(* Author: Tobias Nipkow *)
theory Hoare_Total imports Hoare_Sound_Complete Hoare_Examples begin
subsection "Hoare Logic for Total Correctness"
text{* Note that this definition of total validity @{text"\<Turnstile>\<^sub>t"} only
works if execution is deterministic (which it is in our case). *}
definition hoare_tvalid :: "assn \<Rightarrow> com \<Rightarrow> assn \<Rightarrow> bool"
("\<Turnstile>\<^sub>t {(1_)}/ (_)/ {(1_)}" 50) where
"\<Turnstile>\<^sub>t {P}c{Q} \<longleftrightarrow> (\<forall>s. P s \<longrightarrow> (\<exists>t. (c,s) \<Rightarrow> t \<and> Q t))"
text{* Provability of Hoare triples in the proof system for total
correctness is written @{text"\<turnstile>\<^sub>t {P}c{Q}"} and defined
inductively. The rules for @{text"\<turnstile>\<^sub>t"} differ from those for
@{text"\<turnstile>"} only in the one place where nontermination can arise: the
@{term While}-rule. *}
inductive
hoaret :: "assn \<Rightarrow> com \<Rightarrow> assn \<Rightarrow> bool" ("\<turnstile>\<^sub>t ({(1_)}/ (_)/ {(1_)})" 50)
where
Skip: "\<turnstile>\<^sub>t {P} SKIP {P}" |
Assign: "\<turnstile>\<^sub>t {\<lambda>s. P(s[a/x])} x::=a {P}" |
Seq: "\<lbrakk> \<turnstile>\<^sub>t {P\<^isub>1} c\<^isub>1 {P\<^isub>2}; \<turnstile>\<^sub>t {P\<^isub>2} c\<^isub>2 {P\<^isub>3} \<rbrakk> \<Longrightarrow> \<turnstile>\<^sub>t {P\<^isub>1} c\<^isub>1;;c\<^isub>2 {P\<^isub>3}" |
If: "\<lbrakk> \<turnstile>\<^sub>t {\<lambda>s. P s \<and> bval b s} c\<^isub>1 {Q}; \<turnstile>\<^sub>t {\<lambda>s. P s \<and> \<not> bval b s} c\<^isub>2 {Q} \<rbrakk>
\<Longrightarrow> \<turnstile>\<^sub>t {P} IF b THEN c\<^isub>1 ELSE c\<^isub>2 {Q}" |
While:
"(\<And>n::nat.
\<turnstile>\<^sub>t {\<lambda>s. P s \<and> bval b s \<and> T s n} c {\<lambda>s. P s \<and> (\<exists>n'. T s n' \<and> n' < n)})
\<Longrightarrow> \<turnstile>\<^sub>t {\<lambda>s. P s \<and> (\<exists>n. T s n)} WHILE b DO c {\<lambda>s. P s \<and> \<not>bval b s}" |
conseq: "\<lbrakk> \<forall>s. P' s \<longrightarrow> P s; \<turnstile>\<^sub>t {P}c{Q}; \<forall>s. Q s \<longrightarrow> Q' s \<rbrakk> \<Longrightarrow>
\<turnstile>\<^sub>t {P'}c{Q'}"
text{* The @{term While}-rule is like the one for partial correctness but it
requires additionally that with every execution of the loop body some measure
relation @{term[source]"T :: state \<Rightarrow> nat \<Rightarrow> bool"} decreases.
