ITP001 Axioms: ITP144^7.ax
%------------------------------------------------------------------------------
% File : ITP144^7 : TPTP v8.2.0. 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 : iterate.ax [Gau19]
% : HL4144^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 419 ( 76 unt; 72 typ; 0 def)
% Number of atoms : 1936 ( 420 equ; 77 cnn)
% Maximal formula atoms : 20 ( 4 avg)
% Number of connectives : 8428 ( 77 ~; 6 |; 386 &;7433 @)
% ( 49 <=>; 477 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 10 avg;7433 nst)
% Number of types : 4 ( 3 usr)
% Number of type conns : 1018 (1018 >; 0 *; 0 +; 0 <<)
% Number of symbols : 71 ( 69 usr; 3 con; 0-7 aty)
% Number of variables : 1895 ( 216 ^1576 !; 59 ?;1895 :)
% ( 44 !>; 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_2Earithmetic_2E_2A,type,
c_2Earithmetic_2E_2A: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2E_2B,type,
c_2Earithmetic_2E_2B: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
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_2Eiterate_2E_2E_2E,type,
c_2Eiterate_2E_2E_2E: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
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_2Eprim__rec_2E_3C,type,
c_2Eprim__rec_2E_3C: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Earithmetic_2E_3C_3D,type,
c_2Earithmetic_2E_3C_3D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
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_2Ebool_2E_3F_21,type,
c_2Ebool_2E_3F_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Emin_2E_40,type,
c_2Emin_2E_40:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a ) ).
thf(c_2Epred__set_2EBIGUNION,type,
c_2Epred__set_2EBIGUNION:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > A_27a > $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_2Epred__set_2ECARD,type,
c_2Epred__set_2ECARD:
!>[A_27a: $tType] : ( ( A_27a > $o ) > tyop_2Enum_2Enum ) ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Epred__set_2EDELETE,type,
c_2Epred__set_2EDELETE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Epred__set_2EDIFF,type,
c_2Epred__set_2EDIFF:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EDISJOINT,type,
c_2Epred__set_2EDISJOINT:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Earithmetic_2EDIV,type,
c_2Earithmetic_2EDIV: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epred__set_2EEMPTY,type,
c_2Epred__set_2EEMPTY:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Epred__set_2EFINITE,type,
c_2Epred__set_2EFINITE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Eiterate_2EFINREC,type,
c_2Eiterate_2EFINREC:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > A_27b ) > A_27b > ( A_27a > $o ) > A_27b > tyop_2Enum_2Enum > $o ) ).
thf(c_2Epred__set_2EGSPEC,type,
c_2Epred__set_2EGSPEC:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > ( tyop_2Epair_2Eprod @ A_27a @ $o ) ) > A_27a > $o ) ).
thf(c_2Ecardinal_2EHAS__SIZE,type,
c_2Ecardinal_2EHAS__SIZE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > tyop_2Enum_2Enum > $o ) ).
thf(c_2Epred__set_2EIMAGE,type,
c_2Epred__set_2EIMAGE:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > A_27b > $o ) ).
thf(c_2Ebool_2EIN,type,
c_2Ebool_2EIN:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EINSERT,type,
c_2Epred__set_2EINSERT:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EINTER,type,
c_2Epred__set_2EINTER:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Eiterate_2EITSET,type,
c_2Eiterate_2EITSET:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > A_27a > A_27a ) > ( A_27b > $o ) > A_27a > A_27a ) ).
thf(c_2Earithmetic_2EMOD,type,
c_2Earithmetic_2EMOD: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epred__set_2ESUBSET,type,
c_2Epred__set_2ESUBSET:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Enum_2ESUC,type,
c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Eiterate_2ESum,type,
c_2Eiterate_2ESum:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ) ).
thf(c_2Epair_2EUNCURRY,type,
c_2Epair_2EUNCURRY:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27c ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c ) ).
thf(c_2Epred__set_2EUNION,type,
c_2Epred__set_2EUNION:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EUNIV,type,
c_2Epred__set_2EUNIV:
!>[A_27a: $tType] : ( A_27a > $o ) ).
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_2Eiterate_2Einf,type,
c_2Eiterate_2Einf: ( tyop_2Erealax_2Ereal > $o ) > tyop_2Erealax_2Ereal ).
thf(c_2Eiterate_2Eiterate,type,
c_2Eiterate_2Eiterate:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > A_27b > A_27b ) > ( A_27a > $o ) > ( A_27a > A_27b ) > A_27b ) ).
thf(c_2Ereal_2Emax,type,
c_2Ereal_2Emax: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Emin,type,
c_2Ereal_2Emin: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Eiterate_2Emonoidal,type,
c_2Eiterate_2Emonoidal:
!>[A_27a: $tType] : ( ( A_27a > A_27a > A_27a ) > $o ) ).
thf(c_2Eiterate_2Eneutral,type,
c_2Eiterate_2Eneutral:
!>[A_27a: $tType] : ( ( A_27a > A_27a > A_27a ) > A_27a ) ).
thf(c_2Eiterate_2Ensum,type,
c_2Eiterate_2Ensum:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ) ).
thf(c_2Ecombin_2Eo,type,
c_2Ecombin_2Eo:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27c > A_27b ) > ( A_27a > A_27c ) > A_27a > A_27b ) ).
thf(c_2Eiterate_2Epolynomial__function,type,
c_2Eiterate_2Epolynomial__function: ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) > $o ).
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_2Ereal_2Ereal__gt,type,
c_2Ereal_2Ereal__gt: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).
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_2Esup,type,
c_2Ereal_2Esup: ( tyop_2Erealax_2Ereal > $o ) > tyop_2Erealax_2Ereal ).
thf(c_2Eiterate_2Esupport,type,
c_2Eiterate_2Esupport:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > A_27b > A_27b ) > ( A_27a > A_27b ) > ( A_27a > $o ) > A_27a > $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_2Eiterate_2Esum__def,axiom,
! [A_27a: $tType] :
( ( c_2Eiterate_2ESum @ A_27a )
= ( c_2Eiterate_2Eiterate @ A_27a @ tyop_2Erealax_2Ereal @ c_2Erealax_2Ereal__add ) ) ).
thf(thm_2Eiterate_2EITSET__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27b > A_27a > A_27a,V1s: A_27b > $o,V2b: A_27a] :
( ( c_2Eiterate_2EITSET @ A_27a @ A_27b @ V0f @ V1s @ V2b )
= ( c_2Emin_2E_40 @ ( ( A_27b > $o ) > A_27a )
@ ^ [V3g: ( A_27b > $o ) > A_27a] :
( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ A_27a @ ( V3g @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) @ V2b )
@ ( c_2Ebool_2E_21 @ A_27b
@ ^ [V4x: A_27b] :
( c_2Ebool_2E_21 @ ( A_27b > $o )
@ ^ [V5s: A_27b > $o] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Epred__set_2EFINITE @ A_27b @ V5s ) @ ( c_2Emin_2E_3D @ A_27a @ ( V3g @ ( c_2Epred__set_2EINSERT @ A_27b @ V4x @ V5s ) ) @ ( c_2Ebool_2ECOND @ A_27a @ ( c_2Ebool_2EIN @ A_27b @ V4x @ V5s ) @ ( V3g @ V5s ) @ ( V0f @ V4x @ ( V3g @ V5s ) ) ) ) ) ) ) )
@ V1s ) ) ).
thf(thm_2Eiterate_2EFINREC__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0f: A_27a > A_27b > A_27b,V1b: A_27b,V2s: A_27a > $o,V3a: A_27b] :
( ( c_2Eiterate_2EFINREC @ A_27a @ A_27b @ V0f @ V1b @ V2s @ V3a @ c_2Enum_2E0 )
<=> ( ( V2s
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ( V3a = V1b ) ) )
& ! [V4f: A_27a > A_27b > A_27b,V5b: A_27b,V6s: A_27a > $o,V7a: A_27b,V8n: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2EFINREC @ A_27a @ A_27b @ V4f @ V5b @ V6s @ V7a @ ( c_2Enum_2ESUC @ V8n ) )
<=> ? [V9x: A_27a,V10c: A_27b] :
( ( c_2Ebool_2EIN @ A_27a @ V9x @ V6s )
& ( c_2Eiterate_2EFINREC @ A_27a @ A_27b @ V4f @ V5b @ ( c_2Epred__set_2EDELETE @ A_27a @ V6s @ V9x ) @ V10c @ V8n )
& ( V7a
= ( V4f @ V9x @ V10c ) ) ) ) ) ).
thf(thm_2Eiterate_2Einf,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Eiterate_2Einf @ V0s )
= ( c_2Emin_2E_40 @ tyop_2Erealax_2Ereal
@ ^ [V1a: tyop_2Erealax_2Ereal] :
( c_2Ebool_2E_2F_5C
@ ( c_2Ebool_2E_21 @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s ) @ ( c_2Ereal_2Ereal__lte @ V1a @ V2x ) ) )
@ ( c_2Ebool_2E_21 @ tyop_2Erealax_2Ereal
@ ^ [V3b: tyop_2Erealax_2Ereal] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_21 @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V0s ) @ ( c_2Ereal_2Ereal__lte @ V3b @ V4x ) ) )
@ ( c_2Ereal_2Ereal__lte @ V3b @ V1a ) ) ) ) ) ) ).
thf(thm_2Eiterate_2Enumseg,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2E_2E_2E @ V0m @ V1n )
= ( c_2Epred__set_2EGSPEC @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum
@ ^ [V2x: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ $o @ V2x @ ( c_2Ebool_2E_2F_5C @ ( c_2Earithmetic_2E_3C_3D @ V0m @ V2x ) @ ( c_2Earithmetic_2E_3C_3D @ V2x @ V1n ) ) ) ) ) ).
thf(thm_2Eiterate_2Eneutral,axiom,
! [A_27a: $tType,V0op: A_27a > A_27a > A_27a] :
( ( c_2Eiterate_2Eneutral @ A_27a @ V0op )
= ( c_2Emin_2E_40 @ A_27a
@ ^ [V1x: A_27a] :
( c_2Ebool_2E_21 @ A_27a
@ ^ [V2y: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ A_27a @ ( V0op @ V1x @ V2y ) @ V2y ) @ ( c_2Emin_2E_3D @ A_27a @ ( V0op @ V2y @ V1x ) @ V2y ) ) ) ) ) ).
thf(thm_2Eiterate_2Emonoidal,axiom,
! [A_27a: $tType,V0op: A_27a > A_27a > A_27a] :
( ( c_2Eiterate_2Emonoidal @ A_27a @ V0op )
<=> ( ! [V1x: A_27a,V2y: A_27a] :
( ( V0op @ V1x @ V2y )
= ( V0op @ V2y @ V1x ) )
& ! [V3x: A_27a,V4y: A_27a,V5z: A_27a] :
( ( V0op @ V3x @ ( V0op @ V4y @ V5z ) )
= ( V0op @ ( V0op @ V3x @ V4y ) @ V5z ) )
& ! [V6x: A_27a] :
( ( V0op @ ( c_2Eiterate_2Eneutral @ A_27a @ V0op ) @ V6x )
= V6x ) ) ) ).
thf(thm_2Eiterate_2Esupport,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27b > A_27b > A_27b,V1f: A_27a > A_27b,V2s: A_27a > $o] :
( ( c_2Eiterate_2Esupport @ A_27a @ A_27b @ V0op @ V1f @ V2s )
= ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V3x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V3x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s ) @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ A_27b @ ( V1f @ V3x ) @ ( c_2Eiterate_2Eneutral @ A_27b @ V0op ) ) ) ) ) ) ) ).
thf(thm_2Eiterate_2Eiterate,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27b > A_27b > A_27b,V1s: A_27a > $o,V2f: A_27a > A_27b] :
( ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ V1s @ V2f )
= ( c_2Ebool_2ECOND @ A_27b @ ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Eiterate_2Esupport @ A_27a @ A_27b @ V0op @ V2f @ V1s ) )
@ ( c_2Eiterate_2EITSET @ A_27b @ A_27a
@ ^ [V3x: A_27a,V4a: A_27b] : ( V0op @ ( V2f @ V3x ) @ V4a )
@ ( c_2Eiterate_2Esupport @ A_27a @ A_27b @ V0op @ V2f @ V1s )
@ ( c_2Eiterate_2Eneutral @ A_27b @ V0op ) )
@ ( c_2Eiterate_2Eneutral @ A_27b @ V0op ) ) ) ).
thf(thm_2Eiterate_2Ensum__def,axiom,
! [A_27a: $tType] :
( ( c_2Eiterate_2Ensum @ A_27a )
= ( c_2Eiterate_2Eiterate @ A_27a @ tyop_2Enum_2Enum @ c_2Earithmetic_2E_2B ) ) ).
thf(thm_2Eiterate_2Epolynomial__function,axiom,
! [V0p: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Eiterate_2Epolynomial__function @ V0p )
<=> ? [V1m: tyop_2Enum_2Enum,V2c: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
! [V3x: tyop_2Erealax_2Ereal] :
( ( V0p @ V3x )
= ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ V1m )
@ ^ [V4i: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( V2c @ V4i ) @ ( c_2Ereal_2Epow @ V3x @ V4i ) ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__COMBINE__L,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum,V3p: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V2n )
& ( c_2Earithmetic_2E_3C_3D @ V1m @ V2n )
& ( c_2Earithmetic_2E_3C_3D @ V2n @ ( c_2Earithmetic_2E_2B @ V3p @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( ( c_2Erealax_2Ereal__add @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ ( c_2Earithmetic_2E_2D @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0f ) @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2n @ V3p ) @ V0f ) )
= ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V3p ) @ V0f ) ) ) ).
thf(thm_2Eiterate_2ESUM__COMBINE__R,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum,V3p: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V1m @ ( c_2Earithmetic_2E_2B @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
& ( c_2Earithmetic_2E_3C_3D @ V2n @ V3p ) )
=> ( ( c_2Erealax_2Ereal__add @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f ) @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2E_2B @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V3p ) @ V0f ) )
= ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V3p ) @ V0f ) ) ) ).
thf(thm_2Eiterate_2ESUM__DIFFS__ALT,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n )
@ ^ [V3k: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__sub @ ( V0f @ ( c_2Earithmetic_2E_2B @ V3k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( V0f @ V3k ) ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Earithmetic_2E_3C_3D @ V1m @ V2n ) @ ( c_2Ereal_2Ereal__sub @ ( V0f @ ( c_2Earithmetic_2E_2B @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( V0f @ V1m ) ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eiterate_2ESUM__DIFFS,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n )
@ ^ [V3k: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__sub @ ( V0f @ V3k ) @ ( V0f @ ( c_2Earithmetic_2E_2B @ V3k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Earithmetic_2E_3C_3D @ V1m @ V2n ) @ ( c_2Ereal_2Ereal__sub @ ( V0f @ V1m ) @ ( V0f @ ( c_2Earithmetic_2E_2B @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eiterate_2ESUM__PARTIAL__PRE,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n )
@ ^ [V4k: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( V0f @ V4k ) @ ( c_2Ereal_2Ereal__sub @ ( V1g @ V4k ) @ ( V1g @ ( c_2Earithmetic_2E_2D @ V4k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Earithmetic_2E_3C_3D @ V2m @ V3n )
@ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__sub @ ( c_2Erealax_2Ereal__mul @ ( V0f @ ( c_2Earithmetic_2E_2B @ V3n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( V1g @ V3n ) ) @ ( c_2Erealax_2Ereal__mul @ ( V0f @ V2m ) @ ( V1g @ ( c_2Earithmetic_2E_2D @ V2m @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
@ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n )
@ ^ [V5k: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( V1g @ V5k ) @ ( c_2Ereal_2Ereal__sub @ ( V0f @ ( c_2Earithmetic_2E_2B @ V5k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( V0f @ V5k ) ) ) ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eiterate_2ESUM__PARTIAL__SUC,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n )
@ ^ [V4k: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( V0f @ V4k ) @ ( c_2Ereal_2Ereal__sub @ ( V1g @ ( c_2Earithmetic_2E_2B @ V4k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( V1g @ V4k ) ) ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Earithmetic_2E_3C_3D @ V2m @ V3n )
@ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__sub @ ( c_2Erealax_2Ereal__mul @ ( V0f @ ( c_2Earithmetic_2E_2B @ V3n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( V1g @ ( c_2Earithmetic_2E_2B @ V3n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ ( c_2Erealax_2Ereal__mul @ ( V0f @ V2m ) @ ( V1g @ V2m ) ) )
@ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n )
@ ^ [V5k: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( V1g @ ( c_2Earithmetic_2E_2B @ V5k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Ereal_2Ereal__sub @ ( V0f @ ( c_2Earithmetic_2E_2B @ V5k @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( V0f @ V5k ) ) ) ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eiterate_2EREAL__OF__NUM__SUM__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Ereal_2Ereal__of__num @ ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f ) )
= ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n )
@ ^ [V3i: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__of__num @ ( V0f @ V3i ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__PAIR,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1m ) @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V2n ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0f )
= ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n )
@ ^ [V3i: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__add @ ( V0f @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V3i ) ) @ ( V0f @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V3i ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__CLAUSES__RIGHT,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V2n )
& ( c_2Earithmetic_2E_3C_3D @ V1m @ V2n ) )
=> ( ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f )
= ( c_2Erealax_2Ereal__add @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ ( c_2Earithmetic_2E_2D @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0f ) @ ( V0f @ V2n ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__CLAUSES__LEFT,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1m @ V2n )
=> ( ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f )
= ( c_2Erealax_2Ereal__add @ ( V0f @ V1m ) @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2E_2B @ V1m @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2n ) @ V0f ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__OFFSET__0,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1m @ V2n )
=> ( ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f )
= ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ ( c_2Earithmetic_2E_2D @ V2n @ V1m ) )
@ ^ [V3i: tyop_2Enum_2Enum] : ( V0f @ ( c_2Earithmetic_2E_2B @ V3i @ V1m ) ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__OFFSET,axiom,
! [V0p: tyop_2Enum_2Enum,V1f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2E_2B @ V2m @ V0p ) @ ( c_2Earithmetic_2E_2B @ V3n @ V0p ) ) @ V1f )
= ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n )
@ ^ [V4i: tyop_2Enum_2Enum] : ( V1f @ ( c_2Earithmetic_2E_2B @ V4i @ V0p ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__ADD__SPLIT,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum,V3p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1m @ ( c_2Earithmetic_2E_2B @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
=> ( ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ ( c_2Earithmetic_2E_2B @ V2n @ V3p ) ) @ V0f )
= ( c_2Erealax_2Ereal__add @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f ) @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2E_2B @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2E_2B @ V2n @ V3p ) ) @ V0f ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__SWAP__NUMSEG,axiom,
! [V0a: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum,V2c: tyop_2Enum_2Enum,V3d: tyop_2Enum_2Enum,V4f: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V0a @ V1b )
@ ^ [V5i: tyop_2Enum_2Enum] : ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2c @ V3d ) @ ( V4f @ V5i ) ) )
= ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2c @ V3d )
@ ^ [V6j: tyop_2Enum_2Enum] :
( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V0a @ V1b )
@ ^ [V7i: tyop_2Enum_2Enum] : ( V4f @ V7i @ V6j ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__CLAUSES__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ! [V1m: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ c_2Enum_2E0 ) @ V0f )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V1m @ c_2Enum_2E0 ) @ ( V0f @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
& ! [V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ ( c_2Enum_2ESUC @ V3n ) ) @ V0f )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Earithmetic_2E_3C_3D @ V2m @ ( c_2Enum_2ESUC @ V3n ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V0f ) @ ( V0f @ ( c_2Enum_2ESUC @ V3n ) ) ) @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V0f ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__SING__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1n @ V1n ) @ V0f )
= ( V0f @ V1n ) ) ).
thf(thm_2Eiterate_2ESUM__POS__EQ__0__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( ! [V3p: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V1m @ V3p )
& ( c_2Earithmetic_2E_3C_3D @ V3p @ V2n ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V0f @ V3p ) ) )
& ( ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ! [V4p: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V1m @ V4p )
& ( c_2Earithmetic_2E_3C_3D @ V4p @ V2n ) )
=> ( ( V0f @ V4p )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__POS__LE__NUMSEG,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ! [V3p: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V0m @ V3p )
& ( c_2Earithmetic_2E_3C_3D @ V3p @ V1n ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V2f @ V3p ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V0m @ V1n ) @ V2f ) ) ) ).
thf(thm_2Eiterate_2ESUM__TRIV__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V2n @ V1m )
=> ( ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eiterate_2ESUM__EQ__0__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ! [V3i: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V1m @ V3i )
& ( c_2Earithmetic_2E_3C_3D @ V3i @ V2n ) )
=> ( ( V0f @ V3i )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eiterate_2ESUM__CONST__NUMSEG,axiom,
! [V0c: tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n )
@ ^ [V3n: tyop_2Enum_2Enum] : V0c )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2B @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V1m ) ) @ V0c ) ) ).
thf(thm_2Eiterate_2ESUM__ABS__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f ) )
@ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n )
@ ^ [V3i: tyop_2Enum_2Enum] : ( c_2Ereal_2Eabs @ ( V0f @ V3i ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__EQ__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ! [V4i: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V2m @ V4i )
& ( c_2Earithmetic_2E_3C_3D @ V4i @ V3n ) )
=> ( ( V0f @ V4i )
= ( V1g @ V4i ) ) )
=> ( ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V0f )
= ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V1g ) ) ) ).
thf(thm_2Eiterate_2ESUM__LE__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ! [V4i: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V2m @ V4i )
& ( c_2Earithmetic_2E_3C_3D @ V4i @ V3n ) )
=> ( c_2Ereal_2Ereal__lte @ ( V0f @ V4i ) @ ( V1g @ V4i ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V0f ) @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V1g ) ) ) ).
thf(thm_2Eiterate_2ESUM__SUB__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n )
@ ^ [V4i: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__sub @ ( V0f @ V4i ) @ ( V1g @ V4i ) ) )
= ( c_2Ereal_2Ereal__sub @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V0f ) @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V1g ) ) ) ).
thf(thm_2Eiterate_2ESUM__ADD__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n )
@ ^ [V4i: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__add @ ( V0f @ V4i ) @ ( V1g @ V4i ) ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V0f ) @ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V1g ) ) ) ).
thf(thm_2Eiterate_2ESUM__CLOSED,axiom,
! [A_27a: $tType,V0P: tyop_2Erealax_2Ereal > $o,V1f: A_27a > tyop_2Erealax_2Ereal,V2s: A_27a > $o] :
( ( ( V0P @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
& ! [V3x: tyop_2Erealax_2Ereal,V4y: tyop_2Erealax_2Ereal] :
( ( ( V0P @ V3x )
& ( V0P @ V4y ) )
=> ( V0P @ ( c_2Erealax_2Ereal__add @ V3x @ V4y ) ) )
& ! [V5a: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V5a @ V2s )
=> ( V0P @ ( V1f @ V5a ) ) ) )
=> ( V0P @ ( c_2Eiterate_2ESum @ A_27a @ V2s @ V1f ) ) ) ).
thf(thm_2Eiterate_2ESUM__IMAGE__LE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1g: A_27b > tyop_2Erealax_2Ereal,V2s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V1g @ ( V0f @ V3x ) ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eiterate_2ESum @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V2s ) @ V1g ) @ ( c_2Eiterate_2ESum @ A_27a @ V2s @ ( c_2Ecombin_2Eo @ A_27a @ tyop_2Erealax_2Ereal @ A_27b @ V1g @ V0f ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__LE__INCLUDED,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1g: A_27b > tyop_2Erealax_2Ereal,V2s: A_27a > $o,V3t: A_27b > $o,V4i: A_27b > A_27a] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
& ( c_2Epred__set_2EFINITE @ A_27b @ V3t )
& ! [V5y: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V5y @ V3t )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V1g @ V5y ) ) )
& ! [V6x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V6x @ V2s )
=> ? [V7y: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V7y @ V3t )
& ( ( V4i @ V7y )
= V6x )
& ( c_2Ereal_2Ereal__lte @ ( V0f @ V6x ) @ ( V1g @ V7y ) ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eiterate_2ESum @ A_27a @ V2s @ V0f ) @ ( c_2Eiterate_2ESum @ A_27b @ V3t @ V1g ) ) ) ).
