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(* Title: HOL/Bali/AxSound.thy
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ID: $Id$
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Author: David von Oheimb
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License: GPL (GNU GENERAL PUBLIC LICENSE)
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*)
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header {* Soundness proof for Axiomatic semantics of Java expressions and
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statements
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*}
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theory AxSound = AxSem:
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section "validity"
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consts
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triple_valid2:: "prog \<Rightarrow> nat \<Rightarrow> 'a triple \<Rightarrow> bool"
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( "_\<Turnstile>_\<Colon>_"[61,0, 58] 57)
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ax_valids2:: "prog \<Rightarrow> 'a triples \<Rightarrow> 'a triples \<Rightarrow> bool"
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("_,_|\<Turnstile>\<Colon>_" [61,58,58] 57)
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defs triple_valid2_def: "G\<Turnstile>n\<Colon>t \<equiv> case t of {P} t\<succ> {Q} \<Rightarrow>
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\<forall>Y s Z. P Y s Z \<longrightarrow> (\<forall>L. s\<Colon>\<preceq>(G,L)
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\<longrightarrow> (\<forall>T C. (normal s \<longrightarrow> \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>t\<Colon>T) \<longrightarrow>
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(\<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))))"
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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)"
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lemma triple_valid2_def2: "G\<Turnstile>n\<Colon>{P} t\<succ> {Q} =
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(\<forall>Y s Z. P Y s Z \<longrightarrow> (\<forall>Y' s'. G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (Y',s')\<longrightarrow>
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(\<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>
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Q Y' s' Z \<and> s'\<Colon>\<preceq>(G,L)))))"
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apply (unfold triple_valid2_def)
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apply (simp (no_asm) add: split_paired_All)
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apply blast
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done
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lemma triple_valid2_eq [rule_format (no_asm)]:
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"wf_prog G ==> triple_valid2 G = triple_valid G"
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apply (rule ext)
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apply (rule ext)
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apply (rule triple.induct)
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apply (simp (no_asm) add: triple_valid_def2 triple_valid2_def2)
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apply (rule iffI)
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apply fast
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apply clarify
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apply (tactic "smp_tac 3 1")
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apply (case_tac "normal s")
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apply clarsimp
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apply (blast dest: evaln_eval eval_type_sound [THEN conjunct1])
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apply clarsimp
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done
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lemma ax_valids2_eq: "wf_prog G \<Longrightarrow> G,A|\<Turnstile>\<Colon>ts = G,A|\<Turnstile>ts"
