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theory CSP_T_law_decompo(*-------------------------------------------* | 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) | | March 2007 (modified) | | | | CSP-Prover on Isabelle2009 | | June 2009 (modified) | | | | Yoshinao Isobe (AIST JAPAN) | *-------------------------------------------*) theory CSP_T_law_decompo imports CSP_T_traces begin (*------------------------------------------------* | | | laws for monotonicity and congruence | | | *------------------------------------------------*) (********************************************************* Act_prefix mono *********************************************************) (*------------------* | csp law | *------------------*) lemma cspT_Act_prefix_mono: "[| a = b ; P <=T[M1,M2] Q |] ==> a -> P <=T[M1,M2] 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[M1,M2] Q |] ==> a -> P =T[M1,M2] 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[M1,M2] Qf a |] ==> ? :X -> Pf <=T[M1,M2] ? :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[M1,M2] Qf a |] ==> ? :X -> Pf =T[M1,M2] ? :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[M1,M2] Q1 ; P2 <=T[M1,M2] Q2 |] ==> P1 [+] P2 <=T[M1,M2] 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[M1,M2] Q1 ; P2 =T[M1,M2] Q2 |] ==> P1 [+] P2 =T[M1,M2] 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[M1,M2] Q1 ; P2 <=T[M1,M2] Q2 |] ==> P1 |~| P2 <=T[M1,M2] 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[M1,M2] Q1 ; P2 =T[M1,M2] Q2 |] ==> P1 |~| P2 =T[M1,M2] Q1 |~| Q2" by (simp add: cspT_eq_ref_iff cspT_Int_choice_mono) (********************************************************* replicated internal choice *********************************************************) (*------------------* | csp law | *------------------*) (****** mono ******) lemma cspT_Rep_int_choice_mono_sum: "[| C1 = C2 ; !! c. c: sumset C1 ==> Pf c <=T[M1,M2] Qf c |] ==> !! :C1 .. Pf <=T[M1,M2] !! :C2 .. Qf" apply (simp add: cspT_semantics) apply (simp add: subdomT_iff) apply (intro allI impI) apply (simp add: in_traces) apply (fast) done (* nat *) lemma cspT_Rep_int_choice_mono_nat: "[| N1 = N2 ; !! n. n:N1 ==> Pf n <=T[M1,M2] Qf n |] ==> !nat :N1 .. Pf <=T[M1,M2] !nat :N2 .. Qf" apply (simp add: Rep_int_choice_ss_def) apply (rule cspT_Rep_int_choice_mono_sum) apply (auto) done (* set *) lemma cspT_Rep_int_choice_mono_set: "[| Xs1 = Xs2 ; !! X. X:Xs1 ==> Pf X <=T[M1,M2] Qf X |] ==> !set :Xs1 .. Pf <=T[M1,M2] !set :Xs2 .. Qf" apply (simp add: Rep_int_choice_ss_def) apply (rule cspT_Rep_int_choice_mono_sum) apply (auto) done lemma cspT_Rep_int_choice_mono_com: "[| X1 = X2 ; !! x. x:X1 ==> Pf x <=T[M1,M2] Qf x |] ==> ! :X1 .. Pf <=T[M1,M2] ! :X2 .. Qf" apply (simp add: Rep_int_choice_com_def) apply (rule cspT_Rep_int_choice_mono_set) apply (auto) done lemma cspT_Rep_int_choice_mono_f: "[| inj f ; X1 = X2 ; !! x. x:X1 ==> Pf x <=T[M1,M2] Qf x |] ==> !<f> :X1 .. Pf <=T[M1,M2] !