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(* Title: HOL/IOA/example/Correctness.ML
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ID: $Id$
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Author: Tobias Nipkow & Konrad Slind
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Copyright 1994 TU Muenchen
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*)
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open Correctness; open Abschannel; open Abschannel_finite;
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open Impl; open Impl_finite;
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goal Abschannel.thy "(? x.x=P & Q(x)) = Q(P)";
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by (fast_tac HOL_cs 1);
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qed"exis_elim";
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Delsimps [split_paired_All];
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Addsimps
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([srch_asig_def, rsch_asig_def, rsch_ioa_def, srch_ioa_def, ch_ioa_def,
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ch_asig_def, srch_actions_def, rsch_actions_def, rename_def, asig_of_def,
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actions_def, exis_elim, srch_trans_def, rsch_trans_def, ch_trans_def,
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trans_of_def] @ asig_projections @ set_lemmas);
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val abschannel_fin = [srch_fin_asig_def, rsch_fin_asig_def,
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rsch_fin_ioa_def, srch_fin_ioa_def,
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ch_fin_ioa_def,ch_fin_trans_def,ch_fin_asig_def];
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Addsimps abschannel_fin;
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val impl_ioas = [Sender.sender_ioa_def,Receiver.receiver_ioa_def];
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val impl_trans = [Sender.sender_trans_def,Receiver.receiver_trans_def];
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val impl_asigs = [Sender.sender_asig_def,Receiver.receiver_asig_def];
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Addcongs [let_weak_cong];
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Addsimps [Let_def, ioa_triple_proj, starts_of_par];
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val env_ioas = [Env.env_ioa_def,Env.env_asig_def,Env.env_trans_def];
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val hom_ioas = env_ioas @ impl_ioas @ impl_trans @ impl_asigs @
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asig_projections @ set_lemmas;
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Addsimps hom_ioas;
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Addsimps [reduce_Nil,reduce_Cons];
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(* auxiliary function *)
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fun rotate n i = EVERY(replicate n (etac revcut_rl i));
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(* lemmas about reduce *)
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goal Correctness.thy "(reduce(l)=[]) = (l=[])";
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by (rtac iffI 1);
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by (subgoal_tac "(l~=[]) --> (reduce(l)~=[])" 1);
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by (fast_tac HOL_cs 1);
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by (List.list.induct_tac "l" 1);
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by (Simp_tac 1);
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by (Simp_tac 1);
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by (rtac (expand_list_case RS iffD2) 1);
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by (Asm_full_simp_tac 1);
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by (REPEAT (rtac allI 1));
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by (rtac impI 1);
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by (hyp_subst_tac 1);
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by (rtac (expand_if RS ssubst) 1);
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by (Asm_full_simp_tac 1);
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by (Asm_full_simp_tac 1);
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val l_iff_red_nil = result();
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goal Correctness.thy "s~=[] --> hd(s)=hd(reduce(s))";
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by (List.list.induct_tac "s" 1);
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by (Simp_tac 1);
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by (case_tac "list =[]" 1);
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by (Asm_full_simp_tac 1);
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(* main case *)
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by (rotate 1 1);
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by (asm_full_simp_tac list_ss 1);
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by (Simp_tac 1);
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by (rtac (expand_list_case RS iffD2) 1);
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by (asm_full_simp_tac list_ss 1);
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by (REPEAT (rtac allI 1));
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by (rtac impI 1);
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by (rtac (expand_if RS ssubst) 1);
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by (REPEAT(hyp_subst_tac 1));
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by (etac subst 1);
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by (Simp_tac 1);
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qed"hd_is_reduce_hd";
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(* to be used in the following Lemma *)
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goal Correctness.thy "l~=[] --> reverse(reduce(l))~=[]";
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by (List.list.induct_tac "l" 1);
