TPTP Problem File: ITP003_2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP003_2 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory export of thm_2Earithmetic_2EMOD__2.p, bushy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : thm_2Earithmetic_2EMOD__2.p [Gau19]
% : HL401001_2.p [TPAP]
% Status : Theorem
% Rating : 0.80 v9.0.0, 0.89 v8.2.0, 1.00 v7.5.0
% Syntax : Number of formulae : 174 ( 54 unt; 56 typ; 0 def)
% Number of atoms : 641 ( 45 equ)
% Maximal formula atoms : 15 ( 3 avg)
% Number of connectives : 336 ( 60 ~; 53 |; 61 &)
% ( 72 <=>; 90 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 4 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of FOOLs : 247 ( 247 fml; 0 var)
% Number of types : 5 ( 3 usr)
% Number of type conns : 39 ( 28 >; 11 *; 0 +; 0 <<)
% Number of predicates : 8 ( 5 usr; 3 prp; 0-2 aty)
% Number of functors : 51 ( 51 usr; 25 con; 0-2 aty)
% Number of variables : 200 ( 182 !; 18 ?; 200 :)
% SPC : TF0_THM_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001_2.ax').
%------------------------------------------------------------------------------
tff(tp_ty_2Enum_2Enum,type,
ty_2Enum_2Enum: del ).
tff(stp_ty_2Enum_2Enum,type,
tp__ty_2Enum_2Enum: $tType ).
tff(stp_inj_ty_2Enum_2Enum,type,
inj__ty_2Enum_2Enum: tp__ty_2Enum_2Enum > $i ).
tff(stp_surj_ty_2Enum_2Enum,type,
surj__ty_2Enum_2Enum: $i > tp__ty_2Enum_2Enum ).
tff(stp_inj_surj_ty_2Enum_2Enum,axiom,
! [X: tp__ty_2Enum_2Enum] : ( surj__ty_2Enum_2Enum(inj__ty_2Enum_2Enum(X)) = X ) ).
tff(stp_inj_mem_ty_2Enum_2Enum,axiom,
! [X: tp__ty_2Enum_2Enum] : mem(inj__ty_2Enum_2Enum(X),ty_2Enum_2Enum) ).
tff(stp_iso_mem_ty_2Enum_2Enum,axiom,
! [X: $i] :
( mem(X,ty_2Enum_2Enum)
=> ( X = inj__ty_2Enum_2Enum(surj__ty_2Enum_2Enum(X)) ) ) ).
tff(tp_c_2Earithmetic_2EBIT1,type,
c_2Earithmetic_2EBIT1: $i ).
tff(mem_c_2Earithmetic_2EBIT1,axiom,
mem(c_2Earithmetic_2EBIT1,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) ).
tff(stp_fo_c_2Earithmetic_2EBIT1,type,
fo__c_2Earithmetic_2EBIT1: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
tff(stp_eq_fo_c_2Earithmetic_2EBIT1,axiom,
! [X0: tp__ty_2Enum_2Enum] : ( inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EBIT1(X0)) = ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(X0)) ) ).
tff(stp_o,type,
tp__o: $tType ).
tff(stp_inj_o,type,
inj__o: tp__o > $i ).
tff(stp_surj_o,type,
surj__o: $i > tp__o ).
tff(stp_inj_surj_o,axiom,
! [X: tp__o] : ( surj__o(inj__o(X)) = X ) ).
tff(stp_inj_mem_o,axiom,
! [X: tp__o] : mem(inj__o(X),bool) ).
tff(stp_iso_mem_o,axiom,
! [X: $i] :
( mem(X,bool)
=> ( X = inj__o(surj__o(X)) ) ) ).
tff(tp_c_2Earithmetic_2EEVEN,type,
c_2Earithmetic_2EEVEN: $i ).
tff(mem_c_2Earithmetic_2EEVEN,axiom,
mem(c_2Earithmetic_2EEVEN,arr(ty_2Enum_2Enum,bool)) ).
tff(stp_fo_c_2Earithmetic_2EEVEN,type,
fo__c_2Earithmetic_2EEVEN: tp__ty_2Enum_2Enum > tp__o ).
tff(stp_eq_fo_c_2Earithmetic_2EEVEN,axiom,
! [X0: tp__ty_2Enum_2Enum] : ( inj__o(fo__c_2Earithmetic_2EEVEN(X0)) = ap(c_2Earithmetic_2EEVEN,inj__ty_2Enum_2Enum(X0)) ) ).
tff(tp_c_2Earithmetic_2EZERO,type,
c_2Earithmetic_2EZERO: $i ).
tff(mem_c_2Earithmetic_2EZERO,axiom,
mem(c_2Earithmetic_2EZERO,ty_2Enum_2Enum) ).
tff(stp_fo_c_2Earithmetic_2EZERO,type,
fo__c_2Earithmetic_2EZERO: tp__ty_2Enum_2Enum ).
tff(stp_eq_fo_c_2Earithmetic_2EZERO,axiom,
inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO) = c_2Earithmetic_2EZERO ).
