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<title>HoareAsLogic: Hoare Logic as a Logic</title>
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<h1 class="libtitle">HoareAsLogic<span class="subtitle">Hoare Logic as a Logic</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">Export</span> <span class="id" type="var">Hoare</span>.<br/>
<br/>
</div>
<div class="doc">
The presentation of Hoare logic in chapter <span class="inlinecode"><span class="id" type="var">Hoare</span></span> 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>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">hoare_proof</span> : <span class="id" type="var">Assertion</span> <span style="font-family: arial;">→</span> <span class="id" type="var">com</span> <span style="font-family: arial;">→</span> <span class="id" type="var">Assertion</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Type</span> :=<br/>
| <span class="id" type="var">H_Skip</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">P</span>,<br/>
<span class="id" type="var">hoare_proof</span> <span class="id" type="var">P</span> (<span class="id" type="var">SKIP</span>) <span class="id" type="var">P</span><br/>
| <span class="id" type="var">H_Asgn</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">Q</span> <span class="id" type="var">V</span> <span class="id" type="var">a</span>,<br/>
<span class="id" type="var">hoare_proof</span> (<span class="id" type="var">assn_sub</span> <span class="id" type="var">V</span> <span class="id" type="var">a</span> <span class="id" type="var">Q</span>) (<span class="id" type="var">V</span> ::= <span class="id" type="var">a</span>) <span class="id" type="var">Q</span><br/>
| <span class="id" type="var">H_Seq</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">P</span> <span class="id" type="var">c</span> <span class="id" type="var">Q</span> <span class="id" type="var">d</span> <span class="id" type="var">R</span>,<br/>
<span class="id" type="var">hoare_proof</span> <span class="id" type="var">P</span> <span class="id" type="var">c</span> <span class="id" type="var">Q</span> <span style="font-family: arial;">→</span> <span class="id" type="var">hoare_proof</span> <span class="id" type="var">Q</span> <span class="id" type="var">d</span> <span class="id" type="var">R</span> <span style="font-family: arial;">→</span> <span class="id" type="var">hoare_proof</span> <span class="id" type="var">P</span> (<span class="id" type="var">c</span>;;<span class="id" type="var">d</span>) <span class="id" type="var">R</span><br/>
| <span class="id" type="var">H_If</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">P</span> <span class="id" type="var">Q</span> <span class="id" type="var">b</span> <span class="id" type="var">c1</span> <span class="id" type="var">c2</span>,<br/>
<span class="id" type="var">hoare_proof</span> (<span class="id" type="keyword">fun</span> <span class="id" type="var">st</span> ⇒ <span class="id" type="var">P</span> <span class="id" type="var">st</span> <span style="font-family: arial;">∧</span> <span class="id" type="var">bassn</span> <span class="id" type="var">b</span> <span class="id" type="var">st</span>) <span class="id" type="var">c1</span> <span class="id" type="var">Q</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">hoare_proof</span> (<span class="id" type="keyword">fun</span> <span class="id" type="var">st</span> ⇒ <span class="id" type="var">P</span> <span class="id" type="var">st</span> <span style="font-family: arial;">∧</span> ~(<span class="id" type="var">bassn</span> <span class="id" type="var">b</span> <span class="id" type="var">st</span>)) <span class="id" type="var">c2</span> <span class="id" type="var">Q</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">hoare_proof</span> <span class="id" type="var">P</span> (<span class="id" type="var">IFB</span> <span class="id" type="var">b</span> <span class="id" type="var">THEN</span> <span class="id" type="var">c1</span> <span class="id" type="var">ELSE</span> <span class="id" type="var">c2</span> <span class="id" type="var">FI</span>) <span class="id" type="var">Q</span><br/>
| <span class="id" type="var">H_While</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">P</span> <span class="id" type="var">b</span> <span class="id" type="var">c</span>,<br/>
<span class="id" type="var">hoare_proof</span> (<span class="id" type="keyword">fun</span> <span class="id" type="var">st</span> ⇒ <span class="id" type="var">P</span> <span class="id" type="var">st</span> <span style="font-family: arial;">∧</span> <span class="id" type="var">bassn</span> <span class="id" type="var">b</span> <span class="id" type="var">st</span>) <span class="id" type="var">c</span> <span class="id" type="var">P</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">hoare_proof</span> <span class="id" type="var">P</span> (<span class="id" type="var">WHILE</span> <span class="id" type="var">b</span> <span class="id" type="var">DO</span> <span class="id" type="var">c</span> <span class="id" type="var">END</span>) (<span class="id" type="keyword">fun</span> <span class="id" type="var">st</span> ⇒ <span class="id" type="var">P</span> <span class="id" type="var">st</span> <span style="font-family: arial;">∧</span> ¬ (<span class="id" type="var">bassn</span> <span class="id" type="var">b</span> <span class="id" type="var">st</span>))<br/>
