ITP001 Axioms: ITP138^7.ax
%------------------------------------------------------------------------------
% File : ITP138^7 : 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 : real_sigma.ax [Gau19]
% : HL4138^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 91 ( 9 unt; 50 typ; 0 def)
% Number of atoms : 147 ( 40 equ; 7 cnn)
% Maximal formula atoms : 10 ( 1 avg)
% Number of connectives : 735 ( 7 ~; 1 |; 20 &; 644 @)
% ( 8 <=>; 55 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 8 avg; 644 nst)
% Number of types : 4 ( 3 usr)
% Number of type conns : 199 ( 199 >; 0 *; 0 +; 0 <<)
% Number of symbols : 49 ( 47 usr; 3 con; 0-6 aty)
% Number of variables : 213 ( 26 ^ 150 !; 2 ?; 213 :)
% ( 35 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(tyop_2Erealax_2Ereal,type,
tyop_2Erealax_2Ereal: $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Eseq_2E_2D_2D_3E,type,
c_2Eseq_2E_2D_2D_3E: ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal > $o ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Enum_2E0,type,
c_2Enum_2E0: tyop_2Enum_2Enum ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Earithmetic_2EBIT1,type,
c_2Earithmetic_2EBIT1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EBIT2,type,
c_2Earithmetic_2EBIT2: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_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_2ECROSS,type,
c_2Epred__set_2ECROSS:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > $o ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) ).
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_2EDISJOINT,type,
c_2Epred__set_2EDISJOINT:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
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_2Epair_2EFST,type,
c_2Epair_2EFST:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27a ) ).
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_2EINJ,type,
c_2Epred__set_2EINJ:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > ( A_27b > $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_2Epred__set_2EITSET,type,
c_2Epred__set_2EITSET:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > A_27b ) > ( A_27a > $o ) > A_27b > A_27b ) ).
thf(c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ereal__sigma_2EREAL__SUM__IMAGE,type,
c_2Ereal__sigma_2EREAL__SUM__IMAGE:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Erealax_2Ereal ) > ( A_27a > $o ) > tyop_2Erealax_2Ereal ) ).
thf(c_2Epair_2ESND,type,
c_2Epair_2ESND:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27b ) ).
thf(c_2Epred__set_2ESUBSET,type,
c_2Epred__set_2ESUBSET:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
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_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_2Epred__set_2Ecount,type,
c_2Epred__set_2Ecount: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Erealax_2Einv,type,
c_2Erealax_2Einv: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
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_2Ereal_2Epow,type,
c_2Ereal_2Epow: tyop_2Erealax_2Ereal > tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__add,type,
c_2Erealax_2Ereal__add: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__lt,type,
c_2Erealax_2Ereal__lt: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).
thf(c_2Ereal_2Ereal__lte,type,
c_2Ereal_2Ereal__lte: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).
thf(c_2Erealax_2Ereal__mul,type,
c_2Erealax_2Ereal__mul: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__neg,type,
c_2Erealax_2Ereal__neg: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Ereal__of__num,type,
c_2Ereal_2Ereal__of__num: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Ereal__sub,type,
c_2Ereal_2Ereal__sub: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Esum,type,
c_2Ereal_2Esum: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > ( tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
thf(c_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_2Ereal__sigma_2EREAL__SUM__IMAGE__DEF,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o] :
( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V0f @ V1s )
= ( c_2Epred__set_2EITSET @ A_27a @ tyop_2Erealax_2Ereal
@ ^ [V2e: A_27a,V3acc: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add @ ( V0f @ V2e ) @ V3acc )
@ V1s
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__THM,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal] :
( ( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V0f @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
& ! [V1e: A_27a,V2s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
=> ( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V0f @ ( c_2Epred__set_2EINSERT @ A_27a @ V1e @ V2s ) )
= ( c_2Erealax_2Ereal__add @ ( V0f @ V1e ) @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V0f @ ( c_2Epred__set_2EDELETE @ A_27a @ V2s @ V1e ) ) ) ) ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__SING,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Erealax_2Ereal,V1e: A_27a] :
( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V0f @ ( c_2Epred__set_2EINSERT @ A_27a @ V1e @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
= ( V0f @ V1e ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__POS,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_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V0f @ V1s ) ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__SPOS,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) )
