ITP001 Axioms: ITP127_5.ax
%------------------------------------------------------------------------------
% File : ITP127_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 : seq_2.ax [Gau20]
% : HL4127_5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 198 ( 29 unt; 44 typ; 0 def)
% Number of atoms : 3492 ( 53 equ)
% Maximal formula atoms : 13 ( 17 avg)
% Number of connectives : 352 ( 3 ~; 5 |; 75 &)
% ( 28 <=>; 241 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of FOOLs : 2989 (2989 fml; 0 var)
% Number of types : 2 ( 1 usr)
% Number of type conns : 44 ( 31 >; 13 *; 0 +; 0 <<)
% Number of predicates : 63 ( 62 usr; 35 prp; 0-3 aty)
% Number of functors : 43 ( 43 usr; 12 con; 0-2 aty)
% Number of variables : 384 ( 363 !; 21 ?; 384 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_c_2Eseq_2E_2D_2D_3E,type,
c_2Eseq_2E_2D_2D_3E: $i ).
tff(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))) ).
tff(tp_c_2Eseq_2Ecauchy,type,
c_2Eseq_2Ecauchy: $i ).
tff(mem_c_2Eseq_2Ecauchy,axiom,
mem(c_2Eseq_2Ecauchy,arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),bool)) ).
tff(tp_c_2Eseq_2Econvergent,type,
c_2Eseq_2Econvergent: $i ).
tff(mem_c_2Eseq_2Econvergent,axiom,
mem(c_2Eseq_2Econvergent,arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),bool)) ).
tff(tp_c_2Eseq_2Elim,type,
c_2Eseq_2Elim: $i ).
tff(mem_c_2Eseq_2Elim,axiom,
mem(c_2Eseq_2Elim,arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),ty_2Erealax_2Ereal)) ).
tff(tp_c_2Eseq_2Emono,type,
c_2Eseq_2Emono: $i ).
tff(mem_c_2Eseq_2Emono,axiom,
mem(c_2Eseq_2Emono,arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),bool)) ).
tff(tp_c_2Eseq_2Emono__decreasing,type,
c_2Eseq_2Emono__decreasing: $i ).
tff(mem_c_2Eseq_2Emono__decreasing,axiom,
mem(c_2Eseq_2Emono__decreasing,arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),bool)) ).
tff(tp_c_2Eseq_2Emono__increasing,type,
c_2Eseq_2Emono__increasing: $i ).
tff(mem_c_2Eseq_2Emono__increasing,axiom,
mem(c_2Eseq_2Emono__increasing,arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),bool)) ).
tff(tp_c_2Eseq_2Esubseq,type,
c_2Eseq_2Esubseq: $i ).
tff(mem_c_2Eseq_2Esubseq,axiom,
mem(c_2Eseq_2Esubseq,arr(arr(ty_2Enum_2Enum,ty_2Enum_2Enum),bool)) ).
tff(tp_c_2Eseq_2Esuminf,type,
c_2Eseq_2Esuminf: $i ).
tff(mem_c_2Eseq_2Esuminf,axiom,
mem(c_2Eseq_2Esuminf,arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),ty_2Erealax_2Ereal)) ).
tff(tp_c_2Eseq_2Esummable,type,
c_2Eseq_2Esummable: $i ).
tff(mem_c_2Eseq_2Esummable,axiom,
mem(c_2Eseq_2Esummable,arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),bool)) ).
tff(tp_c_2Eseq_2Esums,type,
c_2Eseq_2Esums: $i ).
tff(mem_c_2Eseq_2Esums,axiom,
mem(c_2Eseq_2Esums,arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,bool))) ).
tff(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))) ) ) ).
tff(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))) ) ) ) ) ).
tff(conj_thm_2Eseq_2ESEQ__CONST,axiom,
! [V0k: tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Eseq_2E_2D_2D_3E,k(ty_2Enum_2Enum,inj__ty_2Erealax_2Ereal(V0k))),inj__ty_2Erealax_2Ereal(V0k))) ).