| <span class="id" type="var">H_Consequence</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">P</span> <span class="id" type="var">Q</span> <span class="id" type="var">P'</span> <span class="id" type="var">Q'</span> : <span class="id" type="var">Assertion</span>) <span class="id" type="var">c</span>,<br/>
<span class="id" type="var">hoare_proof</span> <span class="id" type="var">P'</span> <span class="id" type="var">c</span> <span class="id" type="var">Q'</span> <span style="font-family: arial;">→</span><br/>
(<span style="font-family: arial;">∀</span><span class="id" type="var">st</span>, <span class="id" type="var">P</span> <span class="id" type="var">st</span> <span style="font-family: arial;">→</span> <span class="id" type="var">P'</span> <span class="id" type="var">st</span>) <span style="font-family: arial;">→</span><br/>
(<span style="font-family: arial;">∀</span><span class="id" type="var">st</span>, <span class="id" type="var">Q'</span> <span class="id" type="var">st</span> <span style="font-family: arial;">→</span> <span class="id" type="var">Q</span> <span class="id" type="var">st</span>) <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">hoare_proof</span> <span class="id" type="var">P</span> <span class="id" type="var">c</span> <span class="id" type="var">Q</span>.<br/>
<br/>
</div>
<div class="doc">
We don't need to include axioms corresponding to <span class="inlinecode"><span class="id" type="var">hoare_consequence_pre</span></span>
or <span class="inlinecode"><span class="id" type="var">hoare_consequence_post</span></span>, because these can be proven easily
from <span class="inlinecode"><span class="id" type="var">H_Consequence</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">H_Consequence_pre</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">P</span> <span class="id" type="var">Q</span> <span class="id" type="var">P'</span>: <span class="id" type="var">Assertion</span>) <span class="id" type="var">c</span>,<br/>
<span class="id" type="var">hoare_proof</span> <span class="id" type="var">P'</span> <span class="id" type="var">c</span> <span class="id" type="var">Q</span> <span style="font-family: arial;">→</span><br/>
(<span style="font-family: arial;">∀</span><span class="id" type="var">st</span>, <span class="id" type="var">P</span> <span class="id" type="var">st</span> <span style="font-family: arial;">→</span> <span class="id" type="var">P'</span> <span class="id" type="var">st</span>) <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">hoare_proof</span> <span class="id" type="var">P</span> <span class="id" type="var">c</span> <span class="id" type="var">Q</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">H_Consequence_post</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">P</span> <span class="id" type="var">Q</span> <span class="id" type="var">Q'</span> : <span class="id" type="var">Assertion</span>) <span class="id" type="var">c</span>,<br/>
<span class="id" type="var">hoare_proof</span> <span class="id" type="var">P</span> <span class="id" type="var">c</span> <span class="id" type="var">Q'</span> <span style="font-family: arial;">→</span><br/>
(<span style="font-family: arial;">∀</span><span class="id" type="var">st</span>, <span class="id" type="var">Q'</span> <span class="id" type="var">st</span> <span style="font-family: arial;">→</span> <span class="id" type="var">Q</span> <span class="id" type="var">st</span>) <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">hoare_proof</span> <span class="id" type="var">P</span> <span class="id" type="var">c</span> <span class="id" type="var">Q</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
</div>
<div class="doc">
Now, for example, let's construct a proof object representing a
derivation for the hoare triple
<div class="paragraph"> </div>
<div class="code code-tight">
<span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">assn_sub</span> <span class="id" type="var">X</span> (<span class="id" type="var">X</span>+1) (<span class="id" type="var">assn_sub</span> <span class="id" type="var">X</span> (<span class="id" type="var">X</span>+2) (<span class="id" type="var">X</span>=3))<span style="letter-spacing:-.4em;">}</span>} <span class="id" type="var">X</span>::=<span class="id" type="var">X</span>+1;; <span class="id" type="var">X</span>::=<span class="id" type="var">X</span>+2 <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">X</span>=3<span style="letter-spacing:-.4em;">}</span>}.
