ITP001 Axioms: ITP142_5.ax
%------------------------------------------------------------------------------
% File : ITP142_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 : util_prob_2.ax [Gau20]
% : HL4142_5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 135 ( 32 unt; 29 typ; 0 def)
% Number of atoms : 2399 ( 67 equ)
% Maximal formula atoms : 9 ( 17 avg)
% Number of connectives : 212 ( 15 ~; 1 |; 57 &)
% ( 16 <=>; 123 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 2096 (2096 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 54 ( 25 >; 29 *; 0 +; 0 <<)
% Number of predicates : 70 ( 68 usr; 36 prp; 0-3 aty)
% Number of functors : 29 ( 29 usr; 4 con; 0-5 aty)
% Number of variables : 262 ( 234 !; 28 ?; 262 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_c_2Eutil__prob_2Edisjoint,type,
c_2Eutil__prob_2Edisjoint: del > $i ).
tff(mem_c_2Eutil__prob_2Edisjoint,axiom,
! [A_27a: del] : mem(c_2Eutil__prob_2Edisjoint(A_27a),arr(arr(arr(A_27a,bool),bool),bool)) ).
tff(tp_c_2Eutil__prob_2Elg,type,
c_2Eutil__prob_2Elg: $i ).
tff(mem_c_2Eutil__prob_2Elg,axiom,
mem(c_2Eutil__prob_2Elg,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) ).
tff(stp_fo_c_2Eutil__prob_2Elg,type,
fo__c_2Eutil__prob_2Elg: tp__ty_2Erealax_2Ereal > tp__ty_2Erealax_2Ereal ).
tff(stp_eq_fo_c_2Eutil__prob_2Elg,axiom,
! [X0: tp__ty_2Erealax_2Ereal] : ( inj__ty_2Erealax_2Ereal(fo__c_2Eutil__prob_2Elg(X0)) = ap(c_2Eutil__prob_2Elg,inj__ty_2Erealax_2Ereal(X0)) ) ).
tff(tp_c_2Eutil__prob_2Elogr,type,
c_2Eutil__prob_2Elogr: $i ).
tff(mem_c_2Eutil__prob_2Elogr,axiom,
mem(c_2Eutil__prob_2Elogr,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) ).
tff(stp_fo_c_2Eutil__prob_2Elogr,type,
fo__c_2Eutil__prob_2Elogr: ( tp__ty_2Erealax_2Ereal * tp__ty_2Erealax_2Ereal ) > tp__ty_2Erealax_2Ereal ).
tff(stp_eq_fo_c_2Eutil__prob_2Elogr,axiom,
! [X0: tp__ty_2Erealax_2Ereal,X1: tp__ty_2Erealax_2Ereal] : ( inj__ty_2Erealax_2Ereal(fo__c_2Eutil__prob_2Elogr(X0,X1)) = ap(ap(c_2Eutil__prob_2Elogr,inj__ty_2Erealax_2Ereal(X0)),inj__ty_2Erealax_2Ereal(X1)) ) ).
tff(tp_c_2Eutil__prob_2Eminimal,type,
c_2Eutil__prob_2Eminimal: $i ).
tff(mem_c_2Eutil__prob_2Eminimal,axiom,
mem(c_2Eutil__prob_2Eminimal,arr(arr(ty_2Enum_2Enum,bool),ty_2Enum_2Enum)) ).
tff(tp_c_2Eutil__prob_2Epair,type,
c_2Eutil__prob_2Epair: ( del * del ) > $i ).
tff(mem_c_2Eutil__prob_2Epair,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Eutil__prob_2Epair(A_27a,A_27b),arr(arr(A_27a,bool),arr(arr(A_27b,bool),arr(ty_2Epair_2Eprod(A_27a,A_27b),bool)))) ).
tff(tp_c_2Eutil__prob_2Epowr,type,
c_2Eutil__prob_2Epowr: $i ).
tff(mem_c_2Eutil__prob_2Epowr,axiom,
mem(c_2Eutil__prob_2Epowr,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) ).
tff(stp_fo_c_2Eutil__prob_2Epowr,type,
fo__c_2Eutil__prob_2Epowr: ( tp__ty_2Erealax_2Ereal * tp__ty_2Erealax_2Ereal ) > tp__ty_2Erealax_2Ereal ).
tff(stp_eq_fo_c_2Eutil__prob_2Epowr,axiom,
! [X0: tp__ty_2Erealax_2Ereal,X1: tp__ty_2Erealax_2Ereal] : ( inj__ty_2Erealax_2Ereal(fo__c_2Eutil__prob_2Epowr(X0,X1)) = ap(ap(c_2Eutil__prob_2Epowr,inj__ty_2Erealax_2Ereal(X0)),inj__ty_2Erealax_2Ereal(X1)) ) ).
tff(tp_c_2Eutil__prob_2Eprod__sets,type,
c_2Eutil__prob_2Eprod__sets: ( del * del ) > $i ).
tff(mem_c_2Eutil__prob_2Eprod__sets,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Eutil__prob_2Eprod__sets(A_27a,A_27b),arr(arr(arr(A_27a,bool),bool),arr(arr(arr(A_27b,bool),bool),arr(arr(ty_2Epair_2Eprod(A_27a,A_27b),bool),bool)))) ).
tff(conj_thm_2Eutil__prob_2EEQ__T__IMP,axiom,
! [V0x: tp__o] :
( p(inj__o(V0x))
<=> ( $true
=> p(inj__o(V0x)) ) ) ).
tff(lamtp_f3153,type,
f3153: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3153,axiom,
! [A_27a: del,A_27b: del,V2x: $i] :
( mem(V2x,A_27a)
=> ! [V0X: $i] :
( mem(V0X,arr(A_27a,bool))
=> ! [V1Y: $i] :
( mem(V1Y,arr(A_27b,bool))
=> ! [V3y: $i] : ( ap(f3153(A_27a,A_27b,V2x,V0X,V1Y),V3y) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0X)),ap(ap(c_2Ebool_2EIN(A_27b),V3y),V1Y)) ) ) ) ) ).
