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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
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<div class='booktitleinheader'><a href='index.html'>Volume 2: Programming Language Foundations</a></div>
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<h1 class="libtitle">HoareAsLogic<span class="subtitle">Hoare Logic as a Logic</span></h1>
<div class="doc">
<div class="paragraph"> </div>
The presentation of Hoare logic in chapter <a href="Hoare.html"><span class="inlineref">Hoare</span></a> could be
described as "model-theoretic": the proof rules for each of the
constructors were presented as <i>theorems</i> about the evaluation
behavior of programs, and proofs of program correctness (validity
of Hoare triples) were constructed by combining these theorems
directly in Coq.
<div class="paragraph"> </div>
Another way of presenting Hoare logic is to define a completely
separate proof system -- a set of axioms and inference rules that
talk about commands, Hoare triples, etc. -- and then say that a
proof of a Hoare triple is a valid derivation in <i>that</i> logic. We
can do this by giving an inductive definition of <i>valid
derivations</i> in this new logic.
<div class="paragraph"> </div>
This chapter is optional. Before reading it, you'll want to read
the <a href="https://softwarefoundations.cis.upenn.edu/lf-current/ProofObjects.html"><span class="inlineref">ProofObjects</span></a> chapter in <i>Logical
Foundations</i> (<i>Software Foundations</i>, volume 1).
</div>
<div class="code">
<span class="id" title="keyword">Set</span> <span class="id" title="var">Warnings</span> "-deprecated-hint-without-locality,-deprecated-hint-without-locality".<br/>
<span class="id" title="keyword">From</span> <span class="id" title="var">PLF</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">Maps</span>.<br/>
<span class="id" title="keyword">From</span> <span class="id" title="var">PLF</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Hoare.html#"><span class="id" title="library">Hoare</span></a>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Hint Constructors</span> <a class="idref" href="HoareAsLogic.html#ceval"><span class="id" title="inductive">ceval</span></a> : <span class="id" title="var">core</span>.<br/>
</div>
<div class="doc">
<a id="lab132"></a><h1 class="section">Hoare Logic and Model Theory</h1>
<div class="paragraph"> </div>
In <a href="Hoare.html"><span class="inlineref">Hoare</span></a> we introduced Hoare triples, which contain a
precondition, command, and postcondition. For example (and for
the moment deliberately avoiding the notation we previously
introduced),
<br/>
<span class="inlinecode"> <span class="id" title="var">Pre</span>: <span class="id" title="var">X</span> = 0<br/>
<span class="id" title="var">Com</span>: <span class="id" title="var">X</span> := <span class="id" title="var">X</span> + 1<br/>
<span class="id" title="var">Post</span>: <span class="id" title="var">X</span> = 1
</span> is a Hoare triple, as is
<br/>
<span class="inlinecode"> <span class="id" title="var">Pre</span>: <span class="id" title="var">X</span> = 0<br/>
<span class="id" title="var">Com</span>: <span class="id" title="var">skip</span><br/>
<span class="id" title="var">Post</span>: <span class="id" title="var">X</span> = 1
</span> But there's an important difference between those two triples: the
former expresses a truth about how Imp programs execute, whereas
the latter does not.
<div class="paragraph"> </div>
To capture that difference, we introduced a definition
<span class="inlinecode"><span class="id" title="var">valid_hoare_triple</span></span> that described when a triple expresses such a
truth. Let's repeat that definition, but this time we'll call it
<span class="inlinecode"><span class="id" title="var">valid</span></span>:
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <a id="valid" class="idref" href="#valid"><span class="id" title="definition">valid</span></a> (<a id="P:1" class="idref" href="#P:1"><span class="id" title="binder">P</span></a> : <a class="idref" href="Hoare.html#Assertion"><span class="id" title="definition">Assertion</span></a>) (<a id="c:2" class="idref" href="#c:2"><span class="id" title="binder">c</span></a> : <span class="id" title="inductive">com</span>) (<a id="Q:3" class="idref" href="#Q:3"><span class="id" title="binder">Q</span></a> : <a class="idref" href="Hoare.html#Assertion"><span class="id" title="definition">Assertion</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
<span class="id" title="keyword">∀</span> <a id="st:4" class="idref" href="#st:4"><span class="id" title="binder">st</span></a> <a id="st':5" class="idref" href="#st':5"><span class="id" title="binder">st'</span></a>,<br/>
<a class="idref" href="HoareAsLogic.html#st:4"><span class="id" title="variable">st</span></a> <span class="id" title="notation">=[</span> <a class="idref" href="HoareAsLogic.html#c:2"><span class="id" title="variable">c</span></a> <span class="id" title="notation">]=></span> <a class="idref" href="HoareAsLogic.html#st':5"><span class="id" title="variable">st'</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="HoareAsLogic.html#P:1"><span class="id" title="variable">P</span></a> <a class="idref" href="HoareAsLogic.html#st:4"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="HoareAsLogic.html#Q:3"><span class="id" title="variable">Q</span></a> <a class="idref" href="HoareAsLogic.html#st':5"><span class="id" title="variable">st'</span></a>.<br/>
</div>
<div class="doc">
This notion of <i>validity</i> is based on the underlying model of how
Imp programs execute. That model itself is based on states. So,
<br/>
<span class="inlinecode"> <span class="id" title="var">Pre</span>: <span class="id" title="var">X</span> = 0<br/>
<span class="id" title="var">Com</span>: <span class="id" title="var">X</span> := <span class="id" title="var">X</span> + 1<br/>
<span class="id" title="var">Post</span>: <span class="id" title="var">X</span> = 1
</span> is <i>valid</i>, because starting from any state in which <span class="inlinecode"><span class="id" title="var">X</span></span> is <span class="inlinecode">0</span>,
and executing <span class="inlinecode"><span class="id" title="var">X</span></span> <span class="inlinecode">:=</span> <span class="inlinecode"><span class="id" title="var">X</span></span> <span class="inlinecode">+</span> <span class="inlinecode">1</span>, we are guaranteed to reach a state in
which <span class="inlinecode"><span class="id" title="var">X</span></span> is <span class="inlinecode">1</span>. But,
<br/>
<span class="inlinecode"> <span class="id" title="var">Pre</span>: <span class="id" title="var">X</span> = 0<br/>
<span class="id" title="var">Com</span>: <span class="id" title="var">skip</span><br/>
<span class="id" title="var">Post</span>: <span class="id" title="var">X</span> = 1
</span> is <i>invalid</i>, because starting from any state in which <span class="inlinecode"><span class="id" title="var">X</span></span> is <span class="inlinecode">0</span>,
we are guaranteed not to change the state, so <span class="inlinecode"><span class="id" title="var">X</span></span> cannot be <span class="inlinecode">1</span>.
<div class="paragraph"> </div>
So far, we have punned between the syntax of a Hoare triple,
written <span class="inlinecode"><span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">P</span><span style='letter-spacing:-.4em;'>}</span>}</span> <span class="inlinecode"><span class="id" title="var">c</span></span> <span class="inlinecode"><span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">Q</span><span style='letter-spacing:-.4em;'>}</span>}</span>, and its validity, as expressed by
<span class="inlinecode"><span class="id" title="var">valid</span></span>. In essence, we have said that the semantic meaning of
that syntax is the proposition returned by <span class="inlinecode"><span class="id" title="var">valid</span></span>. This way of
giving semantic meaning to something syntactic is part of the
branch of mathematical logic known as <i>model theory</i>.
<div class="paragraph"> </div>
Our approach to Hoare logic through model theory led us to
state proof rules in terms of that same state-based model, and to
prove program correctness in it, too. But there is another
approach, which is arguably more common in Hoare logic. We turn to
it, next.