=> ! [V1f: A_27a > tyop_2Erealax_2Ereal] :
( ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V2x ) ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V1f @ V0s ) ) ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__NONZERO,axiom,
! [A_27a: $tType,V0P: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0P )
=> ! [V1f: A_27a > tyop_2Erealax_2Ereal] :
( ( ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0P )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V2x ) ) )
& ? [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V0P )
& ( (~)
@ ( ( V1f @ V3x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) )
=> ( ( (~)
@ ( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V1f @ V0P )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
<=> ( (~)
@ ( V0P
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__IF__ELIM,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1P: A_27a > $o,V2f: A_27a > tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V0s )
=> ( V1P @ V3x ) ) )
=> ( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a
@ ^ [V4x: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( V1P @ V4x ) @ ( V2f @ V4x ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
@ V0s )
= ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V2f @ V0s ) ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__FINITE__SAME,axiom,
! [A_27a: $tType,V0P: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0P )
=> ! [V1f: A_27a > tyop_2Erealax_2Ereal,V2p: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V2p @ V0P )
& ! [V3q: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3q @ V0P )
=> ( ( V1f @ V2p )
= ( V1f @ V3q ) ) ) )
=> ( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V1f @ V0P )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Epred__set_2ECARD @ A_27a @ V0P ) ) @ ( V1f @ V2p ) ) ) ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__FINITE__CONST,axiom,
! [A_27a: $tType,V0P: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0P )
=> ! [V1f: A_27a > tyop_2Erealax_2Ereal,V2x: tyop_2Erealax_2Ereal] :
( ! [V3y: A_27a] :
( ( V1f @ V3y )
= V2x )
=> ( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V1f @ V0P )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Epred__set_2ECARD @ A_27a @ V0P ) ) @ V2x ) ) ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__IN__IF,axiom,
! [A_27a: $tType,V0P: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0P )
=> ! [V1f: A_27a > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V1f @ V0P )
= ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a
@ ^ [V2x: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V0P ) @ ( V1f @ V2x ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
@ V0P ) ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__CMUL,axiom,
! [A_27a: $tType,V0P: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0P )
=> ! [V1f: A_27a > tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal] :
( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a
@ ^ [V3x: A_27a] : ( c_2Erealax_2Ereal__mul @ V2c @ ( V1f @ V3x ) )
@ V0P )
= ( c_2Erealax_2Ereal__mul @ V2c @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V1f @ V0P ) ) ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__NEG,axiom,
! [A_27a: $tType,V0P: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0P )
=> ! [V1f: A_27a > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a
@ ^ [V2x: A_27a] : ( c_2Erealax_2Ereal__neg @ ( V1f @ V2x ) )
@ V0P )
= ( c_2Erealax_2Ereal__neg @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V1f @ V0P ) ) ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0P )
=> ! [V1f_27: A_27a > A_27b] :
( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V1f_27 @ V0P @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V1f_27 @ V0P ) )
=> ! [V2f: A_27b > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27b @ V2f @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V1f_27 @ V0P ) )
= ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ ( c_2Ecombin_2Eo @ A_27a @ tyop_2Erealax_2Ereal @ A_27b @ V2f @ V1f_27 ) @ V0P ) ) ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__DISJOINT__UNION,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1P_27: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0P )
& ( c_2Epred__set_2EFINITE @ A_27a @ V1P_27 )
& ( c_2Epred__set_2EDISJOINT @ A_27a @ V0P @ V1P_27 ) )
=> ! [V2f: A_27a > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V2f @ ( c_2Epred__set_2EUNION @ A_27a @ V0P @ V1P_27 ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V2f @ V0P ) @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V2f @ V1P_27 ) ) ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__EQ__CARD,axiom,
! [A_27a: $tType,V0P: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0P )
=> ( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a
@ ^ [V1x: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ebool_2EIN @ A_27a @ V1x @ V0P ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
@ V0P )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Epred__set_2ECARD @ A_27a @ V0P ) ) ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__INV__CARD__EQ__1,axiom,
! [A_27a: $tType,V0P: A_27a > $o] :
( ( ( (~)
