Quelle  Sliding_Choice.thy

(*<*)
―‹ ********************************************************************
 * Project : HOL-CSP - A Shallow Embedding of CSP in Isabelle/HOL
 * Version : 2.0
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 * Author : Benoît Ballenghien, Safouan Taha, Burkhart Wolff, Lina Ye.
 * (Based on HOL-CSP 1.0 by Haykal Tej and Burkhart Wolff)
 *
 * This file : Sliding choice
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(*>*)


section ‹ Sliding Choice ›

(*<*)
theory  Sliding_Choice
  imports Deterministic_Choice Non_Deterministic_Choice
begin
  (*>*)

subsection ‹Definition›

definition Sliding :: ‹('a, 'r)process_ptick ==> ('a, 'r)process_ptick ==> ('a, 'r)process_ptick› (infixl ‹⊳› 80)
  where ‹P ⊳ Q ≡ (P ◻ Q) ⊓ Q›



subsection ‹Projections›

lemma F_Sliding:
  ‹F (P ⊳ Q) = F Q ∪ {(s, X). s ≠ [] ∧ (s, X) ∈ F P ∨
 s = [] ∧ (∃r. s ∈ D P ∨ tick r ∉ X ∧ [tick r] ∈ T P)}›
  by (auto simp add: Sliding_def F_Ndet F_Det intro: is_processT8 is_processT6_TR_notin)

lemma D_Sliding: ‹D (P ⊳ Q) = D P ∪ D Q›
  by (simp add: Sliding_def D_Ndet D_Det)

lemma T_Sliding: ‹T (P ⊳ Q) = T P ∪ T Q›
  by (simp add: Sliding_def T_Ndet T_Det)

lemmas Sliding_projs = F_Sliding D_Sliding T_Sliding



subsection ‹Properties›

lemma Sliding_id: ‹P ⊳ P = P›
  by (simp add: Sliding_def)

(* 
text \<open>Of course, \<^term>\<open>P \<rhd> STOP \<noteq> STOP\<close> and \<^term>\<open>P \<rhd> STOP \<noteq> P\<close> in general.\<close>
lemma \<open>\<exists>P. P \<rhd> STOP \<noteq> STOP \<and> P \<rhd> STOP \<noteq> P\<close>
proof (intro exI)
  show \<open>SKIP undefined \<rhd> STOP \<noteq> STOP \<and> SKIP undefined \<rhd> STOP \<noteq> SKIP undefined\<close>
    by (metis Det_STOP Ndet_commute SKIP_F_iff SKIP_Neq_STOP
            STOP_F_iff Sliding_def mono_Ndet_F_left)
qed
   *)




subsection ‹Continuity›

text ‹From the definition, monotony and continuity is obvious.›

lemma mono_Sliding : ‹P ⊑ P' ==> Q ⊑ Q' ==> P ⊳ Q ⊑ P' ⊳ Q'›
  unfolding Sliding_def by (simp add: mono_Det mono_Ndet)

lemma Sliding_cont[simp] : ‹cont f ==> cont g ==> cont (λx. f x ⊳ g x)›
  by (simp add: Sliding_def)

(*<*)
end
  (*>*)