mathsf{}### \boldsymbol\mathsf{Axiom 2.5: the principle of mathematical induction}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Axiom schema}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{A}_{6}\mathsf{)} - Principle of mathematical induction\mathsf{: Let }\mathit{axiom}\mathsf{ }\mathcal{A}_{6}\mathsf{ }\left\ulcorner\text{\sffamily{\itshape{Let P(n) be any property pertaining to a natural number n. Suppose that P(O) is true, and suppose that whenever P(n) is true, P(n++) is also true. Then P(n) is true for every natural number n.}}}\right\ulcorner\mathsf{ be included (postulated) in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{75}\mathsf{)}\mathsf{: }\left(\left(\left(\mathbf{n}_{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{P}_{1}\left(\mathit{0}\right) \mathit{\land} \left(\mathbf{P}_{1}\left(\mathbf{n}_{5}\right) \mathit{\implies} \mathbf{P}_{1}\left(\left(\mathbf{n}_{5}\right)\mathit{++}\right)\right)\right)\right) \mathit{\implies} \left(\left(\mathbf{m}_{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \mathbf{P}_{1}\left(\mathbf{m}_{3}\right)\right)\right)\mathsf{.}\mathsf{}\mathsf{}### \boldsymbol\mathsf{The number system n}}\mathsf{}\mathsf{}### \boldsymbol\mathsf{Recursive definitions}}\mathsf{} \ No newline at end of file +\mathsf{}\mathsf{}\section{\boldsymbol\mathsf{Theory properties}}}\mathsf{}\boldsymbol\mathsf{Consistency: }}undetermined\mathsf{}\mathsf{}\boldsymbol\mathsf{Stabilized: }}False\mathsf{}\mathsf{}\boldsymbol\mathsf{Extended theory: }}N/A\mathsf{}\section{\boldsymbol\mathsf{Simple-objects declarations}}}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{0}\right\ulcorner, \left\ulcorner\mathit{natural-number}\right\ulcorner, \left\ulcorner\mathit{1}\right\ulcorner, \left\ulcorner\mathit{2}\right\ulcorner, \left\ulcorner\mathit{3}\right\ulcorner, \left\ulcorner\mathit{4}\right\ulcorner, \left\ulcorner\mathit{5}\right\ulcorner, \left\ulcorner\mathit{6}\right\ulcorner\mathsf{ be }\mathit{simple-objects}\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\section{\boldsymbol\mathsf{Relations}}}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{++}\right\ulcorner\mathsf{, }\left\ulcorner\mathit{Inc}\right\ulcorner\mathsf{ be }\mathit{unary}\mathit{-relations}\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{\land}\right\ulcorner\mathsf{, }\left\ulcorner\mathit{\neq}\right\ulcorner\mathsf{, }\left\ulcorner\mathit{is-a}\right\ulcorner\mathsf{, }\left\ulcorner\mathit{=}\right\ulcorner\mathsf{, }\left\ulcorner\mathit{\implies}\right\ulcorner\mathsf{ be }\mathit{binary}\mathit{-relations}\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\section{\boldsymbol\mathsf{Inference rules}}}\mathsf{}\mathsf{The following inference rules are considered valid under this theory:}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{axiom-interpretation}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner๐’œ โŠข P\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{conjunction-introduction}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner(P, Q) โŠข (P โ‹€ Q)\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{definition-interpretation}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner๐’Ÿ โŠข P\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{equal-terms-substitution}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner(P, (Q = R)) โŠข P'\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{equality-commutativity}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner(x = y) โŠข (y = x)\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{inconsistency-by-inequality-introduction}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner((P = Q), (P โ‰  Q)) โŠข Inc(๐’ฏ)\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{modus-ponens}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner((P โŸน Q), P) โŠข Q\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{proof-by-refutation-of-equality}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner(โ„‹(P=Q), Inc(โ„‹)) โŠข (P โ‰  Q)\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{variable-substitution}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner(P, ๐›ท) โŠข P'\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\section{\boldsymbol\mathsf{Theory elaboration sequence}}}\mathsf{}# \boldsymbol\mathsf{2}}: \boldsymbol\mathsf{The natural numbers}}\mathsf{}\mathsf{}## \boldsymbol\mathsf{2.1}}: \boldsymbol\mathsf{The peano axioms}}\mathsf{}\mathsf{}### \boldsymbol\mathsf{Informal definition of natural number}}\mathsf{}\mathsf{}### \boldsymbol\mathsf{Axiom 2.1}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Axiom}} \boldsymbol\mathsf{2.1}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{A}_{1}\mathsf{)}\mathsf{: Let }\mathit{axiom}\mathsf{ }\mathcal{A}_{1}\mathsf{ }\left\ulcorner\text{\sffamily{\itshape{0 is a natural number.}}}\right\ulcorner\mathsf{ be included (postulated) in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{axiom-interpretation}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{axiom-interpretation}\mathsf{ defined as }\left\ulcorner๐’œ โŠข P\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{1}\mathsf{)}\mathsf{: }\left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{}\mathsf{}### \boldsymbol\mathsf{Axiom 2.2}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Axiom}} \boldsymbol\mathsf{2.2}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{A}_{2}\mathsf{)}\mathsf{: Let }\mathit{axiom}\mathsf{ }\mathcal{A}_{2}\mathsf{ }\left\ulcorner\text{\sffamily{\itshape{If n is a natural number, then n++ is a natural number.}}}\right\ulcorner\mathsf{ be included (postulated) in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{2}\mathsf{)}\mathsf{: }\left(\left(\mathbf{n}_{1} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathbf{n}_{1}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{variable-substitution}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{variable-substitution}\mathsf{ defined as }\left\ulcorner(P, ๐›ท) โŠข P'\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{3}\mathsf{)}\mathsf{: }\left(\left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathit{0}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{modus-ponens}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{modus-ponens}\mathsf{ defined as }\left\ulcorner((P โŸน Q), P) โŠข Q\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}} \boldsymbol\mathsf{2.2.3}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{4}\mathsf{)}\mathsf{: }\left(\left(\mathit{0}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Definition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{D}_{1}\mathsf{)}\mathsf{: Let }\mathit{definition}\mathsf{ }\mathcal{D}_{1}\mathsf{ }\left\ulcorner\text{\sffamily{\itshape{We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)}}}\right\ulcorner\mathsf{ be included (postulated) in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{definition-interpretation}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{definition-interpretation}\mathsf{ defined as }\left\ulcorner๐’Ÿ โŠข P\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{5}\mathsf{)}\mathsf{: }\left(\mathit{1} \mathit{=} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{6}\mathsf{)}\mathsf{: }\left(\mathit{2} \mathit{=} \left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{7}\mathsf{)}\mathsf{: }\left(\mathit{3} \mathit{=} \left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{8}\mathsf{)}\mathsf{: }\left(\mathit{4} \mathit{=} \left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{9}\mathsf{)}\mathsf{: }\left(\left(\left(\mathit{0}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{10}\mathsf{)}\mathsf{: }\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{11}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{12}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{13}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{14}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{equality-commutativity}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{equality-commutativity}\mathsf{ defined as }\left\ulcorner(x = y) โŠข (y = x)\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{15}\mathsf{)}\mathsf{: }\left(\left(\mathit{0}\right)\mathit{++} \mathit{=} \mathit{1}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{equal-terms-substitution}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{equal-terms-substitution}\mathsf{ defined as }\left\ulcorner(P, (Q = R)) โŠข P'\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{16}\mathsf{)}\mathsf{: }\left(\mathit{2} \mathit{=} \left(\mathit{1}\right)\mathit{++}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{17}\mathsf{)}\mathsf{: }\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{2}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{18}\mathsf{)}\mathsf{: }\left(\mathit{3} \mathit{=} \left(\mathit{2}\right)\mathit{++}\right)\mathsf{.}\mathsf{}\mathsf{}### \boldsymbol\mathsf{3 is a natural number}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{19}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{3}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{20}\mathsf{)}\mathsf{: }\left(\left(\mathit{2}\right)\mathit{++} \mathit{=} \mathit{3}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}} \boldsymbol\mathsf{2.1.4}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{21}\mathsf{)}\mathsf{: }\left(\mathit{3} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{22}\mathsf{)}\mathsf{: }\left(\mathit{4} \mathit{=} \left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{23}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{4}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{24}\mathsf{)}\mathsf{: }\left(\left(\mathit{3}\right)\mathit{++} \mathit{=} \mathit{4}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{25}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{26}\mathsf{)}\mathsf{: }\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{}\mathsf{}### \boldsymbol\mathsf{Axiom 2.3}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Axiom}} \boldsymbol\mathsf{2.3}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{A}_{3}\mathsf{)}\mathsf{: Let }\mathit{axiom}\mathsf{ }\mathcal{A}_{3}\mathsf{ }\left\ulcorner\text{\sffamily{\itshape{0 is not the successor of any natural number; i.e., we have n++ 0 for every natural number n.}}}\right\ulcorner\mathsf{ be included (postulated) in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{27}\mathsf{)}\mathsf{: }\left(\left(\mathbf{n}_{2} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathbf{n}_{2}\right)\mathit{++} \mathit{\neq} \mathit{0}\right)\right)\mathsf{.}\mathsf{}\mathsf{}### \boldsymbol\mathsf{4 is not equal to 0.}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{28}\mathsf{)}\mathsf{: }\left(\left(\mathit{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathit{3}\right)\mathit{++} \mathit{\neq} \mathit{0}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{29}\mathsf{)}\mathsf{: }\left(\left(\mathit{3}\right)\mathit{++} \mathit{\neq} \mathit{0}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}} \boldsymbol\mathsf{2.1.6}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{30}\mathsf{)}\mathsf{: }\left(\mathit{4} \mathit{\neq} \mathit{0}\right)\mathsf{.}\mathsf{}\mathsf{}### \boldsymbol\mathsf{Axiom 2.4}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Axiom}} \boldsymbol\mathsf{2.4}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{A}_{4}\mathsf{)}\mathsf{: Let }\mathit{axiom}\mathsf{ }\mathcal{A}_{4}\mathsf{ }\left\ulcorner\text{\sffamily{\itshape{Different natural numbers must have different successors; i.e., if n, m are natural numbers and n m, then n++ m++. Equivalently, if n++ = m++, then we must have n = m.}}}\right\ulcorner\mathsf{ be included (postulated) in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{31}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathbf{n}_{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{m}_{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathbf{n}_{3} \mathit{\neq} \mathbf{m}_{1}\right)\right) \mathit{\implies} \left(\left(\mathbf{n}_{3}\right)\mathit{++} \mathit{\neq} \left(\mathbf{m}_{1}\right)\mathit{++}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{32}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathbf{n}_{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{m}_{2} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\left(\mathbf{n}_{4}\right)\mathit{++} \mathit{=} \left(\mathbf{m}_{2}\right)\mathit{++}\right)\right) \mathit{\implies} \left(\mathbf{n}_{4} \mathit{=} \mathbf{m}_{2}\right)\right)\mathsf{.}\mathsf{}\mathsf{}### \boldsymbol\mathsf{6 is not equal to 2.}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{33}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{4} \mathit{\neq} \mathit{0}\right)\right) \mathit{\implies} \left(\left(\mathit{4}\right)\mathit{++} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{conjunction-introduction}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{conjunction-introduction}\mathsf{ defined as }\left\ulcorner(P, Q) โŠข (P โ‹€ Q)\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{34}\mathsf{)}\mathsf{: }\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{35}\mathsf{)}\mathsf{: }\left(\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{4} \mathit{\neq} \mathit{0}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{36}\mathsf{)}\mathsf{: }\left(\left(\mathit{4}\right)\mathit{++} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{37}\mathsf{)}\mathsf{: }\left(\mathit{5} \mathit{=} \left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{38}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{5}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{39}\mathsf{)}\mathsf{: }\left(\left(\mathit{4}\right)\mathit{++} \mathit{=} \mathit{5}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{40}\mathsf{)}\mathsf{: }\left(\mathit{5} \mathit{=} \left(\mathit{4}\right)\mathit{++}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{41}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\left(\mathit{5}\right)\mathit{++} \mathit{\neq} \left(\mathit{1}\right)\mathit{++}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{42}\mathsf{)}\mathsf{: }\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathit{4}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{43}\mathsf{)}\mathsf{: }\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{44}\mathsf{)}\mathsf{: }\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{45}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\left(\mathit{5}\right)\mathit{++} \mathit{\neq} \left(\mathit{1}\right)\mathit{++}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{46}\mathsf{)}\mathsf{: }\left(\left(\mathit{4}\right)\mathit{++} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{47}\mathsf{)}\mathsf{: }\left(\mathit{5} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{48}\mathsf{)}\mathsf{: }\left(\mathit{6} \mathit{=} \left(\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{49}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{6}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{50}\mathsf{)}\mathsf{: }\left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{51}\mathsf{)}\mathsf{: }\left(\left(\mathit{5}\right)\mathit{++} \mathit{=} \mathit{6}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{52}\mathsf{)}\mathsf{: }\left(\mathit{6} \mathit{=} \left(\mathit{5}\right)\mathit{++}\right)\mathsf{.}\mathsf{}\mathsf{}#### \boldsymbol\mathsf{Proof by contradiction}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Hypothesis}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{H}_{1}\mathsf{)}\mathsf{: }\left(\mathit{6} \mathit{=} \mathit{2}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{inconsistency-by-inequality-introduction}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{inconsistency-by-inequality-introduction}\mathsf{ defined as }\left\ulcorner((P = Q), (P โ‰  Q)) โŠข Inc(๐’ฏ)\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{66}\mathsf{)}\mathsf{: }\mathit{Inc}\left(\mathcal{H}_{1}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{proof-by-refutation-of-equality}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{proof-by-refutation-of-equality}\mathsf{ defined as }\left\ulcorner(โ„‹(P=Q), Inc(โ„‹)) โŠข (P โ‰  Q)\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}} \boldsymbol\mathsf{2.1.8}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{67}\mathsf{)}\mathsf{: }\left(\mathit{6} \mathit{\neq} \mathit{2}\right)\mathsf{.}\mathsf{}\mathsf{}#### \boldsymbol\mathsf{Direct proof}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{68}\mathsf{)}\mathsf{: }\left(\left(\mathit{1}\right)\mathit{++} \mathit{=} \mathit{2}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{69}\mathsf{)}\mathsf{: }\left(\mathit{5} \mathit{\neq} \mathit{1}\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{70}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\mathit{6} \mathit{\neq} \left(\mathit{1}\right)\mathit{++}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{71}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\mathit{6} \mathit{\neq} \mathit{2}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{72}\mathsf{)}\mathsf{: }\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{73}\mathsf{)}\mathsf{: }\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right)\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{74}\mathsf{)}\mathsf{: }\left(\mathit{6} \mathit{\neq} \mathit{2}\right)\mathsf{.}\mathsf{}\mathsf{}### \boldsymbol\mathsf{Axiom 2.5: the principle of mathematical induction}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Axiom schema}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{A}_{6}\mathsf{)} - Principle of mathematical induction\mathsf{: Let }\mathit{axiom}\mathsf{ }\mathcal{A}_{6}\mathsf{ }\left\ulcorner\text{\sffamily{\itshape{Let P(n) be any property pertaining to a natural number n. Suppose that P(O) is true, and suppose that whenever P(n) is true, P(n++) is also true. Then P(n) is true for every natural number n.}}}\right\ulcorner\mathsf{ be included (postulated) in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{75}\mathsf{)}\mathsf{: }\left(\left(\left(\mathbf{n}_{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{P}_{1}\left(\mathit{0}\right) \mathit{\land} \left(\mathbf{P}_{1}\left(\mathbf{n}_{5}\right) \mathit{\implies} \mathbf{P}_{1}\left(\left(\mathbf{n}_{5}\right)\mathit{++}\right)\right)\right)\right) \mathit{\implies} \left(\left(\mathbf{m}_{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \mathbf{P}_{1}\left(\mathbf{m}_{3}\right)\right)\right)\mathsf{.}\mathsf{}\mathsf{}### \boldsymbol\mathsf{The number system n}}\mathsf{}\mathsf{}### \boldsymbol\mathsf{Recursive definitions}}\mathsf{} \ No newline at end of file diff --git a/theory/build/tao_2006_2_1_the_peano_axioms_report_noproof_enus_plaintext.txt b/theory/build/tao_2006_2_1_the_peano_axioms_report_noproof_enus_plaintext.txt index 2f1e6e1a..e450e384 100644 --- a/theory/build/tao_2006_2_1_the_peano_axioms_report_noproof_enus_plaintext.txt +++ b/theory/build/tao_2006_2_1_the_peano_axioms_report_noproof_enus_plaintext.txt @@ -10,7 +10,7 @@ Let "0", "natural-number", "1", "2", "3", "4", "5", "6" be simple-objects in U1. # Relations Let "++", "Inc" be unary-relations in U1. -Let "=", "and", "is-a", "neq", "==>" be binary-relations in U1. +Let "and", "neq", "is-a", "=", "==>" be binary-relations in U1. # Inference rules The following inference rules are considered valid under this theory: diff --git a/theory/build/tao_2006_2_1_the_peano_axioms_report_noproof_enus_unicode.txt b/theory/build/tao_2006_2_1_the_peano_axioms_report_noproof_enus_unicode.txt index 3f8de9d4..85bebebb 100644 --- a/theory/build/tao_2006_2_1_the_peano_axioms_report_noproof_enus_unicode.txt +++ b/theory/build/tao_2006_2_1_the_peano_axioms_report_noproof_enus_unicode.txt @@ -10,7 +10,7 @@ # ๐—ฅ๐—ฒ๐—น๐—ฎ๐˜๐—ถ๐—ผ๐—ป๐˜€ ๐–ซ๐–พ๐— โŒœ++โŒ, โŒœ๐ผ๐‘›๐‘โŒ ๐–ป๐–พ ๐‘ข๐‘›๐‘Ž๐‘Ÿ๐‘ฆ-๐‘Ÿ๐‘’๐‘™๐‘Ž๐‘ก๐‘–๐‘œ๐‘›๐‘  ๐—‚๐—‡ ๐’ฐโ‚. -๐–ซ๐–พ๐— โŒœ=โŒ, โŒœโˆงโŒ, โŒœ๐‘–๐‘ -๐‘ŽโŒ, โŒœโ‰ โŒ, โŒœโŸนโŒ ๐–ป๐–พ ๐‘๐‘–๐‘›๐‘Ž๐‘Ÿ๐‘ฆ-๐‘Ÿ๐‘’๐‘™๐‘Ž๐‘ก๐‘–๐‘œ๐‘›๐‘  ๐—‚๐—‡ ๐’ฐโ‚. +๐–ซ๐–พ๐— โŒœโˆงโŒ, โŒœโ‰ โŒ, โŒœ๐‘–๐‘ -๐‘ŽโŒ, โŒœ=โŒ, โŒœโŸนโŒ ๐–ป๐–พ ๐‘๐‘–๐‘›๐‘Ž๐‘Ÿ๐‘ฆ-๐‘Ÿ๐‘’๐‘™๐‘Ž๐‘ก๐‘–๐‘œ๐‘›๐‘  ๐—‚๐—‡ ๐’ฐโ‚. # ๐—œ๐—ป๐—ณ๐—ฒ๐—ฟ๐—ฒ๐—ป๐—ฐ๐—ฒ ๐—ฟ๐˜‚๐—น๐—ฒ๐˜€ ๐–ณ๐—๐–พ ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—‚๐—‡๐—€ ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ๐—Œ ๐–บ๐—‹๐–พ ๐–ผ๐—ˆ๐—‡๐—Œ๐—‚๐–ฝ๐–พ๐—‹๐–พ๐–ฝ ๐—๐–บ๐—…๐—‚๐–ฝ ๐—Ž๐—‡๐–ฝ๐–พ๐—‹ ๐—๐—๐—‚๐—Œ ๐—๐—๐–พ๐—ˆ๐—‹๐—’: diff --git a/theory/build/tao_2006_2_1_the_peano_axioms_report_proof_enus_latex.txt b/theory/build/tao_2006_2_1_the_peano_axioms_report_proof_enus_latex.txt index 408abe37..34b5c790 100644 --- a/theory/build/tao_2006_2_1_the_peano_axioms_report_proof_enus_latex.txt +++ b/theory/build/tao_2006_2_1_the_peano_axioms_report_proof_enus_latex.txt @@ -1 +1 @@ -\mathsf{}\mathsf{}\section{\boldsymbol\mathsf{Theory properties}}}\mathsf{}\boldsymbol\mathsf{Consistency: }}undetermined\mathsf{}\mathsf{}\boldsymbol\mathsf{Stabilized: }}False\mathsf{}\mathsf{}\boldsymbol\mathsf{Extended theory: }}N/A\mathsf{}\section{\boldsymbol\mathsf{Simple-objects declarations}}}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{0}\right\ulcorner, \left\ulcorner\mathit{natural-number}\right\ulcorner, \left\ulcorner\mathit{1}\right\ulcorner, \left\ulcorner\mathit{2}\right\ulcorner, \left\ulcorner\mathit{3}\right\ulcorner, \left\ulcorner\mathit{4}\right\ulcorner, \left\ulcorner\mathit{5}\right\ulcorner, \left\ulcorner\mathit{6}\right\ulcorner\mathsf{ be }\mathit{simple-objects}\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\section{\boldsymbol\mathsf{Relations}}}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{++}\right\ulcorner\mathsf{, }\left\ulcorner\mathit{Inc}\right\ulcorner\mathsf{ be }\mathit{unary}\mathit{-relations}\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{=}\right\ulcorner\mathsf{, }\left\ulcorner\mathit{\land}\right\ulcorner\mathsf{, }\left\ulcorner\mathit{is-a}\right\ulcorner\mathsf{, }\left\ulcorner\mathit{\neq}\right\ulcorner\mathsf{, }\left\ulcorner\mathit{\implies}\right\ulcorner\mathsf{ be }\mathit{binary}\mathit{-relations}\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\section{\boldsymbol\mathsf{Inference rules}}}\mathsf{}\mathsf{The following inference rules are considered valid under this theory:}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{axiom-interpretation}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner๐’œ โŠข P\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{conjunction-introduction}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner(P, Q) โŠข (P โ‹€ Q)\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{definition-interpretation}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner๐’Ÿ โŠข P\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{equal-terms-substitution}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner(P, (Q = R)) โŠข P'\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{equality-commutativity}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner(x = y) โŠข (y = x)\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{inconsistency-by-inequality-introduction}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner((P = Q), (P โ‰  Q)) โŠข Inc(๐’ฏ)\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{modus-ponens}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner((P โŸน Q), P) โŠข Q\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{proof-by-refutation-of-equality}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner(โ„‹(P=Q), Inc(โ„‹)) โŠข (P โ‰  Q)\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{variable-substitution}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner(P, ๐›ท) โŠข P'\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\section{\boldsymbol\mathsf{Theory elaboration sequence}}}\mathsf{}# \boldsymbol\mathsf{2}}: \boldsymbol\mathsf{The natural numbers}}\mathsf{}\mathsf{}## \boldsymbol\mathsf{2.1}}: \boldsymbol\mathsf{The peano axioms}}\mathsf{}\mathsf{}### \boldsymbol\mathsf{Informal definition of natural number}}\mathsf{}\mathsf{}### \boldsymbol\mathsf{Axiom 2.1}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Axiom}} \boldsymbol\mathsf{2.1}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{A}_{1}\mathsf{)}\mathsf{: Let }\mathit{axiom}\mathsf{ }\mathcal{A}_{1}\mathsf{ }\left\ulcorner\text{\sffamily{\itshape{0 is a natural number.}}}\right\ulcorner\mathsf{ be included (postulated) in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{axiom-interpretation}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{axiom-interpretation}\mathsf{ defined as }\left\ulcorner๐’œ โŠข P\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{1}\mathsf{)}\mathsf{: }\left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left\ulcorner\text{\sffamily{\itshape{0 is a natural number.}}}\right\ulcorner\mathsf{ is postulated by }\boldsymbol\mathsf{axiom}} \boldsymbol\mathsf{2.1}} (\mathit{A}_{1})\mathsf{. }\left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\mathsf{ is a valid formula statement interpreted from that axiom.}\mathsf{ Therefore, by the }\mathit{axiom-interpretation}\mathsf{ inference rule, it follows that }\left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}### \boldsymbol\mathsf{Axiom 2.2}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Axiom}} \boldsymbol\mathsf{2.2}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{A}_{2}\mathsf{)}\mathsf{: Let }\mathit{axiom}\mathsf{ }\mathcal{A}_{2}\mathsf{ }\left\ulcorner\text{\sffamily{\itshape{If n is a natural number, then n++ is a natural number.}}}\right\ulcorner\mathsf{ be included (postulated) in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{2}\mathsf{)}\mathsf{: }\left(\left(\mathbf{n}_{1} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathbf{n}_{1}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left\ulcorner\text{\sffamily{\itshape{If n is a natural number, then n++ is a natural number.}}}\right\ulcorner\mathsf{ is postulated by }\boldsymbol\mathsf{axiom}} \boldsymbol\mathsf{2.2}} (\mathit{A}_{2})\mathsf{. }\left(\left(\mathbf{n}_{1} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathbf{n}_{1}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ is a valid formula statement interpreted from that axiom.}\mathsf{ Therefore, by the }\mathit{axiom-interpretation}\mathsf{ inference rule, it follows that }\left(\left(\mathbf{n}_{1} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathbf{n}_{1}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{variable-substitution}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{variable-substitution}\mathsf{ defined as }\left\ulcorner(P, ๐›ท) โŠข P'\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{3}\mathsf{)}\mathsf{: }\left(\left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathit{0}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathbf{n}_{1} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathbf{n}_{1}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{2})\mathsf{.}\mathsf{ Let }\mathbf{n}_{1} = 0\mathsf{.}\mathsf{ Therefore, by the }\mathit{variable-substitution}\mathsf{ inference rule, it follows that }\left(\left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathit{0}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{modus-ponens}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{modus-ponens}\mathsf{ defined as }\left\ulcorner((P โŸน Q), P) โŠข Q\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}} \boldsymbol\mathsf{2.2.3}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{4}\mathsf{)}\mathsf{: }\left(\left(\mathit{0}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathit{0}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{3})\mathsf{.}\left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{1})\mathsf{.}\mathsf{ Therefore, by the }\mathit{modus-ponens}\mathsf{ inference rule, it follows that }\left(\left(\mathit{0}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Definition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{D}_{1}\mathsf{)}\mathsf{: Let }\mathit{definition}\mathsf{ }\mathcal{D}_{1}\mathsf{ }\left\ulcorner\text{\sffamily{\itshape{We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)}}}\right\ulcorner\mathsf{ be included (postulated) in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{definition-interpretation}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{definition-interpretation}\mathsf{ defined as }\left\ulcorner๐’Ÿ โŠข P\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{5}\mathsf{)}\mathsf{: }\left(\mathit{1} \mathit{=} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left\ulcorner\text{\sffamily{\itshape{We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)}}}\right\ulcorner\mathsf{ is postulated by }\boldsymbol\mathsf{def.}} (\mathit{D}_{1})\mathsf{. }\left(\mathit{1} \mathit{=} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{ is an interpretation of that definition.}\mathsf{ Therefore, by the }\mathit{definition-interpretation}\mathsf{ inference rule, it follows that }\left(\mathit{1} \mathit{=} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{6}\mathsf{)}\mathsf{: }\left(\mathit{2} \mathit{=} \left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left\ulcorner\text{\sffamily{\itshape{We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)}}}\right\ulcorner\mathsf{ is postulated by }\boldsymbol\mathsf{def.}} (\mathit{D}_{1})\mathsf{. }\left(\mathit{2} \mathit{=} \left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{ is an interpretation of that definition.}\mathsf{ Therefore, by the }\mathit{definition-interpretation}\mathsf{ inference rule, it follows that }\left(\mathit{2} \mathit{=} \left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{7}\mathsf{)}\mathsf{: }\left(\mathit{3} \mathit{=} \left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left\ulcorner\text{\sffamily{\itshape{We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)}}}\right\ulcorner\mathsf{ is postulated by }\boldsymbol\mathsf{def.}} (\mathit{D}_{1})\mathsf{. }\left(\mathit{3} \mathit{=} \left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{ is an interpretation of that definition.}\mathsf{ Therefore, by the }\mathit{definition-interpretation}\mathsf{ inference rule, it follows that }\left(\mathit{3} \mathit{=} \left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{8}\mathsf{)}\mathsf{: }\left(\mathit{4} \mathit{=} \left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left\ulcorner\text{\sffamily{\itshape{We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)}}}\right\ulcorner\mathsf{ is postulated by }\boldsymbol\mathsf{def.}} (\mathit{D}_{1})\mathsf{. }\left(\mathit{4} \mathit{=} \left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{ is an interpretation of that definition.}\mathsf{ Therefore, by the }\mathit{definition-interpretation}\mathsf{ inference rule, it follows that }\left(\mathit{4} \mathit{=} \left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{9}\mathsf{)}\mathsf{: }\left(\left(\left(\mathit{0}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathbf{n}_{1} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathbf{n}_{1}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{2})\mathsf{.}\mathsf{ Let }\mathbf{n}_{1} = (0)++\mathsf{.}\mathsf{ Therefore, by the }\mathit{variable-substitution}\mathsf{ inference rule, it follows that }\left(\left(\left(\mathit{0}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{10}\mathsf{)}\mathsf{: }\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\mathit{0}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{9})\mathsf{.}\left(\left(\mathit{0}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} \boldsymbol\mathsf{2.2.3}} (\mathit{P}_{4})\mathsf{.}\mathsf{ Therefore, by the }\mathit{modus-ponens}\mathsf{ inference rule, it follows that }\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{11}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathbf{n}_{1} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathbf{n}_{1}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{2})\mathsf{.}\mathsf{ Let }\mathbf{n}_{1} = ((0)++)++\mathsf{.}\mathsf{ Therefore, by the }\mathit{variable-substitution}\mathsf{ inference rule, it follows that }\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{12}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{11})\mathsf{.}\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{10})\mathsf{.}\mathsf{ Therefore, by the }\mathit{modus-ponens}\mathsf{ inference rule, it follows that }\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{13}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathbf{n}_{1} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathbf{n}_{1}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{2})\mathsf{.}\mathsf{ Let }\mathbf{n}_{1} = (((0)++)++)++\mathsf{.}\mathsf{ Therefore, by the }\mathit{variable-substitution}\mathsf{ inference rule, it follows that }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{14}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{13})\mathsf{.}\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{12})\mathsf{.}\mathsf{ Therefore, by the }\mathit{modus-ponens}\mathsf{ inference rule, it follows that }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{equality-commutativity}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{equality-commutativity}\mathsf{ defined as }\left\ulcorner(x = y) โŠข (y = x)\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{15}\mathsf{)}\mathsf{: }\left(\left(\mathit{0}\right)\mathit{++} \mathit{=} \mathit{1}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\mathit{1} \mathit{=} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{5})\mathsf{. }\mathsf{ Therefore, by the }\mathit{equality-commutativity}\mathsf{ inference rule, it follows that }\left(\left(\mathit{0}\right)\mathit{++} \mathit{=} \mathit{1}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{equal-terms-substitution}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{equal-terms-substitution}\mathsf{ defined as }\left\ulcorner(P, (Q = R)) โŠข P'\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{16}\mathsf{)}\mathsf{: }\left(\mathit{2} \mathit{=} \left(\mathit{1}\right)\mathit{++}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\mathit{2} \mathit{=} \left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{6})\mathsf{. }\left(\left(\mathit{0}\right)\mathit{++} \mathit{=} \mathit{1}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{15})\mathsf{.}\mathsf{ Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule, it follows that }\left(\mathit{2} \mathit{=} \left(\mathit{1}\right)\mathit{++}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{17}\mathsf{)}\mathsf{: }\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{2}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\mathit{2} \mathit{=} \left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{6})\mathsf{. }\mathsf{ Therefore, by the }\mathit{equality-commutativity}\mathsf{ inference rule, it follows that }\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{2}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{18}\mathsf{)}\mathsf{: }\left(\mathit{3} \mathit{=} \left(\mathit{2}\right)\mathit{++}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\mathit{3} \mathit{=} \left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{7})\mathsf{. }\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{2}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{17})\mathsf{.}\mathsf{ Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule, it follows that }\left(\mathit{3} \mathit{=} \left(\mathit{2}\right)\mathit{++}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}### \boldsymbol\mathsf{3 is a natural number}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{19}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{3}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\mathit{3} \mathit{=} \left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{7})\mathsf{. }\mathsf{ Therefore, by the }\mathit{equality-commutativity}\mathsf{ inference rule, it follows that }\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{3}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{20}\mathsf{)}\mathsf{: }\left(\left(\mathit{2}\right)\mathit{++} \mathit{=} \mathit{3}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{3}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{19})\mathsf{. }\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{2}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{17})\mathsf{.}\mathsf{ Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule, it follows that }\left(\left(\mathit{2}\right)\mathit{++} \mathit{=} \mathit{3}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}} \boldsymbol\mathsf{2.1.4}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{21}\mathsf{)}\mathsf{: }\left(\mathit{3} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{12})\mathsf{. }\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{3}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{19})\mathsf{.}\mathsf{ Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule, it follows that }\left(\mathit{3} \mathit{is-a} \mathit{natural-number}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{22}\mathsf{)}\mathsf{: }\left(\mathit{4} \mathit{=} \left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left\ulcorner\text{\sffamily{\itshape{We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)}}}\right\ulcorner\mathsf{ is postulated by }\boldsymbol\mathsf{def.}} (\mathit{D}_{1})\mathsf{. }\left(\mathit{4} \mathit{=} \left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{ is an interpretation of that definition.}\mathsf{ Therefore, by the }\mathit{definition-interpretation}\mathsf{ inference rule, it follows that }\left(\mathit{4} \mathit{=} \left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{23}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{4}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\mathit{4} \mathit{=} \left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{8})\mathsf{. }\mathsf{ Therefore, by the }\mathit{equality-commutativity}\mathsf{ inference rule, it follows that }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{4}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{24}\mathsf{)}\mathsf{: }\left(\left(\mathit{3}\right)\mathit{++} \mathit{=} \mathit{4}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{4}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{23})\mathsf{. }\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{3}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{19})\mathsf{.}\mathsf{ Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule, it follows that }\left(\left(\mathit{3}\right)\mathit{++} \mathit{=} \mathit{4}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{25}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{13})\mathsf{. }\left(\left(\mathit{3}\right)\mathit{++} \mathit{=} \mathit{4}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{24})\mathsf{.}\mathsf{ Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule, it follows that }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{26}\mathsf{)}\mathsf{: }\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{14})\mathsf{. }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{4}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{23})\mathsf{.}\mathsf{ Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule, it follows that }\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}### \boldsymbol\mathsf{Axiom 2.3}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Axiom}} \boldsymbol\mathsf{2.3}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{A}_{3}\mathsf{)}\mathsf{: Let }\mathit{axiom}\mathsf{ }\mathcal{A}_{3}\mathsf{ }\left\ulcorner\text{\sffamily{\itshape{0 is not the successor of any natural number; i.e., we have n++ 0 for every natural number n.}}}\right\ulcorner\mathsf{ be included (postulated) in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{27}\mathsf{)}\mathsf{: }\left(\left(\mathbf{n}_{2} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathbf{n}_{2}\right)\mathit{++} \mathit{\neq} \mathit{0}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left\ulcorner\text{\sffamily{\itshape{0 is not the successor of any natural number; i.e., we have n++ 0 for every natural number n.}}}\right\ulcorner\mathsf{ is postulated by }\boldsymbol\mathsf{axiom}} \boldsymbol\mathsf{2.3}} (\mathit{A}_{3})\mathsf{. }\left(\left(\mathbf{n}_{2} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathbf{n}_{2}\right)\mathit{++} \mathit{\neq} \mathit{0}\right)\right)\mathsf{ is a valid formula statement interpreted from that axiom.}\mathsf{ Therefore, by the }\mathit{axiom-interpretation}\mathsf{ inference rule, it follows that }\left(\left(\mathbf{n}_{2} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathbf{n}_{2}\right)\mathit{++} \mathit{\neq} \mathit{0}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}### \boldsymbol\mathsf{4 is not equal to 0.}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{28}\mathsf{)}\mathsf{: }\left(\left(\mathit{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathit{3}\right)\mathit{++} \mathit{\neq} \mathit{0}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathbf{n}_{2} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathbf{n}_{2}\right)\mathit{++} \mathit{\neq} \mathit{0}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{27})\mathsf{.}\mathsf{ Let }\mathbf{n}_{2} = 3\mathsf{.}\mathsf{ Therefore, by the }\mathit{variable-substitution}\mathsf{ inference rule, it follows that }\left(\left(\mathit{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathit{3}\right)\mathit{++} \mathit{\neq} \mathit{0}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{29}\mathsf{)}\mathsf{: }\left(\left(\mathit{3}\right)\mathit{++} \mathit{\neq} \mathit{0}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathit{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathit{3}\right)\mathit{++} \mathit{\neq} \mathit{0}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{28})\mathsf{.}\left(\mathit{3} \mathit{is-a} \mathit{natural-number}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} \boldsymbol\mathsf{2.1.4}} (\mathit{P}_{21})\mathsf{.}\mathsf{ Therefore, by the }\mathit{modus-ponens}\mathsf{ inference rule, it follows that }\left(\left(\mathit{3}\right)\mathit{++} \mathit{\neq} \mathit{0}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}} \boldsymbol\mathsf{2.1.6}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{30}\mathsf{)}\mathsf{: }\left(\mathit{4} \mathit{\neq} \mathit{0}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathit{3}\right)\mathit{++} \mathit{\neq} \mathit{0}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{29})\mathsf{. }\left(\left(\mathit{3}\right)\mathit{++} \mathit{=} \mathit{4}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{24})\mathsf{.}\mathsf{ Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule, it follows that }\left(\mathit{4} \mathit{\neq} \mathit{0}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}### \boldsymbol\mathsf{Axiom 2.4}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Axiom}} \boldsymbol\mathsf{2.4}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{A}_{4}\mathsf{)}\mathsf{: Let }\mathit{axiom}\mathsf{ }\mathcal{A}_{4}\mathsf{ }\left\ulcorner\text{\sffamily{\itshape{Different natural numbers must have different successors; i.e., if n, m are natural numbers and n m, then n++ m++. Equivalently, if n++ = m++, then we must have n = m.}}}\right\ulcorner\mathsf{ be included (postulated) in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{31}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathbf{n}_{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{m}_{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathbf{n}_{3} \mathit{\neq} \mathbf{m}_{1}\right)\right) \mathit{\implies} \left(\left(\mathbf{n}_{3}\right)\mathit{++} \mathit{\neq} \left(\mathbf{m}_{1}\right)\mathit{++}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left\ulcorner\text{\sffamily{\itshape{Different natural numbers must have different successors; i.e., if n, m are natural numbers and n m, then n++ m++. Equivalently, if n++ = m++, then we must have n = m.}}}\right\ulcorner\mathsf{ is postulated by }\boldsymbol\mathsf{axiom}} \boldsymbol\mathsf{2.4}} (\mathit{A}_{4})\mathsf{. }\left(\left(\left(\left(\mathbf{n}_{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{m}_{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathbf{n}_{3} \mathit{\neq} \mathbf{m}_{1}\right)\right) \mathit{\implies} \left(\left(\mathbf{n}_{3}\right)\mathit{++} \mathit{\neq} \left(\mathbf{m}_{1}\right)\mathit{++}\right)\right)\mathsf{ is a valid formula statement interpreted from that axiom.