ITP001 Axioms: ITP138+5.ax
%------------------------------------------------------------------------------
% File : ITP138+5 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : real_sigma+2.ax [Gau20]
% : HL4138+5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 60 ( 0 unt; 0 def)
% Number of atoms : 313 ( 64 equ)
% Maximal formula atoms : 13 ( 5 avg)
% Number of connectives : 259 ( 6 ~; 0 |; 20 &)
% ( 1 <=>; 232 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 9 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of predicates : 4 ( 3 usr; 0 prp; 1-2 aty)
% Number of functors : 72 ( 72 usr; 21 con; 0-4 aty)
% Number of variables : 238 ( 237 !; 1 ?)
% SPC : FOF_SAT_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(mem_c_2Ereal__sigma_2EREAL__SUM__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),arr(arr(A_27a,ty_2Erealax_2Ereal),arr(arr(A_27a,bool),ty_2Erealax_2Ereal))) ) ).
fof(lameq_f3081,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2e] :
( mem(V2e,A_27a)
=> ! [V3acc] : ap(f3081(A_27a,V0f,V2e),V3acc) = ap(ap(c_2Erealax_2Ereal__add,ap(V0f,V2e)),V3acc) ) ) ).
fof(lameq_f3082,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2e] : ap(f3082(A_27a,V0f),V2e) = f3081(A_27a,V0f,V2e) ) ).
fof(ax_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V0f),V1s) = ap(ap(ap(c_2Epred__set_2EITSET(A_27a,ty_2Erealax_2Ereal),f3082(A_27a,V0f)),V1s),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)) ) ) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ( ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V0f),c_2Epred__set_2EEMPTY(A_27a)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)
& ! [V1e] :
( mem(V1e,A_27a)
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
=> ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V0f),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1e),V2s)) = ap(ap(c_2Erealax_2Ereal__add,ap(V0f,V1e)),ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V0f),ap(ap(c_2Epred__set_2EDELETE(A_27a),V2s),V1e))) ) ) ) ) ) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__SING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1e] :
( mem(V1e,A_27a)
=> ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V0f),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1e),c_2Epred__set_2EEMPTY(A_27a))) = ap(V0f,V1e) ) ) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__POS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V0f,V2x))) ) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V0f),V1s))) ) ) ) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__SPOS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& V0s != c_2Epred__set_2EEMPTY(A_27a) )
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s))
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V1f,V2x))) ) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V1f),V0s))) ) ) ) ) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__NONZERO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0P))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ( ( ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0P))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V1f,V2x))) ) )
& ? [V3x] :
( mem(V3x,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0P))
& ap(V1f,V3x) != ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) )
=> ( ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V1f),V0P) != ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)
<=> V0P != c_2Epred__set_2EEMPTY(A_27a) ) ) ) ) ) ) ).
fof(lameq_f3083,axiom,
! [A_27a,V2f] :
( mem(V2f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1P] :
( mem(V1P,arr(A_27a,bool))
=> ! [V4x] : ap(f3083(A_27a,V2f,V1P),V4x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(V1P,V4x)),ap(V2f,V4x)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)) ) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__IF__ELIM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1P] :
( mem(V1P,arr(A_27a,bool))
=> ! [V2f] :
( mem(V2f,arr(A_27a,ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s))
=> p(ap(V1P,V3x)) ) ) )
=> ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),f3083(A_27a,V2f,V1P)),V0s) = ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V2f),V0s) ) ) ) ) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__FINITE__SAME,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0P))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2p] :
( mem(V2p,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2p),V0P))
& ! [V3q] :
( mem(V3q,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3q),V0P))
=> ap(V1f,V2p) = ap(V1f,V3q) ) ) )
=> ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V1f),V0P) = ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V0P))),ap(V1f,V2p)) ) ) ) ) ) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__FINITE__CONST,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0P))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( ! [V3y] :
( mem(V3y,A_27a)
=> ap(V1f,V3y) = V2x )
=> ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V1f),V0P) = ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V0P))),V2x) ) ) ) ) ) ) ).
fof(lameq_f3084,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V2x] : ap(f3084(A_27a,V1f,V0P),V2x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0P)),ap(V1f,V2x)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)) ) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__IN__IF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0P))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V1f),V0P) = ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),f3084(A_27a,V1f,V0P)),V0P) ) ) ) ) ).