thf(thm_2Eiterate_2ESUM__CASES__1,axiom,
! [A_27a: $tType,V0y: tyop_2Erealax_2Ereal,V1f: A_27a > tyop_2Erealax_2Ereal,V2s: A_27a > $o,V3a: A_27a] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
& ( c_2Ebool_2EIN @ A_27a @ V3a @ V2s ) )
=> ( ( c_2Eiterate_2ESum @ A_27a @ V2s
@ ^ [V4x: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ A_27a @ V4x @ V3a ) @ V0y @ ( V1f @ V4x ) ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Eiterate_2ESum @ A_27a @ V2s @ V1f ) @ ( c_2Ereal_2Ereal__sub @ V0y @ ( V1f @ V3a ) ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__CASES,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1P: A_27a > $o,V2f: A_27a > tyop_2Erealax_2Ereal,V3g: A_27a > tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ( ( c_2Eiterate_2ESum @ A_27a @ V0s
@ ^ [V4x: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( V1P @ V4x ) @ ( V2f @ V4x ) @ ( V3g @ V4x ) ) )
= ( c_2Erealax_2Ereal__add
@ ( c_2Eiterate_2ESum @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V5x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V5x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V5x @ V0s ) @ ( V1P @ V5x ) ) ) )
@ V2f )
@ ( c_2Eiterate_2ESum @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V6x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V6x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V6x @ V0s ) @ ( c_2Ebool_2E_7E @ ( V1P @ V6x ) ) ) ) )
@ V3g ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__BIGUNION__NONZERO,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: ( A_27a > $o ) > $o] :
( ( ( c_2Epred__set_2EFINITE @ ( A_27a > $o ) @ V1s )
& ! [V2t: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ V1s )
=> ( c_2Epred__set_2EFINITE @ A_27a @ V2t ) )
& ! [V3t1: A_27a > $o,V4t2: A_27a > $o,V5x: A_27a] :
( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3t1 @ V1s )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V4t2 @ V1s )
& ( (~) @ ( V3t1 = V4t2 ) )
& ( c_2Ebool_2EIN @ A_27a @ V5x @ V3t1 )
& ( c_2Ebool_2EIN @ A_27a @ V5x @ V4t2 ) )
=> ( ( V0f @ V5x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Eiterate_2ESum @ A_27a @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V1s ) @ V0f )
= ( c_2Eiterate_2ESum @ ( A_27a > $o ) @ V1s
@ ^ [V6t: A_27a > $o] : ( c_2Eiterate_2ESum @ A_27a @ V6t @ V0f ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__UNION__NONZERO,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( c_2Epred__set_2EFINITE @ A_27a @ V2t )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Epred__set_2EINTER @ A_27a @ V1s @ V2t ) )
=> ( ( V0f @ V3x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Eiterate_2ESum @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) @ V0f )
= ( c_2Erealax_2Ereal__add @ ( c_2Eiterate_2ESum @ A_27a @ V1s @ V0f ) @ ( c_2Eiterate_2ESum @ A_27a @ V2t @ V0f ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__INJECTION,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1p: A_27a > A_27a,V2s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
=> ( c_2Ebool_2EIN @ A_27a @ ( V1p @ V3x ) @ V2s ) )
& ! [V4x: A_27a,V5y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V4x @ V2s )
& ( c_2Ebool_2EIN @ A_27a @ V5y @ V2s )
& ( ( V1p @ V4x )
= ( V1p @ V5y ) ) )
=> ( V4x = V5y ) ) )
=> ( ( c_2Eiterate_2ESum @ A_27a @ V2s @ ( c_2Ecombin_2Eo @ A_27a @ tyop_2Erealax_2Ereal @ A_27a @ V0f @ V1p ) )
= ( c_2Eiterate_2ESum @ A_27a @ V2s @ V0f ) ) ) ).
thf(thm_2Eiterate_2ESUM__EQ__GENERAL__INVERSES,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o,V2f: A_27a > tyop_2Erealax_2Ereal,V3g: A_27b > tyop_2Erealax_2Ereal,V4h: A_27a > A_27b,V5k: A_27b > A_27a] :
( ( ! [V6y: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V6y @ V1t )
=> ( ( c_2Ebool_2EIN @ A_27a @ ( V5k @ V6y ) @ V0s )
& ( ( V4h @ ( V5k @ V6y ) )
= V6y ) ) )
& ! [V7x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V7x @ V0s )
=> ( ( c_2Ebool_2EIN @ A_27b @ ( V4h @ V7x ) @ V1t )
& ( ( V5k @ ( V4h @ V7x ) )
= V7x )
& ( ( V3g @ ( V4h @ V7x ) )
= ( V2f @ V7x ) ) ) ) )
=> ( ( c_2Eiterate_2ESum @ A_27a @ V0s @ V2f )
= ( c_2Eiterate_2ESum @ A_27b @ V1t @ V3g ) ) ) ).
thf(thm_2Eiterate_2ESUM__EQ__GENERAL,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o,V2f: A_27a > tyop_2Erealax_2Ereal,V3g: A_27b > tyop_2Erealax_2Ereal,V4h: A_27a > A_27b] :
( ( ! [V5y: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V5y @ V1t )
=> ( c_2Ebool_2E_3F_21 @ A_27a
@ ^ [V6x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V6x @ V0s ) @ ( c_2Emin_2E_3D @ A_27b @ ( V4h @ V6x ) @ V5y ) ) ) )
& ! [V7x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V7x @ V0s )
=> ( ( c_2Ebool_2EIN @ A_27b @ ( V4h @ V7x ) @ V1t )
& ( ( V3g @ ( V4h @ V7x ) )
= ( V2f @ V7x ) ) ) ) )
=> ( ( c_2Eiterate_2ESum @ A_27a @ V0s @ V2f )
= ( c_2Eiterate_2ESum @ A_27b @ V1t @ V3g ) ) ) ).
thf(thm_2Eiterate_2ESUM__SUM__PRODUCT,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27a > A_27b > $o,V2x: A_27a > A_27b > tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ! [V3i: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3i @ V0s )
=> ( c_2Epred__set_2EFINITE @ A_27b @ ( V1t @ V3i ) ) ) )
=> ( ( c_2Eiterate_2ESum @ A_27a @ V0s
@ ^ [V4i: A_27a] : ( c_2Eiterate_2ESum @ A_27b @ ( V1t @ V4i ) @ ( V2x @ V4i ) ) )
= ( c_2Eiterate_2ESum @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( tyop_2Epair_2Eprod @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ $o )
@ ^ [V5i: A_27a,V6j: A_27b] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ $o @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V5i @ V6j ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V5i @ V0s ) @ ( c_2Ebool_2EIN @ A_27b @ V6j @ ( V1t @ V5i ) ) ) ) ) )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ tyop_2Erealax_2Ereal
@ ^ [V7i: A_27a,V8j: A_27b] : ( V2x @ V7i @ V8j ) ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__BIJECTION,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1p: A_27a > A_27a,V2s: A_27a > $o] :
( ( ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
=> ( c_2Ebool_2EIN @ A_27a @ ( V1p @ V3x ) @ V2s ) )
& ! [V4y: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4y @ V2s )
=> ( c_2Ebool_2E_3F_21 @ A_27a
@ ^ [V5x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V5x @ V2s ) @ ( c_2Emin_2E_3D @ A_27a @ ( V1p @ V5x ) @ V4y ) ) ) ) )
=> ( ( c_2Eiterate_2ESum @ A_27a @ V2s @ V0f )
= ( c_2Eiterate_2ESum @ A_27a @ V2s @ ( c_2Ecombin_2Eo @ A_27a @ tyop_2Erealax_2Ereal @ A_27a @ V0f @ V1p ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__IMAGE__NONZERO,axiom,
! [A_27a: $tType,A_27b: $tType,V0d: A_27b > tyop_2Erealax_2Ereal,V1i: A_27a > A_27b,V2s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
& ! [V3x: A_27a,V4y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
& ( c_2Ebool_2EIN @ A_27a @ V4y @ V2s )
& ( (~) @ ( V3x = V4y ) )
& ( ( V1i @ V3x )
= ( V1i @ V4y ) ) )
=> ( ( V0d @ ( V1i @ V3x ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Eiterate_2ESum @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V1i @ V2s ) @ V0d )
= ( c_2Eiterate_2ESum @ A_27a @ V2s @ ( c_2Ecombin_2Eo @ A_27a @ tyop_2Erealax_2Ereal @ A_27b @ V0d @ V1i ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__SUBSET__SIMPLE,axiom,
! [A_27a: $tType,V0u: A_27a > $o,V1v: A_27a > $o,V2f: A_27a > tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1v )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V0u @ V1v )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Epred__set_2EDIFF @ A_27a @ V1v @ V0u ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V2f @ V3x ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eiterate_2ESum @ A_27a @ V0u @ V2f ) @ ( c_2Eiterate_2ESum @ A_27a @ V1v @ V2f ) ) ) ).
thf(thm_2Eiterate_2ESUM__SUBSET,axiom,
! [A_27a: $tType,V0u: A_27a > $o,V1v: A_27a > $o,V2f: A_27a > tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0u )
& ( c_2Epred__set_2EFINITE @ A_27a @ V1v )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Epred__set_2EDIFF @ A_27a @ V0u @ V1v ) )
=> ( c_2Ereal_2Ereal__lte @ ( V2f @ V3x ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
& ! [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ ( c_2Epred__set_2EDIFF @ A_27a @ V1v @ V0u ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V2f @ V4x ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eiterate_2ESum @ A_27a @ V0u @ V2f ) @ ( c_2Eiterate_2ESum @ A_27a @ V1v @ V2f ) ) ) ).
thf(thm_2Eiterate_2EREAL__OF__NUM__SUM,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ( ( c_2Ereal_2Ereal__of__num @ ( c_2Eiterate_2Ensum @ A_27a @ V1s @ V0f ) )
= ( c_2Eiterate_2ESum @ A_27a @ V1s
@ ^ [V2x: A_27a] : ( c_2Ereal_2Ereal__of__num @ ( V0f @ V2x ) ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__GROUP,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1g: A_27a > tyop_2Erealax_2Ereal,V2s: A_27a > $o,V3t: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
& ( c_2Epred__set_2ESUBSET @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V2s ) @ V3t ) )
=> ( ( c_2Eiterate_2ESum @ A_27b @ V3t
@ ^ [V4y: A_27b] :
( c_2Eiterate_2ESum @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V5x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V5x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V5x @ V2s ) @ ( c_2Emin_2E_3D @ A_27b @ ( V0f @ V5x ) @ V4y ) ) ) )
@ V1g ) )
= ( c_2Eiterate_2ESum @ A_27a @ V2s @ V1g ) ) ) ).
thf(thm_2Eiterate_2ESUM__IMAGE__GEN,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1g: A_27a > tyop_2Erealax_2Ereal,V2s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
=> ( ( c_2Eiterate_2ESum @ A_27a @ V2s @ V1g )
= ( c_2Eiterate_2ESum @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V2s )
@ ^ [V3y: A_27b] :
( c_2Eiterate_2ESum @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V4x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V4x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V4x @ V2s ) @ ( c_2Emin_2E_3D @ A_27b @ ( V0f @ V4x ) @ V3y ) ) ) )
@ V1g ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__MULTICOUNT,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o,V1s: A_27a > $o,V2t: A_27b > $o,V3k: tyop_2Enum_2Enum] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( c_2Epred__set_2EFINITE @ A_27b @ V2t )
& ! [V4j: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V4j @ V2t )
=> ( ( c_2Epred__set_2ECARD @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V5i: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V5i @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V5i @ V1s ) @ ( V0R @ V5i @ V4j ) ) ) ) )
= V3k ) ) )
=> ( ( c_2Eiterate_2ESum @ A_27a @ V1s
@ ^ [V6i: A_27a] :
( c_2Ereal_2Ereal__of__num
@ ( c_2Epred__set_2ECARD @ A_27b
@ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b
@ ^ [V7j: A_27b] : ( c_2Epair_2E_2C @ A_27b @ $o @ V7j @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27b @ V7j @ V2t ) @ ( V0R @ V6i @ V7j ) ) ) ) ) ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2E_2A @ V3k @ ( c_2Epred__set_2ECARD @ A_27b @ V2t ) ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__MULTICOUNT__GEN,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o,V1s: A_27a > $o,V2t: A_27b > $o,V3k: A_27b > tyop_2Enum_2Enum] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( c_2Epred__set_2EFINITE @ A_27b @ V2t )
& ! [V4j: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V4j @ V2t )
=> ( ( c_2Epred__set_2ECARD @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V5i: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V5i @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V5i @ V1s ) @ ( V0R @ V5i @ V4j ) ) ) ) )
= ( V3k @ V4j ) ) ) )
=> ( ( c_2Eiterate_2ESum @ A_27a @ V1s
@ ^ [V6i: A_27a] :
( c_2Ereal_2Ereal__of__num
@ ( c_2Epred__set_2ECARD @ A_27b
@ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b
@ ^ [V7j: A_27b] : ( c_2Epair_2E_2C @ A_27b @ $o @ V7j @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27b @ V7j @ V2t ) @ ( V0R @ V6i @ V7j ) ) ) ) ) ) )
= ( c_2Eiterate_2ESum @ A_27b @ V2t
@ ^ [V8i: A_27b] : ( c_2Ereal_2Ereal__of__num @ ( V3k @ V8i ) ) ) ) ) ).
thf(thm_2Eiterate_2ECARD__EQ__SUM,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ( ( c_2Ereal_2Ereal__of__num @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) )
= ( c_2Eiterate_2ESum @ A_27a @ V0s
@ ^ [V1x: A_27a] : ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__SUM__RESTRICT,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o,V1f: A_27a > A_27b > tyop_2Erealax_2Ereal,V2s: A_27a > $o,V3t: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
& ( c_2Epred__set_2EFINITE @ A_27b @ V3t ) )
=> ( ( c_2Eiterate_2ESum @ A_27a @ V2s
@ ^ [V4x: A_27a] :
( c_2Eiterate_2ESum @ A_27b
@ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b
@ ^ [V5y: A_27b] : ( c_2Epair_2E_2C @ A_27b @ $o @ V5y @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27b @ V5y @ V3t ) @ ( V0R @ V4x @ V5y ) ) ) )
@ ^ [V6y: A_27b] : ( V1f @ V4x @ V6y ) ) )
= ( c_2Eiterate_2ESum @ A_27b @ V3t
@ ^ [V7y: A_27b] :
( c_2Eiterate_2ESum @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V8x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V8x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V8x @ V2s ) @ ( V0R @ V8x @ V7y ) ) ) )
@ ^ [V9x: A_27a] : ( V1f @ V9x @ V7y ) ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__RESTRICT__SET,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1s: A_27a > $o,V2f: A_27a > tyop_2Erealax_2Ereal] :
( ( c_2Eiterate_2ESum @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V3x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V3x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s ) @ ( V0P @ V3x ) ) ) )
@ V2f )
= ( c_2Eiterate_2ESum @ A_27a @ V1s
@ ^ [V4x: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( V0P @ V4x ) @ ( V2f @ V4x ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__EQ__SUPERSET,axiom,
! [A_27a: $tType,V0g: A_27a > tyop_2Erealax_2Ereal,V1f: A_27a > tyop_2Erealax_2Ereal,V2s: A_27a > $o,V3t: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V3t )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V3t @ V2s )
& ! [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ V3t )
=> ( ( V1f @ V4x )
= ( V0g @ V4x ) ) )
& ! [V5x: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V5x @ V2s )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V5x @ V3t ) ) )
=> ( ( V1f @ V5x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Eiterate_2ESum @ A_27a @ V2s @ V1f )
= ( c_2Eiterate_2ESum @ A_27a @ V3t @ V0g ) ) ) ).
thf(thm_2Eiterate_2ESUM__UNION__EQ,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o,V2t: A_27a > $o,V3u: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V3u )
& ( ( c_2Epred__set_2EINTER @ A_27a @ V1s @ V2t )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ( ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t )
= V3u ) )
=> ( ( c_2Erealax_2Ereal__add @ ( c_2Eiterate_2ESum @ A_27a @ V1s @ V0f ) @ ( c_2Eiterate_2ESum @ A_27a @ V2t @ V0f ) )
= ( c_2Eiterate_2ESum @ A_27a @ V3u @ V0f ) ) ) ).
thf(thm_2Eiterate_2ESUM__BOUND__LT__GEN,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1f: A_27a > tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V0s )
=> ( c_2Erealax_2Ereal__lt @ ( V1f @ V3x ) @ ( c_2Ereal_2E_2F @ V2b @ ( c_2Ereal_2Ereal__of__num @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) ) ) ) ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Eiterate_2ESum @ A_27a @ V0s @ V1f ) @ V2b ) ) ).
thf(thm_2Eiterate_2ESUM__BOUND__LT__ALL,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1f: A_27a > tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V0s )
=> ( c_2Erealax_2Ereal__lt @ ( V1f @ V3x ) @ V2b ) ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Eiterate_2ESum @ A_27a @ V0s @ V1f ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) ) @ V2b ) ) ) ).
thf(thm_2Eiterate_2ESUM__BOUND__LT,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1f: A_27a > tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ ( V1f @ V3x ) @ V2b ) )
& ? [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ V0s )
& ( c_2Erealax_2Ereal__lt @ ( V1f @ V4x ) @ V2b ) ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Eiterate_2ESum @ A_27a @ V0s @ V1f ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) ) @ V2b ) ) ) ).
thf(thm_2Eiterate_2ESUM__ABS__BOUND,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1f: A_27a > tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( V1f @ V3x ) ) @ V2b ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Eiterate_2ESum @ A_27a @ V0s @ V1f ) ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) ) @ V2b ) ) ) ).
thf(thm_2Eiterate_2ESUM__BOUND__GEN,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1f: A_27a > tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ ( V1f @ V3x ) @ ( c_2Ereal_2E_2F @ V2b @ ( c_2Ereal_2Ereal__of__num @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) ) ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eiterate_2ESum @ A_27a @ V0s @ V1f ) @ V2b ) ) ).
thf(thm_2Eiterate_2ESUM__BOUND,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1f: A_27a > tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ ( V1f @ V3x ) @ V2b ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eiterate_2ESum @ A_27a @ V0s @ V1f ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) ) @ V2b ) ) ) ).
thf(thm_2Eiterate_2ESUM__RESTRICT,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ( ( c_2Eiterate_2ESum @ A_27a @ V1s
@ ^ [V2x: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s ) @ ( V0f @ V2x ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
= ( c_2Eiterate_2ESum @ A_27a @ V1s @ V0f ) ) ) ).
thf(thm_2Eiterate_2ESUM__UNION__LZERO,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1u: A_27a > $o,V2v: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2v )
& ! [V3x: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1u )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V2v ) ) )
=> ( ( V0f @ V3x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Eiterate_2ESum @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V1u @ V2v ) @ V0f )
= ( c_2Eiterate_2ESum @ A_27a @ V2v @ V0f ) ) ) ).
thf(thm_2Eiterate_2ESUM__UNION__RZERO,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1u: A_27a > $o,V2v: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1u )
& ! [V3x: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2v )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V1u ) ) )
=> ( ( V0f @ V3x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Eiterate_2ESum @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V1u @ V2v ) @ V0f )
= ( c_2Eiterate_2ESum @ A_27a @ V1u @ V0f ) ) ) ).
thf(thm_2Eiterate_2ESUM__SUPERSET,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1u: A_27a > $o,V2v: A_27a > $o] :
( ( ( c_2Epred__set_2ESUBSET @ A_27a @ V1u @ V2v )
& ! [V3x: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2v )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V1u ) ) )
=> ( ( V0f @ V3x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Eiterate_2ESum @ A_27a @ V2v @ V0f )
= ( c_2Eiterate_2ESum @ A_27a @ V1u @ V0f ) ) ) ).
thf(thm_2Eiterate_2ESUM__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1g: A_27b > tyop_2Erealax_2Ereal,V2s: A_27a > $o] :
( ! [V3x: A_27a,V4y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
& ( c_2Ebool_2EIN @ A_27a @ V4y @ V2s )
& ( ( V0f @ V3x )
= ( V0f @ V4y ) ) )
=> ( V3x = V4y ) )
=> ( ( c_2Eiterate_2ESum @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V2s ) @ V1g )
= ( c_2Eiterate_2ESum @ A_27a @ V2s @ ( c_2Ecombin_2Eo @ A_27a @ tyop_2Erealax_2Ereal @ A_27b @ V1g @ V0f ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__SWAP,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > tyop_2Erealax_2Ereal,V1s: A_27a > $o,V2t: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( c_2Epred__set_2EFINITE @ A_27b @ V2t ) )
=> ( ( c_2Eiterate_2ESum @ A_27a @ V1s
@ ^ [V3i: A_27a] : ( c_2Eiterate_2ESum @ A_27b @ V2t @ ( V0f @ V3i ) ) )
= ( c_2Eiterate_2ESum @ A_27b @ V2t
@ ^ [V4j: A_27b] :
( c_2Eiterate_2ESum @ A_27a @ V1s
@ ^ [V5i: A_27a] : ( V0f @ V5i @ V4j ) ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__DELTA,axiom,
! [A_27a: $tType,V0b: tyop_2Erealax_2Ereal,V1s: A_27a > $o,V2a: A_27a] :
( ( c_2Eiterate_2ESum @ A_27a @ V1s
@ ^ [V3x: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ A_27a @ V3x @ V2a ) @ V0b @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ebool_2EIN @ A_27a @ V2a @ V1s ) @ V0b @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eiterate_2ESUM__SING,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1x: A_27a] :
( ( c_2Eiterate_2ESum @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V0f )
= ( V0f @ V1x ) ) ).
thf(thm_2Eiterate_2ESUM__DELETE__CASES,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o,V2a: A_27a] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ( ( c_2Eiterate_2ESum @ A_27a @ ( c_2Epred__set_2EDELETE @ A_27a @ V1s @ V2a ) @ V0f )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ebool_2EIN @ A_27a @ V2a @ V1s ) @ ( c_2Ereal_2Ereal__sub @ ( c_2Eiterate_2ESum @ A_27a @ V1s @ V0f ) @ ( V0f @ V2a ) ) @ ( c_2Eiterate_2ESum @ A_27a @ V1s @ V0f ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__DELETE,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o,V2a: A_27a] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ V2a @ V1s ) )
=> ( ( c_2Eiterate_2ESum @ A_27a @ ( c_2Epred__set_2EDELETE @ A_27a @ V1s @ V2a ) @ V0f )
= ( c_2Ereal_2Ereal__sub @ ( c_2Eiterate_2ESum @ A_27a @ V1s @ V0f ) @ ( V0f @ V2a ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__ZERO__EXISTS,axiom,
! [A_27a: $tType,V0u: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( ( c_2Eiterate_2ESum @ A_27a @ V1s @ V0u )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ! [V2i: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2i @ V1s )
=> ( ( V0u @ V2i )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
| ? [V3j: A_27a,V4k: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3j @ V1s )
& ( c_2Erealax_2Ereal__lt @ ( V0u @ V3j ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
& ( c_2Ebool_2EIN @ A_27a @ V4k @ V1s )
& ( c_2Ereal_2Ereal__gt @ ( V0u @ V4k ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__POS__EQ__0,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V0f @ V2x ) ) )
& ( ( c_2Eiterate_2ESum @ A_27a @ V1s @ V0f )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s )
=> ( ( V0f @ V3x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__POS__BOUND,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V0f @ V3x ) ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Eiterate_2ESum @ A_27a @ V2s @ V0f ) @ V1b ) )
=> ! [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ V2s )
=> ( c_2Ereal_2Ereal__lte @ ( V0f @ V4x ) @ V1b ) ) ) ).
thf(thm_2Eiterate_2ESUM__POS__LE,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o] :
( ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V0f @ V2x ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Eiterate_2ESum @ A_27a @ V1s @ V0f ) ) ) ).
thf(thm_2Eiterate_2ESUM__CONST,axiom,
! [A_27a: $tType,V0c: tyop_2Erealax_2Ereal,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ( ( c_2Eiterate_2ESum @ A_27a @ V1s
@ ^ [V2n: A_27a] : V0c )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Epred__set_2ECARD @ A_27a @ V1s ) ) @ V0c ) ) ) ).
