ITP001 Axioms: ITP144+5.ax
%------------------------------------------------------------------------------
% File : ITP144+5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : iterate+2.ax [Gau20]
% : HL4144+5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 519 ( 13 unt; 0 def)
% Number of atoms : 3417 ( 580 equ)
% Maximal formula atoms : 36 ( 6 avg)
% Number of connectives : 2974 ( 76 ~; 5 |; 443 &)
% ( 49 <=>;2401 =>; 0 <=; 0 <~>)
% Maximal formula depth : 33 ( 10 avg)
% Maximal term depth : 13 ( 2 avg)
% Number of predicates : 4 ( 3 usr; 0 prp; 1-2 aty)
% Number of functors : 243 ( 243 usr; 36 con; 0-8 aty)
% Number of variables : 2317 (2259 !; 58 ?)
% SPC : FOF_SAT_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(mem_c_2Eiterate_2E_2E_2E,axiom,
mem(c_2Eiterate_2E_2E_2E,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,bool)))) ).
fof(mem_c_2Eiterate_2EFINREC,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Eiterate_2EFINREC(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,A_27b)),arr(A_27b,arr(arr(A_27a,bool),arr(A_27b,arr(ty_2Enum_2Enum,bool)))))) ) ) ).
fof(mem_c_2Eiterate_2EITSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Eiterate_2EITSET(A_27a,A_27b),arr(arr(A_27b,arr(A_27a,A_27a)),arr(arr(A_27b,bool),arr(A_27a,A_27a)))) ) ) ).
fof(mem_c_2Eiterate_2ESum,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eiterate_2ESum(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,ty_2Erealax_2Ereal),ty_2Erealax_2Ereal))) ) ).
fof(mem_c_2Eiterate_2Einf,axiom,
mem(c_2Eiterate_2Einf,arr(arr(ty_2Erealax_2Ereal,bool),ty_2Erealax_2Ereal)) ).
fof(mem_c_2Eiterate_2Eiterate,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Eiterate_2Eiterate(A_27a,A_27b),arr(arr(A_27b,arr(A_27b,A_27b)),arr(arr(A_27a,bool),arr(arr(A_27a,A_27b),A_27b)))) ) ) ).
fof(mem_c_2Eiterate_2Emonoidal,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eiterate_2Emonoidal(A_27a),arr(arr(A_27a,arr(A_27a,A_27a)),bool)) ) ).
fof(mem_c_2Eiterate_2Eneutral,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eiterate_2Eneutral(A_27a),arr(arr(A_27a,arr(A_27a,A_27a)),A_27a)) ) ).
fof(mem_c_2Eiterate_2Ensum,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eiterate_2Ensum(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,ty_2Enum_2Enum),ty_2Enum_2Enum))) ) ).
fof(mem_c_2Eiterate_2Epolynomial__function,axiom,
mem(c_2Eiterate_2Epolynomial__function,arr(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),bool)) ).
fof(mem_c_2Eiterate_2Esupport,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Eiterate_2Esupport(A_27a,A_27b),arr(arr(A_27b,arr(A_27b,A_27b)),arr(arr(A_27a,A_27b),arr(arr(A_27a,bool),arr(A_27a,bool))))) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__LT__BETWEEN,axiom,
! [V0a] :
( mem(V0a,ty_2Erealax_2Ereal)
=> ! [V1b] :
( mem(V1b,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,V0a),V1b))
<=> ? [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Erealax_2Ereal__lt,V0a),V2x))
& p(ap(ap(c_2Erealax_2Ereal__lt,V2x),V1b)) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESIMP__REAL__ARCH,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ? [V1n] :
( mem(V1n,ty_2Enum_2Enum)
& p(ap(ap(c_2Ereal_2Ereal__lte,V0x),ap(c_2Ereal_2Ereal__of__num,V1n))) ) ) ).
fof(conj_thm_2Eiterate_2ELOWER__BOUND__FINITE__SET__REAL,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))
=> ? [V2a] :
( mem(V2a,ty_2Erealax_2Ereal)
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V2a),ap(V0f,V3x))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EUPPER__BOUND__FINITE__SET__REAL,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))
=> ? [V2a] :
( mem(V2a,ty_2Erealax_2Ereal)
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V3x)),V2a)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__WLOG__LT,axiom,
! [V0P] :
( mem(V0P,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,bool)))
=> ( ( ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> p(ap(ap(V0P,V1x),V1x)) )
& ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ! [V3y] :
( mem(V3y,ty_2Erealax_2Ereal)
=> ( p(ap(ap(V0P,V2x),V3y))
<=> p(ap(ap(V0P,V3y),V2x)) ) ) )
& ! [V4x] :
( mem(V4x,ty_2Erealax_2Ereal)
=> ! [V5y] :
( mem(V5y,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,V4x),V5y))
=> p(ap(ap(V0P,V4x),V5y)) ) ) ) )
=> ! [V6x] :
( mem(V6x,ty_2Erealax_2Ereal)
=> ! [V7y] :
( mem(V7y,ty_2Erealax_2Ereal)
=> p(ap(ap(V0P,V6x),V7y)) ) ) ) ) ).
fof(ax_thm_2Eiterate_2EFINREC__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V1b] :
( mem(V1b,A_27b)
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3a] :
( mem(V3a,A_27b)
=> ( p(ap(ap(ap(ap(ap(c_2Eiterate_2EFINREC(A_27a,A_27b),V0f),V1b),V2s),V3a),c_2Enum_2E0))
<=> ( V2s = c_2Epred__set_2EEMPTY(A_27a)
& V3a = V1b ) ) ) ) ) )
& ! [V4f] :
( mem(V4f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V5b] :
( mem(V5b,A_27b)
=> ! [V6s] :
( mem(V6s,arr(A_27a,bool))
=> ! [V7a] :
( mem(V7a,A_27b)
=> ! [V8n] :
( mem(V8n,ty_2Enum_2Enum)
=> ( p(ap(ap(ap(ap(ap(c_2Eiterate_2EFINREC(A_27a,A_27b),V4f),V5b),V6s),V7a),ap(c_2Enum_2ESUC,V8n)))
<=> ? [V9x] :
( mem(V9x,A_27a)
& ? [V10c] :
( mem(V10c,A_27b)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V9x),V6s))
& p(ap(ap(ap(ap(ap(c_2Eiterate_2EFINREC(A_27a,A_27b),V4f),V5b),ap(ap(c_2Epred__set_2EDELETE(A_27a),V6s),V9x)),V10c),V8n))
& V7a = ap(ap(V4f,V9x),V10c) ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EFINREC__def__compute,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V1b] :
( mem(V1b,A_27b)
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3a] :
( mem(V3a,A_27b)
=> ( p(ap(ap(ap(ap(ap(c_2Eiterate_2EFINREC(A_27a,A_27b),V0f),V1b),V2s),V3a),c_2Enum_2E0))
<=> ( V2s = c_2Epred__set_2EEMPTY(A_27a)
& V3a = V1b ) ) ) ) ) )
& ! [V4f] :
( mem(V4f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V5b] :
( mem(V5b,A_27b)
=> ! [V6s] :
( mem(V6s,arr(A_27a,bool))
=> ! [V7a] :
( mem(V7a,A_27b)
=> ! [V8n] :
( mem(V8n,ty_2Enum_2Enum)
=> ( p(ap(ap(ap(ap(ap(c_2Eiterate_2EFINREC(A_27a,A_27b),V4f),V5b),V6s),V7a),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V8n))))
<=> ? [V9x] :
( mem(V9x,A_27a)
& ? [V10c] :
( mem(V10c,A_27b)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V9x),V6s))
& p(ap(ap(ap(ap(ap(c_2Eiterate_2EFINREC(A_27a,A_27b),V4f),V5b),ap(ap(c_2Epred__set_2EDELETE(A_27a),V6s),V9x)),V10c),ap(ap(c_2Earithmetic_2E_2D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V8n))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))
& V7a = ap(ap(V4f,V9x),V10c) ) ) ) ) ) ) ) )
& ! [V11f] :
( mem(V11f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V12b] :
( mem(V12b,A_27b)
=> ! [V13s] :
( mem(V13s,arr(A_27a,bool))
=> ! [V14a] :
( mem(V14a,A_27b)
=> ! [V15n] :
( mem(V15n,ty_2Enum_2Enum)
=> ( p(ap(ap(ap(ap(ap(c_2Eiterate_2EFINREC(A_27a,A_27b),V11f),V12b),V13s),V14a),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,V15n))))
<=> ? [V16x] :
( mem(V16x,A_27a)
& ? [V17c] :
( mem(V17c,A_27b)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V16x),V13s))
& p(ap(ap(ap(ap(ap(c_2Eiterate_2EFINREC(A_27a,A_27b),V11f),V12b),ap(ap(c_2Epred__set_2EDELETE(A_27a),V13s),V16x)),V17c),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,V15n))))
& V14a = ap(ap(V11f,V16x),V17c) ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EFINREC__1__LEMMA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V1b] :
( mem(V1b,A_27b)
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3a] :
( mem(V3a,A_27b)
=> ( p(ap(ap(ap(ap(ap(c_2Eiterate_2EFINREC(A_27a,A_27b),V0f),V1b),V2s),V3a),ap(c_2Enum_2ESUC,c_2Enum_2E0)))
<=> ? [V4x] :
( mem(V4x,A_27a)
& V2s = ap(ap(c_2Epred__set_2EINSERT(A_27a),V4x),c_2Epred__set_2EEMPTY(A_27a))
& V3a = ap(ap(V0f,V4x),V1b) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EFINREC__SUC__LEMMA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V1b] :
( mem(V1b,A_27b)
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] :
( mem(V3y,A_27a)
=> ! [V4s] :
( mem(V4s,A_27b)
=> ( V2x != V3y
=> ap(ap(V0f,V2x),ap(ap(V0f,V3y),V4s)) = ap(ap(V0f,V3y),ap(ap(V0f,V2x),V4s)) ) ) ) )
=> ! [V5n] :
( mem(V5n,ty_2Enum_2Enum)
=> ! [V6s] :
( mem(V6s,arr(A_27a,bool))
=> ! [V7z] :
( mem(V7z,A_27b)
=> ( p(ap(ap(ap(ap(ap(c_2Eiterate_2EFINREC(A_27a,A_27b),V0f),V1b),V6s),V7z),ap(c_2Enum_2ESUC,V5n)))
=> ! [V8x] :
( mem(V8x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V8x),V6s))
=> ? [V9w] :
( mem(V9w,A_27b)
& p(ap(ap(ap(ap(ap(c_2Eiterate_2EFINREC(A_27a,A_27b),V0f),V1b),ap(ap(c_2Epred__set_2EDELETE(A_27a),V6s),V8x)),V9w),V5n))
& V7z = ap(ap(V0f,V8x),V9w) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EFINREC__UNIQUE__LEMMA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V1b] :
( mem(V1b,A_27b)
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] :
( mem(V3y,A_27a)
=> ! [V4s] :
( mem(V4s,A_27b)
=> ( V2x != V3y
=> ap(ap(V0f,V2x),ap(ap(V0f,V3y),V4s)) = ap(ap(V0f,V3y),ap(ap(V0f,V2x),V4s)) ) ) ) )
=> ! [V5n1] :
( mem(V5n1,ty_2Enum_2Enum)
=> ! [V6n2] :
( mem(V6n2,ty_2Enum_2Enum)
=> ! [V7s] :
( mem(V7s,arr(A_27a,bool))
=> ! [V8a1] :
( mem(V8a1,A_27b)
=> ! [V9a2] :
( mem(V9a2,A_27b)
=> ( ( p(ap(ap(ap(ap(ap(c_2Eiterate_2EFINREC(A_27a,A_27b),V0f),V1b),V7s),V8a1),V5n1))
& p(ap(ap(ap(ap(ap(c_2Eiterate_2EFINREC(A_27a,A_27b),V0f),V1b),V7s),V9a2),V6n2)) )
=> ( V8a1 = V9a2
& V5n1 = V6n2 ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EFINREC__EXISTS__LEMMA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V1b] :
( mem(V1b,A_27b)
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
=> ? [V3a] :
( mem(V3a,A_27b)
& ? [V4n] :
( mem(V4n,ty_2Enum_2Enum)
& p(ap(ap(ap(ap(ap(c_2Eiterate_2EFINREC(A_27a,A_27b),V0f),V1b),V2s),V3a),V4n)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EFINREC__FUN__LEMMA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1R] :
( mem(V1R,arr(A_27a,arr(A_27b,arr(A_27c,bool))))
=> ( ( ! [V2s] :
( mem(V2s,A_27a)
=> ( p(ap(V0P,V2s))
=> ? [V3a] :
( mem(V3a,A_27b)
& ? [V4n] :
( mem(V4n,A_27c)
& p(ap(ap(ap(V1R,V2s),V3a),V4n)) ) ) ) )
& ! [V5n1] :
( mem(V5n1,A_27c)
=> ! [V6n2] :
( mem(V6n2,A_27c)
=> ! [V7s] :
( mem(V7s,A_27a)
=> ! [V8a1] :
( mem(V8a1,A_27b)
=> ! [V9a2] :
( mem(V9a2,A_27b)
=> ( ( p(ap(ap(ap(V1R,V7s),V8a1),V5n1))
& p(ap(ap(ap(V1R,V7s),V9a2),V6n2)) )
=> ( V8a1 = V9a2
& V5n1 = V6n2 ) ) ) ) ) ) ) )
=> ? [V10f] :
( mem(V10f,arr(A_27a,A_27b))
& ! [V11s] :
( mem(V11s,A_27a)
=> ! [V12a] :
( mem(V12a,A_27b)
=> ( p(ap(V0P,V11s))
=> ( ? [V13n] :
( mem(V13n,A_27c)
& p(ap(ap(ap(V1R,V11s),V12a),V13n)) )
<=> ap(V10f,V11s) = V12a ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EFINREC__FUN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V1b] :
( mem(V1b,A_27b)
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] :
( mem(V3y,A_27a)
=> ! [V4s] :
( mem(V4s,A_27b)
=> ( V2x != V3y
=> ap(ap(V0f,V2x),ap(ap(V0f,V3y),V4s)) = ap(ap(V0f,V3y),ap(ap(V0f,V2x),V4s)) ) ) ) )
=> ? [V5g] :
( mem(V5g,arr(arr(A_27a,bool),A_27b))
& ap(V5g,c_2Epred__set_2EEMPTY(A_27a)) = V1b
& ! [V6s] :
( mem(V6s,arr(A_27a,bool))
=> ! [V7x] :
( mem(V7x,A_27a)
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V6s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V7x),V6s)) )
=> ap(V5g,V6s) = ap(ap(V0f,V7x),ap(V5g,ap(ap(c_2Epred__set_2EDELETE(A_27a),V6s),V7x))) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESET__RECURSION__LEMMA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V1b] :
( mem(V1b,A_27b)
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] :
( mem(V3y,A_27a)
=> ! [V4s] :
( mem(V4s,A_27b)
=> ( V2x != V3y
=> ap(ap(V0f,V2x),ap(ap(V0f,V3y),V4s)) = ap(ap(V0f,V3y),ap(ap(V0f,V2x),V4s)) ) ) ) )
=> ? [V5g] :
( mem(V5g,arr(arr(A_27a,bool),A_27b))
& ap(V5g,c_2Epred__set_2EEMPTY(A_27a)) = V1b
& ! [V6x] :
( mem(V6x,A_27a)
=> ! [V7s] :
( mem(V7s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V7s))
=> ap(V5g,ap(ap(c_2Epred__set_2EINSERT(A_27a),V6x),V7s)) = ap(ap(ap(c_2Ebool_2ECOND(A_27b),ap(ap(c_2Ebool_2EIN(A_27a),V6x),V7s)),ap(V5g,V7s)),ap(ap(V0f,V6x),ap(V5g,V7s))) ) ) ) ) ) ) ) ) ) ).
fof(lameq_f3173,axiom,
! [A_27a,A_27b,V4x] :
( mem(V4x,A_27b)
=> ! [V3g] :
( mem(V3g,arr(arr(A_27b,bool),A_27a))
=> ! [V0f] :
( mem(V0f,arr(A_27b,arr(A_27a,A_27a)))
=> ! [V5s] : ap(f3173(A_27a,A_27b,V4x,V3g,V0f),V5s) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(c_2Epred__set_2EFINITE(A_27b),V5s)),ap(ap(c_2Emin_2E_3D(A_27a),ap(V3g,ap(ap(c_2Epred__set_2EINSERT(A_27b),V4x),V5s))),ap(ap(ap(c_2Ebool_2ECOND(A_27a),ap(ap(c_2Ebool_2EIN(A_27b),V4x),V5s)),ap(V3g,V5s)),ap(ap(V0f,V4x),ap(V3g,V5s))))) ) ) ) ).
fof(lameq_f3174,axiom,
! [A_27a,A_27b,V0f] :
( mem(V0f,arr(A_27b,arr(A_27a,A_27a)))
=> ! [V3g] :
( mem(V3g,arr(arr(A_27b,bool),A_27a))
=> ! [V4x] : ap(f3174(A_27a,A_27b,V0f,V3g),V4x) = ap(c_2Ebool_2E_21(arr(A_27b,bool)),f3173(A_27a,A_27b,V4x,V3g,V0f)) ) ) ).
fof(lameq_f3175,axiom,
! [A_27a,A_27b,V2b] :
( mem(V2b,A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27b,arr(A_27a,A_27a)))
=> ! [V3g] : ap(f3175(A_27a,A_27b,V2b,V0f),V3g) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Emin_2E_3D(A_27a),ap(V3g,c_2Epred__set_2EEMPTY(A_27b))),V2b)),ap(c_2Ebool_2E_21(A_27b),f3174(A_27a,A_27b,V0f,V3g))) ) ) ).
fof(ax_thm_2Eiterate_2EITSET__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27b,arr(A_27a,A_27a)))
=> ! [V1s] :
( mem(V1s,arr(A_27b,bool))
=> ! [V2b] :
( mem(V2b,A_27a)
=> ap(ap(ap(c_2Eiterate_2EITSET(A_27a,A_27b),V0f),V1s),V2b) = ap(ap(c_2Emin_2E_40(arr(arr(A_27b,bool),A_27a)),f3175(A_27a,A_27b,V2b,V0f)),V1s) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EFINITE__RECURSION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V1b] :
( mem(V1b,A_27b)
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] :
( mem(V3y,A_27a)
=> ! [V4s] :
( mem(V4s,A_27b)
=> ( V2x != V3y
=> ap(ap(V0f,V2x),ap(ap(V0f,V3y),V4s)) = ap(ap(V0f,V3y),ap(ap(V0f,V2x),V4s)) ) ) ) )
=> ( ap(ap(ap(c_2Eiterate_2EITSET(A_27b,A_27a),V0f),c_2Epred__set_2EEMPTY(A_27a)),V1b) = V1b
& ! [V5x] :
( mem(V5x,A_27a)
=> ! [V6s] :
( mem(V6s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V6s))
=> ap(ap(ap(c_2Eiterate_2EITSET(A_27b,A_27a),V0f),ap(ap(c_2Epred__set_2EINSERT(A_27a),V5x),V6s)),V1b) = ap(ap(ap(c_2Ebool_2ECOND(A_27b),ap(ap(c_2Ebool_2EIN(A_27a),V5x),V6s)),ap(ap(ap(c_2Eiterate_2EITSET(A_27b,A_27a),V0f),V6s),V1b)),ap(ap(V0f,V5x),ap(ap(ap(c_2Eiterate_2EITSET(A_27b,A_27a),V0f),V6s),V1b))) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ECARD__UNION__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2u] :
( mem(V2u,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2u))
& ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V1t) = c_2Epred__set_2EEMPTY(A_27a)
& ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1t) = V2u )
=> ap(ap(c_2Earithmetic_2E_2B,ap(c_2Epred__set_2ECARD(A_27a),V0s)),ap(c_2Epred__set_2ECARD(A_27a),V1t)) = ap(c_2Epred__set_2ECARD(A_27a),V2u) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUBSET__RESTRICT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1P] :
( mem(V1P,arr(A_27a,bool))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f1884(A_27a,V1P,V0s))),V0s)) ) ) ) ).
fof(conj_thm_2Eiterate_2EFINITE__RESTRICT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1P] :
( mem(V1P,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f1884(A_27a,V1P,V0s)))) ) ) ) ) ).
fof(lameq_f3176,axiom,
! [V0a] :
( mem(V0a,ty_2Erealax_2Ereal)
=> ! [V1x] : ap(f3176(V0a),V1x) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),V1x),ap(ap(c_2Erealax_2Ereal__lt,V0a),V1x)) ) ).
fof(lameq_f3177,axiom,
! [V2a] :
( mem(V2a,ty_2Erealax_2Ereal)
=> ! [V3x] : ap(f3177(V2a),V3x) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),V3x),ap(ap(c_2Ereal_2Ereal__lte,V2a),V3x)) ) ).
fof(lameq_f3178,axiom,
! [V4b] :
( mem(V4b,ty_2Erealax_2Ereal)
=> ! [V5x] : ap(f3178(V4b),V5x) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),V5x),ap(ap(c_2Erealax_2Ereal__lt,V5x),V4b)) ) ).
fof(lameq_f3179,axiom,
! [V6b] :
( mem(V6b,ty_2Erealax_2Ereal)
=> ! [V7x] : ap(f3179(V6b),V7x) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),V7x),ap(ap(c_2Ereal_2Ereal__lte,V7x),V6b)) ) ).
fof(lameq_f3180,axiom,
! [V9b] :
( mem(V9b,ty_2Erealax_2Ereal)
=> ! [V8a] :
( mem(V8a,ty_2Erealax_2Ereal)
=> ! [V10x] : ap(f3180(V9b,V8a),V10x) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),V10x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Erealax_2Ereal__lt,V8a),V10x)),ap(ap(c_2Erealax_2Ereal__lt,V10x),V9b))) ) ) ).
fof(lameq_f3181,axiom,
! [V12b] :
( mem(V12b,ty_2Erealax_2Ereal)
=> ! [V11a] :
( mem(V11a,ty_2Erealax_2Ereal)
=> ! [V13x] : ap(f3181(V12b,V11a),V13x) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),V13x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ereal_2Ereal__lte,V11a),V13x)),ap(ap(c_2Erealax_2Ereal__lt,V13x),V12b))) ) ) ).
fof(lameq_f3182,axiom,
! [V15b] :
( mem(V15b,ty_2Erealax_2Ereal)
=> ! [V14a] :
( mem(V14a,ty_2Erealax_2Ereal)
=> ! [V16x] : ap(f3182(V15b,V14a),V16x) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),V16x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Erealax_2Ereal__lt,V14a),V16x)),ap(ap(c_2Ereal_2Ereal__lte,V16x),V15b))) ) ) ).
fof(lameq_f3183,axiom,
! [V18b] :
( mem(V18b,ty_2Erealax_2Ereal)
=> ! [V17a] :
( mem(V17a,ty_2Erealax_2Ereal)
=> ! [V19x] : ap(f3183(V18b,V17a),V19x) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),V19x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ereal_2Ereal__lte,V17a),V19x)),ap(ap(c_2Ereal_2Ereal__lte,V19x),V18b))) ) ) ).
fof(conj_thm_2Eiterate_2EFINITE__REAL__INTERVAL,axiom,
( ! [V0a] :
( mem(V0a,ty_2Erealax_2Ereal)
=> ~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3176(V0a)))) )
& ! [V2a] :
( mem(V2a,ty_2Erealax_2Ereal)
=> ~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3177(V2a)))) )
& ! [V4b] :
( mem(V4b,ty_2Erealax_2Ereal)
=> ~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3178(V4b)))) )
& ! [V6b] :
( mem(V6b,ty_2Erealax_2Ereal)
=> ~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3179(V6b)))) )
& ! [V8a] :
( mem(V8a,ty_2Erealax_2Ereal)
=> ! [V9b] :
( mem(V9b,ty_2Erealax_2Ereal)
=> ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3180(V9b,V8a))))
<=> p(ap(ap(c_2Ereal_2Ereal__lte,V9b),V8a)) ) ) )
& ! [V11a] :
( mem(V11a,ty_2Erealax_2Ereal)
=> ! [V12b] :
( mem(V12b,ty_2Erealax_2Ereal)
=> ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3181(V12b,V11a))))
<=> p(ap(ap(c_2Ereal_2Ereal__lte,V12b),V11a)) ) ) )
& ! [V14a] :
( mem(V14a,ty_2Erealax_2Ereal)
=> ! [V15b] :
( mem(V15b,ty_2Erealax_2Ereal)
=> ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3182(V15b,V14a))))
<=> p(ap(ap(c_2Ereal_2Ereal__lte,V15b),V14a)) ) ) )
& ! [V17a] :
( mem(V17a,ty_2Erealax_2Ereal)
=> ! [V18b] :
( mem(V18b,ty_2Erealax_2Ereal)
=> ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3183(V18b,V17a))))
<=> p(ap(ap(c_2Ereal_2Ereal__lte,V18b),V17a)) ) ) ) ) ).