@ ( V0P
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& ( c_2Epred__set_2EFINITE @ A_27a @ V0P ) )
=> ( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a
@ ^ [V1s: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ebool_2EIN @ A_27a @ V1s @ V0P ) @ ( c_2Erealax_2Einv @ ( c_2Ereal_2Ereal__of__num @ ( c_2Epred__set_2ECARD @ A_27a @ V0P ) ) ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
@ V0P )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__INTER__NONZERO,axiom,
! [A_27a: $tType,V0P: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0P )
=> ! [V1f: A_27a > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V1f
@ ( c_2Epred__set_2EINTER @ A_27a @ V0P
@ ^ [V2p: A_27a] : ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ tyop_2Erealax_2Ereal @ ( V1f @ V2p ) @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) )
= ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V1f @ V0P ) ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__INTER__ELIM,axiom,
! [A_27a: $tType,V0P: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0P )
=> ! [V1f: A_27a > tyop_2Erealax_2Ereal,V2P_27: A_27a > $o] :
( ! [V3x: A_27a] :
( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V2P_27 ) )
=> ( ( V1f @ V3x )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V1f @ ( c_2Epred__set_2EINTER @ A_27a @ V0P @ V2P_27 ) )
= ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V1f @ V0P ) ) ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__COUNT,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ tyop_2Enum_2Enum @ V0f @ ( c_2Epred__set_2Ecount @ V1n ) )
= ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V1n ) @ V0f ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__MONO,axiom,
! [A_27a: $tType,V0P: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0P )
=> ! [V1f: A_27a > tyop_2Erealax_2Ereal,V2f_27: A_27a > tyop_2Erealax_2Ereal] :
( ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V0P )
=> ( c_2Ereal_2Ereal__lte @ ( V1f @ V3x ) @ ( V2f_27 @ V3x ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V1f @ V0P ) @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V2f_27 @ V0P ) ) ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__POS__MEM__LE,axiom,
! [A_27a: $tType,V0P: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0P )
=> ! [V1f: A_27a > tyop_2Erealax_2Ereal] :
( ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0P )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V1f @ V2x ) ) )
=> ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V0P )
=> ( c_2Ereal_2Ereal__lte @ ( V1f @ V3x ) @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V1f @ V0P ) ) ) ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__CONST__EQ__1__EQ__INV__CARD,axiom,
! [A_27a: $tType,V0P: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0P )
=> ! [V1f: A_27a > tyop_2Erealax_2Ereal] :
( ( ( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V1f @ V0P )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
& ! [V2x: A_27a,V3y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0P )
& ( c_2Ebool_2EIN @ A_27a @ V3y @ V0P ) )
=> ( ( V1f @ V2x )
= ( V1f @ V3y ) ) ) )
=> ! [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ V0P )
=> ( ( V1f @ V4x )
= ( c_2Erealax_2Einv @ ( c_2Ereal_2Ereal__of__num @ ( c_2Epred__set_2ECARD @ A_27a @ V0P ) ) ) ) ) ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__ADD,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ! [V1f: A_27a > tyop_2Erealax_2Ereal,V2f_27: A_27a > tyop_2Erealax_2Ereal] :
( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a
@ ^ [V3x: A_27a] : ( c_2Erealax_2Ereal__add @ ( V1f @ V3x ) @ ( V2f_27 @ V3x ) )
@ V0s )
= ( c_2Erealax_2Ereal__add @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V1f @ V0s ) @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V2f_27 @ V0s ) ) ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__REAL__SUM__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1s_27: A_27b > $o,V2f: A_27a > A_27b > tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( c_2Epred__set_2EFINITE @ A_27b @ V1s_27 ) )
=> ( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a
@ ^ [V3x: A_27a] : ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27b @ ( V2f @ V3x ) @ V1s_27 )
@ V0s )
= ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V4x: tyop_2Epair_2Eprod @ A_27a @ A_27b] : ( V2f @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V4x ) @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V4x ) )
@ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V0s @ V1s_27 ) ) ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__0,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a
@ ^ [V1x: A_27a] : ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
@ V0s )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ereal__sigma_2ESEQ__REAL__SUM__IMAGE,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ! [V1f: tyop_2Enum_2Enum > A_27a > tyop_2Erealax_2Ereal,V2f_27: A_27a > tyop_2Erealax_2Ereal] :
( ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V0s )
=> ( c_2Eseq_2E_2D_2D_3E
@ ^ [V4n: tyop_2Enum_2Enum] : ( V1f @ V4n @ V3x )
@ ( V2f_27 @ V3x ) ) )
=> ( c_2Eseq_2E_2D_2D_3E
@ ^ [V5n: tyop_2Enum_2Enum] : ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ ( V1f @ V5n ) @ V0s )
@ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V2f_27 @ V0s ) ) ) ) ).
