Theory CSP_T_law_decompo

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

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

theory CSP_T_law_decompo = CSP_T_traces:

(*------------------------------------------------*
 |                                                |
 |      laws for monotonicity and congruence      |
 |                                                |
 *------------------------------------------------*)

(*********************************************************
                        Act_prefix mono
 *********************************************************)

(*------------------*
 |      csp law     |
 *------------------*)

lemma cspT_Act_prefix_mono:
  "[| a = b ; P <=T Q |] ==> a -> P <=T b -> Q"
apply (simp add: cspT_semantics)
apply (simp add: subdomT_iff)
apply (intro allI impI)
apply (simp add: in_traces)
apply (fast)
done

lemma cspT_Act_prefix_cong:
  "[| a = b ; P =T Q |] ==> a -> P =T b -> Q"
apply (simp add: cspT_eq_ref_iff)
apply (simp add: cspT_Act_prefix_mono)
done

(*********************************************************
                   Ext_pre_choice mono 
 *********************************************************)

(*------------------*
 |      csp law     |
 *------------------*)

lemma cspT_Ext_pre_choice_mono:
  "[| X = Y ; !! a. a:Y ==> Pf a <=T Qf a |] ==> ? :X -> Pf <=T ? :Y -> Qf"
apply (simp add: cspT_semantics)
apply (simp add: subdomT_iff)
apply (intro allI impI)
apply (simp add: in_traces)
apply (fast)
done

lemma cspT_Ext_pre_choice_cong:
  "[| X = Y ; !! a. a:Y ==> Pf a =T Qf a |] ==> ? :X -> Pf =T ? :Y -> Qf"
by (simp add: cspT_eq_ref_iff cspT_Ext_pre_choice_mono)

(*********************************************************
                      Ext choice mono
 *********************************************************)

(*------------------*
 |      csp law     |
 *------------------*)

lemma cspT_Ext_choice_mono:
  "[| P1 <=T Q1 ; P2 <=T Q2 |] ==> P1 [+] P2 <=T Q1 [+] Q2"
apply (simp add: cspT_semantics)
apply (simp add: subdomT_iff)
apply (intro allI impI)
apply (simp add: in_traces)
apply (fast)
done

lemma cspT_Ext_choice_cong:
  "[| P1 =T Q1 ; P2 =T Q2 |] ==> P1 [+] P2 =T Q1 [+] Q2"
by (simp add: cspT_eq_ref_iff cspT_Ext_choice_mono)

(*********************************************************
                      Int choice mono
 *********************************************************)

(*------------------*
 |      csp law     |
 *------------------*)

lemma cspT_Int_choice_mono:
  "[| P1 <=T Q1 ; P2 <=T Q2 |] ==> P1 |~| P2 <=T Q1 |~| Q2"
apply (simp add: cspT_semantics)
apply (simp add: subdomT_iff)
apply (intro allI impI)
apply (simp add: in_traces)
apply (fast)
done

lemma cspT_Int_choice_cong:
  "[| P1 =T Q1 ; P2 =T Q2 |] ==> P1 |~| P2 =T Q1 |~| Q2"
by (simp add: cspT_eq_ref_iff cspT_Int_choice_mono)

(*********************************************************
               replicated internal choice
 *********************************************************)

(*------------------*
 |      csp law     |
 *------------------*)

lemma cspT_Rep_int_choice_mono0:
  "[| C1 = C2 ; !! c. c:C2 ==> Pf c <=T Qf c |]
      ==> !! :C1 .. Pf <=T !! :C2 .. Qf"
apply (simp add: cspT_semantics)
apply (simp add: subdomT_iff)
apply (intro allI impI)
apply (simp add: in_traces)
apply (fast)
done

lemma cspT_Rep_int_choice_cong0:
  "[| C1 = C2 ; !! c. c:C2 ==> Pf c =T Qf c |]
      ==> !! :C1 .. Pf =T !! :C2 .. Qf"
by (simp add: cspT_eq_ref_iff cspT_Rep_int_choice_mono0)

(*** fun ***)

lemma cspT_Rep_int_choice_mono_fun:
  "[| inj f ; X = Y ; !! a. a:Y ==> Pf a <=T Qf a |]
      ==> !!<f> :X .. Pf <=T !!<f> :Y .. Qf"
apply (simp add: Rep_int_choice_fun_def)
apply (rule cspT_Rep_int_choice_mono0)
apply (auto)
done

lemma cspT_Rep_int_choice_cong_fun:
  "[| inj f ; X = Y ; !! a. a:Y ==> Pf a =T Qf a |]
      ==> !!<f> :X .. Pf =T !!<f> :Y .. Qf"
by (simp add: cspT_eq_ref_iff cspT_Rep_int_choice_mono_fun)

(*** set ***)

lemma cspT_Rep_int_choice_mono_set:
  "[| Xs = Ys ; !! X. X:Ys ==> Pf X <=T Qf X |]
      ==> !set :Xs .. Pf <=T !set :Ys .. Qf"
by (simp add: cspT_Rep_int_choice_mono_fun)

lemma cspT_Rep_int_choice_cong_set:
  "[| Xs = Ys ; !! X. X:Ys ==> Pf X =T Qf X |]
      ==> !set :Xs .. Pf =T !set :Ys .. Qf"
by (simp add: cspT_Rep_int_choice_cong_fun)

(*** nat ***)

lemma cspT_Rep_int_choice_mono_nat:
  "[| N1 = N2 ; !! n. n:N2 ==> Pf n <=T Qf n |]
      ==> !nat :N1 .. Pf <=T !nat :N2 .. Qf"
by (simp add: cspT_Rep_int_choice_mono_fun)

lemma cspT_Rep_int_choice_cong_nat:
  "[| N1 = N2 ; !! n. n:N2 ==> Pf n =T Qf n |]
      ==> !nat :N1 .. Pf =T !nat :N2 .. Qf"
by (simp add: cspT_Rep_int_choice_cong_fun)

(*** com ***)

lemma cspT_Rep_int_choice_mono_com:
  "[| X = Y ; !! a. a:Y ==> Pf a <=T Qf a |]
      ==> ! :X .. Pf <=T ! :Y .. Qf"
apply (simp add: Rep_int_choice_com_def)
apply (rule cspT_Rep_int_choice_mono_set)
by (auto)

lemma cspT_Rep_int_choice_cong_com:
  "[| X = Y ; !! a. a:Y ==> Pf a =T Qf a |]
      ==> ! :X .. Pf =T ! :Y .. Qf"
apply (simp add: Rep_int_choice_com_def)
apply (rule cspT_Rep_int_choice_cong_set)
by (auto)