}\mathsf{ Therefore, by the }\mathit{axiom-interpretation}\mathsf{ inference rule, it follows that }\left(\left(\left(\left(\mathbf{n}_{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{m}_{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathbf{n}_{3} \mathit{\neq} \mathbf{m}_{1}\right)\right) \mathit{\implies} \left(\left(\mathbf{n}_{3}\right)\mathit{++} \mathit{\neq} \left(\mathbf{m}_{1}\right)\mathit{++}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{32}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathbf{n}_{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{m}_{2} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\left(\mathbf{n}_{4}\right)\mathit{++} \mathit{=} \left(\mathbf{m}_{2}\right)\mathit{++}\right)\right) \mathit{\implies} \left(\mathbf{n}_{4} \mathit{=} \mathbf{m}_{2}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left\ulcorner\text{\sffamily{\itshape{Different natural numbers must have different successors; i.e., if n, m are natural numbers and n m, then n++ m++. Equivalently, if n++ = m++, then we must have n = m.}}}\right\ulcorner\mathsf{ is postulated by }\boldsymbol\mathsf{axiom}} \boldsymbol\mathsf{2.4}} (\mathit{A}_{4})\mathsf{. }\left(\left(\left(\left(\mathbf{n}_{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{m}_{2} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\left(\mathbf{n}_{4}\right)\mathit{++} \mathit{=} \left(\mathbf{m}_{2}\right)\mathit{++}\right)\right) \mathit{\implies} \left(\mathbf{n}_{4} \mathit{=} \mathbf{m}_{2}\right)\right)\mathsf{ is a valid formula statement interpreted from that axiom.}\mathsf{ Therefore, by the }\mathit{axiom-interpretation}\mathsf{ inference rule, it follows that }\left(\left(\left(\left(\mathbf{n}_{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{m}_{2} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\left(\mathbf{n}_{4}\right)\mathit{++} \mathit{=} \left(\mathbf{m}_{2}\right)\mathit{++}\right)\right) \mathit{\implies} \left(\mathbf{n}_{4} \mathit{=} \mathbf{m}_{2}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}### \boldsymbol\mathsf{6 is not equal to 2.}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{33}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{4} \mathit{\neq} \mathit{0}\right)\right) \mathit{\implies} \left(\left(\mathit{4}\right)\mathit{++} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\mathbf{n}_{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{m}_{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathbf{n}_{3} \mathit{\neq} \mathbf{m}_{1}\right)\right) \mathit{\implies} \left(\left(\mathbf{n}_{3}\right)\mathit{++} \mathit{\neq} \left(\mathbf{m}_{1}\right)\mathit{++}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{31})\mathsf{.}\mathsf{ Let }\mathbf{n}_{3} = 4\mathsf{, }\mathbf{m}_{1} = 0\mathsf{.}\mathsf{ Therefore, by the }\mathit{variable-substitution}\mathsf{ inference rule, it follows that }\left(\left(\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{4} \mathit{\neq} \mathit{0}\right)\right) \mathit{\implies} \left(\left(\mathit{4}\right)\mathit{++} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{conjunction-introduction}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{conjunction-introduction}\mathsf{ defined as }\left\ulcorner(P, Q) โŠข (P โ‹€ Q)\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{34}\mathsf{)}\mathsf{: }\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{26})\mathsf{. }\left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{1})\mathsf{.}\mathsf{ Therefore, by the }\mathit{conjunction-introduction}\mathsf{ inference rule, it follows that }\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{35}\mathsf{)}\mathsf{: }\left(\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{4} \mathit{\neq} \mathit{0}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{34})\mathsf{. }\left(\mathit{4} \mathit{\neq} \mathit{0}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} \boldsymbol\mathsf{2.1.6}} (\mathit{P}_{30})\mathsf{.}\mathsf{ Therefore, by the }\mathit{conjunction-introduction}\mathsf{ inference rule, it follows that }\left(\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{4} \mathit{\neq} \mathit{0}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{36}\mathsf{)}\mathsf{: }\left(\left(\mathit{4}\right)\mathit{++} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{4} \mathit{\neq} \mathit{0}\right)\right) \mathit{\implies} \left(\left(\mathit{4}\right)\mathit{++} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{33})\mathsf{.}\left(\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{4} \mathit{\neq} \mathit{0}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{35})\mathsf{.}\mathsf{ Therefore, by the }\mathit{modus-ponens}\mathsf{ inference rule, it follows that }\left(\left(\mathit{4}\right)\mathit{++} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{37}\mathsf{)}\mathsf{: }\left(\mathit{5} \mathit{=} \left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left\ulcorner\text{\sffamily{\itshape{We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)}}}\right\ulcorner\mathsf{ is postulated by }\boldsymbol\mathsf{def.}} (\mathit{D}_{1})\mathsf{. }\left(\mathit{5} \mathit{=} \left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{ is an interpretation of that definition.}\mathsf{ Therefore, by the }\mathit{definition-interpretation}\mathsf{ inference rule, it follows that }\left(\mathit{5} \mathit{=} \left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{38}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{5}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\mathit{5} \mathit{=} \left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{37})\mathsf{. }\mathsf{ Therefore, by the }\mathit{equality-commutativity}\mathsf{ inference rule, it follows that }\left(\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{5}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{39}\mathsf{)}\mathsf{: }\left(\left(\mathit{4}\right)\mathit{++} \mathit{=} \mathit{5}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{5}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{38})\mathsf{. }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{4}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{23})\mathsf{.}\mathsf{ Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule, it follows that }\left(\left(\mathit{4}\right)\mathit{++} \mathit{=} \mathit{5}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{40}\mathsf{)}\mathsf{: }\left(\mathit{5} \mathit{=} \left(\mathit{4}\right)\mathit{++}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathit{4}\right)\mathit{++} \mathit{=} \mathit{5}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{39})\mathsf{. }\mathsf{ Therefore, by the }\mathit{equality-commutativity}\mathsf{ inference rule, it follows that }\left(\mathit{5} \mathit{=} \left(\mathit{4}\right)\mathit{++}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{41}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\left(\mathit{5}\right)\mathit{++} \mathit{\neq} \left(\mathit{1}\right)\mathit{++}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\mathbf{n}_{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{m}_{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathbf{n}_{3} \mathit{\neq} \mathbf{m}_{1}\right)\right) \mathit{\implies} \left(\left(\mathbf{n}_{3}\right)\mathit{++} \mathit{\neq} \left(\mathbf{m}_{1}\right)\mathit{++}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{31})\mathsf{.}\mathsf{ Let }\mathbf{n}_{3} = 5\mathsf{, }\mathbf{m}_{1} = 1\mathsf{.}\mathsf{ Therefore, by the }\mathit{variable-substitution}\mathsf{ inference rule, it follows that }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\left(\mathit{5}\right)\mathit{++} \mathit{\neq} \left(\mathit{1}\right)\mathit{++}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{42}\mathsf{)}\mathsf{: }\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathit{4}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathbf{n}_{1} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathbf{n}_{1}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{2})\mathsf{.}\mathsf{ Let }\mathbf{n}_{1} = 4\mathsf{.}\mathsf{ Therefore, by the }\mathit{variable-substitution}\mathsf{ inference rule, it follows that }\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathit{4}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{43}\mathsf{)}\mathsf{: }\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathit{4}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{42})\mathsf{. }\left(\left(\mathit{4}\right)\mathit{++} \mathit{=} \mathit{5}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{39})\mathsf{.}\mathsf{ Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule, it follows that }\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{44}\mathsf{)}\mathsf{: }\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{43})\mathsf{.}\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{26})\mathsf{.}\mathsf{ Therefore, by the }\mathit{modus-ponens}\mathsf{ inference rule, it follows that }\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{45}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\left(\mathit{5}\right)\mathit{++} \mathit{\neq} \left(\mathit{1}\right)\mathit{++}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\mathbf{n}_{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{m}_{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathbf{n}_{3} \mathit{\neq} \mathbf{m}_{1}\right)\right) \mathit{\implies} \left(\left(\mathbf{n}_{3}\right)\mathit{++} \mathit{\neq} \left(\mathbf{m}_{1}\right)\mathit{++}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{31})\mathsf{.}\mathsf{ Let }\mathbf{n}_{3} = 5\mathsf{, }\mathbf{m}_{1} = 1\mathsf{.}\mathsf{ Therefore, by the }\mathit{variable-substitution}\mathsf{ inference rule, it follows that }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\left(\mathit{5}\right)\mathit{++} \mathit{\neq} \left(\mathit{1}\right)\mathit{++}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{46}\mathsf{)}\mathsf{: }\left(\left(\mathit{4}\right)\mathit{++} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{4} \mathit{\neq} \mathit{0}\right)\right) \mathit{\implies} \left(\left(\mathit{4}\right)\mathit{++} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{33})\mathsf{.}\left(\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{4} \mathit{\neq} \mathit{0}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{35})\mathsf{.}\mathsf{ Therefore, by the }\mathit{modus-ponens}\mathsf{ inference rule, it follows that }\left(\left(\mathit{4}\right)\mathit{++} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{47}\mathsf{)}\mathsf{: }\left(\mathit{5} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathit{4}\right)\mathit{++} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{46})\mathsf{. }\left(\left(\mathit{4}\right)\mathit{++} \mathit{=} \mathit{5}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{39})\mathsf{.}\mathsf{ Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule, it follows that }\left(\mathit{5} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{48}\mathsf{)}\mathsf{: }\left(\mathit{6} \mathit{=} \left(\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left\ulcorner\text{\sffamily{\itshape{We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)}}}\right\ulcorner\mathsf{ is postulated by }\boldsymbol\mathsf{def.}} (\mathit{D}_{1})\mathsf{. }\left(\mathit{6} \mathit{=} \left(\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{ is an interpretation of that definition.}\mathsf{ Therefore, by the }\mathit{definition-interpretation}\mathsf{ inference rule, it follows that }\left(\mathit{6} \mathit{=} \left(\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{49}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{6}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\mathit{6} \mathit{=} \left(\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{48})\mathsf{. }\mathsf{ Therefore, by the }\mathit{equality-commutativity}\mathsf{ inference rule, it follows that }\left(\left(\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{6}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{50}\mathsf{)}\mathsf{: }\left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathit{0}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} \boldsymbol\mathsf{2.2.3}} (\mathit{P}_{4})\mathsf{. }\left(\left(\mathit{0}\right)\mathit{++} \mathit{=} \mathit{1}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{15})\mathsf{.}\mathsf{ Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule, it follows that }\left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{51}\mathsf{)}\mathsf{: }\left(\left(\mathit{5}\right)\mathit{++} \mathit{=} \mathit{6}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{6}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{49})\mathsf{. }\left(\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{5}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{38})\mathsf{.}\mathsf{ Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule, it follows that }\left(\left(\mathit{5}\right)\mathit{++} \mathit{=} \mathit{6}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{52}\mathsf{)}\mathsf{: }\left(\mathit{6} \mathit{=} \left(\mathit{5}\right)\mathit{++}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathit{5}\right)\mathit{++} \mathit{=} \mathit{6}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{51})\mathsf{. }\mathsf{ Therefore, by the }\mathit{equality-commutativity}\mathsf{ inference rule, it follows that }\left(\mathit{6} \mathit{=} \left(\mathit{5}\right)\mathit{++}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}#### \boldsymbol\mathsf{Proof by contradiction}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Hypothesis}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{H}_{1}\mathsf{)}\mathsf{: }\left(\mathit{6} \mathit{=} \mathit{2}\right)\mathsf{.}\mathsf{ This hypothesis is elaborated in theory }\mathcal{H}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{inconsistency-by-inequality-introduction}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{inconsistency-by-inequality-introduction}\mathsf{ defined as }\left\ulcorner((P = Q), (P โ‰  Q)) โŠข Inc(๐’ฏ)\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{66}\mathsf{)}\mathsf{: }\mathit{Inc}\left(\mathcal{H}_{1}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\mathit{P}_{65}\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{65})\mathsf{. }\mathit{P}_{30}\mathsf{ follows from }\boldsymbol\mathsf{prop.}} \boldsymbol\mathsf{2.1.6}} (\mathit{P}_{30})\mathsf{. }\mathsf{ Therefore, by the }\mathit{inconsistency-by-inequality-introduction}\mathsf{ inference rule, it follows that }\mathit{Inc}\left(\mathcal{H}_{1}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{proof-by-refutation-of-equality}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{proof-by-refutation-of-equality}\mathsf{ defined as }\left\ulcorner(โ„‹(P=Q), Inc(โ„‹)) โŠข (P โ‰  Q)\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}} \boldsymbol\mathsf{2.1.8}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{67}\mathsf{)}\mathsf{: }\left(\mathit{6} \mathit{\neq} \mathit{2}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\mathsf{Let }\boldsymbol\mathsf{hyp.}} (\mathit{H}_{1})\mathsf{ be the hypothesis }\left(\mathit{6} \mathit{=} \mathit{2}\right)\mathsf{. }\mathit{Inc}\left(\mathcal{H}_{1}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{66})\mathsf{.}\mathsf{ Therefore, by the }\mathit{proof-by-refutation-of-equality}\mathsf{ inference rule, it follows that }\left(\mathit{6} \mathit{\neq} \mathit{2}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}#### \boldsymbol\mathsf{Direct proof}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{68}\mathsf{)}\mathsf{: }\left(\left(\mathit{1}\right)\mathit{++} \mathit{=} \mathit{2}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{2}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{17})\mathsf{. }\left(\left(\mathit{0}\right)\mathit{++} \mathit{=} \mathit{1}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{15})\mathsf{.}\mathsf{ Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule, it follows that }\left(\left(\mathit{1}\right)\mathit{++} \mathit{=} \mathit{2}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{69}\mathsf{)}\mathsf{: }\left(\mathit{5} \mathit{\neq} \mathit{1}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\mathit{5} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{47})\mathsf{. }\left(\left(\mathit{0}\right)\mathit{++} \mathit{=} \mathit{1}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{15})\mathsf{.}\mathsf{ Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule, it follows that }\left(\mathit{5} \mathit{\neq} \mathit{1}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{70}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\mathit{6} \mathit{\neq} \left(\mathit{1}\right)\mathit{++}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\left(\mathit{5}\right)\mathit{++} \mathit{\neq} \left(\mathit{1}\right)\mathit{++}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{45})\mathsf{. }\left(\left(\mathit{5}\right)\mathit{++} \mathit{=} \mathit{6}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{51})\mathsf{.}\mathsf{ Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule, it follows that }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\mathit{6} \mathit{\neq} \left(\mathit{1}\right)\mathit{++}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{71}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\mathit{6} \mathit{\neq} \mathit{2}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\mathit{6} \mathit{\neq} \left(\mathit{1}\right)\mathit{++}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{70})\mathsf{. }\left(\left(\mathit{1}\right)\mathit{++} \mathit{=} \mathit{2}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{68})\mathsf{.}\mathsf{ Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule, it follows that }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\mathit{6} \mathit{\neq} \mathit{2}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{72}\mathsf{)}\mathsf{: }\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{44})\mathsf{. }\left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{50})\mathsf{.}\mathsf{ Therefore, by the }\mathit{conjunction-introduction}\mathsf{ inference rule, it follows that }\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{73}\mathsf{)}\mathsf{: }\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{72})\mathsf{. }\left(\mathit{5} \mathit{\neq} \mathit{1}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{69})\mathsf{.}\mathsf{ Therefore, by the }\mathit{conjunction-introduction}\mathsf{ inference rule, it follows that }\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{74}\mathsf{)}\mathsf{: }\left(\mathit{6} \mathit{\neq} \mathit{2}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\mathit{6} \mathit{\neq} \mathit{2}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{71})\mathsf{.}\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{73})\mathsf{.}\mathsf{ Therefore, by the }\mathit{modus-ponens}\mathsf{ inference rule, it follows that }\left(\mathit{6} \mathit{\neq} \mathit{2}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}### \boldsymbol\mathsf{Axiom 2.5: the principle of mathematical induction}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Axiom schema}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{A}_{6}\mathsf{)} - Principle of mathematical induction\mathsf{: Let }\mathit{axiom}\mathsf{ }\mathcal{A}_{6}\mathsf{ }\left\ulcorner\text{\sffamily{\itshape{Let P(n) be any property pertaining to a natural number n. Suppose that P(O) is true, and suppose that whenever P(n) is true, P(n++) is also true. Then P(n) is true for every natural number n.}}}\right\ulcorner\mathsf{ be included (postulated) in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{75}\mathsf{)}\mathsf{: }\left(\left(\left(\mathbf{n}_{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{P}_{1}\left(\mathit{0}\right) \mathit{\land} \left(\mathbf{P}_{1}\left(\mathbf{n}_{5}\right) \mathit{\implies} \mathbf{P}_{1}\left(\left(\mathbf{n}_{5}\right)\mathit{++}\right)\right)\right)\right) \mathit{\implies} \left(\left(\mathbf{m}_{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \mathbf{P}_{1}\left(\mathbf{m}_{3}\right)\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left\ulcorner\text{\sffamily{\itshape{Let P(n) be any property pertaining to a natural number n. Suppose that P(O) is true, and suppose that whenever P(n) is true, P(n++) is also true. Then P(n) is true for every natural number n.}}}\right\ulcorner\mathsf{ is postulated by }\boldsymbol\mathsf{axiom schema}} (\mathit{A}_{6})\mathsf{. }\left(\left(\left(\mathbf{n}_{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{P}_{1}\left(\mathit{0}\right) \mathit{\land} \left(\mathbf{P}_{1}\left(\mathbf{n}_{5}\right) \mathit{\implies} \mathbf{P}_{1}\left(\left(\mathbf{n}_{5}\right)\mathit{++}\right)\right)\right)\right) \mathit{\implies} \left(\left(\mathbf{m}_{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \mathbf{P}_{1}\left(\mathbf{m}_{3}\right)\right)\right)\mathsf{ is a valid formula statement interpreted from that axiom.}\mathsf{ Therefore, by the }\mathit{axiom-interpretation}\mathsf{ inference rule, it follows that }\left(\left(\left(\mathbf{n}_{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{P}_{1}\left(\mathit{0}\right) \mathit{\land} \left(\mathbf{P}_{1}\left(\mathbf{n}_{5}\right) \mathit{\implies} \mathbf{P}_{1}\left(\left(\mathbf{n}_{5}\right)\mathit{++}\right)\right)\right)\right) \mathit{\implies} \left(\left(\mathbf{m}_{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \mathbf{P}_{1}\left(\mathbf{m}_{3}\right)\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}### \boldsymbol\mathsf{The number system n}}\mathsf{}\mathsf{}### \boldsymbol\mathsf{Recursive definitions}}\mathsf{} \ No newline at end of file +\mathsf{}\mathsf{}\section{\boldsymbol\mathsf{Theory properties}}}\mathsf{}\boldsymbol\mathsf{Consistency: }}undetermined\mathsf{}\mathsf{}\boldsymbol\mathsf{Stabilized: }}False\mathsf{}\mathsf{}\boldsymbol\mathsf{Extended theory: }}N/A\mathsf{}\section{\boldsymbol\mathsf{Simple-objects declarations}}}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{0}\right\ulcorner, \left\ulcorner\mathit{natural-number}\right\ulcorner, \left\ulcorner\mathit{1}\right\ulcorner, \left\ulcorner\mathit{2}\right\ulcorner, \left\ulcorner\mathit{3}\right\ulcorner, \left\ulcorner\mathit{4}\right\ulcorner, \left\ulcorner\mathit{5}\right\ulcorner, \left\ulcorner\mathit{6}\right\ulcorner\mathsf{ be }\mathit{simple-objects}\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\section{\boldsymbol\mathsf{Relations}}}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{++}\right\ulcorner\mathsf{, }\left\ulcorner\mathit{Inc}\right\ulcorner\mathsf{ be }\mathit{unary}\mathit{-relations}\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{\land}\right\ulcorner\mathsf{, }\left\ulcorner\mathit{\neq}\right\ulcorner\mathsf{, }\left\ulcorner\mathit{is-a}\right\ulcorner\mathsf{, }\left\ulcorner\mathit{=}\right\ulcorner\mathsf{, }\left\ulcorner\mathit{\implies}\right\ulcorner\mathsf{ be }\mathit{binary}\mathit{-relations}\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\section{\boldsymbol\mathsf{Inference rules}}}\mathsf{}\mathsf{The following inference rules are considered valid under this theory:}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{axiom-interpretation}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner๐’œ โŠข P\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{conjunction-introduction}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner(P, Q) โŠข (P โ‹€ Q)\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{definition-interpretation}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner๐’Ÿ โŠข P\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{equal-terms-substitution}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner(P, (Q = R)) โŠข P'\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{equality-commutativity}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner(x = y) โŠข (y = x)\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{inconsistency-by-inequality-introduction}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner((P = Q), (P โ‰  Q)) โŠข Inc(๐’ฏ)\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{modus-ponens}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner((P โŸน Q), P) โŠข Q\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{proof-by-refutation-of-equality}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner(โ„‹(P=Q), Inc(โ„‹)) โŠข (P โ‰  Q)\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\mathsf{}\mathsf{Let }\left\ulcorner\mathit{variable-substitution}\right\ulcorner\mathsf{ be an }\mathit{inference-rule}\mathsf{ defined as }\left\ulcorner(P, ๐›ท) โŠข P'\right\ulcorner\mathsf{ in }\mathcal{U}_{1}\mathsf{.}\mathsf{}\section{\boldsymbol\mathsf{Theory elaboration sequence}}}\mathsf{}# \boldsymbol\mathsf{2}}: \boldsymbol\mathsf{The natural numbers}}\mathsf{}\mathsf{}## \boldsymbol\mathsf{2.1}}: \boldsymbol\mathsf{The peano axioms}}\mathsf{}\mathsf{}### \boldsymbol\mathsf{Informal definition of natural number}}\mathsf{}\mathsf{}### \boldsymbol\mathsf{Axiom 2.1}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Axiom}} \boldsymbol\mathsf{2.1}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{A}_{1}\mathsf{)}\mathsf{: Let }\mathit{axiom}\mathsf{ }\mathcal{A}_{1}\mathsf{ }\left\ulcorner\text{\sffamily{\itshape{0 is a natural number.}}}\right\ulcorner\mathsf{ be included (postulated) in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{axiom-interpretation}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{axiom-interpretation}\mathsf{ defined as }\left\ulcorner๐’œ โŠข P\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{1}\mathsf{)}\mathsf{: }\left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left\ulcorner\text{\sffamily{\itshape{0 is a natural number.}}}\right\ulcorner\mathsf{ is postulated by }\boldsymbol\mathsf{axiom}} \boldsymbol\mathsf{2.1}} (\mathit{A}_{1})\mathsf{. }\left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\mathsf{ is a valid formula statement interpreted from that axiom.}\mathsf{Therefore, by the }\mathit{axiom-interpretation}\mathsf{ inference rule: }๐’œ โŠข P\mathsf{, it follows that }\left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}### \boldsymbol\mathsf{Axiom 2.2}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Axiom}} \boldsymbol\mathsf{2.2}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{A}_{2}\mathsf{)}\mathsf{: Let }\mathit{axiom}\mathsf{ }\mathcal{A}_{2}\mathsf{ }\left\ulcorner\text{\sffamily{\itshape{If n is a natural number, then n++ is a natural number.}}}\right\ulcorner\mathsf{ be included (postulated) in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{2}\mathsf{)}\mathsf{: }\left(\left(\mathbf{n}_{1} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathbf{n}_{1}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left\ulcorner\text{\sffamily{\itshape{If n is a natural number, then n++ is a natural number.}}}\right\ulcorner\mathsf{ is postulated by }\boldsymbol\mathsf{axiom}} \boldsymbol\mathsf{2.2}} (\mathit{A}_{2})\mathsf{. }\left(\left(\mathbf{n}_{1} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathbf{n}_{1}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ is a valid formula statement interpreted from that axiom.}\mathsf{Therefore, by the }\mathit{axiom-interpretation}\mathsf{ inference rule: }๐’œ โŠข P\mathsf{, it follows that }\left(\left(\mathbf{n}_{1} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathbf{n}_{1}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{variable-substitution}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{variable-substitution}\mathsf{ defined as }\left\ulcorner(P, ๐›ท) โŠข P'\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{3}\mathsf{)}\mathsf{: }\left(\left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathit{0}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathbf{n}_{1} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathbf{n}_{1}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{2})\mathsf{.}\mathsf{ Let }\mathbf{n}_{1} = 0\mathsf{.}\mathsf{Therefore, by the }\mathit{variable-substitution}\mathsf{ inference rule: }(P, ๐›ท) โŠข P'\mathsf{, it follows that }\left(\left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathit{0}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{modus-ponens}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{modus-ponens}\mathsf{ defined as }\left\ulcorner((P โŸน Q), P) โŠข Q\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}} \boldsymbol\mathsf{2.2.3}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{4}\mathsf{)}\mathsf{: }\left(\left(\mathit{0}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathit{0}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{3})\mathsf{.}\left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{1})\mathsf{.}\mathsf{Therefore, by the }\mathit{modus-ponens}\mathsf{ inference rule: }((P โŸน Q), P) โŠข Q\mathsf{, it follows that }\left(\left(\mathit{0}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Definition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{D}_{1}\mathsf{)}\mathsf{: Let }\mathit{definition}\mathsf{ }\mathcal{D}_{1}\mathsf{ }\left\ulcorner\text{\sffamily{\itshape{We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)}}}\right\ulcorner\mathsf{ be included (postulated) in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{definition-interpretation}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{definition-interpretation}\mathsf{ defined as }\left\ulcorner๐’Ÿ โŠข P\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{5}\mathsf{)}\mathsf{: }\left(\mathit{1} \mathit{=} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left\ulcorner\text{\sffamily{\itshape{We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)}}}\right\ulcorner\mathsf{ is postulated by }\boldsymbol\mathsf{def.}} (\mathit{D}_{1})\mathsf{. }\left(\mathit{1} \mathit{=} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{ is an interpretation of that definition.}\mathsf{Therefore, by the }\mathit{definition-interpretation}\mathsf{ inference rule: }๐’Ÿ โŠข P\mathsf{, it follows that }\left(\mathit{1} \mathit{=} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{6}\mathsf{)}\mathsf{: }\left(\mathit{2} \mathit{=} \left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left\ulcorner\text{\sffamily{\itshape{We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)}}}\right\ulcorner\mathsf{ is postulated by }\boldsymbol\mathsf{def.}} (\mathit{D}_{1})\mathsf{. }\left(\mathit{2} \mathit{=} \left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{ is an interpretation of that definition.}\mathsf{Therefore, by the }\mathit{definition-interpretation}\mathsf{ inference rule: }๐’Ÿ โŠข P\mathsf{, it follows that }\left(\mathit{2} \mathit{=} \left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{7}\mathsf{)}\mathsf{: }\left(\mathit{3} \mathit{=} \left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left\ulcorner\text{\sffamily{\itshape{We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)}}}\right\ulcorner\mathsf{ is postulated by }\boldsymbol\mathsf{def.}} (\mathit{D}_{1})\mathsf{. }\left(\mathit{3} \mathit{=} \left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{ is an interpretation of that definition.}\mathsf{Therefore, by the }\mathit{definition-interpretation}\mathsf{ inference rule: }๐’Ÿ โŠข P\mathsf{, it follows that }\left(\mathit{3} \mathit{=} \left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{8}\mathsf{)}\mathsf{: }\left(\mathit{4} \mathit{=} \left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left\ulcorner\text{\sffamily{\itshape{We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)}}}\right\ulcorner\mathsf{ is postulated by }\boldsymbol\mathsf{def.}} (\mathit{D}_{1})\mathsf{. }\left(\mathit{4} \mathit{=} \left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{ is an interpretation of that definition.}\mathsf{Therefore, by the }\mathit{definition-interpretation}\mathsf{ inference rule: }๐’Ÿ โŠข P\mathsf{, it follows that }\left(\mathit{4} \mathit{=} \left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{9}\mathsf{)}\mathsf{: }\left(\left(\left(\mathit{0}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathbf{n}_{1} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathbf{n}_{1}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{2})\mathsf{.}\mathsf{ Let }\mathbf{n}_{1} = (0)++\mathsf{.}\mathsf{Therefore, by the }\mathit{variable-substitution}\mathsf{ inference rule: }(P, ๐›ท) โŠข P'\mathsf{, it follows that }\left(\left(\left(\mathit{0}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{10}\mathsf{)}\mathsf{: }\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\mathit{0}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{9})\mathsf{.}\left(\left(\mathit{0}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} \boldsymbol\mathsf{2.2.3}} (\mathit{P}_{4})\mathsf{.}\mathsf{Therefore, by the }\mathit{modus-ponens}\mathsf{ inference rule: }((P โŸน Q), P) โŠข Q\mathsf{, it follows that }\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{11}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathbf{n}_{1} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathbf{n}_{1}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{2})\mathsf{.}\mathsf{ Let }\mathbf{n}_{1} = ((0)++)++\mathsf{.}\mathsf{Therefore, by the }\mathit{variable-substitution}\mathsf{ inference rule: }(P, ๐›ท) โŠข P'\mathsf{, it follows that }\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{12}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{11})\mathsf{.}\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{10})\mathsf{.}\mathsf{Therefore, by the }\mathit{modus-ponens}\mathsf{ inference rule: }((P โŸน Q), P) โŠข Q\mathsf{, it follows that }\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{13}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathbf{n}_{1} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathbf{n}_{1}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{2})\mathsf{.}\mathsf{ Let }\mathbf{n}_{1} = (((0)++)++)++\mathsf{.}\mathsf{Therefore, by the }\mathit{variable-substitution}\mathsf{ inference rule: }(P, ๐›ท) โŠข P'\mathsf{, it follows that }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{14}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{13})\mathsf{.}\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{12})\mathsf{.}\mathsf{Therefore, by the }\mathit{modus-ponens}\mathsf{ inference rule: }((P โŸน Q), P) โŠข Q\mathsf{, it follows that }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{equality-commutativity}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{equality-commutativity}\mathsf{ defined as }\left\ulcorner(x = y) โŠข (y = x)\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{15}\mathsf{)}\mathsf{: }\left(\left(\mathit{0}\right)\mathit{++} \mathit{=} \mathit{1}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\mathit{1} \mathit{=} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{5})\mathsf{. }\mathsf{Therefore, by the }\mathit{equality-commutativity}\mathsf{ inference rule: }(x = y) โŠข (y = x)\mathsf{, it follows that }\left(\left(\mathit{0}\right)\mathit{++} \mathit{=} \mathit{1}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{equal-terms-substitution}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{equal-terms-substitution}\mathsf{ defined as }\left\ulcorner(P, (Q = R)) โŠข P'\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{16}\mathsf{)}\mathsf{: }\left(\mathit{2} \mathit{=} \left(\mathit{1}\right)\mathit{++}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\mathit{2} \mathit{=} \left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{6})\mathsf{. }\left(\left(\mathit{0}\right)\mathit{++} \mathit{=} \mathit{1}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{15})\mathsf{.}\mathsf{Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule: }(P, (Q = R)) โŠข P'\mathsf{, it follows that }\left(\mathit{2} \mathit{=} \left(\mathit{1}\right)\mathit{++}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{17}\mathsf{)}\mathsf{: }\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{2}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\mathit{2} \mathit{=} \left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{6})\mathsf{. }\mathsf{Therefore, by the }\mathit{equality-commutativity}\mathsf{ inference rule: }(x = y) โŠข (y = x)\mathsf{, it follows that }\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{2}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{18}\mathsf{)}\mathsf{: }\left(\mathit{3} \mathit{=} \left(\mathit{2}\right)\mathit{++}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\mathit{3} \mathit{=} \left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{7})\mathsf{. }\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{2}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{17})\mathsf{.}\mathsf{Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule: }(P, (Q = R)) โŠข P'\mathsf{, it follows that }\left(\mathit{3} \mathit{=} \left(\mathit{2}\right)\mathit{++}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}### \boldsymbol\mathsf{3 is a natural number}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{19}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{3}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\mathit{3} \mathit{=} \left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{7})\mathsf{. }\mathsf{Therefore, by the }\mathit{equality-commutativity}\mathsf{ inference rule: }(x = y) โŠข (y = x)\mathsf{, it follows that }\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{3}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{20}\mathsf{)}\mathsf{: }\left(\left(\mathit{2}\right)\mathit{++} \mathit{=} \mathit{3}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{3}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{19})\mathsf{. }\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{2}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{17})\mathsf{.}\mathsf{Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule: }(P, (Q = R)) โŠข P'\mathsf{, it follows that }\left(\left(\mathit{2}\right)\mathit{++} \mathit{=} \mathit{3}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}} \boldsymbol\mathsf{2.1.4}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{21}\mathsf{)}\mathsf{: }\left(\mathit{3} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{12})\mathsf{. }\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{3}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{19})\mathsf{.}\mathsf{Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule: }(P, (Q = R)) โŠข P'\mathsf{, it follows that }\left(\mathit{3} \mathit{is-a} \mathit{natural-number}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{22}\mathsf{)}\mathsf{: }\left(\mathit{4} \mathit{=} \left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left\ulcorner\text{\sffamily{\itshape{We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)}}}\right\ulcorner\mathsf{ is postulated by }\boldsymbol\mathsf{def.}} (\mathit{D}_{1})\mathsf{. }\left(\mathit{4} \mathit{=} \left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{ is an interpretation of that definition.}\mathsf{Therefore, by the }\mathit{definition-interpretation}\mathsf{ inference rule: }๐’Ÿ โŠข P\mathsf{, it follows that }\left(\mathit{4} \mathit{=} \left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{23}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{4}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\mathit{4} \mathit{=} \left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{8})\mathsf{. }\mathsf{Therefore, by the }\mathit{equality-commutativity}\mathsf{ inference rule: }(x = y) โŠข (y = x)\mathsf{, it follows that }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{4}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{24}\mathsf{)}\mathsf{: }\left(\left(\mathit{3}\right)\mathit{++} \mathit{=} \mathit{4}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{4}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{23})\mathsf{. }\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{3}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{19})\mathsf{.}\mathsf{Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule: }(P, (Q = R)) โŠข P'\mathsf{, it follows that }\left(\left(\mathit{3}\right)\mathit{++} \mathit{=} \mathit{4}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{25}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{13})\mathsf{. }\left(\left(\mathit{3}\right)\mathit{++} \mathit{=} \mathit{4}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{24})\mathsf{.}\mathsf{Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule: }(P, (Q = R)) โŠข P'\mathsf{, it follows that }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{26}\mathsf{)}\mathsf{: }\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{14})\mathsf{. }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{4}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{23})\mathsf{.}\mathsf{Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule: }(P, (Q = R)) โŠข P'\mathsf{, it follows that }\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}### \boldsymbol\mathsf{Axiom 2.3}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Axiom}} \boldsymbol\mathsf{2.3}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{A}_{3}\mathsf{)}\mathsf{: Let }\mathit{axiom}\mathsf{ }\mathcal{A}_{3}\mathsf{ }\left\ulcorner\text{\sffamily{\itshape{0 is not the successor of any natural number; i.e., we have n++ 0 for every natural number n.}}}\right\ulcorner\mathsf{ be included (postulated) in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{27}\mathsf{)}\mathsf{: }\left(\left(\mathbf{n}_{2} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathbf{n}_{2}\right)\mathit{++} \mathit{\neq} \mathit{0}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left\ulcorner\text{\sffamily{\itshape{0 is not the successor of any natural number; i.e., we have n++ 0 for every natural number n.}}}\right\ulcorner\mathsf{ is postulated by }\boldsymbol\mathsf{axiom}} \boldsymbol\mathsf{2.3}} (\mathit{A}_{3})\mathsf{. }\left(\left(\mathbf{n}_{2} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathbf{n}_{2}\right)\mathit{++} \mathit{\neq} \mathit{0}\right)\right)\mathsf{ is a valid formula statement interpreted from that axiom.}\mathsf{Therefore, by the }\mathit{axiom-interpretation}\mathsf{ inference rule: }๐’œ โŠข P\mathsf{, it follows that }\left(\left(\mathbf{n}_{2} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathbf{n}_{2}\right)\mathit{++} \mathit{\neq} \mathit{0}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}### \boldsymbol\mathsf{4 is not equal to 0.}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{28}\mathsf{)}\mathsf{: }\left(\left(\mathit{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathit{3}\right)\mathit{++} \mathit{\neq} \mathit{0}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathbf{n}_{2} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathbf{n}_{2}\right)\mathit{++} \mathit{\neq} \mathit{0}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{27})\mathsf{.}\mathsf{ Let }\mathbf{n}_{2} = 3\mathsf{.}\mathsf{Therefore, by the }\mathit{variable-substitution}\mathsf{ inference rule: }(P, ๐›ท) โŠข P'\mathsf{, it follows that }\left(\left(\mathit{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathit{3}\right)\mathit{++} \mathit{\neq} \mathit{0}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{29}\mathsf{)}\mathsf{: }\left(\left(\mathit{3}\right)\mathit{++} \mathit{\neq} \mathit{0}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathit{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathit{3}\right)\mathit{++} \mathit{\neq} \mathit{0}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{28})\mathsf{.}\left(\mathit{3} \mathit{is-a} \mathit{natural-number}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} \boldsymbol\mathsf{2.1.4}} (\mathit{P}_{21})\mathsf{.