Theory Sliding_Choice

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―‹ ********************************************************************
 * Project         : HOL-CSP - A Shallow Embedding of CSP in Isabelle/HOL
 * Version         : 2.0
 *
 * 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 ›

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theory  Sliding_Choice
  imports Deterministic_Choice Non_Deterministic_Choice
begin
  (*>*)

subsection ‹Definition›

definition Sliding :: ‹('a, 'r)processptick ⇒ ('a, 'r)processptick ⇒ ('a, 'r)processptick› (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 ‹Of course, term‹P ⊳ STOP ≠ STOP› and term‹P ⊳ STOP ≠ P› in general.›
lemma ‹∃P. P ⊳ STOP ≠ STOP ∧ P ⊳ STOP ≠ P›
proof (intro exI)
  show ‹SKIP undefined ⊳ STOP ≠ STOP ∧ SKIP undefined ⊳ STOP ≠ SKIP undefined›
    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)

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end
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