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apply (unfold ax_valids_def ax_valids2_def)
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apply (force simp add: triple_valid2_eq)
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done
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lemma triple_valid2_Suc [rule_format (no_asm)]: "G\<Turnstile>Suc n\<Colon>t \<longrightarrow> G\<Turnstile>n\<Colon>t"
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apply (induct_tac "t")
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apply (subst triple_valid2_def2)
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apply (subst triple_valid2_def2)
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apply (fast intro: evaln_nonstrict_Suc)
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done
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lemma Methd_triple_valid2_0: "G\<Turnstile>0\<Colon>{Normal P} Methd C sig-\<succ> {Q}"
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apply (clarsimp elim!: evaln_elim_cases simp add: triple_valid2_def2)
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done
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lemma Methd_triple_valid2_SucI:
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"\<lbrakk>G\<Turnstile>n\<Colon>{Normal P} body G C sig-\<succ>{Q}\<rbrakk>
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\<Longrightarrow> G\<Turnstile>Suc n\<Colon>{Normal P} Methd C sig-\<succ> {Q}"
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apply (simp (no_asm_use) add: triple_valid2_def2)
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apply (intro strip, tactic "smp_tac 3 1", clarify)
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apply (erule wt_elim_cases, erule evaln_elim_cases)
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apply (unfold body_def Let_def)
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apply clarsimp
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apply blast
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done
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lemma triples_valid2_Suc:
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"Ball ts (triple_valid2 G (Suc n)) \<Longrightarrow> Ball ts (triple_valid2 G n)"
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apply (fast intro: triple_valid2_Suc)
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done
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lemma "G|\<Turnstile>n:insert t A = (G\<Turnstile>n:t \<and> G|\<Turnstile>n:A)";
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oops
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section "soundness"
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lemma Methd_sound:
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"\<lbrakk>G,A\<union> {{P} Methd-\<succ> {Q} | ms}|\<Turnstile>\<Colon>{{P} body G-\<succ> {Q} | ms}\<rbrakk> \<Longrightarrow>
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G,A|\<Turnstile>\<Colon>{{P} Methd-\<succ> {Q} | ms}"
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apply (unfold ax_valids2_def mtriples_def)
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apply (rule allI)
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apply (induct_tac "n")
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apply (clarify, tactic {* pair_tac "x" 1 *}, simp (no_asm))
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apply (fast intro: Methd_triple_valid2_0)
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apply (clarify, tactic {* pair_tac "xa" 1 *}, simp (no_asm))
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apply (drule triples_valid2_Suc)
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apply (erule (1) notE impE)
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apply (drule_tac x = na in spec)
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apply (rule Methd_triple_valid2_SucI)
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apply (simp (no_asm_use) add: ball_Un)
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apply auto
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done
