bugfix concerning claset(), added limited support for ALLGOALS + fast_tac etc.
(* Title: HOL/Bali/AxSound.thy
ID: $Id$
Author: David von Oheimb
License: GPL (GNU GENERAL PUBLIC LICENSE)
*)
header {* Soundness proof for Axiomatic semantics of Java expressions and
statements
*}
theory AxSound = AxSem:
section "validity"
consts
triple_valid2:: "prog \<Rightarrow> nat \<Rightarrow> 'a triple \<Rightarrow> bool"
( "_\<Turnstile>_\<Colon>_"[61,0, 58] 57)
ax_valids2:: "prog \<Rightarrow> 'a triples \<Rightarrow> 'a triples \<Rightarrow> bool"
("_,_|\<Turnstile>\<Colon>_" [61,58,58] 57)
defs triple_valid2_def: "G\<Turnstile>n\<Colon>t \<equiv> case t of {P} t\<succ> {Q} \<Rightarrow>
\<forall>Y s Z. P Y s Z \<longrightarrow> (\<forall>L. s\<Colon>\<preceq>(G,L)
\<longrightarrow> (\<forall>T C. (normal s \<longrightarrow> \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>t\<Colon>T) \<longrightarrow>
(\<forall>Y' s'. G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (Y',s') \<longrightarrow> Q Y' s' Z \<and> s'\<Colon>\<preceq>(G,L))))"
defs ax_valids2_def: "G,A|\<Turnstile>\<Colon>ts \<equiv> \<forall>n. (\<forall>t\<in>A. G\<Turnstile>n\<Colon>t) \<longrightarrow> (\<forall>t\<in>ts. G\<Turnstile>n\<Colon>t)"
lemma triple_valid2_def2: "G\<Turnstile>n\<Colon>{P} t\<succ> {Q} =
(\<forall>Y s Z. P Y s Z \<longrightarrow> (\<forall>Y' s'. G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (Y',s')\<longrightarrow>
(\<forall>L. s\<Colon>\<preceq>(G,L) \<longrightarrow> (\<forall>T C. (normal s \<longrightarrow> \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>t\<Colon>T) \<longrightarrow>
Q Y' s' Z \<and> s'\<Colon>\<preceq>(G,L)))))"
apply (unfold triple_valid2_def)
apply (simp (no_asm) add: split_paired_All)
apply blast
done
lemma triple_valid2_eq [rule_format (no_asm)]:
"wf_prog G ==> triple_valid2 G = triple_valid G"
apply (rule ext)
apply (rule ext)
apply (rule triple.induct)
apply (simp (no_asm) add: triple_valid_def2 triple_valid2_def2)
apply (rule iffI)
apply fast
apply clarify
apply (tactic "smp_tac 3 1")
apply (case_tac "normal s")
apply clarsimp
apply (blast dest: evaln_eval eval_type_sound [THEN conjunct1])
apply clarsimp
done
lemma ax_valids2_eq: "wf_prog G \<Longrightarrow> G,A|\<Turnstile>\<Colon>ts = G,A|\<Turnstile>ts"
apply (unfold ax_valids_def ax_valids2_def)
apply (force simp add: triple_valid2_eq)
done
lemma triple_valid2_Suc [rule_format (no_asm)]: "G\<Turnstile>Suc n\<Colon>t \<longrightarrow> G\<Turnstile>n\<Colon>t"
apply (induct_tac "t")
apply (subst triple_valid2_def2)
apply (subst triple_valid2_def2)
