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 : 151 ( 18 unt; 0 def)
% Number of atoms : 705 ( 51 equ)
% Maximal formula atoms : 22 ( 4 avg)
% Number of connectives : 557 ( 3 ~; 5 |; 95 &)
% ( 28 <=>; 426 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Maximal term depth : 11 ( 2 avg)
% Number of predicates : 4 ( 3 usr; 0 prp; 1-2 aty)
% Number of functors : 94 ( 94 usr; 44 con; 0-3 aty)
% Number of variables : 381 ( 360 !; 21 ?)
% SPC : FOF_SAT_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(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))) ).
fof(mem_c_2Eseq_2Ecauchy,axiom,
mem(c_2Eseq_2Ecauchy,arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),bool)) ).
fof(mem_c_2Eseq_2Econvergent,axiom,
mem(c_2Eseq_2Econvergent,arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),bool)) ).
fof(mem_c_2Eseq_2Elim,axiom,
mem(c_2Eseq_2Elim,arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),ty_2Erealax_2Ereal)) ).
fof(mem_c_2Eseq_2Emono,axiom,
mem(c_2Eseq_2Emono,arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),bool)) ).
fof(mem_c_2Eseq_2Emono__decreasing,axiom,
mem(c_2Eseq_2Emono__decreasing,arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),bool)) ).
fof(mem_c_2Eseq_2Emono__increasing,axiom,
mem(c_2Eseq_2Emono__increasing,arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),bool)) ).
fof(mem_c_2Eseq_2Esubseq,axiom,
mem(c_2Eseq_2Esubseq,arr(arr(ty_2Enum_2Enum,ty_2Enum_2Enum),bool)) ).
fof(mem_c_2Eseq_2Esuminf,axiom,
mem(c_2Eseq_2Esuminf,arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),ty_2Erealax_2Ereal)) ).
fof(mem_c_2Eseq_2Esummable,axiom,
mem(c_2Eseq_2Esummable,arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),bool)) ).
fof(mem_c_2Eseq_2Esums,axiom,
mem(c_2Eseq_2Esums,arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Erealax_2Ereal,bool))) ).
fof(ax_thm_2Eseq_2Etends__num__real,axiom,
! [V0x] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x0] :
( mem(V1x0,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Eseq_2E_2D_2D_3E,V0x),V1x0))
<=> p(ap(ap(ap(c_2Enets_2Etends(ty_2Erealax_2Ereal,ty_2Enum_2Enum),V0x),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))) ) ) ) ).
fof(conj_thm_2Eseq_2ESEQ,axiom,
! [V0x] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x0] :
( mem(V1x0,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Eseq_2E_2D_2D_3E,V0x),V1x0))
<=> ! [V2e] :
( mem(V2e,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e))
=> ? [V3N] :
( mem(V3N,ty_2Enum_2Enum)
& ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3E_3D,V4n),V3N))
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(V0x,V4n)),V1x0))),V2e)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eseq_2ESEQ__CONST,axiom,
! [V0k] :
( mem(V0k,ty_2Erealax_2Ereal)
=> p(ap(ap(c_2Eseq_2E_2D_2D_3E,k(ty_2Enum_2Enum,V0k)),V0k)) ) ).
fof(lameq_f2777,axiom,
! [V0x] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2y] :
( mem(V2y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4n] : ap(f2777(V0x,V2y),V4n) = ap(ap(c_2Erealax_2Ereal__add,ap(V0x,V4n)),ap(V2y,V4n)) ) ) ).
fof(conj_thm_2Eseq_2ESEQ__ADD,axiom,
! [V0x] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x0] :
( mem(V1x0,ty_2Erealax_2Ereal)
=> ! [V2y] :
( mem(V2y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3y0] :
( mem(V3y0,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Eseq_2E_2D_2D_3E,V0x),V1x0))
& p(ap(ap(c_2Eseq_2E_2D_2D_3E,V2y),V3y0)) )
=> p(ap(ap(c_2Eseq_2E_2D_2D_3E,f2777(V0x,V2y)),ap(ap(c_2Erealax_2Ereal__add,V1x0),V3y0))) ) ) ) ) ) ).
fof(lameq_f2778,axiom,
! [V0x] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2y] :
( mem(V2y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4n] : ap(f2778(V0x,V2y),V4n) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0x,V4n)),ap(V2y,V4n)) ) ) ).
fof(conj_thm_2Eseq_2ESEQ__MUL,axiom,
! [V0x] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x0] :
( mem(V1x0,ty_2Erealax_2Ereal)
=> ! [V2y] :
( mem(V2y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3y0] :
( mem(V3y0,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Eseq_2E_2D_2D_3E,V0x),V1x0))
& p(ap(ap(c_2Eseq_2E_2D_2D_3E,V2y),V3y0)) )
=> p(ap(ap(c_2Eseq_2E_2D_2D_3E,f2778(V0x,V2y)),ap(ap(c_2Erealax_2Ereal__mul,V1x0),V3y0))) ) ) ) ) ) ).
fof(lameq_f2779,axiom,
! [V0x] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2n] : ap(f2779(V0x),V2n) = ap(c_2Erealax_2Ereal__neg,ap(V0x,V2n)) ) ).
fof(conj_thm_2Eseq_2ESEQ__NEG,axiom,
! [V0x] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x0] :
( mem(V1x0,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Eseq_2E_2D_2D_3E,V0x),V1x0))
<=> p(ap(ap(c_2Eseq_2E_2D_2D_3E,f2779(V0x)),ap(c_2Erealax_2Ereal__neg,V1x0))) ) ) ) ).
fof(lameq_f2780,axiom,
! [V0x] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2n] : ap(f2780(V0x),V2n) = ap(c_2Erealax_2Einv,ap(V0x,V2n)) ) ).