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lemma valids2_inductI: "\<forall>s t n Y' s'. G\<turnstile>s\<midarrow>t\<succ>\<midarrow>n\<rightarrow> (Y',s') \<longrightarrow> t = c \<longrightarrow>
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Ball A (triple_valid2 G n) \<longrightarrow> (\<forall>Y Z. P Y s Z \<longrightarrow>
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(\<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>
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Q Y' s' Z \<and> s'\<Colon>\<preceq>(G, L)))) \<Longrightarrow>
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G,A|\<Turnstile>\<Colon>{ {P} c\<succ> {Q}}"
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apply (simp (no_asm) add: ax_valids2_def triple_valid2_def2)
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apply clarsimp
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done
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ML_setup {*
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Delsimprocs [evaln_expr_proc,evaln_var_proc,evaln_exprs_proc,evaln_stmt_proc]
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*}
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lemma Loop_sound: "\<lbrakk>G,A|\<Turnstile>\<Colon>{ {P} e-\<succ> {P'}};
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G,A|\<Turnstile>\<Colon>{ {Normal (P'\<leftarrow>=True)} .c. {abupd (absorb (Cont l)) .; P}}\<rbrakk> \<Longrightarrow>
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G,A|\<Turnstile>\<Colon>{ {P} .l\<bullet> While(e) c. {(P'\<leftarrow>=False)\<down>=\<diamondsuit>}}"
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apply (rule valids2_inductI)
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apply ((rule allI)+, rule impI, tactic {* pair_tac "s" 1*}, tactic {* pair_tac "s'" 1*})
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apply (erule evaln.induct)
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apply simp_all (* takes half a minute *)
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apply clarify
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apply (erule_tac V = "G,A|\<Turnstile>\<Colon>{ {?P'} .c. {?P}}" in thin_rl)
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apply (simp_all (no_asm_use) add: ax_valids2_def triple_valid2_def2)
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apply (tactic "smp_tac 1 1", tactic "smp_tac 3 1", force)
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apply clarify
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apply (rule wt_elim_cases, assumption)
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apply (tactic "smp_tac 1 1", tactic "smp_tac 1 1", tactic "smp_tac 3 1",
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tactic "smp_tac 2 1", tactic "smp_tac 1 1")
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apply (erule impE,simp (no_asm),blast)
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apply (simp add: imp_conjL split_tupled_all split_paired_All)
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apply (case_tac "the_Bool b")
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apply clarsimp
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apply (case_tac "a")
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apply (simp_all)
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apply clarsimp
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apply (erule_tac V = "c = l\<bullet> While(e) c \<longrightarrow> ?P" in thin_rl)
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apply (blast intro: conforms_absorb)
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apply blast+
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done
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declare subst_Bool_def2 [simp del]
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lemma all_empty: "(!x. P) = P"
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by simp
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lemma sound_valid2_lemma:
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"\<lbrakk>\<forall>v n. Ball A (triple_valid2 G n) \<longrightarrow> P v n; Ball A (triple_valid2 G n)\<rbrakk>
