ITP001 Axioms: ITP127^5.ax
%------------------------------------------------------------------------------
% File : ITP127^5 : TPTP v9.0.0. 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 : seq^2.ax [Gau20]
% : HL4127^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 138 ( 6 unt; 14 typ; 0 def)
% Number of atoms : 3352 ( 23 equ; 0 cnn)
% Maximal formula atoms : 100 ( 24 avg)
% Number of connectives : 4218 ( 3 ~; 5 |; 75 &;3898 @)
% ( 28 <=>; 209 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 13 avg;3898 nst)
% Number of types : 2 ( 1 usr)
% Number of type conns : 2 ( 2 >; 0 *; 0 +; 0 <<)
% Number of symbols : 76 ( 75 usr; 73 con; 0-2 aty)
% Number of variables : 353 ( 41 ^ 291 !; 21 ?; 353 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Eseq_2E_2D_2D_3E,type,
c_2Eseq_2E_2D_2D_3E: $i ).
thf(mem_c_2Eseq_2E_2D_2D_3E,axiom,
mem @ c_2Eseq_2E_2D_2D_3E @ ( arr @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ).
thf(tp_c_2Eseq_2Ecauchy,type,
c_2Eseq_2Ecauchy: $i ).
thf(mem_c_2Eseq_2Ecauchy,axiom,
mem @ c_2Eseq_2Ecauchy @ ( arr @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ bool ) ).
thf(tp_c_2Eseq_2Econvergent,type,
c_2Eseq_2Econvergent: $i ).
thf(mem_c_2Eseq_2Econvergent,axiom,
mem @ c_2Eseq_2Econvergent @ ( arr @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ bool ) ).
thf(tp_c_2Eseq_2Elim,type,
c_2Eseq_2Elim: $i ).
thf(mem_c_2Eseq_2Elim,axiom,
mem @ c_2Eseq_2Elim @ ( arr @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ty_2Erealax_2Ereal ) ).
thf(tp_c_2Eseq_2Emono,type,
c_2Eseq_2Emono: $i ).
thf(mem_c_2Eseq_2Emono,axiom,
mem @ c_2Eseq_2Emono @ ( arr @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ bool ) ).
thf(tp_c_2Eseq_2Emono__decreasing,type,
c_2Eseq_2Emono__decreasing: $i ).
thf(mem_c_2Eseq_2Emono__decreasing,axiom,
mem @ c_2Eseq_2Emono__decreasing @ ( arr @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ bool ) ).
thf(tp_c_2Eseq_2Emono__increasing,type,
c_2Eseq_2Emono__increasing: $i ).
thf(mem_c_2Eseq_2Emono__increasing,axiom,
mem @ c_2Eseq_2Emono__increasing @ ( arr @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ bool ) ).
thf(tp_c_2Eseq_2Esubseq,type,
c_2Eseq_2Esubseq: $i ).
thf(mem_c_2Eseq_2Esubseq,axiom,
mem @ c_2Eseq_2Esubseq @ ( arr @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ bool ) ).
thf(tp_c_2Eseq_2Esuminf,type,
c_2Eseq_2Esuminf: $i ).
thf(mem_c_2Eseq_2Esuminf,axiom,
mem @ c_2Eseq_2Esuminf @ ( arr @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ty_2Erealax_2Ereal ) ).
thf(tp_c_2Eseq_2Esummable,type,
c_2Eseq_2Esummable: $i ).
thf(mem_c_2Eseq_2Esummable,axiom,
mem @ c_2Eseq_2Esummable @ ( arr @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ bool ) ).
thf(tp_c_2Eseq_2Esums,type,
c_2Eseq_2Esums: $i ).
thf(mem_c_2Eseq_2Esums,axiom,
mem @ c_2Eseq_2Esums @ ( arr @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) ).
