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<title>Stlc: The Simply Typed Lambda-Calculus</title>
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<h1 class="libtitle">Stlc<span class="subtitle">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: 2012-07-25 16:43:16 -0400 (Wed, 25 Jul 2012) $ *)</span><br/>
<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Export</span> <span class="id" type="keyword">Types</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab548"></a><h1 class="section">The Simply Typed Lambda-Calculus</h1>
<div class="paragraph"> </div>
The simply typed lambda-calculus (STLC) is a tiny core calculus
embodying the key concept of <i>functional abstraction</i>, which shows
up in pretty much every real-world programming language in some
form (functions, procedures, methods, etc.).
<div class="paragraph"> </div>
We will follow exactly the same pattern as in the previous
chapter when formalizing this calculus (syntax, small-step
semantics, typing rules) and its main properties (progress and
preservation). The new technical challenges (which will take some
work to deal with) all arise from the mechanisms of <i>variable
binding</i> and <i>substitution</i>.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab549"></a><h2 class="section">Overview</h2>
<div class="paragraph"> </div>
The STLC is built on some collection of <i>base types</i> — booleans,
numbers, strings, etc. The exact choice of base types doesn't
matter — the construction of the language and its theoretical
properties work out pretty much the same — so for the sake of
brevity let's take just <span class="inlinecode"><span class="id" type="var">Bool</span></span> for the moment. At the end of the
chapter we'll see how to add more base types, and in later
chapters we'll enrich the pure STLC with other useful constructs
like pairs, records, subtyping, and mutable state.
<div class="paragraph"> </div>
Starting from the booleans, we add three things:
<div class="paragraph"> </div>
<ul class="doclist">
<li> variables
</li>
<li> function abstractions
</li>
<li> application
</li>
</ul>
<div class="paragraph"> </div>
This gives us the following collection of abstract syntax
constructors (written out here in informal BNF notation — we'll
formalize it below).
<div class="paragraph"> </div>
<div class="paragraph"> </div>
Informal concrete syntax:
<pre>
t ::= x variable
| \x:T.t1 abstraction
| t1 t2 application
| true constant true
| false constant false
| if t1 then t2 else t3 conditional
</pre>
<div class="paragraph"> </div>
The <span class="inlinecode">\</span> symbol in a function abstraction <span class="inlinecode">\<span class="id" type="var">x</span>:<span class="id" type="var">T.t1</span></span> is often
written as a greek "lambda" (hence the name of the calculus). The
variable <span class="inlinecode"><span class="id" type="var">x</span></span> is called the <i>parameter</i> to the function; the term
<span class="inlinecode"><span class="id" type="var">t1</span></span> is its <i>body</i>. The annotation <span class="inlinecode">:<span class="id" type="var">T</span></span> specifies the type of
arguments that the function can be applied to.
<div class="paragraph"> </div>
Some examples:
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode">\<span class="id" type="var">x</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">x</span></span>
<div class="paragraph"> </div>
The identity function for booleans.
<div class="paragraph"> </div>
</li>
<li> <span class="inlinecode">(\<span class="id" type="var">x</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">x</span>)</span> <span class="inlinecode"><span class="id" type="var">true</span></span>
<div class="paragraph"> </div>
The identity function for booleans, applied to the boolean <span class="inlinecode"><span class="id" type="var">true</span></span>.
<div class="paragraph"> </div>
</li>
<li> <span class="inlinecode">\<span class="id" type="var">x</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="keyword">if</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="keyword">then</span></span> <span class="inlinecode"><span class="id" type="var">false</span></span> <span class="inlinecode"><span class="id" type="keyword">else</span></span> <span class="inlinecode"><span class="id" type="var">true</span></span>
<div class="paragraph"> </div>
The boolean "not" function.
<div class="paragraph"> </div>
</li>
<li> <span class="inlinecode">\<span class="id" type="var">x</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">true</span></span>
<div class="paragraph"> </div>
The constant function that takes every (boolean) argument to
<span class="inlinecode"><span class="id" type="var">true</span></span>.
<div class="paragraph"> </div>
</li>
<li> <span class="inlinecode">\<span class="id" type="var">x</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode">\<span class="id" type="var">y</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">x</span></span>
<div class="paragraph"> </div>
A two-argument function that takes two booleans and returns
the first one. (Note that, as in Coq, a two-argument function
is really a one-argument function whose body is also a
one-argument function.)
<div class="paragraph"> </div>
</li>
<li> <span class="inlinecode">(\<span class="id" type="var">x</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode">\<span class="id" type="var">y</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">x</span>)</span> <span class="inlinecode"><span class="id" type="var">false</span></span> <span class="inlinecode"><span class="id" type="var">true</span></span>
<div class="paragraph"> </div>
A two-argument function that takes two booleans and returns
the first one, applied to the booleans <span class="inlinecode"><span class="id" type="var">false</span></span> and <span class="inlinecode"><span class="id" type="var">true</span></span>.
Note that, as in Coq, application associates to the left —
i.e., this expression is parsed as <span class="inlinecode">((\<span class="id" type="var">x</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode">\<span class="id" type="var">y</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">x</span>)</span>
<span class="inlinecode"><span class="id" type="var">false</span>)</span> <span class="inlinecode"><span class="id" type="var">true</span></span>.
