(* Title: HOL/HOLCF/IOA/ABP/Correctness.thy
Author: Olaf Müller
*)
section {* The main correctness proof: System_fin implements System *}
theory Correctness
imports "~~/src/HOL/HOLCF/IOA/meta_theory/IOA" Env Impl Impl_finite
begin
ML_file "Check.ML"
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 \<parallel> impl_ioa)"
definition
system_fin_ioa :: "('m action, bool * 'm impl_state)ioa" where
"system_fin_ioa = (env_ioa \<parallel> 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 (put_simpset HOL_ss @{context}
addsimps (@{thms reverse.simps} @ [@{thm hd_append}, @{thm rev_red_not_nil}])
|> Splitter.add_split @{thm list.split}) *})
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 (put_simpset HOL_ss @{context}
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 (put_simpset HOL_ss @{context}
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 \<parallel> 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 \<parallel> 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 \<parallel> srch_ioa \<parallel> 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 \<parallel> srch_fin_ioa \<parallel> 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 \<parallel> receiver_ioa \<parallel> srch_ioa \<parallel> 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 \<parallel> receiver_ioa \<parallel> srch_fin_ioa \<parallel> 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