thf(ax_thm_2Eseq_2Etends__num__real,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1x0: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V0x ) @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Enets_2Etends @ ty_2Erealax_2Ereal @ ty_2Enum_2Enum ) @ V0x ) @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Etopology_2Etopology @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ c_2Earithmetic_2E_3E_3D ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1x0: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V0x ) @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) )
<=> ! [V2e: 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 @ V2e ) ) )
=> ? [V3N: tp__ty_2Enum_2Enum] :
! [V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ ( inj__ty_2Enum_2Enum @ V3N ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0x @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__CONST,axiom,
! [V0k: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap
@ ( ap @ c_2Eseq_2E_2D_2D_3E
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1x: $i] : ( inj__ty_2Erealax_2Ereal @ V0k ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V0k ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__ADD,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1x0: tp__ty_2Erealax_2Ereal,V2y: $i] :
( ( mem @ V2y @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V3y0: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V0x ) @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) )
& ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V2y ) @ ( inj__ty_2Erealax_2Ereal @ V3y0 ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Eseq_2E_2D_2D_3E
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0x @ V4n ) ) @ ( ap @ V2y @ V4n ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__MUL,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1x0: tp__ty_2Erealax_2Ereal,V2y: $i] :
( ( mem @ V2y @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V3y0: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V0x ) @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) )
& ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V2y ) @ ( inj__ty_2Erealax_2Ereal @ V3y0 ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Eseq_2E_2D_2D_3E
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0x @ V4n ) ) @ ( ap @ V2y @ V4n ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__NEG,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1x0: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V0x ) @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) )
<=> ( p
@ ( ap
@ ( ap @ c_2Eseq_2E_2D_2D_3E
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ V0x @ V2n ) ) ) )
@ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__INV,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1x0: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V0x ) @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) )
& ( V1x0
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Eseq_2E_2D_2D_3E
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ c_2Erealax_2Einv @ ( ap @ V0x @ V2n ) ) ) )
@ ( ap @ c_2Erealax_2Einv @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__SUB,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1x0: tp__ty_2Erealax_2Ereal,V2y: $i] :
( ( mem @ V2y @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V3y0: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V0x ) @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) )
& ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V2y ) @ ( inj__ty_2Erealax_2Ereal @ V3y0 ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Eseq_2E_2D_2D_3E
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0x @ V4n ) ) @ ( ap @ V2y @ V4n ) ) ) )
@ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__DIV,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1x0: tp__ty_2Erealax_2Ereal,V2y: $i] :
( ( mem @ V2y @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V3y0: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V0x ) @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) )
& ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V2y ) @ ( inj__ty_2Erealax_2Ereal @ V3y0 ) ) )
& ( V3y0
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Eseq_2E_2D_2D_3E
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ V0x @ V4n ) ) @ ( ap @ V2y @ V4n ) ) ) )
@ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__UNIQ,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1x1: tp__ty_2Erealax_2Ereal,V2x2: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V0x ) @ ( inj__ty_2Erealax_2Ereal @ V1x1 ) ) )
& ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V0x ) @ ( inj__ty_2Erealax_2Ereal @ V2x2 ) ) ) )
=> ( V1x1 = V2x2 ) ) ) ).
thf(ax_thm_2Eseq_2Econvergent,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Eseq_2Econvergent @ V0f ) )
<=> ? [V1l: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) ) ) ).
thf(ax_thm_2Eseq_2Ecauchy,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Eseq_2Ecauchy @ V0f ) )
<=> ! [V1e: 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 @ V1e ) ) )
=> ? [V2N: tp__ty_2Enum_2Enum] :
! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V2N ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ ( inj__ty_2Enum_2Enum @ V2N ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) ) ) ) ).
thf(ax_thm_2Eseq_2Elim,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Eseq_2Elim @ V0f ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( c_2Emin_2E_40 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1l: $i] : ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V0f ) @ V1l ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__LIM,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Eseq_2Econvergent @ V0f ) )
<=> ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V0f ) @ ( ap @ c_2Eseq_2Elim @ V0f ) ) ) ) ) ).