tff(tp_c_2Earithmetic_2EBIT2,type,
c_2Earithmetic_2EBIT2: $i ).
tff(mem_c_2Earithmetic_2EBIT2,axiom,
mem(c_2Earithmetic_2EBIT2,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) ).
tff(stp_fo_c_2Earithmetic_2EBIT2,type,
fo__c_2Earithmetic_2EBIT2: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
tff(stp_eq_fo_c_2Earithmetic_2EBIT2,axiom,
! [X0: tp__ty_2Enum_2Enum] : ( inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EBIT2(X0)) = ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(X0)) ) ).
tff(tp_c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: $i ).
tff(mem_c_2Earithmetic_2ENUMERAL,axiom,
mem(c_2Earithmetic_2ENUMERAL,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) ).
tff(stp_fo_c_2Earithmetic_2ENUMERAL,type,
fo__c_2Earithmetic_2ENUMERAL: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
tff(stp_eq_fo_c_2Earithmetic_2ENUMERAL,axiom,
! [X0: tp__ty_2Enum_2Enum] : ( inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2ENUMERAL(X0)) = ap(c_2Earithmetic_2ENUMERAL,inj__ty_2Enum_2Enum(X0)) ) ).
tff(tp_c_2Earithmetic_2EODD,type,
c_2Earithmetic_2EODD: $i ).
tff(mem_c_2Earithmetic_2EODD,axiom,
mem(c_2Earithmetic_2EODD,arr(ty_2Enum_2Enum,bool)) ).
tff(stp_fo_c_2Earithmetic_2EODD,type,
fo__c_2Earithmetic_2EODD: tp__ty_2Enum_2Enum > tp__o ).
tff(stp_eq_fo_c_2Earithmetic_2EODD,axiom,
! [X0: tp__ty_2Enum_2Enum] : ( inj__o(fo__c_2Earithmetic_2EODD(X0)) = ap(c_2Earithmetic_2EODD,inj__ty_2Enum_2Enum(X0)) ) ).
tff(tp_c_2Earithmetic_2EMOD,type,
c_2Earithmetic_2EMOD: $i ).
tff(mem_c_2Earithmetic_2EMOD,axiom,
mem(c_2Earithmetic_2EMOD,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))) ).
tff(stp_fo_c_2Earithmetic_2EMOD,type,
fo__c_2Earithmetic_2EMOD: ( tp__ty_2Enum_2Enum * tp__ty_2Enum_2Enum ) > tp__ty_2Enum_2Enum ).
tff(stp_eq_fo_c_2Earithmetic_2EMOD,axiom,
! [X0: tp__ty_2Enum_2Enum,X1: tp__ty_2Enum_2Enum] : ( inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EMOD(X0,X1)) = ap(ap(c_2Earithmetic_2EMOD,inj__ty_2Enum_2Enum(X0)),inj__ty_2Enum_2Enum(X1)) ) ).
tff(tp_c_2Earithmetic_2E_2A,type,
c_2Earithmetic_2E_2A: $i ).
tff(mem_c_2Earithmetic_2E_2A,axiom,
mem(c_2Earithmetic_2E_2A,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))) ).
tff(stp_fo_c_2Earithmetic_2E_2A,type,
fo__c_2Earithmetic_2E_2A: ( tp__ty_2Enum_2Enum * tp__ty_2Enum_2Enum ) > tp__ty_2Enum_2Enum ).
tff(stp_eq_fo_c_2Earithmetic_2E_2A,axiom,
! [X0: tp__ty_2Enum_2Enum,X1: tp__ty_2Enum_2Enum] : ( inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2E_2A(X0,X1)) = ap(ap(c_2Earithmetic_2E_2A,inj__ty_2Enum_2Enum(X0)),inj__ty_2Enum_2Enum(X1)) ) ).
tff(tp_c_2Earithmetic_2E_2B,type,
c_2Earithmetic_2E_2B: $i ).
tff(mem_c_2Earithmetic_2E_2B,axiom,
mem(c_2Earithmetic_2E_2B,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))) ).
tff(stp_fo_c_2Earithmetic_2E_2B,type,
fo__c_2Earithmetic_2E_2B: ( tp__ty_2Enum_2Enum * tp__ty_2Enum_2Enum ) > tp__ty_2Enum_2Enum ).
tff(stp_eq_fo_c_2Earithmetic_2E_2B,axiom,
! [X0: tp__ty_2Enum_2Enum,X1: tp__ty_2Enum_2Enum] : ( inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2E_2B(X0,X1)) = ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(X0)),inj__ty_2Enum_2Enum(X1)) ) ).
tff(tp_c_2Ebool_2ET,type,
c_2Ebool_2ET: $i ).
tff(mem_c_2Ebool_2ET,axiom,
mem(c_2Ebool_2ET,bool) ).
tff(stp_fo_c_2Ebool_2ET,type,
fo__c_2Ebool_2ET: tp__o ).
tff(stp_eq_fo_c_2Ebool_2ET,axiom,
inj__o(fo__c_2Ebool_2ET) = c_2Ebool_2ET ).