<div class="paragraph"> </div>
</div>
We can use Coq's tactics to help us construct the proof object.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">sample_proof</span><br/>
: <span class="id" type="var">hoare_proof</span><br/>
(<span class="id" type="var">assn_sub</span> <span class="id" type="var">X</span> (<span class="id" type="var">APlus</span> (<span class="id" type="var">AId</span> <span class="id" type="var">X</span>) (<span class="id" type="var">ANum</span> 1))<br/>
(<span class="id" type="var">assn_sub</span> <span class="id" type="var">X</span> (<span class="id" type="var">APlus</span> (<span class="id" type="var">AId</span> <span class="id" type="var">X</span>) (<span class="id" type="var">ANum</span> 2))<br/>
(<span class="id" type="keyword">fun</span> <span class="id" type="var">st</span> ⇒ <span class="id" type="var">st</span> <span class="id" type="var">X</span> = 3) ))<br/>
(<span class="id" type="var">X</span> ::= <span class="id" type="var">APlus</span> (<span class="id" type="var">AId</span> <span class="id" type="var">X</span>) (<span class="id" type="var">ANum</span> 1);; (<span class="id" type="var">X</span> ::= <span class="id" type="var">APlus</span> (<span class="id" type="var">AId</span> <span class="id" type="var">X</span>) (<span class="id" type="var">ANum</span> 2)))<br/>
(<span class="id" type="keyword">fun</span> <span class="id" type="var">st</span> ⇒ <span class="id" type="var">st</span> <span class="id" type="var">X</span> = 3).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">H_Seq</span>; <span class="id" type="tactic">apply</span> <span class="id" type="var">H_Asgn</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="comment">(*<br/>
Print sample_proof.<br/>
====><br/>
H_Seq<br/>
(assn_sub X (APlus (AId X) (ANum 1))<br/>
(assn_sub X (APlus (AId X) (ANum 2)) (fun st : state => st X = VNat 3)))<br/>
(X ::= APlus (AId X) (ANum 1))<br/>
(assn_sub X (APlus (AId X) (ANum 2)) (fun st : state => st X = VNat 3))<br/>
(X ::= APlus (AId X) (ANum 2)) (fun st : state => st X = VNat 3)<br/>
(H_Asgn<br/>
(assn_sub X (APlus (AId X) (ANum 2)) (fun st : state => st X = VNat 3))<br/>
X (APlus (AId X) (ANum 1)))<br/>
(H_Asgn (fun st : state => st X = VNat 3) X (APlus (AId X) (ANum 2)))<br/>
*)</span><br/>
<br/>
</div>
<div class="doc">
<a name="lab583"></a><h4 class="section">Exercise: 2 stars (hoare_proof_sound)</h4>
Prove that such proof objects represent true claims.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">hoare_proof_sound</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">P</span> <span class="id" type="var">c</span> <span class="id" type="var">Q</span>,<br/>
<span class="id" type="var">hoare_proof</span> <span class="id" type="var">P</span> <span class="id" type="var">c</span> <span class="id" type="var">Q</span> <span style="font-family: arial;">→</span> <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">P</span><span style="letter-spacing:-.4em;">}</span>} <span class="id" type="var">c</span> <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">Q</span><span style="letter-spacing:-.4em;">}</span>}.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
We can also use Coq's reasoning facilities to prove metatheorems
about Hoare Logic. For example, here are the analogs of two
theorems we saw in chapter <span class="inlinecode"><span class="id" type="var">Hoare</span></span> — this time expressed in terms
of the syntax of Hoare Logic derivations (provability) rather than
directly in terms of the semantics of Hoare triples.
