ITP001 Axioms: ITP131+5.ax
%------------------------------------------------------------------------------
% File : ITP131+5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : powser+2.ax [Gau20]
% : HL4131+5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 45 ( 1 unt; 0 def)
% Number of atoms : 174 ( 39 equ)
% Maximal formula atoms : 11 ( 3 avg)
% Number of connectives : 131 ( 2 ~; 0 |; 12 &)
% ( 0 <=>; 117 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 7 avg)
% Maximal term depth : 14 ( 2 avg)
% Number of predicates : 3 ( 2 usr; 0 prp; 1-2 aty)
% Number of functors : 57 ( 57 usr; 26 con; 0-4 aty)
% Number of variables : 135 ( 135 !; 0 ?)
% SPC : FOF_SAT_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(mem_c_2Epowser_2Ediffs,axiom,
mem(c_2Epowser_2Ediffs,arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))) ).
fof(lameq_f2842,axiom,
! [V2y] :
( mem(V2y,ty_2Erealax_2Ereal)
=> ! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V3p] : ap(f2842(V2y,V0n,V1x),V3p) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2Epow,V1x),V3p)),ap(ap(c_2Ereal_2Epow,V2y),ap(ap(c_2Earithmetic_2E_2D,ap(c_2Enum_2ESUC,V0n)),V3p))) ) ) ) ).
fof(lameq_f2843,axiom,
! [V2y] :
( mem(V2y,ty_2Erealax_2Ereal)
=> ! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V4p] : ap(f2843(V2y,V0n,V1x),V4p) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2Epow,V1x),V4p)),ap(ap(c_2Ereal_2Epow,V2y),ap(ap(c_2Earithmetic_2E_2D,V0n),V4p))) ) ) ) ).
fof(conj_thm_2Epowser_2EPOWDIFF__LEMMA,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V2y] :
( mem(V2y,ty_2Erealax_2Ereal)
=> ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Enum_2E0),ap(c_2Enum_2ESUC,V0n))),f2842(V2y,V0n,V1x)) = ap(ap(c_2Erealax_2Ereal__mul,V2y),ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Enum_2E0),ap(c_2Enum_2ESUC,V0n))),f2843(V2y,V0n,V1x))) ) ) ) ).
fof(lameq_f2844,axiom,
! [V2y] :
( mem(V2y,ty_2Erealax_2Ereal)
=> ! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V3p] : ap(f2844(V2y,V0n,V1x),V3p) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2Epow,V1x),V3p)),ap(ap(c_2Ereal_2Epow,V2y),ap(ap(c_2Earithmetic_2E_2D,V0n),V3p))) ) ) ) ).
fof(conj_thm_2Epowser_2EPOWDIFF,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V2y] :
( mem(V2y,ty_2Erealax_2Ereal)
=> ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Ereal_2Epow,V1x),ap(c_2Enum_2ESUC,V0n))),ap(ap(c_2Ereal_2Epow,V2y),ap(c_2Enum_2ESUC,V0n))) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2Ereal__sub,V1x),V2y)),ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Enum_2E0),ap(c_2Enum_2ESUC,V0n))),f2844(V2y,V0n,V1x))) ) ) ) ).
fof(lameq_f2845,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V2y] :
( mem(V2y,ty_2Erealax_2Ereal)
=> ! [V4p] : ap(f2845(V0n,V1x,V2y),V4p) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2Epow,V1x),ap(ap(c_2Earithmetic_2E_2D,V0n),V4p))),ap(ap(c_2Ereal_2Epow,V2y),V4p)) ) ) ) ).
fof(conj_thm_2Epowser_2EPOWREV,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V2y] :
( mem(V2y,ty_2Erealax_2Ereal)
=> ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Enum_2E0),ap(c_2Enum_2ESUC,V0n))),f2844(V2y,V0n,V1x)) = ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Enum_2E0),ap(c_2Enum_2ESUC,V0n))),f2845(V0n,V1x,V2y)) ) ) ) ).