(*** f ***)

lemma cspT_Rep_int_choice_mono_f:
  "[| inj f ; X = Y ; !! a. a:Y ==> Pf a <=T Qf a |]
      ==> !<f> :X .. Pf <=T !<f> :Y .. Qf"
apply (simp add: Rep_int_choice_f_def)
apply (rule cspT_Rep_int_choice_mono_com)
apply (auto)
done

lemma cspT_Rep_int_choice_cong_f:
  "[| inj f ; X = Y ; !! a. a:Y ==> Pf a =T Qf a |]
      ==> !<f> :X .. Pf =T !<f> :Y .. Qf"
apply (simp add: Rep_int_choice_f_def)
apply (rule cspT_Rep_int_choice_cong_com)
apply (auto)
done

lemmas cspT_Rep_int_choice_mono = cspT_Rep_int_choice_mono0
                                  cspT_Rep_int_choice_mono_com
                                  cspT_Rep_int_choice_mono_set
                                  cspT_Rep_int_choice_mono_nat
                                  cspT_Rep_int_choice_mono_f

lemmas cspT_Rep_int_choice_cong = cspT_Rep_int_choice_cong0
                                  cspT_Rep_int_choice_cong_com
                                  cspT_Rep_int_choice_cong_set
                                  cspT_Rep_int_choice_cong_nat
                                  cspT_Rep_int_choice_cong_f

(*********************************************************
                   IF mono
 *********************************************************)

(*------------------*
 |      csp law     |
 *------------------*)

lemma cspT_IF_mono:
  "[| b1 = b2 ; P1 <=T Q1 ; P2 <=T Q2 |]
           ==> IF b1 THEN P1 ELSE P2 <=T
               IF b2 THEN Q1 ELSE Q2"
apply (simp add: cspT_semantics)
apply (simp add: subdomT_iff)
apply (intro allI impI)
apply (simp add: in_traces)
done

lemma cspT_IF_cong:
  "[| b1 = b2 ; P1 =T Q1 ;  P2 =T Q2 |]
           ==> IF b1 THEN P1 ELSE P2 =T
               IF b2 THEN Q1 ELSE Q2"
by (simp add: cspT_eq_ref_iff cspT_IF_mono)

(*********************************************************
                     Parallel mono 
 *********************************************************)

(*------------------*
 |      csp law     |
 *------------------*)

lemma cspT_Parallel_mono:
  "[| X = Y ; P1 <=T Q1 ; P2 <=T Q2 |]
           ==> P1 |[X]| P2 <=T Q1 |[Y]| Q2"
apply (simp add: cspT_semantics)
apply (simp add: subdomT_iff)
apply (intro allI impI)
apply (simp add: in_traces)
apply (fast)
done

lemma cspT_Parallel_cong:
  "[| X = Y ; P1 =T Q1 ; P2 =T Q2 |]
           ==> P1 |[X]| P2 =T Q1 |[Y]| Q2"
by (simp add: cspT_eq_ref_iff cspT_Parallel_mono)

(*********************************************************
                        Hiding mono
 *********************************************************)

(*------------------*
 |      csp law     |
 *------------------*)

lemma cspT_Hiding_mono:
  "[| X = Y ; P <=T Q |] ==> P -- X <=T Q -- Y"
apply (simp add: cspT_semantics)
apply (simp add: subdomT_iff)
apply (intro allI impI)
apply (simp add: in_traces)
apply (fast)
done

lemma cspT_Hiding_cong:
  "[| X = Y ; P =T Q |] ==> P -- X =T Q -- Y"
by (simp add: cspT_eq_ref_iff cspT_Hiding_mono)

(*********************************************************
                        Renaming mono
 *********************************************************)

(*------------------*
 |      csp law     |
 *------------------*)

lemma cspT_Renaming_mono:
  "[| r1 = r2 ; P <=T Q |] ==> P [[r1]] <=T Q [[r2]]"
apply (simp add: cspT_semantics)
apply (simp add: subdomT_iff)
apply (intro allI impI)
apply (simp add: in_traces)
apply (fast)
done

lemma cspT_Renaming_cong:
  "[| r1 = r2 ; P =T Q |] ==> P [[r1]] =T Q [[r2]]"
by (simp add: cspT_eq_ref_iff cspT_Renaming_mono)

(*********************************************************
               Sequential composition mono 
 *********************************************************)

(*------------------*
 |      csp law     |
 *------------------*)

lemma cspT_Seq_compo_mono:
  "[| P1 <=T Q1 ; P2 <=T Q2 |]
           ==> P1 ;; P2 <=T Q1 ;; Q2"
apply (simp add: cspT_semantics)
apply (simp add: subdomT_iff)
apply (intro allI impI)
apply (simp add: in_traces)
apply (fast)
done

lemma cspT_Seq_compo_cong:
  "[| P1 =T Q1 ; P2 =T Q2 |] ==> P1 ;; P2 =T Q1 ;; Q2"
by (simp add: cspT_eq_ref_iff cspT_Seq_compo_mono)

(*********************************************************
                        Depth_rest mono
 *********************************************************)

(*------------------*
 |      csp law     |
 *------------------*)

lemma cspT_Depth_rest_mono:
  "[| n1 = n2 ; P <=T Q |] ==> P |. n1 <=T Q |. n2"
apply (simp add: cspT_semantics)
apply (simp add: subdomT_iff)
apply (intro allI impI)
apply (simp add: in_traces)
done

lemma cspT_Depth_rest_cong:
  "[| n1 = n2 ; P =T Q |] ==> P |. n1 =T Q |. n2"
by (simp add: cspT_eq_ref_iff cspT_Depth_rest_mono)

(*********************************************************
                        Timeout mono
 *********************************************************)

(*------------------*
 |      csp law     |
 *------------------*)

lemma cspT_Timeout_mono:
  "[| P1 <=T Q1 ; P2 <=T Q2 |]
      ==> P1 [> P2 <=T Q1 [> Q2"
apply (rule cspT_Ext_choice_mono)
apply (rule cspT_Int_choice_mono)
by (simp_all)

lemma cspT_Timeout_cong:
  "[| P1 =T Q1 ; P2 =T Q2 |]
      ==> P1 [> P2 =T Q1 [> Q2"
by (simp add: cspT_eq_ref_iff cspT_Timeout_mono)

(*------------------------------------------------------*
 |                       alias                          |
 *------------------------------------------------------*)

lemmas cspT_free_mono =
   cspT_Ext_choice_mono cspT_Int_choice_mono cspT_Parallel_mono
   cspT_Hiding_mono cspT_Renaming_mono cspT_Seq_compo_mono
   cspT_Depth_rest_mono

lemmas cspT_mono = cspT_free_mono
   cspT_Act_prefix_mono cspT_Ext_pre_choice_mono 
   cspT_Rep_int_choice_mono cspT_IF_mono 

lemmas cspT_free_cong =
   cspT_Ext_choice_cong cspT_Int_choice_cong cspT_Parallel_cong
   cspT_Hiding_cong cspT_Renaming_cong cspT_Seq_compo_cong
   cspT_Depth_rest_cong

lemmas cspT_cong = cspT_free_cong
   cspT_Act_prefix_cong cspT_Ext_pre_choice_cong 
   cspT_Rep_int_choice_cong cspT_IF_cong 

lemmas cspT_free_decompo = cspT_free_mono cspT_free_cong
lemmas cspT_decompo = cspT_mono cspT_cong

lemmas cspT_rm_head_mono = cspT_Act_prefix_mono cspT_Ext_pre_choice_mono 
lemmas cspT_rm_head_cong = cspT_Act_prefix_cong cspT_Ext_pre_choice_cong
lemmas cspT_rm_head      = cspT_rm_head_mono cspT_rm_head_cong