tff(lamtp_f2777,type,
f2777: ( $i * $i ) > $i ).
tff(lameq_f2777,axiom,
! [V0x: $i] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2y: $i] :
( mem(V2y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4n: tp__ty_2Enum_2Enum] : ( ap(f2777(V0x,V2y),inj__ty_2Enum_2Enum(V4n)) = ap(ap(c_2Erealax_2Ereal__add,ap(V0x,inj__ty_2Enum_2Enum(V4n))),ap(V2y,inj__ty_2Enum_2Enum(V4n))) ) ) ) ).
tff(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,f2777(V0x,V2y)),ap(ap(c_2Erealax_2Ereal__add,inj__ty_2Erealax_2Ereal(V1x0)),inj__ty_2Erealax_2Ereal(V3y0)))) ) ) ) ).
tff(lamtp_f2778,type,
f2778: ( $i * $i ) > $i ).
tff(lameq_f2778,axiom,
! [V0x: $i] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2y: $i] :
( mem(V2y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4n: tp__ty_2Enum_2Enum] : ( ap(f2778(V0x,V2y),inj__ty_2Enum_2Enum(V4n)) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0x,inj__ty_2Enum_2Enum(V4n))),ap(V2y,inj__ty_2Enum_2Enum(V4n))) ) ) ) ).
tff(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,f2778(V0x,V2y)),ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V1x0)),inj__ty_2Erealax_2Ereal(V3y0)))) ) ) ) ).
tff(lamtp_f2779,type,
f2779: $i > $i ).
tff(lameq_f2779,axiom,
! [V0x: $i] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2n: tp__ty_2Enum_2Enum] : ( ap(f2779(V0x),inj__ty_2Enum_2Enum(V2n)) = ap(c_2Erealax_2Ereal__neg,ap(V0x,inj__ty_2Enum_2Enum(V2n))) ) ) ).
tff(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,f2779(V0x)),ap(c_2Erealax_2Ereal__neg,inj__ty_2Erealax_2Ereal(V1x0)))) ) ) ).
tff(lamtp_f2780,type,
f2780: $i > $i ).
tff(lameq_f2780,axiom,
! [V0x: $i] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2n: tp__ty_2Enum_2Enum] : ( ap(f2780(V0x),inj__ty_2Enum_2Enum(V2n)) = ap(c_2Erealax_2Einv,ap(V0x,inj__ty_2Enum_2Enum(V2n))) ) ) ).
tff(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,f2780(V0x)),ap(c_2Erealax_2Einv,inj__ty_2Erealax_2Ereal(V1x0)))) ) ) ).
tff(lamtp_f2781,type,
f2781: ( $i * $i ) > $i ).
tff(lameq_f2781,axiom,
! [V0x: $i] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2y: $i] :
( mem(V2y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4n: tp__ty_2Enum_2Enum] : ( ap(f2781(V0x,V2y),inj__ty_2Enum_2Enum(V4n)) = ap(ap(c_2Ereal_2Ereal__sub,ap(V0x,inj__ty_2Enum_2Enum(V4n))),ap(V2y,inj__ty_2Enum_2Enum(V4n))) ) ) ) ).
tff(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,f2781(V0x,V2y)),ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V1x0)),inj__ty_2Erealax_2Ereal(V3y0)))) ) ) ) ).
tff(lamtp_f2782,type,
f2782: ( $i * $i ) > $i ).
tff(lameq_f2782,axiom,
! [V0x: $i] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2y: $i] :
( mem(V2y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4n: tp__ty_2Enum_2Enum] : ( ap(f2782(V0x,V2y),inj__ty_2Enum_2Enum(V4n)) = ap(ap(c_2Ereal_2E_2F,ap(V0x,inj__ty_2Enum_2Enum(V4n))),ap(V2y,inj__ty_2Enum_2Enum(V4n))) ) ) ) ).
