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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="comment">(* $Date: 2012-04-30 11:09:42 -0400 (Mon, 30 Apr 2012) $ *)</span><br/>
<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Export</span> <span class="id" type="var">Stlc</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab596"></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 yet 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="lab597"></a><h2 class="section">Numbers</h2>
<div class="paragraph"> </div>
Adding types, constants, and primitive operations for numbers is
easy — just a matter of combining the <span class="inlinecode"><span class="id" type="keyword">Types</span></span> and <span class="inlinecode"><span class="id" type="var">Stlc</span></span>
chapters.
<div class="paragraph"> </div>
<a name="lab598"></a><h2 class="section"><span class="inlinecode"><span class="id" type="keyword">let</span></span>-bindings</h2>
<div class="paragraph"> </div>
When writing a complex expression, it is often useful to give
names to some of its subexpressions: this avoids repetition and
often increases 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">t1</span></span> <span class="inlinecode"><span class="id" type="keyword">in</span></span> <span class="inlinecode"><span class="id" type="var">t2</span></span> to mean ``evaluate the expression <span class="inlinecode"><span class="id" type="var">t1</span></span> and bind
the name <span class="inlinecode"><span class="id" type="var">x</span></span> to the resulting value while evaluating <span class="inlinecode"><span class="id" type="var">t2</span></span>.''
<div class="paragraph"> </div>
Our <span class="inlinecode"><span class="id" type="keyword">let</span></span>-binder follows OCaml's in choosing a call-by-value
evaluation order, where the <span class="inlinecode"><span class="id" type="keyword">let</span></span>-bound term must be fully
evaluated before evaluation 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 course, it's probably easier simply to look
at the rules defining this new feature as 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>
Reduction:
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t1 <span style="font-family: arial;">⇒</span> t1'</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=t1 in t2 <span style="font-family: arial;">⇒</span> let x=t1' in t2</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=v1 in t2 <span style="font-family: arial;">⇒</span> [x:=v1]t2</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> t1 : T1 <span style="font-family: serif; font-size:85%;">Γ</span> , x:T1 <span style="font-family: arial;">⊢</span> t2 : T2</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=t1 in t2 : T2</td>
<td></td>
</td>
</table></center>
<div class="paragraph"> </div>
<a name="lab599"></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 pairs 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>
\x: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">t1</span>,<span class="id" type="var">t2</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">T1</span>*<span class="id" type="var">T2</span></span>, called the <i>product</i> of <span class="inlinecode"><span class="id" type="var">T1</span></span> and <span class="inlinecode"><span class="id" type="var">T2</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 evaluation, we need several new rules specifying how pairs and
projection behave.
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t1 <span style="font-family: arial;">⇒</span> t1'</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">(t1,t2) <span style="font-family: arial;">⇒</span> (t1',t2)</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t2 <span style="font-family: arial;">⇒</span> t2'</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">(v1,t2) <span style="font-family: arial;">⇒</span> (v1,t2')</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t1 <span style="font-family: arial;">⇒</span> t1'</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">t1.fst <span style="font-family: arial;">⇒</span> t1'.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">(v1,v2).fst <span style="font-family: arial;">⇒</span> v1</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t1 <span style="font-family: arial;">⇒</span> t1'</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">t1.snd <span style="font-family: arial;">⇒</span> t1'.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">(v1,v2).snd <span style="font-family: arial;">⇒</span> v2</td>
<td></td>
</td>
</table></center>
<div class="paragraph"> </div>
<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> specify that, when a fully
evaluated 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 evaluated.
<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> evaluate 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">v1</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">v1</span>,<span class="id" type="var">v2</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 evaluated 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> t1 : T1 <span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> t2 : T2</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> (t1,t2) : T1*T2</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> t1 : T11*T12</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> t1.fst : T11</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> t1 : T11*T12</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> t1.snd : T12</td>
<td></td>
</td>
</table></center>
<div class="paragraph"> </div>
The rule <span class="inlinecode"><span class="id" type="var">T_Pair</span></span> says that <span class="inlinecode">(<span class="id" type="var">t1</span>,<span class="id" type="var">t2</span>)</span> has type <span class="inlinecode"><span class="id" type="var">T1</span>*<span class="id" type="var">T2</span></span> if <span class="inlinecode"><span class="id" type="var">t1</span></span> has
type <span class="inlinecode"><span class="id" type="var">T1</span></span> and <span class="inlinecode"><span class="id" type="var">t2</span></span> has type <span class="inlinecode"><span class="id" type="var">T2</span></span>. Conversely, the rules <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">t1</span></span> has a product type
<span class="inlinecode"><span class="id" type="var">T11</span>*<span class="id" type="var">T12</span></span> (i.e., if it will evaluate to a pair), then the types of
the projections from this pair are <span class="inlinecode"><span class="id" type="var">T11</span></span> and <span class="inlinecode"><span class="id" type="var">T12</span></span>.
<div class="paragraph"> </div>
<a name="lab600"></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 result values of computations —
indeed, <span class="inlinecode"><span class="id" type="var">unit</span></span> is the <i>only</i> possible result of evaluating 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
various sorts of <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="lab601"></a><h2 class="section">Sums</h2>
<div class="paragraph"> </div>
<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>,
which describes a set of values drawn from exactly two given types, e.g.
