header "Semantic Equivalence up to a Condition"
theory Sem_Equiv
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
bequiv_up_to :: "assn \<Rightarrow> bexp \<Rightarrow> bexp \<Rightarrow> bool" ("_ \<Turnstile> _ <\<sim>> _" [60,0,10] 60)
where
"P \<Turnstile> b <\<sim>> b' \<equiv> \<forall>s. P s \<longrightarrow> bval b s = bval b' s"
lemma equiv_up_to_True:
"((\<lambda>_. True) \<Turnstile> c \<sim> c') = (c \<sim> c')"
by (simp add: equiv_def equiv_up_to_def)
lemma equiv_up_to_weaken:
"P \<Turnstile> c \<sim> c' \<Longrightarrow> (\<And>s. P' s \<Longrightarrow> P s) \<Longrightarrow> P' \<Turnstile> c \<sim> c'"
by (simp add: equiv_up_to_def)
lemma equiv_up_toI:
"(\<And>s s'. P s \<Longrightarrow> (c, s) \<Rightarrow> s' = (c', s) \<Rightarrow> s') \<Longrightarrow> P \<Turnstile> c \<sim> c'"
by (unfold equiv_up_to_def) blast
lemma equiv_up_toD1:
"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> (c', s) \<Rightarrow> s' \<Longrightarrow> P s \<Longrightarrow> (c, s) \<Rightarrow> s'"
by (unfold equiv_up_to_def) blast
lemma equiv_up_to_refl [simp, intro!]:
"P \<Turnstile> c \<sim> c"
by (auto simp: equiv_up_to_def)
lemma equiv_up_to_sym:
"(P \<Turnstile> c \<sim> c') = (P \<Turnstile> c' \<sim> c)"
by (auto simp: equiv_up_to_def)
lemma equiv_up_to_trans:
"P \<Turnstile> c \<sim> c' \<Longrightarrow> P \<Turnstile> c' \<sim> c'' \<Longrightarrow> P \<Turnstile> c \<sim> c''"
by (auto simp: equiv_up_to_def)
lemma bequiv_up_to_refl [simp, intro!]:
"P \<Turnstile> b <\<sim>> b"
by (auto simp: bequiv_up_to_def)
lemma bequiv_up_to_sym:
"(P \<Turnstile> b <\<sim>> b') = (P \<Turnstile> b' <\<sim>> b)"
by (auto simp: bequiv_up_to_def)
lemma bequiv_up_to_trans:
"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 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>
(\<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) blast
lemma equiv_up_to_while_lemma:
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>
(\<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)
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`
have "bval b' s1" by (simp add: bequiv_up_to_def)
moreover
from WhileTrue.prems
have "(\<lambda>s. P s \<and> bval b s) \<Turnstile> c \<sim> c'" by simp
with `bval b s1` `P s1` `(c, s1) \<Rightarrow> s2`
have "(c', s1) \<Rightarrow> s2" by (simp add: equiv_up_to_def)
moreover
from WhileTrue.prems
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
hence "(WHILE b' DO c', s2) \<Rightarrow> s3" by (rule IH)
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)+
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:
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>
(\<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)
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>
P \<Turnstile> IF b THEN c ELSE d \<sim> IF b' THEN c' ELSE d'"
by (auto simp: bequiv_up_to_def equiv_up_to_def)
lemma equiv_up_to_if_weak:
"P \<Turnstile> b <\<sim>> b' \<Longrightarrow> P \<Turnstile> c \<sim> c' \<Longrightarrow> P \<Turnstile> d \<sim> d' \<Longrightarrow>
P \<Turnstile> IF b THEN c ELSE d \<sim> IF b' THEN c' ELSE d'"
by (fastforce elim!: equiv_up_to_if equiv_up_to_weaken)
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)
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"
by (auto simp: equiv_up_to_def)
lemma equiv_up_to_while_False [intro!]:
"(\<And>s. P s \<Longrightarrow> \<not> bval b s) \<Longrightarrow> P \<Turnstile> WHILE b DO c \<sim> SKIP"
by (auto simp: equiv_up_to_def)
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
by (blast dest: while_never)
end