(* Title: HOL/NanoJava/Equivalence.thy
ID: $Id$
Author: David von Oheimb
Copyright 2001 Technische Universitaet Muenchen
*)
header "Equivalence of Operational and Axiomatic Semantics"
theory Equivalence = OpSem + AxSem:
subsection "Validity"
constdefs
valid :: "[assn,stmt, assn] => bool" ("|= {(1_)}/ (_)/ {(1_)}" [3,90,3] 60)
"|= {P} c {Q} \<equiv> \<forall>s t. P s --> (\<exists>n. s -c -n-> t) --> Q t"
evalid :: "[assn,expr,vassn] => bool" ("|=e {(1_)}/ (_)/ {(1_)}" [3,90,3] 60)
"|=e {P} e {Q} \<equiv> \<forall>s v t. P s --> (\<exists>n. s -e>v-n-> t) --> Q v t"
nvalid :: "[nat, triple ] => bool" ("|=_: _" [61,61] 60)
"|=n: t \<equiv> let (P,c,Q) = t in \<forall>s t. s -c -n-> t --> P s --> Q t"
envalid :: "[nat,etriple ] => bool" ("|=_:e _" [61,61] 60)
"|=n:e t \<equiv> let (P,e,Q) = t in \<forall>s v t. s -e>v-n-> t --> P s --> Q v t"
nvalids :: "[nat, triple set] => bool" ("||=_: _" [61,61] 60)
"||=n: T \<equiv> \<forall>t\<in>T. |=n: t"
cnvalids :: "[triple set,triple set] => bool" ("_ ||=/ _" [61,61] 60)
"A ||= C \<equiv> \<forall>n. ||=n: A --> ||=n: C"
cenvalid :: "[triple set,etriple ] => bool" ("_ ||=e/ _" [61,61] 60)
"A ||=e t \<equiv> \<forall>n. ||=n: A --> |=n:e t"
syntax (xsymbols)
valid :: "[assn,stmt, assn] => bool" ( "\<Turnstile> {(1_)}/ (_)/ {(1_)}" [3,90,3] 60)
evalid :: "[assn,expr,vassn] => bool" ("\<Turnstile>\<^sub>e {(1_)}/ (_)/ {(1_)}" [3,90,3] 60)
nvalid :: "[nat, triple ] => bool" ("\<Turnstile>_: _" [61,61] 60)
envalid :: "[nat,etriple ] => bool" ("\<Turnstile>_:\<^sub>e _" [61,61] 60)
nvalids :: "[nat, triple set] => bool" ("|\<Turnstile>_: _" [61,61] 60)
cnvalids :: "[triple set,triple set] => bool" ("_ |\<Turnstile>/ _" [61,61] 60)
cenvalid :: "[triple set,etriple ] => bool" ("_ |\<Turnstile>\<^sub>e/ _"[61,61] 60)
lemma nvalid_def2: "\<Turnstile>n: (P,c,Q) \<equiv> \<forall>s t. s -c-n\<rightarrow> t \<longrightarrow> P s \<longrightarrow> Q t"
by (simp add: nvalid_def Let_def)
lemma valid_def2: "\<Turnstile> {P} c {Q} = (\<forall>n. \<Turnstile>n: (P,c,Q))"
apply (simp add: valid_def nvalid_def2)
apply blast
done
lemma envalid_def2: "\<Turnstile>n:\<^sub>e (P,e,Q) \<equiv> \<forall>s v t. s -e\<succ>v-n\<rightarrow> t \<longrightarrow> P s \<longrightarrow> Q v t"
by (simp add: envalid_def Let_def)
lemma evalid_def2: "\<Turnstile>\<^sub>e {P} e {Q} = (\<forall>n. \<Turnstile>n:\<^sub>e (P,e,Q))"
apply (simp add: evalid_def envalid_def2)
apply blast
done
lemma cenvalid_def2:
"A|\<Turnstile>\<^sub>e (P,e,Q) = (\<forall>n. |\<Turnstile>n: A \<longrightarrow> (\<forall>s v t. s -e\<succ>v-n\<rightarrow> t \<longrightarrow> P s \<longrightarrow> Q v t))"
by(simp add: cenvalid_def envalid_def2)
subsection "Soundness"
declare exec_elim_cases [elim!] eval_elim_cases [elim!]