tff(ax_true_p,axiom,
p(c_2Ebool_2ET) ).
tff(tp_c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND: del > $i ).
tff(mem_c_2Ebool_2ECOND,axiom,
! [A_27a: del] : mem(c_2Ebool_2ECOND(A_27a),arr(bool,arr(A_27a,arr(A_27a,A_27a)))) ).
tff(tp_c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F: del > $i ).
tff(mem_c_2Ebool_2E_3F,axiom,
! [A_27a: del] : mem(c_2Ebool_2E_3F(A_27a),arr(arr(A_27a,bool),bool)) ).
tff(ax_ex_p,axiom,
! [A: del,Q: $i] :
( mem(Q,arr(A,bool))
=> ( p(ap(c_2Ebool_2E_3F(A),Q))
<=> ? [X: $i] :
( mem(X,A)
& p(ap(Q,X)) ) ) ) ).
tff(tp_c_2Enum_2ESUC,type,
c_2Enum_2ESUC: $i ).
tff(mem_c_2Enum_2ESUC,axiom,
mem(c_2Enum_2ESUC,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) ).
tff(stp_fo_c_2Enum_2ESUC,type,
fo__c_2Enum_2ESUC: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
tff(stp_eq_fo_c_2Enum_2ESUC,axiom,
! [X0: tp__ty_2Enum_2Enum] : ( inj__ty_2Enum_2Enum(fo__c_2Enum_2ESUC(X0)) = ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(X0)) ) ).
tff(tp_c_2Enum_2E0,type,
c_2Enum_2E0: $i ).
tff(mem_c_2Enum_2E0,axiom,
mem(c_2Enum_2E0,ty_2Enum_2Enum) ).
tff(stp_fo_c_2Enum_2E0,type,
fo__c_2Enum_2E0: tp__ty_2Enum_2Enum ).
tff(stp_eq_fo_c_2Enum_2E0,axiom,
inj__ty_2Enum_2Enum(fo__c_2Enum_2E0) = c_2Enum_2E0 ).
tff(tp_c_2Eprim__rec_2E_3C,type,
c_2Eprim__rec_2E_3C: $i ).
tff(mem_c_2Eprim__rec_2E_3C,axiom,
mem(c_2Eprim__rec_2E_3C,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,bool))) ).
tff(stp_fo_c_2Eprim__rec_2E_3C,type,
fo__c_2Eprim__rec_2E_3C: ( tp__ty_2Enum_2Enum * tp__ty_2Enum_2Enum ) > tp__o ).
tff(stp_eq_fo_c_2Eprim__rec_2E_3C,axiom,
! [X0: tp__ty_2Enum_2Enum,X1: tp__ty_2Enum_2Enum] : ( inj__o(fo__c_2Eprim__rec_2E_3C(X0,X1)) = ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(X0)),inj__ty_2Enum_2Enum(X1)) ) ).
tff(tp_c_2Ebool_2EF,type,
c_2Ebool_2EF: $i ).
tff(mem_c_2Ebool_2EF,axiom,
mem(c_2Ebool_2EF,bool) ).
tff(stp_fo_c_2Ebool_2EF,type,
fo__c_2Ebool_2EF: tp__o ).
tff(stp_eq_fo_c_2Ebool_2EF,axiom,
inj__o(fo__c_2Ebool_2EF) = c_2Ebool_2EF ).
tff(ax_false_p,axiom,
~ p(c_2Ebool_2EF) ).
tff(tp_c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $i ).
tff(mem_c_2Ebool_2E_2F_5C,axiom,
mem(c_2Ebool_2E_2F_5C,arr(bool,arr(bool,bool))) ).
tff(stp_fo_c_2Ebool_2E_2F_5C,type,
fo__c_2Ebool_2E_2F_5C: ( tp__o * tp__o ) > tp__o ).
tff(stp_eq_fo_c_2Ebool_2E_2F_5C,axiom,
! [X0: tp__o,X1: tp__o] : ( inj__o(fo__c_2Ebool_2E_2F_5C(X0,X1)) = ap(ap(c_2Ebool_2E_2F_5C,inj__o(X0)),inj__o(X1)) ) ).
tff(ax_and_p,axiom,
! [Q: $i] :
( mem(Q,bool)
=> ! [R: $i] :
( mem(R,bool)
=> ( p(ap(ap(c_2Ebool_2E_2F_5C,Q),R))
<=> ( p(Q)
& p(R) ) ) ) ) ).
tff(tp_c_2Emin_2E_3D,type,
c_2Emin_2E_3D: del > $i ).
tff(mem_c_2Emin_2E_3D,axiom,
! [A_27a: del] : mem(c_2Emin_2E_3D(A_27a),arr(A_27a,arr(A_27a,bool))) ).
tff(ax_eq_p,axiom,
! [A: del,X: $i] :
( mem(X,A)
=> ! [Y: $i] :
( mem(Y,A)
=> ( p(ap(ap(c_2Emin_2E_3D(A),X),Y))
<=> ( X = Y ) ) ) ) ).