(*-------------------------------------------------------*
 |            decomposition with ALL and EX              |
 *-------------------------------------------------------*)

(*** Rep_int_choice ***)

lemma cspT_Rep_int_choice_decompo_ALL_EX_ref:
  "ALL c2:C2. EX c1:C1. Pf c1 <=T Qf c2
   ==> !! :C1 .. Pf <=T !! :C2 .. Qf"
apply (simp add: cspT_semantics)
apply (rule, simp add: in_traces)
apply (erule disjE)
apply (simp)
apply (elim bexE)
apply (drule_tac x="c" in bspec, simp)
apply (erule bexE)
apply (rule disjI2)
apply (rule_tac x="c1" in bexI)
apply (erule subdomTE)
apply (simp_all)
done

lemma cspT_Rep_int_choice_decompo_ALL_EX_eq:
  "[| ALL c1:C1. EX c2:C2. Pf c1 =T Qf c2 ;
      ALL c2:C2. EX c1:C1. Pf c1 =T Qf c2 |]
   ==> !! :C1 .. Pf =T !! :C2 .. Qf"
apply (simp add: cspT_eq_ref_iff)
apply (rule conjI)
apply (rule cspT_Rep_int_choice_decompo_ALL_EX_ref)
apply (force)
apply (rule cspT_Rep_int_choice_decompo_ALL_EX_ref)
apply (force)
done

lemmas cspT_Rep_int_choice_decompo_ALL_EX
     = cspT_Rep_int_choice_decompo_ALL_EX_ref
       cspT_Rep_int_choice_decompo_ALL_EX_eq

end

lemma cspT_Act_prefix_mono: 

  [| a = b; P <=T Q |] ==> a -> P <=T b -> Q

lemma cspT_Act_prefix_cong: 

  [| a = b; P =T Q |] ==> a -> P =T b -> Q

lemma cspT_Ext_pre_choice_mono: 

  [| X = Y; !!a. a ∈ Y ==> Pf a <=T Qf a |] ==> ? :X -> Pf <=T ? :Y -> Qf

lemma cspT_Ext_pre_choice_cong: 

  [| X = Y; !!a. a ∈ Y ==> Pf a =T Qf a |] ==> ? :X -> Pf =T ? :Y -> Qf

lemma cspT_Ext_choice_mono: 

  [| P1.0 <=T Q1.0; P2.0 <=T Q2.0 |] ==> P1.0 [+] P2.0 <=T Q1.0 [+] Q2.0

lemma cspT_Ext_choice_cong: 

  [| P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 [+] P2.0 =T Q1.0 [+] Q2.0

lemma cspT_Int_choice_mono: 

  [| P1.0 <=T Q1.0; P2.0 <=T Q2.0 |] ==> P1.0 |~| P2.0 <=T Q1.0 |~| Q2.0

lemma cspT_Int_choice_cong: 

  [| P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 |~| P2.0 =T Q1.0 |~| Q2.0

lemma cspT_Rep_int_choice_mono0: 

  [| C1.0 = C2.0; !!c. c ∈ C2.0 ==> Pf c <=T Qf c |]
  ==> !! :C1.0 .. Pf <=T !! :C2.0 .. Qf

lemma cspT_Rep_int_choice_cong0: 

  [| C1.0 = C2.0; !!c. c ∈ C2.0 ==> Pf c =T Qf c |]
  ==> !! :C1.0 .. Pf =T !! :C2.0 .. Qf

lemma cspT_Rep_int_choice_mono_fun: 

  [| inj f; X = Y; !!a. a ∈ Y ==> Pf a <=T Qf a |]
  ==> !!<f> :X .. Pf <=T !!<f> :Y .. Qf

lemma cspT_Rep_int_choice_cong_fun: 

  [| inj f; X = Y; !!a. a ∈ Y ==> Pf a =T Qf a |]
  ==> !!<f> :X .. Pf =T !!<f> :Y .. Qf

lemma cspT_Rep_int_choice_mono_set: 

  [| Xs = Ys; !!X. X ∈ Ys ==> Pf X <=T Qf X |]
  ==> !set :Xs .. Pf <=T !set :Ys .. Qf

lemma cspT_Rep_int_choice_cong_set: 

  [| Xs = Ys; !!X. X ∈ Ys ==> Pf X =T Qf X |] ==> !set :Xs .. Pf =T !set :Ys .. Qf

lemma cspT_Rep_int_choice_mono_nat: 

  [| N1.0 = N2.0; !!n. n ∈ N2.0 ==> Pf n <=T Qf n |]
  ==> !nat :N1.0 .. Pf <=T !nat :N2.0 .. Qf

lemma cspT_Rep_int_choice_cong_nat: 

  [| N1.0 = N2.0; !!n. n ∈ N2.0 ==> Pf n =T Qf n |]
  ==> !nat :N1.0 .. Pf =T !nat :N2.0 .. Qf

lemma cspT_Rep_int_choice_mono_com: 

  [| X = Y; !!a. a ∈ Y ==> Pf a <=T Qf a |] ==> ! :X .. Pf <=T ! :Y .. Qf

lemma cspT_Rep_int_choice_cong_com: 

  [| X = Y; !!a. a ∈ Y ==> Pf a =T Qf a |] ==> ! :X .. Pf =T ! :Y .. Qf

lemma cspT_Rep_int_choice_mono_f: 

  [| inj f; X = Y; !!a. a ∈ Y ==> Pf a <=T Qf a |]
  ==> !<f> :X .. Pf <=T !<f> :Y .. Qf

lemma cspT_Rep_int_choice_cong_f: 

  [| inj f; X = Y; !!a. a ∈ Y ==> Pf a =T Qf a |]
  ==> !<f> :X .. Pf =T !<f> :Y .. Qf

lemmas cspT_Rep_int_choice_mono: 

