ITP001 Axioms: ITP003_5.ax
%------------------------------------------------------------------------------
% File : ITP003_5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : bool_2.ax [Gau20]
% : HL4003_5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 394 ( 77 unt; 102 typ; 0 def)
% Number of atoms : 2665 ( 179 equ)
% Maximal formula atoms : 19 ( 6 avg)
% Number of connectives : 908 ( 46 ~; 61 |; 182 &)
% ( 151 <=>; 468 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of FOOLs : 1511 (1511 fml; 0 var)
% Number of types : 2 ( 1 usr)
% Number of type conns : 197 ( 89 >; 108 *; 0 +; 0 <<)
% Number of predicates : 17 ( 14 usr; 7 prp; 0-2 aty)
% Number of functors : 101 ( 101 usr; 12 con; 0-5 aty)
% Number of variables : 892 ( 836 !; 56 ?; 892 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_ty_2Ebool_2Eitself,type,
ty_2Ebool_2Eitself: del > del ).
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(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_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_2Ebool_2E_3F_21,type,
c_2Ebool_2E_3F_21: del > $i ).
tff(mem_c_2Ebool_2E_3F_21,axiom,
! [A_27a: del] : mem(c_2Ebool_2E_3F_21(A_27a),arr(arr(A_27a,bool),bool)) ).
tff(tp_c_2Ebool_2EARB,type,
c_2Ebool_2EARB: del > $i ).
tff(mem_c_2Ebool_2EARB,axiom,
! [A_27a: del] : mem(c_2Ebool_2EARB(A_27a),A_27a) ).
tff(tp_c_2Ebool_2EBOUNDED,type,
c_2Ebool_2EBOUNDED: $i ).
tff(mem_c_2Ebool_2EBOUNDED,axiom,
mem(c_2Ebool_2EBOUNDED,arr(bool,bool)) ).
tff(stp_fo_c_2Ebool_2EBOUNDED,type,
fo__c_2Ebool_2EBOUNDED: tp__o > tp__o ).
tff(stp_eq_fo_c_2Ebool_2EBOUNDED,axiom,
! [X0: tp__o] : ( inj__o(fo__c_2Ebool_2EBOUNDED(X0)) = ap(c_2Ebool_2EBOUNDED,inj__o(X0)) ) ).
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_2EDATATYPE,type,
c_2Ebool_2EDATATYPE: del > $i ).
tff(mem_c_2Ebool_2EDATATYPE,axiom,
! [A_27a: del] : mem(c_2Ebool_2EDATATYPE(A_27a),arr(A_27a,bool)) ).
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_2EIN,type,
c_2Ebool_2EIN: del > $i ).
tff(mem_c_2Ebool_2EIN,axiom,
! [A_27a: del] : mem(c_2Ebool_2EIN(A_27a),arr(A_27a,arr(arr(A_27a,bool),bool))) ).
tff(tp_c_2Ebool_2ELET,type,
c_2Ebool_2ELET: ( del * del ) > $i ).
tff(mem_c_2Ebool_2ELET,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Ebool_2ELET(A_27a,A_27b),arr(arr(A_27a,A_27b),arr(A_27a,A_27b))) ).
tff(tp_c_2Ebool_2EONE__ONE,type,
c_2Ebool_2EONE__ONE: ( del * del ) > $i ).
tff(mem_c_2Ebool_2EONE__ONE,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Ebool_2EONE__ONE(A_27a,A_27b),arr(arr(A_27a,A_27b),bool)) ).
tff(tp_c_2Ebool_2EONTO,type,
c_2Ebool_2EONTO: ( del * del ) > $i ).
tff(mem_c_2Ebool_2EONTO,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Ebool_2EONTO(A_27a,A_27b),arr(arr(A_27a,A_27b),bool)) ).
tff(tp_c_2Ebool_2ERES__ABSTRACT,type,
c_2Ebool_2ERES__ABSTRACT: ( del * del ) > $i ).
tff(mem_c_2Ebool_2ERES__ABSTRACT,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Ebool_2ERES__ABSTRACT(A_27a,A_27b),arr(arr(A_27a,bool),arr(arr(A_27a,A_27b),arr(A_27a,A_27b)))) ).
tff(tp_c_2Ebool_2ERES__EXISTS,type,
c_2Ebool_2ERES__EXISTS: del > $i ).
tff(mem_c_2Ebool_2ERES__EXISTS,axiom,
! [A_27a: del] : mem(c_2Ebool_2ERES__EXISTS(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) ).
tff(tp_c_2Ebool_2ERES__EXISTS__UNIQUE,type,
c_2Ebool_2ERES__EXISTS__UNIQUE: del > $i ).
tff(mem_c_2Ebool_2ERES__EXISTS__UNIQUE,axiom,
! [A_27a: del] : mem(c_2Ebool_2ERES__EXISTS__UNIQUE(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) ).
tff(tp_c_2Ebool_2ERES__FORALL,type,
c_2Ebool_2ERES__FORALL: del > $i ).
tff(mem_c_2Ebool_2ERES__FORALL,axiom,
! [A_27a: del] : mem(c_2Ebool_2ERES__FORALL(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) ).
tff(tp_c_2Ebool_2ERES__SELECT,type,
c_2Ebool_2ERES__SELECT: del > $i ).
tff(mem_c_2Ebool_2ERES__SELECT,axiom,
! [A_27a: del] : mem(c_2Ebool_2ERES__SELECT(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),A_27a))) ).
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_2ETYPE__DEFINITION,type,
c_2Ebool_2ETYPE__DEFINITION: ( del * del ) > $i ).
tff(mem_c_2Ebool_2ETYPE__DEFINITION,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Ebool_2ETYPE__DEFINITION(A_27a,A_27b),arr(arr(A_27a,bool),arr(arr(A_27b,A_27a),bool))) ).
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_2Eitself__case,type,
c_2Ebool_2Eitself__case: ( del * del ) > $i ).
tff(mem_c_2Ebool_2Eitself__case,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Ebool_2Eitself__case(A_27a,A_27b),arr(ty_2Ebool_2Eitself(A_27a),arr(A_27b,A_27b))) ).
tff(tp_c_2Ebool_2Eliteral__case,type,
c_2Ebool_2Eliteral__case: ( del * del ) > $i ).
tff(mem_c_2Ebool_2Eliteral__case,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Ebool_2Eliteral__case(A_27a,A_27b),arr(arr(A_27a,A_27b),arr(A_27a,A_27b))) ).
tff(tp_c_2Ebool_2Ethe__value,type,
c_2Ebool_2Ethe__value: del > $i ).
tff(mem_c_2Ebool_2Ethe__value,axiom,
! [A_27a: del] : mem(c_2Ebool_2Ethe__value(A_27a),ty_2Ebool_2Eitself(A_27a)) ).
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(ax_thm_2Ebool_2ET__DEF,axiom,
( $true
<=> ( i(bool) = i(bool) ) ) ).
tff(lamtp_f1,type,
f1: del > $i ).
tff(lameq_f1,axiom,
! [A_27a: del,V0P: $i] : ( ap(f1(A_27a),V0P) = ap(ap(c_2Emin_2E_3D(arr(A_27a,bool)),V0P),k(A_27a,c_2Ebool_2ET)) ) ).
tff(ax_thm_2Ebool_2EFORALL__DEF,axiom,
! [A_27a: del] : ( c_2Ebool_2E_21(A_27a) = f1(A_27a) ) ).
tff(lamtp_f2,type,
f2: del > $i ).
tff(lameq_f2,axiom,
! [A_27a: del,V0P: $i] : ( ap(f2(A_27a),V0P) = ap(V0P,ap(c_2Emin_2E_40(A_27a),V0P)) ) ).
tff(ax_thm_2Ebool_2EEXISTS__DEF,axiom,
! [A_27a: del] : ( c_2Ebool_2E_3F(A_27a) = f2(A_27a) ) ).
tff(lamtp_f3,type,
f3: ( tp__o * tp__o ) > $i ).
tff(lameq_f3,axiom,
! [V1t2: tp__o,V0t1: tp__o,V2t: tp__o] : ( ap(f3(V1t2,V0t1),inj__o(V2t)) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Emin_2E_3D_3D_3E,inj__o(V0t1)),ap(ap(c_2Emin_2E_3D_3D_3E,inj__o(V1t2)),inj__o(V2t)))),inj__o(V2t)) ) ).
tff(lamtp_f4,type,
f4: tp__o > $i ).
tff(lameq_f4,axiom,
! [V0t1: tp__o,V1t2: tp__o] : ( ap(f4(V0t1),inj__o(V1t2)) = ap(c_2Ebool_2E_21(bool),f3(V1t2,V0t1)) ) ).
tff(lamtp_f5,type,
f5: $i ).
tff(lameq_f5,axiom,
! [V0t1: tp__o] : ( ap(f5,inj__o(V0t1)) = f4(V0t1) ) ).
tff(ax_thm_2Ebool_2EAND__DEF,axiom,
c_2Ebool_2E_2F_5C = f5 ).
tff(lamtp_f6,type,
f6: ( tp__o * tp__o ) > $i ).
tff(lameq_f6,axiom,
! [V0t1: tp__o,V1t2: tp__o,V2t: tp__o] : ( ap(f6(V0t1,V1t2),inj__o(V2t)) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Emin_2E_3D_3D_3E,inj__o(V0t1)),inj__o(V2t))),ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Emin_2E_3D_3D_3E,inj__o(V1t2)),inj__o(V2t))),inj__o(V2t))) ) ).
tff(lamtp_f7,type,
f7: tp__o > $i ).
tff(lameq_f7,axiom,
! [V0t1: tp__o,V1t2: tp__o] : ( ap(f7(V0t1),inj__o(V1t2)) = ap(c_2Ebool_2E_21(bool),f6(V0t1,V1t2)) ) ).
tff(lamtp_f8,type,
f8: $i ).
tff(lameq_f8,axiom,
! [V0t1: tp__o] : ( ap(f8,inj__o(V0t1)) = f7(V0t1) ) ).
tff(ax_thm_2Ebool_2EOR__DEF,axiom,
c_2Ebool_2E_5C_2F = f8 ).
tff(ax_thm_2Ebool_2EF__DEF,axiom,
( $false
<=> ! [V0t: tp__o] : p(inj__o(V0t)) ) ).
tff(lamtp_f9,type,
f9: $i ).
tff(lameq_f9,axiom,
! [V0t: tp__o] : ( ap(f9,inj__o(V0t)) = ap(ap(c_2Emin_2E_3D_3D_3E,inj__o(V0t)),inj__o(fo__c_2Ebool_2EF)) ) ).
tff(ax_thm_2Ebool_2ENOT__DEF,axiom,
c_2Ebool_2E_7E = f9 ).
