Theory CSP_T_law_SKIP

Up to index of Isabelle/HOL/HOL-Complex/CSP/CSP_T

theory CSP_T_law_SKIP
imports CSP_T_law_basic
begin
           (*-------------------------------------------*
            |        CSP-Prover on Isabelle2004         |
            |               December 2004               |
            |                   July 2005  (modified)   |
            |              September 2005  (modified)   |
            |                                           |
            |        CSP-Prover on Isabelle2005         |
            |                October 2005  (modified)   |
            |                  April 2006  (modified)   |
            |                                           |
            |        Yoshinao Isobe (AIST JAPAN)        |
            *-------------------------------------------*)

theory CSP_T_law_SKIP = CSP_T_law_basic:

(*****************************************************************

         1. SKIP |[X]| SKIP
         2. SKIP |[X]| P
         3. P |[X]| SKIP
         4. SKIP -- X
         5. SKIP [[r]]
         6. SKIP ;; P
         7. P ;; SKIP
         8. SKIP |. n

 *****************************************************************)

(*********************************************************
                    SKIP |[X]| SKIP
 *********************************************************)

lemma cspT_Parallel_term:
   "SKIP |[X]| SKIP =T SKIP"
apply (simp add: cspT_semantics)
apply (rule order_antisym)

(* => *)
 apply (rule)
 apply (simp add: in_traces)
 apply (elim disjE conjE exE)
 apply (simp_all)

(* <= *)
 apply (rule)
 apply (simp add: in_traces)
 apply (elim disjE conjE exE)
 apply (simp_all)
done

(*********************************************************
                      SKIP |[X]| P
 *********************************************************)

lemma cspT_Parallel_preterm_l: 
   "SKIP |[X]| (? :Y -> Qf) =T ? x:(Y-X) -> (SKIP |[X]| Qf x)"
apply (simp add: cspT_semantics)
apply (rule order_antisym)

(* => *)
 apply (rule)
 apply (simp add: in_traces)
 apply (insert trace_nil_or_Tick_or_Ev)
 apply (elim disjE conjE exE)
  apply (simp_all)

  apply (drule_tac x="t" in spec)
  apply (erule disjE, simp)
  apply (erule disjE, simp)
  apply (elim conjE exE, simp)
  apply (simp add: par_tr_head)
  apply (rule_tac x="<>" in exI)
  apply (rule_tac x="sa" in exI, simp)

  apply (drule_tac x="t" in spec)
  apply (erule disjE, simp)
  apply (erule disjE, simp)
  apply (elim conjE exE, simp)
  apply (simp add: par_tr_head)
  apply (rule_tac x="<Tick>" in exI)
  apply (rule_tac x="sa" in exI, simp)

(* <= *)

 apply (rule)
 apply (simp add: in_traces)
 apply (elim disjE conjE exE)
  apply (simp_all)

  apply (rule_tac x="<>" in exI)
  apply (rule_tac x="<Ev a> ^^ ta" in exI, simp)
  apply (simp add: par_tr_head)

  apply (rule_tac x="<Tick>" in exI)
  apply (rule_tac x="<Ev a> ^^ ta" in exI, simp)
  apply (simp add: par_tr_head)
done

(*********************************************************
                      P |[X]| SKIP
 *********************************************************)

lemma cspT_Parallel_preterm_r: 
   "(? :Y -> Pf) |[X]| SKIP
     =T ? x:(Y-X) -> (Pf x |[X]| SKIP)"
apply (rule cspT_trans)
apply (rule cspT_Parallel_commut)
apply (rule cspT_trans)
apply (rule cspT_Parallel_preterm_l)
apply (rule cspT_rm_head, simp)
apply (rule cspT_Parallel_commut)
done

lemmas cspT_Parallel_preterm = cspT_Parallel_preterm_l cspT_Parallel_preterm_r

(*********************************************************
                      SKIP and Parallel
 *********************************************************)