tff(lamtp_f3154,type,
f3154: ( del * del * $i * $i ) > $i ).
tff(lameq_f3154,axiom,
! [A_27b: del,A_27a: del,V0X: $i] :
( mem(V0X,arr(A_27a,bool))
=> ! [V1Y: $i] :
( mem(V1Y,arr(A_27b,bool))
=> ! [V2x: $i] : ( ap(f3154(A_27b,A_27a,V0X,V1Y),V2x) = f3153(A_27a,A_27b,V2x,V0X,V1Y) ) ) ) ).
tff(ax_thm_2Eutil__prob_2Epair__def,axiom,
! [A_27a: del,A_27b: del,V0X: $i] :
( mem(V0X,arr(A_27a,bool))
=> ! [V1Y: $i] :
( mem(V1Y,arr(A_27b,bool))
=> ( ap(ap(c_2Eutil__prob_2Epair(A_27a,A_27b),V0X),V1Y) = ap(c_2Epair_2EUNCURRY(A_27a,A_27b,bool),f3154(A_27b,A_27a,V0X,V1Y)) ) ) ) ).
tff(conj_thm_2Eutil__prob_2EIN__PAIR,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( mem(V0x,ty_2Epair_2Eprod(A_27a,A_27b))
=> ! [V1X: $i] :
( mem(V1X,arr(A_27a,bool))
=> ! [V2Y: $i] :
( mem(V2Y,arr(A_27b,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Epair_2Eprod(A_27a,A_27b)),V0x),ap(ap(c_2Eutil__prob_2Epair(A_27a,A_27b),V1X),V2Y)))
<=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),ap(c_2Epair_2EFST(A_27a,A_27b),V0x)),V1X))
& p(ap(ap(c_2Ebool_2EIN(A_27b),ap(c_2Epair_2ESND(A_27a,A_27b),V0x)),V2Y)) ) ) ) ) ) ).
tff(conj_thm_2Eutil__prob_2EPAIR__UNIV,axiom,
! [A_27a: del,A_27b: del] : ( ap(ap(c_2Eutil__prob_2Epair(A_27a,A_27b),c_2Epred__set_2EUNIV(A_27a)),c_2Epred__set_2EUNIV(A_27b)) = c_2Epred__set_2EUNIV(ty_2Epair_2Eprod(A_27a,A_27b)) ) ).
tff(conj_thm_2Eutil__prob_2EPAIRED__BETA__THM,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V1z: $i] :
( mem(V1z,ty_2Epair_2Eprod(A_27a,A_27b))
=> ( ap(ap(c_2Epair_2EUNCURRY(A_27a,A_27b,A_27c),V0f),V1z) = ap(ap(V0f,ap(c_2Epair_2EFST(A_27a,A_27b),V1z)),ap(c_2Epair_2ESND(A_27a,A_27b),V1z)) ) ) ) ).
tff(ax_thm_2Eutil__prob_2Epowr__def,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1a: tp__ty_2Erealax_2Ereal] : ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eutil__prob_2Epowr,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Erealax_2Ereal(V1a))) = surj__ty_2Erealax_2Ereal(ap(c_2Etransc_2Eexp,ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V1a)),ap(c_2Etransc_2Eln,inj__ty_2Erealax_2Ereal(V0x))))) ) ).
tff(ax_thm_2Eutil__prob_2Elogr__def,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1x: tp__ty_2Erealax_2Ereal] : ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eutil__prob_2Elogr,inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1x))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2E_2F,ap(c_2Etransc_2Eln,inj__ty_2Erealax_2Ereal(V1x))),ap(c_2Etransc_2Eln,inj__ty_2Erealax_2Ereal(V0a)))) ) ).
tff(ax_thm_2Eutil__prob_2Elg__def,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] : ( surj__ty_2Erealax_2Ereal(ap(c_2Eutil__prob_2Elg,inj__ty_2Erealax_2Ereal(V0x))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eutil__prob_2Elogr,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),inj__ty_2Erealax_2Ereal(V0x))) ) ).
tff(conj_thm_2Eutil__prob_2Elg__1,axiom,
surj__ty_2Erealax_2Ereal(ap(c_2Eutil__prob_2Elg,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ).
tff(conj_thm_2Eutil__prob_2Elogr__1,axiom,
! [V0b: tp__ty_2Erealax_2Ereal] : ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eutil__prob_2Elogr,inj__ty_2Erealax_2Ereal(V0b)),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ).
tff(conj_thm_2Eutil__prob_2Elg__nonzero,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( ( V0x != surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) )
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V0x))) )
=> ( ( surj__ty_2Erealax_2Ereal(ap(c_2Eutil__prob_2Elg,inj__ty_2Erealax_2Ereal(V0x))) != surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) )
<=> ( V0x != surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) ) ) ) ).
tff(conj_thm_2Eutil__prob_2Elg__mul,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V0x)))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V1y))) )
=> ( surj__ty_2Erealax_2Ereal(ap(c_2Eutil__prob_2Elg,ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Erealax_2Ereal(V1y)))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(c_2Eutil__prob_2Elg,inj__ty_2Erealax_2Ereal(V0x))),ap(c_2Eutil__prob_2Elg,inj__ty_2Erealax_2Ereal(V1y)))) ) ) ).
tff(conj_thm_2Eutil__prob_2Elogr__mul,axiom,
! [V0b: tp__ty_2Erealax_2Ereal,V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V1x)))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V2y))) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eutil__prob_2Elogr,inj__ty_2Erealax_2Ereal(V0b)),ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Erealax_2Ereal(V2y)))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eutil__prob_2Elogr,inj__ty_2Erealax_2Ereal(V0b)),inj__ty_2Erealax_2Ereal(V1x))),ap(ap(c_2Eutil__prob_2Elogr,inj__ty_2Erealax_2Ereal(V0b)),inj__ty_2Erealax_2Ereal(V2y)))) ) ) ).
tff(conj_thm_2Eutil__prob_2Elg__2,axiom,
surj__ty_2Erealax_2Ereal(ap(c_2Eutil__prob_2Elg,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) ).