tff(lamtp_f10,type,
f10: ( del * $i * $i ) > $i ).
tff(lameq_f10,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] : ( ap(f10(A_27a,V0P,V1x),V2y) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2E_2F_5C,ap(V0P,V1x)),ap(V0P,V2y))),ap(ap(c_2Emin_2E_3D(A_27a),V1x),V2y)) ) ) ) ).
tff(lamtp_f11,type,
f11: ( del * $i ) > $i ).
tff(lameq_f11,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1x: $i] : ( ap(f11(A_27a,V0P),V1x) = ap(c_2Ebool_2E_21(A_27a),f10(A_27a,V0P,V1x)) ) ) ).
tff(lamtp_f12,type,
f12: del > $i ).
tff(lameq_f12,axiom,
! [A_27a: del,V0P: $i] : ( ap(f12(A_27a),V0P) = ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Ebool_2E_3F(A_27a),V0P)),ap(c_2Ebool_2E_21(A_27a),f11(A_27a,V0P))) ) ).
tff(ax_thm_2Ebool_2EEXISTS__UNIQUE__DEF,axiom,
! [A_27a: del] : ( c_2Ebool_2E_3F_21(A_27a) = f12(A_27a) ) ).
tff(lamtp_f13,type,
f13: ( del * del * $i ) > $i ).
tff(lameq_f13,axiom,
! [A_27b: del,A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1x: $i] : ( ap(f13(A_27b,A_27a,V0f),V1x) = ap(V0f,V1x) ) ) ).
tff(lamtp_f14,type,
f14: ( del * del ) > $i ).
tff(lameq_f14,axiom,
! [A_27b: del,A_27a: del,V0f: $i] : ( ap(f14(A_27b,A_27a),V0f) = f13(A_27b,A_27a,V0f) ) ).
tff(ax_thm_2Ebool_2ELET__DEF,axiom,
! [A_27a: del,A_27b: del] : ( c_2Ebool_2ELET(A_27a,A_27b) = f14(A_27b,A_27a) ) ).
tff(lamtp_f15,type,
f15: ( del * $i * $i * tp__o ) > $i ).
tff(lameq_f15,axiom,
! [A_27a: del,V1t1: $i] :
( mem(V1t1,A_27a)
=> ! [V2t2: $i] :
( mem(V2t2,A_27a)
=> ! [V0t: tp__o,V3x: $i] : ( ap(f15(A_27a,V1t1,V2t2,V0t),V3x) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Emin_2E_3D(bool),inj__o(V0t)),inj__o(fo__c_2Ebool_2ET))),ap(ap(c_2Emin_2E_3D(A_27a),V3x),V1t1))),ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Emin_2E_3D(bool),inj__o(V0t)),inj__o(fo__c_2Ebool_2EF))),ap(ap(c_2Emin_2E_3D(A_27a),V3x),V2t2))) ) ) ) ).
tff(lamtp_f16,type,
f16: ( del * tp__o * $i ) > $i ).
tff(lameq_f16,axiom,
! [A_27a: del,V0t: tp__o,V1t1: $i] :
( mem(V1t1,A_27a)
=> ! [V2t2: $i] : ( ap(f16(A_27a,V0t,V1t1),V2t2) = ap(c_2Emin_2E_40(A_27a),f15(A_27a,V1t1,V2t2,V0t)) ) ) ).
tff(lamtp_f17,type,
f17: ( del * tp__o ) > $i ).
tff(lameq_f17,axiom,
! [A_27a: del,V0t: tp__o,V1t1: $i] : ( ap(f17(A_27a,V0t),V1t1) = f16(A_27a,V0t,V1t1) ) ).
tff(lamtp_f18,type,
f18: del > $i ).
tff(lameq_f18,axiom,
! [A_27a: del,V0t: tp__o] : ( ap(f18(A_27a),inj__o(V0t)) = f17(A_27a,V0t) ) ).
tff(ax_thm_2Ebool_2ECOND__DEF,axiom,
! [A_27a: del] : ( c_2Ebool_2ECOND(A_27a) = f18(A_27a) ) ).
tff(lamtp_f19,type,
f19: ( del * del * $i * $i ) > $i ).
tff(lameq_f19,axiom,
! [A_27b: del,A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1x1: $i] :
( mem(V1x1,A_27a)
=> ! [V2x2: $i] : ( ap(f19(A_27b,A_27a,V0f,V1x1),V2x2) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Emin_2E_3D(A_27b),ap(V0f,V1x1)),ap(V0f,V2x2))),ap(ap(c_2Emin_2E_3D(A_27a),V1x1),V2x2)) ) ) ) ).
tff(lamtp_f20,type,
f20: ( del * del * $i ) > $i ).
tff(lameq_f20,axiom,
! [A_27b: del,A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1x1: $i] : ( ap(f20(A_27b,A_27a,V0f),V1x1) = ap(c_2Ebool_2E_21(A_27a),f19(A_27b,A_27a,V0f,V1x1)) ) ) ).
tff(lamtp_f21,type,
f21: ( del * del ) > $i ).
tff(lameq_f21,axiom,
! [A_27b: del,A_27a: del,V0f: $i] : ( ap(f21(A_27b,A_27a),V0f) = ap(c_2Ebool_2E_21(A_27a),f20(A_27b,A_27a,V0f)) ) ).
tff(ax_thm_2Ebool_2EONE__ONE__DEF,axiom,
! [A_27a: del,A_27b: del] : ( c_2Ebool_2EONE__ONE(A_27a,A_27b) = f21(A_27b,A_27a) ) ).
tff(lamtp_f22,type,
f22: ( del * del * $i * $i ) > $i ).
tff(lameq_f22,axiom,
! [A_27b: del,A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1y: $i] :
( mem(V1y,A_27b)
=> ! [V2x: $i] : ( ap(f22(A_27b,A_27a,V0f,V1y),V2x) = ap(ap(c_2Emin_2E_3D(A_27b),V1y),ap(V0f,V2x)) ) ) ) ).
tff(lamtp_f23,type,
f23: ( del * del * $i ) > $i ).
tff(lameq_f23,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1y: $i] : ( ap(f23(A_27a,A_27b,V0f),V1y) = ap(c_2Ebool_2E_3F(A_27a),f22(A_27b,A_27a,V0f,V1y)) ) ) ).
tff(lamtp_f24,type,
f24: ( del * del ) > $i ).
tff(lameq_f24,axiom,
! [A_27a: del,A_27b: del,V0f: $i] : ( ap(f24(A_27a,A_27b),V0f) = ap(c_2Ebool_2E_21(A_27b),f23(A_27a,A_27b,V0f)) ) ).
tff(ax_thm_2Ebool_2EONTO__DEF,axiom,
! [A_27a: del,A_27b: del] : ( c_2Ebool_2EONTO(A_27a,A_27b) = f24(A_27a,A_27b) ) ).
tff(lamtp_f25,type,
f25: ( del * del * $i * $i ) > $i ).
tff(lameq_f25,axiom,
! [A_27a: del,A_27b: del,V1rep: $i] :
( mem(V1rep,arr(A_27b,A_27a))
=> ! [V2x_27: $i] :
( mem(V2x_27,A_27b)
=> ! [V3x_27_27: $i] : ( ap(f25(A_27a,A_27b,V1rep,V2x_27),V3x_27_27) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Emin_2E_3D(A_27a),ap(V1rep,V2x_27)),ap(V1rep,V3x_27_27))),ap(ap(c_2Emin_2E_3D(A_27b),V2x_27),V3x_27_27)) ) ) ) ).
tff(lamtp_f26,type,
f26: ( del * del * $i ) > $i ).
tff(lameq_f26,axiom,
! [A_27a: del,A_27b: del,V1rep: $i] :
( mem(V1rep,arr(A_27b,A_27a))
=> ! [V2x_27: $i] : ( ap(f26(A_27a,A_27b,V1rep),V2x_27) = ap(c_2Ebool_2E_21(A_27b),f25(A_27a,A_27b,V1rep,V2x_27)) ) ) ).
tff(lamtp_f27,type,
f27: ( del * del * $i * $i ) > $i ).
tff(lameq_f27,axiom,
! [A_27a: del,A_27b: del,V1rep: $i] :
( mem(V1rep,arr(A_27b,A_27a))
=> ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5x_27: $i] : ( ap(f27(A_27a,A_27b,V1rep,V4x),V5x_27) = ap(ap(c_2Emin_2E_3D(A_27a),V4x),ap(V1rep,V5x_27)) ) ) ) ).
tff(lamtp_f28,type,
f28: ( del * del * $i * $i ) > $i ).
tff(lameq_f28,axiom,
! [A_27b: del,A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1rep: $i] :
( mem(V1rep,arr(A_27b,A_27a))
=> ! [V4x: $i] : ( ap(f28(A_27b,A_27a,V0P,V1rep),V4x) = ap(ap(c_2Emin_2E_3D(bool),ap(V0P,V4x)),ap(c_2Ebool_2E_3F(A_27b),f27(A_27a,A_27b,V1rep,V4x))) ) ) ) ).
tff(lamtp_f29,type,
f29: ( del * del * $i ) > $i ).
tff(lameq_f29,axiom,
! [A_27b: del,A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1rep: $i] : ( ap(f29(A_27b,A_27a,V0P),V1rep) = ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Ebool_2E_21(A_27b),f26(A_27a,A_27b,V1rep))),ap(c_2Ebool_2E_21(A_27a),f28(A_27b,A_27a,V0P,V1rep))) ) ) ).
tff(lamtp_f30,type,
f30: ( del * del ) > $i ).
tff(lameq_f30,axiom,
! [A_27b: del,A_27a: del,V0P: $i] : ( ap(f30(A_27b,A_27a),V0P) = f29(A_27b,A_27a,V0P) ) ).
tff(ax_thm_2Ebool_2ETYPE__DEFINITION,axiom,
! [A_27a: del,A_27b: del] : ( c_2Ebool_2ETYPE__DEFINITION(A_27a,A_27b) = f30(A_27b,A_27a) ) ).
tff(ax_thm_2Ebool_2EBOOL__CASES__AX,axiom,
! [V0t: tp__o] :
( ( p(inj__o(V0t))
<=> $true )
| ( p(inj__o(V0t))
<=> $false ) ) ).
tff(lamtp_f31,type,
f31: ( del * del * $i ) > $i ).
tff(lameq_f31,axiom,
! [A_27b: del,A_27a: del,V0t: $i] :
( mem(V0t,arr(A_27a,A_27b))
=> ! [V1x: $i] : ( ap(f31(A_27b,A_27a,V0t),V1x) = ap(V0t,V1x) ) ) ).
tff(ax_thm_2Ebool_2EETA__AX,axiom,
! [A_27a: del,A_27b: del,V0t: $i] :
( mem(V0t,arr(A_27a,A_27b))
=> ( f31(A_27b,A_27a,V0t) = V0t ) ) ).
