(* Title: HOL/HOLCF/IOA/Simulations.thy
Author: Olaf Müller
*)
section \<open>Simulations in HOLCF/IOA\<close>
theory Simulations
imports RefCorrectness
begin
default_sort type
definition
is_simulation :: "[('s1 * 's2)set,('a,'s1)ioa,('a,'s2)ioa] => bool" where
"is_simulation R C A =
((!s:starts_of C. R``{s} Int starts_of A ~= {}) &
(!s s' t a. reachable C s &
s \<midarrow>a\<midarrow>C\<rightarrow> t &
(s,s') : R
--> (? t' ex. (t,t'):R & move A ex s' a t')))"
definition
is_backward_simulation :: "[('s1 * 's2)set,('a,'s1)ioa,('a,'s2)ioa] => bool" where
"is_backward_simulation R C A =
((!s:starts_of C. R``{s} <= starts_of A) &
(!s t t' a. reachable C s &
s \<midarrow>a\<midarrow>C\<rightarrow> t &
(t,t') : R
--> (? ex s'. (s,s'):R & move A ex s' a t')))"
definition
is_forw_back_simulation :: "[('s1 * 's2 set)set,('a,'s1)ioa,('a,'s2)ioa] => bool" where
"is_forw_back_simulation R C A =
((!s:starts_of C. ? S'. (s,S'):R & S'<= starts_of A) &
(!s S' t a. reachable C s &
s \<midarrow>a\<midarrow>C\<rightarrow> t &
(s,S') : R
--> (? T'. (t,T'):R & (! t':T'. ? s':S'. ? ex. move A ex s' a t'))))"
definition
is_back_forw_simulation :: "[('s1 * 's2 set)set,('a,'s1)ioa,('a,'s2)ioa] => bool" where
"is_back_forw_simulation R C A =
((!s:starts_of C. ! S'. (s,S'):R --> S' Int starts_of A ~={}) &
(!s t T' a. reachable C s &
s \<midarrow>a\<midarrow>C\<rightarrow> t &
(t,T') : R
--> (? S'. (s,S'):R & (! s':S'. ? t':T'. ? ex. move A ex s' a t'))))"
definition
is_history_relation :: "[('s1 * 's2)set,('a,'s1)ioa,('a,'s2)ioa] => bool" where
"is_history_relation R C A = (is_simulation R C A &
is_ref_map (%x.(@y. (x,y):(R^-1))) A C)"
definition
is_prophecy_relation :: "[('s1 * 's2)set,('a,'s1)ioa,('a,'s2)ioa] => bool" where
"is_prophecy_relation R C A = (is_backward_simulation R C A &
is_ref_map (%x.(@y. (x,y):(R^-1))) A C)"
lemma set_non_empty: "(A~={}) = (? x. x:A)"
apply auto
done
lemma Int_non_empty: "(A Int B ~= {}) = (? x. x: A & x:B)"
apply (simp add: set_non_empty)
done
lemma Sim_start_convert:
"(R``{x} Int S ~= {}) = (? y. (x,y):R & y:S)"
apply (unfold Image_def)
apply (simp add: Int_non_empty)
done
declare Sim_start_convert [simp]
lemma ref_map_is_simulation:
"!! f. is_ref_map f C A ==> is_simulation {p. (snd p) = f (fst p)} C A"
apply (unfold is_ref_map_def is_simulation_def)
apply simp
done
end