thf(thm_2Eiterate_2ESUM__ABS__LE,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1g: A_27a > tyop_2Erealax_2Ereal,V2s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( V0f @ V3x ) ) @ ( V1g @ V3x ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Eiterate_2ESum @ A_27a @ V2s @ V0f ) ) @ ( c_2Eiterate_2ESum @ A_27a @ V2s @ V1g ) ) ) ).
thf(thm_2Eiterate_2ESUM__ABS,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Eiterate_2ESum @ A_27a @ V1s @ V0f ) )
@ ( c_2Eiterate_2ESum @ A_27a @ V1s
@ ^ [V2x: A_27a] : ( c_2Ereal_2Eabs @ ( V0f @ V2x ) ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__EQ,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1g: A_27a > tyop_2Erealax_2Ereal,V2s: A_27a > $o] :
( ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
=> ( ( V0f @ V3x )
= ( V1g @ V3x ) ) )
=> ( ( c_2Eiterate_2ESum @ A_27a @ V2s @ V0f )
= ( c_2Eiterate_2ESum @ A_27a @ V2s @ V1g ) ) ) ).
thf(thm_2Eiterate_2ESUM__POS__LT__ALL,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1f: A_27a > tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& ! [V2i: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2i @ V0s )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V2i ) ) ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Eiterate_2ESum @ A_27a @ V0s @ V1f ) ) ) ).
thf(thm_2Eiterate_2ESUM__POS__LT,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V0f @ V2x ) ) )
& ? [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s )
& ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V0f @ V3x ) ) ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Eiterate_2ESum @ A_27a @ V1s @ V0f ) ) ) ).
thf(thm_2Eiterate_2ESUM__LT__ALL,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1g: A_27a > tyop_2Erealax_2Ereal,V2s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
& ( (~)
@ ( V2s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
=> ( c_2Erealax_2Ereal__lt @ ( V0f @ V3x ) @ ( V1g @ V3x ) ) ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Eiterate_2ESum @ A_27a @ V2s @ V0f ) @ ( c_2Eiterate_2ESum @ A_27a @ V2s @ V1g ) ) ) ).
thf(thm_2Eiterate_2ESUM__LT,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1g: A_27a > tyop_2Erealax_2Ereal,V2s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
=> ( c_2Ereal_2Ereal__lte @ ( V0f @ V3x ) @ ( V1g @ V3x ) ) )
& ? [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ V2s )
& ( c_2Erealax_2Ereal__lt @ ( V0f @ V4x ) @ ( V1g @ V4x ) ) ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Eiterate_2ESum @ A_27a @ V2s @ V0f ) @ ( c_2Eiterate_2ESum @ A_27a @ V2s @ V1g ) ) ) ).
thf(thm_2Eiterate_2ESUM__LE,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1g: A_27a > tyop_2Erealax_2Ereal,V2s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
=> ( c_2Ereal_2Ereal__lte @ ( V0f @ V3x ) @ ( V1g @ V3x ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eiterate_2ESum @ A_27a @ V2s @ V0f ) @ ( c_2Eiterate_2ESum @ A_27a @ V2s @ V1g ) ) ) ).
thf(thm_2Eiterate_2ESUM__SUB,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1g: A_27a > tyop_2Erealax_2Ereal,V2s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
=> ( ( c_2Eiterate_2ESum @ A_27a @ V2s
@ ^ [V3x: A_27a] : ( c_2Ereal_2Ereal__sub @ ( V0f @ V3x ) @ ( V1g @ V3x ) ) )
= ( c_2Ereal_2Ereal__sub @ ( c_2Eiterate_2ESum @ A_27a @ V2s @ V0f ) @ ( c_2Eiterate_2ESum @ A_27a @ V2s @ V1g ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__NEG,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o] :
( ( c_2Eiterate_2ESum @ A_27a @ V1s
@ ^ [V2x: A_27a] : ( c_2Erealax_2Ereal__neg @ ( V0f @ V2x ) ) )
= ( c_2Erealax_2Ereal__neg @ ( c_2Eiterate_2ESum @ A_27a @ V1s @ V0f ) ) ) ).
thf(thm_2Eiterate_2ESUM__RMUL,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1c: tyop_2Erealax_2Ereal,V2s: A_27a > $o] :
( ( c_2Eiterate_2ESum @ A_27a @ V2s
@ ^ [V3x: A_27a] : ( c_2Erealax_2Ereal__mul @ ( V0f @ V3x ) @ V1c ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Eiterate_2ESum @ A_27a @ V2s @ V0f ) @ V1c ) ) ).
thf(thm_2Eiterate_2ESUM__LMUL,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1c: tyop_2Erealax_2Ereal,V2s: A_27a > $o] :
( ( c_2Eiterate_2ESum @ A_27a @ V2s
@ ^ [V3x: A_27a] : ( c_2Erealax_2Ereal__mul @ V1c @ ( V0f @ V3x ) ) )
= ( c_2Erealax_2Ereal__mul @ V1c @ ( c_2Eiterate_2ESum @ A_27a @ V2s @ V0f ) ) ) ).
thf(thm_2Eiterate_2ESUM__0,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Eiterate_2ESum @ A_27a @ V0s
@ ^ [V1n: A_27a] : ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Eiterate_2ESUM__EQ__0,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o] :
( ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
=> ( ( V0f @ V2x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Eiterate_2ESum @ A_27a @ V1s @ V0f )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eiterate_2ESUM__ADD__GEN,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1g: A_27a > tyop_2Erealax_2Ereal,V2s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V3x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V3x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s ) @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V0f @ V3x ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ) ) )
& ( c_2Epred__set_2EFINITE @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V4x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V4x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V4x @ V2s ) @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V1g @ V4x ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ) ) ) )
=> ( ( c_2Eiterate_2ESum @ A_27a @ V2s
@ ^ [V5x: A_27a] : ( c_2Erealax_2Ereal__add @ ( V0f @ V5x ) @ ( V1g @ V5x ) ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Eiterate_2ESum @ A_27a @ V2s @ V0f ) @ ( c_2Eiterate_2ESum @ A_27a @ V2s @ V1g ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__ADD,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1g: A_27a > tyop_2Erealax_2Ereal,V2s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
=> ( ( c_2Eiterate_2ESum @ A_27a @ V2s
@ ^ [V3x: A_27a] : ( c_2Erealax_2Ereal__add @ ( V0f @ V3x ) @ ( V1g @ V3x ) ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Eiterate_2ESum @ A_27a @ V2s @ V0f ) @ ( c_2Eiterate_2ESum @ A_27a @ V2s @ V1g ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__SUPPORT,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o] :
( ( c_2Eiterate_2ESum @ A_27a @ ( c_2Eiterate_2Esupport @ A_27a @ tyop_2Erealax_2Ereal @ c_2Erealax_2Ereal__add @ V0f @ V1s ) @ V0f )
= ( c_2Eiterate_2ESum @ A_27a @ V1s @ V0f ) ) ).
thf(thm_2Eiterate_2ESUM__INCL__EXCL,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2f: A_27a > tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( c_2Epred__set_2EFINITE @ A_27a @ V1t ) )
=> ( ( c_2Erealax_2Ereal__add @ ( c_2Eiterate_2ESum @ A_27a @ V0s @ V2f ) @ ( c_2Eiterate_2ESum @ A_27a @ V1t @ V2f ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Eiterate_2ESum @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V0s @ V1t ) @ V2f ) @ ( c_2Eiterate_2ESum @ A_27a @ ( c_2Epred__set_2EINTER @ A_27a @ V0s @ V1t ) @ V2f ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__DIFF,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V2t @ V1s ) )
=> ( ( c_2Eiterate_2ESum @ A_27a @ ( c_2Epred__set_2EDIFF @ A_27a @ V1s @ V2t ) @ V0f )
= ( c_2Ereal_2Ereal__sub @ ( c_2Eiterate_2ESum @ A_27a @ V1s @ V0f ) @ ( c_2Eiterate_2ESum @ A_27a @ V2t @ V0f ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__UNION,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( c_2Epred__set_2EFINITE @ A_27a @ V2t )
& ( c_2Epred__set_2EDISJOINT @ A_27a @ V1s @ V2t ) )
=> ( ( c_2Eiterate_2ESum @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) @ V0f )
= ( c_2Erealax_2Ereal__add @ ( c_2Eiterate_2ESum @ A_27a @ V1s @ V0f ) @ ( c_2Eiterate_2ESum @ A_27a @ V2t @ V0f ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__CLAUSES,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0f: A_27a > tyop_2Erealax_2Ereal] :
( ( c_2Eiterate_2ESum @ A_27a @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V0f )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
& ! [V1x: A_27b,V2f: A_27b > tyop_2Erealax_2Ereal,V3s: A_27b > $o] :
( ( c_2Epred__set_2EFINITE @ A_27b @ V3s )
=> ( ( c_2Eiterate_2ESum @ A_27b @ ( c_2Epred__set_2EINSERT @ A_27b @ V1x @ V3s ) @ V2f )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ebool_2EIN @ A_27b @ V1x @ V3s ) @ ( c_2Eiterate_2ESum @ A_27b @ V3s @ V2f ) @ ( c_2Erealax_2Ereal__add @ ( V2f @ V1x ) @ ( c_2Eiterate_2ESum @ A_27b @ V3s @ V2f ) ) ) ) ) ) ).
thf(thm_2Eiterate_2ESUM__DEGENERATE,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o] :
( ( (~)
@ ( c_2Epred__set_2EFINITE @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V2x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V2x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s ) @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V0f @ V2x ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ) ) ) )
=> ( ( c_2Eiterate_2ESum @ A_27a @ V1s @ V0f )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eiterate_2EMONOIDAL__REAL__MUL,axiom,
c_2Eiterate_2Emonoidal @ tyop_2Erealax_2Ereal @ c_2Erealax_2Ereal__mul ).
thf(thm_2Eiterate_2EMONOIDAL__REAL__ADD,axiom,
c_2Eiterate_2Emonoidal @ tyop_2Erealax_2Ereal @ c_2Erealax_2Ereal__add ).
thf(thm_2Eiterate_2ENEUTRAL__REAL__MUL,axiom,
( ( c_2Eiterate_2Eneutral @ tyop_2Erealax_2Ereal @ c_2Erealax_2Ereal__mul )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Eiterate_2ENEUTRAL__REAL__ADD,axiom,
( ( c_2Eiterate_2Eneutral @ tyop_2Erealax_2Ereal @ c_2Erealax_2Ereal__add )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Eiterate_2ECARD__BIGUNION,axiom,
! [A_27a: $tType,V0s: ( A_27a > $o ) > $o] :
( ( ( c_2Epred__set_2EFINITE @ ( A_27a > $o ) @ V0s )
& ! [V1t: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1t @ V0s )
=> ( c_2Epred__set_2EFINITE @ A_27a @ V1t ) )
& ! [V2t: A_27a > $o,V3u: A_27a > $o] :
( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ V0s )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3u @ V0s )
& ( (~) @ ( V2t = V3u ) ) )
=> ( ( c_2Epred__set_2EINTER @ A_27a @ V2t @ V3u )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) )
=> ( ( c_2Epred__set_2ECARD @ A_27a @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V0s ) )
= ( c_2Eiterate_2Ensum @ ( A_27a > $o ) @ V0s @ ( c_2Epred__set_2ECARD @ A_27a ) ) ) ) ).
thf(thm_2Eiterate_2EINF__UNIQUE__FINITE,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V1s )
& ( (~)
@ ( V1s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( ( c_2Eiterate_2Einf @ V1s )
= V0a )
<=> ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0a @ V1s )
& ! [V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2y @ V1s )
=> ( c_2Ereal_2Ereal__lte @ V0a @ V2y ) ) ) ) ) ).
thf(thm_2Eiterate_2ESUP__UNIQUE__FINITE,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V1s )
& ( (~)
@ ( V1s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( ( c_2Ereal_2Esup @ V1s )
= V0a )
<=> ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V0a @ V1s )
& ! [V2y: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2y @ V1s )
=> ( c_2Ereal_2Ereal__lte @ V2y @ V0a ) ) ) ) ) ).
thf(thm_2Eiterate_2EREAL__SUP__LT__FINITE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Esup @ V0s ) @ V1a )
<=> ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
=> ( c_2Erealax_2Ereal__lt @ V2x @ V1a ) ) ) ) ).
thf(thm_2Eiterate_2EREAL__LT__SUP__FINITE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( c_2Erealax_2Ereal__lt @ V1a @ ( c_2Ereal_2Esup @ V0s ) )
<=> ? [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
& ( c_2Erealax_2Ereal__lt @ V1a @ V2x ) ) ) ) ).
thf(thm_2Eiterate_2EREAL__SUP__LE__FINITE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Esup @ V0s ) @ V1a )
<=> ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V2x @ V1a ) ) ) ) ).
thf(thm_2Eiterate_2EREAL__LE__SUP__FINITE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( c_2Ereal_2Ereal__lte @ V1a @ ( c_2Ereal_2Esup @ V0s ) )
<=> ? [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
& ( c_2Ereal_2Ereal__lte @ V1a @ V2x ) ) ) ) ).
thf(thm_2Eiterate_2ESUP__FINITE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Esup @ V0s ) @ V0s )
& ! [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V1x @ ( c_2Ereal_2Esup @ V0s ) ) ) ) ) ).
thf(thm_2Eiterate_2ESUP__FINITE__LEMMA,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ? [V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1b @ V0s )
& ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V2x @ V1b ) ) ) ) ).
thf(thm_2Eiterate_2ESUP,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ? [V1b: tyop_2Erealax_2Ereal] :
! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V2x @ V1b ) ) )
=> ( ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V3x @ ( c_2Ereal_2Esup @ V0s ) ) )
& ! [V4b: tyop_2Erealax_2Ereal] :
( ! [V5x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V5x @ V4b ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Esup @ V0s ) @ V4b ) ) ) ) ).
thf(thm_2Eiterate_2ESUP__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ! [V2b: tyop_2Erealax_2Ereal] :
( ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V3x @ V2b ) )
<=> ! [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V1t )
=> ( c_2Ereal_2Ereal__lte @ V4x @ V2b ) ) )
=> ( ( c_2Ereal_2Esup @ V0s )
= ( c_2Ereal_2Esup @ V1t ) ) ) ).
thf(thm_2Eiterate_2Esup__alt,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Ereal_2Esup @ V0s )
= ( c_2Emin_2E_40 @ tyop_2Erealax_2Ereal
@ ^ [V1a: tyop_2Erealax_2Ereal] :
( c_2Ebool_2E_2F_5C
@ ( c_2Ebool_2E_21 @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s ) @ ( c_2Ereal_2Ereal__lte @ V2x @ V1a ) ) )
@ ( c_2Ebool_2E_21 @ tyop_2Erealax_2Ereal
@ ^ [V3b: tyop_2Erealax_2Ereal] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_21 @ tyop_2Erealax_2Ereal
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V0s ) @ ( c_2Ereal_2Ereal__lte @ V4x @ V3b ) ) )
@ ( c_2Ereal_2Ereal__lte @ V1a @ V3b ) ) ) ) ) ) ).
thf(thm_2Eiterate_2EREAL__COMPLETE,axiom,
! [V0P: tyop_2Erealax_2Ereal > $o] :
( ( ? [V1x: tyop_2Erealax_2Ereal] : ( V0P @ V1x )
& ? [V2M: tyop_2Erealax_2Ereal] :
! [V3x: tyop_2Erealax_2Ereal] :
( ( V0P @ V3x )
=> ( c_2Ereal_2Ereal__lte @ V3x @ V2M ) ) )
=> ? [V4M: tyop_2Erealax_2Ereal] :
( ! [V5x: tyop_2Erealax_2Ereal] :
( ( V0P @ V5x )
=> ( c_2Ereal_2Ereal__lte @ V5x @ V4M ) )
& ! [V6M_27: tyop_2Erealax_2Ereal] :
( ! [V7x: tyop_2Erealax_2Ereal] :
( ( V0P @ V7x )
=> ( c_2Ereal_2Ereal__lte @ V7x @ V6M_27 ) )
=> ( c_2Ereal_2Ereal__lte @ V4M @ V6M_27 ) ) ) ) ).
thf(thm_2Eiterate_2ECHOOSE__SUBSET,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ! [V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1n @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) )
=> ? [V2t: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V2t @ V0s )
& ( c_2Ecardinal_2EHAS__SIZE @ A_27a @ V2t @ V1n ) ) ) ) ).
thf(thm_2Eiterate_2ECHOOSE__SUBSET__STRONG,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ( c_2Earithmetic_2E_3C_3D @ V0n @ ( c_2Epred__set_2ECARD @ A_27a @ V1s ) ) )
=> ? [V2t: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V2t @ V1s )
& ( c_2Ecardinal_2EHAS__SIZE @ A_27a @ V2t @ V0n ) ) ) ).
thf(thm_2Eiterate_2ESET__PROVE__CASES,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o] :
( ( ( V0P @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ! [V1a: A_27a,V2s: A_27a > $o] :
( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V1a @ V2s ) )
=> ( V0P @ ( c_2Epred__set_2EINSERT @ A_27a @ V1a @ V2s ) ) ) )
=> ! [V3s: A_27a > $o] : ( V0P @ V3s ) ) ).
thf(thm_2Eiterate_2Ereal__INFINITE,axiom,
(~) @ ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ) ).
thf(thm_2Eiterate_2EFINITE__REAL__INTERVAL,axiom,
( ! [V0a: tyop_2Erealax_2Ereal] :
( (~)
@ ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V1x @ ( c_2Erealax_2Ereal__lt @ V0a @ V1x ) ) ) ) )
& ! [V2a: tyop_2Erealax_2Ereal] :
( (~)
@ ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x @ ( c_2Ereal_2Ereal__lte @ V2a @ V3x ) ) ) ) )
& ! [V4b: tyop_2Erealax_2Ereal] :
( (~)
@ ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V5x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V5x @ ( c_2Erealax_2Ereal__lt @ V5x @ V4b ) ) ) ) )
& ! [V6b: tyop_2Erealax_2Ereal] :
( (~)
@ ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V7x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V7x @ ( c_2Ereal_2Ereal__lte @ V7x @ V6b ) ) ) ) )
& ! [V8a: tyop_2Erealax_2Ereal,V9b: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V10x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V10x @ ( c_2Ebool_2E_2F_5C @ ( c_2Erealax_2Ereal__lt @ V8a @ V10x ) @ ( c_2Erealax_2Ereal__lt @ V10x @ V9b ) ) ) ) )
= ( c_2Ereal_2Ereal__lte @ V9b @ V8a ) )
& ! [V11a: tyop_2Erealax_2Ereal,V12b: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V13x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V13x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ V11a @ V13x ) @ ( c_2Erealax_2Ereal__lt @ V13x @ V12b ) ) ) ) )
= ( c_2Ereal_2Ereal__lte @ V12b @ V11a ) )
& ! [V14a: tyop_2Erealax_2Ereal,V15b: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V16x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V16x @ ( c_2Ebool_2E_2F_5C @ ( c_2Erealax_2Ereal__lt @ V14a @ V16x ) @ ( c_2Ereal_2Ereal__lte @ V16x @ V15b ) ) ) ) )
= ( c_2Ereal_2Ereal__lte @ V15b @ V14a ) )
& ! [V17a: tyop_2Erealax_2Ereal,V18b: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V19x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V19x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ereal_2Ereal__lte @ V17a @ V19x ) @ ( c_2Ereal_2Ereal__lte @ V19x @ V18b ) ) ) ) )
= ( c_2Ereal_2Ereal__lte @ V18b @ V17a ) ) ) ).
thf(thm_2Eiterate_2EFINITE__RESTRICT,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1P: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ( c_2Epred__set_2EFINITE @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V2x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V2x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s ) @ ( V1P @ V2x ) ) ) ) ) ) ).
thf(thm_2Eiterate_2ESUBSET__RESTRICT,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1P: A_27a > $o] :
( c_2Epred__set_2ESUBSET @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V2x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V2x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s ) @ ( V1P @ V2x ) ) ) )
@ V0s ) ).
thf(thm_2Eiterate_2ECARD__UNION__EQ,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2u: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2u )
& ( ( c_2Epred__set_2EINTER @ A_27a @ V0s @ V1t )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ( ( c_2Epred__set_2EUNION @ A_27a @ V0s @ V1t )
= V2u ) )
=> ( ( c_2Earithmetic_2E_2B @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) @ ( c_2Epred__set_2ECARD @ A_27a @ V1t ) )
= ( c_2Epred__set_2ECARD @ A_27a @ V2u ) ) ) ).
thf(thm_2Eiterate_2EFINITE__RECURSION,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > A_27b,V1b: A_27b] :
( ! [V2x: A_27a,V3y: A_27a,V4s: A_27b] :
( ( (~) @ ( V2x = V3y ) )
=> ( ( V0f @ V2x @ ( V0f @ V3y @ V4s ) )
= ( V0f @ V3y @ ( V0f @ V2x @ V4s ) ) ) )
=> ( ( ( c_2Eiterate_2EITSET @ A_27b @ A_27a @ V0f @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V1b )
= V1b )
& ! [V5x: A_27a,V6s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V6s )
=> ( ( c_2Eiterate_2EITSET @ A_27b @ A_27a @ V0f @ ( c_2Epred__set_2EINSERT @ A_27a @ V5x @ V6s ) @ V1b )
= ( c_2Ebool_2ECOND @ A_27b @ ( c_2Ebool_2EIN @ A_27a @ V5x @ V6s ) @ ( c_2Eiterate_2EITSET @ A_27b @ A_27a @ V0f @ V6s @ V1b ) @ ( V0f @ V5x @ ( c_2Eiterate_2EITSET @ A_27b @ A_27a @ V0f @ V6s @ V1b ) ) ) ) ) ) ) ).
thf(thm_2Eiterate_2ESET__RECURSION__LEMMA,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > A_27b,V1b: A_27b] :
( ! [V2x: A_27a,V3y: A_27a,V4s: A_27b] :
( ( (~) @ ( V2x = V3y ) )
=> ( ( V0f @ V2x @ ( V0f @ V3y @ V4s ) )
= ( V0f @ V3y @ ( V0f @ V2x @ V4s ) ) ) )
=> ? [V5g: ( A_27a > $o ) > A_27b] :
( ( ( V5g @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= V1b )
& ! [V6x: A_27a,V7s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V7s )
=> ( ( V5g @ ( c_2Epred__set_2EINSERT @ A_27a @ V6x @ V7s ) )
= ( c_2Ebool_2ECOND @ A_27b @ ( c_2Ebool_2EIN @ A_27a @ V6x @ V7s ) @ ( V5g @ V7s ) @ ( V0f @ V6x @ ( V5g @ V7s ) ) ) ) ) ) ) ).
thf(thm_2Eiterate_2EFINREC__FUN,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > A_27b,V1b: A_27b] :
( ! [V2x: A_27a,V3y: A_27a,V4s: A_27b] :
( ( (~) @ ( V2x = V3y ) )
=> ( ( V0f @ V2x @ ( V0f @ V3y @ V4s ) )
= ( V0f @ V3y @ ( V0f @ V2x @ V4s ) ) ) )
=> ? [V5g: ( A_27a > $o ) > A_27b] :
( ( ( V5g @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= V1b )
& ! [V6s: A_27a > $o,V7x: A_27a] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V6s )
& ( c_2Ebool_2EIN @ A_27a @ V7x @ V6s ) )
=> ( ( V5g @ V6s )
= ( V0f @ V7x @ ( V5g @ ( c_2Epred__set_2EDELETE @ A_27a @ V6s @ V7x ) ) ) ) ) ) ) ).