<div class="paragraph"> </div>
The first one says that, for every <span class="inlinecode"><span class="id" type="var">P</span></span> and <span class="inlinecode"><span class="id" type="var">c</span></span>, the assertion
<span class="inlinecode"><span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">P</span><span style="letter-spacing:-.4em;">}</span>}</span> <span class="inlinecode"><span class="id" type="var">c</span></span> <span class="inlinecode"><span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">True</span><span style="letter-spacing:-.4em;">}</span>}</span> is <i>provable</i> in Hoare Logic. Note that the
proof is more complex than the semantic proof in <span class="inlinecode"><span class="id" type="var">Hoare</span></span>: we
actually need to perform an induction over the structure of the
command <span class="inlinecode"><span class="id" type="var">c</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">H_Post_True_deriv</span>:<br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">c</span> <span class="id" type="var">P</span>, <span class="id" type="var">hoare_proof</span> <span class="id" type="var">P</span> <span class="id" type="var">c</span> (<span class="id" type="keyword">fun</span> <span class="id" type="var">_</span> ⇒ <span class="id" type="var">True</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intro</span> <span class="id" type="var">c</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">c</span>; <span class="id" type="tactic">intro</span> <span class="id" type="var">P</span>.<br/>
- <span class="comment">(* SKIP *)</span><br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">H_Consequence</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">H_Skip</span>.<br/>
<span class="id" type="tactic">intros</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">H</span>.<br/>
<span class="comment">(* Proof of True *)</span><br/>
<span class="id" type="tactic">intros</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">I</span>.<br/>
- <span class="comment">(* ::= *)</span><br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">H_Consequence_pre</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">H_Asgn</span>.<br/>
<span class="id" type="tactic">intros</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">I</span>.<br/>
- <span class="comment">(* ;; *)</span><br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">H_Consequence_pre</span>.<br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">H_Seq</span>.<br/>
<span class="id" type="tactic">apply</span> (<span class="id" type="var">IHc1</span> (<span class="id" type="keyword">fun</span> <span class="id" type="var">_</span> ⇒ <span class="id" type="var">True</span>)).<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">IHc2</span>.<br/>
<span class="id" type="tactic">intros</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">I</span>.<br/>
- <span class="comment">(* IFB *)</span><br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">H_Consequence_pre</span> <span class="id" type="keyword">with</span> (<span class="id" type="keyword">fun</span> <span class="id" type="var">_</span> ⇒ <span class="id" type="var">True</span>).<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">H_If</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">IHc1</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">IHc2</span>.<br/>
<span class="id" type="tactic">intros</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">I</span>.<br/>
- <span class="comment">(* WHILE *)</span><br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">H_Consequence</span>.<br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">H_While</span>.<br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">IHc</span>.<br/>
<span class="id" type="tactic">intros</span>; <span class="id" type="tactic">apply</span> <span class="id" type="var">I</span>.<br/>
<span class="id" type="tactic">intros</span>; <span class="id" type="tactic">apply</span> <span class="id" type="var">I</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Similarly, we can show that <span class="inlinecode"><span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">False</span><span style="letter-spacing:-.4em;">}</span>}</span> <span class="inlinecode"><span class="id" type="var">c</span></span> <span class="inlinecode"><span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">Q</span><span style="letter-spacing:-.4em;">}</span>}</span> is provable for