tff(tp_c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $i ).
tff(mem_c_2Ebool_2E_5C_2F,axiom,
mem(c_2Ebool_2E_5C_2F,arr(bool,arr(bool,bool))) ).
tff(stp_fo_c_2Ebool_2E_5C_2F,type,
fo__c_2Ebool_2E_5C_2F: ( tp__o * tp__o ) > tp__o ).
tff(stp_eq_fo_c_2Ebool_2E_5C_2F,axiom,
! [X0: tp__o,X1: tp__o] : ( inj__o(fo__c_2Ebool_2E_5C_2F(X0,X1)) = ap(ap(c_2Ebool_2E_5C_2F,inj__o(X0)),inj__o(X1)) ) ).
tff(ax_or_p,axiom,
! [Q: $i] :
( mem(Q,bool)
=> ! [R: $i] :
( mem(R,bool)
=> ( p(ap(ap(c_2Ebool_2E_5C_2F,Q),R))
<=> ( p(Q)
| p(R) ) ) ) ) ).
tff(tp_c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $i ).
tff(mem_c_2Ebool_2E_7E,axiom,
mem(c_2Ebool_2E_7E,arr(bool,bool)) ).
tff(stp_fo_c_2Ebool_2E_7E,type,
fo__c_2Ebool_2E_7E: tp__o > tp__o ).
tff(stp_eq_fo_c_2Ebool_2E_7E,axiom,
! [X0: tp__o] : ( inj__o(fo__c_2Ebool_2E_7E(X0)) = ap(c_2Ebool_2E_7E,inj__o(X0)) ) ).
tff(ax_neg_p,axiom,
! [Q: $i] :
( mem(Q,bool)
=> ( p(ap(c_2Ebool_2E_7E,Q))
<=> ~ p(Q) ) ) ).
tff(tp_c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $i ).
tff(mem_c_2Emin_2E_3D_3D_3E,axiom,
mem(c_2Emin_2E_3D_3D_3E,arr(bool,arr(bool,bool))) ).
tff(stp_fo_c_2Emin_2E_3D_3D_3E,type,
fo__c_2Emin_2E_3D_3D_3E: ( tp__o * tp__o ) > tp__o ).
tff(stp_eq_fo_c_2Emin_2E_3D_3D_3E,axiom,
! [X0: tp__o,X1: tp__o] : ( inj__o(fo__c_2Emin_2E_3D_3D_3E(X0,X1)) = ap(ap(c_2Emin_2E_3D_3D_3E,inj__o(X0)),inj__o(X1)) ) ).
tff(ax_imp_p,axiom,
! [Q: $i] :
( mem(Q,bool)
=> ! [R: $i] :
( mem(R,bool)
=> ( p(ap(ap(c_2Emin_2E_3D_3D_3E,Q),R))
<=> ( p(Q)
=> p(R) ) ) ) ) ).
tff(tp_c_2Ebool_2E_21,type,
c_2Ebool_2E_21: del > $i ).
tff(mem_c_2Ebool_2E_21,axiom,
! [A_27a: del] : mem(c_2Ebool_2E_21(A_27a),arr(arr(A_27a,bool),bool)) ).
tff(ax_all_p,axiom,
! [A: del,Q: $i] :
( mem(Q,arr(A,bool))
=> ( p(ap(c_2Ebool_2E_21(A),Q))
<=> ! [X: $i] :
( mem(X,A)
=> p(ap(Q,X)) ) ) ) ).
tff(conj_thm_2Earithmetic_2EONE,axiom,
surj__ty_2Enum_2Enum(ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))) = surj__ty_2Enum_2Enum(ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ).
tff(conj_thm_2Earithmetic_2ETWO,axiom,
surj__ty_2Enum_2Enum(ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))) = surj__ty_2Enum_2Enum(ap(c_2Enum_2ESUC,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) ).
tff(conj_thm_2Earithmetic_2EADD__0,axiom,
! [V0m: tp__ty_2Enum_2Enum] : ( surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) = V0m ) ).
tff(conj_thm_2Earithmetic_2ELESS__MONO__EQ,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Eprim__rec_2E_3C,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V0m))),ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V1n))))
<=> p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n))) ) ).
tff(conj_thm_2Earithmetic_2EADD1,axiom,
! [V0m: tp__ty_2Enum_2Enum] : ( surj__ty_2Enum_2Enum(ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V0m))) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V0m)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) ) ).
tff(conj_thm_2Earithmetic_2EMULT__COMM,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] : ( surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2A,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n))) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2A,inj__ty_2Enum_2Enum(V1n)),inj__ty_2Enum_2Enum(V0m))) ) ).
tff(conj_thm_2Earithmetic_2EEVEN__ODD,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( p(ap(c_2Earithmetic_2EEVEN,inj__ty_2Enum_2Enum(V0n)))
<=> ~ p(ap(c_2Earithmetic_2EODD,inj__ty_2Enum_2Enum(V0n))) ) ).