thf(thm_2Ereal__sigma_2ENESTED__REAL__SUM__IMAGE__REVERSE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > tyop_2Erealax_2Ereal,V1s: A_27a > $o,V2s_27: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( c_2Epred__set_2EFINITE @ A_27b @ V2s_27 ) )
=> ( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a
@ ^ [V3x: A_27a] : ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27b @ ( V0f @ V3x ) @ V2s_27 )
@ V1s )
= ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27b
@ ^ [V4x: A_27b] :
( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a
@ ^ [V5y: A_27a] : ( V0f @ V5y @ V4x )
@ V1s )
@ V2s_27 ) ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__EQ__sum,axiom,
! [V0n: tyop_2Enum_2Enum,V1r: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Esum @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V0n ) @ V1r )
= ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ tyop_2Enum_2Enum @ V1r @ ( c_2Epred__set_2Ecount @ V0n ) ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__POW,axiom,
! [A_27a: $tType,V0a: A_27a > tyop_2Erealax_2Ereal,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ( ( c_2Ereal_2Epow @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V0a @ V1s ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) )
= ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27a @ tyop_2Erealax_2Ereal
@ ^ [V2i: A_27a,V3j: A_27a] : ( c_2Erealax_2Ereal__mul @ ( V0a @ V2i ) @ ( V0a @ V3j ) ) )
@ ( c_2Epred__set_2ECROSS @ A_27a @ A_27a @ V1s @ V1s ) ) ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__EQ,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1f: A_27a > tyop_2Erealax_2Ereal,V2f_27: A_27a > tyop_2Erealax_2Ereal] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V0s )
=> ( ( V1f @ V3x )
= ( V2f_27 @ V3x ) ) ) )
=> ( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V1f @ V0s )
= ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V2f_27 @ V0s ) ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__IN__IF__ALT,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1f: A_27a > tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V1f @ V0s )
= ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a
@ ^ [V3x: A_27a] : ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V0s ) @ ( V1f @ V3x ) @ V2z )
@ V0s ) ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__SUB,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1f: A_27a > tyop_2Erealax_2Ereal,V2f_27: A_27a > tyop_2Erealax_2Ereal] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a
@ ^ [V3x: A_27a] : ( c_2Ereal_2Ereal__sub @ ( V1f @ V3x ) @ ( V2f_27 @ V3x ) )
@ V0s )
= ( c_2Ereal_2Ereal__sub @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V1f @ V0s ) @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V2f_27 @ V0s ) ) ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__MONO__SET,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_2ESUBSET @ A_27a @ V1s @ V2t )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2t )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( V0f @ V3x ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V0f @ V1s ) @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a @ V0f @ V2t ) ) ) ).
thf(thm_2Ereal__sigma_2EREAL__SUM__IMAGE__CROSS__SYM,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > tyop_2Erealax_2Ereal,V1s1: A_27a > $o,V2s2: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s1 )
& ( c_2Epred__set_2EFINITE @ A_27b @ V2s2 ) )
=> ( ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ tyop_2Erealax_2Ereal
@ ^ [V3x: A_27a,V4y: A_27b] : ( V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V3x @ V4y ) ) )
@ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V1s1 @ V2s2 ) )
= ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a )
@ ( c_2Epair_2EUNCURRY @ A_27b @ A_27a @ tyop_2Erealax_2Ereal
@ ^ [V5y: A_27b,V6x: A_27a] : ( V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V6x @ V5y ) ) )
@ ( c_2Epred__set_2ECROSS @ A_27b @ A_27a @ V2s2 @ V1s1 ) ) ) ) ).
%------------------------------------------------------------------------------