ITP001 Axioms: ITP131^7.ax
%------------------------------------------------------------------------------
% File : ITP131^7 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : powser.ax [Gau19]
% : HL4131^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 60 ( 13 unt; 37 typ; 0 def)
% Number of atoms : 49 ( 12 equ; 3 cnn)
% Maximal formula atoms : 6 ( 0 avg)
% Number of connectives : 560 ( 3 ~; 1 |; 13 &; 525 @)
% ( 7 <=>; 11 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg; 525 nst)
% Number of types : 4 ( 3 usr)
% Number of type conns : 77 ( 77 >; 0 *; 0 +; 0 <<)
% Number of symbols : 36 ( 34 usr; 3 con; 0-4 aty)
% Number of variables : 98 ( 30 ^ 62 !; 1 ?; 98 :)
% ( 5 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(tyop_2Erealax_2Ereal,type,
tyop_2Erealax_2Ereal: $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Earithmetic_2E_2D,type,
c_2Earithmetic_2E_2D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ereal_2E_2F,type,
c_2Ereal_2E_2F: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Enum_2E0,type,
c_2Enum_2E0: tyop_2Enum_2Enum ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Earithmetic_2EBIT1,type,
c_2Earithmetic_2EBIT1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EBIT2,type,
c_2Earithmetic_2EBIT2: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Enum_2ESUC,type,
c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EZERO,type,
c_2Earithmetic_2EZERO: tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Ereal_2Eabs,type,
c_2Ereal_2Eabs: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Elim_2Ediffl,type,
c_2Elim_2Ediffl: ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).
thf(c_2Epowser_2Ediffs,type,
c_2Epowser_2Ediffs: ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Epow,type,
c_2Ereal_2Epow: tyop_2Erealax_2Ereal > tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__add,type,
c_2Erealax_2Ereal__add: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__lt,type,
c_2Erealax_2Ereal__lt: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).
thf(c_2Ereal_2Ereal__lte,type,
c_2Ereal_2Ereal__lte: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).
thf(c_2Erealax_2Ereal__mul,type,
c_2Erealax_2Ereal__mul: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__neg,type,
c_2Erealax_2Ereal__neg: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Ereal__of__num,type,
c_2Ereal_2Ereal__of__num: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Ereal__sub,type,
c_2Ereal_2Ereal__sub: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Esum,type,
c_2Ereal_2Esum: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
thf(c_2Eseq_2Esuminf,type,
c_2Eseq_2Esuminf: ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
thf(c_2Eseq_2Esummable,type,
c_2Eseq_2Esummable: ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > $o ).
thf(c_2Eseq_2Esums,type,
c_2Eseq_2Esums: ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal > $o ).
thf(c_2Elim_2Etends__real__real,type,
c_2Elim_2Etends__real__real: ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Epowser_2Ediffs,axiom,
! [V0c: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Epowser_2Ediffs @ V0c )
= ( ^ [V1n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Enum_2ESUC @ V1n ) ) @ ( V0c @ ( c_2Enum_2ESUC @ V1n ) ) ) ) ) ).
thf(thm_2Epowser_2EPOWDIFF__LEMMA,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ ( c_2Enum_2ESUC @ V0n ) )
@ ^ [V3p: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Epow @ V1x @ V3p ) @ ( c_2Ereal_2Epow @ V2y @ ( c_2Earithmetic_2E_2D @ ( c_2Enum_2ESUC @ V0n ) @ V3p ) ) ) )
= ( c_2Erealax_2Ereal__mul @ V2y
@ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ ( c_2Enum_2ESUC @ V0n ) )
@ ^ [V4p: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Epow @ V1x @ V4p ) @ ( c_2Ereal_2Epow @ V2y @ ( c_2Earithmetic_2E_2D @ V0n @ V4p ) ) ) ) ) ) ).
thf(thm_2Epowser_2EPOWDIFF,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Epow @ V1x @ ( c_2Enum_2ESUC @ V0n ) ) @ ( c_2Ereal_2Epow @ V2y @ ( c_2Enum_2ESUC @ V0n ) ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__sub @ V1x @ V2y )
@ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ ( c_2Enum_2ESUC @ V0n ) )
@ ^ [V3p: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Epow @ V1x @ V3p ) @ ( c_2Ereal_2Epow @ V2y @ ( c_2Earithmetic_2E_2D @ V0n @ V3p ) ) ) ) ) ) ).