  [| C1.0 = C2.0; !!c. c ∈ C2.0 ==> Pf c <=T Qf c |]
  ==> !! :C1.0 .. Pf <=T !! :C2.0 .. Qf
  [| X = Y; !!a. a ∈ Y ==> Pf a <=T Qf a |] ==> ! :X .. Pf <=T ! :Y .. Qf
  [| Xs = Ys; !!X. X ∈ Ys ==> Pf X <=T Qf X |]
  ==> !set :Xs .. Pf <=T !set :Ys .. Qf
  [| N1.0 = N2.0; !!n. n ∈ N2.0 ==> Pf n <=T Qf n |]
  ==> !nat :N1.0 .. Pf <=T !nat :N2.0 .. Qf
  [| inj f; X = Y; !!a. a ∈ Y ==> Pf a <=T Qf a |]
  ==> !<f> :X .. Pf <=T !<f> :Y .. Qf

lemmas cspT_Rep_int_choice_mono: 

  [| C1.0 = C2.0; !!c. c ∈ C2.0 ==> Pf c <=T Qf c |]
  ==> !! :C1.0 .. Pf <=T !! :C2.0 .. Qf
  [| X = Y; !!a. a ∈ Y ==> Pf a <=T Qf a |] ==> ! :X .. Pf <=T ! :Y .. Qf
  [| Xs = Ys; !!X. X ∈ Ys ==> Pf X <=T Qf X |]
  ==> !set :Xs .. Pf <=T !set :Ys .. Qf
  [| N1.0 = N2.0; !!n. n ∈ N2.0 ==> Pf n <=T Qf n |]
  ==> !nat :N1.0 .. Pf <=T !nat :N2.0 .. Qf
  [| inj f; X = Y; !!a. a ∈ Y ==> Pf a <=T Qf a |]
  ==> !<f> :X .. Pf <=T !<f> :Y .. Qf

lemmas cspT_Rep_int_choice_cong: 

  [| C1.0 = C2.0; !!c. c ∈ C2.0 ==> Pf c =T Qf c |]
  ==> !! :C1.0 .. Pf =T !! :C2.0 .. Qf
  [| X = Y; !!a. a ∈ Y ==> Pf a =T Qf a |] ==> ! :X .. Pf =T ! :Y .. Qf
  [| Xs = Ys; !!X. X ∈ Ys ==> Pf X =T Qf X |] ==> !set :Xs .. Pf =T !set :Ys .. Qf
  [| N1.0 = N2.0; !!n. n ∈ N2.0 ==> Pf n =T Qf n |]
  ==> !nat :N1.0 .. Pf =T !nat :N2.0 .. Qf
  [| inj f; X = Y; !!a. a ∈ Y ==> Pf a =T Qf a |]
  ==> !<f> :X .. Pf =T !<f> :Y .. Qf

lemmas cspT_Rep_int_choice_cong: 

  [| C1.0 = C2.0; !!c. c ∈ C2.0 ==> Pf c =T Qf c |]
  ==> !! :C1.0 .. Pf =T !! :C2.0 .. Qf
  [| X = Y; !!a. a ∈ Y ==> Pf a =T Qf a |] ==> ! :X .. Pf =T ! :Y .. Qf
  [| Xs = Ys; !!X. X ∈ Ys ==> Pf X =T Qf X |] ==> !set :Xs .. Pf =T !set :Ys .. Qf
  [| N1.0 = N2.0; !!n. n ∈ N2.0 ==> Pf n =T Qf n |]
  ==> !nat :N1.0 .. Pf =T !nat :N2.0 .. Qf
  [| inj f; X = Y; !!a. a ∈ Y ==> Pf a =T Qf a |]
  ==> !<f> :X .. Pf =T !<f> :Y .. Qf

lemma cspT_IF_mono: 

  [| b1.0 = b2.0; P1.0 <=T Q1.0; P2.0 <=T Q2.0 |]
  ==> IF b1.0 THEN P1.0 ELSE P2.0 <=T IF b2.0 THEN Q1.0 ELSE Q2.0

lemma cspT_IF_cong: 

  [| b1.0 = b2.0; P1.0 =T Q1.0; P2.0 =T Q2.0 |]
  ==> IF b1.0 THEN P1.0 ELSE P2.0 =T IF b2.0 THEN Q1.0 ELSE Q2.0

lemma cspT_Parallel_mono: 

  [| X = Y; P1.0 <=T Q1.0; P2.0 <=T Q2.0 |]
  ==> P1.0 |[X]| P2.0 <=T Q1.0 |[Y]| Q2.0

lemma cspT_Parallel_cong: 

  [| X = Y; P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 |[X]| P2.0 =T Q1.0 |[Y]| Q2.0

lemma cspT_Hiding_mono: 

  [| X = Y; P <=T Q |] ==> P -- X <=T Q -- Y

lemma cspT_Hiding_cong: 

  [| X = Y; P =T Q |] ==> P -- X =T Q -- Y

lemma cspT_Renaming_mono: 

  [| r1.0 = r2.0; P <=T Q |] ==> P [[r1.0]] <=T Q [[r2.0]]

lemma cspT_Renaming_cong: 

  [| r1.0 = r2.0; P =T Q |] ==> P [[r1.0]] =T Q [[r2.0]]

lemma cspT_Seq_compo_mono: 

  [| P1.0 <=T Q1.0; P2.0 <=T Q2.0 |] ==> P1.0 ;; P2.0 <=T Q1.0 ;; Q2.0

lemma cspT_Seq_compo_cong: 

  [| P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 ;; P2.0 =T Q1.0 ;; Q2.0

lemma cspT_Depth_rest_mono: 

  [| n1.0 = n2.0; P <=T Q |] ==> P |. n1.0 <=T Q |. n2.0

lemma cspT_Depth_rest_cong: 

  [| n1.0 = n2.0; P =T Q |] ==> P |. n1.0 =T Q |. n2.0

lemma cspT_Timeout_mono: 

  [| P1.0 <=T Q1.0; P2.0 <=T Q2.0 |] ==> P1.0 [> P2.0 <=T Q1.0 [> Q2.0

lemma cspT_Timeout_cong: 

  [| P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 [> P2.0 =T Q1.0 [> Q2.0

lemmas cspT_free_mono: 

  [| P1.0 <=T Q1.0; P2.0 <=T Q2.0 |] ==> P1.0 [+] P2.0 <=T Q1.0 [+] Q2.0
  [| P1.0 <=T Q1.0; P2.0 <=T Q2.0 |] ==> P1.0 |~| P2.0 <=T Q1.0 |~| Q2.0
  [| X = Y; P1.0 <=T Q1.0; P2.0 <=T Q2.0 |]
  ==> P1.0 |[X]| P2.0 <=T Q1.0 |[Y]| Q2.0
  [| X = Y; P <=T Q |] ==> P -- X <=T Q -- Y
  [| r1.0 = r2.0; P <=T Q |] ==> P [[r1.0]] <=T Q [[r2.0]]
  [| P1.0 <=T Q1.0; P2.0 <=T Q2.0 |] ==> P1.0 ;; P2.0 <=T Q1.0 ;; Q2.0
  [| n1.0 = n2.0; P <=T Q |] ==> P |. n1.0 <=T Q |. n2.0

lemmas cspT_free_mono: 