thf(ax_thm_2Eseq_2Esubseq,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ c_2Eseq_2Esubseq @ V0f ) )
<=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESUBSEQ__SUC,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ c_2Eseq_2Esubseq @ V0f ) )
<=> ! [V1n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( ap @ V0f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ) ) ).
thf(ax_thm_2Eseq_2Emono,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Eseq_2Emono @ V0f ) )
<=> ( ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) )
| ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__ge @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2EMONO__SUC,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Eseq_2Emono @ V0f ) )
<=> ( ! [V1n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__ge @ ( ap @ V0f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
| ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2EMAX__LEMMA,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1N: tp__ty_2Enum_2Enum] :
? [V2k: tp__ty_2Erealax_2Ereal] :
! [V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( inj__ty_2Enum_2Enum @ V1N ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2k ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__BOUNDED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Enets_2Ebounded @ ty_2Erealax_2Ereal @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Emetric_2Emetric @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ c_2Emetric_2Emr1 ) @ c_2Earithmetic_2E_3E_3D ) ) @ V0s ) )
<=> ? [V1k: tp__ty_2Erealax_2Ereal] :
! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0s @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1k ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__BOUNDED__2,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1k: tp__ty_2Erealax_2Ereal,V2k_27: tp__ty_2Erealax_2Ereal] :
( ! [V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1k ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2k_27 ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Enets_2Ebounded @ ty_2Erealax_2Ereal @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Emetric_2Emetric @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ c_2Emetric_2Emr1 ) @ c_2Earithmetic_2E_3E_3D ) ) @ V0f ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__CBOUNDED,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Eseq_2Ecauchy @ V0f ) )
=> ( p @ ( ap @ ( ap @ ( c_2Enets_2Ebounded @ ty_2Erealax_2Ereal @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Emetric_2Emetric @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ c_2Emetric_2Emr1 ) @ c_2Earithmetic_2E_3E_3D ) ) @ V0f ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__ICONV,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Enets_2Ebounded @ ty_2Erealax_2Ereal @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Emetric_2Emetric @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ c_2Emetric_2Emr1 ) @ c_2Earithmetic_2E_3E_3D ) ) @ V0f ) )
& ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__ge @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) )
=> ( p @ ( ap @ c_2Eseq_2Econvergent @ V0f ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__NEG__CONV,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Eseq_2Econvergent @ V0f ) )
<=> ( p
@ ( ap @ c_2Eseq_2Econvergent
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ V0f @ V1n ) ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__NEG__BOUNDED,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Enets_2Ebounded @ ty_2Erealax_2Ereal @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Emetric_2Emetric @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ c_2Emetric_2Emr1 ) @ c_2Earithmetic_2E_3E_3D ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ V0f @ V1n ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Enets_2Ebounded @ ty_2Erealax_2Ereal @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Emetric_2Emetric @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ c_2Emetric_2Emr1 ) @ c_2Earithmetic_2E_3E_3D ) ) @ V0f ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__BCONV,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Enets_2Ebounded @ ty_2Erealax_2Ereal @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Emetric_2Emetric @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ c_2Emetric_2Emr1 ) @ c_2Earithmetic_2E_3E_3D ) ) @ V0f ) )
& ( p @ ( ap @ c_2Eseq_2Emono @ V0f ) ) )
=> ( p @ ( ap @ c_2Eseq_2Econvergent @ V0f ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__MONOSUB,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ? [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
& ( p @ ( ap @ c_2Eseq_2Esubseq @ V1f ) )
& ( p
@ ( ap @ c_2Eseq_2Emono
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ V0s @ ( ap @ V1f @ V2n ) ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__SBOUNDED,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Enets_2Ebounded @ ty_2Erealax_2Ereal @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Emetric_2Emetric @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ c_2Emetric_2Emr1 ) @ c_2Earithmetic_2E_3E_3D ) ) @ V0s ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Enets_2Ebounded @ ty_2Erealax_2Ereal @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Emetric_2Emetric @ ty_2Erealax_2Ereal ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) @ c_2Emetric_2Emr1 ) @ c_2Earithmetic_2E_3E_3D ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ V0s @ ( ap @ V1f @ V2n ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__SUBLE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ c_2Eseq_2Esubseq @ V0f ) )