thf(thm_2Epowser_2EPOWREV,axiom,
! [V0n: tyop_2Enum_2Enum,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ ( c_2Enum_2ESUC @ V0n ) )
@ ^ [V3p: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Epow @ V1x @ V3p ) @ ( c_2Ereal_2Epow @ V2y @ ( c_2Earithmetic_2E_2D @ V0n @ V3p ) ) ) )
= ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ ( c_2Enum_2ESUC @ V0n ) )
@ ^ [V4p: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Epow @ V1x @ ( c_2Earithmetic_2E_2D @ V0n @ V4p ) ) @ ( c_2Ereal_2Epow @ V2y @ V4p ) ) ) ) ).
thf(thm_2Epowser_2EPOWSER__INSIDEA,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Eseq_2Esummable
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( V0f @ V3n ) @ ( c_2Ereal_2Epow @ V1x @ V3n ) ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ V2z ) @ ( c_2Ereal_2Eabs @ V1x ) ) )
=> ( c_2Eseq_2Esummable
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Eabs @ ( V0f @ V4n ) ) @ ( c_2Ereal_2Epow @ V2z @ V4n ) ) ) ) ).
thf(thm_2Epowser_2EPOWSER__INSIDE,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Eseq_2Esummable
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( V0f @ V3n ) @ ( c_2Ereal_2Epow @ V1x @ V3n ) ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ V2z ) @ ( c_2Ereal_2Eabs @ V1x ) ) )
=> ( c_2Eseq_2Esummable
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( V0f @ V4n ) @ ( c_2Ereal_2Epow @ V2z @ V4n ) ) ) ) ).
thf(thm_2Epowser_2EDIFFS__NEG,axiom,
! [V0c: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Epowser_2Ediffs
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__neg @ ( V0c @ V1n ) ) )
= ( ^ [V2n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__neg @ ( c_2Epowser_2Ediffs @ V0c @ V2n ) ) ) ) ).
thf(thm_2Epowser_2EDIFFS__LEMMA,axiom,
! [V0n: tyop_2Enum_2Enum,V1c: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V0n )
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( c_2Epowser_2Ediffs @ V1c @ V3n ) @ ( c_2Ereal_2Epow @ V2x @ V3n ) ) )
= ( c_2Erealax_2Ereal__add
@ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V0n )
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V4n ) @ ( c_2Erealax_2Ereal__mul @ ( V1c @ V4n ) @ ( c_2Ereal_2Epow @ V2x @ ( c_2Earithmetic_2E_2D @ V4n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) )
@ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V0n ) @ ( c_2Erealax_2Ereal__mul @ ( V1c @ V0n ) @ ( c_2Ereal_2Epow @ V2x @ ( c_2Earithmetic_2E_2D @ V0n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(thm_2Epowser_2EDIFFS__LEMMA2,axiom,
! [V0n: tyop_2Enum_2Enum,V1c: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V0n )
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V3n ) @ ( c_2Erealax_2Ereal__mul @ ( V1c @ V3n ) @ ( c_2Ereal_2Epow @ V2x @ ( c_2Earithmetic_2E_2D @ V3n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) )
= ( c_2Ereal_2Ereal__sub
@ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V0n )
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( c_2Epowser_2Ediffs @ V1c @ V4n ) @ ( c_2Ereal_2Epow @ V2x @ V4n ) ) )
@ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V0n ) @ ( c_2Erealax_2Ereal__mul @ ( V1c @ V0n ) @ ( c_2Ereal_2Epow @ V2x @ ( c_2Earithmetic_2E_2D @ V0n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(thm_2Epowser_2EDIFFS__EQUIV,axiom,
! [V0c: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal] :
( ( c_2Eseq_2Esummable
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( c_2Epowser_2Ediffs @ V0c @ V2n ) @ ( c_2Ereal_2Epow @ V1x @ V2n ) ) )
=> ( c_2Eseq_2Esums
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V3n ) @ ( c_2Erealax_2Ereal__mul @ ( V0c @ V3n ) @ ( c_2Ereal_2Epow @ V1x @ ( c_2Earithmetic_2E_2D @ V3n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
@ ( c_2Eseq_2Esuminf
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( c_2Epowser_2Ediffs @ V0c @ V4n ) @ ( c_2Ereal_2Epow @ V1x @ V4n ) ) ) ) ) ).
thf(thm_2Epowser_2ETERMDIFF__LEMMA1,axiom,
! [V0m: tyop_2Enum_2Enum,V1z: tyop_2Erealax_2Ereal,V2h: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V0m )
@ ^ [V3p: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__sub @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__add @ V1z @ V2h ) @ ( c_2Earithmetic_2E_2D @ V0m @ V3p ) ) @ ( c_2Ereal_2Epow @ V1z @ V3p ) ) @ ( c_2Ereal_2Epow @ V1z @ V0m ) ) )
= ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V0m )
@ ^ [V4p: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Epow @ V1z @ V4p ) @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__add @ V1z @ V2h ) @ ( c_2Earithmetic_2E_2D @ V0m @ V4p ) ) @ ( c_2Ereal_2Epow @ V1z @ ( c_2Earithmetic_2E_2D @ V0m @ V4p ) ) ) ) ) ) ).