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\<Longrightarrow>P v n"
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by blast
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ML {*
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val fullsimptac = full_simp_tac(simpset() delsimps [thm "all_empty"]);
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val sound_prepare_tac = EVERY'[REPEAT o thin_tac "?x \<in> ax_derivs G",
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full_simp_tac (simpset()addsimps[thm "ax_valids2_def",thm "triple_valid2_def2",
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thm "imp_conjL"] delsimps[thm "all_empty"]),
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Clarify_tac];
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val sound_elim_tac = EVERY'[eresolve_tac (thms "evaln_elim_cases"),
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TRY o eresolve_tac (thms "wt_elim_cases"), fullsimptac, Clarify_tac];
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val sound_valid2_tac = REPEAT o FIRST'[smp_tac 1,
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datac (thm "sound_valid2_lemma") 1];
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val sound_forw_hyp_tac =
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EVERY'[smp_tac 3
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ORELSE' EVERY'[dtac spec,dtac spec, dtac spec,etac impE, Fast_tac]
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ORELSE' EVERY'[dtac spec,dtac spec,etac impE, Fast_tac],
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fullsimptac,
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smp_tac 2,TRY o smp_tac 1,
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TRY o EVERY'[etac impE, TRY o rtac impI,
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atac ORELSE' (EVERY' [REPEAT o rtac exI,Blast_tac]),
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fullsimptac, Clarify_tac, TRY o smp_tac 1]]
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*}
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(* ### rtac conjI,rtac HOL.refl *)
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lemma Call_sound:
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"\<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}};
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\<forall>a vs invC declC l. G,A|\<Turnstile>\<Colon>{ {(R a\<leftarrow>Vals vs \<and>.
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(\<lambda>s. declC = invocation_declclass
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G mode (store s) a statT \<lparr>name=mn,parTs=pTs\<rparr> \<and>
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invC = invocation_class mode (store s) a statT \<and>
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l = locals (store s)) ;.
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init_lvars G declC \<lparr>name=mn,parTs=pTs\<rparr> mode a vs) \<and>.
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(\<lambda>s. normal s \<longrightarrow> G\<turnstile>mode\<rightarrow>invC\<preceq>statT)}
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Methd declC \<lparr>name=mn,parTs=pTs\<rparr>-\<succ> {set_lvars l .; S}}\<rbrakk> \<Longrightarrow>
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G,A|\<Turnstile>\<Colon>{ {Normal P} {accC,statT,mode}e\<cdot>mn({pTs}ps)-\<succ> {S}}"
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apply (tactic "EVERY'[sound_prepare_tac, sound_elim_tac, sound_valid2_tac] 1")
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apply (rename_tac x1 s1 x2 s2 ab bb v vs m pTsa statDeclC)
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apply (tactic "smp_tac 6 1")
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apply (tactic "sound_forw_hyp_tac 1")
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apply (tactic "sound_forw_hyp_tac 1")
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apply (drule max_spec2mheads)
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schirmer@12925
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apply (drule (3) evaln_eval, drule (3) eval_ts)