fof(conj_thm_2Eiterate_2Ereal__INFINITE,axiom,
~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))) ).
fof(conj_thm_2Eiterate_2ESET__PROVE__CASES,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(V0P,c_2Epred__set_2EEMPTY(A_27a)))
& ! [V1a] :
( mem(V1a,A_27a)
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V1a),V2s))
=> p(ap(V0P,ap(ap(c_2Epred__set_2EINSERT(A_27a),V1a),V2s))) ) ) ) )
=> ! [V3s] :
( mem(V3s,arr(A_27a,bool))
=> p(ap(V0P,V3s)) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ECHOOSE__SUBSET__STRONG,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,V0n),ap(c_2Epred__set_2ECARD(A_27a),V1s))) )
=> ? [V2t] :
( mem(V2t,arr(A_27a,bool))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2t),V1s))
& p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27a),V2t),V0n)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ECHOOSE__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1n),ap(c_2Epred__set_2ECARD(A_27a),V0s)))
=> ? [V2t] :
( mem(V2t,arr(A_27a,bool))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2t),V0s))
& p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27a),V2t),V1n)) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__COMPLETE,axiom,
! [V0P] :
( mem(V0P,arr(ty_2Erealax_2Ereal,bool))
=> ( ( ? [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
& p(ap(V0P,V1x)) )
& ? [V2M] :
( mem(V2M,ty_2Erealax_2Ereal)
& ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ( p(ap(V0P,V3x))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V2M)) ) ) ) )
=> ? [V4M] :
( mem(V4M,ty_2Erealax_2Ereal)
& ! [V5x] :
( mem(V5x,ty_2Erealax_2Ereal)
=> ( p(ap(V0P,V5x))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V5x),V4M)) ) )
& ! [V6M_27] :
( mem(V6M_27,ty_2Erealax_2Ereal)
=> ( ! [V7x] :
( mem(V7x,ty_2Erealax_2Ereal)
=> ( p(ap(V0P,V7x))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V7x),V6M_27)) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,V4M),V6M_27)) ) ) ) ) ) ).
fof(lameq_f3184,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ! [V2x] : ap(f3184(V0s,V1a),V2x) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)),ap(ap(c_2Ereal_2Ereal__lte,V2x),V1a)) ) ) ).
fof(lameq_f3185,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V3b] :
( mem(V3b,ty_2Erealax_2Ereal)
=> ! [V4x] : ap(f3185(V0s,V3b),V4x) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V0s)),ap(ap(c_2Ereal_2Ereal__lte,V4x),V3b)) ) ) ).
fof(lameq_f3186,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ! [V3b] : ap(f3186(V0s,V1a),V3b) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(c_2Ebool_2E_21(ty_2Erealax_2Ereal),f3185(V0s,V3b))),ap(ap(c_2Ereal_2Ereal__lte,V1a),V3b)) ) ) ).
fof(lameq_f3187,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a] : ap(f3187(V0s),V1a) = ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Ebool_2E_21(ty_2Erealax_2Ereal),f3184(V0s,V1a))),ap(c_2Ebool_2E_21(ty_2Erealax_2Ereal),f3186(V0s,V1a))) ) ).
fof(conj_thm_2Eiterate_2Esup__alt,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ap(c_2Ereal_2Esup,V0s) = ap(c_2Emin_2E_40(ty_2Erealax_2Ereal),f3187(V0s)) ) ).
fof(conj_thm_2Eiterate_2ESUP__EQ,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1t] :
( mem(V1t,arr(ty_2Erealax_2Ereal,bool))
=> ( ! [V2b] :
( mem(V2b,ty_2Erealax_2Ereal)
=> ( ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V2b)) ) )
<=> ! [V4x] :
( mem(V4x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V1t))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V4x),V2b)) ) ) ) )
=> ap(c_2Ereal_2Esup,V0s) = ap(c_2Ereal_2Esup,V1t) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUP,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ( ( V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)
& ? [V1b] :
( mem(V1b,ty_2Erealax_2Ereal)
& ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V2x),V1b)) ) ) ) )
=> ( ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V3x),ap(c_2Ereal_2Esup,V0s))) ) )
& ! [V4b] :
( mem(V4b,ty_2Erealax_2Ereal)
=> ( ! [V5x] :
( mem(V5x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V5x),V4b)) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Esup,V0s)),V4b)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUP__FINITE__LEMMA,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s))
& V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) )
=> ? [V1b] :
( mem(V1b,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1b),V0s))
& ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V2x),V1b)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUP__FINITE,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s))
& V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) )
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(c_2Ereal_2Esup,V0s)),V0s))
& ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V1x),ap(c_2Ereal_2Esup,V0s))) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__LE__SUP__FINITE,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s))
& V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) )
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),ap(c_2Ereal_2Esup,V0s)))
<=> ? [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s))
& p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V2x)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__SUP__LE__FINITE,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s))
& V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) )
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Esup,V0s)),V1a))
<=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V2x),V1a)) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__LT__SUP__FINITE,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s))
& V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) )
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,V1a),ap(c_2Ereal_2Esup,V0s)))
<=> ? [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s))
& p(ap(ap(c_2Erealax_2Ereal__lt,V1a),V2x)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__SUP__LT__FINITE,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s))
& V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) )
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Esup,V0s)),V1a))
<=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s))
=> p(ap(ap(c_2Erealax_2Ereal__lt,V2x),V1a)) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__SUP__LE__EQ,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ( ( V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)
& ? [V2b] :
( mem(V2b,ty_2Erealax_2Ereal)
& ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V2b)) ) ) ) )
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Esup,V0s)),V1y))
<=> ! [V4x] :
( mem(V4x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V4x),V1y)) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__SUP__UNIQUE,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1b] :
( mem(V1b,ty_2Erealax_2Ereal)
=> ( ( ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V2x),V1b)) ) )
& ! [V3b_27] :
( mem(V3b_27,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,V3b_27),V1b))
=> ? [V4x] :
( mem(V4x,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V0s))
& p(ap(ap(c_2Erealax_2Ereal__lt,V3b_27),V4x)) ) ) ) )
=> ap(c_2Ereal_2Esup,V0s) = V1b ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__SUP__LE__S,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1b] :
( mem(V1b,ty_2Erealax_2Ereal)
=> ( ( V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)
& ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V2x),V1b)) ) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Esup,V0s)),V1b)) ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__SUP__LE__SUBSET,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1t] :
( mem(V1t,arr(ty_2Erealax_2Ereal,bool))
=> ( ( V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)
& p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V0s),V1t))
& ? [V2b] :
( mem(V2b,ty_2Erealax_2Ereal)
& ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V1t))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V2b)) ) ) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Esup,V0s)),ap(c_2Ereal_2Esup,V1t))) ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__LE__SUP,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ! [V2b] :
( mem(V2b,ty_2Erealax_2Ereal)
=> ! [V3y] :
( mem(V3y,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3y),V0s))
& p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V3y))
& ! [V4x] :
( mem(V4x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V4x),V2b)) ) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,V1a),ap(c_2Ereal_2Esup,V0s))) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__SUP__BOUNDS,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ! [V2b] :
( mem(V2b,ty_2Erealax_2Ereal)
=> ( ( V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)
& ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s))
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V3x))
& p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V2b)) ) ) ) )
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),ap(c_2Ereal_2Esup,V0s)))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Esup,V0s)),V2b)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__ABS__SUP__LE,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ( ( V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)
& ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,V2x)),V1a)) ) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(c_2Ereal_2Esup,V0s))),V1a)) ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__SUP__ASCLOSE,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
=> ! [V2e] :
( mem(V2e,ty_2Erealax_2Ereal)
=> ( ( V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)
& ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V3x),V1l))),V2e)) ) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(c_2Ereal_2Esup,V0s)),V1l))),V2e)) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUP__SING,axiom,
! [V0a] :
( mem(V0a,ty_2Erealax_2Ereal)
=> ap(c_2Ereal_2Esup,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))) = V0a ) ).
fof(conj_thm_2Eiterate_2ESUP__UNIQUE,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1b] :
( mem(V1b,ty_2Erealax_2Ereal)
=> ( ! [V2c] :
( mem(V2c,ty_2Erealax_2Ereal)
=> ( ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V2c)) ) )
<=> p(ap(ap(c_2Ereal_2Ereal__lte,V1b),V2c)) ) )
=> ap(c_2Ereal_2Esup,V0s) = V1b ) ) ) ).
fof(conj_thm_2Eiterate_2ESUP__UNION,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1t] :
( mem(V1t,arr(ty_2Erealax_2Ereal,bool))
=> ( ( V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)
& V1t != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)
& ? [V2b] :
( mem(V2b,ty_2Erealax_2Ereal)
& ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V2b)) ) ) )
& ? [V4c] :
( mem(V4c,ty_2Erealax_2Ereal)
& ! [V5x] :
( mem(V5x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V1t))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V5x),V4c)) ) ) ) )
=> ap(c_2Ereal_2Esup,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t)) = ap(ap(c_2Ereal_2Emax,ap(c_2Ereal_2Esup,V0s)),ap(c_2Ereal_2Esup,V1t)) ) ) ) ).
fof(lameq_f3188,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ! [V2x] : ap(f3188(V0s,V1a),V2x) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s)),ap(ap(c_2Ereal_2Ereal__lte,V1a),V2x)) ) ) ).
fof(lameq_f3189,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V3b] :
( mem(V3b,ty_2Erealax_2Ereal)
=> ! [V4x] : ap(f3189(V0s,V3b),V4x) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V0s)),ap(ap(c_2Ereal_2Ereal__lte,V3b),V4x)) ) ) ).
fof(lameq_f3190,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ! [V3b] : ap(f3190(V0s,V1a),V3b) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(c_2Ebool_2E_21(ty_2Erealax_2Ereal),f3189(V0s,V3b))),ap(ap(c_2Ereal_2Ereal__lte,V3b),V1a)) ) ) ).
fof(lameq_f3191,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a] : ap(f3191(V0s),V1a) = ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Ebool_2E_21(ty_2Erealax_2Ereal),f3188(V0s,V1a))),ap(c_2Ebool_2E_21(ty_2Erealax_2Ereal),f3190(V0s,V1a))) ) ).
fof(ax_thm_2Eiterate_2Einf,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ap(c_2Eiterate_2Einf,V0s) = ap(c_2Emin_2E_40(ty_2Erealax_2Ereal),f3191(V0s)) ) ).
fof(conj_thm_2Eiterate_2EINF__EQ,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1t] :
( mem(V1t,arr(ty_2Erealax_2Ereal,bool))
=> ( ! [V2a] :
( mem(V2a,ty_2Erealax_2Ereal)
=> ( ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V2a),V3x)) ) )
<=> ! [V4x] :
( mem(V4x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V1t))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V2a),V4x)) ) ) ) )
=> ap(c_2Eiterate_2Einf,V0s) = ap(c_2Eiterate_2Einf,V1t) ) ) ) ).
fof(conj_thm_2Eiterate_2EINF,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ( ( V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)
& ? [V1b] :
( mem(V1b,ty_2Erealax_2Ereal)
& ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V1b),V2x)) ) ) ) )
=> ( ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Eiterate_2Einf,V0s)),V3x)) ) )
& ! [V4b] :
( mem(V4b,ty_2Erealax_2Ereal)
=> ( ! [V5x] :
( mem(V5x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V5x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V4b),V5x)) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,V4b),ap(c_2Eiterate_2Einf,V0s))) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EINF__FINITE__LEMMA,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s))
& V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) )
=> ? [V1b] :
( mem(V1b,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1b),V0s))
& ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V1b),V2x)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EINF__FINITE,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s))
& V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) )
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),ap(c_2Eiterate_2Einf,V0s)),V0s))
& ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V1x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Eiterate_2Einf,V0s)),V1x)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__LE__INF__FINITE,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s))
& V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) )
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),ap(c_2Eiterate_2Einf,V0s)))
<=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V2x)) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__INF__LE__FINITE,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s))
& V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) )
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Eiterate_2Einf,V0s)),V1a))
<=> ? [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s))
& p(ap(ap(c_2Ereal_2Ereal__lte,V2x),V1a)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__LT__INF__FINITE,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s))
& V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) )
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,V1a),ap(c_2Eiterate_2Einf,V0s)))
<=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s))
=> p(ap(ap(c_2Erealax_2Ereal__lt,V1a),V2x)) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__INF__LT__FINITE,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V0s))
& V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) )
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Eiterate_2Einf,V0s)),V1a))
<=> ? [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s))
& p(ap(ap(c_2Erealax_2Ereal__lt,V2x),V1a)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__INF__UNIQUE,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1b] :
( mem(V1b,ty_2Erealax_2Ereal)
=> ( ( ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V1b),V2x)) ) )
& ! [V3b_27] :
( mem(V3b_27,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,V1b),V3b_27))
=> ? [V4x] :
( mem(V4x,ty_2Erealax_2Ereal)
& p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V4x),V0s))
& p(ap(ap(c_2Erealax_2Ereal__lt,V4x),V3b_27)) ) ) ) )
=> ap(c_2Eiterate_2Einf,V0s) = V1b ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__LE__INF,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1b] :
( mem(V1b,ty_2Erealax_2Ereal)
=> ( ( V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)
& ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V1b),V2x)) ) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,V1b),ap(c_2Eiterate_2Einf,V0s))) ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__LE__INF__SUBSET,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1t] :
( mem(V1t,arr(ty_2Erealax_2Ereal,bool))
=> ( ( V1t != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)
& p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Erealax_2Ereal),V1t),V0s))
& ? [V2b] :
( mem(V2b,ty_2Erealax_2Ereal)
& ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V2b),V3x)) ) ) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Eiterate_2Einf,V0s)),ap(c_2Eiterate_2Einf,V1t))) ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__INF__BOUNDS,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ! [V2b] :
( mem(V2b,ty_2Erealax_2Ereal)
=> ( ( V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)
& ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s))
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),V3x))
& p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V2b)) ) ) ) )
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,V1a),ap(c_2Eiterate_2Einf,V0s)))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Eiterate_2Einf,V0s)),V2b)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__ABS__INF__LE,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ( ( V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)
& ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,V2x)),V1a)) ) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(c_2Eiterate_2Einf,V0s))),V1a)) ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__INF__ASCLOSE,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1l] :
( mem(V1l,ty_2Erealax_2Ereal)
=> ! [V2e] :
( mem(V2e,ty_2Erealax_2Ereal)
=> ( ( V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)
& ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V3x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V3x),V1l))),V2e)) ) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(c_2Eiterate_2Einf,V0s)),V1l))),V2e)) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUP__UNIQUE__FINITE,axiom,
! [V0a] :
( mem(V0a,ty_2Erealax_2Ereal)
=> ! [V1s] :
( mem(V1s,arr(ty_2Erealax_2Ereal,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V1s))
& V1s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) )
=> ( ap(c_2Ereal_2Esup,V1s) = V0a
<=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0a),V1s))
& ! [V2y] :
( mem(V2y,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2y),V1s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V2y),V0a)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EINF__UNIQUE__FINITE,axiom,
! [V0a] :
( mem(V0a,ty_2Erealax_2Ereal)
=> ! [V1s] :
( mem(V1s,arr(ty_2Erealax_2Ereal,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V1s))
& V1s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) )
=> ( ap(c_2Eiterate_2Einf,V1s) = V0a
<=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V0a),V1s))
& ! [V2y] :
( mem(V2y,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2y),V1s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V0a),V2y)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUP__INSERT__FINITE,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1s] :
( mem(V1s,arr(ty_2Erealax_2Ereal,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V1s))
=> ap(c_2Ereal_2Esup,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0x),V1s)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Emin_2E_3D(arr(ty_2Erealax_2Ereal,bool)),V1s),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V0x),ap(ap(c_2Ereal_2Emax,V0x),ap(c_2Ereal_2Esup,V1s))) ) ) ) ).
fof(conj_thm_2Eiterate_2EINF__INSERT__FINITE,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1s] :
( mem(V1s,arr(ty_2Erealax_2Ereal,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V1s))
=> ap(c_2Eiterate_2Einf,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0x),V1s)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Emin_2E_3D(arr(ty_2Erealax_2Ereal,bool)),V1s),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))),V0x),ap(ap(c_2Ereal_2Emin,V0x),ap(c_2Eiterate_2Einf,V1s))) ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__SUP__EQ__INF,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ( ( V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)
& ? [V1B] :
( mem(V1B,ty_2Erealax_2Ereal)
& ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),V2x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,V2x)),V1B)) ) ) ) )
=> ( ap(c_2Ereal_2Esup,V0s) = ap(c_2Eiterate_2Einf,V0s)
<=> ? [V3a] :
( mem(V3a,ty_2Erealax_2Ereal)
& V0s = ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V3a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EINF__SING,axiom,
! [V0a] :
( mem(V0a,ty_2Erealax_2Ereal)
=> ap(c_2Eiterate_2Einf,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),V0a),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal))) = V0a ) ).
fof(lameq_f3192,axiom,
! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V2x] : ap(f3192(V1n,V0m),V2x) = ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,bool),V2x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Earithmetic_2E_3C_3D,V0m),V2x)),ap(ap(c_2Earithmetic_2E_3C_3D,V2x),V1n))) ) ) ).
fof(ax_thm_2Eiterate_2Enumseg,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1n) = ap(c_2Epred__set_2EGSPEC(ty_2Enum_2Enum,ty_2Enum_2Enum),f3192(V1n,V0m)) ) ) ).
fof(conj_thm_2Eiterate_2EFINITE__NUMSEG,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1n))) ) ) ).
fof(conj_thm_2Eiterate_2ENUMSEG__COMBINE__R,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1p] :
( mem(V1p,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),ap(ap(c_2Earithmetic_2E_2B,V1p),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,V1p),V2n)) )
=> ap(ap(c_2Epred__set_2EUNION(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1p)),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,V1p),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V2n)) = ap(ap(c_2Eiterate_2E_2E_2E,V0m),V2n) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENUMSEG__COMBINE__L,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1p] :
( mem(V1p,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),V1p))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,V1p),ap(ap(c_2Earithmetic_2E_2B,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) )
=> ap(ap(c_2Epred__set_2EUNION(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V0m),ap(ap(c_2Earithmetic_2E_2D,V1p),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(ap(c_2Eiterate_2E_2E_2E,V1p),V2n)) = ap(ap(c_2Eiterate_2E_2E_2E,V0m),V2n) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENUMSEG__LREC,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),V1n))
=> ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V0m),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,V0m),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V1n)) = ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1n) ) ) ) ).
fof(conj_thm_2Eiterate_2ENUMSEG__RREC,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),V1n))
=> ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V1n),ap(ap(c_2Eiterate_2E_2E_2E,V0m),ap(ap(c_2Earithmetic_2E_2D,V1n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) = ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1n) ) ) ) ).
fof(conj_thm_2Eiterate_2ENUMSEG__REC,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),ap(c_2Enum_2ESUC,V1n)))
=> ap(ap(c_2Eiterate_2E_2E_2E,V0m),ap(c_2Enum_2ESUC,V1n)) = ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),ap(c_2Enum_2ESUC,V1n)),ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1n)) ) ) ) ).
fof(conj_thm_2Eiterate_2EIN__NUMSEG,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ! [V2p] :
( mem(V2p,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V2p),ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1n)))
<=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),V2p))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,V2p),V1n)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EIN__NUMSEG__0,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V0m),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),V1n)))
<=> p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),V1n)) ) ) ) ).
fof(conj_thm_2Eiterate_2ENUMSEG__SING,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ap(ap(c_2Eiterate_2E_2E_2E,V0n),V0n) = ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V0n),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)) ) ).
fof(conj_thm_2Eiterate_2ENUMSEG__EMPTY,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1n) = c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)
<=> p(ap(ap(c_2Eprim__rec_2E_3C,V1n),V0m)) ) ) ) ).
fof(conj_thm_2Eiterate_2ECARD__NUMSEG__LEMMA,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1d] :
( mem(V1d,ty_2Enum_2Enum)
=> ap(c_2Epred__set_2ECARD(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V0m),ap(ap(c_2Earithmetic_2E_2B,V0m),V1d))) = ap(ap(c_2Earithmetic_2E_2B,V1d),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ) ).
fof(conj_thm_2Eiterate_2ECARD__NUMSEG,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ap(c_2Epred__set_2ECARD(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1n)) = ap(ap(c_2Earithmetic_2E_2D,ap(ap(c_2Earithmetic_2E_2B,V1n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V0m) ) ) ).
fof(conj_thm_2Eiterate_2EHAS__SIZE__NUMSEG,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ecardinal_2EHAS__SIZE(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1n)),ap(ap(c_2Earithmetic_2E_2D,ap(ap(c_2Earithmetic_2E_2B,V1n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V0m))) ) ) ).
fof(conj_thm_2Eiterate_2ECARD__NUMSEG__1,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ap(c_2Epred__set_2ECARD(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),V0n)) = V0n ) ).
fof(conj_thm_2Eiterate_2EHAS__SIZE__NUMSEG__1,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ecardinal_2EHAS__SIZE(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),V0n)),V0n)) ) ).
fof(conj_thm_2Eiterate_2ENUMSEG__CLAUSES,axiom,
( ! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ap(ap(c_2Eiterate_2E_2E_2E,V0m),c_2Enum_2E0) = ap(ap(ap(c_2Ebool_2ECOND(arr(ty_2Enum_2Enum,bool)),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),V0m),c_2Enum_2E0)),ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),c_2Enum_2E0),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum))),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)) )
& ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ap(ap(c_2Eiterate_2E_2E_2E,V1m),ap(c_2Enum_2ESUC,V2n)) = ap(ap(ap(c_2Ebool_2ECOND(arr(ty_2Enum_2Enum,bool)),ap(ap(c_2Earithmetic_2E_3C_3D,V1m),ap(c_2Enum_2ESUC,V2n))),ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),ap(c_2Enum_2ESUC,V2n)),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n))),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)) ) ) ) ).
fof(conj_thm_2Eiterate_2EFINITE__INDEX__NUMSEG,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,A_27a))
& ! [V2i] :
( mem(V2i,ty_2Enum_2Enum)
=> ! [V3j] :
( mem(V3j,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V2i),ap(ap(c_2Eiterate_2E_2E_2E,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),ap(c_2Epred__set_2ECARD(A_27a),V0s))))
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V3j),ap(ap(c_2Eiterate_2E_2E_2E,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),ap(c_2Epred__set_2ECARD(A_27a),V0s))))
& ap(V1f,V2i) = ap(V1f,V3j) )
=> V2i = V3j ) ) )
& V0s = ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,A_27a),V1f),ap(ap(c_2Eiterate_2E_2E_2E,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),ap(c_2Epred__set_2ECARD(A_27a),V0s))) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EFINITE__INDEX__NUMBERS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
<=> ? [V1k] :
( mem(V1k,arr(ty_2Enum_2Enum,bool))
& ? [V2f] :
( mem(V2f,arr(ty_2Enum_2Enum,A_27a))
& ! [V3i] :
( mem(V3i,ty_2Enum_2Enum)
=> ! [V4j] :
( mem(V4j,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V3i),V1k))
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V4j),V1k))
& ap(V2f,V3i) = ap(V2f,V4j) )
=> V3i = V4j ) ) )
& p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V1k))
& V0s = ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,A_27a),V2f),V1k) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EDISJOINT__NUMSEG,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ! [V2p] :
( mem(V2p,ty_2Enum_2Enum)
=> ! [V3q] :
( mem(V3q,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Epred__set_2EDISJOINT(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1n)),ap(ap(c_2Eiterate_2E_2E_2E,V2p),V3q)))
<=> ( p(ap(ap(c_2Eprim__rec_2E_3C,V1n),V2p))
| p(ap(ap(c_2Eprim__rec_2E_3C,V3q),V0m))
| p(ap(ap(c_2Eprim__rec_2E_3C,V1n),V0m))
| p(ap(ap(c_2Eprim__rec_2E_3C,V3q),V2p)) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENUMSEG__ADD__SPLIT,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ! [V2p] :
( mem(V2p,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),ap(ap(c_2Earithmetic_2E_2B,V1n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))
=> ap(ap(c_2Eiterate_2E_2E_2E,V0m),ap(ap(c_2Earithmetic_2E_2B,V1n),V2p)) = ap(ap(c_2Epred__set_2EUNION(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1n)),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,V1n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(ap(c_2Earithmetic_2E_2B,V1n),V2p))) ) ) ) ) ).