thf(thm_2Eiterate_2EFINREC__FUN__LEMMA,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0P: A_27a > $o,V1R: A_27a > A_27b > A_27c > $o] :
( ( ! [V2s: A_27a] :
( ( V0P @ V2s )
=> ? [V3a: A_27b,V4n: A_27c] : ( V1R @ V2s @ V3a @ V4n ) )
& ! [V5n1: A_27c,V6n2: A_27c,V7s: A_27a,V8a1: A_27b,V9a2: A_27b] :
( ( ( V1R @ V7s @ V8a1 @ V5n1 )
& ( V1R @ V7s @ V9a2 @ V6n2 ) )
=> ( ( V8a1 = V9a2 )
& ( V5n1 = V6n2 ) ) ) )
=> ? [V10f: A_27a > A_27b] :
! [V11s: A_27a,V12a: A_27b] :
( ( V0P @ V11s )
=> ( ? [V13n: A_27c] : ( V1R @ V11s @ V12a @ V13n )
<=> ( ( V10f @ V11s )
= V12a ) ) ) ) ).
thf(thm_2Eiterate_2EFINREC__EXISTS__LEMMA,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > A_27b,V1b: A_27b,V2s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
=> ? [V3a: A_27b,V4n: tyop_2Enum_2Enum] : ( c_2Eiterate_2EFINREC @ A_27a @ A_27b @ V0f @ V1b @ V2s @ V3a @ V4n ) ) ).
thf(thm_2Eiterate_2EFINREC__UNIQUE__LEMMA,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > A_27b,V1b: A_27b] :
( ! [V2x: A_27a,V3y: A_27a,V4s: A_27b] :
( ( (~) @ ( V2x = V3y ) )
=> ( ( V0f @ V2x @ ( V0f @ V3y @ V4s ) )
= ( V0f @ V3y @ ( V0f @ V2x @ V4s ) ) ) )
=> ! [V5n1: tyop_2Enum_2Enum,V6n2: tyop_2Enum_2Enum,V7s: A_27a > $o,V8a1: A_27b,V9a2: A_27b] :
( ( ( c_2Eiterate_2EFINREC @ A_27a @ A_27b @ V0f @ V1b @ V7s @ V8a1 @ V5n1 )
& ( c_2Eiterate_2EFINREC @ A_27a @ A_27b @ V0f @ V1b @ V7s @ V9a2 @ V6n2 ) )
=> ( ( V8a1 = V9a2 )
& ( V5n1 = V6n2 ) ) ) ) ).
thf(thm_2Eiterate_2EFINREC__SUC__LEMMA,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > A_27b,V1b: A_27b] :
( ! [V2x: A_27a,V3y: A_27a,V4s: A_27b] :
( ( (~) @ ( V2x = V3y ) )
=> ( ( V0f @ V2x @ ( V0f @ V3y @ V4s ) )
= ( V0f @ V3y @ ( V0f @ V2x @ V4s ) ) ) )
=> ! [V5n: tyop_2Enum_2Enum,V6s: A_27a > $o,V7z: A_27b] :
( ( c_2Eiterate_2EFINREC @ A_27a @ A_27b @ V0f @ V1b @ V6s @ V7z @ ( c_2Enum_2ESUC @ V5n ) )
=> ! [V8x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V8x @ V6s )
=> ? [V9w: A_27b] :
( ( c_2Eiterate_2EFINREC @ A_27a @ A_27b @ V0f @ V1b @ ( c_2Epred__set_2EDELETE @ A_27a @ V6s @ V8x ) @ V9w @ V5n )
& ( V7z
= ( V0f @ V8x @ V9w ) ) ) ) ) ) ).
thf(thm_2Eiterate_2EFINREC__1__LEMMA,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > A_27b,V1b: A_27b,V2s: A_27a > $o,V3a: A_27b] :
( ( c_2Eiterate_2EFINREC @ A_27a @ A_27b @ V0f @ V1b @ V2s @ V3a @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) )
<=> ? [V4x: A_27a] :
( ( V2s
= ( c_2Epred__set_2EINSERT @ A_27a @ V4x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& ( V3a
= ( V0f @ V4x @ V1b ) ) ) ) ).
thf(thm_2Eiterate_2EFINREC__def__compute,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0f: A_27a > A_27b > A_27b,V1b: A_27b,V2s: A_27a > $o,V3a: A_27b] :
( ( c_2Eiterate_2EFINREC @ A_27a @ A_27b @ V0f @ V1b @ V2s @ V3a @ c_2Enum_2E0 )
<=> ( ( V2s
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ( V3a = V1b ) ) )
& ! [V4f: A_27a > A_27b > A_27b,V5b: A_27b,V6s: A_27a > $o,V7a: A_27b,V8n: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2EFINREC @ A_27a @ A_27b @ V4f @ V5b @ V6s @ V7a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V8n ) ) )
<=> ? [V9x: A_27a,V10c: A_27b] :
( ( c_2Ebool_2EIN @ A_27a @ V9x @ V6s )
& ( c_2Eiterate_2EFINREC @ A_27a @ A_27b @ V4f @ V5b @ ( c_2Epred__set_2EDELETE @ A_27a @ V6s @ V9x ) @ V10c @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V8n ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
& ( V7a
= ( V4f @ V9x @ V10c ) ) ) )
& ! [V11f: A_27a > A_27b > A_27b,V12b: A_27b,V13s: A_27a > $o,V14a: A_27b,V15n: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2EFINREC @ A_27a @ A_27b @ V11f @ V12b @ V13s @ V14a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V15n ) ) )
<=> ? [V16x: A_27a,V17c: A_27b] :
( ( c_2Ebool_2EIN @ A_27a @ V16x @ V13s )
& ( c_2Eiterate_2EFINREC @ A_27a @ A_27b @ V11f @ V12b @ ( c_2Epred__set_2EDELETE @ A_27a @ V13s @ V16x ) @ V17c @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V15n ) ) )
& ( V14a
= ( V11f @ V16x @ V17c ) ) ) ) ) ).
thf(thm_2Eiterate_2EREAL__WLOG__LT,axiom,
! [V0P: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o] :
( ( ! [V1x: tyop_2Erealax_2Ereal] : ( V0P @ V1x @ V1x )
& ! [V2x: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( V0P @ V2x @ V3y )
= ( V0P @ V3y @ V2x ) )
& ! [V4x: tyop_2Erealax_2Ereal,V5y: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V4x @ V5y )
=> ( V0P @ V4x @ V5y ) ) )
=> ! [V6x: tyop_2Erealax_2Ereal,V7y: tyop_2Erealax_2Ereal] : ( V0P @ V6x @ V7y ) ) ).
thf(thm_2Eiterate_2EUPPER__BOUND__FINITE__SET__REAL,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ? [V2a: tyop_2Erealax_2Ereal] :
! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s )
=> ( c_2Ereal_2Ereal__lte @ ( V0f @ V3x ) @ V2a ) ) ) ).
thf(thm_2Eiterate_2ELOWER__BOUND__FINITE__SET__REAL,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ? [V2a: tyop_2Erealax_2Ereal] :
! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s )
=> ( c_2Ereal_2Ereal__lte @ V2a @ ( V0f @ V3x ) ) ) ) ).
thf(thm_2Eiterate_2ESIMP__REAL__ARCH,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
? [V1n: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__lte @ V0x @ ( c_2Ereal_2Ereal__of__num @ V1n ) ) ).
thf(thm_2Eiterate_2EREAL__LT__BETWEEN,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V0a @ V1b )
<=> ? [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V0a @ V2x )
& ( c_2Erealax_2Ereal__lt @ V2x @ V1b ) ) ) ).
thf(thm_2Eiterate_2EREAL__SUP__LE__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1y: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ? [V2b: tyop_2Erealax_2Ereal] :
! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V3x @ V2b ) ) )
=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Esup @ V0s ) @ V1y )
<=> ! [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V4x @ V1y ) ) ) ) ).
thf(thm_2Eiterate_2EREAL__SUP__UNIQUE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1b: tyop_2Erealax_2Ereal] :
( ( ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V2x @ V1b ) )
& ! [V3b_27: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V3b_27 @ V1b )
=> ? [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V0s )
& ( c_2Erealax_2Ereal__lt @ V3b_27 @ V4x ) ) ) )
=> ( ( c_2Ereal_2Esup @ V0s )
= V1b ) ) ).
thf(thm_2Eiterate_2EREAL__SUP__LE__S,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1b: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V2x @ V1b ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Esup @ V0s ) @ V1b ) ) ).
thf(thm_2Eiterate_2EREAL__SUP__LE__SUBSET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V0s @ V1t )
& ? [V2b: tyop_2Erealax_2Ereal] :
! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V1t )
=> ( c_2Ereal_2Ereal__lte @ V3x @ V2b ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Esup @ V0s ) @ ( c_2Ereal_2Esup @ V1t ) ) ) ).
thf(thm_2Eiterate_2EREAL__LE__SUP,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal,V3y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3y @ V0s )
& ( c_2Ereal_2Ereal__lte @ V1a @ V3y )
& ! [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V4x @ V2b ) ) )
=> ( c_2Ereal_2Ereal__lte @ V1a @ ( c_2Ereal_2Esup @ V0s ) ) ) ).
thf(thm_2Eiterate_2EREAL__SUP__BOUNDS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0s )
=> ( ( c_2Ereal_2Ereal__lte @ V1a @ V3x )
& ( c_2Ereal_2Ereal__lte @ V3x @ V2b ) ) ) )
=> ( ( c_2Ereal_2Ereal__lte @ V1a @ ( c_2Ereal_2Esup @ V0s ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Esup @ V0s ) @ V2b ) ) ) ).
thf(thm_2Eiterate_2EREAL__ABS__SUP__LE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ V2x ) @ V1a ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Esup @ V0s ) ) @ V1a ) ) ).
thf(thm_2Eiterate_2EREAL__SUP__ASCLOSE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1l: tyop_2Erealax_2Ereal,V2e: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V3x @ V1l ) ) @ V2e ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Esup @ V0s ) @ V1l ) ) @ V2e ) ) ).
thf(thm_2Eiterate_2EREAL__INF__ASCLOSE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1l: tyop_2Erealax_2Ereal,V2e: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V3x @ V1l ) ) @ V2e ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ ( c_2Eiterate_2Einf @ V0s ) @ V1l ) ) @ V2e ) ) ).
thf(thm_2Eiterate_2EREAL__ABS__INF__LE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ V2x ) @ V1a ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Eiterate_2Einf @ V0s ) ) @ V1a ) ) ).
thf(thm_2Eiterate_2EREAL__INF__BOUNDS,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal,V2b: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0s )
=> ( ( c_2Ereal_2Ereal__lte @ V1a @ V3x )
& ( c_2Ereal_2Ereal__lte @ V3x @ V2b ) ) ) )
=> ( ( c_2Ereal_2Ereal__lte @ V1a @ ( c_2Eiterate_2Einf @ V0s ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Eiterate_2Einf @ V0s ) @ V2b ) ) ) ).
thf(thm_2Eiterate_2EREAL__LE__INF__SUBSET,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( (~)
@ ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Erealax_2Ereal @ V1t @ V0s )
& ? [V2b: tyop_2Erealax_2Ereal] :
! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V2b @ V3x ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eiterate_2Einf @ V0s ) @ ( c_2Eiterate_2Einf @ V1t ) ) ) ).
thf(thm_2Eiterate_2EREAL__LE__INF,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1b: tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V1b @ V2x ) ) )
=> ( c_2Ereal_2Ereal__lte @ V1b @ ( c_2Eiterate_2Einf @ V0s ) ) ) ).
thf(thm_2Eiterate_2EREAL__INF__UNIQUE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1b: tyop_2Erealax_2Ereal] :
( ( ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V1b @ V2x ) )
& ! [V3b_27: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ V1b @ V3b_27 )
=> ? [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V0s )
& ( c_2Erealax_2Ereal__lt @ V4x @ V3b_27 ) ) ) )
=> ( ( c_2Eiterate_2Einf @ V0s )
= V1b ) ) ).
thf(thm_2Eiterate_2EREAL__INF__LT__FINITE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( c_2Erealax_2Ereal__lt @ ( c_2Eiterate_2Einf @ V0s ) @ V1a )
<=> ? [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
& ( c_2Erealax_2Ereal__lt @ V2x @ V1a ) ) ) ) ).
thf(thm_2Eiterate_2EREAL__LT__INF__FINITE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( c_2Erealax_2Ereal__lt @ V1a @ ( c_2Eiterate_2Einf @ V0s ) )
<=> ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
=> ( c_2Erealax_2Ereal__lt @ V1a @ V2x ) ) ) ) ).
thf(thm_2Eiterate_2EREAL__INF__LE__FINITE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Eiterate_2Einf @ V0s ) @ V1a )
<=> ? [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
& ( c_2Ereal_2Ereal__lte @ V2x @ V1a ) ) ) ) ).
thf(thm_2Eiterate_2EREAL__LE__INF__FINITE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1a: tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( c_2Ereal_2Ereal__lte @ V1a @ ( c_2Eiterate_2Einf @ V0s ) )
<=> ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V1a @ V2x ) ) ) ) ).
thf(thm_2Eiterate_2EINF__FINITE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ ( c_2Eiterate_2Einf @ V0s ) @ V0s )
& ! [V1x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eiterate_2Einf @ V0s ) @ V1x ) ) ) ) ).
thf(thm_2Eiterate_2EINF__FINITE__LEMMA,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) )
=> ? [V1b: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V1b @ V0s )
& ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V1b @ V2x ) ) ) ) ).
thf(thm_2Eiterate_2EINF,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ? [V1b: tyop_2Erealax_2Ereal] :
! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V1b @ V2x ) ) )
=> ( ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Eiterate_2Einf @ V0s ) @ V3x ) )
& ! [V4b: tyop_2Erealax_2Ereal] :
( ! [V5x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V4b @ V5x ) )
=> ( c_2Ereal_2Ereal__lte @ V4b @ ( c_2Eiterate_2Einf @ V0s ) ) ) ) ) ).
thf(thm_2Eiterate_2EINF__EQ,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ! [V2a: tyop_2Erealax_2Ereal] :
( ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V2a @ V3x ) )
<=> ! [V4x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V4x @ V1t )
=> ( c_2Ereal_2Ereal__lte @ V2a @ V4x ) ) )
=> ( ( c_2Eiterate_2Einf @ V0s )
= ( c_2Eiterate_2Einf @ V1t ) ) ) ).
thf(thm_2Eiterate_2ESUP__UNION,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1t: tyop_2Erealax_2Ereal > $o] :
( ( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ( (~)
@ ( V1t
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ? [V2b: tyop_2Erealax_2Ereal] :
! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V3x @ V2b ) )
& ? [V4c: tyop_2Erealax_2Ereal] :
! [V5x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V5x @ V1t )
=> ( c_2Ereal_2Ereal__lte @ V5x @ V4c ) ) )
=> ( ( c_2Ereal_2Esup @ ( c_2Epred__set_2EUNION @ tyop_2Erealax_2Ereal @ V0s @ V1t ) )
= ( c_2Ereal_2Emax @ ( c_2Ereal_2Esup @ V0s ) @ ( c_2Ereal_2Esup @ V1t ) ) ) ) ).
thf(thm_2Eiterate_2ESUP__UNIQUE,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o,V1b: tyop_2Erealax_2Ereal] :
( ! [V2c: tyop_2Erealax_2Ereal] :
( ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V3x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ V3x @ V2c ) )
<=> ( c_2Ereal_2Ereal__lte @ V1b @ V2c ) )
=> ( ( c_2Ereal_2Esup @ V0s )
= V1b ) ) ).
thf(thm_2Eiterate_2ESUP__SING,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Esup @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
= V0a ) ).
thf(thm_2Eiterate_2ESUP__INSERT__FINITE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V1s )
=> ( ( c_2Ereal_2Esup @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0x @ V1s ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V0x @ ( c_2Ereal_2Emax @ V0x @ ( c_2Ereal_2Esup @ V1s ) ) ) ) ) ).
thf(thm_2Eiterate_2EINF__INSERT__FINITE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1s: tyop_2Erealax_2Ereal > $o] :
( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal @ V1s )
=> ( ( c_2Eiterate_2Einf @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0x @ V1s ) )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Emin_2E_3D @ ( tyop_2Erealax_2Ereal > $o ) @ V1s @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) @ V0x @ ( c_2Ereal_2Emin @ V0x @ ( c_2Eiterate_2Einf @ V1s ) ) ) ) ) ).
thf(thm_2Eiterate_2EREAL__SUP__EQ__INF,axiom,
! [V0s: tyop_2Erealax_2Ereal > $o] :
( ( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
& ? [V1B: tyop_2Erealax_2Ereal] :
! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Ebool_2EIN @ tyop_2Erealax_2Ereal @ V2x @ V0s )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ V2x ) @ V1B ) ) )
=> ( ( ( c_2Ereal_2Esup @ V0s )
= ( c_2Eiterate_2Einf @ V0s ) )
<=> ? [V3a: tyop_2Erealax_2Ereal] :
( V0s
= ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V3a @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) ) ) ) ).
thf(thm_2Eiterate_2EINF__SING,axiom,
! [V0a: tyop_2Erealax_2Ereal] :
( ( c_2Eiterate_2Einf @ ( c_2Epred__set_2EINSERT @ tyop_2Erealax_2Ereal @ V0a @ ( c_2Epred__set_2EEMPTY @ tyop_2Erealax_2Ereal ) ) )
= V0a ) ).
thf(thm_2Eiterate_2EFINITE__NUMSEG,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] : ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V0m @ V1n ) ) ).
thf(thm_2Eiterate_2ENUMSEG__COMBINE__R,axiom,
! [V0m: tyop_2Enum_2Enum,V1p: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V0m @ ( c_2Earithmetic_2E_2B @ V1p @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
& ( c_2Earithmetic_2E_3C_3D @ V1p @ V2n ) )
=> ( ( c_2Epred__set_2EUNION @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V0m @ V1p ) @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2E_2B @ V1p @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2n ) )
= ( c_2Eiterate_2E_2E_2E @ V0m @ V2n ) ) ) ).
thf(thm_2Eiterate_2ENUMSEG__COMBINE__L,axiom,
! [V0m: tyop_2Enum_2Enum,V1p: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V0m @ V1p )
& ( c_2Earithmetic_2E_3C_3D @ V1p @ ( c_2Earithmetic_2E_2B @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( ( c_2Epred__set_2EUNION @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V0m @ ( c_2Earithmetic_2E_2D @ V1p @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Eiterate_2E_2E_2E @ V1p @ V2n ) )
= ( c_2Eiterate_2E_2E_2E @ V0m @ V2n ) ) ) ).
thf(thm_2Eiterate_2ENUMSEG__LREC,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n )
=> ( ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ V0m @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2E_2B @ V0m @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V1n ) )
= ( c_2Eiterate_2E_2E_2E @ V0m @ V1n ) ) ) ).
thf(thm_2Eiterate_2ENUMSEG__RREC,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n )
=> ( ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ V1n @ ( c_2Eiterate_2E_2E_2E @ V0m @ ( c_2Earithmetic_2E_2D @ V1n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
= ( c_2Eiterate_2E_2E_2E @ V0m @ V1n ) ) ) ).
thf(thm_2Eiterate_2ENUMSEG__REC,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0m @ ( c_2Enum_2ESUC @ V1n ) )
=> ( ( c_2Eiterate_2E_2E_2E @ V0m @ ( c_2Enum_2ESUC @ V1n ) )
= ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ ( c_2Enum_2ESUC @ V1n ) @ ( c_2Eiterate_2E_2E_2E @ V0m @ V1n ) ) ) ) ).
thf(thm_2Eiterate_2EIN__NUMSEG,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V2p @ ( c_2Eiterate_2E_2E_2E @ V0m @ V1n ) )
<=> ( ( c_2Earithmetic_2E_3C_3D @ V0m @ V2p )
& ( c_2Earithmetic_2E_3C_3D @ V2p @ V1n ) ) ) ).
thf(thm_2Eiterate_2EIN__NUMSEG__0,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V0m @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ V1n ) )
= ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n ) ) ).
thf(thm_2Eiterate_2ENUMSEG__SING,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2E_2E_2E @ V0n @ V0n )
= ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ V0n @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) ) ).
thf(thm_2Eiterate_2ENUMSEG__EMPTY,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Eiterate_2E_2E_2E @ V0m @ V1n )
= ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) )
<=> ( c_2Eprim__rec_2E_3C @ V1n @ V0m ) ) ).
thf(thm_2Eiterate_2ECARD__NUMSEG__LEMMA,axiom,
! [V0m: tyop_2Enum_2Enum,V1d: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2ECARD @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V0m @ ( c_2Earithmetic_2E_2B @ V0m @ V1d ) ) )
= ( c_2Earithmetic_2E_2B @ V1d @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Eiterate_2ECARD__NUMSEG,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2ECARD @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V0m @ V1n ) )
= ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2B @ V1n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0m ) ) ).
thf(thm_2Eiterate_2EHAS__SIZE__NUMSEG,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] : ( c_2Ecardinal_2EHAS__SIZE @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V0m @ V1n ) @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2B @ V1n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0m ) ) ).
thf(thm_2Eiterate_2ECARD__NUMSEG__1,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2ECARD @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V0n ) )
= V0n ) ).
thf(thm_2Eiterate_2EHAS__SIZE__NUMSEG__1,axiom,
! [V0n: tyop_2Enum_2Enum] : ( c_2Ecardinal_2EHAS__SIZE @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V0n ) @ V0n ) ).
thf(thm_2Eiterate_2ENUMSEG__CLAUSES,axiom,
( ! [V0m: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2E_2E_2E @ V0m @ c_2Enum_2E0 )
= ( c_2Ebool_2ECOND @ ( tyop_2Enum_2Enum > $o ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V0m @ c_2Enum_2E0 ) @ ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) )
& ! [V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2E_2E_2E @ V1m @ ( c_2Enum_2ESUC @ V2n ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Enum_2Enum > $o ) @ ( c_2Earithmetic_2E_3C_3D @ V1m @ ( c_2Enum_2ESUC @ V2n ) ) @ ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ ( c_2Enum_2ESUC @ V2n ) @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) ) @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) ) ) ) ).
thf(thm_2Eiterate_2EFINITE__INDEX__NUMSEG,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
<=> ? [V1f: tyop_2Enum_2Enum > A_27a] :
( ! [V2i: tyop_2Enum_2Enum,V3j: tyop_2Enum_2Enum] :
( ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V2i @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) ) )
& ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V3j @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) ) )
& ( ( V1f @ V2i )
= ( V1f @ V3j ) ) )
=> ( V2i = V3j ) )
& ( V0s
= ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ A_27a @ V1f @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) ) ) ) ) ) ).
thf(thm_2Eiterate_2EFINITE__INDEX__NUMBERS,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
<=> ? [V1k: tyop_2Enum_2Enum > $o,V2f: tyop_2Enum_2Enum > A_27a] :
( ! [V3i: tyop_2Enum_2Enum,V4j: tyop_2Enum_2Enum] :
( ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V3i @ V1k )
& ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V4j @ V1k )
& ( ( V2f @ V3i )
= ( V2f @ V4j ) ) )
=> ( V3i = V4j ) )
& ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V1k )
& ( V0s
= ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ A_27a @ V2f @ V1k ) ) ) ) ).
thf(thm_2Eiterate_2EDISJOINT__NUMSEG,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum,V3q: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2EDISJOINT @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V0m @ V1n ) @ ( c_2Eiterate_2E_2E_2E @ V2p @ V3q ) )
<=> ( ( c_2Eprim__rec_2E_3C @ V1n @ V2p )
| ( c_2Eprim__rec_2E_3C @ V3q @ V0m )
| ( c_2Eprim__rec_2E_3C @ V1n @ V0m )
| ( c_2Eprim__rec_2E_3C @ V3q @ V2p ) ) ) ).
thf(thm_2Eiterate_2ENUMSEG__ADD__SPLIT,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0m @ ( c_2Earithmetic_2E_2B @ V1n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
=> ( ( c_2Eiterate_2E_2E_2E @ V0m @ ( c_2Earithmetic_2E_2B @ V1n @ V2p ) )
= ( c_2Epred__set_2EUNION @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V0m @ V1n ) @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2E_2B @ V1n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2E_2B @ V1n @ V2p ) ) ) ) ) ).
thf(thm_2Eiterate_2ENUMSEG__OFFSET__IMAGE,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2E_2B @ V0m @ V2p ) @ ( c_2Earithmetic_2E_2B @ V1n @ V2p ) )
= ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum
@ ^ [V3i: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2B @ V3i @ V2p )
@ ( c_2Eiterate_2E_2E_2E @ V0m @ V1n ) ) ) ).