  [| P1.0 <=T Q1.0; P2.0 <=T Q2.0 |] ==> P1.0 [+] P2.0 <=T Q1.0 [+] Q2.0
  [| P1.0 <=T Q1.0; P2.0 <=T Q2.0 |] ==> P1.0 |~| P2.0 <=T Q1.0 |~| Q2.0
  [| X = Y; P1.0 <=T Q1.0; P2.0 <=T Q2.0 |]
  ==> P1.0 |[X]| P2.0 <=T Q1.0 |[Y]| Q2.0
  [| X = Y; P <=T Q |] ==> P -- X <=T Q -- Y
  [| r1.0 = r2.0; P <=T Q |] ==> P [[r1.0]] <=T Q [[r2.0]]
  [| P1.0 <=T Q1.0; P2.0 <=T Q2.0 |] ==> P1.0 ;; P2.0 <=T Q1.0 ;; Q2.0
  [| n1.0 = n2.0; P <=T Q |] ==> P |. n1.0 <=T Q |. n2.0

lemmas cspT_mono: 

  [| P1.0 <=T Q1.0; P2.0 <=T Q2.0 |] ==> P1.0 [+] P2.0 <=T Q1.0 [+] Q2.0
  [| P1.0 <=T Q1.0; P2.0 <=T Q2.0 |] ==> P1.0 |~| P2.0 <=T Q1.0 |~| Q2.0
  [| X = Y; P1.0 <=T Q1.0; P2.0 <=T Q2.0 |]
  ==> P1.0 |[X]| P2.0 <=T Q1.0 |[Y]| Q2.0
  [| X = Y; P <=T Q |] ==> P -- X <=T Q -- Y
  [| r1.0 = r2.0; P <=T Q |] ==> P [[r1.0]] <=T Q [[r2.0]]
  [| P1.0 <=T Q1.0; P2.0 <=T Q2.0 |] ==> P1.0 ;; P2.0 <=T Q1.0 ;; Q2.0
  [| n1.0 = n2.0; P <=T Q |] ==> P |. n1.0 <=T Q |. n2.0
  [| a = b; P <=T Q |] ==> a -> P <=T b -> Q
  [| X = Y; !!a. a ∈ Y ==> Pf a <=T Qf a |] ==> ? :X -> Pf <=T ? :Y -> Qf
  [| C1.0 = C2.0; !!c. c ∈ C2.0 ==> Pf c <=T Qf c |]
  ==> !! :C1.0 .. Pf <=T !! :C2.0 .. Qf
  [| X = Y; !!a. a ∈ Y ==> Pf a <=T Qf a |] ==> ! :X .. Pf <=T ! :Y .. Qf
  [| Xs = Ys; !!X. X ∈ Ys ==> Pf X <=T Qf X |]
  ==> !set :Xs .. Pf <=T !set :Ys .. Qf
  [| N1.0 = N2.0; !!n. n ∈ N2.0 ==> Pf n <=T Qf n |]
  ==> !nat :N1.0 .. Pf <=T !nat :N2.0 .. Qf
  [| inj f; X = Y; !!a. a ∈ Y ==> Pf a <=T Qf a |]
  ==> !<f> :X .. Pf <=T !<f> :Y .. Qf
  [| b1.0 = b2.0; P1.0 <=T Q1.0; P2.0 <=T Q2.0 |]
  ==> IF b1.0 THEN P1.0 ELSE P2.0 <=T IF b2.0 THEN Q1.0 ELSE Q2.0

lemmas cspT_mono: 

  [| P1.0 <=T Q1.0; P2.0 <=T Q2.0 |] ==> P1.0 [+] P2.0 <=T Q1.0 [+] Q2.0
  [| P1.0 <=T Q1.0; P2.0 <=T Q2.0 |] ==> P1.0 |~| P2.0 <=T Q1.0 |~| Q2.0
  [| X = Y; P1.0 <=T Q1.0; P2.0 <=T Q2.0 |]
  ==> P1.0 |[X]| P2.0 <=T Q1.0 |[Y]| Q2.0
  [| X = Y; P <=T Q |] ==> P -- X <=T Q -- Y
  [| r1.0 = r2.0; P <=T Q |] ==> P [[r1.0]] <=T Q [[r2.0]]
  [| P1.0 <=T Q1.0; P2.0 <=T Q2.0 |] ==> P1.0 ;; P2.0 <=T Q1.0 ;; Q2.0
  [| n1.0 = n2.0; P <=T Q |] ==> P |. n1.0 <=T Q |. n2.0
  [| a = b; P <=T Q |] ==> a -> P <=T b -> Q
  [| X = Y; !!a. a ∈ Y ==> Pf a <=T Qf a |] ==> ? :X -> Pf <=T ? :Y -> Qf
  [| C1.0 = C2.0; !!c. c ∈ C2.0 ==> Pf c <=T Qf c |]
  ==> !! :C1.0 .. Pf <=T !! :C2.0 .. Qf
  [| X = Y; !!a. a ∈ Y ==> Pf a <=T Qf a |] ==> ! :X .. Pf <=T ! :Y .. Qf
  [| Xs = Ys; !!X. X ∈ Ys ==> Pf X <=T Qf X |]
  ==> !set :Xs .. Pf <=T !set :Ys .. Qf
  [| N1.0 = N2.0; !!n. n ∈ N2.0 ==> Pf n <=T Qf n |]
  ==> !nat :N1.0 .. Pf <=T !nat :N2.0 .. Qf
  [| inj f; X = Y; !!a. a ∈ Y ==> Pf a <=T Qf a |]
  ==> !<f> :X .. Pf <=T !<f> :Y .. Qf
  [| b1.0 = b2.0; P1.0 <=T Q1.0; P2.0 <=T Q2.0 |]
  ==> IF b1.0 THEN P1.0 ELSE P2.0 <=T IF b2.0 THEN Q1.0 ELSE Q2.0

lemmas cspT_free_cong: 

  [| P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 [+] P2.0 =T Q1.0 [+] Q2.0
  [| P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 |~| P2.0 =T Q1.0 |~| Q2.0
  [| X = Y; P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 |[X]| P2.0 =T Q1.0 |[Y]| Q2.0
  [| X = Y; P =T Q |] ==> P -- X =T Q -- Y
  [| r1.0 = r2.0; P =T Q |] ==> P [[r1.0]] =T Q [[r2.0]]
  [| P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 ;; P2.0 =T Q1.0 ;; Q2.0
  [| n1.0 = n2.0; P =T Q |] ==> P |. n1.0 =T Q |. n2.0

lemmas cspT_free_cong: 