<div class="paragraph"> </div>
</li>
<li> <span class="inlinecode">\<span class="id" type="var">f</span>:<span class="id" type="var">Bool</span><span style="font-family: arial;">→</span><span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">f</span></span> <span class="inlinecode">(<span class="id" type="var">f</span></span> <span class="inlinecode"><span class="id" type="var">true</span>)</span>
<div class="paragraph"> </div>
A higher-order function that takes a <i>function</i> <span class="inlinecode"><span class="id" type="var">f</span></span> (from
booleans to booleans) as an argument, applies <span class="inlinecode"><span class="id" type="var">f</span></span> to <span class="inlinecode"><span class="id" type="var">true</span></span>,
and applies <span class="inlinecode"><span class="id" type="var">f</span></span> again to the result.
<div class="paragraph"> </div>
</li>
<li> <span class="inlinecode">(\<span class="id" type="var">f</span>:<span class="id" type="var">Bool</span><span style="font-family: arial;">→</span><span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">f</span></span> <span class="inlinecode">(<span class="id" type="var">f</span></span> <span class="inlinecode"><span class="id" type="var">true</span>))</span> <span class="inlinecode">(\<span class="id" type="var">x</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">false</span>)</span>
<div class="paragraph"> </div>
The same higher-order function, applied to the constantly
<span class="inlinecode"><span class="id" type="var">false</span></span> function.
</li>
</ul>
<div class="paragraph"> </div>
As the last several examples show, the STLC is a language of
<i>higher-order</i> functions: we can write down functions that take
other functions as arguments and/or return other functions as
results.
<div class="paragraph"> </div>
Another point to note is that the STLC doesn't provide any
primitive syntax for defining <i>named</i> functions — all functions
are "anonymous." We'll see in chapter <span class="inlinecode"><span class="id" type="var">MoreStlc</span></span> that it is easy
to add named functions to what we've got — indeed, the
fundamental naming and binding mechanisms are exactly the same.
<div class="paragraph"> </div>
The <i>types</i> of the STLC include <span class="inlinecode"><span class="id" type="var">Bool</span></span>, which classifies the
boolean constants <span class="inlinecode"><span class="id" type="var">true</span></span> and <span class="inlinecode"><span class="id" type="var">false</span></span> as well as more complex
computations that yield booleans, plus <i>arrow types</i> that classify
functions.
<div class="paragraph"> </div>
<pre>
T ::= Bool
| T1 -> T2
</pre>
For example:
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode">\<span class="id" type="var">x</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">false</span></span> has type <span class="inlinecode"><span class="id" type="var">Bool</span><span style="font-family: arial;">→</span><span class="id" type="var">Bool</span></span>
<div class="paragraph"> </div>
</li>
<li> <span class="inlinecode">\<span class="id" type="var">x</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">x</span></span> has type <span class="inlinecode"><span class="id" type="var">Bool</span><span style="font-family: arial;">→</span><span class="id" type="var">Bool</span></span>
<div class="paragraph"> </div>
</li>
<li> <span class="inlinecode">(\<span class="id" type="var">x</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">x</span>)</span> <span class="inlinecode"><span class="id" type="var">true</span></span> has type <span class="inlinecode"><span class="id" type="var">Bool</span></span>
<div class="paragraph"> </div>
</li>
<li> <span class="inlinecode">\<span class="id" type="var">x</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode">\<span class="id" type="var">y</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">x</span></span> has type <span class="inlinecode"><span class="id" type="var">Bool</span><span style="font-family: arial;">→</span><span class="id" type="var">Bool</span><span style="font-family: arial;">→</span><span class="id" type="var">Bool</span></span> (i.e. <span class="inlinecode"><span class="id" type="var">Bool</span></span> <span class="inlinecode"><span style="font-family: arial;">→</span></span> <span class="inlinecode">(<span class="id" type="var">Bool</span><span style="font-family: arial;">→</span><span class="id" type="var">Bool</span>)</span>)
<div class="paragraph"> </div>
</li>
<li> <span class="inlinecode">(\<span class="id" type="var">x</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode">\<span class="id" type="var">y</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">x</span>)</span> <span class="inlinecode"><span class="id" type="var">false</span></span> has type <span class="inlinecode"><span class="id" type="var">Bool</span><span style="font-family: arial;">→</span><span class="id" type="var">Bool</span></span>
<div class="paragraph"> </div>
</li>
<li> <span class="inlinecode">(\<span class="id" type="var">x</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode">\<span class="id" type="var">y</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">x</span>)</span> <span class="inlinecode"><span class="id" type="var">false</span></span> <span class="inlinecode"><span class="id" type="var">true</span></span> has type <span class="inlinecode"><span class="id" type="var">Bool</span></span>
<div class="paragraph"> </div>
</li>
<li> <span class="inlinecode">\<span class="id" type="var">f</span>:<span class="id" type="var">Bool</span><span style="font-family: arial;">→</span><span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">f</span></span> <span class="inlinecode">(<span class="id" type="var">f</span></span> <span class="inlinecode"><span class="id" type="var">true</span>)</span> has type <span class="inlinecode">(<span class="id" type="var">Bool</span><span style="font-family: arial;">→</span><span class="id" type="var">Bool</span>)</span> <span class="inlinecode"><span style="font-family: arial;">→</span></span> <span class="inlinecode"><span class="id" type="var">Bool</span></span>
<div class="paragraph"> </div>
</li>
<li> <span class="inlinecode">(\<span class="id" type="var">f</span>:<span class="id" type="var">Bool</span><span style="font-family: arial;">→</span><span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">f</span></span> <span class="inlinecode">(<span class="id" type="var">f</span></span> <span class="inlinecode"><span class="id" type="var">true</span>))</span> <span class="inlinecode">(\<span class="id" type="var">x</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">false</span>)</span> has type <span class="inlinecode"><span class="id" type="var">Bool</span></span>
</li>
</ul>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab550"></a><h2 class="section">Syntax</h2>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Module</span> <span class="id" type="var">STLC</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab551"></a><h3 class="section">Types</h3>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">ty</span> : <span class="id" type="keyword">Type</span> := <br/>
| <span class="id" type="var">TBool</span> : <span class="id" type="var">ty</span> <br/>