<div class="paragraph"> </div>
<a id="lab133"></a><h1 class="section">Hoare Logic and Proof Theory</h1>
<div class="paragraph"> </div>
Instead of using states and evaluation as the basis for reasoning,
let's take the proof rules from <a href="Hoare.html"><span class="inlineref">Hoare</span></a> as the basis. Those
proof rules give us a set of axioms and inference rules that
constitute a logic in their own right. We repeat them here:
<div class="paragraph"> </div>
<div class="paragraph"> </div>
<br/>
<span class="inlinecode"> ---------------- (<span class="id" title="var">hoare_skip</span>)<br/>
<span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">P</span><span style='letter-spacing:-.4em;'>}</span>} <span class="id" title="var">skip</span> <span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">P</span><span style='letter-spacing:-.4em;'>}</span>}<br/>
<br/>
----------------------------- (<span class="id" title="var">hoare_asgn</span>)<br/>
<span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">Q</span> [<span class="id" title="var">X</span> <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>⊢</span><span style='font-size:90%;'>></span></span></span></span> <span class="id" title="var">a</span>]<span style='letter-spacing:-.4em;'>}</span>} <span class="id" title="var">X</span> := <span class="id" title="var">a</span> <span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">Q</span><span style='letter-spacing:-.4em;'>}</span>}<br/>
<br/>
<span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">P</span><span style='letter-spacing:-.4em;'>}</span>} <span class="id" title="var">c<sub>1</sub></span> <span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">Q</span><span style='letter-spacing:-.4em;'>}</span>}<br/>
<span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">Q</span><span style='letter-spacing:-.4em;'>}</span>} <span class="id" title="var">c<sub>2</sub></span> <span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">R</span><span style='letter-spacing:-.4em;'>}</span>}<br/>
------------------ (<span class="id" title="var">hoare_seq</span>)<br/>
<span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">P</span><span style='letter-spacing:-.4em;'>}</span>} <span class="id" title="var">c<sub>1</sub></span>; <span class="id" title="var">c<sub>2</sub></span> <span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">R</span><span style='letter-spacing:-.4em;'>}</span>}<br/>
<br/>
<span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">P</span> ∧ <span class="id" title="var">b</span><span style='letter-spacing:-.4em;'>}</span>} <span class="id" title="var">c<sub>1</sub></span> <span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">Q</span><span style='letter-spacing:-.4em;'>}</span>}<br/>
<span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">P</span> ∧ ¬<span class="id" title="var">b</span><span style='letter-spacing:-.4em;'>}</span>} <span class="id" title="var">c<sub>2</sub></span> <span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">Q</span><span style='letter-spacing:-.4em;'>}</span>}<br/>
------------------------------------ (<span class="id" title="var">hoare_if</span>)<br/>
<span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">P</span><span style='letter-spacing:-.4em;'>}</span>} <span class="id" title="keyword">if</span> <span class="id" title="var">b</span> <span class="id" title="keyword">then</span> <span class="id" title="var">c<sub>1</sub></span> <span class="id" title="keyword">else</span> <span class="id" title="var">c<sub>2</sub></span> <span class="id" title="keyword">end</span> <span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">Q</span><span style='letter-spacing:-.4em;'>}</span>}<br/>
<br/>
<span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">P</span> ∧ <span class="id" title="var">b</span><span style='letter-spacing:-.4em;'>}</span>} <span class="id" title="var">c</span> <span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">P</span><span style='letter-spacing:-.4em;'>}</span>}<br/>
----------------------------- (<span class="id" title="var">hoare_while</span>)<br/>
<span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">P</span>} <span class="id" title="var">while</span> <span class="id" title="var">b</span> <span class="id" title="tactic">do</span> <span class="id" title="var">c</span> <span class="id" title="keyword">end</span> <span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">P</span> ∧ ¬<span class="id" title="var">b</span><span style='letter-spacing:-.4em;'>}</span>}<br/>
<br/>
<span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">P'</span><span style='letter-spacing:-.4em;'>}</span>} <span class="id" title="var">c</span> <span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">Q'</span><span style='letter-spacing:-.4em;'>}</span>}<br/>
<span class="id" title="var">P</span> <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span><span style='letter-spacing:-.2em;'>></span>></span></span> <span class="id" title="var">P'</span><br/>
<span class="id" title="var">Q'</span> <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span><span style='letter-spacing:-.2em;'>></span>></span></span> <span class="id" title="var">Q</span><br/>
----------------------------- (<span class="id" title="var">hoare_consequence</span>)<br/>
<span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">P</span><span style='letter-spacing:-.4em;'>}</span>} <span class="id" title="var">c</span> <span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">Q</span><span style='letter-spacing:-.4em;'>}</span>}
</span>
<div class="paragraph"> </div>
Read the Hoare triples in those rules as devoid of any
meaning other than what the rules give them. Forget about states
and evaluations. They are just syntax that the rules tell us how
to manipulate in legal ways.
<div class="paragraph"> </div>
Through this new lens, triple <span class="inlinecode"><span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">X</span></span> <span class="inlinecode">=</span> <span class="inlinecode">0<span style='letter-spacing:-.4em;'>}</span>}</span> <span class="inlinecode"><span class="id" title="var">X</span></span> <span class="inlinecode">:=</span> <span class="inlinecode"><span class="id" title="var">X</span></span> <span class="inlinecode">+</span> <span class="inlinecode">1</span> <span class="inlinecode"><span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">X</span></span> <span class="inlinecode">=</span> <span class="inlinecode">1<span style='letter-spacing:-.4em;'>}</span>}</span>
is <i>derivable</i>, because we can derive a proof tree using the rules:
<div class="paragraph"> </div>
<div class="paragraph"> </div>
<br/>
<span class="inlinecode"> --------------------------- (<span class="id" title="var">hoare_asgn</span>)<br/>
<span class="id" title="var">X</span>=0 <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span><span style='letter-spacing:-.2em;'>></span>></span></span> <span class="id" title="var">X</span>+1=1 <span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">X</span>+1=1<span style='letter-spacing:-.4em;'>}</span>} <span class="id" title="var">X</span> := <span class="id" title="var">X</span>+1 <span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">X</span>=1<span style='letter-spacing:-.4em;'>}</span>}<br/>
------------------------------------------------------- (<span class="id" title="var">hoare_consequence</span>)<br/>
<span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">X</span>=0<span style='letter-spacing:-.4em;'>}</span>} <span class="id" title="var">X</span> := <span class="id" title="var">X</span>+1 <span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">X</span>=1<span style='letter-spacing:-.4em;'>}</span>}
</span>
<div class="paragraph"> </div>
At each step we have either used one of the rules, or we
have appealed to reasoning about assertions, which do not involve
Hoare triples. (Note that we have left off the trivial part of
<span class="inlinecode"><span class="id" title="var">hoare_consequence</span></span> above, namely X=1 <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span><span style='letter-spacing:-.2em;'>></span>></span></span> X=1, only because of
horizontal space contraints: it's hard to fit that many characters
on a line and have the page still be readable. If you prefer,
think of it as using <span class="inlinecode"><span class="id" title="var">hoare_consequence_pre</span></span> instead.)
<div class="paragraph"> </div>
On the other hand, <span class="inlinecode"><span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">X</span></span> <span class="inlinecode">=</span> <span class="inlinecode">0<span style='letter-spacing:-.4em;'>}</span>}</span> <span class="inlinecode"><span class="id" title="var">skip</span></span> <span class="inlinecode"><span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">X</span></span> <span class="inlinecode">=</span> <span class="inlinecode">1<span style='letter-spacing:-.4em;'>}</span>}</span> is <i>not</i> derivable,
because there is no way to apply the rules to construct a proof
tree with this triple at its root.
<div class="paragraph"> </div>
This approach gives meaning to triples not in terms of a model,
but in terms of how they can be used to construct proof trees.
It's a different way of giving semantic meaning to something
syntactic, and it's part of the branch of mathematical logic known
as <i>proof theory</i>.
<div class="paragraph"> </div>
Our goal for the rest of this chapter is to formalize Hoare logic
using proof theory, and then to prove that the model-theoretic and
proof-theoretic formalizations are consistent with one another.
<div class="paragraph"> </div>
<a id="lab134"></a><h1 class="section">Derivability</h1>
<div class="paragraph"> </div>
To formalize derivability of Hoare triples, we introduce inductive type
<span class="inlinecode"><span class="id" title="var">derivable</span></span>, which describes legal proof trees using the Hoare rules.