tff(ax_thm_2Ebool_2ESELECT__AX,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ( p(ap(V0P,V1x))
=> p(ap(V0P,ap(c_2Emin_2E_40(A_27a),V0P))) ) ) ) ).
tff(stp_i,type,
tp__i: $tType ).
tff(stp_inj_i,type,
inj__i: tp__i > $i ).
tff(stp_surj_i,type,
surj__i: $i > tp__i ).
tff(stp_inj_surj_i,axiom,
! [X: tp__i] : ( surj__i(inj__i(X)) = X ) ).
tff(stp_inj_mem_i,axiom,
! [X: tp__i] : mem(inj__i(X),ind) ).
tff(stp_iso_mem_i,axiom,
! [X: $i] :
( mem(X,ind)
=> ( X = inj__i(surj__i(X)) ) ) ).
tff(ax_thm_2Ebool_2EINFINITY__AX,axiom,
? [V0f: $i] :
( mem(V0f,arr(ind,ind))
& p(ap(c_2Ebool_2EONE__ONE(ind,ind),V0f))
& ~ p(ap(c_2Ebool_2EONTO(ind,ind),V0f)) ) ).
tff(ax_thm_2Ebool_2Eliteral__case__DEF,axiom,
! [A_27a: del,A_27b: del] : ( c_2Ebool_2Eliteral__case(A_27a,A_27b) = f14(A_27b,A_27a) ) ).
tff(lamtp_f32,type,
f32: ( del * $i ) > $i ).
tff(lameq_f32,axiom,
! [A_27a: del,V0x: $i] :
( mem(V0x,A_27a)
=> ! [V1f: $i] : ( ap(f32(A_27a,V0x),V1f) = ap(V1f,V0x) ) ) ).
tff(lamtp_f33,type,
f33: del > $i ).
tff(lameq_f33,axiom,
! [A_27a: del,V0x: $i] : ( ap(f33(A_27a),V0x) = f32(A_27a,V0x) ) ).
tff(ax_thm_2Ebool_2EIN__DEF,axiom,
! [A_27a: del] : ( c_2Ebool_2EIN(A_27a) = f33(A_27a) ) ).
tff(lamtp_f34,type,
f34: ( del * $i * $i ) > $i ).
tff(lameq_f34,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,arr(A_27a,bool))
=> ! [V1m: $i] :
( mem(V1m,arr(A_27a,bool))
=> ! [V2x: $i] : ( ap(f34(A_27a,V0p,V1m),V2x) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0p)),ap(V1m,V2x)) ) ) ) ).
tff(lamtp_f35,type,
f35: ( del * $i ) > $i ).
tff(lameq_f35,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,arr(A_27a,bool))
=> ! [V1m: $i] : ( ap(f35(A_27a,V0p),V1m) = ap(c_2Ebool_2E_21(A_27a),f34(A_27a,V0p,V1m)) ) ) ).
tff(lamtp_f36,type,
f36: del > $i ).
tff(lameq_f36,axiom,
! [A_27a: del,V0p: $i] : ( ap(f36(A_27a),V0p) = f35(A_27a,V0p) ) ).
tff(ax_thm_2Ebool_2ERES__FORALL__DEF,axiom,
! [A_27a: del] : ( c_2Ebool_2ERES__FORALL(A_27a) = f36(A_27a) ) ).
tff(lamtp_f37,type,
f37: ( del * $i * $i ) > $i ).
tff(lameq_f37,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,arr(A_27a,bool))
=> ! [V1m: $i] :
( mem(V1m,arr(A_27a,bool))
=> ! [V2x: $i] : ( ap(f37(A_27a,V0p,V1m),V2x) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0p)),ap(V1m,V2x)) ) ) ) ).
tff(lamtp_f38,type,
f38: ( del * $i ) > $i ).
tff(lameq_f38,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,arr(A_27a,bool))
=> ! [V1m: $i] : ( ap(f38(A_27a,V0p),V1m) = ap(c_2Ebool_2E_3F(A_27a),f37(A_27a,V0p,V1m)) ) ) ).
tff(lamtp_f39,type,
f39: del > $i ).
tff(lameq_f39,axiom,
! [A_27a: del,V0p: $i] : ( ap(f39(A_27a),V0p) = f38(A_27a,V0p) ) ).
tff(ax_thm_2Ebool_2ERES__EXISTS__DEF,axiom,
! [A_27a: del] : ( c_2Ebool_2ERES__EXISTS(A_27a) = f39(A_27a) ) ).
tff(lamtp_f40,type,
f40: ( del * $i ) > $i ).
tff(lameq_f40,axiom,
! [A_27a: del,V1m: $i] :
( mem(V1m,arr(A_27a,bool))
=> ! [V2x: $i] : ( ap(f40(A_27a,V1m),V2x) = ap(V1m,V2x) ) ) ).
tff(lamtp_f41,type,
f41: ( del * $i * $i ) > $i ).
tff(lameq_f41,axiom,
! [A_27a: del,V1m: $i] :
( mem(V1m,arr(A_27a,bool))
=> ! [V3x: $i] :
( mem(V3x,A_27a)
=> ! [V4y: $i] : ( ap(f41(A_27a,V1m,V3x),V4y) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2E_2F_5C,ap(V1m,V3x)),ap(V1m,V4y))),ap(ap(c_2Emin_2E_3D(A_27a),V3x),V4y)) ) ) ) ).
tff(lamtp_f42,type,
f42: ( del * $i * $i ) > $i ).
tff(lameq_f42,axiom,
! [A_27a: del,V1m: $i] :
( mem(V1m,arr(A_27a,bool))
=> ! [V0p: $i] :
( mem(V0p,arr(A_27a,bool))
=> ! [V3x: $i] : ( ap(f42(A_27a,V1m,V0p),V3x) = ap(ap(c_2Ebool_2ERES__FORALL(A_27a),V0p),f41(A_27a,V1m,V3x)) ) ) ) ).
tff(lamtp_f43,type,
f43: ( del * $i ) > $i ).
tff(lameq_f43,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,arr(A_27a,bool))
=> ! [V1m: $i] : ( ap(f43(A_27a,V0p),V1m) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2ERES__EXISTS(A_27a),V0p),f40(A_27a,V1m))),ap(ap(c_2Ebool_2ERES__FORALL(A_27a),V0p),f42(A_27a,V1m,V0p))) ) ) ).
tff(lamtp_f44,type,
f44: del > $i ).
tff(lameq_f44,axiom,
! [A_27a: del,V0p: $i] : ( ap(f44(A_27a),V0p) = f43(A_27a,V0p) ) ).
tff(ax_thm_2Ebool_2ERES__EXISTS__UNIQUE__DEF,axiom,
! [A_27a: del] : ( c_2Ebool_2ERES__EXISTS__UNIQUE(A_27a) = f44(A_27a) ) ).
tff(lamtp_f45,type,
f45: ( del * $i ) > $i ).
tff(lameq_f45,axiom,
! [A_27a: del,V0p: $i] :
( mem(V0p,arr(A_27a,bool))
=> ! [V1m: $i] : ( ap(f45(A_27a,V0p),V1m) = ap(c_2Emin_2E_40(A_27a),f37(A_27a,V0p,V1m)) ) ) ).
tff(lamtp_f46,type,
f46: del > $i ).
tff(lameq_f46,axiom,
! [A_27a: del,V0p: $i] : ( ap(f46(A_27a),V0p) = f45(A_27a,V0p) ) ).
tff(ax_thm_2Ebool_2ERES__SELECT__DEF,axiom,
! [A_27a: del] : ( c_2Ebool_2ERES__SELECT(A_27a) = f46(A_27a) ) ).
tff(ax_thm_2Ebool_2EBOUNDED__DEF,axiom,
c_2Ebool_2EBOUNDED = k(bool,c_2Ebool_2ET) ).
tff(ax_thm_2Ebool_2EDATATYPE__TAG__DEF,axiom,
! [A_27a: del] : ( c_2Ebool_2EDATATYPE(A_27a) = k(A_27a,c_2Ebool_2ET) ) ).
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(lamtp_f47,type,
f47: ( del * del * $i ) > $i ).
tff(lameq_f47,axiom,
! [A_27b: del,A_27a: del,V0M: $i] :
( mem(V0M,arr(A_27a,A_27b))
=> ! [V1x: $i] : ( ap(f47(A_27b,A_27a,V0M),V1x) = ap(V0M,V1x) ) ) ).
tff(conj_thm_2Ebool_2EETA__THM,axiom,
! [A_27a: del,A_27b: del,V0M: $i] :
( mem(V0M,arr(A_27a,A_27b))
=> ( f47(A_27b,A_27a,V0M) = V0M ) ) ).
tff(conj_thm_2Ebool_2EEXCLUDED__MIDDLE,axiom,
! [V0t: tp__o] :
( p(inj__o(V0t))
| ~ p(inj__o(V0t)) ) ).
tff(lamtp_f48,type,
f48: ( del * del * $i ) > $i ).
tff(lameq_f48,axiom,
! [A_27b: del,A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V2x: $i] : ( ap(f48(A_27b,A_27a,V0f),V2x) = ap(V0f,V2x) ) ) ).
tff(conj_thm_2Ebool_2EBETA__THM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1y: $i] :
( mem(V1y,A_27a)
=> ( ap(f48(A_27b,A_27a,V0f),V1y) = ap(V0f,V1y) ) ) ) ).
tff(conj_thm_2Ebool_2ELET__THM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ( ap(ap(c_2Ebool_2ELET(A_27a,A_27b),V0f),V1x) = ap(V0f,V1x) ) ) ) ).
tff(conj_thm_2Ebool_2EFORALL__THM,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,bool))
=> ( p(ap(c_2Ebool_2E_21(A_27a),V0f))
<=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> p(ap(V0f,V1x)) ) ) ) ).
tff(conj_thm_2Ebool_2EEXISTS__THM,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,bool))
=> ( p(ap(c_2Ebool_2E_3F(A_27a),V0f))
<=> ? [V1x: $i] :
( mem(V1x,A_27a)
& p(ap(V0f,V1x)) ) ) ) ).
tff(conj_thm_2Ebool_2EABS__SIMP,axiom,
! [A_27a: del,A_27b: del,V0t1: $i] :
( mem(V0t1,A_27a)
=> ! [V1t2: $i] :
( mem(V1t2,A_27b)
=> ( ap(k(A_27b,V0t1),V1t2) = V0t1 ) ) ) ).
tff(conj_thm_2Ebool_2EFORALL__SIMP,axiom,
! [A_27a: del,V0t: tp__o] :
( ! [V1x: $i] :
( mem(V1x,A_27a)
=> p(inj__o(V0t)) )
<=> p(inj__o(V0t)) ) ).