(* p.288 *)

lemma cspT_SKIP_Parallel_Ext_choice_SKIP_l:
  "((? :Y -> Pf) [+] SKIP) |[X]| SKIP =T 
   (? x:(Y - X) -> (Pf x |[X]| SKIP)) [+] SKIP"
apply (simp add: cspT_semantics)
apply (rule order_antisym)

(* => *)
 apply (rule, simp add: in_traces)
 apply (elim conjE exE disjE)
 apply (simp_all)

  apply (rule disjI2)
  apply (rule disjI1)
  apply (simp add: par_tr_nil_right)
  apply (elim conjE)
  apply (simp add: image_iff)
  apply (rule_tac x="sa" in exI)
  apply (rule_tac x="<>" in exI)
  apply (simp add: par_tr_nil_right)

  apply (rule disjI2)
  apply (rule disjI1)
  apply (simp add: par_tr_Tick_right)
  apply (elim conjE)
  apply (simp add: image_iff)
  apply (rule_tac x="sa" in exI)
  apply (rule_tac x="<Tick>" in exI)
  apply (simp add: par_tr_Tick_right)

(* <= *)
 apply (rule, simp add: in_traces)
 apply (elim conjE exE disjE)
 apply (simp_all)

  apply (simp add: par_tr_nil_right)
  apply (elim conjE)
  apply (rule_tac x="<Ev a> ^^ sa" in exI)
  apply (rule_tac x="<>" in exI)
  apply (simp add: par_tr_nil_right)
  apply (simp add: image_iff)

  apply (simp add: par_tr_Tick_right)
  apply (elim conjE)
  apply (rule_tac x="<Ev a> ^^ sa" in exI)
  apply (rule_tac x="<Tick>" in exI)
  apply (simp add: par_tr_Tick_right)
  apply (simp add: image_iff)
done

lemma cspT_SKIP_Parallel_Ext_choice_SKIP_r:
  "SKIP |[X]| ((? :Y -> Pf) [+] SKIP)  =T 
   (? x:(Y - X) -> (SKIP |[X]| Pf x)) [+] SKIP"
apply (rule cspT_rw_left)
apply (rule cspT_commut)
apply (rule cspT_rw_left)
apply (rule cspT_SKIP_Parallel_Ext_choice_SKIP_l)
apply (rule cspT_rw_left)
apply (rule cspT_decompo)
apply (rule cspT_decompo)
apply (simp)
apply (rule cspT_commut)
apply (rule cspT_reflex)
apply (rule cspT_reflex)
done

lemmas cspT_SKIP_Parallel_Ext_choice_SKIP =
       cspT_SKIP_Parallel_Ext_choice_SKIP_l
       cspT_SKIP_Parallel_Ext_choice_SKIP_r

(*********************************************************
                      SKIP -- X
 *********************************************************)

lemma cspT_SKIP_Hiding_Id: 
   "SKIP -- X =T SKIP"
apply (simp add: cspT_semantics)
apply (rule order_antisym)

(* => *)
 apply (rule)
 apply (simp add: in_traces)
 apply (elim disjE conjE exE)
 apply (simp_all)

(* <= *)
 apply (rule)
 apply (simp add: in_traces)
 apply (elim disjE conjE exE)
 apply (simp_all)
 apply (rule_tac x="<>" in exI)
 apply (simp)
 apply (rule_tac x="<Tick>" in exI)
 apply (simp)
done

(*********************************************************
                      SKIP and Hiding
 *********************************************************)

(*           p.288 version 

  "((? :Y -> Pf) [+] SKIP) -- X =T 
       IF (Y Int X = {}) THEN ((? x:Y -> (Pf x -- X)) [+] SKIP)
                         ELSE (((? x:(Y-X) -> (Pf x -- X)) [+] SKIP)
                               |~| (! x:(Y Int X) .. (Pf x -- X)))"
*)

lemma cspT_SKIP_Hiding_step:
  "((? :Y -> Pf) [+] SKIP) -- X =T 
   ((? x:(Y-X) -> (Pf x -- X)) [+] SKIP) |~| (! x:(Y Int X) .. (Pf x -- X))"
apply (simp add: cspT_semantics)
apply (rule order_antisym)