</div>
<div class="code">
<span class="id" title="keyword">Inductive</span> <a id="derivable" class="idref" href="#derivable"><span class="id" title="definition, inductive"><span id="derivable_rect" class="id"><span id="derivable_ind" class="id"><span id="derivable_rec" class="id"><span id="derivable_sind" class="id">derivable</span></span></span></span></span></a> : <a class="idref" href="Hoare.html#Assertion"><span class="id" title="definition">Assertion</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <span class="id" title="inductive">com</span> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Hoare.html#Assertion"><span class="id" title="definition">Assertion</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span> :=<br/>
| <a id="H_Skip" class="idref" href="#H_Skip"><span class="id" title="constructor">H_Skip</span></a> : <span class="id" title="keyword">∀</span> <a id="P:8" class="idref" href="#P:8"><span class="id" title="binder">P</span></a>,<br/>
<a class="idref" href="HoareAsLogic.html#derivable:6"><span class="id" title="inductive">derivable</span></a> <a class="idref" href="HoareAsLogic.html#P:8"><span class="id" title="variable">P</span></a> <span class="id" title="notation"><{</span><span class="id" title="notation">skip</span><span class="id" title="notation">}></span> <a class="idref" href="HoareAsLogic.html#P:8"><span class="id" title="variable">P</span></a><br/>
| <a id="H_Asgn" class="idref" href="#H_Asgn"><span class="id" title="constructor">H_Asgn</span></a> : <span class="id" title="keyword">∀</span> <a id="Q:9" class="idref" href="#Q:9"><span class="id" title="binder">Q</span></a> <a id="V:10" class="idref" href="#V:10"><span class="id" title="binder">V</span></a> <a id="a:11" class="idref" href="#a:11"><span class="id" title="binder">a</span></a>,<br/>
<a class="idref" href="HoareAsLogic.html#derivable:6"><span class="id" title="inductive">derivable</span></a> (<a class="idref" href="HoareAsLogic.html#Q:9"><span class="id" title="variable">Q</span></a> <a class="idref" href="Hoare.html#510fc37417e4aa55210e4cfe35fb96ae"><span class="id" title="notation">[</span></a><a class="idref" href="HoareAsLogic.html#V:10"><span class="id" title="variable">V</span></a> <a class="idref" href="Hoare.html#510fc37417e4aa55210e4cfe35fb96ae"><span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>⊢</span><span style='font-size:90%;'>></span></span></span></span></span></a> <a class="idref" href="HoareAsLogic.html#a:11"><span class="id" title="variable">a</span></a><a class="idref" href="Hoare.html#510fc37417e4aa55210e4cfe35fb96ae"><span class="id" title="notation">]</span></a>) <span class="id" title="notation"><{</span><a class="idref" href="HoareAsLogic.html#V:10"><span class="id" title="variable">V</span></a> <span class="id" title="notation">:=</span> <a class="idref" href="HoareAsLogic.html#a:11"><span class="id" title="variable">a</span></a><span class="id" title="notation">}></span> <a class="idref" href="HoareAsLogic.html#Q:9"><span class="id" title="variable">Q</span></a><br/>
| <a id="H_Seq" class="idref" href="#H_Seq"><span class="id" title="constructor">H_Seq</span></a> : <span class="id" title="keyword">∀</span> <a id="P:12" class="idref" href="#P:12"><span class="id" title="binder">P</span></a> <a id="c:13" class="idref" href="#c:13"><span class="id" title="binder">c</span></a> <a id="Q:14" class="idref" href="#Q:14"><span class="id" title="binder">Q</span></a> <a id="d:15" class="idref" href="#d:15"><span class="id" title="binder">d</span></a> <a id="R:16" class="idref" href="#R:16"><span class="id" title="binder">R</span></a>,<br/>
<a class="idref" href="HoareAsLogic.html#derivable:6"><span class="id" title="inductive">derivable</span></a> <a class="idref" href="HoareAsLogic.html#Q:14"><span class="id" title="variable">Q</span></a> <a class="idref" href="HoareAsLogic.html#d:15"><span class="id" title="variable">d</span></a> <a class="idref" href="HoareAsLogic.html#R:16"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="HoareAsLogic.html#derivable:6"><span class="id" title="inductive">derivable</span></a> <a class="idref" href="HoareAsLogic.html#P:12"><span class="id" title="variable">P</span></a> <a class="idref" href="HoareAsLogic.html#c:13"><span class="id" title="variable">c</span></a> <a class="idref" href="HoareAsLogic.html#Q:14"><span class="id" title="variable">Q</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="HoareAsLogic.html#derivable:6"><span class="id" title="inductive">derivable</span></a> <a class="idref" href="HoareAsLogic.html#P:12"><span class="id" title="variable">P</span></a> <span class="id" title="notation"><{</span><a class="idref" href="HoareAsLogic.html#c:13"><span class="id" title="variable">c</span></a><span class="id" title="notation">;</span><a class="idref" href="HoareAsLogic.html#d:15"><span class="id" title="variable">d</span></a><span class="id" title="notation">}></span> <a class="idref" href="HoareAsLogic.html#R:16"><span class="id" title="variable">R</span></a><br/>
| <a id="H_If" class="idref" href="#H_If"><span class="id" title="constructor">H_If</span></a> : <span class="id" title="keyword">∀</span> <a id="P:17" class="idref" href="#P:17"><span class="id" title="binder">P</span></a> <a id="Q:18" class="idref" href="#Q:18"><span class="id" title="binder">Q</span></a> <a id="b:19" class="idref" href="#b:19"><span class="id" title="binder">b</span></a> <a id="c<sub>1</sub>:20" class="idref" href="#c<sub>1</sub>:20"><span class="id" title="binder">c<sub>1</sub></span></a> <a id="c<sub>2</sub>:21" class="idref" href="#c<sub>2</sub>:21"><span class="id" title="binder">c<sub>2</sub></span></a>,<br/>
<a class="idref" href="HoareAsLogic.html#derivable:6"><span class="id" title="inductive">derivable</span></a> (<span class="id" title="keyword">fun</span> <a id="st:22" class="idref" href="#st:22"><span class="id" title="binder">st</span></a> ⇒ <a class="idref" href="HoareAsLogic.html#P:17"><span class="id" title="variable">P</span></a> <a class="idref" href="HoareAsLogic.html#st:22"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="Hoare.html#bassertion"><span class="id" title="definition">bassertion</span></a> <a class="idref" href="HoareAsLogic.html#b:19"><span class="id" title="variable">b</span></a> <a class="idref" href="HoareAsLogic.html#st:22"><span class="id" title="variable">st</span></a>) <a class="idref" href="HoareAsLogic.html#c<sub>1</sub>:20"><span class="id" title="variable">c<sub>1</sub></span></a> <a class="idref" href="HoareAsLogic.html#Q:18"><span class="id" title="variable">Q</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="HoareAsLogic.html#derivable:6"><span class="id" title="inductive">derivable</span></a> (<span class="id" title="keyword">fun</span> <a id="st:23" class="idref" href="#st:23"><span class="id" title="binder">st</span></a> ⇒ <a class="idref" href="HoareAsLogic.html#P:17"><span class="id" title="variable">P</span></a> <a