thf(thm_2Eiterate_2ESUBSET__NUMSEG,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2p: tyop_2Enum_2Enum,V3q: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2ESUBSET @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V0m @ V1n ) @ ( c_2Eiterate_2E_2E_2E @ V2p @ V3q ) )
<=> ( ( c_2Eprim__rec_2E_3C @ V1n @ V0m )
| ( ( c_2Earithmetic_2E_3C_3D @ V2p @ V0m )
& ( c_2Earithmetic_2E_3C_3D @ V1n @ V3q ) ) ) ) ).
thf(thm_2Eiterate_2ENUMSEG__LE,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2EGSPEC @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum
@ ^ [V1x: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ $o @ V1x @ ( c_2Earithmetic_2E_3C_3D @ V1x @ V0n ) ) )
= ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ V0n ) ) ).
thf(thm_2Eiterate_2ENUMSEG__LT,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2EGSPEC @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum
@ ^ [V1x: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ $o @ V1x @ ( c_2Eprim__rec_2E_3C @ V1x @ V0n ) ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Enum_2Enum > $o ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V0n @ c_2Enum_2E0 ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ ( c_2Earithmetic_2E_2D @ V0n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Eiterate_2ETOPOLOGICAL__SORT,axiom,
! [A_27a: $tType,V0_3C_3C: A_27a > A_27a > $o] :
( ( ! [V1x: A_27a,V2y: A_27a] :
( ( ( V0_3C_3C @ V1x @ V2y )
& ( V0_3C_3C @ V2y @ V1x ) )
=> ( V1x = V2y ) )
& ! [V3x: A_27a,V4y: A_27a,V5z: A_27a] :
( ( ( V0_3C_3C @ V3x @ V4y )
& ( V0_3C_3C @ V4y @ V5z ) )
=> ( V0_3C_3C @ V3x @ V5z ) ) )
=> ! [V6n: tyop_2Enum_2Enum,V7s: A_27a > $o] :
( ( c_2Ecardinal_2EHAS__SIZE @ A_27a @ V7s @ V6n )
=> ? [V8f: tyop_2Enum_2Enum > A_27a] :
( ( V7s
= ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ A_27a @ V8f @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V6n ) ) )
& ! [V9j: tyop_2Enum_2Enum,V10k: tyop_2Enum_2Enum] :
( ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V9j @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V6n ) )
& ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V10k @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V6n ) )
& ( c_2Eprim__rec_2E_3C @ V9j @ V10k ) )
=> ( (~) @ ( V0_3C_3C @ ( V8f @ V10k ) @ ( V8f @ V9j ) ) ) ) ) ) ) ).
thf(thm_2Eiterate_2EMONOIDAL__AC,axiom,
! [A_27a: $tType,V0op: A_27a > A_27a > A_27a] :
( ( c_2Eiterate_2Emonoidal @ A_27a @ V0op )
=> ( ! [V1a: A_27a] :
( ( V0op @ ( c_2Eiterate_2Eneutral @ A_27a @ V0op ) @ V1a )
= V1a )
& ! [V2a: A_27a] :
( ( V0op @ V2a @ ( c_2Eiterate_2Eneutral @ A_27a @ V0op ) )
= V2a )
& ! [V3a: A_27a,V4b: A_27a] :
( ( V0op @ V3a @ V4b )
= ( V0op @ V4b @ V3a ) )
& ! [V5a: A_27a,V6b: A_27a,V7c: A_27a] :
( ( V0op @ ( V0op @ V5a @ V6b ) @ V7c )
= ( V0op @ V5a @ ( V0op @ V6b @ V7c ) ) )
& ! [V8a: A_27a,V9b: A_27a,V10c: A_27a] :
( ( V0op @ V8a @ ( V0op @ V9b @ V10c ) )
= ( V0op @ V9b @ ( V0op @ V8a @ V10c ) ) ) ) ) ).
thf(thm_2Eiterate_2EIN__SUPPORT,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27a > A_27a > A_27a,V1f: A_27b > A_27a,V2x: A_27b,V3s: A_27b > $o] :
( ( c_2Ebool_2EIN @ A_27b @ V2x @ ( c_2Eiterate_2Esupport @ A_27b @ A_27a @ V0op @ V1f @ V3s ) )
<=> ( ( c_2Ebool_2EIN @ A_27b @ V2x @ V3s )
& ( (~)
@ ( ( V1f @ V2x )
= ( c_2Eiterate_2Eneutral @ A_27a @ V0op ) ) ) ) ) ).
thf(thm_2Eiterate_2ESUPPORT__SUPPORT,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27a > A_27a > A_27a,V1f: A_27b > A_27a,V2s: A_27b > $o] :
( ( c_2Eiterate_2Esupport @ A_27b @ A_27a @ V0op @ V1f @ ( c_2Eiterate_2Esupport @ A_27b @ A_27a @ V0op @ V1f @ V2s ) )
= ( c_2Eiterate_2Esupport @ A_27b @ A_27a @ V0op @ V1f @ V2s ) ) ).
thf(thm_2Eiterate_2ESUPPORT__EMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27a > A_27a > A_27a,V1f: A_27b > A_27a,V2s: A_27b > $o] :
( ! [V3x: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V3x @ V2s )
=> ( ( V1f @ V3x )
= ( c_2Eiterate_2Eneutral @ A_27a @ V0op ) ) )
<=> ( ( c_2Eiterate_2Esupport @ A_27b @ A_27a @ V0op @ V1f @ V2s )
= ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) ).
thf(thm_2Eiterate_2ESUPPORT__SUBSET,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27a > A_27a > A_27a,V1f: A_27b > A_27a,V2s: A_27b > $o] : ( c_2Epred__set_2ESUBSET @ A_27b @ ( c_2Eiterate_2Esupport @ A_27b @ A_27a @ V0op @ V1f @ V2s ) @ V2s ) ).
thf(thm_2Eiterate_2EFINITE__SUPPORT,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27a > A_27a > A_27a,V1f: A_27b > A_27a,V2s: A_27b > $o] :
( ( c_2Epred__set_2EFINITE @ A_27b @ V2s )
=> ( c_2Epred__set_2EFINITE @ A_27b @ ( c_2Eiterate_2Esupport @ A_27b @ A_27a @ V0op @ V1f @ V2s ) ) ) ).
thf(thm_2Eiterate_2ESUPPORT__CLAUSES,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType,A_27g: $tType,A_27h: $tType,A_27i: $tType,V0op: A_27b > A_27b > A_27b] :
( ! [V1f: A_27a > A_27b] :
( ( c_2Eiterate_2Esupport @ A_27a @ A_27b @ V0op @ V1f @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ! [V2f: A_27c > A_27b,V3x: A_27c,V4s: A_27c > $o] :
( ( c_2Eiterate_2Esupport @ A_27c @ A_27b @ V0op @ V2f @ ( c_2Epred__set_2EINSERT @ A_27c @ V3x @ V4s ) )
= ( c_2Ebool_2ECOND @ ( A_27c > $o ) @ ( c_2Emin_2E_3D @ A_27b @ ( V2f @ V3x ) @ ( c_2Eiterate_2Eneutral @ A_27b @ V0op ) ) @ ( c_2Eiterate_2Esupport @ A_27c @ A_27b @ V0op @ V2f @ V4s ) @ ( c_2Epred__set_2EINSERT @ A_27c @ V3x @ ( c_2Eiterate_2Esupport @ A_27c @ A_27b @ V0op @ V2f @ V4s ) ) ) )
& ! [V5f: A_27d > A_27b,V6x: A_27d,V7s: A_27d > $o] :
( ( c_2Eiterate_2Esupport @ A_27d @ A_27b @ V0op @ V5f @ ( c_2Epred__set_2EDELETE @ A_27d @ V7s @ V6x ) )
= ( c_2Epred__set_2EDELETE @ A_27d @ ( c_2Eiterate_2Esupport @ A_27d @ A_27b @ V0op @ V5f @ V7s ) @ V6x ) )
& ! [V8f: A_27e > A_27b,V9s: A_27e > $o,V10t: A_27e > $o] :
( ( c_2Eiterate_2Esupport @ A_27e @ A_27b @ V0op @ V8f @ ( c_2Epred__set_2EUNION @ A_27e @ V9s @ V10t ) )
= ( c_2Epred__set_2EUNION @ A_27e @ ( c_2Eiterate_2Esupport @ A_27e @ A_27b @ V0op @ V8f @ V9s ) @ ( c_2Eiterate_2Esupport @ A_27e @ A_27b @ V0op @ V8f @ V10t ) ) )
& ! [V11f: A_27f > A_27b,V12s: A_27f > $o,V13t: A_27f > $o] :
( ( c_2Eiterate_2Esupport @ A_27f @ A_27b @ V0op @ V11f @ ( c_2Epred__set_2EINTER @ A_27f @ V12s @ V13t ) )
= ( c_2Epred__set_2EINTER @ A_27f @ ( c_2Eiterate_2Esupport @ A_27f @ A_27b @ V0op @ V11f @ V12s ) @ ( c_2Eiterate_2Esupport @ A_27f @ A_27b @ V0op @ V11f @ V13t ) ) )
& ! [V14f: A_27g > A_27b,V15s: A_27g > $o,V16t: A_27g > $o] :
( ( c_2Eiterate_2Esupport @ A_27g @ A_27b @ V0op @ V14f @ ( c_2Epred__set_2EDIFF @ A_27g @ V15s @ V16t ) )
= ( c_2Epred__set_2EDIFF @ A_27g @ ( c_2Eiterate_2Esupport @ A_27g @ A_27b @ V0op @ V14f @ V15s ) @ ( c_2Eiterate_2Esupport @ A_27g @ A_27b @ V0op @ V14f @ V16t ) ) )
& ! [V17f: A_27h > A_27i,V18g: A_27i > A_27b,V19s: A_27h > $o] :
( ( c_2Eiterate_2Esupport @ A_27i @ A_27b @ V0op @ V18g @ ( c_2Epred__set_2EIMAGE @ A_27h @ A_27i @ V17f @ V19s ) )
= ( c_2Epred__set_2EIMAGE @ A_27h @ A_27i @ V17f @ ( c_2Eiterate_2Esupport @ A_27h @ A_27b @ V0op @ ( c_2Ecombin_2Eo @ A_27h @ A_27b @ A_27i @ V18g @ V17f ) @ V19s ) ) ) ) ).
thf(thm_2Eiterate_2ESUPPORT__DELTA,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27a > A_27a > A_27a,V1s: A_27b > $o,V2f: A_27b > A_27a,V3a: A_27b] :
( ( c_2Eiterate_2Esupport @ A_27b @ A_27a @ V0op
@ ^ [V4x: A_27b] : ( c_2Ebool_2ECOND @ A_27a @ ( c_2Emin_2E_3D @ A_27b @ V4x @ V3a ) @ ( V2f @ V4x ) @ ( c_2Eiterate_2Eneutral @ A_27a @ V0op ) )
@ V1s )
= ( c_2Ebool_2ECOND @ ( A_27b > $o ) @ ( c_2Ebool_2EIN @ A_27b @ V3a @ V1s ) @ ( c_2Eiterate_2Esupport @ A_27b @ A_27a @ V0op @ V2f @ ( c_2Epred__set_2EINSERT @ A_27b @ V3a @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) ).
thf(thm_2Eiterate_2EFINITE__SUPPORT__DELTA,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27b > $o,V1op: A_27a > A_27a > A_27a,V2f: A_27b > A_27a,V3a: A_27b] :
( c_2Epred__set_2EFINITE @ A_27b
@ ( c_2Eiterate_2Esupport @ A_27b @ A_27a @ V1op
@ ^ [V4x: A_27b] : ( c_2Ebool_2ECOND @ A_27a @ ( c_2Emin_2E_3D @ A_27b @ V4x @ V3a ) @ ( V2f @ V4x ) @ ( c_2Eiterate_2Eneutral @ A_27a @ V1op ) )
@ V0s ) ) ).
thf(thm_2Eiterate_2EITERATE__SUPPORT,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27a > A_27a > A_27a,V1f: A_27b > A_27a,V2s: A_27b > $o] :
( ( c_2Eiterate_2Eiterate @ A_27b @ A_27a @ V0op @ ( c_2Eiterate_2Esupport @ A_27b @ A_27a @ V0op @ V1f @ V2s ) @ V1f )
= ( c_2Eiterate_2Eiterate @ A_27b @ A_27a @ V0op @ V2s @ V1f ) ) ).
thf(thm_2Eiterate_2EITERATE__EXPAND__CASES,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27a > A_27a > A_27a,V1f: A_27b > A_27a,V2s: A_27b > $o] :
( ( c_2Eiterate_2Eiterate @ A_27b @ A_27a @ V0op @ V2s @ V1f )
= ( c_2Ebool_2ECOND @ A_27a @ ( c_2Epred__set_2EFINITE @ A_27b @ ( c_2Eiterate_2Esupport @ A_27b @ A_27a @ V0op @ V1f @ V2s ) ) @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a @ V0op @ ( c_2Eiterate_2Esupport @ A_27b @ A_27a @ V0op @ V1f @ V2s ) @ V1f ) @ ( c_2Eiterate_2Eneutral @ A_27a @ V0op ) ) ) ).
thf(thm_2Eiterate_2EITERATE__CLAUSES__GEN,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27b > A_27b > A_27b] :
( ( c_2Eiterate_2Emonoidal @ A_27b @ V0op )
=> ( ! [V1f: A_27a > A_27b] :
( ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V1f )
= ( c_2Eiterate_2Eneutral @ A_27b @ V0op ) )
& ! [V2f: A_27a > A_27b,V3x: A_27a,V4s: A_27a > $o] :
( ( ( c_2Eiterate_2Emonoidal @ A_27b @ V0op )
& ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Eiterate_2Esupport @ A_27a @ A_27b @ V0op @ V2f @ V4s ) ) )
=> ( ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ ( c_2Epred__set_2EINSERT @ A_27a @ V3x @ V4s ) @ V2f )
= ( c_2Ebool_2ECOND @ A_27b @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V4s ) @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ V4s @ V2f ) @ ( V0op @ ( V2f @ V3x ) @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ V4s @ V2f ) ) ) ) ) ) ) ).
thf(thm_2Eiterate_2EITERATE__CLAUSES,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27a > A_27a > A_27a] :
( ( c_2Eiterate_2Emonoidal @ A_27a @ V0op )
=> ( ! [V1f: A_27b > A_27a] :
( ( c_2Eiterate_2Eiterate @ A_27b @ A_27a @ V0op @ ( c_2Epred__set_2EEMPTY @ A_27b ) @ V1f )
= ( c_2Eiterate_2Eneutral @ A_27a @ V0op ) )
& ! [V2f: A_27b > A_27a,V3x: A_27b,V4s: A_27b > $o] :
( ( c_2Epred__set_2EFINITE @ A_27b @ V4s )
=> ( ( c_2Eiterate_2Eiterate @ A_27b @ A_27a @ V0op @ ( c_2Epred__set_2EINSERT @ A_27b @ V3x @ V4s ) @ V2f )
= ( c_2Ebool_2ECOND @ A_27a @ ( c_2Ebool_2EIN @ A_27b @ V3x @ V4s ) @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a @ V0op @ V4s @ V2f ) @ ( V0op @ ( V2f @ V3x ) @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a @ V0op @ V4s @ V2f ) ) ) ) ) ) ) ).
thf(thm_2Eiterate_2EITERATE__UNION,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27a > A_27a > A_27a] :
( ( c_2Eiterate_2Emonoidal @ A_27a @ V0op )
=> ! [V1f: A_27b > A_27a,V2s: A_27b > $o,V3t: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27b @ V2s )
& ( c_2Epred__set_2EFINITE @ A_27b @ V3t )
& ( c_2Epred__set_2EDISJOINT @ A_27b @ V2s @ V3t ) )
=> ( ( c_2Eiterate_2Eiterate @ A_27b @ A_27a @ V0op @ ( c_2Epred__set_2EUNION @ A_27b @ V2s @ V3t ) @ V1f )
= ( V0op @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a @ V0op @ V2s @ V1f ) @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a @ V0op @ V3t @ V1f ) ) ) ) ) ).
thf(thm_2Eiterate_2EITERATE__UNION__GEN,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27b > A_27b > A_27b] :
( ( c_2Eiterate_2Emonoidal @ A_27b @ V0op )
=> ! [V1f: A_27a > A_27b,V2s: A_27a > $o,V3t: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Eiterate_2Esupport @ A_27a @ A_27b @ V0op @ V1f @ V2s ) )
& ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Eiterate_2Esupport @ A_27a @ A_27b @ V0op @ V1f @ V3t ) )
& ( c_2Epred__set_2EDISJOINT @ A_27a @ ( c_2Eiterate_2Esupport @ A_27a @ A_27b @ V0op @ V1f @ V2s ) @ ( c_2Eiterate_2Esupport @ A_27a @ A_27b @ V0op @ V1f @ V3t ) ) )
=> ( ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ ( c_2Epred__set_2EUNION @ A_27a @ V2s @ V3t ) @ V1f )
= ( V0op @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ V2s @ V1f ) @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ V3t @ V1f ) ) ) ) ) ).
thf(thm_2Eiterate_2EITERATE__DIFF,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27a > A_27a > A_27a] :
( ( c_2Eiterate_2Emonoidal @ A_27a @ V0op )
=> ! [V1f: A_27b > A_27a,V2s: A_27b > $o,V3t: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27b @ V2s )
& ( c_2Epred__set_2ESUBSET @ A_27b @ V3t @ V2s ) )
=> ( ( V0op @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a @ V0op @ ( c_2Epred__set_2EDIFF @ A_27b @ V2s @ V3t ) @ V1f ) @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a @ V0op @ V3t @ V1f ) )
= ( c_2Eiterate_2Eiterate @ A_27b @ A_27a @ V0op @ V2s @ V1f ) ) ) ) ).
thf(thm_2Eiterate_2EITERATE__DIFF__GEN,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27b > A_27b > A_27b] :
( ( c_2Eiterate_2Emonoidal @ A_27b @ V0op )
=> ! [V1f: A_27a > A_27b,V2s: A_27a > $o,V3t: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Eiterate_2Esupport @ A_27a @ A_27b @ V0op @ V1f @ V2s ) )
& ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Eiterate_2Esupport @ A_27a @ A_27b @ V0op @ V1f @ V3t ) @ ( c_2Eiterate_2Esupport @ A_27a @ A_27b @ V0op @ V1f @ V2s ) ) )
=> ( ( V0op @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ ( c_2Epred__set_2EDIFF @ A_27a @ V2s @ V3t ) @ V1f ) @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ V3t @ V1f ) )
= ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ V2s @ V1f ) ) ) ) ).
thf(thm_2Eiterate_2EITERATE__INCL__EXCL,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27a > A_27a > A_27a] :
( ( c_2Eiterate_2Emonoidal @ A_27a @ V0op )
=> ! [V1s: A_27b > $o,V2t: A_27b > $o,V3f: A_27b > A_27a] :
( ( ( c_2Epred__set_2EFINITE @ A_27b @ V1s )
& ( c_2Epred__set_2EFINITE @ A_27b @ V2t ) )
=> ( ( V0op @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a @ V0op @ V1s @ V3f ) @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a @ V0op @ V2t @ V3f ) )
= ( V0op @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a @ V0op @ ( c_2Epred__set_2EUNION @ A_27b @ V1s @ V2t ) @ V3f ) @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a @ V0op @ ( c_2Epred__set_2EINTER @ A_27b @ V1s @ V2t ) @ V3f ) ) ) ) ) ).
thf(thm_2Eiterate_2EITERATE__CLOSED,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27b > A_27b > A_27b] :
( ( c_2Eiterate_2Emonoidal @ A_27b @ V0op )
=> ! [V1P: A_27b > $o] :
( ( ( V1P @ ( c_2Eiterate_2Eneutral @ A_27b @ V0op ) )
& ! [V2x: A_27b,V3y: A_27b] :
( ( ( V1P @ V2x )
& ( V1P @ V3y ) )
=> ( V1P @ ( V0op @ V2x @ V3y ) ) ) )
=> ! [V4f: A_27a > A_27b,V5s: A_27a > $o] :
( ! [V6x: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V6x @ V5s )
& ( (~)
@ ( ( V4f @ V6x )
= ( c_2Eiterate_2Eneutral @ A_27b @ V0op ) ) ) )
=> ( V1P @ ( V4f @ V6x ) ) )
=> ( V1P @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ V5s @ V4f ) ) ) ) ) ).
thf(thm_2Eiterate_2EITERATE__RELATED,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27b > A_27b > A_27b] :
( ( c_2Eiterate_2Emonoidal @ A_27b @ V0op )
=> ! [V1R: A_27b > A_27b > $o] :
( ( ( V1R @ ( c_2Eiterate_2Eneutral @ A_27b @ V0op ) @ ( c_2Eiterate_2Eneutral @ A_27b @ V0op ) )
& ! [V2x1: A_27b,V3y1: A_27b,V4x2: A_27b,V5y2: A_27b] :
( ( ( V1R @ V2x1 @ V4x2 )
& ( V1R @ V3y1 @ V5y2 ) )
=> ( V1R @ ( V0op @ V2x1 @ V3y1 ) @ ( V0op @ V4x2 @ V5y2 ) ) ) )
=> ! [V6f: A_27a > A_27b,V7g: A_27a > A_27b,V8s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V8s )
& ! [V9x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V9x @ V8s )
=> ( V1R @ ( V6f @ V9x ) @ ( V7g @ V9x ) ) ) )
=> ( V1R @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ V8s @ V6f ) @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ V8s @ V7g ) ) ) ) ) ).
thf(thm_2Eiterate_2EITERATE__EQ__NEUTRAL,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27b > A_27b > A_27b] :
( ( c_2Eiterate_2Emonoidal @ A_27b @ V0op )
=> ! [V1f: A_27a > A_27b,V2s: A_27a > $o] :
( ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
=> ( ( V1f @ V3x )
= ( c_2Eiterate_2Eneutral @ A_27b @ V0op ) ) )
=> ( ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ V2s @ V1f )
= ( c_2Eiterate_2Eneutral @ A_27b @ V0op ) ) ) ) ).
thf(thm_2Eiterate_2EITERATE__SING,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27b > A_27b > A_27b] :
( ( c_2Eiterate_2Emonoidal @ A_27b @ V0op )
=> ! [V1f: A_27a > A_27b,V2x: A_27a] :
( ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ ( c_2Epred__set_2EINSERT @ A_27a @ V2x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V1f )
= ( V1f @ V2x ) ) ) ).
thf(thm_2Eiterate_2EITERATE__DELETE,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27b > A_27b > A_27b] :
( ( c_2Eiterate_2Emonoidal @ A_27b @ V0op )
=> ! [V1f: A_27a > A_27b,V2s: A_27a > $o,V3a: A_27a] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
& ( c_2Ebool_2EIN @ A_27a @ V3a @ V2s ) )
=> ( ( V0op @ ( V1f @ V3a ) @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ ( c_2Epred__set_2EDELETE @ A_27a @ V2s @ V3a ) @ V1f ) )
= ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ V2s @ V1f ) ) ) ) ).
thf(thm_2Eiterate_2EITERATE__DELTA,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27a > A_27a > A_27a] :
( ( c_2Eiterate_2Emonoidal @ A_27a @ V0op )
=> ! [V1f: A_27b > A_27a,V2a: A_27b,V3s: A_27b > $o] :
( ( c_2Eiterate_2Eiterate @ A_27b @ A_27a @ V0op @ V3s
@ ^ [V4x: A_27b] : ( c_2Ebool_2ECOND @ A_27a @ ( c_2Emin_2E_3D @ A_27b @ V4x @ V2a ) @ ( V1f @ V4x ) @ ( c_2Eiterate_2Eneutral @ A_27a @ V0op ) ) )
= ( c_2Ebool_2ECOND @ A_27a @ ( c_2Ebool_2EIN @ A_27b @ V2a @ V3s ) @ ( V1f @ V2a ) @ ( c_2Eiterate_2Eneutral @ A_27a @ V0op ) ) ) ) ).