fof(lameq_f3085,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2c] :
( mem(V2c,ty_2Erealax_2Ereal)
=> ! [V3x] : ap(f3085(A_27a,V1f,V2c),V3x) = ap(ap(c_2Erealax_2Ereal__mul,V2c),ap(V1f,V3x)) ) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__CMUL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0P))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2c] :
( mem(V2c,ty_2Erealax_2Ereal)
=> ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),f3085(A_27a,V1f,V2c)),V0P) = ap(ap(c_2Erealax_2Ereal__mul,V2c),ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V1f),V0P)) ) ) ) ) ) ).
fof(lameq_f3086,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2x] : ap(f3086(A_27a,V1f),V2x) = ap(c_2Erealax_2Ereal__neg,ap(V1f,V2x)) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__NEG,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0P))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),f3086(A_27a,V1f)),V0P) = ap(c_2Erealax_2Ereal__neg,ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V1f),V0P)) ) ) ) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0P))
=> ! [V1f_27] :
( mem(V1f_27,arr(A_27a,A_27b))
=> ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V1f_27),V0P),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1f_27),V0P)))
=> ! [V2f] :
( mem(V2f,arr(A_27b,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27b),V2f),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1f_27),V0P)) = ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Erealax_2Ereal,A_27b),V2f),V1f_27)),V0P) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__DISJOINT__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1P_27] :
( mem(V1P_27,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0P))
& p(ap(c_2Epred__set_2EFINITE(A_27a),V1P_27))
& p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V0P),V1P_27)) )
=> ! [V2f] :
( mem(V2f,arr(A_27a,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V2f),ap(ap(c_2Epred__set_2EUNION(A_27a),V0P),V1P_27)) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V2f),V0P)),ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V2f),V1P_27)) ) ) ) ) ) ).
fof(lameq_f3087,axiom,
! [A_27a,V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1x] : ap(f3087(A_27a,V0P),V1x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0P)),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__EQ__CARD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0P))
=> ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),f3087(A_27a,V0P)),V0P) = ap(c_2Ereal_2Ereal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V0P)) ) ) ) ).
fof(lameq_f3088,axiom,
! [A_27a,V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1s] : ap(f3088(A_27a,V0P),V1s) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Ebool_2EIN(A_27a),V1s),V0P)),ap(c_2Erealax_2Einv,ap(c_2Ereal_2Ereal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V0P)))),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__INV__CARD__EQ__1,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( ( V0P != c_2Epred__set_2EEMPTY(A_27a)
& p(ap(c_2Epred__set_2EFINITE(A_27a),V0P)) )
=> ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),f3088(A_27a,V0P)),V0P) = ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ) ) ).
fof(lameq_f3089,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2p] : ap(f3089(A_27a,V1f),V2p) = ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(ty_2Erealax_2Ereal),ap(V1f,V2p)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__INTER__NONZERO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0P))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V1f),ap(ap(c_2Epred__set_2EINTER(A_27a),V0P),f3089(A_27a,V1f))) = ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V1f),V0P) ) ) ) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__INTER__ELIM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0P))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2P_27] :
( mem(V2P_27,arr(A_27a,bool))
=> ( ! [V3x] :
( mem(V3x,A_27a)
=> ( ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2P_27))
=> ap(V1f,V3x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) )
=> ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V1f),ap(ap(c_2Epred__set_2EINTER(A_27a),V0P),V2P_27)) = ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V1f),V0P) ) ) ) ) ) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__COUNT,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(ty_2Enum_2Enum),V0f),ap(c_2Epred__set_2Ecount,V1n)) = ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Enum_2E0),V1n)),V0f) ) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__MONO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0P))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2f_27] :
( mem(V2f_27,arr(A_27a,ty_2Erealax_2Ereal))
=> ( ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0P))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V1f,V3x)),ap(V2f_27,V3x))) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V1f),V0P)),ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V2f_27),V0P))) ) ) ) ) ) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__POS__MEM__LE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0P))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0P))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V1f,V2x))) ) )
=> ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0P))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V1f,V3x)),ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V1f),V0P))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__CONST__EQ__1__EQ__INV__CARD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0P))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ( ( ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V1f),V0P) = ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))
& ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] :
( mem(V3y,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0P))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V3y),V0P)) )
=> ap(V1f,V2x) = ap(V1f,V3y) ) ) ) )
=> ! [V4x] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V0P))
=> ap(V1f,V4x) = ap(c_2Erealax_2Einv,ap(c_2Ereal_2Ereal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V0P))) ) ) ) ) ) ) ) ).
fof(lameq_f3090,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2f_27] :
( mem(V2f_27,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V3x] : ap(f3090(A_27a,V1f,V2f_27),V3x) = ap(ap(c_2Erealax_2Ereal__add,ap(V1f,V3x)),ap(V2f_27,V3x)) ) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__ADD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2f_27] :
( mem(V2f_27,arr(A_27a,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),f3090(A_27a,V1f,V2f_27)),V0s) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V1f),V0s)),ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V2f_27),V0s)) ) ) ) ) ) ).