fof(lameq_f2846,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V3n] : ap(f2846(V0f,V1x),V3n) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0f,V3n)),ap(ap(c_2Ereal_2Epow,V1x),V3n)) ) ) ).
fof(lameq_f2847,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ! [V4n] : ap(f2847(V0f,V2z),V4n) = ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Eabs,ap(V0f,V4n))),ap(ap(c_2Ereal_2Epow,V2z),V4n)) ) ) ).
fof(conj_thm_2Epowser_2EPOWSER__INSIDEA,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Eseq_2Esummable,f2846(V0f,V1x)))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,V2z)),ap(c_2Ereal_2Eabs,V1x))) )
=> p(ap(c_2Eseq_2Esummable,f2847(V0f,V2z))) ) ) ) ) ).
fof(lameq_f2848,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ! [V4n] : ap(f2848(V0f,V2z),V4n) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0f,V4n)),ap(ap(c_2Ereal_2Epow,V2z),V4n)) ) ) ).
fof(conj_thm_2Epowser_2EPOWSER__INSIDE,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Eseq_2Esummable,f2846(V0f,V1x)))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,V2z)),ap(c_2Ereal_2Eabs,V1x))) )
=> p(ap(c_2Eseq_2Esummable,f2848(V0f,V2z))) ) ) ) ) ).
fof(lameq_f2849,axiom,
! [V0c] :
( mem(V0c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1n] : ap(f2849(V0c),V1n) = ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,ap(c_2Enum_2ESUC,V1n))),ap(V0c,ap(c_2Enum_2ESUC,V1n))) ) ).
fof(ax_thm_2Epowser_2Ediffs,axiom,
! [V0c] :
( mem(V0c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ap(c_2Epowser_2Ediffs,V0c) = f2849(V0c) ) ).
fof(lameq_f2850,axiom,
! [V0c] :
( mem(V0c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1n] : ap(f2850(V0c),V1n) = ap(c_2Erealax_2Ereal__neg,ap(V0c,V1n)) ) ).
fof(lameq_f2851,axiom,
! [V0c] :
( mem(V0c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2n] : ap(f2851(V0c),V2n) = ap(c_2Erealax_2Ereal__neg,ap(ap(c_2Epowser_2Ediffs,V0c),V2n)) ) ).
fof(conj_thm_2Epowser_2EDIFFS__NEG,axiom,
! [V0c] :
( mem(V0c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ap(c_2Epowser_2Ediffs,f2850(V0c)) = f2851(V0c) ) ).
fof(lameq_f2852,axiom,
! [V1c] :
( mem(V1c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ! [V3n] : ap(f2852(V1c,V2x),V3n) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Epowser_2Ediffs,V1c),V3n)),ap(ap(c_2Ereal_2Epow,V2x),V3n)) ) ) ).
fof(lameq_f2853,axiom,
! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ! [V1c] :
( mem(V1c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4n] : ap(f2853(V2x,V1c),V4n) = ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,V4n)),ap(ap(c_2Erealax_2Ereal__mul,ap(V1c,V4n)),ap(ap(c_2Ereal_2Epow,V2x),ap(ap(c_2Earithmetic_2E_2D,V4n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))) ) ) ).
fof(conj_thm_2Epowser_2EDIFFS__LEMMA,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1c] :
( mem(V1c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [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)),f2852(V1c,V2x)) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Enum_2E0),V0n)),f2853(V2x,V1c))),ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,V0n)),ap(ap(c_2Erealax_2Ereal__mul,ap(V1c,V0n)),ap(ap(c_2Ereal_2Epow,V2x),ap(ap(c_2Earithmetic_2E_2D,V0n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))))) ) ) ) ).