fof(lameq_f3193,axiom,
! [V2p] :
( mem(V2p,ty_2Enum_2Enum)
=> ! [V3i] : ap(f3193(V2p),V3i) = ap(ap(c_2Earithmetic_2E_2B,V3i),V2p) ) ).
fof(conj_thm_2Eiterate_2ENUMSEG__OFFSET__IMAGE,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ! [V2p] :
( mem(V2p,ty_2Enum_2Enum)
=> ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,V0m),V2p)),ap(ap(c_2Earithmetic_2E_2B,V1n),V2p)) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Enum_2Enum),f3193(V2p)),ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1n)) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUBSET__NUMSEG,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ! [V2p] :
( mem(V2p,ty_2Enum_2Enum)
=> ! [V3q] :
( mem(V3q,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1n)),ap(ap(c_2Eiterate_2E_2E_2E,V2p),V3q)))
<=> ( p(ap(ap(c_2Eprim__rec_2E_3C,V1n),V0m))
| ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V2p),V0m))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,V1n),V3q)) ) ) ) ) ) ) ) ).
fof(lameq_f3194,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1x] : ap(f3194(V0n),V1x) = ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,bool),V1x),ap(ap(c_2Earithmetic_2E_3C_3D,V1x),V0n)) ) ).
fof(conj_thm_2Eiterate_2ENUMSEG__LE,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ap(c_2Epred__set_2EGSPEC(ty_2Enum_2Enum,ty_2Enum_2Enum),f3194(V0n)) = ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),V0n) ) ).
fof(lameq_f3195,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1x] : ap(f3195(V0n),V1x) = ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,bool),V1x),ap(ap(c_2Eprim__rec_2E_3C,V1x),V0n)) ) ).
fof(conj_thm_2Eiterate_2ENUMSEG__LT,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ap(c_2Epred__set_2EGSPEC(ty_2Enum_2Enum,ty_2Enum_2Enum),f3195(V0n)) = ap(ap(ap(c_2Ebool_2ECOND(arr(ty_2Enum_2Enum,bool)),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),V0n),c_2Enum_2E0)),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),ap(ap(c_2Earithmetic_2E_2D,V0n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ) ).
fof(conj_thm_2Eiterate_2ETOPOLOGICAL__SORT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0_3C_3C] :
( mem(V0_3C_3C,arr(A_27a,arr(A_27a,bool)))
=> ( ( ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] :
( mem(V2y,A_27a)
=> ( ( p(ap(ap(V0_3C_3C,V1x),V2y))
& p(ap(ap(V0_3C_3C,V2y),V1x)) )
=> V1x = V2y ) ) )
& ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4y] :
( mem(V4y,A_27a)
=> ! [V5z] :
( mem(V5z,A_27a)
=> ( ( p(ap(ap(V0_3C_3C,V3x),V4y))
& p(ap(ap(V0_3C_3C,V4y),V5z)) )
=> p(ap(ap(V0_3C_3C,V3x),V5z)) ) ) ) ) )
=> ! [V6n] :
( mem(V6n,ty_2Enum_2Enum)
=> ! [V7s] :
( mem(V7s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27a),V7s),V6n))
=> ? [V8f] :
( mem(V8f,arr(ty_2Enum_2Enum,A_27a))
& V7s = ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,A_27a),V8f),ap(ap(c_2Eiterate_2E_2E_2E,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),V6n))
& ! [V9j] :
( mem(V9j,ty_2Enum_2Enum)
=> ! [V10k] :
( mem(V10k,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V9j),ap(ap(c_2Eiterate_2E_2E_2E,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),V6n)))
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V10k),ap(ap(c_2Eiterate_2E_2E_2E,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),V6n)))
& p(ap(ap(c_2Eprim__rec_2E_3C,V9j),V10k)) )
=> ~ p(ap(ap(V0_3C_3C,ap(V8f,V10k)),ap(V8f,V9j))) ) ) ) ) ) ) ) ) ) ) ).
fof(lameq_f3196,axiom,
! [A_27a,V0op] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] : ap(f3196(A_27a,V0op,V1x),V2y) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Emin_2E_3D(A_27a),ap(ap(V0op,V1x),V2y)),V2y)),ap(ap(c_2Emin_2E_3D(A_27a),ap(ap(V0op,V2y),V1x)),V2y)) ) ) ).
fof(lameq_f3197,axiom,
! [A_27a,V0op] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V1x] : ap(f3197(A_27a,V0op),V1x) = ap(c_2Ebool_2E_21(A_27a),f3196(A_27a,V0op,V1x)) ) ).
fof(ax_thm_2Eiterate_2Eneutral,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0op] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ap(c_2Eiterate_2Eneutral(A_27a),V0op) = ap(c_2Emin_2E_40(A_27a),f3197(A_27a,V0op)) ) ) ).
fof(ax_thm_2Eiterate_2Emonoidal,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0op] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V0op))
<=> ( ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] :
( mem(V2y,A_27a)
=> ap(ap(V0op,V1x),V2y) = ap(ap(V0op,V2y),V1x) ) )
& ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4y] :
( mem(V4y,A_27a)
=> ! [V5z] :
( mem(V5z,A_27a)
=> ap(ap(V0op,V3x),ap(ap(V0op,V4y),V5z)) = ap(ap(V0op,ap(ap(V0op,V3x),V4y)),V5z) ) ) )
& ! [V6x] :
( mem(V6x,A_27a)
=> ap(ap(V0op,ap(c_2Eiterate_2Eneutral(A_27a),V0op)),V6x) = V6x ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EMONOIDAL__AC,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0op] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V0op))
=> ( ! [V1a] :
( mem(V1a,A_27a)
=> ap(ap(V0op,ap(c_2Eiterate_2Eneutral(A_27a),V0op)),V1a) = V1a )
& ! [V2a] :
( mem(V2a,A_27a)
=> ap(ap(V0op,V2a),ap(c_2Eiterate_2Eneutral(A_27a),V0op)) = V2a )
& ! [V3a] :
( mem(V3a,A_27a)
=> ! [V4b] :
( mem(V4b,A_27a)
=> ap(ap(V0op,V3a),V4b) = ap(ap(V0op,V4b),V3a) ) )
& ! [V5a] :
( mem(V5a,A_27a)
=> ! [V6b] :
( mem(V6b,A_27a)
=> ! [V7c] :
( mem(V7c,A_27a)
=> ap(ap(V0op,ap(ap(V0op,V5a),V6b)),V7c) = ap(ap(V0op,V5a),ap(ap(V0op,V6b),V7c)) ) ) )
& ! [V8a] :
( mem(V8a,A_27a)
=> ! [V9b] :
( mem(V9b,A_27a)
=> ! [V10c] :
( mem(V10c,A_27a)
=> ap(ap(V0op,V8a),ap(ap(V0op,V9b),V10c)) = ap(ap(V0op,V9b),ap(ap(V0op,V8a),V10c)) ) ) ) ) ) ) ) ).
fof(lameq_f3198,axiom,
! [A_27b,A_27a,V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V0op] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V3x] : ap(f3198(A_27b,A_27a,V2s,V0op,V1f),V3x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V3x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s)),ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(A_27b),ap(V1f,V3x)),ap(c_2Eiterate_2Eneutral(A_27b),V0op))))) ) ) ) ).
fof(ax_thm_2Eiterate_2Esupport,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ap(ap(ap(c_2Eiterate_2Esupport(A_27a,A_27b),V0op),V1f),V2s) = ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3198(A_27b,A_27a,V2s,V0op,V1f)) ) ) ) ) ) ).
fof(lameq_f3199,axiom,
! [A_27a,A_27b,V0op] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ! [V2f] :
( mem(V2f,arr(A_27a,A_27b))
=> ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4a] : ap(f3199(A_27a,A_27b,V0op,V2f,V3x),V4a) = ap(ap(V0op,ap(V2f,V3x)),V4a) ) ) ) ).
fof(lameq_f3200,axiom,
! [A_27b,A_27a,V0op] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ! [V2f] :
( mem(V2f,arr(A_27a,A_27b))
=> ! [V3x] : ap(f3200(A_27b,A_27a,V0op,V2f),V3x) = f3199(A_27a,A_27b,V0op,V2f,V3x) ) ) ).
fof(ax_thm_2Eiterate_2Eiterate,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2f] :
( mem(V2f,arr(A_27a,A_27b))
=> ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),V1s),V2f) = ap(ap(ap(c_2Ebool_2ECOND(A_27b),ap(c_2Epred__set_2EFINITE(A_27a),ap(ap(ap(c_2Eiterate_2Esupport(A_27a,A_27b),V0op),V2f),V1s))),ap(ap(ap(c_2Eiterate_2EITSET(A_27b,A_27a),f3200(A_27b,A_27a,V0op,V2f)),ap(ap(ap(c_2Eiterate_2Esupport(A_27a,A_27b),V0op),V2f),V1s)),ap(c_2Eiterate_2Eneutral(A_27b),V0op))),ap(c_2Eiterate_2Eneutral(A_27b),V0op)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EIN__SUPPORT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V1f] :
( mem(V1f,arr(A_27b,A_27a))
=> ! [V2x] :
( mem(V2x,A_27b)
=> ! [V3s] :
( mem(V3s,arr(A_27b,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V2x),ap(ap(ap(c_2Eiterate_2Esupport(A_27b,A_27a),V0op),V1f),V3s)))
<=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V2x),V3s))
& ap(V1f,V2x) != ap(c_2Eiterate_2Eneutral(A_27a),V0op) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUPPORT__SUPPORT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V1f] :
( mem(V1f,arr(A_27b,A_27a))
=> ! [V2s] :
( mem(V2s,arr(A_27b,bool))
=> ap(ap(ap(c_2Eiterate_2Esupport(A_27b,A_27a),V0op),V1f),ap(ap(ap(c_2Eiterate_2Esupport(A_27b,A_27a),V0op),V1f),V2s)) = ap(ap(ap(c_2Eiterate_2Esupport(A_27b,A_27a),V0op),V1f),V2s) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUPPORT__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V1f] :
( mem(V1f,arr(A_27b,A_27a))
=> ! [V2s] :
( mem(V2s,arr(A_27b,bool))
=> ( ! [V3x] :
( mem(V3x,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V3x),V2s))
=> ap(V1f,V3x) = ap(c_2Eiterate_2Eneutral(A_27a),V0op) ) )
<=> ap(ap(ap(c_2Eiterate_2Esupport(A_27b,A_27a),V0op),V1f),V2s) = c_2Epred__set_2EEMPTY(A_27b) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUPPORT__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V1f] :
( mem(V1f,arr(A_27b,A_27a))
=> ! [V2s] :
( mem(V2s,arr(A_27b,bool))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27b),ap(ap(ap(c_2Eiterate_2Esupport(A_27b,A_27a),V0op),V1f),V2s)),V2s)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EFINITE__SUPPORT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V1f] :
( mem(V1f,arr(A_27b,A_27a))
=> ! [V2s] :
( mem(V2s,arr(A_27b,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27b),V2s))
=> p(ap(c_2Epred__set_2EFINITE(A_27b),ap(ap(ap(c_2Eiterate_2Esupport(A_27b,A_27a),V0op),V1f),V2s))) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUPPORT__CLAUSES,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [A_27e] :
( ne(A_27e)
=> ! [A_27f] :
( ne(A_27f)
=> ! [A_27g] :
( ne(A_27g)
=> ! [A_27h] :
( ne(A_27h)
=> ! [A_27i] :
( ne(A_27i)
=> ! [V0op] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ap(ap(ap(c_2Eiterate_2Esupport(A_27a,A_27b),V0op),V1f),c_2Epred__set_2EEMPTY(A_27a)) = c_2Epred__set_2EEMPTY(A_27a) )
& ! [V2f] :
( mem(V2f,arr(A_27c,A_27b))
=> ! [V3x] :
( mem(V3x,A_27c)
=> ! [V4s] :
( mem(V4s,arr(A_27c,bool))
=> ap(ap(ap(c_2Eiterate_2Esupport(A_27c,A_27b),V0op),V2f),ap(ap(c_2Epred__set_2EINSERT(A_27c),V3x),V4s)) = ap(ap(ap(c_2Ebool_2ECOND(arr(A_27c,bool)),ap(ap(c_2Emin_2E_3D(A_27b),ap(V2f,V3x)),ap(c_2Eiterate_2Eneutral(A_27b),V0op))),ap(ap(ap(c_2Eiterate_2Esupport(A_27c,A_27b),V0op),V2f),V4s)),ap(ap(c_2Epred__set_2EINSERT(A_27c),V3x),ap(ap(ap(c_2Eiterate_2Esupport(A_27c,A_27b),V0op),V2f),V4s))) ) ) )
& ! [V5f] :
( mem(V5f,arr(A_27d,A_27b))
=> ! [V6x] :
( mem(V6x,A_27d)
=> ! [V7s] :
( mem(V7s,arr(A_27d,bool))
=> ap(ap(ap(c_2Eiterate_2Esupport(A_27d,A_27b),V0op),V5f),ap(ap(c_2Epred__set_2EDELETE(A_27d),V7s),V6x)) = ap(ap(c_2Epred__set_2EDELETE(A_27d),ap(ap(ap(c_2Eiterate_2Esupport(A_27d,A_27b),V0op),V5f),V7s)),V6x) ) ) )
& ! [V8f] :
( mem(V8f,arr(A_27e,A_27b))
=> ! [V9s] :
( mem(V9s,arr(A_27e,bool))
=> ! [V10t] :
( mem(V10t,arr(A_27e,bool))
=> ap(ap(ap(c_2Eiterate_2Esupport(A_27e,A_27b),V0op),V8f),ap(ap(c_2Epred__set_2EUNION(A_27e),V9s),V10t)) = ap(ap(c_2Epred__set_2EUNION(A_27e),ap(ap(ap(c_2Eiterate_2Esupport(A_27e,A_27b),V0op),V8f),V9s)),ap(ap(ap(c_2Eiterate_2Esupport(A_27e,A_27b),V0op),V8f),V10t)) ) ) )
& ! [V11f] :
( mem(V11f,arr(A_27f,A_27b))
=> ! [V12s] :
( mem(V12s,arr(A_27f,bool))
=> ! [V13t] :
( mem(V13t,arr(A_27f,bool))
=> ap(ap(ap(c_2Eiterate_2Esupport(A_27f,A_27b),V0op),V11f),ap(ap(c_2Epred__set_2EINTER(A_27f),V12s),V13t)) = ap(ap(c_2Epred__set_2EINTER(A_27f),ap(ap(ap(c_2Eiterate_2Esupport(A_27f,A_27b),V0op),V11f),V12s)),ap(ap(ap(c_2Eiterate_2Esupport(A_27f,A_27b),V0op),V11f),V13t)) ) ) )
& ! [V14f] :
( mem(V14f,arr(A_27g,A_27b))
=> ! [V15s] :
( mem(V15s,arr(A_27g,bool))
=> ! [V16t] :
( mem(V16t,arr(A_27g,bool))
=> ap(ap(ap(c_2Eiterate_2Esupport(A_27g,A_27b),V0op),V14f),ap(ap(c_2Epred__set_2EDIFF(A_27g),V15s),V16t)) = ap(ap(c_2Epred__set_2EDIFF(A_27g),ap(ap(ap(c_2Eiterate_2Esupport(A_27g,A_27b),V0op),V14f),V15s)),ap(ap(ap(c_2Eiterate_2Esupport(A_27g,A_27b),V0op),V14f),V16t)) ) ) )
& ! [V17f] :
( mem(V17f,arr(A_27h,A_27i))
=> ! [V18g] :
( mem(V18g,arr(A_27i,A_27b))
=> ! [V19s] :
( mem(V19s,arr(A_27h,bool))
=> ap(ap(ap(c_2Eiterate_2Esupport(A_27i,A_27b),V0op),V18g),ap(ap(c_2Epred__set_2EIMAGE(A_27h,A_27i),V17f),V19s)) = ap(ap(c_2Epred__set_2EIMAGE(A_27h,A_27i),V17f),ap(ap(ap(c_2Eiterate_2Esupport(A_27h,A_27b),V0op),ap(ap(c_2Ecombin_2Eo(A_27h,A_27b,A_27i),V18g),V17f)),V19s)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(lameq_f3201,axiom,
! [A_27a,A_27b,V2f] :
( mem(V2f,arr(A_27b,A_27a))
=> ! [V3a] :
( mem(V3a,A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V4x] : ap(f3201(A_27a,A_27b,V2f,V3a,V0op),V4x) = ap(ap(ap(c_2Ebool_2ECOND(A_27a),ap(ap(c_2Emin_2E_3D(A_27b),V4x),V3a)),ap(V2f,V4x)),ap(c_2Eiterate_2Eneutral(A_27a),V0op)) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUPPORT__DELTA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V1s] :
( mem(V1s,arr(A_27b,bool))
=> ! [V2f] :
( mem(V2f,arr(A_27b,A_27a))
=> ! [V3a] :
( mem(V3a,A_27b)
=> ap(ap(ap(c_2Eiterate_2Esupport(A_27b,A_27a),V0op),f3201(A_27a,A_27b,V2f,V3a,V0op)),V1s) = ap(ap(ap(c_2Ebool_2ECOND(arr(A_27b,bool)),ap(ap(c_2Ebool_2EIN(A_27b),V3a),V1s)),ap(ap(ap(c_2Eiterate_2Esupport(A_27b,A_27a),V0op),V2f),ap(ap(c_2Epred__set_2EINSERT(A_27b),V3a),c_2Epred__set_2EEMPTY(A_27b)))),c_2Epred__set_2EEMPTY(A_27b)) ) ) ) ) ) ) ).
fof(lameq_f3202,axiom,
! [A_27a,A_27b,V2f] :
( mem(V2f,arr(A_27b,A_27a))
=> ! [V3a] :
( mem(V3a,A_27b)
=> ! [V1op] :
( mem(V1op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V4x] : ap(f3202(A_27a,A_27b,V2f,V3a,V1op),V4x) = ap(ap(ap(c_2Ebool_2ECOND(A_27a),ap(ap(c_2Emin_2E_3D(A_27b),V4x),V3a)),ap(V2f,V4x)),ap(c_2Eiterate_2Eneutral(A_27a),V1op)) ) ) ) ).