thf(thm_2Eiterate_2EITERATE__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0op: A_27c > A_27c > A_27c] :
( ( c_2Eiterate_2Emonoidal @ A_27c @ V0op )
=> ! [V1f: A_27a > A_27b,V2g: A_27b > A_27c,V3s: A_27a > $o] :
( ! [V4x: A_27a,V5y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V4x @ V3s )
& ( c_2Ebool_2EIN @ A_27a @ V5y @ V3s )
& ( ( V1f @ V4x )
= ( V1f @ V5y ) ) )
=> ( V4x = V5y ) )
=> ( ( c_2Eiterate_2Eiterate @ A_27b @ A_27c @ V0op @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V1f @ V3s ) @ V2g )
= ( c_2Eiterate_2Eiterate @ A_27a @ A_27c @ V0op @ V3s @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b @ V2g @ V1f ) ) ) ) ) ).
thf(thm_2Eiterate_2EITERATE__BIJECTION,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27b > A_27b > A_27b] :
( ( c_2Eiterate_2Emonoidal @ A_27b @ V0op )
=> ! [V1f: A_27a > A_27b,V2p: A_27a > A_27a,V3s: A_27a > $o] :
( ( ! [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ V3s )
=> ( c_2Ebool_2EIN @ A_27a @ ( V2p @ V4x ) @ V3s ) )
& ! [V5y: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V5y @ V3s )
=> ( c_2Ebool_2E_3F_21 @ A_27a
@ ^ [V6x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V6x @ V3s ) @ ( c_2Emin_2E_3D @ A_27a @ ( V2p @ V6x ) @ V5y ) ) ) ) )
=> ( ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ V3s @ V1f )
= ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ V3s @ ( c_2Ecombin_2Eo @ A_27a @ A_27b @ A_27a @ V1f @ V2p ) ) ) ) ) ).
thf(thm_2Eiterate_2EITERATE__ITERATE__PRODUCT,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0op: A_27c > A_27c > A_27c] :
( ( c_2Eiterate_2Emonoidal @ A_27c @ V0op )
=> ! [V1s: A_27a > $o,V2t: A_27a > A_27b > $o,V3x: A_27a > A_27b > A_27c] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ! [V4i: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4i @ V1s )
=> ( c_2Epred__set_2EFINITE @ A_27b @ ( V2t @ V4i ) ) ) )
=> ( ( c_2Eiterate_2Eiterate @ A_27a @ A_27c @ V0op @ V1s
@ ^ [V5i: A_27a] : ( c_2Eiterate_2Eiterate @ A_27b @ A_27c @ V0op @ ( V2t @ V5i ) @ ( V3x @ V5i ) ) )
= ( c_2Eiterate_2Eiterate @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c @ V0op
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( tyop_2Epair_2Eprod @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ $o )
@ ^ [V6i: A_27a,V7j: A_27b] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ $o @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V6i @ V7j ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V6i @ V1s ) @ ( c_2Ebool_2EIN @ A_27b @ V7j @ ( V2t @ V6i ) ) ) ) ) )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c
@ ^ [V8i: A_27a,V9j: A_27b] : ( V3x @ V8i @ V9j ) ) ) ) ) ) ).
thf(thm_2Eiterate_2EITERATE__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27b > A_27b > A_27b] :
( ( c_2Eiterate_2Emonoidal @ A_27b @ V0op )
=> ! [V1f: A_27a > A_27b,V2g: A_27a > A_27b,V3s: A_27a > $o] :
( ! [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ V3s )
=> ( ( V1f @ V4x )
= ( V2g @ V4x ) ) )
=> ( ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ V3s @ V1f )
= ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ V3s @ V2g ) ) ) ) ).
thf(thm_2Eiterate_2EITERATE__EQ__GENERAL,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0op: A_27c > A_27c > A_27c] :
( ( c_2Eiterate_2Emonoidal @ A_27c @ V0op )
=> ! [V1s: A_27a > $o,V2t: A_27b > $o,V3f: A_27a > A_27c,V4g: A_27b > A_27c,V5h: A_27a > A_27b] :
( ( ! [V6y: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V6y @ V2t )
=> ( c_2Ebool_2E_3F_21 @ A_27a
@ ^ [V7x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V7x @ V1s ) @ ( c_2Emin_2E_3D @ A_27b @ ( V5h @ V7x ) @ V6y ) ) ) )
& ! [V8x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V8x @ V1s )
=> ( ( c_2Ebool_2EIN @ A_27b @ ( V5h @ V8x ) @ V2t )
& ( ( V4g @ ( V5h @ V8x ) )
= ( V3f @ V8x ) ) ) ) )
=> ( ( c_2Eiterate_2Eiterate @ A_27a @ A_27c @ V0op @ V1s @ V3f )
= ( c_2Eiterate_2Eiterate @ A_27b @ A_27c @ V0op @ V2t @ V4g ) ) ) ) ).
thf(thm_2Eiterate_2EITERATE__EQ__GENERAL__INVERSES,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0op: A_27c > A_27c > A_27c] :
( ( c_2Eiterate_2Emonoidal @ A_27c @ V0op )
=> ! [V1s: A_27a > $o,V2t: A_27b > $o,V3f: A_27a > A_27c,V4g: A_27b > A_27c,V5h: A_27a > A_27b,V6k: A_27b > A_27a] :
( ( ! [V7y: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V7y @ V2t )
=> ( ( c_2Ebool_2EIN @ A_27a @ ( V6k @ V7y ) @ V1s )
& ( ( V5h @ ( V6k @ V7y ) )
= V7y ) ) )
& ! [V8x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V8x @ V1s )
=> ( ( c_2Ebool_2EIN @ A_27b @ ( V5h @ V8x ) @ V2t )
& ( ( V6k @ ( V5h @ V8x ) )
= V8x )
& ( ( V4g @ ( V5h @ V8x ) )
= ( V3f @ V8x ) ) ) ) )
=> ( ( c_2Eiterate_2Eiterate @ A_27a @ A_27c @ V0op @ V1s @ V3f )
= ( c_2Eiterate_2Eiterate @ A_27b @ A_27c @ V0op @ V2t @ V4g ) ) ) ) ).
thf(thm_2Eiterate_2EITERATE__INJECTION,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27b > A_27b > A_27b] :
( ( c_2Eiterate_2Emonoidal @ A_27b @ V0op )
=> ! [V1f: A_27a > A_27b,V2p: A_27a > A_27a,V3s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V3s )
& ! [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ V3s )
=> ( c_2Ebool_2EIN @ A_27a @ ( V2p @ V4x ) @ V3s ) )
& ! [V5x: A_27a,V6y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V5x @ V3s )
& ( c_2Ebool_2EIN @ A_27a @ V6y @ V3s )
& ( ( V2p @ V5x )
= ( V2p @ V6y ) ) )
=> ( V5x = V6y ) ) )
=> ( ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ V3s @ ( c_2Ecombin_2Eo @ A_27a @ A_27b @ A_27a @ V1f @ V2p ) )
= ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ V3s @ V1f ) ) ) ) ).
thf(thm_2Eiterate_2EITERATE__UNION__NONZERO,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27b > A_27b > A_27b] :
( ( c_2Eiterate_2Emonoidal @ A_27b @ V0op )
=> ! [V1f: A_27a > A_27b,V2s: A_27a > $o,V3t: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
& ( c_2Epred__set_2EFINITE @ A_27a @ V3t )
& ! [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ ( c_2Epred__set_2EINTER @ A_27a @ V2s @ V3t ) )
=> ( ( V1f @ V4x )
= ( c_2Eiterate_2Eneutral @ A_27b @ V0op ) ) ) )
=> ( ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ ( c_2Epred__set_2EUNION @ A_27a @ V2s @ V3t ) @ V1f )
= ( V0op @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ V2s @ V1f ) @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ V3t @ V1f ) ) ) ) ) ).
thf(thm_2Eiterate_2EITERATE__OP,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27a > A_27a > A_27a] :
( ( c_2Eiterate_2Emonoidal @ A_27a @ V0op )
=> ! [V1f: A_27b > A_27a,V2g: A_27b > A_27a,V3s: A_27b > $o] :
( ( c_2Epred__set_2EFINITE @ A_27b @ V3s )
=> ( ( c_2Eiterate_2Eiterate @ A_27b @ A_27a @ V0op @ V3s
@ ^ [V4x: A_27b] : ( V0op @ ( V1f @ V4x ) @ ( V2g @ V4x ) ) )
= ( V0op @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a @ V0op @ V3s @ V1f ) @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a @ V0op @ V3s @ V2g ) ) ) ) ) ).
thf(thm_2Eiterate_2EITERATE__SUPERSET,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27b > A_27b > A_27b] :
( ( c_2Eiterate_2Emonoidal @ A_27b @ V0op )
=> ! [V1f: A_27a > A_27b,V2u: A_27a > $o,V3v: A_27a > $o] :
( ( ( c_2Epred__set_2ESUBSET @ A_27a @ V2u @ V3v )
& ! [V4x: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V4x @ V3v )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V4x @ V2u ) ) )
=> ( ( V1f @ V4x )
= ( c_2Eiterate_2Eneutral @ A_27b @ V0op ) ) ) )
=> ( ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ V3v @ V1f )
= ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ V2u @ V1f ) ) ) ) ).
thf(thm_2Eiterate_2EITERATE__IMAGE__NONZERO,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0op: A_27c > A_27c > A_27c] :
( ( c_2Eiterate_2Emonoidal @ A_27c @ V0op )
=> ! [V1g: A_27b > A_27c,V2f: A_27a > A_27b,V3s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V3s )
& ! [V4x: A_27a,V5y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V4x @ V3s )
& ( c_2Ebool_2EIN @ A_27a @ V5y @ V3s )
& ( (~) @ ( V4x = V5y ) )
& ( ( V2f @ V4x )
= ( V2f @ V5y ) ) )
=> ( ( V1g @ ( V2f @ V4x ) )
= ( c_2Eiterate_2Eneutral @ A_27c @ V0op ) ) ) )
=> ( ( c_2Eiterate_2Eiterate @ A_27b @ A_27c @ V0op @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V2f @ V3s ) @ V1g )
= ( c_2Eiterate_2Eiterate @ A_27a @ A_27c @ V0op @ V3s @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b @ V1g @ V2f ) ) ) ) ) ).
thf(thm_2Eiterate_2EITERATE__CASES,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27b > A_27b > A_27b] :
( ( c_2Eiterate_2Emonoidal @ A_27b @ V0op )
=> ! [V1s: A_27a > $o,V2P: A_27a > $o,V3f: A_27a > A_27b,V4g: A_27a > A_27b] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ( ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ V1s
@ ^ [V5x: A_27a] : ( c_2Ebool_2ECOND @ A_27b @ ( V2P @ V5x ) @ ( V3f @ V5x ) @ ( V4g @ V5x ) ) )
= ( V0op
@ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V6x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V6x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V6x @ V1s ) @ ( V2P @ V6x ) ) ) )
@ V3f )
@ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V7x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V7x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V7x @ V1s ) @ ( c_2Ebool_2E_7E @ ( V2P @ V7x ) ) ) ) )
@ V4g ) ) ) ) ) ).
thf(thm_2Eiterate_2EITERATE__OP__GEN,axiom,
! [A_27a: $tType,A_27b: $tType,V0op: A_27b > A_27b > A_27b] :
( ( c_2Eiterate_2Emonoidal @ A_27b @ V0op )
=> ! [V1f: A_27a > A_27b,V2g: A_27a > A_27b,V3s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Eiterate_2Esupport @ A_27a @ A_27b @ V0op @ V1f @ V3s ) )
& ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Eiterate_2Esupport @ A_27a @ A_27b @ V0op @ V2g @ V3s ) ) )
=> ( ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ V3s
@ ^ [V4x: A_27a] : ( V0op @ ( V1f @ V4x ) @ ( V2g @ V4x ) ) )
= ( V0op @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ V3s @ V1f ) @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b @ V0op @ V3s @ V2g ) ) ) ) ) ).
thf(thm_2Eiterate_2EITERATE__CLAUSES__NUMSEG,axiom,
! [A_27a: $tType,V0f: tyop_2Enum_2Enum > A_27a,V1op: A_27a > A_27a > A_27a] :
( ( c_2Eiterate_2Emonoidal @ A_27a @ V1op )
=> ( ! [V2m: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2Eiterate @ tyop_2Enum_2Enum @ A_27a @ V1op @ ( c_2Eiterate_2E_2E_2E @ V2m @ c_2Enum_2E0 ) @ V0f )
= ( c_2Ebool_2ECOND @ A_27a @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V2m @ c_2Enum_2E0 ) @ ( V0f @ c_2Enum_2E0 ) @ ( c_2Eiterate_2Eneutral @ A_27a @ V1op ) ) )
& ! [V3m: tyop_2Enum_2Enum,V4n: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2Eiterate @ tyop_2Enum_2Enum @ A_27a @ V1op @ ( c_2Eiterate_2E_2E_2E @ V3m @ ( c_2Enum_2ESUC @ V4n ) ) @ V0f )
= ( c_2Ebool_2ECOND @ A_27a @ ( c_2Earithmetic_2E_3C_3D @ V3m @ ( c_2Enum_2ESUC @ V4n ) ) @ ( V1op @ ( c_2Eiterate_2Eiterate @ tyop_2Enum_2Enum @ A_27a @ V1op @ ( c_2Eiterate_2E_2E_2E @ V3m @ V4n ) @ V0f ) @ ( V0f @ ( c_2Enum_2ESUC @ V4n ) ) ) @ ( c_2Eiterate_2Eiterate @ tyop_2Enum_2Enum @ A_27a @ V1op @ ( c_2Eiterate_2E_2E_2E @ V3m @ V4n ) @ V0f ) ) ) ) ) ).
thf(thm_2Eiterate_2EITERATE__PAIR,axiom,
! [A_27a: $tType,V0op: A_27a > A_27a > A_27a] :
( ( c_2Eiterate_2Emonoidal @ A_27a @ V0op )
=> ! [V1f: tyop_2Enum_2Enum > A_27a,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2Eiterate @ tyop_2Enum_2Enum @ A_27a @ V0op @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V2m ) @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V3n ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V1f )
= ( c_2Eiterate_2Eiterate @ tyop_2Enum_2Enum @ A_27a @ V0op @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n )
@ ^ [V4i: tyop_2Enum_2Enum] : ( V0op @ ( V1f @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V4i ) ) @ ( V1f @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V4i ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(thm_2Eiterate_2ENEUTRAL__ADD,axiom,
( ( c_2Eiterate_2Eneutral @ tyop_2Enum_2Enum @ c_2Earithmetic_2E_2B )
= c_2Enum_2E0 ) ).
thf(thm_2Eiterate_2ENEUTRAL__MUL,axiom,
( ( c_2Eiterate_2Eneutral @ tyop_2Enum_2Enum @ c_2Earithmetic_2E_2A )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ).
thf(thm_2Eiterate_2EMONOIDAL__ADD,axiom,
c_2Eiterate_2Emonoidal @ tyop_2Enum_2Enum @ c_2Earithmetic_2E_2B ).
thf(thm_2Eiterate_2EMONOIDAL__MUL,axiom,
c_2Eiterate_2Emonoidal @ tyop_2Enum_2Enum @ c_2Earithmetic_2E_2A ).
thf(thm_2Eiterate_2ENSUM__DEGENERATE,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o] :
( ( (~)
@ ( c_2Epred__set_2EFINITE @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V2x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V2x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s ) @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ ( V0f @ V2x ) @ c_2Enum_2E0 ) ) ) ) ) ) )
=> ( ( c_2Eiterate_2Ensum @ A_27a @ V1s @ V0f )
= c_2Enum_2E0 ) ) ).
thf(thm_2Eiterate_2ENSUM__CLAUSES,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0f: A_27a > tyop_2Enum_2Enum] :
( ( c_2Eiterate_2Ensum @ A_27a @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V0f )
= c_2Enum_2E0 )
& ! [V1x: A_27b,V2f: A_27b > tyop_2Enum_2Enum,V3s: A_27b > $o] :
( ( c_2Epred__set_2EFINITE @ A_27b @ V3s )
=> ( ( c_2Eiterate_2Ensum @ A_27b @ ( c_2Epred__set_2EINSERT @ A_27b @ V1x @ V3s ) @ V2f )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Ebool_2EIN @ A_27b @ V1x @ V3s ) @ ( c_2Eiterate_2Ensum @ A_27b @ V3s @ V2f ) @ ( c_2Earithmetic_2E_2B @ ( V2f @ V1x ) @ ( c_2Eiterate_2Ensum @ A_27b @ V3s @ V2f ) ) ) ) ) ) ).
thf(thm_2Eiterate_2ENSUM__UNION,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( c_2Epred__set_2EFINITE @ A_27a @ V2t )
& ( c_2Epred__set_2EDISJOINT @ A_27a @ V1s @ V2t ) )
=> ( ( c_2Eiterate_2Ensum @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) @ V0f )
= ( c_2Earithmetic_2E_2B @ ( c_2Eiterate_2Ensum @ A_27a @ V1s @ V0f ) @ ( c_2Eiterate_2Ensum @ A_27a @ V2t @ V0f ) ) ) ) ).
thf(thm_2Eiterate_2ENSUM__DIFF,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V2t @ V1s ) )
=> ( ( c_2Eiterate_2Ensum @ A_27a @ ( c_2Epred__set_2EDIFF @ A_27a @ V1s @ V2t ) @ V0f )
= ( c_2Earithmetic_2E_2D @ ( c_2Eiterate_2Ensum @ A_27a @ V1s @ V0f ) @ ( c_2Eiterate_2Ensum @ A_27a @ V2t @ V0f ) ) ) ) ).
thf(thm_2Eiterate_2ENSUM__INCL__EXCL,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2f: A_27a > tyop_2Enum_2Enum] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( c_2Epred__set_2EFINITE @ A_27a @ V1t ) )
=> ( ( c_2Earithmetic_2E_2B @ ( c_2Eiterate_2Ensum @ A_27a @ V0s @ V2f ) @ ( c_2Eiterate_2Ensum @ A_27a @ V1t @ V2f ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Eiterate_2Ensum @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V0s @ V1t ) @ V2f ) @ ( c_2Eiterate_2Ensum @ A_27a @ ( c_2Epred__set_2EINTER @ A_27a @ V0s @ V1t ) @ V2f ) ) ) ) ).
thf(thm_2Eiterate_2ENSUM__SUPPORT,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o] :
( ( c_2Eiterate_2Ensum @ A_27a @ ( c_2Eiterate_2Esupport @ A_27a @ tyop_2Enum_2Enum @ c_2Earithmetic_2E_2B @ V0f @ V1s ) @ V0f )
= ( c_2Eiterate_2Ensum @ A_27a @ V1s @ V0f ) ) ).
thf(thm_2Eiterate_2ENSUM__ADD,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1g: A_27a > tyop_2Enum_2Enum,V2s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
=> ( ( c_2Eiterate_2Ensum @ A_27a @ V2s
@ ^ [V3x: A_27a] : ( c_2Earithmetic_2E_2B @ ( V0f @ V3x ) @ ( V1g @ V3x ) ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Eiterate_2Ensum @ A_27a @ V2s @ V0f ) @ ( c_2Eiterate_2Ensum @ A_27a @ V2s @ V1g ) ) ) ) ).
thf(thm_2Eiterate_2ENSUM__ADD__GEN,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1g: A_27a > tyop_2Enum_2Enum,V2s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V3x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V3x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s ) @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ ( V0f @ V3x ) @ c_2Enum_2E0 ) ) ) ) ) )
& ( c_2Epred__set_2EFINITE @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V4x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V4x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V4x @ V2s ) @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ ( V1g @ V4x ) @ c_2Enum_2E0 ) ) ) ) ) ) )
=> ( ( c_2Eiterate_2Ensum @ A_27a @ V2s
@ ^ [V5x: A_27a] : ( c_2Earithmetic_2E_2B @ ( V0f @ V5x ) @ ( V1g @ V5x ) ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Eiterate_2Ensum @ A_27a @ V2s @ V0f ) @ ( c_2Eiterate_2Ensum @ A_27a @ V2s @ V1g ) ) ) ) ).
thf(thm_2Eiterate_2ENSUM__EQ__0,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o] :
( ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
=> ( ( V0f @ V2x )
= c_2Enum_2E0 ) )
=> ( ( c_2Eiterate_2Ensum @ A_27a @ V1s @ V0f )
= c_2Enum_2E0 ) ) ).
thf(thm_2Eiterate_2ENSUM__0,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Eiterate_2Ensum @ A_27a @ V0s
@ ^ [V1n: A_27a] : c_2Enum_2E0 )
= c_2Enum_2E0 ) ).
thf(thm_2Eiterate_2ENSUM__LMUL,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1c: tyop_2Enum_2Enum,V2s: A_27a > $o] :
( ( c_2Eiterate_2Ensum @ A_27a @ V2s
@ ^ [V3x: A_27a] : ( c_2Earithmetic_2E_2A @ V1c @ ( V0f @ V3x ) ) )
= ( c_2Earithmetic_2E_2A @ V1c @ ( c_2Eiterate_2Ensum @ A_27a @ V2s @ V0f ) ) ) ).
thf(thm_2Eiterate_2ENSUM__RMUL,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1c: tyop_2Enum_2Enum,V2s: A_27a > $o] :
( ( c_2Eiterate_2Ensum @ A_27a @ V2s
@ ^ [V3x: A_27a] : ( c_2Earithmetic_2E_2A @ ( V0f @ V3x ) @ V1c ) )
= ( c_2Earithmetic_2E_2A @ ( c_2Eiterate_2Ensum @ A_27a @ V2s @ V0f ) @ V1c ) ) ).
thf(thm_2Eiterate_2ENSUM__LE,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1g: A_27a > tyop_2Enum_2Enum,V2s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
=> ( c_2Earithmetic_2E_3C_3D @ ( V0f @ V3x ) @ ( V1g @ V3x ) ) ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Eiterate_2Ensum @ A_27a @ V2s @ V0f ) @ ( c_2Eiterate_2Ensum @ A_27a @ V2s @ V1g ) ) ) ).
thf(thm_2Eiterate_2ENSUM__LT,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1g: A_27a > tyop_2Enum_2Enum,V2s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
=> ( c_2Earithmetic_2E_3C_3D @ ( V0f @ V3x ) @ ( V1g @ V3x ) ) )
& ? [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ V2s )
& ( c_2Eprim__rec_2E_3C @ ( V0f @ V4x ) @ ( V1g @ V4x ) ) ) )
=> ( c_2Eprim__rec_2E_3C @ ( c_2Eiterate_2Ensum @ A_27a @ V2s @ V0f ) @ ( c_2Eiterate_2Ensum @ A_27a @ V2s @ V1g ) ) ) ).
thf(thm_2Eiterate_2ENSUM__LT__ALL,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1g: A_27a > tyop_2Enum_2Enum,V2s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
& ( (~)
@ ( V2s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
=> ( c_2Eprim__rec_2E_3C @ ( V0f @ V3x ) @ ( V1g @ V3x ) ) ) )
=> ( c_2Eprim__rec_2E_3C @ ( c_2Eiterate_2Ensum @ A_27a @ V2s @ V0f ) @ ( c_2Eiterate_2Ensum @ A_27a @ V2s @ V1g ) ) ) ).
thf(thm_2Eiterate_2ENSUM__EQ,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1g: A_27a > tyop_2Enum_2Enum,V2s: A_27a > $o] :
( ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
=> ( ( V0f @ V3x )
= ( V1g @ V3x ) ) )
=> ( ( c_2Eiterate_2Ensum @ A_27a @ V2s @ V0f )
= ( c_2Eiterate_2Ensum @ A_27a @ V2s @ V1g ) ) ) ).
thf(thm_2Eiterate_2ENSUM__CONST,axiom,
! [A_27a: $tType,V0c: tyop_2Enum_2Enum,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ( ( c_2Eiterate_2Ensum @ A_27a @ V1s
@ ^ [V2n: A_27a] : V0c )
= ( c_2Earithmetic_2E_2A @ ( c_2Epred__set_2ECARD @ A_27a @ V1s ) @ V0c ) ) ) ).