}\mathsf{Therefore, by the }\mathit{modus-ponens}\mathsf{ inference rule: }((P โŸน Q), P) โŠข Q\mathsf{, it follows that }\left(\left(\mathit{3}\right)\mathit{++} \mathit{\neq} \mathit{0}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}} \boldsymbol\mathsf{2.1.6}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{30}\mathsf{)}\mathsf{: }\left(\mathit{4} \mathit{\neq} \mathit{0}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathit{3}\right)\mathit{++} \mathit{\neq} \mathit{0}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{29})\mathsf{. }\left(\left(\mathit{3}\right)\mathit{++} \mathit{=} \mathit{4}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{24})\mathsf{.}\mathsf{Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule: }(P, (Q = R)) โŠข P'\mathsf{, it follows that }\left(\mathit{4} \mathit{\neq} \mathit{0}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}### \boldsymbol\mathsf{Axiom 2.4}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Axiom}} \boldsymbol\mathsf{2.4}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{A}_{4}\mathsf{)}\mathsf{: Let }\mathit{axiom}\mathsf{ }\mathcal{A}_{4}\mathsf{ }\left\ulcorner\text{\sffamily{\itshape{Different natural numbers must have different successors; i.e., if n, m are natural numbers and n m, then n++ m++. Equivalently, if n++ = m++, then we must have n = m.}}}\right\ulcorner\mathsf{ be included (postulated) in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{31}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathbf{n}_{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{m}_{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathbf{n}_{3} \mathit{\neq} \mathbf{m}_{1}\right)\right) \mathit{\implies} \left(\left(\mathbf{n}_{3}\right)\mathit{++} \mathit{\neq} \left(\mathbf{m}_{1}\right)\mathit{++}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left\ulcorner\text{\sffamily{\itshape{Different natural numbers must have different successors; i.e., if n, m are natural numbers and n m, then n++ m++. Equivalently, if n++ = m++, then we must have n = m.}}}\right\ulcorner\mathsf{ is postulated by }\boldsymbol\mathsf{axiom}} \boldsymbol\mathsf{2.4}} (\mathit{A}_{4})\mathsf{. }\left(\left(\left(\left(\mathbf{n}_{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{m}_{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathbf{n}_{3} \mathit{\neq} \mathbf{m}_{1}\right)\right) \mathit{\implies} \left(\left(\mathbf{n}_{3}\right)\mathit{++} \mathit{\neq} \left(\mathbf{m}_{1}\right)\mathit{++}\right)\right)\mathsf{ is a valid formula statement interpreted from that axiom.}\mathsf{Therefore, by the }\mathit{axiom-interpretation}\mathsf{ inference rule: }๐’œ โŠข P\mathsf{, it follows that }\left(\left(\left(\left(\mathbf{n}_{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{m}_{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathbf{n}_{3} \mathit{\neq} \mathbf{m}_{1}\right)\right) \mathit{\implies} \left(\left(\mathbf{n}_{3}\right)\mathit{++} \mathit{\neq} \left(\mathbf{m}_{1}\right)\mathit{++}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{32}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathbf{n}_{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{m}_{2} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\left(\mathbf{n}_{4}\right)\mathit{++} \mathit{=} \left(\mathbf{m}_{2}\right)\mathit{++}\right)\right) \mathit{\implies} \left(\mathbf{n}_{4} \mathit{=} \mathbf{m}_{2}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left\ulcorner\text{\sffamily{\itshape{Different natural numbers must have different successors; i.e., if n, m are natural numbers and n m, then n++ m++. Equivalently, if n++ = m++, then we must have n = m.}}}\right\ulcorner\mathsf{ is postulated by }\boldsymbol\mathsf{axiom}} \boldsymbol\mathsf{2.4}} (\mathit{A}_{4})\mathsf{. }\left(\left(\left(\left(\mathbf{n}_{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{m}_{2} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\left(\mathbf{n}_{4}\right)\mathit{++} \mathit{=} \left(\mathbf{m}_{2}\right)\mathit{++}\right)\right) \mathit{\implies} \left(\mathbf{n}_{4} \mathit{=} \mathbf{m}_{2}\right)\right)\mathsf{ is a valid formula statement interpreted from that axiom.}\mathsf{Therefore, by the }\mathit{axiom-interpretation}\mathsf{ inference rule: }๐’œ โŠข P\mathsf{, it follows that }\left(\left(\left(\left(\mathbf{n}_{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{m}_{2} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\left(\mathbf{n}_{4}\right)\mathit{++} \mathit{=} \left(\mathbf{m}_{2}\right)\mathit{++}\right)\right) \mathit{\implies} \left(\mathbf{n}_{4} \mathit{=} \mathbf{m}_{2}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}### \boldsymbol\mathsf{6 is not equal to 2.}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{33}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{4} \mathit{\neq} \mathit{0}\right)\right) \mathit{\implies} \left(\left(\mathit{4}\right)\mathit{++} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\mathbf{n}_{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{m}_{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathbf{n}_{3} \mathit{\neq} \mathbf{m}_{1}\right)\right) \mathit{\implies} \left(\left(\mathbf{n}_{3}\right)\mathit{++} \mathit{\neq} \left(\mathbf{m}_{1}\right)\mathit{++}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{31})\mathsf{.}\mathsf{ Let }\mathbf{n}_{3} = 4\mathsf{, }\mathbf{m}_{1} = 0\mathsf{.}\mathsf{Therefore, by the }\mathit{variable-substitution}\mathsf{ inference rule: }(P, ๐›ท) โŠข P'\mathsf{, it follows that }\left(\left(\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{4} \mathit{\neq} \mathit{0}\right)\right) \mathit{\implies} \left(\left(\mathit{4}\right)\mathit{++} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{conjunction-introduction}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{conjunction-introduction}\mathsf{ defined as }\left\ulcorner(P, Q) โŠข (P โ‹€ Q)\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{34}\mathsf{)}\mathsf{: }\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right)\mathsf{ (}\mathit{P}\mathsf{) follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{26})\mathsf{. }\left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\mathsf{ (}\mathit{Q}\mathsf{) follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{1})\mathsf{.}\mathsf{Therefore, by the }\mathit{conjunction-introduction}\mathsf{ inference rule: }(P, Q) โŠข (P โ‹€ Q)\mathsf{, it follows that }\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{35}\mathsf{)}\mathsf{: }\left(\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{4} \mathit{\neq} \mathit{0}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ (}\mathit{P}\mathsf{) follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{34})\mathsf{. }\left(\mathit{4} \mathit{\neq} \mathit{0}\right)\mathsf{ (}\mathit{Q}\mathsf{) follows from }\boldsymbol\mathsf{prop.}} \boldsymbol\mathsf{2.1.6}} (\mathit{P}_{30})\mathsf{.}\mathsf{Therefore, by the }\mathit{conjunction-introduction}\mathsf{ inference rule: }(P, Q) โŠข (P โ‹€ Q)\mathsf{, it follows that }\left(\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{4} \mathit{\neq} \mathit{0}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{36}\mathsf{)}\mathsf{: }\left(\left(\mathit{4}\right)\mathit{++} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{4} \mathit{\neq} \mathit{0}\right)\right) \mathit{\implies} \left(\left(\mathit{4}\right)\mathit{++} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{33})\mathsf{.}\left(\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{4} \mathit{\neq} \mathit{0}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{35})\mathsf{.}\mathsf{Therefore, by the }\mathit{modus-ponens}\mathsf{ inference rule: }((P โŸน Q), P) โŠข Q\mathsf{, it follows that }\left(\left(\mathit{4}\right)\mathit{++} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{37}\mathsf{)}\mathsf{: }\left(\mathit{5} \mathit{=} \left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left\ulcorner\text{\sffamily{\itshape{We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)}}}\right\ulcorner\mathsf{ is postulated by }\boldsymbol\mathsf{def.}} (\mathit{D}_{1})\mathsf{. }\left(\mathit{5} \mathit{=} \left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{ is an interpretation of that definition.}\mathsf{Therefore, by the }\mathit{definition-interpretation}\mathsf{ inference rule: }๐’Ÿ โŠข P\mathsf{, it follows that }\left(\mathit{5} \mathit{=} \left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{38}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{5}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\mathit{5} \mathit{=} \left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{37})\mathsf{. }\mathsf{Therefore, by the }\mathit{equality-commutativity}\mathsf{ inference rule: }(x = y) โŠข (y = x)\mathsf{, it follows that }\left(\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{5}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{39}\mathsf{)}\mathsf{: }\left(\left(\mathit{4}\right)\mathit{++} \mathit{=} \mathit{5}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{5}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{38})\mathsf{. }\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{4}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{23})\mathsf{.}\mathsf{Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule: }(P, (Q = R)) โŠข P'\mathsf{, it follows that }\left(\left(\mathit{4}\right)\mathit{++} \mathit{=} \mathit{5}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{40}\mathsf{)}\mathsf{: }\left(\mathit{5} \mathit{=} \left(\mathit{4}\right)\mathit{++}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathit{4}\right)\mathit{++} \mathit{=} \mathit{5}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{39})\mathsf{. }\mathsf{Therefore, by the }\mathit{equality-commutativity}\mathsf{ inference rule: }(x = y) โŠข (y = x)\mathsf{, it follows that }\left(\mathit{5} \mathit{=} \left(\mathit{4}\right)\mathit{++}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{41}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\left(\mathit{5}\right)\mathit{++} \mathit{\neq} \left(\mathit{1}\right)\mathit{++}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\mathbf{n}_{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{m}_{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathbf{n}_{3} \mathit{\neq} \mathbf{m}_{1}\right)\right) \mathit{\implies} \left(\left(\mathbf{n}_{3}\right)\mathit{++} \mathit{\neq} \left(\mathbf{m}_{1}\right)\mathit{++}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{31})\mathsf{.}\mathsf{ Let }\mathbf{n}_{3} = 5\mathsf{, }\mathbf{m}_{1} = 1\mathsf{.}\mathsf{Therefore, by the }\mathit{variable-substitution}\mathsf{ inference rule: }(P, ๐›ท) โŠข P'\mathsf{, it follows that }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\left(\mathit{5}\right)\mathit{++} \mathit{\neq} \left(\mathit{1}\right)\mathit{++}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{42}\mathsf{)}\mathsf{: }\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathit{4}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathbf{n}_{1} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathbf{n}_{1}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{2})\mathsf{.}\mathsf{ Let }\mathbf{n}_{1} = 4\mathsf{.}\mathsf{Therefore, by the }\mathit{variable-substitution}\mathsf{ inference rule: }(P, ๐›ท) โŠข P'\mathsf{, it follows that }\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathit{4}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{43}\mathsf{)}\mathsf{: }\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\left(\mathit{4}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{42})\mathsf{. }\left(\left(\mathit{4}\right)\mathit{++} \mathit{=} \mathit{5}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{39})\mathsf{.}\mathsf{Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule: }(P, (Q = R)) โŠข P'\mathsf{, it follows that }\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{44}\mathsf{)}\mathsf{: }\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{43})\mathsf{.}\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{26})\mathsf{.}\mathsf{Therefore, by the }\mathit{modus-ponens}\mathsf{ inference rule: }((P โŸน Q), P) โŠข Q\mathsf{, it follows that }\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{45}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\left(\mathit{5}\right)\mathit{++} \mathit{\neq} \left(\mathit{1}\right)\mathit{++}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\mathbf{n}_{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{m}_{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathbf{n}_{3} \mathit{\neq} \mathbf{m}_{1}\right)\right) \mathit{\implies} \left(\left(\mathbf{n}_{3}\right)\mathit{++} \mathit{\neq} \left(\mathbf{m}_{1}\right)\mathit{++}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{31})\mathsf{.}\mathsf{ Let }\mathbf{n}_{3} = 5\mathsf{, }\mathbf{m}_{1} = 1\mathsf{.}\mathsf{Therefore, by the }\mathit{variable-substitution}\mathsf{ inference rule: }(P, ๐›ท) โŠข P'\mathsf{, it follows that }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\left(\mathit{5}\right)\mathit{++} \mathit{\neq} \left(\mathit{1}\right)\mathit{++}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{46}\mathsf{)}\mathsf{: }\left(\left(\mathit{4}\right)\mathit{++} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{4} \mathit{\neq} \mathit{0}\right)\right) \mathit{\implies} \left(\left(\mathit{4}\right)\mathit{++} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{33})\mathsf{.}\left(\left(\left(\mathit{4} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{0} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{4} \mathit{\neq} \mathit{0}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{35})\mathsf{.}\mathsf{Therefore, by the }\mathit{modus-ponens}\mathsf{ inference rule: }((P โŸน Q), P) โŠข Q\mathsf{, it follows that }\left(\left(\mathit{4}\right)\mathit{++} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{47}\mathsf{)}\mathsf{: }\left(\mathit{5} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathit{4}\right)\mathit{++} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{46})\mathsf{. }\left(\left(\mathit{4}\right)\mathit{++} \mathit{=} \mathit{5}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{39})\mathsf{.}\mathsf{Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule: }(P, (Q = R)) โŠข P'\mathsf{, it follows that }\left(\mathit{5} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{48}\mathsf{)}\mathsf{: }\left(\mathit{6} \mathit{=} \left(\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left\ulcorner\text{\sffamily{\itshape{We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)}}}\right\ulcorner\mathsf{ is postulated by }\boldsymbol\mathsf{def.}} (\mathit{D}_{1})\mathsf{. }\left(\mathit{6} \mathit{=} \left(\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{ is an interpretation of that definition.}\mathsf{Therefore, by the }\mathit{definition-interpretation}\mathsf{ inference rule: }๐’Ÿ โŠข P\mathsf{, it follows that }\left(\mathit{6} \mathit{=} \left(\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{49}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{6}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\mathit{6} \mathit{=} \left(\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{48})\mathsf{. }\mathsf{Therefore, by the }\mathit{equality-commutativity}\mathsf{ inference rule: }(x = y) โŠข (y = x)\mathsf{, it follows that }\left(\left(\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{6}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{50}\mathsf{)}\mathsf{: }\left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathit{0}\right)\mathit{++} \mathit{is-a} \mathit{natural-number}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} \boldsymbol\mathsf{2.2.3}} (\mathit{P}_{4})\mathsf{. }\left(\left(\mathit{0}\right)\mathit{++} \mathit{=} \mathit{1}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{15})\mathsf{.}\mathsf{Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule: }(P, (Q = R)) โŠข P'\mathsf{, it follows that }\left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{51}\mathsf{)}\mathsf{: }\left(\left(\mathit{5}\right)\mathit{++} \mathit{=} \mathit{6}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{6}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{49})\mathsf{. }\left(\left(\left(\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{5}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{38})\mathsf{.}\mathsf{Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule: }(P, (Q = R)) โŠข P'\mathsf{, it follows that }\left(\left(\mathit{5}\right)\mathit{++} \mathit{=} \mathit{6}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{52}\mathsf{)}\mathsf{: }\left(\mathit{6} \mathit{=} \left(\mathit{5}\right)\mathit{++}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathit{5}\right)\mathit{++} \mathit{=} \mathit{6}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{51})\mathsf{. }\mathsf{Therefore, by the }\mathit{equality-commutativity}\mathsf{ inference rule: }(x = y) โŠข (y = x)\mathsf{, it follows that }\left(\mathit{6} \mathit{=} \left(\mathit{5}\right)\mathit{++}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}#### \boldsymbol\mathsf{Proof by contradiction}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Hypothesis}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{H}_{1}\mathsf{)}\mathsf{: }\left(\mathit{6} \mathit{=} \mathit{2}\right)\mathsf{.}\mathsf{ This hypothesis is elaborated in theory }\mathcal{H}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{inconsistency-by-inequality-introduction}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{inconsistency-by-inequality-introduction}\mathsf{ defined as }\left\ulcorner((P = Q), (P โ‰  Q)) โŠข Inc(๐’ฏ)\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{66}\mathsf{)}\mathsf{: }\mathit{Inc}\left(\mathcal{H}_{1}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\mathit{4} \mathit{=} \mathit{0}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{65})\mathsf{. }\left(\mathit{4} \mathit{\neq} \mathit{0}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} \boldsymbol\mathsf{2.1.6}} (\mathit{P}_{30})\mathsf{. }\mathsf{Therefore, by the }\mathit{inconsistency-by-inequality-introduction}\mathsf{ inference rule: }((P = Q), (P โ‰  Q)) โŠข Inc(๐’ฏ)\mathsf{, it follows that }\mathit{Inc}\left(\mathcal{H}_{1}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Inference rule}}\mathsf{ (}\mathit{proof-by-refutation-of-equality}\mathsf{)}\mathsf{: Let }\mathit{inference-rule}\mathsf{ }\mathit{proof-by-refutation-of-equality}\mathsf{ defined as }\left\ulcorner(โ„‹(P=Q), Inc(โ„‹)) โŠข (P โ‰  Q)\right\ulcorner\mathsf{ be included and considered valid in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}} \boldsymbol\mathsf{2.1.8}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{67}\mathsf{)}\mathsf{: }\left(\mathit{6} \mathit{\neq} \mathit{2}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\mathsf{Let }\boldsymbol\mathsf{hyp.}} (\mathit{H}_{1})\mathsf{ be the hypothesis }\left(\mathit{6} \mathit{=} \mathit{2}\right)\mathsf{. }\mathit{Inc}\left(\mathcal{H}_{1}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{66})\mathsf{.}\mathsf{Therefore, by the }\mathit{proof-by-refutation-of-equality}\mathsf{ inference rule: }(โ„‹(P=Q), Inc(โ„‹)) โŠข (P โ‰  Q)\mathsf{, it follows that }\left(\mathit{6} \mathit{\neq} \mathit{2}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}#### \boldsymbol\mathsf{Direct proof}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{68}\mathsf{)}\mathsf{: }\left(\left(\mathit{1}\right)\mathit{++} \mathit{=} \mathit{2}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\mathit{0}\right)\mathit{++}\right)\mathit{++} \mathit{=} \mathit{2}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{17})\mathsf{. }\left(\left(\mathit{0}\right)\mathit{++} \mathit{=} \mathit{1}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{15})\mathsf{.}\mathsf{Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule: }(P, (Q = R)) โŠข P'\mathsf{, it follows that }\left(\left(\mathit{1}\right)\mathit{++} \mathit{=} \mathit{2}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{69}\mathsf{)}\mathsf{: }\left(\mathit{5} \mathit{\neq} \mathit{1}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\mathit{5} \mathit{\neq} \left(\mathit{0}\right)\mathit{++}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{47})\mathsf{. }\left(\left(\mathit{0}\right)\mathit{++} \mathit{=} \mathit{1}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{15})\mathsf{.}\mathsf{Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule: }(P, (Q = R)) โŠข P'\mathsf{, it follows that }\left(\mathit{5} \mathit{\neq} \mathit{1}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{70}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\mathit{6} \mathit{\neq} \left(\mathit{1}\right)\mathit{++}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\left(\mathit{5}\right)\mathit{++} \mathit{\neq} \left(\mathit{1}\right)\mathit{++}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{45})\mathsf{. }\left(\left(\mathit{5}\right)\mathit{++} \mathit{=} \mathit{6}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{51})\mathsf{.}\mathsf{Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule: }(P, (Q = R)) โŠข P'\mathsf{, it follows that }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\mathit{6} \mathit{\neq} \left(\mathit{1}\right)\mathit{++}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{71}\mathsf{)}\mathsf{: }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\mathit{6} \mathit{\neq} \mathit{2}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\mathit{6} \mathit{\neq} \left(\mathit{1}\right)\mathit{++}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{70})\mathsf{. }\left(\left(\mathit{1}\right)\mathit{++} \mathit{=} \mathit{2}\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{68})\mathsf{.}\mathsf{Therefore, by the }\mathit{equal-terms-substitution}\mathsf{ inference rule: }(P, (Q = R)) โŠข P'\mathsf{, it follows that }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\mathit{6} \mathit{\neq} \mathit{2}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{72}\mathsf{)}\mathsf{: }\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right)\mathsf{ (}\mathit{P}\mathsf{) follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{44})\mathsf{. }\left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\mathsf{ (}\mathit{Q}\mathsf{) follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{50})\mathsf{.}\mathsf{Therefore, by the }\mathit{conjunction-introduction}\mathsf{ inference rule: }(P, Q) โŠข (P โ‹€ Q)\mathsf{, it follows that }\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{73}\mathsf{)}\mathsf{: }\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right)\mathsf{ (}\mathit{P}\mathsf{) follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{72})\mathsf{. }\left(\mathit{5} \mathit{\neq} \mathit{1}\right)\mathsf{ (}\mathit{Q}\mathsf{) follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{69})\mathsf{.}\mathsf{Therefore, by the }\mathit{conjunction-introduction}\mathsf{ inference rule: }(P, Q) โŠข (P โ‹€ Q)\mathsf{, it follows that }\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{74}\mathsf{)}\mathsf{: }\left(\mathit{6} \mathit{\neq} \mathit{2}\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left(\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right) \mathit{\implies} \left(\mathit{6} \mathit{\neq} \mathit{2}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{71})\mathsf{.}\left(\left(\left(\mathit{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathit{1} \mathit{is-a} \mathit{natural-number}\right)\right) \mathit{\land} \left(\mathit{5} \mathit{\neq} \mathit{1}\right)\right)\mathsf{ follows from }\boldsymbol\mathsf{prop.}} (\mathit{P}_{73})\mathsf{.}\mathsf{Therefore, by the }\mathit{modus-ponens}\mathsf{ inference rule: }((P โŸน Q), P) โŠข Q\mathsf{, it follows that }\left(\mathit{6} \mathit{\neq} \mathit{2}\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}### \boldsymbol\mathsf{Axiom 2.5: the principle of mathematical induction}}\mathsf{}\mathsf{}\boldsymbol\mathsf{Axiom schema}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{A}_{6}\mathsf{)} - Principle of mathematical induction\mathsf{: Let }\mathit{axiom}\mathsf{ }\mathcal{A}_{6}\mathsf{ }\left\ulcorner\text{\sffamily{\itshape{Let P(n) be any property pertaining to a natural number n. Suppose that P(O) is true, and suppose that whenever P(n) is true, P(n++) is also true. Then P(n) is true for every natural number n.}}}\right\ulcorner\mathsf{ be included (postulated) in }\mathcal{T}_{1}\mathsf{.}\mathsf{}\mathsf{}\boldsymbol\mathsf{Proposition}}\mathsf{ (}\mathcal{T}_{1}\mathsf{.}\mathit{P}_{75}\mathsf{)}\mathsf{: }\left(\left(\left(\mathbf{n}_{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{P}_{1}\left(\mathit{0}\right) \mathit{\land} \left(\mathbf{P}_{1}\left(\mathbf{n}_{5}\right) \mathit{\implies} \mathbf{P}_{1}\left(\left(\mathbf{n}_{5}\right)\mathit{++}\right)\right)\right)\right) \mathit{\implies} \left(\left(\mathbf{m}_{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \mathbf{P}_{1}\left(\mathbf{m}_{3}\right)\right)\right)\mathsf{.}\mathsf{ }\boldsymbol\mathsf{Proof}}\mathsf{: }\left\ulcorner\text{\sffamily{\itshape{Let P(n) be any property pertaining to a natural number n. Suppose that P(O) is true, and suppose that whenever P(n) is true, P(n++) is also true. Then P(n) is true for every natural number n.}}}\right\ulcorner\mathsf{ is postulated by }\boldsymbol\mathsf{axiom schema}} (\mathit{A}_{6})\mathsf{. }\left(\left(\left(\mathbf{n}_{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{P}_{1}\left(\mathit{0}\right) \mathit{\land} \left(\mathbf{P}_{1}\left(\mathbf{n}_{5}\right) \mathit{\implies} \mathbf{P}_{1}\left(\left(\mathbf{n}_{5}\right)\mathit{++}\right)\right)\right)\right) \mathit{\implies} \left(\left(\mathbf{m}_{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \mathbf{P}_{1}\left(\mathbf{m}_{3}\right)\right)\right)\mathsf{ is a valid formula statement interpreted from that axiom.}\mathsf{Therefore, by the }\mathit{axiom-interpretation}\mathsf{ inference rule: }๐’œ โŠข P\mathsf{, it follows that }\left(\left(\left(\mathbf{n}_{5} \mathit{is-a} \mathit{natural-number}\right) \mathit{\land} \left(\mathbf{P}_{1}\left(\mathit{0}\right) \mathit{\land} \left(\mathbf{P}_{1}\left(\mathbf{n}_{5}\right) \mathit{\implies} \mathbf{P}_{1}\left(\left(\mathbf{n}_{5}\right)\mathit{++}\right)\right)\right)\right) \mathit{\implies} \left(\left(\mathbf{m}_{3} \mathit{is-a} \mathit{natural-number}\right) \mathit{\implies} \mathbf{P}_{1}\left(\mathbf{m}_{3}\right)\right)\right)\mathsf{. }\mathsf{\qed}\mathsf{}\mathsf{}### \boldsymbol\mathsf{The number system n}}\mathsf{}\mathsf{}### \boldsymbol\mathsf{Recursive definitions}}\mathsf{} \ No newline at end of file diff --git a/theory/build/tao_2006_2_1_the_peano_axioms_report_proof_enus_plaintext.txt b/theory/build/tao_2006_2_1_the_peano_axioms_report_proof_enus_plaintext.txt index a7dfd12f..3d21055a 100644 --- a/theory/build/tao_2006_2_1_the_peano_axioms_report_proof_enus_plaintext.txt +++ b/theory/build/tao_2006_2_1_the_peano_axioms_report_proof_enus_plaintext.txt @@ -10,7 +10,7 @@ Let "0", "natural-number", "1", "2", "3", "4", "5", "6" be simple-objects in U1. # Relations Let "++", "Inc" be unary-relations in U1. -Let "=", "and", "is-a", "neq", "==>" be binary-relations in U1. +Let "and", "neq", "is-a", "=", "==>" be binary-relations in U1. # Inference rules The following inference rules are considered valid under this theory: @@ -31,90 +31,90 @@ Let "variable-substitution" be an inference-rule defined as "(P, ๐›ท) โŠข P'" i ### Axiom 2.1 Axiom 2.1 (T1.A1): Let axiom A1 "0 is a natural number." be included (postulated) in T1. Inference rule (axiom-interpretation): Let inference-rule axiom-interpretation defined as "๐’œ โŠข P" be included and considered valid in T1. -Proposition (T1.P1): (0 is-a natural-number). Proof: "0 is a natural number." is postulated by axiom 2.1 (A1). (0 is-a natural-number) is a valid formula statement interpreted from that axiom. Therefore, by the axiom-interpretation inference rule, it follows that (0 is-a natural-number). QED +Proposition (T1.P1): (0 is-a natural-number). Proof: "0 is a natural number." is postulated by axiom 2.1 (A1). (0 is-a natural-number) is a valid formula statement interpreted from that axiom.Therefore, by the axiom-interpretation inference rule: ๐’œ โŠข P, it follows that (0 is-a natural-number). QED ### Axiom 2.2 Axiom 2.2 (T1.A2): Let axiom A2 "If n is a natural number, then n++ is a natural number." be included (postulated) in T1. -Proposition (T1.P2): ((n1 is-a natural-number) ==> ((n1)++ is-a natural-number)). Proof: "If n is a natural number, then n++ is a natural number." is postulated by axiom 2.2 (A2). ((n1 is-a natural-number) ==> ((n1)++ is-a natural-number)) is a valid formula statement interpreted from that axiom. Therefore, by the axiom-interpretation inference rule, it follows that ((n1 is-a natural-number) ==> ((n1)++ is-a natural-number)). QED +Proposition (T1.P2): ((n1 is-a natural-number) ==> ((n1)++ is-a natural-number)). Proof: "If n is a natural number, then n++ is a natural number." is postulated by axiom 2.2 (A2). ((n1 is-a natural-number) ==> ((n1)++ is-a natural-number)) is a valid formula statement interpreted from that axiom.Therefore, by the axiom-interpretation inference rule: ๐’œ โŠข P, it follows that ((n1 is-a natural-number) ==> ((n1)++ is-a natural-number)). QED Inference rule (variable-substitution): Let inference-rule variable-substitution defined as "(P, ๐›ท) โŠข P'" be included and considered valid in T1. -Proposition (T1.P3): ((0 is-a natural-number) ==> ((0)++ is-a natural-number)). Proof: ((n1 is-a natural-number) ==> ((n1)++ is-a natural-number)) follows from prop. (P2). Let n1 = 0. Therefore, by the variable-substitution inference rule, it follows that ((0 is-a natural-number) ==> ((0)++ is-a natural-number)). QED +Proposition (T1.P3): ((0 is-a natural-number) ==> ((0)++ is-a natural-number)). Proof: ((n1 is-a natural-number) ==> ((n1)++ is-a natural-number)) follows from prop. (P2). Let n1 = 0.Therefore, by the variable-substitution inference rule: (P, ๐›ท) โŠข P', it follows that ((0 is-a natural-number) ==> ((0)++ is-a natural-number)). QED Inference rule (modus-ponens): Let inference-rule modus-ponens defined as "((P โŸน Q), P) โŠข Q" be included and considered valid in T1. -Proposition 2.2.3 (T1.P4): ((0)++ is-a natural-number). Proof: ((0 is-a natural-number) ==> ((0)++ is-a natural-number)) follows from prop. (P3).(0 is-a natural-number) follows from prop. (P1). Therefore, by the modus-ponens inference rule, it follows that ((0)++ is-a natural-number). QED +Proposition 2.2.3 (T1.P4): ((0)++ is-a natural-number). Proof: ((0 is-a natural-number) ==> ((0)++ is-a natural-number)) follows from prop. (P3).(0 is-a natural-number) follows from prop. (P1).Therefore, by the modus-ponens inference rule: ((P โŸน Q), P) โŠข Q, it follows that ((0)++ is-a natural-number). QED Definition (T1.D1): Let definition D1 "We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)" be included (postulated) in T1. Inference rule (definition-interpretation): Let inference-rule definition-interpretation defined as "๐’Ÿ โŠข P" be included and considered valid in T1. -Proposition (T1.P5): (1 = (0)++). Proof: "We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)" is postulated by def. (D1). (1 = (0)++) is an interpretation of that definition. Therefore, by the definition-interpretation inference rule, it follows that (1 = (0)++). QED -Proposition (T1.P6): (2 = ((0)++)++). Proof: "We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)" is postulated by def. (D1). (2 = ((0)++)++) is an interpretation of that definition. Therefore, by the definition-interpretation inference rule, it follows that (2 = ((0)++)++). QED -Proposition (T1.P7): (3 = (((0)++)++)++). Proof: "We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)" is postulated by def. (D1). (3 = (((0)++)++)++) is an interpretation of that definition. Therefore, by the definition-interpretation inference rule, it follows that (3 = (((0)++)++)++). QED -Proposition (T1.P8): (4 = ((((0)++)++)++)++). Proof: "We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)" is postulated by def. (D1). (4 = ((((0)++)++)++)++) is an interpretation of that definition. Therefore, by the definition-interpretation inference rule, it follows that (4 = ((((0)++)++)++)++). QED -Proposition (T1.P9): (((0)++ is-a natural-number) ==> (((0)++)++ is-a natural-number)). Proof: ((n1 is-a natural-number) ==> ((n1)++ is-a natural-number)) follows from prop. (P2). Let n1 = (0)++. Therefore, by the variable-substitution inference rule, it follows that (((0)++ is-a natural-number) ==> (((0)++)++ is-a natural-number)). QED -Proposition (T1.P10): (((0)++)++ is-a natural-number). Proof: (((0)++ is-a natural-number) ==> (((0)++)++ is-a natural-number)) follows from prop. (P9).((0)++ is-a natural-number) follows from prop. 2.2.3 (P4). Therefore, by the modus-ponens inference rule, it follows that (((0)++)++ is-a natural-number). QED -Proposition (T1.P11): ((((0)++)++ is-a natural-number) ==> ((((0)++)++)++ is-a natural-number)). Proof: ((n1 is-a natural-number) ==> ((n1)++ is-a natural-number)) follows from prop. (P2). Let n1 = ((0)++)++. Therefore, by the variable-substitution inference rule, it follows that ((((0)++)++ is-a natural-number) ==> ((((0)++)++)++ is-a natural-number)). QED -Proposition (T1.P12): ((((0)++)++)++ is-a natural-number). Proof: ((((0)++)++ is-a natural-number) ==> ((((0)++)++)++ is-a natural-number)) follows from prop. (P11).(((0)++)++ is-a natural-number) follows from prop. (P10). Therefore, by the modus-ponens inference rule, it follows that ((((0)++)++)++ is-a natural-number). QED -Proposition (T1.P13): (((((0)++)++)++ is-a natural-number) ==> (((((0)++)++)++)++ is-a natural-number)). Proof: ((n1 is-a natural-number) ==> ((n1)++ is-a natural-number)) follows from prop. (P2). Let n1 = (((0)++)++)++. Therefore, by the variable-substitution inference rule, it follows that (((((0)++)++)++ is-a natural-number) ==> (((((0)++)++)++)++ is-a natural-number)). QED -Proposition (T1.P14): (((((0)++)++)++)++ is-a natural-number). Proof: (((((0)++)++)++ is-a natural-number) ==> (((((0)++)++)++)++ is-a natural-number)) follows from prop. (P13).((((0)++)++)++ is-a natural-number) follows from prop. (P12). Therefore, by the modus-ponens inference rule, it follows that (((((0)++)++)++)++ is-a natural-number). QED +Proposition (T1.P5): (1 = (0)++). Proof: "We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)" is postulated by def. (D1). (1 = (0)++) is an interpretation of that definition.Therefore, by the definition-interpretation inference rule: ๐’Ÿ โŠข P, it follows that (1 = (0)++). QED +Proposition (T1.P6): (2 = ((0)++)++). Proof: "We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)" is postulated by def. (D1). (2 = ((0)++)++) is an interpretation of that definition.Therefore, by the definition-interpretation inference rule: ๐’Ÿ โŠข P, it follows that (2 = ((0)++)++). QED +Proposition (T1.P7): (3 = (((0)++)++)++). Proof: "We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)" is postulated by def. (D1). (3 = (((0)++)++)++) is an interpretation of that definition.Therefore, by the definition-interpretation inference rule: ๐’Ÿ โŠข P, it follows that (3 = (((0)++)++)++). QED +Proposition (T1.P8): (4 = ((((0)++)++)++)++). Proof: "We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)" is postulated by def. (D1). (4 = ((((0)++)++)++)++) is an interpretation of that definition.Therefore, by the definition-interpretation inference rule: ๐’Ÿ โŠข P, it follows that (4 = ((((0)++)++)++)++). QED +Proposition (T1.P9): (((0)++ is-a natural-number) ==> (((0)++)++ is-a natural-number)). Proof: ((n1 is-a natural-number) ==> ((n1)++ is-a natural-number)) follows from prop. (P2). Let n1 = (0)++.Therefore, by the variable-substitution inference rule: (P, ๐›ท) โŠข P', it follows that (((0)++ is-a natural-number) ==> (((0)++)++ is-a natural-number)). QED +Proposition (T1.P10): (((0)++)++ is-a natural-number). Proof: (((0)++ is-a natural-number) ==> (((0)++)++ is-a natural-number)) follows from prop. (P9).((0)++ is-a natural-number) follows from prop. 2.2.3 (P4).Therefore, by the modus-ponens inference rule: ((P โŸน Q), P) โŠข Q, it follows that (((0)++)++ is-a natural-number). QED +Proposition (T1.P11): ((((0)++)++ is-a natural-number) ==> ((((0)++)++)++ is-a natural-number)). Proof: ((n1 is-a natural-number) ==> ((n1)++ is-a natural-number)) follows from prop. (P2). Let n1 = ((0)++)++.Therefore, by the variable-substitution inference rule: (P, ๐›ท) โŠข P', it follows that ((((0)++)++ is-a natural-number) ==> ((((0)++)++)++ is-a natural-number)). QED +Proposition (T1.P12): ((((0)++)++)++ is-a natural-number). Proof: ((((0)++)++ is-a natural-number) ==> ((((0)++)++)++ is-a natural-number)) follows from prop. (P11).(((0)++)++ is-a natural-number) follows from prop. (P10).Therefore, by the modus-ponens inference rule: ((P โŸน Q), P) โŠข Q, it follows that ((((0)++)++)++ is-a natural-number). QED +Proposition (T1.P13): (((((0)++)++)++ is-a natural-number) ==> (((((0)++)++)++)++ is-a natural-number)). Proof: ((n1 is-a natural-number) ==> ((n1)++ is-a natural-number)) follows from prop. (P2). Let n1 = (((0)++)++)++.Therefore, by the variable-substitution inference rule: (P, ๐›ท) โŠข P', it follows that (((((0)++)++)++ is-a natural-number) ==> (((((0)++)++)++)++ is-a natural-number)). QED +Proposition (T1.P14): (((((0)++)++)++)++ is-a natural-number). Proof: (((((0)++)++)++ is-a natural-number) ==> (((((0)++)++)++)++ is-a natural-number)) follows from prop. (P13).((((0)++)++)++ is-a natural-number) follows from prop. (P12).Therefore, by the modus-ponens inference rule: ((P โŸน Q), P) โŠข Q, it follows that (((((0)++)++)++)++ is-a natural-number). QED Inference rule (equality-commutativity): Let inference-rule equality-commutativity defined as "(x = y) โŠข (y = x)" be included and considered valid in T1. -Proposition (T1.P15): ((0)++ = 1). Proof: (1 = (0)++) follows from prop. (P5). Therefore, by the equality-commutativity inference rule, it follows that ((0)++ = 1). QED +Proposition (T1.P15): ((0)++ = 1). Proof: (1 = (0)++) follows from prop. (P5). Therefore, by the equality-commutativity inference rule: (x = y) โŠข (y = x), it follows that ((0)++ = 1). QED Inference rule (equal-terms-substitution): Let inference-rule equal-terms-substitution defined as "(P, (Q = R)) โŠข P'" be included and considered valid in T1. -Proposition (T1.P16): (2 = (1)++). Proof: (2 = ((0)++)++) follows from prop. (P6). ((0)++ = 1) follows from prop. (P15). Therefore, by the equal-terms-substitution inference rule, it follows that (2 = (1)++). QED -Proposition (T1.P17): (((0)++)++ = 2). Proof: (2 = ((0)++)++) follows from prop. (P6). Therefore, by the equality-commutativity inference rule, it follows that (((0)++)++ = 2). QED -Proposition (T1.P18): (3 = (2)++). Proof: (3 = (((0)++)++)++) follows from prop. (P7). (((0)++)++ = 2) follows from prop. (P17). Therefore, by the equal-terms-substitution inference rule, it follows that (3 = (2)++). QED +Proposition (T1.P16): (2 = (1)++). Proof: (2 = ((0)++)++) follows from prop. (P6). ((0)++ = 1) follows from prop. (P15).Therefore, by the equal-terms-substitution inference rule: (P, (Q = R)) โŠข P', it follows that (2 = (1)++). QED +Proposition (T1.P17): (((0)++)++ = 2). Proof: (2 = ((0)++)++) follows from prop. (P6). Therefore, by the equality-commutativity inference rule: (x = y) โŠข (y = x), it follows that (((0)++)++ = 2). QED +Proposition (T1.P18): (3 = (2)++). Proof: (3 = (((0)++)++)++) follows from prop. (P7). (((0)++)++ = 2) follows from prop. (P17).Therefore, by the equal-terms-substitution inference rule: (P, (Q = R)) โŠข P', it follows that (3 = (2)++). QED ### 3 is a natural number -Proposition (T1.P19): ((((0)++)++)++ = 3). Proof: (3 = (((0)++)++)++) follows from prop. (P7). Therefore, by the equality-commutativity inference rule, it follows that ((((0)++)++)++ = 3). QED -Proposition (T1.P20): ((2)++ = 3). Proof: ((((0)++)++)++ = 3) follows from prop. (P19). (((0)++)++ = 2) follows from prop. (P17). Therefore, by the equal-terms-substitution inference rule, it follows that ((2)++ = 3). QED -Proposition 2.1.4 (T1.P21): (3 is-a natural-number). Proof: ((((0)++)++)++ is-a natural-number) follows from prop. (P12). ((((0)++)++)++ = 3) follows from prop. (P19). Therefore, by the equal-terms-substitution inference rule, it follows that (3 is-a natural-number). QED -Proposition (T1.P22): (4 = ((((0)++)++)++)++). Proof: "We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)" is postulated by def. (D1). (4 = ((((0)++)++)++)++) is an interpretation of that definition. Therefore, by the definition-interpretation inference rule, it follows that (4 = ((((0)++)++)++)++). QED -Proposition (T1.P23): (((((0)++)++)++)++ = 4). Proof: (4 = ((((0)++)++)++)++) follows from prop. (P8). Therefore, by the equality-commutativity inference rule, it follows that (((((0)++)++)++)++ = 4). QED -Proposition (T1.P24): ((3)++ = 4). Proof: (((((0)++)++)++)++ = 4) follows from prop. (P23). ((((0)++)++)++ = 3) follows from prop. (P19). Therefore, by the equal-terms-substitution inference rule, it follows that ((3)++ = 4). QED -Proposition (T1.P25): (((((0)++)++)++ is-a natural-number) ==> (((((0)++)++)++)++ is-a natural-number)). Proof: (((((0)++)++)++ is-a natural-number) ==> (((((0)++)++)++)++ is-a natural-number)) follows from prop. (P13). ((3)++ = 4) follows from prop. (P24). Therefore, by the equal-terms-substitution inference rule, it follows that (((((0)++)++)++ is-a natural-number) ==> (((((0)++)++)++)++ is-a natural-number)). QED -Proposition (T1.P26): (4 is-a natural-number). Proof: (((((0)++)++)++)++ is-a natural-number) follows from prop. (P14). (((((0)++)++)++)++ = 4) follows from prop. (P23). Therefore, by the equal-terms-substitution inference rule, it follows that (4 is-a natural-number). QED +Proposition (T1.P19): ((((0)++)++)++ = 3). Proof: (3 = (((0)++)++)++) follows from prop. (P7). Therefore, by the equality-commutativity inference rule: (x = y) โŠข (y = x), it follows that ((((0)++)++)++ = 3). QED +Proposition (T1.P20): ((2)++ = 3). Proof: ((((0)++)++)++ = 3) follows from prop. (P19). (((0)++)++ = 2) follows from prop. (P17).Therefore, by the equal-terms-substitution inference rule: (P, (Q = R)) โŠข P', it follows that ((2)++ = 3). QED +Proposition 2.1.4 (T1.P21): (3 is-a natural-number). Proof: ((((0)++)++)++ is-a natural-number) follows from prop. (P12). ((((0)++)++)++ = 3) follows from prop. (P19).Therefore, by the equal-terms-substitution inference rule: (P, (Q = R)) โŠข P', it follows that (3 is-a natural-number). QED +Proposition (T1.P22): (4 = ((((0)++)++)++)++). Proof: "We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)" is postulated by def. (D1). (4 = ((((0)++)++)++)++) is an interpretation of that definition.Therefore, by the definition-interpretation inference rule: ๐’Ÿ โŠข P, it follows that (4 = ((((0)++)++)++)++). QED +Proposition (T1.P23): (((((0)++)++)++)++ = 4). Proof: (4 = ((((0)++)++)++)++) follows from prop. (P8). Therefore, by the equality-commutativity inference rule: (x = y) โŠข (y = x), it follows that (((((0)++)++)++)++ = 4). QED +Proposition (T1.P24): ((3)++ = 4). Proof: (((((0)++)++)++)++ = 4) follows from prop. (P23). ((((0)++)++)++ = 3) follows from prop. (P19).Therefore, by the equal-terms-substitution inference rule: (P, (Q = R)) โŠข P', it follows that ((3)++ = 4). QED +Proposition (T1.P25): (((((0)++)++)++ is-a natural-number) ==> (((((0)++)++)++)++ is-a natural-number)). Proof: (((((0)++)++)++ is-a natural-number) ==> (((((0)++)++)++)++ is-a natural-number)) follows from prop. (P13). ((3)++ = 4) follows from prop. (P24).Therefore, by the equal-terms-substitution inference rule: (P, (Q = R)) โŠข P', it follows that (((((0)++)++)++ is-a natural-number) ==> (((((0)++)++)++)++ is-a natural-number)). QED +Proposition (T1.P26): (4 is-a natural-number). Proof: (((((0)++)++)++)++ is-a natural-number) follows from prop. (P14). (((((0)++)++)++)++ = 4) follows from prop. (P23).Therefore, by the equal-terms-substitution inference rule: (P, (Q = R)) โŠข P', it follows that (4 is-a natural-number). QED ### Axiom 2.3 Axiom 2.3 (T1.A3): Let axiom A3 "0 is not the successor of any natural number; i.e., we have n++ 0 for every natural number n." be included (postulated) in T1. -Proposition (T1.P27): ((n2 is-a natural-number) ==> ((n2)++ neq 0)). Proof: "0 is not the successor of any natural number; i.e., we have n++ 0 for every natural number n." is postulated by axiom 2.3 (A3). ((n2 is-a natural-number) ==> ((n2)++ neq 0)) is a valid formula statement interpreted from that axiom. Therefore, by the axiom-interpretation inference rule, it follows that ((n2 is-a natural-number) ==> ((n2)++ neq 0)). QED +Proposition (T1.P27): ((n2 is-a natural-number) ==> ((n2)++ neq 0)). Proof: "0 is not the successor of any natural number; i.e., we have n++ 0 for every natural number n." is postulated by axiom 2.3 (A3). ((n2 is-a natural-number) ==> ((n2)++ neq 0)) is a valid formula statement interpreted from that axiom.Therefore, by the axiom-interpretation inference rule: ๐’œ โŠข P, it follows that ((n2 is-a natural-number) ==> ((n2)++ neq 0)). QED ### 4 is not equal to 0. -Proposition (T1.P28): ((3 is-a natural-number) ==> ((3)++ neq 0)). Proof: ((n2 is-a natural-number) ==> ((n2)++ neq 0)) follows from prop. (P27). Let n2 = 3. Therefore, by the variable-substitution inference rule, it follows that ((3 is-a natural-number) ==> ((3)++ neq 0)). QED -Proposition (T1.P29): ((3)++ neq 0). Proof: ((3 is-a natural-number) ==> ((3)++ neq 0)) follows from prop. (P28).