Theory Strong_Sim_Pres

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

lemma actPres:
  fixes P    :: ccs
  and   Q    :: ccs
  and   Rel  :: "(ccs × ccs) set"
  and   a    :: name
  and   Rel' :: "(ccs × ccs) set"

  assumes "(P, Q) ∈ Rel"

  shows "α.(P) ↝[Rel] α.(Q)"
using assms
by(fastforce simp add: simulation_def elim: actCases intro: Action)

lemma sumPres:
  fixes P   :: ccs
  and   Q   :: ccs
  and   Rel :: "(ccs × ccs) set"

  assumes "P ↝[Rel] Q"
  and     "Rel ⊆ Rel'"
  and     "Id ⊆ Rel'"

  shows "P ⊕ R ↝[Rel'] Q ⊕ R"
using assms
by(force simp add: simulation_def elim: sumCases intro: Sum1 Sum2)

lemma parPresAux:
  fixes P   :: ccs
  and   Q   :: ccs
  and   Rel :: "(ccs × ccs) set"

  assumes "P ↝[Rel] Q"
  and     "(P, Q) ∈ Rel"
  and     "R ↝[Rel'] T"
  and     "(R, T) ∈ Rel'"
  and     C1: "⋀P' Q' R' T'. ⟦(P', Q') ∈ Rel; (R', T') ∈ Rel'⟧ ⟹ (P' ∥ R', Q' ∥ T') ∈ Rel''"

  shows "P ∥ R ↝[Rel''] Q ∥ T"
proof(induct rule: simI)
  case(Sim a QT)
  from ‹Q ∥ T ⟼a ≺ QT›
  show ?case
  proof(induct rule: parCases)
    case(cPar1 Q')
    from ‹P ↝[Rel] Q› ‹Q ⟼a ≺ Q'› obtain P' where "P ⟼a ≺ P'" and "(P', Q') ∈ Rel" 
      by(rule simE)
    from ‹P ⟼a ≺ P'› have "P ∥ R ⟼a ≺ P' ∥ R" by(rule Par1)
    moreover from ‹(P', Q') ∈ Rel› ‹(R, T) ∈ Rel'› have "(P' ∥ R, Q' ∥ T) ∈ Rel''" by(rule C1)
    ultimately show ?case by blast
  next
    case(cPar2 T')
    from ‹R ↝[Rel'] T› ‹T ⟼a ≺ T'› obtain R' where "R ⟼a ≺ R'" and "(R', T') ∈ Rel'" 
      by(rule simE)
    from ‹R ⟼a ≺ R'› have "P ∥ R ⟼a ≺ P ∥ R'" by(rule Par2)
    moreover from ‹(P, Q) ∈ Rel› ‹(R', T') ∈ Rel'› have "(P ∥ R', Q ∥ T') ∈ Rel''" by(rule C1)
    ultimately show ?case by blast
  next
    case(cComm Q' T' a)
    from ‹P ↝[Rel] Q› ‹Q ⟼a ≺ Q'› obtain P' where "P ⟼a ≺ P'" and "(P', Q') ∈ Rel" 
      by(rule simE)
    from ‹R ↝[Rel'] T› ‹T ⟼(coAction a) ≺ T'› obtain R' where "R ⟼(coAction a) ≺ R'" and "(R', T') ∈ Rel'" 
      by(rule simE)
    from ‹P ⟼a ≺ P'› ‹R ⟼(coAction a) ≺ R'› ‹a ≠ τ› have "P ∥ R ⟼τ ≺ P' ∥ R'" by(rule Comm)
    moreover from ‹(P', Q') ∈ Rel› ‹(R', T') ∈ Rel'› have "(P' ∥ R', Q' ∥ T') ∈ Rel''" by(rule C1)
    ultimately show ?case by blast
  qed
qed

lemma parPres:
  fixes P   :: ccs
  and   Q   :: ccs
  and   Rel :: "(ccs × ccs) set"

  assumes "P ↝[Rel] Q"
  and     "(P, Q) ∈ Rel"
  and     C1: "⋀S T U. (S, T) ∈ Rel ⟹ (S ∥ U, T ∥ U) ∈ Rel'"

  shows "P ∥ R ↝[Rel'] Q ∥ R"
using assms
by(rule_tac parPresAux[where Rel''=Rel' and Rel'=Id]) (auto intro: reflexive)

lemma resPres:
  fixes P   :: ccs
  and   Rel :: "(ccs × ccs) set"
  and   Q   :: ccs
  and   x   :: name

  assumes "P ↝[Rel] Q"
  and     "⋀R S y. (R, S) ∈ Rel ⟹ (⦇νy⦈R, ⦇νy⦈S) ∈ Rel'"

  shows "⦇νx⦈P ↝[Rel'] ⦇νx⦈Q"
using assms
by(fastforce simp add: simulation_def elim: resCases intro: Res)