tff(conj_thm_2Earithmetic_2EODD__EVEN,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( p(ap(c_2Earithmetic_2EODD,inj__ty_2Enum_2Enum(V0n)))
<=> ~ p(ap(c_2Earithmetic_2EEVEN,inj__ty_2Enum_2Enum(V0n))) ) ).
tff(conj_thm_2Earithmetic_2EEVEN__EXISTS,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( p(ap(c_2Earithmetic_2EEVEN,inj__ty_2Enum_2Enum(V0n)))
<=> ? [V1m: tp__ty_2Enum_2Enum] : ( V0n = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),inj__ty_2Enum_2Enum(V1m))) ) ) ).
tff(conj_thm_2Earithmetic_2EODD__EXISTS,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( p(ap(c_2Earithmetic_2EODD,inj__ty_2Enum_2Enum(V0n)))
<=> ? [V1m: tp__ty_2Enum_2Enum] : ( V0n = surj__ty_2Enum_2Enum(ap(c_2Enum_2ESUC,ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),inj__ty_2Enum_2Enum(V1m)))) ) ) ).
tff(conj_thm_2Earithmetic_2EMOD__UNIQUE,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1k: tp__ty_2Enum_2Enum,V2r: tp__ty_2Enum_2Enum] :
( ? [V3q: tp__ty_2Enum_2Enum] :
( ( V1k = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Earithmetic_2E_2A,inj__ty_2Enum_2Enum(V3q)),inj__ty_2Enum_2Enum(V0n))),inj__ty_2Enum_2Enum(V2r))) )
& p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V2r)),inj__ty_2Enum_2Enum(V0n))) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2EMOD,inj__ty_2Enum_2Enum(V1k)),inj__ty_2Enum_2Enum(V0n))) = V2r ) ) ).
tff(ax_thm_2Ebool_2EBOOL__CASES__AX,axiom,
! [V0t: tp__o] :
( ( p(inj__o(V0t))
<=> $true )
| ( p(inj__o(V0t))
<=> $false ) ) ).
tff(conj_thm_2Ebool_2ETRUTH,axiom,
$true ).
tff(conj_thm_2Ebool_2EIMP__ANTISYM__AX,axiom,
! [V0t1: tp__o,V1t2: tp__o] :
( ( p(inj__o(V0t1))
=> p(inj__o(V1t2)) )
=> ( ( p(inj__o(V1t2))
=> p(inj__o(V0t1)) )
=> ( p(inj__o(V0t1))
<=> p(inj__o(V1t2)) ) ) ) ).
tff(conj_thm_2Ebool_2EFALSITY,axiom,
! [V0t: tp__o] :
( $false
=> p(inj__o(V0t)) ) ).
tff(conj_thm_2Ebool_2EEXCLUDED__MIDDLE,axiom,
! [V0t: tp__o] :
( p(inj__o(V0t))
| ~ p(inj__o(V0t)) ) ).
tff(conj_thm_2Ebool_2EIMP__F,axiom,
! [V0t: tp__o] :
( ( p(inj__o(V0t))
=> $false )
=> ~ p(inj__o(V0t)) ) ).
tff(conj_thm_2Ebool_2EF__IMP,axiom,
! [V0t: tp__o] :
( ~ p(inj__o(V0t))
=> ( p(inj__o(V0t))
=> $false ) ) ).
tff(conj_thm_2Ebool_2EAND__CLAUSES,axiom,
! [V0t: tp__o] :
( ( ( $true
& p(inj__o(V0t)) )
<=> p(inj__o(V0t)) )
& ( ( p(inj__o(V0t))
& $true )
<=> p(inj__o(V0t)) )
& ( ( $false
& p(inj__o(V0t)) )
<=> $false )
& ( ( p(inj__o(V0t))
& $false )
<=> $false )
& ( ( p(inj__o(V0t))
& p(inj__o(V0t)) )
<=> p(inj__o(V0t)) ) ) ).
tff(conj_thm_2Ebool_2EOR__CLAUSES,axiom,
! [V0t: tp__o] :
( ( ( $true
| p(inj__o(V0t)) )
<=> $true )
& ( ( p(inj__o(V0t))
| $true )
<=> $true )
& ( ( $false
| p(inj__o(V0t)) )
<=> p(inj__o(V0t)) )
& ( ( p(inj__o(V0t))
| $false )
<=> p(inj__o(V0t)) )
& ( ( p(inj__o(V0t))
| p(inj__o(V0t)) )
<=> p(inj__o(V0t)) ) ) ).
tff(conj_thm_2Ebool_2EIMP__CLAUSES,axiom,
! [V0t: tp__o] :
( ( ( $true
=> p(inj__o(V0t)) )
<=> p(inj__o(V0t)) )
& ( ( p(inj__o(V0t))
=> $true )
<=> $true )
& ( ( $false
=> p(inj__o(V0t)) )
<=> $true )
& ( ( p(inj__o(V0t))
=> p(inj__o(V0t)) )
<=> $true )
& ( ( p(inj__o(V0t))
=> $false )
<=> ~ p(inj__o(V0t)) ) ) ).