apply (fast intro: evaln_nonstrict_Suc)
done
lemma Methd_triple_valid2_0: "G\<Turnstile>0\<Colon>{Normal P} Methd C sig-\<succ> {Q}"
apply (clarsimp elim!: evaln_elim_cases simp add: triple_valid2_def2)
done
lemma Methd_triple_valid2_SucI:
"\<lbrakk>G\<Turnstile>n\<Colon>{Normal P} body G C sig-\<succ>{Q}\<rbrakk>
\<Longrightarrow> G\<Turnstile>Suc n\<Colon>{Normal P} Methd C sig-\<succ> {Q}"
apply (simp (no_asm_use) add: triple_valid2_def2)
apply (intro strip, tactic "smp_tac 3 1", clarify)
apply (erule wt_elim_cases, erule evaln_elim_cases)
apply (unfold body_def Let_def)
apply clarsimp
apply blast
done
lemma triples_valid2_Suc:
"Ball ts (triple_valid2 G (Suc n)) \<Longrightarrow> Ball ts (triple_valid2 G n)"
apply (fast intro: triple_valid2_Suc)
done
lemma "G|\<Turnstile>n:insert t A = (G\<Turnstile>n:t \<and> G|\<Turnstile>n:A)";
oops
section "soundness"
lemma Methd_sound:
"\<lbrakk>G,A\<union> {{P} Methd-\<succ> {Q} | ms}|\<Turnstile>\<Colon>{{P} body G-\<succ> {Q} | ms}\<rbrakk> \<Longrightarrow>
G,A|\<Turnstile>\<Colon>{{P} Methd-\<succ> {Q} | ms}"
apply (unfold ax_valids2_def mtriples_def)
apply (rule allI)
apply (induct_tac "n")
apply (clarify, tactic {* pair_tac "x" 1 *}, simp (no_asm))
apply (fast intro: Methd_triple_valid2_0)
apply (clarify, tactic {* pair_tac "xa" 1 *}, simp (no_asm))
apply (drule triples_valid2_Suc)
apply (erule (1) notE impE)
apply (drule_tac x = na in spec)
apply (tactic {* auto_tac (claset() addSIs [thm "Methd_triple_valid2_SucI"],
simpset() addsimps [ball_Un] addloop ("split_all_tac", split_all_tac)) *})
done
lemma valids2_inductI: "\<forall>s t n Y' s'. G\<turnstile>s\<midarrow>t\<succ>\<midarrow>n\<rightarrow> (Y',s') \<longrightarrow> t = c \<longrightarrow>
Ball A (triple_valid2 G n) \<longrightarrow> (\<forall>Y Z. P Y s Z \<longrightarrow>
(\<forall>L. s\<Colon>\<preceq>(G,L) \<longrightarrow> (\<forall>T C. (normal s \<longrightarrow> \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>t\<Colon>T) \<longrightarrow>
Q Y' s' Z \<and> s'\<Colon>\<preceq>(G, L)))) \<Longrightarrow>
G,A|\<Turnstile>\<Colon>{ {P} c\<succ> {Q}}"
apply (simp (no_asm) add: ax_valids2_def triple_valid2_def2)
apply clarsimp
done
ML_setup {*
Delsimprocs [evaln_expr_proc,evaln_var_proc,evaln_exprs_proc,evaln_stmt_proc]
*}
lemma Loop_sound: "\<lbrakk>G,A|\<Turnstile>\<Colon>{ {P} e-\<succ> {P'}};
G,A|\<Turnstile>\<Colon>{ {Normal (P'\<leftarrow>=True)} .c. {abupd (absorb (Cont l)) .; P}}\<rbrakk> \<Longrightarrow>
G,A|\<Turnstile>\<Colon>{ {P} .l\<bullet> While(e) c. {(P'\<leftarrow>=False)\<down>=\<diamondsuit>}}"
