TPTP Problem File: ITP020_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP020_5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory export of thm_2Eutil__prob_2ENUM__2D__BIJ__INV.p, 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 : thm_2Eutil__prob_2ENUM__2D__BIJ__INV.p [Gau20]
% : HL409501_5.p [TPAP]
% Status : Theorem
% Rating : 0.56 v8.2.0, 0.80 v8.1.0, 0.91 v7.5.0
% Syntax : Number of formulae : 25383 (6067 unt;6463 typ; 0 def)
% Number of atoms : 404898 (20324 equ)
% Maximal formula atoms : 5763 ( 15 avg)
% Number of connectives : 68302 (2237 ~;1027 |;24982 &)
% (6017 <=>;34039 =>; 0 <=; 0 <~>)
% Maximal formula depth : 361 ( 6 avg)
% Maximal term depth : 132 ( 2 avg)
% Number of FOOLs : 319913 (319913 fml; 0 var)
% Number of types : 148 ( 146 usr)
% Number of type conns : 11832 (5151 >;6681 *; 0 +; 0 <<)
% Number of predicates : 40 ( 37 usr; 23 prp; 0-3 aty)
% Number of functors : 6315 (6315 usr;1166 con; 0-11 aty)
% Number of variables : 77170 (63395 !;13775 ?;77170 :)
% SPC : TF0_THM_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001_2.ax').
include('Axioms/ITP001/ITP002_5.ax').
include('Axioms/ITP001/ITP003_5.ax').
include('Axioms/ITP001/ITP004_5.ax').
include('Axioms/ITP001/ITP007_5.ax').
include('Axioms/ITP001/ITP006_5.ax').
include('Axioms/ITP001/ITP005_5.ax').
include('Axioms/ITP001/ITP008_5.ax').
include('Axioms/ITP001/ITP009_5.ax').
include('Axioms/ITP001/ITP010_5.ax').
include('Axioms/ITP001/ITP012_5.ax').
include('Axioms/ITP001/ITP011_5.ax').
include('Axioms/ITP001/ITP013_5.ax').
include('Axioms/ITP001/ITP014_5.ax').
include('Axioms/ITP001/ITP015_5.ax').
include('Axioms/ITP001/ITP017_5.ax').
include('Axioms/ITP001/ITP016_5.ax').
include('Axioms/ITP001/ITP019_5.ax').
include('Axioms/ITP001/ITP018_5.ax').
include('Axioms/ITP001/ITP021_5.ax').
include('Axioms/ITP001/ITP022_5.ax').
include('Axioms/ITP001/ITP020_5.ax').
include('Axioms/ITP001/ITP024_5.ax').
include('Axioms/ITP001/ITP023_5.ax').
include('Axioms/ITP001/ITP025_5.ax').
include('Axioms/ITP001/ITP026_5.ax').
include('Axioms/ITP001/ITP027_5.ax').
include('Axioms/ITP001/ITP028_5.ax').
include('Axioms/ITP001/ITP031_5.ax').
include('Axioms/ITP001/ITP029_5.ax').
include('Axioms/ITP001/ITP033_5.ax').
include('Axioms/ITP001/ITP030_5.ax').
include('Axioms/ITP001/ITP032_5.ax').
include('Axioms/ITP001/ITP038_5.ax').
include('Axioms/ITP001/ITP035_5.ax').
include('Axioms/ITP001/ITP034_5.ax').
include('Axioms/ITP001/ITP036_5.ax').
include('Axioms/ITP001/ITP037_5.ax').
include('Axioms/ITP001/ITP039_5.ax').
include('Axioms/ITP001/ITP041_5.ax').
include('Axioms/ITP001/ITP042_5.ax').
include('Axioms/ITP001/ITP040_5.ax').
include('Axioms/ITP001/ITP044_5.ax').
include('Axioms/ITP001/ITP051_5.ax').
include('Axioms/ITP001/ITP045_5.ax').
include('Axioms/ITP001/ITP056_5.ax').
include('Axioms/ITP001/ITP046_5.ax').
include('Axioms/ITP001/ITP043_5.ax').
include('Axioms/ITP001/ITP052_5.ax').
include('Axioms/ITP001/ITP057_5.ax').
include('Axioms/ITP001/ITP048_5.ax').
include('Axioms/ITP001/ITP047_5.ax').
include('Axioms/ITP001/ITP055_5.ax').
include('Axioms/ITP001/ITP053_5.ax').
include('Axioms/ITP001/ITP054_5.ax').
include('Axioms/ITP001/ITP058_5.ax').
include('Axioms/ITP001/ITP049_5.ax').
include('Axioms/ITP001/ITP050_5.ax').
include('Axioms/ITP001/ITP061_5.ax').