(3 is-a natural-number) follows from prop. 2.1.4 (P21). Therefore, by the modus-ponens inference rule, it follows that ((3)++ neq 0). QED -Proposition 2.1.6 (T1.P30): (4 neq 0). Proof: ((3)++ neq 0) follows from prop. (P29). ((3)++ = 4) follows from prop. (P24). Therefore, by the equal-terms-substitution inference rule, it follows that (4 neq 0). QED +Proposition (T1.P28): ((3 is-a natural-number) ==> ((3)++ neq 0)). Proof: ((n2 is-a natural-number) ==> ((n2)++ neq 0)) follows from prop. (P27). Let n2 = 3.Therefore, by the variable-substitution inference rule: (P, ๐›ท) โŠข P', it follows that ((3 is-a natural-number) ==> ((3)++ neq 0)). QED +Proposition (T1.P29): ((3)++ neq 0). Proof: ((3 is-a natural-number) ==> ((3)++ neq 0)) follows from prop. (P28).(3 is-a natural-number) follows from prop. 2.1.4 (P21).Therefore, by the modus-ponens inference rule: ((P โŸน Q), P) โŠข Q, it follows that ((3)++ neq 0). QED +Proposition 2.1.6 (T1.P30): (4 neq 0). Proof: ((3)++ neq 0) follows from prop. (P29). ((3)++ = 4) follows from prop. (P24).Therefore, by the equal-terms-substitution inference rule: (P, (Q = R)) โŠข P', it follows that (4 neq 0). QED ### Axiom 2.4 Axiom 2.4 (T1.A4): Let axiom A4 "Different natural numbers must have different successors; i.e., if n, m are natural numbers and n m, then n++ m++. Equivalently, if n++ = m++, then we must have n = m." be included (postulated) in T1. -Proposition (T1.P31): ((((n3 is-a natural-number) and (m1 is-a natural-number)) and (n3 neq m1)) ==> ((n3)++ neq (m1)++)). Proof: "Different natural numbers must have different successors; i.e., if n, m are natural numbers and n m, then n++ m++. Equivalently, if n++ = m++, then we must have n = m." is postulated by axiom 2.4 (A4). ((((n3 is-a natural-number) and (m1 is-a natural-number)) and (n3 neq m1)) ==> ((n3)++ neq (m1)++)) is a valid formula statement interpreted from that axiom. Therefore, by the axiom-interpretation inference rule, it follows that ((((n3 is-a natural-number) and (m1 is-a natural-number)) and (n3 neq m1)) ==> ((n3)++ neq (m1)++)). QED -Proposition (T1.P32): ((((n4 is-a natural-number) and (m2 is-a natural-number)) and ((n4)++ = (m2)++)) ==> (n4 = m2)). Proof: "Different natural numbers must have different successors; i.e., if n, m are natural numbers and n m, then n++ m++. Equivalently, if n++ = m++, then we must have n = m." is postulated by axiom 2.4 (A4). ((((n4 is-a natural-number) and (m2 is-a natural-number)) and ((n4)++ = (m2)++)) ==> (n4 = m2)) is a valid formula statement interpreted from that axiom. Therefore, by the axiom-interpretation inference rule, it follows that ((((n4 is-a natural-number) and (m2 is-a natural-number)) and ((n4)++ = (m2)++)) ==> (n4 = m2)). QED +Proposition (T1.P31): ((((n3 is-a natural-number) and (m1 is-a natural-number)) and (n3 neq m1)) ==> ((n3)++ neq (m1)++)). Proof: "Different natural numbers must have different successors; i.e., if n, m are natural numbers and n m, then n++ m++. Equivalently, if n++ = m++, then we must have n = m." is postulated by axiom 2.4 (A4). ((((n3 is-a natural-number) and (m1 is-a natural-number)) and (n3 neq m1)) ==> ((n3)++ neq (m1)++)) is a valid formula statement interpreted from that axiom.Therefore, by the axiom-interpretation inference rule: ๐’œ โŠข P, it follows that ((((n3 is-a natural-number) and (m1 is-a natural-number)) and (n3 neq m1)) ==> ((n3)++ neq (m1)++)). QED +Proposition (T1.P32): ((((n4 is-a natural-number) and (m2 is-a natural-number)) and ((n4)++ = (m2)++)) ==> (n4 = m2)). Proof: "Different natural numbers must have different successors; i.e., if n, m are natural numbers and n m, then n++ m++. Equivalently, if n++ = m++, then we must have n = m." is postulated by axiom 2.4 (A4). ((((n4 is-a natural-number) and (m2 is-a natural-number)) and ((n4)++ = (m2)++)) ==> (n4 = m2)) is a valid formula statement interpreted from that axiom.Therefore, by the axiom-interpretation inference rule: ๐’œ โŠข P, it follows that ((((n4 is-a natural-number) and (m2 is-a natural-number)) and ((n4)++ = (m2)++)) ==> (n4 = m2)). QED ### 6 is not equal to 2. -Proposition (T1.P33): ((((4 is-a natural-number) and (0 is-a natural-number)) and (4 neq 0)) ==> ((4)++ neq (0)++)). Proof: ((((n3 is-a natural-number) and (m1 is-a natural-number)) and (n3 neq m1)) ==> ((n3)++ neq (m1)++)) follows from prop. (P31). Let n3 = 4, m1 = 0. Therefore, by the variable-substitution inference rule, it follows that ((((4 is-a natural-number) and (0 is-a natural-number)) and (4 neq 0)) ==> ((4)++ neq (0)++)). QED +Proposition (T1.P33): ((((4 is-a natural-number) and (0 is-a natural-number)) and (4 neq 0)) ==> ((4)++ neq (0)++)). Proof: ((((n3 is-a natural-number) and (m1 is-a natural-number)) and (n3 neq m1)) ==> ((n3)++ neq (m1)++)) follows from prop. (P31). Let n3 = 4, m1 = 0.Therefore, by the variable-substitution inference rule: (P, ๐›ท) โŠข P', it follows that ((((4 is-a natural-number) and (0 is-a natural-number)) and (4 neq 0)) ==> ((4)++ neq (0)++)). QED Inference rule (conjunction-introduction): Let inference-rule conjunction-introduction defined as "(P, Q) โŠข (P โ‹€ Q)" be included and considered valid in T1. -Proposition (T1.P34): ((4 is-a natural-number) and (0 is-a natural-number)). Proof: (4 is-a natural-number) follows from prop. (P26). (0 is-a natural-number) follows from prop. (P1). Therefore, by the conjunction-introduction inference rule, it follows that ((4 is-a natural-number) and (0 is-a natural-number)). QED -Proposition (T1.P35): (((4 is-a natural-number) and (0 is-a natural-number)) and (4 neq 0)). Proof: ((4 is-a natural-number) and (0 is-a natural-number)) follows from prop. (P34). (4 neq 0) follows from prop. 2.1.6 (P30). Therefore, by the conjunction-introduction inference rule, it follows that (((4 is-a natural-number) and (0 is-a natural-number)) and (4 neq 0)). QED -Proposition (T1.P36): ((4)++ neq (0)++). Proof: ((((4 is-a natural-number) and (0 is-a natural-number)) and (4 neq 0)) ==> ((4)++ neq (0)++)) follows from prop. (P33).(((4 is-a natural-number) and (0 is-a natural-number)) and (4 neq 0)) follows from prop. (P35). Therefore, by the modus-ponens inference rule, it follows that ((4)++ neq (0)++). QED -Proposition (T1.P37): (5 = (((((0)++)++)++)++)++). Proof: "We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)" is postulated by def. (D1). (5 = (((((0)++)++)++)++)++) is an interpretation of that definition. Therefore, by the definition-interpretation inference rule, it follows that (5 = (((((0)++)++)++)++)++). QED -Proposition (T1.P38): ((((((0)++)++)++)++)++ = 5). Proof: (5 = (((((0)++)++)++)++)++) follows from prop. (P37). Therefore, by the equality-commutativity inference rule, it follows that ((((((0)++)++)++)++)++ = 5). QED -Proposition (T1.P39): ((4)++ = 5). Proof: ((((((0)++)++)++)++)++ = 5) follows from prop. (P38). (((((0)++)++)++)++ = 4) follows from prop. (P23). Therefore, by the equal-terms-substitution inference rule, it follows that ((4)++ = 5). QED -Proposition (T1.P40): (5 = (4)++). Proof: ((4)++ = 5) follows from prop. (P39). Therefore, by the equality-commutativity inference rule, it follows that (5 = (4)++). QED -Proposition (T1.P41): ((((5 is-a natural-number) and (1 is-a natural-number)) and (5 neq 1)) ==> ((5)++ neq (1)++)). Proof: ((((n3 is-a natural-number) and (m1 is-a natural-number)) and (n3 neq m1)) ==> ((n3)++ neq (m1)++)) follows from prop. (P31). Let n3 = 5, m1 = 1. Therefore, by the variable-substitution inference rule, it follows that ((((5 is-a natural-number) and (1 is-a natural-number)) and (5 neq 1)) ==> ((5)++ neq (1)++)). QED -Proposition (T1.P42): ((4 is-a natural-number) ==> ((4)++ is-a natural-number)). Proof: ((n1 is-a natural-number) ==> ((n1)++ is-a natural-number)) follows from prop. (P2). Let n1 = 4. Therefore, by the variable-substitution inference rule, it follows that ((4 is-a natural-number) ==> ((4)++ is-a natural-number)). QED -Proposition (T1.P43): ((4 is-a natural-number) ==> (5 is-a natural-number)). Proof: ((4 is-a natural-number) ==> ((4)++ is-a natural-number)) follows from prop. (P42). ((4)++ = 5) follows from prop. (P39). Therefore, by the equal-terms-substitution inference rule, it follows that ((4 is-a natural-number) ==> (5 is-a natural-number)). QED -Proposition (T1.P44): (5 is-a natural-number). Proof: ((4 is-a natural-number) ==> (5 is-a natural-number)) follows from prop. (P43).(4 is-a natural-number) follows from prop. (P26). Therefore, by the modus-ponens inference rule, it follows that (5 is-a natural-number). QED -Proposition (T1.P45): ((((5 is-a natural-number) and (1 is-a natural-number)) and (5 neq 1)) ==> ((5)++ neq (1)++)). Proof: ((((n3 is-a natural-number) and (m1 is-a natural-number)) and (n3 neq m1)) ==> ((n3)++ neq (m1)++)) follows from prop. (P31). Let n3 = 5, m1 = 1. Therefore, by the variable-substitution inference rule, it follows that ((((5 is-a natural-number) and (1 is-a natural-number)) and (5 neq 1)) ==> ((5)++ neq (1)++)). QED -Proposition (T1.P46): ((4)++ neq (0)++). Proof: ((((4 is-a natural-number) and (0 is-a natural-number)) and (4 neq 0)) ==> ((4)++ neq (0)++)) follows from prop. (P33).(((4 is-a natural-number) and (0 is-a natural-number)) and (4 neq 0)) follows from prop. (P35). Therefore, by the modus-ponens inference rule, it follows that ((4)++ neq (0)++). QED -Proposition (T1.P47): (5 neq (0)++). Proof: ((4)++ neq (0)++) follows from prop. (P46). ((4)++ = 5) follows from prop. (P39). Therefore, by the equal-terms-substitution inference rule, it follows that (5 neq (0)++). QED -Proposition (T1.P48): (6 = ((((((0)++)++)++)++)++)++). Proof: "We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)" is postulated by def. (D1). (6 = ((((((0)++)++)++)++)++)++) is an interpretation of that definition. Therefore, by the definition-interpretation inference rule, it follows that (6 = ((((((0)++)++)++)++)++)++). QED -Proposition (T1.P49): (((((((0)++)++)++)++)++)++ = 6). Proof: (6 = ((((((0)++)++)++)++)++)++) follows from prop. (P48). Therefore, by the equality-commutativity inference rule, it follows that (((((((0)++)++)++)++)++)++ = 6). QED -Proposition (T1.P50): (1 is-a natural-number). Proof: ((0)++ is-a natural-number) follows from prop. 2.2.3 (P4). ((0)++ = 1) follows from prop. (P15). Therefore, by the equal-terms-substitution inference rule, it follows that (1 is-a natural-number). QED -Proposition (T1.P51): ((5)++ = 6). Proof: (((((((0)++)++)++)++)++)++ = 6) follows from prop. (P49). ((((((0)++)++)++)++)++ = 5) follows from prop. (P38). Therefore, by the equal-terms-substitution inference rule, it follows that ((5)++ = 6). QED -Proposition (T1.P52): (6 = (5)++). Proof: ((5)++ = 6) follows from prop. (P51). Therefore, by the equality-commutativity inference rule, it follows that (6 = (5)++). QED +Proposition (T1.P34): ((4 is-a natural-number) and (0 is-a natural-number)). Proof: (4 is-a natural-number) (P) follows from prop. (P26). (0 is-a natural-number) (Q) follows from prop. (P1).Therefore, by the conjunction-introduction inference rule: (P, Q) โŠข (P โ‹€ Q), it follows that ((4 is-a natural-number) and (0 is-a natural-number)). QED +Proposition (T1.P35): (((4 is-a natural-number) and (0 is-a natural-number)) and (4 neq 0)). Proof: ((4 is-a natural-number) and (0 is-a natural-number)) (P) follows from prop. (P34). (4 neq 0) (Q) follows from prop. 2.1.6 (P30).Therefore, by the conjunction-introduction inference rule: (P, Q) โŠข (P โ‹€ Q), it follows that (((4 is-a natural-number) and (0 is-a natural-number)) and (4 neq 0)). QED +Proposition (T1.P36): ((4)++ neq (0)++). Proof: ((((4 is-a natural-number) and (0 is-a natural-number)) and (4 neq 0)) ==> ((4)++ neq (0)++)) follows from prop. (P33).(((4 is-a natural-number) and (0 is-a natural-number)) and (4 neq 0)) follows from prop. (P35).Therefore, by the modus-ponens inference rule: ((P โŸน Q), P) โŠข Q, it follows that ((4)++ neq (0)++). QED +Proposition (T1.P37): (5 = (((((0)++)++)++)++)++). Proof: "We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)" is postulated by def. (D1). (5 = (((((0)++)++)++)++)++) is an interpretation of that definition.Therefore, by the definition-interpretation inference rule: ๐’Ÿ โŠข P, it follows that (5 = (((((0)++)++)++)++)++). QED +Proposition (T1.P38): ((((((0)++)++)++)++)++ = 5). Proof: (5 = (((((0)++)++)++)++)++) follows from prop. (P37). Therefore, by the equality-commutativity inference rule: (x = y) โŠข (y = x), it follows that ((((((0)++)++)++)++)++ = 5). QED +Proposition (T1.P39): ((4)++ = 5). Proof: ((((((0)++)++)++)++)++ = 5) follows from prop. (P38). (((((0)++)++)++)++ = 4) follows from prop. (P23).Therefore, by the equal-terms-substitution inference rule: (P, (Q = R)) โŠข P', it follows that ((4)++ = 5). QED +Proposition (T1.P40): (5 = (4)++). Proof: ((4)++ = 5) follows from prop. (P39). Therefore, by the equality-commutativity inference rule: (x = y) โŠข (y = x), it follows that (5 = (4)++). QED +Proposition (T1.P41): ((((5 is-a natural-number) and (1 is-a natural-number)) and (5 neq 1)) ==> ((5)++ neq (1)++)). Proof: ((((n3 is-a natural-number) and (m1 is-a natural-number)) and (n3 neq m1)) ==> ((n3)++ neq (m1)++)) follows from prop. (P31). Let n3 = 5, m1 = 1.Therefore, by the variable-substitution inference rule: (P, ๐›ท) โŠข P', it follows that ((((5 is-a natural-number) and (1 is-a natural-number)) and (5 neq 1)) ==> ((5)++ neq (1)++)). QED +Proposition (T1.P42): ((4 is-a natural-number) ==> ((4)++ is-a natural-number)). Proof: ((n1 is-a natural-number) ==> ((n1)++ is-a natural-number)) follows from prop. (P2). Let n1 = 4.Therefore, by the variable-substitution inference rule: (P, ๐›ท) โŠข P', it follows that ((4 is-a natural-number) ==> ((4)++ is-a natural-number)). QED +Proposition (T1.P43): ((4 is-a natural-number) ==> (5 is-a natural-number)). Proof: ((4 is-a natural-number) ==> ((4)++ is-a natural-number)) follows from prop. (P42). ((4)++ = 5) follows from prop. (P39).Therefore, by the equal-terms-substitution inference rule: (P, (Q = R)) โŠข P', it follows that ((4 is-a natural-number) ==> (5 is-a natural-number)). QED +Proposition (T1.P44): (5 is-a natural-number). Proof: ((4 is-a natural-number) ==> (5 is-a natural-number)) follows from prop. (P43).(4 is-a natural-number) follows from prop. (P26).Therefore, by the modus-ponens inference rule: ((P โŸน Q), P) โŠข Q, it follows that (5 is-a natural-number). QED +Proposition (T1.P45): ((((5 is-a natural-number) and (1 is-a natural-number)) and (5 neq 1)) ==> ((5)++ neq (1)++)). Proof: ((((n3 is-a natural-number) and (m1 is-a natural-number)) and (n3 neq m1)) ==> ((n3)++ neq (m1)++)) follows from prop. (P31). Let n3 = 5, m1 = 1.Therefore, by the variable-substitution inference rule: (P, ๐›ท) โŠข P', it follows that ((((5 is-a natural-number) and (1 is-a natural-number)) and (5 neq 1)) ==> ((5)++ neq (1)++)). QED +Proposition (T1.P46): ((4)++ neq (0)++). Proof: ((((4 is-a natural-number) and (0 is-a natural-number)) and (4 neq 0)) ==> ((4)++ neq (0)++)) follows from prop. (P33).(((4 is-a natural-number) and (0 is-a natural-number)) and (4 neq 0)) follows from prop. (P35).Therefore, by the modus-ponens inference rule: ((P โŸน Q), P) โŠข Q, it follows that ((4)++ neq (0)++). QED +Proposition (T1.P47): (5 neq (0)++). Proof: ((4)++ neq (0)++) follows from prop. (P46). ((4)++ = 5) follows from prop. (P39).Therefore, by the equal-terms-substitution inference rule: (P, (Q = R)) โŠข P', it follows that (5 neq (0)++). QED +Proposition (T1.P48): (6 = ((((((0)++)++)++)++)++)++). Proof: "We define 1 to be the number 0++, 2 to be the number (0++)++, 3 to be the number ((0++)++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. In this text I use "x := y" to denote the statement that x is defined to equal y.)" is postulated by def. (D1). (6 = ((((((0)++)++)++)++)++)++) is an interpretation of that definition.Therefore, by the definition-interpretation inference rule: ๐’Ÿ โŠข P, it follows that (6 = ((((((0)++)++)++)++)++)++). QED +Proposition (T1.P49): (((((((0)++)++)++)++)++)++ = 6). Proof: (6 = ((((((0)++)++)++)++)++)++) follows from prop. (P48). Therefore, by the equality-commutativity inference rule: (x = y) โŠข (y = x), it follows that (((((((0)++)++)++)++)++)++ = 6). QED +Proposition (T1.P50): (1 is-a natural-number). Proof: ((0)++ is-a natural-number) follows from prop. 2.2.3 (P4). ((0)++ = 1) follows from prop. (P15).Therefore, by the equal-terms-substitution inference rule: (P, (Q = R)) โŠข P', it follows that (1 is-a natural-number). QED +Proposition (T1.P51): ((5)++ = 6). Proof: (((((((0)++)++)++)++)++)++ = 6) follows from prop. (P49). ((((((0)++)++)++)++)++ = 5) follows from prop. (P38).Therefore, by the equal-terms-substitution inference rule: (P, (Q = R)) โŠข P', it follows that ((5)++ = 6). QED +Proposition (T1.P52): (6 = (5)++). Proof: ((5)++ = 6) follows from prop. (P51). Therefore, by the equality-commutativity inference rule: (x = y) โŠข (y = x), it follows that (6 = (5)++). QED #### Proof by contradiction Hypothesis (T1.H1): (6 = 2). This hypothesis is elaborated in theory H1. Inference rule (inconsistency-by-inequality-introduction): Let inference-rule inconsistency-by-inequality-introduction defined as "((P = Q), (P โ‰  Q)) โŠข Inc(๐’ฏ)" be included and considered valid in T1. -Proposition (T1.P66): Inc(H1). Proof: P65 follows from prop. (P65). P30 follows from prop. 2.1.6 (P30). Therefore, by the inconsistency-by-inequality-introduction inference rule, it follows that Inc(H1). QED +Proposition (T1.P66): Inc(H1). Proof: (4 = 0) follows from prop. (P65). (4 neq 0) follows from prop. 2.1.6 (P30). Therefore, by the inconsistency-by-inequality-introduction inference rule: ((P = Q), (P โ‰  Q)) โŠข Inc(๐’ฏ), it follows that Inc(H1). QED Inference rule (proof-by-refutation-of-equality): Let inference-rule proof-by-refutation-of-equality defined as "(โ„‹(P=Q), Inc(โ„‹)) โŠข (P โ‰  Q)" be included and considered valid in T1. -Proposition 2.1.8 (T1.P67): (6 neq 2). Proof: Let hyp. (H1) be the hypothesis (6 = 2). Inc(H1) follows from prop. (P66). Therefore, by the proof-by-refutation-of-equality inference rule, it follows that (6 neq 2). QED +Proposition 2.1.8 (T1.P67): (6 neq 2). Proof: Let hyp. (H1) be the hypothesis (6 = 2). Inc(H1) follows from prop. (P66).Therefore, by the proof-by-refutation-of-equality inference rule: (โ„‹(P=Q), Inc(โ„‹)) โŠข (P โ‰  Q), it follows that (6 neq 2). QED #### Direct proof -Proposition (T1.P68): ((1)++ = 2). Proof: (((0)++)++ = 2) follows from prop. (P17). ((0)++ = 1) follows from prop. (P15). Therefore, by the equal-terms-substitution inference rule, it follows that ((1)++ = 2). QED -Proposition (T1.P69): (5 neq 1). Proof: (5 neq (0)++) follows from prop. (P47). ((0)++ = 1) follows from prop. (P15). Therefore, by the equal-terms-substitution inference rule, it follows that (5 neq 1). QED -Proposition (T1.P70): ((((5 is-a natural-number) and (1 is-a natural-number)) and (5 neq 1)) ==> (6 neq (1)++)). Proof: ((((5 is-a natural-number) and (1 is-a natural-number)) and (5 neq 1)) ==> ((5)++ neq (1)++)) follows from prop. (P45). ((5)++ = 6) follows from prop. (P51). Therefore, by the equal-terms-substitution inference rule, it follows that ((((5 is-a natural-number) and (1 is-a natural-number)) and (5 neq 1)) ==> (6 neq (1)++)). QED -Proposition (T1.P71): ((((5 is-a natural-number) and (1 is-a natural-number)) and (5 neq 1)) ==> (6 neq 2)). Proof: ((((5 is-a natural-number) and (1 is-a natural-number)) and (5 neq 1)) ==> (6 neq (1)++)) follows from prop. (P70). ((1)++ = 2) follows from prop. (P68). Therefore, by the equal-terms-substitution inference rule, it follows that ((((5 is-a natural-number) and (1 is-a natural-number)) and (5 neq 1)) ==> (6 neq 2)). QED -Proposition (T1.P72): ((5 is-a natural-number) and (1 is-a natural-number)). Proof: (5 is-a natural-number) follows from prop. (P44). (1 is-a natural-number) follows from prop. (P50). Therefore, by the conjunction-introduction inference rule, it follows that ((5 is-a natural-number) and (1 is-a natural-number)). QED -Proposition (T1.P73): (((5 is-a natural-number) and (1 is-a natural-number)) and (5 neq 1)). Proof: ((5 is-a natural-number) and (1 is-a natural-number)) follows from prop. (P72). (5 neq 1) follows from prop. (P69). Therefore, by the conjunction-introduction inference rule, it follows that (((5 is-a natural-number) and (1 is-a natural-number)) and (5 neq 1)). QED -Proposition (T1.P74): (6 neq 2). Proof: ((((5 is-a natural-number) and (1 is-a natural-number)) and (5 neq 1)) ==> (6 neq 2)) follows from prop. (P71).(((5 is-a natural-number) and (1 is-a natural-number)) and (5 neq 1)) follows from prop. (P73). Therefore, by the modus-ponens inference rule, it follows that (6 neq 2). QED +Proposition (T1.P68): ((1)++ = 2). Proof: (((0)++)++ = 2) follows from prop. (P17). ((0)++ = 1) follows from prop. (P15).Therefore, by the equal-terms-substitution inference rule: (P, (Q = R)) โŠข P', it follows that ((1)++ = 2). QED +Proposition (T1.P69): (5 neq 1). Proof: (5 neq (0)++) follows from prop. (P47). ((0)++ = 1) follows from prop. (P15).Therefore, by the equal-terms-substitution inference rule: (P, (Q = R)) โŠข P', it follows that (5 neq 1). QED +Proposition (T1.P70): ((((5 is-a natural-number) and (1 is-a natural-number)) and (5 neq 1)) ==> (6 neq (1)++)). Proof: ((((5 is-a natural-number) and (1 is-a natural-number)) and (5 neq 1)) ==> ((5)++ neq (1)++)) follows from prop. (P45). ((5)++ = 6) follows from prop. (P51).Therefore, by the equal-terms-substitution inference rule: (P, (Q = R)) โŠข P', it follows that ((((5 is-a natural-number) and (1 is-a natural-number)) and (5 neq 1)) ==> (6 neq (1)++)). QED +Proposition (T1.P71): ((((5 is-a natural-number) and (1 is-a natural-number)) and (5 neq 1)) ==> (6 neq 2)). Proof: ((((5 is-a natural-number) and (1 is-a natural-number)) and (5 neq 1)) ==> (6 neq (1)++)) follows from prop. (P70). ((1)++ = 2) follows from prop. (P68).Therefore, by the equal-terms-substitution inference rule: (P, (Q = R)) โŠข P', it follows that ((((5 is-a natural-number) and (1 is-a natural-number)) and (5 neq 1)) ==> (6 neq 2)). QED +Proposition (T1.P72): ((5 is-a natural-number) and (1 is-a natural-number)). Proof: (5 is-a natural-number) (P) follows from prop. (P44). (1 is-a natural-number) (Q) follows from prop. (P50).Therefore, by the conjunction-introduction inference rule: (P, Q) โŠข (P โ‹€ Q), it follows that ((5 is-a natural-number) and (1 is-a natural-number)). QED +Proposition (T1.P73): (((5 is-a natural-number) and (1 is-a natural-number)) and (5 neq 1)). Proof: ((5 is-a natural-number) and (1 is-a natural-number)) (P) follows from prop. (P72). (5 neq 1) (Q) follows from prop. (P69).Therefore, by the conjunction-introduction inference rule: (P, Q) โŠข (P โ‹€ Q), it follows that (((5 is-a natural-number) and (1 is-a natural-number)) and (5 neq 1)). QED +Proposition (T1.P74): (6 neq 2). Proof: ((((5 is-a natural-number) and (1 is-a natural-number)) and (5 neq 1)) ==> (6 neq 2)) follows from prop. (P71).(((5 is-a natural-number) and (1 is-a natural-number)) and (5 neq 1)) follows from prop. (P73).Therefore, by the modus-ponens inference rule: ((P โŸน Q), P) โŠข Q, it follows that (6 neq 2). QED ### Axiom 2.5: the principle of mathematical induction Axiom schema (T1.A6) - Principle of mathematical induction: Let axiom A6 "Let P(n) be any property pertaining to a natural number n. Suppose that P(O) is true, and suppose that whenever P(n) is true, P(n++) is also true. Then P(n) is true for every natural number n." be included (postulated) in T1. -Proposition (T1.P75): (((n5 is-a natural-number) and (P1(0) and (P1(n5) ==> P1((n5)++)))) ==> ((m3 is-a natural-number) ==> P1(m3))). Proof: "Let P(n) be any property pertaining to a natural number n. Suppose that P(O) is true, and suppose that whenever P(n) is true, P(n++) is also true. Then P(n) is true for every natural number n." is postulated by axiom schema (A6). (((n5 is-a natural-number) and (P1(0) and (P1(n5) ==> P1((n5)++)))) ==> ((m3 is-a natural-number) ==> P1(m3))) is a valid formula statement interpreted from that axiom. Therefore, by the axiom-interpretation inference rule, it follows that (((n5 is-a natural-number) and (P1(0) and (P1(n5) ==> P1((n5)++)))) ==> ((m3 is-a natural-number) ==> P1(m3))). QED +Proposition (T1.P75): (((n5 is-a natural-number) and (P1(0) and (P1(n5) ==> P1((n5)++)))) ==> ((m3 is-a natural-number) ==> P1(m3))). Proof: "Let P(n) be any property pertaining to a natural number n. Suppose that P(O) is true, and suppose that whenever P(n) is true, P(n++) is also true. Then P(n) is true for every natural number n." is postulated by axiom schema (A6). (((n5 is-a natural-number) and (P1(0) and (P1(n5) ==> P1((n5)++)))) ==> ((m3 is-a natural-number) ==> P1(m3))) is a valid formula statement interpreted from that axiom.Therefore, by the axiom-interpretation inference rule: ๐’œ โŠข P, it follows that (((n5 is-a natural-number) and (P1(0) and (P1(n5) ==> P1((n5)++)))) ==> ((m3 is-a natural-number) ==> P1(m3))). QED ### The number system n ### Recursive definitions diff --git a/theory/build/tao_2006_2_1_the_peano_axioms_report_proof_enus_unicode.txt b/theory/build/tao_2006_2_1_the_peano_axioms_report_proof_enus_unicode.txt index dbc0c9b5..2ade0260 100644 --- a/theory/build/tao_2006_2_1_the_peano_axioms_report_proof_enus_unicode.txt +++ b/theory/build/tao_2006_2_1_the_peano_axioms_report_proof_enus_unicode.txt @@ -10,7 +10,7 @@ # ๐—ฅ๐—ฒ๐—น๐—ฎ๐˜๐—ถ๐—ผ๐—ป๐˜€ ๐–ซ๐–พ๐— โŒœ++โŒ, โŒœ๐ผ๐‘›๐‘โŒ ๐–ป๐–พ ๐‘ข๐‘›๐‘Ž๐‘Ÿ๐‘ฆ-๐‘Ÿ๐‘’๐‘™๐‘Ž๐‘ก๐‘–๐‘œ๐‘›๐‘  ๐—‚๐—‡ ๐’ฐโ‚. -๐–ซ๐–พ๐— โŒœ=โŒ, โŒœโˆงโŒ, โŒœ๐‘–๐‘ -๐‘ŽโŒ, โŒœโ‰ โŒ, โŒœโŸนโŒ ๐–ป๐–พ ๐‘๐‘–๐‘›๐‘Ž๐‘Ÿ๐‘ฆ-๐‘Ÿ๐‘’๐‘™๐‘Ž๐‘ก๐‘–๐‘œ๐‘›๐‘  ๐—‚๐—‡ ๐’ฐโ‚. +๐–ซ๐–พ๐— โŒœโˆงโŒ, โŒœโ‰ โŒ, โŒœ๐‘–๐‘ -๐‘ŽโŒ, โŒœ=โŒ, โŒœโŸนโŒ ๐–ป๐–พ ๐‘๐‘–๐‘›๐‘Ž๐‘Ÿ๐‘ฆ-๐‘Ÿ๐‘’๐‘™๐‘Ž๐‘ก๐‘–๐‘œ๐‘›๐‘  ๐—‚๐—‡ ๐’ฐโ‚. # ๐—œ๐—ป๐—ณ๐—ฒ๐—ฟ๐—ฒ๐—ป๐—ฐ๐—ฒ ๐—ฟ๐˜‚๐—น๐—ฒ๐˜€ ๐–ณ๐—๐–พ ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—‚๐—‡๐—€ ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ๐—Œ ๐–บ๐—‹๐–พ ๐–ผ๐—ˆ๐—‡๐—Œ๐—‚๐–ฝ๐–พ๐—‹๐–พ๐–ฝ ๐—๐–บ๐—…๐—‚๐–ฝ ๐—Ž๐—‡๐–ฝ๐–พ๐—‹ ๐—๐—๐—‚๐—Œ ๐—๐—๐–พ๐—ˆ๐—‹๐—’: @@ -31,90 +31,90 @@ ### ๐—”๐˜…๐—ถ๐—ผ๐—บ ๐Ÿฎ.๐Ÿญ ๐—”๐˜…๐—ถ๐—ผ๐—บ ๐Ÿฎ.๐Ÿญ (๐’ฏโ‚.๐ดโ‚): ๐–ซ๐–พ๐— ๐‘Ž๐‘ฅ๐‘–๐‘œ๐‘š ๐’œโ‚ โŒœ๐Ÿข ๐˜ช๐˜ด ๐˜ข ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ.โŒ ๐–ป๐–พ ๐—‚๐—‡๐–ผ๐—…๐—Ž๐–ฝ๐–พ๐–ฝ (๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ) ๐—‚๐—‡ ๐’ฏโ‚. ๐—œ๐—ป๐—ณ๐—ฒ๐—ฟ๐—ฒ๐—ป๐—ฐ๐—ฒ ๐—ฟ๐˜‚๐—น๐—ฒ (๐‘Ž๐‘ฅ๐‘–๐‘œ๐‘š-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘›): ๐–ซ๐–พ๐— ๐‘–๐‘›๐‘“๐‘’๐‘Ÿ๐‘’๐‘›๐‘๐‘’-๐‘Ÿ๐‘ข๐‘™๐‘’ ๐‘Ž๐‘ฅ๐‘–๐‘œ๐‘š-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐–ฝ๐–พ๐–ฟ๐—‚๐—‡๐–พ๐–ฝ ๐–บ๐—Œ โŒœ๐’œ โŠข PโŒ ๐–ป๐–พ ๐—‚๐—‡๐–ผ๐—…๐—Ž๐–ฝ๐–พ๐–ฝ ๐–บ๐—‡๐–ฝ ๐–ผ๐—ˆ๐—‡๐—Œ๐—‚๐–ฝ๐–พ๐—‹๐–พ๐–ฝ ๐—๐–บ๐—…๐—‚๐–ฝ ๐—‚๐—‡ ๐’ฏโ‚. -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚): (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: โŒœ๐Ÿข ๐˜ช๐˜ด ๐˜ข ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ.โŒ ๐—‚๐—Œ ๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ ๐–ป๐—’ ๐—ฎ๐˜…๐—ถ๐—ผ๐—บ ๐Ÿฎ.๐Ÿญ (๐ดโ‚). (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) ๐—‚๐—Œ ๐–บ ๐—๐–บ๐—…๐—‚๐–ฝ ๐–ฟ๐—ˆ๐—‹๐—†๐—Ž๐—…๐–บ ๐—Œ๐—๐–บ๐—๐–พ๐—†๐–พ๐—‡๐— ๐—‚๐—‡๐—๐–พ๐—‹๐—‰๐—‹๐–พ๐—๐–พ๐–ฝ ๐–ฟ๐—‹๐—ˆ๐—† ๐—๐—๐–บ๐— ๐–บ๐—‘๐—‚๐—ˆ๐—†. ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘Ž๐‘ฅ๐‘–๐‘œ๐‘š-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚): (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: โŒœ๐Ÿข ๐˜ช๐˜ด ๐˜ข ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ.โŒ ๐—‚๐—Œ ๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ ๐–ป๐—’ ๐—ฎ๐˜…๐—ถ๐—ผ๐—บ ๐Ÿฎ.๐Ÿญ (๐ดโ‚). (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) ๐—‚๐—Œ ๐–บ ๐—๐–บ๐—…๐—‚๐–ฝ ๐–ฟ๐—ˆ๐—‹๐—†๐—Ž๐—…๐–บ ๐—Œ๐—๐–บ๐—๐–พ๐—†๐–พ๐—‡๐— ๐—‚๐—‡๐—๐–พ๐—‹๐—‰๐—‹๐–พ๐—๐–พ๐–ฝ ๐–ฟ๐—‹๐—ˆ๐—† ๐—๐—๐–บ๐— ๐–บ๐—‘๐—‚๐—ˆ๐—†.๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘Ž๐‘ฅ๐‘–๐‘œ๐‘š-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: ๐’œ โŠข P, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). โˆŽ ### ๐—”๐˜…๐—ถ๐—ผ๐—บ ๐Ÿฎ.๐Ÿฎ ๐—”๐˜…๐—ถ๐—ผ๐—บ ๐Ÿฎ.๐Ÿฎ (๐’ฏโ‚.๐ดโ‚‚): ๐–ซ๐–พ๐— ๐‘Ž๐‘ฅ๐‘–๐‘œ๐‘š ๐’œโ‚‚ โŒœ๐˜๐˜ง ๐˜ฏ ๐˜ช๐˜ด ๐˜ข ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ, ๐˜ต๐˜ฉ๐˜ฆ๐˜ฏ ๐˜ฏ++ ๐˜ช๐˜ด ๐˜ข ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ.โŒ ๐–ป๐–พ ๐—‚๐—‡๐–ผ๐—…๐—Ž๐–ฝ๐–พ๐–ฝ (๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ) ๐—‚๐—‡ ๐’ฏโ‚. -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‚): ((๐งโ‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((๐งโ‚)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: โŒœ๐˜๐˜ง ๐˜ฏ ๐˜ช๐˜ด ๐˜ข ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ, ๐˜ต๐˜ฉ๐˜ฆ๐˜ฏ ๐˜ฏ++ ๐˜ช๐˜ด ๐˜ข ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ.โŒ ๐—‚๐—Œ ๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ ๐–ป๐—’ ๐—ฎ๐˜…๐—ถ๐—ผ๐—บ ๐Ÿฎ.๐Ÿฎ (๐ดโ‚‚). ((๐งโ‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((๐งโ‚)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) ๐—‚๐—Œ ๐–บ ๐—๐–บ๐—…๐—‚๐–ฝ ๐–ฟ๐—ˆ๐—‹๐—†๐—Ž๐—…๐–บ ๐—Œ๐—๐–บ๐—๐–พ๐—†๐–พ๐—‡๐— ๐—‚๐—‡๐—๐–พ๐—‹๐—‰๐—‹๐–พ๐—๐–พ๐–ฝ ๐–ฟ๐—‹๐—ˆ๐—† ๐—๐—๐–บ๐— ๐–บ๐—‘๐—‚๐—ˆ๐—†. ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘Ž๐‘ฅ๐‘–๐‘œ๐‘š-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((๐งโ‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((๐งโ‚)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‚): ((๐งโ‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((๐งโ‚)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: โŒœ๐˜๐˜ง ๐˜ฏ ๐˜ช๐˜ด ๐˜ข ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ, ๐˜ต๐˜ฉ๐˜ฆ๐˜ฏ ๐˜ฏ++ ๐˜ช๐˜ด ๐˜ข ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ.โŒ ๐—‚๐—Œ ๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ ๐–ป๐—’ ๐—ฎ๐˜…๐—ถ๐—ผ๐—บ ๐Ÿฎ.๐Ÿฎ (๐ดโ‚‚). ((๐งโ‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((๐งโ‚)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) ๐—‚๐—Œ ๐–บ ๐—๐–บ๐—…๐—‚๐–ฝ ๐–ฟ๐—ˆ๐—‹๐—†๐—Ž๐—…๐–บ ๐—Œ๐—๐–บ๐—๐–พ๐—†๐–พ๐—‡๐— ๐—‚๐—‡๐—๐–พ๐—‹๐—‰๐—‹๐–พ๐—๐–พ๐–ฝ ๐–ฟ๐—‹๐—ˆ๐—† ๐—๐—๐–บ๐— ๐–บ๐—‘๐—‚๐—ˆ๐—†.๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘Ž๐‘ฅ๐‘–๐‘œ๐‘š-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: ๐’œ โŠข P, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((๐งโ‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((๐งโ‚)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). โˆŽ ๐—œ๐—ป๐—ณ๐—ฒ๐—ฟ๐—ฒ๐—ป๐—ฐ๐—ฒ ๐—ฟ๐˜‚๐—น๐—ฒ (๐‘ฃ๐‘Ž๐‘Ÿ๐‘–๐‘Ž๐‘๐‘™๐‘’-๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘›): ๐–ซ๐–พ๐— ๐‘–๐‘›๐‘“๐‘’๐‘Ÿ๐‘’๐‘›๐‘๐‘’-๐‘Ÿ๐‘ข๐‘™๐‘’ ๐‘ฃ๐‘Ž๐‘Ÿ๐‘–๐‘Ž๐‘๐‘™๐‘’-๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐–ฝ๐–พ๐–ฟ๐—‚๐—‡๐–พ๐–ฝ ๐–บ๐—Œ โŒœ(P, ๐›ท) โŠข P'โŒ ๐–ป๐–พ ๐—‚๐—‡๐–ผ๐—…๐—Ž๐–ฝ๐–พ๐–ฝ ๐–บ๐—‡๐–ฝ ๐–ผ๐—ˆ๐—‡๐—Œ๐—‚๐–ฝ๐–พ๐—‹๐–พ๐–ฝ ๐—๐–บ๐—…๐—‚๐–ฝ ๐—‚๐—‡ ๐’ฏโ‚. -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚ƒ): ((0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((0)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((๐งโ‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((๐งโ‚)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚). ๐–ซ๐–พ๐— ๐งโ‚ = 0. ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘ฃ๐‘Ž๐‘Ÿ๐‘–๐‘Ž๐‘๐‘™๐‘’-๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((0)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚ƒ): ((0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((0)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((๐งโ‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((๐งโ‚)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚). ๐–ซ๐–พ๐— ๐งโ‚ = 0.๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘ฃ๐‘Ž๐‘Ÿ๐‘–๐‘Ž๐‘๐‘™๐‘’-๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, ๐›ท) โŠข P', ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((0)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). โˆŽ ๐—œ๐—ป๐—ณ๐—ฒ๐—ฟ๐—ฒ๐—ป๐—ฐ๐—ฒ ๐—ฟ๐˜‚๐—น๐—ฒ (๐‘š๐‘œ๐‘‘๐‘ข๐‘ -๐‘๐‘œ๐‘›๐‘’๐‘›๐‘ ): ๐–ซ๐–พ๐— ๐‘–๐‘›๐‘“๐‘’๐‘Ÿ๐‘’๐‘›๐‘๐‘’-๐‘Ÿ๐‘ข๐‘™๐‘’ ๐‘š๐‘œ๐‘‘๐‘ข๐‘ -๐‘๐‘œ๐‘›๐‘’๐‘›๐‘  ๐–ฝ๐–พ๐–ฟ๐—‚๐—‡๐–พ๐–ฝ ๐–บ๐—Œ โŒœ((P โŸน Q), P) โŠข QโŒ ๐–ป๐–พ ๐—‚๐—‡๐–ผ๐—…๐—Ž๐–ฝ๐–พ๐–ฝ ๐–บ๐—‡๐–ฝ ๐–ผ๐—ˆ๐—‡๐—Œ๐—‚๐–ฝ๐–พ๐—‹๐–พ๐–ฝ ๐—๐–บ๐—…๐—‚๐–ฝ ๐—‚๐—‡ ๐’ฏโ‚. -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป ๐Ÿฎ.๐Ÿฎ.๐Ÿฏ (๐’ฏโ‚.๐‘ƒโ‚„): ((0)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((0)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒ).(0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘š๐‘œ๐‘‘๐‘ข๐‘ -๐‘๐‘œ๐‘›๐‘’๐‘›๐‘  ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((0)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป ๐Ÿฎ.๐Ÿฎ.๐Ÿฏ (๐’ฏโ‚.๐‘ƒโ‚„): ((0)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((0)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒ).(0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘š๐‘œ๐‘‘๐‘ข๐‘ -๐‘๐‘œ๐‘›๐‘’๐‘›๐‘  ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: ((P โŸน Q), P) โŠข Q, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((0)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). โˆŽ ๐——๐—ฒ๐—ณ๐—ถ๐—ป๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐ทโ‚): ๐–ซ๐–พ๐— ๐‘‘๐‘’๐‘“๐‘–๐‘›๐‘–๐‘ก๐‘–๐‘œ๐‘› ๐’Ÿโ‚ โŒœ๐˜ž๐˜ฆ ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ ๐Ÿฃ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ๐Ÿข++, ๐Ÿค ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ (๐Ÿข++)++, ๐Ÿฅ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ((๐Ÿข++)++)++,๐˜ฆ๐˜ต๐˜ค. (๐˜๐˜ฏ ๐˜ฐ๐˜ต๐˜ฉ๐˜ฆ๐˜ณ ๐˜ธ๐˜ฐ๐˜ณ๐˜ฅ๐˜ด, ๐Ÿฃ := ๐Ÿข++, ๐Ÿค := ๐Ÿฃ++, ๐Ÿฅ := ๐Ÿค++, ๐˜ฆ๐˜ต๐˜ค. ๐˜๐˜ฏ ๐˜ต๐˜ฉ๐˜ช๐˜ด ๐˜ต๐˜ฆ๐˜น๐˜ต ๐˜ ๐˜ถ๐˜ด๐˜ฆ "๐˜น := ๐˜บ" ๐˜ต๐˜ฐ ๐˜ฅ๐˜ฆ๐˜ฏ๐˜ฐ๐˜ต๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ด๐˜ต๐˜ข๐˜ต๐˜ฆ๐˜ฎ๐˜ฆ๐˜ฏ๐˜ต ๐˜ต๐˜ฉ๐˜ข๐˜ต ๐˜น ๐˜ช๐˜ด ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ๐˜ฅ ๐˜ต๐˜ฐ ๐˜ฆ๐˜ฒ๐˜ถ๐˜ข๐˜ญ ๐˜บ.)โŒ ๐–ป๐–พ ๐—‚๐—‡๐–ผ๐—…๐—Ž๐–ฝ๐–พ๐–ฝ (๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ) ๐—‚๐—‡ ๐’ฏโ‚. ๐—œ๐—ป๐—ณ๐—ฒ๐—ฟ๐—ฒ๐—ป๐—ฐ๐—ฒ ๐—ฟ๐˜‚๐—น๐—ฒ (๐‘‘๐‘’๐‘“๐‘–๐‘›๐‘–๐‘ก๐‘–๐‘œ๐‘›-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘›): ๐–ซ๐–พ๐— ๐‘–๐‘›๐‘“๐‘’๐‘Ÿ๐‘’๐‘›๐‘๐‘’-๐‘Ÿ๐‘ข๐‘™๐‘’ ๐‘‘๐‘’๐‘“๐‘–๐‘›๐‘–๐‘ก๐‘–๐‘œ๐‘›-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐–ฝ๐–พ๐–ฟ๐—‚๐—‡๐–พ๐–ฝ ๐–บ๐—Œ โŒœ๐’Ÿ โŠข PโŒ ๐–ป๐–พ ๐—‚๐—‡๐–ผ๐—…๐—Ž๐–ฝ๐–พ๐–ฝ ๐–บ๐—‡๐–ฝ ๐–ผ๐—ˆ๐—‡๐—Œ๐—‚๐–ฝ๐–พ๐—‹๐–พ๐–ฝ ๐—๐–บ๐—…๐—‚๐–ฝ ๐—‚๐—‡ ๐’ฏโ‚. -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚…): (1 = (0)++). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: โŒœ๐˜ž๐˜ฆ ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ ๐Ÿฃ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ๐Ÿข++, ๐Ÿค ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ (๐Ÿข++)++, ๐Ÿฅ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ((๐Ÿข++)++)++,๐˜ฆ๐˜ต๐˜ค. (๐˜๐˜ฏ ๐˜ฐ๐˜ต๐˜ฉ๐˜ฆ๐˜ณ ๐˜ธ๐˜ฐ๐˜ณ๐˜ฅ๐˜ด, ๐Ÿฃ := ๐Ÿข++, ๐Ÿค := ๐Ÿฃ++, ๐Ÿฅ := ๐Ÿค++, ๐˜ฆ๐˜ต๐˜ค. ๐˜๐˜ฏ ๐˜ต๐˜ฉ๐˜ช๐˜ด ๐˜ต๐˜ฆ๐˜น๐˜ต ๐˜ ๐˜ถ๐˜ด๐˜ฆ "๐˜น := ๐˜บ" ๐˜ต๐˜ฐ ๐˜ฅ๐˜ฆ๐˜ฏ๐˜ฐ๐˜ต๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ด๐˜ต๐˜ข๐˜ต๐˜ฆ๐˜ฎ๐˜ฆ๐˜ฏ๐˜ต ๐˜ต๐˜ฉ๐˜ข๐˜ต ๐˜น ๐˜ช๐˜ด ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ๐˜ฅ ๐˜ต๐˜ฐ ๐˜ฆ๐˜ฒ๐˜ถ๐˜ข๐˜ญ ๐˜บ.)โŒ ๐—‚๐—Œ ๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ ๐–ป๐—’ ๐—ฑ๐—ฒ๐—ณ. (๐ทโ‚). (1 = (0)++) ๐—‚๐—Œ ๐–บ๐—‡ ๐—‚๐—‡๐—๐–พ๐—‹๐—‰๐—‹๐–พ๐—๐–บ๐—๐—‚๐—ˆ๐—‡ ๐—ˆ๐–ฟ ๐—๐—๐–บ๐— ๐–ฝ๐–พ๐–ฟ๐—‚๐—‡๐—‚๐—๐—‚๐—ˆ๐—‡. ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘‘๐‘’๐‘“๐‘–๐‘›๐‘–๐‘ก๐‘–๐‘œ๐‘›-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (1 = (0)++). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚†): (2 = ((0)++)++). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: โŒœ๐˜ž๐˜ฆ ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ ๐Ÿฃ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ๐Ÿข++, ๐Ÿค ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ (๐Ÿข++)++, ๐Ÿฅ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ((๐Ÿข++)++)++,๐˜ฆ๐˜ต๐˜ค. (๐˜๐˜ฏ ๐˜ฐ๐˜ต๐˜ฉ๐˜ฆ๐˜ณ ๐˜ธ๐˜ฐ๐˜ณ๐˜ฅ๐˜ด, ๐Ÿฃ := ๐Ÿข++, ๐Ÿค := ๐Ÿฃ++, ๐Ÿฅ := ๐Ÿค++, ๐˜ฆ๐˜ต๐˜ค. ๐˜๐˜ฏ ๐˜ต๐˜ฉ๐˜ช๐˜ด ๐˜ต๐˜ฆ๐˜น๐˜ต ๐˜ ๐˜ถ๐˜ด๐˜ฆ "๐˜น := ๐˜บ" ๐˜ต๐˜ฐ ๐˜ฅ๐˜ฆ๐˜ฏ๐˜ฐ๐˜ต๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ด๐˜ต๐˜ข๐˜ต๐˜ฆ๐˜ฎ๐˜ฆ๐˜ฏ๐˜ต ๐˜ต๐˜ฉ๐˜ข๐˜ต ๐˜น ๐˜ช๐˜ด ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ๐˜ฅ ๐˜ต๐˜ฐ ๐˜ฆ๐˜ฒ๐˜ถ๐˜ข๐˜ญ ๐˜บ.)โŒ ๐—‚๐—Œ ๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ ๐–ป๐—’ ๐—ฑ๐—ฒ๐—ณ. (๐ทโ‚). (2 = ((0)++)++) ๐—‚๐—Œ ๐–บ๐—‡ ๐—‚๐—‡๐—๐–พ๐—‹๐—‰๐—‹๐–พ๐—๐–บ๐—๐—‚๐—ˆ๐—‡ ๐—ˆ๐–ฟ ๐—๐—๐–บ๐— ๐–ฝ๐–พ๐–ฟ๐—‚๐—‡๐—‚๐—๐—‚๐—ˆ๐—‡. ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘‘๐‘’๐‘“๐‘–๐‘›๐‘–๐‘ก๐‘–๐‘œ๐‘›-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (2 = ((0)++)++). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‡): (3 = (((0)++)++)++). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: โŒœ๐˜ž๐˜ฆ ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ ๐Ÿฃ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ๐Ÿข++, ๐Ÿค ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ (๐Ÿข++)++, ๐Ÿฅ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ((๐Ÿข++)++)++,๐˜ฆ๐˜ต๐˜ค. (๐˜๐˜ฏ ๐˜ฐ๐˜ต๐˜ฉ๐˜ฆ๐˜ณ ๐˜ธ๐˜ฐ๐˜ณ๐˜ฅ๐˜ด, ๐Ÿฃ := ๐Ÿข++, ๐Ÿค := ๐Ÿฃ++, ๐Ÿฅ := ๐Ÿค++, ๐˜ฆ๐˜ต๐˜ค. ๐˜๐˜ฏ ๐˜ต๐˜ฉ๐˜ช๐˜ด ๐˜ต๐˜ฆ๐˜น๐˜ต ๐˜ ๐˜ถ๐˜ด๐˜ฆ "๐˜น := ๐˜บ" ๐˜ต๐˜ฐ ๐˜ฅ๐˜ฆ๐˜ฏ๐˜ฐ๐˜ต๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ด๐˜ต๐˜ข๐˜ต๐˜ฆ๐˜ฎ๐˜ฆ๐˜ฏ๐˜ต ๐˜ต๐˜ฉ๐˜ข๐˜ต ๐˜น ๐˜ช๐˜ด ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ๐˜ฅ ๐˜ต๐˜ฐ ๐˜ฆ๐˜ฒ๐˜ถ๐˜ข๐˜ญ ๐˜บ.)โŒ ๐—‚๐—Œ ๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ ๐–ป๐—’ ๐—ฑ๐—ฒ๐—ณ. (๐ทโ‚). (3 = (((0)++)++)++) ๐—‚๐—Œ ๐–บ๐—‡ ๐—‚๐—‡๐—๐–พ๐—‹๐—‰๐—‹๐–พ๐—๐–บ๐—๐—‚๐—ˆ๐—‡ ๐—ˆ๐–ฟ ๐—๐—๐–บ๐— ๐–ฝ๐–พ๐–ฟ๐—‚๐—‡๐—‚๐—๐—‚๐—ˆ๐—‡. ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘‘๐‘’๐‘“๐‘–๐‘›๐‘–๐‘ก๐‘–๐‘œ๐‘›-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (3 = (((0)++)++)++). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚ˆ): (4 = ((((0)++)++)++)++). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: โŒœ๐˜ž๐˜ฆ ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ ๐Ÿฃ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ๐Ÿข++, ๐Ÿค ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ (๐Ÿข++)++, ๐Ÿฅ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ((๐Ÿข++)++)++,๐˜ฆ๐˜ต๐˜ค. (๐˜๐˜ฏ ๐˜ฐ๐˜ต๐˜ฉ๐˜ฆ๐˜ณ ๐˜ธ๐˜ฐ๐˜ณ๐˜ฅ๐˜ด, ๐Ÿฃ := ๐Ÿข++, ๐Ÿค := ๐Ÿฃ++, ๐Ÿฅ := ๐Ÿค++, ๐˜ฆ๐˜ต๐˜ค. ๐˜๐˜ฏ ๐˜ต๐˜ฉ๐˜ช๐˜ด ๐˜ต๐˜ฆ๐˜น๐˜ต ๐˜ ๐˜ถ๐˜ด๐˜ฆ "๐˜น := ๐˜บ" ๐˜ต๐˜ฐ ๐˜ฅ๐˜ฆ๐˜ฏ๐˜ฐ๐˜ต๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ด๐˜ต๐˜ข๐˜ต๐˜ฆ๐˜ฎ๐˜ฆ๐˜ฏ๐˜ต ๐˜ต๐˜ฉ๐˜ข๐˜ต ๐˜น ๐˜ช๐˜ด ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ๐˜ฅ ๐˜ต๐˜ฐ ๐˜ฆ๐˜ฒ๐˜ถ๐˜ข๐˜ญ ๐˜บ.)โŒ ๐—‚๐—Œ ๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ ๐–ป๐—’ ๐—ฑ๐—ฒ๐—ณ. (๐ทโ‚). (4 = ((((0)++)++)++)++) ๐—‚๐—Œ ๐–บ๐—‡ ๐—‚๐—‡๐—๐–พ๐—‹๐—‰๐—‹๐–พ๐—๐–บ๐—๐—‚๐—ˆ๐—‡ ๐—ˆ๐–ฟ ๐—๐—๐–บ๐— ๐–ฝ๐–พ๐–ฟ๐—‚๐—‡๐—‚๐—๐—‚๐—ˆ๐—‡. ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘‘๐‘’๐‘“๐‘–๐‘›๐‘–๐‘ก๐‘–๐‘œ๐‘›-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (4 = ((((0)++)++)++)++). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‰): (((0)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน (((0)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((๐งโ‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((๐งโ‚)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚). ๐–ซ๐–พ๐— ๐งโ‚ = (0)++. ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘ฃ๐‘Ž๐‘Ÿ๐‘–๐‘Ž๐‘๐‘™๐‘’-๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (((0)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน (((0)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚โ‚€): (((0)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (((0)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน (((0)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‰).((0)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. ๐Ÿฎ.๐Ÿฎ.๐Ÿฏ (๐‘ƒโ‚„). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘š๐‘œ๐‘‘๐‘ข๐‘ -๐‘๐‘œ๐‘›๐‘’๐‘›๐‘  ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (((0)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚โ‚): ((((0)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((((0)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((๐งโ‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((๐งโ‚)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚). ๐–ซ๐–พ๐— ๐งโ‚ = ((0)++)++. ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘ฃ๐‘Ž๐‘Ÿ๐‘–๐‘Ž๐‘๐‘™๐‘’-๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((((0)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((((0)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚โ‚‚): ((((0)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((((0)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((((0)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚).(((0)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚€). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘š๐‘œ๐‘‘๐‘ข๐‘ -๐‘๐‘œ๐‘›๐‘’๐‘›๐‘  ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((((0)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚โ‚ƒ): (((((0)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน (((((0)++)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((๐งโ‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((๐งโ‚)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚). ๐–ซ๐–พ๐— ๐งโ‚ = (((0)++)++)++. ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘ฃ๐‘Ž๐‘Ÿ๐‘–๐‘Ž๐‘๐‘™๐‘’-๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (((((0)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน (((((0)++)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚โ‚„): (((((0)++)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (((((0)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน (((((0)++)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚ƒ).((((0)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚‚). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘š๐‘œ๐‘‘๐‘ข๐‘ -๐‘๐‘œ๐‘›๐‘’๐‘›๐‘  ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (((((0)++)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚…): (1 = (0)++). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: โŒœ๐˜ž๐˜ฆ ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ ๐Ÿฃ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ๐Ÿข++, ๐Ÿค ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ (๐Ÿข++)++, ๐Ÿฅ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ((๐Ÿข++)++)++,๐˜ฆ๐˜ต๐˜ค. (๐˜๐˜ฏ ๐˜ฐ๐˜ต๐˜ฉ๐˜ฆ๐˜ณ ๐˜ธ๐˜ฐ๐˜ณ๐˜ฅ๐˜ด, ๐Ÿฃ := ๐Ÿข++, ๐Ÿค := ๐Ÿฃ++, ๐Ÿฅ := ๐Ÿค++, ๐˜ฆ๐˜ต๐˜ค. ๐˜๐˜ฏ ๐˜ต๐˜ฉ๐˜ช๐˜ด ๐˜ต๐˜ฆ๐˜น๐˜ต ๐˜ ๐˜ถ๐˜ด๐˜ฆ "๐˜น := ๐˜บ" ๐˜ต๐˜ฐ ๐˜ฅ๐˜ฆ๐˜ฏ๐˜ฐ๐˜ต๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ด๐˜ต๐˜ข๐˜ต๐˜ฆ๐˜ฎ๐˜ฆ๐˜ฏ๐˜ต ๐˜ต๐˜ฉ๐˜ข๐˜ต ๐˜น ๐˜ช๐˜ด ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ๐˜ฅ ๐˜ต๐˜ฐ ๐˜ฆ๐˜ฒ๐˜ถ๐˜ข๐˜ญ ๐˜บ.)โŒ ๐—‚๐—Œ ๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ ๐–ป๐—’ ๐—ฑ๐—ฒ๐—ณ. (๐ทโ‚). (1 = (0)++) ๐—‚๐—Œ ๐–บ๐—‡ ๐—‚๐—‡๐—๐–พ๐—‹๐—‰๐—‹๐–พ๐—๐–บ๐—๐—‚๐—ˆ๐—‡ ๐—ˆ๐–ฟ ๐—๐—๐–บ๐— ๐–ฝ๐–พ๐–ฟ๐—‚๐—‡๐—‚๐—๐—‚๐—ˆ๐—‡.๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘‘๐‘’๐‘“๐‘–๐‘›๐‘–๐‘ก๐‘–๐‘œ๐‘›-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: ๐’Ÿ โŠข P, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (1 = (0)++). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚†): (2 = ((0)++)++). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: โŒœ๐˜ž๐˜ฆ ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ ๐Ÿฃ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ๐Ÿข++, ๐Ÿค ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ (๐Ÿข++)++, ๐Ÿฅ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ((๐Ÿข++)++)++,๐˜ฆ๐˜ต๐˜ค. (๐˜๐˜ฏ ๐˜ฐ๐˜ต๐˜ฉ๐˜ฆ๐˜ณ ๐˜ธ๐˜ฐ๐˜ณ๐˜ฅ๐˜ด, ๐Ÿฃ := ๐Ÿข++, ๐Ÿค := ๐Ÿฃ++, ๐Ÿฅ := ๐Ÿค++, ๐˜ฆ๐˜ต๐˜ค. ๐˜๐˜ฏ ๐˜ต๐˜ฉ๐˜ช๐˜ด ๐˜ต๐˜ฆ๐˜น๐˜ต ๐˜ ๐˜ถ๐˜ด๐˜ฆ "๐˜น := ๐˜บ" ๐˜ต๐˜ฐ ๐˜ฅ๐˜ฆ๐˜ฏ๐˜ฐ๐˜ต๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ด๐˜ต๐˜ข๐˜ต๐˜ฆ๐˜ฎ๐˜ฆ๐˜ฏ๐˜ต ๐˜ต๐˜ฉ๐˜ข๐˜ต ๐˜น ๐˜ช๐˜ด ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ๐˜ฅ ๐˜ต๐˜ฐ ๐˜ฆ๐˜ฒ๐˜ถ๐˜ข๐˜ญ ๐˜บ.)