--- a/src/HOL/IMP/Sem_Equiv.thy Wed Aug 22 23:45:49 2012 +0200
+++ b/src/HOL/IMP/Sem_Equiv.thy Thu Aug 23 15:32:22 2012 +0200
@@ -1,17 +1,19 @@
header "Semantic Equivalence up to a Condition"
theory Sem_Equiv
-imports Hoare_Sound_Complete
+imports Big_Step
begin
+type_synonym assn = "state \<Rightarrow> bool"
+
definition
equiv_up_to :: "assn \<Rightarrow> com \<Rightarrow> com \<Rightarrow> bool" ("_ \<Turnstile> _ \<sim> _" [60,0,10] 60)
where
"P \<Turnstile> c \<sim> c' \<equiv> \<forall>s s'. P s \<longrightarrow> (c,s) \<Rightarrow> s' \<longleftrightarrow> (c',s) \<Rightarrow> s'"
-definition
+definition
bequiv_up_to :: "assn \<Rightarrow> bexp \<Rightarrow> bexp \<Rightarrow> bool" ("_ \<Turnstile> _ <\<sim>> _" [60,0,10] 60)
-where
+where
"P \<Turnstile> b <\<sim>> b' \<equiv> \<forall>s. P s \<longrightarrow> bval b s = bval b' s"
lemma equiv_up_to_True:
@@ -27,11 +29,11 @@
by (unfold equiv_up_to_def) blast
lemma equiv_up_toD1:
- "P \<Turnstile> c \<sim> c' \<Longrightarrow> P s \<Longrightarrow> (c, s) \<Rightarrow> s' \<Longrightarrow> (c', s) \<Rightarrow> s'"
+ "P \<Turnstile> c \<sim> c' \<Longrightarrow> (c, s) \<Rightarrow> s' \<Longrightarrow> P s \<Longrightarrow> (c', s) \<Rightarrow> s'"
by (unfold equiv_up_to_def) blast
lemma equiv_up_toD2:
- "P \<Turnstile> c \<sim> c' \<Longrightarrow> P s \<Longrightarrow> (c', s) \<Rightarrow> s' \<Longrightarrow> (c, s) \<Rightarrow> s'"
+ "P \<Turnstile> c \<sim> c' \<Longrightarrow> (c', s) \<Rightarrow> s' \<Longrightarrow> P s \<Longrightarrow> (c, s) \<Rightarrow> s'"
by (unfold equiv_up_to_def) blast
@@ -60,32 +62,28 @@
"P \<Turnstile> b <\<sim>> b' \<Longrightarrow> P \<Turnstile> b' <\<sim>> b'' \<Longrightarrow> P \<Turnstile> b <\<sim>> b''"
by (auto simp: bequiv_up_to_def)
-
-lemma equiv_up_to_hoare:
- "P' \<Turnstile> c \<sim> c' \<Longrightarrow> (\<And>s. P s \<Longrightarrow> P' s) \<Longrightarrow> (\<Turnstile> {P} c {Q}) = (\<Turnstile> {P} c' {Q})"
- unfolding hoare_valid_def equiv_up_to_def by blast
-
-lemma equiv_up_to_hoare_eq:
- "P \<Turnstile> c \<sim> c' \<Longrightarrow> (\<Turnstile> {P} c {Q}) = (\<Turnstile> {P} c' {Q})"
- by (rule equiv_up_to_hoare)
+lemma bequiv_up_to_subst:
+ "P \<Turnstile> b <\<sim>> b' \<Longrightarrow> P s \<Longrightarrow> bval b s = bval b' s"
+ by (simp add: bequiv_up_to_def)
lemma equiv_up_to_seq:
- "P \<Turnstile> c \<sim> c' \<Longrightarrow> Q \<Turnstile> d \<sim> d' \<Longrightarrow> \<Turnstile> {P} c {Q} \<Longrightarrow>
+ "P \<Turnstile> c \<sim> c' \<Longrightarrow> Q \<Turnstile> d \<sim> d' \<Longrightarrow>