class="idref" href="HoareAsLogic.html#st:23"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">~(</span></a><a class="idref" href="Hoare.html#bassertion"><span class="id" title="definition">bassertion</span></a> <a class="idref" href="HoareAsLogic.html#b:19"><span class="id" title="variable">b</span></a> <a class="idref" href="HoareAsLogic.html#st:23"><span class="id" title="variable">st</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>) <a class="idref" href="HoareAsLogic.html#c<sub>2</sub>:21"><span class="id" title="variable">c<sub>2</sub></span></a> <a class="idref" href="HoareAsLogic.html#Q:18"><span class="id" title="variable">Q</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="HoareAsLogic.html#derivable:6"><span class="id" title="inductive">derivable</span></a> <a class="idref" href="HoareAsLogic.html#P:17"><span class="id" title="variable">P</span></a> <span class="id" title="notation"><{</span><span class="id" title="notation">if</span> <a class="idref" href="HoareAsLogic.html#b:19"><span class="id" title="variable">b</span></a> <span class="id" title="notation">then</span> <a class="idref" href="HoareAsLogic.html#c<sub>1</sub>:20"><span class="id" title="variable">c<sub>1</sub></span></a> <span class="id" title="notation">else</span> <a class="idref" href="HoareAsLogic.html#c<sub>2</sub>:21"><span class="id" title="variable">c<sub>2</sub></span></a> <span class="id" title="notation">end</span><span class="id" title="notation">}></span> <a class="idref" href="HoareAsLogic.html#Q:18"><span class="id" title="variable">Q</span></a><br/>
| <a id="H_While" class="idref" href="#H_While"><span class="id" title="constructor">H_While</span></a> : <span class="id" title="keyword">∀</span> <a id="P:24" class="idref" href="#P:24"><span class="id" title="binder">P</span></a> <a id="b:25" class="idref" href="#b:25"><span class="id" title="binder">b</span></a> <a id="c:26" class="idref" href="#c:26"><span class="id" title="binder">c</span></a>,<br/>
<a class="idref" href="HoareAsLogic.html#derivable:6"><span class="id" title="inductive">derivable</span></a> (<span class="id" title="keyword">fun</span> <a id="st:27" class="idref" href="#st:27"><span class="id" title="binder">st</span></a> ⇒ <a class="idref" href="HoareAsLogic.html#P:24"><span class="id" title="variable">P</span></a> <a class="idref" href="HoareAsLogic.html#st:27"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="Hoare.html#bassertion"><span class="id" title="definition">bassertion</span></a> <a class="idref" href="HoareAsLogic.html#b:25"><span class="id" title="variable">b</span></a> <a class="idref" href="HoareAsLogic.html#st:27"><span class="id" title="variable">st</span></a>) <a class="idref" href="HoareAsLogic.html#c:26"><span class="id" title="variable">c</span></a> <a class="idref" href="HoareAsLogic.html#P:24"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="HoareAsLogic.html#derivable:6"><span class="id" title="inductive">derivable</span></a> <a class="idref" href="HoareAsLogic.html#P:24"><span class="id" title="variable">P</span></a> <span class="id" title="notation"><{</span><span class="id" title="notation">while</span> <a class="idref" href="HoareAsLogic.html#b:25"><span class="id" title="variable">b</span></a> <span class="id" title="notation">do</span> <a class="idref" href="HoareAsLogic.html#c:26"><span class="id" title="variable">c</span></a> <span class="id" title="notation">end</span><span class="id" title="notation">}></span> (<span class="id" title="keyword">fun</span> <a id="st:28" class="idref" href="#st:28"><span class="id" title="binder">st</span></a> ⇒ <a class="idref" href="HoareAsLogic.html#P:24"><span class="id" title="variable">P</span></a> <a class="idref" href="HoareAsLogic.html#st:28"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">(</span></a><a class="idref" href="Hoare.html#bassertion"><span class="id" title="definition">bassertion</span></a> <a class="idref" href="HoareAsLogic.html#b:25"><span class="id" title="variable">b</span></a> <a class="idref" href="HoareAsLogic.html#st:28"><span class="id" title="variable">st</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>)<br/>
| <a id="H_Consequence" class="idref" href="#H_Consequence"><span class="id" title="constructor">H_Consequence</span></a> : <span class="id" title="keyword">∀</span> (<a id="P:29" class="idref" href="#P:29"><span class="id" title="binder">P</span></a> <a id="Q:30" class="idref" href="#Q:30"><span class="id" title="binder">Q</span></a> <a id="P':31" class="idref" href="#P':31"><span class="id" title="binder">P'</span></a> <a id="Q':32" class="idref" href="#Q':32"><span class="id" title="binder">Q'</span></a> : <a class="idref" href="Hoare.html#Assertion"><span class="id" title="definition">Assertion</span></a>) <a id="c:33" class="idref" href="#c:33"><span class="id" title="binder">c</span></a>,<br/>
<a class="idref" href="HoareAsLogic.html#derivable:6"><span class="id" title="inductive">derivable</span></a> <a class="idref" href="HoareAsLogic.html#P':31"><span class="id" title="variable">P'</span></a> <a class="idref" href="HoareAsLogic.html#c:33"><span class="id" title="variable">c</span></a> <a class="idref" href="HoareAsLogic.html#Q':32"><span class="id" title="variable">Q'</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="st:34" class="idref" href="#st:34"><span class="id" title="binder">st</span></a>, <a class="idref" href="HoareAsLogic.html#P:29"><span class="id" title="variable">P</span></a> <a class="idref" href="HoareAsLogic.html#st:34"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="HoareAsLogic.html#P':31"><span class="id" title="variable">P'</span></a> <a class="idref" href="HoareAsLogic.html#st:34"><span class="id" title="variable">st</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="st:35" class="idref" href="#st:35"><span class="id" title="binder">st</span></a>, <a class="idref" href="HoareAsLogic.html#Q':32"><span class="id" title="variable">Q'</span></a> <a class="idref" href="HoareAsLogic.html#st:35"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="HoareAsLogic.html#Q:30"><span class="id" title="variable">Q</span></a> <a class="idref" href="HoareAsLogic.html#st:35"><span class="id" title="variable">st</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="HoareAsLogic.html#derivable:6"><span class="id" title="inductive">derivable</span></a> <a class="idref" href="HoareAsLogic.html#P:29"><span class="id" title="variable">P</span></a> <a class="idref" href="HoareAsLogic.html#c:33"><span class="id" title="variable">c</span></a> <a class="idref" href="HoareAsLogic.html#Q:30"><span class="id" title="variable">Q</span></a>.<br/>
</div>
<div class="doc">
We don't need to include axioms corresponding to
<span class="inlinecode"><span class="id" title="var">hoare_consequence_pre</span></span> or <span class="inlinecode"><span class="id" title="var">hoare_consequence_post</span></span>, because these
can be proven easily from <span class="inlinecode"><span class="id" title="var">H_Consequence</span></span>.