โŒ ๐—‚๐—Œ ๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ ๐–ป๐—’ ๐—ฑ๐—ฒ๐—ณ. (๐ทโ‚). (2 = ((0)++)++) ๐—‚๐—Œ ๐–บ๐—‡ ๐—‚๐—‡๐—๐–พ๐—‹๐—‰๐—‹๐–พ๐—๐–บ๐—๐—‚๐—ˆ๐—‡ ๐—ˆ๐–ฟ ๐—๐—๐–บ๐— ๐–ฝ๐–พ๐–ฟ๐—‚๐—‡๐—‚๐—๐—‚๐—ˆ๐—‡.๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘‘๐‘’๐‘“๐‘–๐‘›๐‘–๐‘ก๐‘–๐‘œ๐‘›-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: ๐’Ÿ โŠข P, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (2 = ((0)++)++). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‡): (3 = (((0)++)++)++). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: โŒœ๐˜ž๐˜ฆ ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ ๐Ÿฃ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ๐Ÿข++, ๐Ÿค ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ (๐Ÿข++)++, ๐Ÿฅ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ((๐Ÿข++)++)++,๐˜ฆ๐˜ต๐˜ค. (๐˜๐˜ฏ ๐˜ฐ๐˜ต๐˜ฉ๐˜ฆ๐˜ณ ๐˜ธ๐˜ฐ๐˜ณ๐˜ฅ๐˜ด, ๐Ÿฃ := ๐Ÿข++, ๐Ÿค := ๐Ÿฃ++, ๐Ÿฅ := ๐Ÿค++, ๐˜ฆ๐˜ต๐˜ค. ๐˜๐˜ฏ ๐˜ต๐˜ฉ๐˜ช๐˜ด ๐˜ต๐˜ฆ๐˜น๐˜ต ๐˜ ๐˜ถ๐˜ด๐˜ฆ "๐˜น := ๐˜บ" ๐˜ต๐˜ฐ ๐˜ฅ๐˜ฆ๐˜ฏ๐˜ฐ๐˜ต๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ด๐˜ต๐˜ข๐˜ต๐˜ฆ๐˜ฎ๐˜ฆ๐˜ฏ๐˜ต ๐˜ต๐˜ฉ๐˜ข๐˜ต ๐˜น ๐˜ช๐˜ด ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ๐˜ฅ ๐˜ต๐˜ฐ ๐˜ฆ๐˜ฒ๐˜ถ๐˜ข๐˜ญ ๐˜บ.)โŒ ๐—‚๐—Œ ๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ ๐–ป๐—’ ๐—ฑ๐—ฒ๐—ณ. (๐ทโ‚). (3 = (((0)++)++)++) ๐—‚๐—Œ ๐–บ๐—‡ ๐—‚๐—‡๐—๐–พ๐—‹๐—‰๐—‹๐–พ๐—๐–บ๐—๐—‚๐—ˆ๐—‡ ๐—ˆ๐–ฟ ๐—๐—๐–บ๐— ๐–ฝ๐–พ๐–ฟ๐—‚๐—‡๐—‚๐—๐—‚๐—ˆ๐—‡.๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘‘๐‘’๐‘“๐‘–๐‘›๐‘–๐‘ก๐‘–๐‘œ๐‘›-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: ๐’Ÿ โŠข P, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (3 = (((0)++)++)++). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚ˆ): (4 = ((((0)++)++)++)++). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: โŒœ๐˜ž๐˜ฆ ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ ๐Ÿฃ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ๐Ÿข++, ๐Ÿค ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ (๐Ÿข++)++, ๐Ÿฅ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ((๐Ÿข++)++)++,๐˜ฆ๐˜ต๐˜ค. (๐˜๐˜ฏ ๐˜ฐ๐˜ต๐˜ฉ๐˜ฆ๐˜ณ ๐˜ธ๐˜ฐ๐˜ณ๐˜ฅ๐˜ด, ๐Ÿฃ := ๐Ÿข++, ๐Ÿค := ๐Ÿฃ++, ๐Ÿฅ := ๐Ÿค++, ๐˜ฆ๐˜ต๐˜ค. ๐˜๐˜ฏ ๐˜ต๐˜ฉ๐˜ช๐˜ด ๐˜ต๐˜ฆ๐˜น๐˜ต ๐˜ ๐˜ถ๐˜ด๐˜ฆ "๐˜น := ๐˜บ" ๐˜ต๐˜ฐ ๐˜ฅ๐˜ฆ๐˜ฏ๐˜ฐ๐˜ต๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ด๐˜ต๐˜ข๐˜ต๐˜ฆ๐˜ฎ๐˜ฆ๐˜ฏ๐˜ต ๐˜ต๐˜ฉ๐˜ข๐˜ต ๐˜น ๐˜ช๐˜ด ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ๐˜ฅ ๐˜ต๐˜ฐ ๐˜ฆ๐˜ฒ๐˜ถ๐˜ข๐˜ญ ๐˜บ.)โŒ ๐—‚๐—Œ ๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ ๐–ป๐—’ ๐—ฑ๐—ฒ๐—ณ. (๐ทโ‚). (4 = ((((0)++)++)++)++) ๐—‚๐—Œ ๐–บ๐—‡ ๐—‚๐—‡๐—๐–พ๐—‹๐—‰๐—‹๐–พ๐—๐–บ๐—๐—‚๐—ˆ๐—‡ ๐—ˆ๐–ฟ ๐—๐—๐–บ๐— ๐–ฝ๐–พ๐–ฟ๐—‚๐—‡๐—‚๐—๐—‚๐—ˆ๐—‡.๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘‘๐‘’๐‘“๐‘–๐‘›๐‘–๐‘ก๐‘–๐‘œ๐‘›-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: ๐’Ÿ โŠข P, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (4 = ((((0)++)++)++)++). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‰): (((0)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน (((0)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((๐งโ‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((๐งโ‚)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚). ๐–ซ๐–พ๐— ๐งโ‚ = (0)++.๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘ฃ๐‘Ž๐‘Ÿ๐‘–๐‘Ž๐‘๐‘™๐‘’-๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, ๐›ท) โŠข P', ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (((0)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน (((0)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚โ‚€): (((0)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (((0)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน (((0)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‰).((0)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. ๐Ÿฎ.๐Ÿฎ.๐Ÿฏ (๐‘ƒโ‚„).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘š๐‘œ๐‘‘๐‘ข๐‘ -๐‘๐‘œ๐‘›๐‘’๐‘›๐‘  ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: ((P โŸน Q), P) โŠข Q, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (((0)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚โ‚): ((((0)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((((0)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((๐งโ‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((๐งโ‚)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚). ๐–ซ๐–พ๐— ๐งโ‚ = ((0)++)++.๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘ฃ๐‘Ž๐‘Ÿ๐‘–๐‘Ž๐‘๐‘™๐‘’-๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, ๐›ท) โŠข P', ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((((0)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((((0)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚โ‚‚): ((((0)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((((0)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((((0)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚).(((0)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚€).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘š๐‘œ๐‘‘๐‘ข๐‘ -๐‘๐‘œ๐‘›๐‘’๐‘›๐‘  ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: ((P โŸน Q), P) โŠข Q, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((((0)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚โ‚ƒ): (((((0)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน (((((0)++)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((๐งโ‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((๐งโ‚)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚). ๐–ซ๐–พ๐— ๐งโ‚ = (((0)++)++)++.๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘ฃ๐‘Ž๐‘Ÿ๐‘–๐‘Ž๐‘๐‘™๐‘’-๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, ๐›ท) โŠข P', ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (((((0)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน (((((0)++)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚โ‚„): (((((0)++)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (((((0)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน (((((0)++)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚ƒ).((((0)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚‚).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘š๐‘œ๐‘‘๐‘ข๐‘ -๐‘๐‘œ๐‘›๐‘’๐‘›๐‘  ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: ((P โŸน Q), P) โŠข Q, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (((((0)++)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). โˆŽ ๐—œ๐—ป๐—ณ๐—ฒ๐—ฟ๐—ฒ๐—ป๐—ฐ๐—ฒ ๐—ฟ๐˜‚๐—น๐—ฒ (๐‘’๐‘ž๐‘ข๐‘Ž๐‘™๐‘–๐‘ก๐‘ฆ-๐‘๐‘œ๐‘š๐‘š๐‘ข๐‘ก๐‘Ž๐‘ก๐‘–๐‘ฃ๐‘–๐‘ก๐‘ฆ): ๐–ซ๐–พ๐— ๐‘–๐‘›๐‘“๐‘’๐‘Ÿ๐‘’๐‘›๐‘๐‘’-๐‘Ÿ๐‘ข๐‘™๐‘’ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™๐‘–๐‘ก๐‘ฆ-๐‘๐‘œ๐‘š๐‘š๐‘ข๐‘ก๐‘Ž๐‘ก๐‘–๐‘ฃ๐‘–๐‘ก๐‘ฆ ๐–ฝ๐–พ๐–ฟ๐—‚๐—‡๐–พ๐–ฝ ๐–บ๐—Œ โŒœ(x = y) โŠข (y = x)โŒ ๐–ป๐–พ ๐—‚๐—‡๐–ผ๐—…๐—Ž๐–ฝ๐–พ๐–ฝ ๐–บ๐—‡๐–ฝ ๐–ผ๐—ˆ๐—‡๐—Œ๐—‚๐–ฝ๐–พ๐—‹๐–พ๐–ฝ ๐—๐–บ๐—…๐—‚๐–ฝ ๐—‚๐—‡ ๐’ฏโ‚. -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚โ‚…): ((0)++ = 1). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (1 = (0)++) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚…). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™๐‘–๐‘ก๐‘ฆ-๐‘๐‘œ๐‘š๐‘š๐‘ข๐‘ก๐‘Ž๐‘ก๐‘–๐‘ฃ๐‘–๐‘ก๐‘ฆ ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((0)++ = 1). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚โ‚…): ((0)++ = 1). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (1 = (0)++) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚…). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™๐‘–๐‘ก๐‘ฆ-๐‘๐‘œ๐‘š๐‘š๐‘ข๐‘ก๐‘Ž๐‘ก๐‘–๐‘ฃ๐‘–๐‘ก๐‘ฆ ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (x = y) โŠข (y = x), ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((0)++ = 1). โˆŽ ๐—œ๐—ป๐—ณ๐—ฒ๐—ฟ๐—ฒ๐—ป๐—ฐ๐—ฒ ๐—ฟ๐˜‚๐—น๐—ฒ (๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘›): ๐–ซ๐–พ๐— ๐‘–๐‘›๐‘“๐‘’๐‘Ÿ๐‘’๐‘›๐‘๐‘’-๐‘Ÿ๐‘ข๐‘™๐‘’ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐–ฝ๐–พ๐–ฟ๐—‚๐—‡๐–พ๐–ฝ ๐–บ๐—Œ โŒœ(P, (Q = R)) โŠข P'โŒ ๐–ป๐–พ ๐—‚๐—‡๐–ผ๐—…๐—Ž๐–ฝ๐–พ๐–ฝ ๐–บ๐—‡๐–ฝ ๐–ผ๐—ˆ๐—‡๐—Œ๐—‚๐–ฝ๐–พ๐—‹๐–พ๐–ฝ ๐—๐–บ๐—…๐—‚๐–ฝ ๐—‚๐—‡ ๐’ฏโ‚. -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚โ‚†): (2 = (1)++). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (2 = ((0)++)++) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚†). ((0)++ = 1) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚…). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (2 = (1)++). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚โ‚‡): (((0)++)++ = 2). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (2 = ((0)++)++) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚†). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™๐‘–๐‘ก๐‘ฆ-๐‘๐‘œ๐‘š๐‘š๐‘ข๐‘ก๐‘Ž๐‘ก๐‘–๐‘ฃ๐‘–๐‘ก๐‘ฆ ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (((0)++)++ = 2). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚โ‚ˆ): (3 = (2)++). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (3 = (((0)++)++)++) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‡). (((0)++)++ = 2) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚‡). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (3 = (2)++). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚โ‚†): (2 = (1)++). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (2 = ((0)++)++) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚†). ((0)++ = 1) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚…).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, (Q = R)) โŠข P', ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (2 = (1)++). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚โ‚‡): (((0)++)++ = 2). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (2 = ((0)++)++) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚†). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™๐‘–๐‘ก๐‘ฆ-๐‘๐‘œ๐‘š๐‘š๐‘ข๐‘ก๐‘Ž๐‘ก๐‘–๐‘ฃ๐‘–๐‘ก๐‘ฆ ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (x = y) โŠข (y = x), ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (((0)++)++ = 2). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚โ‚ˆ): (3 = (2)++). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (3 = (((0)++)++)++) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‡). (((0)++)++ = 2) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚‡).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, (Q = R)) โŠข P', ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (3 = (2)++). โˆŽ ### ๐Ÿฏ ๐—ถ๐˜€ ๐—ฎ ๐—ป๐—ฎ๐˜๐˜‚๐—ฟ๐—ฎ๐—น ๐—ป๐˜‚๐—บ๐—ฏ๐—ฒ๐—ฟ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚โ‚‰): ((((0)++)++)++ = 3). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (3 = (((0)++)++)++) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‡). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™๐‘–๐‘ก๐‘ฆ-๐‘๐‘œ๐‘š๐‘š๐‘ข๐‘ก๐‘Ž๐‘ก๐‘–๐‘ฃ๐‘–๐‘ก๐‘ฆ ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((((0)++)++)++ = 3). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‚โ‚€): ((2)++ = 3). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((((0)++)++)++ = 3) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚‰). (((0)++)++ = 2) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚‡). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((2)++ = 3). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป ๐Ÿฎ.๐Ÿญ.๐Ÿฐ (๐’ฏโ‚.๐‘ƒโ‚‚โ‚): (3 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((((0)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚‚). ((((0)++)++)++ = 3) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚‰). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (3 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‚โ‚‚): (4 = ((((0)++)++)++)++). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: โŒœ๐˜ž๐˜ฆ ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ ๐Ÿฃ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ๐Ÿข++, ๐Ÿค ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ (๐Ÿข++)++, ๐Ÿฅ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ((๐Ÿข++)++)++,๐˜ฆ๐˜ต๐˜ค. (๐˜๐˜ฏ ๐˜ฐ๐˜ต๐˜ฉ๐˜ฆ๐˜ณ ๐˜ธ๐˜ฐ๐˜ณ๐˜ฅ๐˜ด, ๐Ÿฃ := ๐Ÿข++, ๐Ÿค := ๐Ÿฃ++, ๐Ÿฅ := ๐Ÿค++, ๐˜ฆ๐˜ต๐˜ค. ๐˜๐˜ฏ ๐˜ต๐˜ฉ๐˜ช๐˜ด ๐˜ต๐˜ฆ๐˜น๐˜ต ๐˜ ๐˜ถ๐˜ด๐˜ฆ "๐˜น := ๐˜บ" ๐˜ต๐˜ฐ ๐˜ฅ๐˜ฆ๐˜ฏ๐˜ฐ๐˜ต๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ด๐˜ต๐˜ข๐˜ต๐˜ฆ๐˜ฎ๐˜ฆ๐˜ฏ๐˜ต ๐˜ต๐˜ฉ๐˜ข๐˜ต ๐˜น ๐˜ช๐˜ด ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ๐˜ฅ ๐˜ต๐˜ฐ ๐˜ฆ๐˜ฒ๐˜ถ๐˜ข๐˜ญ ๐˜บ.)โŒ ๐—‚๐—Œ ๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ ๐–ป๐—’ ๐—ฑ๐—ฒ๐—ณ. (๐ทโ‚). (4 = ((((0)++)++)++)++) ๐—‚๐—Œ ๐–บ๐—‡ ๐—‚๐—‡๐—๐–พ๐—‹๐—‰๐—‹๐–พ๐—๐–บ๐—๐—‚๐—ˆ๐—‡ ๐—ˆ๐–ฟ ๐—๐—๐–บ๐— ๐–ฝ๐–พ๐–ฟ๐—‚๐—‡๐—‚๐—๐—‚๐—ˆ๐—‡. ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘‘๐‘’๐‘“๐‘–๐‘›๐‘–๐‘ก๐‘–๐‘œ๐‘›-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (4 = ((((0)++)++)++)++). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‚โ‚ƒ): (((((0)++)++)++)++ = 4). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (4 = ((((0)++)++)++)++) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ˆ). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™๐‘–๐‘ก๐‘ฆ-๐‘๐‘œ๐‘š๐‘š๐‘ข๐‘ก๐‘Ž๐‘ก๐‘–๐‘ฃ๐‘–๐‘ก๐‘ฆ ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (((((0)++)++)++)++ = 4). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‚โ‚„): ((3)++ = 4). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (((((0)++)++)++)++ = 4) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚โ‚ƒ). ((((0)++)++)++ = 3) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚‰). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((3)++ = 4). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‚โ‚…): (((((0)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน (((((0)++)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (((((0)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน (((((0)++)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚ƒ). ((3)++ = 4) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚โ‚„). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (((((0)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน (((((0)++)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‚โ‚†): (4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (((((0)++)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚„). (((((0)++)++)++)++ = 4) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚โ‚ƒ). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚โ‚‰): ((((0)++)++)++ = 3). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (3 = (((0)++)++)++) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‡). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™๐‘–๐‘ก๐‘ฆ-๐‘๐‘œ๐‘š๐‘š๐‘ข๐‘ก๐‘Ž๐‘ก๐‘–๐‘ฃ๐‘–๐‘ก๐‘ฆ ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (x = y) โŠข (y = x), ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((((0)++)++)++ = 3). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‚โ‚€): ((2)++ = 3). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((((0)++)++)++ = 3) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚‰). (((0)++)++ = 2) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚‡).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, (Q = R)) โŠข P', ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((2)++ = 3). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป ๐Ÿฎ.๐Ÿญ.๐Ÿฐ (๐’ฏโ‚.๐‘ƒโ‚‚โ‚): (3 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((((0)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚‚). ((((0)++)++)++ = 3) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚‰).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, (Q = R)) โŠข P', ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (3 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‚โ‚‚): (4 = ((((0)++)++)++)++). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: โŒœ๐˜ž๐˜ฆ ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ ๐Ÿฃ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ๐Ÿข++, ๐Ÿค ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ (๐Ÿข++)++, ๐Ÿฅ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ((๐Ÿข++)++)++,๐˜ฆ๐˜ต๐˜ค. (๐˜๐˜ฏ ๐˜ฐ๐˜ต๐˜ฉ๐˜ฆ๐˜ณ ๐˜ธ๐˜ฐ๐˜ณ๐˜ฅ๐˜ด, ๐Ÿฃ := ๐Ÿข++, ๐Ÿค := ๐Ÿฃ++, ๐Ÿฅ := ๐Ÿค++, ๐˜ฆ๐˜ต๐˜ค. ๐˜๐˜ฏ ๐˜ต๐˜ฉ๐˜ช๐˜ด ๐˜ต๐˜ฆ๐˜น๐˜ต ๐˜ ๐˜ถ๐˜ด๐˜ฆ "๐˜น := ๐˜บ" ๐˜ต๐˜ฐ ๐˜ฅ๐˜ฆ๐˜ฏ๐˜ฐ๐˜ต๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ด๐˜ต๐˜ข๐˜ต๐˜ฆ๐˜ฎ๐˜ฆ๐˜ฏ๐˜ต ๐˜ต๐˜ฉ๐˜ข๐˜ต ๐˜น ๐˜ช๐˜ด ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ๐˜ฅ ๐˜ต๐˜ฐ ๐˜ฆ๐˜ฒ๐˜ถ๐˜ข๐˜ญ ๐˜บ.)โŒ ๐—‚๐—Œ ๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ ๐–ป๐—’ ๐—ฑ๐—ฒ๐—ณ. (๐ทโ‚). (4 = ((((0)++)++)++)++) ๐—‚๐—Œ ๐–บ๐—‡ ๐—‚๐—‡๐—๐–พ๐—‹๐—‰๐—‹๐–พ๐—๐–บ๐—๐—‚๐—ˆ๐—‡ ๐—ˆ๐–ฟ ๐—๐—๐–บ๐— ๐–ฝ๐–พ๐–ฟ๐—‚๐—‡๐—‚๐—๐—‚๐—ˆ๐—‡.๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘‘๐‘’๐‘“๐‘–๐‘›๐‘–๐‘ก๐‘–๐‘œ๐‘›-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: ๐’Ÿ โŠข P, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (4 = ((((0)++)++)++)++). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‚โ‚ƒ): (((((0)++)++)++)++ = 4). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (4 = ((((0)++)++)++)++) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ˆ). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™๐‘–๐‘ก๐‘ฆ-๐‘๐‘œ๐‘š๐‘š๐‘ข๐‘ก๐‘Ž๐‘ก๐‘–๐‘ฃ๐‘–๐‘ก๐‘ฆ ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (x = y) โŠข (y = x), ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (((((0)++)++)++)++ = 4). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‚โ‚„): ((3)++ = 4). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (((((0)++)++)++)++ = 4) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚โ‚ƒ). ((((0)++)++)++ = 3) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚‰).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, (Q = R)) โŠข P', ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((3)++ = 4). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‚โ‚…): (((((0)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน (((((0)++)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (((((0)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน (((((0)++)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚ƒ). ((3)++ = 4) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚โ‚„).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, (Q = R)) โŠข P', ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (((((0)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน (((((0)++)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‚โ‚†): (4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (((((0)++)++)++)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚„). (((((0)++)++)++)++ = 4) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚โ‚ƒ).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, (Q = R)) โŠข P', ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). โˆŽ ### ๐—”๐˜…๐—ถ๐—ผ๐—บ ๐Ÿฎ.๐Ÿฏ ๐—”๐˜…๐—ถ๐—ผ๐—บ ๐Ÿฎ.๐Ÿฏ (๐’ฏโ‚.๐ดโ‚ƒ): ๐–ซ๐–พ๐— ๐‘Ž๐‘ฅ๐‘–๐‘œ๐‘š ๐’œโ‚ƒ โŒœ๐Ÿข ๐˜ช๐˜ด ๐˜ฏ๐˜ฐ๐˜ต ๐˜ต๐˜ฉ๐˜ฆ ๐˜ด๐˜ถ๐˜ค๐˜ค๐˜ฆ๐˜ด๐˜ด๐˜ฐ๐˜ณ ๐˜ฐ๐˜ง ๐˜ข๐˜ฏ๐˜บ ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ; ๐˜ช.๐˜ฆ., ๐˜ธ๐˜ฆ ๐˜ฉ๐˜ข๐˜ท๐˜ฆ ๐˜ฏ++ โ‰  ๐Ÿข ๐˜ง๐˜ฐ๐˜ณ ๐˜ฆ๐˜ท๐˜ฆ๐˜ณ๐˜บ ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ๐˜ฏ.โŒ ๐–ป๐–พ ๐—‚๐—‡๐–ผ๐—…๐—Ž๐–ฝ๐–พ๐–ฝ (๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ) ๐—‚๐—‡ ๐’ฏโ‚. -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‚โ‚‡): ((๐งโ‚‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((๐งโ‚‚)++ โ‰  0)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: โŒœ๐Ÿข ๐˜ช๐˜ด ๐˜ฏ๐˜ฐ๐˜ต ๐˜ต๐˜ฉ๐˜ฆ ๐˜ด๐˜ถ๐˜ค๐˜ค๐˜ฆ๐˜ด๐˜ด๐˜ฐ๐˜ณ ๐˜ฐ๐˜ง ๐˜ข๐˜ฏ๐˜บ ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ; ๐˜ช.๐˜ฆ., ๐˜ธ๐˜ฆ ๐˜ฉ๐˜ข๐˜ท๐˜ฆ ๐˜ฏ++ โ‰  ๐Ÿข ๐˜ง๐˜ฐ๐˜ณ ๐˜ฆ๐˜ท๐˜ฆ๐˜ณ๐˜บ ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ๐˜ฏ.โŒ ๐—‚๐—Œ ๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ ๐–ป๐—’ ๐—ฎ๐˜…๐—ถ๐—ผ๐—บ ๐Ÿฎ.๐Ÿฏ (๐ดโ‚ƒ). ((๐งโ‚‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((๐งโ‚‚)++ โ‰  0)) ๐—‚๐—Œ ๐–บ ๐—๐–บ๐—…๐—‚๐–ฝ ๐–ฟ๐—ˆ๐—‹๐—†๐—Ž๐—…๐–บ ๐—Œ๐—๐–บ๐—๐–พ๐—†๐–พ๐—‡๐— ๐—‚๐—‡๐—๐–พ๐—‹๐—‰๐—‹๐–พ๐—๐–พ๐–ฝ ๐–ฟ๐—‹๐—ˆ๐—† ๐—๐—๐–บ๐— ๐–บ๐—‘๐—‚๐—ˆ๐—†. ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘Ž๐‘ฅ๐‘–๐‘œ๐‘š-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((๐งโ‚‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((๐งโ‚‚)++ โ‰  0)). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‚โ‚‡): ((๐งโ‚‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((๐งโ‚‚)++ โ‰  0)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: โŒœ๐Ÿข ๐˜ช๐˜ด ๐˜ฏ๐˜ฐ๐˜ต ๐˜ต๐˜ฉ๐˜ฆ ๐˜ด๐˜ถ๐˜ค๐˜ค๐˜ฆ๐˜ด๐˜ด๐˜ฐ๐˜ณ ๐˜ฐ๐˜ง ๐˜ข๐˜ฏ๐˜บ ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ; ๐˜ช.๐˜ฆ., ๐˜ธ๐˜ฆ ๐˜ฉ๐˜ข๐˜ท๐˜ฆ ๐˜ฏ++ โ‰  ๐Ÿข ๐˜ง๐˜ฐ๐˜ณ ๐˜ฆ๐˜ท๐˜ฆ๐˜ณ๐˜บ ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ๐˜ฏ.โŒ ๐—‚๐—Œ ๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ ๐–ป๐—’ ๐—ฎ๐˜…๐—ถ๐—ผ๐—บ ๐Ÿฎ.๐Ÿฏ (๐ดโ‚ƒ). ((๐งโ‚‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((๐งโ‚‚)++ โ‰  0)) ๐—‚๐—Œ ๐–บ ๐—๐–บ๐—…๐—‚๐–ฝ ๐–ฟ๐—ˆ๐—‹๐—†๐—Ž๐—…๐–บ ๐—Œ๐—๐–บ๐—๐–พ๐—†๐–พ๐—‡๐— ๐—‚๐—‡๐—๐–พ๐—‹๐—‰๐—‹๐–พ๐—๐–พ๐–ฝ ๐–ฟ๐—‹๐—ˆ๐—† ๐—๐—๐–บ๐— ๐–บ๐—‘๐—‚๐—ˆ๐—†.๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘Ž๐‘ฅ๐‘–๐‘œ๐‘š-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: ๐’œ โŠข P, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((๐งโ‚‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((๐งโ‚‚)++ โ‰  0)). โˆŽ ### ๐Ÿฐ ๐—ถ๐˜€ ๐—ป๐—ผ๐˜ ๐—ฒ๐—พ๐˜‚๐—ฎ๐—น ๐˜๐—ผ ๐Ÿฌ. -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‚โ‚ˆ): ((3 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((3)++ โ‰  0)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((๐งโ‚‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((๐งโ‚‚)++ โ‰  0)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚โ‚‡). ๐–ซ๐–พ๐— ๐งโ‚‚ = 3. ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘ฃ๐‘Ž๐‘Ÿ๐‘–๐‘Ž๐‘๐‘™๐‘’-๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((3 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((3)++ โ‰  0)). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‚โ‚‰): ((3)++ โ‰  0). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((3 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((3)++ โ‰  0)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚โ‚ˆ).(3 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. ๐Ÿฎ.๐Ÿญ.๐Ÿฐ (๐‘ƒโ‚‚โ‚). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘š๐‘œ๐‘‘๐‘ข๐‘ -๐‘๐‘œ๐‘›๐‘’๐‘›๐‘  ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((3)++ โ‰  0). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป ๐Ÿฎ.๐Ÿญ.๐Ÿฒ (๐’ฏโ‚.๐‘ƒโ‚ƒโ‚€): (4 โ‰  0). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((3)++ โ‰  0) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚โ‚‰). ((3)++ = 4) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚โ‚„). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (4 โ‰  0). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‚โ‚ˆ): ((3 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((3)++ โ‰  0)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((๐งโ‚‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((๐งโ‚‚)++ โ‰  0)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚โ‚‡). ๐–ซ๐–พ๐— ๐งโ‚‚ = 3.๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘ฃ๐‘Ž๐‘Ÿ๐‘–๐‘Ž๐‘๐‘™๐‘’-๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, ๐›ท) โŠข P', ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((3 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((3)++ โ‰  0)). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‚โ‚‰): ((3)++ โ‰  0). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((3 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((3)++ โ‰  0)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚โ‚ˆ).(3 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. ๐Ÿฎ.๐Ÿญ.๐Ÿฐ (๐‘ƒโ‚‚โ‚).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘š๐‘œ๐‘‘๐‘ข๐‘ -๐‘๐‘œ๐‘›๐‘’๐‘›๐‘  ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: ((P โŸน Q), P) โŠข Q, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((3)++ โ‰  0). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป ๐Ÿฎ.๐Ÿญ.๐Ÿฒ (๐’ฏโ‚.๐‘ƒโ‚ƒโ‚€): (4 โ‰  0). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((3)++ โ‰  0) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚โ‚‰). ((3)++ = 4) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚โ‚„).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, (Q = R)) โŠข P', ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (4 โ‰  0). โˆŽ ### ๐—”๐˜…๐—ถ๐—ผ๐—บ ๐Ÿฎ.๐Ÿฐ ๐—”๐˜…๐—ถ๐—ผ๐—บ ๐Ÿฎ.๐Ÿฐ (๐’ฏโ‚.๐ดโ‚„): ๐–ซ๐–พ๐— ๐‘Ž๐‘ฅ๐‘–๐‘œ๐‘š ๐’œโ‚„ โŒœ๐˜‹๐˜ช๐˜ง๐˜ง๐˜ฆ๐˜ณ๐˜ฆ๐˜ฏ๐˜ต ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ๐˜ด ๐˜ฎ๐˜ถ๐˜ด๐˜ต ๐˜ฉ๐˜ข๐˜ท๐˜ฆ ๐˜ฅ๐˜ช๐˜ง๐˜ง๐˜ฆ๐˜ณ๐˜ฆ๐˜ฏ๐˜ต ๐˜ด๐˜ถ๐˜ค๐˜ค๐˜ฆ๐˜ด๐˜ด๐˜ฐ๐˜ณ๐˜ด; ๐˜ช.๐˜ฆ., ๐˜ช๐˜ง ๐˜ฏ, ๐˜ฎ ๐˜ข๐˜ณ๐˜ฆ ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ๐˜ด ๐˜ข๐˜ฏ๐˜ฅ ๐˜ฏ โ‰  ๐˜ฎ, ๐˜ต๐˜ฉ๐˜ฆ๐˜ฏ ๐˜ฏ++ โ‰  ๐˜ฎ++. ๐˜Œ๐˜ฒ๐˜ถ๐˜ช๐˜ท๐˜ข๐˜ญ๐˜ฆ๐˜ฏ๐˜ต๐˜ญ๐˜บ, ๐˜ช๐˜ง ๐˜ฏ++ = ๐˜ฎ++, ๐˜ต๐˜ฉ๐˜ฆ๐˜ฏ ๐˜ธ๐˜ฆ ๐˜ฎ๐˜ถ๐˜ด๐˜ต ๐˜ฉ๐˜ข๐˜ท๐˜ฆ ๐˜ฏ = ๐˜ฎ.โŒ ๐–ป๐–พ ๐—‚๐—‡๐–ผ๐—…๐—Ž๐–ฝ๐–พ๐–ฝ (๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ) ๐—‚๐—‡ ๐’ฏโ‚. -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚ƒโ‚): ((((๐งโ‚ƒ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (๐ฆโ‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (๐งโ‚ƒ โ‰  ๐ฆโ‚)) โŸน ((๐งโ‚ƒ)++ โ‰  (๐ฆโ‚)++)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: โŒœ๐˜‹๐˜ช๐˜ง๐˜ง๐˜ฆ๐˜ณ๐˜ฆ๐˜ฏ๐˜ต ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ๐˜ด ๐˜ฎ๐˜ถ๐˜ด๐˜ต ๐˜ฉ๐˜ข๐˜ท๐˜ฆ ๐˜ฅ๐˜ช๐˜ง๐˜ง๐˜ฆ๐˜ณ๐˜ฆ๐˜ฏ๐˜ต ๐˜ด๐˜ถ๐˜ค๐˜ค๐˜ฆ๐˜ด๐˜ด๐˜ฐ๐˜ณ๐˜ด; ๐˜ช.๐˜ฆ., ๐˜ช๐˜ง ๐˜ฏ, ๐˜ฎ ๐˜ข๐˜ณ๐˜ฆ ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ๐˜ด ๐˜ข๐˜ฏ๐˜ฅ ๐˜ฏ โ‰  ๐˜ฎ, ๐˜ต๐˜ฉ๐˜ฆ๐˜ฏ ๐˜ฏ++ โ‰  ๐˜ฎ++. ๐˜Œ๐˜ฒ๐˜ถ๐˜ช๐˜ท๐˜ข๐˜ญ๐˜ฆ๐˜ฏ๐˜ต๐˜ญ๐˜บ, ๐˜ช๐˜ง ๐˜ฏ++ = ๐˜ฎ++, ๐˜ต๐˜ฉ๐˜ฆ๐˜ฏ ๐˜ธ๐˜ฆ ๐˜ฎ๐˜ถ๐˜ด๐˜ต ๐˜ฉ๐˜ข๐˜ท๐˜ฆ ๐˜ฏ = ๐˜ฎ.โŒ ๐—‚๐—Œ ๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ ๐–ป๐—’ ๐—ฎ๐˜…๐—ถ๐—ผ๐—บ ๐Ÿฎ.๐Ÿฐ (๐ดโ‚„). ((((๐งโ‚ƒ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (๐ฆโ‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (๐งโ‚ƒ โ‰  ๐ฆโ‚)) โŸน ((๐งโ‚ƒ)++ โ‰  (๐ฆโ‚)++)) ๐—‚๐—Œ ๐–บ ๐—๐–บ๐—…๐—‚๐–ฝ ๐–ฟ๐—ˆ๐—‹๐—†๐—Ž๐—…๐–บ ๐—Œ๐—๐–บ๐—๐–พ๐—†๐–พ๐—‡๐— ๐—‚๐—‡๐—๐–พ๐—‹๐—‰๐—‹๐–พ๐—๐–พ๐–ฝ ๐–ฟ๐—‹๐—ˆ๐—† ๐—๐—๐–บ๐— ๐–บ๐—‘๐—‚๐—ˆ๐—†. ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘Ž๐‘ฅ๐‘–๐‘œ๐‘š-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((((๐งโ‚ƒ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (๐ฆโ‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (๐งโ‚ƒ โ‰  ๐ฆโ‚)) โŸน ((๐งโ‚ƒ)++ โ‰  (๐ฆโ‚)++)). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚ƒโ‚‚): ((((๐งโ‚„ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (๐ฆโ‚‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง ((๐งโ‚„)++ = (๐ฆโ‚‚)++)) โŸน (๐งโ‚„ = ๐ฆโ‚‚)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: โŒœ๐˜‹๐˜ช๐˜ง๐˜ง๐˜ฆ๐˜ณ๐˜ฆ๐˜ฏ๐˜ต ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ๐˜ด ๐˜ฎ๐˜ถ๐˜ด๐˜ต ๐˜ฉ๐˜ข๐˜ท๐˜ฆ ๐˜ฅ๐˜ช๐˜ง๐˜ง๐˜ฆ๐˜ณ๐˜ฆ๐˜ฏ๐˜ต ๐˜ด๐˜ถ๐˜ค๐˜ค๐˜ฆ๐˜ด๐˜ด๐˜ฐ๐˜ณ๐˜ด; ๐˜ช.๐˜ฆ., ๐˜ช๐˜ง ๐˜ฏ, ๐˜ฎ ๐˜ข๐˜ณ๐˜ฆ ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ๐˜ด ๐˜ข๐˜ฏ๐˜ฅ ๐˜ฏ โ‰  ๐˜ฎ, ๐˜ต๐˜ฉ๐˜ฆ๐˜ฏ ๐˜ฏ++ โ‰  ๐˜ฎ++. ๐˜Œ๐˜ฒ๐˜ถ๐˜ช๐˜ท๐˜ข๐˜ญ๐˜ฆ๐˜ฏ๐˜ต๐˜ญ๐˜บ, ๐˜ช๐˜ง ๐˜ฏ++ = ๐˜ฎ++, ๐˜ต๐˜ฉ๐˜ฆ๐˜ฏ ๐˜ธ๐˜ฆ ๐˜ฎ๐˜ถ๐˜ด๐˜ต ๐˜ฉ๐˜ข๐˜ท๐˜ฆ ๐˜ฏ = ๐˜ฎ.โŒ ๐—‚๐—Œ ๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ ๐–ป๐—’ ๐—ฎ๐˜…๐—ถ๐—ผ๐—บ ๐Ÿฎ.๐Ÿฐ (๐ดโ‚„). ((((๐งโ‚„ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (๐ฆโ‚‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง ((๐งโ‚„)++ = (๐ฆโ‚‚)++)) โŸน (๐งโ‚„ = ๐ฆโ‚‚)) ๐—‚๐—Œ ๐–บ ๐—๐–บ๐—…๐—‚๐–ฝ ๐–ฟ๐—ˆ๐—‹๐—†๐—Ž๐—…๐–บ ๐—Œ๐—๐–บ๐—๐–พ๐—†๐–พ๐—‡๐— ๐—‚๐—‡๐—๐–พ๐—‹๐—‰๐—‹๐–พ๐—๐–พ๐–ฝ ๐–ฟ๐—‹๐—ˆ๐—† ๐—๐—๐–บ๐— ๐–บ๐—‘๐—‚๐—ˆ๐—†. ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘Ž๐‘ฅ๐‘–๐‘œ๐‘š-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((((๐งโ‚„ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (๐ฆโ‚‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง ((๐งโ‚„)++ = (๐ฆโ‚‚)++)) โŸน (๐งโ‚„ = ๐ฆโ‚‚)). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚ƒโ‚): ((((๐งโ‚ƒ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (๐ฆโ‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (๐งโ‚ƒ โ‰  ๐ฆโ‚)) โŸน ((๐งโ‚ƒ)++ โ‰  (๐ฆโ‚)++)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: โŒœ๐˜‹๐˜ช๐˜ง๐˜ง๐˜ฆ๐˜ณ๐˜ฆ๐˜ฏ๐˜ต ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ๐˜ด ๐˜ฎ๐˜ถ๐˜ด๐˜ต ๐˜ฉ๐˜ข๐˜ท๐˜ฆ ๐˜ฅ๐˜ช๐˜ง๐˜ง๐˜ฆ๐˜ณ๐˜ฆ๐˜ฏ๐˜ต ๐˜ด๐˜ถ๐˜ค๐˜ค๐˜ฆ๐˜ด๐˜ด๐˜ฐ๐˜ณ๐˜ด; ๐˜ช.๐˜ฆ., ๐˜ช๐˜ง ๐˜ฏ, ๐˜ฎ ๐˜ข๐˜ณ๐˜ฆ ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ๐˜ด ๐˜ข๐˜ฏ๐˜ฅ ๐˜ฏ โ‰  ๐˜ฎ, ๐˜ต๐˜ฉ๐˜ฆ๐˜ฏ ๐˜ฏ++ โ‰  ๐˜ฎ++. ๐˜Œ๐˜ฒ๐˜ถ๐˜ช๐˜ท๐˜ข๐˜ญ๐˜ฆ๐˜ฏ๐˜ต๐˜ญ๐˜บ, ๐˜ช๐˜ง ๐˜ฏ++ = ๐˜ฎ++, ๐˜ต๐˜ฉ๐˜ฆ๐˜ฏ ๐˜ธ๐˜ฆ ๐˜ฎ๐˜ถ๐˜ด๐˜ต ๐˜ฉ๐˜ข๐˜ท๐˜ฆ ๐˜ฏ = ๐˜ฎ.โŒ ๐—‚๐—Œ ๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ ๐–ป๐—’ ๐—ฎ๐˜…๐—ถ๐—ผ๐—บ ๐Ÿฎ.๐Ÿฐ (๐ดโ‚„). ((((๐งโ‚ƒ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (๐ฆโ‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (๐งโ‚ƒ โ‰  ๐ฆโ‚)) โŸน ((๐งโ‚ƒ)++ โ‰  (๐ฆโ‚)++)) ๐—‚๐—Œ ๐–บ ๐—๐–บ๐—…๐—‚๐–ฝ ๐–ฟ๐—ˆ๐—‹๐—†๐—Ž๐—…๐–บ ๐—Œ๐—๐–บ๐—๐–พ๐—†๐–พ๐—‡๐— ๐—‚๐—‡๐—๐–พ๐—‹๐—‰๐—‹๐–พ๐—๐–พ๐–ฝ ๐–ฟ๐—‹๐—ˆ๐—† ๐—๐—๐–บ๐— ๐–บ๐—‘๐—‚๐—ˆ๐—†.๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘Ž๐‘ฅ๐‘–๐‘œ๐‘š-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: ๐’œ โŠข P, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((((๐งโ‚ƒ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (๐ฆโ‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (๐งโ‚ƒ โ‰  ๐ฆโ‚)) โŸน ((๐งโ‚ƒ)++ โ‰  (๐ฆโ‚)++)). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚ƒโ‚‚): ((((๐งโ‚„ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (๐ฆโ‚‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง ((๐งโ‚„)++ = (๐ฆโ‚‚)++)) โŸน (๐งโ‚„ = ๐ฆโ‚‚)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: โŒœ๐˜‹๐˜ช๐˜ง๐˜ง๐˜ฆ๐˜ณ๐˜ฆ๐˜ฏ๐˜ต ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ๐˜ด ๐˜ฎ๐˜ถ๐˜ด๐˜ต ๐˜ฉ๐˜ข๐˜ท๐˜ฆ ๐˜ฅ๐˜ช๐˜ง๐˜ง๐˜ฆ๐˜ณ๐˜ฆ๐˜ฏ๐˜ต ๐˜ด๐˜ถ๐˜ค๐˜ค๐˜ฆ๐˜ด๐˜ด๐˜ฐ๐˜ณ๐˜ด; ๐˜ช.๐˜ฆ., ๐˜ช๐˜ง ๐˜ฏ, ๐˜ฎ ๐˜ข๐˜ณ๐˜ฆ ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ๐˜ด ๐˜ข๐˜ฏ๐˜ฅ ๐˜ฏ โ‰  ๐˜ฎ, ๐˜ต๐˜ฉ๐˜ฆ๐˜ฏ ๐˜ฏ++ โ‰  ๐˜ฎ++. ๐˜Œ๐˜ฒ๐˜ถ๐˜ช๐˜ท๐˜ข๐˜ญ๐˜ฆ๐˜ฏ๐˜ต๐˜ญ๐˜บ, ๐˜ช๐˜ง ๐˜ฏ++ = ๐˜ฎ++, ๐˜ต๐˜ฉ๐˜ฆ๐˜ฏ ๐˜ธ๐˜ฆ ๐˜ฎ๐˜ถ๐˜ด๐˜ต ๐˜ฉ๐˜ข๐˜ท๐˜ฆ ๐˜ฏ = ๐˜ฎ.โŒ ๐—‚๐—Œ ๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ ๐–ป๐—’ ๐—ฎ๐˜…๐—ถ๐—ผ๐—บ ๐Ÿฎ.๐Ÿฐ (๐ดโ‚„). ((((๐งโ‚„ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (๐ฆโ‚‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง ((๐งโ‚„)++ = (๐ฆโ‚‚)++)) โŸน (๐งโ‚„ = ๐ฆโ‚‚)) ๐—‚๐—Œ ๐–บ ๐—๐–บ๐—…๐—‚๐–ฝ ๐–ฟ๐—ˆ๐—‹๐—†๐—Ž๐—…๐–บ ๐—Œ๐—๐–บ๐—๐–พ๐—†๐–พ๐—‡๐— ๐—‚๐—‡๐—๐–พ๐—‹๐—‰๐—‹๐–พ๐—๐–พ๐–ฝ ๐–ฟ๐—‹๐—ˆ๐—† ๐—๐—๐–บ๐— ๐–บ๐—‘๐—‚๐—ˆ๐—†.๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘Ž๐‘ฅ๐‘–๐‘œ๐‘š-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: ๐’œ โŠข P, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((((๐งโ‚„ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (๐ฆโ‚‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง ((๐งโ‚„)++ = (๐ฆโ‚‚)++)) โŸน (๐งโ‚„ = ๐ฆโ‚‚)). โˆŽ ### ๐Ÿฒ ๐—ถ๐˜€ ๐—ป๐—ผ๐˜ ๐—ฒ๐—พ๐˜‚๐—ฎ๐—น ๐˜๐—ผ ๐Ÿฎ. -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚ƒโ‚ƒ): ((((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (4 โ‰  0)) โŸน ((4)++ โ‰  (0)++)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((((๐งโ‚ƒ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (๐ฆโ‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (๐งโ‚ƒ โ‰  ๐ฆโ‚)) โŸน ((๐งโ‚ƒ)++ โ‰  (๐ฆโ‚)++)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒโ‚). ๐–ซ๐–พ๐— ๐งโ‚ƒ = 4, ๐ฆโ‚ = 0. ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘ฃ๐‘Ž๐‘Ÿ๐‘–๐‘Ž๐‘๐‘™๐‘’-๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (4 โ‰  0)) โŸน ((4)++ โ‰  (0)++)). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚ƒโ‚ƒ): ((((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (4 โ‰  0)) โŸน ((4)++ โ‰  (0)++)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((((๐งโ‚ƒ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (๐ฆโ‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (๐งโ‚ƒ โ‰  ๐ฆโ‚)) โŸน ((๐งโ‚ƒ)++ โ‰  (๐ฆโ‚)++)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒโ‚). ๐–ซ๐–พ๐— ๐งโ‚ƒ = 4, ๐ฆโ‚ = 0.๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘ฃ๐‘Ž๐‘Ÿ๐‘–๐‘Ž๐‘๐‘™๐‘’-๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, ๐›ท) โŠข P', ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (4 โ‰  0)) โŸน ((4)++ โ‰  (0)++)). โˆŽ ๐—œ๐—ป๐—ณ๐—ฒ๐—ฟ๐—ฒ๐—ป๐—ฐ๐—ฒ ๐—ฟ๐˜‚๐—น๐—ฒ (๐‘๐‘œ๐‘›๐‘—๐‘ข๐‘›๐‘๐‘ก๐‘–๐‘œ๐‘›-๐‘–๐‘›๐‘ก๐‘Ÿ๐‘œ๐‘‘๐‘ข๐‘๐‘ก๐‘–๐‘œ๐‘›): ๐–ซ๐–พ๐— ๐‘–๐‘›๐‘“๐‘’๐‘Ÿ๐‘’๐‘›๐‘๐‘’-๐‘Ÿ๐‘ข๐‘™๐‘’ ๐‘๐‘œ๐‘›๐‘—๐‘ข๐‘›๐‘๐‘ก๐‘–๐‘œ๐‘›-๐‘–๐‘›๐‘ก๐‘Ÿ๐‘œ๐‘‘๐‘ข๐‘๐‘ก๐‘–๐‘œ๐‘› ๐–ฝ๐–พ๐–ฟ๐—‚๐—‡๐–พ๐–ฝ ๐–บ๐—Œ โŒœ(P, Q) โŠข (P โ‹€ Q)โŒ ๐–ป๐–พ ๐—‚๐—‡๐–ผ๐—…๐—Ž๐–ฝ๐–พ๐–ฝ ๐–บ๐—‡๐–ฝ ๐–ผ๐—ˆ๐—‡๐—Œ๐—‚๐–ฝ๐–พ๐—‹๐–พ๐–ฝ ๐—๐–บ๐—…๐—‚๐–ฝ ๐—‚๐—‡ ๐’ฏโ‚. -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚ƒโ‚„): ((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚โ‚†). (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘๐‘œ๐‘›๐‘—๐‘ข๐‘›๐‘๐‘ก๐‘–๐‘œ๐‘›-๐‘–๐‘›๐‘ก๐‘Ÿ๐‘œ๐‘‘๐‘ข๐‘๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚ƒโ‚…): (((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (4 โ‰  0)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒโ‚„). (4 โ‰  0) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. ๐Ÿฎ.๐Ÿญ.๐Ÿฒ (๐‘ƒโ‚ƒโ‚€). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘๐‘œ๐‘›๐‘—๐‘ข๐‘›๐‘๐‘ก๐‘–๐‘œ๐‘›-๐‘–๐‘›๐‘ก๐‘Ÿ๐‘œ๐‘‘๐‘ข๐‘๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (4 โ‰  0)). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚ƒโ‚†): ((4)++ โ‰  (0)++). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (4 โ‰  0)) โŸน ((4)++ โ‰  (0)++)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒโ‚ƒ).(((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (4 โ‰  0)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒโ‚…). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘š๐‘œ๐‘‘๐‘ข๐‘ -๐‘๐‘œ๐‘›๐‘’๐‘›๐‘  ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((4)++ โ‰  (0)++). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚ƒโ‚‡): (5 = (((((0)++)++)++)++)++). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: โŒœ๐˜ž๐˜ฆ ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ ๐Ÿฃ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ๐Ÿข++, ๐Ÿค ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ (๐Ÿข++)++, ๐Ÿฅ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ((๐Ÿข++)++)++,๐˜ฆ๐˜ต๐˜ค. (๐˜๐˜ฏ ๐˜ฐ๐˜ต๐˜ฉ๐˜ฆ๐˜ณ ๐˜ธ๐˜ฐ๐˜ณ๐˜ฅ๐˜ด, ๐Ÿฃ := ๐Ÿข++, ๐Ÿค := ๐Ÿฃ++, ๐Ÿฅ := ๐Ÿค++, ๐˜ฆ๐˜ต๐˜ค. ๐˜๐˜ฏ ๐˜ต๐˜ฉ๐˜ช๐˜ด ๐˜ต๐˜ฆ๐˜น๐˜ต ๐˜ ๐˜ถ๐˜ด๐˜ฆ "๐˜น := ๐˜บ" ๐˜ต๐˜ฐ ๐˜ฅ๐˜ฆ๐˜ฏ๐˜ฐ๐˜ต๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ด๐˜ต๐˜ข๐˜ต๐˜ฆ๐˜ฎ๐˜ฆ๐˜ฏ๐˜ต ๐˜ต๐˜ฉ๐˜ข๐˜ต ๐˜น ๐˜ช๐˜ด ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ๐˜ฅ ๐˜ต๐˜ฐ ๐˜ฆ๐˜ฒ๐˜ถ๐˜ข๐˜ญ ๐˜บ.)โŒ ๐—‚๐—Œ ๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ ๐–ป๐—’ ๐—ฑ๐—ฒ๐—ณ. (๐ทโ‚). (5 = (((((0)++)++)++)++)++) ๐—‚๐—Œ ๐–บ๐—‡ ๐—‚๐—‡๐—๐–พ๐—‹๐—‰๐—‹๐–พ๐—๐–บ๐—๐—‚๐—ˆ๐—‡ ๐—ˆ๐–ฟ ๐—๐—๐–บ๐— ๐–ฝ๐–พ๐–ฟ๐—‚๐—‡๐—‚๐—๐—‚๐—ˆ๐—‡. ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘‘๐‘’๐‘“๐‘–๐‘›๐‘–๐‘ก๐‘–๐‘œ๐‘›-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (5 = (((((0)++)++)++)++)++). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚ƒโ‚ˆ): ((((((0)++)++)++)++)++ = 5). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (5 = (((((0)++)++)++)++)++) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒโ‚‡). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™๐‘–๐‘ก๐‘ฆ-๐‘๐‘œ๐‘š๐‘š๐‘ข๐‘ก๐‘Ž๐‘ก๐‘–๐‘ฃ๐‘–๐‘ก๐‘ฆ ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((((((0)++)++)++)++)++ = 5). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚ƒโ‚‰): ((4)++ = 5). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((((((0)++)++)++)++)++ = 5) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒโ‚ˆ). (((((0)++)++)++)++ = 4) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚โ‚ƒ). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((4)++ = 5). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚„โ‚€): (5 = (4)++). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((4)++ = 5) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒโ‚‰). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™๐‘–๐‘ก๐‘ฆ-๐‘๐‘œ๐‘š๐‘š๐‘ข๐‘ก๐‘Ž๐‘ก๐‘–๐‘ฃ๐‘–๐‘ก๐‘ฆ ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (5 = (4)++). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚„โ‚): ((((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)) โŸน ((5)++ โ‰  (1)++)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((((๐งโ‚ƒ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (๐ฆโ‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (๐งโ‚ƒ โ‰  ๐ฆโ‚)) โŸน ((๐งโ‚ƒ)++ โ‰  (๐ฆโ‚)++)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒโ‚). ๐–ซ๐–พ๐— ๐งโ‚ƒ = 5, ๐ฆโ‚ = 1. ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘ฃ๐‘Ž๐‘Ÿ๐‘–๐‘Ž๐‘๐‘™๐‘’-๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)) โŸน ((5)++ โ‰  (1)++)). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚„โ‚‚): ((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((4)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((๐งโ‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((๐งโ‚)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚). ๐–ซ๐–พ๐— ๐งโ‚ = 4. ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘ฃ๐‘Ž๐‘Ÿ๐‘–๐‘Ž๐‘๐‘™๐‘’-๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((4)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚„โ‚ƒ): ((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน (5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((4)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚„โ‚‚). ((4)++ = 5) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒโ‚‰). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน (5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚„โ‚„): (5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน (5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚„โ‚ƒ).(4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚โ‚†). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘š๐‘œ๐‘‘๐‘ข๐‘ -๐‘๐‘œ๐‘›๐‘’๐‘›๐‘  ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚„โ‚…): ((((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)) โŸน ((5)++ โ‰  (1)++)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((((๐งโ‚ƒ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (๐ฆโ‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (๐งโ‚ƒ โ‰  ๐ฆโ‚)) โŸน ((๐งโ‚ƒ)++ โ‰  (๐ฆโ‚)++)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒโ‚). ๐–ซ๐–พ๐— ๐งโ‚ƒ = 5, ๐ฆโ‚ = 1. ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘ฃ๐‘Ž๐‘Ÿ๐‘–๐‘Ž๐‘๐‘™๐‘’-๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)) โŸน ((5)++ โ‰  (1)++)). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚„โ‚†): ((4)++ โ‰  (0)++). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (4 โ‰  0)) โŸน ((4)++ โ‰  (0)++)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒโ‚ƒ).(((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (4 โ‰  0)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒโ‚…). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘š๐‘œ๐‘‘๐‘ข๐‘ -๐‘๐‘œ๐‘›๐‘’๐‘›๐‘  ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((4)++ โ‰  (0)++). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚„โ‚‡): (5 โ‰  (0)++). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((4)++ โ‰  (0)++) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚„โ‚†). ((4)++ = 5) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒโ‚‰). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (5 โ‰  (0)++). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚„โ‚ˆ): (6 = ((((((0)++)++)++)++)++)++). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: โŒœ๐˜ž๐˜ฆ ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ ๐Ÿฃ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ๐Ÿข++, ๐Ÿค ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ (๐Ÿข++)++, ๐Ÿฅ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ((๐Ÿข++)++)++,๐˜ฆ๐˜ต๐˜ค. (๐˜๐˜ฏ ๐˜ฐ๐˜ต๐˜ฉ๐˜ฆ๐˜ณ ๐˜ธ๐˜ฐ๐˜ณ๐˜ฅ๐˜ด, ๐Ÿฃ := ๐Ÿข++, ๐Ÿค := ๐Ÿฃ++, ๐Ÿฅ := ๐Ÿค++, ๐˜ฆ๐˜ต๐˜ค. ๐˜๐˜ฏ ๐˜ต๐˜ฉ๐˜ช๐˜ด ๐˜ต๐˜ฆ๐˜น๐˜ต ๐˜ ๐˜ถ๐˜ด๐˜ฆ "๐˜น := ๐˜บ" ๐˜ต๐˜ฐ ๐˜ฅ๐˜ฆ๐˜ฏ๐˜ฐ๐˜ต๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ด๐˜ต๐˜ข๐˜ต๐˜ฆ๐˜ฎ๐˜ฆ๐˜ฏ๐˜ต ๐˜ต๐˜ฉ๐˜ข๐˜ต ๐˜น ๐˜ช๐˜ด ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ๐˜ฅ ๐˜ต๐˜ฐ ๐˜ฆ๐˜ฒ๐˜ถ๐˜ข๐˜ญ ๐˜บ.)โŒ ๐—‚๐—Œ ๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ ๐–ป๐—’ ๐—ฑ๐—ฒ๐—ณ. (๐ทโ‚). (6 = ((((((0)++)++)++)++)++)++) ๐—‚๐—Œ ๐–บ๐—‡ ๐—‚๐—‡๐—๐–พ๐—‹๐—‰๐—‹๐–พ๐—๐–บ๐—๐—‚๐—ˆ๐—‡ ๐—ˆ๐–ฟ ๐—๐—๐–บ๐— ๐–ฝ๐–พ๐–ฟ๐—‚๐—‡๐—‚๐—๐—‚๐—ˆ๐—‡. ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘‘๐‘’๐‘“๐‘–๐‘›๐‘–๐‘ก๐‘–๐‘œ๐‘›-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (6 = ((((((0)++)++)++)++)++)++). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚„โ‚‰): (((((((0)++)++)++)++)++)++ = 6). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (6 = ((((((0)++)++)++)++)++)++) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚„โ‚ˆ). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™๐‘–๐‘ก๐‘ฆ-๐‘๐‘œ๐‘š๐‘š๐‘ข๐‘ก๐‘Ž๐‘ก๐‘–๐‘ฃ๐‘–๐‘ก๐‘ฆ ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (((((((0)++)++)++)++)++)++ = 6). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚…โ‚€): (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((0)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. ๐Ÿฎ.๐Ÿฎ.๐Ÿฏ (๐‘ƒโ‚„). ((0)++ = 1) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚…). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚…โ‚): ((5)++ = 6). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (((((((0)++)++)++)++)++)++ = 6) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚„โ‚‰). ((((((0)++)++)++)++)++ = 5) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒโ‚ˆ). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((5)++ = 6). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚…โ‚‚): (6 = (5)++). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((5)++ = 6) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚…โ‚). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™๐‘–๐‘ก๐‘ฆ-๐‘๐‘œ๐‘š๐‘š๐‘ข๐‘ก๐‘Ž๐‘ก๐‘–๐‘ฃ๐‘–๐‘ก๐‘ฆ ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (6 = (5)++). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚ƒโ‚„): ((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) (๐‘ƒ) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚โ‚†). (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) (๐‘„) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘๐‘œ๐‘›๐‘—๐‘ข๐‘›๐‘๐‘ก๐‘–๐‘œ๐‘›-๐‘–๐‘›๐‘ก๐‘Ÿ๐‘œ๐‘‘๐‘ข๐‘๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, Q) โŠข (P โ‹€ Q), ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚ƒโ‚…): (((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (4 โ‰  0)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) (๐‘ƒ) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒโ‚„). (4 โ‰  0) (๐‘„) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. ๐Ÿฎ.๐Ÿญ.๐Ÿฒ (๐‘ƒโ‚ƒโ‚€).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘๐‘œ๐‘›๐‘—๐‘ข๐‘›๐‘๐‘ก๐‘–๐‘œ๐‘›-๐‘–๐‘›๐‘ก๐‘Ÿ๐‘œ๐‘‘๐‘ข๐‘๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, Q) โŠข (P โ‹€ Q), ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (4 โ‰  0)). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚ƒโ‚†): ((4)++ โ‰  (0)++). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (4 โ‰  0)) โŸน ((4)++ โ‰  (0)++)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒโ‚ƒ).(((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (4 โ‰  0)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒโ‚…).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘š๐‘œ๐‘‘๐‘ข๐‘ -๐‘๐‘œ๐‘›๐‘’๐‘›๐‘  ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: ((P โŸน Q), P) โŠข Q, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((4)++ โ‰  (0)++). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚ƒโ‚‡): (5 = (((((0)++)++)++)++)++). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: โŒœ๐˜ž๐˜ฆ ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ ๐Ÿฃ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ๐Ÿข++, ๐Ÿค ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ (๐Ÿข++)++, ๐Ÿฅ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ((๐Ÿข++)++)++,๐˜ฆ๐˜ต๐˜ค. (๐˜๐˜ฏ ๐˜ฐ๐˜ต๐˜ฉ๐˜ฆ๐˜ณ ๐˜ธ๐˜ฐ๐˜ณ๐˜ฅ๐˜ด, ๐Ÿฃ := ๐Ÿข++, ๐Ÿค := ๐Ÿฃ++, ๐Ÿฅ := ๐Ÿค++, ๐˜ฆ๐˜ต๐˜ค. ๐˜๐˜ฏ ๐˜ต๐˜ฉ๐˜ช๐˜ด ๐˜ต๐˜ฆ๐˜น๐˜ต ๐˜ ๐˜ถ๐˜ด๐˜ฆ "๐˜น := ๐˜บ" ๐˜ต๐˜ฐ ๐˜ฅ๐˜ฆ๐˜ฏ๐˜ฐ๐˜ต๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ด๐˜ต๐˜ข๐˜ต๐˜ฆ๐˜ฎ๐˜ฆ๐˜ฏ๐˜ต ๐˜ต๐˜ฉ๐˜ข๐˜ต ๐˜น ๐˜ช๐˜ด ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ๐˜ฅ ๐˜ต๐˜ฐ ๐˜ฆ๐˜ฒ๐˜ถ๐˜ข๐˜ญ ๐˜บ.)โŒ ๐—‚๐—Œ ๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ ๐–ป๐—’ ๐—ฑ๐—ฒ๐—ณ. (๐ทโ‚). (5 = (((((0)++)++)++)++)++) ๐—‚๐—Œ ๐–บ๐—‡ ๐—‚๐—‡๐—๐–พ๐—‹๐—‰๐—‹๐–พ๐—๐–บ๐—๐—‚๐—ˆ๐—‡ ๐—ˆ๐–ฟ ๐—๐—๐–บ๐— ๐–ฝ๐–พ๐–ฟ๐—‚๐—‡๐—‚๐—๐—‚๐—ˆ๐—‡.๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘‘๐‘’๐‘“๐‘–๐‘›๐‘–๐‘ก๐‘–๐‘œ๐‘›-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: ๐’Ÿ โŠข P, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (5 = (((((0)++)++)++)++)++). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚ƒโ‚ˆ): ((((((0)++)++)++)++)++ = 5). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (5 = (((((0)++)++)++)++)++) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒโ‚‡). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™๐‘–๐‘ก๐‘ฆ-๐‘๐‘œ๐‘š๐‘š๐‘ข๐‘ก๐‘Ž๐‘ก๐‘–๐‘ฃ๐‘–๐‘ก๐‘ฆ ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (x = y) โŠข (y = x), ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((((((0)++)++)++)++)++ = 5). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚ƒโ‚‰): ((4)++ = 5). