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MoreStlc.html
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
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<title>MoreStlc: More on the Simply Typed Lambda-Calculus</title>
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<h1 class="libtitle">MoreStlc<span class="subtitle">More on the Simply Typed Lambda-Calculus</span></h1>
<div class="code code-tight">
</div>
<div class="doc">
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <a class="idref" href="Maps.html#"><span class="id" type="library">Maps</span></a>.<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <span class="id" type="keyword">Types</span>.<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <a class="idref" href="Smallstep.html#"><span class="id" type="library">Smallstep</span></a>.<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <a class="idref" href="Stlc.html#"><span class="id" type="library">Stlc</span></a>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab601"></a><h1 class="section">Simple Extensions to STLC</h1>
<div class="paragraph"> </div>
The simply typed lambda-calculus has enough structure to make its
theoretical properties interesting, but it is not much of a
programming language. In this chapter, we begin to close the gap
with real-world languages by introducing a number of familiar
features that have straightforward treatments at the level of
typing.
<div class="paragraph"> </div>
<a name="lab602"></a><h2 class="section">Numbers</h2>
<div class="paragraph"> </div>
As we saw in exercise <span class="inlinecode"><span class="id" type="var">stlc_arith</span></span> at the end of the <span class="inlinecode"><span class="id" type="var">StlcProp</span></span>
chapter, adding types, constants, and primitive operations for
numbers is easy — basically just a matter of combining the
<a href="Types.html"><span class="inlineref">Types</span></a> and <a href="Stlc.html"><span class="inlineref">Stlc</span></a> chapters.
<div class="paragraph"> </div>
<a name="lab603"></a><h2 class="section">Let Bindings</h2>
<div class="paragraph"> </div>
When writing a complex expression, it is useful to be able
to give names to some of its subexpressions to avoid repetition
and increase readability. Most languages provide one or more ways
of doing this. In OCaml (and Coq), for example, we can write <span class="inlinecode"><span class="id" type="keyword">let</span></span>
<span class="inlinecode"><span class="id" type="var">x</span>=<span class="id" type="var">t<sub>1</sub></span></span> <span class="inlinecode"><span class="id" type="keyword">in</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> to mean "reduce the expression <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> to a value and
bind the name <span class="inlinecode"><span class="id" type="var">x</span></span> to this value while reducing <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span>."
<div class="paragraph"> </div>
Our <span class="inlinecode"><span class="id" type="keyword">let</span></span>-binder follows OCaml in choosing a standard
<i>call-by-value</i> evaluation order, where the <span class="inlinecode"><span class="id" type="keyword">let</span></span>-bound term must
be fully reduced before reduction of the <span class="inlinecode"><span class="id" type="keyword">let</span></span>-body can begin.
The typing rule <span class="inlinecode"><span class="id" type="var">T_Let</span></span> tells us that the type of a <span class="inlinecode"><span class="id" type="keyword">let</span></span> can be
calculated by calculating the type of the <span class="inlinecode"><span class="id" type="keyword">let</span></span>-bound term,
extending the context with a binding with this type, and in this
enriched context calculating the type of the body (which is then
the type of the whole <span class="inlinecode"><span class="id" type="keyword">let</span></span> expression).
<div class="paragraph"> </div>
At this point in the book, it's probably easier simply to look at
the rules defining this new feature than to wade through a lot of
English text conveying the same information. Here they are:
<div class="paragraph"> </div>
Syntax:
<pre>
t ::= Terms
| ... (other terms same as before)
| let x=t in t let-binding
</pre>
<div class="paragraph"> </div>
<div class="paragraph"> </div>
Reduction:
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t<sub>1</sub> <span style="font-family: arial;">⇒</span> t<sub>1</sub>'</td>
<td class="infrulenamecol" rowspan="3">
(ST_Let1)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">let x=t<sub>1</sub> in t<sub>2</sub> <span style="font-family: arial;">⇒</span> let x=t<sub>1</sub>' in t<sub>2</sub></td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(ST_LetValue)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">let x=v<sub>1</sub> in t<sub>2</sub> <span style="font-family: arial;">⇒</span> [x:=v<sub>1</sub>]t<sub>2</sub></td>
<td></td>
</td>
</table></center> Typing:
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> t<sub>1</sub> : T<sub>1</sub> <span style="font-family: serif; font-size:85%;">Γ</span> , x:T<sub>1</sub> <span style="font-family: arial;">⊢</span> t<sub>2</sub> : T<sub>2</sub></td>
<td class="infrulenamecol" rowspan="3">
(T_Let)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> let x=t<sub>1</sub> in t<sub>2</sub> : T<sub>2</sub></td>
<td></td>
</td>
</table></center>
<div class="paragraph"> </div>
<a name="lab604"></a><h2 class="section">Pairs</h2>
<div class="paragraph"> </div>
Our functional programming examples in Coq have made
frequent use of <i>pairs</i> of values. The type of such a pair is
called a <i>product type</i>.