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by (Simp_tac 1);
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by (case_tac "list =[]" 1);
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by (asm_full_simp_tac (!simpset addsimps [reverse_Cons]) 1);
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(* main case *)
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by (rotate 1 1);
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by (Asm_full_simp_tac 1);
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by (cut_inst_tac [("l","list")] cons_not_nil 1);
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by (Asm_full_simp_tac 1);
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by (REPEAT (etac exE 1));
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by (Asm_simp_tac 1);
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by (rtac (expand_if RS ssubst) 1);
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by (hyp_subst_tac 1);
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by (asm_full_simp_tac (!simpset addsimps [reverse_Cons]) 1);
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qed"rev_red_not_nil";
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(* shows applicability of the induction hypothesis of the following Lemma 1 *)
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goal Correctness.thy "!!l.[| l~=[] |] ==> \
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\ hd(reverse(reduce(a#l))) = hd(reverse(reduce(l)))";
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by (Simp_tac 1);
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by (rtac (expand_list_case RS iffD2) 1);
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by (asm_full_simp_tac list_ss 1);
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by (REPEAT (rtac allI 1));
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by (rtac impI 1);
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by (rtac (expand_if RS ssubst) 1);
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by (asm_full_simp_tac (list_ss addsimps [reverse_Cons,hd_append,
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(rev_red_not_nil RS mp)]) 1);
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qed"last_ind_on_first";
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val impl_ss = !simpset delsimps [reduce_Cons];
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(* Main Lemma 1 for S_pkt in showing that reduce is refinement *)
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goal Correctness.thy
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"if x=hd(reverse(reduce(l))) & reduce(l)~=[] then \
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\ reduce(l@[x])=reduce(l) else \
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\ reduce(l@[x])=reduce(l)@[x]";
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by (rtac (expand_if RS ssubst) 1);
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by (rtac conjI 1);
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(* --> *)
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by (List.list.induct_tac "l" 1);
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by (Simp_tac 1);
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by (case_tac "list=[]" 1);
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by (asm_full_simp_tac (!simpset addsimps [reverse_Nil,reverse_Cons]) 1);
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by (rtac impI 1);
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by (Simp_tac 1);
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by (cut_inst_tac [("l","list")] cons_not_nil 1);
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by (asm_full_simp_tac impl_ss 1);
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by (REPEAT (etac exE 1));
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by (hyp_subst_tac 1);
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by (asm_full_simp_tac (impl_ss addsimps [last_ind_on_first,l_iff_red_nil]) 1);
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(* <-- *)
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by (simp_tac (!simpset addsimps [and_de_morgan_and_absorbe,l_iff_red_nil]) 1);
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by (List.list.induct_tac "l" 1);
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by (Simp_tac 1);
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by (case_tac "list=[]" 1);
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by (asm_full_simp_tac (!simpset addsimps [reverse_Nil,reverse_Cons]) 1);
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by (rtac (expand_if RS ssubst) 1);
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by (fast_tac HOL_cs 1);
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by (rtac impI 1);
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by (Simp_tac 1);
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by (cut_inst_tac [("l","list")] cons_not_nil 1);
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by (asm_full_simp_tac impl_ss 1);
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by (REPEAT (etac exE 1));
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by (hyp_subst_tac 1);
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by (asm_full_simp_tac (impl_ss addsimps [last_ind_on_first,l_iff_red_nil]) 1);
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by (rtac (expand_if RS ssubst) 1);
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by (rtac (expand_if RS ssubst) 1);
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by (asm_full_simp_tac impl_ss 1);
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qed"reduce_hd";
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(* Main Lemma 2 for R_pkt in showing that reduce is refinement *)
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goal Correctness.thy
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"!! s. [| s~=[] |] ==> \
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\ if hd(s)=hd(tl(s)) & tl(s)~=[] then \
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\ reduce(tl(s))=reduce(s) else \
