ITP001 Axioms: ITP131^5.ax
%------------------------------------------------------------------------------
% File : ITP131^5 : 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 : powser^2.ax [Gau20]
% : HL4131^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 18 ( 4 unt; 1 typ; 0 def)
% Number of atoms : 1128 ( 11 equ; 0 cnn)
% Maximal formula atoms : 112 ( 62 avg)
% Number of connectives : 1331 ( 2 ~; 0 |; 12 &;1296 @)
% ( 0 <=>; 21 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 11 avg;1296 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 38 ( 37 usr; 37 con; 0-2 aty)
% Number of variables : 77 ( 30 ^ 47 !; 0 ?; 77 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Epowser_2Ediffs,type,
c_2Epowser_2Ediffs: $i ).
thf(mem_c_2Epowser_2Ediffs,axiom,
mem @ c_2Epowser_2Ediffs @ ( arr @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) ).
thf(conj_thm_2Epowser_2EPOWDIFF__LEMMA,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3p: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V3p ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ V3p ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2y ) )
@ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4p: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V4p ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V4p ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epowser_2EPOWDIFF,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) )
@ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3p: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V3p ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V3p ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epowser_2EPOWREV,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3p: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V3p ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V3p ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4p: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V4p ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ V4p ) ) ) ) ) ) ).
thf(conj_thm_2Epowser_2EPOWSER__INSIDEA,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2z: tp__ty_2Erealax_2Ereal] :
( ( ( p
@ ( ap @ c_2Eseq_2Esummable
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ V3n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V3n ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) )
=> ( p
@ ( ap @ c_2Eseq_2Esummable
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ V4n ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) @ V4n ) ) ) ) ) ) ) ).
thf(conj_thm_2Epowser_2EPOWSER__INSIDE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2z: tp__ty_2Erealax_2Ereal] :
( ( ( p
@ ( ap @ c_2Eseq_2Esummable
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ V3n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V3n ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) )
=> ( p
@ ( ap @ c_2Eseq_2Esummable
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ V4n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) @ V4n ) ) ) ) ) ) ) ).
thf(ax_thm_2Epowser_2Ediffs,axiom,
! [V0c: $i] :
( ( mem @ V0c @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ap @ c_2Epowser_2Ediffs @ V0c )
= ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Enum_2ESUC @ V1n ) ) ) @ ( ap @ V0c @ ( ap @ c_2Enum_2ESUC @ V1n ) ) ) ) ) ) ).
thf(conj_thm_2Epowser_2EDIFFS__NEG,axiom,
! [V0c: $i] :
( ( mem @ V0c @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ap @ c_2Epowser_2Ediffs
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ V0c @ V1n ) ) ) )
= ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ c_2Epowser_2Ediffs @ V0c ) @ V2n ) ) ) ) ) ).
thf(conj_thm_2Epowser_2EDIFFS__LEMMA,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1c: $i] :
( ( mem @ V1c @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Epowser_2Ediffs @ V1c ) @ V3n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V3n ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ c_2Erealax_2Ereal__add
@ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ V4n ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V1c @ V4n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V4n ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V1c @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epowser_2EDIFFS__LEMMA2,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1c: $i] :
( ( mem @ V1c @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ V3n ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V1c @ V3n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V3n ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ c_2Ereal_2Ereal__sub
@ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Epowser_2Ediffs @ V1c ) @ V4n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V4n ) ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V1c @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epowser_2EDIFFS__EQUIV,axiom,
! [V0c: $i] :
( ( mem @ V0c @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p
@ ( ap @ c_2Eseq_2Esummable
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Epowser_2Ediffs @ V0c ) @ V2n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V2n ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Eseq_2Esums
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ V3n ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0c @ V3n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V3n ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) )
@ ( ap @ c_2Eseq_2Esuminf
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Epowser_2Ediffs @ V0c ) @ V4n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V4n ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epowser_2ETERMDIFF__LEMMA1,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1z: tp__ty_2Erealax_2Ereal,V2h: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3p: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1z ) ) @ ( inj__ty_2Erealax_2Ereal @ V2h ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ V3p ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1z ) ) @ V3p ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1z ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4p: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1z ) ) @ V4p ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1z ) ) @ ( inj__ty_2Erealax_2Ereal @ V2h ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ V4p ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1z ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ V4p ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epowser_2ETERMDIFF__LEMMA2,axiom,
! [V0z: tp__ty_2Erealax_2Ereal,V1h: tp__ty_2Erealax_2Ereal,V2n: tp__ty_2Enum_2Enum] :
( ( V1h
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__ty_2Erealax_2Ereal @ V1h ) ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1h ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1h ) )
@ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3p: $i] :
( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0z ) ) @ V3p ) )
@ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V3p ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4q: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__ty_2Erealax_2Ereal @ V1h ) ) ) @ V4q ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V3p ) ) @ V4q ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epowser_2ETERMDIFF__LEMMA3,axiom,
! [V0z: tp__ty_2Erealax_2Ereal,V1h: tp__ty_2Erealax_2Ereal,V2n: tp__ty_2Enum_2Enum,V3k_27: tp__ty_2Erealax_2Ereal] :
( ( ( V1h
!= ( 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_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0z ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3k_27 ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__ty_2Erealax_2Ereal @ V1h ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3k_27 ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__ty_2Erealax_2Ereal @ V1h ) ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0z ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1h ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0z ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V3k_27 ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1h ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epowser_2ETERMDIFF__LEMMA4,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1k_27: tp__ty_2Erealax_2Ereal,V2k: 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 @ V2k ) ) )
& ! [V3h: 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 @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V3h ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V3h ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2k ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ ( inj__ty_2Erealax_2Ereal @ V3h ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1k_27 ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V3h ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ c_2Elim_2Etends__real__real @ V0f ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Epowser_2ETERMDIFF__LEMMA5,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) )
=> ! [V2k: 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 @ V2k ) ) )
& ( p @ ( ap @ c_2Eseq_2Esummable @ V0f ) )
& ! [V3h: 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 @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V3h ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V3h ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2k ) ) ) )
=> ! [V4n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ V1g @ ( inj__ty_2Erealax_2Ereal @ V3h ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V3h ) ) ) ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Etends__real__real
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5h: $i] : ( ap @ c_2Eseq_2Esuminf @ ( ap @ V1g @ V5h ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Epowser_2ETERMDIFF,axiom,
! [V0c: $i] :
( ( mem @ V0c @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1k_27: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] :
( ( ( p
@ ( ap @ c_2Eseq_2Esummable
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0c @ V3n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1k_27 ) ) @ V3n ) ) ) ) )
& ( p
@ ( ap @ c_2Eseq_2Esummable
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Epowser_2Ediffs @ V0c ) @ V4n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1k_27 ) ) @ V4n ) ) ) ) )
& ( p
@ ( ap @ c_2Eseq_2Esummable
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Epowser_2Ediffs @ ( ap @ c_2Epowser_2Ediffs @ V0c ) ) @ V5n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1k_27 ) ) @ V5n ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1k_27 ) ) ) ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ c_2Elim_2Ediffl
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] :
( ap @ c_2Eseq_2Esuminf
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0c @ V7n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ V6x ) @ V7n ) ) ) ) ) )
@ ( ap @ c_2Eseq_2Esuminf
@ ( lam @ ty_2Enum_2Enum
@ ^ [V8n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Epowser_2Ediffs @ V0c ) @ V8n ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V8n ) ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ).
%------------------------------------------------------------------------------