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apply (drule (3) evaln_eval, frule (3) evals_ts)
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apply (drule spec,erule impE,rule exI, blast)
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(* apply (drule spec,drule spec,drule spec,erule impE,rule exI,blast) *)
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apply (case_tac "if is_static m then x2 else (np a') x2")
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defer 1
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apply (rename_tac x, subgoal_tac "(Some x, s2)\<Colon>\<preceq>(G, L)" (* used two times *))
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prefer 2
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apply (force split add: split_if_asm)
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apply (simp del: if_raise_if_None)
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apply (tactic "smp_tac 2 1")
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apply (simp only: init_lvars_def2 invmode_Static_eq)
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apply (clarsimp simp del: resTy_mthd)
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apply (drule spec,erule swap,erule conforms_set_locals [OF _ lconf_empty])
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apply clarsimp
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apply (drule Null_staticD)
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apply (drule eval_gext', drule (1) conf_gext, frule (3) DynT_propI)
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apply (erule impE) apply blast
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apply (subgoal_tac
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"G\<turnstile>invmode (mhd (statDeclC,m)) e
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\<rightarrow>invocation_class (invmode m e) s2 a' statT\<preceq>statT")
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defer apply simp
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apply (drule (3) DynT_mheadsD,simp,simp)
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apply (clarify, drule wf_mdeclD1, clarify)
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apply (frule ty_expr_is_type) apply simp
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apply (subgoal_tac "invmode (mhd (statDeclC,m)) e = IntVir \<longrightarrow> a' \<noteq> Null")
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defer apply simp
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apply (frule (2) wt_MethdI)
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apply clarify
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apply (drule (2) conforms_init_lvars)
|
schirmer@12854
|
226 |
apply (simp)
|
schirmer@12854
|
227 |
apply (assumption)+
|
schirmer@12854
|
228 |
apply simp
|
schirmer@12854
|
229 |
apply (assumption)+
|
schirmer@12854
|
230 |
apply (rule impI) apply simp
|
schirmer@12854
|
231 |
apply simp
|
schirmer@12854
|
232 |
apply simp
|
schirmer@12854
|
233 |
apply (rule Ball_weaken)
|
schirmer@12854
|
234 |
apply assumption
|
schirmer@12854
|
235 |
apply (force simp add: is_acc_type_def)
|
schirmer@12854
|
236 |
apply (tactic "smp_tac 2 1")
|
schirmer@12854
|
237 |
apply simp
|
schirmer@12854
|
238 |
apply (tactic "smp_tac 1 1")
|
schirmer@12854
|
239 |
apply (erule_tac V = "?P \<longrightarrow> ?Q" in thin_rl)
|
schirmer@12854
|
240 |
apply (erule impE)
|
schirmer@12854
|
241 |
apply (rule exI)+
|
schirmer@12854
|
242 |
apply (subgoal_tac "is_static dm = (static m)")
|
schirmer@12854
|
243 |
prefer 2 apply (simp add: member_is_static_simp)
|
schirmer@12854
|
244 |
apply (simp only: )
|
schirmer@12854
|
245 |
apply (simp only: sig.simps)
|
schirmer@12854
|
246 |
apply (force dest!: evaln_eval eval_gext' elim: conforms_return
|
schirmer@12854
|
247 |
del: impCE simp add: init_lvars_def2)
|
schirmer@12854
|
248 |
done
|
schirmer@12854
|
249 |
|
schirmer@12925
|
250 |
corollary evaln_type_sound:
|
wenzelm@12937
|
251 |
assumes evaln: "G\<turnstile>s0 \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (v,s1)" and
|
wenzelm@12937
|
252 |
wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>t\<Colon>T" and
|