(* => *)
 apply (rule, simp add: in_traces)
 apply (elim conjE exE disjE)
 apply (simp_all)

  apply (case_tac "a : X", force)
  apply (force)

(* <= *)
 apply (rule)

  apply (simp add: in_traces)
  apply (elim conjE exE bexE disjE)
  apply (simp_all)

   apply (force)

   apply (rule_tac x="<Ev a> ^^ sa" in exI)
   apply (simp)

   apply (force)

   apply (rule_tac x="<Tick>" in exI)
   apply (simp)

   apply (force)

   apply (rule_tac x="<Ev a> ^^ s" in exI)
   apply (simp)
done

(*********************************************************
                      SKIP [[r]]
 *********************************************************)

lemma cspT_SKIP_Renaming_Id: 
   "SKIP [[r]] =T SKIP"
apply (simp add: cspT_semantics)
apply (rule order_antisym)

(* => *)
 apply (rule)
 apply (simp add: in_traces)
 apply (force)

(* <= *)
 apply (rule)
 apply (simp add: in_traces)
 apply (force)
done

(*********************************************************
                       SKIP ;; P
 *********************************************************)

lemma cspT_Seq_compo_unit_l: "SKIP ;; P =T P"
apply (simp add: cspT_semantics)
apply (rule order_antisym)

(* => *)
 apply (rule, simp add: in_traces)
 apply (force)

(* <= *)
 apply (rule, simp add: in_traces)
 apply (rule disjI2)
 apply (rule_tac x="<>" in exI)
 apply (rule_tac x="t" in exI)
 apply (simp)
done

(*********************************************************
                       P ;; SKIP
 *********************************************************)

lemma cspT_Seq_compo_unit_r: "P ;; SKIP =T P"
apply (simp add: cspT_semantics)
apply (rule order_antisym)

(* => *)
 apply (rule, simp add: in_traces)
 apply (elim conjE exE disjE)
 apply (simp_all)
 apply (rule memT_prefix_closed, simp)
 apply (simp add: rmTick_prefix_rev)
 apply (rule memT_prefix_closed, simp, simp)

(* <= *)
 apply (rule, simp add: in_traces)
 apply (insert trace_last_noTick_or_Tick)
 apply (drule_tac x="t" in spec)
 apply (erule disjE)
  apply (rule disjI1)
  apply (rule_tac x="t" in exI, simp)
  (* *)
  apply (rule disjI2)
  apply (elim conjE exE)
  apply (rule_tac x="s" in exI)
  apply (rule_tac x="<Tick>" in exI)
  apply (simp)
done

lemmas cspT_Seq_compo_unit = cspT_Seq_compo_unit_l cspT_Seq_compo_unit_r

(*********************************************************
               SKIP and Sequential composition
 *********************************************************)

(* p.141 *)

lemma cspT_SKIP_Seq_compo_step:
  "((? :X -> Pf) [> SKIP) ;; Q =T (? x:X -> (Pf x ;; Q)) [> Q"
apply (simp add: cspT_semantics)
apply (rule order_antisym)

(* => *)
 apply (rule, simp add: in_traces)

 apply (elim conjE exE disjE)
 apply (simp_all)

  apply (rule disjI1)
  apply (fast)

  apply (rule disjI2)
  apply (rule disjI1)
  apply (insert trace_nil_or_Tick_or_Ev)
  apply (drule_tac x="s" in spec)

  apply (elim disjE conjE exE)
  apply (simp_all)

  apply (simp add: appt_assoc)
  apply (rule disjI2)

  apply (rule_tac x="sb" in exI)
  apply (rule_tac x="ta" in exI)
  apply (simp)