<div class="paragraph"> </div>
The formalization of pairs is almost too simple to be worth
discussing. However, let's look briefly at the various parts of
the definition to emphasize the common pattern.
<div class="paragraph"> </div>
In Coq, the primitive way of extracting the components of a pair
is <i>pattern matching</i>. An alternative style is to take <span class="inlinecode"><span class="id" type="var">fst</span></span> and
<span class="inlinecode"><span class="id" type="var">snd</span></span> — the first- and second-projection operators — as
primitives. Just for fun, let's do our products this way. For
example, here's how we'd write a function that takes a pair of
numbers and returns the pair of their sum and difference:
<pre>
λx:Nat*Nat.
let sum = x.fst + x.snd in
let diff = x.fst - x.snd in
(sum,diff)
</pre>
<div class="paragraph"> </div>
Adding pairs to the simply typed lambda-calculus, then, involves
adding two new forms of term — pairing, written <span class="inlinecode">(<span class="id" type="var">t<sub>1</sub></span>,<span class="id" type="var">t<sub>2</sub></span>)</span>, and
projection, written <span class="inlinecode"><span class="id" type="var">t.fst</span></span> for the first projection from <span class="inlinecode"><span class="id" type="var">t</span></span> and
<span class="inlinecode"><span class="id" type="var">t.snd</span></span> for the second projection — plus one new type constructor,
<span class="inlinecode"><span class="id" type="var">T<sub>1</sub></span>*<span class="id" type="var">T<sub>2</sub></span></span>, called the <i>product</i> of <span class="inlinecode"><span class="id" type="var">T<sub>1</sub></span></span> and <span class="inlinecode"><span class="id" type="var">T<sub>2</sub></span></span>.
<div class="paragraph"> </div>
Syntax:
<pre>
t ::= Terms
| (t,t) pair
| t.fst first projection
| t.snd second projection
| ...
v ::= Values
| (v,v) pair value
| ...
T ::= Types
| T * T product type
| ...
</pre>
<div class="paragraph"> </div>
For reduction, we need several new rules specifying how pairs and
projection behave.
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t<sub>1</sub> <span style="font-family: arial;">⇒</span> t<sub>1</sub>'</td>
<td class="infrulenamecol" rowspan="3">
(ST_Pair1)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">(t<sub>1</sub>,t<sub>2</sub>) <span style="font-family: arial;">⇒</span> (t<sub>1</sub>',t<sub>2</sub>)</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t<sub>2</sub> <span style="font-family: arial;">⇒</span> t<sub>2</sub>'</td>
<td class="infrulenamecol" rowspan="3">
(ST_Pair2)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">(v<sub>1</sub>,t<sub>2</sub>) <span style="font-family: arial;">⇒</span> (v<sub>1</sub>,t<sub>2</sub>')</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t<sub>1</sub> <span style="font-family: arial;">⇒</span> t<sub>1</sub>'</td>
<td class="infrulenamecol" rowspan="3">
(ST_Fst1)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">t<sub>1</sub>.fst <span style="font-family: arial;">⇒</span> t<sub>1</sub>'.fst</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(ST_FstPair)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">(v<sub>1</sub>,v<sub>2</sub>).fst <span style="font-family: arial;">⇒</span> v<sub>1</sub></td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t<sub>1</sub> <span style="font-family: arial;">⇒</span> t<sub>1</sub>'</td>
<td class="infrulenamecol" rowspan="3">
(ST_Snd1)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">t<sub>1</sub>.snd <span style="font-family: arial;">⇒</span> t<sub>1</sub>'.snd</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(ST_SndPair)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">(v<sub>1</sub>,v<sub>2</sub>).snd <span style="font-family: arial;">⇒</span> v<sub>2</sub></td>
<td></td>
</td>
</table></center>
<div class="paragraph"> </div>
Rules <span class="inlinecode"><span class="id" type="var">ST_FstPair</span></span> and <span class="inlinecode"><span class="id" type="var">ST_SndPair</span></span> say that, when a fully
reduced pair meets a first or second projection, the result is
the appropriate component. The congruence rules <span class="inlinecode"><span class="id" type="var">ST_Fst1</span></span> and
<span class="inlinecode"><span class="id" type="var">ST_Snd1</span></span> allow reduction to proceed under projections, when the
term being projected from has not yet been fully reduced.
