Theory Weak_Late_Cong_Subst_SC

(* 
   Title: The pi-calculus   
   Author/Maintainer: Jesper Bengtson (jebe.dk), 2012
*)
theory Weak_Late_Cong_Subst_SC
  imports Weak_Late_Cong_Subst Strong_Late_Bisim_Subst_SC
begin

(******** Structural Congruence **********)

(******** The ν-operator *****************)

lemma resComm:
  fixes P :: pi
  
  shows "<νa><νb>P ≃⇧s <νb><νa>P"
proof -
  have "<νa><νb>P ∼⇧s <νb><νa>P"
    by(rule Strong_Late_Bisim_Subst_SC.resComm)
  thus ?thesis by(rule strongEqWeakCong)
qed

(******** Match *********)

lemma matchId:
  fixes a :: name
  and   P :: pi

  shows "[a⌢a]P ≃⇧s P"
proof -
  have "[a⌢a]P ∼⇧s P" by(rule Strong_Late_Bisim_Subst_SC.matchId)
  thus ?thesis by(rule strongEqWeakCong)
qed

(******** Mismatch *********)

lemma matchNil:
  fixes a :: name
  and   P :: pi

  shows "[a≠a]P ≃⇧s 𝟬"
proof -
  have "[a≠a]P ∼⇧s 𝟬" by(rule Strong_Late_Bisim_Subst_SC.mismatchNil)
  thus ?thesis by(rule strongEqWeakCong)
qed

(******** The +-operator *********)

lemma sumSym:
  fixes P :: pi
  and   Q :: pi
  
  shows "P ⊕ Q ≃⇧s Q ⊕ P"
proof -
  have "P ⊕ Q ∼⇧s Q ⊕ P" by(rule Strong_Late_Bisim_Subst_SC.sumSym)
  thus ?thesis by(rule strongEqWeakCong)
qed

lemma sumAssoc:
  fixes P :: pi
  and   Q :: pi
  and   R :: pi
  
  shows "(P ⊕ Q) ⊕ R ≃⇧s P ⊕ (Q ⊕ R)"
proof -
  have "(P ⊕ Q) ⊕ R ∼⇧s P ⊕ (Q ⊕ R)" by(rule Strong_Late_Bisim_Subst_SC.sumAssoc)
  thus ?thesis by(rule strongEqWeakCong)
qed

lemma sumZero:
  fixes P :: pi
  
  shows "P ⊕ 𝟬 ≃⇧s P"
proof -
  have "P ⊕ 𝟬 ∼⇧s P" by(rule Strong_Late_Bisim_Subst_SC.sumZero)
  thus ?thesis by(rule strongEqWeakCong)
qed

(******** The |-operator *********)

lemma parZero:
  fixes P :: pi

  shows "P ∥ 𝟬 ≃⇧s P"
proof -
  have "P ∥ 𝟬 ∼⇧s P" by(rule Strong_Late_Bisim_Subst_SC.parZero)
  thus ?thesis by(rule strongEqWeakCong)
qed

lemma parSym:
  fixes P :: pi
  and   Q :: pi

  shows "P ∥ Q ≃⇧s Q ∥ P"
proof -
  have "P ∥ Q ∼⇧s Q ∥ P" by(rule Strong_Late_Bisim_Subst_SC.parSym)
  thus ?thesis by(rule strongEqWeakCong)
qed

lemma scopeExtPar:
  fixes P :: pi
  and   Q :: pi
  and   x :: name

  assumes "x ♯ P"

  shows "<νx>(P ∥ Q) ≃⇧s P ∥ <νx>Q"
proof -
  from assms have "<νx>(P ∥ Q) ∼⇧s P ∥ <νx>Q" by(rule Strong_Late_Bisim_Subst_SC.scopeExtPar)
  thus ?thesis by(rule strongEqWeakCong)
qed

lemma scopeExtPar':
  fixes P :: pi
  and   Q :: pi
  and   x :: name

  assumes xFreshQ: "x ♯ Q"

  shows "<νx>(P ∥ Q) ≃⇧s (<νx>P) ∥ Q"
proof -
  from assms have "<νx>(P ∥ Q) ∼⇧s (<νx>P) ∥ Q" by(rule Strong_Late_Bisim_Subst_SC.scopeExtPar')
  thus ?thesis by(rule strongEqWeakCong)
qed

lemma parAssoc:
  fixes P :: pi
  and   Q :: pi
  and   R :: pi

  shows "(P ∥ Q) ∥ R ≃⇧s P ∥ (Q ∥ R)"
proof -
  have "(P ∥ Q) ∥ R ∼⇧s P ∥ (Q ∥ R)" by(rule Strong_Late_Bisim_Subst_SC.parAssoc)
  thus ?thesis by(rule strongEqWeakCong)
qed

lemma scopeFresh:
  fixes P :: pi
  and   a :: name

  assumes aFreshP: "a ♯ P"

  shows "<νa>P ≃⇧s P"
proof -
  from assms have "<νa>P ∼⇧s P" by(rule Strong_Late_Bisim_Subst_SC.scopeFresh)
  thus ?thesis by(rule strongEqWeakCong)
qed

lemma scopeExtSum:
  fixes P :: pi
  and   Q :: pi
  and   x :: name
  
  assumes "x ♯ P"

  shows "<νx>(P ⊕ Q) ≃⇧s P ⊕ <νx>Q"
proof -
  from assms have "<νx>(P ⊕ Q) ∼⇧s P ⊕ <νx>Q" by(rule Strong_Late_Bisim_Subst_SC.scopeExtSum)
  thus ?thesis by(rule strongEqWeakCong)
qed

lemma bangSC:
  fixes P

  shows "!P ≃⇧s P ∥ !P"
proof -
  have "!P ∼⇧s P ∥ !P" by(rule Strong_Late_Bisim_Subst_SC.bangSC)
  thus ?thesis by(rule strongEqWeakCong)
qed

end