fof(conj_thm_2Eiterate_2EFINITE__SUPPORT__DELTA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27b,bool))
=> ! [V1op] :
( mem(V1op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V2f] :
( mem(V2f,arr(A_27b,A_27a))
=> ! [V3a] :
( mem(V3a,A_27b)
=> p(ap(c_2Epred__set_2EFINITE(A_27b),ap(ap(ap(c_2Eiterate_2Esupport(A_27b,A_27a),V1op),f3202(A_27a,A_27b,V2f,V3a,V1op)),V0s))) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EITERATE__SUPPORT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V1f] :
( mem(V1f,arr(A_27b,A_27a))
=> ! [V2s] :
( mem(V2s,arr(A_27b,bool))
=> ap(ap(ap(c_2Eiterate_2Eiterate(A_27b,A_27a),V0op),ap(ap(ap(c_2Eiterate_2Esupport(A_27b,A_27a),V0op),V1f),V2s)),V1f) = ap(ap(ap(c_2Eiterate_2Eiterate(A_27b,A_27a),V0op),V2s),V1f) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EITERATE__EXPAND__CASES,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V1f] :
( mem(V1f,arr(A_27b,A_27a))
=> ! [V2s] :
( mem(V2s,arr(A_27b,bool))
=> ap(ap(ap(c_2Eiterate_2Eiterate(A_27b,A_27a),V0op),V2s),V1f) = ap(ap(ap(c_2Ebool_2ECOND(A_27a),ap(c_2Epred__set_2EFINITE(A_27b),ap(ap(ap(c_2Eiterate_2Esupport(A_27b,A_27a),V0op),V1f),V2s))),ap(ap(ap(c_2Eiterate_2Eiterate(A_27b,A_27a),V0op),ap(ap(ap(c_2Eiterate_2Esupport(A_27b,A_27a),V0op),V1f),V2s)),V1f)),ap(c_2Eiterate_2Eneutral(A_27a),V0op)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EITERATE__CLAUSES__GEN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ( ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),c_2Epred__set_2EEMPTY(A_27a)),V1f) = ap(c_2Eiterate_2Eneutral(A_27b),V0op) )
& ! [V2f] :
( mem(V2f,arr(A_27a,A_27b))
=> ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4s] :
( mem(V4s,arr(A_27a,bool))
=> ( ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
& p(ap(c_2Epred__set_2EFINITE(A_27a),ap(ap(ap(c_2Eiterate_2Esupport(A_27a,A_27b),V0op),V2f),V4s))) )
=> ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),ap(ap(c_2Epred__set_2EINSERT(A_27a),V3x),V4s)),V2f) = ap(ap(ap(c_2Ebool_2ECOND(A_27b),ap(ap(c_2Ebool_2EIN(A_27a),V3x),V4s)),ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),V4s),V2f)),ap(ap(V0op,ap(V2f,V3x)),ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),V4s),V2f))) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EITERATE__CLAUSES,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V0op))
=> ( ! [V1f] :
( mem(V1f,arr(A_27b,A_27a))
=> ap(ap(ap(c_2Eiterate_2Eiterate(A_27b,A_27a),V0op),c_2Epred__set_2EEMPTY(A_27b)),V1f) = ap(c_2Eiterate_2Eneutral(A_27a),V0op) )
& ! [V2f] :
( mem(V2f,arr(A_27b,A_27a))
=> ! [V3x] :
( mem(V3x,A_27b)
=> ! [V4s] :
( mem(V4s,arr(A_27b,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27b),V4s))
=> ap(ap(ap(c_2Eiterate_2Eiterate(A_27b,A_27a),V0op),ap(ap(c_2Epred__set_2EINSERT(A_27b),V3x),V4s)),V2f) = ap(ap(ap(c_2Ebool_2ECOND(A_27a),ap(ap(c_2Ebool_2EIN(A_27b),V3x),V4s)),ap(ap(ap(c_2Eiterate_2Eiterate(A_27b,A_27a),V0op),V4s),V2f)),ap(ap(V0op,ap(V2f,V3x)),ap(ap(ap(c_2Eiterate_2Eiterate(A_27b,A_27a),V0op),V4s),V2f))) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EITERATE__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V0op))
=> ! [V1f] :
( mem(V1f,arr(A_27b,A_27a))
=> ! [V2s] :
( mem(V2s,arr(A_27b,bool))
=> ! [V3t] :
( mem(V3t,arr(A_27b,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27b),V2s))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V3t))
& p(ap(ap(c_2Epred__set_2EDISJOINT(A_27b),V2s),V3t)) )
=> ap(ap(ap(c_2Eiterate_2Eiterate(A_27b,A_27a),V0op),ap(ap(c_2Epred__set_2EUNION(A_27b),V2s),V3t)),V1f) = ap(ap(V0op,ap(ap(ap(c_2Eiterate_2Eiterate(A_27b,A_27a),V0op),V2s),V1f)),ap(ap(ap(c_2Eiterate_2Eiterate(A_27b,A_27a),V0op),V3t),V1f)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EITERATE__UNION__GEN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t] :
( mem(V3t,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(ap(ap(c_2Eiterate_2Esupport(A_27a,A_27b),V0op),V1f),V2s)))
& p(ap(c_2Epred__set_2EFINITE(A_27a),ap(ap(ap(c_2Eiterate_2Esupport(A_27a,A_27b),V0op),V1f),V3t)))
& p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(ap(ap(c_2Eiterate_2Esupport(A_27a,A_27b),V0op),V1f),V2s)),ap(ap(ap(c_2Eiterate_2Esupport(A_27a,A_27b),V0op),V1f),V3t))) )
=> ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),ap(ap(c_2Epred__set_2EUNION(A_27a),V2s),V3t)),V1f) = ap(ap(V0op,ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),V2s),V1f)),ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),V3t),V1f)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EITERATE__DIFF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V0op))
=> ! [V1f] :
( mem(V1f,arr(A_27b,A_27a))
=> ! [V2s] :
( mem(V2s,arr(A_27b,bool))
=> ! [V3t] :
( mem(V3t,arr(A_27b,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27b),V2s))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27b),V3t),V2s)) )
=> ap(ap(V0op,ap(ap(ap(c_2Eiterate_2Eiterate(A_27b,A_27a),V0op),ap(ap(c_2Epred__set_2EDIFF(A_27b),V2s),V3t)),V1f)),ap(ap(ap(c_2Eiterate_2Eiterate(A_27b,A_27a),V0op),V3t),V1f)) = ap(ap(ap(c_2Eiterate_2Eiterate(A_27b,A_27a),V0op),V2s),V1f) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EITERATE__DIFF__GEN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t] :
( mem(V3t,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(ap(ap(c_2Eiterate_2Esupport(A_27a,A_27b),V0op),V1f),V2s)))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(ap(ap(c_2Eiterate_2Esupport(A_27a,A_27b),V0op),V1f),V3t)),ap(ap(ap(c_2Eiterate_2Esupport(A_27a,A_27b),V0op),V1f),V2s))) )
=> ap(ap(V0op,ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),ap(ap(c_2Epred__set_2EDIFF(A_27a),V2s),V3t)),V1f)),ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),V3t),V1f)) = ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),V2s),V1f) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EITERATE__INCL__EXCL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V0op))
=> ! [V1s] :
( mem(V1s,arr(A_27b,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ! [V3f] :
( mem(V3f,arr(A_27b,A_27a))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27b),V1s))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V2t)) )
=> ap(ap(V0op,ap(ap(ap(c_2Eiterate_2Eiterate(A_27b,A_27a),V0op),V1s),V3f)),ap(ap(ap(c_2Eiterate_2Eiterate(A_27b,A_27a),V0op),V2t),V3f)) = ap(ap(V0op,ap(ap(ap(c_2Eiterate_2Eiterate(A_27b,A_27a),V0op),ap(ap(c_2Epred__set_2EUNION(A_27b),V1s),V2t)),V3f)),ap(ap(ap(c_2Eiterate_2Eiterate(A_27b,A_27a),V0op),ap(ap(c_2Epred__set_2EINTER(A_27b),V1s),V2t)),V3f)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EITERATE__CLOSED,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ! [V1P] :
( mem(V1P,arr(A_27b,bool))
=> ( ( p(ap(V1P,ap(c_2Eiterate_2Eneutral(A_27b),V0op)))
& ! [V2x] :
( mem(V2x,A_27b)
=> ! [V3y] :
( mem(V3y,A_27b)
=> ( ( p(ap(V1P,V2x))
& p(ap(V1P,V3y)) )
=> p(ap(V1P,ap(ap(V0op,V2x),V3y))) ) ) ) )
=> ! [V4f] :
( mem(V4f,arr(A_27a,A_27b))
=> ! [V5s] :
( mem(V5s,arr(A_27a,bool))
=> ( ! [V6x] :
( mem(V6x,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V6x),V5s))
& ap(V4f,V6x) != ap(c_2Eiterate_2Eneutral(A_27b),V0op) )
=> p(ap(V1P,ap(V4f,V6x))) ) )
=> p(ap(V1P,ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),V5s),V4f))) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EITERATE__RELATED,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ! [V1R] :
( mem(V1R,arr(A_27b,arr(A_27b,bool)))
=> ( ( p(ap(ap(V1R,ap(c_2Eiterate_2Eneutral(A_27b),V0op)),ap(c_2Eiterate_2Eneutral(A_27b),V0op)))
& ! [V2x1] :
( mem(V2x1,A_27b)
=> ! [V3y1] :
( mem(V3y1,A_27b)
=> ! [V4x2] :
( mem(V4x2,A_27b)
=> ! [V5y2] :
( mem(V5y2,A_27b)
=> ( ( p(ap(ap(V1R,V2x1),V4x2))
& p(ap(ap(V1R,V3y1),V5y2)) )
=> p(ap(ap(V1R,ap(ap(V0op,V2x1),V3y1)),ap(ap(V0op,V4x2),V5y2))) ) ) ) ) ) )
=> ! [V6f] :
( mem(V6f,arr(A_27a,A_27b))
=> ! [V7g] :
( mem(V7g,arr(A_27a,A_27b))
=> ! [V8s] :
( mem(V8s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V8s))
& ! [V9x] :
( mem(V9x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V9x),V8s))
=> p(ap(ap(V1R,ap(V6f,V9x)),ap(V7g,V9x))) ) ) )
=> p(ap(ap(V1R,ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),V8s),V6f)),ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),V8s),V7g))) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EITERATE__EQ__NEUTRAL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s))
=> ap(V1f,V3x) = ap(c_2Eiterate_2Eneutral(A_27b),V0op) ) )
=> ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),V2s),V1f) = ap(c_2Eiterate_2Eneutral(A_27b),V0op) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EITERATE__SING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2x] :
( mem(V2x,A_27a)
=> ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),ap(ap(c_2Epred__set_2EINSERT(A_27a),V2x),c_2Epred__set_2EEMPTY(A_27a))),V1f) = ap(V1f,V2x) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EITERATE__DELETE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3a] :
( mem(V3a,A_27a)
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V3a),V2s)) )
=> ap(ap(V0op,ap(V1f,V3a)),ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),ap(ap(c_2Epred__set_2EDELETE(A_27a),V2s),V3a)),V1f)) = ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),V2s),V1f) ) ) ) ) ) ) ) ) ).
fof(lameq_f3203,axiom,
! [A_27a,A_27b,V1f] :
( mem(V1f,arr(A_27b,A_27a))
=> ! [V2a] :
( mem(V2a,A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V4x] : ap(f3203(A_27a,A_27b,V1f,V2a,V0op),V4x) = ap(ap(ap(c_2Ebool_2ECOND(A_27a),ap(ap(c_2Emin_2E_3D(A_27b),V4x),V2a)),ap(V1f,V4x)),ap(c_2Eiterate_2Eneutral(A_27a),V0op)) ) ) ) ).
fof(conj_thm_2Eiterate_2EITERATE__DELTA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V0op))
=> ! [V1f] :
( mem(V1f,arr(A_27b,A_27a))
=> ! [V2a] :
( mem(V2a,A_27b)
=> ! [V3s] :
( mem(V3s,arr(A_27b,bool))
=> ap(ap(ap(c_2Eiterate_2Eiterate(A_27b,A_27a),V0op),V3s),f3203(A_27a,A_27b,V1f,V2a,V0op)) = ap(ap(ap(c_2Ebool_2ECOND(A_27a),ap(ap(c_2Ebool_2EIN(A_27b),V2a),V3s)),ap(V1f,V2a)),ap(c_2Eiterate_2Eneutral(A_27a),V0op)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EITERATE__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0op] :
( mem(V0op,arr(A_27c,arr(A_27c,A_27c)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27c),V0op))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2g] :
( mem(V2g,arr(A_27b,A_27c))
=> ! [V3s] :
( mem(V3s,arr(A_27a,bool))
=> ( ! [V4x] :
( mem(V4x,A_27a)
=> ! [V5y] :
( mem(V5y,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V3s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V5y),V3s))
& ap(V1f,V4x) = ap(V1f,V5y) )
=> V4x = V5y ) ) )
=> ap(ap(ap(c_2Eiterate_2Eiterate(A_27b,A_27c),V0op),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1f),V3s)),V2g) = ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27c),V0op),V3s),ap(ap(c_2Ecombin_2Eo(A_27a,A_27c,A_27b),V2g),V1f)) ) ) ) ) ) ) ) ) ) ).
fof(lameq_f3204,axiom,
! [A_27a,V5y] :
( mem(V5y,A_27a)
=> ! [V2p] :
( mem(V2p,arr(A_27a,A_27a))
=> ! [V3s] :
( mem(V3s,arr(A_27a,bool))
=> ! [V6x] : ap(f3204(A_27a,V5y,V2p,V3s),V6x) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V6x),V3s)),ap(ap(c_2Emin_2E_3D(A_27a),ap(V2p,V6x)),V5y)) ) ) ) ).
fof(conj_thm_2Eiterate_2EITERATE__BIJECTION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2p] :
( mem(V2p,arr(A_27a,A_27a))
=> ! [V3s] :
( mem(V3s,arr(A_27a,bool))
=> ( ( ! [V4x] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V3s))
=> p(ap(ap(c_2Ebool_2EIN(A_27a),ap(V2p,V4x)),V3s)) ) )
& ! [V5y] :
( mem(V5y,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5y),V3s))
=> p(ap(c_2Ebool_2E_3F_21(A_27a),f3204(A_27a,V5y,V2p,V3s))) ) ) )
=> ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),V3s),V1f) = ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),V3s),ap(ap(c_2Ecombin_2Eo(A_27a,A_27b,A_27a),V1f),V2p)) ) ) ) ) ) ) ) ) ).
fof(lameq_f3205,axiom,
! [A_27c,A_27b,A_27a,V0op] :
( mem(V0op,arr(A_27c,arr(A_27c,A_27c)))
=> ! [V2t] :
( mem(V2t,arr(A_27a,arr(A_27b,bool)))
=> ! [V3x] :
( mem(V3x,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V5i] : ap(f3205(A_27c,A_27b,A_27a,V0op,V2t,V3x),V5i) = ap(ap(ap(c_2Eiterate_2Eiterate(A_27b,A_27c),V0op),ap(V2t,V5i)),ap(V3x,V5i)) ) ) ) ).
fof(lameq_f3206,axiom,
! [A_27a,A_27b,V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,arr(A_27b,bool)))
=> ! [V6i] :
( mem(V6i,A_27a)
=> ! [V7j] : ap(f3206(A_27a,A_27b,V1s,V2t,V6i),V7j) = ap(ap(c_2Epair_2E_2C(ty_2Epair_2Eprod(A_27a,A_27b),bool),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V6i),V7j)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V6i),V1s)),ap(ap(c_2Ebool_2EIN(A_27b),V7j),ap(V2t,V6i)))) ) ) ) ).
fof(lameq_f3207,axiom,
! [A_27b,A_27a,V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,arr(A_27b,bool)))
=> ! [V6i] : ap(f3207(A_27b,A_27a,V1s,V2t),V6i) = f3206(A_27a,A_27b,V1s,V2t,V6i) ) ) ).
fof(lameq_f3208,axiom,
! [A_27c,A_27a,A_27b,V8i] :
( mem(V8i,A_27a)
=> ! [V3x] :
( mem(V3x,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V9j] : ap(f3208(A_27c,A_27a,A_27b,V8i,V3x),V9j) = ap(ap(V3x,V8i),V9j) ) ) ).
fof(lameq_f3209,axiom,
! [A_27b,A_27c,A_27a,V3x] :
( mem(V3x,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V8i] : ap(f3209(A_27b,A_27c,A_27a,V3x),V8i) = f3208(A_27c,A_27a,A_27b,V8i,V3x) ) ).
fof(conj_thm_2Eiterate_2EITERATE__ITERATE__PRODUCT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0op] :
( mem(V0op,arr(A_27c,arr(A_27c,A_27c)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27c),V0op))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,arr(A_27b,bool)))
=> ! [V3x] :
( mem(V3x,arr(A_27a,arr(A_27b,A_27c)))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& ! [V4i] :
( mem(V4i,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4i),V1s))
=> p(ap(c_2Epred__set_2EFINITE(A_27b),ap(V2t,V4i))) ) ) )
=> ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27c),V0op),V1s),f3205(A_27c,A_27b,A_27a,V0op,V2t,V3x)) = ap(ap(ap(c_2Eiterate_2Eiterate(ty_2Epair_2Eprod(A_27a,A_27b),A_27c),V0op),ap(c_2Epred__set_2EGSPEC(ty_2Epair_2Eprod(A_27a,A_27b),ty_2Epair_2Eprod(A_27a,A_27b)),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,ty_2Epair_2Eprod(ty_2Epair_2Eprod(A_27a,A_27b),bool)),f3207(A_27b,A_27a,V1s,V2t)))),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,A_27c),f3209(A_27b,A_27c,A_27a,V3x))) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EITERATE__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2g] :
( mem(V2g,arr(A_27a,A_27b))
=> ! [V3s] :
( mem(V3s,arr(A_27a,bool))
=> ( ! [V4x] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V3s))
=> ap(V1f,V4x) = ap(V2g,V4x) ) )
=> ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),V3s),V1f) = ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),V3s),V2g) ) ) ) ) ) ) ) ) ).
fof(lameq_f3210,axiom,
! [A_27b,A_27a,V6y] :
( mem(V6y,A_27b)
=> ! [V5h] :
( mem(V5h,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V7x] : ap(f3210(A_27b,A_27a,V6y,V5h,V1s),V7x) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V7x),V1s)),ap(ap(c_2Emin_2E_3D(A_27b),ap(V5h,V7x)),V6y)) ) ) ) ).
fof(conj_thm_2Eiterate_2EITERATE__EQ__GENERAL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0op] :
( mem(V0op,arr(A_27c,arr(A_27c,A_27c)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27c),V0op))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ! [V3f] :
( mem(V3f,arr(A_27a,A_27c))
=> ! [V4g] :
( mem(V4g,arr(A_27b,A_27c))
=> ! [V5h] :
( mem(V5h,arr(A_27a,A_27b))
=> ( ( ! [V6y] :
( mem(V6y,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V6y),V2t))
=> p(ap(c_2Ebool_2E_3F_21(A_27a),f3210(A_27b,A_27a,V6y,V5h,V1s))) ) )
& ! [V8x] :
( mem(V8x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V8x),V1s))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V5h,V8x)),V2t))
& ap(V4g,ap(V5h,V8x)) = ap(V3f,V8x) ) ) ) )
=> ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27c),V0op),V1s),V3f) = ap(ap(ap(c_2Eiterate_2Eiterate(A_27b,A_27c),V0op),V2t),V4g) ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EITERATE__EQ__GENERAL__INVERSES,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0op] :
( mem(V0op,arr(A_27c,arr(A_27c,A_27c)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27c),V0op))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ! [V3f] :
( mem(V3f,arr(A_27a,A_27c))
=> ! [V4g] :
( mem(V4g,arr(A_27b,A_27c))
=> ! [V5h] :
( mem(V5h,arr(A_27a,A_27b))
=> ! [V6k] :
( mem(V6k,arr(A_27b,A_27a))
=> ( ( ! [V7y] :
( mem(V7y,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V7y),V2t))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),ap(V6k,V7y)),V1s))
& ap(V5h,ap(V6k,V7y)) = V7y ) ) )
& ! [V8x] :
( mem(V8x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V8x),V1s))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V5h,V8x)),V2t))
& ap(V6k,ap(V5h,V8x)) = V8x
& ap(V4g,ap(V5h,V8x)) = ap(V3f,V8x) ) ) ) )
=> ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27c),V0op),V1s),V3f) = ap(ap(ap(c_2Eiterate_2Eiterate(A_27b,A_27c),V0op),V2t),V4g) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EITERATE__INJECTION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2p] :
( mem(V2p,arr(A_27a,A_27a))
=> ! [V3s] :
( mem(V3s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V3s))
& ! [V4x] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V3s))
=> p(ap(ap(c_2Ebool_2EIN(A_27a),ap(V2p,V4x)),V3s)) ) )
& ! [V5x] :
( mem(V5x,A_27a)
=> ! [V6y] :
( mem(V6y,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V3s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V6y),V3s))
& ap(V2p,V5x) = ap(V2p,V6y) )
=> V5x = V6y ) ) ) )
=> ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),V3s),ap(ap(c_2Ecombin_2Eo(A_27a,A_27b,A_27a),V1f),V2p)) = ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),V3s),V1f) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EITERATE__UNION__NONZERO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t] :
( mem(V3t,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& p(ap(c_2Epred__set_2EFINITE(A_27a),V3t))
& ! [V4x] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),ap(ap(c_2Epred__set_2EINTER(A_27a),V2s),V3t)))
=> ap(V1f,V4x) = ap(c_2Eiterate_2Eneutral(A_27b),V0op) ) ) )
=> ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),ap(ap(c_2Epred__set_2EUNION(A_27a),V2s),V3t)),V1f) = ap(ap(V0op,ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),V2s),V1f)),ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),V3t),V1f)) ) ) ) ) ) ) ) ) ).
fof(lameq_f3211,axiom,
! [A_27a,A_27b,V0op] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V1f] :
( mem(V1f,arr(A_27b,A_27a))
=> ! [V2g] :
( mem(V2g,arr(A_27b,A_27a))
=> ! [V4x] : ap(f3211(A_27a,A_27b,V0op,V1f,V2g),V4x) = ap(ap(V0op,ap(V1f,V4x)),ap(V2g,V4x)) ) ) ) ).
fof(conj_thm_2Eiterate_2EITERATE__OP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V0op))
=> ! [V1f] :
( mem(V1f,arr(A_27b,A_27a))
=> ! [V2g] :
( mem(V2g,arr(A_27b,A_27a))
=> ! [V3s] :
( mem(V3s,arr(A_27b,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27b),V3s))
=> ap(ap(ap(c_2Eiterate_2Eiterate(A_27b,A_27a),V0op),V3s),f3211(A_27a,A_27b,V0op,V1f,V2g)) = ap(ap(V0op,ap(ap(ap(c_2Eiterate_2Eiterate(A_27b,A_27a),V0op),V3s),V1f)),ap(ap(ap(c_2Eiterate_2Eiterate(A_27b,A_27a),V0op),V3s),V2g)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EITERATE__SUPERSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2u] :
( mem(V2u,arr(A_27a,bool))
=> ! [V3v] :
( mem(V3v,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2u),V3v))
& ! [V4x] :
( mem(V4x,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V3v))
& ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V2u)) )
=> ap(V1f,V4x) = ap(c_2Eiterate_2Eneutral(A_27b),V0op) ) ) )
=> ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),V3v),V1f) = ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),V2u),V1f) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EITERATE__IMAGE__NONZERO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0op] :
( mem(V0op,arr(A_27c,arr(A_27c,A_27c)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27c),V0op))
=> ! [V1g] :
( mem(V1g,arr(A_27b,A_27c))
=> ! [V2f] :
( mem(V2f,arr(A_27a,A_27b))
=> ! [V3s] :
( mem(V3s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V3s))
& ! [V4x] :
( mem(V4x,A_27a)
=> ! [V5y] :
( mem(V5y,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V3s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V5y),V3s))
& V4x != V5y
& ap(V2f,V4x) = ap(V2f,V5y) )
=> ap(V1g,ap(V2f,V4x)) = ap(c_2Eiterate_2Eneutral(A_27c),V0op) ) ) ) )
=> ap(ap(ap(c_2Eiterate_2Eiterate(A_27b,A_27c),V0op),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V2f),V3s)),V1g) = ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27c),V0op),V3s),ap(ap(c_2Ecombin_2Eo(A_27a,A_27c,A_27b),V1g),V2f)) ) ) ) ) ) ) ) ) ) ).
fof(lameq_f3212,axiom,
! [A_27b,A_27a,V3f] :
( mem(V3f,arr(A_27a,A_27b))
=> ! [V2P] :
( mem(V2P,arr(A_27a,bool))
=> ! [V4g] :
( mem(V4g,arr(A_27a,A_27b))
=> ! [V5x] : ap(f3212(A_27b,A_27a,V3f,V2P,V4g),V5x) = ap(ap(ap(c_2Ebool_2ECOND(A_27b),ap(V2P,V5x)),ap(V3f,V5x)),ap(V4g,V5x)) ) ) ) ).
fof(lameq_f3213,axiom,
! [A_27a,V2P] :
( mem(V2P,arr(A_27a,bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V6x] : ap(f3213(A_27a,V2P,V1s),V6x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V6x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V6x),V1s)),ap(V2P,V6x))) ) ) ).
fof(lameq_f3214,axiom,
! [A_27a,V2P] :
( mem(V2P,arr(A_27a,bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V7x] : ap(f3214(A_27a,V2P,V1s),V7x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V7x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V7x),V1s)),ap(c_2Ebool_2E_7E,ap(V2P,V7x)))) ) ) ).
fof(conj_thm_2Eiterate_2EITERATE__CASES,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2P] :
( mem(V2P,arr(A_27a,bool))
=> ! [V3f] :
( mem(V3f,arr(A_27a,A_27b))
=> ! [V4g] :
( mem(V4g,arr(A_27a,A_27b))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
=> ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),V1s),f3212(A_27b,A_27a,V3f,V2P,V4g)) = ap(ap(V0op,ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3213(A_27a,V2P,V1s))),V3f)),ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3214(A_27a,V2P,V1s))),V4g)) ) ) ) ) ) ) ) ) ) ).
fof(lameq_f3215,axiom,
! [A_27b,A_27a,V0op] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2g] :
( mem(V2g,arr(A_27a,A_27b))
=> ! [V4x] : ap(f3215(A_27b,A_27a,V0op,V1f,V2g),V4x) = ap(ap(V0op,ap(V1f,V4x)),ap(V2g,V4x)) ) ) ) ).
fof(conj_thm_2Eiterate_2EITERATE__OP__GEN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0op] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2g] :
( mem(V2g,arr(A_27a,A_27b))
=> ! [V3s] :
( mem(V3s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(ap(ap(c_2Eiterate_2Esupport(A_27a,A_27b),V0op),V1f),V3s)))
& p(ap(c_2Epred__set_2EFINITE(A_27a),ap(ap(ap(c_2Eiterate_2Esupport(A_27a,A_27b),V0op),V2g),V3s))) )
=> ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),V3s),f3215(A_27b,A_27a,V0op,V1f,V2g)) = ap(ap(V0op,ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),V3s),V1f)),ap(ap(ap(c_2Eiterate_2Eiterate(A_27a,A_27b),V0op),V3s),V2g)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EITERATE__CLAUSES__NUMSEG,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,A_27a))
=> ! [V1op] :
( mem(V1op,arr(A_27a,arr(A_27a,A_27a)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V1op))
=> ( ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ap(ap(ap(c_2Eiterate_2Eiterate(ty_2Enum_2Enum,A_27a),V1op),ap(ap(c_2Eiterate_2E_2E_2E,V2m),c_2Enum_2E0)),V0f) = ap(ap(ap(c_2Ebool_2ECOND(A_27a),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),V2m),c_2Enum_2E0)),ap(V0f,c_2Enum_2E0)),ap(c_2Eiterate_2Eneutral(A_27a),V1op)) )
& ! [V3m] :
( mem(V3m,ty_2Enum_2Enum)
=> ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> ap(ap(ap(c_2Eiterate_2Eiterate(ty_2Enum_2Enum,A_27a),V1op),ap(ap(c_2Eiterate_2E_2E_2E,V3m),ap(c_2Enum_2ESUC,V4n))),V0f) = ap(ap(ap(c_2Ebool_2ECOND(A_27a),ap(ap(c_2Earithmetic_2E_3C_3D,V3m),ap(c_2Enum_2ESUC,V4n))),ap(ap(V1op,ap(ap(ap(c_2Eiterate_2Eiterate(ty_2Enum_2Enum,A_27a),V1op),ap(ap(c_2Eiterate_2E_2E_2E,V3m),V4n)),V0f)),ap(V0f,ap(c_2Enum_2ESUC,V4n)))),ap(ap(ap(c_2Eiterate_2Eiterate(ty_2Enum_2Enum,A_27a),V1op),ap(ap(c_2Eiterate_2E_2E_2E,V3m),V4n)),V0f)) ) ) ) ) ) ) ) ).