tff(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,f2782(V0x,V2y)),ap(ap(c_2Ereal_2E_2F,inj__ty_2Erealax_2Ereal(V1x0)),inj__ty_2Erealax_2Ereal(V3y0)))) ) ) ) ).
tff(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 ) ) ) ).
tff(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))) ) ) ).
tff(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))) ) ) ) ) ).
tff(lamtp_f2783,type,
f2783: $i > $i ).
tff(lameq_f2783,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1l: tp__ty_2Erealax_2Ereal] : ( ap(f2783(V0f),inj__ty_2Erealax_2Ereal(V1l)) = ap(ap(c_2Eseq_2E_2D_2D_3E,V0f),inj__ty_2Erealax_2Ereal(V1l)) ) ) ).
tff(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),f2783(V0f))) ) ) ).
tff(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))) ) ) ).
tff(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)))) ) ) ) ).
tff(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))))) ) ) ).
tff(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)))) ) ) ) ) ).
tff(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)))) ) ) ) ).
tff(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))) ) ) ).
tff(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))) ) ) ).
tff(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)) ) ) ).
tff(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)) ) ) ).
tff(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)) ) ) ).
tff(lamtp_f2784,type,
f2784: $i > $i ).
tff(lameq_f2784,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1n: tp__ty_2Enum_2Enum] : ( ap(f2784(V0f),inj__ty_2Enum_2Enum(V1n)) = ap(c_2Erealax_2Ereal__neg,ap(V0f,inj__ty_2Enum_2Enum(V1n))) ) ) ).
tff(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,f2784(V0f))) ) ) ).
tff(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)),f2784(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)) ) ) ).
tff(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)) ) ) ).
tff(lamtp_f2785,type,
f2785: ( $i * $i ) > $i ).
tff(lameq_f2785,axiom,
! [V1f: $i] :
( mem(V1f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V0s: $i] :
( mem(V0s,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2n: tp__ty_2Enum_2Enum] : ( ap(f2785(V1f,V0s),inj__ty_2Enum_2Enum(V2n)) = ap(V0s,ap(V1f,inj__ty_2Enum_2Enum(V2n))) ) ) ) ).
tff(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,f2785(V1f,V0s))) ) ) ).
tff(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)),f2785(V1f,V0s))) ) ) ) ).
tff(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)))) ) ) ).
tff(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))) ) ) ) ).
tff(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)) ) ) ).
tff(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))) ) ) ) ).
tff(lamtp_f2786,type,
f2786: $i > $i ).
tff(lameq_f2786,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2n: tp__ty_2Enum_2Enum] : ( ap(f2786(V0f),inj__ty_2Enum_2Enum(V2n)) = ap(V0f,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V2n))) ) ) ).
tff(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,f2786(V0f)),inj__ty_2Erealax_2Ereal(V1l))) ) ) ).
tff(lamtp_f2787,type,
f2787: $i > $i ).
tff(lameq_f2787,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1n: tp__ty_2Enum_2Enum] : ( ap(f2787(V0f),inj__ty_2Enum_2Enum(V1n)) = ap(c_2Ereal_2Eabs,ap(V0f,inj__ty_2Enum_2Enum(V1n))) ) ) ).
tff(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,f2787(V0f)),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)))) ) ) ).
tff(lamtp_f2788,type,
f2788: $i > $i ).
tff(lameq_f2788,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2n: tp__ty_2Enum_2Enum] : ( ap(f2788(V0f),inj__ty_2Enum_2Enum(V2n)) = ap(c_2Ereal_2Eabs,ap(V0f,inj__ty_2Enum_2Enum(V2n))) ) ) ).
tff(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,f2788(V0f)),ap(c_2Ereal_2Eabs,inj__ty_2Erealax_2Ereal(V1l)))) ) ) ).
tff(lamtp_f2789,type,
f2789: $i > $i ).