tff(conj_thm_2Eutil__prob_2Elg__inv,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V0x)))
=> ( surj__ty_2Erealax_2Ereal(ap(c_2Eutil__prob_2Elg,ap(c_2Erealax_2Einv,inj__ty_2Erealax_2Ereal(V0x)))) = surj__ty_2Erealax_2Ereal(ap(c_2Erealax_2Ereal__neg,ap(c_2Eutil__prob_2Elg,inj__ty_2Erealax_2Ereal(V0x)))) ) ) ).
tff(conj_thm_2Eutil__prob_2Elogr__inv,axiom,
! [V0b: tp__ty_2Erealax_2Ereal,V1x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V1x)))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eutil__prob_2Elogr,inj__ty_2Erealax_2Ereal(V0b)),ap(c_2Erealax_2Einv,inj__ty_2Erealax_2Ereal(V1x)))) = surj__ty_2Erealax_2Ereal(ap(c_2Erealax_2Ereal__neg,ap(ap(c_2Eutil__prob_2Elogr,inj__ty_2Erealax_2Ereal(V0b)),inj__ty_2Erealax_2Ereal(V1x)))) ) ) ).
tff(conj_thm_2Eutil__prob_2Elogr__div,axiom,
! [V0b: tp__ty_2Erealax_2Ereal,V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V1x)))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V2y))) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eutil__prob_2Elogr,inj__ty_2Erealax_2Ereal(V0b)),ap(ap(c_2Ereal_2E_2F,inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Erealax_2Ereal(V2y)))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Eutil__prob_2Elogr,inj__ty_2Erealax_2Ereal(V0b)),inj__ty_2Erealax_2Ereal(V1x))),ap(ap(c_2Eutil__prob_2Elogr,inj__ty_2Erealax_2Ereal(V0b)),inj__ty_2Erealax_2Ereal(V2y)))) ) ) ).
tff(conj_thm_2Eutil__prob_2Eneg__lg,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V0x)))
=> ( surj__ty_2Erealax_2Ereal(ap(c_2Erealax_2Ereal__neg,ap(c_2Eutil__prob_2Elg,inj__ty_2Erealax_2Ereal(V0x)))) = surj__ty_2Erealax_2Ereal(ap(c_2Eutil__prob_2Elg,ap(c_2Erealax_2Einv,inj__ty_2Erealax_2Ereal(V0x)))) ) ) ).
tff(conj_thm_2Eutil__prob_2Eneg__logr,axiom,
! [V0b: tp__ty_2Erealax_2Ereal,V1x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V1x)))
=> ( surj__ty_2Erealax_2Ereal(ap(c_2Erealax_2Ereal__neg,ap(ap(c_2Eutil__prob_2Elogr,inj__ty_2Erealax_2Ereal(V0b)),inj__ty_2Erealax_2Ereal(V1x)))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eutil__prob_2Elogr,inj__ty_2Erealax_2Ereal(V0b)),ap(c_2Erealax_2Einv,inj__ty_2Erealax_2Ereal(V1x)))) ) ) ).
tff(conj_thm_2Eutil__prob_2Elg__pow,axiom,
! [V0n: tp__ty_2Enum_2Enum] : ( surj__ty_2Erealax_2Ereal(ap(c_2Eutil__prob_2Elg,ap(ap(c_2Ereal_2Epow,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),inj__ty_2Enum_2Enum(V0n)))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(V0n))) ) ).
tff(conj_thm_2Eutil__prob_2ENUM__2D__BIJ,axiom,
? [V0f: $i] :
( mem(V0f,arr(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum),ty_2Enum_2Enum))
& p(ap(ap(ap(c_2Epred__set_2EBIJ(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum),ty_2Enum_2Enum),V0f),ap(ap(c_2Epred__set_2ECROSS(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) ).
tff(conj_thm_2Eutil__prob_2ENUM__2D__BIJ__INV,axiom,
? [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum)))
& p(ap(ap(ap(c_2Epred__set_2EBIJ(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum)),V0f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Epred__set_2ECROSS(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))) ) ).
tff(conj_thm_2Eutil__prob_2ENUM__2D__BIJ__NZ,axiom,
? [V0f: $i] :
( mem(V0f,arr(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum),ty_2Enum_2Enum))
& p(ap(ap(ap(c_2Epred__set_2EBIJ(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum),ty_2Enum_2Enum),V0f),ap(ap(c_2Epred__set_2ECROSS(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Epred__set_2EDIFF(ty_2Enum_2Enum),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum))))),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) ).
tff(conj_thm_2Eutil__prob_2ENUM__2D__BIJ__NZ__INV,axiom,
? [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum)))
& p(ap(ap(ap(c_2Epred__set_2EBIJ(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum)),V0f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Epred__set_2ECROSS(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Epred__set_2EDIFF(ty_2Enum_2Enum),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)))))) ) ).
tff(conj_thm_2Eutil__prob_2ENUM__2D__BIJ__NZ__ALT,axiom,
? [V0f: $i] :
( mem(V0f,arr(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum),ty_2Enum_2Enum))
& p(ap(ap(ap(c_2Epred__set_2EBIJ(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum),ty_2Enum_2Enum),V0f),ap(ap(c_2Epred__set_2ECROSS(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))),ap(ap(c_2Epred__set_2EDIFF(ty_2Enum_2Enum),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum))))) ) ).
tff(conj_thm_2Eutil__prob_2ENUM__2D__BIJ__NZ__ALT__INV,axiom,
? [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum)))
& p(ap(ap(ap(c_2Epred__set_2EBIJ(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum)),V0f),ap(ap(c_2Epred__set_2EDIFF(ty_2Enum_2Enum),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)))),ap(ap(c_2Epred__set_2ECROSS(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))) ) ).
tff(conj_thm_2Eutil__prob_2ENUM__2D__BIJ__NZ__ALT2,axiom,
? [V0f: $i] :
( mem(V0f,arr(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum),ty_2Enum_2Enum))
& p(ap(ap(ap(c_2Epred__set_2EBIJ(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum),ty_2Enum_2Enum),V0f),ap(ap(c_2Epred__set_2ECROSS(ty_2Enum_2Enum,ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EDIFF(ty_2Enum_2Enum),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)))),ap(ap(c_2Epred__set_2EDIFF(ty_2Enum_2Enum),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum))))),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) ).