  [| P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 [+] P2.0 =T Q1.0 [+] Q2.0
  [| P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 |~| P2.0 =T Q1.0 |~| Q2.0
  [| X = Y; P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 |[X]| P2.0 =T Q1.0 |[Y]| Q2.0
  [| X = Y; P =T Q |] ==> P -- X =T Q -- Y
  [| r1.0 = r2.0; P =T Q |] ==> P [[r1.0]] =T Q [[r2.0]]
  [| P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 ;; P2.0 =T Q1.0 ;; Q2.0
  [| n1.0 = n2.0; P =T Q |] ==> P |. n1.0 =T Q |. n2.0

lemmas cspT_cong: 

  [| P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 [+] P2.0 =T Q1.0 [+] Q2.0
  [| P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 |~| P2.0 =T Q1.0 |~| Q2.0
  [| X = Y; P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 |[X]| P2.0 =T Q1.0 |[Y]| Q2.0
  [| X = Y; P =T Q |] ==> P -- X =T Q -- Y
  [| r1.0 = r2.0; P =T Q |] ==> P [[r1.0]] =T Q [[r2.0]]
  [| P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 ;; P2.0 =T Q1.0 ;; Q2.0
  [| n1.0 = n2.0; P =T Q |] ==> P |. n1.0 =T Q |. n2.0
  [| a = b; P =T Q |] ==> a -> P =T b -> Q
  [| X = Y; !!a. a ∈ Y ==> Pf a =T Qf a |] ==> ? :X -> Pf =T ? :Y -> Qf
  [| C1.0 = C2.0; !!c. c ∈ C2.0 ==> Pf c =T Qf c |]
  ==> !! :C1.0 .. Pf =T !! :C2.0 .. Qf
  [| X = Y; !!a. a ∈ Y ==> Pf a =T Qf a |] ==> ! :X .. Pf =T ! :Y .. Qf
  [| Xs = Ys; !!X. X ∈ Ys ==> Pf X =T Qf X |] ==> !set :Xs .. Pf =T !set :Ys .. Qf
  [| N1.0 = N2.0; !!n. n ∈ N2.0 ==> Pf n =T Qf n |]
  ==> !nat :N1.0 .. Pf =T !nat :N2.0 .. Qf
  [| inj f; X = Y; !!a. a ∈ Y ==> Pf a =T Qf a |]
  ==> !<f> :X .. Pf =T !<f> :Y .. Qf
  [| b1.0 = b2.0; P1.0 =T Q1.0; P2.0 =T Q2.0 |]
  ==> IF b1.0 THEN P1.0 ELSE P2.0 =T IF b2.0 THEN Q1.0 ELSE Q2.0

lemmas cspT_cong: 

  [| P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 [+] P2.0 =T Q1.0 [+] Q2.0
  [| P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 |~| P2.0 =T Q1.0 |~| Q2.0
  [| X = Y; P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 |[X]| P2.0 =T Q1.0 |[Y]| Q2.0
  [| X = Y; P =T Q |] ==> P -- X =T Q -- Y
  [| r1.0 = r2.0; P =T Q |] ==> P [[r1.0]] =T Q [[r2.0]]
  [| P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 ;; P2.0 =T Q1.0 ;; Q2.0
  [| n1.0 = n2.0; P =T Q |] ==> P |. n1.0 =T Q |. n2.0
  [| a = b; P =T Q |] ==> a -> P =T b -> Q
  [| X = Y; !!a. a ∈ Y ==> Pf a =T Qf a |] ==> ? :X -> Pf =T ? :Y -> Qf
  [| C1.0 = C2.0; !!c. c ∈ C2.0 ==> Pf c =T Qf c |]
  ==> !! :C1.0 .. Pf =T !! :C2.0 .. Qf
  [| X = Y; !!a. a ∈ Y ==> Pf a =T Qf a |] ==> ! :X .. Pf =T ! :Y .. Qf
  [| Xs = Ys; !!X. X ∈ Ys ==> Pf X =T Qf X |] ==> !set :Xs .. Pf =T !set :Ys .. Qf
  [| N1.0 = N2.0; !!n. n ∈ N2.0 ==> Pf n =T Qf n |]
  ==> !nat :N1.0 .. Pf =T !nat :N2.0 .. Qf
  [| inj f; X = Y; !!a. a ∈ Y ==> Pf a =T Qf a |]
  ==> !<f> :X .. Pf =T !<f> :Y .. Qf
  [| b1.0 = b2.0; P1.0 =T Q1.0; P2.0 =T Q2.0 |]
  ==> IF b1.0 THEN P1.0 ELSE P2.0 =T IF b2.0 THEN Q1.0 ELSE Q2.0

lemmas cspT_free_decompo: 

  [| P1.0 <=T Q1.0; P2.0 <=T Q2.0 |] ==> P1.0 [+] P2.0 <=T Q1.0 [+] Q2.0
  [| P1.0 <=T Q1.0; P2.0 <=T Q2.0 |] ==> P1.0 |~| P2.0 <=T Q1.0 |~| Q2.0
  [| X = Y; P1.0 <=T Q1.0; P2.0 <=T Q2.0 |]
  ==> P1.0 |[X]| P2.0 <=T Q1.0 |[Y]| Q2.0
  [| X = Y; P <=T Q |] ==> P -- X <=T Q -- Y
  [| r1.0 = r2.0; P <=T Q |] ==> P [[r1.0]] <=T Q [[r2.0]]
  [| P1.0 <=T Q1.0; P2.0 <=T Q2.0 |] ==> P1.0 ;; P2.0 <=T Q1.0 ;; Q2.0
  [| n1.0 = n2.0; P <=T Q |] ==> P |. n1.0 <=T Q |. n2.0
  [| P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 [+] P2.0 =T Q1.0 [+] Q2.0
  [| P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 |~| P2.0 =T Q1.0 |~| Q2.0
  [| X = Y; P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 |[X]| P2.0 =T Q1.0 |[Y]| Q2.0
  [| X = Y; P =T Q |] ==> P -- X =T Q -- Y
  [| r1.0 = r2.0; P =T Q |] ==> P [[r1.0]] =T Q [[r2.0]]
  [| P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 ;; P2.0 =T Q1.0 ;; Q2.0
  [| n1.0 = n2.0; P =T Q |] ==> P |. n1.0 =T Q |. n2.0

lemmas cspT_free_decompo: 