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\ reduce(tl(s))=tl(reduce(s))";
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by (cut_inst_tac [("l","s")] cons_not_nil 1);
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by (Asm_full_simp_tac 1);
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by (REPEAT (etac exE 1));
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by (Asm_full_simp_tac 1);
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by (rtac (expand_if RS ssubst) 1);
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by (rtac conjI 1);
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by (simp_tac (!simpset addsimps [and_de_morgan_and_absorbe]) 2);
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by (ALLGOALS (EVERY'[rtac impI,etac conjE,cut_inst_tac [("l","xs")] cons_not_nil,Asm_full_simp_tac]));
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by (REPEAT (etac exE 1));
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by (REPEAT (etac exE 2));
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by (REPEAT(hyp_subst_tac 2));
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by (ALLGOALS (Asm_full_simp_tac));
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val reduce_tl =result();
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(*******************************************************************
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C h a n n e l A b s t r a c t i o n
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*******************************************************************)
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goal Correctness.thy
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"is_weak_pmap reduce ch_ioa ch_fin_ioa";
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by (simp_tac (!simpset addsimps [Solve.is_weak_pmap_def]) 1);
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by (rtac conjI 1);
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(* -------------- start states ----------------- *)
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by (Simp_tac 1);
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by (rtac ballI 1);
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by (Asm_full_simp_tac 1);
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(* ---------------- main-part ------------------- *)
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by (REPEAT (rtac allI 1));
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by (rtac imp_conj_lemma 1);
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by (act.induct_tac "a" 1);
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(* ----------------- 2 cases ---------------------*)
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by (ALLGOALS (simp_tac (!simpset addsimps [externals_def])));
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(* fst case --------------------------------------*)
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by (rtac impI 1);
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by (rtac disjI2 1);
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by (rtac reduce_hd 1);
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(* snd case --------------------------------------*)
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by (rtac impI 1);
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by (REPEAT (etac conjE 1));
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by (etac disjE 1);
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by (asm_full_simp_tac (!simpset addsimps [l_iff_red_nil]) 1);
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by (etac (hd_is_reduce_hd RS mp) 1);
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by (asm_full_simp_tac (!simpset addsimps [l_iff_red_nil]) 1);
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by (rtac conjI 1);
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by (etac (hd_is_reduce_hd RS mp) 1);
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by (rtac (bool_if_impl_or RS mp) 1);
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by (etac reduce_tl 1);
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qed"channel_abstraction";
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goal Correctness.thy
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"is_weak_pmap reduce srch_ioa srch_fin_ioa";
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by (simp_tac (list_ss addsimps [srch_fin_ioa_def,rsch_fin_ioa_def,
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srch_ioa_def,rsch_ioa_def,rename_through_pmap,channel_abstraction]) 1);
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qed"sender_abstraction";
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goal Correctness.thy
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"is_weak_pmap reduce rsch_ioa rsch_fin_ioa";
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by (simp_tac (list_ss addsimps [srch_fin_ioa_def,rsch_fin_ioa_def,
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srch_ioa_def,rsch_ioa_def,rename_through_pmap,channel_abstraction]) 1);
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qed"receiver_abstraction";
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(* 3 thms that do not hold generally! The lucky restriction here is
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the absence of internal actions. *)
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goal Correctness.thy
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"is_weak_pmap (%id.id) sender_ioa sender_ioa";
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by (simp_tac (!simpset addsimps [Solve.is_weak_pmap_def]) 1);
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by (rtac conjI 1);
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(* start states *)
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by (rtac ballI 1);
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by (Simp_tac 1);
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(* main-part *)
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by (REPEAT (rtac allI 1));