tff(conj_thm_2Eutil__prob_2ENUM__2D__BIJ__NZ__ALT2__INV,axiom,
? [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum)))
& p(ap(ap(ap(c_2Epred__set_2EBIJ(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum)),V0f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Epred__set_2ECROSS(ty_2Enum_2Enum,ty_2Enum_2Enum),ap(ap(c_2Epred__set_2EDIFF(ty_2Enum_2Enum),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)))),ap(ap(c_2Epred__set_2EDIFF(ty_2Enum_2Enum),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)))))) ) ).
tff(lamtp_f3155,type,
f3155: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3155,axiom,
! [A_27a: del,A_27b: del,V1b: $i] :
( mem(V1b,arr(arr(A_27b,bool),bool))
=> ! [V0a: $i] :
( mem(V0a,arr(arr(A_27a,bool),bool))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t: $i] : ( ap(f3155(A_27a,A_27b,V1b,V0a,V2s),V3t) = ap(ap(c_2Epair_2E_2C(arr(ty_2Epair_2Eprod(A_27a,A_27b),bool),bool),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V2s),V3t)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2s),V0a)),ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V3t),V1b))) ) ) ) ) ).
tff(lamtp_f3156,type,
f3156: ( del * del * $i * $i ) > $i ).
tff(lameq_f3156,axiom,
! [A_27b: del,A_27a: del,V1b: $i] :
( mem(V1b,arr(arr(A_27b,bool),bool))
=> ! [V0a: $i] :
( mem(V0a,arr(arr(A_27a,bool),bool))
=> ! [V2s: $i] : ( ap(f3156(A_27b,A_27a,V1b,V0a),V2s) = f3155(A_27a,A_27b,V1b,V0a,V2s) ) ) ) ).
tff(ax_thm_2Eutil__prob_2Eprod__sets__def,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( mem(V0a,arr(arr(A_27a,bool),bool))
=> ! [V1b: $i] :
( mem(V1b,arr(arr(A_27b,bool),bool))
=> ( ap(ap(c_2Eutil__prob_2Eprod__sets(A_27a,A_27b),V0a),V1b) = ap(c_2Epred__set_2EGSPEC(arr(ty_2Epair_2Eprod(A_27a,A_27b),bool),ty_2Epair_2Eprod(arr(A_27a,bool),arr(A_27b,bool))),ap(c_2Epair_2EUNCURRY(arr(A_27a,bool),arr(A_27b,bool),ty_2Epair_2Eprod(arr(ty_2Epair_2Eprod(A_27a,A_27b),bool),bool)),f3156(A_27b,A_27a,V1b,V0a))) ) ) ) ).
tff(conj_thm_2Eutil__prob_2EIN__o,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( mem(V0x,A_27a)
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27b,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),ap(ap(c_2Ecombin_2Eo(A_27a,bool,A_27b),V2s),V1f)))
<=> p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V1f,V0x)),V2s)) ) ) ) ) ).
tff(conj_thm_2Eutil__prob_2EIN__PROD__SETS,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( mem(V0s,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ! [V1a: $i] :
( mem(V1a,arr(arr(A_27a,bool),bool))
=> ! [V2b: $i] :
( mem(V2b,arr(arr(A_27b,bool),bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(ty_2Epair_2Eprod(A_27a,A_27b),bool)),V0s),ap(ap(c_2Eutil__prob_2Eprod__sets(A_27a,A_27b),V1a),V2b)))
<=> ? [V3t: $i] :
( mem(V3t,arr(A_27a,bool))
& ? [V4u: $i] :
( mem(V4u,arr(A_27b,bool))
& ( V0s = ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V3t),V4u) )
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3t),V1a))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27b,bool)),V4u),V2b)) ) ) ) ) ) ) ).
tff(conj_thm_2Eutil__prob_2Efinite__enumeration__of__sets__has__max__non__empty,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V1s: $i] :
( mem(V1s,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(arr(A_27a,bool)),V1s))
& ! [V2x: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),ap(V0f,inj__ty_2Enum_2Enum(V2x))),V1s))
& ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( V3m != V4n )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V0f,inj__ty_2Enum_2Enum(V3m))),ap(V0f,inj__ty_2Enum_2Enum(V4n)))) ) )
=> ? [V5N: tp__ty_2Enum_2Enum] :
! [V6n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3E_3D,inj__ty_2Enum_2Enum(V6n)),inj__ty_2Enum_2Enum(V5N)))
=> ( ap(V0f,inj__ty_2Enum_2Enum(V6n)) = c_2Epred__set_2EEMPTY(A_27a) ) ) ) ) ) ).
tff(lamtp_f3157,type,
f3157: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f3157,axiom,
! [V0c: tp__ty_2Erealax_2Ereal,V1x: tp__ty_2Erealax_2Ereal] : ( ap(f3157(V0c),inj__ty_2Erealax_2Ereal(V1x)) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),inj__ty_2Erealax_2Ereal(V1x)),ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V0c)),inj__ty_2Erealax_2Ereal(V1x))) ) ).
tff(lamtp_f3158,type,
f3158: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f3158,axiom,
! [V0c: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] : ( ap(f3158(V0c),inj__ty_2Erealax_2Ereal(V2x)) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),inj__ty_2Erealax_2Ereal(V2x)),ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Erealax_2Ereal(V0c))) ) ).
tff(conj_thm_2Eutil__prob_2EPREIMAGE__REAL__COMPL1,axiom,
! [V0c: tp__ty_2Erealax_2Ereal] : ( ap(c_2Epred__set_2ECOMPL(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3157(V0c))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3158(V0c)) ) ).
tff(lamtp_f3159,type,
f3159: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f3159,axiom,
! [V0c: tp__ty_2Erealax_2Ereal,V1x: tp__ty_2Erealax_2Ereal] : ( ap(f3159(V0c),inj__ty_2Erealax_2Ereal(V1x)) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),inj__ty_2Erealax_2Ereal(V1x)),ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V0c)),inj__ty_2Erealax_2Ereal(V1x))) ) ).