  [| P1.0 <=T Q1.0; P2.0 <=T Q2.0 |] ==> P1.0 [+] P2.0 <=T Q1.0 [+] Q2.0
  [| P1.0 <=T Q1.0; P2.0 <=T Q2.0 |] ==> P1.0 |~| P2.0 <=T Q1.0 |~| Q2.0
  [| X = Y; P1.0 <=T Q1.0; P2.0 <=T Q2.0 |]
  ==> P1.0 |[X]| P2.0 <=T Q1.0 |[Y]| Q2.0
  [| X = Y; P <=T Q |] ==> P -- X <=T Q -- Y
  [| r1.0 = r2.0; P <=T Q |] ==> P [[r1.0]] <=T Q [[r2.0]]
  [| P1.0 <=T Q1.0; P2.0 <=T Q2.0 |] ==> P1.0 ;; P2.0 <=T Q1.0 ;; Q2.0
  [| n1.0 = n2.0; P <=T Q |] ==> P |. n1.0 <=T Q |. n2.0
  [| P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 [+] P2.0 =T Q1.0 [+] Q2.0
  [| P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 |~| P2.0 =T Q1.0 |~| Q2.0
  [| X = Y; P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 |[X]| P2.0 =T Q1.0 |[Y]| Q2.0
  [| X = Y; P =T Q |] ==> P -- X =T Q -- Y
  [| r1.0 = r2.0; P =T Q |] ==> P [[r1.0]] =T Q [[r2.0]]
  [| P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 ;; P2.0 =T Q1.0 ;; Q2.0
  [| n1.0 = n2.0; P =T Q |] ==> P |. n1.0 =T Q |. n2.0

lemmas cspT_decompo: 

  [| P1.0 <=T Q1.0; P2.0 <=T Q2.0 |] ==> P1.0 [+] P2.0 <=T Q1.0 [+] Q2.0
  [| P1.0 <=T Q1.0; P2.0 <=T Q2.0 |] ==> P1.0 |~| P2.0 <=T Q1.0 |~| Q2.0
  [| X = Y; P1.0 <=T Q1.0; P2.0 <=T Q2.0 |]
  ==> P1.0 |[X]| P2.0 <=T Q1.0 |[Y]| Q2.0
  [| X = Y; P <=T Q |] ==> P -- X <=T Q -- Y
  [| r1.0 = r2.0; P <=T Q |] ==> P [[r1.0]] <=T Q [[r2.0]]
  [| P1.0 <=T Q1.0; P2.0 <=T Q2.0 |] ==> P1.0 ;; P2.0 <=T Q1.0 ;; Q2.0
  [| n1.0 = n2.0; P <=T Q |] ==> P |. n1.0 <=T Q |. n2.0
  [| a = b; P <=T Q |] ==> a -> P <=T b -> Q
  [| X = Y; !!a. a ∈ Y ==> Pf a <=T Qf a |] ==> ? :X -> Pf <=T ? :Y -> Qf
  [| C1.0 = C2.0; !!c. c ∈ C2.0 ==> Pf c <=T Qf c |]
  ==> !! :C1.0 .. Pf <=T !! :C2.0 .. Qf
  [| X = Y; !!a. a ∈ Y ==> Pf a <=T Qf a |] ==> ! :X .. Pf <=T ! :Y .. Qf
  [| Xs = Ys; !!X. X ∈ Ys ==> Pf X <=T Qf X |]
  ==> !set :Xs .. Pf <=T !set :Ys .. Qf
  [| N1.0 = N2.0; !!n. n ∈ N2.0 ==> Pf n <=T Qf n |]
  ==> !nat :N1.0 .. Pf <=T !nat :N2.0 .. Qf
  [| inj f; X = Y; !!a. a ∈ Y ==> Pf a <=T Qf a |]
  ==> !<f> :X .. Pf <=T !<f> :Y .. Qf
  [| b1.0 = b2.0; P1.0 <=T Q1.0; P2.0 <=T Q2.0 |]
  ==> IF b1.0 THEN P1.0 ELSE P2.0 <=T IF b2.0 THEN Q1.0 ELSE Q2.0
  [| P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 [+] P2.0 =T Q1.0 [+] Q2.0
  [| P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 |~| P2.0 =T Q1.0 |~| Q2.0
  [| X = Y; P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 |[X]| P2.0 =T Q1.0 |[Y]| Q2.0
  [| X = Y; P =T Q |] ==> P -- X =T Q -- Y
  [| r1.0 = r2.0; P =T Q |] ==> P [[r1.0]] =T Q [[r2.0]]
  [| P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 ;; P2.0 =T Q1.0 ;; Q2.0
  [| n1.0 = n2.0; P =T Q |] ==> P |. n1.0 =T Q |. n2.0
  [| a = b; P =T Q |] ==> a -> P =T b -> Q
  [| X = Y; !!a. a ∈ Y ==> Pf a =T Qf a |] ==> ? :X -> Pf =T ? :Y -> Qf
  [| C1.0 = C2.0; !!c. c ∈ C2.0 ==> Pf c =T Qf c |]
  ==> !! :C1.0 .. Pf =T !! :C2.0 .. Qf
  [| X = Y; !!a. a ∈ Y ==> Pf a =T Qf a |] ==> ! :X .. Pf =T ! :Y .. Qf
  [| Xs = Ys; !!X. X ∈ Ys ==> Pf X =T Qf X |] ==> !set :Xs .. Pf =T !set :Ys .. Qf
  [| N1.0 = N2.0; !!n. n ∈ N2.0 ==> Pf n =T Qf n |]
  ==> !nat :N1.0 .. Pf =T !nat :N2.0 .. Qf
  [| inj f; X = Y; !!a. a ∈ Y ==> Pf a =T Qf a |]
  ==> !<f> :X .. Pf =T !<f> :Y .. Qf
  [| b1.0 = b2.0; P1.0 =T Q1.0; P2.0 =T Q2.0 |]
  ==> IF b1.0 THEN P1.0 ELSE P2.0 =T IF b2.0 THEN Q1.0 ELSE Q2.0

lemmas cspT_decompo: 

