diff --git a/knuckledragger/__init__.py b/knuckledragger/__init__.py
index 86c3d04..45df83b 100644
--- a/knuckledragger/__init__.py
+++ b/knuckledragger/__init__.py
@@ -1,8 +1,9 @@
 from . import kernel
 from . import notation
 from . import tactics
+import z3
 
-lemma = kernel.lemma
+lemma = tactics.lemma
 axiom = kernel.axiom
 define = kernel.define
 
@@ -10,3 +11,8 @@
 QExists = notation.QExists
 
 Calc = tactics.Calc
+
+R = z3.RealSort()
+Z = z3.IntSort()
+RSeq = Z >> R
+RFun = R >> R
diff --git a/knuckledragger/kernel.py b/knuckledragger/kernel.py
index 6fe8089..d8417fa 100644
--- a/knuckledragger/kernel.py
+++ b/knuckledragger/kernel.py
@@ -3,7 +3,7 @@
 from typing import Any
 import logging
 
-logger = logging.getLogger("knuckeldragger")
+logger = logging.getLogger("knuckledragger")
 
 
 @dataclass(frozen=True)
@@ -65,11 +65,11 @@ def lemma(
     else:
         s = solver()
         s.set("timeout", timeout)
-        for n, p in enumerate(by):
+        for p in by:
             if not isinstance(p, __Proof):
                 raise LemmaError("In by reasons:", p, "is not a Proof object")
-            s.assert_and_track(p.thm, "by_{}".format(n))
-        s.assert_and_track(z3.Not(thm), "knuckledragger_goal")
+            s.add(p.thm)
+        s.add(z3.Not(thm))
         if dump:
             print(s.sexpr())
         res = s.check()
@@ -78,10 +78,6 @@ def lemma(
                 raise LemmaError(thm, "Countermodel", s.model())
             raise LemmaError("lemma", thm, res)
         else:
-            core = s.unsat_core()
-            assert z3.Bool("knuckledragger_goal") in core
-            if len(core) < len(by) + 1:
-                print("WARNING: Unneeded assumptions. Used", core)
             return __Proof(thm, by, False)
 
 
diff --git a/knuckledragger/notation.py b/knuckledragger/notation.py
index 4982b73..17f0582 100644
--- a/knuckledragger/notation.py
+++ b/knuckledragger/notation.py
@@ -75,16 +75,16 @@ def define(self, args, body):
         return defn
 
 
-add = SortDispatch(z3.ArithRef.__add__)
+add = SortDispatch(z3.ArithRef.__add__, name="add")
 z3.ExprRef.__add__ = lambda x, y: add(x, y)
 
-sub = SortDispatch(z3.ArithRef.__sub__)
+sub = SortDispatch(z3.ArithRef.__sub__, name="sub")
 z3.ExprRef.__sub__ = lambda x, y: sub(x, y)
 
-mul = SortDispatch(z3.ArithRef.__mul__)
+mul = SortDispatch(z3.ArithRef.__mul__, name="mul")
 z3.ExprRef.__mul__ = lambda x, y: mul(x, y)
 
-div = SortDispatch(z3.ArithRef.__div__)
+div = SortDispatch(z3.ArithRef.__div__, name="div")
 z3.ExprRef.__truediv__ = lambda x, y: div(x, y)
 
 and_ = SortDispatch()
@@ -93,10 +93,10 @@ def define(self, args, body):
 or_ = SortDispatch()
 z3.ExprRef.__or__ = lambda x, y: or_(x, y)
 
-lt = SortDispatch(z3.ArithRef.__lt__)
+lt = SortDispatch(z3.ArithRef.__lt__, name="lt")
 z3.ExprRef.__lt__ = lambda x, y: lt(x, y)
 