tff(lamtp_f3160,type,
f3160: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f3160,axiom,
! [V0c: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] : ( ap(f3160(V0c),inj__ty_2Erealax_2Ereal(V2x)) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),inj__ty_2Erealax_2Ereal(V2x)),ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Erealax_2Ereal(V0c))) ) ).
tff(conj_thm_2Eutil__prob_2EPREIMAGE__REAL__COMPL2,axiom,
! [V0c: tp__ty_2Erealax_2Ereal] : ( ap(c_2Epred__set_2ECOMPL(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3159(V0c))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3160(V0c)) ) ).
tff(lamtp_f3161,type,
f3161: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f3161,axiom,
! [V0c: tp__ty_2Erealax_2Ereal,V1x: tp__ty_2Erealax_2Ereal] : ( ap(f3161(V0c),inj__ty_2Erealax_2Ereal(V1x)) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),inj__ty_2Erealax_2Ereal(V1x)),ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Erealax_2Ereal(V0c))) ) ).
tff(lamtp_f3162,type,
f3162: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f3162,axiom,
! [V0c: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] : ( ap(f3162(V0c),inj__ty_2Erealax_2Ereal(V2x)) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),inj__ty_2Erealax_2Ereal(V2x)),ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V0c)),inj__ty_2Erealax_2Ereal(V2x))) ) ).
tff(conj_thm_2Eutil__prob_2EPREIMAGE__REAL__COMPL3,axiom,
! [V0c: tp__ty_2Erealax_2Ereal] : ( ap(c_2Epred__set_2ECOMPL(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3161(V0c))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3162(V0c)) ) ).
tff(lamtp_f3163,type,
f3163: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f3163,axiom,
! [V0c: tp__ty_2Erealax_2Ereal,V1x: tp__ty_2Erealax_2Ereal] : ( ap(f3163(V0c),inj__ty_2Erealax_2Ereal(V1x)) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),inj__ty_2Erealax_2Ereal(V1x)),ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Erealax_2Ereal(V0c))) ) ).
tff(lamtp_f3164,type,
f3164: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f3164,axiom,
! [V0c: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] : ( ap(f3164(V0c),inj__ty_2Erealax_2Ereal(V2x)) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),inj__ty_2Erealax_2Ereal(V2x)),ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V0c)),inj__ty_2Erealax_2Ereal(V2x))) ) ).
tff(conj_thm_2Eutil__prob_2EPREIMAGE__REAL__COMPL4,axiom,
! [V0c: tp__ty_2Erealax_2Ereal] : ( ap(c_2Epred__set_2ECOMPL(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3163(V0c))) = ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3164(V0c)) ) ).
tff(lamtp_f3165,type,
f3165: ( del * del * $i * $i ) > $i ).
tff(lameq_f3165,axiom,
! [A_27b: del,A_27c: del,V3s: $i] :
( mem(V3s,A_27b)
=> ! [V1p: $i] :
( mem(V1p,arr(A_27b,arr(A_27c,bool)))
=> ! [V4n: $i] : ( ap(f3165(A_27b,A_27c,V3s,V1p),V4n) = ap(ap(V1p,V3s),V4n) ) ) ) ).
tff(lamtp_f3166,type,
f3166: ( del * del * $i ) > $i ).
tff(lameq_f3166,axiom,
! [A_27c: del,A_27b: del,V1p: $i] :
( mem(V1p,arr(A_27b,arr(A_27c,bool)))
=> ! [V3s: $i] : ( ap(f3166(A_27c,A_27b,V1p),V3s) = ap(ap(c_2Epair_2E_2C(A_27b,bool),V3s),ap(c_2Ebool_2E_3F(A_27c),f3165(A_27b,A_27c,V3s,V1p))) ) ) ).
tff(lamtp_f3167,type,
f3167: ( del * del * $i * $i ) > $i ).
tff(lameq_f3167,axiom,
! [A_27c: del,A_27b: del,V5n: $i] :
( mem(V5n,A_27c)
=> ! [V1p: $i] :
( mem(V1p,arr(A_27b,arr(A_27c,bool)))
=> ! [V6s: $i] : ( ap(f3167(A_27c,A_27b,V5n,V1p),V6s) = ap(ap(c_2Epair_2E_2C(A_27b,bool),V6s),ap(ap(V1p,V6s),V5n)) ) ) ) ).
tff(lamtp_f3168,type,
f3168: ( del * del * $i ) > $i ).
tff(lameq_f3168,axiom,
! [A_27b: del,A_27c: del,V1p: $i] :
( mem(V1p,arr(A_27b,arr(A_27c,bool)))
=> ! [V5n: $i] : ( ap(f3168(A_27b,A_27c,V1p),V5n) = ap(c_2Epred__set_2EGSPEC(A_27b,A_27b),f3167(A_27c,A_27b,V5n,V1p)) ) ) ).
tff(conj_thm_2Eutil__prob_2EGBIGUNION__IMAGE,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0s: $i] :
( mem(V0s,A_27a)
=> ! [V1p: $i] :
( mem(V1p,arr(A_27b,arr(A_27c,bool)))
=> ! [V2n: $i] :
( mem(V2n,A_27d)
=> ( ap(c_2Epred__set_2EGSPEC(A_27b,A_27b),f3166(A_27c,A_27b,V1p)) = ap(c_2Epred__set_2EBIGUNION(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27c,arr(A_27b,bool)),f3168(A_27b,A_27c,V1p)),c_2Epred__set_2EUNIV(A_27c))) ) ) ) ) ).
tff(conj_thm_2Eutil__prob_2EPOW__HALF__POS,axiom,
! [V0n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Ereal_2Epow,ap(ap(c_2Ereal_2E_2F,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),inj__ty_2Enum_2Enum(V0n)))) ).
tff(conj_thm_2Eutil__prob_2EPOW__HALF__SMALL,axiom,
! [V0e: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V0e)))
=> ? [V1n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Ereal_2Epow,ap(ap(c_2Ereal_2E_2F,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),inj__ty_2Enum_2Enum(V1n))),inj__ty_2Erealax_2Ereal(V0e))) ) ).
