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Stlc.html
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
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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="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <a class="idref" href="Maps.html#"><span class="id" type="library">Maps</span></a>.<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <a class="idref" href="Smallstep.html#"><span class="id" type="library">Smallstep</span></a>.<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <span class="id" type="keyword">Types</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab557"></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 arise from the mechanisms of
<i>variable binding</i> and <i>substitution</i>. It which will take some
work to deal with these.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab558"></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 much — the construction of the language and its
theoretical properties work out the same no matter what we
choose — 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 boolean constants and conditionals, 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 first in informal BNF notation — we'll
formalize it below).
<div class="paragraph"> </div>
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">t</span> ::= <span class="id" type="var">x</span> <span class="id" type="var">variable</span><br/>
| \<span class="id" type="var">x</span>:<span class="id" type="var">T<sub>1</sub>.t2</span> <span class="id" type="var">abstraction</span><br/>
| <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span> <span class="id" type="var">application</span><br/>
| <span class="id" type="var">true</span> <span class="id" type="var">constant</span> <span class="id" type="var">true</span><br/>
| <span class="id" type="var">false</span> <span class="id" type="var">constant</span> <span class="id" type="var">false</span><br/>
| <span class="id" type="keyword">if</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="keyword">then</span> <span class="id" type="var">t<sub>2</sub></span> <span class="id" type="keyword">else</span> <span class="id" type="var">t<sub>3</sub></span> <span class="id" type="var">conditional</span>
<div class="paragraph"> </div>
</div>
<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<sub>1</sub>.t2</span></span> is generally
written as a Greek letter "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">t<sub>2</sub></span></span> is its <i>body</i>. The annotation <span class="inlinecode">:<span class="id" type="var">T<sub>1</sub></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>.
</li>
</ul>
<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">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. (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>.
<div class="paragraph"> </div>
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>
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>
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">T</span> ::= <span class="id" type="var">Bool</span><br/>
| <span class="id" type="var">T<sub>1</sub></span> <span style="font-family: arial;">→</span> <span class="id" type="var">T<sub>2</sub></span>
<div class="paragraph"> </div>
</div>
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>
</li>
</ul>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab559"></a><h2 class="section">Syntax</h2>
<div class="paragraph"> </div>
We begin by formalizing the syntax of the STLC.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Module</span> <a name="STLC"><span class="id" type="module">STLC</span></a>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab560"></a><h3 class="section">Types</h3>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Inductive</span> <a name="STLC.ty"><span class="id" type="inductive">ty</span></a> : <span class="id" type="keyword">Type</span> :=<br/>
| <a name="STLC.TBool"><span class="id" type="constructor">TBool</span></a> : <a class="idref" href="Stlc.html#ty"><span class="id" type="inductive">ty</span></a><br/>
| <a name="STLC.TArrow"><span class="id" type="constructor">TArrow</span></a> : <a class="idref" href="Stlc.html#ty"><span class="id" type="inductive">ty</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="Stlc.html#ty"><span class="id" type="inductive">ty</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="Stlc.html#ty"><span class="id" type="inductive">ty</span></a>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab561"></a><h3 class="section">Terms</h3>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Inductive</span> <a name="STLC.tm"><span class="id" type="inductive">tm</span></a> : <span class="id" type="keyword">Type</span> :=<br/>
| <a name="STLC.tvar"><span class="id" type="constructor">tvar</span></a> : <a class="idref" href="Maps.html#id"><span class="id" type="inductive">id</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="Stlc.html#tm"><span class="id" type="inductive">tm</span></a><br/>
| <a name="STLC.tapp"><span class="id" type="constructor">tapp</span></a> : <a class="idref" href="Stlc.html#tm"><span class="id" type="inductive">tm</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="Stlc.html#tm"><span class="id" type="inductive">tm</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="Stlc.html#tm"><span class="id" type="inductive">tm</span></a><br/>
| <a name="STLC.tabs"><span class="id" type="constructor">tabs</span></a> : <a class="idref" href="Maps.html#id"><span class="id" type="inductive">id</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="Stlc.html#STLC.ty"><span class="id" type="inductive">ty</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="Stlc.html#tm"><span class="id" type="inductive">tm</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="Stlc.html#tm"><span class="id" type="inductive">tm</span></a><br/>
| <a name="STLC.ttrue"><span class="id" type="constructor">ttrue</span></a> : <a class="idref" href="Stlc.html#tm"><span class="id" type="inductive">tm</span></a><br/>
| <a name="STLC.tfalse"><span class="id" type="constructor">tfalse</span></a> : <a class="idref" href="Stlc.html#tm"><span class="id" type="inductive">tm</span></a><br/>
| <a name="STLC.tif"><span class="id" type="constructor">tif</span></a> : <a class="idref" href="Stlc.html#tm"><span class="id" type="inductive">tm</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="Stlc.html#tm"><span class="id" type="inductive">tm</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="Stlc.html#tm"><span class="id" type="inductive">tm</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="Stlc.html#tm"><span class="id" type="inductive">tm</span></a>.<br/>
<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.