-le = SortDispatch(z3.ArithRef.__le__)
+le = SortDispatch(z3.ArithRef.__le__, name="le")
 z3.ExprRef.__le__ = lambda x, y: le(x, y)
 
 
diff --git a/knuckledragger/tactics.py b/knuckledragger/tactics.py
index f0bafe0..be51bac 100644
--- a/knuckledragger/tactics.py
+++ b/knuckledragger/tactics.py
@@ -32,3 +32,63 @@ def __repr__(self):
 
     def qed(self):
         return kd.lemma(self.ForAll(self.terms[0] == self.terms[-1]), by=self.lemmas)
+
+
+def lemma(
+    thm: z3.BoolRef,
+    by: list[kd.kernel.Proof] = [],
+    admit=False,
+    timeout=1000,
+    dump=False,
+    solver=z3.Solver,
+) -> kd.kernel.Proof:
+    """Prove a theorem using a list of previously proved lemmas.
+
+    In essence `prove(Implies(by, thm))`.
+
+    :param thm: The theorem to prove.
+    Args:
+        thm (z3.BoolRef): The theorem to prove.
+        by (list[Proof]): A list of previously proved lemmas.
+        admit     (bool): If True, admit the theorem without proof.
+
+    Returns:
+        Proof: A proof object of thm
+
+    >>> lemma(BoolVal(True))
+
+    >>> lemma(RealVal(1) >= RealVal(0))
+
+    """
+    if admit:
+        return kd.kernel.lemma(
+            thm, by, admit=admit, timeout=timeout, dump=dump, solver=solver
+        )
+    else:
+        s = solver()
+        s.set("timeout", timeout)
+        for n, p in enumerate(by):
+            if not kd.kernel.is_proof(p):
+                raise kd.kernel.LemmaError("In by reasons:", p, "is not a Proof object")
+            s.assert_and_track(p.thm, "by_{}".format(n))
+        s.assert_and_track(z3.Not(thm), "knuckledragger_goal")
+        if dump:
+            print(s.sexpr())
+        res = s.check()
+        if res != z3.unsat:
+            if res == z3.sat:
+                raise kd.kernel.LemmaError(thm, "Countermodel", s.model())
+            raise kd.kernel.LemmaError("lemma", thm, res)
+        else:
+            core = s.unsat_core()
+            if not z3.Bool("knuckledragger_goal") in core:
+                raise kd.kernel.LemmaError(
+                    thm,
+                    core,
+                    "Inconsistent lemmas. Goal is not used for proof. Something has gone awry.",
+                )
+            if len(core) < len(by) + 1:
+                print("WARNING: Unneeded assumptions. Used", core, thm)
+            return kd.kernel.lemma(
+                thm, by, admit=admit, timeout=timeout, dump=dump, solver=solver
+            )
diff --git a/knuckledragger/theories/Complex.py b/knuckledragger/theories/Complex.py
new file mode 100644
index 0000000..42329c0
--- /dev/null
+++ b/knuckledragger/theories/Complex.py
@@ -0,0 +1,48 @@
+import knuckledragger as kd
+import z3
+
+C = kd.notation.Record("C", ("re", z3.RealSort()), ("im", z3.RealSort()))
+
+z, w, u, z1, z2 = z3.Consts("z w u z1 z2", C)
+add = kd.define("add", [z1, z2], C.mk(z1.re + z2.re, z1.im + z2.im))
+kd.notation.add.register(C, add)
+mul = kd.define(
+    "mul", [z1, z2], C.mk(z1.re * z2.re - z1.im * z2.im, z1.re * z2.im + z1.im * z2.re)
+)
+kd.notation.mul.register(C, mul)
+conj = kd.define("conj", [z], C.mk(z.re, -z.im))
+
+
+div = kd.define(
+    "div",
+    [z1, z2],
+    C.mk(
+        (z1.re * z2.re + z1.im * z2.im) / (z2.re**2 + z2.im**2),
+        (z1.im * z2.re - z1.re * z2.im) / (z2.re**2 + z2.im**2),
+    ),
+)
+kd.notation.div.register(C, div)
+
+C0 = C.mk(0, 0)
+C1 = C.mk(1, 0)
+
+add_zero = kd.lemma(z3.ForAll([z], z + C0 == z), by=[add.defn])