tff(conj_thm_2Eutil__prob_2EPOW__HALF__MONO,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n)))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Ereal_2Epow,ap(ap(c_2Ereal_2E_2F,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),inj__ty_2Enum_2Enum(V1n))),ap(ap(c_2Ereal_2Epow,ap(ap(c_2Ereal_2E_2F,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),inj__ty_2Enum_2Enum(V0m)))) ) ).
tff(conj_thm_2Eutil__prob_2EREAL__LE__LT__MUL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V0x)))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V1y))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Erealax_2Ereal(V1y)))) ) ).
tff(conj_thm_2Eutil__prob_2EREAL__LT__LE__MUL,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V0x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V1y))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Erealax_2Ereal(V1y)))) ) ).
tff(conj_thm_2Eutil__prob_2EREAL__MUL__IDEMPOT,axiom,
! [V0r: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V0r)),inj__ty_2Erealax_2Ereal(V0r))) = V0r )
<=> ( ( V0r = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) )
| ( V0r = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) ) ) ) ).
tff(conj_thm_2Eutil__prob_2EREAL__SUP__LE__X,axiom,
! [V0P: $i] :
( mem(V0P,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( ? [V2r: tp__ty_2Erealax_2Ereal] : p(ap(V0P,inj__ty_2Erealax_2Ereal(V2r)))
& ! [V3r: tp__ty_2Erealax_2Ereal] :
( p(ap(V0P,inj__ty_2Erealax_2Ereal(V3r)))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3r)),inj__ty_2Erealax_2Ereal(V1x))) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Esup,V0P)),inj__ty_2Erealax_2Ereal(V1x))) ) ) ).
tff(conj_thm_2Eutil__prob_2EREAL__X__LE__SUP,axiom,
! [V0P: $i] :
( mem(V0P,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( ? [V2r: tp__ty_2Erealax_2Ereal] : p(ap(V0P,inj__ty_2Erealax_2Ereal(V2r)))
& ? [V3z: tp__ty_2Erealax_2Ereal] :
! [V4r: tp__ty_2Erealax_2Ereal] :
( p(ap(V0P,inj__ty_2Erealax_2Ereal(V4r)))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V4r)),inj__ty_2Erealax_2Ereal(V3z))) )
& ? [V5r: tp__ty_2Erealax_2Ereal] :
( p(ap(V0P,inj__ty_2Erealax_2Ereal(V5r)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Erealax_2Ereal(V5r))) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1x)),ap(c_2Ereal_2Esup,V0P))) ) ) ).
tff(lamtp_f3169,type,
f3169: $i > $i ).
tff(lameq_f3169,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1r: tp__ty_2Erealax_2Ereal] : ( ap(f3169(V0p),inj__ty_2Erealax_2Ereal(V1r)) = ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(c_2Erealax_2Ereal__neg,inj__ty_2Erealax_2Ereal(V1r))),V0p) ) ) ).
tff(conj_thm_2Eutil__prob_2EINF__DEF__ALT,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Erealax_2Ereal,bool))
=> ( surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Einf,V0p)) = surj__ty_2Erealax_2Ereal(ap(c_2Erealax_2Ereal__neg,ap(c_2Ereal_2Esup,f3169(V0p)))) ) ) ).
tff(conj_thm_2Eutil__prob_2ELE__INF,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1r: tp__ty_2Erealax_2Ereal] :
( ( ? [V2x: tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0p))
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),V0p))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1r)),inj__ty_2Erealax_2Ereal(V3x))) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1r)),ap(c_2Ereal_2Einf,V0p))) ) ) ).
tff(conj_thm_2Eutil__prob_2EINF__LE,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1r: tp__ty_2Erealax_2Ereal] :
( ( ? [V2z: tp__ty_2Erealax_2Ereal] :
! [V3x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),V0p))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2z)),inj__ty_2Erealax_2Ereal(V3x))) )
& ? [V4x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),V0p))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V4x)),inj__ty_2Erealax_2Ereal(V1r))) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Einf,V0p)),inj__ty_2Erealax_2Ereal(V1r))) ) ) ).
tff(conj_thm_2Eutil__prob_2EINF__GREATER,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1z: tp__ty_2Erealax_2Ereal] :
( ( ? [V2x: tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0p))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Einf,V0p)),inj__ty_2Erealax_2Ereal(V1z))) )
=> ? [V3x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),V0p))
& p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V3x)),inj__ty_2Erealax_2Ereal(V1z))) ) ) ) ).
tff(conj_thm_2Eutil__prob_2EINF__CLOSE,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1e: tp__ty_2Erealax_2Ereal] :
( ( ? [V2x: tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0p))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V1e))) )
=> ? [V3x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),V0p))
& p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V3x)),ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal_2Einf,V0p)),inj__ty_2Erealax_2Ereal(V1e)))) ) ) ) ).
tff(conj_thm_2Eutil__prob_2EREAL__NEG__NZ,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V0x)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))
=> ( V0x != surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ).
tff(conj_thm_2Eutil__prob_2EREAL__LT__LMUL__0__NEG,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Erealax_2Ereal(V1y))))
& p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V0x)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V1y)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) ) ).
tff(conj_thm_2Eutil__prob_2EREAL__LT__RMUL__0__NEG,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Erealax_2Ereal(V1y))))
& p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V1y)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V0x)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) ) ).
tff(conj_thm_2Eutil__prob_2EREAL__LT__LMUL__NEG__0,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Erealax_2Ereal(V1y))),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V0x))) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V1y)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) ) ).
tff(conj_thm_2Eutil__prob_2EREAL__LT__RMUL__NEG__0,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Erealax_2Ereal(V1y))),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V1y))) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V0x)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) ) ).
tff(conj_thm_2Eutil__prob_2EREAL__LT__LMUL__NEG__0__NEG,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Erealax_2Ereal(V1y))),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))
& p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V0x)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V1y))) ) ).