The following functional version is more intuitive: *}
lemma While_fun:
"\<lbrakk> \<And>n::nat. \<turnstile>\<^sub>t {\<lambda>s. P s \<and> bval b s \<and> n = f s} c {\<lambda>s. P s \<and> f s < n}\<rbrakk>
\<Longrightarrow> \<turnstile>\<^sub>t {P} WHILE b DO c {\<lambda>s. P s \<and> \<not>bval b s}"
by (rule While [where T="\<lambda>s n. n = f s", simplified])
text{* Building in the consequence rule: *}
lemma strengthen_pre:
"\<lbrakk> \<forall>s. P' s \<longrightarrow> P s; \<turnstile>\<^sub>t {P} c {Q} \<rbrakk> \<Longrightarrow> \<turnstile>\<^sub>t {P'} c {Q}"
by (metis conseq)
lemma weaken_post:
"\<lbrakk> \<turnstile>\<^sub>t {P} c {Q}; \<forall>s. Q s \<longrightarrow> Q' s \<rbrakk> \<Longrightarrow> \<turnstile>\<^sub>t {P} c {Q'}"
by (metis conseq)
lemma Assign': "\<forall>s. P s \<longrightarrow> Q(s[a/x]) \<Longrightarrow> \<turnstile>\<^sub>t {P} x ::= a {Q}"
by (simp add: strengthen_pre[OF _ Assign])
lemma While_fun':
assumes "\<And>n::nat. \<turnstile>\<^sub>t {\<lambda>s. P s \<and> bval b s \<and> n = f s} c {\<lambda>s. P s \<and> f s < n}"
and "\<forall>s. P s \<and> \<not> bval b s \<longrightarrow> Q s"
shows "\<turnstile>\<^sub>t {P} WHILE b DO c {Q}"
by(blast intro: assms(1) weaken_post[OF While_fun assms(2)])
text{* Our standard example: *}
lemma "\<turnstile>\<^sub>t {\<lambda>s. s ''x'' = i} ''y'' ::= N 0;; wsum {\<lambda>s. s ''y'' = sum i}"
apply(rule Seq)
prefer 2
apply(rule While_fun' [where P = "\<lambda>s. (s ''y'' = sum i - sum(s ''x''))"
and f = "\<lambda>s. nat(s ''x'')"])
apply(rule Seq)
prefer 2
apply(rule Assign)
apply(rule Assign')
apply simp
apply(simp add: minus_numeral_simps(1)[symmetric] del: minus_numeral_simps)
apply(simp)
apply(rule Assign')
apply simp
done
text{* The soundness theorem: *}
theorem hoaret_sound: "\<turnstile>\<^sub>t {P}c{Q} \<Longrightarrow> \<Turnstile>\<^sub>t {P}c{Q}"
proof(unfold hoare_tvalid_def, induction rule: hoaret.induct)
case (While P b T c)
{
fix s n
have "\<lbrakk> P s; T s n \<rbrakk> \<Longrightarrow> \<exists>t. (WHILE b DO c, s) \<Rightarrow> t \<and> P t \<and> \<not> bval b t"
proof(induction "n" arbitrary: s rule: less_induct)
case (less n)
thus ?case by (metis While.IH WhileFalse WhileTrue)
qed
}
thus ?case by auto
next
case If thus ?case by auto blast
qed fastforce+
text{*
The completeness proof proceeds along the same lines as the one for partial
correctness. First we have to strengthen our notion of weakest precondition
to take termination into account: *}
definition wpt :: "com \<Rightarrow> assn \<Rightarrow> assn" ("wp\<^sub>t") where
"wp\<^sub>t c Q \<equiv> \<lambda>s. \<exists>t. (c,s) \<Rightarrow> t \<and> Q t"
lemma [simp]: "wp\<^sub>t SKIP Q = Q"
by(auto intro!: ext simp: wpt_def)
lemma [simp]: "wp\<^sub>t (x ::= e) Q = (\<lambda>s. Q(s(x := aval e s)))"
by(auto intro!: ext simp: wpt_def)
lemma [simp]: "wp\<^sub>t (c\<^isub>1;;c\<^isub>2) Q = wp\<^sub>t c\<^isub>1 (wp\<^sub>t c\<^isub>2 Q)"
unfolding wpt_def
apply(rule ext)
apply auto
done
lemma [simp]:
"wp\<^sub>t (IF b THEN c\<^isub>1 ELSE c\<^isub>2) Q = (\<lambda>s. wp\<^sub>t (if bval b s then c\<^isub>1 else c\<^isub>2) Q s)"
apply(unfold wpt_def)
apply(rule ext)
apply auto
done