tff(conj_thm_2Ebool_2EEXISTS__SIMP,axiom,
! [A_27a: del,V0t: tp__o] :
( ? [V1x: $i] :
( mem(V1x,A_27a)
& p(inj__o(V0t)) )
<=> p(inj__o(V0t)) ) ).
tff(conj_thm_2Ebool_2EAND__INTRO__THM,axiom,
! [V0t1: tp__o,V1t2: tp__o] :
( p(inj__o(V0t1))
=> ( p(inj__o(V1t2))
=> ( p(inj__o(V0t1))
& p(inj__o(V1t2)) ) ) ) ).
tff(conj_thm_2Ebool_2EAND1__THM,axiom,
! [V0t1: tp__o,V1t2: tp__o] :
( ( p(inj__o(V0t1))
& p(inj__o(V1t2)) )
=> p(inj__o(V0t1)) ) ).
tff(conj_thm_2Ebool_2EAND2__THM,axiom,
! [V0t1: tp__o,V1t2: tp__o] :
( ( p(inj__o(V0t1))
& p(inj__o(V1t2)) )
=> p(inj__o(V1t2)) ) ).
tff(conj_thm_2Ebool_2ECONJ__SYM,axiom,
! [V0t1: tp__o,V1t2: tp__o] :
( ( p(inj__o(V0t1))
& p(inj__o(V1t2)) )
<=> ( p(inj__o(V1t2))
& p(inj__o(V0t1)) ) ) ).
tff(conj_thm_2Ebool_2ECONJ__COMM,axiom,
! [V0t1: tp__o,V1t2: tp__o] :
( ( p(inj__o(V0t1))
& p(inj__o(V1t2)) )
<=> ( p(inj__o(V1t2))
& p(inj__o(V0t1)) ) ) ).
tff(conj_thm_2Ebool_2ECONJ__ASSOC,axiom,
! [V0t1: tp__o,V1t2: tp__o,V2t3: tp__o] :
( ( p(inj__o(V0t1))
& p(inj__o(V1t2))
& p(inj__o(V2t3)) )
<=> ( p(inj__o(V0t1))
& p(inj__o(V1t2))
& p(inj__o(V2t3)) ) ) ).
tff(conj_thm_2Ebool_2EOR__INTRO__THM1,axiom,
! [V0t1: tp__o,V1t2: tp__o] :
( p(inj__o(V0t1))
=> ( p(inj__o(V0t1))
| p(inj__o(V1t2)) ) ) ).
tff(conj_thm_2Ebool_2EOR__INTRO__THM2,axiom,
! [V0t1: tp__o,V1t2: tp__o] :
( p(inj__o(V1t2))
=> ( p(inj__o(V0t1))
| p(inj__o(V1t2)) ) ) ).
tff(conj_thm_2Ebool_2EOR__ELIM__THM,axiom,
! [V0t: tp__o,V1t1: tp__o,V2t2: tp__o] :
( ( p(inj__o(V1t1))
| p(inj__o(V2t2)) )
=> ( ( p(inj__o(V1t1))
=> p(inj__o(V0t)) )
=> ( ( p(inj__o(V2t2))
=> 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_2ENOT__F,axiom,
! [V0t: tp__o] :
( ~ p(inj__o(V0t))
=> ( p(inj__o(V0t))
<=> $false ) ) ).
tff(conj_thm_2Ebool_2ENOT__AND,axiom,
! [V0t: tp__o] :
~ ( p(inj__o(V0t))
& ~ p(inj__o(V0t)) ) ).
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_2EEQ__REFL,axiom,
! [A_27a: del,V0x: $i] :
( mem(V0x,A_27a)
=> ( V0x = V0x ) ) ).
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,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__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__EXT,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,A_27b))
=> ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( ap(V0f,V2x) = ap(V1g,V2x) ) )
=> ( V0f = V1g ) ) ) ) ).
tff(conj_thm_2Ebool_2EFUN__EQ__THM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,A_27b))
=> ( ( V0f = V1g )
<=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( ap(V0f,V2x) = ap(V1g,V2x) ) ) ) ) ) ).
tff(conj_thm_2Ebool_2EEQ__TRANS,axiom,
! [A_27a: del,V0x: $i] :
( mem(V0x,A_27a)
=> ! [V1y: $i] :
( mem(V1y,A_27a)
=> ! [V2z: $i] :
( mem(V2z,A_27a)
=> ( ( ( V0x = V1y )
& ( V1y = V2z ) )
=> ( V0x = V2z ) ) ) ) ) ).
tff(conj_thm_2Ebool_2EBOOL__EQ__DISTINCT,axiom,
( ~ ( $true
<=> $false )
& ~ ( $false
<=> $true ) ) ).
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_2ECOND__ID,axiom,
! [A_27a: del,V0b: tp__o,V1t: $i] :
( mem(V1t,A_27a)
=> ( ap(ap(ap(c_2Ebool_2ECOND(A_27a),inj__o(V0b)),V1t),V1t) = V1t ) ) ).
tff(lamtp_f49,type,
f49: ( del * $i ) > $i ).
tff(lameq_f49,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1x: $i] : ( ap(f49(A_27a,V0P),V1x) = ap(V0P,V1x) ) ) ).
tff(conj_thm_2Ebool_2ESELECT__THM,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ( p(ap(V0P,ap(c_2Emin_2E_40(A_27a),f49(A_27a,V0P))))
<=> ? [V2x: $i] :
( mem(V2x,A_27a)
& p(ap(V0P,V2x)) ) ) ) ).
tff(lamtp_f50,type,
f50: ( del * $i ) > $i ).
tff(lameq_f50,axiom,
! [A_27a: del,V0x: $i] :
( mem(V0x,A_27a)
=> ! [V1y: $i] : ( ap(f50(A_27a,V0x),V1y) = ap(ap(c_2Emin_2E_3D(A_27a),V1y),V0x) ) ) ).
tff(conj_thm_2Ebool_2ESELECT__REFL,axiom,
! [A_27a: del,V0x: $i] :
( mem(V0x,A_27a)
=> ( ap(c_2Emin_2E_40(A_27a),f50(A_27a,V0x)) = V0x ) ) ).
tff(lamtp_f51,type,
f51: ( del * $i ) > $i ).
tff(lameq_f51,axiom,
! [A_27a: del,V0x: $i] :
( mem(V0x,A_27a)
=> ! [V1y: $i] : ( ap(f51(A_27a,V0x),V1y) = ap(ap(c_2Emin_2E_3D(A_27a),V0x),V1y) ) ) ).
tff(conj_thm_2Ebool_2ESELECT__REFL__2,axiom,
! [A_27a: del,V0x: $i] :
( mem(V0x,A_27a)
=> ( ap(c_2Emin_2E_40(A_27a),f51(A_27a,V0x)) = V0x ) ) ).
tff(conj_thm_2Ebool_2ESELECT__UNIQUE,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ( ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( p(ap(V0P,V2y))
<=> ( V2y = V1x ) ) )
=> ( ap(c_2Emin_2E_40(A_27a),V0P) = V1x ) ) ) ) ).
tff(conj_thm_2Ebool_2ESELECT__ELIM__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)) )
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(V0P,V3x))
=> p(ap(V1Q,V3x)) ) ) )
=> p(ap(V1Q,ap(c_2Emin_2E_40(A_27a),V0P))) ) ) ) ).
tff(conj_thm_2Ebool_2ENOT__FORALL__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_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_2EFORALL__AND__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__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_2ERIGHT__AND__FORALL__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_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_2EBOTH__EXISTS__AND__THM,axiom,
! [A_27a: del,V0P: tp__o,V1Q: tp__o] :
( ? [V2x: $i] :
( mem(V2x,A_27a)
& p(inj__o(V0P))
& p(inj__o(V1Q)) )
<=> ( ? [V3x: $i] :
( mem(V3x,A_27a)
& p(inj__o(V0P)) )
& ? [V4x: $i] :
( mem(V4x,A_27a)
& p(inj__o(V1Q)) ) ) ) ).
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_2EBOTH__FORALL__OR__THM,axiom,
! [A_27a: del,V0P: tp__o,V1Q: tp__o] :
( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( p(inj__o(V0P))
| p(inj__o(V1Q)) ) )
<=> ( ! [V3x: $i] :
( mem(V3x,A_27a)
=> p(inj__o(V0P)) )
| ! [V4x: $i] :
( mem(V4x,A_27a)
=> p(inj__o(V1Q)) ) ) ) ).
tff(conj_thm_2Ebool_2ELEFT__FORALL__OR__THM,axiom,
! [A_27a: del,V0Q: tp__o,V1P: $i] :
( mem(V1P,arr(A_27a,bool))
=> ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( p(ap(V1P,V2x))
| p(inj__o(V0Q)) ) )
<=> ( ! [V3x: $i] :
( mem(V3x,A_27a)
=> p(ap(V1P,V3x)) )
| p(inj__o(V0Q)) ) ) ) ).
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_2EBOTH__FORALL__IMP__THM,axiom,
! [A_27a: del,V0P: tp__o,V1Q: tp__o] :
( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( p(inj__o(V0P))
=> p(inj__o(V1Q)) ) )
<=> ( ? [V3x: $i] :
( mem(V3x,A_27a)
& p(inj__o(V0P)) )
=> ! [V4x: $i] :
( mem(V4x,A_27a)
=> p(inj__o(V1Q)) ) ) ) ).
tff(conj_thm_2Ebool_2ELEFT__FORALL__IMP__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__FORALL__IMP__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_2EBOTH__EXISTS__IMP__THM,axiom,
! [A_27a: del,V0P: tp__o,V1Q: tp__o] :
( ? [V2x: $i] :
( mem(V2x,A_27a)
& ( p(inj__o(V0P))
=> p(inj__o(V1Q)) ) )
<=> ( ! [V3x: $i] :
( mem(V3x,A_27a)
=> p(inj__o(V0P)) )
=> ? [V4x: $i] :
( mem(V4x,A_27a)
& p(inj__o(V1Q)) ) ) ) ).
tff(conj_thm_2Ebool_2ELEFT__EXISTS__IMP__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__IMP__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_2EOR__IMP__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)) ) ) ).
tff(conj_thm_2Ebool_2ENOT__IMP,axiom,
! [V0A: tp__o,V1B: tp__o] :
( ~ ( p(inj__o(V0A))
=> p(inj__o(V1B)) )
<=> ( p(inj__o(V0A))
& ~ p(inj__o(V1B)) ) ) ).