thf(thm_2Epowser_2ETERMDIFF__LEMMA2,axiom,
! [V0z: tyop_2Erealax_2Ereal,V1h: tyop_2Erealax_2Ereal,V2n: tyop_2Enum_2Enum] :
( ( (~)
@ ( V1h
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__add @ V0z @ V1h ) @ V2n ) @ ( c_2Ereal_2Epow @ V0z @ V2n ) ) @ V1h ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V2n ) @ ( c_2Ereal_2Epow @ V0z @ ( c_2Earithmetic_2E_2D @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
= ( c_2Erealax_2Ereal__mul @ V1h
@ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ ( c_2Earithmetic_2E_2D @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
@ ^ [V3p: tyop_2Enum_2Enum] :
( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Epow @ V0z @ V3p )
@ ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2D @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V3p ) )
@ ^ [V4q: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__add @ V0z @ V1h ) @ V4q ) @ ( c_2Ereal_2Epow @ V0z @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2D @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V3p ) @ V4q ) ) ) ) ) ) ) ) ) ).
thf(thm_2Epowser_2ETERMDIFF__LEMMA3,axiom,
! [V0z: tyop_2Erealax_2Ereal,V1h: tyop_2Erealax_2Ereal,V2n: tyop_2Enum_2Enum,V3k_27: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V1h
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ V0z ) @ V3k_27 )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Erealax_2Ereal__add @ V0z @ V1h ) ) @ V3k_27 ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2E_2F @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__add @ V0z @ V1h ) @ V2n ) @ ( c_2Ereal_2Epow @ V0z @ V2n ) ) @ V1h ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V2n ) @ ( c_2Ereal_2Epow @ V0z @ ( c_2Earithmetic_2E_2D @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V2n ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2E_2D @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Epow @ V3k_27 @ ( c_2Earithmetic_2E_2D @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Ereal_2Eabs @ V1h ) ) ) ) ) ) ).
thf(thm_2Epowser_2ETERMDIFF__LEMMA4,axiom,
! [V0f: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1k_27: tyop_2Erealax_2Ereal,V2k: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2k )
& ! [V3h: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Eabs @ V3h ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ V3h ) @ V2k ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( V0f @ V3h ) ) @ ( c_2Erealax_2Ereal__mul @ V1k_27 @ ( c_2Ereal_2Eabs @ V3h ) ) ) ) )
=> ( c_2Elim_2Etends__real__real @ V0f @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Epowser_2ETERMDIFF__LEMMA5,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Erealax_2Ereal > tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2k: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2k )
& ( c_2Eseq_2Esummable @ V0f )
& ! [V3h: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Eabs @ V3h ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ V3h ) @ V2k ) )
=> ! [V4n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( V1g @ V3h @ V4n ) ) @ ( c_2Erealax_2Ereal__mul @ ( V0f @ V4n ) @ ( c_2Ereal_2Eabs @ V3h ) ) ) ) )
=> ( c_2Elim_2Etends__real__real
@ ^ [V5h: tyop_2Erealax_2Ereal] : ( c_2Eseq_2Esuminf @ ( V1g @ V5h ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Epowser_2ETERMDIFF,axiom,
! [V0c: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1k_27: tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ( ( c_2Eseq_2Esummable
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( V0c @ V3n ) @ ( c_2Ereal_2Epow @ V1k_27 @ V3n ) ) )
& ( c_2Eseq_2Esummable
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( c_2Epowser_2Ediffs @ V0c @ V4n ) @ ( c_2Ereal_2Epow @ V1k_27 @ V4n ) ) )
& ( c_2Eseq_2Esummable
@ ^ [V5n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( c_2Epowser_2Ediffs @ ( c_2Epowser_2Ediffs @ V0c ) @ V5n ) @ ( c_2Ereal_2Epow @ V1k_27 @ V5n ) ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ V2x ) @ ( c_2Ereal_2Eabs @ V1k_27 ) ) )
=> ( c_2Elim_2Ediffl
@ ^ [V6x: tyop_2Erealax_2Ereal] :
( c_2Eseq_2Esuminf
@ ^ [V7n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( V0c @ V7n ) @ ( c_2Ereal_2Epow @ V6x @ V7n ) ) )
@ ( c_2Eseq_2Esuminf
@ ^ [V8n: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( c_2Epowser_2Ediffs @ V0c @ V8n ) @ ( c_2Ereal_2Epow @ V2x @ V8n ) ) )
@ V2x ) ) ).
%------------------------------------------------------------------------------