<div class="paragraph"> </div>
Some examples...
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <a name="STLC.x"><span class="id" type="definition">x</span></a> := (<a class="idref" href="Maps.html#Id"><span class="id" type="constructor">Id</span></a> 0).<br/>
<span class="id" type="keyword">Definition</span> <a name="STLC.y"><span class="id" type="definition">y</span></a> := (<a class="idref" href="Maps.html#Id"><span class="id" type="constructor">Id</span></a> 1).<br/>
<span class="id" type="keyword">Definition</span> <a name="STLC.z"><span class="id" type="definition">z</span></a> := (<a class="idref" href="Maps.html#Id"><span class="id" type="constructor">Id</span></a> 2).<br/>
<span class="id" type="keyword">Hint</span> <span class="id" type="keyword">Unfold</span> <a class="idref" href="Stlc.html#STLC.x"><span class="id" type="definition">x</span></a>.<br/>
<span class="id" type="keyword">Hint</span> <span class="id" type="keyword">Unfold</span> <a class="idref" href="Stlc.html#STLC.y"><span class="id" type="definition">y</span></a>.<br/>
<span class="id" type="keyword">Hint</span> <span class="id" type="keyword">Unfold</span> <a class="idref" href="Stlc.html#STLC.z"><span class="id" type="definition">z</span></a>.<br/>
<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> <a name="STLC.idB"><span class="id" type="abbreviation">idB</span></a> :=<br/>
(<a class="idref" href="Stlc.html#STLC.tabs"><span class="id" type="constructor">tabs</span></a> <a class="idref" href="Stlc.html#STLC.x"><span class="id" type="definition">x</span></a> <a class="idref" href="Stlc.html#STLC.TBool"><span class="id" type="constructor">TBool</span></a> (<a class="idref" href="Stlc.html#STLC.tvar"><span class="id" type="constructor">tvar</span></a> <a class="idref" href="Stlc.html#STLC.x"><span class="id" type="definition">x</span></a>)).<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> <a name="STLC.idBB"><span class="id" type="abbreviation">idBB</span></a> :=<br/>
(<a class="idref" href="Stlc.html#STLC.tabs"><span class="id" type="constructor">tabs</span></a> <a class="idref" href="Stlc.html#STLC.x"><span class="id" type="definition">x</span></a> (<a class="idref" href="Stlc.html#STLC.TArrow"><span class="id" type="constructor">TArrow</span></a> <a class="idref" href="Stlc.html#STLC.TBool"><span class="id" type="constructor">TBool</span></a> <a class="idref" href="Stlc.html#STLC.TBool"><span class="id" type="constructor">TBool</span></a>) (<a class="idref" href="Stlc.html#STLC.tvar"><span class="id" type="constructor">tvar</span></a> <a class="idref" href="Stlc.html#STLC.x"><span class="id" type="definition">x</span></a>)).<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> <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> <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> <a name="STLC.idBBBB"><span class="id" type="abbreviation">idBBBB</span></a> :=<br/>
(<a class="idref" href="Stlc.html#STLC.tabs"><span class="id" type="constructor">tabs</span></a> <a class="idref" href="Stlc.html#STLC.x"><span class="id" type="definition">x</span></a> (<a class="idref" href="Stlc.html#STLC.TArrow"><span class="id" type="constructor">TArrow</span></a> (<a class="idref" href="Stlc.html#STLC.TArrow"><span class="id" type="constructor">TArrow</span></a> <a class="idref" href="Stlc.html#STLC.TBool"><span class="id" type="constructor">TBool</span></a> <a class="idref" href="Stlc.html#STLC.TBool"><span class="id" type="constructor">TBool</span></a>)<br/>
(<a class="idref" href="Stlc.html#STLC.TArrow"><span class="id" type="constructor">TArrow</span></a> <a class="idref" href="Stlc.html#STLC.TBool"><span class="id" type="constructor">TBool</span></a> <a class="idref" href="Stlc.html#STLC.TBool"><span class="id" type="constructor">TBool</span></a>))<br/>
(<a class="idref" href="Stlc.html#STLC.tvar"><span class="id" type="constructor">tvar</span></a> <a class="idref" href="Stlc.html#STLC.x"><span class="id" type="definition">x</span></a>)).<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> <a name="STLC.k"><span class="id" type="abbreviation">k</span></a> := (<a class="idref" href="Stlc.html#STLC.tabs"><span class="id" type="constructor">tabs</span></a> <a class="idref" href="Stlc.html#STLC.x"><span class="id" type="definition">x</span></a> <a class="idref" href="Stlc.html#STLC.TBool"><span class="id" type="constructor">TBool</span></a> (<a class="idref" href="Stlc.html#STLC.tabs"><span class="id" type="constructor">tabs</span></a> <a class="idref" href="Stlc.html#STLC.y"><span class="id" type="definition">y</span></a> <a class="idref" href="Stlc.html#STLC.TBool"><span class="id" type="constructor">TBool</span></a> (<a class="idref" href="Stlc.html#STLC.tvar"><span class="id" type="constructor">tvar</span></a> <a class="idref" href="Stlc.html#STLC.x"><span class="id" type="definition">x</span></a>))).<br/>
