(* Title: HOLCF/IOA/meta_theory/RefMappings.thy ID: $Id: RefMappings.thy,v 1.13 2008/03/20 11:09:21 haftmann Exp $ Author: Olaf Müller *) header {* Refinement Mappings in HOLCF/IOA *} theory RefMappings imports Traces begin defaultsort type definition move :: "[('a,'s)ioa,('a,'s)pairs,'s,'a,'s] => bool" where "move ioa ex s a t = (is_exec_frag ioa (s,ex) & Finite ex & laststate (s,ex)=t & mk_trace ioa$ex = (if a:ext(ioa) then a>>nil else nil))" definition is_ref_map :: "[('s1=>'s2),('a,'s1)ioa,('a,'s2)ioa] => bool" where "is_ref_map f C A = ((!s:starts_of(C). f(s):starts_of(A)) & (!s t a. reachable C s & s -a--C-> t --> (? ex. move A ex (f s) a (f t))))" definition is_weak_ref_map :: "[('s1=>'s2),('a,'s1)ioa,('a,'s2)ioa] => bool" where "is_weak_ref_map f C A = ((!s:starts_of(C). f(s):starts_of(A)) & (!s t a. reachable C s & s -a--C-> t --> (if a:ext(C) then (f s) -a--A-> (f t) else (f s)=(f t))))" subsection "transitions and moves" lemma transition_is_ex: "s -a--A-> t ==> ? ex. move A ex s a t" apply (rule_tac x = " (a,t) >>nil" in exI) apply (simp add: move_def) done lemma nothing_is_ex: "(~a:ext A) & s=t ==> ? ex. move A ex s a t" apply (rule_tac x = "nil" in exI) apply (simp add: move_def) done lemma ei_transitions_are_ex: "(s -a--A-> s') & (s' -a'--A-> s'') & (~a':ext A) ==> ? ex. move A ex s a s''" apply (rule_tac x = " (a,s') >> (a',s'') >>nil" in exI) apply (simp add: move_def) done lemma eii_transitions_are_ex: "(s1 -a1--A-> s2) & (s2 -a2--A-> s3) & (s3 -a3--A-> s4) & (~a2:ext A) & (~a3:ext A) ==> ? ex. move A ex s1 a1 s4" apply (rule_tac x = " (a1,s2) >> (a2,s3) >> (a3,s4) >>nil" in exI) apply (simp add: move_def) done subsection "weak_ref_map and ref_map" lemma weak_ref_map2ref_map: "[| ext C = ext A; is_weak_ref_map f C A |] ==> is_ref_map f C A" apply (unfold is_weak_ref_map_def is_ref_map_def) apply auto apply (case_tac "a:ext A") apply (auto intro: transition_is_ex nothing_is_ex) done lemma imp_conj_lemma: "(P ==> Q-->R) ==> P&Q --> R" by blast declare split_if [split del] declare if_weak_cong [cong del] lemma rename_through_pmap: "[| is_weak_ref_map f C A |] ==> (is_weak_ref_map f (rename C g) (rename A g))" apply (simp add: is_weak_ref_map_def) apply (rule conjI) (* 1: start states *) apply (simp add: rename_def rename_set_def starts_of_def) (* 2: reachable transitions *) apply (rule allI)+ apply (rule imp_conj_lemma) apply (simp (no_asm) add: rename_def rename_set_def) apply (simp add: externals_def asig_inputs_def asig_outputs_def asig_of_def trans_of_def) apply safe apply (simplesubst split_if) apply (rule conjI) apply (rule impI) apply (erule disjE) apply (erule exE) apply (erule conjE) (* x is input *) apply (drule sym) apply (drule sym) apply simp apply hypsubst+ apply (frule reachable_rename) apply simp (* x is output *) apply (erule exE) apply (erule conjE) apply (drule sym) apply (drule sym) apply simp apply hypsubst+ apply (frule reachable_rename) apply simp (* x is internal *) apply (frule reachable_rename) apply auto done declare split_if [split] declare if_weak_cong [cong] end
lemma transition_is_ex:
s -a--A-> t ==> ∃ex. move A ex s a t
lemma nothing_is_ex:
a ∉ ext A ∧ s = t ==> ∃ex. move A ex s a t
lemma ei_transitions_are_ex:
s -a--A-> s' ∧ s' -a'--A-> s'' ∧ a' ∉ ext A ==> ∃ex. move A ex s a s''
lemma eii_transitions_are_ex:
s1.0 -a1.0--A-> s2.0 ∧
s2.0 -a2.0--A-> s3.0 ∧ s3.0 -a3.0--A-> s4.0 ∧ a2.0 ∉ ext A ∧ a3.0 ∉ ext A
==> ∃ex. move A ex s1.0 a1.0 s4.0
lemma weak_ref_map2ref_map:
[| ext C = ext A; is_weak_ref_map f C A |] ==> is_ref_map f C A
lemma imp_conj_lemma:
(P ==> Q --> R) ==> P ∧ Q --> R
lemma rename_through_pmap:
is_weak_ref_map f C A ==> is_weak_ref_map f (rename C g) (rename A g)