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_2EDISJ__COMM,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_2ELEFT__AND__OVER__OR,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(V0A))
& p(inj__o(V2C)) ) ) ) ).
tff(conj_thm_2Ebool_2ERIGHT__AND__OVER__OR,axiom,
! [V0A: tp__o,V1B: tp__o,V2C: tp__o] :
( ( ( p(inj__o(V1B))
| p(inj__o(V2C)) )
& p(inj__o(V0A)) )
<=> ( ( p(inj__o(V1B))
& p(inj__o(V0A)) )
| ( p(inj__o(V2C))
& p(inj__o(V0A)) ) ) ) ).
tff(conj_thm_2Ebool_2ELEFT__OR__OVER__AND,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(V0A))
| p(inj__o(V2C)) ) ) ) ).
tff(conj_thm_2Ebool_2ERIGHT__OR__OVER__AND,axiom,
! [V0A: tp__o,V1B: tp__o,V2C: tp__o] :
( ( ( p(inj__o(V1B))
& p(inj__o(V2C)) )
| p(inj__o(V0A)) )
<=> ( ( p(inj__o(V1B))
| p(inj__o(V0A)) )
& ( p(inj__o(V2C))
| p(inj__o(V0A)) ) ) ) ).
tff(conj_thm_2Ebool_2EIMP__DISJ__THM,axiom,
! [V0A: tp__o,V1B: tp__o] :
( ( p(inj__o(V0A))
=> p(inj__o(V1B)) )
<=> ( ~ p(inj__o(V0A))
| p(inj__o(V1B)) ) ) ).
tff(conj_thm_2Ebool_2EDISJ__IMP__THM,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(V2R)) )
& ( p(inj__o(V1Q))
=> p(inj__o(V2R)) ) ) ) ).
tff(conj_thm_2Ebool_2EIMP__CONJ__THM,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)) ) ) ) ).
tff(conj_thm_2Ebool_2EIMP__F__EQ__F,axiom,
! [V0t: tp__o] :
( ( p(inj__o(V0t))
=> $false )
<=> ( p(inj__o(V0t))
<=> $false ) ) ).
tff(conj_thm_2Ebool_2EAND__IMP__INTRO,axiom,
! [V0t1: tp__o,V1t2: tp__o,V2t3: tp__o] :
( ( p(inj__o(V0t1))
=> ( p(inj__o(V1t2))
=> p(inj__o(V2t3)) ) )
<=> ( ( p(inj__o(V0t1))
& p(inj__o(V1t2)) )
=> p(inj__o(V2t3)) ) ) ).
tff(conj_thm_2Ebool_2EEQ__IMP__THM,axiom,
! [V0t1: tp__o,V1t2: tp__o] :
( ( p(inj__o(V0t1))
<=> p(inj__o(V1t2)) )
<=> ( ( p(inj__o(V0t1))
=> p(inj__o(V1t2)) )
& ( p(inj__o(V1t2))
=> p(inj__o(V0t1)) ) ) ) ).
tff(conj_thm_2Ebool_2EEQ__EXPAND,axiom,
! [V0t1: tp__o,V1t2: tp__o] :
( ( p(inj__o(V0t1))
<=> p(inj__o(V1t2)) )
<=> ( ( p(inj__o(V0t1))
& p(inj__o(V1t2)) )
| ( ~ p(inj__o(V0t1))
& ~ p(inj__o(V1t2)) ) ) ) ).
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(lamtp_f52,type,
f52: ( del * del * tp__o * $i * $i ) > $i ).
tff(lameq_f52,axiom,
! [A_27b: del,A_27a: del,V0b: tp__o,V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2g: $i] :
( mem(V2g,arr(A_27a,A_27b))
=> ! [V3x: $i] : ( ap(f52(A_27b,A_27a,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__ABS,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))
=> ( f52(A_27b,A_27a,V0b,V1f,V2g) = ap(ap(ap(c_2Ebool_2ECOND(arr(A_27a,A_27b)),inj__o(V0b)),V1f),V2g) ) ) ) ).
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_2ECOND__EXPAND__IMP,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_2ECOND__EXPAND__OR,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_2ETYPE__DEFINITION__THM,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1rep: $i] :
( mem(V1rep,arr(A_27b,A_27a))
=> ( p(ap(ap(c_2Ebool_2ETYPE__DEFINITION(A_27a,A_27b),V0P),V1rep))
<=> ( ! [V2x_27: $i] :
( mem(V2x_27,A_27b)
=> ! [V3x_27_27: $i] :
( mem(V3x_27_27,A_27b)
=> ( ( ap(V1rep,V2x_27) = ap(V1rep,V3x_27_27) )
=> ( V2x_27 = V3x_27_27 ) ) ) )
& ! [V4x: $i] :
( mem(V4x,A_27a)
=> ( p(ap(V0P,V4x))
<=> ? [V5x_27: $i] :
( mem(V5x_27,A_27b)
& ( V4x = ap(V1rep,V5x_27) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Ebool_2EONTO__THM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ( p(ap(c_2Ebool_2EONTO(A_27a,A_27b),V0f))
<=> ! [V1y: $i] :
( mem(V1y,A_27b)
=> ? [V2x: $i] :
( mem(V2x,A_27a)
& ( V1y = ap(V0f,V2x) ) ) ) ) ) ).
tff(conj_thm_2Ebool_2EONE__ONE__THM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ( p(ap(c_2Ebool_2EONE__ONE(A_27a,A_27b),V0f))
<=> ! [V1x1: $i] :
( mem(V1x1,A_27a)
=> ! [V2x2: $i] :
( mem(V2x2,A_27a)
=> ( ( ap(V0f,V1x1) = ap(V0f,V2x2) )
=> ( V1x1 = V2x2 ) ) ) ) ) ) ).
tff(conj_thm_2Ebool_2EABS__REP__THM,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ( ? [V1rep: $i] :
( mem(V1rep,arr(A_27b,A_27a))
& p(ap(ap(c_2Ebool_2ETYPE__DEFINITION(A_27a,A_27b),V0P),V1rep)) )
=> ? [V2rep: $i] :
( mem(V2rep,arr(A_27b,A_27a))
& ? [V3abs: $i] :
( mem(V3abs,arr(A_27a,A_27b))
& ! [V4a: $i] :
( mem(V4a,A_27b)
=> ( ap(V3abs,ap(V2rep,V4a)) = V4a ) )
& ! [V5r: $i] :
( mem(V5r,A_27a)
=> ( p(ap(V0P,V5r))
<=> ( ap(V2rep,ap(V3abs,V5r)) = V5r ) ) ) ) ) ) ) ).
tff(lamtp_f53,type,
f53: ( del * del * $i ) > $i ).
tff(lameq_f53,axiom,
! [A_27b: del,A_27a: del,V1N: $i] :
( mem(V1N,arr(A_27a,A_27b))
=> ! [V3x: $i] : ( ap(f53(A_27b,A_27a,V1N),V3x) = ap(V1N,V3x) ) ) ).
tff(lamtp_f54,type,
f54: ( del * del * $i * $i ) > $i ).
tff(lameq_f54,axiom,
! [A_27b: del,A_27a: del,V1N: $i] :
( mem(V1N,arr(A_27a,A_27b))
=> ! [V0P: $i] :
( mem(V0P,arr(A_27b,bool))
=> ! [V4x: $i] : ( ap(f54(A_27b,A_27a,V1N,V0P),V4x) = ap(V0P,ap(V1N,V4x)) ) ) ) ).
tff(conj_thm_2Ebool_2ELET__RAND,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( mem(V0P,arr(A_27b,bool))
=> ! [V1N: $i] :
( mem(V1N,arr(A_27a,A_27b))
=> ! [V2M: $i] :
( mem(V2M,A_27a)
=> ( p(ap(V0P,ap(ap(c_2Ebool_2ELET(A_27a,A_27b),f53(A_27b,A_27a,V1N)),V2M)))
<=> p(ap(ap(c_2Ebool_2ELET(A_27a,bool),f54(A_27b,A_27a,V1N,V0P)),V2M)) ) ) ) ) ).
tff(lamtp_f55,type,
f55: ( del * del * del * $i ) > $i ).
tff(lameq_f55,axiom,
! [A_27c: del,A_27b: del,A_27a: del,V0N: $i] :
( mem(V0N,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V3x: $i] : ( ap(f55(A_27c,A_27b,A_27a,V0N),V3x) = ap(V0N,V3x) ) ) ).
tff(lamtp_f56,type,
f56: ( del * del * del * $i * $i ) > $i ).
tff(lameq_f56,axiom,
! [A_27b: del,A_27c: del,A_27a: del,V0N: $i] :
( mem(V0N,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V2b: $i] :
( mem(V2b,A_27b)
=> ! [V4x: $i] : ( ap(f56(A_27b,A_27c,A_27a,V0N,V2b),V4x) = ap(ap(V0N,V4x),V2b) ) ) ) ).
tff(conj_thm_2Ebool_2ELET__RATOR,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0N: $i] :
( mem(V0N,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V1M: $i] :
( mem(V1M,A_27a)
=> ! [V2b: $i] :
( mem(V2b,A_27b)
=> ( ap(ap(ap(c_2Ebool_2ELET(A_27a,arr(A_27b,A_27c)),f55(A_27c,A_27b,A_27a,V0N)),V1M),V2b) = ap(ap(c_2Ebool_2ELET(A_27a,A_27c),f56(A_27b,A_27c,A_27a,V0N,V2b)),V1M) ) ) ) ) ).
tff(conj_thm_2Ebool_2ESWAP__FORALL__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)) ) )
<=> ! [V3y: $i] :
( mem(V3y,A_27b)
=> ! [V4x: $i] :
( mem(V4x,A_27a)
=> p(ap(ap(V0P,V4x),V3y)) ) ) ) ) ).
tff(conj_thm_2Ebool_2ESWAP__EXISTS__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)) ) )
<=> ? [V3y: $i] :
( mem(V3y,A_27b)
& ? [V4x: $i] :
( mem(V4x,A_27a)
& p(ap(ap(V0P,V4x),V3y)) ) ) ) ) ).
tff(conj_thm_2Ebool_2EEXISTS__UNIQUE__THM,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ( p(ap(c_2Ebool_2E_3F_21(A_27a),f49(A_27a,V0P)))
<=> ( ? [V2x: $i] :
( mem(V2x,A_27a)
& p(ap(V0P,V2x)) )
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ! [V4y: $i] :
( mem(V4y,A_27a)
=> ( ( p(ap(V0P,V3x))
& p(ap(V0P,V4y)) )
=> ( V3x = V4y ) ) ) ) ) ) ) ).
tff(conj_thm_2Ebool_2ELET__CONG,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,A_27b))
=> ! [V2M: $i] :
( mem(V2M,A_27a)
=> ! [V3N: $i] :
( mem(V3N,A_27a)
=> ( ( ( V2M = V3N )
& ! [V4x: $i] :
( mem(V4x,A_27a)
=> ( ( V4x = V3N )
=> ( ap(V0f,V4x) = ap(V1g,V4x) ) ) ) )
=> ( ap(ap(c_2Ebool_2ELET(A_27a,A_27b),V0f),V2M) = ap(ap(c_2Ebool_2ELET(A_27a,A_27b),V1g),V3N) ) ) ) ) ) ) ).