| <span class="id" type="var">TArrow</span> : <span class="id" type="var">ty</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ty</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ty</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab552"></a><h3 class="section">Terms</h3>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">tm</span> : <span class="id" type="keyword">Type</span> :=<br/>
| <span class="id" type="var">tvar</span> : <span class="id" type="var">id</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span><br/>
| <span class="id" type="var">tapp</span> : <span class="id" type="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span><br/>
| <span class="id" type="var">tabs</span> : <span class="id" type="var">id</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ty</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span><br/>
| <span class="id" type="var">ttrue</span> : <span class="id" type="var">tm</span><br/>
| <span class="id" type="var">tfalse</span> : <span class="id" type="var">tm</span><br/>
| <span class="id" type="var">tif</span> : <span class="id" type="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span>.<br/>
<br/>
<div class="togglescript" id="proofcontrol1" onclick="toggleDisplay('proof1');toggleDisplay('proofcontrol1')"><span class="show"></span></div>
<div class="proofscript" id="proof1" onclick="toggleDisplay('proof1');toggleDisplay('proofcontrol1')">
<span class="id" type="keyword">Tactic Notation</span> "t_cases" <span class="id" type="var">tactic</span>(<span class="id" type="var">first</span>) <span class="id" type="var">ident</span>(<span class="id" type="var">c</span>) :=<br/>
<span class="id" type="var">first</span>;<br/>
[ <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "tvar" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "tapp" <br/>
| <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "tabs" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ttrue" <br/>
| <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "tfalse" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "tif" ].<br/>
</div>
<br/>
</div>
<div class="doc">
Note that an abstraction <span class="inlinecode">\<span class="id" type="var">x</span>:<span class="id" type="var">T.t</span></span> (formally, <span class="inlinecode"><span class="id" type="var">tabs</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">T</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span>) is
always annotated with the type <span class="inlinecode"><span class="id" type="var">T</span></span> of its parameter, in contrast
to Coq (and other functional languages like ML, Haskell, etc.),
which use <i>type inference</i> to fill in missing annotations. We're
not considering type inference here, to keep things simple.
<div class="paragraph"> </div>
Some examples...
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">x</span> := (<span class="id" type="var">Id</span> 0).<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">y</span> := (<span class="id" type="var">Id</span> 1).<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">z</span> := (<span class="id" type="var">Id</span> 2).<br/>
<div class="togglescript" id="proofcontrol2" onclick="toggleDisplay('proof2');toggleDisplay('proofcontrol2')"><span class="show"></span></div>
<div class="proofscript" id="proof2" onclick="toggleDisplay('proof2');toggleDisplay('proofcontrol2')">
<span class="id" type="keyword">Hint</span> <span class="id" type="keyword">Unfold</span> <span class="id" type="var">x</span>.<br/>
<span class="id" type="keyword">Hint</span> <span class="id" type="keyword">Unfold</span> <span class="id" type="var">y</span>.<br/>
<span class="id" type="keyword">Hint</span> <span class="id" type="keyword">Unfold</span> <span class="id" type="var">z</span>.<br/>
</div>
<br/>
</div>
<div class="doc">
<span class="inlinecode"><span class="id" type="var">idB</span></span> <span class="inlinecode">=</span> <span class="inlinecode">\<span class="id" type="var">x</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">x</span></span>
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Notation</span> <span class="id" type="var">idB</span> := <br/>
(<span class="id" type="var">tabs</span> <span class="id" type="var">x</span> <span class="id" type="var">TBool</span> (<span class="id" type="var">tvar</span> <span class="id" type="var">x</span>)).<br/>
<br/>
</div>
<div class="doc">
<span class="inlinecode"><span class="id" type="var">idBB</span></span> <span class="inlinecode">=</span> <span class="inlinecode">\<span class="id" type="var">x</span>:<span class="id" type="var">Bool</span><span style="font-family: arial;">→</span><span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">x</span></span>
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Notation</span> <span class="id" type="var">idBB</span> := <br/>
(<span class="id" type="var">tabs</span> <span class="id" type="var">x</span> (<span class="id" type="var">TArrow</span> <span class="id" type="var">TBool</span> <span class="id" type="var">TBool</span>) (<span class="id" type="var">tvar</span> <span class="id" type="var">x</span>)).<br/>
<br/>
</div>
<div class="doc">
<span class="inlinecode"><span class="id" type="var">idBBBB</span></span> <span class="inlinecode">=</span> <span class="inlinecode">\<span class="id" type="var">x</span>:(<span class="id" type="var">Bool</span><span style="font-family: arial;">→</span><span class="id" type="var">Bool</span>)->(<span class="id" type="var">Bool</span><span style="font-family: arial;">→</span><span class="id" type="var">Bool</span>).</span> <span class="inlinecode"><span class="id" type="var">x</span></span>
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Notation</span> <span class="id" type="var">idBBBB</span> :=<br/>
(<span class="id" type="var">tabs</span> <span class="id" type="var">x</span> (<span class="id" type="var">TArrow</span> (<span class="id" type="var">TArrow</span> <span class="id" type="var">TBool</span> <span class="id" type="var">TBool</span>) <br/>
(<span class="id" type="var">TArrow</span> <span class="id" type="var">TBool</span> <span class="id" type="var">TBool</span>)) <br/>
(<span class="id" type="var">tvar</span> <span class="id" type="var">x</span>)).<br/>
<br/>
</div>
<div class="doc">
<span class="inlinecode"><span class="id" type="var">k</span></span> <span class="inlinecode">=</span> <span class="inlinecode">\<span class="id" type="var">x</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode">\<span class="id" type="var">y</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">x</span></span>
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Notation</span> <span class="id" type="var">k</span> := (<span class="id" type="var">tabs</span> <span class="id" type="var">x</span> <span class="id" type="var">TBool</span> (<span class="id" type="var">tabs</span> <span class="id" type="var">y</span> <span class="id" type="var">TBool</span> (<span class="id" type="var">tvar</span> <span class="id" type="var">x</span>))).<br/>
<br/>
</div>
<div class="doc">
(We write these as <span class="inlinecode"><span class="id" type="keyword">Notation</span></span>s rather than <span class="inlinecode"><span class="id" type="keyword">Definition</span></span>s to make
things easier for <span class="inlinecode"><span class="id" type="tactic">auto</span></span>.)