</div>
<div class="code">
<span class="id" title="keyword">Lemma</span> <a id="H_Consequence_pre" class="idref" href="#H_Consequence_pre"><span class="id" title="lemma">H_Consequence_pre</span></a> : <span class="id" title="keyword">∀</span> (<a id="P:36" class="idref" href="#P:36"><span class="id" title="binder">P</span></a> <a id="Q:37" class="idref" href="#Q:37"><span class="id" title="binder">Q</span></a> <a id="P':38" class="idref" href="#P':38"><span class="id" title="binder">P'</span></a>: <a class="idref" href="Hoare.html#Assertion"><span class="id" title="definition">Assertion</span></a>) <a id="c:39" class="idref" href="#c:39"><span class="id" title="binder">c</span></a>,<br/>
<a class="idref" href="HoareAsLogic.html#derivable"><span class="id" title="inductive">derivable</span></a> <a class="idref" href="HoareAsLogic.html#P':38"><span class="id" title="variable">P'</span></a> <a class="idref" href="HoareAsLogic.html#c:39"><span class="id" title="variable">c</span></a> <a class="idref" href="HoareAsLogic.html#Q:37"><span class="id" title="variable">Q</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="st:40" class="idref" href="#st:40"><span class="id" title="binder">st</span></a>, <a class="idref" href="HoareAsLogic.html#P:36"><span class="id" title="variable">P</span></a> <a class="idref" href="HoareAsLogic.html#st:40"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="HoareAsLogic.html#P':38"><span class="id" title="variable">P'</span></a> <a class="idref" href="HoareAsLogic.html#st:40"><span class="id" title="variable">st</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="HoareAsLogic.html#derivable"><span class="id" title="inductive">derivable</span></a> <a class="idref" href="HoareAsLogic.html#P:36"><span class="id" title="variable">P</span></a> <a class="idref" href="HoareAsLogic.html#c:39"><span class="id" title="variable">c</span></a> <a class="idref" href="HoareAsLogic.html#Q:37"><span class="id" title="variable">Q</span></a>.<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" title="keyword">Proof</span>. <span class="id" title="tactic">eauto</span> <span class="id" title="keyword">using</span> <a class="idref" href="HoareAsLogic.html#H_Consequence"><span class="id" title="constructor">H_Consequence</span></a>. <span class="id" title="keyword">Qed</span>.<br/>
</div>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="H_Consequence_post" class="idref" href="#H_Consequence_post"><span class="id" title="lemma">H_Consequence_post</span></a> : <span class="id" title="keyword">∀</span> (<a id="P:41" class="idref" href="#P:41"><span class="id" title="binder">P</span></a> <a id="Q:42" class="idref" href="#Q:42"><span class="id" title="binder">Q</span></a> <a id="Q':43" class="idref" href="#Q':43"><span class="id" title="binder">Q'</span></a> : <a class="idref" href="Hoare.html#Assertion"><span class="id" title="definition">Assertion</span></a>) <a id="c:44" class="idref" href="#c:44"><span class="id" title="binder">c</span></a>,<br/>
<a class="idref" href="HoareAsLogic.html#derivable"><span class="id" title="inductive">derivable</span></a> <a class="idref" href="HoareAsLogic.html#P:41"><span class="id" title="variable">P</span></a> <a class="idref" href="HoareAsLogic.html#c:44"><span class="id" title="variable">c</span></a> <a class="idref" href="HoareAsLogic.html#Q':43"><span class="id" title="variable">Q'</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="st:45" class="idref" href="#st:45"><span class="id" title="binder">st</span></a>, <a class="idref" href="HoareAsLogic.html#Q':43"><span class="id" title="variable">Q'</span></a> <a class="idref" href="HoareAsLogic.html#st:45"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="HoareAsLogic.html#Q:42"><span class="id" title="variable">Q</span></a> <a class="idref" href="HoareAsLogic.html#st:45"><span class="id" title="variable">st</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="HoareAsLogic.html#derivable"><span class="id" title="inductive">derivable</span></a> <a class="idref" href="HoareAsLogic.html#P:41"><span class="id" title="variable">P</span></a> <a class="idref" href="HoareAsLogic.html#c:44"><span class="id" title="variable">c</span></a> <a class="idref" href="HoareAsLogic.html#Q:42"><span class="id" title="variable">Q</span></a>.<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" title="keyword">Proof</span>. <span class="id" title="tactic">eauto</span> <span class="id" title="keyword">using</span> <a class="idref" href="HoareAsLogic.html#H_Consequence"><span class="id" title="constructor">H_Consequence</span></a>. <span class="id" title="keyword">Qed</span>.<br/>
</div>
</div>
<div class="doc">
As an example, let's construct a proof tree for
<br/>
<span class="inlinecode"> <span style='letter-spacing:-.4em;'>{</span>{(<span class="id" title="var">X</span>=3) [<span class="id" title="var">X</span> <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>⊢</span><span style='font-size:90%;'>></span></span></span></span> <span class="id" title="var">X</span> + 2] [<span class="id" title="var">X</span> <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>⊢</span><span style='font-size:90%;'>></span></span></span></span> <span class="id" title="var">X</span> + 1]<span style='letter-spacing:-.4em;'>}</span>}<br/>
<span class="id" title="var">X</span> := <span class="id" title="var">X</span> + 1;<br/>
<span class="id" title="var">X</span> := <span class="id" title="var">X</span> + 2<br/>
<span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">X</span>=3<span style='letter-spacing:-.4em;'>}</span>}
</span>
</div>
<div class="code">
<span class="id" title="keyword">Example</span> <a id="sample_proof" class="idref" href="#sample_proof"><span class="id" title="definition">sample_proof</span></a> :<br/>
<a class="idref" href="HoareAsLogic.html#derivable"><span class="id" title="inductive">derivable</span></a><br/>
(<a class="idref" href="Hoare.html#510fc37417e4aa55210e4cfe35fb96ae"><span class="id" title="notation">(</span></a><span class="id" title="keyword">fun</span> <a id="st:46" class="idref" href="#st:46"><span class="id" title="binder">st</span></a>:<span class="id" title="definition">state</span> ⇒ <a class="idref" href="HoareAsLogic.html#st:46"><span class="id" title="variable">st</span></a> <span class="id" title="definition">X</span> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 3<a class="idref" href="Hoare.html#510fc37417e4aa55210e4cfe35fb96ae"><span class="id" title="notation">)</span></a> <a class="idref" href="Hoare.html#510fc37417e4aa55210e4cfe35fb96ae"><span class="id" title="notation">[</span></a><span class="id" title="definition">X</span> <a class="idref" href="Hoare.html#510fc37417e4aa55210e4cfe35fb96ae"><span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>⊢</span><span style='font-size:90%;'>></span></span></span></span></span></a> <span class="id" title="definition">X</span> <span class="id" title="notation">+</span> 2<a class="idref" href="Hoare.html#510fc37417e4aa55210e4cfe35fb96ae"><span class="id" title="notation">]</span></a> <a class="idref" href="Hoare.html#510fc37417e4aa55210e4cfe35fb96ae"><span class="id" title="notation">[</span></a><span class="id" title="definition">X</span> <a class="idref" href="Hoare.html#510fc37417e4aa55210e4cfe35fb96ae"><span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>⊢</span><span style='font-size:90%;'>></span></span></span></span></span></a> <span class="id" title="definition">X</span> <span class="id" title="notation">+</span> 1<a class="idref" href="Hoare.html#510fc37417e4aa55210e4cfe35fb96ae"><span class="id" title="notation">]</span></a>)<br/>
<span class="id" title="notation"><{</span> <span class="id" title="definition">X</span> <span class="id" title="notation">:=</span> <span class="id" title="definition">X</span> <span class="id" title="notation">+</span> 1<span class="id" title="notation">;</span> <span class="id" title="definition">X</span> <span class="id" title="notation">:=</span> <span class="id" title="definition">X</span> <span class="id" title="notation">+</span> 2<span class="id" title="notation">}></span><br/>
(<span class="id" title="keyword">fun</span> <a id="st:47" class="idref" href="#st:47"><span class="id" title="binder">st</span></a>:<span class="id" title="definition">state</span> ⇒ <a class="idref" href="HoareAsLogic.html#st:47"><span class="id" title="variable">st</span></a> <span class="id" title="definition">X</span> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 3).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">eapply</span> <a class="idref" href="HoareAsLogic.html#H_Seq"><span class="id" title="constructor">H_Seq</span></a>.<br/>
- <span class="id" title="tactic">apply</span> <a class="idref" href="HoareAsLogic.html#H_Asgn"><span class="id" title="constructor">H_Asgn</span></a>.<br/>
- <span class="id" title="tactic">apply</span> <a class="idref" href="HoareAsLogic.html#H_Asgn"><span class="id" title="constructor">H_Asgn</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
You can see how the structure of the proof script mirrors the structure
of the proof tree: at the root there is a use of the sequence rule; and
at the leaves, the assignment rule.
<div class="paragraph"> </div>
<a id="lab135"></a><h4 class="section">Exercise: 3 stars, standard (provable_true_post)</h4>
<div class="paragraph"> </div>
Show that any Hoare triple whose postcondition is <span class="inlinecode"><span class="id" title="var">True</span></span> is derivable. Proceed
by induction on <span class="inlinecode"><span class="id" title="var">c</span></span>.