fof(lameq_f2854,axiom,
! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ! [V1c] :
( mem(V1c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3n] : ap(f2854(V2x,V1c),V3n) = ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,V3n)),ap(ap(c_2Erealax_2Ereal__mul,ap(V1c,V3n)),ap(ap(c_2Ereal_2Epow,V2x),ap(ap(c_2Earithmetic_2E_2D,V3n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))) ) ) ).
fof(lameq_f2855,axiom,
! [V1c] :
( mem(V1c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ! [V4n] : ap(f2855(V1c,V2x),V4n) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Epowser_2Ediffs,V1c),V4n)),ap(ap(c_2Ereal_2Epow,V2x),V4n)) ) ) ).
fof(conj_thm_2Epowser_2EDIFFS__LEMMA2,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1c] :
( mem(V1c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [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)),f2854(V2x,V1c)) = ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Enum_2E0),V0n)),f2855(V1c,V2x))),ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,V0n)),ap(ap(c_2Erealax_2Ereal__mul,ap(V1c,V0n)),ap(ap(c_2Ereal_2Epow,V2x),ap(ap(c_2Earithmetic_2E_2D,V0n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))))) ) ) ) ).
fof(lameq_f2856,axiom,
! [V0c] :
( mem(V0c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V2n] : ap(f2856(V0c,V1x),V2n) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Epowser_2Ediffs,V0c),V2n)),ap(ap(c_2Ereal_2Epow,V1x),V2n)) ) ) ).
fof(lameq_f2857,axiom,
! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V0c] :
( mem(V0c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3n] : ap(f2857(V1x,V0c),V3n) = ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,V3n)),ap(ap(c_2Erealax_2Ereal__mul,ap(V0c,V3n)),ap(ap(c_2Ereal_2Epow,V1x),ap(ap(c_2Earithmetic_2E_2D,V3n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))) ) ) ).
fof(lameq_f2858,axiom,
! [V0c] :
( mem(V0c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V4n] : ap(f2858(V0c,V1x),V4n) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Epowser_2Ediffs,V0c),V4n)),ap(ap(c_2Ereal_2Epow,V1x),V4n)) ) ) ).
fof(conj_thm_2Epowser_2EDIFFS__EQUIV,axiom,
! [V0c] :
( mem(V0c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ( p(ap(c_2Eseq_2Esummable,f2856(V0c,V1x)))
=> p(ap(ap(c_2Eseq_2Esums,f2857(V1x,V0c)),ap(c_2Eseq_2Esuminf,f2858(V0c,V1x)))) ) ) ) ).
fof(lameq_f2859,axiom,
! [V2h] :
( mem(V2h,ty_2Erealax_2Ereal)
=> ! [V1z] :
( mem(V1z,ty_2Erealax_2Ereal)
=> ! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V3p] : ap(f2859(V2h,V1z,V0m),V3p) = ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2Epow,ap(ap(c_2Erealax_2Ereal__add,V1z),V2h)),ap(ap(c_2Earithmetic_2E_2D,V0m),V3p))),ap(ap(c_2Ereal_2Epow,V1z),V3p))),ap(ap(c_2Ereal_2Epow,V1z),V0m)) ) ) ) ).
fof(lameq_f2860,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V2h] :
( mem(V2h,ty_2Erealax_2Ereal)
=> ! [V1z] :
( mem(V1z,ty_2Erealax_2Ereal)
=> ! [V4p] : ap(f2860(V0m,V2h,V1z),V4p) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2Epow,V1z),V4p)),ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Ereal_2Epow,ap(ap(c_2Erealax_2Ereal__add,V1z),V2h)),ap(ap(c_2Earithmetic_2E_2D,V0m),V4p))),ap(ap(c_2Ereal_2Epow,V1z),ap(ap(c_2Earithmetic_2E_2D,V0m),V4p)))) ) ) ) ).
fof(conj_thm_2Epowser_2ETERMDIFF__LEMMA1,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1z] :
( mem(V1z,ty_2Erealax_2Ereal)
=> ! [V2h] :
( mem(V2h,ty_2Erealax_2Ereal)
=> ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Enum_2E0),V0m)),f2859(V2h,V1z,V0m)) = ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Enum_2E0),V0m)),f2860(V0m,V2h,V1z)) ) ) ) ).