=> ! [V1n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__DIRECT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ c_2Eseq_2Esubseq @ V0f ) )
=> ! [V1N1: tp__ty_2Enum_2Enum,V2N2: tp__ty_2Enum_2Enum] :
? [V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( inj__ty_2Enum_2Enum @ V1N1 ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( inj__ty_2Enum_2Enum @ V2N2 ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__CAUCHY,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Eseq_2Ecauchy @ V0f ) )
<=> ( p @ ( ap @ c_2Eseq_2Econvergent @ V0f ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__LE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2l: tp__ty_2Erealax_2Ereal,V3m: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) )
& ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V3m ) ) )
& ? [V4N: tp__ty_2Enum_2Enum] :
! [V5n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V5n ) ) @ ( inj__ty_2Enum_2Enum @ V4N ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2l ) ) @ ( inj__ty_2Erealax_2Ereal @ V3m ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__SUC,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) )
<=> ( p
@ ( ap
@ ( ap @ c_2Eseq_2E_2D_2D_3E
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ V0f @ ( ap @ c_2Enum_2ESUC @ V2n ) ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__ABS,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p
@ ( ap
@ ( ap @ c_2Eseq_2E_2D_2D_3E
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ V1n ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V0f ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__ABS__IMP,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Eseq_2E_2D_2D_3E
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ V2n ) ) ) )
@ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__INV0,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ! [V1y: tp__ty_2Erealax_2Ereal] :
? [V2N: tp__ty_2Enum_2Enum] :
! [V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( inj__ty_2Enum_2Enum @ V2N ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__gt @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Eseq_2E_2D_2D_3E
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ c_2Erealax_2Einv @ ( ap @ V0f @ V4n ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__POWER__ABS,axiom,
! [V0c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Eseq_2E_2D_2D_3E
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) ) @ V1n ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__POWER,axiom,
! [V0c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Eseq_2E_2D_2D_3E
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ V1n ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ENEST__LEMMA,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__ge @ ( ap @ V0f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
& ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V1g @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
& ! [V4n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) )
=> ? [V5l: tp__ty_2Erealax_2Ereal,V6m: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V5l ) ) @ ( inj__ty_2Erealax_2Ereal @ V6m ) ) )
& ! [V7n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V7n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5l ) ) )
& ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V5l ) ) )
& ! [V8n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V6m ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V8n ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V6m ) ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ENEST__LEMMA__UNIQ,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__ge @ ( ap @ V0f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
& ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V1g @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
& ! [V4n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) )
& ( p
@ ( ap
@ ( ap @ c_2Eseq_2E_2D_2D_3E
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5n: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ V5n ) ) @ ( ap @ V1g @ V5n ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ? [V6l: tp__ty_2Erealax_2Ereal] :
( ! [V7n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V7n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V6l ) ) )
& ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V6l ) ) )
& ! [V8n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V6l ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V8n ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V6l ) ) ) ) ) ) ) ).
thf(stp_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,type,
tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal: $tType ).
thf(stp_inj_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,type,
inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal > $i ).
thf(stp_surj_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,type,
surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal: $i > tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal ).
thf(stp_inj_surj_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,axiom,
! [X: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ X ) )
= X ) ).
thf(stp_inj_mem_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,axiom,
! [X: tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal] : ( mem @ ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ X ) @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) ) ).
thf(stp_iso_mem_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,axiom,
! [X: $i] :
( ( mem @ X @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( X
= ( inj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ ( surj__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal @ X ) ) ) ) ).