thf(thm_2Eiterate_2ENSUM__POS__BOUND,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum,V2s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
& ( c_2Earithmetic_2E_3C_3D @ ( c_2Eiterate_2Ensum @ A_27a @ V2s @ V0f ) @ V1b ) )
=> ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
=> ( c_2Earithmetic_2E_3C_3D @ ( V0f @ V3x ) @ V1b ) ) ) ).
thf(thm_2Eiterate_2ENSUM__EQ__0__IFF,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ( ( ( c_2Eiterate_2Ensum @ A_27a @ V1s @ V0f )
= c_2Enum_2E0 )
<=> ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
=> ( ( V0f @ V2x )
= c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Eiterate_2ENSUM__POS__LT,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ? [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ ( V0f @ V2x ) ) ) )
=> ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ ( c_2Eiterate_2Ensum @ A_27a @ V1s @ V0f ) ) ) ).
thf(thm_2Eiterate_2ENSUM__POS__LT__ALL,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1f: A_27a > tyop_2Enum_2Enum] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& ! [V2i: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2i @ V0s )
=> ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ ( V1f @ V2i ) ) ) )
=> ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ ( c_2Eiterate_2Ensum @ A_27a @ V0s @ V1f ) ) ) ).
thf(thm_2Eiterate_2ENSUM__DELETE,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o,V2a: A_27a] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ V2a @ V1s ) )
=> ( ( c_2Earithmetic_2E_2B @ ( V0f @ V2a ) @ ( c_2Eiterate_2Ensum @ A_27a @ ( c_2Epred__set_2EDELETE @ A_27a @ V1s @ V2a ) @ V0f ) )
= ( c_2Eiterate_2Ensum @ A_27a @ V1s @ V0f ) ) ) ).
thf(thm_2Eiterate_2ENSUM__SING,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1x: A_27a] :
( ( c_2Eiterate_2Ensum @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V0f )
= ( V0f @ V1x ) ) ).
thf(thm_2Eiterate_2ENSUM__DELTA,axiom,
! [A_27a: $tType,V0b: tyop_2Enum_2Enum,V1s: A_27a > $o,V2a: A_27a] :
( ( c_2Eiterate_2Ensum @ A_27a @ V1s
@ ^ [V3x: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Emin_2E_3D @ A_27a @ V3x @ V2a ) @ V0b @ c_2Enum_2E0 ) )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Ebool_2EIN @ A_27a @ V2a @ V1s ) @ V0b @ c_2Enum_2E0 ) ) ).
thf(thm_2Eiterate_2ENSUM__SWAP,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > tyop_2Enum_2Enum,V1s: A_27a > $o,V2t: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( c_2Epred__set_2EFINITE @ A_27b @ V2t ) )
=> ( ( c_2Eiterate_2Ensum @ A_27a @ V1s
@ ^ [V3i: A_27a] : ( c_2Eiterate_2Ensum @ A_27b @ V2t @ ( V0f @ V3i ) ) )
= ( c_2Eiterate_2Ensum @ A_27b @ V2t
@ ^ [V4j: A_27b] :
( c_2Eiterate_2Ensum @ A_27a @ V1s
@ ^ [V5i: A_27a] : ( V0f @ V5i @ V4j ) ) ) ) ) ).
thf(thm_2Eiterate_2ENSUM__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1g: A_27b > tyop_2Enum_2Enum,V2s: A_27a > $o] :
( ! [V3x: A_27a,V4y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
& ( c_2Ebool_2EIN @ A_27a @ V4y @ V2s )
& ( ( V0f @ V3x )
= ( V0f @ V4y ) ) )
=> ( V3x = V4y ) )
=> ( ( c_2Eiterate_2Ensum @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V2s ) @ V1g )
= ( c_2Eiterate_2Ensum @ A_27a @ V2s @ ( c_2Ecombin_2Eo @ A_27a @ tyop_2Enum_2Enum @ A_27b @ V1g @ V0f ) ) ) ) ).
thf(thm_2Eiterate_2ENSUM__SUPERSET,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1u: A_27a > $o,V2v: A_27a > $o] :
( ( ( c_2Epred__set_2ESUBSET @ A_27a @ V1u @ V2v )
& ! [V3x: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2v )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V1u ) ) )
=> ( ( V0f @ V3x )
= c_2Enum_2E0 ) ) )
=> ( ( c_2Eiterate_2Ensum @ A_27a @ V2v @ V0f )
= ( c_2Eiterate_2Ensum @ A_27a @ V1u @ V0f ) ) ) ).
thf(thm_2Eiterate_2ENSUM__UNION__RZERO,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1u: A_27a > $o,V2v: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1u )
& ! [V3x: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2v )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V1u ) ) )
=> ( ( V0f @ V3x )
= c_2Enum_2E0 ) ) )
=> ( ( c_2Eiterate_2Ensum @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V1u @ V2v ) @ V0f )
= ( c_2Eiterate_2Ensum @ A_27a @ V1u @ V0f ) ) ) ).
thf(thm_2Eiterate_2ENSUM__UNION__LZERO,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1u: A_27a > $o,V2v: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2v )
& ! [V3x: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1u )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V2v ) ) )
=> ( ( V0f @ V3x )
= c_2Enum_2E0 ) ) )
=> ( ( c_2Eiterate_2Ensum @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V1u @ V2v ) @ V0f )
= ( c_2Eiterate_2Ensum @ A_27a @ V2v @ V0f ) ) ) ).
thf(thm_2Eiterate_2ENSUM__RESTRICT,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ( ( c_2Eiterate_2Ensum @ A_27a @ V1s
@ ^ [V2x: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s ) @ ( V0f @ V2x ) @ c_2Enum_2E0 ) )
= ( c_2Eiterate_2Ensum @ A_27a @ V1s @ V0f ) ) ) ).
thf(thm_2Eiterate_2ENSUM__BOUND,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1f: A_27a > tyop_2Enum_2Enum,V2b: tyop_2Enum_2Enum] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V0s )
=> ( c_2Earithmetic_2E_3C_3D @ ( V1f @ V3x ) @ V2b ) ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Eiterate_2Ensum @ A_27a @ V0s @ V1f ) @ ( c_2Earithmetic_2E_2A @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) @ V2b ) ) ) ).
thf(thm_2Eiterate_2ENSUM__BOUND__GEN,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1f: A_27a > tyop_2Enum_2Enum,V2b: tyop_2Enum_2Enum] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V0s )
=> ( c_2Earithmetic_2E_3C_3D @ ( V1f @ V3x ) @ ( c_2Earithmetic_2EDIV @ V2b @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) ) ) ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Eiterate_2Ensum @ A_27a @ V0s @ V1f ) @ V2b ) ) ).
thf(thm_2Eiterate_2ENSUM__BOUND__LT,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1f: A_27a > tyop_2Enum_2Enum,V2b: tyop_2Enum_2Enum] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V0s )
=> ( c_2Earithmetic_2E_3C_3D @ ( V1f @ V3x ) @ V2b ) )
& ? [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ V0s )
& ( c_2Eprim__rec_2E_3C @ ( V1f @ V4x ) @ V2b ) ) )
=> ( c_2Eprim__rec_2E_3C @ ( c_2Eiterate_2Ensum @ A_27a @ V0s @ V1f ) @ ( c_2Earithmetic_2E_2A @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) @ V2b ) ) ) ).
thf(thm_2Eiterate_2ENSUM__BOUND__LT__ALL,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1f: A_27a > tyop_2Enum_2Enum,V2b: tyop_2Enum_2Enum] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V0s )
=> ( c_2Eprim__rec_2E_3C @ ( V1f @ V3x ) @ V2b ) ) )
=> ( c_2Eprim__rec_2E_3C @ ( c_2Eiterate_2Ensum @ A_27a @ V0s @ V1f ) @ ( c_2Earithmetic_2E_2A @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) @ V2b ) ) ) ).
thf(thm_2Eiterate_2ENSUM__BOUND__LT__GEN,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1f: A_27a > tyop_2Enum_2Enum,V2b: tyop_2Enum_2Enum] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V0s )
=> ( c_2Eprim__rec_2E_3C @ ( V1f @ V3x ) @ ( c_2Earithmetic_2EDIV @ V2b @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) ) ) ) )
=> ( c_2Eprim__rec_2E_3C @ ( c_2Eiterate_2Ensum @ A_27a @ V0s @ V1f ) @ V2b ) ) ).
thf(thm_2Eiterate_2ENSUM__UNION__EQ,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o,V2t: A_27a > $o,V3u: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V3u )
& ( ( c_2Epred__set_2EINTER @ A_27a @ V1s @ V2t )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ( ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t )
= V3u ) )
=> ( ( c_2Earithmetic_2E_2B @ ( c_2Eiterate_2Ensum @ A_27a @ V1s @ V0f ) @ ( c_2Eiterate_2Ensum @ A_27a @ V2t @ V0f ) )
= ( c_2Eiterate_2Ensum @ A_27a @ V3u @ V0f ) ) ) ).
thf(thm_2Eiterate_2ENSUM__EQ__SUPERSET,axiom,
! [A_27a: $tType,V0g: A_27a > tyop_2Enum_2Enum,V1f: A_27a > tyop_2Enum_2Enum,V2s: A_27a > $o,V3t: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V3t )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V3t @ V2s )
& ! [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ V3t )
=> ( ( V1f @ V4x )
= ( V0g @ V4x ) ) )
& ! [V5x: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V5x @ V2s )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V5x @ V3t ) ) )
=> ( ( V1f @ V5x )
= c_2Enum_2E0 ) ) )
=> ( ( c_2Eiterate_2Ensum @ A_27a @ V2s @ V1f )
= ( c_2Eiterate_2Ensum @ A_27a @ V3t @ V0g ) ) ) ).
thf(thm_2Eiterate_2ENSUM__RESTRICT__SET,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1s: A_27a > $o,V2f: A_27a > tyop_2Enum_2Enum] :
( ( c_2Eiterate_2Ensum @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V3x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V3x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s ) @ ( V0P @ V3x ) ) ) )
@ V2f )
= ( c_2Eiterate_2Ensum @ A_27a @ V1s
@ ^ [V4x: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( V0P @ V4x ) @ ( V2f @ V4x ) @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Eiterate_2ENSUM__NSUM__RESTRICT,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o,V1f: A_27a > A_27b > tyop_2Enum_2Enum,V2s: A_27a > $o,V3t: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
& ( c_2Epred__set_2EFINITE @ A_27b @ V3t ) )
=> ( ( c_2Eiterate_2Ensum @ A_27a @ V2s
@ ^ [V4x: A_27a] :
( c_2Eiterate_2Ensum @ A_27b
@ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b
@ ^ [V5y: A_27b] : ( c_2Epair_2E_2C @ A_27b @ $o @ V5y @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27b @ V5y @ V3t ) @ ( V0R @ V4x @ V5y ) ) ) )
@ ^ [V6y: A_27b] : ( V1f @ V4x @ V6y ) ) )
= ( c_2Eiterate_2Ensum @ A_27b @ V3t
@ ^ [V7y: A_27b] :
( c_2Eiterate_2Ensum @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V8x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V8x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V8x @ V2s ) @ ( V0R @ V8x @ V7y ) ) ) )
@ ^ [V9x: A_27a] : ( V1f @ V9x @ V7y ) ) ) ) ) ).
thf(thm_2Eiterate_2ECARD__EQ__NSUM,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ( ( c_2Epred__set_2ECARD @ A_27a @ V0s )
= ( c_2Eiterate_2Ensum @ A_27a @ V0s
@ ^ [V1x: A_27a] : ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Eiterate_2ENSUM__MULTICOUNT__GEN,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o,V1s: A_27a > $o,V2t: A_27b > $o,V3k: A_27b > tyop_2Enum_2Enum] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( c_2Epred__set_2EFINITE @ A_27b @ V2t )
& ! [V4j: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V4j @ V2t )
=> ( ( c_2Epred__set_2ECARD @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V5i: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V5i @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V5i @ V1s ) @ ( V0R @ V5i @ V4j ) ) ) ) )
= ( V3k @ V4j ) ) ) )
=> ( ( c_2Eiterate_2Ensum @ A_27a @ V1s
@ ^ [V6i: A_27a] :
( c_2Epred__set_2ECARD @ A_27b
@ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b
@ ^ [V7j: A_27b] : ( c_2Epair_2E_2C @ A_27b @ $o @ V7j @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27b @ V7j @ V2t ) @ ( V0R @ V6i @ V7j ) ) ) ) ) )
= ( c_2Eiterate_2Ensum @ A_27b @ V2t
@ ^ [V8i: A_27b] : ( V3k @ V8i ) ) ) ) ).
thf(thm_2Eiterate_2ENSUM__MULTICOUNT,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o,V1s: A_27a > $o,V2t: A_27b > $o,V3k: tyop_2Enum_2Enum] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( c_2Epred__set_2EFINITE @ A_27b @ V2t )
& ! [V4j: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V4j @ V2t )
=> ( ( c_2Epred__set_2ECARD @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V5i: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V5i @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V5i @ V1s ) @ ( V0R @ V5i @ V4j ) ) ) ) )
= V3k ) ) )
=> ( ( c_2Eiterate_2Ensum @ A_27a @ V1s
@ ^ [V6i: A_27a] :
( c_2Epred__set_2ECARD @ A_27b
@ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b
@ ^ [V7j: A_27b] : ( c_2Epair_2E_2C @ A_27b @ $o @ V7j @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27b @ V7j @ V2t ) @ ( V0R @ V6i @ V7j ) ) ) ) ) )
= ( c_2Earithmetic_2E_2A @ V3k @ ( c_2Epred__set_2ECARD @ A_27b @ V2t ) ) ) ) ).
thf(thm_2Eiterate_2ENSUM__IMAGE__GEN,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1g: A_27a > tyop_2Enum_2Enum,V2s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
=> ( ( c_2Eiterate_2Ensum @ A_27a @ V2s @ V1g )
= ( c_2Eiterate_2Ensum @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V2s )
@ ^ [V3y: A_27b] :
( c_2Eiterate_2Ensum @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V4x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V4x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V4x @ V2s ) @ ( c_2Emin_2E_3D @ A_27b @ ( V0f @ V4x ) @ V3y ) ) ) )
@ V1g ) ) ) ) ).
thf(thm_2Eiterate_2ENSUM__GROUP,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1g: A_27a > tyop_2Enum_2Enum,V2s: A_27a > $o,V3t: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
& ( c_2Epred__set_2ESUBSET @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V2s ) @ V3t ) )
=> ( ( c_2Eiterate_2Ensum @ A_27b @ V3t
@ ^ [V4y: A_27b] :
( c_2Eiterate_2Ensum @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V5x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V5x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V5x @ V2s ) @ ( c_2Emin_2E_3D @ A_27b @ ( V0f @ V5x ) @ V4y ) ) ) )
@ V1g ) )
= ( c_2Eiterate_2Ensum @ A_27a @ V2s @ V1g ) ) ) ).
thf(thm_2Eiterate_2ENSUM__SUBSET,axiom,
! [A_27a: $tType,V0u: A_27a > $o,V1v: A_27a > $o,V2f: A_27a > tyop_2Enum_2Enum] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0u )
& ( c_2Epred__set_2EFINITE @ A_27a @ V1v )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Epred__set_2EDIFF @ A_27a @ V0u @ V1v ) )
=> ( ( V2f @ V3x )
= c_2Enum_2E0 ) ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Eiterate_2Ensum @ A_27a @ V0u @ V2f ) @ ( c_2Eiterate_2Ensum @ A_27a @ V1v @ V2f ) ) ) ).
thf(thm_2Eiterate_2ENSUM__SUBSET__SIMPLE,axiom,
! [A_27a: $tType,V0u: A_27a > $o,V1v: A_27a > $o,V2f: A_27a > tyop_2Enum_2Enum] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1v )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V0u @ V1v ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Eiterate_2Ensum @ A_27a @ V0u @ V2f ) @ ( c_2Eiterate_2Ensum @ A_27a @ V1v @ V2f ) ) ) ).
thf(thm_2Eiterate_2ENSUM__LE__GEN,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1g: A_27a > tyop_2Enum_2Enum,V2s: A_27a > $o] :
( ( ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
=> ( c_2Earithmetic_2E_3C_3D @ ( V0f @ V3x ) @ ( V1g @ V3x ) ) )
& ( c_2Epred__set_2EFINITE @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V4x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V4x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V4x @ V2s ) @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ ( V1g @ V4x ) @ c_2Enum_2E0 ) ) ) ) ) ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Eiterate_2Ensum @ A_27a @ V2s @ V0f ) @ ( c_2Eiterate_2Ensum @ A_27a @ V2s @ V1g ) ) ) ).
thf(thm_2Eiterate_2ENSUM__IMAGE__NONZERO,axiom,
! [A_27a: $tType,A_27b: $tType,V0d: A_27b > tyop_2Enum_2Enum,V1i: A_27a > A_27b,V2s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
& ! [V3x: A_27a,V4y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
& ( c_2Ebool_2EIN @ A_27a @ V4y @ V2s )
& ( (~) @ ( V3x = V4y ) )
& ( ( V1i @ V3x )
= ( V1i @ V4y ) ) )
=> ( ( V0d @ ( V1i @ V3x ) )
= c_2Enum_2E0 ) ) )
=> ( ( c_2Eiterate_2Ensum @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V1i @ V2s ) @ V0d )
= ( c_2Eiterate_2Ensum @ A_27a @ V2s @ ( c_2Ecombin_2Eo @ A_27a @ tyop_2Enum_2Enum @ A_27b @ V0d @ V1i ) ) ) ) ).
thf(thm_2Eiterate_2ENSUM__BIJECTION,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1p: A_27a > A_27a,V2s: A_27a > $o] :
( ( ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
=> ( c_2Ebool_2EIN @ A_27a @ ( V1p @ V3x ) @ V2s ) )
& ! [V4y: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4y @ V2s )
=> ( c_2Ebool_2E_3F_21 @ A_27a
@ ^ [V5x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V5x @ V2s ) @ ( c_2Emin_2E_3D @ A_27a @ ( V1p @ V5x ) @ V4y ) ) ) ) )
=> ( ( c_2Eiterate_2Ensum @ A_27a @ V2s @ V0f )
= ( c_2Eiterate_2Ensum @ A_27a @ V2s @ ( c_2Ecombin_2Eo @ A_27a @ tyop_2Enum_2Enum @ A_27a @ V0f @ V1p ) ) ) ) ).
thf(thm_2Eiterate_2ENSUM__NSUM__PRODUCT,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27a > A_27b > $o,V2x: A_27a > A_27b > tyop_2Enum_2Enum] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ! [V3i: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3i @ V0s )
=> ( c_2Epred__set_2EFINITE @ A_27b @ ( V1t @ V3i ) ) ) )
=> ( ( c_2Eiterate_2Ensum @ A_27a @ V0s
@ ^ [V4i: A_27a] : ( c_2Eiterate_2Ensum @ A_27b @ ( V1t @ V4i ) @ ( V2x @ V4i ) ) )
= ( c_2Eiterate_2Ensum @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( tyop_2Epair_2Eprod @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ $o )
@ ^ [V5i: A_27a,V6j: A_27b] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ $o @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V5i @ V6j ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V5i @ V0s ) @ ( c_2Ebool_2EIN @ A_27b @ V6j @ ( V1t @ V5i ) ) ) ) ) )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ tyop_2Enum_2Enum
@ ^ [V7i: A_27a,V8j: A_27b] : ( V2x @ V7i @ V8j ) ) ) ) ) ).
thf(thm_2Eiterate_2ENSUM__EQ__GENERAL,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o,V2f: A_27a > tyop_2Enum_2Enum,V3g: A_27b > tyop_2Enum_2Enum,V4h: A_27a > A_27b] :
( ( ! [V5y: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V5y @ V1t )
=> ( c_2Ebool_2E_3F_21 @ A_27a
@ ^ [V6x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V6x @ V0s ) @ ( c_2Emin_2E_3D @ A_27b @ ( V4h @ V6x ) @ V5y ) ) ) )
& ! [V7x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V7x @ V0s )
=> ( ( c_2Ebool_2EIN @ A_27b @ ( V4h @ V7x ) @ V1t )
& ( ( V3g @ ( V4h @ V7x ) )
= ( V2f @ V7x ) ) ) ) )
=> ( ( c_2Eiterate_2Ensum @ A_27a @ V0s @ V2f )
= ( c_2Eiterate_2Ensum @ A_27b @ V1t @ V3g ) ) ) ).
thf(thm_2Eiterate_2ENSUM__EQ__GENERAL__INVERSES,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o,V2f: A_27a > tyop_2Enum_2Enum,V3g: A_27b > tyop_2Enum_2Enum,V4h: A_27a > A_27b,V5k: A_27b > A_27a] :
( ( ! [V6y: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V6y @ V1t )
=> ( ( c_2Ebool_2EIN @ A_27a @ ( V5k @ V6y ) @ V0s )
& ( ( V4h @ ( V5k @ V6y ) )
= V6y ) ) )
& ! [V7x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V7x @ V0s )
=> ( ( c_2Ebool_2EIN @ A_27b @ ( V4h @ V7x ) @ V1t )
& ( ( V5k @ ( V4h @ V7x ) )
= V7x )
& ( ( V3g @ ( V4h @ V7x ) )
= ( V2f @ V7x ) ) ) ) )
=> ( ( c_2Eiterate_2Ensum @ A_27a @ V0s @ V2f )
= ( c_2Eiterate_2Ensum @ A_27b @ V1t @ V3g ) ) ) ).
thf(thm_2Eiterate_2ENSUM__INJECTION,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1p: A_27a > A_27a,V2s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
=> ( c_2Ebool_2EIN @ A_27a @ ( V1p @ V3x ) @ V2s ) )
& ! [V4x: A_27a,V5y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V4x @ V2s )
& ( c_2Ebool_2EIN @ A_27a @ V5y @ V2s )
& ( ( V1p @ V4x )
= ( V1p @ V5y ) ) )
=> ( V4x = V5y ) ) )
=> ( ( c_2Eiterate_2Ensum @ A_27a @ V2s @ ( c_2Ecombin_2Eo @ A_27a @ tyop_2Enum_2Enum @ A_27a @ V0f @ V1p ) )
= ( c_2Eiterate_2Ensum @ A_27a @ V2s @ V0f ) ) ) ).
thf(thm_2Eiterate_2ENSUM__UNION__NONZERO,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( c_2Epred__set_2EFINITE @ A_27a @ V2t )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Epred__set_2EINTER @ A_27a @ V1s @ V2t ) )
=> ( ( V0f @ V3x )
= c_2Enum_2E0 ) ) )
=> ( ( c_2Eiterate_2Ensum @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) @ V0f )
= ( c_2Earithmetic_2E_2B @ ( c_2Eiterate_2Ensum @ A_27a @ V1s @ V0f ) @ ( c_2Eiterate_2Ensum @ A_27a @ V2t @ V0f ) ) ) ) ).
thf(thm_2Eiterate_2ENSUM__BIGUNION__NONZERO,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: ( A_27a > $o ) > $o] :
( ( ( c_2Epred__set_2EFINITE @ ( A_27a > $o ) @ V1s )
& ! [V2t: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ V1s )
=> ( c_2Epred__set_2EFINITE @ A_27a @ V2t ) )
& ! [V3t1: A_27a > $o,V4t2: A_27a > $o,V5x: A_27a] :
( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3t1 @ V1s )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V4t2 @ V1s )
& ( (~) @ ( V3t1 = V4t2 ) )
& ( c_2Ebool_2EIN @ A_27a @ V5x @ V3t1 )
& ( c_2Ebool_2EIN @ A_27a @ V5x @ V4t2 ) )
=> ( ( V0f @ V5x )
= c_2Enum_2E0 ) ) )
=> ( ( c_2Eiterate_2Ensum @ A_27a @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V1s ) @ V0f )
= ( c_2Eiterate_2Ensum @ ( A_27a > $o ) @ V1s
@ ^ [V6t: A_27a > $o] : ( c_2Eiterate_2Ensum @ A_27a @ V6t @ V0f ) ) ) ) ).