tff(lameq_f2789,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4n: tp__ty_2Enum_2Enum] : ( ap(f2789(V0f),inj__ty_2Enum_2Enum(V4n)) = ap(c_2Erealax_2Einv,ap(V0f,inj__ty_2Enum_2Enum(V4n))) ) ) ).
tff(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,f2789(V0f)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) ) ) ).
tff(lamtp_f2790,type,
f2790: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f2790,axiom,
! [V0c: tp__ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] : ( ap(f2790(V0c),inj__ty_2Enum_2Enum(V1n)) = ap(ap(c_2Ereal_2Epow,ap(c_2Ereal_2Eabs,inj__ty_2Erealax_2Ereal(V0c))),inj__ty_2Enum_2Enum(V1n)) ) ).
tff(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,f2790(V0c)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) ) ).
tff(lamtp_f2791,type,
f2791: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f2791,axiom,
! [V0c: tp__ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] : ( ap(f2791(V0c),inj__ty_2Enum_2Enum(V1n)) = ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V0c)),inj__ty_2Enum_2Enum(V1n)) ) ).
tff(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,f2791(V0c)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) ) ).
tff(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))) ) ) ) ) ).
tff(lamtp_f2792,type,
f2792: ( $i * $i ) > $i ).
tff(lameq_f2792,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V5n: tp__ty_2Enum_2Enum] : ( ap(f2792(V0f,V1g),inj__ty_2Enum_2Enum(V5n)) = ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,inj__ty_2Enum_2Enum(V5n))),ap(V1g,inj__ty_2Enum_2Enum(V5n))) ) ) ) ).
tff(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,f2792(V0f,V1g)),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))) ) ) ) ) ).
tff(stp_c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal,type,
tp__c_ty_2Epair_2Eprod_ty_2Erealax_2Ereal_ty_2Erealax_2Ereal: $tType ).
tff(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 ).
tff(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 ).
tff(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 ) ).
tff(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)) ).
tff(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)) ) ) ).
tff(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)))) ) ) ) ).
tff(lamtp_f2793,type,
f2793: $i > $i ).
tff(lameq_f2793,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2n: tp__ty_2Enum_2Enum] : ( ap(f2793(V0f),inj__ty_2Enum_2Enum(V2n)) = 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) ) ) ).
tff(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,f2793(V0f)),inj__ty_2Erealax_2Ereal(V1s))) ) ) ).
tff(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))) ) ) ).
tff(lamtp_f2794,type,
f2794: $i > $i ).
tff(lameq_f2794,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1s: tp__ty_2Erealax_2Ereal] : ( ap(f2794(V0f),inj__ty_2Erealax_2Ereal(V1s)) = ap(ap(c_2Eseq_2Esums,V0f),inj__ty_2Erealax_2Ereal(V1s)) ) ) ).
tff(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),f2794(V0f))) ) ) ).
tff(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)) ) ) ).
tff(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))) ) ) ).
tff(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)) ) ) ) ).
tff(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))) ) ) ).
tff(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))) ) ) ).
tff(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))) ) ) ).
tff(lamtp_f2795,type,
f2795: ( tp__ty_2Enum_2Enum * $i ) > $i ).
tff(lameq_f2795,axiom,
! [V1k: tp__ty_2Enum_2Enum,V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2n: tp__ty_2Enum_2Enum] : ( ap(f2795(V1k,V0f),inj__ty_2Enum_2Enum(V2n)) = ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),ap(ap(c_2Earithmetic_2E_2A,inj__ty_2Enum_2Enum(V2n)),inj__ty_2Enum_2Enum(V1k))),inj__ty_2Enum_2Enum(V1k))),V0f) ) ) ).
tff(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,f2795(V1k,V0f)),ap(c_2Eseq_2Esuminf,V0f))) ) ) ).
tff(lamtp_f2796,type,
f2796: $i > $i ).