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab553"></a><h2 class="section">Operational Semantics</h2>
<div class="paragraph"> </div>
To define the small-step semantics of STLC terms, we begin — as
always — by defining the set of values. Next, we define the
critical notions of <i>free variables</i> and <i>substitution</i>, which are
used in the reduction rule for application expressions. And
finally we give the small-step relation itself.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab554"></a><h3 class="section">Values</h3>
<div class="paragraph"> </div>
To define the values of the STLC, we have a few cases to consider.
<div class="paragraph"> </div>
First, for the boolean part of the language, the situation is
clear: <span class="inlinecode"><span class="id" type="var">true</span></span> and <span class="inlinecode"><span class="id" type="var">false</span></span> are the only values. (An <span class="inlinecode"><span class="id" type="keyword">if</span></span>
expression is never a value.)
<div class="paragraph"> </div>
Second, an application is clearly not a value: It represents a
function being invoked on some argument, which clearly still has
work left to do.
<div class="paragraph"> </div>
Third, for abstractions, we have a choice:
<div class="paragraph"> </div>
<ul class="doclist">
<li> We can say that <span class="inlinecode">\<span class="id" type="var">x</span>:<span class="id" type="var">T.t1</span></span> is a value only when <span class="inlinecode"><span class="id" type="var">t1</span></span> is a
value — i.e., only if the function's body has been
reduced (as much as it can be without knowing what argument it
is going to be applied to).
<div class="paragraph"> </div>
</li>
<li> Or we can say that <span class="inlinecode">\<span class="id" type="var">x</span>:<span class="id" type="var">T.t1</span></span> is always a value, no matter
whether <span class="inlinecode"><span class="id" type="var">t1</span></span> is one or not — in other words, we can say that
reduction stops at abstractions.
</li>
</ul>
<div class="paragraph"> </div>
Coq (in its built-in functional programming langauge) makes the
first choice — for example,
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="keyword">Eval</span> <span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> (<span class="id" type="keyword">fun</span> <span class="id" type="var">x</span>:<span class="id" type="var">bool</span> => 3 + 4)
<div class="paragraph"> </div>
</div>
yields <span class="inlinecode"><span class="id" type="keyword">fun</span></span> <span class="inlinecode"><span class="id" type="var">x</span>:<span class="id" type="var">bool</span></span> <span class="inlinecode">=></span> <span class="inlinecode">7</span>. But most real-world functional
programming languages make the second choice — reduction of a
function's body only begins when the function is actually applied
to an argument. We also make the second choice here.
<div class="paragraph"> </div>
Finally, having made the choice not to reduce under abstractions,
we don't need to worry about whether variables are values, since
we'll always be reducing programs "from the outside in," and that
means the <span class="inlinecode"><span class="id" type="var">step</span></span> relation will always be working with closed
terms (ones with no free variables).
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">value</span> : <span class="id" type="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
| <span class="id" type="var">v_abs</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">x</span> <span class="id" type="var">T</span> <span class="id" type="var">t</span>,<br/>
<span class="id" type="var">value</span> (<span class="id" type="var">tabs</span> <span class="id" type="var">x</span> <span class="id" type="var">T</span> <span class="id" type="var">t</span>)<br/>
| <span class="id" type="var">t_true</span> : <br/>
<span class="id" type="var">value</span> <span class="id" type="var">ttrue</span><br/>
| <span class="id" type="var">t_false</span> : <br/>
<span class="id" type="var">value</span> <span class="id" type="var">tfalse</span>.<br/>
<br/>
<span class="id" type="keyword">Hint</span> <span class="id" type="var">Constructors</span> <span class="id" type="var">value</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab555"></a><h3 class="section">Free Variables and Substitution</h3>
<div class="paragraph"> </div>
Now we come to the heart of the matter: the operation of
substituting one term for a variable in another term.
<div class="paragraph"> </div>
This operation will be used below to define the operational
semantics of function application, where we will need to
substitute the argument term for the function parameter in the
function's body. For example, we reduce
<div class="paragraph"> </div>
<div class="code code-tight">
(\<span class="id" type="var">x</span>:<span class="id" type="var">Bool</span>. <span class="id" type="keyword">if</span> <span class="id" type="var">x</span> <span class="id" type="keyword">then</span> <span class="id" type="var">true</span> <span class="id" type="keyword">else</span> <span class="id" type="var">x</span>) <span class="id" type="var">false</span>
<div class="paragraph"> </div>
</div>
to <span class="inlinecode"><span class="id" type="var">false</span></span> by substituting <span class="inlinecode"><span class="id" type="var">false</span></span> for the parameter <span class="inlinecode"><span class="id" type="var">x</span></span> in the
body of the function. In general, we need to be able to
substitute some given term <span class="inlinecode"><span class="id" type="var">s</span></span> for occurrences of some variable
<span class="inlinecode"><span class="id" type="var">x</span></span> in another term <span class="inlinecode"><span class="id" type="var">t</span></span>. In informal discussions, this is usually
written <span class="inlinecode"></span> <span class="inlinecode">[<span class="id" type="var">x</span>:=<span class="id" type="var">s</span>]<span class="id" type="var">t</span></span> <span class="inlinecode"></span> and pronounced "substitute <span class="inlinecode"><span class="id" type="var">x</span></span> with <span class="inlinecode"><span class="id" type="var">s</span></span> in <span class="inlinecode"><span class="id" type="var">t</span></span>."