tff(conj_thm_2Ebool_2EIMP__CONG,axiom,
! [V0x: tp__o,V1x_27: tp__o,V2y: tp__o,V3y_27: tp__o] :
( ( ( p(inj__o(V0x))
<=> p(inj__o(V1x_27)) )
& ( p(inj__o(V1x_27))
=> ( p(inj__o(V2y))
<=> p(inj__o(V3y_27)) ) ) )
=> ( ( p(inj__o(V0x))
=> p(inj__o(V2y)) )
<=> ( p(inj__o(V1x_27))
=> p(inj__o(V3y_27)) ) ) ) ).
tff(conj_thm_2Ebool_2EAND__CONG,axiom,
! [V0P: tp__o,V1P_27: tp__o,V2Q: tp__o,V3Q_27: tp__o] :
( ( ( p(inj__o(V2Q))
=> ( p(inj__o(V0P))
<=> p(inj__o(V1P_27)) ) )
& ( p(inj__o(V1P_27))
=> ( p(inj__o(V2Q))
<=> p(inj__o(V3Q_27)) ) ) )
=> ( ( p(inj__o(V0P))
& p(inj__o(V2Q)) )
<=> ( p(inj__o(V1P_27))
& p(inj__o(V3Q_27)) ) ) ) ).
tff(conj_thm_2Ebool_2ELEFT__AND__CONG,axiom,
! [V0P: tp__o,V1P_27: tp__o,V2Q: tp__o,V3Q_27: tp__o] :
( ( ( p(inj__o(V0P))
<=> p(inj__o(V1P_27)) )
& ( p(inj__o(V1P_27))
=> ( p(inj__o(V2Q))
<=> p(inj__o(V3Q_27)) ) ) )
=> ( ( p(inj__o(V0P))
& p(inj__o(V2Q)) )
<=> ( p(inj__o(V1P_27))
& p(inj__o(V3Q_27)) ) ) ) ).
tff(conj_thm_2Ebool_2EOR__CONG,axiom,
! [V0P: tp__o,V1P_27: tp__o,V2Q: tp__o,V3Q_27: tp__o] :
( ( ( ~ p(inj__o(V2Q))
=> ( p(inj__o(V0P))
<=> p(inj__o(V1P_27)) ) )
& ( ~ p(inj__o(V1P_27))
=> ( p(inj__o(V2Q))
<=> p(inj__o(V3Q_27)) ) ) )
=> ( ( p(inj__o(V0P))
| p(inj__o(V2Q)) )
<=> ( p(inj__o(V1P_27))
| p(inj__o(V3Q_27)) ) ) ) ).
tff(conj_thm_2Ebool_2ELEFT__OR__CONG,axiom,
! [V0P: tp__o,V1P_27: tp__o,V2Q: tp__o,V3Q_27: tp__o] :
( ( ( p(inj__o(V0P))
<=> p(inj__o(V1P_27)) )
& ( ~ p(inj__o(V1P_27))
=> ( p(inj__o(V2Q))
<=> p(inj__o(V3Q_27)) ) ) )
=> ( ( p(inj__o(V0P))
| p(inj__o(V2Q)) )
<=> ( p(inj__o(V1P_27))
| p(inj__o(V3Q_27)) ) ) ) ).
tff(conj_thm_2Ebool_2ECOND__CONG,axiom,
! [A_27a: del,V0P: tp__o,V1Q: tp__o,V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3x_27: $i] :
( mem(V3x_27,A_27a)
=> ! [V4y: $i] :
( mem(V4y,A_27a)
=> ! [V5y_27: $i] :
( mem(V5y_27,A_27a)
=> ( ( ( p(inj__o(V0P))
<=> p(inj__o(V1Q)) )
& ( p(inj__o(V1Q))
=> ( V2x = V3x_27 ) )
& ( ~ p(inj__o(V1Q))
=> ( V4y = V5y_27 ) ) )
=> ( ap(ap(ap(c_2Ebool_2ECOND(A_27a),inj__o(V0P)),V2x),V4y) = ap(ap(ap(c_2Ebool_2ECOND(A_27a),inj__o(V1Q)),V3x_27),V5y_27) ) ) ) ) ) ) ).
tff(conj_thm_2Ebool_2ERES__FORALL__CONG,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q: $i] :
( mem(V1Q,arr(A_27a,bool))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,bool))
=> ! [V3g: $i] :
( mem(V3g,arr(A_27a,bool))
=> ( ( V0P = V1Q )
=> ( ! [V4x: $i] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V1Q))
=> ( p(ap(V2f,V4x))
<=> p(ap(V3g,V4x)) ) ) )
=> ( p(ap(ap(c_2Ebool_2ERES__FORALL(A_27a),V0P),V2f))
<=> p(ap(ap(c_2Ebool_2ERES__FORALL(A_27a),V1Q),V3g)) ) ) ) ) ) ) ) ).
tff(conj_thm_2Ebool_2ERES__EXISTS__CONG,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q: $i] :
( mem(V1Q,arr(A_27a,bool))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,bool))
=> ! [V3g: $i] :
( mem(V3g,arr(A_27a,bool))
=> ( ( V0P = V1Q )
=> ( ! [V4x: $i] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V1Q))
=> ( p(ap(V2f,V4x))
<=> p(ap(V3g,V4x)) ) ) )
=> ( p(ap(ap(c_2Ebool_2ERES__EXISTS(A_27a),V0P),V2f))
<=> p(ap(ap(c_2Ebool_2ERES__EXISTS(A_27a),V1Q),V3g)) ) ) ) ) ) ) ) ).
tff(conj_thm_2Ebool_2EMONO__AND,axiom,
! [V0x: tp__o,V1y: tp__o,V2z: tp__o,V3w: tp__o] :
( ( ( p(inj__o(V0x))
=> p(inj__o(V1y)) )
& ( p(inj__o(V2z))
=> p(inj__o(V3w)) ) )
=> ( ( p(inj__o(V0x))
& p(inj__o(V2z)) )
=> ( p(inj__o(V1y))
& p(inj__o(V3w)) ) ) ) ).
tff(conj_thm_2Ebool_2EMONO__OR,axiom,
! [V0x: tp__o,V1y: tp__o,V2z: tp__o,V3w: tp__o] :
( ( ( p(inj__o(V0x))
=> p(inj__o(V1y)) )
& ( p(inj__o(V2z))
=> p(inj__o(V3w)) ) )
=> ( ( p(inj__o(V0x))
| p(inj__o(V2z)) )
=> ( p(inj__o(V1y))
| p(inj__o(V3w)) ) ) ) ).
tff(conj_thm_2Ebool_2EMONO__IMP,axiom,
! [V0y: tp__o,V1x: tp__o,V2z: tp__o,V3w: tp__o] :
( ( ( p(inj__o(V0y))
=> p(inj__o(V1x)) )
& ( p(inj__o(V2z))
=> p(inj__o(V3w)) ) )
=> ( ( p(inj__o(V1x))
=> p(inj__o(V2z)) )
=> ( p(inj__o(V0y))
=> p(inj__o(V3w)) ) ) ) ).
tff(conj_thm_2Ebool_2EMONO__NOT,axiom,
! [V0y: tp__o,V1x: tp__o] :
( ( p(inj__o(V0y))
=> p(inj__o(V1x)) )
=> ( ~ p(inj__o(V1x))
=> ~ p(inj__o(V0y)) ) ) ).
tff(conj_thm_2Ebool_2EMONO__NOT__EQ,axiom,
! [V0y: tp__o,V1x: tp__o] :
( ( p(inj__o(V0y))
=> p(inj__o(V1x)) )
<=> ( ~ p(inj__o(V1x))
=> ~ p(inj__o(V0y)) ) ) ).
tff(conj_thm_2Ebool_2EMONO__ALL,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_2EMONO__EXISTS,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_2EMONO__COND,axiom,
! [V0x: tp__o,V1y: tp__o,V2z: tp__o,V3w: tp__o,V4b: tp__o] :
( ( p(inj__o(V0x))
=> p(inj__o(V1y)) )
=> ( ( p(inj__o(V2z))
=> p(inj__o(V3w)) )
=> ( p(ap(ap(ap(c_2Ebool_2ECOND(bool),inj__o(V4b)),inj__o(V0x)),inj__o(V2z)))
=> p(ap(ap(ap(c_2Ebool_2ECOND(bool),inj__o(V4b)),inj__o(V1y)),inj__o(V3w))) ) ) ) ).
tff(conj_thm_2Ebool_2EEXISTS__REFL,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,A_27a)
=> ? [V1x: $i] :
( mem(V1x,A_27a)
& ( V1x = V0a ) ) ) ).
tff(lamtp_f57,type,
f57: ( del * $i ) > $i ).
tff(lameq_f57,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,A_27a)
=> ! [V1x: $i] : ( ap(f57(A_27a,V0a),V1x) = ap(ap(c_2Emin_2E_3D(A_27a),V1x),V0a) ) ) ).
tff(conj_thm_2Ebool_2EEXISTS__UNIQUE__REFL,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,A_27a)
=> p(ap(c_2Ebool_2E_3F_21(A_27a),f57(A_27a,V0a))) ) ).
tff(conj_thm_2Ebool_2EUNWIND__THM1,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1a: $i] :
( mem(V1a,A_27a)
=> ( ? [V2x: $i] :
( mem(V2x,A_27a)
& ( V1a = V2x )
& p(ap(V0P,V2x)) )
<=> p(ap(V0P,V1a)) ) ) ) ).
tff(conj_thm_2Ebool_2EUNWIND__THM2,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1a: $i] :
( mem(V1a,A_27a)
=> ( ? [V2x: $i] :
( mem(V2x,A_27a)
& ( V2x = V1a )
& p(ap(V0P,V2x)) )
<=> p(ap(V0P,V1a)) ) ) ) ).
tff(conj_thm_2Ebool_2EUNWIND__FORALL__THM1,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,bool))
=> ! [V1v: $i] :
( mem(V1v,A_27a)
=> ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( ( V1v = V2x )
=> p(ap(V0f,V2x)) ) )
<=> p(ap(V0f,V1v)) ) ) ) ).
tff(conj_thm_2Ebool_2EUNWIND__FORALL__THM2,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,bool))
=> ! [V1v: $i] :
( mem(V1v,A_27a)
=> ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( ( V2x = V1v )
=> p(ap(V0f,V2x)) ) )
<=> p(ap(V0f,V1v)) ) ) ) ).