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((((((0)++)++)++)++)++ = 5) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒโ‚ˆ). (((((0)++)++)++)++ = 4) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚โ‚ƒ).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, (Q = R)) โŠข P', ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((4)++ = 5). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚„โ‚€): (5 = (4)++). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((4)++ = 5) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒโ‚‰). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™๐‘–๐‘ก๐‘ฆ-๐‘๐‘œ๐‘š๐‘š๐‘ข๐‘ก๐‘Ž๐‘ก๐‘–๐‘ฃ๐‘–๐‘ก๐‘ฆ ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (x = y) โŠข (y = x), ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (5 = (4)++). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚„โ‚): ((((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)) โŸน ((5)++ โ‰  (1)++)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((((๐งโ‚ƒ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (๐ฆโ‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (๐งโ‚ƒ โ‰  ๐ฆโ‚)) โŸน ((๐งโ‚ƒ)++ โ‰  (๐ฆโ‚)++)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒโ‚). ๐–ซ๐–พ๐— ๐งโ‚ƒ = 5, ๐ฆโ‚ = 1.๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘ฃ๐‘Ž๐‘Ÿ๐‘–๐‘Ž๐‘๐‘™๐‘’-๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, ๐›ท) โŠข P', ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)) โŸน ((5)++ โ‰  (1)++)). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚„โ‚‚): ((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((4)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((๐งโ‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((๐งโ‚)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚). ๐–ซ๐–พ๐— ๐งโ‚ = 4.๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘ฃ๐‘Ž๐‘Ÿ๐‘–๐‘Ž๐‘๐‘™๐‘’-๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, ๐›ท) โŠข P', ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((4)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚„โ‚ƒ): ((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน (5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((4)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚„โ‚‚). ((4)++ = 5) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒโ‚‰).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, (Q = R)) โŠข P', ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน (5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚„โ‚„): (5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน (5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚„โ‚ƒ).(4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‚โ‚†).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘š๐‘œ๐‘‘๐‘ข๐‘ -๐‘๐‘œ๐‘›๐‘’๐‘›๐‘  ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: ((P โŸน Q), P) โŠข Q, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚„โ‚…): ((((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)) โŸน ((5)++ โ‰  (1)++)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((((๐งโ‚ƒ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (๐ฆโ‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (๐งโ‚ƒ โ‰  ๐ฆโ‚)) โŸน ((๐งโ‚ƒ)++ โ‰  (๐ฆโ‚)++)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒโ‚). ๐–ซ๐–พ๐— ๐งโ‚ƒ = 5, ๐ฆโ‚ = 1.๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘ฃ๐‘Ž๐‘Ÿ๐‘–๐‘Ž๐‘๐‘™๐‘’-๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, ๐›ท) โŠข P', ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)) โŸน ((5)++ โ‰  (1)++)). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚„โ‚†): ((4)++ โ‰  (0)++). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (4 โ‰  0)) โŸน ((4)++ โ‰  (0)++)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒโ‚ƒ).(((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (4 โ‰  0)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒโ‚…).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘š๐‘œ๐‘‘๐‘ข๐‘ -๐‘๐‘œ๐‘›๐‘’๐‘›๐‘  ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: ((P โŸน Q), P) โŠข Q, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((4)++ โ‰  (0)++). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚„โ‚‡): (5 โ‰  (0)++). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((4)++ โ‰  (0)++) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚„โ‚†). ((4)++ = 5) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒโ‚‰).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, (Q = R)) โŠข P', ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (5 โ‰  (0)++). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚„โ‚ˆ): (6 = ((((((0)++)++)++)++)++)++). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: โŒœ๐˜ž๐˜ฆ ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ ๐Ÿฃ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ๐Ÿข++, ๐Ÿค ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ (๐Ÿข++)++, ๐Ÿฅ ๐˜ต๐˜ฐ ๐˜ฃ๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ((๐Ÿข++)++)++,๐˜ฆ๐˜ต๐˜ค. (๐˜๐˜ฏ ๐˜ฐ๐˜ต๐˜ฉ๐˜ฆ๐˜ณ ๐˜ธ๐˜ฐ๐˜ณ๐˜ฅ๐˜ด, ๐Ÿฃ := ๐Ÿข++, ๐Ÿค := ๐Ÿฃ++, ๐Ÿฅ := ๐Ÿค++, ๐˜ฆ๐˜ต๐˜ค. ๐˜๐˜ฏ ๐˜ต๐˜ฉ๐˜ช๐˜ด ๐˜ต๐˜ฆ๐˜น๐˜ต ๐˜ ๐˜ถ๐˜ด๐˜ฆ "๐˜น := ๐˜บ" ๐˜ต๐˜ฐ ๐˜ฅ๐˜ฆ๐˜ฏ๐˜ฐ๐˜ต๐˜ฆ ๐˜ต๐˜ฉ๐˜ฆ ๐˜ด๐˜ต๐˜ข๐˜ต๐˜ฆ๐˜ฎ๐˜ฆ๐˜ฏ๐˜ต ๐˜ต๐˜ฉ๐˜ข๐˜ต ๐˜น ๐˜ช๐˜ด ๐˜ฅ๐˜ฆ๐˜ง๐˜ช๐˜ฏ๐˜ฆ๐˜ฅ ๐˜ต๐˜ฐ ๐˜ฆ๐˜ฒ๐˜ถ๐˜ข๐˜ญ ๐˜บ.)โŒ ๐—‚๐—Œ ๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ ๐–ป๐—’ ๐—ฑ๐—ฒ๐—ณ. (๐ทโ‚). (6 = ((((((0)++)++)++)++)++)++) ๐—‚๐—Œ ๐–บ๐—‡ ๐—‚๐—‡๐—๐–พ๐—‹๐—‰๐—‹๐–พ๐—๐–บ๐—๐—‚๐—ˆ๐—‡ ๐—ˆ๐–ฟ ๐—๐—๐–บ๐— ๐–ฝ๐–พ๐–ฟ๐—‚๐—‡๐—‚๐—๐—‚๐—ˆ๐—‡.๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘‘๐‘’๐‘“๐‘–๐‘›๐‘–๐‘ก๐‘–๐‘œ๐‘›-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: ๐’Ÿ โŠข P, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (6 = ((((((0)++)++)++)++)++)++). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚„โ‚‰): (((((((0)++)++)++)++)++)++ = 6). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (6 = ((((((0)++)++)++)++)++)++) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚„โ‚ˆ). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™๐‘–๐‘ก๐‘ฆ-๐‘๐‘œ๐‘š๐‘š๐‘ข๐‘ก๐‘Ž๐‘ก๐‘–๐‘ฃ๐‘–๐‘ก๐‘ฆ ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (x = y) โŠข (y = x), ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (((((((0)++)++)++)++)++)++ = 6). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚…โ‚€): (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((0)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. ๐Ÿฎ.๐Ÿฎ.๐Ÿฏ (๐‘ƒโ‚„). ((0)++ = 1) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚…).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, (Q = R)) โŠข P', ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚…โ‚): ((5)++ = 6). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (((((((0)++)++)++)++)++)++ = 6) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚„โ‚‰). ((((((0)++)++)++)++)++ = 5) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚ƒโ‚ˆ).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, (Q = R)) โŠข P', ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((5)++ = 6). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚…โ‚‚): (6 = (5)++). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((5)++ = 6) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚…โ‚). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™๐‘–๐‘ก๐‘ฆ-๐‘๐‘œ๐‘š๐‘š๐‘ข๐‘ก๐‘Ž๐‘ก๐‘–๐‘ฃ๐‘–๐‘ก๐‘ฆ ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (x = y) โŠข (y = x), ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (6 = (5)++). โˆŽ #### ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ ๐—ฏ๐˜† ๐—ฐ๐—ผ๐—ป๐˜๐—ฟ๐—ฎ๐—ฑ๐—ถ๐—ฐ๐˜๐—ถ๐—ผ๐—ป ๐—›๐˜†๐—ฝ๐—ผ๐˜๐—ต๐—ฒ๐˜€๐—ถ๐˜€ (๐’ฏโ‚.๐ปโ‚): (6 = 2). ๐–ณ๐—๐—‚๐—Œ ๐—๐—’๐—‰๐—ˆ๐—๐—๐–พ๐—Œ๐—‚๐—Œ ๐—‚๐—Œ ๐–พ๐—…๐–บ๐–ป๐—ˆ๐—‹๐–บ๐—๐–พ๐–ฝ ๐—‚๐—‡ ๐—๐—๐–พ๐—ˆ๐—‹๐—’ โ„‹โ‚. ๐—œ๐—ป๐—ณ๐—ฒ๐—ฟ๐—ฒ๐—ป๐—ฐ๐—ฒ ๐—ฟ๐˜‚๐—น๐—ฒ (๐‘–๐‘›๐‘๐‘œ๐‘›๐‘ ๐‘–๐‘ ๐‘ก๐‘’๐‘›๐‘๐‘ฆ-๐‘๐‘ฆ-๐‘–๐‘›๐‘’๐‘ž๐‘ข๐‘Ž๐‘™๐‘–๐‘ก๐‘ฆ-๐‘–๐‘›๐‘ก๐‘Ÿ๐‘œ๐‘‘๐‘ข๐‘๐‘ก๐‘–๐‘œ๐‘›): ๐–ซ๐–พ๐— ๐‘–๐‘›๐‘“๐‘’๐‘Ÿ๐‘’๐‘›๐‘๐‘’-๐‘Ÿ๐‘ข๐‘™๐‘’ ๐‘–๐‘›๐‘๐‘œ๐‘›๐‘ ๐‘–๐‘ ๐‘ก๐‘’๐‘›๐‘๐‘ฆ-๐‘๐‘ฆ-๐‘–๐‘›๐‘’๐‘ž๐‘ข๐‘Ž๐‘™๐‘–๐‘ก๐‘ฆ-๐‘–๐‘›๐‘ก๐‘Ÿ๐‘œ๐‘‘๐‘ข๐‘๐‘ก๐‘–๐‘œ๐‘› ๐–ฝ๐–พ๐–ฟ๐—‚๐—‡๐–พ๐–ฝ ๐–บ๐—Œ โŒœ((P = Q), (P โ‰  Q)) โŠข Inc(๐’ฏ)โŒ ๐–ป๐–พ ๐—‚๐—‡๐–ผ๐—…๐—Ž๐–ฝ๐–พ๐–ฝ ๐–บ๐—‡๐–ฝ ๐–ผ๐—ˆ๐—‡๐—Œ๐—‚๐–ฝ๐–พ๐—‹๐–พ๐–ฝ ๐—๐–บ๐—…๐—‚๐–ฝ ๐—‚๐—‡ ๐’ฏโ‚. -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚†โ‚†): ๐ผ๐‘›๐‘(โ„‹โ‚). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ๐‘ƒโ‚†โ‚… ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚†โ‚…). ๐‘ƒโ‚ƒโ‚€ ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. ๐Ÿฎ.๐Ÿญ.๐Ÿฒ (๐‘ƒโ‚ƒโ‚€). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘–๐‘›๐‘๐‘œ๐‘›๐‘ ๐‘–๐‘ ๐‘ก๐‘’๐‘›๐‘๐‘ฆ-๐‘๐‘ฆ-๐‘–๐‘›๐‘’๐‘ž๐‘ข๐‘Ž๐‘™๐‘–๐‘ก๐‘ฆ-๐‘–๐‘›๐‘ก๐‘Ÿ๐‘œ๐‘‘๐‘ข๐‘๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ๐ผ๐‘›๐‘(โ„‹โ‚). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚†โ‚†): ๐ผ๐‘›๐‘(โ„‹โ‚). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (4 = 0) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚†โ‚…). (4 โ‰  0) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. ๐Ÿฎ.๐Ÿญ.๐Ÿฒ (๐‘ƒโ‚ƒโ‚€). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘–๐‘›๐‘๐‘œ๐‘›๐‘ ๐‘–๐‘ ๐‘ก๐‘’๐‘›๐‘๐‘ฆ-๐‘๐‘ฆ-๐‘–๐‘›๐‘’๐‘ž๐‘ข๐‘Ž๐‘™๐‘–๐‘ก๐‘ฆ-๐‘–๐‘›๐‘ก๐‘Ÿ๐‘œ๐‘‘๐‘ข๐‘๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: ((P = Q), (P โ‰  Q)) โŠข Inc(๐’ฏ), ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ๐ผ๐‘›๐‘(โ„‹โ‚). โˆŽ ๐—œ๐—ป๐—ณ๐—ฒ๐—ฟ๐—ฒ๐—ป๐—ฐ๐—ฒ ๐—ฟ๐˜‚๐—น๐—ฒ (๐‘๐‘Ÿ๐‘œ๐‘œ๐‘“-๐‘๐‘ฆ-๐‘Ÿ๐‘’๐‘“๐‘ข๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘›-๐‘œ๐‘“-๐‘’๐‘ž๐‘ข๐‘Ž๐‘™๐‘–๐‘ก๐‘ฆ): ๐–ซ๐–พ๐— ๐‘–๐‘›๐‘“๐‘’๐‘Ÿ๐‘’๐‘›๐‘๐‘’-๐‘Ÿ๐‘ข๐‘™๐‘’ ๐‘๐‘Ÿ๐‘œ๐‘œ๐‘“-๐‘๐‘ฆ-๐‘Ÿ๐‘’๐‘“๐‘ข๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘›-๐‘œ๐‘“-๐‘’๐‘ž๐‘ข๐‘Ž๐‘™๐‘–๐‘ก๐‘ฆ ๐–ฝ๐–พ๐–ฟ๐—‚๐—‡๐–พ๐–ฝ ๐–บ๐—Œ โŒœ(โ„‹(P=Q), Inc(โ„‹)) โŠข (P โ‰  Q)โŒ ๐–ป๐–พ ๐—‚๐—‡๐–ผ๐—…๐—Ž๐–ฝ๐–พ๐–ฝ ๐–บ๐—‡๐–ฝ ๐–ผ๐—ˆ๐—‡๐—Œ๐—‚๐–ฝ๐–พ๐—‹๐–พ๐–ฝ ๐—๐–บ๐—…๐—‚๐–ฝ ๐—‚๐—‡ ๐’ฏโ‚. -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป ๐Ÿฎ.๐Ÿญ.๐Ÿด (๐’ฏโ‚.๐‘ƒโ‚†โ‚‡): (6 โ‰  2). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ๐–ซ๐–พ๐— ๐—ต๐˜†๐—ฝ. (๐ปโ‚) ๐–ป๐–พ ๐—๐—๐–พ ๐—๐—’๐—‰๐—ˆ๐—๐—๐–พ๐—Œ๐—‚๐—Œ (6 = 2). ๐ผ๐‘›๐‘(โ„‹โ‚) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚†โ‚†). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘๐‘Ÿ๐‘œ๐‘œ๐‘“-๐‘๐‘ฆ-๐‘Ÿ๐‘’๐‘“๐‘ข๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘›-๐‘œ๐‘“-๐‘’๐‘ž๐‘ข๐‘Ž๐‘™๐‘–๐‘ก๐‘ฆ ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (6 โ‰  2). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป ๐Ÿฎ.๐Ÿญ.๐Ÿด (๐’ฏโ‚.๐‘ƒโ‚†โ‚‡): (6 โ‰  2). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ๐–ซ๐–พ๐— ๐—ต๐˜†๐—ฝ. (๐ปโ‚) ๐–ป๐–พ ๐—๐—๐–พ ๐—๐—’๐—‰๐—ˆ๐—๐—๐–พ๐—Œ๐—‚๐—Œ (6 = 2). ๐ผ๐‘›๐‘(โ„‹โ‚) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚†โ‚†).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘๐‘Ÿ๐‘œ๐‘œ๐‘“-๐‘๐‘ฆ-๐‘Ÿ๐‘’๐‘“๐‘ข๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘›-๐‘œ๐‘“-๐‘’๐‘ž๐‘ข๐‘Ž๐‘™๐‘–๐‘ก๐‘ฆ ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (โ„‹(P=Q), Inc(โ„‹)) โŠข (P โ‰  Q), ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (6 โ‰  2). โˆŽ #### ๐——๐—ถ๐—ฟ๐—ฒ๐—ฐ๐˜ ๐—ฝ๐—ฟ๐—ผ๐—ผ๐—ณ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚†โ‚ˆ): ((1)++ = 2). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (((0)++)++ = 2) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚‡). ((0)++ = 1) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚…). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((1)++ = 2). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚†โ‚‰): (5 โ‰  1). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (5 โ‰  (0)++) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚„โ‚‡). ((0)++ = 1) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚…). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (5 โ‰  1). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‡โ‚€): ((((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)) โŸน (6 โ‰  (1)++)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)) โŸน ((5)++ โ‰  (1)++)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚„โ‚…). ((5)++ = 6) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚…โ‚). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)) โŸน (6 โ‰  (1)++)). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‡โ‚): ((((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)) โŸน (6 โ‰  2)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)) โŸน (6 โ‰  (1)++)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‡โ‚€). ((1)++ = 2) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚†โ‚ˆ). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)) โŸน (6 โ‰  2)). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‡โ‚‚): ((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚„โ‚„). (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚…โ‚€). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘๐‘œ๐‘›๐‘—๐‘ข๐‘›๐‘๐‘ก๐‘–๐‘œ๐‘›-๐‘–๐‘›๐‘ก๐‘Ÿ๐‘œ๐‘‘๐‘ข๐‘๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‡โ‚ƒ): (((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‡โ‚‚). (5 โ‰  1) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚†โ‚‰). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘๐‘œ๐‘›๐‘—๐‘ข๐‘›๐‘๐‘ก๐‘–๐‘œ๐‘›-๐‘–๐‘›๐‘ก๐‘Ÿ๐‘œ๐‘‘๐‘ข๐‘๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)). โˆŽ -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‡โ‚„): (6 โ‰  2). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)) โŸน (6 โ‰  2)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‡โ‚).(((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‡โ‚ƒ). ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘š๐‘œ๐‘‘๐‘ข๐‘ -๐‘๐‘œ๐‘›๐‘’๐‘›๐‘  ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (6 โ‰  2). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚†โ‚ˆ): ((1)++ = 2). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (((0)++)++ = 2) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚‡). ((0)++ = 1) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚…).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, (Q = R)) โŠข P', ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((1)++ = 2). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚†โ‚‰): (5 โ‰  1). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (5 โ‰  (0)++) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚„โ‚‡). ((0)++ = 1) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚โ‚…).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, (Q = R)) โŠข P', ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (5 โ‰  1). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‡โ‚€): ((((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)) โŸน (6 โ‰  (1)++)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)) โŸน ((5)++ โ‰  (1)++)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚„โ‚…). ((5)++ = 6) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚…โ‚).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, (Q = R)) โŠข P', ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)) โŸน (6 โ‰  (1)++)). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‡โ‚): ((((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)) โŸน (6 โ‰  2)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)) โŸน (6 โ‰  (1)++)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‡โ‚€). ((1)++ = 2) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚†โ‚ˆ).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘’๐‘ž๐‘ข๐‘Ž๐‘™-๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ -๐‘ ๐‘ข๐‘๐‘ ๐‘ก๐‘–๐‘ก๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, (Q = R)) โŠข P', ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)) โŸน (6 โ‰  2)). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‡โ‚‚): ((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: (5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) (๐‘ƒ) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚„โ‚„). (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) (๐‘„) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚…โ‚€).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘๐‘œ๐‘›๐‘—๐‘ข๐‘›๐‘๐‘ก๐‘–๐‘œ๐‘›-๐‘–๐‘›๐‘ก๐‘Ÿ๐‘œ๐‘‘๐‘ข๐‘๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, Q) โŠข (P โ‹€ Q), ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— ((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‡โ‚ƒ): (((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) (๐‘ƒ) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‡โ‚‚). (5 โ‰  1) (๐‘„) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚†โ‚‰).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘๐‘œ๐‘›๐‘—๐‘ข๐‘›๐‘๐‘ก๐‘–๐‘œ๐‘›-๐‘–๐‘›๐‘ก๐‘Ÿ๐‘œ๐‘‘๐‘ข๐‘๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: (P, Q) โŠข (P โ‹€ Q), ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‡โ‚„): (6 โ‰  2). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: ((((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)) โŸน (6 โ‰  2)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‡โ‚).(((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)) ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐–ฟ๐—‹๐—ˆ๐—† ๐—ฝ๐—ฟ๐—ผ๐—ฝ. (๐‘ƒโ‚‡โ‚ƒ).๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘š๐‘œ๐‘‘๐‘ข๐‘ -๐‘๐‘œ๐‘›๐‘’๐‘›๐‘  ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: ((P โŸน Q), P) โŠข Q, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (6 โ‰  2). โˆŽ ### ๐—”๐˜…๐—ถ๐—ผ๐—บ ๐Ÿฎ.๐Ÿฑ: ๐˜๐—ต๐—ฒ ๐—ฝ๐—ฟ๐—ถ๐—ป๐—ฐ๐—ถ๐—ฝ๐—น๐—ฒ ๐—ผ๐—ณ ๐—บ๐—ฎ๐˜๐—ต๐—ฒ๐—บ๐—ฎ๐˜๐—ถ๐—ฐ๐—ฎ๐—น ๐—ถ๐—ป๐—ฑ๐˜‚๐—ฐ๐˜๐—ถ๐—ผ๐—ป ๐—”๐˜…๐—ถ๐—ผ๐—บ ๐˜€๐—ฐ๐—ต๐—ฒ๐—บ๐—ฎ (๐’ฏโ‚.๐ดโ‚†) - Principle of mathematical induction: ๐–ซ๐–พ๐— ๐‘Ž๐‘ฅ๐‘–๐‘œ๐‘š ๐’œโ‚† โŒœ๐˜“๐˜ฆ๐˜ต ๐˜—(๐˜ฏ) ๐˜ฃ๐˜ฆ ๐˜ข๐˜ฏ๐˜บ ๐˜ฑ๐˜ณ๐˜ฐ๐˜ฑ๐˜ฆ๐˜ณ๐˜ต๐˜บ ๐˜ฑ๐˜ฆ๐˜ณ๐˜ต๐˜ข๐˜ช๐˜ฏ๐˜ช๐˜ฏ๐˜จ ๐˜ต๐˜ฐ ๐˜ข ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ๐˜ฏ. ๐˜š๐˜ถ๐˜ฑ๐˜ฑ๐˜ฐ๐˜ด๐˜ฆ ๐˜ต๐˜ฉ๐˜ข๐˜ต ๐˜—(๐˜–) ๐˜ช๐˜ด ๐˜ต๐˜ณ๐˜ถ๐˜ฆ, ๐˜ข๐˜ฏ๐˜ฅ ๐˜ด๐˜ถ๐˜ฑ๐˜ฑ๐˜ฐ๐˜ด๐˜ฆ ๐˜ต๐˜ฉ๐˜ข๐˜ต ๐˜ธ๐˜ฉ๐˜ฆ๐˜ฏ๐˜ฆ๐˜ท๐˜ฆ๐˜ณ ๐˜—(๐˜ฏ) ๐˜ช๐˜ด ๐˜ต๐˜ณ๐˜ถ๐˜ฆ, ๐˜—(๐˜ฏ++) ๐˜ช๐˜ด ๐˜ข๐˜ญ๐˜ด๐˜ฐ ๐˜ต๐˜ณ๐˜ถ๐˜ฆ. ๐˜›๐˜ฉ๐˜ฆ๐˜ฏ ๐˜—(๐˜ฏ) ๐˜ช๐˜ด ๐˜ต๐˜ณ๐˜ถ๐˜ฆ ๐˜ง๐˜ฐ๐˜ณ ๐˜ฆ๐˜ท๐˜ฆ๐˜ณ๐˜บ ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ๐˜ฏ.โŒ ๐–ป๐–พ ๐—‚๐—‡๐–ผ๐—…๐—Ž๐–ฝ๐–พ๐–ฝ (๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ) ๐—‚๐—‡ ๐’ฏโ‚. -๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‡โ‚…): (((๐งโ‚… ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (๐โ‚(0) โˆง (๐โ‚(๐งโ‚…) โŸน ๐โ‚((๐งโ‚…)++)))) โŸน ((๐ฆโ‚ƒ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ๐โ‚(๐ฆโ‚ƒ))). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: โŒœ๐˜“๐˜ฆ๐˜ต ๐˜—(๐˜ฏ) ๐˜ฃ๐˜ฆ ๐˜ข๐˜ฏ๐˜บ ๐˜ฑ๐˜ณ๐˜ฐ๐˜ฑ๐˜ฆ๐˜ณ๐˜ต๐˜บ ๐˜ฑ๐˜ฆ๐˜ณ๐˜ต๐˜ข๐˜ช๐˜ฏ๐˜ช๐˜ฏ๐˜จ ๐˜ต๐˜ฐ ๐˜ข ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ๐˜ฏ. ๐˜š๐˜ถ๐˜ฑ๐˜ฑ๐˜ฐ๐˜ด๐˜ฆ ๐˜ต๐˜ฉ๐˜ข๐˜ต ๐˜—(๐˜–) ๐˜ช๐˜ด ๐˜ต๐˜ณ๐˜ถ๐˜ฆ, ๐˜ข๐˜ฏ๐˜ฅ ๐˜ด๐˜ถ๐˜ฑ๐˜ฑ๐˜ฐ๐˜ด๐˜ฆ ๐˜ต๐˜ฉ๐˜ข๐˜ต ๐˜ธ๐˜ฉ๐˜ฆ๐˜ฏ๐˜ฆ๐˜ท๐˜ฆ๐˜ณ ๐˜—(๐˜ฏ) ๐˜ช๐˜ด ๐˜ต๐˜ณ๐˜ถ๐˜ฆ, ๐˜—(๐˜ฏ++) ๐˜ช๐˜ด ๐˜ข๐˜ญ๐˜ด๐˜ฐ ๐˜ต๐˜ณ๐˜ถ๐˜ฆ. ๐˜›๐˜ฉ๐˜ฆ๐˜ฏ ๐˜—(๐˜ฏ) ๐˜ช๐˜ด ๐˜ต๐˜ณ๐˜ถ๐˜ฆ ๐˜ง๐˜ฐ๐˜ณ ๐˜ฆ๐˜ท๐˜ฆ๐˜ณ๐˜บ ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ๐˜ฏ.โŒ ๐—‚๐—Œ ๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ ๐–ป๐—’ ๐—ฎ๐˜…๐—ถ๐—ผ๐—บ ๐˜€๐—ฐ๐—ต๐—ฒ๐—บ๐—ฎ (๐ดโ‚†). (((๐งโ‚… ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (๐โ‚(0) โˆง (๐โ‚(๐งโ‚…) โŸน ๐โ‚((๐งโ‚…)++)))) โŸน ((๐ฆโ‚ƒ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ๐โ‚(๐ฆโ‚ƒ))) ๐—‚๐—Œ ๐–บ ๐—๐–บ๐—…๐—‚๐–ฝ ๐–ฟ๐—ˆ๐—‹๐—†๐—Ž๐—…๐–บ ๐—Œ๐—๐–บ๐—๐–พ๐—†๐–พ๐—‡๐— ๐—‚๐—‡๐—๐–พ๐—‹๐—‰๐—‹๐–พ๐—๐–พ๐–ฝ ๐–ฟ๐—‹๐—ˆ๐—† ๐—๐—๐–บ๐— ๐–บ๐—‘๐—‚๐—ˆ๐—†. ๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘Ž๐‘ฅ๐‘–๐‘œ๐‘š-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (((๐งโ‚… ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (๐โ‚(0) โˆง (๐โ‚(๐งโ‚…) โŸน ๐โ‚((๐งโ‚…)++)))) โŸน ((๐ฆโ‚ƒ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ๐โ‚(๐ฆโ‚ƒ))). โˆŽ +๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (๐’ฏโ‚.๐‘ƒโ‚‡โ‚…): (((๐งโ‚… ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (๐โ‚(0) โˆง (๐โ‚(๐งโ‚…) โŸน ๐โ‚((๐งโ‚…)++)))) โŸน ((๐ฆโ‚ƒ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ๐โ‚(๐ฆโ‚ƒ))). ๐—ฃ๐—ฟ๐—ผ๐—ผ๐—ณ: โŒœ๐˜“๐˜ฆ๐˜ต ๐˜—(๐˜ฏ) ๐˜ฃ๐˜ฆ ๐˜ข๐˜ฏ๐˜บ ๐˜ฑ๐˜ณ๐˜ฐ๐˜ฑ๐˜ฆ๐˜ณ๐˜ต๐˜บ ๐˜ฑ๐˜ฆ๐˜ณ๐˜ต๐˜ข๐˜ช๐˜ฏ๐˜ช๐˜ฏ๐˜จ ๐˜ต๐˜ฐ ๐˜ข ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ๐˜ฏ. ๐˜š๐˜ถ๐˜ฑ๐˜ฑ๐˜ฐ๐˜ด๐˜ฆ ๐˜ต๐˜ฉ๐˜ข๐˜ต ๐˜—(๐˜–) ๐˜ช๐˜ด ๐˜ต๐˜ณ๐˜ถ๐˜ฆ, ๐˜ข๐˜ฏ๐˜ฅ ๐˜ด๐˜ถ๐˜ฑ๐˜ฑ๐˜ฐ๐˜ด๐˜ฆ ๐˜ต๐˜ฉ๐˜ข๐˜ต ๐˜ธ๐˜ฉ๐˜ฆ๐˜ฏ๐˜ฆ๐˜ท๐˜ฆ๐˜ณ ๐˜—(๐˜ฏ) ๐˜ช๐˜ด ๐˜ต๐˜ณ๐˜ถ๐˜ฆ, ๐˜—(๐˜ฏ++) ๐˜ช๐˜ด ๐˜ข๐˜ญ๐˜ด๐˜ฐ ๐˜ต๐˜ณ๐˜ถ๐˜ฆ. ๐˜›๐˜ฉ๐˜ฆ๐˜ฏ ๐˜—(๐˜ฏ) ๐˜ช๐˜ด ๐˜ต๐˜ณ๐˜ถ๐˜ฆ ๐˜ง๐˜ฐ๐˜ณ ๐˜ฆ๐˜ท๐˜ฆ๐˜ณ๐˜บ ๐˜ฏ๐˜ข๐˜ต๐˜ถ๐˜ณ๐˜ข๐˜ญ ๐˜ฏ๐˜ถ๐˜ฎ๐˜ฃ๐˜ฆ๐˜ณ ๐˜ฏ.โŒ ๐—‚๐—Œ ๐—‰๐—ˆ๐—Œ๐—๐—Ž๐—…๐–บ๐—๐–พ๐–ฝ ๐–ป๐—’ ๐—ฎ๐˜…๐—ถ๐—ผ๐—บ ๐˜€๐—ฐ๐—ต๐—ฒ๐—บ๐—ฎ (๐ดโ‚†). (((๐งโ‚… ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (๐โ‚(0) โˆง (๐โ‚(๐งโ‚…) โŸน ๐โ‚((๐งโ‚…)++)))) โŸน ((๐ฆโ‚ƒ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ๐โ‚(๐ฆโ‚ƒ))) ๐—‚๐—Œ ๐–บ ๐—๐–บ๐—…๐—‚๐–ฝ ๐–ฟ๐—ˆ๐—‹๐—†๐—Ž๐—…๐–บ ๐—Œ๐—๐–บ๐—๐–พ๐—†๐–พ๐—‡๐— ๐—‚๐—‡๐—๐–พ๐—‹๐—‰๐—‹๐–พ๐—๐–พ๐–ฝ ๐–ฟ๐—‹๐—ˆ๐—† ๐—๐—๐–บ๐— ๐–บ๐—‘๐—‚๐—ˆ๐—†.๐–ณ๐—๐–พ๐—‹๐–พ๐–ฟ๐—ˆ๐—‹๐–พ, ๐–ป๐—’ ๐—๐—๐–พ ๐‘Ž๐‘ฅ๐‘–๐‘œ๐‘š-๐‘–๐‘›๐‘ก๐‘’๐‘Ÿ๐‘๐‘Ÿ๐‘’๐‘ก๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐—‚๐—‡๐–ฟ๐–พ๐—‹๐–พ๐—‡๐–ผ๐–พ ๐—‹๐—Ž๐—…๐–พ: ๐’œ โŠข P, ๐—‚๐— ๐–ฟ๐—ˆ๐—…๐—…๐—ˆ๐—๐—Œ ๐—๐—๐–บ๐— (((๐งโ‚… ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (๐โ‚(0) โˆง (๐โ‚(๐งโ‚…) โŸน ๐โ‚((๐งโ‚…)++)))) โŸน ((๐ฆโ‚ƒ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ๐โ‚(๐ฆโ‚ƒ))). โˆŽ ### ๐—ง๐—ต๐—ฒ ๐—ป๐˜‚๐—บ๐—ฏ๐—ฒ๐—ฟ ๐˜€๐˜†๐˜€๐˜๐—ฒ๐—บ ๐—ป ### ๐—ฅ๐—ฒ๐—ฐ๐˜‚๐—ฟ๐˜€๐—ถ๐˜ƒ๐—ฒ ๐—ฑ๐—ฒ๐—ณ๐—ถ๐—ป๐—ถ๐˜๐—ถ๐—ผ๐—ป๐˜€ From abd2a419414ad2e87c68861289e6c9bab58d8c01 Mon Sep 17 00:00:00 2001 From: David Doret Date: Thu, 24 Aug 2023 05:56:19 +0200 Subject: [PATCH 05/21] tests --- punctilious/locale_en_us.py | 1 - tests/test_axiom_declaration.py | 12 +++++++----- tests/test_note.py | 6 +++--- 3 files changed, 10 insertions(+), 9 deletions(-) diff --git a/punctilious/locale_en_us.py b/punctilious/locale_en_us.py index daea8a10..fd096705 100644 --- a/punctilious/locale_en_us.py +++ b/punctilious/locale_en_us.py @@ -434,7 +434,6 @@ def compose_note_report(self, o: InferredStatement, **kwargs) -> collections.abc yield o.compose_title(cap=True) yield SansSerifNormal(': ') yield from o.compose_content() - yield SansSerifNormal('.') return True def compose_parent_hypothesis_statement_report(self, o: Hypothesis, diff --git a/tests/test_axiom_declaration.py b/tests/test_axiom_declaration.py index 6c8c50df..3cb56d0b 100644 --- a/tests/test_axiom_declaration.py +++ b/tests/test_axiom_declaration.py @@ -20,6 +20,8 @@ def test_axiom_declaration(self): pu.configuration.echo_default = False pu.configuration.encoding = pu.encodings.plaintext u = pu.UniverseOfDiscourse() + u_plaintext = u.nameset.rep_symbol(encoding=pu.encodings.plaintext) + u_unicode = u.nameset.rep_symbol(encoding=pu.encodings.unicode) content1 = '๐’ทโ„ดโ„ด๐“‡โ„ด๐’ถ ๐’ทโ„ฏโ„ฏ๐’น๐“Š ๐“‡๐’พโ„ด๐’ท๐“โ„ด ๐’ท๐“Ž๐“Š๐’น๐’พโ„ฏ๐“€๐“Š.' # random_data.random_sentence() content2 = '๐“‡โ„ฏ๐’น๐“Š ๐’นโ„ด๐’ทโ„ดโ„ฏ ๐’น๐’พ๐“ƒ๐“Š ๐’ท๐’ถ๐’น๐“‡๐“Ž๐“Š ๐’ฝ๐“Ž๐“‹๐’พ๐“‹โ„ด๐’พ ๐’ทโ„ฏ๐’ท๐’พ ๐“๐“Ž๐’ถโ„Š๐“ƒ๐“Š ๐’น๐“Ž๐’ถ๐“€โ„ฏ ๐“ˆ๐’พ๐“โ„ดโ„ด ๐’ท๐’ถ๐“โ„ด ๐“…โ„ด๐’ถ๐’นโ„ดโ„ฏ ๐’ท๐’พ๐“ƒโ„ด๐’ท๐“‡๐“Š ๐’ป๐“โ„ด๐’น๐’พโ„ด ๐“๐’พ๐’ฝ๐’ถ ๐“‡๐“Žโ„ฏ๐“…โ„ฏ ๐“…โ„ด๐’พ๐“ˆ๐“‰๐’พ๐’ป๐“๐“Žโ„ด ๐“‰โ„ฏ๐’ท๐“โ„ดโ„ฏ ๐’นโ„ด๐’ท๐“‡โ„ฏ ๐’ฟโ„ดโ„ฏ๐’ท๐’พ ๐’ท๐’ถ๐’ฟ๐“Ž๐“Š ๐“ˆ๐“‰๐’ถ๐’ท๐“Ž๐’ถ๐“๐’ถ โ„Š๐“ƒ๐’ถ๐’ท๐“‡๐’พ ๐“‚โ„ด๐’ถ๐’ปโ„ด๐“Š ๐’นโ„ด๐“Š๐“‰๐“Š ๐“ˆ๐“Š๐’น๐’พ โ„Š๐“ƒโ„ด๐’ท๐“Š ๐“‚๐“Š๐’นโ„ฏ๐’ท๐’พ ๐’ทโ„ฏ๐’นโ„ด๐’ทโ„ฏ ๐’ป๐“๐’พโ„ด๐’ท๐’พ ๐’น๐’ถ๐“๐’ถโ„ฏ.' # random_data.random_sentence(min_words=30) content3 = '๐’ท๐“‡๐“Žโ„ฏ๐’ท๐“‡๐’พ ๐’ท๐“Ž๐’ถ๐’ท๐“‡๐“Žโ„ด๐’ทโ„ด๐“Š ๐’ท๐“Š๐’ท๐’พ.' # random_data.random_sentence() @@ -31,17 +33,17 @@ def test_axiom_declaration(self): a4 = u.declare_axiom(content4, symbol='C', name='the axiom of test') a5 = u.declare_axiom(content5, acronym='oaot', symbol='d', name='the other axiom of test') self.assertEqual( - f'๐–ซ๐–พ๐— โŒœ๐’œโ‚โŒ ๐–ป๐–พ ๐—๐—๐–พ ๐‘Ž๐‘ฅ๐‘–๐‘œ๐‘š โŒœ๐’ทโ„ดโ„ด๐“‡โ„ด๐’ถ ๐’ทโ„ฏโ„ฏ๐’น๐“Š ๐“‡๐’พโ„ด๐’ท๐“โ„ด ๐’ท๐“Ž๐“Š๐’น๐’พโ„ฏ๐“€๐“Š.โŒ ๐—‚๐—‡ ๐’ฐโ‚.', + f'๐–ซ๐–พ๐— โŒœ๐’œโ‚โŒ ๐–ป๐–พ ๐—๐—๐–พ ๐‘Ž๐‘ฅ๐‘–๐‘œ๐‘š โŒœ๐’ทโ„ดโ„ด๐“‡โ„ด๐’ถ ๐’ทโ„ฏโ„ฏ๐’น๐“Š ๐“‡๐’พโ„ด๐’ท๐“โ„ด ๐’ท๐“Ž๐“Š๐’น๐’พโ„ฏ๐“€๐“Š.โŒ ๐—‚๐—‡ {u_unicode}.', a1.rep_report(encoding=pu.encodings.unicode, wrap=False)) self.assertEqual( - f'๐–ซ๐–พ๐— โŒœ๐’œโ‚‚โŒ ๐–ป๐–พ ๐—๐—๐–พ ๐‘Ž๐‘ฅ๐‘–๐‘œ๐‘š โŒœ๐“‡โ„ฏ๐’น๐“Š ๐’นโ„ด๐’ทโ„ดโ„ฏ ๐’น๐’พ๐“ƒ๐“Š ๐’ท๐’ถ๐’น๐“‡๐“Ž๐“Š ๐’ฝ๐“Ž๐“‹๐’พ๐“‹โ„ด๐’พ ๐’ทโ„ฏ๐’ท๐’พ ๐“๐“Ž๐’ถโ„Š๐“ƒ๐“Š ๐’น๐“Ž๐’ถ๐“€โ„ฏ ๐“ˆ๐’พ๐“โ„ดโ„ด ๐’ท๐’ถ๐“โ„ด ๐“…โ„ด๐’ถ๐’นโ„ดโ„ฏ ๐’ท๐’พ๐“ƒโ„ด๐’ท๐“‡๐“Š ๐’ป๐“โ„ด๐’น๐’พโ„ด ๐“๐’พ๐’ฝ๐’ถ ๐“‡๐“Žโ„ฏ๐“…โ„ฏ ๐“…โ„ด๐’พ๐“ˆ๐“‰๐’พ๐’ป๐“๐“Žโ„ด ๐“‰โ„ฏ๐’ท๐“โ„ดโ„ฏ ๐’นโ„ด๐’ท๐“‡โ„ฏ ๐’ฟโ„ดโ„ฏ๐’ท๐’พ ๐’ท๐’ถ๐’ฟ๐“Ž๐“Š ๐“ˆ๐“‰๐’ถ๐’ท๐“Ž๐’ถ๐“๐’ถ โ„Š๐“ƒ๐’ถ๐’ท๐“‡๐’พ ๐“‚โ„ด๐’ถ๐’ปโ„ด๐“Š ๐’นโ„ด๐“Š๐“‰๐“Š ๐“ˆ๐“Š๐’น๐’พ โ„Š๐“ƒโ„ด๐’ท๐“Š ๐“‚๐“Š๐’นโ„ฏ๐’ท๐’พ ๐’ทโ„ฏ๐’นโ„ด๐’ทโ„ฏ ๐’ป๐“๐’พโ„ด๐’ท๐’พ ๐’น๐’ถ๐“๐’ถโ„ฏ.โŒ ๐—‚๐—‡ ๐’ฐโ‚.', + f'๐–ซ๐–พ๐— โŒœ๐’œโ‚‚โŒ ๐–ป๐–พ ๐—๐—๐–พ ๐‘Ž๐‘ฅ๐‘–๐‘œ๐‘š โŒœ๐“‡โ„ฏ๐’น๐“Š ๐’นโ„ด๐’ทโ„ดโ„ฏ ๐’น๐’พ๐“ƒ๐“Š ๐’ท๐’ถ๐’น๐“‡๐“Ž๐“Š ๐’ฝ๐“Ž๐“‹๐’พ๐“‹โ„ด๐’พ ๐’ทโ„ฏ๐’ท๐’พ ๐“๐“Ž๐’ถโ„Š๐“ƒ๐“Š ๐’น๐“Ž๐’ถ๐“€โ„ฏ ๐“ˆ๐’พ๐“โ„ดโ„ด ๐’ท๐’ถ๐“โ„ด ๐“…โ„ด๐’ถ๐’นโ„ดโ„ฏ ๐’ท๐’พ๐“ƒโ„ด๐’ท๐“‡๐“Š ๐’ป๐“โ„ด๐’น๐’พโ„ด ๐“๐’พ๐’ฝ๐’ถ ๐“‡๐“Žโ„ฏ๐“…โ„ฏ ๐“…โ„ด๐’พ๐“ˆ๐“‰๐’พ๐’ป๐“๐“Žโ„ด ๐“‰โ„ฏ๐’ท๐“โ„ดโ„ฏ ๐’นโ„ด๐’ท๐“‡โ„ฏ ๐’ฟโ„ดโ„ฏ๐’ท๐’พ ๐’ท๐’ถ๐’ฟ๐“Ž๐“Š ๐“ˆ๐“‰๐’ถ๐’ท๐“Ž๐’ถ๐“๐’ถ โ„Š๐“ƒ๐’ถ๐’ท๐“‡๐’พ ๐“‚โ„ด๐’ถ๐’ปโ„ด๐“Š ๐’นโ„ด๐“Š๐“‰๐“Š ๐“ˆ๐“Š๐’น๐’พ โ„Š๐“ƒโ„ด๐’ท๐“Š ๐“‚๐“Š๐’นโ„ฏ๐’ท๐’พ ๐’ทโ„ฏ๐’นโ„ด๐’ทโ„ฏ ๐’ป๐“๐’พโ„ด๐’ท๐’พ ๐’น๐’ถ๐“๐’ถโ„ฏ.โŒ ๐—‚๐—‡ {u_unicode}.', a2.rep_report(encoding=pu.encodings.unicode, wrap=False)) self.assertEqual( - f'๐–ซ๐–พ๐— โŒœ๐ตโ‚โŒ ๐–ป๐–พ ๐—๐—๐–พ ๐‘Ž๐‘ฅ๐‘–๐‘œ๐‘š โŒœ๐’ท๐“‡๐“Žโ„ฏ๐’ท๐“‡๐’พ ๐’ท๐“Ž๐’ถ๐’ท๐“‡๐“Žโ„ด๐’ทโ„ด๐“Š ๐’ท๐“Š๐’ท๐’พ.โŒ ๐—‚๐—‡ ๐’ฐโ‚.', + f'๐–ซ๐–พ๐— โŒœ๐ตโ‚โŒ ๐–ป๐–พ ๐—๐—๐–พ ๐‘Ž๐‘ฅ๐‘–๐‘œ๐‘š โŒœ๐’ท๐“‡๐“Žโ„ฏ๐’ท๐“‡๐’พ ๐’ท๐“Ž๐’ถ๐’ท๐“‡๐“Žโ„ด๐’ทโ„ด๐“Š ๐’ท๐“Š๐’ท๐’พ.โŒ ๐—‚๐—‡ {u_unicode}.', a3.rep_report(encoding=pu.encodings.unicode, wrap=False)) self.assertEqual( - f'๐–ซ๐–พ๐— โŒœ๐ถโ‚โŒ ๐–ป๐–พ ๐—๐—๐–พ ๐‘Ž๐‘ฅ๐‘–๐‘œ๐‘š โŒœ๐’ท๐“โ„ด๐“โ„ฏ ๐’ท๐“โ„ด๐’ฟโ„ฏ ๐’ฝ๐“Ž๐“Šโ„Š๐“ƒโ„ฏโ„ฏ.โŒ ๐—‚๐—‡ ๐’ฐโ‚, ๐—„๐—‡๐—ˆ๐—๐—‡ ๐–บ๐—Œ ๐—๐—๐–พ ๐–บ๐—‘๐—‚๐—ˆ๐—† ๐—ˆ๐–ฟ ๐—๐–พ๐—Œ๐—.', + f'๐–ซ๐–พ๐— โŒœ๐ถโ‚โŒ ๐–ป๐–พ ๐—๐—๐–พ ๐‘Ž๐‘ฅ๐‘–๐‘œ๐‘š โŒœ๐’ท๐“โ„ด๐“โ„ฏ ๐’ท๐“โ„ด๐’ฟโ„ฏ ๐’ฝ๐“Ž๐“Šโ„Š๐“ƒโ„ฏโ„ฏ.โŒ ๐—‚๐—‡ {u_unicode}, ๐—„๐—‡๐—ˆ๐—๐—‡ ๐–บ๐—Œ ๐—๐—๐–พ ๐–บ๐—‘๐—‚๐—ˆ๐—† ๐—ˆ๐–ฟ ๐—๐–พ๐—Œ๐—.', a4.rep_report(encoding=pu.encodings.unicode, wrap=False)) self.assertEqual( - f'๐–ซ๐–พ๐— โŒœ๐‘‘โ‚โŒ ๐–ป๐–พ ๐—๐—๐–พ ๐‘Ž๐‘ฅ๐‘–๐‘œ๐‘š โŒœ๐’น๐’พ๐“…๐’พ๐“‹๐’พ ๐“ˆ๐“‰๐’พ๐“‚โ„ด๐“‹โ„ด ๐’ป๐“‡๐’พ๐’ป๐“Š๐’ท๐“Š ๐’นโ„ด๐’ทโ„ฏ ๐’น๐“‡๐“Š๐“‚๐“Š๐’นโ„ด๐’พ๐“‰๐“Š.โŒ ๐—‚๐—‡ ๐’ฐโ‚, ๐—„๐—‡๐—ˆ๐—๐—‡ ๐–บ๐—Œ ๐—๐—๐–พ ๐—ˆ๐—๐—๐–พ๐—‹ ๐–บ๐—‘๐—‚๐—ˆ๐—† ๐—ˆ๐–ฟ ๐—๐–พ๐—Œ๐—, ๐—ˆ๐—‹ ๐—Œ๐—‚๐—†๐—‰๐—…๐—’ ๐—ˆ๐–บ๐—ˆ๐—.', + f'๐–ซ๐–พ๐— โŒœ๐‘‘โ‚โŒ ๐–ป๐–พ ๐—๐—๐–พ ๐‘Ž๐‘ฅ๐‘–๐‘œ๐‘š โŒœ๐’น๐’พ๐“…๐’พ๐“‹๐’พ ๐“ˆ๐“‰๐’พ๐“‚โ„ด๐“‹โ„ด ๐’ป๐“‡๐’พ๐’ป๐“Š๐’ท๐“Š ๐’นโ„ด๐’ทโ„ฏ ๐’น๐“‡๐“Š๐“‚๐“Š๐’นโ„ด๐’พ๐“‰๐“Š.โŒ ๐—‚๐—‡ {u_unicode}, ๐—„๐—‡๐—ˆ๐—๐—‡ ๐–บ๐—Œ ๐—๐—๐–พ ๐—ˆ๐—๐—๐–พ๐—‹ ๐–บ๐—‘๐—‚๐—ˆ๐—† ๐—ˆ๐–ฟ ๐—๐–พ๐—Œ๐—, ๐—ˆ๐—‹ ๐—Œ๐—‚๐—†๐—‰๐—…๐—’ ๐—ˆ๐–บ๐—ˆ๐—.', a5.rep_report(encoding=pu.encodings.unicode, wrap=False)) diff --git a/tests/test_note.py b/tests/test_note.py index b44c10f5..d9dcd449 100644 --- a/tests/test_note.py +++ b/tests/test_note.py @@ -17,6 +17,6 @@ def test_note_introduction(self): note = t.take_note('Hello world!', ref='1.1.1') self.assertEqual('๐—ก๐—ผ๐˜๐—ฒ ๐Ÿญ.๐Ÿญ.๐Ÿญ (๐Ÿ—…โ‚): ๐–ง๐–พ๐—…๐—…๐—ˆ ๐—๐—ˆ๐—‹๐—…๐–ฝ!', note.rep_report()) comment = t.take_note('Foo', ref='1.1.2', paragraph_header=pu.paragraph_headers.comment) - self.assertEqual('๐—–๐—ผ๐—บ๐—บ๐—ฒ๐—ป๐˜ ๐Ÿญ.๐Ÿญ.๐Ÿฎ (๐Ÿ—…โ‚ƒ): Foo', comment.rep_report()) - remark = t.take_note('Bar', ref='1.1.3', paragraph_header=pu.paragraph_headers.remark) - self.assertEqual('๐—ฅ๐—ฒ๐—บ๐—ฎ๐—ฟ๐—ธ ๐Ÿญ.๐Ÿญ.๐Ÿฏ (๐Ÿ—…โ‚„): Bar', remark.rep_report()) + self.assertEqual('๐—–๐—ผ๐—บ๐—บ๐—ฒ๐—ป๐˜ ๐Ÿญ.๐Ÿญ.๐Ÿฎ (๐Ÿ—…โ‚‚): ๐–ฅ๐—ˆ๐—ˆ', comment.rep_report()) + remark = t.take_note('Bar', paragraph_header=pu.paragraph_headers.remark) + self.assertEqual('๐—ฅ๐—ฒ๐—บ๐—ฎ๐—ฟ๐—ธ (๐Ÿ—…โ‚ƒ): ๐–ก๐–บ๐—‹', remark.rep_report()) From 43917a3a9c2f735a9ce162625972303bc98a7062 Mon Sep 17 00:00:00 2001 From: David Doret Date: Thu, 24 Aug 2023 12:34:52 +0200 Subject: [PATCH 06/21] tests --- punctilious/core.py | 18 +++++++++--------- punctilious/locale_en_us.py | 13 +++++++++++++ tests/test_conjunction_elimination_left.py | 9 ++++++--- tests/test_universe_of_discourse.py | 16 ++++++++-------- 4 files changed, 36 insertions(+), 20 deletions(-) diff --git a/punctilious/core.py b/punctilious/core.py index b05902e9..052e560f 100644 --- a/punctilious/core.py +++ b/punctilious/core.py @@ -4057,11 +4057,12 @@ def __init__(self, universe_of_discourse: UniverseOfDiscourse, echo: (None, bool explicit_name=explicit_name, echo=echo) def infer_formula(self, p_land_q: FormulaStatement = None, t: TheoryElaborationSequence = None, - echo: (None, bool) = None) -> Formula: + echo: (None, bool) = None, **kwargs) -> Formula: """ :param p_land_q: A formula-statement of the form: (P โ‹€ Q). :param t: The current theory-elaboration-sequence. + :param echo: :return: The (proven) formula: P. """ p = unpack_formula(p_land_q).parameters[0] @@ -7155,18 +7156,17 @@ def __init__(self, t: TheoryElaborationSequence, echo: (None, bool) = None, def infer_formula(self, p_land_q: (None, FormulaStatement) = None, echo: (None, bool) = None): return super().infer_formula(p_land_q, echo=echo) - def infer_statement(self, p_land_q: (None, FormulaStatement) = None, - nameset: (None, str, NameSet) = None, ref: (None, str) = None, + def infer_statement(self, p_land_q: (None, FormulaStatement) = None, ref: (None, str) = None, paragraph_header: (None, ParagraphHeader) = None, subtitle: (None, str) = None, echo: (None, bool) = None) -> InferredStatement: """Apply the conjunction elimination (left) inference-rule and return the inferred-statement. - :param p_iff_q: (mandatory) The conjunction statement. + :param p_land_q: :return: The proven inferred-statement p implies q in the current theory. """ - return super().infer_statement(p_land_q, nameset=nameset, ref=ref, - paragraph_header=paragraph_header, subtitle=subtitle, echo=echo) + return super().infer_statement(p_land_q, ref=ref, paragraph_header=paragraph_header, + subtitle=subtitle, echo=echo) class ConjunctionEliminationRightInclusion(InferenceRuleInclusion): @@ -8049,7 +8049,7 @@ def conjunction_elimination_left(self) -> ConjunctionEliminationLeftDeclaration: return self._conjunction_elimination_left @property - def conjunction_elimination_right(self) -> InferenceRuleInclusion: + def conjunction_elimination_right(self) -> ConjunctionEliminationRightInclusion: """The well-known conjunction-elimination (right) inference-rule: P โˆง Q โŠข Q. Abridged property: t.i.cer @@ -8062,7 +8062,7 @@ def conjunction_elimination_right(self) -> InferenceRuleInclusion: return self._conjunction_elimination_right @property - def cel(self) -> InferenceRuleInclusion: + def cel(self) -> ConjunctionEliminationLeftDeclaration: """The well-known conjunction-elimination (left) inference-rule: (P โˆง Q) โŠข P. Unabridged property: universe_of_discourse.inference_rules.conjunction_elimination_left() @@ -8073,7 +8073,7 @@ def cel(self) -> InferenceRuleInclusion: return self.conjunction_elimination_left @property - def cer(self) -> InferenceRuleInclusion: + def cer(self) -> ConjunctionEliminationRightInclusion: """The well-known conjunction-elimination (right) inference-rule: P โˆง Q โŠข Q. Unabridged property: universe_of_discourse.inference_rules.conjunction_elimination_right() diff --git a/punctilious/locale_en_us.py b/punctilious/locale_en_us.py index fd096705..c7a9d5e4 100644 --- a/punctilious/locale_en_us.py +++ b/punctilious/locale_en_us.py @@ -121,6 +121,19 @@ def compose_biconditional_introduction_paragraph_proof(self, o: InferredStatemen yield SansSerifNormal('. ') return True + def compose_conjunction_elimination_left_paragraph_proof(self, o: InferredStatement) -> \ + collections.abc.Generator[Composable, Composable, bool]: + global text_dict + # Retrieve the parameters from the statement + p_and_q: FormulaStatement = o.parameters[0] + yield from p_and_q.valid_proposition.compose_formula() + yield SansSerifNormal(', is of the form ') + yield '(๐‘ท โˆง ๐‘ธ)' # TODO: Replace hardcoded formula with real formula + yield SansSerifNormal(', and follows from ') + yield from p_and_q.compose_ref_link() + yield SansSerifNormal('. ') + return True + def compose_conjunction_introduction_paragraph_proof(self, o: InferredStatement) -> \ collections.abc.Generator[Composable, Composable, bool]: global text_dict diff --git a/tests/test_conjunction_elimination_left.py b/tests/test_conjunction_elimination_left.py index 77159c74..378552e2 100644 --- a/tests/test_conjunction_elimination_left.py +++ b/tests/test_conjunction_elimination_left.py @@ -5,7 +5,10 @@ class TestConjunctionEliminationLeft(TestCase): def test_cel(self): + pu.configuration.echo_default = True u = pu.UniverseOfDiscourse() + u_plaintext = u.nameset.rep_symbol(encoding=pu.encodings.plaintext) + u_unicode = u.nameset.rep_symbol(encoding=pu.encodings.unicode) o1 = u.o.declare() o2 = u.o.declare() o3 = u.o.declare() @@ -14,8 +17,8 @@ def test_cel(self): t = u.t() a = u.declare_axiom(random_data.random_sentence()) ap = t.include_axiom(a) - phi1 = t.i.axiom_interpretation.infer_statement(ap, u.f(u.r.conjunction, u.f(r1, o1, o2), - u.f(r2, o3))) + phi1 = t.i.axiom_interpretation.infer_statement(axiom=ap, + formula=u.f(u.r.conjunction, u.f(r1, o1, o2), u.f(r2, o3))) self.assertEqual('(๐‘Ÿโ‚(๐‘œโ‚, ๐‘œโ‚‚) โˆง ๐‘Ÿโ‚‚(๐‘œโ‚ƒ))', phi1.rep_formula(pu.encodings.unicode)) - phi2 = t.i.cel.infer_statement(phi1) + phi2 = t.i.conjunction_elimination_left.infer_statement(p_and_q=phi1) self.assertEqual('๐‘Ÿโ‚(๐‘œโ‚, ๐‘œโ‚‚)', phi2.rep_formula(pu.encodings.unicode)) diff --git a/tests/test_universe_of_discourse.py b/tests/test_universe_of_discourse.py index 9bcfcdc2..1101e697 100644 --- a/tests/test_universe_of_discourse.py +++ b/tests/test_universe_of_discourse.py @@ -6,16 +6,16 @@ class TestUniverseOfDiscourse(TestCase): def test_universe_of_discourse(self): u1 = pu.create_universe_of_discourse() - self.assertEqual('U1', u1.rep(encoding=pu.encodings.plaintext)) - self.assertEqual('๐’ฐโ‚', u1.rep(encoding=pu.encodings.unicode)) - self.assertEqual('Let "U1" be a universe-of-discourse.', + u1_plaintext = u1.nameset.rep_symbol(encoding=pu.encodings.plaintext) + u1_unicode = u1.nameset.rep_symbol(encoding=pu.encodings.unicode) + self.assertEqual(f'Let "{u1_plaintext}" be a universe-of-discourse.', u1.rep_creation(encoding=pu.encodings.plaintext)) - self.assertEqual('๐–ซ๐–พ๐— โŒœ๐’ฐโ‚โŒ ๐–ป๐–พ ๐–บ ๐‘ข๐‘›๐‘–๐‘ฃ๐‘’๐‘Ÿ๐‘ ๐‘’-๐‘œ๐‘“-๐‘‘๐‘–๐‘ ๐‘๐‘œ๐‘ข๐‘Ÿ๐‘ ๐‘’.', + self.assertEqual(f'๐–ซ๐–พ๐— โŒœ{u1_unicode}โŒ ๐–ป๐–พ ๐–บ ๐‘ข๐‘›๐‘–๐‘ฃ๐‘’๐‘Ÿ๐‘ ๐‘’-๐‘œ๐‘“-๐‘‘๐‘–๐‘ ๐‘๐‘œ๐‘ข๐‘Ÿ๐‘ ๐‘’.', u1.rep_creation(encoding=pu.encodings.unicode)) u2 = pu.create_universe_of_discourse() - self.assertEqual('U2', u2.rep(encoding=pu.encodings.plaintext)) - self.assertEqual('๐’ฐโ‚‚', u2.rep(encoding=pu.encodings.unicode)) - self.assertEqual('Let "U2" be a universe-of-discourse.', + u2_plaintext = u2.nameset.rep_symbol(encoding=pu.encodings.plaintext) + u2_unicode = u2.nameset.rep_symbol(encoding=pu.encodings.unicode) + self.assertEqual(f'Let "{u2_plaintext}" be a universe-of-discourse.', u2.rep_creation(encoding=pu.encodings.plaintext)) - self.assertEqual('๐–ซ๐–พ๐— โŒœ๐’ฐโ‚‚โŒ ๐–ป๐–พ ๐–บ ๐‘ข๐‘›๐‘–๐‘ฃ๐‘’๐‘Ÿ๐‘ ๐‘’-๐‘œ๐‘“-๐‘‘๐‘–๐‘ ๐‘๐‘œ๐‘ข๐‘Ÿ๐‘ ๐‘’.', + self.assertEqual(f'๐–ซ๐–พ๐— โŒœ{u2_unicode}โŒ ๐–ป๐–พ ๐–บ ๐‘ข๐‘›๐‘–๐‘ฃ๐‘’๐‘Ÿ๐‘ ๐‘’-๐‘œ๐‘“-๐‘‘๐‘–๐‘ ๐‘๐‘œ๐‘ข๐‘Ÿ๐‘ ๐‘’.', u2.rep_creation(encoding=pu.encodings.unicode)) From 388df72545d7d4e217e2693757438e3a1eb44b29 Mon Sep 17 00:00:00 2001 From: David Doret Date: Thu, 24 Aug 2023 13:30:57 +0200 Subject: [PATCH 07/21] tests --- punctilious/__init__.py | 11 +- punctilious/core.py | 155 +++++++++--------- punctilious/locale_en_us.py | 13 ++ tests/test_conjunction_elimination_right.py | 6 +- tests/test_definition_declaration.py | 12 +- tests/test_objct_header.py | 42 ----- ...on.py => test_proof_by_contradiction_1.py} | 6 +- tests/test_proof_by_refutation.py | 2 +- tests/test_theory_elaboration_sequence.py | 7 +- tests/test_theory_elaboration_stabilized.py | 31 ++-- .../package/tao_2006_2_1_the_peano_axioms.py | 3 +- 11 files changed, 127 insertions(+), 161 deletions(-) delete mode 100644 tests/test_objct_header.py rename tests/{test_proof_by_contradiction.py => test_proof_by_contradiction_1.py} (89%) diff --git a/punctilious/__init__.py b/punctilious/__init__.py index a43d656c..8a0a9576 100644 --- a/punctilious/__init__.py +++ b/punctilious/__init__.py @@ -8,11 +8,12 @@ configuration, consistency_values, DashedName, create_universe_of_discourse, \ DefinitionInclusion, Encoding, encodings, FailedVerificationException, Formula, FreeVariable, \ Header, InconsistencyWarning, interpret_formula, interpret_statement_formula, is_in_class, \ - InferredStatement, NoteInclusion, Paragraph, prioritize_value, QuasiQuotation, Relation, \ - rep_two_columns_proof_item, ScriptNormal, SansSerifBold, SansSerifNormal, SerifBoldItalic, \ - SerifItalic, SerifNormal, SimpleObjct, Statement, Subscript, subscriptify, paragraph_headers, \ - ComposableText, NameSet, SymbolicObject, TextStyle, text_styles, TheoreticalObject, \ - TheoryElaborationSequence, TheoryPackage, paragraph_headers, UniverseOfDiscourse + InferredStatement, NoteInclusion, Paragraph, ParagraphHeader, paragraph_headers, \ + prioritize_value, QuasiQuotation, Relation, rep_two_columns_proof_item, ScriptNormal, \ + SansSerifBold, SansSerifNormal, SerifBoldItalic, SerifItalic, SerifNormal, SimpleObjct, \ + Statement, Subscript, subscriptify, paragraph_headers, ComposableText, NameSet, SymbolicObject, \ + TextStyle, text_styles, TheoreticalObject, TheoryElaborationSequence, TheoryPackage, \ + paragraph_headers, UniverseOfDiscourse # from foundation_system_1 import foundation_system_1, ft, u diff --git a/punctilious/core.py b/punctilious/core.py index 052e560f..44b561d5 100644 --- a/punctilious/core.py +++ b/punctilious/core.py @@ -2571,8 +2571,8 @@ class Formula(TheoreticalObject): def __init__(self, relation: (Relation, FreeVariable), parameters: tuple, universe_of_discourse: UniverseOfDiscourse, nameset: (None, str, NameSet) = None, - lock_variable_scope: bool = False, title: (None, str, TitleOBSOLETE) = None, - dashed_name: (None, str, DashedName) = None, echo: (None, bool) = None): + lock_variable_scope: bool = False, dashed_name: (None, str, DashedName) = None, + echo: (None, bool) = None): """ """ echo = prioritize_value(echo, configuration.echo_formula_declaration, @@ -4105,8 +4105,8 @@ def __init__(self, universe_of_discourse: UniverseOfDiscourse, echo: (None, bool name = 'conjunction elimination (right)' # Assure backward-compatibility with the parent class, # which received these methods as __init__ arguments. - infer_formula = BiconditionalEliminationRightDeclaration.infer_formula - verify_args = BiconditionalEliminationRightDeclaration.verify_args + infer_formula = ConjunctionEliminationRightDeclaration.infer_formula + verify_args = ConjunctionEliminationRightDeclaration.verify_args super().__init__(infer_formula=infer_formula, verify_args=verify_args, universe_of_discourse=universe_of_discourse, symbol=symbol, auto_index=auto_index, dashed_name=dashed_name, acronym=acronym, abridged_name=abridged_name, name=name, @@ -4120,7 +4120,8 @@ def infer_formula(self, p_land_q: FormulaStatement = None, t: TheoryElaborationS :param t: The current theory-elaboration-sequence. :return: The (proven) formula: Q. """ - q = unpack_formula(p_land_q).parameters[1] + p_land_q = interpret_formula(u=t.u, arity=2, flexible_formula=p_land_q) + q = p_land_q.parameters[1] return q def compose_paragraph_proof(self, o: InferredStatement) -> collections.abc.Generator[ @@ -4819,7 +4820,7 @@ def verify_args(self, p_implies_q: FormulaStatement, p: FormulaStatement, return True -class ProofByContradictionDeclaration(InferenceRuleDeclaration): +class ProofByContradiction1Declaration(InferenceRuleDeclaration): def __init__(self, universe_of_discourse: UniverseOfDiscourse, echo: (None, bool) = None): symbol = 'proof-by-contradiction' acronym = 'pbc' @@ -4829,8 +4830,8 @@ def __init__(self, universe_of_discourse: UniverseOfDiscourse, echo: (None, bool name = 'proof by contradiction' # Assure backward-compatibility with the parent class, # which received these methods as __init__ arguments. - infer_formula = ProofByContradictionDeclaration.infer_formula - verify_args = ProofByContradictionDeclaration.verify_args + infer_formula = ProofByContradiction1Declaration.infer_formula + verify_args = ProofByContradiction1Declaration.verify_args super().__init__(infer_formula=infer_formula, verify_args=verify_args, universe_of_discourse=universe_of_discourse, symbol=symbol, auto_index=auto_index, dashed_name=dashed_name, acronym=acronym, name=name, explicit_name=explicit_name, @@ -4882,7 +4883,7 @@ def verify_args(self, not_p: Hypothesis, inc_not_p: InferredStatement, return True -class ProofByContradictionOfInequalityDeclaration(InferenceRuleDeclaration): +class ProofByContradiction2Declaration(InferenceRuleDeclaration): def __init__(self, universe_of_discourse: UniverseOfDiscourse, echo: (None, bool) = None): symbol = 'proof-by-contradiction-of-inequality' acronym = 'pbci' @@ -4892,8 +4893,8 @@ def __init__(self, universe_of_discourse: UniverseOfDiscourse, echo: (None, bool name = 'proof by contradiction of inequality' # Assure backward-compatibility with the parent class, # which received these methods as __init__ arguments. - infer_formula = ProofByContradictionOfInequalityDeclaration.infer_formula - verify_args = ProofByContradictionOfInequalityDeclaration.verify_args + infer_formula = ProofByContradiction2Declaration.infer_formula + verify_args = ProofByContradiction2Declaration.verify_args super().__init__(infer_formula=infer_formula, verify_args=verify_args, universe_of_discourse=universe_of_discourse, symbol=symbol, auto_index=auto_index, dashed_name=dashed_name, acronym=acronym, name=name, explicit_name=explicit_name, @@ -4941,7 +4942,7 @@ def verify_args(self, p_neq_q: Hypothesis, inc_p_neq_q: InferredStatement, return True -class ProofByRefutationDeclaration(InferenceRuleDeclaration): +class ProofByRefutation1Declaration(InferenceRuleDeclaration): def __init__(self, universe_of_discourse: UniverseOfDiscourse, echo: (None, bool) = None): symbol = 'proof-by-refutation' acronym = 'pbr' @@ -4951,8 +4952,8 @@ def __init__(self, universe_of_discourse: UniverseOfDiscourse, echo: (None, bool name = 'proof by refutation' # Assure backward-compatibility with the parent class, # which received these methods as __init__ arguments. - infer_formula = ProofByContradictionDeclaration.infer_formula - verify_args = ProofByContradictionDeclaration.verify_args + infer_formula = ProofByContradiction1Declaration.infer_formula + verify_args = ProofByContradiction1Declaration.verify_args super().__init__(infer_formula=infer_formula, verify_args=verify_args, universe_of_discourse=universe_of_discourse, symbol=symbol, auto_index=auto_index, dashed_name=dashed_name, acronym=acronym, name=name, explicit_name=explicit_name, @@ -4997,7 +4998,7 @@ def verify_args(self, p: Hypothesis, inc_p: InferredStatement, t: TheoryElaborat return True -class ProofByRefutationOfEqualityDeclaration(InferenceRuleDeclaration): +class ProofByRefutation2Declaration(InferenceRuleDeclaration): def __init__(self, universe_of_discourse: UniverseOfDiscourse, echo: (None, bool) = None): symbol = 'proof-by-refutation-of-equality' acronym = 'pbre' @@ -5008,8 +5009,8 @@ def __init__(self, universe_of_discourse: UniverseOfDiscourse, echo: (None, bool definition = '(โ„‹(P=Q), Inc(โ„‹)) โŠข (P โ‰  Q)' # Assure backward-compatibility with the parent class, # which received these methods as __init__ arguments. - infer_formula = ProofByRefutationOfEqualityDeclaration.infer_formula - verify_args = ProofByRefutationOfEqualityDeclaration.verify_args + infer_formula = ProofByRefutation2Declaration.infer_formula + verify_args = ProofByRefutation2Declaration.verify_args super().