thf(conj_thm_2Eseq_2EBOLZANO__LEMMA,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ bool ) )
=> ( ( ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) )
& ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
& ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3c ) ) ) ) )
& ! [V4x: tp__ty_2Erealax_2Ereal] :
? [V5d: 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 @ V5d ) ) )
& ! [V6a: tp__ty_2Erealax_2Ereal,V7b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V6a ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V7b ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V7b ) ) @ ( inj__ty_2Erealax_2Ereal @ V6a ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V5d ) ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V6a ) ) @ ( inj__ty_2Erealax_2Ereal @ V7b ) ) ) ) ) ) )
=> ! [V8a: tp__ty_2Erealax_2Ereal,V9b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V8a ) ) @ ( inj__ty_2Erealax_2Ereal @ V9b ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V8a ) ) @ ( inj__ty_2Erealax_2Ereal @ V9b ) ) ) ) ) ) ) ).
thf(ax_thm_2Eseq_2Esums,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1s: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Eseq_2Esums @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1s ) ) )
<=> ( p
@ ( ap
@ ( ap @ c_2Eseq_2E_2D_2D_3E
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( 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 ) ) @ V2n ) ) @ V0f ) ) )
@ ( inj__ty_2Erealax_2Ereal @ V1s ) ) ) ) ) ).
thf(ax_thm_2Eseq_2Esummable,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Eseq_2Esummable @ V0f ) )
<=> ? [V1s: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ c_2Eseq_2Esums @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1s ) ) ) ) ) ).
thf(ax_thm_2Eseq_2Esuminf,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Eseq_2Esuminf @ V0f ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( c_2Emin_2E_40 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1s: $i] : ( ap @ ( ap @ c_2Eseq_2Esums @ V0f ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESUM__SUMMABLE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Eseq_2Esums @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) )
=> ( p @ ( ap @ c_2Eseq_2Esummable @ V0f ) ) ) ) ).
thf(conj_thm_2Eseq_2ESUMMABLE__SUM,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Eseq_2Esummable @ V0f ) )
=> ( p @ ( ap @ ( ap @ c_2Eseq_2Esums @ V0f ) @ ( ap @ c_2Eseq_2Esuminf @ V0f ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESUM__UNIQ,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Eseq_2Esums @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
=> ( V1x
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Eseq_2Esuminf @ V0f ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESER__0,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ! [V2m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2m ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eseq_2Esums @ V0f ) @ ( 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 @ V1n ) ) ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESER__POS__LE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ c_2Eseq_2Esummable @ V0f ) )
& ! [V2m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( 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 @ V1n ) ) ) @ V0f ) ) @ ( ap @ c_2Eseq_2Esuminf @ V0f ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESER__POS__LT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ c_2Eseq_2Esummable @ V0f ) )
& ! [V2m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( 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 @ V1n ) ) ) @ V0f ) ) @ ( ap @ c_2Eseq_2Esuminf @ V0f ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESER__GROUP,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1k: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ c_2Eseq_2Esummable @ V0f ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Eseq_2Esums
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ V2n ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) @ V0f ) ) )
@ ( ap @ c_2Eseq_2Esuminf @ V0f ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESER__PAIR,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Eseq_2Esummable @ V0f ) )
=> ( p
@ ( ap
@ ( ap @ c_2Eseq_2Esums
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ V1n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V0f ) ) )
@ ( ap @ c_2Eseq_2Esuminf @ V0f ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESER__OFFSET,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Eseq_2Esummable @ V0f ) )
=> ! [V1k: tp__ty_2Enum_2Enum] :
( p
@ ( ap
@ ( ap @ c_2Eseq_2Esums
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V2n ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) ) )
@ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Eseq_2Esuminf @ V0f ) ) @ ( 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 @ V1k ) ) ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESER__POS__LT__PAIR,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ c_2Eseq_2Esummable @ V0f ) )
