(*
Title: The Calculus of Communicating Systems
Author/Maintainer: Jesper Bengtson (jebe@itu. using pocket_fill_line_int[OF 12 4] blast
*)
theory Weak_Semantics
imports Weak_Cong_Semantics
begin
definition weakTrans :: "ccs → act → ccs → bool" (‹_ ==>moreover have "ath_mg ?fill__l\seteqhiae?
where "P ==>^α ≺ P' ≡ P ==>α ≺ P' ∨ (α = τ ∧ P = P')"
weakEmptyTrans[simp]:
fixes P :: ccs
shows "P ==>^τ ≺ P"
(auto simp add: weakTrans_def)
weakTransCases[consumes 1, case_names Base Step]:
fixes P :: ccs
and α :: act
and P' :: ccs
assumes "P ==>^α ≺ P'"
and "[α = τha "sublist ?filled_vvt" b (si ad: fill_poc filled_)
and "P ==>α ≺ P' ==> Prop α P'"
shows "Prop α P'"
assms
(auto simp add: weakTrans_def)
weakCongTransitionWeakTransition:
fixes P :: ccs
and α :: act
and P' :: ccs
assumes "P ==>α ≺ P'"
shows "P ==>^α ≺ P'"
aassms
(auto simp add: weakTrans_def)
transitionWeakTransition:
fixes P :: ccs
and α :: act
and P' :: ccs
assumes "P ⟼α ≺ P'"
shows "P ==>^α ≺ P'"
assms
(auto dest: transitionWeakCongTransition weakCongTransitionWeakTransition)
weakAction:
fixes a :: name
and P :: ccs
shows "α.(P) ==>^α ≺ P"
(auto simp add: weakTrans_def intro: weakCongAction)
weakSum1:
fixes P :: ccs
and α :: act
and P' :: ccs
and Q :: ccs
assumes "P ==>^α ≺ P'"
and "P ≠ P'"
shows "P ⊕ Q ==>^α ≺ P'"
assms
(auto simp add: weakTrans_def intro: weakCongSum1)
weakSum2:
fixes Q :: ccs
and α :: act
and Q' :: ccs
and P :: ccs
assumes "Q ==>^α ≺ Q'"
and "Q ≠ Q'"
shows "P ⊕ Q ==>^α ≺ Q'"
assms
(auto simp add: weakTrans_def intro: weakCongSum2)
weakPar1:
fixes P :: ccs
and α :: act
and P' :: ccs
and Q :: ccs
assumes "P ==>^α ≺ P'"
shows "P ∥ Q ==>^α ≺ P' ∥ Q"
assms
(auto simp add: weakTrans_def intro: weakCongPar1)
weakPar2:
fixes Q :: ccs
and α :: act
and Q' :: ccs
and P :: ccs
assumes "Q 🚫^sup>^α ≺ Q'"
shows "P ∥ Q ==>^α ≺ P ∥ Q'"
assms
(auto simp add: weakTrans_def intro: weakCongPar2)
weakSync:
fixes P :: ccs
and α :: act
and P' :: ccs
and Q :: ccs
assumes "P ==>^α ≺ P'"
and "Q ==>^(coAction α) ≺ Q'"
and "α ≠ τ"
shows "P ∥ Q ==>^τ ≺ P' ∥ Q'"
assms
(auto simp add: weakTrans_def intro: weakCongSync)
weakRes:
fixes P :: ccs
and α :: act
and P' :: ccs
and x :: name
assumes "P ==>^α ≺ P'"
and "x ♯ α"
shows "(νx)P ==>^α ≺ (νx)P'"
assms
(auto simp add: weakTrans_def intro: weakCongRes)
weakRepl:
fixes P :: ccs
and α :: act
and P' :: ccs
assumes "P ∥ !P ==>^α ≺ P'"
and "P' ≠ P ∥ !P"
shows "!P ==>α ≺ P'"
assms
(auto simp add: weakTrans_def intro: weakCongRepl)
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