fof(conj_thm_2Eseq_2ESEQ__INV,axiom,
! [V0x] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x0] :
( mem(V1x0,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Eseq_2E_2D_2D_3E,V0x),V1x0))
& V1x0 != ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) )
=> p(ap(ap(c_2Eseq_2E_2D_2D_3E,f2780(V0x)),ap(c_2Erealax_2Einv,V1x0))) ) ) ) ).
fof(lameq_f2781,axiom,
! [V0x] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2y] :
( mem(V2y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4n] : ap(f2781(V0x,V2y),V4n) = ap(ap(c_2Ereal_2Ereal__sub,ap(V0x,V4n)),ap(V2y,V4n)) ) ) ).
fof(conj_thm_2Eseq_2ESEQ__SUB,axiom,
! [V0x] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x0] :
( mem(V1x0,ty_2Erealax_2Ereal)
=> ! [V2y] :
( mem(V2y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3y0] :
( mem(V3y0,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Eseq_2E_2D_2D_3E,V0x),V1x0))
& p(ap(ap(c_2Eseq_2E_2D_2D_3E,V2y),V3y0)) )
=> p(ap(ap(c_2Eseq_2E_2D_2D_3E,f2781(V0x,V2y)),ap(ap(c_2Ereal_2Ereal__sub,V1x0),V3y0))) ) ) ) ) ) ).
fof(lameq_f2782,axiom,
! [V0x] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2y] :
( mem(V2y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4n] : ap(f2782(V0x,V2y),V4n) = ap(ap(c_2Ereal_2E_2F,ap(V0x,V4n)),ap(V2y,V4n)) ) ) ).
fof(conj_thm_2Eseq_2ESEQ__DIV,axiom,
! [V0x] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x0] :
( mem(V1x0,ty_2Erealax_2Ereal)
=> ! [V2y] :
( mem(V2y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3y0] :
( mem(V3y0,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Eseq_2E_2D_2D_3E,V0x),V1x0))
& p(ap(ap(c_2Eseq_2E_2D_2D_3E,V2y),V3y0))
& V3y0 != ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) )
=> p(ap(ap(c_2Eseq_2E_2D_2D_3E,f2782(V0x,V2y)),ap(ap(c_2Ereal_2E_2F,V1x0),V3y0))) ) ) ) ) ) ).
fof(conj_thm_2Eseq_2ESEQ__UNIQ,axiom,
! [V0x] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x1] :
( mem(V1x1,ty_2Erealax_2Ereal)
=> ! [V2x2] :
( mem(V2x2,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Eseq_2E_2D_2D_3E,V0x),V1x1))
& p(ap(ap(c_2Eseq_2E_2D_2D_3E,V0x),V2x2)) )
=> V1x1 = V2x2 ) ) ) ) ).
fof(ax_thm_2Eseq_2Econvergent,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Eseq_2Econvergent,V0f))
<=> ? [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Eseq_2E_2D_2D_3E,V0f),V1l)) ) ) ) ).
fof(ax_thm_2Eseq_2Ecauchy,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Eseq_2Ecauchy,V0f))
<=> ! [V1e] :
( mem(V1e,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1e))
=> ? [V2N] :
( mem(V2N,ty_2Enum_2Enum)
& ! [V3m] :
( mem(V3m,ty_2Enum_2Enum)
=> ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Earithmetic_2E_3E_3D,V3m),V2N))
& p(ap(ap(c_2Earithmetic_2E_3E_3D,V4n),V2N)) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,V3m)),ap(V0f,V4n)))),V1e)) ) ) ) ) ) ) ) ) ).
fof(lameq_f2783,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1l] : ap(f2783(V0f),V1l) = ap(ap(c_2Eseq_2E_2D_2D_3E,V0f),V1l) ) ).
fof(ax_thm_2Eseq_2Elim,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ap(c_2Eseq_2Elim,V0f) = ap(c_2Emin_2E_40(ty_2Erealax_2Ereal),f2783(V0f)) ) ).
fof(conj_thm_2Eseq_2ESEQ__LIM,axiom,
! [V0f] :
( 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))) ) ) ).
fof(ax_thm_2Eseq_2Esubseq,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ( p(ap(c_2Eseq_2Esubseq,V0f))
<=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Eprim__rec_2E_3C,V1m),V2n))
=> p(ap(ap(c_2Eprim__rec_2E_3C,ap(V0f,V1m)),ap(V0f,V2n))) ) ) ) ) ) ).
fof(conj_thm_2Eseq_2ESUBSEQ__SUC,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ( p(ap(c_2Eseq_2Esubseq,V0f))
<=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Eprim__rec_2E_3C,ap(V0f,V1n)),ap(V0f,ap(c_2Enum_2ESUC,V1n)))) ) ) ) ).
fof(ax_thm_2Eseq_2Emono,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Eseq_2Emono,V0f))
<=> ( ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V2n))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V1m)),ap(V0f,V2n))) ) ) )
| ! [V3m] :
( mem(V3m,ty_2Enum_2Enum)
=> ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V3m),V4n))
=> p(ap(ap(c_2Ereal_2Ereal__ge,ap(V0f,V3m)),ap(V0f,V4n))) ) ) ) ) ) ) ).
fof(conj_thm_2Eseq_2EMONO__SUC,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Eseq_2Emono,V0f))
<=> ( ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__ge,ap(V0f,ap(c_2Enum_2ESUC,V1n))),ap(V0f,V1n))) )
| ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,ap(c_2Enum_2ESUC,V2n))),ap(V0f,V2n))) ) ) ) ) ).
fof(conj_thm_2Eseq_2EMAX__LEMMA,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1N] :
( mem(V1N,ty_2Enum_2Enum)
=> ? [V2k] :
( mem(V2k,ty_2Erealax_2Ereal)
& ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Eprim__rec_2E_3C,V3n),V1N))
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(V0s,V3n))),V2k)) ) ) ) ) ) ).
