author | berghofe |
Tue, 30 Jul 1996 17:33:26 +0200 | |
changeset 1894 | c2c8279d40f0 |
parent 1778 | 6c942cf3bf68 |
child 2083 | b56425a385b9 |
permissions | -rw-r--r-- |
1138 | 1 |
(* Title: HOL/IOA/example/Correctness.ML |
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ID: $Id$ |
1138 | 3 |
Author: Tobias Nipkow & Konrad Slind |
4 |
Copyright 1994 TU Muenchen |
|
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|
6 |
*) |
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7 |
||
8 |
open Correctness; open Abschannel; open Abschannel_finite; |
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open Impl; open Impl_finite; |
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|
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goal Abschannel.thy "(? x.x=P & Q(x)) = Q(P)"; |
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by (Fast_tac 1); |
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qed"exis_elim"; |
14 |
||
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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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|
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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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|
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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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|
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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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|
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Addsimps [reduce_Nil,reduce_Cons]; |
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||
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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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44 |
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(* lemmas about reduce *) |
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46 |
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47 |
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 1); |
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by (List.list.induct_tac "l" 1); |
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by (Simp_tac 1); |
53 |
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)); |
57 |
by (rtac impI 1); |
|
58 |
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); |
61 |
by (Asm_full_simp_tac 1); |
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val l_iff_red_nil = result(); |
63 |
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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"; |
82 |
||
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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 *) |
90 |
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"; |
100 |
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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); |
106 |
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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113 |
||
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val impl_ss = !simpset delsimps [reduce_Cons]; |
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|
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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 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); |
146 |
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); |
167 |
by (rtac conjI 1); |
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by (simp_tac (!simpset addsimps [and_de_morgan_and_absorbe]) 2); |
169 |
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(); |
175 |
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176 |
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(******************************************************************* |