apply (rule valids2_inductI)
apply ((rule allI)+, rule impI, tactic {* pair_tac "s" 1*}, tactic {* pair_tac "s'" 1*})
apply (erule evaln.induct)
apply simp_all (* takes half a minute *)
apply clarify
apply (erule_tac V = "G,A|\<Turnstile>\<Colon>{ {?P'} .c. {?P}}" in thin_rl)
apply (simp_all (no_asm_use) add: ax_valids2_def triple_valid2_def2)
apply (tactic "smp_tac 1 1", tactic "smp_tac 3 1", force)
apply clarify
apply (rule wt_elim_cases, assumption)
apply (tactic "smp_tac 1 1", tactic "smp_tac 1 1", tactic "smp_tac 3 1",
tactic "smp_tac 2 1", tactic "smp_tac 1 1")
apply (erule impE,simp (no_asm),blast)
apply (simp add: imp_conjL split_tupled_all split_paired_All)
apply (case_tac "the_Bool b")
apply clarsimp
apply (case_tac "a")
apply (simp_all)
apply clarsimp
apply (erule_tac V = "c = l\<bullet> While(e) c \<longrightarrow> ?P" in thin_rl)
apply (blast intro: conforms_absorb)
apply blast+
done
declare subst_Bool_def2 [simp del]
lemma all_empty: "(!x. P) = P"
by simp
lemma sound_valid2_lemma:
"\<lbrakk>\<forall>v n. Ball A (triple_valid2 G n) \<longrightarrow> P v n; Ball A (triple_valid2 G n)\<rbrakk>
\<Longrightarrow>P v n"
by blast
ML {*
val fullsimptac = full_simp_tac(simpset() delsimps [thm "all_empty"]);
val sound_prepare_tac = EVERY'[REPEAT o thin_tac "?x \<in> ax_derivs G",
full_simp_tac (simpset()addsimps[thm "ax_valids2_def",thm "triple_valid2_def2",
thm "imp_conjL"] delsimps[thm "all_empty"]),
Clarify_tac];
val sound_elim_tac = EVERY'[eresolve_tac (thms "evaln_elim_cases"),
TRY o eresolve_tac (thms "wt_elim_cases"), fullsimptac, Clarify_tac];
val sound_valid2_tac = REPEAT o FIRST'[smp_tac 1,
datac (thm "sound_valid2_lemma") 1];
val sound_forw_hyp_tac =
EVERY'[smp_tac 3
ORELSE' EVERY'[dtac spec,dtac spec, dtac spec,etac impE, Fast_tac]
ORELSE' EVERY'[dtac spec,dtac spec,etac impE, Fast_tac],
fullsimptac,
smp_tac 2,TRY o smp_tac 1,
TRY o EVERY'[etac impE, TRY o rtac impI,
atac ORELSE' (EVERY' [REPEAT o rtac exI,Blast_tac]),
fullsimptac, Clarify_tac, TRY o smp_tac 1]]
*}
(* ### rtac conjI,rtac HOL.refl *)
lemma Call_sound:
"\<lbrakk>wf_prog G; G,A|\<Turnstile>\<Colon>{ {Normal P} e-\<succ> {Q}}; \<forall>a. G,A|\<Turnstile>\<Colon>{ {Q\<leftarrow>Val a} ps\<doteq>\<succ> {R a}};
\<forall>a vs invC declC l. G,A|\<Turnstile>\<Colon>{ {(R a\<leftarrow>Vals vs \<and>.
(\<lambda>s. declC = invocation_declclass
G mode (store s) a statT \<lparr>name=mn,parTs=pTs\<rparr> \<and>
invC = invocation_class mode (store s) a statT \<and>
l = locals (store s)) ;.
init_lvars G declC \<lparr>name=mn,parTs=pTs\<rparr> mode a vs) \<and>.