<br/>
</div>
<div class="doc">
<span class="inlinecode"><span class="id" type="var">notB</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="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>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Notation</span> <a name="STLC.notB"><span class="id" type="abbreviation">notB</span></a> := (<a class="idref" href="Stlc.html#STLC.tabs"><span class="id" type="constructor">tabs</span></a> <a class="idref" href="Stlc.html#STLC.x"><span class="id" type="definition">x</span></a> <a class="idref" href="Stlc.html#STLC.TBool"><span class="id" type="constructor">TBool</span></a> (<a class="idref" href="Stlc.html#STLC.tif"><span class="id" type="constructor">tif</span></a> (<a class="idref" href="Stlc.html#STLC.tvar"><span class="id" type="constructor">tvar</span></a> <a class="idref" href="Stlc.html#STLC.x"><span class="id" type="definition">x</span></a>) <a class="idref" href="Stlc.html#STLC.tfalse"><span class="id" type="constructor">tfalse</span></a> <a class="idref" href="Stlc.html#STLC.ttrue"><span class="id" type="constructor">ttrue</span></a>)).<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="lab562"></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="lab563"></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</span>.</span> <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> is a value only when <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></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</span>.</span> <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> is always a value, no matter
whether <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></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 Gallina,
makes the first choice — for example,
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">Compute</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>.
<div class="paragraph"> </div>
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>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <a name="STLC.value"><span class="id" type="inductive">value</span></a> : <a class="idref" href="Stlc.html#STLC.tm"><span class="id" type="inductive">tm</span></a> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
| <a name="STLC.v_abs"><span class="id" type="constructor">v_abs</span></a> : <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/>
<a class="idref" href="Stlc.html#value"><span class="id" type="inductive">value</span></a> (<a class="idref" href="Stlc.html#STLC.tabs"><span class="id" type="constructor">tabs</span></a> <a class="idref" href="Stlc.html#x"><span class="id" type="variable">x</span></a> <a class="idref" href="Stlc.html#T"><span class="id" type="variable">T</span></a> <a class="idref" href="Stlc.html#t"><span class="id" type="variable">t</span></a>)<br/>
| <a name="STLC.v_true"><span class="id" type="constructor">v_true</span></a> :<br/>
<a class="idref" href="Stlc.html#value"><span class="id" type="inductive">value</span></a> <a class="idref" href="Stlc.html#STLC.ttrue"><span class="id" type="constructor">ttrue</span></a><br/>
| <a name="STLC.v_false"><span class="id" type="constructor">v_false</span></a> :<br/>
<a class="idref" href="Stlc.html#value"><span class="id" type="inductive">value</span></a> <a class="idref" href="Stlc.html#STLC.tfalse"><span class="id" type="constructor">tfalse</span></a>.<br/>
<br/>
<span class="id" type="keyword">Hint</span> <span class="id" type="var">Constructors</span> <a class="idref" href="Stlc.html#value"><span class="id" type="inductive">value</span></a>.<br/>
<br/>
</div>
<div class="doc">
Finally, we must consider what constitutes a <i>complete</i> program.
<div class="paragraph"> </div>
Intuitively, a "complete program" must not refer to any undefined
variables. We'll see shortly how to define the <i>free</i> variables
in a STLC term. A complete program is <i>closed</i> — that is, it
contains no free variables.
<div class="paragraph"> </div>
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.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab564"></a><h3 class="section">Substitution</h3>
<div class="paragraph"> </div>
Now we come to the heart of the STLC: the operation of
substituting one term for a variable in another term. This
operation is 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
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="keyword">if</span> <span class="id" type="var">false</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">false</span>
<div class="paragraph"> </div>
</div>
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.