<f> :X2 .. Qf" apply (simp add: Rep_int_choice_f_def) apply (rule cspT_Rep_int_choice_mono_com) apply (auto) done lemmas cspT_Rep_int_choice_mono = cspT_Rep_int_choice_mono_sum cspT_Rep_int_choice_mono_set cspT_Rep_int_choice_mono_nat cspT_Rep_int_choice_mono_com cspT_Rep_int_choice_mono_f (****** cong ******) lemma cspT_Rep_int_choice_cong_sum: "[| C1 = C2 ; !! c. c: sumset C1 ==> Pf c =T[M1,M2] Qf c |] ==> !! :C1 .. Pf =T[M1,M2] !! :C2 .. Qf" by (simp add: cspT_eq_ref_iff cspT_Rep_int_choice_mono) lemma cspT_Rep_int_choice_cong_nat: "[| N1 = N2 ; !! n. n:N1 ==> Pf n =T[M1,M2] Qf n |] ==> !nat :N1 .. Pf =T[M1,M2] !nat :N2 .. Qf" by (simp add: cspT_eq_ref_iff cspT_Rep_int_choice_mono) lemma cspT_Rep_int_choice_cong_set: "[| Xs1 = Xs2 ; !! X. X:Xs1 ==> Pf X =T[M1,M2] Qf X |] ==> !set :Xs1 .. Pf =T[M1,M2] !set :Xs2 .. Qf" by (simp add: cspT_eq_ref_iff cspT_Rep_int_choice_mono) lemma cspT_Rep_int_choice_cong_com: "[| X1 = X2 ; !! x. x:X1 ==> Pf x =T[M1,M2] Qf x |] ==> ! :X1 .. Pf =T[M1,M2] ! :X2 .. Qf" by (simp add: cspT_eq_ref_iff cspT_Rep_int_choice_mono) lemma cspT_Rep_int_choice_cong_f: "[| inj f ; X1 = X2 ; !! x. x:X1 ==> Pf x =T[M1,M2] Qf x |] ==> !<f> :X1 .. Pf =T[M1,M2] !<f> :X2 .. Qf" by (simp add: cspT_eq_ref_iff cspT_Rep_int_choice_mono) lemmas cspT_Rep_int_choice_cong = cspT_Rep_int_choice_cong_sum cspT_Rep_int_choice_cong_set cspT_Rep_int_choice_cong_nat cspT_Rep_int_choice_cong_com cspT_Rep_int_choice_cong_f (********************************************************* IF mono *********************************************************) (*------------------* | csp law | *------------------*) lemma cspT_IF_mono: "[| b1 = b2 ; P1 <=T[M1,M2] Q1 ; P2 <=T[M1,M2] Q2 |] ==> IF b1 THEN P1 ELSE P2 <=T[M1,M2] 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[M1,M2] Q1 ; P2 =T[M1,M2] Q2 |] ==> IF b1 THEN P1 ELSE P2 =T[M1,M2] 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[M1,M2] Q1 ; P2 <=T[M1,M2] Q2 |] ==> P1 |[X]| P2 <=T[M1,M2] 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[M1,M2] Q1 ; P2 =T[M1,M2] Q2 |] ==> P1 |[X]| P2 =T[M1,M2] 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[M1,M2] Q |] ==> P -- X <=T[M1,M2] 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[M1,M2] Q |] ==> P -- X =T[M1,M2] Q -- Y" by (simp add: cspT_eq_ref_iff cspT_Hiding_mono) (********************************************************* Renaming mono *********************************************************) (*------------------* | csp law | *------------------*) lemma cspT_Renaming_mono: "[| r1 = r2 ; P <=T[M1,M2] Q |] ==> P [[r1]] <=T[M1,M2] 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[M1,M2] Q |] ==> P [[r1]] =T[M1,M2] Q [[r2]]" by (simp add: cspT_eq_ref_iff cspT_Renaming_mono) (********************************************************* Sequential composition mono *********************************************************) (*------------------* | csp law | *------------------*) lemma cspT_Seq_compo_mono: "[| P1 <=T[M1,M2] Q1 ; P2 <=T[M1,M2] Q2 |] ==> P1 ;; P2 <=T[M1,M2] 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[M1,M2] Q1 ; P2 =T[M1,M2] Q2 |] ==> P1 ;; P2 =T[M1,M2] 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[M1,M2] Q |] ==> P |. n1 <=T[M1,M2] 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[M1,M2] Q |] ==> P |. n1 =T[M1,M2] Q |. n2" by (simp add: cspT_eq_ref_iff cspT_Depth_rest_mono) (********************************************************* Timeout mono *********************************************************) (*------------------* | csp law | *------------------*) lemma cspT_Timeout_mono: "[| P1 <=T[M1,M2] Q1 ; P2 <=T[M1,M2] Q2 |] ==> P1 [> P2 <=T[M1,M2] Q1 [> Q2" apply (rule cspT_Ext_choice_mono) apply (rule