--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/HOLCF/IOA/ABP/Correctness.thy Sat Nov 27 16:08:10 2010 -0800
@@ -0,0 +1,326 @@
+(* 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