thf(thm_2Eiterate_2ENSUM__CASES,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1P: A_27a > $o,V2f: A_27a > tyop_2Enum_2Enum,V3g: A_27a > tyop_2Enum_2Enum] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ( ( c_2Eiterate_2Ensum @ A_27a @ V0s
@ ^ [V4x: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( V1P @ V4x ) @ ( V2f @ V4x ) @ ( V3g @ V4x ) ) )
= ( c_2Earithmetic_2E_2B
@ ( c_2Eiterate_2Ensum @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V5x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V5x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V5x @ V0s ) @ ( V1P @ V5x ) ) ) )
@ V2f )
@ ( c_2Eiterate_2Ensum @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V6x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V6x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V6x @ V0s ) @ ( c_2Ebool_2E_7E @ ( V1P @ V6x ) ) ) ) )
@ V3g ) ) ) ) ).
thf(thm_2Eiterate_2ENSUM__CLOSED,axiom,
! [A_27a: $tType,V0P: tyop_2Enum_2Enum > $o,V1f: A_27a > tyop_2Enum_2Enum,V2s: A_27a > $o] :
( ( ( V0P @ c_2Enum_2E0 )
& ! [V3x: tyop_2Enum_2Enum,V4y: tyop_2Enum_2Enum] :
( ( ( V0P @ V3x )
& ( V0P @ V4y ) )
=> ( V0P @ ( c_2Earithmetic_2E_2B @ V3x @ V4y ) ) )
& ! [V5a: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V5a @ V2s )
=> ( V0P @ ( V1f @ V5a ) ) ) )
=> ( V0P @ ( c_2Eiterate_2Ensum @ A_27a @ V2s @ V1f ) ) ) ).
thf(thm_2Eiterate_2ENSUM__ADD__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V1g: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n )
@ ^ [V4i: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2B @ ( V0f @ V4i ) @ ( V1g @ V4i ) ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V0f ) @ ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V1g ) ) ) ).
thf(thm_2Eiterate_2ENSUM__LE__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V1g: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ! [V4i: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V2m @ V4i )
& ( c_2Earithmetic_2E_3C_3D @ V4i @ V3n ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( V0f @ V4i ) @ ( V1g @ V4i ) ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V0f ) @ ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V1g ) ) ) ).
thf(thm_2Eiterate_2ENSUM__EQ__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V1g: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ! [V4i: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V2m @ V4i )
& ( c_2Earithmetic_2E_3C_3D @ V4i @ V3n ) )
=> ( ( V0f @ V4i )
= ( V1g @ V4i ) ) )
=> ( ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V0f )
= ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V1g ) ) ) ).
thf(thm_2Eiterate_2ENSUM__CONST__NUMSEG,axiom,
! [V0c: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n )
@ ^ [V3n: tyop_2Enum_2Enum] : V0c )
= ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2B @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V1m ) @ V0c ) ) ).
thf(thm_2Eiterate_2ENSUM__EQ__0__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ! [V3i: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V1m @ V3i )
& ( c_2Earithmetic_2E_3C_3D @ V3i @ V2n ) )
=> ( ( V0f @ V3i )
= c_2Enum_2E0 ) )
=> ( ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f )
= c_2Enum_2E0 ) ) ).
thf(thm_2Eiterate_2ENSUM__EQ__0__IFF__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f )
= c_2Enum_2E0 )
<=> ! [V3i: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V1m @ V3i )
& ( c_2Earithmetic_2E_3C_3D @ V3i @ V2n ) )
=> ( ( V0f @ V3i )
= c_2Enum_2E0 ) ) ) ).
thf(thm_2Eiterate_2ENSUM__TRIV__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V2n @ V1m )
=> ( ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f )
= c_2Enum_2E0 ) ) ).
thf(thm_2Eiterate_2ENSUM__SING__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1n @ V1n ) @ V0f )
= ( V0f @ V1n ) ) ).
thf(thm_2Eiterate_2ENSUM__CLAUSES__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ! [V1m: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ c_2Enum_2E0 ) @ V0f )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V1m @ c_2Enum_2E0 ) @ ( V0f @ c_2Enum_2E0 ) @ c_2Enum_2E0 ) )
& ! [V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ ( c_2Enum_2ESUC @ V3n ) ) @ V0f )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2E_3C_3D @ V2m @ ( c_2Enum_2ESUC @ V3n ) ) @ ( c_2Earithmetic_2E_2B @ ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V0f ) @ ( V0f @ ( c_2Enum_2ESUC @ V3n ) ) ) @ ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n ) @ V0f ) ) ) ) ).
thf(thm_2Eiterate_2ENSUM__SWAP__NUMSEG,axiom,
! [V0a: tyop_2Enum_2Enum,V1b: tyop_2Enum_2Enum,V2c: tyop_2Enum_2Enum,V3d: tyop_2Enum_2Enum,V4f: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V0a @ V1b )
@ ^ [V5i: tyop_2Enum_2Enum] : ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2c @ V3d ) @ ( V4f @ V5i ) ) )
= ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2c @ V3d )
@ ^ [V6j: tyop_2Enum_2Enum] :
( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V0a @ V1b )
@ ^ [V7i: tyop_2Enum_2Enum] : ( V4f @ V7i @ V6j ) ) ) ) ).
thf(thm_2Eiterate_2ENSUM__ADD__SPLIT,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum,V3p: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1m @ ( c_2Earithmetic_2E_2B @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
=> ( ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ ( c_2Earithmetic_2E_2B @ V2n @ V3p ) ) @ V0f )
= ( c_2Earithmetic_2E_2B @ ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f ) @ ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2E_2B @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2E_2B @ V2n @ V3p ) ) @ V0f ) ) ) ) ).
thf(thm_2Eiterate_2ENSUM__OFFSET,axiom,
! [V0p: tyop_2Enum_2Enum,V1f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2E_2B @ V2m @ V0p ) @ ( c_2Earithmetic_2E_2B @ V3n @ V0p ) ) @ V1f )
= ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V2m @ V3n )
@ ^ [V4i: tyop_2Enum_2Enum] : ( V1f @ ( c_2Earithmetic_2E_2B @ V4i @ V0p ) ) ) ) ).
thf(thm_2Eiterate_2ENSUM__OFFSET__0,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1m @ V2n )
=> ( ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f )
= ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ ( c_2Earithmetic_2E_2D @ V2n @ V1m ) )
@ ^ [V3i: tyop_2Enum_2Enum] : ( V0f @ ( c_2Earithmetic_2E_2B @ V3i @ V1m ) ) ) ) ) ).
thf(thm_2Eiterate_2ENSUM__CLAUSES__LEFT,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1m @ V2n )
=> ( ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f )
= ( c_2Earithmetic_2E_2B @ ( V0f @ V1m ) @ ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2E_2B @ V1m @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2n ) @ V0f ) ) ) ) ).
thf(thm_2Eiterate_2ENSUM__CLAUSES__RIGHT,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V2n )
& ( c_2Earithmetic_2E_3C_3D @ V1m @ V2n ) )
=> ( ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n ) @ V0f )
= ( c_2Earithmetic_2E_2B @ ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ ( c_2Earithmetic_2E_2D @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0f ) @ ( V0f @ V2n ) ) ) ) ).
thf(thm_2Eiterate_2ENSUM__PAIR,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V1m ) @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V2n ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0f )
= ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1m @ V2n )
@ ^ [V3i: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2B @ ( V0f @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V3i ) ) @ ( V0f @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ V3i ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(thm_2Eiterate_2EMOD__NSUM__MOD,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
& ( (~) @ ( V1n = c_2Enum_2E0 ) ) )
=> ( ( c_2Earithmetic_2EMOD @ ( c_2Eiterate_2Ensum @ A_27a @ V2s @ V0f ) @ V1n )
= ( c_2Earithmetic_2EMOD
@ ( c_2Eiterate_2Ensum @ A_27a @ V2s
@ ^ [V3i: A_27a] : ( c_2Earithmetic_2EMOD @ ( V0f @ V3i ) @ V1n ) )
@ V1n ) ) ) ).
thf(thm_2Eiterate_2EMOD__NSUM__MOD__NUMSEG,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V1a: tyop_2Enum_2Enum,V2b: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( (~) @ ( V3n = c_2Enum_2E0 ) )
=> ( ( c_2Earithmetic_2EMOD @ ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1a @ V2b ) @ V0f ) @ V3n )
= ( c_2Earithmetic_2EMOD
@ ( c_2Eiterate_2Ensum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V1a @ V2b )
@ ^ [V4i: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EMOD @ ( V0f @ V4i ) @ V3n ) )
@ V3n ) ) ) ).
thf(thm_2Eiterate_2ETH,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0f: A_27a > tyop_2Erealax_2Ereal,V1g: A_27a > tyop_2Erealax_2Ereal,V2s: A_27a > $o] :
( ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
=> ( ( V0f @ V3x )
= ( V1g @ V3x ) ) )
=> ( ( c_2Eiterate_2ESum @ A_27a @ V2s
@ ^ [V4i: A_27a] : ( V0f @ V4i ) )
= ( c_2Eiterate_2ESum @ A_27a @ V2s @ V1g ) ) )
& ! [V5f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V6g: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V7a: tyop_2Enum_2Enum,V8b: tyop_2Enum_2Enum] :
( ! [V9i: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V7a @ V9i )
& ( c_2Earithmetic_2E_3C_3D @ V9i @ V8b ) )
=> ( ( V5f @ V9i )
= ( V6g @ V9i ) ) )
=> ( ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V7a @ V8b )
@ ^ [V10i: tyop_2Enum_2Enum] : ( V5f @ V10i ) )
= ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ V7a @ V8b ) @ V6g ) ) )
& ! [V11f: A_27b > tyop_2Erealax_2Ereal,V12g: A_27b > tyop_2Erealax_2Ereal,V13p: A_27b > $o] :
( ! [V14x: A_27b] :
( ( V13p @ V14x )
=> ( ( V11f @ V14x )
= ( V12g @ V14x ) ) )
=> ( ( c_2Eiterate_2ESum @ A_27b
@ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b
@ ^ [V15y: A_27b] : ( c_2Epair_2E_2C @ A_27b @ $o @ V15y @ ( V13p @ V15y ) ) )
@ ^ [V16i: A_27b] : ( V11f @ V16i ) )
= ( c_2Eiterate_2ESum @ A_27b
@ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b
@ ^ [V17y: A_27b] : ( c_2Epair_2E_2C @ A_27b @ $o @ V17y @ ( V13p @ V17y ) ) )
@ V12g ) ) ) ) ).
thf(thm_2Eiterate_2EREAL__SUB__POW,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V2n )
=> ( ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Epow @ V0x @ V2n ) @ ( c_2Ereal_2Epow @ V1y @ V2n ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y )
@ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ ( c_2Earithmetic_2E_2D @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
@ ^ [V3i: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Epow @ V0x @ V3i ) @ ( c_2Ereal_2Epow @ V1y @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2D @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V3i ) ) ) ) ) ) ) ).
thf(thm_2Eiterate_2EREAL__SUB__POW__R1,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V1n )
=> ( ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Epow @ V0x @ V1n ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__sub @ V0x @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
@ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ ( c_2Earithmetic_2E_2D @ V1n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
@ ^ [V2i: tyop_2Enum_2Enum] : ( c_2Ereal_2Epow @ V0x @ V2i ) ) ) ) ) ).
thf(thm_2Eiterate_2EREAL__SUB__POW__L1,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V1n )
=> ( ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ V0x @ V1n ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__sub @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0x )
@ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ ( c_2Earithmetic_2E_2D @ V1n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
@ ^ [V2i: tyop_2Enum_2Enum] : ( c_2Ereal_2Epow @ V0x @ V2i ) ) ) ) ) ).
thf(thm_2Eiterate_2EFORALL__IN__GSPEC,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType,A_27g: $tType,V0Q: A_27b > $o] :
( ! [V1P: A_27a > $o,V2f: A_27a > A_27b] :
( ! [V3z: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V3z
@ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27a
@ ^ [V4x: A_27a] : ( c_2Epair_2E_2C @ A_27b @ $o @ ( V2f @ V4x ) @ ( V1P @ V4x ) ) ) )
=> ( V0Q @ V3z ) )
<=> ! [V5x: A_27a] :
( ( V1P @ V5x )
=> ( V0Q @ ( V2f @ V5x ) ) ) )
& ! [V6P: A_27c > A_27d > $o,V7f: A_27c > A_27d > A_27b] :
( ! [V8z: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V8z
@ ( c_2Epred__set_2EGSPEC @ A_27b @ ( tyop_2Epair_2Eprod @ A_27c @ A_27d )
@ ( c_2Epair_2EUNCURRY @ A_27c @ A_27d @ ( tyop_2Epair_2Eprod @ A_27b @ $o )
@ ^ [V9x: A_27c,V10y: A_27d] : ( c_2Epair_2E_2C @ A_27b @ $o @ ( V7f @ V9x @ V10y ) @ ( V6P @ V9x @ V10y ) ) ) ) )
=> ( V0Q @ V8z ) )
<=> ! [V11x: A_27c,V12y: A_27d] :
( ( V6P @ V11x @ V12y )
=> ( V0Q @ ( V7f @ V11x @ V12y ) ) ) )
& ! [V13P: A_27e > A_27f > A_27g > $o,V14f: A_27e > A_27f > A_27g > A_27b] :
( ! [V15z: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V15z
@ ( c_2Epred__set_2EGSPEC @ A_27b @ ( tyop_2Epair_2Eprod @ A_27e @ ( tyop_2Epair_2Eprod @ A_27f @ A_27g ) )
@ ( c_2Epair_2EUNCURRY @ A_27e @ ( tyop_2Epair_2Eprod @ A_27f @ A_27g ) @ ( tyop_2Epair_2Eprod @ A_27b @ $o )
@ ^ [V16w: A_27e] :
( c_2Epair_2EUNCURRY @ A_27f @ A_27g @ ( tyop_2Epair_2Eprod @ A_27b @ $o )
@ ^ [V17x: A_27f,V18y: A_27g] : ( c_2Epair_2E_2C @ A_27b @ $o @ ( V14f @ V16w @ V17x @ V18y ) @ ( V13P @ V16w @ V17x @ V18y ) ) ) ) ) )
=> ( V0Q @ V15z ) )
<=> ! [V19w: A_27e,V20x: A_27f,V21y: A_27g] :
( ( V13P @ V19w @ V20x @ V21y )
=> ( V0Q @ ( V14f @ V19w @ V20x @ V21y ) ) ) ) ) ).
thf(thm_2Eiterate_2EREAL__SUB__POLYFUN,axiom,
! [V0a: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal,V3n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V3n )
=> ( ( c_2Ereal_2Ereal__sub
@ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ V3n )
@ ^ [V4i: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( V0a @ V4i ) @ ( c_2Ereal_2Epow @ V1x @ V4i ) ) )
@ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ V3n )
@ ^ [V5i: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( V0a @ V5i ) @ ( c_2Ereal_2Epow @ V2y @ V5i ) ) ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__sub @ V1x @ V2y )
@ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ ( c_2Earithmetic_2E_2D @ V3n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
@ ^ [V6j: tyop_2Enum_2Enum] :
( c_2Erealax_2Ereal__mul
@ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2E_2B @ V6j @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V3n )
@ ^ [V7i: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( V0a @ V7i ) @ ( c_2Ereal_2Epow @ V2y @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2D @ V7i @ V6j ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
@ ( c_2Ereal_2Epow @ V1x @ V6j ) ) ) ) ) ) ).
thf(thm_2Eiterate_2EREAL__SUB__POLYFUN__ALT,axiom,
! [V0a: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1x: tyop_2Erealax_2Ereal,V2y: tyop_2Erealax_2Ereal,V3n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V3n )
=> ( ( c_2Ereal_2Ereal__sub
@ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ V3n )
@ ^ [V4i: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( V0a @ V4i ) @ ( c_2Ereal_2Epow @ V1x @ V4i ) ) )
@ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ V3n )
@ ^ [V5i: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( V0a @ V5i ) @ ( c_2Ereal_2Epow @ V2y @ V5i ) ) ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__sub @ V1x @ V2y )
@ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ ( c_2Earithmetic_2E_2D @ V3n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
@ ^ [V6j: tyop_2Enum_2Enum] :
( c_2Erealax_2Ereal__mul
@ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2D @ V3n @ V6j ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
@ ^ [V7k: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( V0a @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2B @ V6j @ V7k ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Ereal_2Epow @ V2y @ V7k ) ) )
@ ( c_2Ereal_2Epow @ V1x @ V6j ) ) ) ) ) ) ).
thf(thm_2Eiterate_2EREAL__POLYFUN__ROOTBOUND,axiom,
! [V0n: tyop_2Enum_2Enum,V1c: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( (~)
@ ! [V2i: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V2i @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ V0n ) )
=> ( ( V1c @ V2i )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V3x: tyop_2Erealax_2Ereal] :
( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V3x
@ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal
@ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ V0n )
@ ^ [V4i: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( V1c @ V4i ) @ ( c_2Ereal_2Epow @ V3x @ V4i ) ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) )
& ( c_2Earithmetic_2E_3C_3D
@ ( c_2Epred__set_2ECARD @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V5x: tyop_2Erealax_2Ereal] :
( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V5x
@ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal
@ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ V0n )
@ ^ [V6i: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( V1c @ V6i ) @ ( c_2Ereal_2Epow @ V5x @ V6i ) ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) )
@ V0n ) ) ) ).
thf(thm_2Eiterate_2EREAL__POLYFUN__FINITE__ROOTS,axiom,
! [V0n: tyop_2Enum_2Enum,V1c: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] :
( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x
@ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal
@ ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ V0n )
@ ^ [V3i: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( V1c @ V3i ) @ ( c_2Ereal_2Epow @ V2x @ V3i ) ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) )
<=> ? [V4i: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V4i @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ V0n ) )
& ( (~)
@ ( ( V1c @ V4i )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ) ).
thf(thm_2Eiterate_2EREAL__POLYFUN__EQ__0,axiom,
! [V0n: tyop_2Enum_2Enum,V1c: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ! [V2x: tyop_2Erealax_2Ereal] :
( ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ V0n )
@ ^ [V3i: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( V1c @ V3i ) @ ( c_2Ereal_2Epow @ V2x @ V3i ) ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
<=> ! [V4i: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V4i @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ V0n ) )
=> ( ( V1c @ V4i )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Eiterate_2EREAL__POLYFUN__EQ__CONST,axiom,
! [V0n: tyop_2Enum_2Enum,V1c: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V2k: tyop_2Erealax_2Ereal] :
( ! [V3x: tyop_2Erealax_2Ereal] :
( ( c_2Eiterate_2ESum @ tyop_2Enum_2Enum @ ( c_2Eiterate_2E_2E_2E @ c_2Enum_2E0 @ V0n )
@ ^ [V4i: tyop_2Enum_2Enum] : ( c_2Erealax_2Ereal__mul @ ( V1c @ V4i ) @ ( c_2Ereal_2Epow @ V3x @ V4i ) ) )
= V2k )
<=> ( ( ( V1c @ c_2Enum_2E0 )
= V2k )
& ! [V5i: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V5i @ ( c_2Eiterate_2E_2E_2E @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V0n ) )
=> ( ( V1c @ V5i )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ) ).
thf(thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__CONST,axiom,
! [V0c: tyop_2Erealax_2Ereal] :
( c_2Eiterate_2Epolynomial__function
@ ^ [V1x: tyop_2Erealax_2Ereal] : V0c ) ).
thf(thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__ID,axiom,
( c_2Eiterate_2Epolynomial__function
@ ^ [V0x: tyop_2Erealax_2Ereal] : V0x ) ).
thf(thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__ADD,axiom,
! [V0p: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1q: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Eiterate_2Epolynomial__function @ V0p )
& ( c_2Eiterate_2Epolynomial__function @ V1q ) )
=> ( c_2Eiterate_2Epolynomial__function
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ ( V0p @ V2x ) @ ( V1q @ V2x ) ) ) ) ).
thf(thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__LMUL,axiom,
! [V0p: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1c: tyop_2Erealax_2Ereal] :
( ( c_2Eiterate_2Epolynomial__function @ V0p )
=> ( c_2Eiterate_2Epolynomial__function
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ V1c @ ( V0p @ V2x ) ) ) ) ).
thf(thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__RMUL,axiom,
! [V0p: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1c: tyop_2Erealax_2Ereal] :
( ( c_2Eiterate_2Epolynomial__function @ V0p )
=> ( c_2Eiterate_2Epolynomial__function
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ ( V0p @ V2x ) @ V1c ) ) ) ).
thf(thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__NEG,axiom,
! [V0p: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Eiterate_2Epolynomial__function
@ ^ [V1x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg @ ( V0p @ V1x ) ) )
= ( c_2Eiterate_2Epolynomial__function @ V0p ) ) ).
thf(thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__SUB,axiom,
! [V0p: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1q: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Eiterate_2Epolynomial__function @ V0p )
& ( c_2Eiterate_2Epolynomial__function @ V1q ) )
=> ( c_2Eiterate_2Epolynomial__function
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__sub @ ( V0p @ V2x ) @ ( V1q @ V2x ) ) ) ) ).
thf(thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__MUL,axiom,
! [V0p: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1q: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Eiterate_2Epolynomial__function @ V0p )
& ( c_2Eiterate_2Epolynomial__function @ V1q ) )
=> ( c_2Eiterate_2Epolynomial__function
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ ( V0p @ V2x ) @ ( V1q @ V2x ) ) ) ) ).
thf(thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__SUM,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1p: tyop_2Erealax_2Ereal > A_27a > tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ! [V2i: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2i @ V0s )
=> ( c_2Eiterate_2Epolynomial__function
@ ^ [V3x: tyop_2Erealax_2Ereal] : ( V1p @ V3x @ V2i ) ) ) )
=> ( c_2Eiterate_2Epolynomial__function
@ ^ [V4x: tyop_2Erealax_2Ereal] : ( c_2Eiterate_2ESum @ A_27a @ V0s @ ( V1p @ V4x ) ) ) ) ).
thf(thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__POW,axiom,
! [V0p: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( c_2Eiterate_2Epolynomial__function @ V0p )
=> ( c_2Eiterate_2Epolynomial__function
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Epow @ ( V0p @ V2x ) @ V1n ) ) ) ).
thf(thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__INDUCT,axiom,
! [V0P: ( tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ) > $o] :
( ( ( V0P
@ ^ [V1x: tyop_2Erealax_2Ereal] : V1x )
& ! [V2c: tyop_2Erealax_2Ereal] :
( V0P
@ ^ [V3x: tyop_2Erealax_2Ereal] : V2c )
& ! [V4p: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V5q: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( V0P @ V4p )
& ( V0P @ V5q ) )
=> ( V0P
@ ^ [V6x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ ( V4p @ V6x ) @ ( V5q @ V6x ) ) ) )
& ! [V7p: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V8q: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( V0P @ V7p )
& ( V0P @ V8q ) )
=> ( V0P
@ ^ [V9x: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul @ ( V7p @ V9x ) @ ( V8q @ V9x ) ) ) ) )
=> ! [V10p: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( c_2Eiterate_2Epolynomial__function @ V10p )
=> ( V0P @ V10p ) ) ) ).
thf(thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__o,axiom,
! [V0p: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1q: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal] :
( ( ( c_2Eiterate_2Epolynomial__function @ V0p )
& ( c_2Eiterate_2Epolynomial__function @ V1q ) )
=> ( c_2Eiterate_2Epolynomial__function @ ( c_2Ecombin_2Eo @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0p @ V1q ) ) ) ).
thf(thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__FINITE__ROOTS,axiom,
! [V0p: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal,V1a: tyop_2Erealax_2Ereal] :
( ( c_2Eiterate_2Epolynomial__function @ V0p )
=> ( ( c_2Epred__set_2EFINITE @ tyop_2Erealax_2Ereal
@ ( c_2Epred__set_2EGSPEC @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V2x: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ $o @ V2x @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V0p @ V2x ) @ V1a ) ) ) )
<=> ( (~)
@ ! [V3x: tyop_2Erealax_2Ereal] :
( ( V0p @ V3x )
= V1a ) ) ) ) ).
%------------------------------------------------------------------------------