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_2Ebool_2Ebool__case__thm,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 ) ) )
& ! [V2t1: $i] :
( mem(V2t1,A_27a)
=> ! [V3t2: $i] :
( mem(V3t2,A_27a)
=> ( ap(ap(ap(c_2Ebool_2ECOND(A_27a),inj__o(fo__c_2Ebool_2EF)),V2t1),V3t2) = V3t2 ) ) ) ) ).
tff(conj_thm_2Ebool_2Ebool__case__ID,axiom,
! [A_27a: del,V0b: tp__o,V1t: $i] :
( mem(V1t,A_27a)
=> ( ap(ap(ap(c_2Ebool_2ECOND(A_27a),inj__o(V0b)),V1t),V1t) = V1t ) ) ).
tff(conj_thm_2Ebool_2EboolAxiom,axiom,
! [A_27a: del,V0t1: $i] :
( mem(V0t1,A_27a)
=> ! [V1t2: $i] :
( mem(V1t2,A_27a)
=> ? [V2fn: $i] :
( mem(V2fn,arr(bool,A_27a))
& ( ap(V2fn,inj__o(fo__c_2Ebool_2ET)) = V0t1 )
& ( ap(V2fn,inj__o(fo__c_2Ebool_2EF)) = V1t2 ) ) ) ) ).
tff(conj_thm_2Ebool_2Ebool__INDUCT,axiom,
! [V0P: $i] :
( mem(V0P,arr(bool,bool))
=> ( ( p(ap(V0P,inj__o(fo__c_2Ebool_2ET)))
& p(ap(V0P,inj__o(fo__c_2Ebool_2EF))) )
=> ! [V1b: tp__o] : p(ap(V0P,inj__o(V1b))) ) ) ).
tff(conj_thm_2Ebool_2Ebool__case__CONG,axiom,
! [A_27a: del,V0P: tp__o,V1Q: tp__o,V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3x_27: $i] :
( mem(V3x_27,A_27a)
=> ! [V4y: $i] :
( mem(V4y,A_27a)
=> ! [V5y_27: $i] :
( mem(V5y_27,A_27a)
=> ( ( ( p(inj__o(V0P))
<=> p(inj__o(V1Q)) )
& ( p(inj__o(V1Q))
=> ( V2x = V3x_27 ) )
& ( ~ p(inj__o(V1Q))
=> ( V4y = V5y_27 ) ) )
=> ( ap(ap(ap(c_2Ebool_2ECOND(A_27a),inj__o(V0P)),V2x),V4y) = ap(ap(ap(c_2Ebool_2ECOND(A_27a),inj__o(V1Q)),V3x_27),V5y_27) ) ) ) ) ) ) ).
tff(conj_thm_2Ebool_2EFORALL__BOOL,axiom,
! [V0P: $i] :
( mem(V0P,arr(bool,bool))
=> ( ! [V1b: tp__o] : p(ap(V0P,inj__o(V1b)))
<=> ( p(ap(V0P,inj__o(fo__c_2Ebool_2ET)))
& p(ap(V0P,inj__o(fo__c_2Ebool_2EF))) ) ) ) ).
tff(lamtp_f58,type,
f58: ( del * $i * $i ) > $i ).
tff(lameq_f58,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q: $i] :
( mem(V1Q,arr(A_27a,bool))
=> ! [V2x: $i] : ( ap(f58(A_27a,V0P,V1Q),V2x) = ap(ap(c_2Ebool_2E_5C_2F,ap(V0P,V2x)),ap(V1Q,V2x)) ) ) ) ).
tff(lamtp_f59,type,
f59: ( del * $i ) > $i ).
tff(lameq_f59,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V3x: $i] : ( ap(f59(A_27a,V0P),V3x) = ap(V0P,V3x) ) ) ).
tff(lamtp_f60,type,
f60: ( del * $i ) > $i ).
tff(lameq_f60,axiom,
! [A_27a: del,V1Q: $i] :
( mem(V1Q,arr(A_27a,bool))
=> ! [V4x: $i] : ( ap(f60(A_27a,V1Q),V4x) = ap(V1Q,V4x) ) ) ).
tff(conj_thm_2Ebool_2EUEXISTS__OR__THM,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q: $i] :
( mem(V1Q,arr(A_27a,bool))
=> ( p(ap(c_2Ebool_2E_3F_21(A_27a),f58(A_27a,V0P,V1Q)))
=> ( p(ap(c_2Ebool_2E_3F_21(A_27a),f59(A_27a,V0P)))
| p(ap(c_2Ebool_2E_3F_21(A_27a),f60(A_27a,V1Q))) ) ) ) ) ).
tff(conj_thm_2Ebool_2EUEXISTS__SIMP,axiom,
! [A_27a: del,V0t: tp__o] :
( p(ap(c_2Ebool_2E_3F_21(A_27a),k(A_27a,inj__o(V0t))))
<=> ( p(inj__o(V0t))
& ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( V2x = V3y ) ) ) ) ) ).
tff(ax_thm_2Ebool_2ERES__ABSTRACT__DEF,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0p: $i] :
( mem(V0p,arr(A_27a,bool))
=> ! [V1m: $i] :
( mem(V1m,arr(A_27a,A_27b))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0p))
=> ( ap(ap(ap(c_2Ebool_2ERES__ABSTRACT(A_27a,A_27b),V0p),V1m),V2x) = ap(V1m,V2x) ) ) ) ) )
& ! [V3p: $i] :
( mem(V3p,arr(A_27a,bool))
=> ! [V4m1: $i] :
( mem(V4m1,arr(A_27a,A_27b))
=> ! [V5m2: $i] :
( mem(V5m2,arr(A_27a,A_27b))
=> ( ! [V6x: $i] :
( mem(V6x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V6x),V3p))
=> ( ap(V4m1,V6x) = ap(V5m2,V6x) ) ) )
=> ( ap(ap(c_2Ebool_2ERES__ABSTRACT(A_27a,A_27b),V3p),V4m1) = ap(ap(c_2Ebool_2ERES__ABSTRACT(A_27a,A_27b),V3p),V5m2) ) ) ) ) ) ) ).
tff(conj_thm_2Ebool_2ERES__FORALL__THM,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2ERES__FORALL(A_27a),V0P),V1f))
<=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0P))
=> p(ap(V1f,V2x)) ) ) ) ) ) ).
tff(conj_thm_2Ebool_2ERES__EXISTS__THM,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2ERES__EXISTS(A_27a),V0P),V1f))
<=> ? [V2x: $i] :
( mem(V2x,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0P))
& p(ap(V1f,V2x)) ) ) ) ) ).
tff(lamtp_f61,type,
f61: ( del * $i ) > $i ).
tff(lameq_f61,axiom,
! [A_27a: del,V1f: $i] :
( mem(V1f,arr(A_27a,bool))
=> ! [V2x: $i] : ( ap(f61(A_27a,V1f),V2x) = ap(V1f,V2x) ) ) ).
tff(lamtp_f62,type,
f62: ( del * $i * $i ) > $i ).
tff(lameq_f62,axiom,
! [A_27a: del,V1f: $i] :
( mem(V1f,arr(A_27a,bool))
=> ! [V3x: $i] :
( mem(V3x,A_27a)
=> ! [V4y: $i] : ( ap(f62(A_27a,V1f,V3x),V4y) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2E_2F_5C,ap(V1f,V3x)),ap(V1f,V4y))),ap(ap(c_2Emin_2E_3D(A_27a),V3x),V4y)) ) ) ) ).
tff(lamtp_f63,type,
f63: ( del * $i * $i ) > $i ).
tff(lameq_f63,axiom,
! [A_27a: del,V1f: $i] :
( mem(V1f,arr(A_27a,bool))
=> ! [V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V3x: $i] : ( ap(f63(A_27a,V1f,V0P),V3x) = ap(ap(c_2Ebool_2ERES__FORALL(A_27a),V0P),f62(A_27a,V1f,V3x)) ) ) ) ).
tff(conj_thm_2Ebool_2ERES__EXISTS__UNIQUE__THM,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2ERES__EXISTS__UNIQUE(A_27a),V0P),V1f))
<=> ( p(ap(ap(c_2Ebool_2ERES__EXISTS(A_27a),V0P),f61(A_27a,V1f)))
& p(ap(ap(c_2Ebool_2ERES__FORALL(A_27a),V0P),f63(A_27a,V1f,V0P))) ) ) ) ) ).
tff(lamtp_f64,type,
f64: ( del * $i * $i ) > $i ).
tff(lameq_f64,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,bool))
=> ! [V2x: $i] : ( ap(f64(A_27a,V0P,V1f),V2x) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0P)),ap(V1f,V2x)) ) ) ) ).
tff(conj_thm_2Ebool_2ERES__SELECT__THM,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,bool))
=> ( ap(ap(c_2Ebool_2ERES__SELECT(A_27a),V0P),V1f) = ap(c_2Emin_2E_40(A_27a),f64(A_27a,V0P,V1f)) ) ) ) ).
tff(conj_thm_2Ebool_2ERES__FORALL__TRUE,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2ERES__FORALL(A_27a),V0P),k(A_27a,c_2Ebool_2ET)))
<=> $true ) ) ).
tff(conj_thm_2Ebool_2ERES__EXISTS__FALSE,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2ERES__EXISTS(A_27a),V0P),k(A_27a,c_2Ebool_2EF)))
<=> $false ) ) ).
tff(lamtp_f65,type,
f65: $i ).
tff(lameq_f65,axiom,
! [V4b: tp__o] : ( ap(f65,inj__o(V4b)) = ap(c_2Ebool_2E_7E,inj__o(V4b)) ) ).
tff(conj_thm_2Ebool_2EBOOL__FUN__CASES__THM,axiom,
! [V0f: $i] :
( mem(V0f,arr(bool,bool))
=> ( ( V0f = k(bool,c_2Ebool_2ET) )
| ( V0f = k(bool,c_2Ebool_2EF) )
| ( V0f = i(bool) )
| ( V0f = f65 ) ) ) ).
tff(conj_thm_2Ebool_2EBOOL__FUN__INDUCT,axiom,
! [V0P: $i] :
( mem(V0P,arr(arr(bool,bool),bool))
=> ( ( p(ap(V0P,k(bool,c_2Ebool_2ET)))
& p(ap(V0P,k(bool,c_2Ebool_2EF)))
& p(ap(V0P,i(bool)))
& p(ap(V0P,f65)) )
=> ! [V5f: $i] :
( mem(V5f,arr(bool,bool))
=> p(ap(V0P,V5f)) ) ) ) ).
tff(conj_thm_2Ebool_2Eliteral__case__THM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ( ap(ap(c_2Ebool_2Eliteral__case(A_27a,A_27b),V0f),V1x) = ap(V0f,V1x) ) ) ) ).
tff(lamtp_f66,type,
f66: ( del * del * del * $i * $i ) > $i ).