178 |
C h a n n e l A b s t r a c t i o n |
|
179 |
*******************************************************************) |
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|
181 |
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)); |
191 |
by (rtac imp_conj_lemma 1); |
|
192 |
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); |
197 |
by (rtac disjI2 1); |
|
198 |
by (rtac reduce_hd 1); |
|
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(* snd case --------------------------------------*) |
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by (rtac impI 1); |
201 |
by (REPEAT (etac conjE 1)); |
|
202 |
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); |
207 |
by (etac (hd_is_reduce_hd RS mp) 1); |
|
208 |
by (rtac (bool_if_impl_or RS mp) 1); |
|
209 |
by (etac reduce_tl 1); |
|
210 |
qed"channel_abstraction"; |
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211 |
||
212 |
goal Correctness.thy |
|
213 |
"is_weak_pmap reduce srch_ioa srch_fin_ioa"; |
|
214 |
by (simp_tac (list_ss addsimps [srch_fin_ioa_def,rsch_fin_ioa_def, |
|
215 |
srch_ioa_def,rsch_ioa_def,rename_through_pmap,channel_abstraction]) 1); |
|
216 |
qed"sender_abstraction"; |
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217 |
||
218 |
goal Correctness.thy |
|
219 |
"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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223 |
||
224 |
||
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(* 3 thms that do not hold generally! The lucky restriction here is |
|
226 |
the absence of internal actions. *) |
|
227 |
goal Correctness.thy |
|
228 |
"is_weak_pmap (%id.id) sender_ioa sender_ioa"; |
|
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by (simp_tac (!simpset addsimps [Solve.is_weak_pmap_def]) 1); |
1778 | 230 |
by (TRY( |
231 |
(rtac conjI 1) THEN |
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1673
d22110ddd0af
repaired critical proofs depending on the order inside non-confluent SimpSets
oheimb
parents:
1667
diff
changeset
|
232 |
(* start states *) |
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(Fast_tac 1))); |
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(* main-part *) |
235 |
by (REPEAT (rtac allI 1)); |
|
236 |
by (rtac imp_conj_lemma 1); |
|
237 |
by (Action.action.induct_tac "a" 1); |
|
238 |
(* 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"; |
241 |
||
242 |
(* 2 copies of before *) |
|
243 |
goal Correctness.thy |
|
244 |
"is_weak_pmap (%id.id) receiver_ioa receiver_ioa"; |
|
1266 | 245 |
by (simp_tac (!simpset addsimps [Solve.is_weak_pmap_def]) 1); |
1778 | 246 |
by (TRY( |
247 |
(rtac conjI 1) THEN |
|
1673
d22110ddd0af
repaired critical proofs depending on the order inside non-confluent SimpSets
oheimb
parents:
1667
diff
changeset
|
248 |
(* start states *) |
1894 | 249 |
(Fast_tac 1))); |
1050 | 250 |
(* main-part *) |
251 |
by (REPEAT (rtac allI 1)); |
|
252 |
by (rtac imp_conj_lemma 1); |
|
253 |
by (Action.action.induct_tac "a" 1); |
|
254 |
(* 7 cases *) |
|
1266 | 255 |
by (ALLGOALS (simp_tac ((!simpset addsimps [externals_def]) setloop (split_tac [expand_if])))); |
1050 | 256 |
qed"receiver_unchanged"; |
257 |
||
258 |
goal Correctness.thy |
|
259 |
"is_weak_pmap (%id.id) env_ioa env_ioa"; |
|
1266 | 260 |
by (simp_tac (!simpset addsimps [Solve.is_weak_pmap_def]) 1); |
1778 | 261 |
by (TRY( |
262 |
(rtac conjI 1) THEN |
|
1673
d22110ddd0af
repaired critical proofs depending on the order inside non-confluent SimpSets
oheimb
parents:
1667
diff
changeset
|
263 |
(* start states *) |
1894 | 264 |
(Fast_tac 1))); |
1050 | 265 |
(* main-part *) |
266 |
by (REPEAT (rtac allI 1)); |
|
267 |
by (rtac imp_conj_lemma 1); |
|
268 |
by (Action.action.induct_tac "a" 1); |
|
269 |
(* 7 cases *) |
|
1266 | 270 |
by (ALLGOALS (simp_tac ((!simpset addsimps [externals_def]) setloop (split_tac [expand_if])))); |
1050 | 271 |