__init__(definition=definition, infer_formula=infer_formula, verify_args=verify_args, universe_of_discourse=universe_of_discourse, symbol=symbol, auto_index=auto_index, dashed_name=dashed_name, acronym=acronym, name=name, @@ -6389,10 +6390,10 @@ def __init__(self, u: UniverseOfDiscourse): self._inconsistency_by_inequality_introduction = None self._inconsistency_by_negation_introduction = None self._modus_ponens = None - self._proof_by_contradiction = None - self._proof_by_contradiction_of_non_equality = None - self._proof_by_refutation = None - self._proof_by_refutation_of_equality = None + self._proof_by_contradiction_1 = None + self._proof_by_contradiction_2 = None + self._proof_by_refutation_1 = None + self._proof_by_refutation_2 = None self._variable_substitution = None @property @@ -6851,7 +6852,7 @@ def mp(self) -> InferenceRuleDeclaration: return self.modus_ponens @property - def pbc(self) -> ProofByContradictionDeclaration: + def pbc(self) -> ProofByContradiction1Declaration: """ Unabridged property: u.i.proof_by_contradiction @@ -6859,10 +6860,10 @@ def pbc(self) -> ProofByContradictionDeclaration: If the well-known inference-rule does not exist in the universe-of-discourse, the inference-rule is automatically declared. """ - return self.proof_by_contradiction + return self.proof_by_contradiction_1 @property - def pbr(self) -> ProofByRefutationDeclaration: + def pbr(self) -> ProofByRefutation1Declaration: """ Unabridged property: u.i.proof_by_refutation @@ -6870,10 +6871,10 @@ def pbr(self) -> ProofByRefutationDeclaration: If the well-known inference-rule does not exist in the universe-of-discourse, the inference-rule is automatically declared. """ - return self.proof_by_refutation + return self.proof_by_refutation_1 @property - def proof_by_contradiction(self) -> ProofByContradictionDeclaration: + def proof_by_contradiction_1(self) -> ProofByContradiction1Declaration: """ Abridged property: u.i.pbc @@ -6882,20 +6883,20 @@ def proof_by_contradiction(self) -> ProofByContradictionDeclaration: If the well-known inference-rule does not exist in the universe-of-discourse, the inference-rule is automatically declared. """ - if self._proof_by_contradiction is None: - self._proof_by_contradiction = ProofByContradictionDeclaration( + if self._proof_by_contradiction_1 is None: + self._proof_by_contradiction_1 = ProofByContradiction1Declaration( universe_of_discourse=self.u) - return self._proof_by_contradiction + return self._proof_by_contradiction_1 @property - def proof_by_contradiction_of_non_equality(self) -> ProofByContradictionOfInequalityDeclaration: - if self._proof_by_contradiction_of_non_equality is None: - self._proof_by_contradiction_of_non_equality = ProofByContradictionOfInequalityDeclaration( + def proof_by_contradiction_2(self) -> ProofByContradiction2Declaration: + if self._proof_by_contradiction_2 is None: + self._proof_by_contradiction_2 = ProofByContradiction2Declaration( universe_of_discourse=self.u) - return self._proof_by_contradiction_of_non_equality + return self._proof_by_contradiction_2 @property - def proof_by_refutation(self) -> ProofByRefutationDeclaration: + def proof_by_refutation_1(self) -> ProofByRefutation1Declaration: """ Abridged property: u.i.pbr @@ -6904,16 +6905,17 @@ def proof_by_refutation(self) -> ProofByRefutationDeclaration: If the well-known inference-rule does not exist in the universe-of-discourse, the inference-rule is automatically declared. """ - if self._proof_by_refutation is None: - self._proof_by_refutation = ProofByRefutationDeclaration(universe_of_discourse=self.u) - return self._proof_by_refutation + if self._proof_by_refutation_1 is None: + self._proof_by_refutation_1 = ProofByRefutation1Declaration( + universe_of_discourse=self.u) + return self._proof_by_refutation_1 @property - def proof_by_refutation_of_equality(self) -> ProofByRefutationOfEqualityDeclaration: - if self._proof_by_refutation_of_equality is None: - self._proof_by_refutation_of_equality = ProofByRefutationOfEqualityDeclaration( + def proof_by_refutation_2(self) -> ProofByRefutation2Declaration: + if self._proof_by_refutation_2 is None: + self._proof_by_refutation_2 = ProofByRefutation2Declaration( universe_of_discourse=self.u) - return self._proof_by_refutation_of_equality + return self._proof_by_refutation_2 @property def variable_substitution(self) -> VariableSubstitutionDeclaration: @@ -7153,19 +7155,19 @@ def __init__(self, t: TheoryElaborationSequence, echo: (None, bool) = None, abridged_name=abridged_name, name=name, explicit_name=explicit_name, echo=echo, proof=proof) - def infer_formula(self, p_land_q: (None, FormulaStatement) = None, echo: (None, bool) = None): - return super().infer_formula(p_land_q, echo=echo) + def infer_formula(self, p_and_q: (None, FormulaStatement) = None, echo: (None, bool) = None): + return super().infer_formula(p_and_q, echo=echo) - def infer_statement(self, p_land_q: (None, FormulaStatement) = None, ref: (None, str) = None, + def infer_statement(self, p_and_q: (None, FormulaStatement) = None, ref: (None, str) = None, paragraph_header: (None, ParagraphHeader) = None, subtitle: (None, str) = None, echo: (None, bool) = None) -> InferredStatement: """Apply the conjunction elimination (left) inference-rule and return the inferred-statement. - :param p_land_q: + :param p_and_q: :return: The proven inferred-statement p implies q in the current theory. """ - return super().infer_statement(p_land_q, ref=ref, paragraph_header=paragraph_header, + return super().infer_statement(p_and_q, ref=ref, paragraph_header=paragraph_header, subtitle=subtitle, echo=echo) @@ -7186,20 +7188,20 @@ def __init__(self, t: TheoryElaborationSequence, echo: (None, bool) = None, abridged_name=abridged_name, name=name, explicit_name=explicit_name, echo=echo, proof=proof) - def infer_formula(self, p_iff_q: (None, FormulaStatement) = None, echo: (None, bool) = None): - return super().infer_formula(p_iff_q, echo=echo) + def infer_formula(self, p_and_q: (None, FormulaStatement) = None, echo: (None, bool) = None): + return super().infer_formula(p_and_q, echo=echo) - def infer_statement(self, p_iff_q: (None, FormulaStatement) = None, + def infer_statement(self, p_and_q: (None, FormulaStatement) = None, nameset: (None, str, NameSet) = None, ref: (None, str) = None, paragraph_header: (None, ParagraphHeader) = None, subtitle: (None, str) = None, echo: (None, bool) = None) -> InferredStatement: """Apply the conjunction elimination (right) inference-rule and return the inferred-statement. - :param p_iff_q: (mandatory) The conjunction statement. + :param p_and_q: (mandatory) The conjunction statement. :return: The proven inferred-statement p implies q in the current theory. """ - return super().infer_statement(p_iff_q, nameset=nameset, ref=ref, + return super().infer_statement(p_and_q, nameset=nameset, ref=ref, paragraph_header=paragraph_header, subtitle=subtitle, echo=echo) @@ -7685,25 +7687,25 @@ def infer_statement(self, p_implies_q: (None, FormulaStatement) = None, paragraph_header=paragraph_header, subtitle=subtitle, echo=echo) -class ProofByContradictionInclusion(InferenceRuleInclusion): +class ProofByContradiction1Inclusion(InferenceRuleInclusion): """ """ def __init__(self, t: TheoryElaborationSequence, echo: (None, bool) = None, proof: (None, bool) = None): - i = t.universe_of_discourse.inference_rules.proof_by_contradiction - dashed_name = 'proof-by-contradiction' - acronym = 'pbc' + i = t.universe_of_discourse.inference_rules.proof_by_contradiction_1 + dashed_name = 'proof-by-contradiction-1' + acronym = 'pbc1' abridged_name = None - name = 'proof by contradiction' - explicit_name = 'proof by contradiction inference rule' + name = 'proof by contradiction #1' + explicit_name = 'proof by contradiction #1 inference rule' super().__init__(t=t, i=i, dashed_name=dashed_name, acronym=acronym, abridged_name=abridged_name, name=name, explicit_name=explicit_name, echo=echo, proof=proof) def infer_formula(self, not_p: (None, Hypothesis) = None, - inc_not_p: (None, FormulaStatement) = None, echo: (None, bool) = None): + inc_not_p: (None, FormulaStatement) = None, echo: (None, bool) = None) -> Formula: """Apply the proof-by-contradiction inference-rule and return the inferred-formula. :param not_p: (mandatory) The (ยฌP) hypothesis-statement. @@ -7727,19 +7729,19 @@ def infer_statement(self, not_p: (None, FormulaStatement) = None, paragraph_header=paragraph_header, subtitle=subtitle, echo=echo) -class ProofByContradictionOfInequalityInclusion(InferenceRuleInclusion): +class ProofByContradiction2Inclusion(InferenceRuleInclusion): """ """ def __init__(self, t: TheoryElaborationSequence, echo: (None, bool) = None, proof: (None, bool) = None): - i = t.universe_of_discourse.inference_rules.proof_by_contradiction_of_non_equality - dashed_name = 'proof-by-contradiction-of-inequality' - acronym = 'pbci' + i = t.universe_of_discourse.inference_rules.proof_by_contradiction_2 + dashed_name = 'proof-by-contradiction-2' + acronym = 'pbc2' abridged_name = None - name = 'proof by contradiction of inequality' - explicit_name = 'proof by contradiction of inequality inference rule' + name = 'proof by contradiction #2' + explicit_name = 'proof by contradiction #2 inference rule' super().__init__(t=t, i=i, dashed_name=dashed_name, acronym=acronym, abridged_name=abridged_name, name=name, explicit_name=explicit_name, echo=echo, proof=proof) @@ -7763,7 +7765,7 @@ class ProofByRefutationInclusion(InferenceRuleInclusion): def __init__(self, t: TheoryElaborationSequence, echo: (None, bool) = None, proof: (None, bool) = None): - i = t.universe_of_discourse.inference_rules.proof_by_refutation + i = t.universe_of_discourse.inference_rules.proof_by_refutation_1 dashed_name = 'proof-by-refutation' acronym = 'pbr' abridged_name = None @@ -7805,7 +7807,7 @@ class ProofByRefutationOfEqualityInclusion(InferenceRuleInclusion): def __init__(self, t: TheoryElaborationSequence, echo: (None, bool) = None, proof: (None, bool) = None): - i = t.universe_of_discourse.inference_rules.proof_by_refutation_of_equality + i = t.universe_of_discourse.inference_rules.proof_by_refutation_2 dashed_name = 'proof-by-refutation-of-equality' acronym = 'pbre' abridged_name = None @@ -7914,7 +7916,7 @@ def __init__(self, t: TheoryElaborationSequence): self._inconsistency_by_inequality_introduction = None self._inconsistency_by_negation_introduction = None self._modus_ponens = None - self._proof_by_contradiction = None + self._proof_by_contradiction_1 = None self._proof_by_contradiction_of_non_equality = None self._proof_by_refutation = None self._proof_by_refutation_of_equality = None @@ -8036,7 +8038,7 @@ def biconditional_introduction(self) -> BiconditionalIntroductionInclusion: return self._biconditional_introduction @property - def conjunction_elimination_left(self) -> ConjunctionEliminationLeftDeclaration: + def conjunction_elimination_left(self) -> ConjunctionEliminationLeftInclusion: """The well-known conjunction-elimination (left) inference-rule: P โˆง Q โŠข P. Abridged property: t.i.cel() @@ -8265,7 +8267,7 @@ def mp(self) -> ModusPonensInclusion: return self.modus_ponens @property - def pbc(self) -> ProofByContradictionInclusion: + def pbc(self) -> ProofByContradiction1Inclusion: """ Unabridged property: u.i.proof_by_contradiction @@ -8273,7 +8275,7 @@ def pbc(self) -> ProofByContradictionInclusion: If the well-known inference-rule does not exist in the universe-of-discourse, the inference-rule is automatically declared. """ - return self.proof_by_contradiction + return self.proof_by_contradiction_1 @property def pbr(self) -> ProofByRefutationInclusion: @@ -8287,7 +8289,7 @@ def pbr(self) -> ProofByRefutationInclusion: return self.proof_by_refutation @property - def proof_by_contradiction(self) -> ProofByContradictionInclusion: + def proof_by_contradiction_1(self) -> ProofByContradiction1Inclusion: """ Abridged property: u.i.pbc @@ -8295,15 +8297,14 @@ def proof_by_contradiction(self) -> ProofByContradictionInclusion: If the well-known inference-rule does not exist in the universe-of-discourse, the inference-rule is automatically declared. """ - if self._proof_by_contradiction is None: - self._proof_by_contradiction = ProofByContradictionInclusion(t=self.t) - return self._proof_by_contradiction + if self._proof_by_contradiction_1 is None: + self._proof_by_contradiction_1 = ProofByContradiction1Inclusion(t=self.t) + return self._proof_by_contradiction_1 @property - def proof_by_contradiction_of_non_equality(self) -> ProofByContradictionOfInequalityInclusion: + def proof_by_contradiction_of_non_equality(self) -> ProofByContradiction2Inclusion: if self._proof_by_contradiction_of_non_equality is None: - self._proof_by_contradiction_of_non_equality = ProofByContradictionOfInequalityInclusion( - t=self.t) + self._proof_by_contradiction_of_non_equality = ProofByContradiction2Inclusion(t=self.t) return self._proof_by_contradiction_of_non_equality @property diff --git a/punctilious/locale_en_us.py b/punctilious/locale_en_us.py index c7a9d5e4..61dbee2d 100644 --- a/punctilious/locale_en_us.py +++ b/punctilious/locale_en_us.py @@ -134,6 +134,19 @@ def compose_conjunction_elimination_left_paragraph_proof(self, o: InferredStatem yield SansSerifNormal('. ') return True + def compose_conjunction_elimination_right_paragraph_proof(self, o: InferredStatement) -> \ + collections.abc.Generator[Composable, Composable, bool]: + global text_dict + # Retrieve the parameters from the statement + p_and_q: FormulaStatement = o.parameters[0] + yield from p_and_q.valid_proposition.compose_formula() + yield SansSerifNormal(', is of the form ') + yield '(๐‘ท โˆง ๐‘ธ)' # TODO: Replace hardcoded formula with real formula + yield SansSerifNormal(', and follows from ') + yield from p_and_q.compose_ref_link() + yield SansSerifNormal('. ') + return True + def compose_conjunction_introduction_paragraph_proof(self, o: InferredStatement) -> \ collections.abc.Generator[Composable, Composable, bool]: global text_dict diff --git a/tests/test_conjunction_elimination_right.py b/tests/test_conjunction_elimination_right.py index b48c49d0..aaa2d478 100644 --- a/tests/test_conjunction_elimination_right.py +++ b/tests/test_conjunction_elimination_right.py @@ -14,8 +14,8 @@ def test_cer(self): t = u.t() a = u.declare_axiom(random_data.random_sentence()) ap = t.include_axiom(a) - phi1 = t.i.axiom_interpretation.infer_statement(ap, u.f(u.r.conjunction, u.f(r1, o1, o2), - u.f(r2, o3))) + phi1 = t.i.axiom_interpretation.infer_statement(ap, + u.f(u.r.conjunction, u.f(r1, o1, o2), u.f(r2, o3))) self.assertEqual('(๐‘Ÿโ‚(๐‘œโ‚, ๐‘œโ‚‚) โˆง ๐‘Ÿโ‚‚(๐‘œโ‚ƒ))', phi1.rep_formula(pu.encodings.unicode)) - phi2 = t.i.cer.infer_statement(phi1) + phi2 = t.i.cer.infer_statement(p_and_q=phi1) self.assertEqual('๐‘Ÿโ‚‚(๐‘œโ‚ƒ)', phi2.rep_formula(encoding=pu.encodings.unicode)) diff --git a/tests/test_definition_declaration.py b/tests/test_definition_declaration.py index 9a93db86..2fbc26ae 100644 --- a/tests/test_definition_declaration.py +++ b/tests/test_definition_declaration.py @@ -21,6 +21,8 @@ def test_definition_declaration(self): pu.configuration.echo_default = False pu.configuration.encoding = pu.encodings.plaintext u = pu.UniverseOfDiscourse() + u_plaintext = u.nameset.rep_symbol(encoding=pu.encodings.plaintext) + u_unicode = u.nameset.rep_symbol(encoding=pu.encodings.unicode) content1 = '๐’ทโ„ดโ„ด๐“‡โ„ด๐’ถ ๐’ทโ„ฏโ„ฏ๐’น๐“Š ๐“‡๐’พโ„ด๐’ท๐“โ„ด ๐’ท๐“Ž๐“Š๐’น๐’พโ„ฏ๐“€๐“Š.' # random_data.random_sentence() content2 = '๐“‡โ„ฏ๐’น๐“Š ๐’นโ„ด๐’ทโ„ดโ„ฏ ๐’น๐’พ๐“ƒ๐“Š ๐’ท๐’ถ๐’น๐“‡๐“Ž๐“Š ๐’ฝ๐“Ž๐“‹๐’พ๐“‹โ„ด๐’พ ๐’ทโ„ฏ๐’ท๐’พ ๐“๐“Ž๐’ถโ„Š๐“ƒ๐“Š ๐’น๐“Ž๐’ถ๐“€โ„ฏ ๐“ˆ๐’พ๐“โ„ดโ„ด ๐’ท๐’ถ๐“โ„ด ๐“…โ„ด๐’ถ๐’นโ„ดโ„ฏ ๐’ท๐’พ๐“ƒโ„ด๐’ท๐“‡๐“Š ๐’ป๐“โ„ด๐’น๐’พโ„ด ๐“๐’พ๐’ฝ๐’ถ ๐“‡๐“Žโ„ฏ๐“…โ„ฏ ๐“…โ„ด๐’พ๐“ˆ๐“‰๐’พ๐’ป๐“๐“Žโ„ด ๐“‰โ„ฏ๐’ท๐“โ„ดโ„ฏ ๐’นโ„ด๐’ท๐“‡โ„ฏ ๐’ฟโ„ดโ„ฏ๐’ท๐’พ ๐’ท๐’ถ๐’ฟ๐“Ž๐“Š ๐“ˆ๐“‰๐’ถ๐’ท๐“Ž๐’ถ๐“๐’ถ โ„Š๐“ƒ๐’ถ๐’ท๐“‡๐’พ ๐“‚โ„ด๐’ถ๐’ปโ„ด๐“Š ๐’นโ„ด๐“Š๐“‰๐“Š ๐“ˆ๐“Š๐’น๐’พ โ„Š๐“ƒโ„ด๐’ท๐“Š ๐“‚๐“Š๐’นโ„ฏ๐’ท๐’พ ๐’ทโ„ฏ๐’นโ„ด๐’ทโ„ฏ ๐’ป๐“๐’พโ„ด๐’ท๐’พ ๐’น๐’ถ๐“๐’ถโ„ฏ.' # random_data.random_sentence(min_words=30) content3 = '๐’ท๐“‡๐“Žโ„ฏ๐’ท๐“‡๐’พ ๐’ท๐“Ž๐’ถ๐’ท๐“‡๐“Žโ„ด๐’ทโ„ด๐“Š ๐’ท๐“Š๐’ท๐’พ.' # random_data.random_sentence() @@ -33,17 +35,17 @@ def test_definition_declaration(self): a5 = u.declare_definition(content5, acronym='oaot', symbol='d', name='the other definition of test') self.assertEqual( - f'๐–ซ๐–พ๐— โŒœ๐’Ÿโ‚โŒ ๐–ป๐–พ ๐—๐—๐–พ ๐‘‘๐‘’๐‘“๐‘–๐‘›๐‘–๐‘ก๐‘–๐‘œ๐‘› โŒœ๐’ทโ„ดโ„ด๐“‡โ„ด๐’ถ ๐’ทโ„ฏโ„ฏ๐’น๐“Š ๐“‡๐’พโ„ด๐’ท๐“โ„ด ๐’ท๐“Ž๐“Š๐’น๐’พโ„ฏ๐“€๐“Š.โŒ ๐—‚๐—‡ ๐’ฐโ‚.', + f'๐–ซ๐–พ๐— โŒœ๐’Ÿโ‚โŒ ๐–ป๐–พ ๐—๐—๐–พ ๐‘‘๐‘’๐‘“๐‘–๐‘›๐‘–๐‘ก๐‘–๐‘œ๐‘› โŒœ๐’ทโ„ดโ„ด๐“‡โ„ด๐’ถ ๐’ทโ„ฏโ„ฏ๐’น๐“Š ๐“‡๐’พโ„ด๐’ท๐“โ„ด ๐’ท๐“Ž๐“Š๐’น๐’พโ„ฏ๐“€๐“Š.โŒ ๐—‚๐—‡ {u_unicode}.', a1.rep_report(encoding=pu.encodings.unicode, wrap=False)) self.assertEqual( - f'๐–ซ๐–พ๐— โŒœ๐’Ÿโ‚‚โŒ ๐–ป๐–พ ๐—๐—๐–พ ๐‘‘๐‘’๐‘“๐‘–๐‘›๐‘–๐‘ก๐‘–๐‘œ๐‘› โŒœ๐“‡โ„ฏ๐’น๐“Š ๐’นโ„ด๐’ทโ„ดโ„ฏ ๐’น๐’พ๐“ƒ๐“Š ๐’ท๐’ถ๐’น๐“‡๐“Ž๐“Š ๐’ฝ๐“Ž๐“‹๐’พ๐“‹โ„ด๐’พ ๐’ทโ„ฏ๐’ท๐’พ ๐“๐“Ž๐’ถโ„Š๐“ƒ๐“Š ๐’น๐“Ž๐’ถ๐“€โ„ฏ ๐“ˆ๐’พ๐“โ„ดโ„ด ๐’ท๐’ถ๐“โ„ด ๐“…โ„ด๐’ถ๐’นโ„ดโ„ฏ ๐’ท๐’พ๐“ƒโ„ด๐’ท๐“‡๐“Š ๐’ป๐“โ„ด๐’น๐’พโ„ด ๐“๐’พ๐’ฝ๐’ถ ๐“‡๐“Žโ„ฏ๐“…โ„ฏ ๐“…โ„ด๐’พ๐“ˆ๐“‰๐’พ๐’ป๐“๐“Žโ„ด ๐“‰โ„ฏ๐’ท๐“โ„ดโ„ฏ ๐’นโ„ด๐’ท๐“‡โ„ฏ ๐’ฟโ„ดโ„ฏ๐’ท๐’พ ๐’ท๐’ถ๐’ฟ๐“Ž๐“Š ๐“ˆ๐“‰๐’ถ๐’ท๐“Ž๐’ถ๐“๐’ถ โ„Š๐“ƒ๐’ถ๐’ท๐“‡๐’พ ๐“‚โ„ด๐’ถ๐’ปโ„ด๐“Š ๐’นโ„ด๐“Š๐“‰๐“Š ๐“ˆ๐“Š๐’น๐’พ โ„Š๐“ƒโ„ด๐’ท๐“Š ๐“‚๐“Š๐’นโ„ฏ๐’ท๐’พ ๐’ทโ„ฏ๐’นโ„ด๐’ทโ„ฏ ๐’ป๐“๐’พโ„ด๐’ท๐’พ ๐’น๐’ถ๐“๐’ถโ„ฏ.โŒ ๐—‚๐—‡ ๐’ฐโ‚.', + f'๐–ซ๐–พ๐— โŒœ๐’Ÿโ‚‚โŒ ๐–ป๐–พ ๐—๐—๐–พ ๐‘‘๐‘’๐‘“๐‘–๐‘›๐‘–๐‘ก๐‘–๐‘œ๐‘› โŒœ๐“‡โ„ฏ๐’น๐“Š ๐’นโ„ด๐’ทโ„ดโ„ฏ ๐’น๐’พ๐“ƒ๐“Š ๐’ท๐’ถ๐’น๐“‡๐“Ž๐“Š ๐’ฝ๐“Ž๐“‹๐’พ๐“‹โ„ด๐’พ ๐’ทโ„ฏ๐’ท๐’พ ๐“๐“Ž๐’ถโ„Š๐“ƒ๐“Š ๐’น๐“Ž๐’ถ๐“€โ„ฏ ๐“ˆ๐’พ๐“โ„ดโ„ด ๐’ท๐’ถ๐“โ„ด ๐“…โ„ด๐’ถ๐’นโ„ดโ„ฏ ๐’ท๐’พ๐“ƒโ„ด๐’ท๐“‡๐“Š ๐’ป๐“โ„ด๐’น๐’พโ„ด ๐“๐’พ๐’ฝ๐’ถ ๐“‡๐“Žโ„ฏ๐“…โ„ฏ ๐“…โ„ด๐’พ๐“ˆ๐“‰๐’พ๐’ป๐“๐“Žโ„ด ๐“‰โ„ฏ๐’ท๐“โ„ดโ„ฏ ๐’นโ„ด๐’ท๐“‡โ„ฏ ๐’ฟโ„ดโ„ฏ๐’ท๐’พ ๐’ท๐’ถ๐’ฟ๐“Ž๐“Š ๐“ˆ๐“‰๐’ถ๐’ท๐“Ž๐’ถ๐“๐’ถ โ„Š๐“ƒ๐’ถ๐’ท๐“‡๐’พ ๐“‚โ„ด๐’ถ๐’ปโ„ด๐“Š ๐’นโ„ด๐“Š๐“‰๐“Š ๐“ˆ๐“Š๐’น๐’พ โ„Š๐“ƒโ„ด๐’ท๐“Š ๐“‚๐“Š๐’นโ„ฏ๐’ท๐’พ ๐’ทโ„ฏ๐’นโ„ด๐’ทโ„ฏ ๐’ป๐“๐’พโ„ด๐’ท๐’พ ๐’น๐’ถ๐“๐’ถโ„ฏ.โŒ ๐—‚๐—‡ {u_unicode}.', a2.rep_report(encoding=pu.encodings.unicode, wrap=False)) self.assertEqual( - f'๐–ซ๐–พ๐— โŒœ๐‘โ‚โŒ ๐–ป๐–พ ๐—๐—๐–พ ๐‘‘๐‘’๐‘“๐‘–๐‘›๐‘–๐‘ก๐‘–๐‘œ๐‘› โŒœ๐’ท๐“‡๐“Žโ„ฏ๐’ท๐“‡๐’พ ๐’ท๐“Ž๐’ถ๐’ท๐“‡๐“Žโ„ด๐’ทโ„ด๐“Š ๐’ท๐“Š๐’ท๐’พ.โŒ ๐—‚๐—‡ ๐’ฐโ‚.', + f'๐–ซ๐–พ๐— โŒœ๐‘โ‚โŒ ๐–ป๐–พ ๐—๐—๐–พ ๐‘‘๐‘’๐‘“๐‘–๐‘›๐‘–๐‘ก๐‘–๐‘œ๐‘› โŒœ๐’ท๐“‡๐“Žโ„ฏ๐’ท๐“‡๐’พ ๐’ท๐“Ž๐’ถ๐’ท๐“‡๐“Žโ„ด๐’ทโ„ด๐“Š ๐’ท๐“Š๐’ท๐’พ.โŒ ๐—‚๐—‡ {u_unicode}.', a3.rep_report(encoding=pu.encodings.unicode, wrap=False)) self.assertEqual( - f'๐–ซ๐–พ๐— โŒœ๐‘โ‚โŒ ๐–ป๐–พ ๐—๐—๐–พ ๐‘‘๐‘’๐‘“๐‘–๐‘›๐‘–๐‘ก๐‘–๐‘œ๐‘› โŒœ๐’ท๐“โ„ด๐“โ„ฏ ๐’ท๐“โ„ด๐’ฟโ„ฏ ๐’ฝ๐“Ž๐“Šโ„Š๐“ƒโ„ฏโ„ฏ.โŒ ๐—‚๐—‡ ๐’ฐโ‚, ๐—„๐—‡๐—ˆ๐—๐—‡ ๐–บ๐—Œ ๐—๐—๐–พ ๐–ฝ๐–พ๐–ฟ๐—‚๐—‡๐—‚๐—๐—‚๐—ˆ๐—‡ ๐—ˆ๐–ฟ ๐—๐–พ๐—Œ๐—.', + f'๐–ซ๐–พ๐— โŒœ๐‘โ‚โŒ ๐–ป๐–พ ๐—๐—๐–พ ๐‘‘๐‘’๐‘“๐‘–๐‘›๐‘–๐‘ก๐‘–๐‘œ๐‘› โŒœ๐’ท๐“โ„ด๐“โ„ฏ ๐’ท๐“โ„ด๐’ฟโ„ฏ ๐’ฝ๐“Ž๐“Šโ„Š๐“ƒโ„ฏโ„ฏ.โŒ ๐—‚๐—‡ {u_unicode}, ๐—„๐—‡๐—ˆ๐—๐—‡ ๐–บ๐—Œ ๐—๐—๐–พ ๐–ฝ๐–พ๐–ฟ๐—‚๐—‡๐—‚๐—๐—‚๐—ˆ๐—‡ ๐—ˆ๐–ฟ ๐—๐–พ๐—Œ๐—.', a4.rep_report(encoding=pu.encodings.unicode, wrap=False)) self.assertEqual( - f'๐–ซ๐–พ๐— โŒœ๐‘‘โ‚โŒ ๐–ป๐–พ ๐—๐—๐–พ ๐‘‘๐‘’๐‘“๐‘–๐‘›๐‘–๐‘ก๐‘–๐‘œ๐‘› โŒœ๐’น๐’พ๐“…๐’พ๐“‹๐’พ ๐“ˆ๐“‰๐’พ๐“‚โ„ด๐“‹โ„ด ๐’ป๐“‡๐’พ๐’ป๐“Š๐’ท๐“Š ๐’นโ„ด๐’ทโ„ฏ ๐’น๐“‡๐“Š๐“‚๐“Š๐’นโ„ด๐’พ๐“‰๐“Š.โŒ ๐—‚๐—‡ ๐’ฐโ‚, ๐—„๐—‡๐—ˆ๐—๐—‡ ๐–บ๐—Œ ๐—๐—๐–พ ๐—ˆ๐—๐—๐–พ๐—‹ ๐–ฝ๐–พ๐–ฟ๐—‚๐—‡๐—‚๐—๐—‚๐—ˆ๐—‡ ๐—ˆ๐–ฟ ๐—๐–พ๐—Œ๐—, ๐—ˆ๐—‹ ๐—Œ๐—‚๐—†๐—‰๐—…๐—’ ๐—ˆ๐–บ๐—ˆ๐—.', + f'๐–ซ๐–พ๐— โŒœ๐‘‘โ‚โŒ ๐–ป๐–พ ๐—๐—๐–พ ๐‘‘๐‘’๐‘“๐‘–๐‘›๐‘–๐‘ก๐‘–๐‘œ๐‘› โŒœ๐’น๐’พ๐“…๐’พ๐“‹๐’พ ๐“ˆ๐“‰๐’พ๐“‚โ„ด๐“‹โ„ด ๐’ป๐“‡๐’พ๐’ป๐“Š๐’ท๐“Š ๐’นโ„ด๐’ทโ„ฏ ๐’น๐“‡๐“Š๐“‚๐“Š๐’นโ„ด๐’พ๐“‰๐“Š.โŒ ๐—‚๐—‡ {u_unicode}, ๐—„๐—‡๐—ˆ๐—๐—‡ ๐–บ๐—Œ ๐—๐—๐–พ ๐—ˆ๐—๐—๐–พ๐—‹ ๐–ฝ๐–พ๐–ฟ๐—‚๐—‡๐—‚๐—๐—‚๐—ˆ๐—‡ ๐—ˆ๐–ฟ ๐—๐–พ๐—Œ๐—, ๐—ˆ๐—‹ ๐—Œ๐—‚๐—†๐—‰๐—…๐—’ ๐—ˆ๐–บ๐—ˆ๐—.', a5.rep_report(encoding=pu.encodings.unicode, wrap=False)) diff --git a/tests/test_objct_header.py b/tests/test_objct_header.py deleted file mode 100644 index 8b00c5c4..00000000 --- a/tests/test_objct_header.py +++ /dev/null @@ -1,42 +0,0 @@ -from unittest import TestCase -import punctilious as pu -import random_data - - -class TestHeader(TestCase): - def test_header_1(self): - header_reference = random_data.random_word() - header = pu.TitleOBSOLETE(header_reference, cat=pu.paragraph_headers.axiom_declaration) - self.assertEqual(f'axiom {header_reference}', str(header)) - self.assertEqual(f'axiom {header_reference}', - header.rep(encoding=pu.encodings.plaintext)) - self.assertEqual(f'axiom {header_reference}', - header.rep_title(encoding=pu.encodings.plaintext)) - self.assertEqual(f'axiom {header_reference}', - header.rep_ref(encoding=pu.encodings.plaintext)) - self.assertEqual(f'Axiom {header_reference}', - header.rep(encoding=pu.encodings.plaintext, cap=True)) - self.assertEqual(f'Axiom {header_reference}', - header.rep_title(encoding=pu.encodings.plaintext, cap=True)) - self.assertEqual(f'Axiom {header_reference}', - header.rep_ref(encoding=pu.encodings.plaintext, cap=True)) - - def test_header_2(self): - header_reference = random_data.random_word() - header_title = random_data.random_sentence() - header = pu.TitleOBSOLETE(header_reference, - cat=pu.paragraph_headers._hypothesis_statement_in_child_theory, - subtitle=header_title) - self.assertEqual(f'prop. {header_reference}', str(header)) - self.assertEqual(f'prop. {header_reference}', - header.rep(encoding=pu.encodings.plaintext)) - self.assertEqual(f'proposition {header_reference} - {header_title}', - header.rep_title(encoding=pu.encodings.plaintext)) - self.assertEqual(f'prop. {header_reference}', - header.rep_ref(encoding=pu.encodings.plaintext)) - self.assertEqual(f'Prop. {header_reference}', - header.rep(encoding=pu.encodings.plaintext, cap=True)) - self.assertEqual(f'Proposition {header_reference} - {header_title}', - header.rep_title(encoding=pu.encodings.plaintext, cap=True)) - self.assertEqual(f'Prop. {header_reference}', - header.rep_ref(encoding=pu.encodings.plaintext, cap=True)) diff --git a/tests/test_proof_by_contradiction.py b/tests/test_proof_by_contradiction_1.py similarity index 89% rename from tests/test_proof_by_contradiction.py rename to tests/test_proof_by_contradiction_1.py index 50cea2b3..b4bc7f0a 100644 --- a/tests/test_proof_by_contradiction.py +++ b/tests/test_proof_by_contradiction_1.py @@ -5,7 +5,7 @@ # TODO: Proof by contradiction: design test class TestProofByContradiction(TestCase): - def test_proof_by_contradiction(self): + def test_proof_by_contradiction_1(self): pu.configuration.echo_default = False pu.configuration.echo_inferred_statement = True # Prepare the universe of discourse @@ -36,11 +36,11 @@ def test_proof_by_contradiction(self): t2 = t1_h1.hypothesis_child_theory t2_a1 = t1_h1.hypothesis_statement_in_child_theory t2_p5 = t2.i.conjunction_introduction.infer_statement(p=t1_p1, q=t1_p2) - t2_p6 = t2.i.variable_substitution.infer_statement(t1_p3_implication, o1, o2, o3) + t2_p6 = t2.i.variable_substitution.infer_statement(p=t1_p3_implication, phi=(o1, o2, o3)) # p7: ๐‘Ÿโ‚(๐‘œโ‚, ๐‘œโ‚ƒ) by modus ponens t2_p7 = t2.i.modus_ponens.infer_statement(p_implies_q=t2_p6, p=t2_p5) # p7 is in contradiction with the hypothetical_formula t1_p8 = t1.i.inconsistency_by_negation_introduction.infer_statement(p=t2_p7, not_p=t2_a1, inconsistent_theory=t2) - t1_p9 = t1.i.proof_by_contradiction.infer_statement(not_p=t1_h1, inc_not_p=t1_p8) + t1_p9 = t1.i.proof_by_contradiction_1.infer_statement(not_p=t1_h1, inc_not_p=t1_p8) pass diff --git a/tests/test_proof_by_refutation.py b/tests/test_proof_by_refutation.py index c9cf94bf..162a450f 100644 --- a/tests/test_proof_by_refutation.py +++ b/tests/test_proof_by_refutation.py @@ -42,5 +42,5 @@ def test_proof_by_refutation(self): # p7 is in refutation with the hypothetical_formula t1_p8 = t1.i.inconsistency_by_negation_introduction.infer_statement(p=t2_a1, not_p=t2_p7, inconsistent_theory=t2) - t1_p9 = t1.i.proof_by_refutation.infer_statement(p=t1_h1, inc_p=t1_p8) + t1_p9 = t1.i.proof_by_refutation_1.infer_statement(p=t1_h1, inc_p=t1_p8) pass diff --git a/tests/test_theory_elaboration_sequence.py b/tests/test_theory_elaboration_sequence.py index 8f9accd4..954dc52b 100644 --- a/tests/test_theory_elaboration_sequence.py +++ b/tests/test_theory_elaboration_sequence.py @@ -6,10 +6,13 @@ class TestTheoryElaborationSequence(TestCase): def test_theory_elaboration_sequence(self): u1 = pu.UniverseOfDiscourse() + u_plaintext = u1.nameset.rep_symbol(encoding=pu.encodings.plaintext) + u_unicode = u1.nameset.rep_symbol(encoding=pu.encodings.unicode) t1 = u1.declare_theory() self.assertEqual('T1', t1.rep(encoding=pu.encodings.plaintext)) self.assertEqual('๐’ฏโ‚', t1.rep(encoding=pu.encodings.unicode)) - self.assertEqual('Let T1 be a theory-elaboration-sequence in U1.', + self.assertEqual(f'Let "T1" be a theory-elaboration-sequence in {u_plaintext}.', t1.rep_report(encoding=pu.encodings.plaintext)) - self.assertEqual('Let ๐’ฏโ‚ be a theory-elaboration-sequence in ๐’ฐโ‚.', + self.assertEqual( + f'๐–ซ๐–พ๐— โŒœ๐’ฏโ‚โŒ ๐–ป๐–พ ๐–บ ๐‘กโ„Ž๐‘’๐‘œ๐‘Ÿ๐‘ฆ-๐‘’๐‘™๐‘Ž๐‘๐‘œ๐‘Ÿ๐‘Ž๐‘ก๐‘–๐‘œ๐‘›-๐‘ ๐‘’๐‘ž๐‘ข๐‘’๐‘›๐‘๐‘’ ๐—‚๐—‡ {u_unicode}.', t1.rep_report(encoding=pu.encodings.unicode)) diff --git a/tests/test_theory_elaboration_stabilized.py b/tests/test_theory_elaboration_stabilized.py index d70d7f65..937f4d85 100644 --- a/tests/test_theory_elaboration_stabilized.py +++ b/tests/test_theory_elaboration_stabilized.py @@ -17,28 +17,17 @@ def test_theory_elaboration_stabilized(self): robust_theory = u.t() ap1 = robust_theory.include_axiom(a=a1) with u.v() as x, u.v() as y, u.v() as z: - implication = robust_theory.i.axiom_interpretation.infer_statement( - ap1, - u.f(u.r.implies, - u.f(u.r.land, u.f(r1, x, y), - u.f(r1, y, z)), - u.f(r1, x, z))) - r1o1o2 = robust_theory.i.axiom_interpretation.infer_statement( - ap1, u.f(r1, o1, o2)) - r1o2o3 = robust_theory.i.axiom_interpretation.infer_statement( - ap1, - u.f(r1, o2, o3)) - r1o1o2_and_r1o2o3 = robust_theory.i.ci.infer_statement( - r1o1o2, - r1o2o3) - implication_2 = robust_theory.i.vs.infer_statement(implication, o1, o2, o3) - robust_theory.i.mp.infer_statement( - implication_2, - r1o1o2_and_r1o2o3) + implication = robust_theory.i.axiom_interpretation.infer_statement(ap1, + u.f(u.r.implies, u.f(u.r.land, u.f(r1, x, y), u.f(r1, y, z)), u.f(r1, x, z))) + r1o1o2 = robust_theory.i.axiom_interpretation.infer_statement(ap1, u.f(r1, o1, o2)) + r1o2o3 = robust_theory.i.axiom_interpretation.infer_statement(ap1, u.f(r1, o2, o3)) + r1o1o2_and_r1o2o3 = robust_theory.i.ci.infer_statement(r1o1o2, r1o2o3) + implication_2 = robust_theory.i.variable_substitution.infer_statement(p=implication, + phi=(o1, o2, o3)) + robust_theory.i.mp.infer_statement(p_implies_q=implication_2, p=r1o1o2_and_r1o2o3) robust_theory.stabilize() self.assertTrue(robust_theory.stabilized, - 'The stabilized property of the original-theory is not True.') + 'The stabilized property of the original-theory is not True.') - hypothesis = robust_theory.pose_hypothesis( - hypothesis_formula=u.f(r1, o2, o3)) + hypothesis = robust_theory.pose_hypothesis(hypothesis_formula=u.f(r1, o2, o3)) hypothesis_theory = hypothesis.hypothesis_child_theory diff --git a/theory/package/tao_2006_2_1_the_peano_axioms.py b/theory/package/tao_2006_2_1_the_peano_axioms.py index 756c02c6..691ece90 100644 --- a/theory/package/tao_2006_2_1_the_peano_axioms.py +++ b/theory/package/tao_2006_2_1_the_peano_axioms.py @@ -258,8 +258,7 @@ def develop_theory(self, t: pu.TheoryElaborationSequence) -> pu.TheoryElaboratio p=h1_p11) p072 = t.i.inconsistency_by_inequality_introduction.infer_statement(p_eq_q=h1_p071, p_neq_q=p031, inconsistent_theory=h1.hypothesis_child_theory) - p073 = t.i.proof_by_refutation_of_equality.infer_statement(p_eq_q=h1, inc_p_eq_q=p072, - ref='2.1.8') + p073 = t.i.proof_by_refutation_2.infer_statement(p_eq_q=h1, inc_p_eq_q=p072, ref='2.1.8') t.open_section('Direct proof', section_parent=s55, numbering=False) From ae18653fa662d0ef789410bc979e063e7562e3d3 Mon Sep 17 00:00:00 2001 From: David Doret Date: Thu, 24 Aug 2023 14:02:29 +0200 Subject: [PATCH 08/21] tests --- docs/source/proof_by_contradiction_1.rst | 35 +++ punctilious/core.py | 268 +++++++++++++---------- punctilious/locale_en_us.py | 53 ++++- 3 files changed, 236 insertions(+), 120 deletions(-) create mode 100644 docs/source/proof_by_contradiction_1.rst diff --git a/docs/source/proof_by_contradiction_1.rst b/docs/source/proof_by_contradiction_1.rst new file mode 100644 index 00000000..cf06634a --- /dev/null +++ b/docs/source/proof_by_contradiction_1.rst @@ -0,0 +1,35 @@ +proof-by-contradiction-1 +========================= + +Definition +---------- + +*proof-by-contradiction-1* is the :doc:`inference_rule`: + +.. math:: + + \left( \left(P \implies Q\right) \land P \right) \vdash Q + +Python implementation +--------------------- + +ProofByContradiction1Declaration +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +.. module:: core + :noindex: +.. autoclass:: ProofByContradiction1Declaration + :members: + :special-members: __init__ + +ProofByContradiction1Inclusion +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +.. autoclass:: ProofByContradiction1Inclusion + :members: + :special-members: __init__ + +Bibliography +------------ + +.. footbibliography:: \ No newline at end of file diff --git a/punctilious/core.py b/punctilious/core.py index 44b561d5..2abae74f 100644 --- a/punctilious/core.py +++ b/punctilious/core.py @@ -4769,6 +4769,11 @@ def __init__(self, universe_of_discourse: UniverseOfDiscourse, echo: (None, bool auto_index=auto_index, dashed_name=dashed_name, acronym=acronym, abridged_name=abridged_name, name=name, explicit_name=explicit_name, echo=echo) + def compose_paragraph_proof(self, o: InferredStatement) -> collections.abc.Generator[ + Composable, Composable, bool]: + output = yield from configuration.locale.compose_modus_ponens_paragraph_proof(o=o) + return output + def infer_formula(self, p_implies_q: FormulaStatement, p: FormulaStatement, t: TheoryElaborationSequence, echo: (None, bool) = None) -> Formula: """ @@ -4784,11 +4789,6 @@ def infer_formula(self, p_implies_q: FormulaStatement, p: FormulaStatement, # is true in t. return q # TODO: Provide support for statements that are atomic propositional formula, # that is # without relation or where the objct is a 0-ary relation. - def compose_paragraph_proof(self, o: InferredStatement) -> collections.abc.Generator[ - Composable, Composable, bool]: - output = yield from configuration.locale.compose_modus_ponens_paragraph_proof(o=o) - return output - def verify_args(self, p_implies_q: FormulaStatement, p: FormulaStatement, t: TheoryElaborationSequence) -> bool: """ @@ -4828,54 +4828,57 @@ def __init__(self, universe_of_discourse: UniverseOfDiscourse, echo: (None, bool dashed_name = 'proof-by-contradiction' explicit_name = 'proof by contradiction inference rule' name = 'proof by contradiction' + definition = '(๐“— ๐‘Ž๐‘ ๐‘ ๐‘ข๐‘š๐‘’ ยฌ๐‘ท, ๐ผ๐‘›๐‘(๐“—)) โŠข ๐‘ท' # Assure backward-compatibility with the parent class, # which received these methods as __init__ arguments. infer_formula = ProofByContradiction1Declaration.infer_formula verify_args = ProofByContradiction1Declaration.verify_args - super().__init__(infer_formula=infer_formula, verify_args=verify_args, - universe_of_discourse=universe_of_discourse, symbol=symbol, auto_index=auto_index, - dashed_name=dashed_name, acronym=acronym, name=name, explicit_name=explicit_name, - echo=echo) + super().__init__(definition=definition, infer_formula=infer_formula, + verify_args=verify_args, universe_of_discourse=universe_of_discourse, symbol=symbol, + auto_index=auto_index, dashed_name=dashed_name, acronym=acronym, name=name, + explicit_name=explicit_name, echo=echo) - def infer_formula(self, not_p: Hypothesis, inc_not_p: FormulaStatement, + def compose_paragraph_proof(self, o: InferredStatement) -> collections.abc.Generator[ + Composable, Composable, bool]: + output = yield from configuration.locale.compose_proof_by_contradiction_1_paragraph_proof( + o=o) + return output + + def infer_formula(self, not_p_hypothesis: Hypothesis, inc_hypothesis: FormulaStatement, t: TheoryElaborationSequence, echo: (None, bool) = None) -> Formula: - not_p = not_p.hypothesis_formula + not_p = not_p_hypothesis.hypothesis_formula p = not_p.parameters[0] return p - # def compose_paragraph_proof(self, o: InferredStatement) -> collections.abc.Generator[ - # Composable, Composable, bool]: - # output = yield from configuration.locale.compose_modus_ponens_paragraph_proof( - # o=o) - # return output - - def verify_args(self, not_p: Hypothesis, inc_not_p: InferredStatement, + def verify_args(self, not_p_hypothesis: Hypothesis, inc_hypothesis: InferredStatement, t: TheoryElaborationSequence, echo: (None, bool) = None) -> bool: """ - :param not_p: The hypothesis-statement in the parent theory. - :param inc_not_p: The contradiction-statement Inc(Tn) where Tn is the hypothesis-theory. + :param not_p_hypothesis: The hypothesis-statement in the parent theory. + :param inc_hypothesis: The contradiction-statement Inc(Tn) where Tn is the hypothesis-theory. :param t: The current (parent) theory. :param echo: :return: """ - verify(is_in_class(not_p, classes.hypothesis), 'โŒœnot_pโŒ must be an hypothesis.', - not_p=not_p, slf=self) - verify(is_in_class(inc_not_p, classes.inferred_proposition), - 'โŒœinc_not_pโŒ must be an inferred-statement.', inc_not_p=inc_not_p, slf=self) - verify(t.contains_theoretical_objct(not_p), 'โŒœnot_pโŒ must be in theory โŒœtโŒ.', not_p=not_p, - t=t, slf=self) - verify(not_p.hypothesis_child_theory.extended_theory is t, + verify(is_in_class(not_p_hypothesis, classes.hypothesis), 'โŒœnot_pโŒ must be an hypothesis.', + not_p=not_p_hypothesis, slf=self) + verify(is_in_class(inc_hypothesis, classes.inferred_proposition), + 'โŒœinc_not_pโŒ must be an inferred-statement.', inc_not_p=inc_hypothesis, slf=self) + verify(t.contains_theoretical_objct(not_p_hypothesis), 'โŒœnot_pโŒ must be in theory โŒœtโŒ.', + not_p=not_p_hypothesis, t=t, slf=self) + verify(not_p_hypothesis.hypothesis_child_theory.extended_theory is t, 'โŒœnot_p.hypothetical_theoryโŒ must extend the parent theory โŒœtโŒ.', - not_p_hypothetical_theory=not_p.hypothesis_child_theory, not_p=not_p, t=t, slf=self) - verify(inc_not_p.relation is t.u.relations.inconsistency, + not_p_hypothetical_theory=not_p_hypothesis.hypothesis_child_theory, + not_p=not_p_hypothesis, t=t, slf=self) + verify(inc_hypothesis.relation is t.u.relations.inconsistency, 'โŒœinc_not_p.relationโŒ must be of form (Inc(Hn)).', - inc_not_p_relation=inc_not_p.relation, inc_not_p=inc_not_p, t=t, slf=self) - verify(inc_not_p.valid_proposition.parameters[0] is not_p.hypothesis_child_theory, + inc_not_p_relation=inc_hypothesis.relation, inc_not_p=inc_hypothesis, t=t, slf=self) + verify(inc_hypothesis.valid_proposition.parameters[ + 0] is not_p_hypothesis.hypothesis_child_theory, 'โŒœinc_not_pโŒ must be of form (Inc(Hn)) where parameter[0] Hn is the ' 'hypothetical-theory.', - inc_not_p_parameters_0=inc_not_p.valid_proposition.parameters[0], - not_p_hypothetical_theory=not_p.hypothesis_child_theory, t=t, slf=self) + inc_not_p_parameters_0=inc_hypothesis.valid_proposition.parameters[0], + not_p_hypothetical_theory=not_p_hypothesis.hypothesis_child_theory, t=t, slf=self) # TODO: ProofByContradictionDeclaration.verify_args: check that the parent theory is # stable??? # TODO: ProofByContradictionDeclaration.verify_args: check that the hypothetical-theory @@ -4885,56 +4888,63 @@ def verify_args(self, not_p: Hypothesis, inc_not_p: InferredStatement, class ProofByContradiction2Declaration(InferenceRuleDeclaration): def __init__(self, universe_of_discourse: UniverseOfDiscourse, echo: (None, bool) = None): - symbol = 'proof-by-contradiction-of-inequality' - acronym = 'pbci' + symbol = 'proof-by-contradiction-2' + acronym = 'pbc2' auto_index = False - dashed_name = 'proof-by-contradiction-of-inequality' - explicit_name = 'proof by contradiction of inequality inference rule' - name = 'proof by contradiction of inequality' + dashed_name = 'proof-by-contradiction-2' + explicit_name = 'proof by contradiction #2 inference rule' + name = 'proof by contradiction #2' + definition = '(๐“— ๐‘Ž๐‘ ๐‘ ๐‘ข๐‘š๐‘’ (๐‘ท โ‰  ๐‘ธ), ๐ผ๐‘›๐‘(๐“—)) โŠข (๐‘ท = ๐‘ธ)' # Assure backward-compatibility with the parent class, # which received these methods as __init__ arguments. infer_formula = ProofByContradiction2Declaration.infer_formula verify_args = ProofByContradiction2Declaration.verify_args - super().__init__(infer_formula=infer_formula, verify_args=verify_args, - universe_of_discourse=universe_of_discourse, symbol=symbol, auto_index=auto_index, - dashed_name=dashed_name, acronym=acronym, name=name, explicit_name=explicit_name, - echo=echo) + super().__init__(definition=definition, infer_formula=infer_formula, + verify_args=verify_args, universe_of_discourse=universe_of_discourse, symbol=symbol, + auto_index=auto_index, dashed_name=dashed_name, acronym=acronym, name=name, + explicit_name=explicit_name, echo=echo) + + def compose_paragraph_proof(self, o: InferredStatement) -> collections.abc.Generator[ + Composable, Composable, bool]: + output = yield from configuration.locale.compose_proof_by_contradiction_2_paragraph_proof( + o=o) + return output - def infer_formula(self, p_neq_q: Hypothesis, inc_p_neq_q: FormulaStatement, + def infer_formula(self, p_neq_q_hypothesis: Hypothesis, inc_hypothesis: FormulaStatement, t: TheoryElaborationSequence, echo: (None, bool) = None) -> Formula: - p_neq_q = p_neq_q.hypothesis_formula + p_neq_q = p_neq_q_hypothesis.hypothesis_formula p = p_neq_q.parameters[0] q = p_neq_q.parameters[1] return t.u.f(t.u.r.equality, p, q) - def verify_args(self, p_neq_q: Hypothesis, inc_p_neq_q: InferredStatement, + def verify_args(self, p_neq_q_hypothesis: Hypothesis, inc_hypothesis: InferredStatement, t: TheoryElaborationSequence, echo: (None, bool) = None) -> bool: """ - :param p_neq_q: The hypothesis-statement in the parent theory. - :param inc_p_neq_q: The contradiction-statement Inc(Tn) where Tn is the hypothesis-theory. + :param p_neq_q_hypothesis: The hypothesis-statement in the parent theory. + :param inc_hypothesis: The contradiction-statement Inc(Tn) where Tn is the hypothesis-theory. :param t: The current (parent) theory. :param echo: :return: """ - verify(is_in_class(p_neq_q, classes.hypothesis), 'โŒœp_neq_qโŒ must be an hypothesis.', - p_neq_q=p_neq_q, slf=self) - verify(is_in_class(inc_p_neq_q, classes.inferred_proposition), - 'โŒœinc_p_neq_qโŒ must be an inferred-statement.', inc_p_neq_q=inc_p_neq_q, slf=self) - verify(t.contains_theoretical_objct(p_neq_q), 'โŒœp_neq_qโŒ must be in theory โŒœtโŒ.', - p_neq_q=p_neq_q, t=t, slf=self) - verify(p_neq_q.hypothesis_child_theory.extended_theory is t, + verify(is_in_class(p_neq_q_hypothesis, classes.hypothesis), + 'โŒœp_neq_qโŒ must be an hypothesis.', p_neq_q=p_neq_q_hypothesis, slf=self) + verify(is_in_class(inc_hypothesis, classes.inferred_proposition), + 'โŒœinc_p_neq_qโŒ must be an inferred-statement.', inc_p_neq_q=inc_hypothesis, slf=self) + verify(t.contains_theoretical_objct(p_neq_q_hypothesis), 'โŒœp_neq_qโŒ must be in theory โŒœtโŒ.', + p_neq_q=p_neq_q_hypothesis, t=t, slf=self) + verify(p_neq_q_hypothesis.hypothesis_child_theory.extended_theory is t, 'โŒœp_neq_q.hypothetical_theoryโŒ must extend the parent theory โŒœtโŒ.', - p_neq_q_hypothetical_theory=p_neq_q.hypothesis_child_theory, p_neq_q=p_neq_q, t=t, - slf=self) - verify(inc_p_neq_q.relation is t.u.relations.inequality, + p_neq_q_hypothetical_theory=p_neq_q_hypothesis.hypothesis_child_theory, + p_neq_q=p_neq_q_hypothesis, t=t, slf=self) + verify(inc_hypothesis.relation is t.u.relations.inequality, 'โŒœinc_p_neq_q.relationโŒ must be the inequality relation.', - inc_p_neq_q_relation=inc_p_neq_q.relation, inc_p_neq_q=inc_p_neq_q, t=t, slf=self) + inc_p_neq_q_relation=inc_hypothesis.relation, inc_p_neq_q=inc_hypothesis, t=t, slf=self) p_neq_q_prime: FormulaStatement - p_neq_q_prime = inc_p_neq_q.parameters[0] - verify(p_neq_q.is_formula_syntactically_equivalent_to(p_neq_q_prime), + p_neq_q_prime = inc_hypothesis.parameters[0] + verify(p_neq_q_hypothesis.is_formula_syntactically_equivalent_to(p_neq_q_prime), 'โŒœp_neq_qโŒ must be formula-syntactically-equivalent to โŒœp_neq_qโŒ in โŒœinc_p_neq_qโŒ.', - p_neq_q=p_neq_q, p_neq_q_prime=p_neq_q_prime, t=t, slf=self) + p_neq_q=p_neq_q_hypothesis, p_neq_q_prime=p_neq_q_prime, t=t, slf=self) # TODO: ProofByContradictionDeclaration.verify_args: check that the parent theory is # stable??? # TODO: ProofByContradictionDeclaration.verify_args: check that the hypothetical-theory @@ -4950,47 +4960,55 @@ def __init__(self, universe_of_discourse: UniverseOfDiscourse, echo: (None, bool dashed_name = 'proof-by-refutation' explicit_name = 'proof by refutation inference rule' name = 'proof by refutation' + definition = '(๐“— ๐‘Ž๐‘ ๐‘ ๐‘ข๐‘š๐‘’ ๐‘ท, ๐ผ๐‘›๐‘(๐“—)) โŠข ยฌ๐‘ท' # Assure backward-compatibility with the parent class, # which received these methods as __init__ arguments. infer_formula = ProofByContradiction1Declaration.infer_formula verify_args = ProofByContradiction1Declaration.verify_args - super().__init__(infer_formula=infer_formula, verify_args=verify_args, - universe_of_discourse=universe_of_discourse, symbol=symbol, auto_index=auto_index, - dashed_name=dashed_name, acronym=acronym, name=name, explicit_name=explicit_name, - echo=echo) + super().