tff(conj_thm_2Ebool_2ENOT__CLAUSES,axiom,
( ! [V0t: tp__o] :
( ~ ~ p(inj__o(V0t))
<=> p(inj__o(V0t)) )
& ( ~ $true
<=> $false )
& ( ~ $false
<=> $true ) ) ).
tff(conj_thm_2Ebool_2EREFL__CLAUSE,axiom,
! [A_27a: del,V0x: $i] :
( mem(V0x,A_27a)
=> ( ( V0x = V0x )
<=> $true ) ) ).
tff(conj_thm_2Ebool_2EEQ__SYM__EQ,axiom,
! [A_27a: del,V0x: $i] :
( mem(V0x,A_27a)
=> ! [V1y: $i] :
( mem(V1y,A_27a)
=> ( ( V0x = V1y )
<=> ( V1y = V0x ) ) ) ) ).
tff(conj_thm_2Ebool_2EEQ__CLAUSES,axiom,
! [V0t: tp__o] :
( ( ( $true
<=> p(inj__o(V0t)) )
<=> p(inj__o(V0t)) )
& ( ( p(inj__o(V0t))
<=> $true )
<=> p(inj__o(V0t)) )
& ( ( $false
<=> p(inj__o(V0t)) )
<=> ~ p(inj__o(V0t)) )
& ( ( p(inj__o(V0t))
<=> $false )
<=> ~ p(inj__o(V0t)) ) ) ).
tff(conj_thm_2Ebool_2ECOND__CLAUSES,axiom,
! [A_27a: del,V0t1: $i] :
( mem(V0t1,A_27a)
=> ! [V1t2: $i] :
( mem(V1t2,A_27a)
=> ( ( ap(ap(ap(c_2Ebool_2ECOND(A_27a),inj__o(fo__c_2Ebool_2ET)),V0t1),V1t2) = V0t1 )
& ( ap(ap(ap(c_2Ebool_2ECOND(A_27a),inj__o(fo__c_2Ebool_2EF)),V0t1),V1t2) = V1t2 ) ) ) ) ).
tff(conj_thm_2Ebool_2ENOT__EXISTS__THM,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ( ~ ? [V1x: $i] :
( mem(V1x,A_27a)
& p(ap(V0P,V1x)) )
<=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ~ p(ap(V0P,V2x)) ) ) ) ).
tff(conj_thm_2Ebool_2ELEFT__AND__FORALL__THM,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q: tp__o] :
( ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> p(ap(V0P,V2x)) )
& p(inj__o(V1Q)) )
<=> ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(V0P,V3x))
& p(inj__o(V1Q)) ) ) ) ) ).
tff(conj_thm_2Ebool_2EEXISTS__OR__THM,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q: $i] :
( mem(V1Q,arr(A_27a,bool))
=> ( ? [V2x: $i] :
( mem(V2x,A_27a)
& ( p(ap(V0P,V2x))
| p(ap(V1Q,V2x)) ) )
<=> ( ? [V3x: $i] :
( mem(V3x,A_27a)
& p(ap(V0P,V3x)) )
| ? [V4x: $i] :
( mem(V4x,A_27a)
& p(ap(V1Q,V4x)) ) ) ) ) ) ).
tff(conj_thm_2Ebool_2ELEFT__OR__EXISTS__THM,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q: tp__o] :
( ( ? [V2x: $i] :
( mem(V2x,A_27a)
& p(ap(V0P,V2x)) )
| p(inj__o(V1Q)) )
<=> ? [V3x: $i] :
( mem(V3x,A_27a)
& ( p(ap(V0P,V3x))
| p(inj__o(V1Q)) ) ) ) ) ).
tff(conj_thm_2Ebool_2ERIGHT__OR__EXISTS__THM,axiom,
! [A_27a: del,V0P: tp__o,V1Q: $i] :
( mem(V1Q,arr(A_27a,bool))
=> ( ( p(inj__o(V0P))
| ? [V2x: $i] :
( mem(V2x,A_27a)
& p(ap(V1Q,V2x)) ) )
<=> ? [V3x: $i] :
( mem(V3x,A_27a)
& ( p(inj__o(V0P))
| p(ap(V1Q,V3x)) ) ) ) ) ).
tff(conj_thm_2Ebool_2ELEFT__EXISTS__AND__THM,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q: tp__o] :
( ? [V2x: $i] :
( mem(V2x,A_27a)
& p(ap(V0P,V2x))
& p(inj__o(V1Q)) )
<=> ( ? [V3x: $i] :
( mem(V3x,A_27a)
& p(ap(V0P,V3x)) )
& p(inj__o(V1Q)) ) ) ) ).
tff(conj_thm_2Ebool_2ERIGHT__EXISTS__AND__THM,axiom,
! [A_27a: del,V0P: tp__o,V1Q: $i] :
( mem(V1Q,arr(A_27a,bool))
=> ( ? [V2x: $i] :
( mem(V2x,A_27a)
& p(inj__o(V0P))
& p(ap(V1Q,V2x)) )
<=> ( p(inj__o(V0P))
& ? [V3x: $i] :
( mem(V3x,A_27a)
& p(ap(V1Q,V3x)) ) ) ) ) ).
