(* Title: HOL/IMP/Hoare_Def.thy
Author: Tobias Nipkow
*)
header "Soundness and Completeness wrt Denotational Semantics"
theory Hoare_Den imports Hoare Denotation begin
definition
hoare_valid :: "[assn,com,assn] => bool" ("|= {(1_)}/ (_)/ {(1_)}" 50) where
"|= {P}c{Q} = (!s t. (s,t) : C(c) --> P s --> Q t)"
lemma hoare_sound: "|- {P}c{Q} ==> |= {P}c{Q}"
proof(induct rule: hoare.induct)
case (While P b c)
{ fix s t
let ?G = "Gamma b (C c)"
assume "(s,t) \<in> lfp ?G"
hence "P s \<longrightarrow> P t \<and> \<not> b t"
proof(rule lfp_induct2)
show "mono ?G" by(rule Gamma_mono)
next
fix s t assume "(s,t) \<in> ?G (lfp ?G \<inter> {(s,t). P s \<longrightarrow> P t \<and> \<not> b t})"
thus "P s \<longrightarrow> P t \<and> \<not> b t" using While.hyps
by(auto simp: hoare_valid_def Gamma_def)
qed
}
thus ?case by(simp add:hoare_valid_def)
qed (auto simp: hoare_valid_def)
definition
wp :: "com => assn => assn" where
"wp c Q = (%s. !t. (s,t) : C(c) --> Q t)"
lemma wp_SKIP: "wp \<SKIP> Q = Q"
by (simp add: wp_def)
lemma wp_Ass: "wp (x:==a) Q = (%s. Q(s[x\<mapsto>a s]))"
by (simp add: wp_def)
lemma wp_Semi: "wp (c;d) Q = wp c (wp d Q)"
by (rule ext) (auto simp: wp_def)
lemma wp_If:
"wp (\<IF> b \<THEN> c \<ELSE> d) Q = (%s. (b s --> wp c Q s) & (~b s --> wp d 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"
by(simp only: wp_def C_While_If)
(*Not suitable for rewriting: LOOPS!*)
lemma wp_While_if:
"wp (\<WHILE> b \<DO> c) Q s = (if b s then wp (c;\<WHILE> b \<DO> c) Q s else Q s)"
by(simp add:wp_While_If wp_If wp_SKIP)
lemma wp_While_True: "b s ==>
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: "~b s ==> wp (\<WHILE> b \<DO> c) Q s = Q s"
by(simp add: wp_While_if)
lemmas [simp] = wp_SKIP wp_Ass wp_Semi wp_If wp_While_True wp_While_False
lemma wp_While: "wp (\<WHILE> b \<DO> c) Q s =
(s : gfp(%S.{s. if b s then wp c (%s. s:S) s else Q s}))"
apply (simp (no_asm))
apply (rule iffI)
apply (rule weak_coinduct)
apply (erule CollectI)
apply safe
apply simp
apply simp
apply (simp add: wp_def Gamma_def)
apply (intro strip)
apply (rule mp)
prefer 2 apply (assumption)
apply (erule lfp_induct2)
apply (fast intro!: monoI)
apply (subst gfp_unfold)
apply (fast intro!: monoI)
apply fast
done
declare C_while [simp del]
lemma wp_is_pre: "|- {wp c Q} c {Q}"
proof(induct c arbitrary: Q)
case SKIP show ?case by auto
next
case Assign show ?case by auto
next
case Semi thus ?case by auto
next
case (Cond b c1 c2)
let ?If = "IF b THEN c1 ELSE c2"
show ?case
proof(rule If)
show "|- {\<lambda>s. wp ?If Q s \<and> b s} c1 {Q}"
proof(rule strengthen_pre[OF _ Cond(1)])
show "\<forall>s. wp ?If Q s \<and> b s \<longrightarrow> wp c1 Q s" by auto
qed
show "|- {\<lambda>s. wp ?If Q s \<and> \<not> b s} c2 {Q}"
proof(rule strengthen_pre[OF _ Cond(2)])
show "\<forall>s. wp ?If Q s \<and> \<not> b s \<longrightarrow> wp c2 Q s" by auto
qed
qed
next
case (While b c)
let ?w = "WHILE b DO c"
show ?case
proof(rule While')
show "|- {\<lambda>s. wp ?w Q s \<and> b s} c {wp ?w Q}"
proof(rule strengthen_pre[OF _ While(1)])
show "\<forall>s. wp ?w Q s \<and> b s \<longrightarrow> wp c (wp ?w Q) s" by auto
qed
show "\<forall>s. wp ?w Q s \<and> \<not> b s \<longrightarrow> Q s" by auto
qed
qed
lemma hoare_relative_complete: assumes "|= {P}c{Q}" shows "|- {P}c{Q}"
proof(rule conseq)
show "\<forall>s. P s \<longrightarrow> wp c Q s" using assms
by (auto simp: hoare_valid_def wp_def)
show "|- {wp c Q} c {Q}" by(rule wp_is_pre)
show "\<forall>s. Q s \<longrightarrow> Q s" by auto
qed
end