text{* Now we define the number of iterations @{term "WHILE b DO c"} needs to
terminate when started in state @{text s}. Because this is a truly partial
function, we define it as an (inductive) relation first: *}
inductive Its :: "bexp \<Rightarrow> com \<Rightarrow> state \<Rightarrow> nat \<Rightarrow> bool" where
Its_0: "\<not> bval b s \<Longrightarrow> Its b c s 0" |
Its_Suc: "\<lbrakk> bval b s; (c,s) \<Rightarrow> s'; Its b c s' n \<rbrakk> \<Longrightarrow> Its b c s (Suc n)"
text{* The relation is in fact a function: *}
lemma Its_fun: "Its b c s n \<Longrightarrow> Its b c s n' \<Longrightarrow> n=n'"
proof(induction arbitrary: n' rule:Its.induct)
case Its_0 thus ?case by(metis Its.cases)
next
case Its_Suc thus ?case by(metis Its.cases big_step_determ)
qed
text{* For all terminating loops, @{const Its} yields a result: *}
lemma WHILE_Its: "(WHILE b DO c,s) \<Rightarrow> t \<Longrightarrow> \<exists>n. Its b c s n"
proof(induction "WHILE b DO c" s t rule: big_step_induct)
case WhileFalse thus ?case by (metis Its_0)
next
case WhileTrue thus ?case by (metis Its_Suc)
qed
lemma wpt_is_pre: "\<turnstile>\<^sub>t {wp\<^sub>t c Q} c {Q}"
proof (induction c arbitrary: Q)
case SKIP show ?case by simp (blast intro:hoaret.Skip)
next
case Assign show ?case by simp (blast intro:hoaret.Assign)
next
case Seq thus ?case by simp (blast intro:hoaret.Seq)
next
case If thus ?case by simp (blast intro:hoaret.If hoaret.conseq)
next
case (While b c)
let ?w = "WHILE b DO c"
let ?T = "Its b c"
have "\<forall>s. wp\<^sub>t (WHILE b DO c) Q s \<longrightarrow> wp\<^sub>t (WHILE b DO c) Q s \<and> (\<exists>n. Its b c s n)"
unfolding wpt_def by (metis WHILE_Its)
moreover
{ fix n
{ fix s t
assume "bval b s" "?T s n" "(?w, s) \<Rightarrow> t" "Q t"
from `bval b s` `(?w, s) \<Rightarrow> t` obtain s' where
"(c,s) \<Rightarrow> s'" "(?w,s') \<Rightarrow> t" by auto
from `(?w, s') \<Rightarrow> t` obtain n'' where "?T s' n''" by (blast dest: WHILE_Its)
with `bval b s` `(c, s) \<Rightarrow> s'`
have "?T s (Suc n'')" by (rule Its_Suc)
with `?T s n` have "n = Suc n''" by (rule Its_fun)
with `(c,s) \<Rightarrow> s'` `(?w,s') \<Rightarrow> t` `Q t` `?T s' n''`
have "wp\<^sub>t c (\<lambda>s'. wp\<^sub>t ?w Q s' \<and> (\<exists>n'. ?T s' n' \<and> n' < n)) s"
by (auto simp: wpt_def)
}
hence "\<forall>s. wp\<^sub>t ?w Q s \<and> bval b s \<and> ?T s n \<longrightarrow>
wp\<^sub>t c (\<lambda>s'. wp\<^sub>t ?w Q s' \<and> (\<exists>n'. ?T s' n' \<and> n' < n)) s"
unfolding wpt_def by auto
(* by (metis WhileE Its_Suc Its_fun WHILE_Its lessI) *)
note strengthen_pre[OF this While]
} note hoaret.While[OF this]
moreover have "\<forall>s. wp\<^sub>t ?w Q s \<and> \<not> bval b s \<longrightarrow> Q s" by (auto simp add:wpt_def)
ultimately show ?case by (rule conseq)
qed
text{*\noindent In the @{term While}-case, @{const Its} provides the obvious
termination argument.
The actual completeness theorem follows directly, in the same manner
as for partial correctness: *}
theorem hoaret_complete: "\<Turnstile>\<^sub>t {P}c{Q} \<Longrightarrow> \<turnstile>\<^sub>t {P}c{Q}"
apply(rule strengthen_pre[OF _ wpt_is_pre])
apply(auto simp: hoare_tvalid_def hoare_valid_def wpt_def)
done
end