fof(lameq_f2861,axiom,
! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ! [V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ! [V1h] :
( mem(V1h,ty_2Erealax_2Ereal)
=> ! [V0z] :
( mem(V0z,ty_2Erealax_2Ereal)
=> ! [V4q] : ap(f2861(V2n,V3p,V1h,V0z),V4q) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2Epow,ap(ap(c_2Erealax_2Ereal__add,V0z),V1h)),V4q)),ap(ap(c_2Ereal_2Epow,V0z),ap(ap(c_2Earithmetic_2E_2D,ap(ap(c_2Earithmetic_2E_2D,ap(ap(c_2Earithmetic_2E_2D,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V3p)),V4q))) ) ) ) ) ).
fof(lameq_f2862,axiom,
! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ! [V1h] :
( mem(V1h,ty_2Erealax_2Ereal)
=> ! [V0z] :
( mem(V0z,ty_2Erealax_2Ereal)
=> ! [V3p] : ap(f2862(V2n,V1h,V0z),V3p) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2Epow,V0z),V3p)),ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Enum_2E0),ap(ap(c_2Earithmetic_2E_2D,ap(ap(c_2Earithmetic_2E_2D,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V3p))),f2861(V2n,V3p,V1h,V0z))) ) ) ) ).
fof(conj_thm_2Epowser_2ETERMDIFF__LEMMA2,axiom,
! [V0z] :
( mem(V0z,ty_2Erealax_2Ereal)
=> ! [V1h] :
( mem(V1h,ty_2Erealax_2Ereal)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( V1h != ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)
=> ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Ereal_2E_2F,ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Ereal_2Epow,ap(ap(c_2Erealax_2Ereal__add,V0z),V1h)),V2n)),ap(ap(c_2Ereal_2Epow,V0z),V2n))),V1h)),ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,V2n)),ap(ap(c_2Ereal_2Epow,V0z),ap(ap(c_2Earithmetic_2E_2D,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))) = ap(ap(c_2Erealax_2Ereal__mul,V1h),ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Enum_2E0),ap(ap(c_2Earithmetic_2E_2D,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),f2862(V2n,V1h,V0z))) ) ) ) ) ).
fof(conj_thm_2Epowser_2ETERMDIFF__LEMMA3,axiom,
! [V0z] :
( mem(V0z,ty_2Erealax_2Ereal)
=> ! [V1h] :
( mem(V1h,ty_2Erealax_2Ereal)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ! [V3k_27] :
( mem(V3k_27,ty_2Erealax_2Ereal)
=> ( ( V1h != ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,V0z)),V3k_27))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Erealax_2Ereal__add,V0z),V1h))),V3k_27)) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Ereal_2E_2F,ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Ereal_2Epow,ap(ap(c_2Erealax_2Ereal__add,V0z),V1h)),V2n)),ap(ap(c_2Ereal_2Epow,V0z),V2n))),V1h)),ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,V2n)),ap(ap(c_2Ereal_2Epow,V0z),ap(ap(c_2Earithmetic_2E_2D,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))))),ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,V2n)),ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,ap(ap(c_2Earithmetic_2E_2D,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2Epow,V3k_27),ap(ap(c_2Earithmetic_2E_2D,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))))),ap(c_2Ereal_2Eabs,V1h)))))) ) ) ) ) ) ).
fof(conj_thm_2Epowser_2ETERMDIFF__LEMMA4,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1k_27] :
( mem(V1k_27,ty_2Erealax_2Ereal)
=> ! [V2k] :
( mem(V2k,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2k))
& ! [V3h] :
( mem(V3h,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal_2Eabs,V3h)))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,V3h)),V2k)) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(V0f,V3h))),ap(ap(c_2Erealax_2Ereal__mul,V1k_27),ap(c_2Ereal_2Eabs,V3h)))) ) ) )
=> p(ap(ap(ap(c_2Elim_2Etends__real__real,V0f),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ) ) ).