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by (rtac imp_conj_lemma 1);
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by (Action.action.induct_tac "a" 1);
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(* 7 cases *)
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by (ALLGOALS (simp_tac ((!simpset addsimps [externals_def]) setloop (split_tac [expand_if]))));
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qed"sender_unchanged";
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(* 2 copies of before *)
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goal Correctness.thy
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"is_weak_pmap (%id.id) receiver_ioa receiver_ioa";
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by (simp_tac (!simpset addsimps [Solve.is_weak_pmap_def]) 1);
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by (rtac conjI 1);
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(* start states *)
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by (rtac ballI 1);
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by (Simp_tac 1);
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(* main-part *)
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by (REPEAT (rtac allI 1));
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by (rtac imp_conj_lemma 1);
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by (Action.action.induct_tac "a" 1);
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(* 7 cases *)
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by (ALLGOALS (simp_tac ((!simpset addsimps [externals_def]) setloop (split_tac [expand_if]))));
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qed"receiver_unchanged";
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goal Correctness.thy
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"is_weak_pmap (%id.id) env_ioa env_ioa";
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by (simp_tac (!simpset addsimps [Solve.is_weak_pmap_def]) 1);
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by (rtac conjI 1);
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(* start states *)
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by (rtac ballI 1);
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by (Simp_tac 1);
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(* main-part *)
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by (REPEAT (rtac allI 1));
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by (rtac imp_conj_lemma 1);
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by (Action.action.induct_tac "a" 1);
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(* 7 cases *)
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by (ALLGOALS (simp_tac ((!simpset addsimps [externals_def]) setloop (split_tac [expand_if]))));
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qed"env_unchanged";
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Delsimps [Collect_False_empty];
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goal Correctness.thy "compat_ioas srch_ioa rsch_ioa";
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by (simp_tac (!simpset addsimps [compat_ioas_def,compat_asigs_def,Int_def,empty_def]) 1);
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by (rtac set_ext 1);
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by (Action.action.induct_tac "x" 1);
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by (ALLGOALS(Simp_tac));
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val compat_single_ch = result();
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(* totally the same as before *)
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goal Correctness.thy "compat_ioas srch_fin_ioa rsch_fin_ioa";
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by (simp_tac (!simpset addsimps [compat_ioas_def,compat_asigs_def,Int_def,empty_def]) 1);
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by (rtac set_ext 1);
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by (Action.action.induct_tac "x" 1);
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by (ALLGOALS(Simp_tac));
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val compat_single_fin_ch = result();
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val ss =
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!simpset delsimps ([trans_of_def, srch_asig_def,rsch_asig_def,
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asig_of_def, actions_def, srch_trans_def, rsch_trans_def,
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srch_ioa_def, srch_fin_ioa_def, rsch_fin_ioa_def,
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rsch_ioa_def, Sender.sender_trans_def,
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Receiver.receiver_trans_def] @ set_lemmas);
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goal Correctness.thy "compat_ioas receiver_ioa (srch_ioa || rsch_ioa)";
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by (simp_tac (ss addsimps [empty_def,compat_ioas_def,compat_asigs_def,
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asig_of_par,asig_comp_def,actions_def,
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Int_def]) 1);
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by (Simp_tac 1);
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by (rtac set_ext 1);
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by (Action.action.induct_tac "x" 1);
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by (ALLGOALS Simp_tac);
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by (ALLGOALS(simp_tac (!simpset addsimps [insert_def,Un_def])));
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by (ALLGOALS(simp_tac (!simpset addsimps [de_morgan,singleton_set])));
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val compat_rec = result();
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(* 5 proofs totally the same as before *)
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310 |
goal Correctness.thy "compat_ioas receiver_ioa (srch_fin_ioa || rsch_fin_ioa)";