any <span class="inlinecode"><span class="id" type="var">c</span></span> and <span class="inlinecode"><span class="id" type="var">Q</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">False_and_P_imp</span>: <span style="font-family: arial;">∀</span><span class="id" type="var">P</span> <span class="id" type="var">Q</span>,<br/>
<span class="id" type="var">False</span> <span style="font-family: arial;">∧</span> <span class="id" type="var">P</span> <span style="font-family: arial;">→</span> <span class="id" type="var">Q</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">P</span> <span class="id" type="var">Q</span> [<span class="id" type="var">CONTRA</span> <span class="id" type="var">HP</span>].<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">CONTRA</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Tactic Notation</span> "pre_false_helper" <span class="id" type="var">constr</span>(<span class="id" type="var">CONSTR</span>) :=<br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">H_Consequence_pre</span>;<br/>
[<span class="id" type="tactic">eapply</span> <span class="id" type="var">CONSTR</span> | <span class="id" type="tactic">intros</span> ? <span class="id" type="var">CONTRA</span>; <span class="id" type="tactic">destruct</span> <span class="id" type="var">CONTRA</span>].<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">H_Pre_False_deriv</span>:<br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">c</span> <span class="id" type="var">Q</span>, <span class="id" type="var">hoare_proof</span> (<span class="id" type="keyword">fun</span> <span class="id" type="var">_</span> ⇒ <span class="id" type="var">False</span>) <span class="id" type="var">c</span> <span class="id" type="var">Q</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">c</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">c</span>; <span class="id" type="tactic">intro</span> <span class="id" type="var">Q</span>.<br/>
- <span class="comment">(* SKIP *)</span> <span class="id" type="var">pre_false_helper</span> <span class="id" type="var">H_Skip</span>.<br/>
- <span class="comment">(* ::= *)</span> <span class="id" type="var">pre_false_helper</span> <span class="id" type="var">H_Asgn</span>.<br/>
- <span class="comment">(* ;; *)</span> <span class="id" type="var">pre_false_helper</span> <span class="id" type="var">H_Seq</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">IHc1</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">IHc2</span>.<br/>
- <span class="comment">(* IFB *)</span><br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">H_If</span>; <span class="id" type="tactic">eapply</span> <span class="id" type="var">H_Consequence_pre</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">IHc1</span>. <span class="id" type="tactic">intro</span>. <span class="id" type="tactic">eapply</span> <span class="id" type="var">False_and_P_imp</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">IHc2</span>. <span class="id" type="tactic">intro</span>. <span class="id" type="tactic">eapply</span> <span class="id" type="var">False_and_P_imp</span>.<br/>
- <span class="comment">(* WHILE *)</span><br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">H_Consequence_post</span>.<br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">H_While</span>.<br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">H_Consequence_pre</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">IHc</span>.<br/>
<span class="id" type="tactic">intro</span>. <span class="id" type="tactic">eapply</span> <span class="id" type="var">False_and_P_imp</span>.<br/>
<span class="id" type="tactic">intro</span>. <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">eapply</span> <span class="id" type="var">False_and_P_imp</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
As a last step, we can show that the set of <span class="inlinecode"><span class="id" type="var">hoare_proof</span></span> axioms is
sufficient to prove any true fact about (partial) correctness.
More precisely, any semantic Hoare triple that we can prove can
also be proved from these axioms. Such a set of axioms is said
to be <i>relatively complete</i>.
<div class="paragraph"> </div>
This proof is inspired by the one at
http://www.ps.uni-saarland.de/courses/sem-ws11/script/Hoare.html
<div class="paragraph"> </div>
To prove this fact, we'll need to invent some intermediate
assertions using a technical device known as <i>weakest preconditions</i>.