tff(conj_thm_2Eutil__prob_2EREAL__LT__RMUL__NEG__0__NEG,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Erealax_2Ereal(V1y))),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))
& p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V1y)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V0x))) ) ).
tff(conj_thm_2Eutil__prob_2EREAL__LT__RDIV__EQ__NEG,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2z: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V2z)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Ereal_2E_2F,inj__ty_2Erealax_2Ereal(V1y)),inj__ty_2Erealax_2Ereal(V2z))),inj__ty_2Erealax_2Ereal(V0x)))
<=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Erealax_2Ereal(V2z))),inj__ty_2Erealax_2Ereal(V1y))) ) ) ).
tff(conj_thm_2Eutil__prob_2EREAL__LE__RDIV__EQ__NEG,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2z: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V2z)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Ereal_2E_2F,inj__ty_2Erealax_2Ereal(V1y)),inj__ty_2Erealax_2Ereal(V2z))),inj__ty_2Erealax_2Ereal(V0x)))
<=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Erealax_2Ereal(V2z))),inj__ty_2Erealax_2Ereal(V1y))) ) ) ).
tff(conj_thm_2Eutil__prob_2EPOW__POS__EVEN,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V1x)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Enum_2Enum(V0n))))
<=> p(ap(c_2Earithmetic_2EEVEN,inj__ty_2Enum_2Enum(V0n))) ) ) ).
tff(conj_thm_2Eutil__prob_2EPOW__NEG__ODD,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V1x)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Enum_2Enum(V0n))),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))
<=> p(ap(c_2Earithmetic_2EODD,inj__ty_2Enum_2Enum(V0n))) ) ) ).
tff(conj_thm_2Eutil__prob_2ELOGR__MONO__LE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V0x)))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V1y)))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),inj__ty_2Erealax_2Ereal(V2b))) )
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eutil__prob_2Elogr,inj__ty_2Erealax_2Ereal(V2b)),inj__ty_2Erealax_2Ereal(V0x))),ap(ap(c_2Eutil__prob_2Elogr,inj__ty_2Erealax_2Ereal(V2b)),inj__ty_2Erealax_2Ereal(V1y))))
<=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Erealax_2Ereal(V1y))) ) ) ).
tff(conj_thm_2Eutil__prob_2ELOGR__MONO__LE__IMP,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V0x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Erealax_2Ereal(V1y)))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),inj__ty_2Erealax_2Ereal(V2b))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eutil__prob_2Elogr,inj__ty_2Erealax_2Ereal(V2b)),inj__ty_2Erealax_2Ereal(V0x))),ap(ap(c_2Eutil__prob_2Elogr,inj__ty_2Erealax_2Ereal(V2b)),inj__ty_2Erealax_2Ereal(V1y)))) ) ).
tff(lamtp_f3170,type,
f3170: ( tp__ty_2Enum_2Enum * $i ) > $i ).
tff(lameq_f3170,axiom,
! [V1n: tp__ty_2Enum_2Enum,V0p: $i] :
( mem(V0p,arr(ty_2Enum_2Enum,bool))
=> ! [V2m: tp__ty_2Enum_2Enum] : ( ap(f3170(V1n,V0p),inj__ty_2Enum_2Enum(V2m)) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V1n))),ap(c_2Ebool_2E_7E,ap(V0p,inj__ty_2Enum_2Enum(V2m)))) ) ) ).
tff(lamtp_f3171,type,
f3171: $i > $i ).
tff(lameq_f3171,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Enum_2Enum,bool))
=> ! [V1n: tp__ty_2Enum_2Enum] : ( ap(f3171(V0p),inj__ty_2Enum_2Enum(V1n)) = ap(ap(c_2Ebool_2E_2F_5C,ap(V0p,inj__ty_2Enum_2Enum(V1n))),ap(c_2Ebool_2E_21(ty_2Enum_2Enum),f3170(V1n,V0p))) ) ) ).
tff(ax_thm_2Eutil__prob_2Eminimal__def,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Enum_2Enum,bool))
=> ( surj__ty_2Enum_2Enum(ap(c_2Eutil__prob_2Eminimal,V0p)) = surj__ty_2Enum_2Enum(ap(c_2Emin_2E_40(ty_2Enum_2Enum),f3171(V0p))) ) ) ).
tff(conj_thm_2Eutil__prob_2EMINIMAL__EXISTS0,axiom,
! [V0P: $i] :
( mem(V0P,arr(ty_2Enum_2Enum,bool))
=> ( ? [V1n: tp__ty_2Enum_2Enum] : p(ap(V0P,inj__ty_2Enum_2Enum(V1n)))
<=> ? [V2n: tp__ty_2Enum_2Enum] :
( p(ap(V0P,inj__ty_2Enum_2Enum(V2n)))
& ! [V3m: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V3m)),inj__ty_2Enum_2Enum(V2n)))
=> ~ p(ap(V0P,inj__ty_2Enum_2Enum(V3m))) ) ) ) ) ).
tff(conj_thm_2Eutil__prob_2EMINIMAL__EXISTS,axiom,
! [V0P: $i] :
( mem(V0P,arr(ty_2Enum_2Enum,bool))
=> ( ? [V1n: tp__ty_2Enum_2Enum] : p(ap(V0P,inj__ty_2Enum_2Enum(V1n)))
<=> ( p(ap(V0P,ap(c_2Eutil__prob_2Eminimal,V0P)))
& ! [V2n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V2n)),ap(c_2Eutil__prob_2Eminimal,V0P)))
=> ~ p(ap(V0P,inj__ty_2Enum_2Enum(V2n))) ) ) ) ) ).
tff(conj_thm_2Eutil__prob_2EMINIMAL__EXISTS__IMP,axiom,
! [V0P: $i] :
( mem(V0P,arr(ty_2Enum_2Enum,bool))
=> ( ? [V1n: tp__ty_2Enum_2Enum] : p(ap(V0P,inj__ty_2Enum_2Enum(V1n)))
=> ? [V2m: tp__ty_2Enum_2Enum] :
( p(ap(V0P,inj__ty_2Enum_2Enum(V2m)))
& ! [V3n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V3n)),inj__ty_2Enum_2Enum(V2m)))
=> ~ p(ap(V0P,inj__ty_2Enum_2Enum(V3n))) ) ) ) ) ).