+ (\<And>s s'. (c,s) \<Rightarrow> s' \<Longrightarrow> P s \<Longrightarrow> Q s') \<Longrightarrow>
P \<Turnstile> (c; d) \<sim> (c'; d')"
- by (clarsimp simp: equiv_up_to_def hoare_valid_def) blast
+ by (clarsimp simp: equiv_up_to_def) blast
lemma equiv_up_to_while_lemma:
- shows "(d,s) \<Rightarrow> s' \<Longrightarrow>
+ shows "(d,s) \<Rightarrow> s' \<Longrightarrow>
P \<Turnstile> b <\<sim>> b' \<Longrightarrow>
- (\<lambda>s. P s \<and> bval b s) \<Turnstile> c \<sim> c' \<Longrightarrow>
- \<Turnstile> {\<lambda>s. P s \<and> bval b s} c {P} \<Longrightarrow>
- P s \<Longrightarrow>
- d = WHILE b DO c \<Longrightarrow>
- (WHILE b' DO c', s) \<Rightarrow> s'"
+ (\<lambda>s. P s \<and> bval b s) \<Turnstile> c \<sim> c' \<Longrightarrow>
+ (\<And>s s'. (c, s) \<Rightarrow> s' \<Longrightarrow> P s \<Longrightarrow> bval b s \<Longrightarrow> P s') \<Longrightarrow>
+ P s \<Longrightarrow>
+ d = WHILE b DO c \<Longrightarrow>
+ (WHILE b' DO c', s) \<Rightarrow> s'"
proof (induction rule: big_step_induct)
case (WhileTrue b s1 c s2 s3)
- note IH = WhileTrue.IH(2) [OF WhileTrue.prems(1-3) _ WhileTrue.prems(5)]
+ hence IH: "P s2 \<Longrightarrow> (WHILE b' DO c', s2) \<Rightarrow> s3" by auto
from WhileTrue.prems
have "P \<Turnstile> b <\<sim>> b'" by simp
with `bval b s1` `P s1`
@@ -97,38 +95,46 @@
have "(c', s1) \<Rightarrow> s2" by (simp add: equiv_up_to_def)
moreover
from WhileTrue.prems
- have "\<Turnstile> {\<lambda>s. P s \<and> bval b s} c {P}" by simp
+ have "\<And>s s'. (c,s) \<Rightarrow> s' \<Longrightarrow> P s \<Longrightarrow> bval b s \<Longrightarrow> P s'" by simp
with `P s1` `bval b s1` `(c, s1) \<Rightarrow> s2`
- have "P s2" by (simp add: hoare_valid_def)
+ have "P s2" by simp
hence "(WHILE b' DO c', s2) \<Rightarrow> s3" by (rule IH)
- ultimately
+ ultimately
show ?case by blast
next
case WhileFalse
thus ?case by (auto simp: bequiv_up_to_def)
-qed (fastforce simp: equiv_up_to_def bequiv_up_to_def hoare_valid_def)+
+qed (fastforce simp: equiv_up_to_def bequiv_up_to_def)+
lemma bequiv_context_subst:
"P \<Turnstile> b <\<sim>> b' \<Longrightarrow> (P s \<and> bval b s) = (P s \<and> bval b' s)"
by (auto simp: bequiv_up_to_def)
lemma equiv_up_to_while:
- "P \<Turnstile> b <\<sim>> b' \<Longrightarrow> (\<lambda>s. P s \<and> bval b s) \<Turnstile> c \<sim> c' \<Longrightarrow>
- \<Turnstile> {\<lambda>s. P s \<and> bval b s} c {P} \<Longrightarrow>
- P \<Turnstile> WHILE b DO c \<sim> WHILE b' DO c'"
- apply (safe intro!: equiv_up_toI)
- apply (auto intro: equiv_up_to_while_lemma)[1]