fof(conj_thm_2Eseq_2ESEQ__BOUNDED,axiom,
! [V0s] :
( 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] :
( mem(V1k,ty_2Erealax_2Ereal)
& ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(V0s,V2n))),V1k)) ) ) ) ) ).
fof(conj_thm_2Eseq_2ESEQ__BOUNDED__2,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1k] :
( mem(V1k,ty_2Erealax_2Ereal)
=> ! [V2k_27] :
( mem(V2k_27,ty_2Erealax_2Ereal)
=> ( ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,V1k),ap(V0f,V3n)))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V3n)),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)) ) ) ) ) ).
fof(conj_thm_2Eseq_2ESEQ__CBOUNDED,axiom,
! [V0f] :
( 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)) ) ) ).
fof(conj_thm_2Eseq_2ESEQ__ICONV,axiom,
! [V0f] :
( 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] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3E_3D,V1m),V2n))
=> p(ap(ap(c_2Ereal_2Ereal__ge,ap(V0f,V1m)),ap(V0f,V2n))) ) ) ) )
=> p(ap(c_2Eseq_2Econvergent,V0f)) ) ) ).
fof(lameq_f2784,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1n] : ap(f2784(V0f),V1n) = ap(c_2Erealax_2Ereal__neg,ap(V0f,V1n)) ) ).
fof(conj_thm_2Eseq_2ESEQ__NEG__CONV,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Eseq_2Econvergent,V0f))
<=> p(ap(c_2Eseq_2Econvergent,f2784(V0f))) ) ) ).
fof(conj_thm_2Eseq_2ESEQ__NEG__BOUNDED,axiom,
! [V0f] :
( 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)) ) ) ).
fof(conj_thm_2Eseq_2ESEQ__BCONV,axiom,
! [V0f] :
( 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)) ) ) ).
fof(lameq_f2785,axiom,
! [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V0s] :
( mem(V0s,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2n] : ap(f2785(V1f,V0s),V2n) = ap(V0s,ap(V1f,V2n)) ) ) ).
fof(conj_thm_2Eseq_2ESEQ__MONOSUB,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ? [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
& p(ap(c_2Eseq_2Esubseq,V1f))
& p(ap(c_2Eseq_2Emono,f2785(V1f,V0s))) ) ) ).
fof(conj_thm_2Eseq_2ESEQ__SBOUNDED,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1f] :
( 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))) ) ) ) ).
fof(conj_thm_2Eseq_2ESEQ__SUBLE,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ( p(ap(c_2Eseq_2Esubseq,V0f))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,V1n),ap(V0f,V1n))) ) ) ) ).
fof(conj_thm_2Eseq_2ESEQ__DIRECT,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ( p(ap(c_2Eseq_2Esubseq,V0f))
=> ! [V1N1] :
( mem(V1N1,ty_2Enum_2Enum)
=> ! [V2N2] :
( mem(V2N2,ty_2Enum_2Enum)
=> ? [V3n] :
( mem(V3n,ty_2Enum_2Enum)
& p(ap(ap(c_2Earithmetic_2E_3E_3D,V3n),V1N1))
& p(ap(ap(c_2Earithmetic_2E_3E_3D,ap(V0f,V3n)),V2N2)) ) ) ) ) ) ).
fof(conj_thm_2Eseq_2ESEQ__CAUCHY,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Eseq_2Ecauchy,V0f))
<=> p(ap(c_2Eseq_2Econvergent,V0f)) ) ) ).
fof(conj_thm_2Eseq_2ESEQ__LE,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2l] :
( mem(V2l,ty_2Erealax_2Ereal)
=> ! [V3m] :
( mem(V3m,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Eseq_2E_2D_2D_3E,V0f),V2l))
& p(ap(ap(c_2Eseq_2E_2D_2D_3E,V1g),V3m))
& ? [V4N] :
( mem(V4N,ty_2Enum_2Enum)
& ! [V5n] :
( mem(V5n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3E_3D,V5n),V4N))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V5n)),ap(V1g,V5n))) ) ) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,V2l),V3m)) ) ) ) ) ) ).
fof(lameq_f2786,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2n] : ap(f2786(V0f),V2n) = ap(V0f,ap(c_2Enum_2ESUC,V2n)) ) ).
fof(conj_thm_2Eseq_2ESEQ__SUC,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Eseq_2E_2D_2D_3E,V0f),V1l))
<=> p(ap(ap(c_2Eseq_2E_2D_2D_3E,f2786(V0f)),V1l)) ) ) ) ).
fof(lameq_f2787,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1n] : ap(f2787(V0f),V1n) = ap(c_2Ereal_2Eabs,ap(V0f,V1n)) ) ).
fof(conj_thm_2Eseq_2ESEQ__ABS,axiom,
! [V0f] :
( 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,c_2Enum_2E0)))
<=> p(ap(ap(c_2Eseq_2E_2D_2D_3E,V0f),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ).
fof(lameq_f2788,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2n] : ap(f2788(V0f),V2n) = ap(c_2Ereal_2Eabs,ap(V0f,V2n)) ) ).
fof(conj_thm_2Eseq_2ESEQ__ABS__IMP,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Eseq_2E_2D_2D_3E,V0f),V1l))
=> p(ap(ap(c_2Eseq_2E_2D_2D_3E,f2788(V0f)),ap(c_2Ereal_2Eabs,V1l))) ) ) ) ).
fof(lameq_f2789,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4n] : ap(f2789(V0f),V4n) = ap(c_2Erealax_2Einv,ap(V0f,V4n)) ) ).
fof(conj_thm_2Eseq_2ESEQ__INV0,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ? [V2N] :
( mem(V2N,ty_2Enum_2Enum)
& ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3E_3D,V3n),V2N))
=> p(ap(ap(c_2Ereal_2Ereal__gt,ap(V0f,V3n)),V1y)) ) ) ) )
=> p(ap(ap(c_2Eseq_2E_2D_2D_3E,f2789(V0f)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ).