<div class="paragraph"> </div>
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>
<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">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">T<sub>11</sub></span>. <span class="id" type="var">t<sub>12</sub></span>) = \<span class="id" type="var">x</span>:<span class="id" type="var">T<sub>11</sub></span>. <span class="id" type="var">t<sub>12</sub></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">T<sub>11</sub></span>. <span class="id" type="var">t<sub>12</sub></span>) = \<span class="id" type="var">y</span>:<span class="id" type="var">T<sub>11</sub></span>. [<span class="id" type="var">x</span>:=<span class="id" type="var">s</span>]<span class="id" type="var">t<sub>12</sub></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">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span>) = ([<span class="id" type="var">x</span>:=<span class="id" type="var">s</span>]<span class="id" type="var">t<sub>1</sub></span>) ([<span class="id" type="var">x</span>:=<span class="id" type="var">s</span>]<span class="id" type="var">t<sub>2</sub></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">t<sub>1</sub></span> <span class="id" type="keyword">then</span> <span class="id" type="var">t<sub>2</sub></span> <span class="id" type="keyword">else</span> <span class="id" type="var">t<sub>3</sub></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">t<sub>1</sub></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">t<sub>2</sub></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">t<sub>3</sub></span>
<div class="paragraph"> </div>
</div>
<div class="paragraph"> </div>
... and formally:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Reserved Notation</span> "'[' x ':=' s ']' t" (<span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 20).<br/>
<br/>
<span class="id" type="keyword">Fixpoint</span> <a name="STLC.subst"><span class="id" type="definition">subst</span></a> (<span class="id" type="var">x</span>:<a class="idref" href="Maps.html#id"><span class="id" type="inductive">id</span></a>) (<span class="id" type="var">s</span>:<a class="idref" href="Stlc.html#STLC.tm"><span class="id" type="inductive">tm</span></a>) (<span class="id" type="var">t</span>:<a class="idref" href="Stlc.html#STLC.tm"><span class="id" type="inductive">tm</span></a>) : <a class="idref" href="Stlc.html#STLC.tm"><span class="id" type="inductive">tm</span></a> :=<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="Stlc.html#t"><span class="id" type="variable">t</span></a> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="Stlc.html#STLC.tvar"><span class="id" type="constructor">tvar</span></a> <span class="id" type="var">x'</span> ⇒<br/>
<span class="id" type="keyword">if</span> <a class="idref" href="Maps.html#beq_id"><span class="id" type="definition">beq_id</span></a> <a class="idref" href="Stlc.html#x"><span class="id" type="variable">x</span></a> <span class="id" type="var">x'</span> <span class="id" type="keyword">then</span> <a class="idref" href="Stlc.html#s"><span class="id" type="variable">s</span></a> <span class="id" type="keyword">else</span> <a class="idref" href="Stlc.html#t"><span class="id" type="variable">t</span></a><br/>
| <a class="idref" href="Stlc.html#STLC.tabs"><span class="id" type="constructor">tabs</span></a> <span class="id" type="var">x'</span> <span class="id" type="var">T</span> <span class="id" type="var">t<sub>1</sub></span> ⇒<br/>
<a class="idref" href="Stlc.html#STLC.tabs"><span class="id" type="constructor">tabs</span></a> <span class="id" type="var">x'</span> <span class="id" type="var">T</span> (<span class="id" type="keyword">if</span> <a class="idref" href="Maps.html#beq_id"><span class="id" type="definition">beq_id</span></a> <a class="idref" href="Stlc.html#x"><span class="id" type="variable">x</span></a> <span class="id" type="var">x'</span> <span class="id" type="keyword">then</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="keyword">else</span> ([<a class="idref" href="Stlc.html#x"><span class="id" type="variable">x</span></a>:=<a class="idref" href="Stlc.html#s"><span class="id" type="variable">s</span></a>] <span class="id" type="var">t<sub>1</sub></span>))<br/>
| <a class="idref" href="Stlc.html#STLC.tapp"><span class="id" type="constructor">tapp</span></a> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span> ⇒<br/>
<a class="idref" href="Stlc.html#STLC.tapp"><span class="id" type="constructor">tapp</span></a> ([<a class="idref" href="Stlc.html#x"><span class="id" type="variable">x</span></a>:=<a class="idref" href="Stlc.html#s"><span class="id" type="variable">s</span></a>] <span class="id" type="var">t<sub>1</sub></span>) ([<a class="idref" href="Stlc.html#x"><span class="id" type="variable">x</span></a>:=<a class="idref" href="Stlc.html#s"><span class="id" type="variable">s</span></a>] <span class="id" type="var">t<sub>2</sub></span>)<br/>
