--- a/src/HOLCF/IOA/ABP/Correctness.thy Sat Nov 27 14:34:54 2010 -0800
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,326 +0,0 @@
-(* Title: HOLCF/IOA/ABP/Correctness.thy
- Author: Olaf Müller
-*)
-
-header {* The main correctness proof: System_fin implements System *}
-
-theory Correctness
-imports IOA Env Impl Impl_finite
-uses "Check.ML"
-begin
-
-primrec reduce :: "'a list => 'a list"
-where
- reduce_Nil: "reduce [] = []"
-| reduce_Cons: "reduce(x#xs) =
- (case xs of
- [] => [x]
- | y#ys => (if (x=y)
- then reduce xs
- else (x#(reduce xs))))"
-
-definition
- abs where
- "abs =
- (%p.(fst(p),(fst(snd(p)),(fst(snd(snd(p))),
- (reduce(fst(snd(snd(snd(p))))),reduce(snd(snd(snd(snd(p))))))))))"
-
-definition
- system_ioa :: "('m action, bool * 'm impl_state)ioa" where
- "system_ioa = (env_ioa || impl_ioa)"
-
-definition
- system_fin_ioa :: "('m action, bool * 'm impl_state)ioa" where
- "system_fin_ioa = (env_ioa || impl_fin_ioa)"
-
-
-axiomatization where
- sys_IOA: "IOA system_ioa" and
- sys_fin_IOA: "IOA system_fin_ioa"
-
-
-
-declare split_paired_All [simp del] Collect_empty_eq [simp del]
-
-lemmas [simp] =
- srch_asig_def rsch_asig_def rsch_ioa_def srch_ioa_def ch_ioa_def
- ch_asig_def srch_actions_def rsch_actions_def rename_def rename_set_def asig_of_def
- actions_def exis_elim srch_trans_def rsch_trans_def ch_trans_def
- trans_of_def asig_projections set_lemmas
-
-lemmas abschannel_fin [simp] =
- srch_fin_asig_def rsch_fin_asig_def
- rsch_fin_ioa_def srch_fin_ioa_def
- ch_fin_ioa_def ch_fin_trans_def ch_fin_asig_def
-
-lemmas impl_ioas = sender_ioa_def receiver_ioa_def
- and impl_trans = sender_trans_def receiver_trans_def
- and impl_asigs = sender_asig_def receiver_asig_def
-
-declare let_weak_cong [cong]
-declare ioa_triple_proj [simp] starts_of_par [simp]
-
-lemmas env_ioas = env_ioa_def env_asig_def env_trans_def
-lemmas hom_ioas =
- env_ioas [simp] impl_ioas [simp] impl_trans [simp] impl_asigs [simp]
- asig_projections set_lemmas
-
-
-subsection {* lemmas about reduce *}
-
-lemma l_iff_red_nil: "(reduce l = []) = (l = [])"
- by (induct l) (auto split: list.split)
-
-lemma hd_is_reduce_hd: "s ~= [] --> hd s = hd (reduce s)"
- by (induct s) (auto split: list.split)
-
-text {* to be used in the following Lemma *}
-lemma rev_red_not_nil [rule_format]:
- "l ~= [] --> reverse (reduce l) ~= []"
- by (induct l) (auto split: list.split)
-
-text {* shows applicability of the induction hypothesis of the following Lemma 1 *}
-lemma last_ind_on_first:
- "l ~= [] ==> hd (reverse (reduce (a # l))) = hd (reverse (reduce l))"
- apply simp
- apply (tactic {* auto_tac (@{claset},
- HOL_ss addsplits [@{thm list.split}]
- addsimps (@{thms reverse.simps} @ [@{thm hd_append}, @{thm rev_red_not_nil}])) *})
- done
-
-text {* Main Lemma 1 for @{text "S_pkt"} in showing that reduce is refinement. *}
-lemma reduce_hd:
- "if x=hd(reverse(reduce(l))) & reduce(l)~=[] then
- reduce(l@[x])=reduce(l) else
- reduce(l@[x])=reduce(l)@[x]"
-apply (simplesubst split_if)
-apply (rule conjI)
-txt {* @{text "-->"} *}
-apply (induct_tac "l")
-apply (simp (no_asm))
-apply (case_tac "list=[]")
- apply simp
- apply (rule impI)
-apply (simp (no_asm))
-apply (cut_tac l = "list" in cons_not_nil)
- apply (simp del: reduce_Cons)
- apply (erule exE)+