fof(lameq_f2790,axiom,
! [V0c] :
( mem(V0c,ty_2Erealax_2Ereal)
=> ! [V1n] : ap(f2790(V0c),V1n) = ap(ap(c_2Ereal_2Epow,ap(c_2Ereal_2Eabs,V0c)),V1n) ) ).
fof(conj_thm_2Eseq_2ESEQ__POWER__ABS,axiom,
! [V0c] :
( mem(V0c,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,V0c)),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))
=> p(ap(ap(c_2Eseq_2E_2D_2D_3E,f2790(V0c)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ).
fof(lameq_f2791,axiom,
! [V0c] :
( mem(V0c,ty_2Erealax_2Ereal)
=> ! [V1n] : ap(f2791(V0c),V1n) = ap(ap(c_2Ereal_2Epow,V0c),V1n) ) ).
fof(conj_thm_2Eseq_2ESEQ__POWER,axiom,
! [V0c] :
( mem(V0c,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,V0c)),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))
=> p(ap(ap(c_2Eseq_2E_2D_2D_3E,f2791(V0c)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ).
fof(conj_thm_2Eseq_2ENEST__LEMMA,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ( ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__ge,ap(V0f,ap(c_2Enum_2ESUC,V2n))),ap(V0f,V2n))) )
& ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V1g,ap(c_2Enum_2ESUC,V3n))),ap(V1g,V3n))) )
& ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V4n)),ap(V1g,V4n))) ) )
=> ? [V5l] :
( mem(V5l,ty_2Erealax_2Ereal)
& ? [V6m] :
( mem(V6m,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Ereal_2Ereal__lte,V5l),V6m))
& ! [V7n] :
( mem(V7n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V7n)),V5l)) )
& p(ap(ap(c_2Eseq_2E_2D_2D_3E,V0f),V5l))
& ! [V8n] :
( mem(V8n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,V6m),ap(V1g,V8n))) )
& p(ap(ap(c_2Eseq_2E_2D_2D_3E,V1g),V6m)) ) ) ) ) ) ).
fof(lameq_f2792,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V5n] : ap(f2792(V0f,V1g),V5n) = ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,V5n)),ap(V1g,V5n)) ) ) ).
fof(conj_thm_2Eseq_2ENEST__LEMMA__UNIQ,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ( ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__ge,ap(V0f,ap(c_2Enum_2ESUC,V2n))),ap(V0f,V2n))) )
& ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V1g,ap(c_2Enum_2ESUC,V3n))),ap(V1g,V3n))) )
& ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V4n)),ap(V1g,V4n))) )
& p(ap(ap(c_2Eseq_2E_2D_2D_3E,f2792(V0f,V1g)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) )
=> ? [V6l] :
( mem(V6l,ty_2Erealax_2Ereal)
& ! [V7n] :
( mem(V7n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V7n)),V6l)) )
& p(ap(ap(c_2Eseq_2E_2D_2D_3E,V0f),V6l))
& ! [V8n] :
( mem(V8n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,V6l),ap(V1g,V8n))) )
& p(ap(ap(c_2Eseq_2E_2D_2D_3E,V1g),V6l)) ) ) ) ) ).
fof(conj_thm_2Eseq_2EBOLZANO__LEMMA,axiom,
! [V0P] :
( mem(V0P,arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),bool))
=> ( ( ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ! [V2b] :
( mem(V2b,ty_2Erealax_2Ereal)
=> ! [V3c] :
( mem(V3c,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V2b))
& p(ap(ap(c_2Ereal_2Ereal__lte,V2b),V3c))
& p(ap(V0P,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V2b)))
& p(ap(V0P,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V2b),V3c))) )
=> p(ap(V0P,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V1a),V3c))) ) ) ) )
& ! [V4x] :
( mem(V4x,ty_2Erealax_2Ereal)
=> ? [V5d] :
( mem(V5d,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V5d))
& ! [V6a] :
( mem(V6a,ty_2Erealax_2Ereal)
=> ! [V7b] :
( mem(V7b,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V6a),V4x))
& p(ap(ap(c_2Ereal_2Ereal__lte,V4x),V7b))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Ereal_2Ereal__sub,V7b),V6a)),V5d)) )
=> p(ap(V0P,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V6a),V7b))) ) ) ) ) ) )
=> ! [V8a] :
( mem(V8a,ty_2Erealax_2Ereal)
=> ! [V9b] :
( mem(V9b,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,V8a),V9b))
=> p(ap(V0P,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V8a),V9b))) ) ) ) ) ) ).
fof(lameq_f2793,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2n] : ap(f2793(V0f),V2n) = ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Enum_2E0),V2n)),V0f) ) ).
fof(ax_thm_2Eseq_2Esums,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1s] :
( mem(V1s,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Eseq_2Esums,V0f),V1s))
<=> p(ap(ap(c_2Eseq_2E_2D_2D_3E,f2793(V0f)),V1s)) ) ) ) ).
fof(ax_thm_2Eseq_2Esummable,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Eseq_2Esummable,V0f))
<=> ? [V1s] :
( mem(V1s,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Eseq_2Esums,V0f),V1s)) ) ) ) ).
fof(lameq_f2794,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1s] : ap(f2794(V0f),V1s) = ap(ap(c_2Eseq_2Esums,V0f),V1s) ) ).
fof(ax_thm_2Eseq_2Esuminf,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ap(c_2Eseq_2Esuminf,V0f) = ap(c_2Emin_2E_40(ty_2Erealax_2Ereal),f2794(V0f)) ) ).
fof(conj_thm_2Eseq_2ESUM__SUMMABLE,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Eseq_2Esums,V0f),V1l))
=> p(ap(c_2Eseq_2Esummable,V0f)) ) ) ) ).