<span class="inlinecode"><span class="id" type="var">ST_Pair1</span></span> and <span class="inlinecode"><span class="id" type="var">ST_Pair2</span></span> reduce the parts of pairs: first the
left part, and then — when a value appears on the left — the right
part. The ordering arising from the use of the metavariables <span class="inlinecode"><span class="id" type="var">v</span></span>
and <span class="inlinecode"><span class="id" type="var">t</span></span> in these rules enforces a left-to-right evaluation
strategy for pairs. (Note the implicit convention that
metavariables like <span class="inlinecode"><span class="id" type="var">v</span></span> and <span class="inlinecode"><span class="id" type="var">v<sub>1</sub></span></span> can only denote values.) We've
also added a clause to the definition of values, above, specifying
that <span class="inlinecode">(<span class="id" type="var">v<sub>1</sub></span>,<span class="id" type="var">v<sub>2</sub></span>)</span> is a value. The fact that the components of a pair
value must themselves be values ensures that a pair passed as an
argument to a function will be fully reduced before the function
body starts executing.
<div class="paragraph"> </div>
The typing rules for pairs and projections are straightforward.
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> t<sub>1</sub> : T<sub>1</sub> <span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> t<sub>2</sub> : T<sub>2</sub></td>
<td class="infrulenamecol" rowspan="3">
(T_Pair)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> (t<sub>1</sub>,t<sub>2</sub>) : T<sub>1</sub>*T<sub>2</sub></td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> t<sub>1</sub> : T<sub>11</sub>*T<sub>12</sub></td>
<td class="infrulenamecol" rowspan="3">
(T_Fst)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> t<sub>1</sub>.fst : T<sub>11</sub></td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> t<sub>1</sub> : T<sub>11</sub>*T<sub>12</sub></td>
<td class="infrulenamecol" rowspan="3">
(T_Snd)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> t<sub>1</sub>.snd : T<sub>12</sub></td>
<td></td>
</td>
</table></center>
<div class="paragraph"> </div>
<span class="inlinecode"><span class="id" type="var">T_Pair</span></span> says that <span class="inlinecode">(<span class="id" type="var">t<sub>1</sub></span>,<span class="id" type="var">t<sub>2</sub></span>)</span> has type <span class="inlinecode"><span class="id" type="var">T<sub>1</sub></span>*<span class="id" type="var">T<sub>2</sub></span></span> if <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> has
type <span class="inlinecode"><span class="id" type="var">T<sub>1</sub></span></span> and <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> has type <span class="inlinecode"><span class="id" type="var">T<sub>2</sub></span></span>. Conversely, <span class="inlinecode"><span class="id" type="var">T_Fst</span></span> and <span class="inlinecode"><span class="id" type="var">T_Snd</span></span>
tell us that, if <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> has a product type <span class="inlinecode"><span class="id" type="var">T<sub>11</sub></span>*<span class="id" type="var">T<sub>12</sub></span></span> (i.e., if it
will reduce to a pair), then the types of the projections from
this pair are <span class="inlinecode"><span class="id" type="var">T<sub>11</sub></span></span> and <span class="inlinecode"><span class="id" type="var">T<sub>12</sub></span></span>.
<div class="paragraph"> </div>
<a name="lab605"></a><h2 class="section">Unit</h2>
<div class="paragraph"> </div>
Another handy base type, found especially in languages in
the ML family, is the singleton type <span class="inlinecode"><span class="id" type="var">Unit</span></span>. It has a single element — the term constant <span class="inlinecode"><span class="id" type="var">unit</span></span> (with a small
<span class="inlinecode"><span class="id" type="var">u</span></span>) — and a typing rule making <span class="inlinecode"><span class="id" type="var">unit</span></span> an element of <span class="inlinecode"><span class="id" type="var">Unit</span></span>. We
also add <span class="inlinecode"><span class="id" type="var">unit</span></span> to the set of possible values — indeed, <span class="inlinecode"><span class="id" type="var">unit</span></span> is
the <i>only</i> possible result of reducing an expression of type
<span class="inlinecode"><span class="id" type="var">Unit</span></span>.
<div class="paragraph"> </div>
Syntax:
<pre>
t ::= Terms
| unit unit value
| ...
v ::= Values
| unit unit
| ...
T ::= Types
| Unit Unit type
| ...
</pre>
Typing:
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(T_Unit)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> unit : Unit</td>
<td></td>
</td>
</table></center>
<div class="paragraph"> </div>
It may seem a little strange to bother defining a type that
has just one element — after all, wouldn't every computation
living in such a type be trivial?