<div class="paragraph"> </div>
Here are some examples:
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode">[<span class="id" type="var">x</span>:=<span class="id" type="var">true</span>]</span> <span class="inlinecode">(<span class="id" type="keyword">if</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="keyword">then</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="keyword">else</span></span> <span class="inlinecode"><span class="id" type="var">false</span>)</span> yields <span class="inlinecode"><span class="id" type="keyword">if</span></span> <span class="inlinecode"><span class="id" type="var">true</span></span> <span class="inlinecode"><span class="id" type="keyword">then</span></span> <span class="inlinecode"><span class="id" type="var">true</span></span> <span class="inlinecode"><span class="id" type="keyword">else</span></span> <span class="inlinecode"><span class="id" type="var">false</span></span>
<div class="paragraph"> </div>
</li>
<li> <span class="inlinecode">[<span class="id" type="var">x</span>:=<span class="id" type="var">true</span>]</span> <span class="inlinecode"><span class="id" type="var">x</span></span> yields <span class="inlinecode"><span class="id" type="var">true</span></span>
<div class="paragraph"> </div>
</li>
<li> <span class="inlinecode">[<span class="id" type="var">x</span>:=<span class="id" type="var">true</span>]</span> <span class="inlinecode">(<span class="id" type="keyword">if</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="keyword">then</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="keyword">else</span></span> <span class="inlinecode"><span class="id" type="var">y</span>)</span> yields <span class="inlinecode"><span class="id" type="keyword">if</span></span> <span class="inlinecode"><span class="id" type="var">true</span></span> <span class="inlinecode"><span class="id" type="keyword">then</span></span> <span class="inlinecode"><span class="id" type="var">true</span></span> <span class="inlinecode"><span class="id" type="keyword">else</span></span> <span class="inlinecode"><span class="id" type="var">y</span></span>
<div class="paragraph"> </div>
</li>
<li> <span class="inlinecode">[<span class="id" type="var">x</span>:=<span class="id" type="var">true</span>]</span> <span class="inlinecode"><span class="id" type="var">y</span></span> yields <span class="inlinecode"><span class="id" type="var">y</span></span>
<div class="paragraph"> </div>
</li>
<li> <span class="inlinecode">[<span class="id" type="var">x</span>:=<span class="id" type="var">true</span>]</span> <span class="inlinecode"><span class="id" type="var">false</span></span> yields <span class="inlinecode"><span class="id" type="var">false</span></span> (vacuous substitution)
<div class="paragraph"> </div>
</li>
<li> <span class="inlinecode">[<span class="id" type="var">x</span>:=<span class="id" type="var">true</span>]</span> <span class="inlinecode">(\<span class="id" type="var">y</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="keyword">if</span></span> <span class="inlinecode"><span class="id" type="var">y</span></span> <span class="inlinecode"><span class="id" type="keyword">then</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="keyword">else</span></span> <span class="inlinecode"><span class="id" type="var">false</span>)</span> yields <span class="inlinecode">\<span class="id" type="var">y</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="keyword">if</span></span> <span class="inlinecode"><span class="id" type="var">y</span></span> <span class="inlinecode"><span class="id" type="keyword">then</span></span> <span class="inlinecode"><span class="id" type="var">true</span></span> <span class="inlinecode"><span class="id" type="keyword">else</span></span> <span class="inlinecode"><span class="id" type="var">false</span></span>
</li>
<li> <span class="inlinecode">[<span class="id" type="var">x</span>:=<span class="id" type="var">true</span>]</span> <span class="inlinecode">(\<span class="id" type="var">y</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">x</span>)</span> yields <span class="inlinecode">\<span class="id" type="var">y</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">true</span></span>
<div class="paragraph"> </div>
</li>
<li> <span class="inlinecode">[<span class="id" type="var">x</span>:=<span class="id" type="var">true</span>]</span> <span class="inlinecode">(\<span class="id" type="var">y</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">y</span>)</span> yields <span class="inlinecode">\<span class="id" type="var">y</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">y</span></span>
<div class="paragraph"> </div>
</li>
<li> <span class="inlinecode">[<span class="id" type="var">x</span>:=<span class="id" type="var">true</span>]</span> <span class="inlinecode">(\<span class="id" type="var">x</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">x</span>)</span> yields <span class="inlinecode">\<span class="id" type="var">x</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">x</span></span>
</li>
</ul>
<div class="paragraph"> </div>
The last example is very important: substituting <span class="inlinecode"><span class="id" type="var">x</span></span> with <span class="inlinecode"><span class="id" type="var">true</span></span> in
<span class="inlinecode">\<span class="id" type="var">x</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">x</span></span> does <i>not</i> yield <span class="inlinecode">\<span class="id" type="var">x</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">true</span></span>! The reason for
this is that the <span class="inlinecode"><span class="id" type="var">x</span></span> in the body of <span class="inlinecode">\<span class="id" type="var">x</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">x</span></span> is <i>bound</i> by the
abstraction: it is a new, local name that just happens to be
spelled the same as some global name <span class="inlinecode"><span class="id" type="var">x</span></span>.