fof(conj_thm_2Eseq_2ESUMMABLE__SUM,axiom,
! [V0f] :
( 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))) ) ) ).
fof(conj_thm_2Eseq_2ESUM__UNIQ,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Eseq_2Esums,V0f),V1x))
=> V1x = ap(c_2Eseq_2Esuminf,V0f) ) ) ) ).
fof(conj_thm_2Eseq_2ESER__0,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1n),V2m))
=> ap(V0f,V2m) = ap(c_2Ereal_2Ereal__of__num,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),c_2Enum_2E0),V1n)),V0f))) ) ) ) ).
fof(conj_thm_2Eseq_2ESER__POS__LE,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( ( p(ap(c_2Eseq_2Esummable,V0f))
& ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1n),V2m))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V0f,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),c_2Enum_2E0),V1n)),V0f)),ap(c_2Eseq_2Esuminf,V0f))) ) ) ) ).
fof(conj_thm_2Eseq_2ESER__POS__LT,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( ( p(ap(c_2Eseq_2Esummable,V0f))
& ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1n),V2m))
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V0f,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),c_2Enum_2E0),V1n)),V0f)),ap(c_2Eseq_2Esuminf,V0f))) ) ) ) ).
fof(lameq_f2795,axiom,
! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2n] : ap(f2795(V1k,V0f),V2n) = ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),ap(ap(c_2Earithmetic_2E_2A,V2n),V1k)),V1k)),V0f) ) ) ).
fof(conj_thm_2Eseq_2ESER__GROUP,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ( ( p(ap(c_2Eseq_2Esummable,V0f))
& p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V1k)) )
=> p(ap(ap(c_2Eseq_2Esums,f2795(V1k,V0f)),ap(c_2Eseq_2Esuminf,V0f))) ) ) ) ).
fof(lameq_f2796,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1n] : ap(f2796(V0f),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,c_2Earithmetic_2EZERO))),V1n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V0f) ) ).
fof(conj_thm_2Eseq_2ESER__PAIR,axiom,
! [V0f] :
( 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))) ) ) ).
fof(lameq_f2797,axiom,
! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2n] : ap(f2797(V1k,V0f),V2n) = ap(V0f,ap(ap(c_2Earithmetic_2E_2B,V2n),V1k)) ) ) ).
fof(conj_thm_2Eseq_2ESER__OFFSET,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Eseq_2Esummable,V0f))
=> ! [V1k] :
( mem(V1k,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),c_2Enum_2E0),V1k)),V0f)))) ) ) ) ).
fof(conj_thm_2Eseq_2ESER__POS__LT__PAIR,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( ( p(ap(c_2Eseq_2Esummable,V0f))
& ! [V2d] :
( mem(V2d,ty_2Enum_2Enum)
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Erealax_2Ereal__add,ap(V0f,ap(ap(c_2Earithmetic_2E_2B,V1n),ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))),V2d)))),ap(V0f,ap(ap(c_2Earithmetic_2E_2B,V1n),ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))),V2d)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,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),c_2Enum_2E0),V1n)),V0f)),ap(c_2Eseq_2Esuminf,V0f))) ) ) ) ).
fof(conj_thm_2Eseq_2ESER__ADD,axiom,
! [V0x] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x0] :
( mem(V1x0,ty_2Erealax_2Ereal)
=> ! [V2y] :
( mem(V2y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3y0] :
( mem(V3y0,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Eseq_2Esums,V0x),V1x0))
& p(ap(ap(c_2Eseq_2Esums,V2y),V3y0)) )
=> p(ap(ap(c_2Eseq_2Esums,f2777(V0x,V2y)),ap(ap(c_2Erealax_2Ereal__add,V1x0),V3y0))) ) ) ) ) ) ).
fof(lameq_f2798,axiom,
! [V0x] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2c] :
( mem(V2c,ty_2Erealax_2Ereal)
=> ! [V3n] : ap(f2798(V0x,V2c),V3n) = ap(ap(c_2Erealax_2Ereal__mul,V2c),ap(V0x,V3n)) ) ) ).
fof(conj_thm_2Eseq_2ESER__CMUL,axiom,
! [V0x] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x0] :
( mem(V1x0,ty_2Erealax_2Ereal)
=> ! [V2c] :
( mem(V2c,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Eseq_2Esums,V0x),V1x0))
=> p(ap(ap(c_2Eseq_2Esums,f2798(V0x,V2c)),ap(ap(c_2Erealax_2Ereal__mul,V2c),V1x0))) ) ) ) ) ).
fof(conj_thm_2Eseq_2ESER__NEG,axiom,
! [V0x] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x0] :
( mem(V1x0,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Eseq_2Esums,V0x),V1x0))
=> p(ap(ap(c_2Eseq_2Esums,f2779(V0x)),ap(c_2Erealax_2Ereal__neg,V1x0))) ) ) ) ).
fof(conj_thm_2Eseq_2ESER__SUB,axiom,
! [V0x] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x0] :
( mem(V1x0,ty_2Erealax_2Ereal)
=> ! [V2y] :
( mem(V2y,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3y0] :
( mem(V3y0,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Eseq_2Esums,V0x),V1x0))
& p(ap(ap(c_2Eseq_2Esums,V2y),V3y0)) )
=> p(ap(ap(c_2Eseq_2Esums,f2781(V0x,V2y)),ap(ap(c_2Ereal_2Ereal__sub,V1x0),V3y0))) ) ) ) ) ) ).
fof(lameq_f2799,axiom,
! [V0x] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2c] :
( mem(V2c,ty_2Erealax_2Ereal)
=> ! [V3n] : ap(f2799(V0x,V2c),V3n) = ap(ap(c_2Ereal_2E_2F,ap(V0x,V3n)),V2c) ) ) ).