<div class="paragraph"> </div>
This is a fair question, and indeed in the STLC the <span class="inlinecode"><span class="id" type="var">Unit</span></span> type is
not especially critical (though we'll see two uses for it below).
Where <span class="inlinecode"><span class="id" type="var">Unit</span></span> really comes in handy is in richer languages with
<i>side effects</i> — e.g., assignment statements that mutate
variables or pointers, exceptions and other sorts of nonlocal
control structures, etc. In such languages, it is convenient to
have a type for the (trivial) result of an expression that is
evaluated only for its effect.
<div class="paragraph"> </div>
<a name="lab606"></a><h2 class="section">Sums</h2>
<div class="paragraph"> </div>
Many programs need to deal with values that can take two distinct
forms. For example, we might identify employees in an accounting
application using using <i>either</i> their name <i>or</i> their id number.
A search function might return <i>either</i> a matching value <i>or</i> an
error code.
<div class="paragraph"> </div>
These are specific examples of a binary <i>sum type</i> (sometimes called
a <i>disjoint union</i>), which describes a set of values drawn from
one of two given types, e.g.:
<pre>
Nat + Bool
</pre>
<div class="paragraph"> </div>
We create elements of these types by <i>tagging</i> elements of
the component types. For example, if <span class="inlinecode"><span class="id" type="var">n</span></span> is a <span class="inlinecode"><span class="id" type="var">Nat</span></span> then <span class="inlinecode"><span class="id" type="var">inl</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span>
is an element of <span class="inlinecode"><span class="id" type="var">Nat</span>+<span class="id" type="var">Bool</span></span>; similarly, if <span class="inlinecode"><span class="id" type="var">b</span></span> is a <span class="inlinecode"><span class="id" type="var">Bool</span></span> then
<span class="inlinecode"><span class="id" type="var">inr</span></span> <span class="inlinecode"><span class="id" type="var">b</span></span> is a <span class="inlinecode"><span class="id" type="var">Nat</span>+<span class="id" type="var">Bool</span></span>. The names of the tags <span class="inlinecode"><span class="id" type="var">inl</span></span> and <span class="inlinecode"><span class="id" type="var">inr</span></span>
arise from thinking of them as functions
<pre>
inl : Nat -> Nat + Bool
inr : Bool -> Nat + Bool
</pre>
that "inject" elements of <span class="inlinecode"><span class="id" type="var">Nat</span></span> or <span class="inlinecode"><span class="id" type="var">Bool</span></span> into the left and right
components of the sum type <span class="inlinecode"><span class="id" type="var">Nat</span>+<span class="id" type="var">Bool</span></span>. (But note that we don't
actually treat them as functions in the way we formalize them:
<span class="inlinecode"><span class="id" type="var">inl</span></span> and <span class="inlinecode"><span class="id" type="var">inr</span></span> are keywords, and <span class="inlinecode"><span class="id" type="var">inl</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> and <span class="inlinecode"><span class="id" type="var">inr</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> are primitive
syntactic forms, not function applications.)
<div class="paragraph"> </div>
In general, the elements of a type <span class="inlinecode"><span class="id" type="var">T<sub>1</sub></span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">T<sub>2</sub></span></span> consist of the
elements of <span class="inlinecode"><span class="id" type="var">T<sub>1</sub></span></span> tagged with the token <span class="inlinecode"><span class="id" type="var">inl</span></span>, plus the elements of
<span class="inlinecode"><span class="id" type="var">T<sub>2</sub></span></span> tagged with <span class="inlinecode"><span class="id" type="var">inr</span></span>.
<div class="paragraph"> </div>
One important usage of sums is signaling errors:
<pre>
div : Nat -> Nat -> (Nat + Unit) =
div =
λx:Nat. λy:Nat.
if iszero y then
inr unit
else
inl ...
</pre>
The type <span class="inlinecode"><span class="id" type="var">Nat</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">Unit</span></span> above is in fact isomorphic to <span class="inlinecode"><span class="id" type="var">option</span></span> <span class="inlinecode"><span class="id" type="var">nat</span></span>
in Coq, and we've already seen how to signal errors with options.
<div class="paragraph"> </div>
To <i>use</i> elements of sum types, we introduce a <span class="inlinecode"><span class="id" type="tactic">case</span></span>
construct (a very simplified form of Coq's <span class="inlinecode"><span class="id" type="keyword">match</span></span>) to destruct
them. For example, the following procedure converts a <span class="inlinecode"><span class="id" type="var">Nat</span>+<span class="id" type="var">Bool</span></span>
into a <span class="inlinecode"><span class="id" type="var">Nat</span></span>: <<
getNat =
λx:Nat+Bool.
case x of
inl n => n
| inr b => if b then 1 else 0
>>
<div class="paragraph"> </div>
More formally...