| <a class="idref" href="Stlc.html#STLC.ttrue"><span class="id" type="constructor">ttrue</span></a> ⇒<br/>
<a class="idref" href="Stlc.html#STLC.ttrue"><span class="id" type="constructor">ttrue</span></a><br/>
| <a class="idref" href="Stlc.html#STLC.tfalse"><span class="id" type="constructor">tfalse</span></a> ⇒<br/>
<a class="idref" href="Stlc.html#STLC.tfalse"><span class="id" type="constructor">tfalse</span></a><br/>
| <a class="idref" href="Stlc.html#STLC.tif"><span class="id" type="constructor">tif</span></a> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span> <span class="id" type="var">t<sub>3</sub></span> ⇒<br/>
<a class="idref" href="Stlc.html#STLC.tif"><span class="id" type="constructor">tif</span></a> ([<a class="idref" href="Stlc.html#x"><span class="id" type="variable">x</span></a>:=<a class="idref" href="Stlc.html#s"><span class="id" type="variable">s</span></a>] <span class="id" type="var">t<sub>1</sub></span>) ([<a class="idref" href="Stlc.html#x"><span class="id" type="variable">x</span></a>:=<a class="idref" href="Stlc.html#s"><span class="id" type="variable">s</span></a>] <span class="id" type="var">t<sub>2</sub></span>) ([<a class="idref" href="Stlc.html#x"><span class="id" type="variable">x</span></a>:=<a class="idref" href="Stlc.html#s"><span class="id" type="variable">s</span></a>] <span class="id" type="var">t<sub>3</sub></span>)<br/>
<span class="id" type="keyword">end</span><br/>
<br/>
<span class="id" type="keyword">where</span> "'[' x ':=' s ']' t" := (<a class="idref" href="Stlc.html#subst"><span class="id" type="definition">subst</span></a> <span class="id" type="var">x</span> <span class="id" type="var">s</span> <span class="id" type="var">t</span>).<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> that include
binders for all of the variables they mention), we can avoid this
extra complexity here, but it must be dealt with when formalizing
richer languages.
<div class="paragraph"> </div>
<a name="lab565"></a><h4 class="section">Exercise: 3 stars (substi)</h4>
The definition that we gave above uses Coq's <span class="inlinecode"><span class="id" type="keyword">Fixpoint</span></span> facility
to define substitution as a <i>function</i>. Suppose, instead, we
wanted to define substitution as an inductive <i>relation</i> <span class="inlinecode"><span class="id" type="var">substi</span></span>.
We've begun the definition by providing the <span class="inlinecode"><span class="id" type="keyword">Inductive</span></span> header and
one of the constructors; your job is to fill in the rest of the
constructors and prove that the relation you've defined coincides
with the function given above.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <a name="STLC.substi"><span class="id" type="inductive">substi</span></a> (<span class="id" type="var">s</span>:<a class="idref" href="Stlc.html#STLC.tm"><span class="id" type="inductive">tm</span></a>) (<span class="id" type="var">x</span>:<a class="idref" href="Maps.html#id"><span class="id" type="inductive">id</span></a>) : <a class="idref" href="Stlc.html#STLC.tm"><span class="id" type="inductive">tm</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="Stlc.html#STLC.tm"><span class="id" type="inductive">tm</span></a> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
| <a name="STLC.s_var1"><span class="id" type="constructor">s_var1</span></a> :<br/>
<a class="idref" href="Stlc.html#substi"><span class="id" type="inductive">substi</span></a> <span class="id" type="var">s</span> <span class="id" type="var">x</span> (<a class="idref" href="Stlc.html#STLC.tvar"><span class="id" type="constructor">tvar</span></a> <span class="id" type="var">x</span>) <span class="id" type="var">s</span><br/>
<span class="comment">(* FILL IN HERE *)</span><br/>
.<br/>
<br/>
<span class="id" type="keyword">Hint</span> <span class="id" type="var">Constructors</span> <a class="idref" href="Stlc.html#substi"><span class="id" type="inductive">substi</span></a>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <a name="STLC.substi_correct"><span class="id" type="lemma">substi_correct</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">s</span> <span class="id" type="var">x</span> <span class="id" type="var">t</span> <span class="id" type="var">t'</span>,<br/>
