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theory Simulations(* Title: HOLCF/IOA/meta_theory/Simulations.thy ID: $Id: Simulations.thy,v 1.7 2005/09/02 15:24:02 wenzelm Exp $ Author: Olaf Müller *) header {* Simulations in HOLCF/IOA *} theory Simulations imports RefCorrectness begin defaultsort type consts is_simulation ::"[('s1 * 's2)set,('a,'s1)ioa,('a,'s2)ioa] => bool" is_backward_simulation ::"[('s1 * 's2)set,('a,'s1)ioa,('a,'s2)ioa] => bool" is_forw_back_simulation ::"[('s1 * 's2 set)set,('a,'s1)ioa,('a,'s2)ioa] => bool" is_back_forw_simulation ::"[('s1 * 's2 set)set,('a,'s1)ioa,('a,'s2)ioa] => bool" is_history_relation ::"[('s1 * 's2)set,('a,'s1)ioa,('a,'s2)ioa] => bool" is_prophecy_relation ::"[('s1 * 's2)set,('a,'s1)ioa,('a,'s2)ioa] => bool" defs is_simulation_def: "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'))" is_backward_simulation_def: "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'))" is_forw_back_simulation_def: "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')))" is_back_forw_simulation_def: "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')))" is_history_relation_def: "is_history_relation R C A == is_simulation R C A & is_ref_map (%x.(@y. (x,y):(R^-1))) A C" is_prophecy_relation_def: "is_prophecy_relation R C A == is_backward_simulation R C A & is_ref_map (%x.(@y. (x,y):(R^-1))) A C" ML {* use_legacy_bindings (the_context ()) *} end
theorem set_non_empty:
(A ≠ {}) = (∃x. x ∈ A)
theorem Int_non_empty:
(A ∩ B ≠ {}) = (∃x. x ∈ A ∧ x ∈ B)
theorem Sim_start_convert:
(R `` {x} ∩ S ≠ {}) = (∃y. (x, y) ∈ R ∧ y ∈ S)
theorem ref_map_is_simulation:
is_ref_map f C A ==> is_simulation {p. snd p = f (fst p)} C A