<div class="paragraph"> </div>
Syntax:
<pre>
t ::= Terms
| inl T t tagging (left)
| inr T t tagging (right)
| case t of case
inl x => t
| inr x => t
| ...
v ::= Values
| inl T v tagged value (left)
| inr T v tagged value (right)
| ...
T ::= Types
| T + T sum type
| ...
</pre>
<div class="paragraph"> </div>
Reduction:
<div class="paragraph"> </div>
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t<sub>1</sub> <span style="font-family: arial;">⇒</span> t<sub>1</sub>'</td>
<td class="infrulenamecol" rowspan="3">
(ST_Inl)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">inl T t<sub>1</sub> <span style="font-family: arial;">⇒</span> inl T t<sub>1</sub>'</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t<sub>1</sub> <span style="font-family: arial;">⇒</span> t<sub>1</sub>'</td>
<td class="infrulenamecol" rowspan="3">
(ST_Inr)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">inr T t<sub>1</sub> <span style="font-family: arial;">⇒</span> inr T t<sub>1</sub>'</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t<sub>0</sub> <span style="font-family: arial;">⇒</span> t<sub>0</sub>'</td>
<td class="infrulenamecol" rowspan="3">
(ST_Case)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">case t<sub>0</sub> of inl x<sub>1</sub> ⇒ t<sub>1</sub> | inr x<sub>2</sub> ⇒ t<sub>2</sub> <span style="font-family: arial;">⇒</span></td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule">case t<sub>0</sub>' of inl x<sub>1</sub> ⇒ t<sub>1</sub> | inr x<sub>2</sub> ⇒ t<sub>2</sub></td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(ST_CaseInl)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">case (inl T v<sub>0</sub>) of inl x<sub>1</sub> ⇒ t<sub>1</sub> | inr x<sub>2</sub> ⇒ t<sub>2</sub></td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: arial;">⇒</span> [x<sub>1</sub>:=v<sub>0</sub>]t<sub>1</sub></td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(ST_CaseInr)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">case (inr T v<sub>0</sub>) of inl x<sub>1</sub> ⇒ t<sub>1</sub> | inr x<sub>2</sub> ⇒ t<sub>2</sub></td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: arial;">⇒</span> [x<sub>2</sub>:=v<sub>0</sub>]t<sub>2</sub></td>
<td></td>
</td>
</table></center>
<div class="paragraph"> </div>
Typing:
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> t<sub>1</sub> : T<sub>1</sub></td>
<td class="infrulenamecol" rowspan="3">
(T_Inl)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> inl T<sub>2</sub> t<sub>1</sub> : T<sub>1</sub> + T<sub>2</sub></td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> t<sub>1</sub> : T<sub>2</sub></td>
<td class="infrulenamecol" rowspan="3">
(T_Inr)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> inr T<sub>1</sub> t<sub>1</sub> : T<sub>1</sub> + T<sub>2</sub></td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> t<sub>0</sub> : T<sub>1</sub>+T<sub>2</sub></td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> , x<sub>1</sub>:T<sub>1</sub> <span style="font-family: arial;">⊢</span> t<sub>1</sub> : T</td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> , x<sub>2</sub>:T<sub>2</sub> <span style="font-family: arial;">⊢</span> t<sub>2</sub> : T</td>
<td class="infrulenamecol" rowspan="3">
(T_Case)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> case t<sub>0</sub> of inl x<sub>1</sub> ⇒ t<sub>1</sub> | inr x<sub>2</sub> ⇒ t<sub>2</sub> : T</td>
<td></td>
</td>
</table></center>
<div class="paragraph"> </div>
We use the type annotation in <span class="inlinecode"><span class="id" type="var">inl</span></span> and <span class="inlinecode"><span class="id" type="var">inr</span></span> to make the typing
relation simpler, similarly to what we did for functions.