+mul_zero = kd.lemma(z3.ForAll([z], z * C0 == C0), by=[mul.defn])
+mul_one = kd.lemma(z3.ForAll([z], z * C1 == z), by=[mul.defn])
+add_comm = kd.lemma(z3.ForAll([z, w], z + w == w + z), by=[add.defn])
+add_assoc = kd.lemma(
+    z3.ForAll([z, w, u], (z + (w + u)) == ((z + w) + u)), by=[add.defn]
+)
+mul_comm = kd.lemma(z3.ForAll([z, w], z * w == w * z), by=[mul.defn])
+
+# unstable perfoamnce.
+# mul_div = kd.lemma(ForAll([z,w], Implies(w != C0, z == z * w / w)), by=[div.defn, mul.defn], timeout=1000)
+##mul_div = Calc()
+div_one = kd.lemma(z3.ForAll([z], z / C1 == z), by=[div.defn])
+div_inv = kd.lemma(z3.ForAll([z], z3.Implies(z != C0, z / z == C1)), by=[div.defn])
+
+# inv = kd.define("inv", [z], )
+
+# conjugate
+# polar
+norm2 = kd.define("norm2", [z], z * conj(z))
diff --git a/knuckledragger/theories/Interval.py b/knuckledragger/theories/Interval.py
new file mode 100644
index 0000000..bce370c
--- /dev/null
+++ b/knuckledragger/theories/Interval.py
@@ -0,0 +1,33 @@
+import knuckledragger as kd
+import knuckledragger.theories.Real as R
+import z3
+
+Interval = kd.notation.Record("Interval", ("lo", kd.R), ("hi", kd.R))
+x, y, z = z3.Reals("x y z")
+i, j, k = z3.Consts("i j k", Interval)
+
+setof = kd.define("setof", [i], z3.Lambda([x], z3.And(i.lo <= x, x <= i.hi)))
+
+meet = kd.define("meet", [i, j], Interval.mk(R.max(i.lo, j.lo), R.min(i.hi, j.hi)))
+meet_intersect = kd.lemma(
+    z3.ForAll([i, j], z3.SetIntersect(setof(i), setof(j)) == setof(meet(i, j))),
+    by=[setof.defn, meet.defn, R.min.defn, R.max.defn],
+)
+
+join = kd.define("join", [i, j], Interval.mk(R.min(i.lo, j.lo), R.max(i.hi, j.hi)))
+join_union = kd.lemma(
+    z3.ForAll([i, j], z3.IsSubset(z3.SetUnion(setof(i), setof(j)), setof(join(i, j)))),
+    by=[setof.defn, join.defn, R.min.defn, R.max.defn],
+)
+
+
+width = kd.define("width", [i], i.hi - i.lo)
+mid = kd.define("mid", [i], (i.lo + i.hi) / 2)
+
+add = kd.notation.add.define([i, j], Interval.mk(i.lo + j.lo, i.hi + j.hi))
+add_set = kd.lemma(
+    z3.ForAll([x, y, i, j], z3.Implies(setof(i)[x] & setof(j)[y], setof(i + j)[x + y])),
+    by=[add.defn, setof.defn],
+)
+
+sub = kd.notation.sub.define([i, j], Interval.mk(i.lo - j.hi, i.hi - j.lo))
diff --git a/knuckledragger/theories/Real.py b/knuckledragger/theories/Real.py
index 68367d6..499236e 100644
--- a/knuckledragger/theories/Real.py
+++ b/knuckledragger/theories/Real.py
@@ -50,3 +50,12 @@
 max_idem = kd.lemma(ForAll([x], max(x, x) == x), by=[max.defn])
 max_ge = kd.lemma(ForAll([x, y], max(x, y) >= x), by=[max.defn])
 max_ge_2 = kd.lemma(ForAll([x, y], max(x, y) >= y), by=[max.defn])
+
+min = kd.define("min", [x, y], z3.If(x <= y, x, y))
+min_comm = kd.lemma(ForAll([x, y], min(x, y) == min(y, x)), by=[min.defn])
+min_assoc = kd.lemma(
+    ForAll([x, y, z], min(x, min(y, z)) == min(min(x, y), z)), by=[min.defn]
+)
+min_idem = kd.lemma(ForAll([x], min(x, x) == x), by=[min.defn])
+min_le = kd.lemma(ForAll([x, y], min(x, y) <= x), by=[min.defn])
+min_le_2 = kd.lemma(ForAll([x, y], min(x, y) <= y), by=[min.defn])
diff --git a/knuckledragger/theories/Seq.py b/knuckledragger/theories/Seq.py
index 5e23d37..1d0b78c 100644
--- a/knuckledragger/theories/Seq.py
+++ b/knuckledragger/theories/Seq.py
@@ -1,6 +1,18 @@
-from knuckledragger.theories.Nat import Nat
-from knuckledragger.theories.Real import R
-from z3 import ArraySort
+import knuckledragger as kd
+import z3
 