[<a class="idref" href="Stlc.html#x"><span class="id" type="variable">x</span></a>:=<a class="idref" href="Stlc.html#s"><span class="id" type="variable">s</span></a>]<a class="idref" href="Stlc.html#t"><span class="id" type="variable">t</span></a> = <a class="idref" href="Stlc.html#t'"><span class="id" type="variable">t'</span></a> <span style="font-family: arial;">↔</span> <a class="idref" href="Stlc.html#STLC.substi"><span class="id" type="inductive">substi</span></a> <a class="idref" href="Stlc.html#s"><span class="id" type="variable">s</span></a> <a class="idref" href="Stlc.html#x"><span class="id" type="variable">x</span></a> <a class="idref" href="Stlc.html#t"><span class="id" type="variable">t</span></a> <a class="idref" href="Stlc.html#t'"><span class="id" type="variable">t'</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab566"></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 (the function) until it becomes an abstraction; 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 abstraction. 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">v<sub>2</sub></span> <span style="font-family: arial;">⇒</span> [<span class="id" type="var">x</span>:=<span class="id" type="var">v<sub>2</sub></span>]<span class="id" type="var">t<sub>12</sub></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 v<sub>2</sub></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) v<sub>2</sub> <span style="font-family: arial;">⇒</span> [x:=v<sub>2</sub>]t<sub>12</sub></td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t<sub>1</sub> <span style="font-family: arial;">⇒</span> t<sub>1</sub>'</td>
<td class="infrulenamecol" rowspan="3">
(ST_App1)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">t<sub>1</sub> t<sub>2</sub> <span style="font-family: arial;">⇒</span> t<sub>1</sub>' t<sub>2</sub></td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">value v<sub>1</sub></td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule">t<sub>2</sub> <span style="font-family: arial;">⇒</span> t<sub>2</sub>'</td>
<td class="infrulenamecol" rowspan="3">
(ST_App2)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">v<sub>1</sub> t<sub>2</sub> <span style="font-family: arial;">⇒</span> v<sub>1</sub> t<sub>2</sub>'</td>
<td></td>
</td>
</table></center> ... 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 t<sub>1</sub> else t<sub>2</sub>) <span style="font-family: arial;">⇒</span> t<sub>1</sub></td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(ST_IfFalse)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">(if false then t<sub>1</sub> else t<sub>2</sub>) <span style="font-family: arial;">⇒</span> t<sub>2</sub></td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t<sub>1</sub> <span style="font-family: arial;">⇒</span> t<sub>1</sub>'</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 t<sub>1</sub> then t<sub>2</sub> else t<sub>3</sub>) <span style="font-family: arial;">⇒</span> (if t<sub>1</sub>' then t<sub>2</sub> else t<sub>3</sub>)</td>
<td></td>
</td>
</table></center>
<div class="paragraph"> </div>
Formally:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Reserved Notation</span> "t<sub>1</sub> '<span style="font-family: arial;">⇒</span>' t<sub>2</sub>" (<span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 40).<br/>
<br/>
<span class="id" type="keyword">Inductive</span> <a name="STLC.step"><span class="id" type="inductive">step</span></a> : <a class="idref" href="Stlc.html#STLC.tm"><span class="id" type="inductive">tm</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="Stlc.html#STLC.tm"><span class="id" type="inductive">tm</span></a> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
| <a name="STLC.ST_AppAbs"><span class="id" type="constructor">ST_AppAbs</span></a> : <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<sub>12</sub></span> <span class="id" type="var">v<sub>2</sub></span>,<br/>
<a class="idref" href="Stlc.html#STLC.value"><span class="id" type="inductive">value</span></a> <a class="idref" href="Stlc.html#v<sub>2</sub>"><span class="id" type="variable">v<sub>2</sub></span></a> <span style="font-family: arial;">→</span><br/>
(<a class="idref" href="Stlc.html#STLC.tapp"><span class="id" type="constructor">tapp</span></a> (<a class="idref" href="Stlc.html#STLC.tabs"><span class="id" type="constructor">tabs</span></a> <a class="idref" href="Stlc.html#x"><span class="id" type="variable">x</span></a> <a class="idref" href="Stlc.html#T"><span class="id" type="variable">T</span></a> <a class="idref" href="Stlc.html#t<sub>12</sub>"><span class="id" type="variable">t<sub>12</sub></span></a>) <a class="idref" href="Stlc.html#v<sub>2</sub>"><span class="id" type="variable">v<sub>2</sub></span></a>) <span style="font-family: arial;">⇒</span> [<a class="idref" href="Stlc.html#x"><span class="id" type="variable">x</span></a>:=<a class="idref" href="Stlc.html#v<sub>2</sub>"><span class="id" type="variable">v<sub>2</sub></span></a>]<a class="idref" href="Stlc.html#t<sub>12</sub>"><span class="id" type="variable">t<sub>12</sub></span></a><br/>
| <a name="STLC.ST_App1"><span class="id" type="constructor">ST_App1</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>1</sub>'</span> <span class="id" type="var">t<sub>2</sub></span>,<br/>
<a class="idref" href="Stlc.html#t<sub>1</sub>"><span class="id" type="variable">t<sub>1</sub></span></a> <span style="font-family: arial;">⇒</span> <a class="idref" href="Stlc.html#t<sub>1</sub>'"><span class="id" type="variable">t<sub>1</sub>'</span></a> <span style="font-family: arial;">→</span><br/>