wenzelm@12937
|
253 |
conf_s0: "s0\<Colon>\<preceq>(G,L)" and
|
wenzelm@12937
|
254 |
wf: "wf_prog G"
|
wenzelm@12937
|
255 |
shows "s1\<Colon>\<preceq>(G,L) \<and> (normal s1 \<longrightarrow> G,L,store s1\<turnstile>t\<succ>v\<Colon>\<preceq>T) \<and>
|
schirmer@12925
|
256 |
(error_free s0 = error_free s1)"
|
schirmer@12925
|
257 |
proof -
|
schirmer@12925
|
258 |
from evaln wt conf_s0 wf
|
schirmer@12925
|
259 |
show ?thesis
|
schirmer@12925
|
260 |
by (blast dest: evaln_eval eval_type_sound)
|
schirmer@12925
|
261 |
qed
|
schirmer@12925
|
262 |
|
schirmer@12854
|
263 |
lemma Init_sound: "\<lbrakk>wf_prog G; the (class G C) = c;
|
schirmer@12854
|
264 |
G,A|\<Turnstile>\<Colon>{ {Normal ((P \<and>. Not \<circ> initd C) ;. supd (init_class_obj G C))}
|
schirmer@12854
|
265 |
.(if C = Object then Skip else Init (super c)). {Q}};
|
schirmer@12854
|
266 |
\<forall>l. G,A|\<Turnstile>\<Colon>{ {Q \<and>. (\<lambda>s. l = locals (store s)) ;. set_lvars empty}
|
schirmer@12854
|
267 |
.init c. {set_lvars l .; R}}\<rbrakk> \<Longrightarrow>
|
schirmer@12854
|
268 |
G,A|\<Turnstile>\<Colon>{ {Normal (P \<and>. Not \<circ> initd C)} .Init C. {R}}"
|
schirmer@12854
|
269 |
apply (tactic "EVERY'[sound_prepare_tac, sound_elim_tac,sound_valid2_tac] 1")
|
schirmer@12854
|
270 |
apply (tactic {* instantiate_tac [("l24","\<lambda> n Y Z sa Y' s' L y a b aa ba ab bb. locals b")]*})
|
schirmer@12854
|
271 |
apply (clarsimp simp add: split_paired_Ex)
|
schirmer@12854
|
272 |
apply (drule spec, drule spec, drule spec, erule impE)
|
schirmer@12854
|
273 |
apply (erule_tac V = "All ?P" in thin_rl, fast)
|
schirmer@12854
|
274 |
apply clarsimp
|
schirmer@12854
|
275 |
apply (tactic "smp_tac 2 1", drule spec, erule impE,
|
schirmer@12854
|
276 |
erule (3) conforms_init_class_obj)
|
schirmer@12925
|
277 |
apply (frule (1) wf_prog_cdecl)
|
schirmer@12854
|
278 |
apply (erule impE, rule exI,erule_tac V = "All ?P" in thin_rl,
|
schirmer@12854
|
279 |
force dest: wf_cdecl_supD split add: split_if simp add: is_acc_class_def)
|
schirmer@12854
|
280 |
apply clarify
|
schirmer@12854
|
281 |
apply (drule spec)
|
schirmer@12854
|
282 |
apply (drule spec)
|
schirmer@12854
|
283 |
apply (drule spec)
|
schirmer@12854
|
284 |
apply (erule impE)
|
schirmer@12854
|
285 |
apply ( fast)
|
schirmer@12854
|
286 |
apply (simp (no_asm_use) del: empty_def2)
|
schirmer@12854
|
287 |
apply (tactic "smp_tac 2 1")
|
schirmer@12854
|
288 |
apply (drule spec, erule impE, erule conforms_set_locals, rule lconf_empty)
|
schirmer@12925
|
289 |
apply (erule impE,rule impI,rule exI,erule wf_cdecl_wt_init)
|
schirmer@12854
|
290 |
apply clarsimp
|
schirmer@12925
|
291 |
apply (erule (1) conforms_return)
|
schirmer@12925
|
292 |
apply (frule wf_cdecl_wt_init)
|
schirmer@12925
|
293 |
apply (subgoal_tac "(a, set_locals empty b)\<Colon>\<preceq>(G, empty)")
|
schirmer@12925
|
294 |
apply (frule (3) evaln_eval)
|
schirmer@12925
|
295 |
apply (drule eval_gext')
|
schirmer@12925
|
296 |
apply force
|
schirmer@12925
|
297 |
|
schirmer@12925
|
298 |
(* refer to case Init in eval_type_sound proof, to see whats going on*)
|
schirmer@12925
|
299 |
apply (subgoal_tac "(a,b)\<Colon>\<preceq>(G, L)")
|
schirmer@12925
|
300 |
apply (blast intro: conforms_set_locals)
|
schirmer@12925
|
301 |
|
schirmer@12925
|
302 |
apply (drule evaln_type_sound)
|
schirmer@12925
|
303 |
apply (cases "C=Object")
|
schirmer@12925
|
304 |
apply force
|
schirmer@12925
|
305 |
apply (force dest: wf_cdecl_supD is_acc_classD)
|
schirmer@12925
|
306 |
apply (erule (4) conforms_init_class_obj)
|
schirmer@12925
|
307 |
apply blast
|
schirmer@12854
|
308 |
done
|
schirmer@12854
|
309 |
|
schirmer@12854
|
310 |
lemma all_conjunct2: "\<forall>l. P' l \<and> P l \<Longrightarrow> \<forall>l. P l"
|
schirmer@12854
|
311 |
by fast
|
schirmer@12854
|
312 |
|
schirmer@12854
|
313 |
lemma all4_conjunct2:
|
schirmer@12854
|
314 |
"\<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"
|
schirmer@12854
|
315 |
by fast
|
schirmer@12854
|
316 |
|
schirmer@13337
|
317 |
|
schirmer@12854
|
318 |
lemmas sound_lemmas = Init_sound Loop_sound Methd_sound
|
schirmer@12854
|
319 |
|
schirmer@12854
|
320 |