& ! [V2d: 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_2Erealax_2Ereal__add @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2d ) ) ) ) ) @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2d ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( 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 @ V1n ) ) ) @ V0f ) ) @ ( ap @ c_2Eseq_2Esuminf @ V0f ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESER__ADD,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1x0: tp__ty_2Erealax_2Ereal,V2y: $i] :
( ( mem @ V2y @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V3y0: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Eseq_2Esums @ V0x ) @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) )
& ( p @ ( ap @ ( ap @ c_2Eseq_2Esums @ V2y ) @ ( inj__ty_2Erealax_2Ereal @ V3y0 ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Eseq_2Esums
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0x @ V4n ) ) @ ( ap @ V2y @ V4n ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESER__CMUL,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1x0: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Eseq_2Esums @ V0x ) @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Eseq_2Esums
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( ap @ V0x @ V3n ) ) ) )
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESER__NEG,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1x0: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Eseq_2Esums @ V0x ) @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Eseq_2Esums
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ V0x @ V2n ) ) ) )
@ ( ap @ c_2Erealax_2Ereal__neg @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESER__SUB,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1x0: tp__ty_2Erealax_2Ereal,V2y: $i] :
( ( mem @ V2y @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V3y0: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Eseq_2Esums @ V0x ) @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) )
& ( p @ ( ap @ ( ap @ c_2Eseq_2Esums @ V2y ) @ ( inj__ty_2Erealax_2Ereal @ V3y0 ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Eseq_2Esums
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0x @ V4n ) ) @ ( ap @ V2y @ V4n ) ) ) )
@ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESER__CDIV,axiom,
! [V0x: $i] :
( ( mem @ V0x @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1x0: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Eseq_2Esums @ V0x ) @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Eseq_2Esums
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ V0x @ V3n ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) )
@ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V1x0 ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESER__CAUCHY,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Eseq_2Esummable @ V0f ) )
<=> ! [V1e: 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 @ V1e ) ) )
=> ? [V2N: tp__ty_2Enum_2Enum] :
! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V2N ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ V0f ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1e ) ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESER__ZERO,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Eseq_2Esummable @ V0f ) )
=> ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V0f ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESER__COMPAR,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ? [V2N: tp__ty_2Enum_2Enum] :
! [V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( inj__ty_2Enum_2Enum @ V2N ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) )
& ( p @ ( ap @ c_2Eseq_2Esummable @ V1g ) ) )
=> ( p @ ( ap @ c_2Eseq_2Esummable @ V0f ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESER__COMPARA,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ? [V2N: tp__ty_2Enum_2Enum] :
! [V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( inj__ty_2Enum_2Enum @ V2N ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) )
& ( p @ ( ap @ c_2Eseq_2Esummable @ V1g ) ) )
=> ( p
@ ( ap @ c_2Eseq_2Esummable
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4k: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ V4k ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESER__LE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
& ( p @ ( ap @ c_2Eseq_2Esummable @ V0f ) )
& ( p @ ( ap @ c_2Eseq_2Esummable @ V1g ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Eseq_2Esuminf @ V0f ) ) @ ( ap @ c_2Eseq_2Esuminf @ V1g ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESER__LE2,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
& ( p @ ( ap @ c_2Eseq_2Esummable @ V1g ) ) )
=> ( ( p @ ( ap @ c_2Eseq_2Esummable @ V0f ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Eseq_2Esuminf @ V0f ) ) @ ( ap @ c_2Eseq_2Esuminf @ V1g ) ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESER__ACONV,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p
@ ( ap @ c_2Eseq_2Esummable
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ V1n ) ) ) ) )
=> ( p @ ( ap @ c_2Eseq_2Esummable @ V0f ) ) ) ) ).
