Replaced { } proofs by local lemmas; added Hoare logic with logical variables.
(* Author: Tobias Nipkow *)
subsection \<open>Soundness and Completeness\<close>
theory Hoare_Sound_Complete
imports Hoare
begin
subsubsection "Soundness"
lemma hoare_sound: "\<turnstile> {P}c{Q} \<Longrightarrow> \<Turnstile> {P}c{Q}"
proof(induction rule: hoare.induct)
case (While P b c)
have "(WHILE b DO c,s) \<Rightarrow> t \<Longrightarrow> P s \<Longrightarrow> P t \<and> \<not> bval b t" for s t
proof(induction "WHILE b DO c" s t rule: big_step_induct)
case WhileFalse thus ?case by blast
next
case WhileTrue thus ?case
using While.IH unfolding hoare_valid_def by blast
qed
thus ?case unfolding hoare_valid_def by blast
qed (auto simp: hoare_valid_def)
subsubsection "Weakest Precondition"
definition wp :: "com \<Rightarrow> assn \<Rightarrow> assn" where
"wp c Q = (\<lambda>s. \<forall>t. (c,s) \<Rightarrow> t \<longrightarrow> Q t)"
lemma wp_SKIP[simp]: "wp SKIP Q = Q"
by (rule ext) (auto simp: wp_def)
lemma wp_Ass[simp]: "wp (x::=a) Q = (\<lambda>s. Q(s[a/x]))"
by (rule ext) (auto simp: wp_def)
lemma wp_Seq[simp]: "wp (c\<^sub>1;;c\<^sub>2) Q = wp c\<^sub>1 (wp c\<^sub>2 Q)"
by (rule ext) (auto simp: wp_def)
lemma wp_If[simp]:
"wp (IF b THEN c\<^sub>1 ELSE c\<^sub>2) Q =
(\<lambda>s. if bval b s then wp c\<^sub>1 Q s else wp c\<^sub>2 Q s)"
by (rule ext) (auto simp: wp_def)
lemma wp_While_If:
"wp (WHILE b DO c) Q s =
wp (IF b THEN c;;WHILE b DO c ELSE SKIP) Q s"
unfolding wp_def by (metis unfold_while)
lemma wp_While_True[simp]: "bval b s \<Longrightarrow>
wp (WHILE b DO c) Q s = wp (c;; WHILE b DO c) Q s"
by(simp add: wp_While_If)
lemma wp_While_False[simp]: "\<not> bval b s \<Longrightarrow> wp (WHILE b DO c) Q s = Q s"
by(simp add: wp_While_If)
subsubsection "Completeness"
lemma wp_is_pre: "\<turnstile> {wp c Q} c {Q}"
proof(induction c arbitrary: Q)
case If thus ?case by(auto intro: conseq)
next
case (While b c)
let ?w = "WHILE b DO c"
show "\<turnstile> {wp ?w Q} ?w {Q}"
proof(rule While')
show "\<turnstile> {\<lambda>s. wp ?w Q s \<and> bval b s} c {wp ?w Q}"
proof(rule strengthen_pre[OF _ While.IH])
show "\<forall>s. wp ?w Q s \<and> bval b s \<longrightarrow> wp c (wp ?w Q) s" by auto
qed
show "\<forall>s. wp ?w Q s \<and> \<not> bval b s \<longrightarrow> Q s" by auto
qed
qed auto
lemma hoare_complete: assumes "\<Turnstile> {P}c{Q}" shows "\<turnstile> {P}c{Q}"
proof(rule strengthen_pre)
show "\<forall>s. P s \<longrightarrow> wp c Q s" using assms
by (auto simp: hoare_valid_def wp_def)
show "\<turnstile> {wp c Q} c {Q}" by(rule wp_is_pre)
qed
corollary hoare_sound_complete: "\<turnstile> {P}c{Q} \<longleftrightarrow> \<Turnstile> {P}c{Q}"
by (metis hoare_complete hoare_sound)
end