(* <= *)
 apply (rule, simp add: in_traces)

 apply (elim conjE exE disjE)
 apply (simp_all)

 apply (rule disjI1)
 apply (rule_tac x="<>" in exI)
 apply (simp)

 apply (rule disjI1)
 apply (rule_tac x="<Ev a> ^^ sa" in exI)
 apply (simp)

 apply (rule disjI2)
 apply (rule_tac x="<Ev a> ^^ sa" in exI)
 apply (rule_tac x="ta" in exI)
 apply (simp add: appt_assoc)

 apply (rule disjI1)
 apply (rule_tac x="<>" in exI)
 apply (simp)

 apply (rule disjI2)
 apply (rule_tac x="<>" in exI)
 apply (rule_tac x="t" in exI)
 apply (simp)
done

(*********************************************************
                      SKIP |. n
 *********************************************************)

lemma cspT_SKIP_Depth_rest: 
   "SKIP |. (Suc n) =T SKIP"
apply (simp add: cspT_semantics)
apply (rule order_antisym)

(* => *)
 apply (rule)
 apply (simp add: in_traces)

(* <= *)
 apply (rule)
 apply (simp add: in_traces)
 apply (force)
done

(*********************************************************
                      cspT_SKIP
 *********************************************************)

lemmas cspT_SKIP =
       cspT_Parallel_term
       cspT_Parallel_preterm
       cspT_SKIP_Parallel_Ext_choice_SKIP
       cspT_SKIP_Hiding_Id
       cspT_SKIP_Hiding_step
       cspT_SKIP_Renaming_Id
       cspT_Seq_compo_unit
       cspT_SKIP_Seq_compo_step
       cspT_SKIP_Depth_rest

(*********************************************************
                       P [+] SKIP
 *********************************************************)

(* p.141 *)

lemma cspT_Ext_choice_SKIP_resolve: "P [+] SKIP =T P [> SKIP"
apply (simp add: cspT_semantics)
apply (rule order_antisym)

(* => *)
 apply (rule, simp add: in_traces)

(* <= *)
 apply (rule, simp add: in_traces)
done

lemma cspT_Ext_choice_SKIP_resolve_sym: "P [> SKIP =T P [+] SKIP"
apply (rule cspT_sym)
apply (simp add: cspT_Ext_choice_SKIP_resolve)
done

end

lemma cspT_Parallel_term: 

  SKIP |[X]| SKIP =T SKIP

lemma cspT_Parallel_preterm_l: 

  SKIP |[X]| ? :Y -> Qf =T ? x:(Y - X) -> (SKIP |[X]| Qf x)

lemma cspT_Parallel_preterm_r: 

  ? :Y -> Pf |[X]| SKIP =T ? x:(Y - X) -> (Pf x |[X]| SKIP)

lemmas cspT_Parallel_preterm: 

  SKIP |[X]| ? :Y -> Qf =T ? x:(Y - X) -> (SKIP |[X]| Qf x)
  ? :Y -> Pf |[X]| SKIP =T ? x:(Y - X) -> (Pf x |[X]| SKIP)

lemmas cspT_Parallel_preterm: 

  SKIP |[X]| ? :Y -> Qf =T ? x:(Y - X) -> (SKIP |[X]| Qf x)
  ? :Y -> Pf |[X]| SKIP =T ? x:(Y - X) -> (Pf x |[X]| SKIP)

lemma cspT_SKIP_Parallel_Ext_choice_SKIP_l: 

  (? :Y -> Pf [+] SKIP) |[X]| SKIP =T ? x:(Y - X) -> (Pf x |[X]| SKIP) [+] SKIP

lemma cspT_SKIP_Parallel_Ext_choice_SKIP_r: 

  SKIP |[X]| (? :Y -> Pf [+] SKIP) =T ? x:(Y - X) -> (SKIP |[X]| Pf x) [+] SKIP

lemmas cspT_SKIP_Parallel_Ext_choice_SKIP: 

  (? :Y -> Pf [+] SKIP) |[X]| SKIP =T ? x:(Y - X) -> (Pf x |[X]| SKIP) [+] SKIP
  SKIP |[X]| (? :Y -> Pf [+] SKIP) =T ? x:(Y - X) -> (SKIP |[X]| Pf x) [+] SKIP

lemmas cspT_SKIP_Parallel_Ext_choice_SKIP: 