tff(lameq_f2796,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1n: tp__ty_2Enum_2Enum] : ( ap(f2796(V0f),inj__ty_2Enum_2Enum(V1n)) = 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)))),inj__ty_2Enum_2Enum(V1n))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),V0f) ) ) ).
tff(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,f2796(V0f)),ap(c_2Eseq_2Esuminf,V0f))) ) ) ).
tff(lamtp_f2797,type,
f2797: ( tp__ty_2Enum_2Enum * $i ) > $i ).
tff(lameq_f2797,axiom,
! [V1k: tp__ty_2Enum_2Enum,V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2n: tp__ty_2Enum_2Enum] : ( ap(f2797(V1k,V0f),inj__ty_2Enum_2Enum(V2n)) = ap(V0f,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V2n)),inj__ty_2Enum_2Enum(V1k))) ) ) ).
tff(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,f2797(V1k,V0f)),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)))) ) ) ).
tff(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))) ) ) ).
tff(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,f2777(V0x,V2y)),ap(ap(c_2Erealax_2Ereal__add,inj__ty_2Erealax_2Ereal(V1x0)),inj__ty_2Erealax_2Ereal(V3y0)))) ) ) ) ).
tff(lamtp_f2798,type,
f2798: ( $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f2798,axiom,
! [V0x: $i] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2c: tp__ty_2Erealax_2Ereal,V3n: tp__ty_2Enum_2Enum] : ( ap(f2798(V0x,V2c),inj__ty_2Enum_2Enum(V3n)) = ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V2c)),ap(V0x,inj__ty_2Enum_2Enum(V3n))) ) ) ).
tff(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,f2798(V0x,V2c)),ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V2c)),inj__ty_2Erealax_2Ereal(V1x0)))) ) ) ).
tff(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,f2779(V0x)),ap(c_2Erealax_2Ereal__neg,inj__ty_2Erealax_2Ereal(V1x0)))) ) ) ).
tff(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,f2781(V0x,V2y)),ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V1x0)),inj__ty_2Erealax_2Ereal(V3y0)))) ) ) ) ).
tff(lamtp_f2799,type,
f2799: ( $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f2799,axiom,
! [V0x: $i] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2c: tp__ty_2Erealax_2Ereal,V3n: tp__ty_2Enum_2Enum] : ( ap(f2799(V0x,V2c),inj__ty_2Enum_2Enum(V3n)) = ap(ap(c_2Ereal_2E_2F,ap(V0x,inj__ty_2Enum_2Enum(V3n))),inj__ty_2Erealax_2Ereal(V2c)) ) ) ).
tff(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,f2799(V0x,V2c)),ap(ap(c_2Ereal_2E_2F,inj__ty_2Erealax_2Ereal(V1x0)),inj__ty_2Erealax_2Ereal(V2c)))) ) ) ).
tff(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))) ) ) ) ) ).
tff(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)))) ) ) ).
tff(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)) ) ) ) ).
tff(lamtp_f2800,type,
f2800: $i > $i ).
tff(lameq_f2800,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4k: tp__ty_2Enum_2Enum] : ( ap(f2800(V0f),inj__ty_2Enum_2Enum(V4k)) = ap(c_2Ereal_2Eabs,ap(V0f,inj__ty_2Enum_2Enum(V4k))) ) ) ).
tff(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,f2800(V0f))) ) ) ) ).
tff(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))) ) ) ) ).
tff(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))) ) ) ) ) ).
tff(conj_thm_2Eseq_2ESER__ACONV,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Eseq_2Esummable,f2787(V0f)))
=> p(ap(c_2Eseq_2Esummable,V0f)) ) ) ).
tff(conj_thm_2Eseq_2ESER__ABS,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Eseq_2Esummable,f2787(V0f)))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(c_2Eseq_2Esuminf,V0f))),ap(c_2Eseq_2Esuminf,f2788(V0f)))) ) ) ).
tff(lamtp_f2801,type,
f2801: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f2801,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V2n: tp__ty_2Enum_2Enum] : ( ap(f2801(V0x),inj__ty_2Enum_2Enum(V2n)) = ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Enum_2Enum(V2n)) ) ).