(\<lambda>s. normal s \<longrightarrow> G\<turnstile>mode\<rightarrow>invC\<preceq>statT)}
Methd declC \<lparr>name=mn,parTs=pTs\<rparr>-\<succ> {set_lvars l .; S}}\<rbrakk> \<Longrightarrow>
G,A|\<Turnstile>\<Colon>{ {Normal P} {accC,statT,mode}e\<cdot>mn({pTs}ps)-\<succ> {S}}"
apply (tactic "EVERY'[sound_prepare_tac, sound_elim_tac, sound_valid2_tac] 1")
apply (rename_tac x1 s1 x2 s2 ab bb v vs m pTsa statDeclC)
apply (tactic "smp_tac 6 1")
apply (tactic "sound_forw_hyp_tac 1")
apply (tactic "sound_forw_hyp_tac 1")
apply (drule max_spec2mheads)
apply (drule (3) evaln_eval, drule (3) eval_ts)
apply (drule (3) evaln_eval, frule (3) evals_ts)
apply (drule spec,erule impE,rule exI, blast)
(* apply (drule spec,drule spec,drule spec,erule impE,rule exI,blast) *)
apply (case_tac "if is_static m then x2 else (np a') x2")
defer 1
apply (rename_tac x, subgoal_tac "(Some x, s2)\<Colon>\<preceq>(G, L)" (* used two times *))
prefer 2
apply (force split add: split_if_asm)
apply (simp del: if_raise_if_None)
apply (tactic "smp_tac 2 1")
apply (simp only: init_lvars_def2 invmode_Static_eq)
apply (clarsimp simp del: resTy_mthd)
apply (drule spec,erule swap,erule conforms_set_locals [OF _ lconf_empty])
apply clarsimp
apply (drule Null_staticD)
apply (drule eval_gext', drule (1) conf_gext, frule (3) DynT_propI)
apply (erule impE) apply blast
apply (subgoal_tac
"G\<turnstile>invmode (mhd (statDeclC,m)) e
\<rightarrow>invocation_class (invmode m e) s2 a' statT\<preceq>statT")
defer apply simp
apply (drule (3) DynT_mheadsD,simp,simp)
apply (clarify, drule wf_mdeclD1, clarify)
apply (frule ty_expr_is_type) apply simp
apply (subgoal_tac "invmode (mhd (statDeclC,m)) e = IntVir \<longrightarrow> a' \<noteq> Null")
defer apply simp
apply (frule (2) wt_MethdI)
apply clarify
apply (drule (2) conforms_init_lvars)
apply (simp)
apply (assumption)+
apply simp
apply (assumption)+
apply (rule impI) apply simp
apply simp
apply simp
apply (rule Ball_weaken)
apply assumption
apply (force simp add: is_acc_type_def)
apply (tactic "smp_tac 2 1")
apply simp
apply (tactic "smp_tac 1 1")
apply (erule_tac V = "?P \<longrightarrow> ?Q" in thin_rl)
apply (erule impE)
apply (rule exI)+
apply (subgoal_tac "is_static dm = (static m)")
prefer 2 apply (simp add: member_is_static_simp)
apply (simp only: )
apply (simp only: sig.simps)
apply (force dest!: evaln_eval eval_gext' elim: conforms_return
del: impCE simp add: init_lvars_def2)
done
corollary evaln_type_sound:
assumes evaln: "G\<turnstile>s0 \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (v,s1)" and
wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>t\<Colon>T" and
conf_s0: "s0\<Colon>\<preceq>(G,L)" and
wf: "wf_prog G"
shows "s1\<Colon>\<preceq>(G,L) \<and> (normal s1 \<longrightarrow> G,L,store s1\<turnstile>t\<succ>v\<Colon>\<preceq>T) \<and>
(error_free s0 = error_free s1)"
proof -
from evaln wt conf_s0 wf
show ?thesis
by (blast dest: evaln_eval eval_type_sound)
qed
lemma Init_sound: "\<lbrakk>wf_prog G; the (class G C) = c;
G,A|\<Turnstile>\<Colon>{ {Normal ((P \<and>. Not \<circ> initd C) ;. supd (init_class_obj G C))}