thf(conj_thm_2Eseq_2ESER__ABS,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p
@ ( ap @ c_2Eseq_2Esummable
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ V1n ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ c_2Eseq_2Esuminf @ V0f ) ) )
@ ( ap @ c_2Eseq_2Esuminf
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ V2n ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2EGP__FINITE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( 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 ) ) ) ) ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( 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 @ V1n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V2n ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2EGP,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Eseq_2Esums
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V1n ) ) )
@ ( ap @ c_2Erealax_2Einv @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( 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 @ V0x ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2EABS__NEG__LEMMA,axiom,
! [V0c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) ) )
=> ( V1x
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESER__RATIO,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1c: tp__ty_2Erealax_2Ereal,V2N: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
& ! [V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( inj__ty_2Enum_2Enum @ V2N ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) ) ) )
=> ( p @ ( ap @ c_2Eseq_2Esummable @ V0f ) ) ) ) ).
thf(conj_thm_2Eseq_2ELE__SEQ__IMP__LE__LIM,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V2f ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__LE__IMP__LIM__LE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
& ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V2f ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__MONO__LE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2n: tp__ty_2Enum_2Enum] :
( ( ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__LE__MONO,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2n: tp__ty_2Enum_2Enum] :
( ( ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ).
thf(ax_thm_2Eseq_2Emono__increasing__def,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Eseq_2Emono__increasing @ V0f ) )
<=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2Emono__increasing__suc,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Eseq_2Emono__increasing @ V0f ) )
<=> ! [V1n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( ap @ V0f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ) ) ).
thf(ax_thm_2Eseq_2Emono__decreasing__def,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Eseq_2Emono__decreasing @ V0f ) )
<=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2Emono__decreasing__suc,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Eseq_2Emono__decreasing @ V0f ) )
<=> ! [V1n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2Emono__increasing__converges__to__sup,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1r: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Eseq_2Emono__increasing @ V0f ) )
& ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) )
=> ( V1r
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Esup @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ V0f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2EINCREASING__SEQ,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal] :
( ( ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ V0f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) )
& ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) )
& ! [V4e: 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 @ V4e ) ) )
=> ? [V5n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V4e ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) ) ) ).
thf(conj_thm_2Eseq_2EMAX__LE__X,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum,V2k: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ c_2Earithmetic_2EMAX @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) ) ) ) ).
thf(conj_thm_2Eseq_2EX__LE__MAX,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum,V2k: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EMAX @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
| ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ETRANSFORM__2D__NUM,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) )
=> ( ( ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
& ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) )
=> ! [V5m: tp__ty_2Enum_2Enum,V6n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ETRIANGLE__2D__NUM,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) )
=> ( ! [V1d: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1d ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
=> ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESEQ__SANDWICH,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V3l: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) )
& ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V2h ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) )
& ! [V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ ( ap @ V2h @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eseq_2E_2D_2D_3E @ V1g ) @ ( inj__ty_2Erealax_2Ereal @ V3l ) ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESER__POS,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ c_2Eseq_2Esummable @ V0f ) )
& ! [V1n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Eseq_2Esuminf @ V0f ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESER__POS__MONO,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ! [V1n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
=> ( p
@ ( ap @ c_2Eseq_2Emono
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( 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 ) ) @ V2n ) ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2EPOS__SUMMABLE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ! [V1n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
& ? [V2x: tp__ty_2Erealax_2Ereal] :
! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( 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 @ V3n ) ) ) @ V0f ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
=> ( p @ ( ap @ c_2Eseq_2Esummable @ V0f ) ) ) ) ).
thf(conj_thm_2Eseq_2ESUMMABLE__LE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ c_2Eseq_2Esummable @ V0f ) )
& ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( 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 @ V2n ) ) ) @ V0f ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Eseq_2Esuminf @ V0f ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESUMS__EQ,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Eseq_2Esums @ V0f ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
<=> ( ( p @ ( ap @ c_2Eseq_2Esummable @ V0f ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Eseq_2Esuminf @ V0f ) )
= V1x ) ) ) ) ).