qed"env_unchanged"; |
272 |
||
1532 | 273 |
Delsimps [Collect_False_empty]; |
1050 | 274 |
|
275 |
goal Correctness.thy "compat_ioas srch_ioa rsch_ioa"; |
|
1266 | 276 |
by (simp_tac (!simpset addsimps [compat_ioas_def,compat_asigs_def,Int_def,empty_def]) 1); |
1050 | 277 |
by (rtac set_ext 1); |
278 |
by (Action.action.induct_tac "x" 1); |
|
1266 | 279 |
by (ALLGOALS(Simp_tac)); |
1050 | 280 |
val compat_single_ch = result(); |
281 |
||
282 |
(* totally the same as before *) |
|
283 |
goal Correctness.thy "compat_ioas srch_fin_ioa rsch_fin_ioa"; |
|
1266 | 284 |
by (simp_tac (!simpset addsimps [compat_ioas_def,compat_asigs_def,Int_def,empty_def]) 1); |
1050 | 285 |
by (rtac set_ext 1); |
286 |
by (Action.action.induct_tac "x" 1); |
|
1266 | 287 |
by (ALLGOALS(Simp_tac)); |
1050 | 288 |
val compat_single_fin_ch = result(); |
289 |
||
1266 | 290 |
val ss = |
291 |
!simpset delsimps ([trans_of_def, srch_asig_def,rsch_asig_def, |
|
292 |
asig_of_def, actions_def, srch_trans_def, rsch_trans_def, |
|
293 |
srch_ioa_def, srch_fin_ioa_def, rsch_fin_ioa_def, |
|
294 |
rsch_ioa_def, Sender.sender_trans_def, |
|
295 |
Receiver.receiver_trans_def] @ set_lemmas); |
|
296 |
||
1050 | 297 |
goal Correctness.thy "compat_ioas receiver_ioa (srch_ioa || rsch_ioa)"; |
1266 | 298 |
by (simp_tac (ss addsimps [empty_def,compat_ioas_def,compat_asigs_def, |
299 |
asig_of_par,asig_comp_def,actions_def, |
|
300 |
Int_def]) 1); |
|
301 |
by (Simp_tac 1); |
|
1050 | 302 |
by (rtac set_ext 1); |
303 |
by (Action.action.induct_tac "x" 1); |
|
1266 | 304 |
by (ALLGOALS Simp_tac); |
305 |
by (ALLGOALS(simp_tac (!simpset addsimps [insert_def,Un_def]))); |
|
306 |
by (ALLGOALS(simp_tac (!simpset addsimps [de_morgan,singleton_set]))); |
|
307 |
val compat_rec = result(); |
|
1050 | 308 |
|
309 |
(* 5 proofs totally the same as before *) |
|
310 |
goal Correctness.thy "compat_ioas receiver_ioa (srch_fin_ioa || rsch_fin_ioa)"; |
|
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); |
312 |
by (Simp_tac 1); |
|
1050 | 313 |
by (rtac set_ext 1); |
314 |
by (Action.action.induct_tac "x" 1); |
|
1266 | 315 |
by (ALLGOALS Simp_tac); |
316 |
by (ALLGOALS(simp_tac (!simpset addsimps [insert_def,Un_def]))); |
|
317 |
by (ALLGOALS(simp_tac (!simpset addsimps [de_morgan,singleton_set]))); |
|
1050 | 318 |
val compat_rec_fin =result(); |
319 |
||
320 |
goal Correctness.thy "compat_ioas sender_ioa \ |
|
321 |
\ (receiver_ioa || srch_ioa || rsch_ioa)"; |
|
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); |
323 |
by (Simp_tac 1); |
|
1050 | 324 |
by (rtac set_ext 1); |
325 |
by (Action.action.induct_tac "x" 1); |
|
1266 | 326 |
by (ALLGOALS(Simp_tac)); |
327 |
by (ALLGOALS(simp_tac (!simpset addsimps [insert_def,Un_def]))); |
|
328 |
by (ALLGOALS(simp_tac (!simpset addsimps [de_morgan,singleton_set]))); |
|
1050 | 329 |
val compat_sen=result(); |
330 |
||
331 |
goal Correctness.thy "compat_ioas sender_ioa\ |
|
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); |
334 |
by (Simp_tac 1); |
|
1050 | 335 |
by (rtac set_ext 1); |
336 |
by (Action.action.induct_tac "x" 1); |
|
1266 | 337 |
by (ALLGOALS(Simp_tac)); |
338 |
by (ALLGOALS(simp_tac (!simpset addsimps [insert_def,Un_def]))); |
|
339 |
by (ALLGOALS(simp_tac (!simpset addsimps [de_morgan,singleton_set]))); |
|
1050 | 340 |
val compat_sen_fin =result(); |
341 |
||
342 |
goal Correctness.thy "compat_ioas env_ioa\ |
|
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) |
|
1570 | 396 |
addsimps [system_def, system_fin_def, abs_def, |
397 |
impl_ioa_def, impl_fin_ioa_def, sys_IOA, |
|
398 |
sys_fin_IOA]) 1); |
|
1138 | 399 |
|
400 |
by (REPEAT (EVERY[rtac fxg_is_weak_pmap_of_product_IOA 1, |
|
1266 | 401 |
simp_tac (ss addsimps abstractions) 1, |
1138 | 402 |
rtac conjI 1])); |
403 |
||
1266 | 404 |
by (ALLGOALS (simp_tac (ss addsimps ext_ss @ compat_ss))); |
1138 | 405 |
|
1050 | 406 |
qed "system_refinement"; |