.(if C = Object then Skip else Init (super c)). {Q}};
\<forall>l. G,A|\<Turnstile>\<Colon>{ {Q \<and>. (\<lambda>s. l = locals (store s)) ;. set_lvars empty}
.init c. {set_lvars l .; R}}\<rbrakk> \<Longrightarrow>
G,A|\<Turnstile>\<Colon>{ {Normal (P \<and>. Not \<circ> initd C)} .Init C. {R}}"
apply (tactic "EVERY'[sound_prepare_tac, sound_elim_tac,sound_valid2_tac] 1")
apply (tactic {* instantiate_tac [("l24","\<lambda> n Y Z sa Y' s' L y a b aa ba ab bb. locals b")]*})
apply (clarsimp simp add: split_paired_Ex)
apply (drule spec, drule spec, drule spec, erule impE)
apply (erule_tac V = "All ?P" in thin_rl, fast)
apply clarsimp
apply (tactic "smp_tac 2 1", drule spec, erule impE,
erule (3) conforms_init_class_obj)
apply (frule (1) wf_prog_cdecl)
apply (erule impE, rule exI,erule_tac V = "All ?P" in thin_rl,
force dest: wf_cdecl_supD split add: split_if simp add: is_acc_class_def)
apply clarify
apply (drule spec)
apply (drule spec)
apply (drule spec)
apply (erule impE)
apply ( fast)
apply (simp (no_asm_use) del: empty_def2)
apply (tactic "smp_tac 2 1")
apply (drule spec, erule impE, erule conforms_set_locals, rule lconf_empty)
apply (erule impE,rule impI,rule exI,erule wf_cdecl_wt_init)
apply clarsimp
apply (erule (1) conforms_return)
apply (frule wf_cdecl_wt_init)
apply (subgoal_tac "(a, set_locals empty b)\<Colon>\<preceq>(G, empty)")
apply (frule (3) evaln_eval)
apply (drule eval_gext')
apply force
(* refer to case Init in eval_type_sound proof, to see whats going on*)
apply (subgoal_tac "(a,b)\<Colon>\<preceq>(G, L)")
apply (blast intro: conforms_set_locals)
apply (drule evaln_type_sound)
apply (cases "C=Object")
apply force
apply (force dest: wf_cdecl_supD is_acc_classD)
apply (erule (4) conforms_init_class_obj)
apply blast
done
lemma all_conjunct2: "\<forall>l. P' l \<and> P l \<Longrightarrow> \<forall>l. P l"
by fast
lemma all4_conjunct2:
"\<forall>a vs D l. (P' a vs D l \<and> P a vs D l) \<Longrightarrow> \<forall>a vs D l. P a vs D l"
by fast
lemmas sound_lemmas = Init_sound Loop_sound Methd_sound
lemma ax_sound2: "wf_prog G \<Longrightarrow> G,A|\<turnstile>ts \<Longrightarrow> G,A|\<Turnstile>\<Colon>ts"
apply (erule ax_derivs.induct)
prefer 20 (* Call *)
apply (erule (1) Call_sound) apply simp apply force apply force
apply (tactic {* TRYALL (eresolve_tac (thms "sound_lemmas") THEN_ALL_NEW
eresolve_tac [asm_rl, thm "all_conjunct2", thm "all4_conjunct2"]) *})
apply(tactic "COND (has_fewer_prems(30+3)) (ALLGOALS sound_prepare_tac) no_tac")
(*empty*)
apply fast (* insert *)
apply fast (* asm *)
(*apply fast *) (* cut *)
apply fast (* weaken *)
apply (tactic "smp_tac 3 1", clarify, tactic "smp_tac 1 1",
case_tac"fst s",clarsimp,erule (3) evaln_type_sound [THEN conjunct1],
clarsimp) (* conseq *)
apply (simp (no_asm_use) add: type_ok_def,fast)(* hazard *)
apply force (* Abrupt *)
(* 25 subgoals *)
apply (tactic {* ALLGOALS sound_elim_tac *})(* LVar, Lit, Super, Nil, Skip,Do *)
apply (tactic {* ALLGOALS (asm_simp_tac (noAll_simpset()
delsimps [thm "all_empty"])) *}) (* Done *)
(* for FVar *)
apply (frule wf_ws_prog)
apply (frule ty_expr_is_type [THEN type_is_class,