lemma Impl_nvalid_0: "\<Turnstile>0: (P,Impl C m,Q)"
by (clarsimp simp add: nvalid_def2)
lemma Impl_nvalid_Suc: "\<Turnstile>n: (P,body C m,Q) \<Longrightarrow> \<Turnstile>Suc n: (P,Impl C m,Q)"
by (clarsimp simp add: nvalid_def2)
lemma nvalid_SucD: "\<And>t. \<Turnstile>Suc n:t \<Longrightarrow> \<Turnstile>n:t"
by (force simp add: split_paired_all nvalid_def2 intro: exec_mono)
lemma nvalids_SucD: "Ball A (nvalid (Suc n)) \<Longrightarrow> Ball A (nvalid n)"
by (fast intro: nvalid_SucD)
lemma Loop_sound_lemma [rule_format (no_asm)]:
"\<forall>s t. s -c-n\<rightarrow> t \<longrightarrow> P s \<and> s<x> \<noteq> Null \<longrightarrow> P t \<Longrightarrow>
(s -c0-n0\<rightarrow> t \<longrightarrow> P s \<longrightarrow> c0 = While (x) c \<longrightarrow> n0 = n \<longrightarrow> P t \<and> t<x> = Null)"
apply (rule_tac "P2.1"="%s e v n t. True" in exec_eval.induct [THEN conjunct1])
apply clarsimp+
done
lemma Impl_sound_lemma:
"\<lbrakk>\<forall>n. Ball (A \<union> B) (nvalid n) \<longrightarrow> Ball (\<Union>z. split (f z) ` ms) (nvalid n);
(C, m) \<in> ms; Ball A (nvalid na); Ball B (nvalid na)\<rbrakk> \<Longrightarrow> nvalid na (f z C m)"
by blast
lemma all_conjunct2: "\<forall>l. P' l \<and> P l \<Longrightarrow> \<forall>l. P l"
by fast
lemma all3_conjunct2:
"\<forall>a p l. (P' a p l \<and> P a p l) \<Longrightarrow> \<forall>a p l. P a p l"
by fast
lemma cnvalid1_eq:
"A |\<Turnstile> {(P,c,Q)} \<equiv> \<forall>n. |\<Turnstile>n: A \<longrightarrow> (\<forall>s t. s -c-n\<rightarrow> t \<longrightarrow> P s \<longrightarrow> Q t)"
by(simp add: cnvalids_def nvalids_def nvalid_def2)
lemma hoare_sound_main:"\<And>t. (A |\<turnstile> C \<longrightarrow> A |\<Turnstile> C) \<and> (A |\<turnstile>\<^sub>e t \<longrightarrow> A |\<Turnstile>\<^sub>e t)"
apply (tactic "split_all_tac 1", rename_tac P e Q)
apply (rule hoare_ehoare.induct)
apply (tactic {* ALLGOALS (REPEAT o dresolve_tac [thm "all_conjunct2", thm "all3_conjunct2"]) *})
apply (tactic {* ALLGOALS (REPEAT o thin_tac "?x : hoare") *})
apply (tactic {* ALLGOALS (REPEAT o thin_tac "?x : ehoare") *})
apply (simp_all only: cnvalid1_eq cenvalid_def2)
apply fast
apply fast
apply fast
apply (clarify,tactic "smp_tac 1 1",erule(2) Loop_sound_lemma,(rule HOL.refl)+)