tff(conj_thm_2Ebool_2ERIGHT__FORALL__OR__THM,axiom,
! [A_27a: del,V0P: tp__o,V1Q: $i] :
( mem(V1Q,arr(A_27a,bool))
=> ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( p(inj__o(V0P))
| p(ap(V1Q,V2x)) ) )
<=> ( p(inj__o(V0P))
| ! [V3x: $i] :
( mem(V3x,A_27a)
=> p(ap(V1Q,V3x)) ) ) ) ) ).
tff(conj_thm_2Ebool_2EDISJ__ASSOC,axiom,
! [V0A: tp__o,V1B: tp__o,V2C: tp__o] :
( ( p(inj__o(V0A))
| p(inj__o(V1B))
| p(inj__o(V2C)) )
<=> ( p(inj__o(V0A))
| p(inj__o(V1B))
| p(inj__o(V2C)) ) ) ).
tff(conj_thm_2Ebool_2EDISJ__SYM,axiom,
! [V0A: tp__o,V1B: tp__o] :
( ( p(inj__o(V0A))
| p(inj__o(V1B)) )
<=> ( p(inj__o(V1B))
| p(inj__o(V0A)) ) ) ).
tff(conj_thm_2Ebool_2EDE__MORGAN__THM,axiom,
! [V0A: tp__o,V1B: tp__o] :
( ( ~ ( p(inj__o(V0A))
& p(inj__o(V1B)) )
<=> ( ~ p(inj__o(V0A))
| ~ p(inj__o(V1B)) ) )
& ( ~ ( p(inj__o(V0A))
| p(inj__o(V1B)) )
<=> ( ~ p(inj__o(V0A))
& ~ p(inj__o(V1B)) ) ) ) ).
tff(conj_thm_2Ebool_2ECOND__RATOR,axiom,
! [A_27a: del,A_27b: del,V0b: tp__o,V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2g: $i] :
( mem(V2g,arr(A_27a,A_27b))
=> ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( ap(ap(ap(ap(c_2Ebool_2ECOND(arr(A_27a,A_27b)),inj__o(V0b)),V1f),V2g),V3x) = ap(ap(ap(c_2Ebool_2ECOND(A_27b),inj__o(V0b)),ap(V1f,V3x)),ap(V2g,V3x)) ) ) ) ) ).
tff(conj_thm_2Ebool_2ECOND__RAND,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1b: tp__o,V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( ap(V0f,ap(ap(ap(c_2Ebool_2ECOND(A_27a),inj__o(V1b)),V2x),V3y)) = ap(ap(ap(c_2Ebool_2ECOND(A_27b),inj__o(V1b)),ap(V0f,V2x)),ap(V0f,V3y)) ) ) ) ) ).
tff(conj_thm_2Ebool_2ECOND__EXPAND,axiom,
! [V0b: tp__o,V1t1: tp__o,V2t2: tp__o] :
( p(ap(ap(ap(c_2Ebool_2ECOND(bool),inj__o(V0b)),inj__o(V1t1)),inj__o(V2t2)))
<=> ( ( ~ p(inj__o(V0b))
| p(inj__o(V1t1)) )
& ( p(inj__o(V0b))
| p(inj__o(V2t2)) ) ) ) ).
tff(conj_thm_2Ebool_2ESKOLEM__THM,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( mem(V0P,arr(A_27a,arr(A_27b,bool)))
=> ( ! [V1x: $i] :
( mem(V1x,A_27a)
=> ? [V2y: $i] :
( mem(V2y,A_27b)
& p(ap(ap(V0P,V1x),V2y)) ) )
<=> ? [V3f: $i] :
( mem(V3f,arr(A_27a,A_27b))
& ! [V4x: $i] :
( mem(V4x,A_27a)
=> p(ap(ap(V0P,V4x),ap(V3f,V4x))) ) ) ) ) ).
tff(conj_thm_2Eprim__rec_2ELESS__0,axiom,
! [V0n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V0n)))) ).
tff(conj_thm_2Esat_2ENOT__NOT,axiom,
! [V0t: tp__o] :
( ~ ~ p(inj__o(V0t))
<=> p(inj__o(V0t)) ) ).
tff(conj_thm_2Esat_2EAND__INV__IMP,axiom,
! [V0A: tp__o] :
( p(inj__o(V0A))
=> ( ~ p(inj__o(V0A))
=> $false ) ) ).
tff(conj_thm_2Esat_2EOR__DUAL2,axiom,
! [V0A: tp__o,V1B: tp__o] :
( ( ~ ( p(inj__o(V0A))
| p(inj__o(V1B)) )
=> $false )
<=> ( ( p(inj__o(V0A))
=> $false )
=> ( ~ p(inj__o(V1B))
=> $false ) ) ) ).
tff(conj_thm_2Esat_2EOR__DUAL3,axiom,
! [V0A: tp__o,V1B: tp__o] :
( ( ~ ( ~ p(inj__o(V0A))
| p(inj__o(V1B)) )
=> $false )
<=> ( p(inj__o(V0A))
=> ( ~ p(inj__o(V1B))
=> $false ) ) ) ).