fof(lameq_f3091,axiom,
! [A_27b,A_27a,V2f] :
( mem(V2f,arr(A_27a,arr(A_27b,ty_2Erealax_2Ereal)))
=> ! [V1s_27] :
( mem(V1s_27,arr(A_27b,bool))
=> ! [V3x] : ap(f3091(A_27b,A_27a,V2f,V1s_27),V3x) = ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27b),ap(V2f,V3x)),V1s_27) ) ) ).
fof(lameq_f3092,axiom,
! [A_27a,A_27b,V2f] :
( mem(V2f,arr(A_27a,arr(A_27b,ty_2Erealax_2Ereal)))
=> ! [V4x] : ap(f3092(A_27a,A_27b,V2f),V4x) = ap(ap(V2f,ap(c_2Epair_2EFST(A_27a,A_27b),V4x)),ap(c_2Epair_2ESND(A_27a,A_27b),V4x)) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__REAL__SUM__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1s_27] :
( mem(V1s_27,arr(A_27b,bool))
=> ! [V2f] :
( mem(V2f,arr(A_27a,arr(A_27b,ty_2Erealax_2Ereal)))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V1s_27)) )
=> ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),f3091(A_27b,A_27a,V2f,V1s_27)),V0s) = ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(ty_2Epair_2Eprod(A_27a,A_27b)),f3092(A_27a,A_27b,V2f)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0s),V1s_27)) ) ) ) ) ) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__0,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),k(A_27a,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))),V0s) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ).
fof(lameq_f3093,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Erealax_2Ereal)))
=> ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4n] : ap(f3093(A_27a,V1f,V3x),V4n) = ap(ap(V1f,V4n),V3x) ) ) ).
fof(lameq_f3094,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Erealax_2Ereal)))
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V5n] : ap(f3094(A_27a,V1f,V0s),V5n) = ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),ap(V1f,V5n)),V0s) ) ) ).
fof(conj_thm_2Ereal__sigma_2ESEQ__REAL__SUM__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,ty_2Erealax_2Ereal)))
=> ! [V2f_27] :
( mem(V2f_27,arr(A_27a,ty_2Erealax_2Ereal))
=> ( ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s))
=> p(ap(ap(c_2Eseq_2E_2D_2D_3E,f3093(A_27a,V1f,V3x)),ap(V2f_27,V3x))) ) )
=> p(ap(ap(c_2Eseq_2E_2D_2D_3E,f3094(A_27a,V1f,V0s)),ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V2f_27),V0s))) ) ) ) ) ) ) ).
fof(lameq_f3095,axiom,
! [A_27b,A_27a,V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,ty_2Erealax_2Ereal)))
=> ! [V2s_27] :
( mem(V2s_27,arr(A_27b,bool))
=> ! [V3x] : ap(f3095(A_27b,A_27a,V0f,V2s_27),V3x) = ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27b),ap(V0f,V3x)),V2s_27) ) ) ).
fof(lameq_f3096,axiom,
! [A_27b,A_27a,V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,ty_2Erealax_2Ereal)))
=> ! [V4x] :
( mem(V4x,A_27b)
=> ! [V5y] : ap(f3096(A_27b,A_27a,V0f,V4x),V5y) = ap(ap(V0f,V5y),V4x) ) ) ).
fof(lameq_f3097,axiom,
! [A_27a,A_27b,V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,ty_2Erealax_2Ereal)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V4x] : ap(f3097(A_27a,A_27b,V0f,V1s),V4x) = ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),f3096(A_27b,A_27a,V0f,V4x)),V1s) ) ) ).
fof(conj_thm_2Ereal__sigma_2ENESTED__REAL__SUM__IMAGE__REVERSE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,ty_2Erealax_2Ereal)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2s_27] :
( mem(V2s_27,arr(A_27b,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V2s_27)) )
=> ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),f3095(A_27b,A_27a,V0f,V2s_27)),V1s) = ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27b),f3097(A_27a,A_27b,V0f,V1s)),V2s_27) ) ) ) ) ) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__EQ__sum,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1r] :
( mem(V1r,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Enum_2E0),V0n)),V1r) = ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(ty_2Enum_2Enum),V1r),ap(c_2Epred__set_2Ecount,V0n)) ) ) ).