apply fast
apply fast
apply fast
apply fast
apply fast
apply fast
apply (clarsimp del: Meth_elim_cases) (* Call *)
apply (tactic "smp_tac 1 1", tactic "smp_tac 3 1", tactic "smp_tac 0 1")
apply (tactic "smp_tac 2 1", tactic "smp_tac 3 1", tactic "smp_tac 0 1")
apply (tactic "smp_tac 4 1", tactic "smp_tac 2 1", fast)
apply (clarsimp del: Impl_elim_cases) (* Meth *)
defer
prefer 4 apply blast (* Conseq *)
prefer 4 apply blast (* eConseq *)
apply (simp_all (no_asm_use) only: cnvalids_def nvalids_def)
apply blast
apply blast
apply blast
(* Impl *)
apply (rule allI)
apply (induct_tac "n")
apply (clarify intro!: Impl_nvalid_0)
apply (clarify intro!: Impl_nvalid_Suc)
apply (drule nvalids_SucD)
apply (erule (1) impE)
apply (drule (4) Impl_sound_lemma)
done
theorem hoare_sound: "{} \<turnstile> {P} c {Q} \<Longrightarrow> \<Turnstile> {P} c {Q}"
apply (simp only: valid_def2)
apply (drule hoare_sound_main [THEN conjunct1, rule_format])
apply (unfold cnvalids_def nvalids_def)
apply fast
done
theorem ehoare_sound: "{} \<turnstile>\<^sub>e {P} e {Q} \<Longrightarrow> \<Turnstile>\<^sub>e {P} e {Q}"
apply (simp only: evalid_def2)
apply (drule hoare_sound_main [THEN conjunct2, rule_format])
apply (unfold cenvalid_def nvalids_def)
apply fast
done
subsection "(Relative) Completeness"
constdefs MGT :: "stmt => state => triple"
"MGT c z \<equiv> (\<lambda>s. z = s, c, \<lambda> t. \<exists>n. z -c- n-> t)"
MGTe :: "expr => state => etriple"
"MGTe e z \<equiv> (\<lambda>s. z = s, e, \<lambda>v t. \<exists>n. z -e>v-n-> t)"
syntax (xsymbols)
MGTe :: "expr => state => etriple" ("MGT\<^sub>e")
lemma MGF_implies_complete:
"\<forall>z. {} |\<turnstile> { MGT c z} \<Longrightarrow> \<Turnstile> {P} c {Q} \<Longrightarrow> {} \<turnstile> {P} c {Q}"
apply (simp only: valid_def2)
apply (unfold MGT_def)
apply (erule hoare_ehoare.Conseq)
apply (clarsimp simp add: nvalid_def2)
done
lemma eMGF_implies_complete:
"\<forall>z. {} |\<turnstile>\<^sub>e MGT\<^sub>e e z \<Longrightarrow> \<Turnstile>\<^sub>e {P} e {Q} \<Longrightarrow> {} \<turnstile>\<^sub>e {P} e {Q}"
apply (simp only: evalid_def2)
apply (unfold MGTe_def)
apply (erule hoare_ehoare.eConseq)
apply (clarsimp simp add: envalid_def2)
done
declare exec_eval.intros[intro!]