fof(lameq_f2863,axiom,
! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)))
=> ! [V5h] : ap(f2863(V1g),V5h) = ap(c_2Eseq_2Esuminf,ap(V1g,V5h)) ) ).
fof(conj_thm_2Epowser_2ETERMDIFF__LEMMA5,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Erealax_2Ereal,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)))
=> ! [V2k] :
( mem(V2k,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2k))
& p(ap(c_2Eseq_2Esummable,V0f))
& ! [V3h] :
( mem(V3h,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal_2Eabs,V3h)))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,V3h)),V2k)) )
=> ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(V1g,V3h),V4n))),ap(ap(c_2Erealax_2Ereal__mul,ap(V0f,V4n)),ap(c_2Ereal_2Eabs,V3h)))) ) ) ) )
=> p(ap(ap(ap(c_2Elim_2Etends__real__real,f2863(V1g)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ) ) ).
fof(lameq_f2864,axiom,
! [V0c] :
( mem(V0c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1k_27] :
( mem(V1k_27,ty_2Erealax_2Ereal)
=> ! [V3n] : ap(f2864(V0c,V1k_27),V3n) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0c,V3n)),ap(ap(c_2Ereal_2Epow,V1k_27),V3n)) ) ) ).
fof(lameq_f2865,axiom,
! [V0c] :
( mem(V0c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1k_27] :
( mem(V1k_27,ty_2Erealax_2Ereal)
=> ! [V4n] : ap(f2865(V0c,V1k_27),V4n) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Epowser_2Ediffs,V0c),V4n)),ap(ap(c_2Ereal_2Epow,V1k_27),V4n)) ) ) ).
fof(lameq_f2866,axiom,
! [V0c] :
( mem(V0c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1k_27] :
( mem(V1k_27,ty_2Erealax_2Ereal)
=> ! [V5n] : ap(f2866(V0c,V1k_27),V5n) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Epowser_2Ediffs,ap(c_2Epowser_2Ediffs,V0c)),V5n)),ap(ap(c_2Ereal_2Epow,V1k_27),V5n)) ) ) ).
fof(lameq_f2867,axiom,
! [V0c] :
( mem(V0c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V6x] :
( mem(V6x,ty_2Erealax_2Ereal)
=> ! [V7n] : ap(f2867(V0c,V6x),V7n) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0c,V7n)),ap(ap(c_2Ereal_2Epow,V6x),V7n)) ) ) ).
fof(lameq_f2868,axiom,
! [V0c] :
( mem(V0c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V6x] : ap(f2868(V0c),V6x) = ap(c_2Eseq_2Esuminf,f2867(V0c,V6x)) ) ).
fof(lameq_f2869,axiom,
! [V0c] :
( mem(V0c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ! [V8n] : ap(f2869(V0c,V2x),V8n) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Epowser_2Ediffs,V0c),V8n)),ap(ap(c_2Ereal_2Epow,V2x),V8n)) ) ) ).
fof(conj_thm_2Epowser_2ETERMDIFF,axiom,
! [V0c] :
( mem(V0c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1k_27] :
( mem(V1k_27,ty_2Erealax_2Ereal)
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Eseq_2Esummable,f2864(V0c,V1k_27)))
& p(ap(c_2Eseq_2Esummable,f2865(V0c,V1k_27)))
& p(ap(c_2Eseq_2Esummable,f2866(V0c,V1k_27)))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,V2x)),ap(c_2Ereal_2Eabs,V1k_27))) )
=> p(ap(ap(ap(c_2Elim_2Ediffl,f2868(V0c)),ap(c_2Eseq_2Esuminf,f2869(V0c,V2x))),V2x)) ) ) ) ) ).
%------------------------------------------------------------------------------