<div class="paragraph"> </div>
Here is the definition, informally...
<div class="paragraph"> </div>
<div class="code code-tight">
[<span class="id" type="var">x</span>:=<span class="id" type="var">s</span>]<span class="id" type="var">x</span> = <span class="id" type="var">s</span><br/>
[<span class="id" type="var">x</span>:=<span class="id" type="var">s</span>]<span class="id" type="var">y</span> = <span class="id" type="var">y</span> <span class="id" type="keyword">if</span> <span class="id" type="var">x</span> <> <span class="id" type="var">y</span><br/>
[<span class="id" type="var">x</span>:=<span class="id" type="var">s</span>](\<span class="id" type="var">x</span>:<span class="id" type="var">T11.t12</span>) = \<span class="id" type="var">x</span>:<span class="id" type="var">T11</span>. <span class="id" type="var">t12</span> <br/>
[<span class="id" type="var">x</span>:=<span class="id" type="var">s</span>](\<span class="id" type="var">y</span>:<span class="id" type="var">T11.t12</span>) = \<span class="id" type="var">y</span>:<span class="id" type="var">T11</span>. [<span class="id" type="var">x</span>:=<span class="id" type="var">s</span>]<span class="id" type="var">t12</span> <span class="id" type="keyword">if</span> <span class="id" type="var">x</span> <> <span class="id" type="var">y</span><br/>
[<span class="id" type="var">x</span>:=<span class="id" type="var">s</span>](<span class="id" type="var">t1</span> <span class="id" type="var">t2</span>) = ([<span class="id" type="var">x</span>:=<span class="id" type="var">s</span>]<span class="id" type="var">t1</span>) ([<span class="id" type="var">x</span>:=<span class="id" type="var">s</span>]<span class="id" type="var">t2</span>) <br/>
[<span class="id" type="var">x</span>:=<span class="id" type="var">s</span>]<span class="id" type="var">true</span> = <span class="id" type="var">true</span><br/>
[<span class="id" type="var">x</span>:=<span class="id" type="var">s</span>]<span class="id" type="var">false</span> = <span class="id" type="var">false</span><br/>
[<span class="id" type="var">x</span>:=<span class="id" type="var">s</span>](<span class="id" type="keyword">if</span> <span class="id" type="var">t1</span> <span class="id" type="keyword">then</span> <span class="id" type="var">t2</span> <span class="id" type="keyword">else</span> <span class="id" type="var">t3</span>) = <br/>
<span class="id" type="keyword">if</span> [<span class="id" type="var">x</span>:=<span class="id" type="var">s</span>]<span class="id" type="var">t1</span> <span class="id" type="keyword">then</span> [<span class="id" type="var">x</span>:=<span class="id" type="var">s</span>]<span class="id" type="var">t2</span> <span class="id" type="keyword">else</span> [<span class="id" type="var">x</span>:=<span class="id" type="var">s</span>]<span class="id" type="var">t3</span>
<div class="paragraph"> </div>
</div>
... and formally:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="tactic">subst</span> (<span class="id" type="var">x</span>:<span class="id" type="var">id</span>) (<span class="id" type="var">s</span>:<span class="id" type="var">tm</span>) (<span class="id" type="var">t</span>:<span class="id" type="var">tm</span>) : <span class="id" type="var">tm</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">t</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">tvar</span> <span class="id" type="var">x'</span> => <br/>
<span class="id" type="keyword">if</span> <span class="id" type="var">beq_id</span> <span class="id" type="var">x</span> <span class="id" type="var">x'</span> <span class="id" type="keyword">then</span> <span class="id" type="var">s</span> <span class="id" type="keyword">else</span> <span class="id" type="var">t</span><br/>
| <span class="id" type="var">tabs</span> <span class="id" type="var">x'</span> <span class="id" type="var">T</span> <span class="id" type="var">t1</span> => <br/>
<span class="id" type="var">tabs</span> <span class="id" type="var">x'</span> <span class="id" type="var">T</span> (<span class="id" type="keyword">if</span> <span class="id" type="var">beq_id</span> <span class="id" type="var">x</span> <span class="id" type="var">x'</span> <span class="id" type="keyword">then</span> <span class="id" type="var">t1</span> <span class="id" type="keyword">else</span> (<span class="id" type="tactic">subst</span> <span class="id" type="var">x</span> <span class="id" type="var">s</span> <span class="id" type="var">t1</span>)) <br/>
| <span class="id" type="var">tapp</span> <span class="id" type="var">t1</span> <span class="id" type="var">t2</span> => <br/>
<span class="id" type="var">tapp</span> (<span class="id" type="tactic">subst</span> <span class="id" type="var">x</span> <span class="id" type="var">s</span> <span class="id" type="var">t1</span>) (<span class="id" type="tactic">subst</span> <span class="id" type="var">x</span> <span class="id" type="var">s</span> <span class="id" type="var">t2</span>)<br/>
| <span class="id" type="var">ttrue</span> => <br/>
<span class="id" type="var">ttrue</span><br/>
| <span class="id" type="var">tfalse</span> => <br/>
<span class="id" type="var">tfalse</span><br/>
| <span class="id" type="var">tif</span> <span class="id" type="var">t1</span> <span class="id" type="var">t2</span> <span class="id" type="var">t3</span> => <br/>
<span class="id" type="var">tif</span> (<span class="id" type="tactic">subst</span> <span class="id" type="var">x</span> <span class="id" type="var">s</span> <span class="id" type="var">t1</span>) (<span class="id" type="tactic">subst</span> <span class="id" type="var">x</span> <span class="id" type="var">s</span> <span class="id" type="var">t2</span>) (<span class="id" type="tactic">subst</span> <span class="id" type="var">x</span> <span class="id" type="var">s</span> <span class="id" type="var">t3</span>)<br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Notation</span> "'[' x ':=' s ']' t" := (<span class="id" type="tactic">subst</span> <span class="id" type="var">x</span> <span class="id" type="var">s</span> <span class="id" type="var">t</span>) (<span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 20).<br/>
<br/>
</div>
<div class="doc">
<i>Technical note</i>: Substitution becomes trickier to define if we
consider the case where <span class="inlinecode"><span class="id" type="var">s</span></span>, the term being substituted for a
variable in some other term, may itself contain free variables.