fof(lameq_f3216,axiom,
! [A_27a,V0op] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,A_27a))
=> ! [V4i] : ap(f3216(A_27a,V0op,V1f),V4i) = ap(ap(V0op,ap(V1f,ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))),V4i))),ap(V1f,ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))),V4i)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ) ) ).
fof(conj_thm_2Eiterate_2EITERATE__PAIR,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0op] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V0op))
=> ! [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,A_27a))
=> ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ap(ap(ap(c_2Eiterate_2Eiterate(ty_2Enum_2Enum,A_27a),V0op),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))),V2m)),ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))),V3n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),V1f) = ap(ap(ap(c_2Eiterate_2Eiterate(ty_2Enum_2Enum,A_27a),V0op),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),f3216(A_27a,V0op,V1f)) ) ) ) ) ) ) ).
fof(ax_thm_2Eiterate_2Ensum__def,axiom,
! [A_27a] :
( ne(A_27a)
=> c_2Eiterate_2Ensum(A_27a) = ap(c_2Eiterate_2Eiterate(A_27a,ty_2Enum_2Enum),c_2Earithmetic_2E_2B) ) ).
fof(conj_thm_2Eiterate_2ENEUTRAL__ADD,axiom,
ap(c_2Eiterate_2Eneutral(ty_2Enum_2Enum),c_2Earithmetic_2E_2B) = c_2Enum_2E0 ).
fof(conj_thm_2Eiterate_2ENEUTRAL__MUL,axiom,
ap(c_2Eiterate_2Eneutral(ty_2Enum_2Enum),c_2Earithmetic_2E_2A) = ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)) ).
fof(conj_thm_2Eiterate_2EMONOIDAL__ADD,axiom,
p(ap(c_2Eiterate_2Emonoidal(ty_2Enum_2Enum),c_2Earithmetic_2E_2B)) ).
fof(conj_thm_2Eiterate_2EMONOIDAL__MUL,axiom,
p(ap(c_2Eiterate_2Emonoidal(ty_2Enum_2Enum),c_2Earithmetic_2E_2A)) ).
fof(lameq_f3217,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2x] : ap(f3217(A_27a,V0f,V1s),V2x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V2x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s)),ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),ap(V0f,V2x)),c_2Enum_2E0)))) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__DEGENERATE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ~ p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3217(A_27a,V0f,V1s))))
=> ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f) = c_2Enum_2E0 ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__CLAUSES,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ap(ap(c_2Eiterate_2Ensum(A_27a),c_2Epred__set_2EEMPTY(A_27a)),V0f) = c_2Enum_2E0 )
& ! [V1x] :
( mem(V1x,A_27b)
=> ! [V2f] :
( mem(V2f,arr(A_27b,ty_2Enum_2Enum))
=> ! [V3s] :
( mem(V3s,arr(A_27b,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27b),V3s))
=> ap(ap(c_2Eiterate_2Ensum(A_27b),ap(ap(c_2Epred__set_2EINSERT(A_27b),V1x),V3s)),V2f) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Enum_2Enum),ap(ap(c_2Ebool_2EIN(A_27b),V1x),V3s)),ap(ap(c_2Eiterate_2Ensum(A_27b),V3s),V2f)),ap(ap(c_2Earithmetic_2E_2B,ap(V2f,V1x)),ap(ap(c_2Eiterate_2Ensum(A_27b),V3s),V2f))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [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_2EDISJOINT(A_27a),V1s),V2t)) )
=> ap(ap(c_2Eiterate_2Ensum(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)),V0f) = ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f)),ap(ap(c_2Eiterate_2Ensum(A_27a),V2t),V0f)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__DIFF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [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(ap(c_2Epred__set_2ESUBSET(A_27a),V2t),V1s)) )
=> ap(ap(c_2Eiterate_2Ensum(A_27a),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1s),V2t)),V0f) = ap(ap(c_2Earithmetic_2E_2D,ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f)),ap(ap(c_2Eiterate_2Ensum(A_27a),V2t),V0f)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__INCL__EXCL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2f] :
( mem(V2f,arr(A_27a,ty_2Enum_2Enum))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& p(ap(c_2Epred__set_2EFINITE(A_27a),V1t)) )
=> ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(A_27a),V0s),V2f)),ap(ap(c_2Eiterate_2Ensum(A_27a),V1t),V2f)) = ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1t)),V2f)),ap(ap(c_2Eiterate_2Ensum(A_27a),ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V1t)),V2f)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__SUPPORT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ap(ap(c_2Eiterate_2Ensum(A_27a),ap(ap(ap(c_2Eiterate_2Esupport(A_27a,ty_2Enum_2Enum),c_2Earithmetic_2E_2B),V0f),V1s)),V0f) = ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__ADD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
=> ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),f453(A_27a,V0f,V1g)) = ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V0f)),ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V1g)) ) ) ) ) ) ).
fof(lameq_f3218,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3x] : ap(f3218(A_27a,V0f,V2s),V3x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V3x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s)),ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),ap(V0f,V3x)),c_2Enum_2E0)))) ) ) ).
fof(lameq_f3219,axiom,
! [A_27a,V1g] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V4x] : ap(f3219(A_27a,V1g,V2s),V4x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V4x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V4x),V2s)),ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),ap(V1g,V4x)),c_2Enum_2E0)))) ) ) ).
fof(lameq_f3220,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V5x] : ap(f3220(A_27a,V0f,V1g),V5x) = ap(ap(c_2Earithmetic_2E_2B,ap(V0f,V5x)),ap(V1g,V5x)) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__ADD__GEN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3218(A_27a,V0f,V2s))))
& p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3219(A_27a,V1g,V2s)))) )
=> ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),f3220(A_27a,V0f,V1g)) = ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V0f)),ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V1g)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__EQ__0,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
=> ap(V0f,V2x) = c_2Enum_2E0 ) )
=> ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f) = c_2Enum_2E0 ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__0,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ap(ap(c_2Eiterate_2Ensum(A_27a),V0s),k(A_27a,c_2Enum_2E0)) = c_2Enum_2E0 ) ) ).
fof(lameq_f3221,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1c] :
( mem(V1c,ty_2Enum_2Enum)
=> ! [V3x] : ap(f3221(A_27a,V0f,V1c),V3x) = ap(ap(c_2Earithmetic_2E_2A,V1c),ap(V0f,V3x)) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__LMUL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1c] :
( mem(V1c,ty_2Enum_2Enum)
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),f3221(A_27a,V0f,V1c)) = ap(ap(c_2Earithmetic_2E_2A,V1c),ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V0f)) ) ) ) ) ).
fof(lameq_f3222,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1c] :
( mem(V1c,ty_2Enum_2Enum)
=> ! [V3x] : ap(f3222(A_27a,V0f,V1c),V3x) = ap(ap(c_2Earithmetic_2E_2A,ap(V0f,V3x)),V1c) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__RMUL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1c] :
( mem(V1c,ty_2Enum_2Enum)
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),f3222(A_27a,V0f,V1c)) = ap(ap(c_2Earithmetic_2E_2A,ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V0f)),V1c) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__LE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s))
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(V0f,V3x)),ap(V1g,V3x))) ) ) )
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V0f)),ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V1g))) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__LT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s))
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(V0f,V3x)),ap(V1g,V3x))) ) )
& ? [V4x] :
( mem(V4x,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V2s))
& p(ap(ap(c_2Eprim__rec_2E_3C,ap(V0f,V4x)),ap(V1g,V4x))) ) )
=> p(ap(ap(c_2Eprim__rec_2E_3C,ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V0f)),ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V1g))) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__LT__ALL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& V2s != c_2Epred__set_2EEMPTY(A_27a)
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s))
=> p(ap(ap(c_2Eprim__rec_2E_3C,ap(V0f,V3x)),ap(V1g,V3x))) ) ) )
=> p(ap(ap(c_2Eprim__rec_2E_3C,ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V0f)),ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V1g))) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s))
=> ap(V0f,V3x) = ap(V1g,V3x) ) )
=> ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V0f) = ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V1g) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__CONST,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0c] :
( mem(V0c,ty_2Enum_2Enum)
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
=> ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),k(A_27a,V0c)) = ap(ap(c_2Earithmetic_2E_2A,ap(c_2Epred__set_2ECARD(A_27a),V1s)),V0c) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__POS__BOUND,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1b] :
( mem(V1b,ty_2Enum_2Enum)
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V0f)),V1b)) )
=> ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s))
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(V0f,V3x)),V1b)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__EQ__0__IFF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
=> ( ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f) = c_2Enum_2E0
<=> ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
=> ap(V0f,V2x) = c_2Enum_2E0 ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__POS__LT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [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_2Eprim__rec_2E_3C,c_2Enum_2E0),ap(V0f,V2x))) ) )
=> p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f))) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__POS__LT__ALL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Enum_2Enum))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& V0s != c_2Epred__set_2EEMPTY(A_27a)
& ! [V2i] :
( mem(V2i,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2i),V0s))
=> p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),ap(V1f,V2i))) ) ) )
=> p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),ap(ap(c_2Eiterate_2Ensum(A_27a),V0s),V1f))) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__DELETE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2a] :
( mem(V2a,A_27a)
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V2a),V1s)) )
=> ap(ap(c_2Earithmetic_2E_2B,ap(V0f,V2a)),ap(ap(c_2Eiterate_2Ensum(A_27a),ap(ap(c_2Epred__set_2EDELETE(A_27a),V1s),V2a)),V0f)) = ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__SING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ap(ap(c_2Eiterate_2Ensum(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),c_2Epred__set_2EEMPTY(A_27a))),V0f) = ap(V0f,V1x) ) ) ) ).
fof(lameq_f3223,axiom,
! [A_27a,V0b] :
( mem(V0b,ty_2Enum_2Enum)
=> ! [V2a] :
( mem(V2a,A_27a)
=> ! [V3x] : ap(f3223(A_27a,V0b,V2a),V3x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Enum_2Enum),ap(ap(c_2Emin_2E_3D(A_27a),V3x),V2a)),V0b),c_2Enum_2E0) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__DELTA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0b] :
( mem(V0b,ty_2Enum_2Enum)
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2a] :
( mem(V2a,A_27a)
=> ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),f3223(A_27a,V0b,V2a)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Enum_2Enum),ap(ap(c_2Ebool_2EIN(A_27a),V2a),V1s)),V0b),c_2Enum_2E0) ) ) ) ) ).
fof(lameq_f3224,axiom,
! [A_27b,A_27a,V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,ty_2Enum_2Enum)))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ! [V3i] : ap(f3224(A_27b,A_27a,V0f,V2t),V3i) = ap(ap(c_2Eiterate_2Ensum(A_27b),V2t),ap(V0f,V3i)) ) ) ).
fof(lameq_f3225,axiom,
! [A_27b,A_27a,V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,ty_2Enum_2Enum)))
=> ! [V4j] :
( mem(V4j,A_27b)
=> ! [V5i] : ap(f3225(A_27b,A_27a,V0f,V4j),V5i) = ap(ap(V0f,V5i),V4j) ) ) ).
fof(lameq_f3226,axiom,
! [A_27a,A_27b,V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,ty_2Enum_2Enum)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V4j] : ap(f3226(A_27a,A_27b,V0f,V1s),V4j) = ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),f3225(A_27b,A_27a,V0f,V4j)) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__SWAP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,ty_2Enum_2Enum)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V2t)) )
=> ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),f3224(A_27b,A_27a,V0f,V2t)) = ap(ap(c_2Eiterate_2Ensum(A_27b),V2t),f3226(A_27a,A_27b,V0f,V1s)) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27b,ty_2Enum_2Enum))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4y] :
( mem(V4y,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V4y),V2s))
& ap(V0f,V3x) = ap(V0f,V4y) )
=> V3x = V4y ) ) )
=> ap(ap(c_2Eiterate_2Ensum(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V2s)),V1g) = ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Enum_2Enum,A_27b),V1g),V0f)) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__SUPERSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1u] :
( mem(V1u,arr(A_27a,bool))
=> ! [V2v] :
( mem(V2v,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1u),V2v))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2v))
& ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1u)) )
=> ap(V0f,V3x) = c_2Enum_2E0 ) ) )
=> ap(ap(c_2Eiterate_2Ensum(A_27a),V2v),V0f) = ap(ap(c_2Eiterate_2Ensum(A_27a),V1u),V0f) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__UNION__RZERO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1u] :
( mem(V1u,arr(A_27a,bool))
=> ! [V2v] :
( mem(V2v,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1u))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2v))
& ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1u)) )
=> ap(V0f,V3x) = c_2Enum_2E0 ) ) )
=> ap(ap(c_2Eiterate_2Ensum(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1u),V2v)),V0f) = ap(ap(c_2Eiterate_2Ensum(A_27a),V1u),V0f) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__UNION__LZERO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1u] :
( mem(V1u,arr(A_27a,bool))
=> ! [V2v] :
( mem(V2v,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2v))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1u))
& ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2v)) )
=> ap(V0f,V3x) = c_2Enum_2E0 ) ) )
=> ap(ap(c_2Eiterate_2Ensum(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1u),V2v)),V0f) = ap(ap(c_2Eiterate_2Ensum(A_27a),V2v),V0f) ) ) ) ) ) ).
fof(lameq_f3227,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2x] : ap(f3227(A_27a,V0f,V1s),V2x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Enum_2Enum),ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s)),ap(V0f,V2x)),c_2Enum_2E0) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__RESTRICT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
=> ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),f3227(A_27a,V0f,V1s)) = ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__BOUND,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2b] :
( mem(V2b,ty_2Enum_2Enum)
=> ( ( 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(ap(c_2Earithmetic_2E_3C_3D,ap(V1f,V3x)),V2b)) ) ) )
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(ap(c_2Eiterate_2Ensum(A_27a),V0s),V1f)),ap(ap(c_2Earithmetic_2E_2A,ap(c_2Epred__set_2ECARD(A_27a),V0s)),V2b))) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__BOUND__GEN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2b] :
( mem(V2b,ty_2Enum_2Enum)
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& V0s != c_2Epred__set_2EEMPTY(A_27a)
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s))
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(V1f,V3x)),ap(ap(c_2Earithmetic_2EDIV,V2b),ap(c_2Epred__set_2ECARD(A_27a),V0s)))) ) ) )
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(ap(c_2Eiterate_2Ensum(A_27a),V0s),V1f)),V2b)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__BOUND__LT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2b] :
( mem(V2b,ty_2Enum_2Enum)
=> ( ( 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(ap(c_2Earithmetic_2E_3C_3D,ap(V1f,V3x)),V2b)) ) )
& ? [V4x] :
( mem(V4x,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V0s))
& p(ap(ap(c_2Eprim__rec_2E_3C,ap(V1f,V4x)),V2b)) ) )
=> p(ap(ap(c_2Eprim__rec_2E_3C,ap(ap(c_2Eiterate_2Ensum(A_27a),V0s),V1f)),ap(ap(c_2Earithmetic_2E_2A,ap(c_2Epred__set_2ECARD(A_27a),V0s)),V2b))) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__BOUND__LT__ALL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2b] :
( mem(V2b,ty_2Enum_2Enum)
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& V0s != c_2Epred__set_2EEMPTY(A_27a)
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s))
=> p(ap(ap(c_2Eprim__rec_2E_3C,ap(V1f,V3x)),V2b)) ) ) )
=> p(ap(ap(c_2Eprim__rec_2E_3C,ap(ap(c_2Eiterate_2Ensum(A_27a),V0s),V1f)),ap(ap(c_2Earithmetic_2E_2A,ap(c_2Epred__set_2ECARD(A_27a),V0s)),V2b))) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__BOUND__LT__GEN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2b] :
( mem(V2b,ty_2Enum_2Enum)
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& V0s != c_2Epred__set_2EEMPTY(A_27a)
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s))
=> p(ap(ap(c_2Eprim__rec_2E_3C,ap(V1f,V3x)),ap(ap(c_2Earithmetic_2EDIV,V2b),ap(c_2Epred__set_2ECARD(A_27a),V0s)))) ) ) )
=> p(ap(ap(c_2Eprim__rec_2E_3C,ap(ap(c_2Eiterate_2Ensum(A_27a),V0s),V1f)),V2b)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__UNION__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,bool))
=> ! [V3u] :
( mem(V3u,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V3u))
& ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t) = c_2Epred__set_2EEMPTY(A_27a)
& ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t) = V3u )
=> ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f)),ap(ap(c_2Eiterate_2Ensum(A_27a),V2t),V0f)) = ap(ap(c_2Eiterate_2Ensum(A_27a),V3u),V0f) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__EQ__SUPERSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0g] :
( mem(V0g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t] :
( mem(V3t,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V3t))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V3t),V2s))
& ! [V4x] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V3t))
=> ap(V1f,V4x) = ap(V0g,V4x) ) )
& ! [V5x] :
( mem(V5x,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V2s))
& ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V3t)) )
=> ap(V1f,V5x) = c_2Enum_2E0 ) ) )
=> ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V1f) = ap(ap(c_2Eiterate_2Ensum(A_27a),V3t),V0g) ) ) ) ) ) ) ).
fof(lameq_f3228,axiom,
! [A_27a,V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V3x] : ap(f3228(A_27a,V0P,V1s),V3x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V3x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s)),ap(V0P,V3x))) ) ) ).
fof(lameq_f3229,axiom,
! [A_27a,V2f] :
( mem(V2f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V4x] : ap(f3229(A_27a,V2f,V0P),V4x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Enum_2Enum),ap(V0P,V4x)),ap(V2f,V4x)),c_2Enum_2E0) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__RESTRICT__SET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2f] :
( mem(V2f,arr(A_27a,ty_2Enum_2Enum))
=> ap(ap(c_2Eiterate_2Ensum(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3228(A_27a,V0P,V1s))),V2f) = ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),f3229(A_27a,V2f,V0P)) ) ) ) ) ).
fof(lameq_f3230,axiom,
! [A_27a,A_27b,V3t] :
( mem(V3t,arr(A_27b,bool))
=> ! [V4x] :
( mem(V4x,A_27a)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V5y] : ap(f3230(A_27a,A_27b,V3t,V4x,V0R),V5y) = ap(ap(c_2Epair_2E_2C(A_27b,bool),V5y),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27b),V5y),V3t)),ap(ap(V0R,V4x),V5y))) ) ) ) ).
fof(lameq_f3231,axiom,
! [A_27a,A_27b,V4x] :
( mem(V4x,A_27a)
=> ! [V1f] :
( mem(V1f,arr(A_27a,arr(A_27b,ty_2Enum_2Enum)))
=> ! [V6y] : ap(f3231(A_27a,A_27b,V4x,V1f),V6y) = ap(ap(V1f,V4x),V6y) ) ) ).
fof(lameq_f3232,axiom,
! [A_27b,A_27a,V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V3t] :
( mem(V3t,arr(A_27b,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,arr(A_27b,ty_2Enum_2Enum)))
=> ! [V4x] : ap(f3232(A_27b,A_27a,V0R,V3t,V1f),V4x) = ap(ap(c_2Eiterate_2Ensum(A_27b),ap(c_2Epred__set_2EGSPEC(A_27b,A_27b),f3230(A_27a,A_27b,V3t,V4x,V0R))),f3231(A_27a,A_27b,V4x,V1f)) ) ) ) ).
fof(lameq_f3233,axiom,
! [A_27b,A_27a,V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V7y] :
( mem(V7y,A_27b)
=> ! [V8x] : ap(f3233(A_27b,A_27a,V2s,V0R,V7y),V8x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V8x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V8x),V2s)),ap(ap(V0R,V8x),V7y))) ) ) ) ).
fof(lameq_f3234,axiom,
! [A_27b,A_27a,V1f] :
( mem(V1f,arr(A_27a,arr(A_27b,ty_2Enum_2Enum)))
=> ! [V7y] :
( mem(V7y,A_27b)
=> ! [V9x] : ap(f3234(A_27b,A_27a,V1f,V7y),V9x) = ap(ap(V1f,V9x),V7y) ) ) ).
fof(lameq_f3235,axiom,
! [A_27a,A_27b,V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,arr(A_27b,ty_2Enum_2Enum)))
=> ! [V7y] : ap(f3235(A_27a,A_27b,V0R,V2s,V1f),V7y) = ap(ap(c_2Eiterate_2Ensum(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3233(A_27b,A_27a,V2s,V0R,V7y))),f3234(A_27b,A_27a,V1f,V7y)) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__NSUM__RESTRICT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V1f] :
( mem(V1f,arr(A_27a,arr(A_27b,ty_2Enum_2Enum)))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t] :
( mem(V3t,arr(A_27b,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V3t)) )
=> ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),f3232(A_27b,A_27a,V0R,V3t,V1f)) = ap(ap(c_2Eiterate_2Ensum(A_27b),V3t),f3235(A_27a,A_27b,V0R,V2s,V1f)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ECARD__EQ__NSUM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ap(c_2Epred__set_2ECARD(A_27a),V0s) = ap(ap(c_2Eiterate_2Ensum(A_27a),V0s),k(A_27a,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))) ) ) ) ).
fof(lameq_f3236,axiom,
! [A_27b,A_27a,V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V4j] :
( mem(V4j,A_27b)
=> ! [V5i] : ap(f3236(A_27b,A_27a,V1s,V0R,V4j),V5i) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V5i),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V5i),V1s)),ap(ap(V0R,V5i),V4j))) ) ) ) ).
fof(lameq_f3237,axiom,
! [A_27a,A_27b,V2t] :
( mem(V2t,arr(A_27b,bool))
=> ! [V6i] :
( mem(V6i,A_27a)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V7j] : ap(f3237(A_27a,A_27b,V2t,V6i,V0R),V7j) = ap(ap(c_2Epair_2E_2C(A_27b,bool),V7j),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27b),V7j),V2t)),ap(ap(V0R,V6i),V7j))) ) ) ) ).
fof(lameq_f3238,axiom,
! [A_27b,A_27a,V2t] :
( mem(V2t,arr(A_27b,bool))
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V6i] : ap(f3238(A_27b,A_27a,V2t,V0R),V6i) = ap(c_2Epred__set_2ECARD(A_27b),ap(c_2Epred__set_2EGSPEC(A_27b,A_27b),f3237(A_27a,A_27b,V2t,V6i,V0R))) ) ) ).
fof(lameq_f3239,axiom,
! [A_27b,V3k] :
( mem(V3k,arr(A_27b,ty_2Enum_2Enum))
=> ! [V8i] : ap(f3239(A_27b,V3k),V8i) = ap(V3k,V8i) ) ).
fof(conj_thm_2Eiterate_2ENSUM__MULTICOUNT__GEN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ! [V3k] :
( mem(V3k,arr(A_27b,ty_2Enum_2Enum))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V2t))
& ! [V4j] :
( mem(V4j,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4j),V2t))
=> ap(c_2Epred__set_2ECARD(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3236(A_27b,A_27a,V1s,V0R,V4j))) = ap(V3k,V4j) ) ) )
=> ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),f3238(A_27b,A_27a,V2t,V0R)) = ap(ap(c_2Eiterate_2Ensum(A_27b),V2t),f3239(A_27b,V3k)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__MULTICOUNT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ! [V3k] :
( mem(V3k,ty_2Enum_2Enum)
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V2t))
& ! [V4j] :
( mem(V4j,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4j),V2t))
=> ap(c_2Epred__set_2ECARD(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3236(A_27b,A_27a,V1s,V0R,V4j))) = V3k ) ) )
=> ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),f3238(A_27b,A_27a,V2t,V0R)) = ap(ap(c_2Earithmetic_2E_2A,V3k),ap(c_2Epred__set_2ECARD(A_27b),V2t)) ) ) ) ) ) ) ) ).
fof(lameq_f3240,axiom,
! [A_27b,A_27a,V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V3y] :
( mem(V3y,A_27b)
=> ! [V4x] : ap(f3240(A_27b,A_27a,V2s,V0f,V3y),V4x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V4x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V4x),V2s)),ap(ap(c_2Emin_2E_3D(A_27b),ap(V0f,V4x)),V3y))) ) ) ) ).
fof(lameq_f3241,axiom,
! [A_27a,A_27b,V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V3y] : ap(f3241(A_27a,A_27b,V0f,V2s,V1g),V3y) = ap(ap(c_2Eiterate_2Ensum(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3240(A_27b,A_27a,V2s,V0f,V3y))),V1g) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__IMAGE__GEN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
=> ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V1g) = ap(ap(c_2Eiterate_2Ensum(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V2s)),f3241(A_27a,A_27b,V0f,V2s,V1g)) ) ) ) ) ) ) ).