fof(lameq_f3098,axiom,
! [A_27a,V2i] :
( mem(V2i,A_27a)
=> ! [V0a] :
( mem(V0a,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V3j] : ap(f3098(A_27a,V2i,V0a),V3j) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0a,V2i)),ap(V0a,V3j)) ) ) ).
fof(lameq_f3099,axiom,
! [A_27a,V0a] :
( mem(V0a,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2i] : ap(f3099(A_27a,V0a),V2i) = f3098(A_27a,V2i,V0a) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__POW,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
=> ap(ap(c_2Ereal_2Epow,ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V0a),V1s)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))) = ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(ty_2Epair_2Eprod(A_27a,A_27a)),ap(c_2Epair_2EUNCURRY(A_27a,A_27a,ty_2Erealax_2Ereal),f3099(A_27a,V0a))),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27a),V1s),V1s)) ) ) ) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2f_27] :
( mem(V2f_27,arr(A_27a,ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s))
=> ap(V1f,V3x) = ap(V2f_27,V3x) ) ) )
=> ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V1f),V0s) = ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V2f_27),V0s) ) ) ) ) ) ).
fof(lameq_f3100,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ! [V3x] : ap(f3100(A_27a,V1f,V0s,V2z),V3x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s)),ap(V1f,V3x)),V2z) ) ) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__IN__IF__ALT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V1f),V0s) = ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),f3100(A_27a,V1f,V0s,V2z)),V0s) ) ) ) ) ) ).
fof(lameq_f3101,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2f_27] :
( mem(V2f_27,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V3x] : ap(f3101(A_27a,V1f,V2f_27),V3x) = ap(ap(c_2Ereal_2Ereal__sub,ap(V1f,V3x)),ap(V2f_27,V3x)) ) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__SUB,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2f_27] :
( mem(V2f_27,arr(A_27a,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),f3101(A_27a,V1f,V2f_27)),V0s) = ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V1f),V0s)),ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V2f_27),V0s)) ) ) ) ) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__MONO__SET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& p(ap(c_2Epred__set_2EFINITE(A_27a),V2t))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),V2t))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2t))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V0f,V3x))) ) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V0f),V1s)),ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V0f),V2t))) ) ) ) ) ) ).
fof(lameq_f3102,axiom,
! [A_27a,A_27b,V3x] :
( mem(V3x,A_27a)
=> ! [V0f] :
( mem(V0f,arr(ty_2Epair_2Eprod(A_27a,A_27b),ty_2Erealax_2Ereal))
=> ! [V4y] : ap(f3102(A_27a,A_27b,V3x,V0f),V4y) = ap(V0f,ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V3x),V4y)) ) ) ).
fof(lameq_f3103,axiom,
! [A_27b,A_27a,V0f] :
( mem(V0f,arr(ty_2Epair_2Eprod(A_27a,A_27b),ty_2Erealax_2Ereal))
=> ! [V3x] : ap(f3103(A_27b,A_27a,V0f),V3x) = f3102(A_27a,A_27b,V3x,V0f) ) ).
fof(lameq_f3104,axiom,
! [A_27b,A_27a,V5y] :
( mem(V5y,A_27b)
=> ! [V0f] :
( mem(V0f,arr(ty_2Epair_2Eprod(A_27a,A_27b),ty_2Erealax_2Ereal))
=> ! [V6x] : ap(f3104(A_27b,A_27a,V5y,V0f),V6x) = ap(V0f,ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V6x),V5y)) ) ) ).
fof(lameq_f3105,axiom,
! [A_27a,A_27b,V0f] :
( mem(V0f,arr(ty_2Epair_2Eprod(A_27a,A_27b),ty_2Erealax_2Ereal))
=> ! [V5y] : ap(f3105(A_27a,A_27b,V0f),V5y) = f3104(A_27b,A_27a,V5y,V0f) ) ).
fof(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__CROSS__SYM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(ty_2Epair_2Eprod(A_27a,A_27b),ty_2Erealax_2Ereal))
=> ! [V1s1] :
( mem(V1s1,arr(A_27a,bool))
=> ! [V2s2] :
( mem(V2s2,arr(A_27b,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s1))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V2s2)) )
=> ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(ty_2Epair_2Eprod(A_27a,A_27b)),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,ty_2Erealax_2Ereal),f3103(A_27b,A_27a,V0f))),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V1s1),V2s2)) = ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(ty_2Epair_2Eprod(A_27b,A_27a)),ap(c_2Epair_2EUNCURRY(A_27b,A_27a,ty_2Erealax_2Ereal),f3105(A_27a,A_27b,V0f))),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27a),V2s2),V1s1)) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------