Given a command <span class="inlinecode"><span class="id" type="var">c</span></span> and a desired postcondition assertion <span class="inlinecode"><span class="id" type="var">Q</span></span>,
the weakest precondition <span class="inlinecode"><span class="id" type="var">wp</span></span> <span class="inlinecode"><span class="id" type="var">c</span></span> <span class="inlinecode"><span class="id" type="var">Q</span></span> is an assertion <span class="inlinecode"><span class="id" type="var">P</span></span> such that
<span class="inlinecode"><span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">P</span><span style="letter-spacing:-.4em;">}</span>}</span> <span class="inlinecode"><span class="id" type="var">c</span></span> <span class="inlinecode"><span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">Q</span><span style="letter-spacing:-.4em;">}</span>}</span> holds, and moreover, for any other assertion <span class="inlinecode"><span class="id" type="var">P'</span></span>,
if <span class="inlinecode"><span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">P'</span><span style="letter-spacing:-.4em;">}</span>}</span> <span class="inlinecode"><span class="id" type="var">c</span></span> <span class="inlinecode"><span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">Q</span><span style="letter-spacing:-.4em;">}</span>}</span> holds then <span class="inlinecode"><span class="id" type="var">P'</span></span> <span class="inlinecode"><span style="font-family: arial;">→</span></span> <span class="inlinecode"><span class="id" type="var">P</span></span>. We can more directly
define this as follows:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">wp</span> (<span class="id" type="var">c</span>:<span class="id" type="var">com</span>) (<span class="id" type="var">Q</span>:<span class="id" type="var">Assertion</span>) : <span class="id" type="var">Assertion</span> :=<br/>
<span class="id" type="keyword">fun</span> <span class="id" type="var">s</span> ⇒ <span style="font-family: arial;">∀</span><span class="id" type="var">s'</span>, <span class="id" type="var">c</span> / <span class="id" type="var">s</span> <span style="font-family: arial;">⇓</span> <span class="id" type="var">s'</span> <span style="font-family: arial;">→</span> <span class="id" type="var">Q</span> <span class="id" type="var">s'</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab584"></a><h4 class="section">Exercise: 1 star (wp_is_precondition)</h4>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">wp_is_precondition</span>: <span style="font-family: arial;">∀</span><span class="id" type="var">c</span> <span class="id" type="var">Q</span>,<br/>
<span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">wp</span> <span class="id" type="var">c</span> <span class="id" type="var">Q</span><span style="letter-spacing:-.4em;">}</span>} <span class="id" type="var">c</span> <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">Q</span><span style="letter-spacing:-.4em;">}</span>}.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab585"></a><h4 class="section">Exercise: 1 star (wp_is_weakest)</h4>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">wp_is_weakest</span>: <span style="font-family: arial;">∀</span><span class="id" type="var">c</span> <span class="id" type="var">Q</span> <span class="id" type="var">P'</span>,<br/>
<span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">P'</span><span style="letter-spacing:-.4em;">}</span>} <span class="id" type="var">c</span> <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">Q</span><span style="letter-spacing:-.4em;">}</span>} <span style="font-family: arial;">→</span> <span style="font-family: arial;">∀</span><span class="id" type="var">st</span>, <span class="id" type="var">P'</span> <span class="id" type="var">st</span> <span style="font-family: arial;">→</span> <span class="id" type="var">wp</span> <span class="id" type="var">c</span> <span class="id" type="var">Q</span> <span class="id" type="var">st</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
</div>
<div class="doc">
The following utility lemma will also be useful.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">bassn_eval_false</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">b</span> <span class="id" type="var">st</span>, ¬ <span class="id" type="var">bassn</span> <span class="id" type="var">b</span> <span class="id" type="var">st</span> <span style="font-family: arial;">→</span> <span class="id" type="var">beval</span> <span class="id" type="var">st</span> <span class="id" type="var">b</span> = <span class="id" type="var">false</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">b</span> <span class="id" type="var">st</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">unfold</span> <span class="id" type="var">bassn</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">destruct</span> (<span class="id" type="var">beval</span> <span class="id" type="var">st</span> <span class="id" type="var">b</span>).<br/>
<span class="id" type="var">exfalso</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab586"></a><h4 class="section">Exercise: 4 stars (hoare_proof_complete)</h4>
Complete the proof of the theorem.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">hoare_proof_complete</span>: <span style="font-family: arial;">∀</span><span class="id" type="var">P</span> <span class="id" type="var">c</span> <span class="id" type="var">Q</span>,<br/>