-"""A Sequence type of Nat -> R"""
-Seq = ArraySort(Nat, R)
+# TODO: seq needs well formedness condition inherited from elements
+
+
+def induct(T: z3.SortRef, P) -> kd.kernel.Proof:
+    z = z3.FreshConst(T, prefix="z")
+    sort = z3.SeqSort(T)
+    x, y = z3.FreshConst(sort), z3.FreshConst(sort)
+    return kd.axiom(
+        z3.And(
+            P(z3.Empty(sort)),
+            kd.QForAll([z], P(z3.Unit(z))),
+            kd.QForAll([x, y], P(x), P(y), P(z3.Concat(x, y))),
+        )  # -------------------------------------------------
+        == kd.QForAll([x], P(x))
+    )
diff --git a/knuckledragger/theories/List.py b/knuckledragger/theories/Vec.py
similarity index 100%
rename from knuckledragger/theories/List.py
rename to knuckledragger/theories/Vec.py
diff --git a/knuckledragger/utils.py b/knuckledragger/utils.py
index b35cb82..d544975 100644
--- a/knuckledragger/utils.py
+++ b/knuckledragger/utils.py
@@ -12,6 +12,7 @@ def simp(t: z3.ExprRef) -> z3.ExprRef:
         G.add(v.ax.thm)
     G.add(expr == t)
     G2 = z3.Then(z3.Tactic("demodulator"), z3.Tactic("simplify")).apply(G)[0]
+    # TODO make this extraction more robust
     return G2[len(G2) - 1].children()[1]
 
 
diff --git a/tests/test_kernel.py b/tests/test_kernel.py
index 6b25863..b84d27e 100644
--- a/tests/test_kernel.py
+++ b/tests/test_kernel.py
@@ -1,38 +1,38 @@
 import pytest
-from knuckledragger import lemma, axiom
 import z3
 
 import knuckledragger as kd
 import knuckledragger.theories.Nat
 import knuckledragger.theories.Int
 import knuckledragger.theories.Real as R
+import knuckledragger.theories.Complex as C
+import knuckledragger.theories.Interval
 
-import knuckledragger.theories.List
-import knuckledragger.theories.Seq
+import knuckledragger.theories.Seq as ThSeq
 
 from knuckledragger import Calc
 import knuckledragger.utils
 
 
 def test_true_infer():
-    lemma(z3.BoolVal(True))
+    kd.lemma(z3.BoolVal(True))
 
 
 def test_false_infer():
     with pytest.raises(Exception) as _:
-        lemma(z3.BoolVal(False))
+        kd.lemma(z3.BoolVal(False))
 
 
 def test_explosion():
-    a = axiom(z3.BoolVal(False), "False axiom")
+    a = kd.axiom(z3.BoolVal(False), "False axiom")
     with pytest.raises(Exception) as _:
-        lemma(z3.BoolVal(True), by=[a])
+        kd.lemma(z3.BoolVal(True), by=[a])
 
 
 def test_calc():
     x, y, z = z3.Ints("x y z")
-    l1 = axiom(x == y)
-    l2 = axiom(y == z)
+    l1 = kd.axiom(x == y)
+    l2 = kd.axiom(y == z)
     Calc([], x).eq(y, by=[l1]).eq(z, by=[l2]).qed()
 
 
@@ -77,3 +77,7 @@ def test_simp():
 def test_record():
     foo = kd.notation.Record("foo", ("bar", z3.IntSort()), ("baz", z3.BoolSort()))
     assert z3.simplify(foo.mk(1, True).bar).eq(z3.IntVal(1))
+
+
+def test_seq():
+    ThSeq.induct(z3.IntSort(), lambda x: x == x)