- apply (simp add: equiv_up_to_hoare_eq bequiv_context_subst)
- apply (drule equiv_up_to_sym [THEN iffD1])
- apply (drule bequiv_up_to_sym [THEN iffD1])
- apply (auto intro: equiv_up_to_while_lemma)[1]
- done
+ assumes b: "P \<Turnstile> b <\<sim>> b'"
+ assumes c: "(\<lambda>s. P s \<and> bval b s) \<Turnstile> c \<sim> c'"
+ assumes I: "\<And>s s'. (c, s) \<Rightarrow> s' \<Longrightarrow> P s \<Longrightarrow> bval b s \<Longrightarrow> P s'"
+ shows "P \<Turnstile> WHILE b DO c \<sim> WHILE b' DO c'"
+proof -
+ from b have b': "P \<Turnstile> b' <\<sim>> b" by (simp add: bequiv_up_to_sym)
+
+ from c b have c': "(\<lambda>s. P s \<and> bval b' s) \<Turnstile> c' \<sim> c"
+ by (simp add: equiv_up_to_sym bequiv_context_subst)
+
+ from I
+ have I': "\<And>s s'. (c', s) \<Rightarrow> s' \<Longrightarrow> P s \<Longrightarrow> bval b' s \<Longrightarrow> P s'"
+ by (auto dest!: equiv_up_toD1 [OF c'] simp: bequiv_up_to_subst [OF b'])
+
+ note equiv_up_to_while_lemma [OF _ b c]
+ equiv_up_to_while_lemma [OF _ b' c']
+ thus ?thesis using I I' by (auto intro!: equiv_up_toI)
+qed
lemma equiv_up_to_while_weak:
- "P \<Turnstile> b <\<sim>> b' \<Longrightarrow> P \<Turnstile> c \<sim> c' \<Longrightarrow> \<Turnstile> {P} c {P} \<Longrightarrow>
+ "P \<Turnstile> b <\<sim>> b' \<Longrightarrow> P \<Turnstile> c \<sim> c' \<Longrightarrow>
+ (\<And>s s'. (c, s) \<Rightarrow> s' \<Longrightarrow> P s \<Longrightarrow> bval b s \<Longrightarrow> P s') \<Longrightarrow>
P \<Turnstile> WHILE b DO c \<sim> WHILE b' DO c'"
- by (fastforce elim!: equiv_up_to_while equiv_up_to_weaken
- simp: hoare_valid_def)
+ by (fastforce elim!: equiv_up_to_while equiv_up_to_weaken)
lemma equiv_up_to_if:
"P \<Turnstile> b <\<sim>> b' \<Longrightarrow> (\<lambda>s. P s \<and> bval b s) \<Turnstile> c \<sim> c' \<Longrightarrow> (\<lambda>s. P s \<and> \<not>bval b s) \<Turnstile> d \<sim> d' \<Longrightarrow>
@@ -142,7 +148,7 @@
lemma equiv_up_to_if_True [intro!]:
"(\<And>s. P s \<Longrightarrow> bval b s) \<Longrightarrow> P \<Turnstile> IF b THEN c1 ELSE c2 \<sim> c1"
- by (auto simp: equiv_up_to_def)
+ by (auto simp: equiv_up_to_def)
lemma equiv_up_to_if_False [intro!]:
"(\<And>s. P s \<Longrightarrow> \<not> bval b s) \<Longrightarrow> P \<Turnstile> IF b THEN c1 ELSE c2 \<sim> c2"
@@ -154,7 +160,7 @@
lemma while_never: "(c, s) \<Rightarrow> u \<Longrightarrow> c \<noteq> WHILE (Bc True) DO c'"
by (induct rule: big_step_induct) auto
-
+
lemma equiv_up_to_while_True [intro!,simp]:
"P \<Turnstile> WHILE Bc True DO c \<sim> WHILE Bc True DO SKIP"
unfolding equiv_up_to_def