fof(lameq_f3242,axiom,
! [A_27b,A_27a,V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V4y] :
( mem(V4y,A_27b)
=> ! [V5x] : ap(f3242(A_27b,A_27a,V2s,V0f,V4y),V5x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V5x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V5x),V2s)),ap(ap(c_2Emin_2E_3D(A_27b),ap(V0f,V5x)),V4y))) ) ) ) ).
fof(lameq_f3243,axiom,
! [A_27a,A_27b,V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V4y] : ap(f3243(A_27a,A_27b,V0f,V2s,V1g),V4y) = ap(ap(c_2Eiterate_2Ensum(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3242(A_27b,A_27a,V2s,V0f,V4y))),V1g) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__GROUP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t] :
( mem(V3t,arr(A_27b,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V2s)),V3t)) )
=> ap(ap(c_2Eiterate_2Ensum(A_27b),V3t),f3243(A_27a,A_27b,V0f,V2s,V1g)) = ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V1g) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0u] :
( mem(V0u,arr(A_27a,bool))
=> ! [V1v] :
( mem(V1v,arr(A_27a,bool))
=> ! [V2f] :
( mem(V2f,arr(A_27a,ty_2Enum_2Enum))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0u))
& p(ap(c_2Epred__set_2EFINITE(A_27a),V1v))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0u),V1v)))
=> ap(V2f,V3x) = c_2Enum_2E0 ) ) )
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(ap(c_2Eiterate_2Ensum(A_27a),V0u),V2f)),ap(ap(c_2Eiterate_2Ensum(A_27a),V1v),V2f))) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__SUBSET__SIMPLE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0u] :
( mem(V0u,arr(A_27a,bool))
=> ! [V1v] :
( mem(V1v,arr(A_27a,bool))
=> ! [V2f] :
( mem(V2f,arr(A_27a,ty_2Enum_2Enum))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1v))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0u),V1v)) )
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(ap(c_2Eiterate_2Ensum(A_27a),V0u),V2f)),ap(ap(c_2Eiterate_2Ensum(A_27a),V1v),V2f))) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__LE__GEN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s))
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(V0f,V3x)),ap(V1g,V3x))) ) )
& p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3219(A_27a,V1g,V2s)))) )
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V0f)),ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V1g))) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__IMAGE__NONZERO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0d] :
( mem(V0d,arr(A_27b,ty_2Enum_2Enum))
=> ! [V1i] :
( mem(V1i,arr(A_27a,A_27b))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4y] :
( mem(V4y,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V4y),V2s))
& V3x != V4y
& ap(V1i,V3x) = ap(V1i,V4y) )
=> ap(V0d,ap(V1i,V3x)) = c_2Enum_2E0 ) ) ) )
=> ap(ap(c_2Eiterate_2Ensum(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1i),V2s)),V0d) = ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Enum_2Enum,A_27b),V0d),V1i)) ) ) ) ) ) ) ).
fof(lameq_f3244,axiom,
! [A_27a,V4y] :
( mem(V4y,A_27a)
=> ! [V1p] :
( mem(V1p,arr(A_27a,A_27a))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V5x] : ap(f3244(A_27a,V4y,V1p,V2s),V5x) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V5x),V2s)),ap(ap(c_2Emin_2E_3D(A_27a),ap(V1p,V5x)),V4y)) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__BIJECTION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1p] :
( mem(V1p,arr(A_27a,A_27a))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s))
=> p(ap(ap(c_2Ebool_2EIN(A_27a),ap(V1p,V3x)),V2s)) ) )
& ! [V4y] :
( mem(V4y,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4y),V2s))
=> p(ap(c_2Ebool_2E_3F_21(A_27a),f3244(A_27a,V4y,V1p,V2s))) ) ) )
=> ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V0f) = ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Enum_2Enum,A_27a),V0f),V1p)) ) ) ) ) ) ).
fof(lameq_f3245,axiom,
! [A_27b,A_27a,V1t] :
( mem(V1t,arr(A_27a,arr(A_27b,bool)))
=> ! [V2x] :
( mem(V2x,arr(A_27a,arr(A_27b,ty_2Enum_2Enum)))
=> ! [V4i] : ap(f3245(A_27b,A_27a,V1t,V2x),V4i) = ap(ap(c_2Eiterate_2Ensum(A_27b),ap(V1t,V4i)),ap(V2x,V4i)) ) ) ).
fof(lameq_f3246,axiom,
! [A_27a,A_27b,V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,arr(A_27b,bool)))
=> ! [V5i] :
( mem(V5i,A_27a)
=> ! [V6j] : ap(f3246(A_27a,A_27b,V0s,V1t,V5i),V6j) = ap(ap(c_2Epair_2E_2C(ty_2Epair_2Eprod(A_27a,A_27b),bool),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V5i),V6j)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V5i),V0s)),ap(ap(c_2Ebool_2EIN(A_27b),V6j),ap(V1t,V5i)))) ) ) ) ).
fof(lameq_f3247,axiom,
! [A_27b,A_27a,V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,arr(A_27b,bool)))
=> ! [V5i] : ap(f3247(A_27b,A_27a,V0s,V1t),V5i) = f3246(A_27a,A_27b,V0s,V1t,V5i) ) ) ).
fof(lameq_f3248,axiom,
! [A_27a,A_27b,V7i] :
( mem(V7i,A_27a)
=> ! [V2x] :
( mem(V2x,arr(A_27a,arr(A_27b,ty_2Enum_2Enum)))
=> ! [V8j] : ap(f3248(A_27a,A_27b,V7i,V2x),V8j) = ap(ap(V2x,V7i),V8j) ) ) ).
fof(lameq_f3249,axiom,
! [A_27b,A_27a,V2x] :
( mem(V2x,arr(A_27a,arr(A_27b,ty_2Enum_2Enum)))
=> ! [V7i] : ap(f3249(A_27b,A_27a,V2x),V7i) = f3248(A_27a,A_27b,V7i,V2x) ) ).
fof(conj_thm_2Eiterate_2ENSUM__NSUM__PRODUCT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,arr(A_27b,bool)))
=> ! [V2x] :
( mem(V2x,arr(A_27a,arr(A_27b,ty_2Enum_2Enum)))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& ! [V3i] :
( mem(V3i,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3i),V0s))
=> p(ap(c_2Epred__set_2EFINITE(A_27b),ap(V1t,V3i))) ) ) )
=> ap(ap(c_2Eiterate_2Ensum(A_27a),V0s),f3245(A_27b,A_27a,V1t,V2x)) = ap(ap(c_2Eiterate_2Ensum(ty_2Epair_2Eprod(A_27a,A_27b)),ap(c_2Epred__set_2EGSPEC(ty_2Epair_2Eprod(A_27a,A_27b),ty_2Epair_2Eprod(A_27a,A_27b)),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,ty_2Epair_2Eprod(ty_2Epair_2Eprod(A_27a,A_27b),bool)),f3247(A_27b,A_27a,V0s,V1t)))),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,ty_2Enum_2Enum),f3249(A_27b,A_27a,V2x))) ) ) ) ) ) ) ).
fof(lameq_f3250,axiom,
! [A_27b,A_27a,V5y] :
( mem(V5y,A_27b)
=> ! [V4h] :
( mem(V4h,arr(A_27a,A_27b))
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V6x] : ap(f3250(A_27b,A_27a,V5y,V4h,V0s),V6x) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V6x),V0s)),ap(ap(c_2Emin_2E_3D(A_27b),ap(V4h,V6x)),V5y)) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__EQ__GENERAL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ! [V2f] :
( mem(V2f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V3g] :
( mem(V3g,arr(A_27b,ty_2Enum_2Enum))
=> ! [V4h] :
( mem(V4h,arr(A_27a,A_27b))
=> ( ( ! [V5y] :
( mem(V5y,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V5y),V1t))
=> p(ap(c_2Ebool_2E_3F_21(A_27a),f3250(A_27b,A_27a,V5y,V4h,V0s))) ) )
& ! [V7x] :
( mem(V7x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V7x),V0s))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V4h,V7x)),V1t))
& ap(V3g,ap(V4h,V7x)) = ap(V2f,V7x) ) ) ) )
=> ap(ap(c_2Eiterate_2Ensum(A_27a),V0s),V2f) = ap(ap(c_2Eiterate_2Ensum(A_27b),V1t),V3g) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__EQ__GENERAL__INVERSES,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ! [V2f] :
( mem(V2f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V3g] :
( mem(V3g,arr(A_27b,ty_2Enum_2Enum))
=> ! [V4h] :
( mem(V4h,arr(A_27a,A_27b))
=> ! [V5k] :
( mem(V5k,arr(A_27b,A_27a))
=> ( ( ! [V6y] :
( mem(V6y,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V6y),V1t))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),ap(V5k,V6y)),V0s))
& ap(V4h,ap(V5k,V6y)) = V6y ) ) )
& ! [V7x] :
( mem(V7x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V7x),V0s))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V4h,V7x)),V1t))
& ap(V5k,ap(V4h,V7x)) = V7x
& ap(V3g,ap(V4h,V7x)) = ap(V2f,V7x) ) ) ) )
=> ap(ap(c_2Eiterate_2Ensum(A_27a),V0s),V2f) = ap(ap(c_2Eiterate_2Ensum(A_27b),V1t),V3g) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__INJECTION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1p] :
( mem(V1p,arr(A_27a,A_27a))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s))
=> p(ap(ap(c_2Ebool_2EIN(A_27a),ap(V1p,V3x)),V2s)) ) )
& ! [V4x] :
( mem(V4x,A_27a)
=> ! [V5y] :
( mem(V5y,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V2s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V5y),V2s))
& ap(V1p,V4x) = ap(V1p,V5y) )
=> V4x = V5y ) ) ) )
=> ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Enum_2Enum,A_27a),V0f),V1p)) = ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V0f) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__UNION__NONZERO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [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))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t)))
=> ap(V0f,V3x) = c_2Enum_2E0 ) ) )
=> ap(ap(c_2Eiterate_2Ensum(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)),V0f) = ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f)),ap(ap(c_2Eiterate_2Ensum(A_27a),V2t),V0f)) ) ) ) ) ) ).
fof(lameq_f3251,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V6t] : ap(f3251(A_27a,V0f),V6t) = ap(ap(c_2Eiterate_2Ensum(A_27a),V6t),V0f) ) ).
fof(conj_thm_2Eiterate_2ENSUM__BIGUNION__NONZERO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s] :
( mem(V1s,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(arr(A_27a,bool)),V1s))
& ! [V2t] :
( mem(V2t,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),V1s))
=> p(ap(c_2Epred__set_2EFINITE(A_27a),V2t)) ) )
& ! [V3t1] :
( mem(V3t1,arr(A_27a,bool))
=> ! [V4t2] :
( mem(V4t2,arr(A_27a,bool))
=> ! [V5x] :
( mem(V5x,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3t1),V1s))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V4t2),V1s))
& V3t1 != V4t2
& p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V3t1))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V4t2)) )
=> ap(V0f,V5x) = c_2Enum_2E0 ) ) ) ) )
=> ap(ap(c_2Eiterate_2Ensum(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),V1s)),V0f) = ap(ap(c_2Eiterate_2Ensum(arr(A_27a,bool)),V1s),f3251(A_27a,V0f)) ) ) ) ) ).
fof(lameq_f3252,axiom,
! [A_27a,V2f] :
( mem(V2f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1P] :
( mem(V1P,arr(A_27a,bool))
=> ! [V3g] :
( mem(V3g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V4x] : ap(f3252(A_27a,V2f,V1P,V3g),V4x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Enum_2Enum),ap(V1P,V4x)),ap(V2f,V4x)),ap(V3g,V4x)) ) ) ) ).
fof(lameq_f3253,axiom,
! [A_27a,V1P] :
( mem(V1P,arr(A_27a,bool))
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V5x] : ap(f3253(A_27a,V1P,V0s),V5x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V5x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V5x),V0s)),ap(V1P,V5x))) ) ) ).
fof(lameq_f3254,axiom,
! [A_27a,V1P] :
( mem(V1P,arr(A_27a,bool))
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V6x] : ap(f3254(A_27a,V1P,V0s),V6x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V6x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V6x),V0s)),ap(c_2Ebool_2E_7E,ap(V1P,V6x)))) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__CASES,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_2Enum_2Enum))
=> ! [V3g] :
( mem(V3g,arr(A_27a,ty_2Enum_2Enum))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ap(ap(c_2Eiterate_2Ensum(A_27a),V0s),f3252(A_27a,V2f,V1P,V3g)) = ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3253(A_27a,V1P,V0s))),V2f)),ap(ap(c_2Eiterate_2Ensum(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3254(A_27a,V1P,V0s))),V3g)) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__CLOSED,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(ty_2Enum_2Enum,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(V0P,c_2Enum_2E0))
& ! [V3x] :
( mem(V3x,ty_2Enum_2Enum)
=> ! [V4y] :
( mem(V4y,ty_2Enum_2Enum)
=> ( ( p(ap(V0P,V3x))
& p(ap(V0P,V4y)) )
=> p(ap(V0P,ap(ap(c_2Earithmetic_2E_2B,V3x),V4y))) ) ) )
& ! [V5a] :
( mem(V5a,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5a),V2s))
=> p(ap(V0P,ap(V1f,V5a))) ) ) )
=> p(ap(V0P,ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V1f))) ) ) ) ) ) ).
fof(lameq_f3255,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V4i] : ap(f3255(V0f,V1g),V4i) = ap(ap(c_2Earithmetic_2E_2B,ap(V0f,V4i)),ap(V1g,V4i)) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__ADD__NUMSEG,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),f3255(V0f,V1g)) = ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),V0f)),ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),V1g)) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__LE__NUMSEG,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ( ! [V4i] :
( mem(V4i,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V2m),V4i))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,V4i),V3n)) )
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(V0f,V4i)),ap(V1g,V4i))) ) )
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),V0f)),ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),V1g))) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__EQ__NUMSEG,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ( ! [V4i] :
( mem(V4i,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V2m),V4i))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,V4i),V3n)) )
=> ap(V0f,V4i) = ap(V1g,V4i) ) )
=> ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),V0f) = ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),V1g) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__CONST__NUMSEG,axiom,
! [V0c] :
( mem(V0c,ty_2Enum_2Enum)
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),k(ty_2Enum_2Enum,V0c)) = ap(ap(c_2Earithmetic_2E_2A,ap(ap(c_2Earithmetic_2E_2D,ap(ap(c_2Earithmetic_2E_2B,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V1m)),V0c) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__EQ__0__NUMSEG,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( ! [V3i] :
( mem(V3i,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V3i))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,V3i),V2n)) )
=> ap(V0f,V3i) = c_2Enum_2E0 ) )
=> ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),V0f) = c_2Enum_2E0 ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__EQ__0__IFF__NUMSEG,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),V0f) = c_2Enum_2E0
<=> ! [V3i] :
( mem(V3i,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V3i))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,V3i),V2n)) )
=> ap(V0f,V3i) = c_2Enum_2E0 ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__TRIV__NUMSEG,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Eprim__rec_2E_3C,V2n),V1m))
=> ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),V0f) = c_2Enum_2E0 ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__SING__NUMSEG,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1n),V1n)),V0f) = ap(V0f,V1n) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__CLAUSES__NUMSEG,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ( ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),c_2Enum_2E0)),V0f) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Enum_2Enum),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),V1m),c_2Enum_2E0)),ap(V0f,c_2Enum_2E0)),c_2Enum_2E0) )
& ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),ap(c_2Enum_2ESUC,V3n))),V0f) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Enum_2Enum),ap(ap(c_2Earithmetic_2E_3C_3D,V2m),ap(c_2Enum_2ESUC,V3n))),ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),V0f)),ap(V0f,ap(c_2Enum_2ESUC,V3n)))),ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),V0f)) ) ) ) ) ).
fof(lameq_f3256,axiom,
! [V2c] :
( mem(V2c,ty_2Enum_2Enum)
=> ! [V3d] :
( mem(V3d,ty_2Enum_2Enum)
=> ! [V4f] :
( mem(V4f,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)))
=> ! [V5i] : ap(f3256(V2c,V3d,V4f),V5i) = ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2c),V3d)),ap(V4f,V5i)) ) ) ) ).
fof(lameq_f3257,axiom,
! [V4f] :
( mem(V4f,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)))
=> ! [V6j] :
( mem(V6j,ty_2Enum_2Enum)
=> ! [V7i] : ap(f3257(V4f,V6j),V7i) = ap(ap(V4f,V7i),V6j) ) ) ).
fof(lameq_f3258,axiom,
! [V0a] :
( mem(V0a,ty_2Enum_2Enum)
=> ! [V1b] :
( mem(V1b,ty_2Enum_2Enum)
=> ! [V4f] :
( mem(V4f,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)))
=> ! [V6j] : ap(f3258(V0a,V1b,V4f),V6j) = ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V0a),V1b)),f3257(V4f,V6j)) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__SWAP__NUMSEG,axiom,
! [V0a] :
( mem(V0a,ty_2Enum_2Enum)
=> ! [V1b] :
( mem(V1b,ty_2Enum_2Enum)
=> ! [V2c] :
( mem(V2c,ty_2Enum_2Enum)
=> ! [V3d] :
( mem(V3d,ty_2Enum_2Enum)
=> ! [V4f] :
( mem(V4f,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)))
=> ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V0a),V1b)),f3256(V2c,V3d,V4f)) = ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2c),V3d)),f3258(V0a,V1b,V4f)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__ADD__SPLIT,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ! [V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),ap(ap(c_2Earithmetic_2E_2B,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))
=> ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),ap(ap(c_2Earithmetic_2E_2B,V2n),V3p))),V0f) = ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),V0f)),ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(ap(c_2Earithmetic_2E_2B,V2n),V3p))),V0f)) ) ) ) ) ) ).
fof(lameq_f3259,axiom,
! [V0p] :
( mem(V0p,ty_2Enum_2Enum)
=> ! [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V4i] : ap(f3259(V0p,V1f),V4i) = ap(V1f,ap(ap(c_2Earithmetic_2E_2B,V4i),V0p)) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__OFFSET,axiom,
! [V0p] :
( mem(V0p,ty_2Enum_2Enum)
=> ! [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,V2m),V0p)),ap(ap(c_2Earithmetic_2E_2B,V3n),V0p))),V1f) = ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),f3259(V0p,V1f)) ) ) ) ) ).
fof(lameq_f3260,axiom,
! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V3i] : ap(f3260(V1m,V0f),V3i) = ap(V0f,ap(ap(c_2Earithmetic_2E_2B,V3i),V1m)) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__OFFSET__0,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V2n))
=> ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),V0f) = ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),ap(ap(c_2Earithmetic_2E_2D,V2n),V1m))),f3260(V1m,V0f)) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__CLAUSES__LEFT,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V2n))
=> ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),V0f) = ap(ap(c_2Earithmetic_2E_2B,ap(V0f,V1m)),ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,V1m),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V2n)),V0f)) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ENSUM__CLAUSES__RIGHT,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V2n))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V2n)) )
=> ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),V0f) = ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),ap(ap(c_2Earithmetic_2E_2D,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),V0f)),ap(V0f,V2n)) ) ) ) ) ).
fof(lameq_f3261,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V3i] : ap(f3261(V0f),V3i) = ap(ap(c_2Earithmetic_2E_2B,ap(V0f,ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))),V3i))),ap(V0f,ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))),V3i)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ) ).
fof(conj_thm_2Eiterate_2ENSUM__PAIR,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))),V1m)),ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))),V2n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),V0f) = ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),f3261(V0f)) ) ) ) ).
fof(lameq_f3262,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ! [V3i] : ap(f3262(A_27a,V0f,V1n),V3i) = ap(ap(c_2Earithmetic_2EMOD,ap(V0f,V3i)),V1n) ) ) ).
fof(conj_thm_2Eiterate_2EMOD__NSUM__MOD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& V1n != c_2Enum_2E0 )
=> ap(ap(c_2Earithmetic_2EMOD,ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V0f)),V1n) = ap(ap(c_2Earithmetic_2EMOD,ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),f3262(A_27a,V0f,V1n))),V1n) ) ) ) ) ) ).
fof(lameq_f3263,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ! [V4i] : ap(f3263(V0f,V3n),V4i) = ap(ap(c_2Earithmetic_2EMOD,ap(V0f,V4i)),V3n) ) ) ).
fof(conj_thm_2Eiterate_2EMOD__NSUM__MOD__NUMSEG,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1a] :
( mem(V1a,ty_2Enum_2Enum)
=> ! [V2b] :
( mem(V2b,ty_2Enum_2Enum)
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ( V3n != c_2Enum_2E0
=> ap(ap(c_2Earithmetic_2EMOD,ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1a),V2b)),V0f)),V3n) = ap(ap(c_2Earithmetic_2EMOD,ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1a),V2b)),f3263(V0f,V3n))),V3n) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ECARD__BIGUNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(arr(A_27a,bool)),V0s))
& ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V1t),V0s))
=> p(ap(c_2Epred__set_2EFINITE(A_27a),V1t)) ) )
& ! [V2t] :
( mem(V2t,arr(A_27a,bool))
=> ! [V3u] :
( mem(V3u,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),V0s))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3u),V0s))
& V2t != V3u )
=> ap(ap(c_2Epred__set_2EINTER(A_27a),V2t),V3u) = c_2Epred__set_2EEMPTY(A_27a) ) ) ) )
=> ap(c_2Epred__set_2ECARD(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),V0s)) = ap(ap(c_2Eiterate_2Ensum(arr(A_27a,bool)),V0s),c_2Epred__set_2ECARD(A_27a)) ) ) ) ).
fof(ax_thm_2Eiterate_2Esum__def,axiom,
! [A_27a] :
( ne(A_27a)
=> c_2Eiterate_2ESum(A_27a) = ap(c_2Eiterate_2Eiterate(A_27a,ty_2Erealax_2Ereal),c_2Erealax_2Ereal__add) ) ).
fof(conj_thm_2Eiterate_2ENEUTRAL__REAL__ADD,axiom,
ap(c_2Eiterate_2Eneutral(ty_2Erealax_2Ereal),c_2Erealax_2Ereal__add) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ).
fof(conj_thm_2Eiterate_2ENEUTRAL__REAL__MUL,axiom,
ap(c_2Eiterate_2Eneutral(ty_2Erealax_2Ereal),c_2Erealax_2Ereal__mul) = ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ).
fof(conj_thm_2Eiterate_2EMONOIDAL__REAL__ADD,axiom,
p(ap(c_2Eiterate_2Emonoidal(ty_2Erealax_2Ereal),c_2Erealax_2Ereal__add)) ).
fof(conj_thm_2Eiterate_2EMONOIDAL__REAL__MUL,axiom,
p(ap(c_2Eiterate_2Emonoidal(ty_2Erealax_2Ereal),c_2Erealax_2Ereal__mul)) ).
fof(lameq_f3264,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2x] : ap(f3264(A_27a,V0f,V1s),V2x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V2x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s)),ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(ty_2Erealax_2Ereal),ap(V0f,V2x)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))))) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__DEGENERATE,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),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3264(A_27a,V0f,V1s))))
=> ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__CLAUSES,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ap(ap(c_2Eiterate_2ESum(A_27a),c_2Epred__set_2EEMPTY(A_27a)),V0f) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) )
& ! [V1x] :
( mem(V1x,A_27b)
=> ! [V2f] :
( mem(V2f,arr(A_27b,ty_2Erealax_2Ereal))
=> ! [V3s] :
( mem(V3s,arr(A_27b,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27b),V3s))
=> ap(ap(c_2Eiterate_2ESum(A_27b),ap(ap(c_2Epred__set_2EINSERT(A_27b),V1x),V3s)),V2f) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Ebool_2EIN(A_27b),V1x),V3s)),ap(ap(c_2Eiterate_2ESum(A_27b),V3s),V2f)),ap(ap(c_2Erealax_2Ereal__add,ap(V2f,V1x)),ap(ap(c_2Eiterate_2ESum(A_27b),V3s),V2f))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__UNION,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_2EDISJOINT(A_27a),V1s),V2t)) )
=> ap(ap(c_2Eiterate_2ESum(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)),V0f) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f)),ap(ap(c_2Eiterate_2ESum(A_27a),V2t),V0f)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__DIFF,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(ap(c_2Epred__set_2ESUBSET(A_27a),V2t),V1s)) )
=> ap(ap(c_2Eiterate_2ESum(A_27a),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1s),V2t)),V0f) = ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f)),ap(ap(c_2Eiterate_2ESum(A_27a),V2t),V0f)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__INCL__EXCL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2f] :
( mem(V2f,arr(A_27a,ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& p(ap(c_2Epred__set_2EFINITE(A_27a),V1t)) )
=> ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(A_27a),V0s),V2f)),ap(ap(c_2Eiterate_2ESum(A_27a),V1t),V2f)) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1t)),V2f)),ap(ap(c_2Eiterate_2ESum(A_27a),ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V1t)),V2f)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__SUPPORT,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_2Eiterate_2ESum(A_27a),ap(ap(ap(c_2Eiterate_2Esupport(A_27a,ty_2Erealax_2Ereal),c_2Erealax_2Ereal__add),V0f),V1s)),V0f) = ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f) ) ) ) ).