include('Axioms/ITP001/ITP069_5.ax').
include('Axioms/ITP001/ITP062_5.ax').
include('Axioms/ITP001/ITP068_5.ax').
include('Axioms/ITP001/ITP078_5.ax').
include('Axioms/ITP001/ITP064_5.ax').
include('Axioms/ITP001/ITP060_5.ax').
include('Axioms/ITP001/ITP067_5.ax').
include('Axioms/ITP001/ITP075_5.ax').
include('Axioms/ITP001/ITP074_5.ax').
include('Axioms/ITP001/ITP063_5.ax').
include('Axioms/ITP001/ITP059_5.ax').
include('Axioms/ITP001/ITP065_5.ax').
include('Axioms/ITP001/ITP076_5.ax').
include('Axioms/ITP001/ITP066_5.ax').
include('Axioms/ITP001/ITP077_5.ax').
include('Axioms/ITP001/ITP070_5.ax').
include('Axioms/ITP001/ITP071_5.ax').
include('Axioms/ITP001/ITP072_5.ax').
include('Axioms/ITP001/ITP073_5.ax').
include('Axioms/ITP001/ITP080_5.ax').
include('Axioms/ITP001/ITP083_5.ax').
include('Axioms/ITP001/ITP088_5.ax').
include('Axioms/ITP001/ITP082_5.ax').
include('Axioms/ITP001/ITP081_5.ax').
include('Axioms/ITP001/ITP079_5.ax').
include('Axioms/ITP001/ITP087_5.ax').
include('Axioms/ITP001/ITP085_5.ax').
include('Axioms/ITP001/ITP086_5.ax').
include('Axioms/ITP001/ITP090_5.ax').
include('Axioms/ITP001/ITP089_5.ax').
include('Axioms/ITP001/ITP084_5.ax').
include('Axioms/ITP001/ITP104_5.ax').
include('Axioms/ITP001/ITP097_5.ax').
include('Axioms/ITP001/ITP094_5.ax').
include('Axioms/ITP001/ITP095_5.ax').
include('Axioms/ITP001/ITP093_5.ax').
include('Axioms/ITP001/ITP102_5.ax').
include('Axioms/ITP001/ITP091_5.ax').
include('Axioms/ITP001/ITP098_5.ax').
include('Axioms/ITP001/ITP096_5.ax').
include('Axioms/ITP001/ITP100_5.ax').
include('Axioms/ITP001/ITP092_5.ax').
include('Axioms/ITP001/ITP105_5.ax').
include('Axioms/ITP001/ITP101_5.ax').
include('Axioms/ITP001/ITP103_5.ax').
include('Axioms/ITP001/ITP099_5.ax').
include('Axioms/ITP001/ITP111_5.ax').
include('Axioms/ITP001/ITP108_5.ax').
include('Axioms/ITP001/ITP106_5.ax').
include('Axioms/ITP001/ITP107_5.ax').
include('Axioms/ITP001/ITP114_5.ax').
include('Axioms/ITP001/ITP110_5.ax').
include('Axioms/ITP001/ITP109_5.ax').
include('Axioms/ITP001/ITP112_5.ax').
include('Axioms/ITP001/ITP113_5.ax').
include('Axioms/ITP001/ITP117_5.ax').
include('Axioms/ITP001/ITP116_5.ax').
include('Axioms/ITP001/ITP118_5.ax').
include('Axioms/ITP001/ITP120_5.ax').
include('Axioms/ITP001/ITP119_5.ax').
include('Axioms/ITP001/ITP121_5.ax').
include('Axioms/ITP001/ITP123_5.ax').
include('Axioms/ITP001/ITP124_5.ax').
include('Axioms/ITP001/ITP126_5.ax').
include('Axioms/ITP001/ITP127_5.ax').
include('Axioms/ITP001/ITP129_5.ax').
include('Axioms/ITP001/ITP128_5.ax').
include('Axioms/ITP001/ITP130_5.ax').
include('Axioms/ITP001/ITP131_5.ax').
include('Axioms/ITP001/ITP132_5.ax').
include('Axioms/ITP001/ITP134_5.ax').
include('Axioms/ITP001/ITP133_5.ax').
include('Axioms/ITP001/ITP135_5.ax').
include('Axioms/ITP001/ITP136_5.ax').
include('Axioms/ITP001/ITP138_5.ax').
include('Axioms/ITP001/ITP137_5.ax').
include('Axioms/ITP001/ITP140_5.ax').
include('Axioms/ITP001/ITP141_5.ax').
include('Axioms/ITP001/ITP139_5.ax').
%------------------------------------------------------------------------------
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,conjecture,
? [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)))) ) ).
%------------------------------------------------------------------------------