<div class="paragraph"> </div>
Without this extra information, the typing rule <span class="inlinecode"><span class="id" type="var">T_Inl</span></span>, for
example, would have to say that, once we have shown that <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> is
an element of type <span class="inlinecode"><span class="id" type="var">T<sub>1</sub></span></span>, we can derive that <span class="inlinecode"><span class="id" type="var">inl</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> is an element
of <span class="inlinecode"><span class="id" type="var">T<sub>1</sub></span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">T<sub>2</sub></span></span> for <i>any</i> type T<sub>2</sub>. For example, we could derive both
<span class="inlinecode"><span class="id" type="var">inl</span></span> <span class="inlinecode">5</span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" type="var">Nat</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">Nat</span></span> and <span class="inlinecode"><span class="id" type="var">inl</span></span> <span class="inlinecode">5</span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" type="var">Nat</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">Bool</span></span> (and infinitely many
other types). This peculiarity (technically, a failure of
uniqueness of types) would mean that we cannot build a
typechecking algorithm simply by "reading the rules from bottom to
top" as we could for all the other features seen so far.
<div class="paragraph"> </div>
There are various ways to deal with this difficulty. One simple
one — which we've adopted here — forces the programmer to
explicitly annotate the "other side" of a sum type when performing
an injection. This is a bit heavy for programmers (so real
languages adopt other solutions), but it is easy to understand and
formalize.
<div class="paragraph"> </div>
<a name="lab607"></a><h2 class="section">Lists</h2>
<div class="paragraph"> </div>
The typing features we have seen can be classified into <i>base
types</i> like <span class="inlinecode"><span class="id" type="var">Bool</span></span>, and <i>type constructors</i> like <span class="inlinecode"><span style="font-family: arial;">→</span></span> and <span class="inlinecode">*</span> that
build new types from old ones. Another useful type constructor is
<span class="inlinecode"><span class="id" type="var">List</span></span>. For every type <span class="inlinecode"><span class="id" type="var">T</span></span>, the type <span class="inlinecode"><span class="id" type="var">List</span></span> <span class="inlinecode"><span class="id" type="var">T</span></span> describes
finite-length lists whose elements are drawn from <span class="inlinecode"><span class="id" type="var">T</span></span>.
<div class="paragraph"> </div>
In principle, we could encode lists using pairs, sums and
<i>recursive</i> types. But giving semantics to recursive types is
non-trivial. Instead, we'll just discuss the special case of lists
directly.
<div class="paragraph"> </div>
Below we give the syntax, semantics, and typing rules for lists.
Except for the fact that explicit type annotations are mandatory
on <span class="inlinecode"><span class="id" type="var">nil</span></span> and cannot appear on <span class="inlinecode"><span class="id" type="var">cons</span></span>, these lists are essentially
identical to those we built in Coq. We use <span class="inlinecode"><span class="id" type="var">lcase</span></span> to destruct
lists, to avoid dealing with questions like "what is the <span class="inlinecode"><span class="id" type="var">head</span></span> of
the empty list?"
<div class="paragraph"> </div>
For example, here is a function that calculates the sum of
the first two elements of a list of numbers:
<pre>
λx:List Nat.
lcase x of nil -> 0
| a::x' -> lcase x' of nil -> a
| b::x'' -> a+b
</pre>
<div class="paragraph"> </div>
Syntax:
<pre>
t ::= Terms
| nil T
| cons t t
| lcase t of nil -> t
| x::x -> t
| ...
v ::= Values
| nil T nil value
| cons v v cons value
| ...
T ::= Types
| List T list of Ts
| ...
</pre>
<div class="paragraph"> </div>
Reduction:
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t<sub>1</sub> <span style="font-family: arial;">⇒</span> t<sub>1</sub>'</td>
<td class="infrulenamecol" rowspan="3">
(ST_Cons1)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">cons t<sub>1</sub> t<sub>2</sub> <span style="font-family: arial;">⇒</span> cons t<sub>1</sub>' t<sub>2</sub></td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t<sub>2</sub> <span style="font-family: arial;">⇒</span> t<sub>2</sub>'</td>
<td class="infrulenamecol" rowspan="3">
(ST_Cons2)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">cons v<sub>1</sub> t<sub>2</sub> <span style="font-family: arial;">⇒</span> cons v<sub>1</sub> t<sub>2</sub>'</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t<sub>1</sub> <span style="font-family: arial;">⇒</span> t<sub>1</sub>'</td>
<td class="infrulenamecol" rowspan="3">