tff(conj_thm_2Eutil__prob_2EMINIMAL__EQ__IMP,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1p: $i] :
( mem(V1p,arr(ty_2Enum_2Enum,bool))
=> ( ( p(ap(V1p,inj__ty_2Enum_2Enum(V0m)))
& ! [V2n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V2n)),inj__ty_2Enum_2Enum(V0m)))
=> ~ p(ap(V1p,inj__ty_2Enum_2Enum(V2n))) ) )
=> ( V0m = surj__ty_2Enum_2Enum(ap(c_2Eutil__prob_2Eminimal,V1p)) ) ) ) ).
tff(conj_thm_2Eutil__prob_2EMINIMAL__SUC,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1p: $i] :
( mem(V1p,arr(ty_2Enum_2Enum,bool))
=> ( ( ( surj__ty_2Enum_2Enum(ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V0n))) = surj__ty_2Enum_2Enum(ap(c_2Eutil__prob_2Eminimal,V1p)) )
& p(ap(V1p,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V0n)))) )
<=> ( ~ p(ap(V1p,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))
& ( V0n = surj__ty_2Enum_2Enum(ap(c_2Eutil__prob_2Eminimal,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,bool,ty_2Enum_2Enum),V1p),c_2Enum_2ESUC))) )
& p(ap(V1p,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V0n)))) ) ) ) ).
tff(conj_thm_2Eutil__prob_2EMINIMAL__EQ,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Enum_2Enum,bool))
=> ! [V1m: tp__ty_2Enum_2Enum] :
( ( p(ap(V0p,inj__ty_2Enum_2Enum(V1m)))
& ( V1m = surj__ty_2Enum_2Enum(ap(c_2Eutil__prob_2Eminimal,V0p)) ) )
<=> ( p(ap(V0p,inj__ty_2Enum_2Enum(V1m)))
& ! [V2n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V2n)),inj__ty_2Enum_2Enum(V1m)))
=> ~ p(ap(V0p,inj__ty_2Enum_2Enum(V2n))) ) ) ) ) ).
tff(conj_thm_2Eutil__prob_2EMINIMAL__SUC__IMP,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1p: $i] :
( mem(V1p,arr(ty_2Enum_2Enum,bool))
=> ( ( p(ap(V1p,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V0n))))
& ~ p(ap(V1p,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))
& ( V0n = surj__ty_2Enum_2Enum(ap(c_2Eutil__prob_2Eminimal,ap(ap(c_2Ecombin_2Eo(ty_2Enum_2Enum,bool,ty_2Enum_2Enum),V1p),c_2Enum_2ESUC))) ) )
=> ( surj__ty_2Enum_2Enum(ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V0n))) = surj__ty_2Enum_2Enum(ap(c_2Eutil__prob_2Eminimal,V1p)) ) ) ) ).
tff(ax_thm_2Eutil__prob_2Edisjoint__def,axiom,
! [A_27a: del,V0A: $i] :
( mem(V0A,arr(arr(A_27a,bool),bool))
=> ( p(ap(c_2Eutil__prob_2Edisjoint(A_27a),V0A))
<=> ! [V1a: $i] :
( mem(V1a,arr(A_27a,bool))
=> ! [V2b: $i] :
( mem(V2b,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1a),V0A))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2b),V0A))
& ( V1a != V2b ) )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1a),V2b)) ) ) ) ) ) ).
tff(conj_thm_2Eutil__prob_2EdisjointI,axiom,
! [A_27a: del,V0A: $i] :
( mem(V0A,arr(arr(A_27a,bool),bool))
=> ( ! [V1a: $i] :
( mem(V1a,arr(A_27a,bool))
=> ! [V2b: $i] :
( mem(V2b,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1a),V0A))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2b),V0A))
=> ( ( V1a != V2b )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1a),V2b)) ) ) ) ) )
=> p(ap(c_2Eutil__prob_2Edisjoint(A_27a),V0A)) ) ) ).
tff(conj_thm_2Eutil__prob_2EdisjointD,axiom,
! [A_27a: del,V0A: $i] :
( mem(V0A,arr(arr(A_27a,bool),bool))
=> ! [V1a: $i] :
( mem(V1a,arr(A_27a,bool))
=> ! [V2b: $i] :
( mem(V2b,arr(A_27a,bool))
=> ( p(ap(c_2Eutil__prob_2Edisjoint(A_27a),V0A))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1a),V0A))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2b),V0A))
=> ( ( V1a != V2b )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V1a),V2b)) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eutil__prob_2Edisjoint__empty,axiom,
! [A_27a: del] : p(ap(c_2Eutil__prob_2Edisjoint(A_27a),c_2Epred__set_2EEMPTY(arr(A_27a,bool)))) ).
tff(conj_thm_2Eutil__prob_2Edisjoint__union,axiom,
! [A_27a: del,V0A: $i] :
( mem(V0A,arr(arr(A_27a,bool),bool))
=> ! [V1B: $i] :
( mem(V1B,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Eutil__prob_2Edisjoint(A_27a),V0A))
& p(ap(c_2Eutil__prob_2Edisjoint(A_27a),V1B))
& ( ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),V0A)),ap(c_2Epred__set_2EBIGUNION(A_27a),V1B)) = c_2Epred__set_2EEMPTY(A_27a) ) )
=> p(ap(c_2Eutil__prob_2Edisjoint(A_27a),ap(ap(c_2Epred__set_2EUNION(arr(A_27a,bool)),V0A),V1B))) ) ) ) ).
tff(conj_thm_2Eutil__prob_2Edisjoint__sing,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,arr(A_27a,bool))
=> p(ap(c_2Eutil__prob_2Edisjoint(A_27a),ap(ap(c_2Epred__set_2EINSERT(arr(A_27a,bool)),V0a),c_2Epred__set_2EEMPTY(arr(A_27a,bool))))) ) ).
%------------------------------------------------------------------------------