THEN accfield_declC_is_class])
apply (simp,simp,simp)
apply (frule_tac [4] wt_init_comp_ty) (* for NewA*)
apply (tactic "ALLGOALS sound_valid2_tac")
apply (tactic "TRYALL sound_forw_hyp_tac") (* Cast, Inst, Acc, Expr *)
apply (tactic {* TRYALL (EVERY'[dtac spec, TRY o EVERY'[rotate_tac ~1,
dtac spec], dtac conjunct2, smp_tac 1,
TRY o dres_inst_tac [("P","P'")] (thm "subst_Bool_the_BoolI")]) *})
apply (frule_tac [14] x = x1 in conforms_NormI) (* for Fin *)
(* 15 subgoals *)
(* FVar *)
apply (tactic "sound_forw_hyp_tac 1")
apply (clarsimp simp add: fvar_def2 Let_def split add: split_if_asm)
(* AVar *)
(*
apply (drule spec, drule spec, erule impE, fast)
apply (simp)
apply (tactic "smp_tac 2 1")
apply (tactic "smp_tac 1 1")
apply (erule impE)
apply (rule impI)
apply (rule exI)+
apply blast
apply (clarsimp simp add: avar_def2)
*)
apply (tactic "sound_forw_hyp_tac 1")
apply (clarsimp simp add: avar_def2)
(* NewC *)
apply (clarsimp simp add: is_acc_class_def)
apply (erule (1) halloc_conforms, simp, simp)
(* NewA *)
apply (tactic "sound_forw_hyp_tac 1")
apply (rule conjI,blast)
apply (erule (1) halloc_conforms, simp, simp, simp add: is_acc_type_def)
(* Ass *)
apply (tactic "sound_forw_hyp_tac 1")
apply (case_tac "aa")
prefer 2
apply clarsimp
apply (drule (3) evaln_type_sound)
apply (drule (3) evaln_eval)
apply (frule (3) eval_type_sound)
apply clarsimp
apply (frule wf_ws_prog)
apply (drule (2) conf_widen)
apply (drule_tac "s1.0" = b in eval_gext')
apply (clarsimp simp add: assign_conforms_def)
(* Cond *)
apply (tactic "smp_tac 3 1") apply (tactic "smp_tac 2 1")
apply (tactic "smp_tac 1 1") apply (erule impE)
apply (rule impI,rule exI)
apply (rule_tac x = "if the_Bool b then T1 else T2" in exI)
apply (force split add: split_if)
apply assumption
(* Body *)
apply (tactic "sound_forw_hyp_tac 1")
apply (rule conforms_absorb,assumption)
(* Lab *)
apply (tactic "sound_forw_hyp_tac 1")
apply (rule conforms_absorb,assumption)
(* If *)
apply (tactic "sound_forw_hyp_tac 1")
apply (tactic "sound_forw_hyp_tac 1")
apply (force split add: split_if)
(* Throw *)
apply (drule (3) evaln_type_sound)
apply clarsimp
apply (drule (3) Throw_lemma)
apply clarsimp
(* Try *)
apply (frule (1) sxalloc_type_sound)
apply (erule sxalloc_elim_cases2)
apply (tactic "smp_tac 3 1")
apply (clarsimp split add: option.split_asm)
apply (clarsimp split add: option.split_asm)
apply (tactic "smp_tac 1 1")
apply (simp only: split add: split_if_asm)
prefer 2
apply (tactic "smp_tac 3 1", erule_tac V = "All ?P" in thin_rl, clarsimp)
apply (drule spec, erule_tac V = "All ?P" in thin_rl, drule spec, drule spec,
erule impE, force)
apply (frule (2) Try_lemma)
apply clarsimp
apply (fast elim!: conforms_deallocL)
(* Fin *)
apply (tactic "sound_forw_hyp_tac 1")
apply (case_tac "x1", force)
apply clarsimp
apply (drule (3) evaln_eval, drule (4) Fin_lemma)
done
declare subst_Bool_def2 [simp]
theorem ax_sound:
"wf_prog G \<Longrightarrow> G,(A::'a triple set)|\<turnstile>(ts::'a triple set) \<Longrightarrow> G,A|\<Turnstile>ts"
apply (subst ax_valids2_eq [symmetric])
apply assumption
apply (erule (1) ax_sound2)
done
end