(ST_Lcase1)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">(lcase t<sub>1</sub> of nil <span style="font-family: arial;">→</span> t<sub>2</sub> | xh::xt <span style="font-family: arial;">→</span> t<sub>3</sub>) <span style="font-family: arial;">⇒</span></td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule">(lcase t<sub>1</sub>' of nil <span style="font-family: arial;">→</span> t<sub>2</sub> | xh::xt <span style="font-family: arial;">→</span> t<sub>3</sub>)</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(ST_LcaseNil)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">(lcase nil T of nil <span style="font-family: arial;">→</span> t<sub>2</sub> | xh::xt <span style="font-family: arial;">→</span> t<sub>3</sub>)</td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: arial;">⇒</span> t<sub>2</sub></td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(ST_LcaseCons)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">(lcase (cons vh vt) of nil <span style="font-family: arial;">→</span> t<sub>2</sub> | xh::xt <span style="font-family: arial;">→</span> t<sub>3</sub>)</td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: arial;">⇒</span> [xh:=vh,xt:=vt]t<sub>3</sub></td>
<td></td>
</td>
</table></center>
<div class="paragraph"> </div>
Typing:
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(T_Nil)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> nil T : List T</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> t<sub>1</sub> : T <span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> t<sub>2</sub> : List T</td>
<td class="infrulenamecol" rowspan="3">
(T_Cons)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> cons t<sub>1</sub> t<sub>2</sub>: List T</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> t<sub>1</sub> : List T<sub>1</sub></td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> t<sub>2</sub> : T</td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> , h:T<sub>1</sub>, t:List T<sub>1</sub> <span style="font-family: arial;">⊢</span> t<sub>3</sub> : T</td>
<td class="infrulenamecol" rowspan="3">
(T_Lcase)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> (lcase t<sub>1</sub> of nil <span style="font-family: arial;">→</span> t<sub>2</sub> | h::t <span style="font-family: arial;">→</span> t<sub>3</sub>) : T</td>
<td></td>
</td>
</table></center>
<div class="paragraph"> </div>
<a name="lab608"></a><h2 class="section">General Recursion</h2>
<div class="paragraph"> </div>
Another facility found in most programming languages (including
Coq) is the ability to define recursive functions. For example,
we might like to be able to define the factorial function like
this:
<pre>
fact = λx:Nat.
if x=0 then 1 else x * (fact (pred x)))
</pre>
Formalizing such a definition mechanism can be done, but it requires
some effort: we'd have to introduce a notion of "function definitions"
and carry around an "environment" of such definitions in the definition
of the <span class="inlinecode"><span class="id" type="var">step</span></span> relation.
<div class="paragraph"> </div>
Here is another way that is straightforward to formalize: instead
of writing recursive definitions where the right-hand side can
contain the identifier being defined, we can define a <i>fixed-point
operator</i> that performs the "unfolding" of the recursive definition
in the right-hand side as needed, during reduction.
<pre>
fact =
fix
(\f:Nat->Nat.
λx:Nat.
if x=0 then 1 else x * (f (pred x)))
</pre>
<div class="paragraph"> </div>
The intuition is that the higher-order function <span class="inlinecode"><span class="id" type="var">f</span></span> passed
to <span class="inlinecode"><span class="id" type="var">fix</span></span> is a <i>generator</i> for the <span class="inlinecode"><span class="id" type="var">fact</span></span> function: if <span class="inlinecode"><span class="id" type="var">f</span></span> is
applied to a function that approximates the desired behavior of
<span class="inlinecode"><span class="id" type="var">fact</span></span> up to some number <span class="inlinecode"><span class="id" type="var">n</span></span> (that is, a function that returns
correct results on inputs less than or equal to <span class="inlinecode"><span class="id" type="var">n</span></span>), then it
returns a better approximation to <span class="inlinecode"><span class="id" type="var">fact</span></span> — a function that
returns correct results for inputs up to <span class="inlinecode"><span class="id" type="var">n</span>+1</span>. Applying <span class="inlinecode"><span class="id" type="var">fix</span></span> to
this generator returns its <i>fixed point</i> — a function that gives
the desired behavior for all inputs <span class="inlinecode"><span class="id" type="var">n</span></span>.
<div class="paragraph"> </div>
(The term "fixed point" has exactly the same sense as in ordinary
mathematics, where a fixed point of a function <span class="inlinecode"><span class="id" type="var">f</span></span> is an input <span class="inlinecode"><span class="id" type="var">x</span></span>
such that <span class="inlinecode"><span class="id" type="var">f</span>(<span class="id" type="var">x</span>)</span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">x</span></span>. Here, a fixed point of a function <span class="inlinecode"><span class="id" type="var">F</span></span> of
type (say) <span class="inlinecode">(<span class="id" type="var">Nat</span><span style="font-family: arial;">→</span><span class="id" type="var">Nat</span>)->(<span class="id" type="var">Nat</span><span style="font-family: arial;">→</span><span class="id" type="var">Nat</span>)</span> is a function <span class="inlinecode"><span class="id" type="var">f</span></span> of type
<span class="inlinecode"><span class="id" type="var">Nat</span><span style="font-family: arial;">→</span><span class="id" type="var">Nat</span></span> such that <span class="inlinecode"><span class="id" type="var">F</span></span> <span class="inlinecode"><span class="id" type="var">f</span></span> is behaviorally equivalent to <span class="inlinecode"><span class="id" type="var">f</span></span>.)