Since we are only interested here in defining the <span class="inlinecode"><span class="id" type="var">step</span></span> relation
on closed terms (i.e., terms like <span class="inlinecode">\<span class="id" type="var">x</span>:<span class="id" type="var">Bool</span>.</span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">y</span></span> that mention
variables are not bound by some enclosing lambda), we can avoid
this extra complexity here.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab556"></a><h3 class="section">Reduction</h3>
<div class="paragraph"> </div>
The small-step reduction relation for STLC now follows the same
pattern as the ones we have seen before. Intuitively, to reduce a
function application, we first reduce its left-hand side until it
becomes a literal function; then we reduce its right-hand
side (the argument) until it is also a value; and finally we
substitute the argument for the bound variable in the body of the
function. This last rule, written informally as
<div class="paragraph"> </div>
<div class="code code-tight">
(\<span class="id" type="var">x</span>:<span class="id" type="var">T.t12</span>) <span class="id" type="var">v2</span> <span style="font-family: arial;">⇒</span> [<span class="id" type="var">x</span>:=<span class="id" type="var">v2</span>]<span class="id" type="var">t12</span>
<div class="paragraph"> </div>
</div>
is traditionally called "beta-reduction".
<div class="paragraph"> </div>
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">value v2</td>
<td class="infrulenamecol" rowspan="3">
(ST_AppAbs)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">(\x:T.t12) v2 <span style="font-family: arial;">⇒</span> [x:=v2]t12</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_App1)
</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">value v1</td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule">t2 <span style="font-family: arial;">⇒</span> t2'</td>
<td class="infrulenamecol" rowspan="3">
(ST_App2)
</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> ... plus the usual rules for booleans:
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(ST_IfTrue)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">(if true then t1 else t2) <span style="font-family: arial;">⇒</span> t1</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(ST_IfFalse)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">(if false then t1 else t2) <span style="font-family: arial;">⇒</span> 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_If)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">(if t1 then t2 else t3) <span style="font-family: arial;">⇒</span> (if t1' then t2 else t3)</td>
<td></td>
</td>
</table></center>
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Reserved Notation</span> "t1 '<span style="font-family: arial;">⇒</span>' t2" (<span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 40).<br/>
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">step</span> : <span class="id" type="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
| <span class="id" type="var">ST_AppAbs</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">x</span> <span class="id" type="var">T</span> <span class="id" type="var">t12</span> <span class="id" type="var">v2</span>,<br/>
<span class="id" type="var">value</span> <span class="id" type="var">v2</span> <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">tapp</span> (<span class="id" type="var">tabs</span> <span class="id" type="var">x</span> <span class="id" type="var">T</span> <span class="id" type="var">t12</span>) <span class="id" type="var">v2</span>) <span style="font-family: arial;">⇒</span> [<span class="id" type="var">x</span>:=<span class="id" type="var">v2</span>]<span class="id" type="var">t12</span><br/>
| <span class="id" type="var">ST_App1</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t1</span> <span class="id" type="var">t1'</span> <span class="id" type="var">t2</span>,<br/>
<span class="id" type="var">t1</span> <span style="font-family: arial;">⇒</span> <span class="id" type="var">t1'</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">tapp</span> <span class="id" type="var">t1</span> <span class="id" type="var">t2</span> <span style="font-family: arial;">⇒</span> <span class="id" type="var">tapp</span> <span class="id" type="var">t1'</span> <span class="id" type="var">t2</span><br/>
| <span class="id" type="var">ST_App2</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">v1</span> <span class="id" type="var">t2</span> <span class="id" type="var">t2'</span>,<br/>
<span class="id" type="var">value</span> <span class="id" type="var">v1</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">t2</span> <span style="font-family: arial;">⇒</span> <span class="id" type="var">t2'</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">tapp</span> <span class="id" type="var">v1</span> <span class="id" type="var">t2</span> <span style="font-family: arial;">⇒</span> <span class="id" type="var">tapp</span> <span class="id" type="var">v1</span> <span class="id" type="var">t2'</span><br/>
| <span class="id" type="var">ST_IfTrue</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t1</span> <span class="id" type="var">t2</span>,<br/>
(<span class="id" type="var">tif</span> <span class="id" type="var">ttrue</span> <span class="id" type="var">t1</span> <span class="id" type="var">t2</span>) <span style="font-family: arial;">⇒</span> <span class="id" type="var">t1</span><br/>
| <span class="id" type="var">ST_IfFalse</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t1</span> <span class="id" type="var">t2</span>,<br/>
(<span class="id" type="var">tif</span> <span class="id" type="var">tfalse</span> <span class="id" type="var">t1</span> <span class="id" type="var">t2</span>) <span style="font-family: arial;">⇒</span> <span class="id" type="var">t2</span><br/>
| <span class="id" type="var">ST_If</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t1</span> <span class="id" type="var">t1'</span> <span class="id" type="var">t2</span> <span class="id" type="var">t3</span>,<br/>
<span class="id" type="var">t1</span> <span style="font-family: arial;">⇒</span> <span class="id" type="var">t1'</span> <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">tif</span> <span class="id" type="var">t1</span> <span class="id" type="var">t2</span> <span class="id" type="var">t3</span>) <span style="font-family: arial;">⇒</span> (<span class="id" type="var">tif</span> <span class="id" type="var">t1'</span> <span class="id" type="var">t2</span> <span class="id" type="var">t3</span>)<br/>
<br/>