fof(conj_thm_2Eseq_2ESER__CDIV,axiom,
! [V0x] :
( mem(V0x,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x0] :
( mem(V1x0,ty_2Erealax_2Ereal)
=> ! [V2c] :
( mem(V2c,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Eseq_2Esums,V0x),V1x0))
=> p(ap(ap(c_2Eseq_2Esums,f2799(V0x,V2c)),ap(ap(c_2Ereal_2E_2F,V1x0),V2c))) ) ) ) ) ).
fof(conj_thm_2Eseq_2ESER__CAUCHY,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Eseq_2Esummable,V0f))
<=> ! [V1e] :
( mem(V1e,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1e))
=> ? [V2N] :
( mem(V2N,ty_2Enum_2Enum)
& ! [V3m] :
( mem(V3m,ty_2Enum_2Enum)
=> ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3E_3D,V3m),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),V3m),V4n)),V0f))),V1e)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eseq_2ESER__ZERO,axiom,
! [V0f] :
( 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,c_2Enum_2E0))) ) ) ).
fof(conj_thm_2Eseq_2ESER__COMPAR,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ( ? [V2N] :
( mem(V2N,ty_2Enum_2Enum)
& ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3E_3D,V3n),V2N))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(V0f,V3n))),ap(V1g,V3n))) ) ) )
& p(ap(c_2Eseq_2Esummable,V1g)) )
=> p(ap(c_2Eseq_2Esummable,V0f)) ) ) ) ).
fof(lameq_f2800,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4k] : ap(f2800(V0f),V4k) = ap(c_2Ereal_2Eabs,ap(V0f,V4k)) ) ).
fof(conj_thm_2Eseq_2ESER__COMPARA,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ( ? [V2N] :
( mem(V2N,ty_2Enum_2Enum)
& ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3E_3D,V3n),V2N))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(V0f,V3n))),ap(V1g,V3n))) ) ) )
& p(ap(c_2Eseq_2Esummable,V1g)) )
=> p(ap(c_2Eseq_2Esummable,f2800(V0f))) ) ) ) ).
fof(conj_thm_2Eseq_2ESER__LE,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ( ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V2n)),ap(V1g,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))) ) ) ) ).
fof(conj_thm_2Eseq_2ESER__LE2,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ( ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(V0f,V2n))),ap(V1g,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))) ) ) ) ) ).
fof(conj_thm_2Eseq_2ESER__ACONV,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Eseq_2Esummable,f2787(V0f)))
=> p(ap(c_2Eseq_2Esummable,V0f)) ) ) ).
fof(conj_thm_2Eseq_2ESER__ABS,axiom,
! [V0f] :
( 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)))) ) ) ).
fof(lameq_f2801,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V2n] : ap(f2801(V0x),V2n) = ap(ap(c_2Ereal_2Epow,V0x),V2n) ) ).
fof(conj_thm_2Eseq_2EGP__FINITE,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ( V0x != ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Enum_2E0),V1n)),f2801(V0x)) = ap(ap(c_2Ereal_2E_2F,ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Ereal_2Epow,V0x),V1n)),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(ap(c_2Ereal_2Ereal__sub,V0x),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ) ) ) ).
fof(lameq_f2802,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1n] : ap(f2802(V0x),V1n) = ap(ap(c_2Ereal_2Epow,V0x),V1n) ) ).
fof(conj_thm_2Eseq_2EGP,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,V0x)),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,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,c_2Earithmetic_2EZERO)))),V0x)))) ) ) ).
fof(conj_thm_2Eseq_2EABS__NEG__LEMMA,axiom,
! [V0c] :
( mem(V0c,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,V0c),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)))
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V2y] :
( mem(V2y,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,V1x)),ap(ap(c_2Erealax_2Ereal__mul,V0c),ap(c_2Ereal_2Eabs,V2y))))
=> V1x = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) ).
fof(conj_thm_2Eseq_2ESER__RATIO,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1c] :
( mem(V1c,ty_2Erealax_2Ereal)
=> ! [V2N] :
( mem(V2N,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,V1c),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))
& ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3E_3D,V3n),V2N))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(V0f,ap(c_2Enum_2ESUC,V3n)))),ap(ap(c_2Erealax_2Ereal__mul,V1c),ap(c_2Ereal_2Eabs,ap(V0f,V3n))))) ) ) )
=> p(ap(c_2Eseq_2Esummable,V0f)) ) ) ) ) ).
fof(conj_thm_2Eseq_2ELE__SEQ__IMP__LE__LIM,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ! [V2f] :
( mem(V2f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ( ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,V0x),ap(V2f,V3n))) )
& p(ap(ap(c_2Eseq_2E_2D_2D_3E,V2f),V1y)) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,V0x),V1y)) ) ) ) ) ).
fof(conj_thm_2Eseq_2ESEQ__LE__IMP__LIM__LE,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ! [V2f] :
( mem(V2f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ( ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V2f,V3n)),V0x)) )
& p(ap(ap(c_2Eseq_2E_2D_2D_3E,V2f),V1y)) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,V1y),V0x)) ) ) ) ) ).
fof(conj_thm_2Eseq_2ESEQ__MONO__LE,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( ( ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V3n)),ap(V0f,ap(ap(c_2Earithmetic_2E_2B,V3n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))) )
& p(ap(ap(c_2Eseq_2E_2D_2D_3E,V0f),V1x)) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V2n)),V1x)) ) ) ) ) ).
fof(conj_thm_2Eseq_2ESEQ__LE__MONO,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( ( ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,ap(ap(c_2Earithmetic_2E_2B,V3n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(V0f,V3n))) )
& p(ap(ap(c_2Eseq_2E_2D_2D_3E,V0f),V1x)) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,V1x),ap(V0f,V2n))) ) ) ) ) ).
fof(ax_thm_2Eseq_2Emono__increasing__def,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Eseq_2Emono__increasing,V0f))
<=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V2n))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V1m)),ap(V0f,V2n))) ) ) ) ) ) ).