<div class="paragraph"> </div>
Syntax:
<pre>
t ::= Terms
| fix t fixed-point operator
| ...
</pre>
Reduction:
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t<sub>1</sub> <span style="font-family: arial;">⇒</span> t<sub>1</sub>'</td>
<td class="infrulenamecol" rowspan="3">
(ST_Fix1)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">fix t<sub>1</sub> <span style="font-family: arial;">⇒</span> fix t<sub>1</sub>'</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">F = \xf:T<sub>1</sub>.t2</td>
<td class="infrulenamecol" rowspan="3">
(ST_FixAbs)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">fix F <span style="font-family: arial;">⇒</span> [xf:=fix F]t<sub>2</sub></td>
<td></td>
</td>
</table></center> Typing:
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> t<sub>1</sub> : T<sub>1</sub>->T<sub>1</sub></td>
<td class="infrulenamecol" rowspan="3">
(T_Fix)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> fix t<sub>1</sub> : T<sub>1</sub></td>
<td></td>
</td>
</table></center>
<div class="paragraph"> </div>
Let's see how <span class="inlinecode"><span class="id" type="var">ST_FixAbs</span></span> works by reducing <span class="inlinecode"><span class="id" type="var">fact</span></span> <span class="inlinecode">3</span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">fix</span></span> <span class="inlinecode"><span class="id" type="var">F</span></span> <span class="inlinecode">3</span>,
where <span class="inlinecode"><span class="id" type="var">F</span></span> <span class="inlinecode">=</span> <span class="inlinecode">(\<span class="id" type="var">f</span>.</span> <span class="inlinecode">\<span class="id" type="var">x</span>.</span> <span class="inlinecode"><span class="id" type="keyword">if</span></span> <span class="inlinecode"><span class="id" type="var">x</span>=0</span> <span class="inlinecode"><span class="id" type="keyword">then</span></span> <span class="inlinecode">1</span> <span class="inlinecode"><span class="id" type="keyword">else</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode">*</span> <span class="inlinecode">(<span class="id" type="var">f</span></span> <span class="inlinecode">(<span class="id" type="var">pred</span></span> <span class="inlinecode"><span class="id" type="var">x</span>)))</span> (we are
omitting type annotations for brevity here).
<pre>
fix F 3
</pre>
<span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode"><span class="id" type="var">ST_FixAbs</span></span>
<pre>
(\x. if x=0 then 1 else x * (fix F (pred x))) 3
</pre>
<span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode"><span class="id" type="var">ST_AppAbs</span></span>
<pre>
if 3=0 then 1 else 3 * (fix F (pred 3))
</pre>
<span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode"><span class="id" type="var">ST_If0_Nonzero</span></span>
<pre>
3 * (fix F (pred 3))
</pre>
<span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode"><span class="id" type="var">ST_FixAbs</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">ST_Mult2</span></span>
<pre>
3 * ((\x. if x=0 then 1 else x * (fix F (pred x))) (pred 3))
</pre>
<span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode"><span class="id" type="var">ST_PredNat</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">ST_Mult2</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">ST_App2</span></span>
<pre>
3 * ((\x. if x=0 then 1 else x * (fix F (pred x))) 2)
</pre>
<span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode"><span class="id" type="var">ST_AppAbs</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">ST_Mult2</span></span>
<pre>
3 * (if 2=0 then 1 else 2 * (fix F (pred 2)))
</pre>
<span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode"><span class="id" type="var">ST_If0_Nonzero</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">ST_Mult2</span></span>
<pre>
3 * (2 * (fix F (pred 2)))
</pre>
<span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode"><span class="id" type="var">ST_FixAbs</span></span> <span class="inlinecode">+</span> <span class="inlinecode">2</span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">ST_Mult2</span></span>
<pre>
3 * (2 * ((\x. if x=0 then 1 else x * (fix F (pred x))) (pred 2)))
</pre>
<span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode"><span class="id" type="var">ST_PredNat</span></span> <span class="inlinecode">+</span> <span class="inlinecode">2</span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">ST_Mult2</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">ST_App2</span></span>
<pre>
3 * (2 * ((\x. if x=0 then 1 else x * (fix F (pred x))) 1))