Theory Weak_Sim

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

definition weakSimulation :: "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: weakSimulation_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: weakSimulation_def)

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

  assumes "Q τ Q'"
  and     "(P, Q)  Rel"
  and     Sim: "R S. (R, S)  Rel  R ^<Rel> S"

  obtains P' where "P τ P'" and "(P', Q')  Rel"
using Q τ Q' (P, Q)  Rel
proof(induct arbitrary: thesis rule: tauChainInduct)
  case Base
  from (P, Q)  Rel show ?case
    by(force intro: Base)
next
  case(Step Q'' Q')
  from (P, Q)  Rel obtain P'' where "P τ P''" and "(P'', Q'')  Rel"
    by(blast intro: Step)
  from (P'', Q'')  Rel have "P'' ^<Rel> Q''" by(rule Sim)
  then obtain P' where "P'' ^τ  P'" and "(P', Q')  Rel" using Q'' τ  Q' by(rule weakSimE)
  with P τ P'' show thesis
    by(force simp add: weakTrans_def weakCongTrans_def intro: Step)
qed

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

  assumes "(P, Q)  Rel"
  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"
  moreover from Q ^α  Q' have "P'. P ^α  P'  (P', Q')  Rel"
  proof(induct rule: weakTransCases)
    case Base
    from (P, Q)  Rel show ?case by force
  next
    case Step
    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 (P, Q)  Rel Sim obtain P''' where PChain: "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 weakSimE)
    from Q''Chain (P'', Q'')  Rel Sim obtain P' where P''Chain: "P'' τ P'" and "(P', Q')  Rel"
      by(rule simTauChain)
    from P'''Trans P''Chain Step show ?thesis
    proof(induct rule: weakTransCases)
      case Base
      from PChain P''' τ P' have "P ^τ  P'"
      proof(induct rule: tauChainInduct)
        case Base
        from P τ P' show ?case
        proof(induct rule: tauChainInduct)
          case Base
          show ?case by simp
        next
          case(Step P' P'')
          thus ?case by(fastforce simp add: weakTrans_def weakCongTrans_def)
        qed
      next
        case(Step P''' P'')
        thus ?case by(fastforce simp add: weakTrans_def weakCongTrans_def)
      qed
      with (P', Q')  Rel show ?case by blast
    next
      case Step
      thus ?case using (P', Q')  Rel PChain
        by(rule_tac x=P' in exI) (force simp add: weakTrans_def weakCongTrans_def)
    qed
  qed
  ultimately show ?thesis
    by blast
qed

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

  shows "P ^<Rel> P"
using assms
by(auto simp add: weakSimulation_def intro: transitionWeakCongTransition weakCongTransitionWeakTransition)
  
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, Q)  Rel"
  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: weakSimulation_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 transitionWeakTransition)

end