<a class="idref" href="Stlc.html#STLC.tapp"><span class="id" type="constructor">tapp</span></a> <a class="idref" href="Stlc.html#t<sub>1</sub>"><span class="id" type="variable">t<sub>1</sub></span></a> <a class="idref" href="Stlc.html#t<sub>2</sub>"><span class="id" type="variable">t<sub>2</sub></span></a> <span style="font-family: arial;">⇒</span> <a class="idref" href="Stlc.html#STLC.tapp"><span class="id" type="constructor">tapp</span></a> <a class="idref" href="Stlc.html#t<sub>1</sub>'"><span class="id" type="variable">t<sub>1</sub>'</span></a> <a class="idref" href="Stlc.html#t<sub>2</sub>"><span class="id" type="variable">t<sub>2</sub></span></a><br/>
| <a name="STLC.ST_App2"><span class="id" type="constructor">ST_App2</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">v<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span> <span class="id" type="var">t<sub>2</sub>'</span>,<br/>
<a class="idref" href="Stlc.html#STLC.value"><span class="id" type="inductive">value</span></a> <a class="idref" href="Stlc.html#v<sub>1</sub>"><span class="id" type="variable">v<sub>1</sub></span></a> <span style="font-family: arial;">→</span><br/>
<a class="idref" href="Stlc.html#t<sub>2</sub>"><span class="id" type="variable">t<sub>2</sub></span></a> <span style="font-family: arial;">⇒</span> <a class="idref" href="Stlc.html#t<sub>2</sub>'"><span class="id" type="variable">t<sub>2</sub>'</span></a> <span style="font-family: arial;">→</span><br/>
<a class="idref" href="Stlc.html#STLC.tapp"><span class="id" type="constructor">tapp</span></a> <a class="idref" href="Stlc.html#v<sub>1</sub>"><span class="id" type="variable">v<sub>1</sub></span></a> <a class="idref" href="Stlc.html#t<sub>2</sub>"><span class="id" type="variable">t<sub>2</sub></span></a> <span style="font-family: arial;">⇒</span> <a class="idref" href="Stlc.html#STLC.tapp"><span class="id" type="constructor">tapp</span></a> <a class="idref" href="Stlc.html#v<sub>1</sub>"><span class="id" type="variable">v<sub>1</sub></span></a> <a class="idref" href="Stlc.html#t<sub>2</sub>'"><span class="id" type="variable">t<sub>2</sub>'</span></a><br/>
| <a name="STLC.ST_IfTrue"><span class="id" type="constructor">ST_IfTrue</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span>,<br/>
(<a class="idref" href="Stlc.html#STLC.tif"><span class="id" type="constructor">tif</span></a> <a class="idref" href="Stlc.html#STLC.ttrue"><span class="id" type="constructor">ttrue</span></a> <a class="idref" href="Stlc.html#t<sub>1</sub>"><span class="id" type="variable">t<sub>1</sub></span></a> <a class="idref" href="Stlc.html#t<sub>2</sub>"><span class="id" type="variable">t<sub>2</sub></span></a>) <span style="font-family: arial;">⇒</span> <a class="idref" href="Stlc.html#t<sub>1</sub>"><span class="id" type="variable">t<sub>1</sub></span></a><br/>
| <a name="STLC.ST_IfFalse"><span class="id" type="constructor">ST_IfFalse</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span>,<br/>
(<a class="idref" href="Stlc.html#STLC.tif"><span class="id" type="constructor">tif</span></a> <a class="idref" href="Stlc.html#STLC.tfalse"><span class="id" type="constructor">tfalse</span></a> <a class="idref" href="Stlc.html#t<sub>1</sub>"><span class="id" type="variable">t<sub>1</sub></span></a> <a class="idref" href="Stlc.html#t<sub>2</sub>"><span class="id" type="variable">t<sub>2</sub></span></a>) <span style="font-family: arial;">⇒</span> <a class="idref" href="Stlc.html#t<sub>2</sub>"><span class="id" type="variable">t<sub>2</sub></span></a><br/>
| <a name="STLC.ST_If"><span class="id" type="constructor">ST_If</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>1</sub>'</span> <span class="id" type="var">t<sub>2</sub></span> <span class="id" type="var">t<sub>3</sub></span>,<br/>
<a class="idref" href="Stlc.html#t<sub>1</sub>"><span class="id" type="variable">t<sub>1</sub></span></a> <span style="font-family: arial;">⇒</span> <a class="idref" href="Stlc.html#t<sub>1</sub>'"><span class="id" type="variable">t<sub>1</sub>'</span></a> <span style="font-family: arial;">→</span><br/>
(<a class="idref" href="Stlc.html#STLC.tif"><span class="id" type="constructor">tif</span></a> <a class="idref" href="Stlc.html#t<sub>1</sub>"><span class="id" type="variable">t<sub>1</sub></span></a> <a class="idref" href="Stlc.html#t<sub>2</sub>"><span class="id" type="variable">t<sub>2</sub></span></a> <a class="idref" href="Stlc.html#t<sub>3</sub>"><span class="id" type="variable">t<sub>3</sub></span></a>) <span style="font-family: arial;">⇒</span> (<a class="idref" href="Stlc.html#STLC.tif"><span class="id" type="constructor">tif</span></a> <a class="idref" href="Stlc.html#t<sub>1</sub>'"><span class="id" type="variable">t<sub>1</sub>'</span></a> <a class="idref" href="Stlc.html#t<sub>2</sub>"><span class="id" type="variable">t<sub>2</sub></span></a> <a class="idref" href="Stlc.html#t<sub>3</sub>"><span class="id" type="variable">t<sub>3</sub></span></a>)<br/>