cspT_Int_choice_mono) apply (simp_all) done lemma cspT_Timeout_cong: "[| P1 =T[M1,M2] Q1 ; P2 =T[M1,M2] Q2 |] ==> P1 [> P2 =T[M1,M2] 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_sum_decompo_ALL_EX_ref: "ALL c2: sumset C2. EX c1: sumset C1. (Pf c1) <=T[M1,M2] (Qf c2) ==> !! :C1 .. Pf <=T[M1,M2] !! :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_nat_decompo_ALL_EX_ref: "ALL n2:N2. EX n1:N1. (Pf n1) <=T[M1,M2] (Qf n2) ==> !nat :N1 .. Pf <=T[M1,M2] !nat :N2 .. Qf" apply (simp add: Rep_int_choice_ss_def) apply (rule cspT_Rep_int_choice_sum_decompo_ALL_EX_ref) apply (auto) apply (drule_tac x="a" in bspec, simp) apply (auto) done lemma cspT_Rep_int_choice_set_decompo_ALL_EX_ref: "ALL X2:Xs2. EX X1:Xs1. (Pf X1) <=T[M1,M2] (Qf X2) ==> !set :Xs1 .. Pf <=T[M1,M2] !set :Xs2 .. Qf" apply (simp add: Rep_int_choice_ss_def) apply (rule cspT_Rep_int_choice_sum_decompo_ALL_EX_ref) apply (auto) apply (drule_tac x="a" in bspec, simp) apply (auto) done lemmas cspT_Rep_int_choice_decompo_ALL_EX_ref = cspT_Rep_int_choice_sum_decompo_ALL_EX_ref cspT_Rep_int_choice_nat_decompo_ALL_EX_ref cspT_Rep_int_choice_set_decompo_ALL_EX_ref lemma cspT_Rep_int_choice_sum_decompo_ALL_EX_eq: "[| ALL c1: sumset C1. EX c2: sumset C2. (Pf c1) =T[M1,M2] (Qf c2) ; ALL c2: sumset C2. EX c1: sumset C1. (Pf c1) =T[M1,M2] (Qf c2) |] ==> !! :C1 .. Pf =T[M1,M2] !! :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 lemma cspT_Rep_int_choice_nat_decompo_ALL_EX_eq: "[| ALL n1:N1. EX n2:N2. (Pf n1) =T[M1,M2] (Qf n2) ; ALL n2:N2. EX n1:N1. (Pf n1) =T[M1,M2] (Qf n2) |] ==> !nat :N1 .. Pf =T[M1,M2] !nat :N2 .. 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 lemma cspT_Rep_int_choice_set_decompo_ALL_EX_eq: "[| ALL X1:Xs1. EX X2:Xs2. (Pf X1) =T[M1,M2] (Qf X2) ; ALL X2:Xs2. EX X1:Xs1. (Pf X1) =T[M1,M2] (Qf X2) |] ==> !set :Xs1 .. Pf =T[M1,M2] !set :Xs2 .. 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_eq = cspT_Rep_int_choice_sum_decompo_ALL_EX_eq cspT_Rep_int_choice_nat_decompo_ALL_EX_eq cspT_Rep_int_choice_set_decompo_ALL_EX_eq lemmas cspT_Rep_int_choice_decompo_ALL_EX = cspT_Rep_int_choice_decompo_ALL_EX_ref cspT_Rep_int_choice_decompo_ALL_EX_eq (* =================================================== * | addition for CSP-Prover 5 | * =================================================== *) (********************************************************* Act_prefix mono *********************************************************) (*------------------* | csp law | *------------------*) (** mono **) lemma cspT_Send_prefix_mono: "[| a = b ; P <=T[M1,M2] Q |] ==> a ! v -> P <=T[M1,M2] b ! v -> Q" apply (simp add: csp_prefix_ss_def) apply (simp add: cspT_decompo) done lemma cspT_Rec_prefix_mono: "[| inj a; a = b ; X = Y ; !! x. x:Y ==> Pf x <=T[M1,M2] Qf x |] ==> a ? x:X -> Pf x <=T[M1,M2] b ? x:Y -> Qf x" apply (simp add: csp_prefix_ss_def) apply (rule cspT_Ext_pre_choice_mono) apply (simp) apply (simp add: image_iff) apply (erule bexE) apply (simp) done lemma cspT_Int_pre_choice_mono: "[| X = Y ; !! x. x:Y ==> Pf x <=T[M1,M2] Qf x |] ==> ! :X -> Pf <=T[M1,M2] ! :Y -> Qf" apply (simp add: csp_prefix_ss_def) apply (simp add: cspT_decompo) done lemma cspT_Nondet_send_prefix_mono: "[| inj a; a = b ; X = Y ; !! x. x:Y ==> Pf x <=T[M1,M2] Qf x |] ==> a !? x:X -> Pf x <=T[M1,M2] b !? x:Y -> Qf x" apply (simp add: csp_prefix_ss_def) apply (rule cspT_mono) apply (simp) apply (rule cspT_mono) apply (simp) apply (simp add: image_iff) apply (erule bexE) apply (simp) done (** cong **) lemma cspT_Send_prefix_cong: "[| a = b ; P =T[M1,M2] Q |] ==> a ! v -> P =T[M1,M2] b ! v -> Q" apply (simp add: csp_prefix_ss_def) apply (simp add: cspT_decompo) done lemma cspT_Rec_prefix_cong: "[| inj a; a = b ; X = Y ; !! x. x:Y ==> Pf x =T[M1,M2] Qf x |] ==> a ? x:X -> Pf x =T[M1,M2] b ? x:Y -> Qf x" by (simp add: cspT_eq_ref_iff cspT_Rec_prefix_mono) lemma cspT_Int_pre_choice_cong: "[| X = Y ; !! x. x:Y ==> Pf x =T[M1,M2] Qf x |] ==> ! :X -> Pf =T[M1,M2] ! :Y -> Qf" apply (simp add: csp_prefix_ss_def) apply (simp add: cspT_decompo) done lemma cspT_Nondet_send_prefix_cong: "[| inj a; a = b ; X = Y ; !! x. x:Y ==> Pf x =T[M1,M2] Qf x |] ==> a !? x:X -> Pf x =T[M1,M2] b !? x:Y -> Qf x" by (simp add: cspT_eq_ref_iff cspT_Nondet_send_prefix_mono) lemmas cspT_prefix_ss_mono = cspT_Send_prefix_mono cspT_Rec_prefix_mono cspT_Int_pre_choice_mono cspT_Nondet_send_prefix_mono lemmas cspT_prefx_ss_cong = cspT_Send_prefix_cong cspT_Rec_prefix_cong cspT_Int_pre_choice_cong cspT_Nondet_send_prefix_cong (* ------------------------------------------------------ * decomposition with ss * ------------------------------------------------------ *) lemmas cspT_mono_ss = cspT_mono cspT_prefix_ss_mono lemmas cspT_cong_ss = cspT_cong cspT_prefx_ss_cong lemmas cspT_decompo_ss = cspT_mono_ss cspT_cong_ss (********************************************************* Rep_internal_choice for UNIV this is useful for tactic *********************************************************) (*------------------* | csp law | *------------------*) (* mono_UNIV *) lemma cspT_Rep_int_choice_mono_UNIV_nat: "[| !! n. Pf n <=T[M1,M2] Qf n |] ==> !nat n .. Pf n <=T[M1,M2] !nat n .. Qf n" by (simp add: cspT_Rep_int_choice_mono) lemma cspT_Rep_int_choice_mono_UNIV_set: "[| !! X. Pf X <=T[M1,M2] Qf X |] ==> !set X .. Pf X <=T[M1,M2] !set X .. Qf X" by (simp add: cspT_Rep_int_choice_mono) lemma cspT_Rep_int_choice_mono_UNIV_com: "[| !! x. Pf x <=T[M1,M2] Qf x |] ==> ! x .. Pf x <=T[M1,M2] ! x .. Qf x" by (simp add: cspT_Rep_int_choice_mono) lemma cspT_Rep_int_choice_mono_UNIV_f: "[| inj f ; !! x. Pf x <=T[M1,M2] Qf x |] ==> !<f> x .. Pf x <=T[M1,M2] !<f> x .. Qf x" by (simp add: cspT_Rep_int_choice_mono) lemmas cspT_Rep_int_choice_mono_UNIV = cspT_Rep_int_choice_mono_UNIV_nat cspT_Rep_int_choice_mono_UNIV_set cspT_Rep_int_choice_mono_UNIV_com cspT_Rep_int_choice_mono_UNIV_f (* cong *) lemma cspT_Rep_int_choice_cong_UNIV_nat: "[| !! n. Pf n =T[M1,M2] Qf n |] ==> !nat n .. Pf n =T[M1,M2] !nat n .. Qf n" by (simp add: cspT_Rep_int_choice_cong) lemma cspT_Rep_int_choice_cong_UNIV_set: "[| !! X. Pf X =T[M1,M2] Qf X |] ==> !set X .. Pf X =T[M1,M2] !set X .. Qf X" by (simp add: cspT_Rep_int_choice_cong) lemma cspT_Rep_int_choice_cong_UNIV_com: "[| !! x. Pf x =T[M1,M2] Qf x |] ==> ! x .. Pf x =T[M1,M2] ! x .. Qf x" by (simp add: cspT_Rep_int_choice_cong) lemma cspT_Rep_int_choice_cong_UNIV_f: "[| inj f ; !! x. Pf x =T[M1,M2] Qf x |] ==> !<f> x .. Pf x =T[M1,M2] !<f> x .. Qf x" by (simp add: cspT_Rep_int_choice_cong) lemmas cspT_Rep_int_choice_cong_UNIV = cspT_Rep_int_choice_cong_UNIV_nat cspT_Rep_int_choice_cong_UNIV_set cspT_Rep_int_choice_cong_UNIV_com cspT_Rep_int_choice_cong_UNIV_f end