tff(conj_thm_2Esat_2EAND__INV2,axiom,
! [V0A: tp__o] :
( ( ~ p(inj__o(V0A))
=> $false )
=> ( ( p(inj__o(V0A))
=> $false )
=> $false ) ) ).
tff(conj_thm_2Esat_2Edc__eq,axiom,
! [V0p: tp__o,V1q: tp__o,V2r: tp__o] :
( ( p(inj__o(V0p))
<=> ( p(inj__o(V1q))
<=> p(inj__o(V2r)) ) )
<=> ( ( p(inj__o(V0p))
| p(inj__o(V1q))
| p(inj__o(V2r)) )
& ( p(inj__o(V0p))
| ~ p(inj__o(V2r))
| ~ p(inj__o(V1q)) )
& ( p(inj__o(V1q))
| ~ p(inj__o(V2r))
| ~ p(inj__o(V0p)) )
& ( p(inj__o(V2r))
| ~ p(inj__o(V1q))
| ~ p(inj__o(V0p)) ) ) ) ).
tff(conj_thm_2Esat_2Edc__conj,axiom,
! [V0p: tp__o,V1q: tp__o,V2r: tp__o] :
( ( p(inj__o(V0p))
<=> ( p(inj__o(V1q))
& p(inj__o(V2r)) ) )
<=> ( ( p(inj__o(V0p))
| ~ p(inj__o(V1q))
| ~ p(inj__o(V2r)) )
& ( p(inj__o(V1q))
| ~ p(inj__o(V0p)) )
& ( p(inj__o(V2r))
| ~ p(inj__o(V0p)) ) ) ) ).
tff(conj_thm_2Esat_2Edc__disj,axiom,
! [V0p: tp__o,V1q: tp__o,V2r: tp__o] :
( ( p(inj__o(V0p))
<=> ( p(inj__o(V1q))
| p(inj__o(V2r)) ) )
<=> ( ( p(inj__o(V0p))
| ~ p(inj__o(V1q)) )
& ( p(inj__o(V0p))
| ~ p(inj__o(V2r)) )
& ( p(inj__o(V1q))
| p(inj__o(V2r))
| ~ p(inj__o(V0p)) ) ) ) ).
tff(conj_thm_2Esat_2Edc__imp,axiom,
! [V0p: tp__o,V1q: tp__o,V2r: tp__o] :
( ( p(inj__o(V0p))
<=> ( p(inj__o(V1q))
=> p(inj__o(V2r)) ) )
<=> ( ( p(inj__o(V0p))
| p(inj__o(V1q)) )
& ( p(inj__o(V0p))
| ~ p(inj__o(V2r)) )
& ( ~ p(inj__o(V1q))
| p(inj__o(V2r))
| ~ p(inj__o(V0p)) ) ) ) ).
tff(conj_thm_2Esat_2Edc__neg,axiom,
! [V0p: tp__o,V1q: tp__o] :
( ( p(inj__o(V0p))
<=> ~ p(inj__o(V1q)) )
<=> ( ( p(inj__o(V0p))
| p(inj__o(V1q)) )
& ( ~ p(inj__o(V1q))
| ~ p(inj__o(V0p)) ) ) ) ).
tff(conj_thm_2Esat_2Epth__ni1,axiom,
! [V0p: tp__o,V1q: tp__o] :
( ~ ( p(inj__o(V0p))
=> p(inj__o(V1q)) )
=> p(inj__o(V0p)) ) ).
tff(conj_thm_2Esat_2Epth__ni2,axiom,
! [V0p: tp__o,V1q: tp__o] :
( ~ ( p(inj__o(V0p))
=> p(inj__o(V1q)) )
=> ~ p(inj__o(V1q)) ) ).
tff(conj_thm_2Esat_2Epth__no1,axiom,
! [V0p: tp__o,V1q: tp__o] :
( ~ ( p(inj__o(V0p))
| p(inj__o(V1q)) )
=> ~ p(inj__o(V0p)) ) ).
tff(conj_thm_2Esat_2Epth__no2,axiom,
! [V0p: tp__o,V1q: tp__o] :
( ~ ( p(inj__o(V0p))
| p(inj__o(V1q)) )
=> ~ p(inj__o(V1q)) ) ).
tff(conj_thm_2Esat_2Epth__nn,axiom,
! [V0p: tp__o] :
( ~ ~ p(inj__o(V0p))
=> p(inj__o(V0p)) ) ).
tff(conj_thm_2Earithmetic_2EMOD__2,conjecture,
! [V0n: tp__ty_2Enum_2Enum] : ( surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2EMOD,inj__ty_2Enum_2Enum(V0n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) = surj__ty_2Enum_2Enum(ap(ap(ap(c_2Ebool_2ECOND(ty_2Enum_2Enum),ap(c_2Earithmetic_2EEVEN,inj__ty_2Enum_2Enum(V0n))),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) ) ).
%------------------------------------------------------------------------------