fof(lameq_f3265,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V3x] : ap(f3265(A_27a,V0f,V1g),V3x) = ap(ap(c_2Erealax_2Ereal__add,ap(V0f,V3x)),ap(V1g,V3x)) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__ADD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
=> ap(ap(c_2Eiterate_2ESum(A_27a),V2s),f3265(A_27a,V0f,V1g)) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f)),ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1g)) ) ) ) ) ) ).
fof(lameq_f3266,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3x] : ap(f3266(A_27a,V0f,V2s),V3x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V3x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s)),ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(ty_2Erealax_2Ereal),ap(V0f,V3x)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))))) ) ) ).
fof(lameq_f3267,axiom,
! [A_27a,V1g] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V4x] : ap(f3267(A_27a,V1g,V2s),V4x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V4x),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V4x),V2s)),ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(ty_2Erealax_2Ereal),ap(V1g,V4x)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))))) ) ) ).
fof(lameq_f3268,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V5x] : ap(f3268(A_27a,V0f,V1g),V5x) = ap(ap(c_2Erealax_2Ereal__add,ap(V0f,V5x)),ap(V1g,V5x)) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__ADD__GEN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3266(A_27a,V0f,V2s))))
& p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3267(A_27a,V1g,V2s)))) )
=> ap(ap(c_2Eiterate_2ESum(A_27a),V2s),f3268(A_27a,V0f,V1g)) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f)),ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1g)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__EQ__0,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
=> ap(V0f,V2x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) )
=> ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__0,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ap(ap(c_2Eiterate_2ESum(A_27a),V0s),k(A_27a,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ).
fof(lameq_f3269,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1c] :
( mem(V1c,ty_2Erealax_2Ereal)
=> ! [V3x] : ap(f3269(A_27a,V0f,V1c),V3x) = ap(ap(c_2Erealax_2Ereal__mul,V1c),ap(V0f,V3x)) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__LMUL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1c] :
( mem(V1c,ty_2Erealax_2Ereal)
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ap(ap(c_2Eiterate_2ESum(A_27a),V2s),f3269(A_27a,V0f,V1c)) = ap(ap(c_2Erealax_2Ereal__mul,V1c),ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f)) ) ) ) ) ).
fof(lameq_f3270,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1c] :
( mem(V1c,ty_2Erealax_2Ereal)
=> ! [V3x] : ap(f3270(A_27a,V0f,V1c),V3x) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0f,V3x)),V1c) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__RMUL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1c] :
( mem(V1c,ty_2Erealax_2Ereal)
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ap(ap(c_2Eiterate_2ESum(A_27a),V2s),f3270(A_27a,V0f,V1c)) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f)),V1c) ) ) ) ) ).
fof(lameq_f3271,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2x] : ap(f3271(A_27a,V0f),V2x) = ap(c_2Erealax_2Ereal__neg,ap(V0f,V2x)) ) ).
fof(conj_thm_2Eiterate_2ESUM__NEG,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_2Eiterate_2ESum(A_27a),V1s),f3271(A_27a,V0f)) = ap(c_2Erealax_2Ereal__neg,ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f)) ) ) ) ).
fof(lameq_f3272,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V3x] : ap(f3272(A_27a,V0f,V1g),V3x) = ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,V3x)),ap(V1g,V3x)) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__SUB,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
=> ap(ap(c_2Eiterate_2ESum(A_27a),V2s),f3272(A_27a,V0f,V1g)) = ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f)),ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1g)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__LE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V3x)),ap(V1g,V3x))) ) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f)),ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1g))) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__LT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V3x)),ap(V1g,V3x))) ) )
& ? [V4x] :
( mem(V4x,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V2s))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(V0f,V4x)),ap(V1g,V4x))) ) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f)),ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1g))) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__LT__ALL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& V2s != c_2Epred__set_2EEMPTY(A_27a)
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s))
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(V0f,V3x)),ap(V1g,V3x))) ) ) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f)),ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1g))) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__POS__LT,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))) ) )
& ? [V3x] :
( mem(V3x,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V0f,V3x))) ) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f))) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__POS__LT__ALL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& V0s != c_2Epred__set_2EEMPTY(A_27a)
& ! [V2i] :
( mem(V2i,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2i),V0s))
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V1f,V2i))) ) ) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Eiterate_2ESum(A_27a),V0s),V1f))) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s))
=> ap(V0f,V3x) = ap(V1g,V3x) ) )
=> ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f) = ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1g) ) ) ) ) ) ).
fof(lameq_f3273,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2x] : ap(f3273(A_27a,V0f),V2x) = ap(c_2Ereal_2Eabs,ap(V0f,V2x)) ) ).
fof(conj_thm_2Eiterate_2ESUM__ABS,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))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f))),ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3273(A_27a,V0f)))) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__ABS__LE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(V0f,V3x))),ap(V1g,V3x))) ) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f))),ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1g))) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__CONST,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0c] :
( mem(V0c,ty_2Erealax_2Ereal)
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
=> ap(ap(c_2Eiterate_2ESum(A_27a),V1s),k(A_27a,V0c)) = ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V1s))),V0c) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__POS__LE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ! [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_2Eiterate_2ESum(A_27a),V1s),V0f))) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__POS__BOUND,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1b] :
( mem(V1b,ty_2Erealax_2Ereal)
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s))
=> 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_2Eiterate_2ESum(A_27a),V2s),V0f)),V1b)) )
=> ! [V4x] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V2s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V4x)),V1b)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__POS__EQ__0,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))) ) )
& ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) )
=> ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s))
=> ap(V0f,V3x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__ZERO__EXISTS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0u] :
( mem(V0u,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_2Eiterate_2ESum(A_27a),V1s),V0u) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) )
=> ( ! [V2i] :
( mem(V2i,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2i),V1s))
=> ap(V0u,V2i) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) )
| ? [V3j] :
( mem(V3j,A_27a)
& ? [V4k] :
( mem(V4k,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V3j),V1s))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(V0u,V3j)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V4k),V1s))
& p(ap(ap(c_2Ereal_2Ereal__gt,ap(V0u,V4k)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__DELETE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2a] :
( mem(V2a,A_27a)
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V2a),V1s)) )
=> ap(ap(c_2Eiterate_2ESum(A_27a),ap(ap(c_2Epred__set_2EDELETE(A_27a),V1s),V2a)),V0f) = ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f)),ap(V0f,V2a)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__DELETE__CASES,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2a] :
( mem(V2a,A_27a)
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
=> ap(ap(c_2Eiterate_2ESum(A_27a),ap(ap(c_2Epred__set_2EDELETE(A_27a),V1s),V2a)),V0f) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Ebool_2EIN(A_27a),V2a),V1s)),ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f)),ap(V0f,V2a))),ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__SING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ap(ap(c_2Eiterate_2ESum(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),c_2Epred__set_2EEMPTY(A_27a))),V0f) = ap(V0f,V1x) ) ) ) ).
fof(lameq_f3274,axiom,
! [A_27a,V0b] :
( mem(V0b,ty_2Erealax_2Ereal)
=> ! [V2a] :
( mem(V2a,A_27a)
=> ! [V3x] : ap(f3274(A_27a,V0b,V2a),V3x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Emin_2E_3D(A_27a),V3x),V2a)),V0b),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__DELTA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0b] :
( mem(V0b,ty_2Erealax_2Ereal)
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2a] :
( mem(V2a,A_27a)
=> ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3274(A_27a,V0b,V2a)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Ebool_2EIN(A_27a),V2a),V1s)),V0b),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)) ) ) ) ) ).
fof(lameq_f3275,axiom,
! [A_27b,A_27a,V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,ty_2Erealax_2Ereal)))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ! [V3i] : ap(f3275(A_27b,A_27a,V0f,V2t),V3i) = ap(ap(c_2Eiterate_2ESum(A_27b),V2t),ap(V0f,V3i)) ) ) ).
fof(lameq_f3276,axiom,
! [A_27a,A_27b,V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,ty_2Erealax_2Ereal)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V4j] : ap(f3276(A_27a,A_27b,V0f,V1s),V4j) = ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3225(A_27b,A_27a,V0f,V4j)) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__SWAP,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))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V2t)) )
=> ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3275(A_27b,A_27a,V0f,V2t)) = ap(ap(c_2Eiterate_2ESum(A_27b),V2t),f3276(A_27a,A_27b,V0f,V1s)) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27b,ty_2Erealax_2Ereal))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4y] :
( mem(V4y,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V4y),V2s))
& ap(V0f,V3x) = ap(V0f,V4y) )
=> V3x = V4y ) ) )
=> ap(ap(c_2Eiterate_2ESum(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V2s)),V1g) = ap(ap(c_2Eiterate_2ESum(A_27a),V2s),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Erealax_2Ereal,A_27b),V1g),V0f)) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__SUPERSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1u] :
( mem(V1u,arr(A_27a,bool))
=> ! [V2v] :
( mem(V2v,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1u),V2v))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2v))
& ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1u)) )
=> ap(V0f,V3x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) )
=> ap(ap(c_2Eiterate_2ESum(A_27a),V2v),V0f) = ap(ap(c_2Eiterate_2ESum(A_27a),V1u),V0f) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__UNION__RZERO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1u] :
( mem(V1u,arr(A_27a,bool))
=> ! [V2v] :
( mem(V2v,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1u))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2v))
& ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1u)) )
=> ap(V0f,V3x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) )
=> ap(ap(c_2Eiterate_2ESum(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1u),V2v)),V0f) = ap(ap(c_2Eiterate_2ESum(A_27a),V1u),V0f) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__UNION__LZERO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1u] :
( mem(V1u,arr(A_27a,bool))
=> ! [V2v] :
( mem(V2v,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2v))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1u))
& ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2v)) )
=> ap(V0f,V3x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) )
=> ap(ap(c_2Eiterate_2ESum(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1u),V2v)),V0f) = ap(ap(c_2Eiterate_2ESum(A_27a),V2v),V0f) ) ) ) ) ) ).
fof(lameq_f3277,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2x] : ap(f3277(A_27a,V0f,V1s),V2x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s)),ap(V0f,V2x)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__RESTRICT,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))
=> ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3277(A_27a,V0f,V1s)) = ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__BOUND,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2b] :
( mem(V2b,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(ap(c_2Ereal_2Ereal__lte,ap(V1f,V3x)),V2b)) ) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eiterate_2ESum(A_27a),V0s),V1f)),ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V0s))),V2b))) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__BOUND__GEN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2b] :
( mem(V2b,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& V0s != c_2Epred__set_2EEMPTY(A_27a)
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V1f,V3x)),ap(ap(c_2Ereal_2E_2F,V2b),ap(c_2Ereal_2Ereal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V0s))))) ) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eiterate_2ESum(A_27a),V0s),V1f)),V2b)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__ABS__BOUND,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2b] :
( mem(V2b,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(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(V1f,V3x))),V2b)) ) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Eiterate_2ESum(A_27a),V0s),V1f))),ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V0s))),V2b))) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__BOUND__LT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2b] :
( mem(V2b,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(ap(c_2Ereal_2Ereal__lte,ap(V1f,V3x)),V2b)) ) )
& ? [V4x] :
( mem(V4x,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V0s))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(V1f,V4x)),V2b)) ) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Eiterate_2ESum(A_27a),V0s),V1f)),ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V0s))),V2b))) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__BOUND__LT__ALL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2b] :
( mem(V2b,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& V0s != c_2Epred__set_2EEMPTY(A_27a)
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s))
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(V1f,V3x)),V2b)) ) ) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Eiterate_2ESum(A_27a),V0s),V1f)),ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V0s))),V2b))) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__BOUND__LT__GEN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2b] :
( mem(V2b,ty_2Erealax_2Ereal)
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& V0s != c_2Epred__set_2EEMPTY(A_27a)
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s))
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(V1f,V3x)),ap(ap(c_2Ereal_2E_2F,V2b),ap(c_2Ereal_2Ereal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V0s))))) ) ) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Eiterate_2ESum(A_27a),V0s),V1f)),V2b)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__UNION__EQ,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))
=> ! [V3u] :
( mem(V3u,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V3u))
& ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t) = c_2Epred__set_2EEMPTY(A_27a)
& ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t) = V3u )
=> ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f)),ap(ap(c_2Eiterate_2ESum(A_27a),V2t),V0f)) = ap(ap(c_2Eiterate_2ESum(A_27a),V3u),V0f) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__EQ__SUPERSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0g] :
( mem(V0g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t] :
( mem(V3t,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V3t))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V3t),V2s))
& ! [V4x] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V3t))
=> ap(V1f,V4x) = ap(V0g,V4x) ) )
& ! [V5x] :
( mem(V5x,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V2s))
& ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V3t)) )
=> ap(V1f,V5x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) )
=> ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1f) = ap(ap(c_2Eiterate_2ESum(A_27a),V3t),V0g) ) ) ) ) ) ) ).
fof(lameq_f3278,axiom,
! [A_27a,V2f] :
( mem(V2f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V4x] : ap(f3278(A_27a,V2f,V0P),V4x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(V0P,V4x)),ap(V2f,V4x)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__RESTRICT__SET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2f] :
( mem(V2f,arr(A_27a,ty_2Erealax_2Ereal))
=> ap(ap(c_2Eiterate_2ESum(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3228(A_27a,V0P,V1s))),V2f) = ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3278(A_27a,V2f,V0P)) ) ) ) ) ).
fof(lameq_f3279,axiom,
! [A_27b,A_27a,V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V3t] :
( mem(V3t,arr(A_27b,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,arr(A_27b,ty_2Erealax_2Ereal)))
=> ! [V4x] : ap(f3279(A_27b,A_27a,V0R,V3t,V1f),V4x) = ap(ap(c_2Eiterate_2ESum(A_27b),ap(c_2Epred__set_2EGSPEC(A_27b,A_27b),f3230(A_27a,A_27b,V3t,V4x,V0R))),f3231(A_27a,A_27b,V4x,V1f)) ) ) ) ).
fof(lameq_f3280,axiom,
! [A_27a,A_27b,V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,arr(A_27b,ty_2Erealax_2Ereal)))
=> ! [V7y] : ap(f3280(A_27a,A_27b,V0R,V2s,V1f),V7y) = ap(ap(c_2Eiterate_2ESum(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3233(A_27b,A_27a,V2s,V0R,V7y))),f3234(A_27b,A_27a,V1f,V7y)) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__SUM__RESTRICT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V1f] :
( mem(V1f,arr(A_27a,arr(A_27b,ty_2Erealax_2Ereal)))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t] :
( mem(V3t,arr(A_27b,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V3t)) )
=> ap(ap(c_2Eiterate_2ESum(A_27a),V2s),f3279(A_27b,A_27a,V0R,V3t,V1f)) = ap(ap(c_2Eiterate_2ESum(A_27b),V3t),f3280(A_27a,A_27b,V0R,V2s,V1f)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ECARD__EQ__SUM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ap(c_2Ereal_2Ereal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V0s)) = ap(ap(c_2Eiterate_2ESum(A_27a),V0s),k(A_27a,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ) ) ) ).
fof(lameq_f3281,axiom,
! [A_27b,A_27a,V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ! [V6i] : ap(f3281(A_27b,A_27a,V0R,V2t),V6i) = ap(c_2Ereal_2Ereal__of__num,ap(c_2Epred__set_2ECARD(A_27b),ap(c_2Epred__set_2EGSPEC(A_27b,A_27b),f3237(A_27a,A_27b,V2t,V6i,V0R)))) ) ) ).
fof(lameq_f3282,axiom,
! [A_27b,V3k] :
( mem(V3k,arr(A_27b,ty_2Enum_2Enum))
=> ! [V8i] : ap(f3282(A_27b,V3k),V8i) = ap(c_2Ereal_2Ereal__of__num,ap(V3k,V8i)) ) ).
fof(conj_thm_2Eiterate_2ESUM__MULTICOUNT__GEN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ! [V3k] :
( mem(V3k,arr(A_27b,ty_2Enum_2Enum))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V2t))
& ! [V4j] :
( mem(V4j,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4j),V2t))
=> ap(c_2Epred__set_2ECARD(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3236(A_27b,A_27a,V1s,V0R,V4j))) = ap(V3k,V4j) ) ) )
=> ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3281(A_27b,A_27a,V0R,V2t)) = ap(ap(c_2Eiterate_2ESum(A_27b),V2t),f3282(A_27b,V3k)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__MULTICOUNT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ! [V3k] :
( mem(V3k,ty_2Enum_2Enum)
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V2t))
& ! [V4j] :
( mem(V4j,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4j),V2t))
=> ap(c_2Epred__set_2ECARD(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3236(A_27b,A_27a,V1s,V0R,V4j))) = V3k ) ) )
=> ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3281(A_27b,A_27a,V0R,V2t)) = ap(c_2Ereal_2Ereal__of__num,ap(ap(c_2Earithmetic_2E_2A,V3k),ap(c_2Epred__set_2ECARD(A_27b),V2t))) ) ) ) ) ) ) ) ).
fof(lameq_f3283,axiom,
! [A_27a,A_27b,V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V3y] : ap(f3283(A_27a,A_27b,V0f,V2s,V1g),V3y) = ap(ap(c_2Eiterate_2ESum(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3240(A_27b,A_27a,V2s,V0f,V3y))),V1g) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__IMAGE__GEN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
=> ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1g) = ap(ap(c_2Eiterate_2ESum(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V2s)),f3283(A_27a,A_27b,V0f,V2s,V1g)) ) ) ) ) ) ) ).
fof(lameq_f3284,axiom,
! [A_27a,A_27b,V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V4y] : ap(f3284(A_27a,A_27b,V0f,V2s,V1g),V4y) = ap(ap(c_2Eiterate_2ESum(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3242(A_27b,A_27a,V2s,V0f,V4y))),V1g) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__GROUP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t] :
( mem(V3t,arr(A_27b,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V2s)),V3t)) )
=> ap(ap(c_2Eiterate_2ESum(A_27b),V3t),f3284(A_27a,A_27b,V0f,V2s,V1g)) = ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1g) ) ) ) ) ) ) ) ).
fof(lameq_f3285,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2x] : ap(f3285(A_27a,V0f),V2x) = ap(c_2Ereal_2Ereal__of__num,ap(V0f,V2x)) ) ).
fof(conj_thm_2Eiterate_2EREAL__OF__NUM__SUM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
=> ap(c_2Ereal_2Ereal__of__num,ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f)) = ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3285(A_27a,V0f)) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0u] :
( mem(V0u,arr(A_27a,bool))
=> ! [V1v] :
( mem(V1v,arr(A_27a,bool))
=> ! [V2f] :
( mem(V2f,arr(A_27a,ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0u))
& p(ap(c_2Epred__set_2EFINITE(A_27a),V1v))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0u),V1v)))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V2f,V3x)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) )
& ! [V4x] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1v),V0u)))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V2f,V4x))) ) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eiterate_2ESum(A_27a),V0u),V2f)),ap(ap(c_2Eiterate_2ESum(A_27a),V1v),V2f))) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__SUBSET__SIMPLE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0u] :
( mem(V0u,arr(A_27a,bool))
=> ! [V1v] :
( mem(V1v,arr(A_27a,bool))
=> ! [V2f] :
( mem(V2f,arr(A_27a,ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1v))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0u),V1v))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1v),V0u)))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V2f,V3x))) ) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eiterate_2ESum(A_27a),V0u),V2f)),ap(ap(c_2Eiterate_2ESum(A_27a),V1v),V2f))) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__IMAGE__NONZERO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0d] :
( mem(V0d,arr(A_27b,ty_2Erealax_2Ereal))
=> ! [V1i] :
( mem(V1i,arr(A_27a,A_27b))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4y] :
( mem(V4y,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V4y),V2s))
& V3x != V4y
& ap(V1i,V3x) = ap(V1i,V4y) )
=> ap(V0d,ap(V1i,V3x)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) )
=> ap(ap(c_2Eiterate_2ESum(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1i),V2s)),V0d) = ap(ap(c_2Eiterate_2ESum(A_27a),V2s),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Erealax_2Ereal,A_27b),V0d),V1i)) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__BIJECTION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1p] :
( mem(V1p,arr(A_27a,A_27a))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s))
=> p(ap(ap(c_2Ebool_2EIN(A_27a),ap(V1p,V3x)),V2s)) ) )
& ! [V4y] :
( mem(V4y,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4y),V2s))
=> p(ap(c_2Ebool_2E_3F_21(A_27a),f3244(A_27a,V4y,V1p,V2s))) ) ) )
=> ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f) = ap(ap(c_2Eiterate_2ESum(A_27a),V2s),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Erealax_2Ereal,A_27a),V0f),V1p)) ) ) ) ) ) ).
fof(lameq_f3286,axiom,
! [A_27b,A_27a,V1t] :
( mem(V1t,arr(A_27a,arr(A_27b,bool)))
=> ! [V2x] :
( mem(V2x,arr(A_27a,arr(A_27b,ty_2Erealax_2Ereal)))
=> ! [V4i] : ap(f3286(A_27b,A_27a,V1t,V2x),V4i) = ap(ap(c_2Eiterate_2ESum(A_27b),ap(V1t,V4i)),ap(V2x,V4i)) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__SUM__PRODUCT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,arr(A_27b,bool)))
=> ! [V2x] :
( mem(V2x,arr(A_27a,arr(A_27b,ty_2Erealax_2Ereal)))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& ! [V3i] :
( mem(V3i,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3i),V0s))
=> p(ap(c_2Epred__set_2EFINITE(A_27b),ap(V1t,V3i))) ) ) )
=> ap(ap(c_2Eiterate_2ESum(A_27a),V0s),f3286(A_27b,A_27a,V1t,V2x)) = ap(ap(c_2Eiterate_2ESum(ty_2Epair_2Eprod(A_27a,A_27b)),ap(c_2Epred__set_2EGSPEC(ty_2Epair_2Eprod(A_27a,A_27b),ty_2Epair_2Eprod(A_27a,A_27b)),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,ty_2Epair_2Eprod(ty_2Epair_2Eprod(A_27a,A_27b),bool)),f3247(A_27b,A_27a,V0s,V1t)))),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,ty_2Erealax_2Ereal),f3249(A_27b,A_27a,V2x))) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__EQ__GENERAL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ! [V2f] :
( mem(V2f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V3g] :
( mem(V3g,arr(A_27b,ty_2Erealax_2Ereal))
=> ! [V4h] :
( mem(V4h,arr(A_27a,A_27b))
=> ( ( ! [V5y] :
( mem(V5y,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V5y),V1t))
=> p(ap(c_2Ebool_2E_3F_21(A_27a),f3250(A_27b,A_27a,V5y,V4h,V0s))) ) )
& ! [V7x] :
( mem(V7x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V7x),V0s))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V4h,V7x)),V1t))
& ap(V3g,ap(V4h,V7x)) = ap(V2f,V7x) ) ) ) )
=> ap(ap(c_2Eiterate_2ESum(A_27a),V0s),V2f) = ap(ap(c_2Eiterate_2ESum(A_27b),V1t),V3g) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__EQ__GENERAL__INVERSES,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ! [V2f] :
( mem(V2f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V3g] :
( mem(V3g,arr(A_27b,ty_2Erealax_2Ereal))
=> ! [V4h] :
( mem(V4h,arr(A_27a,A_27b))
=> ! [V5k] :
( mem(V5k,arr(A_27b,A_27a))
=> ( ( ! [V6y] :
( mem(V6y,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V6y),V1t))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),ap(V5k,V6y)),V0s))
& ap(V4h,ap(V5k,V6y)) = V6y ) ) )
& ! [V7x] :
( mem(V7x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V7x),V0s))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),ap(V4h,V7x)),V1t))
& ap(V5k,ap(V4h,V7x)) = V7x
& ap(V3g,ap(V4h,V7x)) = ap(V2f,V7x) ) ) ) )
=> ap(ap(c_2Eiterate_2ESum(A_27a),V0s),V2f) = ap(ap(c_2Eiterate_2ESum(A_27b),V1t),V3g) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__INJECTION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1p] :
( mem(V1p,arr(A_27a,A_27a))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s))
=> p(ap(ap(c_2Ebool_2EIN(A_27a),ap(V1p,V3x)),V2s)) ) )
& ! [V4x] :
( mem(V4x,A_27a)
=> ! [V5y] :
( mem(V5y,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V2s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V5y),V2s))
& ap(V1p,V4x) = ap(V1p,V5y) )
=> V4x = V5y ) ) ) )
=> ap(ap(c_2Eiterate_2ESum(A_27a),V2s),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Erealax_2Ereal,A_27a),V0f),V1p)) = ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__UNION__NONZERO,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))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t)))
=> ap(V0f,V3x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) )
=> ap(ap(c_2Eiterate_2ESum(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)),V0f) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f)),ap(ap(c_2Eiterate_2ESum(A_27a),V2t),V0f)) ) ) ) ) ) ).