- apply hypsubst
-apply (simp del: reduce_Cons add: last_ind_on_first l_iff_red_nil)
-txt {* @{text "<--"} *}
-apply (simp (no_asm) add: and_de_morgan_and_absorbe l_iff_red_nil)
-apply (induct_tac "l")
-apply (simp (no_asm))
-apply (case_tac "list=[]")
-apply (cut_tac [2] l = "list" in cons_not_nil)
-apply simp
-apply (auto simp del: reduce_Cons simp add: last_ind_on_first l_iff_red_nil split: split_if)
-apply simp
-done
-
-
-text {* Main Lemma 2 for R_pkt in showing that reduce is refinement. *}
-lemma reduce_tl: "s~=[] ==>
- if hd(s)=hd(tl(s)) & tl(s)~=[] then
- reduce(tl(s))=reduce(s) else
- reduce(tl(s))=tl(reduce(s))"
-apply (cut_tac l = "s" in cons_not_nil)
-apply simp
-apply (erule exE)+
-apply (auto split: list.split)
-done
-
-
-subsection {* Channel Abstraction *}
-
-declare split_if [split del]
-
-lemma channel_abstraction: "is_weak_ref_map reduce ch_ioa ch_fin_ioa"
-apply (simp (no_asm) add: is_weak_ref_map_def)
-txt {* main-part *}
-apply (rule allI)+
-apply (rule imp_conj_lemma)
-apply (induct_tac "a")
-txt {* 2 cases *}
-apply (simp_all (no_asm) cong del: if_weak_cong add: externals_def)
-txt {* fst case *}
- apply (rule impI)
- apply (rule disjI2)
-apply (rule reduce_hd)
-txt {* snd case *}
- apply (rule impI)
- apply (erule conjE)+
- apply (erule disjE)
-apply (simp add: l_iff_red_nil)
-apply (erule hd_is_reduce_hd [THEN mp])
-apply (simp add: l_iff_red_nil)
-apply (rule conjI)
-apply (erule hd_is_reduce_hd [THEN mp])
-apply (rule bool_if_impl_or [THEN mp])
-apply (erule reduce_tl)
-done
-
-declare split_if [split]
-
-lemma sender_abstraction: "is_weak_ref_map reduce srch_ioa srch_fin_ioa"
-apply (tactic {*
- simp_tac (HOL_ss addsimps [@{thm srch_fin_ioa_def}, @{thm rsch_fin_ioa_def},
- @{thm srch_ioa_def}, @{thm rsch_ioa_def}, @{thm rename_through_pmap},
- @{thm channel_abstraction}]) 1 *})
-done
-
-lemma receiver_abstraction: "is_weak_ref_map reduce rsch_ioa rsch_fin_ioa"
-apply (tactic {*
- simp_tac (HOL_ss addsimps [@{thm srch_fin_ioa_def}, @{thm rsch_fin_ioa_def},
- @{thm srch_ioa_def}, @{thm rsch_ioa_def}, @{thm rename_through_pmap},
- @{thm channel_abstraction}]) 1 *})
-done
-
-
-text {* 3 thms that do not hold generally! The lucky restriction here is
- the absence of internal actions. *}
-lemma sender_unchanged: "is_weak_ref_map (%id. id) sender_ioa sender_ioa"
-apply (simp (no_asm) add: is_weak_ref_map_def)
-txt {* main-part *}
-apply (rule allI)+
-apply (induct_tac a)
-txt {* 7 cases *}
-apply (simp_all (no_asm) add: externals_def)
-done
-
-text {* 2 copies of before *}
-lemma receiver_unchanged: "is_weak_ref_map (%id. id) receiver_ioa receiver_ioa"
-apply (simp (no_asm) add: is_weak_ref_map_def)
-txt {* main-part *}
-apply (rule allI)+
-apply (induct_tac a)
-txt {* 7 cases *}
-apply (simp_all (no_asm) add: externals_def)
-done
-
-lemma env_unchanged: "is_weak_ref_map (%id. id) env_ioa env_ioa"
-apply (simp (no_asm) add: is_weak_ref_map_def)
-txt {* main-part *}
-apply (rule allI)+
-apply (induct_tac a)
-txt {* 7 cases *}
-apply (simp_all (no_asm) add: externals_def)
-done
-
-
-lemma compat_single_ch: "compatible srch_ioa rsch_ioa"
-apply (simp add: compatible_def Int_def)
-apply (rule set_eqI)
-apply (induct_tac x)
-apply simp_all
-done
-
-text {* totally the same as before *}
