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theory RefMappings(* Title: HOLCF/IOA/meta_theory/RefMappings.thy ID: $Id: RefMappings.thy,v 1.9 2005/09/02 15:24:02 wenzelm Exp $ Author: Olaf Müller *) header {* Refinement Mappings in HOLCF/IOA *} theory RefMappings imports Traces begin defaultsort type consts move ::"[('a,'s)ioa,('a,'s)pairs,'s,'a,'s] => bool" is_ref_map ::"[('s1=>'s2),('a,'s1)ioa,('a,'s2)ioa] => bool" is_weak_ref_map ::"[('s1=>'s2),('a,'s1)ioa,('a,'s2)ioa] => bool" defs move_def: "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))" is_ref_map_def: "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)))" is_weak_ref_map_def: "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)))" ML {* use_legacy_bindings (the_context ()) *} end
theorem transition_is_ex:
s -a--A-> t ==> ∃ex. move A ex s a t
theorem nothing_is_ex:
a ∉ ext A ∧ s = t ==> ∃ex. move A ex s a t
theorem ei_transitions_are_ex:
s -a--A-> s' ∧ s' -a'--A-> s'' ∧ a' ∉ ext A ==> ∃ex. move A ex s a s''
theorem 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
theorem weak_ref_map2ref_map:
[| ext C = ext A; is_weak_ref_map f C A |] ==> is_ref_map f C A
theorem imp_conj_lemma:
(P ==> Q --> R) ==> P ∧ Q --> R
theorem rename_through_pmap:
is_weak_ref_map f C A ==> is_weak_ref_map f (rename C g) (rename A g)