Up to index of Isabelle/HOLCF/IOA
theory Simulations(* Title: HOLCF/IOA/meta_theory/Simulations.thy ID: $Id: Simulations.thy,v 1.9 2007/10/21 14:27:43 wenzelm Exp $ Author: Olaf Müller *) header {* Simulations in HOLCF/IOA *} theory Simulations imports RefCorrectness begin defaultsort 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 -a--C-> 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 -a--C-> 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 -a--C-> 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 -a--C-> 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
lemma set_non_empty:
(A ≠ {}) = (∃x. x ∈ A)
lemma Int_non_empty:
(A ∩ B ≠ {}) = (∃x. x ∈ A ∧ x ∈ B)
lemma Sim_start_convert:
(R `` {x} ∩ S ≠ {}) = (∃y. (x, y) ∈ R ∧ y ∈ S)
lemma ref_map_is_simulation:
is_ref_map f C A ==> is_simulation {p. snd p = f (fst p)} C A