<br/>
<span class="id" type="keyword">where</span> "t<sub>1</sub> '<span style="font-family: arial;">⇒</span>' t<sub>2</sub>" := (<a class="idref" href="Stlc.html#step"><span class="id" type="inductive">step</span></a> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span>).<br/>
<br/>
<span class="id" type="keyword">Hint</span> <span class="id" type="var">Constructors</span> <a class="idref" href="Stlc.html#step"><span class="id" type="inductive">step</span></a>.<br/>
<br/>
<span class="id" type="keyword">Notation</span> <a name="STLC.multistep"><span class="id" type="abbreviation">multistep</span></a> := (<a class="idref" href="Smallstep.html#multi"><span class="id" type="inductive">multi</span></a> <a class="idref" href="Stlc.html#STLC.step"><span class="id" type="inductive">step</span></a>).<br/>
<span class="id" type="keyword">Notation</span> "t<sub>1</sub> '<span style="font-family: arial;">⇒*</span>' t<sub>2</sub>" := (<a class="idref" href="Stlc.html#STLC.multistep"><span class="id" type="abbreviation">multistep</span></a> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span>) (<span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 40).<br/>
<br/>
</div>
<div class="doc">
<a name="lab567"></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> <a name="STLC.step_example1"><span class="id" type="lemma">step_example1</span></a> :<br/>
(<a class="idref" href="Stlc.html#STLC.tapp"><span class="id" type="constructor">tapp</span></a> <a class="idref" href="Stlc.html#STLC.idBB"><span class="id" type="abbreviation">idBB</span></a> <a class="idref" href="Stlc.html#STLC.idB"><span class="id" type="abbreviation">idB</span></a>) <span style="font-family: arial;">⇒*</span> <a class="idref" href="Stlc.html#STLC.idB"><span class="id" type="abbreviation">idB</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">eapply</span> <a class="idref" href="Smallstep.html#multi_step"><span class="id" type="constructor">multi_step</span></a>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="Stlc.html#STLC.ST_AppAbs"><span class="id" type="constructor">ST_AppAbs</span></a>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="Stlc.html#STLC.v_abs"><span class="id" type="constructor">v_abs</span></a>.<br/>
<span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="Smallstep.html#multi_refl"><span class="id" type="constructor">multi_refl</span></a>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
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 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>)))<br/>
<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">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> <a name="STLC.step_example2"><span class="id" type="lemma">step_example2</span></a> :<br/>
(<a class="idref" href="Stlc.html#STLC.tapp"><span class="id" type="constructor">tapp</span></a> <a class="idref" href="Stlc.html#STLC.idBB"><span class="id" type="abbreviation">idBB</span></a> (<a class="idref" href="Stlc.html#STLC.tapp"><span class="id" type="constructor">tapp</span></a> <a class="idref" href="Stlc.html#STLC.idBB"><span class="id" type="abbreviation">idBB</span></a> <a class="idref" href="Stlc.html#STLC.idB"><span class="id" type="abbreviation">idB</span></a>)) <span style="font-family: arial;">⇒*</span> <a class="idref" href="Stlc.html#STLC.idB"><span class="id" type="abbreviation">idB</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">eapply</span> <a class="idref" href="Smallstep.html#multi_step"><span class="id" type="constructor">multi_step</span></a>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="Stlc.html#STLC.ST_App2"><span class="id" type="constructor">ST_App2</span></a>. <span class="id" type="tactic">auto</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="Stlc.html#STLC.ST_AppAbs"><span class="id" type="constructor">ST_AppAbs</span></a>. <span class="id" type="tactic">auto</span>.<br/>
<span class="id" type="tactic">eapply</span> <a class="idref" href="Smallstep.html#multi_step"><span class="id" type="constructor">multi_step</span></a>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="Stlc.html#STLC.ST_AppAbs"><span class="id" type="constructor">ST_AppAbs</span></a>. <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> <a class="idref" href="Smallstep.html#multi_refl"><span class="id" type="constructor">multi_refl</span></a>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
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="keyword">if</span> <span class="id" type="var">x</span> <span class="id" type="keyword">then</span> <span class="id" type="var">false</span><br/>
<span class="id" type="keyword">else</span> <span class="id" type="var">true</span>)) <span class="id" type="var">true</span>)<br/>
<span style="font-family: arial;">⇒*</span> <span class="id" type="var">false</span>
<div class="paragraph"> </div>
</div>