tff(lameq_f66,axiom,
! [A_27b: del,A_27c: del,A_27a: del,V1N: $i] :
( mem(V1N,arr(A_27a,A_27b))
=> ! [V0P: $i] :
( mem(V0P,arr(A_27b,A_27c))
=> ! [V4x: $i] : ( ap(f66(A_27b,A_27c,A_27a,V1N,V0P),V4x) = ap(V0P,ap(V1N,V4x)) ) ) ) ).
tff(conj_thm_2Ebool_2Eliteral__case__RAND,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0P: $i] :
( mem(V0P,arr(A_27b,A_27c))
=> ! [V1N: $i] :
( mem(V1N,arr(A_27a,A_27b))
=> ! [V2M: $i] :
( mem(V2M,A_27a)
=> ( ap(V0P,ap(ap(c_2Ebool_2Eliteral__case(A_27a,A_27b),f53(A_27b,A_27a,V1N)),V2M)) = ap(ap(c_2Ebool_2Eliteral__case(A_27a,A_27c),f66(A_27b,A_27c,A_27a,V1N,V0P)),V2M) ) ) ) ) ).
tff(conj_thm_2Ebool_2Eliteral__case__RATOR,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0N: $i] :
( mem(V0N,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V1M: $i] :
( mem(V1M,A_27a)
=> ! [V2b: $i] :
( mem(V2b,A_27b)
=> ( ap(ap(ap(c_2Ebool_2Eliteral__case(A_27a,arr(A_27b,A_27c)),f55(A_27c,A_27b,A_27a,V0N)),V1M),V2b) = ap(ap(c_2Ebool_2Eliteral__case(A_27a,A_27c),f56(A_27b,A_27c,A_27a,V0N,V2b)),V1M) ) ) ) ) ).
tff(conj_thm_2Ebool_2Eliteral__case__CONG,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,A_27b))
=> ! [V2M: $i] :
( mem(V2M,A_27a)
=> ! [V3N: $i] :
( mem(V3N,A_27a)
=> ( ( ( V2M = V3N )
& ! [V4x: $i] :
( mem(V4x,A_27a)
=> ( ( V4x = V3N )
=> ( ap(V0f,V4x) = ap(V1g,V4x) ) ) ) )
=> ( ap(ap(c_2Ebool_2Eliteral__case(A_27a,A_27b),V0f),V2M) = ap(ap(c_2Ebool_2Eliteral__case(A_27a,A_27b),V1g),V3N) ) ) ) ) ) ) ).
tff(lamtp_f67,type,
f67: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f67,axiom,
! [A_27b: del,A_27a: del,V1t: $i] :
( mem(V1t,A_27b)
=> ! [V0a: $i] :
( mem(V0a,A_27a)
=> ! [V2u: $i] :
( mem(V2u,A_27b)
=> ! [V3x: $i] : ( ap(f67(A_27b,A_27a,V1t,V0a,V2u),V3x) = ap(ap(ap(c_2Ebool_2ECOND(A_27b),ap(ap(c_2Emin_2E_3D(A_27a),V3x),V0a)),V1t),V2u) ) ) ) ) ).
tff(conj_thm_2Ebool_2Eliteral__case__id,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( mem(V0a,A_27a)
=> ! [V1t: $i] :
( mem(V1t,A_27b)
=> ! [V2u: $i] :
( mem(V2u,A_27b)
=> ( ap(ap(c_2Ebool_2Eliteral__case(A_27a,A_27b),f67(A_27b,A_27a,V1t,V0a,V2u)),V0a) = V1t ) ) ) ) ).
tff(conj_thm_2Ebool_2EBOUNDED__THM,axiom,
! [V0v: tp__o] :
( p(ap(c_2Ebool_2EBOUNDED,inj__o(V0v)))
<=> $true ) ).
tff(conj_thm_2Ebool_2ELCOMM__THM,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27a,A_27a)))
=> ( ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ! [V3z: $i] :
( mem(V3z,A_27a)
=> ( ap(ap(V0f,V1x),ap(ap(V0f,V2y),V3z)) = ap(ap(V0f,ap(ap(V0f,V1x),V2y)),V3z) ) ) ) )
=> ( ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( mem(V5y,A_27a)
=> ( ap(ap(V0f,V4x),V5y) = ap(ap(V0f,V5y),V4x) ) ) )
=> ! [V6x: $i] :
( mem(V6x,A_27a)
=> ! [V7y: $i] :
( mem(V7y,A_27a)
=> ! [V8z: $i] :
( mem(V8z,A_27a)
=> ( ap(ap(V0f,V6x),ap(ap(V0f,V7y),V8z)) = ap(ap(V0f,V7y),ap(ap(V0f,V6x),V8z)) ) ) ) ) ) ) ) ).
tff(conj_thm_2Ebool_2EDATATYPE__TAG__THM,axiom,
! [A_27a: del,V0x: $i] :
( mem(V0x,A_27a)
=> ( p(ap(c_2Ebool_2EDATATYPE(A_27a),V0x))
<=> $true ) ) ).
tff(conj_thm_2Ebool_2EDATATYPE__BOOL,axiom,
! [V0bool: $i] :
( mem(V0bool,arr(bool,arr(bool,bool)))
=> ( p(ap(c_2Ebool_2EDATATYPE(bool),ap(ap(V0bool,inj__o(fo__c_2Ebool_2ET)),inj__o(fo__c_2Ebool_2EF))))
<=> $true ) ) ).
tff(ax_thm_2Ebool_2Eitself__TY__DEF,axiom,
! [A_27a: del] :
? [V0rep: $i] :
( mem(V0rep,arr(ty_2Ebool_2Eitself(A_27a),A_27a))
& p(ap(ap(c_2Ebool_2ETYPE__DEFINITION(A_27a,ty_2Ebool_2Eitself(A_27a)),ap(c_2Emin_2E_3D(A_27a),c_2Ebool_2EARB(A_27a))),V0rep)) ) ).
tff(conj_thm_2Ebool_2EITSELF__UNIQUE,axiom,
! [A_27a: del,V0i: $i] :
( mem(V0i,ty_2Ebool_2Eitself(A_27a))
=> ( V0i = c_2Ebool_2Ethe__value(A_27a) ) ) ).
tff(conj_thm_2Ebool_2Eitself__Axiom,axiom,
! [A_27a: del,A_27b: del,V0e: $i] :
( mem(V0e,A_27b)
=> ? [V1f: $i] :
( mem(V1f,arr(ty_2Ebool_2Eitself(A_27a),A_27b))
& ( ap(V1f,c_2Ebool_2Ethe__value(A_27a)) = V0e ) ) ) ).
tff(conj_thm_2Ebool_2Eitself__induction,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(ty_2Ebool_2Eitself(A_27a),bool))
=> ( p(ap(V0P,c_2Ebool_2Ethe__value(A_27a)))
=> ! [V1i: $i] :
( mem(V1i,ty_2Ebool_2Eitself(A_27a))
=> p(ap(V0P,V1i)) ) ) ) ).
tff(ax_thm_2Ebool_2Eitself__case__thm,axiom,
! [A_27a: del,A_27b: del,V0b: $i] :
( mem(V0b,A_27b)
=> ( ap(ap(c_2Ebool_2Eitself__case(A_27a,A_27b),c_2Ebool_2Ethe__value(A_27a)),V0b) = V0b ) ) ).
tff(conj_thm_2Ebool_2EFORALL__itself,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(ty_2Ebool_2Eitself(A_27a),bool))
=> ( ! [V1x: $i] :
( mem(V1x,ty_2Ebool_2Eitself(A_27a))
=> p(ap(V0P,V1x)) )
<=> p(ap(V0P,c_2Ebool_2Ethe__value(A_27a))) ) ) ).
tff(conj_thm_2Ebool_2EEXISTS__itself,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(ty_2Ebool_2Eitself(A_27a),bool))
=> ( ? [V1x: $i] :
( mem(V1x,ty_2Ebool_2Eitself(A_27a))
& p(ap(V0P,V1x)) )
<=> p(ap(V0P,c_2Ebool_2Ethe__value(A_27a))) ) ) ).
tff(conj_thm_2Ebool_2EPULL__EXISTS,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)) ) ) )
& ( ( ? [V4x: $i] :
( mem(V4x,A_27a)
& p(ap(V0P,V4x)) )
& p(inj__o(V1Q)) )
<=> ? [V5x: $i] :
( mem(V5x,A_27a)
& p(ap(V0P,V5x))
& p(inj__o(V1Q)) ) )
& ( ( p(inj__o(V1Q))
& ? [V6x: $i] :
( mem(V6x,A_27a)
& p(ap(V0P,V6x)) ) )
<=> ? [V7x: $i] :
( mem(V7x,A_27a)
& p(inj__o(V1Q))
& p(ap(V0P,V7x)) ) ) ) ) ).
tff(conj_thm_2Ebool_2EPULL__FORALL,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q: tp__o] :
( ( ( p(inj__o(V1Q))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> p(ap(V0P,V2x)) ) )
<=> ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(inj__o(V1Q))
=> p(ap(V0P,V3x)) ) ) )
& ( ( ! [V4x: $i] :
( mem(V4x,A_27a)
=> p(ap(V0P,V4x)) )
& p(inj__o(V1Q)) )
<=> ! [V5x: $i] :
( mem(V5x,A_27a)
=> ( p(ap(V0P,V5x))
& p(inj__o(V1Q)) ) ) )
& ( ( p(inj__o(V1Q))
& ! [V6x: $i] :
( mem(V6x,A_27a)
=> p(ap(V0P,V6x)) ) )
<=> ! [V7x: $i] :
( mem(V7x,A_27a)
=> ( p(inj__o(V1Q))
& p(ap(V0P,V7x)) ) ) ) ) ) ).
tff(conj_thm_2Ebool_2EPEIRCE,axiom,
! [V0P: tp__o,V1Q: tp__o] :
( ( ( p(inj__o(V0P))
=> p(inj__o(V1Q)) )
=> p(inj__o(V0P)) )
=> p(inj__o(V0P)) ) ).
tff(conj_thm_2Ebool_2EJRH__INDUCT__UTIL,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1t: $i] :
( mem(V1t,A_27a)
=> ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( ( V2x = V1t )
=> p(ap(V0P,V2x)) ) )
=> p(ap(c_2Ebool_2E_3F(A_27a),V0P)) ) ) ) ).
tff(conj_thm_2Ebool_2EDISJ__EQ__IMP,axiom,
! [V0A: tp__o,V1B: tp__o] :
( ( p(inj__o(V0A))
| p(inj__o(V1B)) )
<=> ( ~ p(inj__o(V0A))
=> p(inj__o(V1B)) ) ) ).
%------------------------------------------------------------------------------