  (? :Y -> Pf [+] SKIP) |[X]| SKIP =T ? x:(Y - X) -> (Pf x |[X]| SKIP) [+] SKIP
  SKIP |[X]| (? :Y -> Pf [+] SKIP) =T ? x:(Y - X) -> (SKIP |[X]| Pf x) [+] SKIP

lemma cspT_SKIP_Hiding_Id: 

  SKIP -- X =T SKIP

lemma cspT_SKIP_Hiding_step: 

  (? :Y -> Pf [+] SKIP) -- X =T 
  ? x:(Y - X) -> Pf x -- X [+] SKIP |~| ! x:(Y ∩ X) .. Pf x -- X

lemma cspT_SKIP_Renaming_Id: 

  SKIP [[r]] =T SKIP

lemma cspT_Seq_compo_unit_l: 

  SKIP ;; P =T P

lemma cspT_Seq_compo_unit_r: 

  P ;; SKIP =T P

lemmas cspT_Seq_compo_unit: 

  SKIP ;; P =T P
  P ;; SKIP =T P

lemmas cspT_Seq_compo_unit: 

  SKIP ;; P =T P
  P ;; SKIP =T P

lemma cspT_SKIP_Seq_compo_step: 

  (? :X -> Pf [> SKIP) ;; Q =T ? x:X -> (Pf x ;; Q) [> Q

lemma cspT_SKIP_Depth_rest: 

  SKIP |. Suc n =T SKIP

lemmas cspT_SKIP: 

  SKIP |[X]| SKIP =T SKIP
  SKIP |[X]| ? :Y -> Qf =T ? x:(Y - X) -> (SKIP |[X]| Qf x)
  ? :Y -> Pf |[X]| SKIP =T ? x:(Y - X) -> (Pf x |[X]| SKIP)
  (? :Y -> Pf [+] SKIP) |[X]| SKIP =T ? x:(Y - X) -> (Pf x |[X]| SKIP) [+] SKIP
  SKIP |[X]| (? :Y -> Pf [+] SKIP) =T ? x:(Y - X) -> (SKIP |[X]| Pf x) [+] SKIP
  SKIP -- X =T SKIP
  (? :Y -> Pf [+] SKIP) -- X =T 
  ? x:(Y - X) -> Pf x -- X [+] SKIP |~| ! x:(Y ∩ X) .. Pf x -- X
  SKIP [[r]] =T SKIP
  SKIP ;; P =T P
  P ;; SKIP =T P
  (? :X -> Pf [> SKIP) ;; Q =T ? x:X -> (Pf x ;; Q) [> Q
  SKIP |. Suc n =T SKIP

lemmas cspT_SKIP: 

  SKIP |[X]| SKIP =T SKIP
  SKIP |[X]| ? :Y -> Qf =T ? x:(Y - X) -> (SKIP |[X]| Qf x)
  ? :Y -> Pf |[X]| SKIP =T ? x:(Y - X) -> (Pf x |[X]| SKIP)
  (? :Y -> Pf [+] SKIP) |[X]| SKIP =T ? x:(Y - X) -> (Pf x |[X]| SKIP) [+] SKIP
  SKIP |[X]| (? :Y -> Pf [+] SKIP) =T ? x:(Y - X) -> (SKIP |[X]| Pf x) [+] SKIP
  SKIP -- X =T SKIP
  (? :Y -> Pf [+] SKIP) -- X =T 
  ? x:(Y - X) -> Pf x -- X [+] SKIP |~| ! x:(Y ∩ X) .. Pf x -- X
  SKIP [[r]] =T SKIP
  SKIP ;; P =T P
  P ;; SKIP =T P
  (? :X -> Pf [> SKIP) ;; Q =T ? x:X -> (Pf x ;; Q) [> Q
  SKIP |. Suc n =T SKIP

lemma cspT_Ext_choice_SKIP_resolve: 

  P [+] SKIP =T P [> SKIP

lemma cspT_Ext_choice_SKIP_resolve_sym: 

  P [> SKIP =T P [+] SKIP