lemma ax_sound2: "wf_prog G \<Longrightarrow> G,A|\<turnstile>ts \<Longrightarrow> G,A|\<Turnstile>\<Colon>ts"
|
schirmer@12854
|
321 |
apply (erule ax_derivs.induct)
|
schirmer@13337
|
322 |
prefer 22 (* Call *)
|
schirmer@12854
|
323 |
apply (erule (1) Call_sound) apply simp apply force apply force
|
schirmer@12854
|
324 |
|
schirmer@12854
|
325 |
apply (tactic {* TRYALL (eresolve_tac (thms "sound_lemmas") THEN_ALL_NEW
|
schirmer@12854
|
326 |
eresolve_tac [asm_rl, thm "all_conjunct2", thm "all4_conjunct2"]) *})
|
schirmer@12854
|
327 |
|
schirmer@13337
|
328 |
apply(tactic "COND (has_fewer_prems(30+9)) (ALLGOALS sound_prepare_tac) no_tac")
|
schirmer@12854
|
329 |
|
schirmer@12854
|
330 |
(*empty*)
|
schirmer@12854
|
331 |
apply fast (* insert *)
|
schirmer@12854
|
332 |
apply fast (* asm *)
|
schirmer@12854
|
333 |
(*apply fast *) (* cut *)
|
schirmer@12854
|
334 |
apply fast (* weaken *)
|
schirmer@12925
|
335 |
apply (tactic "smp_tac 3 1", clarify, tactic "smp_tac 1 1",
|
schirmer@12925
|
336 |
case_tac"fst s",clarsimp,erule (3) evaln_type_sound [THEN conjunct1],
|
schirmer@12925
|
337 |
clarsimp) (* conseq *)
|
schirmer@12925
|
338 |
apply (simp (no_asm_use) add: type_ok_def,fast)(* hazard *)
|
schirmer@12854
|
339 |
apply force (* Abrupt *)
|
schirmer@12854
|
340 |
|
schirmer@13337
|
341 |
prefer 28 apply (simp add: evaln_InsInitV)
|
schirmer@13337
|
342 |
prefer 28 apply (simp add: evaln_InsInitE)
|
schirmer@13337
|
343 |
prefer 28 apply (simp add: evaln_Callee)
|
schirmer@13337
|
344 |
prefer 28 apply (simp add: evaln_FinA)
|
schirmer@13337
|
345 |
|
schirmer@13337
|
346 |
(* 27 subgoals *)
|
schirmer@13337
|
347 |
apply (tactic {* sound_elim_tac 1 *})
|
schirmer@12854
|
348 |
apply (tactic {* ALLGOALS sound_elim_tac *})(* LVar, Lit, Super, Nil, Skip,Do *)
|
schirmer@12854
|
349 |
apply (tactic {* ALLGOALS (asm_simp_tac (noAll_simpset()
|
schirmer@12854
|
350 |
delsimps [thm "all_empty"])) *}) (* Done *)
|
schirmer@12854
|
351 |
(* for FVar *)
|
schirmer@12854
|
352 |
apply (frule wf_ws_prog)
|
schirmer@12854
|
353 |
apply (frule ty_expr_is_type [THEN type_is_class,
|
schirmer@12854
|
354 |
THEN accfield_declC_is_class])
|
berghofe@13601
|
355 |
apply (simp (no_asm_use), simp (no_asm_use), simp (no_asm_use))
|
schirmer@12854
|
356 |
apply (frule_tac [4] wt_init_comp_ty) (* for NewA*)
|
schirmer@12854
|
357 |
apply (tactic "ALLGOALS sound_valid2_tac")
|
schirmer@13337
|
358 |
apply (tactic "TRYALL sound_forw_hyp_tac") (* UnOp, Cast, Inst, Acc, Expr *)
|
schirmer@12854
|
359 |
apply (tactic {* TRYALL (EVERY'[dtac spec, TRY o EVERY'[rotate_tac ~1,
|
schirmer@12854
|
360 |
dtac spec], dtac conjunct2, smp_tac 1,
|
schirmer@12854
|
361 |
TRY o dres_inst_tac [("P","P'")] (thm "subst_Bool_the_BoolI")]) *})
|
schirmer@13337
|
362 |
apply (frule_tac [15] x = x1 in conforms_NormI) (* for Fin *)
|
schirmer@12854
|
363 |
|
schirmer@12854
|
364 |
(* 15 subgoals *)
|
schirmer@12854
|
365 |
(* FVar *)
|
schirmer@12854
|
366 |
apply (tactic "sound_forw_hyp_tac 1")
|
schirmer@12854
|
367 |
apply (clarsimp simp add: fvar_def2 Let_def split add: split_if_asm)
|
schirmer@12854
|
368 |
|
schirmer@12854
|
369 |
(* AVar *)
|
schirmer@12854
|
370 |
(*
|
schirmer@12854
|
371 |
apply (drule spec, drule spec, erule impE, fast)
|
schirmer@12854
|
372 |
apply (simp)
|
schirmer@12854
|
373 |
apply (tactic "smp_tac 2 1")
|
schirmer@12854
|
374 |
apply (tactic "smp_tac 1 1")
|
schirmer@12854
|
375 |
apply (erule impE)
|
schirmer@12854
|
376 |
apply (rule impI)
|
schirmer@12854
|
377 |
apply (rule exI)+
|
schirmer@12854
|
378 |
apply blast
|
schirmer@12854
|
379 |
apply (clarsimp simp add: avar_def2)
|
schirmer@12854
|
380 |
*)
|
schirmer@12854
|
381 |
apply (tactic "sound_forw_hyp_tac 1")
|
schirmer@12854
|
382 |
apply (clarsimp simp add: avar_def2)
|
schirmer@12854
|
383 |
|
schirmer@12854
|
384 |
(* NewC *)
|
schirmer@12854
|
385 |
apply (clarsimp simp add: is_acc_class_def)
|
schirmer@12854
|
386 |
apply (erule (1) halloc_conforms, simp, simp)
|
schirmer@12854
|
387 |
|
schirmer@12854
|
388 |
(* NewA *)
|
schirmer@12854
|
389 |
apply (tactic "sound_forw_hyp_tac 1")
|
schirmer@12854
|
390 |
apply (rule conjI,blast)
|
schirmer@12854
|
391 |
apply (erule (1) halloc_conforms, simp, simp, simp add: is_acc_type_def)