<pre>
Nat + Bool
</pre>
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">v</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
<div class="paragraph"> </div>
<pre>
inl : Nat -> Nat + Bool
inr : Bool -> Nat + Bool
</pre>
<div class="paragraph"> </div>
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. This allows us to give
them their own special typing rules.)
<div class="paragraph"> </div>
In general, the elements of a type <span class="inlinecode"><span class="id" type="var">T1</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">T2</span></span> consist of the
elements of <span class="inlinecode"><span class="id" type="var">T1</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">T2</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 =
\x:Nat. \y:Nat.
if iszero y then
inr Nat unit
else
inl String ...
</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>:
<div class="paragraph"> </div>
<div class="paragraph"> </div>
<pre>
getNat =
\x:Nat+Bool.
case x of
inl n => n
| inr b => if b then 1 else 0
</pre>
<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>
Evaluation:
<div class="paragraph"> </div>
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t1 <span style="font-family: arial;">⇒</span> t1'</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 t1 <span style="font-family: arial;">⇒</span> inl T t1'</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t1 <span style="font-family: arial;">⇒</span> t1'</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 t1 <span style="font-family: arial;">⇒</span> inr T t1'</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t0 <span style="font-family: arial;">⇒</span> t0'</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 t0 of inl x1 => t1 | inr x2 => t2 <span style="font-family: arial;">⇒</span></td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule">case t0' of inl x1 => t1 | inr x2 => t2</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 v0) of inl x1 => t1 | inr x2 => t2</td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: arial;">⇒</span> [x1:=v0]t1</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 v0) of inl x1 => t1 | inr x2 => t2</td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: arial;">⇒</span> [x2:=v0]t2</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> t1 : T1</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 T2 t1 : T1 + T2</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> t1 : T2</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 T1 t1 : T1 + T2</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> t0 : T1+T2</td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> , x1:T1 <span style="font-family: arial;">⊢</span> t1 : T</td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> , x2:T2 <span style="font-family: arial;">⊢</span> t2 : 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 t0 of inl x1 => t1 | inr x2 => t2 : T</td>
<td></td>
</td>
</table></center>
<div class="paragraph"> </div>
The reason for the type annotation on the <span class="inlinecode"><span class="id" type="var">inl</span></span> and <span class="inlinecode"><span class="id" type="var">inr</span></span> forms
has to do with uniqueness of types. 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">t1</span></span> is an element of type <span class="inlinecode"><span class="id" type="var">T1</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">t1</span></span> is an element of <span class="inlinecode"><span class="id" type="var">T1</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">T2</span></span> for <i>any</i>
type T2. 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 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 rather heavyweight for programmers (and so
real languages adopt other solutions), but it is easy to
understand and formalize.
<div class="paragraph"> </div>
<a name="lab602"></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>
\x:List Nat.
lcase x of nil -> 0
| a::x' -> lcase x' of nil -> a
| b::x'' -> a+b
</pre>
<div class="paragraph"> </div>
While we say that <span class="inlinecode"><span class="id" type="var">cons</span></span> <span class="inlinecode"><span class="id" type="var">v1</span></span> <span class="inlinecode"><span class="id" type="var">v2</span></span> is a value, we really mean that
<span class="inlinecode"><span class="id" type="var">v2</span></span> should also be a list — we'll have to enforce this in the
formal definition of value.
<div class="paragraph"> </div>
<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>
Reduction:
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t1 <span style="font-family: arial;">⇒</span> t1'</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 t1 t2 <span style="font-family: arial;">⇒</span> cons t1' t2</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t2 <span style="font-family: arial;">⇒</span> t2'</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 v1 t2 <span style="font-family: arial;">⇒</span> cons v1 t2'</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t1 <span style="font-family: arial;">⇒</span> t1'</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 t1 of nil <span style="font-family: arial;">→</span> t2 | xh::xt <span style="font-family: arial;">→</span> t3) <span style="font-family: arial;">⇒</span></td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule">(lcase t1' of nil <span style="font-family: arial;">→</span> t2 | xh::xt <span style="font-family: arial;">→</span> t3)</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> t2 | xh::xt <span style="font-family: arial;">→</span> t3)</td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: arial;">⇒</span> t2</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> t2 | xh::xt <span style="font-family: arial;">→</span> t3)</td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: arial;">⇒</span> [xh:=vh,xt:=vt]t3</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> t1 : T <span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> t2 : 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 t1 t2: 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> t1 : List T1</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> t2 : T</td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> , h:T1, t:List T1 <span style="font-family: arial;">⊢</span> t3 : 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 t1 of nil <span style="font-family: arial;">→</span> t2 | h::t <span style="font-family: arial;">→</span> t3) : T</td>
<td></td>
</td>
</table></center>
<div class="paragraph"> </div>
<a name="lab603"></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 = \x:Nat.
if x=0 then 1 else x * (fact (pred x)))
</pre>
But this would require quite a bit of work to formalize: 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 lazily during reduction.
<pre>
fact =
fix
(\f:Nat->Nat.
\x: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">fact</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> 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">t1 <span style="font-family: arial;">⇒</span> t1'</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 t1 <span style="font-family: arial;">⇒</span> fix t1'</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">F = \xf:T1.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]t2</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> t1 : T1->T1</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 t1 : T1</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