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <a id="provable_true_post" class="idref" href="#provable_true_post"><span class="id" title="lemma">provable_true_post</span></a> : <span class="id" title="keyword">∀</span> <a id="c:48" class="idref" href="#c:48"><span class="id" title="binder">c</span></a> <a id="P:49" class="idref" href="#P:49"><span class="id" title="binder">P</span></a>,<br/>
<a class="idref" href="HoareAsLogic.html#derivable"><span class="id" title="inductive">derivable</span></a> <a class="idref" href="HoareAsLogic.html#P:49"><span class="id" title="variable">P</span></a> <a class="idref" href="HoareAsLogic.html#c:48"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" title="var">Admitted</span>.<br/>
<font size=-2>☐</font>
</div>
<div class="doc">
<div class="paragraph"> </div>
<a id="lab136"></a><h4 class="section">Exercise: 3 stars, standard (provable_false_pre)</h4>
<div class="paragraph"> </div>
Show that any Hoare triple whose precondition is <span class="inlinecode"><span class="id" title="var">False</span></span> is derivable. Again,
proceed by induction on <span class="inlinecode"><span class="id" title="var">c</span></span>.
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <a id="provable_false_pre" class="idref" href="#provable_false_pre"><span class="id" title="lemma">provable_false_pre</span></a> : <span class="id" title="keyword">∀</span> <a id="c:50" class="idref" href="#c:50"><span class="id" title="binder">c</span></a> <a id="Q:51" class="idref" href="#Q:51"><span class="id" title="binder">Q</span></a>,<br/>
<a class="idref" href="HoareAsLogic.html#derivable"><span class="id" title="inductive">derivable</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a> <a class="idref" href="HoareAsLogic.html#c:50"><span class="id" title="variable">c</span></a> <a class="idref" href="HoareAsLogic.html#Q:51"><span class="id" title="variable">Q</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" title="var">Admitted</span>.<br/>
<font size=-2>☐</font>
</div>
<div class="doc">
<div class="paragraph"> </div>
<a id="lab137"></a><h1 class="section">Soundness and Completeness</h1>
<div class="paragraph"> </div>
We now have two approaches to formulating Hoare logic:
<div class="paragraph"> </div>
<ul class="doclist">
<li> The model-theoretic approach uses <span class="inlinecode"><span class="id" title="var">valid</span></span> to characterize when a Hoare
triple holds in a model, which is based on states.
<div class="paragraph"> </div>
</li>
<li> The proof-theoretic approach uses <span class="inlinecode"><span class="id" title="var">derivable</span></span> to characterize when a Hoare
triple is derivable as the end of a proof tree.
</li>
</ul>
<div class="paragraph"> </div>
Do these two approaches agree? That is, are the valid Hoare triples exactly
the derivable ones? This is a standard question investigated in
mathematical logic. There are two pieces to answering it:
<div class="paragraph"> </div>
<ul class="doclist">
<li> A logic is <i>sound</i> if everything that is derivable is valid.
<div class="paragraph"> </div>
</li>
<li> A logic is <i>complete</i> if everything that is valid is derivable.
</li>
</ul>
<div class="paragraph"> </div>
We can prove that Hoare logic is sound and complete.
<div class="paragraph"> </div>
<div class="paragraph"> </div>
<a id="lab138"></a><h4 class="section">Exercise: 3 stars, standard (hoare_sound)</h4>
<div class="paragraph"> </div>
Prove that if a Hoare triple is derivable, then it is valid.
Nearly all the work for this was already done in <a href="Hoare.html"><span class="inlineref">Hoare</span></a> as
theorems <span class="inlinecode"><span class="id" title="var">hoare_skip</span></span>, <span class="inlinecode"><span class="id" title="var">hoare_asgn</span></span>, etc.; leverage those
proofs. Proceed by induction on the derivation of the triple.
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <a id="hoare_sound" class="idref" href="#hoare_sound"><span class="id" title="lemma">hoare_sound</span></a> : <span class="id" title="keyword">∀</span> <a id="P:52" class="idref" href="#P:52"><span class="id" title="binder">P</span></a> <a id="c:53" class="idref" href="#c:53"><span class="id" title="binder">c</span></a> <a id="Q:54" class="idref" href="#Q:54"><span class="id" title="binder">Q</span></a>,<br/>
<a class="idref" href="HoareAsLogic.html#derivable"><span class="id" title="inductive">derivable</span></a> <a class="idref" href="HoareAsLogic.html#P:52"><span class="id" title="variable">P</span></a> <a class="idref" href="HoareAsLogic.html#c:53"><span class="id" title="variable">c</span></a> <a class="idref" href="HoareAsLogic.html#Q:54"><span class="id" title="variable">Q</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="HoareAsLogic.html#valid"><span class="id" title="definition">valid</span></a> <a class="idref" href="HoareAsLogic.html#P:52"><span class="id" title="variable">P</span></a> <a class="idref" href="HoareAsLogic.html#c:53"><span class="id" title="variable">c</span></a> <a class="idref" href="HoareAsLogic.html#Q:54"><span class="id" title="variable">Q</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" title="var">Admitted</span>.<br/>
<font size=-2>☐</font>
</div>
<div class="doc">
<div class="paragraph"> </div>
The proof of completeness is more challenging. To carry out the
proof, we need to invent some intermediate assertions using a
technical device known as <i>weakest preconditions</i> (which are also
discussed in <a href="Hoare2.html"><span class="inlineref">Hoare2</span></a>). Given a command <span class="inlinecode"><span class="id" title="var">c</span></span> and a desired
postcondition assertion <span class="inlinecode"><span class="id" title="var">Q</span></span>, the weakest precondition <span class="inlinecode"><span class="id" title="var">wp</span></span> <span class="inlinecode"><span class="id" title="var">c</span></span> <span class="inlinecode"><span class="id" title="var">Q</span></span> is
an assertion <span class="inlinecode"><span class="id" title="var">P</span></span> such that <span class="inlinecode"><span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">P</span><span style='letter-spacing:-.4em;'>}</span>}</span> <span class="inlinecode"><span class="id" title="var">c</span></span> <span class="inlinecode"><span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">Q</span><span style='letter-spacing:-.4em;'>}</span>}</span> holds, and moreover,
for any other assertion <span class="inlinecode"><span class="id" title="var">P'</span></span>, if <span class="inlinecode"><span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">P'</span><span style='letter-spacing:-.4em;'>}</span>}</span> <span class="inlinecode"><span class="id" title="var">c</span></span> <span class="inlinecode"><span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">Q</span><span style='letter-spacing:-.4em;'>}</span>}</span> holds then <span class="inlinecode"><span class="id" title="var">P'</span></span>
<span class="inlinecode"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span><span style='letter-spacing:-.2em;'>></span>></span></span></span> <span class="inlinecode"><span class="id" title="var">P</span></span>.
<div class="paragraph"> </div>
Another way of stating that idea is that <span class="inlinecode"><span class="id" title="var">wp</span></span> <span class="inlinecode"><span class="id" title="var">c</span></span> <span class="inlinecode"><span class="id" title="var">Q</span></span> is the following
assertion:
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <a id="wp" class="idref" href="#wp"><span class="id" title="definition">wp</span></a> (<a id="c:55" class="idref" href="#c:55"><span class="id" title="binder">c</span></a>:<span class="id" title="inductive">com</span>) (<a id="Q:56" class="idref" href="#Q:56"><span class="id" title="binder">Q</span></a>:<a class="idref" href="Hoare.html#Assertion"><span class="id" title="definition">Assertion</span></a>) : <a class="idref" href="Hoare.html#Assertion"><span class="id" title="definition">Assertion</span></a> :=<br/>
<span class="id" title="keyword">fun</span> <a id="s:57" class="idref" href="#s:57"><span class="id" title="binder">s</span></a> ⇒ <span class="id" title="keyword">∀</span> <a id="s':58" class="idref" href="#s':58"><span class="id" title="binder">s'</span></a>, <a class="idref" href="HoareAsLogic.html#s:57"><span class="id" title="variable">s</span></a> <span class="id" title="notation">=[</span> <a class="idref" href="HoareAsLogic.html#c:55"><span class="id" title="variable">c</span></a> <span class="id" title="notation">]=></span> <a class="idref" href="HoareAsLogic.html#s':58"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="HoareAsLogic.html#Q:56"><span class="id" title="variable">Q</span></a> <a class="idref" href="HoareAsLogic.html#s':58"><span class="id" title="variable">s'</span></a>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Hint Unfold</span> <a class="idref" href="HoareAsLogic.html#wp"><span class="id" title="definition">wp</span></a> : <span class="id" title="var">core</span>.<br/>
</div>
<div class="doc">
The following two theorems show that the two ways of thinking
about <span class="inlinecode"><span class="id" title="var">wp</span></span> are the same.