lemma MGF_Loop: "\<forall>z. A \<turnstile> {op = z} c {\<lambda>t. \<exists>n. z -c-n\<rightarrow> t} \<Longrightarrow>
A \<turnstile> {op = z} While (x) c {\<lambda>t. \<exists>n. z -While (x) c-n\<rightarrow> t}"
apply (rule_tac P' = "\<lambda>z s. (z,s) \<in> ({(s,t). \<exists>n. s<x> \<noteq> Null \<and> s -c-n\<rightarrow> t})^*"
in hoare_ehoare.Conseq)
apply (rule allI)
apply (rule hoare_ehoare.Loop)
apply (erule hoare_ehoare.Conseq)
apply clarsimp
apply (blast intro:rtrancl_into_rtrancl)
apply (erule thin_rl)
apply clarsimp
apply (erule_tac x = z in allE)
apply clarsimp
apply (erule converse_rtrancl_induct)
apply blast
apply clarsimp
apply (drule (1) exec_exec_max)
apply (blast del: exec_elim_cases)
done
lemma MGF_lemma: "\<forall>C m z. A |\<turnstile> {MGT (Impl C m) z} \<Longrightarrow>
(\<forall>z. A |\<turnstile> {MGT c z}) \<and> (\<forall>z. A |\<turnstile>\<^sub>e MGT\<^sub>e e z)"
apply (simp add: MGT_def MGTe_def)
apply (rule stmt_expr.induct)
apply (rule_tac [!] allI)
apply (rule Conseq1 [OF hoare_ehoare.Skip])
apply blast
apply (rule hoare_ehoare.Comp)
apply (erule spec)
apply (erule hoare_ehoare.Conseq)
apply clarsimp
apply (drule (1) exec_exec_max)
apply blast
apply (erule thin_rl)
apply (rule hoare_ehoare.Cond)
apply (erule spec)
apply (rule allI)
apply (simp)
apply (rule conjI)
apply (rule impI, erule hoare_ehoare.Conseq, clarsimp, drule (1) eval_exec_max,
erule thin_rl, erule thin_rl, force)+
apply (erule MGF_Loop)
apply (erule hoare_ehoare.eConseq [THEN hoare_ehoare.LAss])
apply fast
apply (erule thin_rl)
apply (rule_tac Q = "\<lambda>a s. \<exists>n. z -expr1\<succ>Addr a-n\<rightarrow> s" in hoare_ehoare.FAss)
apply (drule spec)
apply (erule eConseq2)
apply fast
apply (rule allI)
apply (erule hoare_ehoare.eConseq)
apply clarsimp
apply (drule (1) eval_eval_max)
apply blast
apply (rule hoare_ehoare.Meth)
apply (rule allI)
apply (drule spec, drule spec, erule hoare_ehoare.Conseq)
apply blast
apply blast
apply (rule eConseq1 [OF hoare_ehoare.NewC])
apply blast
apply (erule hoare_ehoare.eConseq [THEN hoare_ehoare.Cast])
apply fast
apply (rule eConseq1 [OF hoare_ehoare.LAcc])
apply blast
apply (erule hoare_ehoare.eConseq [THEN hoare_ehoare.FAcc])
apply fast
apply (rule_tac R = "\<lambda>a v s. \<exists>n1 n2 t. z -expr1\<succ>a-n1\<rightarrow> t \<and> t -expr2\<succ>v-n2\<rightarrow> s" in
hoare_ehoare.Call)
apply (erule spec)
apply (rule allI)
apply (erule hoare_ehoare.eConseq)
apply clarsimp
apply blast
apply (rule allI)+
apply (rule hoare_ehoare.Meth)
apply (rule allI)
apply (drule spec, drule spec, erule hoare_ehoare.Conseq)
apply (erule thin_rl, erule thin_rl)
apply (clarsimp del: Impl_elim_cases)
apply (drule (2) eval_eval_exec_max)
apply (fast del: Impl_elim_cases)
done
lemma MGF_Impl: "{} |\<turnstile> {MGT (Impl C m) z}"
apply (unfold MGT_def)
apply (rule Impl1)
apply (rule_tac [2] UNIV_I)
apply clarsimp
apply (rule hoare_ehoare.ConjI)
apply clarsimp
apply (rule ssubst [OF Impl_body_eq])
apply (fold MGT_def)
apply (rule MGF_lemma [THEN conjunct1, rule_format])
apply (rule hoare_ehoare.Asm)
apply force
done
theorem hoare_relative_complete: "\<Turnstile> {P} c {Q} \<Longrightarrow> {} \<turnstile> {P} c {Q}"
apply (rule MGF_implies_complete)
apply (erule_tac [2] asm_rl)
apply (rule allI)
apply (rule MGF_lemma [THEN conjunct1, rule_format])
apply (rule MGF_Impl)
done
theorem ehoare_relative_complete: "\<Turnstile>\<^sub>e {P} e {Q} \<Longrightarrow> {} \<turnstile>\<^sub>e {P} e {Q}"
apply (rule eMGF_implies_complete)
apply (erule_tac [2] asm_rl)
apply (rule allI)
apply (rule MGF_lemma [THEN conjunct2, rule_format])
apply (rule MGF_Impl)
done
end