|
schirmer@12854
|
392 |
|
schirmer@13337
|
393 |
(* BinOp *)
|
schirmer@13337
|
394 |
apply (tactic "sound_forw_hyp_tac 1")
|
schirmer@13384
|
395 |
apply (case_tac "need_second_arg binop v1")
|
schirmer@13384
|
396 |
apply fastsimp
|
schirmer@13384
|
397 |
apply simp
|
schirmer@13337
|
398 |
|
schirmer@12854
|
399 |
(* Ass *)
|
schirmer@12854
|
400 |
apply (tactic "sound_forw_hyp_tac 1")
|
schirmer@12854
|
401 |
apply (case_tac "aa")
|
schirmer@12854
|
402 |
prefer 2
|
schirmer@12854
|
403 |
apply clarsimp
|
schirmer@12925
|
404 |
apply (drule (3) evaln_type_sound)
|
schirmer@12925
|
405 |
apply (drule (3) evaln_eval)
|
schirmer@12925
|
406 |
apply (frule (3) eval_type_sound)
|
schirmer@12854
|
407 |
apply clarsimp
|
schirmer@12854
|
408 |
apply (frule wf_ws_prog)
|
schirmer@12854
|
409 |
apply (drule (2) conf_widen)
|
schirmer@12854
|
410 |
apply (drule_tac "s1.0" = b in eval_gext')
|
schirmer@12854
|
411 |
apply (clarsimp simp add: assign_conforms_def)
|
schirmer@12854
|
412 |
|
schirmer@12925
|
413 |
|
schirmer@12854
|
414 |
(* Cond *)
|
schirmer@12854
|
415 |
apply (tactic "smp_tac 3 1") apply (tactic "smp_tac 2 1")
|
schirmer@12854
|
416 |
apply (tactic "smp_tac 1 1") apply (erule impE)
|
schirmer@12854
|
417 |
apply (rule impI,rule exI)
|
schirmer@12854
|
418 |
apply (rule_tac x = "if the_Bool b then T1 else T2" in exI)
|
schirmer@12854
|
419 |
apply (force split add: split_if)
|
schirmer@12854
|
420 |
apply assumption
|
schirmer@12854
|
421 |
|
schirmer@12854
|
422 |
(* Body *)
|
schirmer@12854
|
423 |
apply (tactic "sound_forw_hyp_tac 1")
|
schirmer@12854
|
424 |
apply (rule conforms_absorb,assumption)
|
schirmer@12854
|
425 |
|
schirmer@12854
|
426 |
(* Lab *)
|
schirmer@12854
|
427 |
apply (tactic "sound_forw_hyp_tac 1")
|
schirmer@12854
|
428 |
apply (rule conforms_absorb,assumption)
|
schirmer@12854
|
429 |
|
schirmer@12854
|
430 |
(* If *)
|
schirmer@12854
|
431 |
apply (tactic "sound_forw_hyp_tac 1")
|
schirmer@12854
|
432 |
apply (tactic "sound_forw_hyp_tac 1")
|
schirmer@12854
|
433 |
apply (force split add: split_if)
|
schirmer@12854
|
434 |
|
schirmer@12854
|
435 |
(* Throw *)
|
schirmer@12925
|
436 |
apply (drule (3) evaln_type_sound)
|
schirmer@12854
|
437 |
apply clarsimp
|
schirmer@12854
|
438 |
apply (drule (3) Throw_lemma)
|
schirmer@12854
|
439 |
apply clarsimp
|
schirmer@12854
|
440 |
|
schirmer@12854
|
441 |
(* Try *)
|
schirmer@12854
|
442 |
apply (frule (1) sxalloc_type_sound)
|
schirmer@12854
|
443 |
apply (erule sxalloc_elim_cases2)
|
schirmer@12854
|
444 |
apply (tactic "smp_tac 3 1")
|
schirmer@12854
|
445 |
apply (clarsimp split add: option.split_asm)
|
schirmer@12854
|
446 |
apply (clarsimp split add: option.split_asm)
|
schirmer@12854
|
447 |
apply (tactic "smp_tac 1 1")
|
schirmer@12854
|
448 |
apply (simp only: split add: split_if_asm)
|
schirmer@12854
|
449 |
prefer 2
|
schirmer@12854
|
450 |
apply (tactic "smp_tac 3 1", erule_tac V = "All ?P" in thin_rl, clarsimp)
|
schirmer@12854
|
451 |
apply (drule spec, erule_tac V = "All ?P" in thin_rl, drule spec, drule spec,
|
schirmer@12854
|
452 |
erule impE, force)
|
schirmer@12854
|
453 |
apply (frule (2) Try_lemma)
|
schirmer@12854
|
454 |
apply clarsimp
|
schirmer@12854
|
455 |
apply (fast elim!: conforms_deallocL)
|
schirmer@12854
|
456 |
|
schirmer@12854
|
457 |
(* Fin *)
|
schirmer@12854
|
458 |
apply (tactic "sound_forw_hyp_tac 1")
|
schirmer@12854
|
459 |
apply (case_tac "x1", force)
|
schirmer@12854
|
460 |
apply clarsimp
|
schirmer@12925
|
461 |
apply (drule (3) evaln_eval, drule (4) Fin_lemma)
|
schirmer@12854
|
462 |
done
|
schirmer@12854
|
463 |
|
schirmer@12854
|
464 |
|
schirmer@12854
|
465 |
|
schirmer@12854
|
466 |
declare subst_Bool_def2 [simp]
|
schirmer@12854
|
467 |
|
schirmer@12854
|
468 |
theorem ax_sound:
|
schirmer@12854
|
469 |
"wf_prog G \<Longrightarrow> G,(A::'a triple set)|\<turnstile>(ts::'a triple set) \<Longrightarrow> G,A|\<Turnstile>ts"
|
schirmer@12854
|
470 |
apply (subst ax_valids2_eq [symmetric])
|
schirmer@12854
|
471 |
apply assumption
|
schirmer@12854
|
472 |
apply (erule (1) ax_sound2)
|
schirmer@12854
|
473 |
done
|
schirmer@12854
|
474 |
|
schirmer@12854
|
475 |
|
schirmer@12854
|
476 |
end
|