Theory Weak_Cong_Sim

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

definition weakCongSimulation :: "ccs  (ccs × ccs) set  ccs  bool"   ("_ ↝<_> _" [80, 80, 80] 80)
where
  "P ↝<Rel> Q  a Q'. Q a  Q'  (P'. P a  P'  (P', Q')  Rel)"

lemma weakSimI[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"
using assms
by(auto simp add: weakCongSimulation_def)

lemma weakSimE:
  fixes P   :: ccs
  and   Rel :: "(ccs × ccs) set"
  and   Q   :: ccs
  and   α   :: act
  and   Q'  :: ccs

  assumes "P ↝<Rel> Q"
  and     "Q α  Q'"

  obtains P' where "P α  P'" and "(P', Q')  Rel"
using assms
by(auto simp add: weakCongSimulation_def)

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

  assumes "P ↝[Rel] Q"
  
  shows "P ↝<Rel> Q"
using assms
by(rule_tac weakSimI, auto)
  (blast dest: simE transitionWeakCongTransition)

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

  assumes "P ↝<Rel> Q"
  
  shows "P ^<Rel> Q"
using assms
by(rule_tac Weak_Sim.weakSimI, auto)
  (blast dest: weakSimE weakCongTransitionWeakTransition)

lemma test:
  assumes "P τ P'"

  shows "P = P'  (P''. P τ  P''  P'' τ P')"
using assms
by(induct rule: tauChainInduct) auto

lemma tauChainCasesSym[consumes 1, case_names cTauNil cTauStep]:
  assumes "P τ P'"
  and     "Prop P"
  and     "P''. P τ  P''; P'' τ P'  Prop P'"

  shows "Prop P'"
using assms
by(blast dest: test)

lemma simE2:
  fixes P   :: ccs
  and   Rel :: "(ccs × ccs) set"
  and   Q   :: ccs
  and   α   :: act
  and   Q'  :: ccs

  assumes "P ↝<Rel> Q"
  and     "Q α  Q'"
  and     Sim: "R S. (R, S)  Rel  R ^<Rel> S"
  
  obtains P' where "P α  P'" and "(P', Q')  Rel"
proof -
  assume Goal: "P'. P α  P'; (P', Q')  Rel  thesis"
  from Q α  Q' obtain Q''' Q''
    where QChain: "Q τ Q'''" and Q'''Trans: "Q''' α  Q''" and Q''Chain: "Q'' τ Q'"
    by(rule weakCongTransE)
  from QChain Q'''Trans show ?thesis
  proof(induct rule: tauChainCasesSym)
    case cTauNil
    from P ↝<Rel> Q Q α  Q'' obtain P''' where PTrans: "P α  P'''" and "(P''', Q'')  Rel"
      by(blast dest: weakSimE)
    moreover from Q''Chain (P''', Q'')  Rel Sim obtain P' where P''Chain: "P''' τ P'" and "(P', Q')  Rel"
      by(rule simTauChain)
    with PTrans P''Chain show ?thesis
      by(force intro: Goal simp add: weakCongTrans_def weakTrans_def)
  next
    case(cTauStep Q'''')
    from P ↝<Rel> Q Q τ  Q'''' obtain P'''' where  PChain: "P τ  P''''" and "(P'''', Q'''')  Rel"
      by(drule_tac weakSimE) auto
    from Q'''' τ Q''' (P'''', Q'''')  Rel Sim obtain P''' where P''''Chain: "P'''' τ P'''" and "(P''', Q''')  Rel"
      by(rule simTauChain)
    from (P''', Q''')  Rel have "P''' ^<Rel> Q'''" by(rule Sim)
    then obtain P'' where P'''Trans: "P''' ^α  P''" and "(P'', Q'')  Rel" using Q'''Trans by(rule Weak_Sim.weakSimE)
    from Q''Chain (P'', Q'')  Rel Sim obtain P' where P''Chain: "P'' τ P'" and "(P', Q')  Rel"
      by(rule simTauChain)
    from PChain P''''Chain P'''Trans P''Chain
    have "P α  P'"
      apply(auto simp add: weakCongTrans_def weakTrans_def)
      apply(rule_tac x=P''aa in exI)
      apply auto
      defer
      apply blast
      by(auto simp add: tauChain_def)

    with (P', Q')  Rel show ?thesis
      by(force intro: Goal simp add: weakCongTrans_def weakTrans_def)
  qed
qed

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

  shows "P ↝<Rel> P"
using assms
by(auto simp add: weakCongSimulation_def intro: transitionWeakCongTransition)
  
lemma 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 ↝<Rel> Q"
  and     "Q ↝<Rel'> R"
  and     "Rel O Rel'  Rel''"
  and     "S T. (S, T)  Rel  S ^<Rel> T"
  
  shows "P ↝<Rel''> R"
proof(induct rule: weakSimI)
  case(Sim α R')
  thus ?case using assms
    apply(drule_tac Q=R in weakSimE, auto)
    by(drule_tac Q=Q in simE2) auto
qed

lemma weakMonotonic:
  fixes P :: ccs
  and   A :: "(ccs × ccs) set"
  and   Q :: ccs
  and   B :: "(ccs × ccs) set"

  assumes "P ↝<A> Q"
  and     "A  B"

  shows "P ↝<B> Q"
using assms
by(fastforce simp add: weakCongSimulation_def)

end