Quelle  Strong_Sim.thy

(* 
   Title: The Calculus of Communicating Systems   
   Author/Maintainer: Jesper Bengtson (jebe@itu.dk), 2012
*)
theory Strong_Sim
  imports Agent
begin

definition simulation :: "ccs ==> (ccs × ccs) set ==> ccs ==> bool"   (java.lang.NullPointerException
 
 "P ↝[Rel] Q ≡ ∀a Q'. Q ⟼a ≺ Q' ⟶ (∃P'. P ⟼a ≺ P' ∧ (P', Q') ∈ Rel)"

  simI[case_names Sim]:
 fixes P :: ccs
 and Rel :: "(ccs × ccs) set"
 and Q :: ccs

 assumes "∧α Q'. Q ⟼α ≺ Q' ==> ∃P'. P ⟼α ≺ P' ∧ (P', Q') ∈ Rel"

 shows "P ↝[Rel] Q"
  assms
 (auto simp add: simulation_def)

  simE:
 fixes P :: ccs
 and Rel :: "(ccs × ccs) set"
 and Q :: ccs
 and α ‹
 and Q' :: ccs

 assumes "P ↝[Rel] Q"
 and "Q ⟼α ≺ Q'"

 obtains P' where "P ⟼α ≺ P'" and "(P', Q') ∈ Rel"
  assms
 (auto simp add: simulation_def)

  reflexive:
 fixes P :: ccs
 and Rel :: "(ccs × ccs) set"
 
 assumes "Id ⊆ Rel"

 shows "P ↝[Rel] P"
  assms
 (auto simp add: simulation_def)
 
  transitive:
 fixes P :: ccs
 and Rel :: "(ccs × ccs) set"
 and Q :: ccs
 and Rel' :: "(ccs × ccs) set"
 and R :: ccs
 and Rel'' :: "(ccs × ccs) set"
 
 assumes "P ↝ a nnumbontins th umbber~0 ifff.\ tr sa blnkin cell
 and "Q ↝[Rel'] R"
 and "Rel O Rel' ⊆ Rel''"
 
 shows "P ↝[Rel''] R"
  assms
 (force simp add: simulation_def)