<span class="id" type="keyword">where</span> "t1 '<span style="font-family: arial;">⇒</span>' t2" := (<span class="id" type="var">step</span> <span class="id" type="var">t1</span> <span class="id" type="var">t2</span>).<br/>
<br/>
<span class="id" type="keyword">Tactic Notation</span> "step_cases" <span class="id" type="var">tactic</span>(<span class="id" type="var">first</span>) <span class="id" type="var">ident</span>(<span class="id" type="var">c</span>) :=<br/>
<span class="id" type="var">first</span>;<br/>
[ <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_AppAbs" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_App1" <br/>
| <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_App2" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_IfTrue" <br/>
| <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_IfFalse" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_If" ].<br/>
<br/>
<span class="id" type="keyword">Hint</span> <span class="id" type="var">Constructors</span> <span class="id" type="var">step</span>.<br/>
<br/>
<span class="id" type="keyword">Notation</span> <span class="id" type="var">multistep</span> := (<span class="id" type="var">multi</span> <span class="id" type="var">step</span>).<br/>
<span class="id" type="keyword">Notation</span> "t1 '<span style="font-family: arial;">⇒*</span>' t2" := (<span class="id" type="var">multistep</span> <span class="id" type="var">t1</span> <span class="id" type="var">t2</span>) (<span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 40).<br/>
<br/>
</div>
<div class="doc">
<a name="lab557"></a><h3 class="section">Examples</h3>
<div class="paragraph"> </div>
Example:
<div class="paragraph"> </div>
<div class="code code-tight">
((\<span class="id" type="var">x</span>:<span class="id" type="var">Bool</span><span style="font-family: arial;">→</span><span class="id" type="var">Bool</span>. <span class="id" type="var">x</span>) (\<span class="id" type="var">x</span>:<span class="id" type="var">Bool</span>. <span class="id" type="var">x</span>)) <span style="font-family: arial;">⇒*</span> (\<span class="id" type="var">x</span>:<span class="id" type="var">Bool</span>. <span class="id" type="var">x</span>)
<div class="paragraph"> </div>
</div>
i.e.
<div class="paragraph"> </div>
<div class="code code-tight">
(<span class="id" type="var">idBB</span> <span class="id" type="var">idB</span>) <span style="font-family: arial;">⇒*</span> <span class="id" type="var">idB</span>
<div class="paragraph"> </div>
</div>
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">step_example1</span> :<br/>
(<span class="id" type="var">tapp</span> <span class="id" type="var">idBB</span> <span class="id" type="var">idB</span>) <span style="font-family: arial;">⇒*</span> <span class="id" type="var">idB</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">multi_step</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">ST_AppAbs</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">v_abs</span>.<br/>
<span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">multi_refl</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
A more automatic proof
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">step_example1'</span> :<br/>
(<span class="id" type="var">tapp</span> <span class="id" type="var">idBB</span> <span class="id" type="var">idB</span>) <span style="font-family: arial;">⇒*</span> <span class="id" type="var">idB</span>.<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="var">normalize</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">step_example2</span> :<br/>
(<span class="id" type="var">tapp</span> <span class="id" type="var">idBB</span> (<span class="id" type="var">tapp</span> <span class="id" type="var">idBB</span> <span class="id" type="var">idB</span>)) <span style="font-family: arial;">⇒*</span> <span class="id" type="var">idB</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">multi_step</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">ST_App2</span>. <span class="id" type="tactic">auto</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">ST_AppAbs</span>. <span class="id" type="tactic">auto</span>.<br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">multi_step</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">ST_AppAbs</span>. <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">auto</span>.<br/>
<span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">multi_refl</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Again, we can use the <span class="inlinecode"><span class="id" type="var">normalize</span></span> tactic from above to simplify
the proof.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">step_example2'</span> :<br/>
(<span class="id" type="var">tapp</span> <span class="id" type="var">idBB</span> (<span class="id" type="var">tapp</span> <span class="id" type="var">idBB</span> <span class="id" type="var">idB</span>)) <span style="font-family: arial;">⇒*</span> <span class="id" type="var">idB</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="var">normalize</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
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<a name="lab558"></a><h4 class="section">Exercise: 2 stars (step_example3)</h4>
Try to do this one both with and without <span class="inlinecode"><span class="id" type="var">normalize</span></span>.
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<span class="id" type="keyword">Lemma</span> <span class="id" type="var">step_example3</span> :<br/>
(<span class="id" type="var">tapp</span> (<span class="id" type="var">tapp</span> <span class="id" type="var">idBBBB</span> <span class="id" type="var">idBB</span>) <span class="id" type="var">idB</span>)<br/>
<span style="font-family: arial;">⇒*</span> <span class="id" type="var">idB</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
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<span class="comment">(* FILL IN HERE *)</span><br/>
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<font size=-2>☐</font>
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<a name="lab559"></a><h2 class="section">Typing</h2>
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<a name="lab560"></a><h3 class="section">Contexts</h3>
<div class="paragraph"> </div>
<i>Question</i>: What is the type of the term "<span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">y</span></span>"?
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<i>Answer</i>: It depends on the types of <span class="inlinecode"><span class="id" type="var">x</span></span> and <span class="inlinecode"><span class="id" type="var">y</span></span>!
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