__init__(definition=definition, infer_formula=infer_formula, + verify_args=verify_args, universe_of_discourse=universe_of_discourse, symbol=symbol, + auto_index=auto_index, dashed_name=dashed_name, acronym=acronym, name=name, + explicit_name=explicit_name, echo=echo) - def infer_formula(self, p: Hypothesis, inc_p: FormulaStatement, t: TheoryElaborationSequence, - echo: (None, bool) = None) -> Formula: - p = p.hypothesis_formula + def compose_paragraph_proof(self, o: InferredStatement) -> collections.abc.Generator[ + Composable, Composable, bool]: + output = yield from configuration.locale.compose_proof_by_refutation_1_paragraph_proof(o=o) + return output + + def infer_formula(self, p_hypothesis: Hypothesis, inc_hypothesis: FormulaStatement, + t: TheoryElaborationSequence, echo: (None, bool) = None) -> Formula: + p = p_hypothesis.hypothesis_formula not_p = t.u.f(t.u.r.negation, p) return not_p - # def compose_paragraph_proof(self, o: InferredStatement) -> collections.abc.Generator[ # Composable, Composable, bool]: # output = yield from # # # # # # configuration.locale.compose_modus_ponens_paragraph_proof( # o=o) # # # # return output - - def verify_args(self, p: Hypothesis, inc_p: InferredStatement, t: TheoryElaborationSequence, - echo: (None, bool) = None) -> bool: + def verify_args(self, p_hypothesis: Hypothesis, inc_hypothesis: InferredStatement, + t: TheoryElaborationSequence, echo: (None, bool) = None) -> bool: """ - :param p: The hypothesis-statement in the parent theory. - :param inc_p: The refutation-statement Inc(Tn) where Tn is the hypothesis-theory. + :param p_hypothesis: The hypothesis-statement in the parent theory. + :param inc_hypothesis: The refutation-statement Inc(Tn) where Tn is the hypothesis-theory. :param t: The current (parent) theory. :param echo: :return: """ - verify(is_in_class(p, classes.hypothesis), 'โŒœpโŒ must be an hypothesis.', p=p, slf=self) - verify(is_in_class(inc_p, classes.inferred_proposition), - 'โŒœinc_pโŒ must be an inferred-statement.', inc_p=inc_p, slf=self) - verify(t.contains_theoretical_objct(p), 'โŒœpโŒ must be in theory โŒœtโŒ.', p=p, t=t, slf=self) - verify(p.hypothesis_child_theory.extended_theory is t, + verify(is_in_class(p_hypothesis, classes.hypothesis), 'โŒœpโŒ must be an hypothesis.', + p=p_hypothesis, slf=self) + verify(is_in_class(inc_hypothesis, classes.inferred_proposition), + 'โŒœinc_pโŒ must be an inferred-statement.', inc_p=inc_hypothesis, slf=self) + verify(t.contains_theoretical_objct(p_hypothesis), 'โŒœpโŒ must be in theory โŒœtโŒ.', + p=p_hypothesis, t=t, slf=self) + verify(p_hypothesis.hypothesis_child_theory.extended_theory is t, 'โŒœp.hypothetical_theoryโŒ must extend the parent theory โŒœtโŒ.', - p_hypothetical_theory=p.hypothesis_child_theory, p=p, t=t, slf=self) - verify(inc_p.relation is t.u.relations.inconsistency, - 'โŒœinc_p.relationโŒ must be of form (Inc(Hn)).', inc_p_relation=inc_p.relation, - inc_p=inc_p, t=t, slf=self) - verify(inc_p.valid_proposition.parameters[0] is p.hypothesis_child_theory, + p_hypothetical_theory=p_hypothesis.hypothesis_child_theory, p=p_hypothesis, t=t, + slf=self) + verify(inc_hypothesis.relation is t.u.relations.inconsistency, + 'โŒœinc_p.relationโŒ must be of form (Inc(Hn)).', inc_p_relation=inc_hypothesis.relation, + inc_p=inc_hypothesis, t=t, slf=self) + verify( + inc_hypothesis.valid_proposition.parameters[0] is p_hypothesis.hypothesis_child_theory, 'โŒœinc_pโŒ must be of form (Inc(Hn)) where parameter[0] Hn is the hypothetical-theory.', - inc_p_parameters_0=inc_p.valid_proposition.parameters[0], - p_hypothetical_theory=p.hypothesis_child_theory, t=t, slf=self) + inc_p_parameters_0=inc_hypothesis.valid_proposition.parameters[0], + p_hypothetical_theory=p_hypothesis.hypothesis_child_theory, t=t, slf=self) # TODO: ProofByContradictionDeclaration.verify_args: check that the parent theory is # stable??? # TODO: ProofByContradictionDeclaration.verify_args: check that the hypothetical-theory @@ -5000,13 +5018,13 @@ def verify_args(self, p: Hypothesis, inc_p: InferredStatement, t: TheoryElaborat class ProofByRefutation2Declaration(InferenceRuleDeclaration): def __init__(self, universe_of_discourse: UniverseOfDiscourse, echo: (None, bool) = None): - symbol = 'proof-by-refutation-of-equality' - acronym = 'pbre' + symbol = 'proof-by-refutation-2' + acronym = 'pbr2' auto_index = False - dashed_name = 'proof-by-refutation-of-equality' - explicit_name = 'proof by refutation of equality inference rule' - name = 'proof by refutation of equality' - definition = '(โ„‹(P=Q), Inc(โ„‹)) โŠข (P โ‰  Q)' + dashed_name = 'proof-by-refutation-2' + explicit_name = 'proof by refutation #2 inference rule' + name = 'proof by refutation #2' + definition = '(๐“— ๐‘Ž๐‘ ๐‘ ๐‘ข๐‘š๐‘’ (๐‘ท = ๐‘ธ), ๐ผ๐‘›๐‘(๐“—)) โŠข (๐‘ท โ‰  ๐‘ธ)' # Assure backward-compatibility with the parent class, # which received these methods as __init__ arguments. infer_formula = ProofByRefutation2Declaration.infer_formula @@ -5018,40 +5036,41 @@ def __init__(self, universe_of_discourse: UniverseOfDiscourse, echo: (None, bool def compose_paragraph_proof(self, o: InferredStatement) -> collections.abc.Generator[ Composable, Composable, bool]: - output = yield from configuration.locale.compose_proof_by_reputation_of_equality_paragraph_proof( - o=o) + output = yield from configuration.locale.compose_proof_by_refutation_2_paragraph_proof(o=o) return output - def infer_formula(self, p_eq_q: Hypothesis, inc_p_eq_q: FormulaStatement, + def infer_formula(self, p_eq_q_hypothesis: Hypothesis, inc_hypothesis: FormulaStatement, t: TheoryElaborationSequence, echo: (None, bool) = None) -> Formula: - p_eq_q = p_eq_q.hypothesis_formula + p_eq_q = p_eq_q_hypothesis.hypothesis_formula p_neq_q = t.u.f(t.u.r.inequality, p_eq_q.parameters[0], p_eq_q.parameters[1]) return p_neq_q # def compose_paragraph_proof(self, o: InferredStatement) -> collections.abc.Generator[ # Composable, Composable, bool]: # output = yield from # # # # # # configuration.locale.compose_modus_ponens_paragraph_proof( # o=o) # # # # return output - def verify_args(self, p_eq_q: Hypothesis, inc_p_eq_q: InferredStatement, + def verify_args(self, p_eq_q_hypothesis: Hypothesis, inc_hypothesis: InferredStatement, t: TheoryElaborationSequence, echo: (None, bool) = None) -> bool: """ - :param p_eq_q: The hypothesis-statement in the parent theory. - :param inc_p_eq_q: The refutation-statement Inc(Tn) where Tn is the hypothesis-theory. + :param p_eq_q_hypothesis: The hypothesis-statement in the parent theory. + :param inc_hypothesis: The refutation-statement Inc(Tn) where Tn is the hypothesis-theory. :param t: The current (parent) theory. :param echo: :return: """ - verify(is_in_class(p_eq_q, classes.hypothesis), 'โŒœpโŒ must be an hypothesis.', p=p_eq_q, - slf=self) - verify(is_in_class(inc_p_eq_q, classes.inferred_proposition), - 'โŒœinc_pโŒ must be an inferred-statement.', inc_p=inc_p_eq_q, slf=self) - verify(t.contains_theoretical_objct(p_eq_q), 'โŒœpโŒ must be in theory โŒœtโŒ.', p=p_eq_q, t=t, - slf=self) - verify(p_eq_q.hypothesis_child_theory.extended_theory is t, + verify(is_in_class(p_eq_q_hypothesis, classes.hypothesis), 'โŒœpโŒ must be an hypothesis.', + p=p_eq_q_hypothesis, slf=self) + verify(is_in_class(inc_hypothesis, classes.inferred_proposition), + 'โŒœinc_pโŒ must be an inferred-statement.', inc_p=inc_hypothesis, slf=self) + verify(t.contains_theoretical_objct(p_eq_q_hypothesis), 'โŒœpโŒ must be in theory โŒœtโŒ.', + p=p_eq_q_hypothesis, t=t, slf=self) + verify(p_eq_q_hypothesis.hypothesis_child_theory.extended_theory is t, 'โŒœp.hypothetical_theoryโŒ must extend the parent theory โŒœtโŒ.', - p_hypothetical_theory=p_eq_q.hypothesis_child_theory, p=p_eq_q, t=t, slf=self) - verify(inc_p_eq_q.valid_proposition.relation is t.u.relations.inconsistency, + p_hypothetical_theory=p_eq_q_hypothesis.hypothesis_child_theory, p=p_eq_q_hypothesis, + t=t, slf=self) + verify(inc_hypothesis.valid_proposition.relation is t.u.relations.inconsistency, 'โŒœinc_p.relationโŒ must be of form (Inc(Hn)).', - inc_p_relation=inc_p_eq_q.valid_proposition.relation, inc_p=inc_p_eq_q, t=t, slf=self) + inc_p_relation=inc_hypothesis.valid_proposition.relation, inc_p=inc_hypothesis, t=t, + slf=self) # TODO: ProofByContradictionDeclaration.verify_args: check that the parent theory is # stable??? # TODO: ProofByContradictionDeclaration.verify_args: check that the hypothetical-theory @@ -6852,7 +6871,7 @@ def mp(self) -> InferenceRuleDeclaration: return self.modus_ponens @property - def pbc(self) -> ProofByContradiction1Declaration: + def pbc1(self) -> ProofByContradiction1Declaration: """ Unabridged property: u.i.proof_by_contradiction @@ -6862,6 +6881,17 @@ def pbc(self) -> ProofByContradiction1Declaration: """ return self.proof_by_contradiction_1 + @property + def pbc2(self) -> ProofByContradiction2Declaration: + """ + + Unabridged property: u.i.proof_by_contradiction + + If the well-known inference-rule does not exist in the universe-of-discourse, + the inference-rule is automatically declared. + """ + return self.proof_by_contradiction_2 + @property def pbr(self) -> ProofByRefutation1Declaration: """ @@ -7758,7 +7788,7 @@ def infer_statement(self, p_neq_q: (None, FormulaStatement) = None, paragraph_header=paragraph_header, subtitle=subtitle, echo=echo) -class ProofByRefutationInclusion(InferenceRuleInclusion): +class ProofByRefutation1Inclusion(InferenceRuleInclusion): """ """ @@ -7800,7 +7830,7 @@ def infer_statement(self, p: (None, FormulaStatement) = None, paragraph_header=paragraph_header, subtitle=subtitle, echo=echo) -class ProofByRefutationOfEqualityInclusion(InferenceRuleInclusion): +class ProofByRefutation2Inclusion(InferenceRuleInclusion): """ """ @@ -8278,7 +8308,7 @@ def pbc(self) -> ProofByContradiction1Inclusion: return self.proof_by_contradiction_1 @property - def pbr(self) -> ProofByRefutationInclusion: + def pbr(self) -> ProofByRefutation1Inclusion: """ Unabridged property: u.i.proof_by_refutation @@ -8286,7 +8316,7 @@ def pbr(self) -> ProofByRefutationInclusion: If the well-known inference-rule does not exist in the universe-of-discourse, the inference-rule is automatically declared. """ - return self.proof_by_refutation + return self.proof_by_refutation_1 @property def proof_by_contradiction_1(self) -> ProofByContradiction1Inclusion: @@ -8302,13 +8332,13 @@ def proof_by_contradiction_1(self) -> ProofByContradiction1Inclusion: return self._proof_by_contradiction_1 @property - def proof_by_contradiction_of_non_equality(self) -> ProofByContradiction2Inclusion: + def proof_by_contradiction_2(self) -> ProofByContradiction2Inclusion: if self._proof_by_contradiction_of_non_equality is None: self._proof_by_contradiction_of_non_equality = ProofByContradiction2Inclusion(t=self.t) return self._proof_by_contradiction_of_non_equality @property - def proof_by_refutation(self) -> ProofByRefutationInclusion: + def proof_by_refutation_1(self) -> ProofByRefutation1Inclusion: """ Abridged property: u.i.pbr @@ -8317,13 +8347,13 @@ def proof_by_refutation(self) -> ProofByRefutationInclusion: the inference-rule is automatically declared. """ if self._proof_by_refutation is None: - self._proof_by_refutation = ProofByRefutationInclusion(t=self.t) + self._proof_by_refutation = ProofByRefutation1Inclusion(t=self.t) return self._proof_by_refutation @property - def proof_by_refutation_of_equality(self) -> ProofByRefutationOfEqualityInclusion: + def proof_by_refutation_2(self) -> ProofByRefutation2Inclusion: if self._proof_by_refutation_of_equality is None: - self._proof_by_refutation_of_equality = ProofByRefutationOfEqualityInclusion(t=self.t) + self._proof_by_refutation_of_equality = ProofByRefutation2Inclusion(t=self.t) return self._proof_by_refutation_of_equality @property diff --git a/punctilious/locale_en_us.py b/punctilious/locale_en_us.py index 61dbee2d..e253b3bd 100644 --- a/punctilious/locale_en_us.py +++ b/punctilious/locale_en_us.py @@ -475,7 +475,58 @@ def compose_parent_hypothesis_statement_report(self, o: Hypothesis, yield SansSerifNormal('.') return True - def compose_proof_by_reputation_of_equality_paragraph_proof(self, o: InferredStatement) -> \ + def compose_proof_by_contradiction_1_paragraph_proof(self, o: InferredStatement) -> \ + collections.abc.Generator[Composable, Composable, bool]: + global text_dict + # Retrieve the parameters from the statement + p_eq_q: Hypothesis = o.parameters[0] + yield SansSerifNormal('Let ') + yield from p_eq_q.compose_ref_link() + yield SansSerifNormal(' be the hypothesis ') + yield from p_eq_q.hypothesis_formula.compose_formula() + yield SansSerifNormal('. ') + inc_p_eq_q: FormulaStatement = o.parameters[1] + yield from inc_p_eq_q.valid_proposition.compose_formula() + yield SansSerifNormal(' follows from ') + yield from inc_p_eq_q.compose_ref_link() + yield SansSerifNormal('.') + return True + + def compose_proof_by_contradiction_2_paragraph_proof(self, o: InferredStatement) -> \ + collections.abc.Generator[Composable, Composable, bool]: + global text_dict + # Retrieve the parameters from the statement + p_eq_q: Hypothesis = o.parameters[0] + yield SansSerifNormal('Let ') + yield from p_eq_q.compose_ref_link() + yield SansSerifNormal(' be the hypothesis ') + yield from p_eq_q.hypothesis_formula.compose_formula() + yield SansSerifNormal('. ') + inc_p_eq_q: FormulaStatement = o.parameters[1] + yield from inc_p_eq_q.valid_proposition.compose_formula() + yield SansSerifNormal(' follows from ') + yield from inc_p_eq_q.compose_ref_link() + yield SansSerifNormal('.') + return True + + def compose_proof_by_refutation_1_paragraph_proof(self, o: InferredStatement) -> \ + collections.abc.Generator[Composable, Composable, bool]: + global text_dict + # Retrieve the parameters from the statement + p_eq_q: Hypothesis = o.parameters[0] + yield SansSerifNormal('Let ') + yield from p_eq_q.compose_ref_link() + yield SansSerifNormal(' be the hypothesis ') + yield from p_eq_q.hypothesis_formula.compose_formula() + yield SansSerifNormal('. ') + inc_p_eq_q: FormulaStatement = o.parameters[1] + yield from inc_p_eq_q.valid_proposition.compose_formula() + yield SansSerifNormal(' follows from ') + yield from inc_p_eq_q.compose_ref_link() + yield SansSerifNormal('.') + return True + + def compose_proof_by_refutation_2_paragraph_proof(self, o: InferredStatement) -> \ collections.abc.Generator[Composable, Composable, bool]: global text_dict # Retrieve the parameters from the statement From 0dcfc299546f6ef47bc00ec90ba722baf14fd4ef Mon Sep 17 00:00:00 2001 From: David Doret Date: Thu, 24 Aug 2023 22:53:27 +0200 Subject: [PATCH 09/21] cleanup --- punctilious/core.py | 68 +++++++++--------- tests/test_proof_by_contradiction_1.py | 3 +- ...ation.py => test_proof_by_refutation_1.py} | 8 +-- tests/test_section.py | 16 ++--- tests/test_tao_2006_the_peano_axioms.py | 4 +- tests/test_theory_elaboration_stabilized.py | 7 +- tests/test_variable_substitution.py | 4 +- .../package/tao_2006_2_1_the_peano_axioms.py | 70 ++++++++++--------- 8 files changed, 92 insertions(+), 88 deletions(-) rename tests/{test_proof_by_refutation.py => test_proof_by_refutation_1.py} (88%) diff --git a/punctilious/core.py b/punctilious/core.py index 2abae74f..4c704d36 100644 --- a/punctilious/core.py +++ b/punctilious/core.py @@ -7734,28 +7734,28 @@ def __init__(self, t: TheoryElaborationSequence, echo: (None, bool) = None, abridged_name=abridged_name, name=name, explicit_name=explicit_name, echo=echo, proof=proof) - def infer_formula(self, not_p: (None, Hypothesis) = None, - inc_not_p: (None, FormulaStatement) = None, echo: (None, bool) = None) -> Formula: + def infer_formula(self, not_p_hypothesis: (None, Hypothesis) = None, + inc_hypothesis: (None, FormulaStatement) = None, echo: (None, bool) = None) -> Formula: """Apply the proof-by-contradiction inference-rule and return the inferred-formula. - :param not_p: (mandatory) The (ยฌP) hypothesis-statement. - :param inc_not_p: (mandatory) The proof of inconsistency of the not_p + :param not_p_hypothesis: (mandatory) The (ยฌP) hypothesis-statement. + :param inc_hypothesis: (mandatory) The proof of inconsistency of the not_p hypothetical-theory: Inc(ยฌP). :return: The inferred formula . """ - return super().infer_formula(not_p, inc_not_p, echo=echo) + return super().infer_formula(not_p_hypothesis, inc_hypothesis, echo=echo) - def infer_statement(self, not_p: (None, FormulaStatement) = None, - inc_not_p: (None, FormulaStatement) = None, nameset: (None, str, NameSet) = None, + def infer_statement(self, not_p_hypothesis: (None, FormulaStatement) = None, + inc_hypothesis: (None, FormulaStatement) = None, nameset: (None, str, NameSet) = None, ref: (None, str) = None, paragraph_header: (None, ParagraphHeader) = None, subtitle: (None, str) = None, echo: (None, bool) = None) -> InferredStatement: """Apply the modus-ponens inference-rule and return the inferred-statement. - :param not_p: (mandatory) The implication statement. - :param inc_not_p: (mandatory) The p statement, proving that p is true in the current theory. + :param not_p_hypothesis: (mandatory) The implication statement. + :param inc_hypothesis: (mandatory) The p statement, proving that p is true in the current theory. :return: An inferred-statement proving p in the current theory. """ - return super().infer_statement(not_p, inc_not_p, nameset=nameset, ref=ref, + return super().infer_statement(not_p_hypothesis, inc_hypothesis, nameset=nameset, ref=ref, paragraph_header=paragraph_header, subtitle=subtitle, echo=echo) @@ -7776,15 +7776,15 @@ def __init__(self, t: TheoryElaborationSequence, echo: (None, bool) = None, abridged_name=abridged_name, name=name, explicit_name=explicit_name, echo=echo, proof=proof) - def infer_formula(self, p_neq_q: (None, Hypothesis) = None, - inc_p_neq_q: (None, FormulaStatement) = None, echo: (None, bool) = None): - return super().infer_formula(p_neq_q, inc_p_neq_q, echo=echo) + def infer_formula(self, p_neq_q_hypothesis: (None, Hypothesis) = None, + inc_hypothesis: (None, FormulaStatement) = None, echo: (None, bool) = None): + return super().infer_formula(p_neq_q_hypothesis, inc_hypothesis, echo=echo) - def infer_statement(self, p_neq_q: (None, FormulaStatement) = None, - inc_p_neq_q: (None, FormulaStatement) = None, nameset: (None, str, NameSet) = None, + def infer_statement(self, p_neq_q_hypothesis: (None, FormulaStatement) = None, + inc_hypothesis: (None, FormulaStatement) = None, nameset: (None, str, NameSet) = None, ref: (None, str) = None, paragraph_header: (None, ParagraphHeader) = None, subtitle: (None, str) = None, echo: (None, bool) = None) -> InferredStatement: - return super().infer_statement(p_neq_q, inc_p_neq_q, nameset=nameset, ref=ref, + return super().infer_statement(p_neq_q_hypothesis, inc_hypothesis, nameset=nameset, ref=ref, paragraph_header=paragraph_header, subtitle=subtitle, echo=echo) @@ -7805,28 +7805,28 @@ def __init__(self, t: TheoryElaborationSequence, echo: (None, bool) = None, abridged_name=abridged_name, name=name, explicit_name=explicit_name, echo=echo, proof=proof) - def infer_formula(self, p: (None, Hypothesis) = None, inc_p: (None, FormulaStatement) = None, - echo: (None, bool) = None): + def infer_formula(self, p_hypothesis: (None, Hypothesis) = None, + inc_hypothesis: (None, FormulaStatement) = None, echo: (None, bool) = None): """Apply the proof-by-refutation inference-rule and return the inferred-formula. - :param p: (mandatory) The (ยฌP) hypothesis-statement. - :param inc_p: (mandatory) The proof of inconsistency of the not_p hypothetical-theory: + :param p_hypothesis: (mandatory) The (ยฌP) hypothesis-statement. + :param inc_hypothesis: (mandatory) The proof of inconsistency of the not_p hypothetical-theory: Inc(ยฌP). :return: The inferred formula . """ - return super().infer_formula(p, inc_p, echo=echo) + return super().infer_formula(p_hypothesis, inc_hypothesis, echo=echo) - def infer_statement(self, p: (None, FormulaStatement) = None, - inc_p: (None, FormulaStatement) = None, nameset: (None, str, NameSet) = None, + def infer_statement(self, p_hypothesis: (None, FormulaStatement) = None, + inc_hypothesis: (None, FormulaStatement) = None, nameset: (None, str, NameSet) = None, ref: (None, str) = None, paragraph_header: (None, ParagraphHeader) = None, subtitle: (None, str) = None, echo: (None, bool) = None) -> InferredStatement: """Apply the modus-ponens inference-rule and return the inferred-statement. - :param p: (mandatory) The implication statement. - :param inc_p: (mandatory) The p statement, proving that p is true in the current theory. + :param p_hypothesis: (mandatory) The implication statement. + :param inc_hypothesis: (mandatory) The p statement, proving that p is true in the current theory. :return: An inferred-statement proving p in the current theory. """ - return super().infer_statement(p, inc_p, nameset=nameset, ref=ref, + return super().infer_statement(p_hypothesis, inc_hypothesis, nameset=nameset, ref=ref, paragraph_header=paragraph_header, subtitle=subtitle, echo=echo) @@ -7847,19 +7847,19 @@ def __init__(self, t: TheoryElaborationSequence, echo: (None, bool) = None, abridged_name=abridged_name, name=name, explicit_name=explicit_name, echo=echo, proof=proof) - def infer_formula(self, p_eq_q: (None, Hypothesis) = None, - inc_p_eq_q: (None, FormulaStatement) = None, echo: (None, bool) = None): + def infer_formula(self, p_eq_q_hypothesis: (None, Hypothesis) = None, + inc_hypothesis: (None, FormulaStatement) = None, echo: (None, bool) = None): """Apply the proof-by-refutation inference-rule and return the inferred-formula. - :param p_eq_q: (mandatory) The (ยฌP) hypothesis-statement. - :param inc_p_eq_q: (mandatory) The proof of inconsistency of the not_p hypothetical-theory: + :param p_eq_q_hypothesis: (mandatory) The (ยฌP) hypothesis-statement. + :param inc_hypothesis: (mandatory) The proof of inconsistency of the not_p hypothetical-theory: Inc(ยฌP). :return: The inferred formula . """ - return super().infer_formula(p_eq_q, inc_p_eq_q, echo=echo) + return super().infer_formula(p_eq_q_hypothesis, inc_hypothesis, echo=echo) - def infer_statement(self, p_eq_q: (None, FormulaStatement) = None, - inc_p_eq_q: (None, FormulaStatement) = None, nameset: (None, str, NameSet) = None, + def infer_statement(self, p_eq_q_hypothesis: (None, FormulaStatement) = None, + inc_hypothesis: (None, FormulaStatement) = None, nameset: (None, str, NameSet) = None, ref: (None, str) = None, paragraph_header: (None, ParagraphHeader) = None, subtitle: (None, str) = None, echo: (None, bool) = None) -> InferredStatement: """Apply the modus-ponens inference-rule and return the inferred-statement. @@ -7868,7 +7868,7 @@ def infer_statement(self, p_eq_q: (None, FormulaStatement) = None, :param inc_p: (mandatory) The p statement, proving that p is true in the current theory. :return: An inferred-statement proving p in the current theory. """ - return super().infer_statement(p_eq_q, inc_p_eq_q, nameset=nameset, ref=ref, + return super().infer_statement(p_eq_q_hypothesis, inc_hypothesis, nameset=nameset, ref=ref, paragraph_header=paragraph_header, subtitle=subtitle, echo=echo) diff --git a/tests/test_proof_by_contradiction_1.py b/tests/test_proof_by_contradiction_1.py index b4bc7f0a..dfb6b037 100644 --- a/tests/test_proof_by_contradiction_1.py +++ b/tests/test_proof_by_contradiction_1.py @@ -42,5 +42,6 @@ def test_proof_by_contradiction_1(self): # p7 is in contradiction with the hypothetical_formula t1_p8 = t1.i.inconsistency_by_negation_introduction.infer_statement(p=t2_p7, not_p=t2_a1, inconsistent_theory=t2) - t1_p9 = t1.i.proof_by_contradiction_1.infer_statement(not_p=t1_h1, inc_not_p=t1_p8) + t1_p9 = t1.i.proof_by_contradiction_1.infer_statement(not_p_hypothesis=t1_h1, + inc_hypothesis=t1_p8) pass diff --git a/tests/test_proof_by_refutation.py b/tests/test_proof_by_refutation_1.py similarity index 88% rename from tests/test_proof_by_refutation.py rename to tests/test_proof_by_refutation_1.py index 162a450f..02f6d35c 100644 --- a/tests/test_proof_by_refutation.py +++ b/tests/test_proof_by_refutation_1.py @@ -4,8 +4,8 @@ # TODO: Proof by refutation: design test -class TestProofByRefutation(TestCase): - def test_proof_by_refutation(self): +class TestProofByRefutation1(TestCase): + def test_proof_by_refutation_1(self): pu.configuration.echo_default = False pu.configuration.echo_inferred_statement = True # Prepare the universe of discourse @@ -36,11 +36,11 @@ def test_proof_by_refutation(self): t2 = t1_h1.hypothesis_child_theory t2_a1 = t1_h1.hypothesis_statement_in_child_theory t2_p5 = t2.i.conjunction_introduction.infer_statement(p=t1_p1, q=t1_p2) - t2_p6 = t2.i.variable_substitution.infer_statement(t1_p3_implication, o1, o2, o3) + t2_p6 = t2.i.variable_substitution.infer_statement(p=t1_p3_implication, phi=(o1, o2, o3)) # p7: ๐‘Ÿโ‚(๐‘œโ‚, ๐‘œโ‚ƒ) by modus ponens t2_p7 = t2.i.modus_ponens.infer_statement(p_implies_q=t2_p6, p=t2_p5) # p7 is in refutation with the hypothetical_formula t1_p8 = t1.i.inconsistency_by_negation_introduction.infer_statement(p=t2_a1, not_p=t2_p7, inconsistent_theory=t2) - t1_p9 = t1.i.proof_by_refutation_1.infer_statement(p=t1_h1, inc_p=t1_p8) + t1_p9 = t1.i.proof_by_refutation_1.infer_statement(p_hypothesis=t1_h1, inc_hypothesis=t1_p8) pass diff --git a/tests/test_section.py b/tests/test_section.py index 61e00f98..b787a709 100644 --- a/tests/test_section.py +++ b/tests/test_section.py @@ -10,17 +10,17 @@ def test_section(self): u = p.UniverseOfDiscourse() t = u.t(subtitle='A test theory') s_1 = t.open_section('something interesting', echo=True) - self.assertEqual('# ๐Ÿญ ๐—ฆ๐—ผ๐—บ๐—ฒ๐˜๐—ต๐—ถ๐—ป๐—ด ๐—ถ๐—ป๐˜๐—ฒ๐—ฟ๐—ฒ๐˜€๐˜๐—ถ๐—ป๐—ด', s_1.rep_report()) + self.assertEqual('# ๐Ÿญ: ๐—ฆ๐—ผ๐—บ๐—ฒ๐˜๐—ต๐—ถ๐—ป๐—ด ๐—ถ๐—ป๐˜๐—ฒ๐—ฟ๐—ฒ๐˜€๐˜๐—ถ๐—ป๐—ด', s_1.rep_report()) s_2 = t.open_section('something else', echo=True) - self.assertEqual('# ๐Ÿฎ ๐—ฆ๐—ผ๐—บ๐—ฒ๐˜๐—ต๐—ถ๐—ป๐—ด ๐—ฒ๐—น๐˜€๐—ฒ', s_2.rep_report()) + self.assertEqual('# ๐Ÿฎ: ๐—ฆ๐—ผ๐—บ๐—ฒ๐˜๐—ต๐—ถ๐—ป๐—ด ๐—ฒ๐—น๐˜€๐—ฒ', s_2.rep_report()) s_2_1 = t.open_section('some details on this topic', section_parent=s_2, echo=True) - self.assertEqual('## ๐Ÿฎ.๐Ÿญ ๐—ฆ๐—ผ๐—บ๐—ฒ ๐—ฑ๐—ฒ๐˜๐—ฎ๐—ถ๐—น๐˜€ ๐—ผ๐—ป ๐˜๐—ต๐—ถ๐˜€ ๐˜๐—ผ๐—ฝ๐—ถ๐—ฐ', s_2_1.rep_report()) + self.assertEqual('## ๐Ÿฎ.๐Ÿญ: ๐—ฆ๐—ผ๐—บ๐—ฒ ๐—ฑ๐—ฒ๐˜๐—ฎ๐—ถ๐—น๐˜€ ๐—ผ๐—ป ๐˜๐—ต๐—ถ๐˜€ ๐˜๐—ผ๐—ฝ๐—ถ๐—ฐ', s_2_1.rep_report()) s_2_2 = t.open_section('some more details on this topic', section_parent=s_2, echo=True) - self.assertEqual('## ๐Ÿฎ.๐Ÿฎ ๐—ฆ๐—ผ๐—บ๐—ฒ ๐—บ๐—ผ๐—ฟ๐—ฒ ๐—ฑ๐—ฒ๐˜๐—ฎ๐—ถ๐—น๐˜€ ๐—ผ๐—ป ๐˜๐—ต๐—ถ๐˜€ ๐˜๐—ผ๐—ฝ๐—ถ๐—ฐ', s_2_2.rep_report()) + self.assertEqual('## ๐Ÿฎ.๐Ÿฎ: ๐—ฆ๐—ผ๐—บ๐—ฒ ๐—บ๐—ผ๐—ฟ๐—ฒ ๐—ฑ๐—ฒ๐˜๐—ฎ๐—ถ๐—น๐˜€ ๐—ผ๐—ป ๐˜๐—ต๐—ถ๐˜€ ๐˜๐—ผ๐—ฝ๐—ถ๐—ฐ', s_2_2.rep_report()) s_2_2_1 = t.open_section('zooming in', section_parent=s_2_2, echo=True) - self.assertEqual('### ๐Ÿฎ.๐Ÿฎ.๐Ÿญ ๐—ญ๐—ผ๐—ผ๐—บ๐—ถ๐—ป๐—ด ๐—ถ๐—ป', s_2_2_1.rep_report()) + self.assertEqual('### ๐Ÿฎ.๐Ÿฎ.๐Ÿญ: ๐—ญ๐—ผ๐—ผ๐—บ๐—ถ๐—ป๐—ด ๐—ถ๐—ป', s_2_2_1.rep_report()) s_2_2_7 = t.open_section('jumping sections', section_number=7, section_parent=s_2_2, - echo=True) - self.assertEqual('### ๐Ÿฎ.๐Ÿฎ.๐Ÿณ ๐—๐˜‚๐—บ๐—ฝ๐—ถ๐—ป๐—ด ๐˜€๐—ฒ๐—ฐ๐˜๐—ถ๐—ผ๐—ป๐˜€', s_2_2_7.rep_report()) + echo=True) + self.assertEqual('### ๐Ÿฎ.๐Ÿฎ.๐Ÿณ: ๐—๐˜‚๐—บ๐—ฝ๐—ถ๐—ป๐—ด ๐˜€๐—ฒ๐—ฐ๐˜๐—ถ๐—ผ๐—ป๐˜€', s_2_2_7.rep_report()) s_2_3 = t.open_section('yet more details on this topic', section_parent=s_2, echo=True) - self.assertEqual('## ๐Ÿฎ.๐Ÿฏ ๐—ฌ๐—ฒ๐˜ ๐—บ๐—ผ๐—ฟ๐—ฒ ๐—ฑ๐—ฒ๐˜๐—ฎ๐—ถ๐—น๐˜€ ๐—ผ๐—ป ๐˜๐—ต๐—ถ๐˜€ ๐˜๐—ผ๐—ฝ๐—ถ๐—ฐ', s_2_3.rep_report()) + self.assertEqual('## ๐Ÿฎ.๐Ÿฏ: ๐—ฌ๐—ฒ๐˜ ๐—บ๐—ผ๐—ฟ๐—ฒ ๐—ฑ๐—ฒ๐˜๐—ฎ๐—ถ๐—น๐˜€ ๐—ผ๐—ป ๐˜๐—ต๐—ถ๐˜€ ๐˜๐—ผ๐—ฝ๐—ถ๐—ฐ', s_2_3.rep_report()) diff --git a/tests/test_tao_2006_the_peano_axioms.py b/tests/test_tao_2006_the_peano_axioms.py index 461fbf11..21978a48 100644 --- a/tests/test_tao_2006_the_peano_axioms.py +++ b/tests/test_tao_2006_the_peano_axioms.py @@ -3,8 +3,8 @@ # from package.tao_2006_the_peano_axioms import p -u = tpa.u -t = tpa.t +t = tpa.Tao2006ThePeanoAxioms.develop_theory() +u = t.u class TestTao2006ThePeanoAxioms(unittest.TestCase): diff --git a/tests/test_theory_elaboration_stabilized.py b/tests/test_theory_elaboration_stabilized.py index 937f4d85..6bc23f04 100644 --- a/tests/test_theory_elaboration_stabilized.py +++ b/tests/test_theory_elaboration_stabilized.py @@ -22,9 +22,10 @@ def test_theory_elaboration_stabilized(self): r1o1o2 = robust_theory.i.axiom_interpretation.infer_statement(ap1, u.f(r1, o1, o2)) r1o2o3 = robust_theory.i.axiom_interpretation.infer_statement(ap1, u.f(r1, o2, o3)) r1o1o2_and_r1o2o3 = robust_theory.i.ci.infer_statement(r1o1o2, r1o2o3) - implication_2 = robust_theory.i.variable_substitution.infer_statement(p=implication, - phi=(o1, o2, o3)) - robust_theory.i.mp.infer_statement(p_implies_q=implication_2, p=r1o1o2_and_r1o2o3) + implication_2 = robust_theory.i.variable_substitution.infer_statement( + p_hypothesis=implication, phi=(o1, o2, o3)) + robust_theory.i.mp.infer_statement(p_implies_q=implication_2, + p_hypothesis=r1o1o2_and_r1o2o3) robust_theory.stabilize() self.assertTrue(robust_theory.stabilized, 'The stabilized property of the original-theory is not True.') diff --git a/tests/test_variable_substitution.py b/tests/test_variable_substitution.py index 073b64e0..8324ee98 100644 --- a/tests/test_variable_substitution.py +++ b/tests/test_variable_substitution.py @@ -19,7 +19,7 @@ def test_variable_substitution_without_variable(self): p_statement = t.i.axiom_interpretation.infer_statement(axiom=ap, formula=p_formula, echo=True) # y_sequence = tuple() - p_prime = t.i.vs.infer_statement(p=p_statement, phi=(), echo=True) + p_prime = t.i.vs.infer_statement(p_hypothesis=p_statement, phi=(), echo=True) self.assertTrue(p_prime.is_formula_syntactically_equivalent_to(p_statement)) self.assertEqual('๐‘Ÿโ‚(๐‘Ÿโ‚‚(๐‘œโ‚, ๐‘œโ‚‚))', p_prime.rep_formula(encoding=pu.encodings.unicode)) @@ -43,7 +43,7 @@ def test_variable_substitution_with_free_variables(self): formula=f(g(g(z, g(f(x), y)), g(x, y))), echo=True) self.assertEqual('๐‘“(๐‘”(๐‘”(๐ณ, ๐‘”(๐‘“(๐ฑ), ๐ฒ)), ๐‘”(๐ฑ, ๐ฒ)))', p_statement.rep_formula(encoding=pu.encodings.unicode)) - p_prime = t.i.vs.infer_statement(p=p_statement, phi=(o4, o6, o5), echo=True) + p_prime = t.i.vs.infer_statement(p_hypothesis=p_statement, phi=(o4, o6, o5), echo=True) self.assertEqual('๐‘“(๐‘”(๐‘”(๐‘œโ‚„, ๐‘”(๐‘“(๐‘œโ‚†), ๐‘œโ‚…)), ๐‘”(๐‘œโ‚†, ๐‘œโ‚…)))', p_prime.rep_formula(encoding=pu.encodings.unicode)) p_prime.is_formula_syntactically_equivalent_to(o2=f(g(g(o4, g(f(o6), o5)), g(o6, o5)))) diff --git a/theory/package/tao_2006_2_1_the_peano_axioms.py b/theory/package/tao_2006_2_1_the_peano_axioms.py index 691ece90..f128dc97 100644 --- a/theory/package/tao_2006_2_1_the_peano_axioms.py +++ b/theory/package/tao_2006_2_1_the_peano_axioms.py @@ -53,7 +53,7 @@ def develop_theory(self, t: pu.TheoryElaborationSequence) -> pu.TheoryElaboratio p002 = t.i.axiom_interpretation.infer_statement(a04, ( (n | u.r.is_a | natural_number) | u.r.implies | ( (n & plusplus) | u.r.is_a | natural_number))) - p003 = t.i.variable_substitution.infer_statement(p=p002, phi=tuple([zero])) + p003 = t.i.variable_substitution.infer_statement(p_hypothesis=p002, phi=tuple([zero])) p004 = t.i.mp.infer_statement(p003, p001, ref='2.2.3') # DEFINITION 2.1.3 @@ -78,7 +78,7 @@ def develop_theory(self, t: pu.TheoryElaborationSequence) -> pu.TheoryElaboratio u.f(u.r.equal, four, ((((zero & plusplus) & plusplus) & plusplus) & plusplus))) zero_plusplus = (zero & plusplus) - p009 = t.i.variable_substitution.infer_statement(p=p002, phi=zero_plusplus) + p009 = t.i.variable_substitution.infer_statement(p_hypothesis=p002, phi=zero_plusplus) p010 = t.i.mp.infer_statement(p009, p004) zero_plus_plus_plusplus = u.f(plusplus, zero_plusplus) p011 = t.i.variable_substitution.infer_statement(p002, zero_plus_plus_plusplus) @@ -161,13 +161,13 @@ def develop_theory(self, t: pu.TheoryElaborationSequence) -> pu.TheoryElaboratio # Take ๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (Pโ‚ƒโ‚): ((((๐งโ‚ƒ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (๐ฆโ‚ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (๐งโ‚ƒ โ‰  ๐ฆโ‚)) โŸน ((๐งโ‚ƒ)++ โ‰  (๐ฆโ‚)++)). # Substitute ๐งโ‚ƒ with 4, and ๐ฆโ‚ with 0. # ๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (Pโ‚ƒโ‚‚): ((((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (4 โ‰  0)) โŸน ((4)++ โ‰  (0)++)). - p033 = t.i.variable_substitution.infer_statement(p=p032, phi=(four, zero)) + p033 = t.i.variable_substitution.infer_statement(p_hypothesis=p032, phi=(four, zero)) # It follows that ((((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (4 โ‰  0)) โŸน ((4)++ โ‰  (0)++)). # Pair two true propositions (4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) and (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). # ๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป(Pโ‚ƒโ‚„): ((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). - p034 = t.i.conjunction_introduction.infer_statement(p=p027, q=p001) + p034 = t.i.conjunction_introduction.infer_statement(p_hypothesis=p027, q=p001) # ๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (Pโ‚ƒโ‚…): (((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (4 โ‰  0)). - p035 = t.i.conjunction_introduction.infer_statement(p=p034, q=p031) + p035 = t.i.conjunction_introduction.infer_statement(p_hypothesis=p034, q=p031) # ๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (Pโ‚ƒโ‚†): ((4)++ โ‰  (0)++). p036 = t.i.modus_ponens.infer_statement(p033, p035) five = u.o.declare(symbol='5', auto_index=False) @@ -181,7 +181,7 @@ def develop_theory(self, t: pu.TheoryElaborationSequence) -> pu.TheoryElaboratio # ๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (Pโ‚„โ‚€): (5 = (4)++). p040 = t.i.equality_commutativity.infer_statement(p039) # ๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (Pโ‚„โ‚): ((((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)) โŸน ((5)++ โ‰  (1)++)). - p041 = t.i.variable_substitution.infer_statement(p=p032, phi=(five, one)) + p041 = t.i.variable_substitution.infer_statement(p_hypothesis=p032, phi=(five, one)) # ๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (Pโ‚„โ‚‚): ((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน ((4)++ ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). p042 = t.i.variable_substitution.infer_statement(p002, four) # ๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (Pโ‚„โ‚ƒ): ((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โŸน (5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). @@ -189,11 +189,11 @@ def develop_theory(self, t: pu.TheoryElaborationSequence) -> pu.TheoryElaboratio # ๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป(Pโ‚„โ‚„): (5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ). p044 = t.i.modus_ponens.infer_statement(p043, p027) # ๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (Pโ‚„โ‚…): ((((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง (5 โ‰  1)) โŸน ((5)++ โ‰  (1)++)). - p045 = t.i.variable_substitution.infer_statement(p=p032, phi=(five, one)) + p045 = t.i.variable_substitution.infer_statement(p_hypothesis=p032, phi=(five, one)) # ๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (Pโ‚„โ‚†): ((4)++ โ‰  (0)++). p046 = t.i.modus_ponens.infer_statement(p033, p035) # ๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป (Pโ‚„โ‚‡): (5 โ‰  (0)++). - p047 = t.i.equal_terms_substitution.infer_statement(p=p046, q_equal_r=p039) + p047 = t.i.equal_terms_substitution.infer_statement(p_hypothesis=p046, q_equal_r=p039) six = u.o.declare(symbol='6', auto_index=False) # ๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป: (6 = ((((((0)++)++)++)++)++)++). p049 = t.i.definition_interpretation.infer_statement(d02, u.f(u.r.equal, six, @@ -204,7 +204,7 @@ def develop_theory(self, t: pu.TheoryElaborationSequence) -> pu.TheoryElaboratio p054 = t.i.equal_terms_substitution.infer_statement(p004, p015) p051 = t.i.equal_terms_substitution.infer_statement(p050, p038) # (6 = (5)++) - p057 = t.i.equality_commutativity.infer_statement(p_eq_q=p051) + p057 = t.i.equality_commutativity.infer_statement(p_eq_q_hypothesis=p051) t.open_section('Proof by contradiction', section_parent=s55, numbering=False) @@ -218,47 +218,49 @@ def develop_theory(self, t: pu.TheoryElaborationSequence) -> pu.TheoryElaboratio # Then 5++ = 1++, # ((5)++ = 2) h1_p2 = h1.hypothesis_child_theory.i.equal_terms_substitution.infer_statement( - p=hypothesis_statement, q_equal_r=p057) + p_hypothesis=hypothesis_statement, q_equal_r=p057) # ((5)++ = (1)++) - h1_p3 = h1.hypothesis_child_theory.i.equal_terms_substitution.infer_statement(p=h1_p2, - q_equal_r=p016) + h1_p3 = h1.hypothesis_child_theory.i.equal_terms_substitution.infer_statement( + p_hypothesis=h1_p2, q_equal_r=p016) # so by Axiom 2.4 we have 5 = 1 # ((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) - h1_p4 = h1.hypothesis_child_theory.i.conjunction_introduction.infer_statement(p=p044, - q=p054) + h1_p4 = h1.hypothesis_child_theory.i.conjunction_introduction.infer_statement( + p_hypothesis=p044, q=p054) # (((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง ((5)++ = (1)++)) - h1_p5 = h1.hypothesis_child_theory.i.conjunction_introduction.infer_statement(p=h1_p4, - q=h1_p3) + h1_p5 = h1.hypothesis_child_theory.i.conjunction_introduction.infer_statement( + p_hypothesis=h1_p4, q=h1_p3) # ((((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง ((5)++ = (1)++)) โŸน (5 = 1)) - h1_p6 = h1.hypothesis_child_theory.i.variable_substitution.infer_statement(p=p032b, - phi=tuple([five, one])) + h1_p6 = h1.hypothesis_child_theory.i.variable_substitution.infer_statement( + p_hypothesis=p032b, phi=tuple([five, one])) # (5 = 1) h1_p7 = h1.hypothesis_child_theory.i.modus_ponens.infer_statement(p_implies_q=h1_p6, - p=h1_p5) + p_hypothesis=h1_p5) # so that 4++ = 0++. # ((4)++ = 1) - h1_p8 = h1.hypothesis_child_theory.i.equal_terms_substitution.infer_statement(p=h1_p7, - q_equal_r=p040) + h1_p8 = h1.hypothesis_child_theory.i.equal_terms_substitution.infer_statement( + p_hypothesis=h1_p7, q_equal_r=p040) # ((4)++ = (0)++) - h1_p9 = h1.hypothesis_child_theory.i.equal_terms_substitution.infer_statement(p=h1_p8, - q_equal_r=p005) + h1_p9 = h1.hypothesis_child_theory.i.equal_terms_substitution.infer_statement( + p_hypothesis=h1_p8, q_equal_r=p005) # ((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) - h1_p10 = h1.hypothesis_child_theory.i.conjunction_introduction.infer_statement(p=p027, - q=p001) + h1_p10 = h1.hypothesis_child_theory.i.conjunction_introduction.infer_statement( + p_hypothesis=p027, q=p001) # (((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง ((4)++ = (0)++)) - h1_p11 = h1.hypothesis_child_theory.i.conjunction_introduction.infer_statement(p=h1_p10, - q=h1_p9) + h1_p11 = h1.hypothesis_child_theory.i.conjunction_introduction.infer_statement( + p_hypothesis=h1_p10, q=h1_p9) # ((((4 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (0 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)) โˆง ((4)++ = (0)++)) โŸน (4 = 0)) - h1_p12 = h1.hypothesis_child_theory.i.variable_substitution.infer_statement(p=p032b, - phi=tuple([four, zero])) + h1_p12 = h1.hypothesis_child_theory.i.variable_substitution.infer_statement( + p_hypothesis=p032b, phi=tuple([four, zero])) # (4 = 0) # By Axiom 2.4 again we then have 4 = 0, which contradicts our previous proposition. h1_p071 = h1.hypothesis_child_theory.i.modus_ponens.infer_statement(p_implies_q=h1_p12, - p=h1_p11) - p072 = t.i.inconsistency_by_inequality_introduction.infer_statement(p_eq_q=h1_p071, - p_neq_q=p031, inconsistent_theory=h1.hypothesis_child_theory) - p073 = t.i.proof_by_refutation_2.infer_statement(p_eq_q=h1, inc_p_eq_q=p072, ref='2.1.8') + p_hypothesis=h1_p11) + p072 = t.i.inconsistency_by_inequality_introduction.infer_statement( + p_eq_q_hypothesis=h1_p071, p_neq_q_hypothesis=p031, + inconsistent_theory=h1.hypothesis_child_theory) + p073 = t.i.proof_by_refutation_2.infer_statement(p_eq_q_hypothesis=h1, inc_hypothesis=p072, + ref='2.1.8') t.open_section('Direct proof', section_parent=s55, numbering=False) @@ -274,7 +276,7 @@ def develop_theory(self, t: pu.TheoryElaborationSequence) -> pu.TheoryElaboratio # ๐—ฃ๐—ฟ๐—ผ๐—ฝ๐—ผ๐˜€๐—ถ๐˜๐—ถ๐—ผ๐—ป: ((5 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ) โˆง (1 ๐‘–๐‘ -๐‘Ž ๐‘›๐‘Ž๐‘ก๐‘ข๐‘Ÿ๐‘Ž๐‘™-๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ)). p055 = t.i.conjunction_introduction.infer_statement(p044, p054) p056 = t.i.conjunction_introduction.infer_statement(p055, p048) - p057 = t.i.modus_ponens.infer_statement(p_implies_q=p053, p=p056) + p057 = t.i.modus_ponens.infer_statement(p_implies_q=p053, p_hypothesis=p056) t.open_section('Axiom 2.5: The principle of mathematical induction', section_parent=section_2_1, numbering=False) From 60cb3122d55d50138c12e8315eed1fa9d634eb6e Mon Sep 17 00:00:00 2001 From: David Doret Date: Thu, 24 Aug 2023 23:10:22 +0200 Subject: [PATCH 10/21] cleanup --- tests/test_proof_by_contradiction_2.py | 45 +++++++++++++++++++++ tests/test_text.py | 31 +++++--------- tests/test_theory_elaboration_stabilized.py | 7 ++-- tests/test_variable_substitution.py | 4 +- 4 files changed, 61 insertions(+), 26 deletions(-) create mode 100644 tests/test_proof_by_contradiction_2.py diff --git a/tests/test_proof_by_contradiction_2.py b/tests/test_proof_by_contradiction_2.py new file mode 100644 index 00000000..cb64e662 --- /dev/null +++ b/tests/test_proof_by_contradiction_2.py @@ -0,0 +1,45 @@ +from unittest import TestCase +import punctilious as pu +import random_data + + +# TODO: Proof by contradiction: design test +class TestProofByContradiction1(TestCase): + def test_proof_by_contradiction_2(self): + pu.configuration.echo_default = False + pu.configuration.echo_inferred_statement = True + # Prepare the universe of discourse + u = pu.UniverseOfDiscourse() + blah_blah_blah = random_data.random_sentence(min_words=3) + a1 = u.declare_axiom(blah_blah_blah) + o1 = u.o.declare() + o2 = u.o.declare() + o3 = u.o.declare() + r1 = u.r.declare(2, signal_proposition=True) + # Elaborate the parent theory + t1 = u.t() + t1_a2 = t1.include_axiom(a=a1) + t1_p1 = t1.i.axiom_interpretation.infer_statement(axiom=t1_a2, formula=(o1 | u.r.eq | o2)) + t1_p2 = t1.i.axiom_interpretation.infer_statement(axiom=t1_a2, formula=(o2 | u.r.eq | o3)) + with u.v('x') as x, u.v('y') as y, u.v('z') as z: + t1_p3_implication = t1.i.axiom_interpretation.infer_statement(axiom=t1_a2, + formula=((x | u.r.eq | y) | u.r.land | (y | u.r.eq | z))) + t1.stabilize() + hypothetical_formula = (o1 | u.r.neq | o3) + t1_h1 = t1.pose_hypothesis(hypothesis_formula=hypothetical_formula) + # TODO: The hypothetical-theory must be stabilized immediately, + # otherwise new axioms or definitions may be introduced, + # leading to inconsistent results from the perspective of the + # base theory. + t2 = t1_h1.hypothesis_child_theory + t2_a1 = t1_h1.hypothesis_statement_in_child_theory + t2_p5 = t2.i.conjunction_introduction.infer_statement(p=t1_p1, q=t1_p2) + t2_p6 = t2.i.variable_substitution.infer_statement(p=t1_p3_implication, phi=(o1, o2, o3)) + # p7: ๐‘Ÿโ‚(๐‘œโ‚, ๐‘œโ‚ƒ) by modus ponens + t2_p7 = t2.i.modus_ponens.infer_statement(p_implies_q=t2_p6, p=t2_p5) + # p7 is in contradiction with the hypothetical_formula + t1_p8 = t1.i.inconsistency_by_negation_introduction.infer_statement(p=t2_a1, not_p=t2_p7, + inconsistent_theory=t2) + t1_p9 = t1.i.proof_by_contradiction_2.infer_statement(p_hypothesis=t1_h1, + inc_hypothesis=t1_p8) + pass diff --git a/tests/test_text.py b/tests/test_text.py index d18b35d9..338580c1 100644 --- a/tests/test_text.py +++ b/tests/test_text.py @@ -12,38 +12,29 @@ def test_1(self): def test_styled_text(self): pangram = 'The quick brown fox jumps over the lazy dog 0123456789!' - x = pu.ComposableText(pangram, text_style=pu.text_styles.serif_bold) + x = pu.SansSerifBold(pangram) x_plaintext = x.rep(encoding=pu.encodings.plaintext) - self.assertEqual( - 'The quick brown fox jumps over the lazy dog 0123456789!', - x_plaintext) + self.assertEqual('The quick brown fox jumps over the lazy dog 0123456789!', x_plaintext) x_unicode = x.rep(encoding=pu.encodings.unicode) self.assertEqual( - '๐“๐ก๐ž ๐ช๐ฎ๐ข๐œ๐ค ๐›๐ซ๐จ๐ฐ๐ง ๐Ÿ๐จ๐ฑ ๐ฃ๐ฎ๐ฆ๐ฉ๐ฌ ๐จ๐ฏ๐ž๐ซ ๐ญ๐ก๐ž ๐ฅ๐š๐ณ๐ฒ ๐๐จ๐  ๐ŸŽ๐Ÿ๐Ÿ๐Ÿ‘๐Ÿ’๐Ÿ“๐Ÿ”๐Ÿ•๐Ÿ–๐Ÿ—!', - x_unicode) + '๐—ง๐—ต๐—ฒ ๐—พ๐˜‚๐—ถ๐—ฐ๐—ธ ๐—ฏ๐—ฟ๐—ผ๐˜„๐—ป ๐—ณ๐—ผ๐˜… ๐—ท๐˜‚๐—บ๐—ฝ๐˜€ ๐—ผ๐˜ƒ๐—ฒ๐—ฟ ๐˜๐—ต๐—ฒ ๐—น๐—ฎ๐˜‡๐˜† ๐—ฑ๐—ผ๐—ด ๐Ÿฌ๐Ÿญ๐Ÿฎ๐Ÿฏ๐Ÿฐ๐Ÿฑ๐Ÿฒ๐Ÿณ๐Ÿด๐Ÿต!', x_unicode) x_latex = x.rep(encoding=pu.encodings.latex) self.assertEqual( - '\\mathbf{The quick brown fox jumps over the lazy dog 0123456789!}', + '\\boldsymbol\\mathsf{The quick brown fox jumps over the lazy dog 0123456789!}}', x_latex) - x = pu.ComposableText(pangram, text_style=pu.text_styles.serif_normal) - self.assertEqual( - 'The quick brown fox jumps over the lazy dog 0123456789!', + x = pu.ComposableText(pangram) + self.assertEqual('The quick brown fox jumps over the lazy dog 0123456789!', x.rep(pu.encodings.plaintext)) - self.assertEqual( - 'The quick brown fox jumps over the lazy dog 0123456789!', + self.assertEqual('The quick brown fox jumps over the lazy dog 0123456789!', x.rep(pu.encodings.unicode)) - self.assertEqual( - '\\mathnormal{The quick brown fox jumps over the lazy dog 0123456789!}', + self.assertEqual('\\mathnormal{The quick brown fox jumps over the lazy dog 0123456789!}', x.rep(pu.encodings.latex)) x = pu.ComposableText(pangram, pu.text_styles.double_struck) - self.assertEqual( - 'The quick brown fox jumps over the lazy dog 0123456789!', + self.assertEqual('The quick brown fox jumps over the lazy dog 0123456789!', x.rep(pu.encodings.plaintext)) self.assertEqual( - '๐•‹๐•™๐•– ๐•ข๐•ฆ๐•š๐•”๐•œ ๐•“๐•ฃ๐• ๐•จ๐•Ÿ ๐•—๐• ๐•ฉ ๐•›๐•ฆ๐•ž๐•ก๐•ค ๐• ๐•ง๐•–๐•ฃ ๐•ฅ๐•™๐•– ๐•๐•’๐•ซ๐•ช ๐••๐• ๐•˜ ๐Ÿ˜๐Ÿ™๐Ÿš๐Ÿ›๐Ÿœ๐Ÿ๐Ÿž๐ŸŸ๐Ÿ ๐Ÿก!', - x.rep(pu.encodings.unicode)) - self.assertEqual( - '\\mathbb{The quick brown fox jumps over the lazy dog 0123456789!}', + '๐•‹๐•™๐•– ๐•ข๐•ฆ๐•š๐•”๐•œ ๐•“๐•ฃ๐• ๐•จ๐•Ÿ ๐•—๐• ๐•ฉ ๐•›๐•ฆ๐•ž๐•ก๐•ค ๐• ๐•ง๐•–๐•ฃ ๐•ฅ๐•™๐•– ๐•๐•’๐•ซ๐•ช ๐••๐• ๐•˜ ๐Ÿ˜๐Ÿ™๐Ÿš๐Ÿ›๐Ÿœ๐Ÿ๐Ÿž๐ŸŸ๐Ÿ ๐Ÿก!', x.rep(pu.encodings.unicode)) + self.assertEqual('\\mathbb{The quick brown fox jumps over the lazy dog 0123456789!}', x.rep(pu.encodings.latex)) def test_equality(self): diff --git a/tests/test_theory_elaboration_stabilized.py b/tests/test_theory_elaboration_stabilized.py index 6bc23f04..937f4d85 100644 --- a/tests/test_theory_elaboration_stabilized.py +++ b/tests/test_theory_elaboration_stabilized.py @@ -22,10 +22,9 @@ def test_theory_elaboration_stabilized(self): r1o1o2 = robust_theory.i.axiom_interpretation.infer_statement(ap1, u.f(r1, o1, o2)) r1o2o3 = robust_theory.i.axiom_interpretation.infer_statement(ap1, u.f(r1, o2, o3)) r1o1o2_and_r1o2o3 = robust_theory.i.ci.infer_statement(r1o1o2, r1o2o3) - implication_2 = robust_theory.i.variable_substitution.infer_statement( - p_hypothesis=implication, phi=(o1, o2, o3)) - robust_theory.i.mp.infer_statement(p_implies_q=implication_2, - p_hypothesis=r1o1o2_and_r1o2o3) + implication_2 = robust_theory.i.variable_substitution.infer_statement(p=implication, + phi=(o1, o2, o3)) + robust_theory.i.mp.infer_statement(p_implies_q=implication_2, p=r1o1o2_and_r1o2o3) robust_theory.stabilize() self.assertTrue(robust_theory.stabilized, 'The stabilized property of the original-theory is not True.') diff --git a/tests/test_variable_substitution.py b/tests/test_variable_substitution.py index 8324ee98..073b64e0 100644 --- a/tests/test_variable_substitution.py +++ b/tests/test_variable_substitution.py @@ -19,7 +19,7 @@ def test_variable_substitution_without_variable(self): p_statement = t.i.axiom_interpretation.infer_statement(axiom=ap, formula=p_formula, echo=True) # y_sequence = tuple() - p_prime = t.i.vs.infer_statement(p_hypothesis=p_statement, phi=(), echo=True) + p_prime = t.i.vs.infer_statement(p=p_statement, phi=(), echo=True) self.assertTrue(p_prime.is_formula_syntactically_equivalent_to(p_statement)) self.assertEqual('๐‘Ÿโ‚(๐‘Ÿโ‚‚(๐‘œโ‚, ๐‘œโ‚‚))', p_prime.rep_formula(encoding=pu.encodings.unicode)) @@ -43,7 +43,7 @@ def test_variable_substitution_with_free_variables(self): formula=f(g(g(z, g(f(x), y)), g(x, y))), echo=True) self.assertEqual('๐‘“(๐‘”(๐‘”(๐ณ, ๐‘”(๐‘“(๐ฑ), ๐ฒ)), ๐‘”(๐ฑ, ๐ฒ)))', p_statement.rep_formula(encoding=pu.encodings.unicode)) - p_prime = t.i.vs.infer_statement(p_hypothesis=p_statement, phi=(o4, o6, o5), echo=True) + p_prime = t.i.vs.infer_statement(p=p_statement, phi=(o4, o6, o5), echo=True) self.assertEqual('๐‘“(๐‘”(๐‘”(๐‘œโ‚„, ๐‘”(๐‘“(๐‘œโ‚†), ๐‘œโ‚…)), ๐‘”(๐‘œโ‚†, ๐‘œโ‚…)))', p_prime.rep_formula(encoding=pu.encodings.unicode)) p_prime.is_formula_syntactically_equivalent_to(o2=f(g(g(o4, g(f(o6), o5)), g(o6, o5)))) From 5e8fb9c95132884d19c8f5ac04e5251a173ef7e4 Mon Sep 17 00:00:00 2001 From: David Doret Date: Thu, 24 Aug 2023 23:29:34 +0200 Subject: 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+
  • conjunction-elimination-left
    • +
  • conjunction-elimination-right
  • conjunction-introduction
    • +
  • double-negation-elimination
  • double-negation-introduction
    • +
  • elimination-rule
    +
  • inconsistency-by-inequality-introduction
    • +
  • introduction-rule
    +
  • is-a
  • Bibliography