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <a id="wp_is_precondition" class="idref" href="#wp_is_precondition"><span class="id" title="lemma">wp_is_precondition</span></a> : <span class="id" title="keyword">∀</span> <a id="c:59" class="idref" href="#c:59"><span class="id" title="binder">c</span></a> <a id="Q:60" class="idref" href="#Q:60"><span class="id" title="binder">Q</span></a>,<br/>
<a class="idref" href="Hoare.html#a18cbce9fe584a9130c3f34d4f14555f"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>{</span>{</span></a><a class="idref" href="HoareAsLogic.html#wp"><span class="id" title="definition">wp</span></a> <a class="idref" href="HoareAsLogic.html#c:59"><span class="id" title="variable">c</span></a> <a class="idref" href="HoareAsLogic.html#Q:60"><span class="id" title="variable">Q</span></a><a class="idref" href="Hoare.html#a18cbce9fe584a9130c3f34d4f14555f"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}</span></a> <a class="idref" href="HoareAsLogic.html#c:59"><span class="id" title="variable">c</span></a> <a class="idref" href="Hoare.html#a18cbce9fe584a9130c3f34d4f14555f"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>{</span>{</span></a><a class="idref" href="HoareAsLogic.html#Q:60"><span class="id" title="variable">Q</span></a><a class="idref" href="Hoare.html#a18cbce9fe584a9130c3f34d4f14555f"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}</span></a>.<br/>
<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">auto</span>. <span class="id" title="keyword">Qed</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Theorem</span> <a id="wp_is_weakest" class="idref" href="#wp_is_weakest"><span class="id" title="lemma">wp_is_weakest</span></a> : <span class="id" title="keyword">∀</span> <a id="c:61" class="idref" href="#c:61"><span class="id" title="binder">c</span></a> <a id="Q:62" class="idref" href="#Q:62"><span class="id" title="binder">Q</span></a> <a id="P':63" class="idref" href="#P':63"><span class="id" title="binder">P'</span></a>,<br/>
<a class="idref" href="Hoare.html#a18cbce9fe584a9130c3f34d4f14555f"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>{</span>{</span></a><a class="idref" href="HoareAsLogic.html#P':63"><span class="id" title="variable">P'</span></a><a class="idref" href="Hoare.html#a18cbce9fe584a9130c3f34d4f14555f"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}</span></a> <a class="idref" href="HoareAsLogic.html#c:61"><span class="id" title="variable">c</span></a> <a class="idref" href="Hoare.html#a18cbce9fe584a9130c3f34d4f14555f"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>{</span>{</span></a><a class="idref" href="HoareAsLogic.html#Q:62"><span class="id" title="variable">Q</span></a><a class="idref" href="Hoare.html#a18cbce9fe584a9130c3f34d4f14555f"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="HoareAsLogic.html#P':63"><span class="id" title="variable">P'</span></a> <a class="idref" href="Hoare.html#::hoare_spec_scope:x_'->>'_x"><span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span><span style='letter-spacing:-.2em;'>></span>></span></span></span></a> <a class="idref" href="Hoare.html#::hoare_spec_scope:x_'->>'_x"><span class="id" title="notation">(</span></a><a class="idref" href="HoareAsLogic.html#wp"><span class="id" title="definition">wp</span></a> <a class="idref" href="HoareAsLogic.html#c:61"><span class="id" title="variable">c</span></a> <a class="idref" href="HoareAsLogic.html#Q:62"><span class="id" title="variable">Q</span></a><a class="idref" href="Hoare.html#::hoare_spec_scope:x_'->>'_x"><span class="id" title="notation">)</span></a>.<br/>
<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">eauto</span>. <span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
Weakest preconditions are useful because they enable us to
identify assertions that otherwise would require clever thinking.
The next two lemmas show that in action.
<div class="paragraph"> </div>
<a id="lab139"></a><h4 class="section">Exercise: 1 star, standard (wp_seq)</h4>
<div class="paragraph"> </div>
What if we have a sequence <span class="inlinecode"><span class="id" title="var">c<sub>1</sub></span>;</span> <span class="inlinecode"><span class="id" title="var">c<sub>2</sub></span></span>, but not an intermediate assertion for
what should hold in between <span class="inlinecode"><span class="id" title="var">c<sub>1</sub></span></span> and <span class="inlinecode"><span class="id" title="var">c<sub>2</sub></span></span>? No problem. Prove that <span class="inlinecode"><span class="id" title="var">wp</span></span> <span class="inlinecode"><span class="id" title="var">c<sub>2</sub></span></span> <span class="inlinecode"><span class="id" title="var">Q</span></span>
suffices as such an assertion.
</div>
<div class="code">
<span class="id" title="keyword">Lemma</span> <a id="wp_seq" class="idref" href="#wp_seq"><span class="id" title="lemma">wp_seq</span></a> : <span class="id" title="keyword">∀</span> <a id="P:64" class="idref" href="#P:64"><span class="id" title="binder">P</span></a> <a id="Q:65" class="idref" href="#Q:65"><span class="id" title="binder">Q</span></a> <a id="c<sub>1</sub>:66" class="idref" href="#c<sub>1</sub>:66"><span class="id" title="binder">c<sub>1</sub></span></a> <a id="c<sub>2</sub>:67" class="idref" href="#c<sub>2</sub>:67"><span class="id" title="binder">c<sub>2</sub></span></a>,<br/>
<a class="idref" href="HoareAsLogic.html#derivable"><span class="id" title="inductive">derivable</span></a> <a class="idref" href="HoareAsLogic.html#P:64"><span class="id" title="variable">P</span></a> <a class="idref" href="HoareAsLogic.html#c<sub>1</sub>:66"><span class="id" title="variable">c<sub>1</sub></span></a> (<a class="idref" href="HoareAsLogic.html#wp"><span class="id" title="definition">wp</span></a> <a class="idref" href="HoareAsLogic.html#c<sub>2</sub>:67"><span class="id" title="variable">c<sub>2</sub></span></a> <a class="idref" href="HoareAsLogic.html#Q:65"><span class="id" title="variable">Q</span></a>) <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="HoareAsLogic.html#derivable"><span class="id" title="inductive">derivable</span></a> (<a class="idref" href="HoareAsLogic.html#wp"><span class="id" title="definition">wp</span></a> <a class="idref" href="HoareAsLogic.html#c<sub>2</sub>:67"><span class="id" title="variable">c<sub>2</sub></span></a> <a class="idref" href="HoareAsLogic.html#Q:65"><span class="id" title="variable">Q</span></a>) <a class="idref" href="HoareAsLogic.html#c<sub>2</sub>:67"><span class="id" title="variable">c<sub>2</sub></span></a> <a class="idref" href="HoareAsLogic.html#Q:65"><span class="id" title="variable">Q</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="HoareAsLogic.html#derivable"><span class="id" title="inductive">derivable</span></a> <a class="idref" href="HoareAsLogic.html#P:64"><span class="id" title="variable">P</span></a> <span class="id" title="notation"><{</span><a class="idref" href="HoareAsLogic.html#c<sub>1</sub>:66"><span class="id" title="variable">c<sub>1</sub></span></a><span class="id" title="notation">;</span> <a class="idref" href="HoareAsLogic.html#c<sub>2</sub>:67"><span class="id" title="variable">c<sub>2</sub></span></a><span class="id" title="notation">}></span> <a class="idref" href="HoareAsLogic.html#Q:65"><span class="id" title="variable">Q</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" title="var">Admitted</span>.<br/>
<font size=-2>☐</font>
</div>
<div class="doc">
<div class="paragraph"> </div>
<a id="lab140"></a><h4 class="section">Exercise: 2 stars, standard (wp_invariant)</h4>
<div class="paragraph"> </div>
What if we have a while loop, but not an invariant for it? No
problem. Prove that for any <span class="inlinecode"><span class="id" title="var">Q</span></span>, assertion <span class="inlinecode"><span class="id" title="var">wp</span></span> <span class="inlinecode">(<span class="id" title="var">while</span></span> <span class="inlinecode"><span class="id" title="var">b</span></span> <span class="inlinecode"><span class="id" title="tactic">do</span></span> <span class="inlinecode"><span class="id" title="var">c</span></span> <span class="inlinecode"><span class="id" title="keyword">end</span>)</span>
<span class="inlinecode"><span class="id" title="var">Q</span></span> is a loop invariant of <span class="inlinecode"><span class="id" title="var">while</span></span> <span class="inlinecode"><span class="id" title="var">b</span></span> <span class="inlinecode"><span class="id" title="tactic">do</span></span> <span class="inlinecode"><span class="id" title="var">c</span></span> <span class="inlinecode"><span class="id" title="keyword">end</span></span>.