lemma bangPres:
  fixes P   :: ccs
  and   Rel :: "(ccs × ccs) set"
  and   Q   :: ccs

  assumes "(P, Q) ∈ Rel"
  and     C1: "⋀R S. (R, S) ∈ Rel ⟹ R ↝[Rel] S"

  shows "!P ↝[bangRel Rel] !Q"
proof(induct rule: simI)
  case(Sim α Q')
  {
    fix Pa α Q'
    assume "!Q ⟼α ≺ Q'" and "(Pa, !Q) ∈ bangRel Rel"
    hence "∃P'. Pa ⟼α ≺ P' ∧ (P', Q') ∈ bangRel Rel"
    proof(nominal_induct arbitrary: Pa rule: bangInduct)
      case(cPar1 α Q')
      from ‹(Pa, Q ∥ !Q) ∈ bangRel Rel› 
      show ?case
      proof(induct rule: BRParCases)
        case(BRPar P R)
        from ‹(P, Q) ∈ Rel› have "P ↝[Rel] Q" by(rule C1)
        with ‹Q ⟼α ≺ Q'› obtain P' where "P ⟼α ≺ P'" and "(P', Q') ∈ Rel"
          by(blast dest: simE)
        from ‹P ⟼α ≺ P'› have "P ∥ R ⟼α ≺ P' ∥ R" by(rule Par1)
        moreover from ‹(P', Q') ∈ Rel› ‹(R, !Q) ∈ bangRel Rel› have "(P' ∥ R, Q' ∥ !Q) ∈ bangRel Rel"
          by(rule bangRel.BRPar)
        ultimately show ?case by blast
      qed
    next
      case(cPar2 α Q')
      from ‹(Pa, Q ∥ !Q) ∈ bangRel Rel›
      show ?case
      proof(induct rule: BRParCases)
        case(BRPar P R)
        from ‹(R, !Q) ∈ bangRel Rel› obtain R' where "R ⟼α ≺ R'" and "(R', Q') ∈ bangRel Rel" using cPar2
          by blast
        from ‹R ⟼α ≺ R'› have "P ∥ R ⟼α ≺ P ∥ R'" by(rule Par2)
        moreover from ‹(P, Q) ∈ Rel› ‹(R', Q') ∈ bangRel Rel› have "(P ∥ R', Q ∥ Q') ∈ bangRel Rel" by(rule bangRel.BRPar)
        ultimately show ?case by blast
      qed
    next
      case(cComm a Q' Q'' Pa)
      from ‹(Pa, Q ∥ !Q) ∈ bangRel Rel›
      show ?case
      proof(induct rule: BRParCases)
        case(BRPar P R)
        from ‹(P, Q) ∈ Rel› have "P ↝[Rel] Q" by(rule C1)
        with ‹Q ⟼a ≺ Q'› obtain P' where "P ⟼a ≺ P'" and "(P', Q') ∈ Rel"
          by(blast dest: simE)
        from ‹(R, !Q) ∈ bangRel Rel› obtain R' where "R ⟼(coAction a) ≺ R'" and "(R', Q'') ∈ bangRel Rel" using cComm
          by blast
        from ‹P ⟼a ≺ P'› ‹R ⟼(coAction a) ≺ R'› ‹a ≠ τ› have "P ∥ R ⟼τ ≺ P' ∥ R'" by(rule Comm)
        moreover from ‹(P', Q') ∈ Rel› ‹(R', Q'') ∈ bangRel Rel› have "(P' ∥ R', Q' ∥ Q'') ∈ bangRel Rel" by(rule bangRel.BRPar)
        ultimately show ?case by blast
      qed
    next
      case(cBang α Q' Pa)
      from ‹(Pa, !Q) ∈ bangRel Rel›
      show ?case
      proof(induct rule: BRBangCases)
        case(BRBang P)
        from ‹(P, Q) ∈ Rel› have "(!P, !Q) ∈ bangRel Rel" by(rule bangRel.BRBang)
        with ‹(P, Q) ∈ Rel› have "(P ∥ !P, Q ∥ !Q) ∈ bangRel Rel" by(rule bangRel.BRPar)
        then obtain P' where "P ∥ !P ⟼α ≺ P'" and "(P', Q') ∈ bangRel Rel" using cBang
          by blast
        from ‹P ∥ !P ⟼α ≺ P'› have "!P ⟼α ≺ P'" by(rule Bang)
        thus ?case using ‹(P', Q') ∈ bangRel Rel› by blast
      qed
    qed
  }

  moreover from ‹(P, Q) ∈ Rel› have "(!P, !Q) ∈ bangRel Rel" by(rule BRBang) 
  ultimately show ?case using ‹!Q ⟼ α ≺ Q'› by blast
qed

end