<span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">P</span><span style="letter-spacing:-.4em;">}</span>} <span class="id" type="var">c</span> <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">Q</span><span style="letter-spacing:-.4em;">}</span>} <span style="font-family: arial;">→</span> <span class="id" type="var">hoare_proof</span> <span class="id" type="var">P</span> <span class="id" type="var">c</span> <span class="id" type="var">Q</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">P</span> <span class="id" type="var">c</span>. <span class="id" type="tactic">generalize</span> <span class="id" type="tactic">dependent</span> <span class="id" type="var">P</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">c</span>; <span class="id" type="tactic">intros</span> <span class="id" type="var">P</span> <span class="id" type="var">Q</span> <span class="id" type="var">HT</span>.<br/>
- <span class="comment">(* SKIP *)</span><br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">H_Consequence</span>.<br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">H_Skip</span>.<br/>
<span class="id" type="tactic">intros</span>. <span class="id" type="var">eassumption</span>.<br/>
<span class="id" type="tactic">intro</span> <span class="id" type="var">st</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">HT</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">E_Skip</span>.<br/>
- <span class="comment">(* ::= *)</span><br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">H_Consequence</span>.<br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">H_Asgn</span>.<br/>
<span class="id" type="tactic">intro</span> <span class="id" type="var">st</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">HT</span>. <span class="id" type="var">econstructor</span>. <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="tactic">intros</span>; <span class="id" type="tactic">assumption</span>.<br/>
- <span class="comment">(* ;; *)</span><br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">H_Seq</span> <span class="id" type="keyword">with</span> (<span class="id" type="var">wp</span> <span class="id" type="var">c2</span> <span class="id" type="var">Q</span>).<br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">IHc1</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">st</span> <span class="id" type="var">st'</span> <span class="id" type="var">E1</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">unfold</span> <span class="id" type="var">wp</span>. <span class="id" type="tactic">intros</span> <span class="id" type="var">st''</span> <span class="id" type="var">E2</span>.<br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">HT</span>. <span class="id" type="var">econstructor</span>; <span class="id" type="var">eassumption</span>. <span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">IHc2</span>. <span class="id" type="tactic">intros</span> <span class="id" type="var">st</span> <span class="id" type="var">st'</span> <span class="id" type="var">E1</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">H</span>; <span class="id" type="tactic">assumption</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
Finally, we might hope that our axiomatic Hoare logic is <i>decidable</i>;
that is, that there is an (terminating) algorithm (a <i>decision procedure</i>)
that can determine whether or not a given Hoare triple is valid (derivable).
But 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" type="var">True</span><span style="letter-spacing:-.4em;">}</span>}</span> <span class="inlinecode"><span class="id" type="var">c</span></span> <span class="inlinecode"><span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">False</span><span style="letter-spacing:-.4em;">}</span>}</span>. This triple is valid
if and only if <span class="inlinecode"><span class="id" type="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" type="var">True</span>}</span> <span class="inlinecode"><span class="id" type="var">SKIP</span></span> <span class="inlinecode"><span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">P</span><span style="letter-spacing:-.4em;">}</span>}</span> is valid if and only if
<span class="inlinecode"><span style="font-family: arial;">∀</span></span> <span class="inlinecode"><span class="id" type="var">s</span>,</span> <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">s</span></span> is valid, where <span class="inlinecode"><span class="id" type="var">P</span></span> is an arbitrary assertion of Coq's
logic. But it is known that there can be no decision procedure for
this logic.
<div class="paragraph"> </div>
<div class="paragraph"> </div>
Overall, this axiomatic style of presentation gives a clearer picture of what it
means to "give a proof in Hoare logic." However, it is not
entirely satisfactory from the point of view of writing down such
proofs in practice: it is quite verbose. The section of chapter
<span class="inlinecode"><span class="id" type="var">Hoare2</span></span> on formalizing decorated programs shows how we can do even
better.
<div class="paragraph"> </div>
</div>
<div class="code code-tight">
<br/>
</div>
</div>
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