</div>
<div class="code">
<span class="id" title="keyword">Lemma</span> <a id="wp_invariant" class="idref" href="#wp_invariant"><span class="id" title="lemma">wp_invariant</span></a> : <span class="id" title="keyword">∀</span> <a id="b:68" class="idref" href="#b:68"><span class="id" title="binder">b</span></a> <a id="c:69" class="idref" href="#c:69"><span class="id" title="binder">c</span></a> <a id="Q:70" class="idref" href="#Q:70"><span class="id" title="binder">Q</span></a>,<br/>
<a class="idref" href="HoareAsLogic.html#valid"><span class="id" title="definition">valid</span></a> (<a class="idref" href="HoareAsLogic.html#wp"><span class="id" title="definition">wp</span></a> <span class="id" title="notation"><{</span><span class="id" title="notation">while</span> <a class="idref" href="HoareAsLogic.html#b:68"><span class="id" title="variable">b</span></a> <span class="id" title="notation">do</span> <a class="idref" href="HoareAsLogic.html#c:69"><span class="id" title="variable">c</span></a> <span class="id" title="notation">end</span><span class="id" title="notation">}></span> <a class="idref" href="HoareAsLogic.html#Q:70"><span class="id" title="variable">Q</span></a> <a class="idref" href="Hoare.html#cce4ad113420d0c25c6ce4891bbe5028"><span class="id" title="notation">∧</span></a> <a class="idref" href="HoareAsLogic.html#b:68"><span class="id" title="variable">b</span></a>) <a class="idref" href="HoareAsLogic.html#c:69"><span class="id" title="variable">c</span></a> (<a class="idref" href="HoareAsLogic.html#wp"><span class="id" title="definition">wp</span></a> <span class="id" title="notation"><{</span><span class="id" title="notation">while</span> <a class="idref" href="HoareAsLogic.html#b:68"><span class="id" title="variable">b</span></a> <span class="id" title="notation">do</span> <a class="idref" href="HoareAsLogic.html#c:69"><span class="id" title="variable">c</span></a> <span class="id" title="notation">end</span><span class="id" title="notation">}></span> <a class="idref" href="HoareAsLogic.html#Q:70"><span class="id" title="variable">Q</span></a>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" title="var">Admitted</span>.<br/>
<font size=-2>☐</font>
</div>
<div class="doc">
<div class="paragraph"> </div>
<a id="lab141"></a><h4 class="section">Exercise: 4 stars, standard (hoare_complete)</h4>
<div class="paragraph"> </div>
Now we are ready to prove the completeness of Hoare logic. Finish
the proof of the theorem below.
<div class="paragraph"> </div>
Hint: for the <span class="inlinecode"><span class="id" title="var">while</span></span> case, use the invariant suggested by
<span class="inlinecode"><span class="id" title="var">wp_invariant</span></span>.
<div class="paragraph"> </div>
Acknowledgment: Our approach to this proof is inspired by:
<div class="paragraph"> </div>
<a href="https://www.ps.uni-saarland.de/courses/sem-ws<sub>11</sub>/script/Hoare.html"><span class="inlineref" <a href='https://www.ps.uni-saarland.de/courses/sem-ws<sub>11</sub>/script/Hoare.html</span></a>'>https://www.ps.uni-saarland.de/courses/sem-ws<sub>11</sub>/script/Hoare.html</span></a></a>
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <a id="hoare_complete" class="idref" href="#hoare_complete"><span class="id" title="lemma">hoare_complete</span></a>: <span class="id" title="keyword">∀</span> <a id="P:71" class="idref" href="#P:71"><span class="id" title="binder">P</span></a> <a id="c:72" class="idref" href="#c:72"><span class="id" title="binder">c</span></a> <a id="Q:73" class="idref" href="#Q:73"><span class="id" title="binder">Q</span></a>,<br/>
<a class="idref" href="HoareAsLogic.html#valid"><span class="id" title="definition">valid</span></a> <a class="idref" href="HoareAsLogic.html#P:71"><span class="id" title="variable">P</span></a> <a class="idref" href="HoareAsLogic.html#c:72"><span class="id" title="variable">c</span></a> <a class="idref" href="HoareAsLogic.html#Q:73"><span class="id" title="variable">Q</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="HoareAsLogic.html#derivable"><span class="id" title="inductive">derivable</span></a> <a class="idref" href="HoareAsLogic.html#P:71"><span class="id" title="variable">P</span></a> <a class="idref" href="HoareAsLogic.html#c:72"><span class="id" title="variable">c</span></a> <a class="idref" href="HoareAsLogic.html#Q:73"><span class="id" title="variable">Q</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="HoareAsLogic.html#valid"><span class="id" title="definition">valid</span></a>. <span class="id" title="tactic">intros</span> <span class="id" title="var">P</span> <span class="id" title="var">c</span>. <span class="id" title="tactic">generalize</span> <span class="id" title="tactic">dependent</span> <span class="id" title="var">P</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">c</span>; <span class="id" title="tactic">intros</span> <span class="id" title="var">P</span> <span class="id" title="var">Q</span> <span class="id" title="var">HT</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" title="var">Admitted</span>.<br/>
<font size=-2>☐</font>
</div>
<div class="doc">
<div class="paragraph"> </div>
<a id="lab142"></a><h1 class="section">Postscript: Decidability</h1>
<div class="paragraph"> </div>
We might hope that Hoare logic would be <i>decidable</i>; that is, that
there would be an (terminating) algorithm (a <i>decision procedure</i>)
that can determine whether or not a given Hoare triple is valid or
derivable. Sadly, such a decision procedure cannot exist.
<div class="paragraph"> </div>
Consider the triple <span class="inlinecode"><span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">True</span><span style='letter-spacing:-.4em;'>}</span>}</span> <span class="inlinecode"><span class="id" title="var">c</span></span> <span class="inlinecode"><span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">False</span><span style='letter-spacing:-.4em;'>}</span>}</span>. This triple is valid
if and only if <span class="inlinecode"><span class="id" title="var">c</span></span> is non-terminating. So any algorithm that
could determine validity of arbitrary triples could solve the
Halting Problem.
<div class="paragraph"> </div>
Similarly, the triple <span class="inlinecode"><span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">True</span><span style='letter-spacing:-.4em;'>}</span>}</span> <span class="inlinecode"><span class="id" title="var">skip</span></span> <span class="inlinecode"><span style='letter-spacing:-.4em;'>{</span>{<span class="id" title="var">P</span><span style='letter-spacing:-.4em;'>}</span>}</span> is valid if and only
if <span class="inlinecode"><span class="id" title="keyword">∀</span></span> <span class="inlinecode"><span class="id" title="var">s</span>,</span> <span class="inlinecode"><span class="id" title="var">P</span></span> <span class="inlinecode"><span class="id" title="var">s</span></span> is valid, where <span class="inlinecode"><span class="id" title="var">P</span></span> is an arbitrary assertion
of Coq's logic. But this logic is far too powerful to be
decidable.
</div>
<div class="code">
<span class="comment">(* 2024-08-25 08:25 *)</span><br/>
</div>
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