thf(conj_thm_2Eseq_2ESUMINF__POS,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ! [V1n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
& ( p @ ( ap @ c_2Eseq_2Esummable @ V0f ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Eseq_2Esuminf @ V0f ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESUM__PICK,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1k: tp__ty_2Enum_2Enum,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
@ ^ [V3m: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V3m ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESUM__LT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( ! [V4r: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V4r ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V4r ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V4r ) ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V4r ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V0f ) ) @ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V1g ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESUM__CONST__R,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1r: 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 ) ) ) @ ( ap @ ( c_2Ecombin_2EK @ ty_2Erealax_2Ereal @ ty_2Enum_2Enum ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1r ) ) ) ) ).
thf(conj_thm_2Eseq_2ESUMS__ZERO,axiom,
p @ ( ap @ ( ap @ c_2Eseq_2Esums @ ( ap @ ( c_2Ecombin_2EK @ ty_2Erealax_2Ereal @ ty_2Enum_2Enum ) @ ( 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_2Eseq_2ELT__SUC,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
| ( V0a = V1b ) ) ) ).
thf(conj_thm_2Eseq_2ELE__SUC,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
| ( V0a
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2EK__PARTIAL,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27b ) @ V0x )
= ( lam @ A_27b
@ ^ [V1z: $i] : V0x ) ) ) ).
thf(conj_thm_2Eseq_2EHALF__POS,axiom,
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_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 ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2EHALF__LT__1,axiom,
p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( 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 ) ) ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(conj_thm_2Eseq_2EHALF__CANCEL,axiom,
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( 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 ) ) ) ) ) ) )
= ( 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 ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2EX__HALF__HALF,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( 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_2Erealax_2Ereal @ V0x ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( 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_2Erealax_2Ereal @ V0x ) ) ) )
= V0x ) ).
thf(conj_thm_2Eseq_2EONE__MINUS__HALF,axiom,
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( 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 ) ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( 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 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ENUM__2D__BIJ__BIG__SQUARE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) )
=> ! [V1N: tp__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 ) ) ) )
=> ? [V2k: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) @ V0f ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V1N ) ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V2k ) ) ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V2k ) ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ENUM__2D__BIJ__SMALL__SQUARE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) )
=> ! [V1k: tp__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 ) ) ) )
=> ? [V2N: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) @ V0f ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V2N ) ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESUMINF__ADD,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ c_2Eseq_2Esummable @ V0f ) )
& ( p @ ( ap @ c_2Eseq_2Esummable @ V1g ) ) )
=> ( ( p
@ ( ap @ c_2Eseq_2Esummable
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0f @ V2n ) ) @ ( ap @ V1g @ V2n ) ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Eseq_2Esuminf @ V0f ) ) @ ( ap @ c_2Eseq_2Esuminf @ V1g ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ c_2Eseq_2Esuminf
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0f @ V3n ) ) @ ( ap @ V1g @ V3n ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESUMINF__2D,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ ( arr @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) )
=> ( ( ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) )
& ! [V5n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Eseq_2Esums @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) )
& ( p @ ( ap @ c_2Eseq_2Esummable @ V1g ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) @ V2h ) @ ( 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 ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eseq_2Esums @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) @ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Enum_2Enum @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) @ V0f ) ) @ V2h ) ) @ ( ap @ c_2Eseq_2Esuminf @ V1g ) ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2EPOW__HALF__SER,axiom,
( p
@ ( ap
@ ( ap @ c_2Eseq_2Esums
@ ( lam @ ty_2Enum_2Enum
@ ^ [V0n: $i] : ( 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 ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V0n ) @ ( 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_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Eseq_2ESER__POS__COMPARE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
& ( p @ ( ap @ c_2Eseq_2Esummable @ V1g ) )
& ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) )
=> ( ( p @ ( ap @ c_2Eseq_2Esummable @ V0f ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Eseq_2Esuminf @ V0f ) ) @ ( ap @ c_2Eseq_2Esuminf @ V1g ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------