-lemma compat_single_fin_ch: "compatible srch_fin_ioa rsch_fin_ioa"
-apply (simp add: compatible_def Int_def)
-apply (rule set_eqI)
-apply (induct_tac x)
-apply simp_all
-done
-
-lemmas del_simps = trans_of_def srch_asig_def rsch_asig_def
- asig_of_def actions_def srch_trans_def rsch_trans_def srch_ioa_def
- srch_fin_ioa_def rsch_fin_ioa_def rsch_ioa_def sender_trans_def
- receiver_trans_def set_lemmas
-
-lemma compat_rec: "compatible receiver_ioa (srch_ioa || rsch_ioa)"
-apply (simp del: del_simps
- add: compatible_def asig_of_par asig_comp_def actions_def Int_def)
-apply simp
-apply (rule set_eqI)
-apply (induct_tac x)
-apply simp_all
-done
-
-text {* 5 proofs totally the same as before *}
-lemma compat_rec_fin: "compatible receiver_ioa (srch_fin_ioa || rsch_fin_ioa)"
-apply (simp del: del_simps
- add: compatible_def asig_of_par asig_comp_def actions_def Int_def)
-apply simp
-apply (rule set_eqI)
-apply (induct_tac x)
-apply simp_all
-done
-
-lemma compat_sen: "compatible sender_ioa
- (receiver_ioa || srch_ioa || rsch_ioa)"
-apply (simp del: del_simps
- add: compatible_def asig_of_par asig_comp_def actions_def Int_def)
-apply simp
-apply (rule set_eqI)
-apply (induct_tac x)
-apply simp_all
-done
-
-lemma compat_sen_fin: "compatible sender_ioa
- (receiver_ioa || srch_fin_ioa || rsch_fin_ioa)"
-apply (simp del: del_simps
- add: compatible_def asig_of_par asig_comp_def actions_def Int_def)
-apply simp
-apply (rule set_eqI)
-apply (induct_tac x)
-apply simp_all
-done
-
-lemma compat_env: "compatible env_ioa
- (sender_ioa || receiver_ioa || srch_ioa || rsch_ioa)"
-apply (simp del: del_simps
- add: compatible_def asig_of_par asig_comp_def actions_def Int_def)
-apply simp
-apply (rule set_eqI)
-apply (induct_tac x)
-apply simp_all
-done
-
-lemma compat_env_fin: "compatible env_ioa
- (sender_ioa || receiver_ioa || srch_fin_ioa || rsch_fin_ioa)"
-apply (simp del: del_simps
- add: compatible_def asig_of_par asig_comp_def actions_def Int_def)
-apply simp
-apply (rule set_eqI)
-apply (induct_tac x)
-apply simp_all
-done
-
-
-text {* lemmata about externals of channels *}
-lemma ext_single_ch: "externals(asig_of(srch_fin_ioa)) = externals(asig_of(srch_ioa)) &
- externals(asig_of(rsch_fin_ioa)) = externals(asig_of(rsch_ioa))"
- by (simp add: externals_def)
-
-
-subsection {* Soundness of Abstraction *}
-
-lemmas ext_simps = externals_of_par ext_single_ch
- and compat_simps = compat_single_ch compat_single_fin_ch compat_rec
- compat_rec_fin compat_sen compat_sen_fin compat_env compat_env_fin
- and abstractions = env_unchanged sender_unchanged
- receiver_unchanged sender_abstraction receiver_abstraction
-
-
-(* FIX: this proof should be done with compositionality on trace level, not on
- weak_ref_map level, as done here with fxg_is_weak_ref_map_of_product_IOA
-
-Goal "is_weak_ref_map abs system_ioa system_fin_ioa"
-
-by (simp_tac (impl_ss delsimps ([srch_ioa_def, rsch_ioa_def, srch_fin_ioa_def,
- rsch_fin_ioa_def] @ env_ioas @ impl_ioas)
- addsimps [system_def, system_fin_def, abs_def,
- impl_ioa_def, impl_fin_ioa_def, sys_IOA,
- sys_fin_IOA]) 1);
-
-by (REPEAT (EVERY[rtac fxg_is_weak_ref_map_of_product_IOA 1,
- simp_tac (ss addsimps abstractions) 1,
- rtac conjI 1]));
-
-by (ALLGOALS (simp_tac (ss addsimps ext_ss @ compat_ss)));
-
-qed "system_refinement";
-*)
-
-end