tff(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))),f2801(V0x))) = 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))))))) ) ) ).
tff(lamtp_f2802,type,
f2802: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f2802,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] : ( ap(f2802(V0x),inj__ty_2Enum_2Enum(V1n)) = ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Enum_2Enum(V1n)) ) ).
tff(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,f2802(V0x)),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))))) ) ).
tff(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))) ) ) ) ).
tff(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)) ) ) ).
tff(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))) ) ) ).
tff(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))) ) ) ).
tff(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))) ) ) ).
tff(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)))) ) ) ).
tff(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)))) ) ) ) ).
tff(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))))) ) ) ).
tff(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)))) ) ) ) ).
tff(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)))) ) ) ).
tff(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)))) ) ) ) ).
tff(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))) ) ) ).
tff(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))) ) ) ).
tff(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))) ) ) ).
tff(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))) ) ) ).
tff(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))) ) ) ) ).
tff(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))) ) ) ) ) ).
tff(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))) ) ) ).
tff(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,f2793(V0f))) ) ) ).
tff(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)) ) ) ).
tff(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))) ) ) ).
tff(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 ) ) ) ) ).
tff(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))) ) ) ).
tff(lamtp_f2803,type,
f2803: ( tp__ty_2Erealax_2Ereal * tp__ty_2Enum_2Enum ) > $i ).
tff(lameq_f2803,axiom,
! [V2x: tp__ty_2Erealax_2Ereal,V1k: tp__ty_2Enum_2Enum,V3m: tp__ty_2Enum_2Enum] : ( ap(f2803(V2x,V1k),inj__ty_2Enum_2Enum(V3m)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),inj__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))) ) ).
tff(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))),f2803(V2x,V1k))) = 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)))) ) ).
tff(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))) ) ) ) ).
tff(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))) ) ).
tff(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)))) ).
tff(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 ) ) ) ).
tff(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))) ) ) ) ).
tff(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) = k(A_27b,V0x) ) ) ).
tff(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))))))) ).
tff(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)))))) ).
tff(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))))) ).
tff(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 ) ).
tff(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)))))) ).
tff(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))))) ) ) ).
tff(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))))) ) ) ).
tff(lamtp_f2804,type,
f2804: ( $i * $i ) > $i ).
tff(lameq_f2804,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] : ( ap(f2804(V0f,V1g),inj__ty_2Enum_2Enum(V2n)) = ap(ap(c_2Erealax_2Ereal__add,ap(V0f,inj__ty_2Enum_2Enum(V2n))),ap(V1g,inj__ty_2Enum_2Enum(V2n))) ) ) ) ).
tff(lamtp_f2805,type,
f2805: ( $i * $i ) > $i ).
tff(lameq_f2805,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3n: tp__ty_2Enum_2Enum] : ( ap(f2805(V0f,V1g),inj__ty_2Enum_2Enum(V3n)) = ap(ap(c_2Erealax_2Ereal__add,ap(V0f,inj__ty_2Enum_2Enum(V3n))),ap(V1g,inj__ty_2Enum_2Enum(V3n))) ) ) ) ).
tff(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,f2804(V0f,V1g)))
& ( 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,f2805(V0f,V1g))) ) ) ) ) ) ).
tff(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))) ) ) ) ) ).
tff(lamtp_f2806,type,
f2806: $i ).
tff(lameq_f2806,axiom,
! [V0n: tp__ty_2Enum_2Enum] : ( ap(f2806,inj__ty_2Enum_2Enum(V0n)) = 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,inj__ty_2Enum_2Enum(V0n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) ) ).
tff(conj_thm_2Eseq_2EPOW__HALF__SER,axiom,
p(ap(ap(c_2Eseq_2Esums,f2806),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))) ).
tff(conj_thm_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))) ) ) ) ) ).
%------------------------------------------------------------------------------