fof(conj_thm_2Eseq_2Emono__increasing__suc,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Eseq_2Emono__increasing,V0f))
<=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V1n)),ap(V0f,ap(c_2Enum_2ESUC,V1n)))) ) ) ) ).
fof(ax_thm_2Eseq_2Emono__decreasing__def,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Eseq_2Emono__decreasing,V0f))
<=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V2n))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V2n)),ap(V0f,V1m))) ) ) ) ) ) ).
fof(conj_thm_2Eseq_2Emono__decreasing__suc,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Eseq_2Emono__decreasing,V0f))
<=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,ap(c_2Enum_2ESUC,V1n))),ap(V0f,V1n))) ) ) ) ).
fof(conj_thm_2Eseq_2Emono__increasing__converges__to__sup,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1r] :
( mem(V1r,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Eseq_2Emono__increasing,V0f))
& p(ap(ap(c_2Eseq_2E_2D_2D_3E,V0f),V1r)) )
=> V1r = ap(c_2Ereal_2Esup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Erealax_2Ereal),V0f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) ) ) ).
fof(conj_thm_2Eseq_2EINCREASING__SEQ,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
=> ( ( ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V2n)),ap(V0f,ap(c_2Enum_2ESUC,V2n)))) )
& ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V3n)),V1l)) )
& ! [V4e] :
( mem(V4e,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V4e))
=> ? [V5n] :
( mem(V5n,ty_2Enum_2Enum)
& p(ap(ap(c_2Erealax_2Ereal__lt,V1l),ap(ap(c_2Erealax_2Ereal__add,ap(V0f,V5n)),V4e))) ) ) ) )
=> p(ap(ap(c_2Eseq_2E_2D_2D_3E,V0f),V1l)) ) ) ) ).
fof(conj_thm_2Eseq_2EMAX__LE__X,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ! [V2k] :
( mem(V2k,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(ap(c_2Earithmetic_2EMAX,V0m),V1n)),V2k))
<=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),V2k))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,V1n),V2k)) ) ) ) ) ) ).
fof(conj_thm_2Eseq_2EX__LE__MAX,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ! [V2k] :
( mem(V2k,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V2k),ap(ap(c_2Earithmetic_2EMAX,V0m),V1n)))
<=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V2k),V0m))
| p(ap(ap(c_2Earithmetic_2E_3C_3D,V2k),V1n)) ) ) ) ) ) ).
fof(conj_thm_2Eseq_2ETRANSFORM__2D__NUM,axiom,
! [V0P] :
( mem(V0P,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,bool)))
=> ( ( ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(V0P,V1m),V2n))
=> p(ap(ap(V0P,V2n),V1m)) ) ) )
& ! [V3m] :
( mem(V3m,ty_2Enum_2Enum)
=> ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> p(ap(ap(V0P,V3m),ap(ap(c_2Earithmetic_2E_2B,V3m),V4n))) ) ) )
=> ! [V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V6n] :
( mem(V6n,ty_2Enum_2Enum)
=> p(ap(ap(V0P,V5m),V6n)) ) ) ) ) ).
fof(conj_thm_2Eseq_2ETRIANGLE__2D__NUM,axiom,
! [V0P] :
( mem(V0P,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,bool)))
=> ( ! [V1d] :
( mem(V1d,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> p(ap(ap(V0P,V2n),ap(ap(c_2Earithmetic_2E_2B,V1d),V2n))) ) )
=> ! [V3m] :
( mem(V3m,ty_2Enum_2Enum)
=> ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V3m),V4n))
=> p(ap(ap(V0P,V3m),V4n)) ) ) ) ) ) ).
fof(conj_thm_2Eseq_2ESEQ__SANDWICH,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2h] :
( mem(V2h,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3l] :
( mem(V3l,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Eseq_2E_2D_2D_3E,V0f),V3l))
& p(ap(ap(c_2Eseq_2E_2D_2D_3E,V2h),V3l))
& ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V4n)),ap(V1g,V4n)))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(V1g,V4n)),ap(V2h,V4n))) ) ) )
=> p(ap(ap(c_2Eseq_2E_2D_2D_3E,V1g),V3l)) ) ) ) ) ) ).
fof(conj_thm_2Eseq_2ESER__POS,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Eseq_2Esummable,V0f))
& ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V0f,V1n))) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Eseq_2Esuminf,V0f))) ) ) ).
fof(conj_thm_2Eseq_2ESER__POS__MONO,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V0f,V1n))) )
=> p(ap(c_2Eseq_2Emono,f2793(V0f))) ) ) ).
fof(conj_thm_2Eseq_2EPOS__SUMMABLE,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ( ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V0f,V1n))) )
& ? [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
& ! [V3n] :
( mem(V3n,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),c_2Enum_2E0),V3n)),V0f)),V2x)) ) ) )
=> p(ap(c_2Eseq_2Esummable,V0f)) ) ) ).
fof(conj_thm_2Eseq_2ESUMMABLE__LE,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Eseq_2Esummable,V0f))
& ! [V2n] :
( mem(V2n,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),c_2Enum_2E0),V2n)),V0f)),V1x)) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Eseq_2Esuminf,V0f)),V1x)) ) ) ) ).
fof(conj_thm_2Eseq_2ESUMS__EQ,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Eseq_2Esums,V0f),V1x))
<=> ( p(ap(c_2Eseq_2Esummable,V0f))
& ap(c_2Eseq_2Esuminf,V0f) = V1x ) ) ) ) ).
fof(conj_thm_2Eseq_2ESUMINF__POS,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ( ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V0f,V1n))) )
& p(ap(c_2Eseq_2Esummable,V0f)) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Eseq_2Esuminf,V0f))) ) ) ).
fof(lameq_f2803,axiom,
! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ! [V3m] : ap(f2803(V2x,V1k),V3m) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),V3m),V1k)),V2x),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)) ) ) ).