  [| P1.0 <=T Q1.0; P2.0 <=T Q2.0 |] ==> P1.0 [+] P2.0 <=T Q1.0 [+] Q2.0
  [| P1.0 <=T Q1.0; P2.0 <=T Q2.0 |] ==> P1.0 |~| P2.0 <=T Q1.0 |~| Q2.0
  [| X = Y; P1.0 <=T Q1.0; P2.0 <=T Q2.0 |]
  ==> P1.0 |[X]| P2.0 <=T Q1.0 |[Y]| Q2.0
  [| X = Y; P <=T Q |] ==> P -- X <=T Q -- Y
  [| r1.0 = r2.0; P <=T Q |] ==> P [[r1.0]] <=T Q [[r2.0]]
  [| P1.0 <=T Q1.0; P2.0 <=T Q2.0 |] ==> P1.0 ;; P2.0 <=T Q1.0 ;; Q2.0
  [| n1.0 = n2.0; P <=T Q |] ==> P |. n1.0 <=T Q |. n2.0
  [| a = b; P <=T Q |] ==> a -> P <=T b -> Q
  [| X = Y; !!a. a ∈ Y ==> Pf a <=T Qf a |] ==> ? :X -> Pf <=T ? :Y -> Qf
  [| C1.0 = C2.0; !!c. c ∈ C2.0 ==> Pf c <=T Qf c |]
  ==> !! :C1.0 .. Pf <=T !! :C2.0 .. Qf
  [| X = Y; !!a. a ∈ Y ==> Pf a <=T Qf a |] ==> ! :X .. Pf <=T ! :Y .. Qf
  [| Xs = Ys; !!X. X ∈ Ys ==> Pf X <=T Qf X |]
  ==> !set :Xs .. Pf <=T !set :Ys .. Qf
  [| N1.0 = N2.0; !!n. n ∈ N2.0 ==> Pf n <=T Qf n |]
  ==> !nat :N1.0 .. Pf <=T !nat :N2.0 .. Qf
  [| inj f; X = Y; !!a. a ∈ Y ==> Pf a <=T Qf a |]
  ==> !<f> :X .. Pf <=T !<f> :Y .. Qf
  [| b1.0 = b2.0; P1.0 <=T Q1.0; P2.0 <=T Q2.0 |]
  ==> IF b1.0 THEN P1.0 ELSE P2.0 <=T IF b2.0 THEN Q1.0 ELSE Q2.0
  [| P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 [+] P2.0 =T Q1.0 [+] Q2.0
  [| P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 |~| P2.0 =T Q1.0 |~| Q2.0
  [| X = Y; P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 |[X]| P2.0 =T Q1.0 |[Y]| Q2.0
  [| X = Y; P =T Q |] ==> P -- X =T Q -- Y
  [| r1.0 = r2.0; P =T Q |] ==> P [[r1.0]] =T Q [[r2.0]]
  [| P1.0 =T Q1.0; P2.0 =T Q2.0 |] ==> P1.0 ;; P2.0 =T Q1.0 ;; Q2.0
  [| n1.0 = n2.0; P =T Q |] ==> P |. n1.0 =T Q |. n2.0
  [| a = b; P =T Q |] ==> a -> P =T b -> Q
  [| X = Y; !!a. a ∈ Y ==> Pf a =T Qf a |] ==> ? :X -> Pf =T ? :Y -> Qf
  [| C1.0 = C2.0; !!c. c ∈ C2.0 ==> Pf c =T Qf c |]
  ==> !! :C1.0 .. Pf =T !! :C2.0 .. Qf
  [| X = Y; !!a. a ∈ Y ==> Pf a =T Qf a |] ==> ! :X .. Pf =T ! :Y .. Qf
  [| Xs = Ys; !!X. X ∈ Ys ==> Pf X =T Qf X |] ==> !set :Xs .. Pf =T !set :Ys .. Qf
  [| N1.0 = N2.0; !!n. n ∈ N2.0 ==> Pf n =T Qf n |]
  ==> !nat :N1.0 .. Pf =T !nat :N2.0 .. Qf
  [| inj f; X = Y; !!a. a ∈ Y ==> Pf a =T Qf a |]
  ==> !<f> :X .. Pf =T !<f> :Y .. Qf
  [| b1.0 = b2.0; P1.0 =T Q1.0; P2.0 =T Q2.0 |]
  ==> IF b1.0 THEN P1.0 ELSE P2.0 =T IF b2.0 THEN Q1.0 ELSE Q2.0

lemmas cspT_rm_head_mono: 

  [| a = b; P <=T Q |] ==> a -> P <=T b -> Q
  [| X = Y; !!a. a ∈ Y ==> Pf a <=T Qf a |] ==> ? :X -> Pf <=T ? :Y -> Qf

lemmas cspT_rm_head_mono: 

  [| a = b; P <=T Q |] ==> a -> P <=T b -> Q
  [| X = Y; !!a. a ∈ Y ==> Pf a <=T Qf a |] ==> ? :X -> Pf <=T ? :Y -> Qf

lemmas cspT_rm_head_cong: 

  [| a = b; P =T Q |] ==> a -> P =T b -> Q
  [| X = Y; !!a. a ∈ Y ==> Pf a =T Qf a |] ==> ? :X -> Pf =T ? :Y -> Qf

lemmas cspT_rm_head_cong: 

  [| a = b; P =T Q |] ==> a -> P =T b -> Q
  [| X = Y; !!a. a ∈ Y ==> Pf a =T Qf a |] ==> ? :X -> Pf =T ? :Y -> Qf

lemmas cspT_rm_head: 

  [| a = b; P <=T Q |] ==> a -> P <=T b -> Q
  [| X = Y; !!a. a ∈ Y ==> Pf a <=T Qf a |] ==> ? :X -> Pf <=T ? :Y -> Qf
  [| a = b; P =T Q |] ==> a -> P =T b -> Q
  [| X = Y; !!a. a ∈ Y ==> Pf a =T Qf a |] ==> ? :X -> Pf =T ? :Y -> Qf

lemmas cspT_rm_head: 

  [| a = b; P <=T Q |] ==> a -> P <=T b -> Q
  [| X = Y; !!a. a ∈ Y ==> Pf a <=T Qf a |] ==> ? :X -> Pf <=T ? :Y -> Qf
  [| a = b; P =T Q |] ==> a -> P =T b -> Q
  [| X = Y; !!a. a ∈ Y ==> Pf a =T Qf a |] ==> ? :X -> Pf =T ? :Y -> Qf

lemma cspT_Rep_int_choice_decompo_ALL_EX_ref: 

  ∀c2∈C2.0. ∃c1∈C1.0. Pf c1 <=T Qf c2 ==> !! :C1.0 .. Pf <=T !! :C2.0 .. Qf

lemma cspT_Rep_int_choice_decompo_ALL_EX_eq: 

  [| ∀c1∈C1.0. ∃c2∈C2.0. Pf c1 =T Qf c2; ∀c2∈C2.0. ∃c1∈C1.0. Pf c1 =T Qf c2 |]
  ==> !! :C1.0 .. Pf =T !! :C2.0 .. Qf

lemmas cspT_Rep_int_choice_decompo_ALL_EX: 

  ∀c2∈C2.0. ∃c1∈C1.0. Pf c1 <=T Qf c2 ==> !! :C1.0 .. Pf <=T !! :C2.0 .. Qf
  [| ∀c1∈C1.0. ∃c2∈C2.0. Pf c1 =T Qf c2; ∀c2∈C2.0. ∃c1∈C1.0. Pf c1 =T Qf c2 |]
  ==> !! :C1.0 .. Pf =T !! :C2.0 .. Qf

lemmas cspT_Rep_int_choice_decompo_ALL_EX: 

  ∀c2∈C2.0. ∃c1∈C1.0. Pf c1 <=T Qf c2 ==> !! :C1.0 .. Pf <=T !! :C2.0 .. Qf
  [| ∀c1∈C1.0. ∃c2∈C2.0. Pf c1 =T Qf c2; ∀c2∈C2.0. ∃c1∈C1.0. Pf c1 =T Qf c2 |]
  ==> !! :C1.0 .. Pf =T !! :C2.0 .. Qf