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1266
|
311 |
by (simp_tac (ss addsimps [empty_def,compat_ioas_def,compat_asigs_def,asig_of_par,asig_comp_def,actions_def,Int_def]) 1);
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|
312 |
by (Simp_tac 1);
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1050
|
313 |
by (rtac set_ext 1);
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|
314 |
by (Action.action.induct_tac "x" 1);
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1266
|
315 |
by (ALLGOALS Simp_tac);
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316 |
by (ALLGOALS(simp_tac (!simpset addsimps [insert_def,Un_def])));
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|
317 |
by (ALLGOALS(simp_tac (!simpset addsimps [de_morgan,singleton_set])));
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1050
|
318 |
val compat_rec_fin =result();
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|
319 |
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|
320 |
goal Correctness.thy "compat_ioas sender_ioa \
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|
321 |
\ (receiver_ioa || srch_ioa || rsch_ioa)";
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1266
|
322 |
by (simp_tac (ss addsimps [empty_def,compat_ioas_def,compat_asigs_def,asig_of_par,asig_comp_def,actions_def,Int_def]) 1);
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|
323 |
by (Simp_tac 1);
|
1050
|
324 |
by (rtac set_ext 1);
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|
325 |
by (Action.action.induct_tac "x" 1);
|
1266
|
326 |
by (ALLGOALS(Simp_tac));
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|
327 |
by (ALLGOALS(simp_tac (!simpset addsimps [insert_def,Un_def])));
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|
328 |
by (ALLGOALS(simp_tac (!simpset addsimps [de_morgan,singleton_set])));
|
1050
|
329 |
val compat_sen=result();
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|
330 |
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|
331 |
goal Correctness.thy "compat_ioas sender_ioa\
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|
332 |
\ (receiver_ioa || srch_fin_ioa || rsch_fin_ioa)";
|
1266
|
333 |
by (simp_tac (ss addsimps [empty_def, compat_ioas_def,compat_asigs_def,asig_of_par,asig_comp_def,actions_def,Int_def]) 1);
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|
334 |
by (Simp_tac 1);
|
1050
|
335 |
by (rtac set_ext 1);
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|
336 |
by (Action.action.induct_tac "x" 1);
|
1266
|
337 |
by (ALLGOALS(Simp_tac));
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|
338 |
by (ALLGOALS(simp_tac (!simpset addsimps [insert_def,Un_def])));
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|
339 |
by (ALLGOALS(simp_tac (!simpset addsimps [de_morgan,singleton_set])));
|
1050
|
340 |
val compat_sen_fin =result();
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|
341 |
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|
342 |
goal Correctness.thy "compat_ioas env_ioa\
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|
343 |
\ (sender_ioa || receiver_ioa || srch_ioa || rsch_ioa)";
|
1266
|
344 |
by (simp_tac (ss addsimps [empty_def,compat_ioas_def,compat_asigs_def,asig_of_par,asig_comp_def,actions_def,Int_def]) 1);
|
|
345 |
by (Simp_tac 1);
|
1050
|
346 |
by (rtac set_ext 1);
|
|
347 |
by (Action.action.induct_tac "x" 1);
|
1266
|
348 |
by (ALLGOALS(Simp_tac));
|
|
349 |
by (ALLGOALS(simp_tac (!simpset addsimps [insert_def,Un_def])));
|
|
350 |
by (ALLGOALS(simp_tac (!simpset addsimps [de_morgan,singleton_set])));
|
1050
|
351 |
val compat_env=result();
|
|
352 |
|
|
353 |
goal Correctness.thy "compat_ioas env_ioa\
|
|
354 |
\ (sender_ioa || receiver_ioa || srch_fin_ioa || rsch_fin_ioa)";
|
1266
|
355 |
by (simp_tac (ss addsimps [empty_def,compat_ioas_def,compat_asigs_def,asig_of_par,asig_comp_def,actions_def,Int_def]) 1);
|
|
356 |
by (Simp_tac 1);
|
1050
|
357 |
by (rtac set_ext 1);
|
|
358 |
by (Action.action.induct_tac "x" 1);
|
1266
|
359 |
by (ALLGOALS Simp_tac);
|
|
360 |
by (ALLGOALS(simp_tac (!simpset addsimps [insert_def,Un_def])));
|
|
361 |
by (ALLGOALS(simp_tac (!simpset addsimps [de_morgan,singleton_set])));
|
1050
|
362 |
val compat_env_fin=result();
|
|
363 |
|
|
364 |
|
|
365 |
(* lemmata about externals of channels *)
|
|
366 |
goal Correctness.thy
|
|
367 |
"externals(asig_of(srch_fin_ioa)) = externals(asig_of(srch_ioa)) & \
|
|
368 |
\ externals(asig_of(rsch_fin_ioa)) = externals(asig_of(rsch_ioa))";
|
1266
|
369 |
by (simp_tac (!simpset addsimps [externals_def]) 1);
|
1050
|
370 |
val ext_single_ch = result();
|
|
371 |
|
1138
|
372 |
|
|
373 |
|
|
374 |
|
|
375 |
|
|
376 |
val ext_ss = [externals_of_par,ext_single_ch];
|
|
377 |
val compat_ss = [compat_single_ch,compat_single_fin_ch,compat_rec,
|
|
378 |
compat_rec_fin,compat_sen,compat_sen_fin,compat_env,compat_env_fin];
|
|
379 |
val abstractions = [env_unchanged,sender_unchanged,
|
|
380 |
receiver_unchanged,sender_abstraction,receiver_abstraction];
|
|
381 |
|
|
382 |
|
|
383 |
(************************************************************************
|
|
384 |
S o u n d n e s s o f A b s t r a c t i o n
|
|
385 |
*************************************************************************)
|
|
386 |
|
1266
|
387 |
val ss = !simpset delsimps ([asig_of_def, srch_ioa_def, rsch_ioa_def,
|
|
388 |
srch_fin_ioa_def, rsch_fin_ioa_def] @
|
|
389 |
impl_ioas @ env_ioas);
|
1138
|
390 |
|
|
391 |
goal Correctness.thy
|
|
392 |
"is_weak_pmap abs system_ioa system_fin_ioa";
|
|
393 |
|
1266
|
394 |
by (simp_tac (impl_ss delsimps ([srch_ioa_def, rsch_ioa_def, srch_fin_ioa_def,
|
|
395 |
rsch_fin_ioa_def] @ env_ioas @ impl_ioas)
|
|
396 |
addsimps [system_def, system_fin_def, abs_def, impl_def,
|
|
397 |
impl_fin_def, sys_IOA, sys_fin_IOA]) 1);
|
1138
|
398 |
|
|
399 |
by (REPEAT (EVERY[rtac fxg_is_weak_pmap_of_product_IOA 1,
|
1266
|
400 |
simp_tac (ss addsimps abstractions) 1,
|
1138
|
401 |
rtac conjI 1]));
|
|
402 |
|
1266
|
403 |
by (ALLGOALS (simp_tac (ss addsimps ext_ss @ compat_ss)));
|
1138
|
404 |
|
1050
|
405 |
qed "system_refinement";
|