fof(conj_thm_2Eseq_2ESUM__PICK,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1k] :
( mem(V1k,ty_2Enum_2Enum)
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Enum_2E0),V0n)),f2803(V2x,V1k)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Eprim__rec_2E_3C,V1k),V0n)),V2x),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)) ) ) ) ).
fof(conj_thm_2Eseq_2ESUM__LT,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ( ( ! [V4r] :
( mem(V4r,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V2m),V4r))
& p(ap(ap(c_2Eprim__rec_2E_3C,V4r),ap(ap(c_2Earithmetic_2E_2B,V3n),V2m))) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(V0f,V4r)),ap(V1g,V4r))) ) )
& p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),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),V2m),V3n)),V0f)),ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),V2m),V3n)),V1g))) ) ) ) ) ) ).
fof(conj_thm_2Eseq_2ESUM__CONST__R,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1r] :
( mem(V1r,ty_2Erealax_2Ereal)
=> ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Enum_2E0),V0n)),ap(c_2Ecombin_2EK(ty_2Erealax_2Ereal,ty_2Enum_2Enum),V1r)) = ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,V0n)),V1r) ) ) ).
fof(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,c_2Enum_2E0))),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ).
fof(conj_thm_2Eseq_2ELT__SUC,axiom,
! [V0a] :
( mem(V0a,ty_2Enum_2Enum)
=> ! [V1b] :
( mem(V1b,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Eprim__rec_2E_3C,V0a),ap(c_2Enum_2ESUC,V1b)))
<=> ( p(ap(ap(c_2Eprim__rec_2E_3C,V0a),V1b))
| V0a = V1b ) ) ) ) ).
fof(conj_thm_2Eseq_2ELE__SUC,axiom,
! [V0a] :
( mem(V0a,ty_2Enum_2Enum)
=> ! [V1b] :
( mem(V1b,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V0a),ap(c_2Enum_2ESUC,V1b)))
<=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V0a),V1b))
| V0a = ap(c_2Enum_2ESUC,V1b) ) ) ) ) ).
fof(conj_thm_2Eseq_2EK__PARTIAL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ap(c_2Ecombin_2EK(A_27a,A_27b),V0x) = k(A_27b,V0x) ) ) ) ).
fof(conj_thm_2Eseq_2EHALF__POS,axiom,
p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Ereal_2E_2F,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))))) ).
fof(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,c_2Earithmetic_2EZERO)))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ).
fof(conj_thm_2Eseq_2EHALF__CANCEL,axiom,
ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),ap(ap(c_2Ereal_2E_2F,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))) = ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ).
fof(conj_thm_2Eseq_2EX__HALF__HALF,axiom,
! [V0x] :
( mem(V0x,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,c_2Earithmetic_2EZERO)))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))),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,c_2Earithmetic_2EZERO)))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))),V0x)) = V0x ) ).
fof(conj_thm_2Eseq_2EONE__MINUS__HALF,axiom,
ap(ap(c_2Ereal_2Ereal__sub,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(ap(c_2Ereal_2E_2F,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))) = ap(ap(c_2Ereal_2E_2F,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))) ).
fof(conj_thm_2Eseq_2ENUM__2D__BIJ__BIG__SQUARE,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum)))
=> ! [V1N] :
( mem(V1N,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] :
( mem(V2k,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,V1N))),ap(ap(c_2Epred__set_2ECROSS(ty_2Enum_2Enum,ty_2Enum_2Enum),ap(c_2Epred__set_2Ecount,V2k)),ap(c_2Epred__set_2Ecount,V2k)))) ) ) ) ) ).
fof(conj_thm_2Eseq_2ENUM__2D__BIJ__SMALL__SQUARE,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum)))
=> ! [V1k] :
( mem(V1k,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] :
( mem(V2N,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,V1k)),ap(c_2Epred__set_2Ecount,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,V2N)))) ) ) ) ) ).
fof(lameq_f2804,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2n] : ap(f2804(V0f,V1g),V2n) = ap(ap(c_2Erealax_2Ereal__add,ap(V0f,V2n)),ap(V1g,V2n)) ) ) ).
fof(lameq_f2805,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3n] : ap(f2805(V0f,V1g),V3n) = ap(ap(c_2Erealax_2Ereal__add,ap(V0f,V3n)),ap(V1g,V3n)) ) ) ).
fof(conj_thm_2Eseq_2ESUMINF__ADD,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g] :
( 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)))
& ap(ap(c_2Erealax_2Ereal__add,ap(c_2Eseq_2Esuminf,V0f)),ap(c_2Eseq_2Esuminf,V1g)) = ap(c_2Eseq_2Esuminf,f2805(V0f,V1g)) ) ) ) ) ).
fof(conj_thm_2Eseq_2ESUMINF__2D,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2h] :
( mem(V2h,arr(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum)))
=> ( ( ! [V3m] :
( mem(V3m,ty_2Enum_2Enum)
=> ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(V0f,V3m),V4n))) ) )
& ! [V5n] :
( mem(V5n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Eseq_2Esums,ap(V0f,V5n)),ap(V1g,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))) ) ) ) ) ).
fof(lameq_f2806,axiom,
! [V0n] : ap(f2806,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,c_2Earithmetic_2EZERO)))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))),ap(ap(c_2Earithmetic_2E_2B,V0n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))) ).
fof(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,c_2Earithmetic_2EZERO))))) ).
fof(conj_thm_2Eseq_2ESER__POS__COMPARE,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ( ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V0f,V2n))) )
& p(ap(c_2Eseq_2Esummable,V1g))
& ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V3n)),ap(V1g,V3n))) ) )
=> ( p(ap(c_2Eseq_2Esummable,V0f))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Eseq_2Esuminf,V0f)),ap(c_2Eseq_2Esuminf,V1g))) ) ) ) ) ).
%------------------------------------------------------------------------------