fof(lameq_f3287,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V6t] : ap(f3287(A_27a,V0f),V6t) = ap(ap(c_2Eiterate_2ESum(A_27a),V6t),V0f) ) ).
fof(conj_thm_2Eiterate_2ESUM__BIGUNION__NONZERO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s] :
( mem(V1s,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(arr(A_27a,bool)),V1s))
& ! [V2t] :
( mem(V2t,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V2t),V1s))
=> p(ap(c_2Epred__set_2EFINITE(A_27a),V2t)) ) )
& ! [V3t1] :
( mem(V3t1,arr(A_27a,bool))
=> ! [V4t2] :
( mem(V4t2,arr(A_27a,bool))
=> ! [V5x] :
( mem(V5x,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V3t1),V1s))
& p(ap(ap(c_2Ebool_2EIN(arr(A_27a,bool)),V4t2),V1s))
& V3t1 != V4t2
& p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V3t1))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V4t2)) )
=> ap(V0f,V5x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) )
=> ap(ap(c_2Eiterate_2ESum(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),V1s)),V0f) = ap(ap(c_2Eiterate_2ESum(arr(A_27a,bool)),V1s),f3287(A_27a,V0f)) ) ) ) ) ).
fof(lameq_f3288,axiom,
! [A_27a,V2f] :
( mem(V2f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1P] :
( mem(V1P,arr(A_27a,bool))
=> ! [V3g] :
( mem(V3g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V4x] : ap(f3288(A_27a,V2f,V1P,V3g),V4x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(V1P,V4x)),ap(V2f,V4x)),ap(V3g,V4x)) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__CASES,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))
=> ! [V3g] :
( mem(V3g,arr(A_27a,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ap(ap(c_2Eiterate_2ESum(A_27a),V0s),f3288(A_27a,V2f,V1P,V3g)) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3253(A_27a,V1P,V0s))),V2f)),ap(ap(c_2Eiterate_2ESum(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3254(A_27a,V1P,V0s))),V3g)) ) ) ) ) ) ) ).
fof(lameq_f3289,axiom,
! [A_27a,V0y] :
( mem(V0y,ty_2Erealax_2Ereal)
=> ! [V3a] :
( mem(V3a,A_27a)
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V4x] : ap(f3289(A_27a,V0y,V3a,V1f),V4x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Emin_2E_3D(A_27a),V4x),V3a)),V0y),ap(V1f,V4x)) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__CASES__1,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0y] :
( mem(V0y,ty_2Erealax_2Ereal)
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3a] :
( mem(V3a,A_27a)
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V3a),V2s)) )
=> ap(ap(c_2Eiterate_2ESum(A_27a),V2s),f3289(A_27a,V0y,V3a,V1f)) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1f)),ap(ap(c_2Ereal_2Ereal__sub,V0y),ap(V1f,V3a))) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__LE__INCLUDED,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(A_27b,ty_2Erealax_2Ereal))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t] :
( mem(V3t,arr(A_27b,bool))
=> ! [V4i] :
( mem(V4i,arr(A_27b,A_27a))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V3t))
& ! [V5y] :
( mem(V5y,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V5y),V3t))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V1g,V5y))) ) )
& ! [V6x] :
( mem(V6x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V6x),V2s))
=> ? [V7y] :
( mem(V7y,A_27b)
& p(ap(ap(c_2Ebool_2EIN(A_27b),V7y),V3t))
& ap(V4i,V7y) = V6x
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V6x)),ap(V1g,V7y))) ) ) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f)),ap(ap(c_2Eiterate_2ESum(A_27b),V3t),V1g))) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__IMAGE__LE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27b,ty_2Erealax_2Ereal))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V1g,ap(V0f,V3x)))) ) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eiterate_2ESum(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V2s)),V1g)),ap(ap(c_2Eiterate_2ESum(A_27a),V2s),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Erealax_2Ereal,A_27b),V1g),V0f)))) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__CLOSED,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(V0P,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)))
& ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ! [V4y] :
( mem(V4y,ty_2Erealax_2Ereal)
=> ( ( p(ap(V0P,V3x))
& p(ap(V0P,V4y)) )
=> p(ap(V0P,ap(ap(c_2Erealax_2Ereal__add,V3x),V4y))) ) ) )
& ! [V5a] :
( mem(V5a,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5a),V2s))
=> p(ap(V0P,ap(V1f,V5a))) ) ) )
=> p(ap(V0P,ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1f))) ) ) ) ) ) ).
fof(lameq_f3290,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4i] : ap(f3290(V0f,V1g),V4i) = ap(ap(c_2Erealax_2Ereal__add,ap(V0f,V4i)),ap(V1g,V4i)) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__ADD__NUMSEG,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),f3290(V0f,V1g)) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),V0f)),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),V1g)) ) ) ) ) ).
fof(lameq_f3291,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4i] : ap(f3291(V0f,V1g),V4i) = ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,V4i)),ap(V1g,V4i)) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__SUB__NUMSEG,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),f3291(V0f,V1g)) = ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),V0f)),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),V1g)) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__LE__NUMSEG,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ( ! [V4i] :
( mem(V4i,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V2m),V4i))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,V4i),V3n)) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V4i)),ap(V1g,V4i))) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),V0f)),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),V1g))) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__EQ__NUMSEG,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ( ! [V4i] :
( mem(V4i,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V2m),V4i))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,V4i),V3n)) )
=> ap(V0f,V4i) = ap(V1g,V4i) ) )
=> ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),V0f) = ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),V1g) ) ) ) ) ) ).
fof(lameq_f3292,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3i] : ap(f3292(V0f),V3i) = ap(c_2Ereal_2Eabs,ap(V0f,V3i)) ) ).
fof(conj_thm_2Eiterate_2ESUM__ABS__NUMSEG,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),V0f))),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),f3292(V0f)))) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__CONST__NUMSEG,axiom,
! [V0c] :
( mem(V0c,ty_2Erealax_2Ereal)
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),k(ty_2Enum_2Enum,V0c)) = ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,ap(ap(c_2Earithmetic_2E_2D,ap(ap(c_2Earithmetic_2E_2B,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V1m))),V0c) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__EQ__0__NUMSEG,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( ! [V3i] :
( mem(V3i,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V3i))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,V3i),V2n)) )
=> ap(V0f,V3i) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) )
=> ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),V0f) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__TRIV__NUMSEG,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Eprim__rec_2E_3C,V2n),V1m))
=> ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),V0f) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__POS__LE__NUMSEG,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ! [V2f] :
( mem(V2f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ! [V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),V3p))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,V3p),V1n)) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V2f,V3p))) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V0m),V1n)),V2f))) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__POS__EQ__0__NUMSEG,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( ( ! [V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V3p))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,V3p),V2n)) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(V0f,V3p))) ) )
& ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),V0f) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) )
=> ! [V4p] :
( mem(V4p,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V4p))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,V4p),V2n)) )
=> ap(V0f,V4p) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__SING__NUMSEG,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1n),V1n)),V0f) = ap(V0f,V1n) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__CLAUSES__NUMSEG,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),c_2Enum_2E0)),V0f) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),V1m),c_2Enum_2E0)),ap(V0f,c_2Enum_2E0)),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)) )
& ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),ap(c_2Enum_2ESUC,V3n))),V0f) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Earithmetic_2E_3C_3D,V2m),ap(c_2Enum_2ESUC,V3n))),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),V0f)),ap(V0f,ap(c_2Enum_2ESUC,V3n)))),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),V0f)) ) ) ) ) ).
fof(lameq_f3293,axiom,
! [V2c] :
( mem(V2c,ty_2Enum_2Enum)
=> ! [V3d] :
( mem(V3d,ty_2Enum_2Enum)
=> ! [V4f] :
( mem(V4f,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)))
=> ! [V5i] : ap(f3293(V2c,V3d,V4f),V5i) = ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2c),V3d)),ap(V4f,V5i)) ) ) ) ).
fof(lameq_f3294,axiom,
! [V0a] :
( mem(V0a,ty_2Enum_2Enum)
=> ! [V1b] :
( mem(V1b,ty_2Enum_2Enum)
=> ! [V4f] :
( mem(V4f,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)))
=> ! [V6j] : ap(f3294(V0a,V1b,V4f),V6j) = ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V0a),V1b)),f3257(V4f,V6j)) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__SWAP__NUMSEG,axiom,
! [V0a] :
( mem(V0a,ty_2Enum_2Enum)
=> ! [V1b] :
( mem(V1b,ty_2Enum_2Enum)
=> ! [V2c] :
( mem(V2c,ty_2Enum_2Enum)
=> ! [V3d] :
( mem(V3d,ty_2Enum_2Enum)
=> ! [V4f] :
( mem(V4f,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)))
=> ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V0a),V1b)),f3293(V2c,V3d,V4f)) = ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2c),V3d)),f3294(V0a,V1b,V4f)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__ADD__SPLIT,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ! [V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),ap(ap(c_2Earithmetic_2E_2B,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))
=> ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),ap(ap(c_2Earithmetic_2E_2B,V2n),V3p))),V0f) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),V0f)),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(ap(c_2Earithmetic_2E_2B,V2n),V3p))),V0f)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__OFFSET,axiom,
! [V0p] :
( mem(V0p,ty_2Enum_2Enum)
=> ! [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,V2m),V0p)),ap(ap(c_2Earithmetic_2E_2B,V3n),V0p))),V1f) = ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),f3259(V0p,V1f)) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__OFFSET__0,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V2n))
=> ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),V0f) = ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),ap(ap(c_2Earithmetic_2E_2D,V2n),V1m))),f3260(V1m,V0f)) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__CLAUSES__LEFT,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V2n))
=> ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),V0f) = ap(ap(c_2Erealax_2Ereal__add,ap(V0f,V1m)),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,V1m),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V2n)),V0f)) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__CLAUSES__RIGHT,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V2n))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V2n)) )
=> ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),V0f) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),ap(ap(c_2Earithmetic_2E_2D,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),V0f)),ap(V0f,V2n)) ) ) ) ) ).
fof(lameq_f3295,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3i] : ap(f3295(V0f),V3i) = ap(ap(c_2Erealax_2Ereal__add,ap(V0f,ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))),V3i))),ap(V0f,ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))),V3i)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ) ).
fof(conj_thm_2Eiterate_2ESUM__PAIR,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))),V1m)),ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))),V2n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),V0f) = ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),f3295(V0f)) ) ) ) ).
fof(lameq_f3296,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V3i] : ap(f3296(V0f),V3i) = ap(c_2Ereal_2Ereal__of__num,ap(V0f,V3i)) ) ).
fof(conj_thm_2Eiterate_2EREAL__OF__NUM__SUM__NUMSEG,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ap(c_2Ereal_2Ereal__of__num,ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),V0f)) = ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),f3296(V0f)) ) ) ) ).
fof(lameq_f3297,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4k] : ap(f3297(V0f,V1g),V4k) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0f,V4k)),ap(ap(c_2Ereal_2Ereal__sub,ap(V1g,ap(ap(c_2Earithmetic_2E_2B,V4k),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(V1g,V4k))) ) ) ).
fof(lameq_f3298,axiom,
! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V5k] : ap(f3298(V1g,V0f),V5k) = ap(ap(c_2Erealax_2Ereal__mul,ap(V1g,ap(ap(c_2Earithmetic_2E_2B,V5k),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,ap(ap(c_2Earithmetic_2E_2B,V5k),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(V0f,V5k))) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__PARTIAL__SUC,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),f3297(V0f,V1g)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Earithmetic_2E_3C_3D,V2m),V3n)),ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Erealax_2Ereal__mul,ap(V0f,ap(ap(c_2Earithmetic_2E_2B,V3n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(V1g,ap(ap(c_2Earithmetic_2E_2B,V3n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))),ap(ap(c_2Erealax_2Ereal__mul,ap(V0f,V2m)),ap(V1g,V2m)))),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),f3298(V1g,V0f)))),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)) ) ) ) ) ).
fof(lameq_f3299,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4k] : ap(f3299(V0f,V1g),V4k) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0f,V4k)),ap(ap(c_2Ereal_2Ereal__sub,ap(V1g,V4k)),ap(V1g,ap(ap(c_2Earithmetic_2E_2D,V4k),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))) ) ) ).
fof(lameq_f3300,axiom,
! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V5k] : ap(f3300(V1g,V0f),V5k) = ap(ap(c_2Erealax_2Ereal__mul,ap(V1g,V5k)),ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,ap(ap(c_2Earithmetic_2E_2B,V5k),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(V0f,V5k))) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__PARTIAL__PRE,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),f3299(V0f,V1g)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Earithmetic_2E_3C_3D,V2m),V3n)),ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Erealax_2Ereal__mul,ap(V0f,ap(ap(c_2Earithmetic_2E_2B,V3n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(V1g,V3n))),ap(ap(c_2Erealax_2Ereal__mul,ap(V0f,V2m)),ap(V1g,ap(ap(c_2Earithmetic_2E_2D,V2m),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))))),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2m),V3n)),f3300(V1g,V0f)))),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)) ) ) ) ) ).
fof(lameq_f3301,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3k] : ap(f3301(V0f),V3k) = ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,V3k)),ap(V0f,ap(ap(c_2Earithmetic_2E_2B,V3k),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ) ).
fof(conj_thm_2Eiterate_2ESUM__DIFFS,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),f3301(V0f)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V2n)),ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,V1m)),ap(V0f,ap(ap(c_2Earithmetic_2E_2B,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)) ) ) ) ).
fof(lameq_f3302,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3k] : ap(f3302(V0f),V3k) = ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,ap(ap(c_2Earithmetic_2E_2B,V3k),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(V0f,V3k)) ) ).
fof(conj_thm_2Eiterate_2ESUM__DIFFS__ALT,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),f3302(V0f)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V2n)),ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,ap(ap(c_2Earithmetic_2E_2B,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(V0f,V1m))),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__COMBINE__R,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ! [V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),ap(ap(c_2Earithmetic_2E_2B,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,V2n),V3p)) )
=> ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V2n)),V0f)),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V3p)),V0f)) = ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V3p)),V0f) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2ESUM__COMBINE__L,axiom,
! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ! [V3p] :
( mem(V3p,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V2n))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V2n))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,V2n),ap(ap(c_2Earithmetic_2E_2B,V3p),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) )
=> ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),ap(ap(c_2Earithmetic_2E_2D,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),V0f)),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V2n),V3p)),V0f)) = ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V1m),V3p)),V0f) ) ) ) ) ) ).
fof(lameq_f3303,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V4i] : ap(f3303(A_27a,V0f),V4i) = ap(V0f,V4i) ) ).
fof(lameq_f3304,axiom,
! [V5f] :
( mem(V5f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V10i] : ap(f3304(V5f),V10i) = ap(V5f,V10i) ) ).
fof(lameq_f3305,axiom,
! [A_27b,V13p] :
( mem(V13p,arr(A_27b,bool))
=> ! [V15y] : ap(f3305(A_27b,V13p),V15y) = ap(ap(c_2Epair_2E_2C(A_27b,bool),V15y),ap(V13p,V15y)) ) ).
fof(lameq_f3306,axiom,
! [A_27b,V11f] :
( mem(V11f,arr(A_27b,ty_2Erealax_2Ereal))
=> ! [V16i] : ap(f3306(A_27b,V11f),V16i) = ap(V11f,V16i) ) ).
fof(lameq_f3307,axiom,
! [A_27b,V13p] :
( mem(V13p,arr(A_27b,bool))
=> ! [V17y] : ap(f3307(A_27b,V13p),V17y) = ap(ap(c_2Epair_2E_2C(A_27b,bool),V17y),ap(V13p,V17y)) ) ).
fof(conj_thm_2Eiterate_2ETH,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s))
=> ap(V0f,V3x) = ap(V1g,V3x) ) )
=> ap(ap(c_2Eiterate_2ESum(A_27a),V2s),f3303(A_27a,V0f)) = ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1g) ) ) ) )
& ! [V5f] :
( mem(V5f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V6g] :
( mem(V6g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V7a] :
( mem(V7a,ty_2Enum_2Enum)
=> ! [V8b] :
( mem(V8b,ty_2Enum_2Enum)
=> ( ! [V9i] :
( mem(V9i,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V7a),V9i))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,V9i),V8b)) )
=> ap(V5f,V9i) = ap(V6g,V9i) ) )
=> ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V7a),V8b)),f3304(V5f)) = ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,V7a),V8b)),V6g) ) ) ) ) )
& ! [V11f] :
( mem(V11f,arr(A_27b,ty_2Erealax_2Ereal))
=> ! [V12g] :
( mem(V12g,arr(A_27b,ty_2Erealax_2Ereal))
=> ! [V13p] :
( mem(V13p,arr(A_27b,bool))
=> ( ! [V14x] :
( mem(V14x,A_27b)
=> ( p(ap(V13p,V14x))
=> ap(V11f,V14x) = ap(V12g,V14x) ) )
=> ap(ap(c_2Eiterate_2ESum(A_27b),ap(c_2Epred__set_2EGSPEC(A_27b,A_27b),f3305(A_27b,V13p))),f3306(A_27b,V11f)) = ap(ap(c_2Eiterate_2ESum(A_27b),ap(c_2Epred__set_2EGSPEC(A_27b,A_27b),f3307(A_27b,V13p))),V12g) ) ) ) ) ) ) ) ).
fof(lameq_f3308,axiom,
! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V3i] : ap(f3308(V1y,V2n,V0x),V3i) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2Epow,V0x),V3i)),ap(ap(c_2Ereal_2Epow,V1y),ap(ap(c_2Earithmetic_2E_2D,ap(ap(c_2Earithmetic_2E_2D,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V3i))) ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__SUB__POW,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),V2n))
=> ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Ereal_2Epow,V0x),V2n)),ap(ap(c_2Ereal_2Epow,V1y),V2n)) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2Ereal__sub,V0x),V1y)),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),ap(ap(c_2Earithmetic_2E_2D,V2n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),f3308(V1y,V2n,V0x))) ) ) ) ) ).
fof(lameq_f3309,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V2i] : ap(f3309(V0x),V2i) = ap(ap(c_2Ereal_2Epow,V0x),V2i) ) ).
fof(conj_thm_2Eiterate_2EREAL__SUB__POW__R1,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),V1n))
=> ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Ereal_2Epow,V0x),V1n)),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2Ereal__sub,V0x),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),ap(ap(c_2Earithmetic_2E_2D,V1n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),f3309(V0x))) ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__SUB__POW__L1,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),V1n))
=> ap(ap(c_2Ereal_2Ereal__sub,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(ap(c_2Ereal_2Epow,V0x),V1n)) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2Ereal__sub,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V0x)),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),ap(ap(c_2Earithmetic_2E_2D,V1n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),f3309(V0x))) ) ) ) ).
fof(lameq_f3310,axiom,
! [A_27b,A_27a,V2f] :
( mem(V2f,arr(A_27a,A_27b))
=> ! [V1P] :
( mem(V1P,arr(A_27a,bool))
=> ! [V4x] : ap(f3310(A_27b,A_27a,V2f,V1P),V4x) = ap(ap(c_2Epair_2E_2C(A_27b,bool),ap(V2f,V4x)),ap(V1P,V4x)) ) ) ).
fof(lameq_f3311,axiom,
! [A_27c,A_27b,A_27d,V7f] :
( mem(V7f,arr(A_27c,arr(A_27d,A_27b)))
=> ! [V9x] :
( mem(V9x,A_27c)
=> ! [V6P] :
( mem(V6P,arr(A_27c,arr(A_27d,bool)))
=> ! [V10y] : ap(f3311(A_27c,A_27b,A_27d,V7f,V9x,V6P),V10y) = ap(ap(c_2Epair_2E_2C(A_27b,bool),ap(ap(V7f,V9x),V10y)),ap(ap(V6P,V9x),V10y)) ) ) ) ).
fof(lameq_f3312,axiom,
! [A_27d,A_27b,A_27c,V7f] :
( mem(V7f,arr(A_27c,arr(A_27d,A_27b)))
=> ! [V6P] :
( mem(V6P,arr(A_27c,arr(A_27d,bool)))
=> ! [V9x] : ap(f3312(A_27d,A_27b,A_27c,V7f,V6P),V9x) = f3311(A_27c,A_27b,A_27d,V7f,V9x,V6P) ) ) ).
fof(lameq_f3313,axiom,
! [A_27e,A_27f,A_27b,A_27g,V14f] :
( mem(V14f,arr(A_27e,arr(A_27f,arr(A_27g,A_27b))))
=> ! [V17x] :
( mem(V17x,A_27f)
=> ! [V13P] :
( mem(V13P,arr(A_27e,arr(A_27f,arr(A_27g,bool))))
=> ! [V16w] :
( mem(V16w,A_27e)
=> ! [V18y] : ap(f3313(A_27e,A_27f,A_27b,A_27g,V14f,V17x,V13P,V16w),V18y) = ap(ap(c_2Epair_2E_2C(A_27b,bool),ap(ap(ap(V14f,V16w),V17x),V18y)),ap(ap(ap(V13P,V16w),V17x),V18y)) ) ) ) ) ).
fof(lameq_f3314,axiom,
! [A_27g,A_27b,A_27e,A_27f,V14f] :
( mem(V14f,arr(A_27e,arr(A_27f,arr(A_27g,A_27b))))
=> ! [V13P] :
( mem(V13P,arr(A_27e,arr(A_27f,arr(A_27g,bool))))
=> ! [V16w] :
( mem(V16w,A_27e)
=> ! [V17x] : ap(f3314(A_27g,A_27b,A_27e,A_27f,V14f,V13P,V16w),V17x) = f3313(A_27e,A_27f,A_27b,A_27g,V14f,V17x,V13P,V16w) ) ) ) ).
fof(lameq_f3315,axiom,
! [A_27f,A_27b,A_27g,A_27e,V13P] :
( mem(V13P,arr(A_27e,arr(A_27f,arr(A_27g,bool))))
=> ! [V14f] :
( mem(V14f,arr(A_27e,arr(A_27f,arr(A_27g,A_27b))))
=> ! [V16w] : ap(f3315(A_27f,A_27b,A_27g,A_27e,V13P,V14f),V16w) = ap(c_2Epair_2EUNCURRY(A_27f,A_27g,ty_2Epair_2Eprod(A_27b,bool)),f3314(A_27g,A_27b,A_27e,A_27f,V14f,V13P,V16w)) ) ) ).
fof(conj_thm_2Eiterate_2EFORALL__IN__GSPEC,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [A_27e] :
( ne(A_27e)
=> ! [A_27f] :
( ne(A_27f)
=> ! [A_27g] :
( ne(A_27g)
=> ! [V0Q] :
( mem(V0Q,arr(A_27b,bool))
=> ( ! [V1P] :
( mem(V1P,arr(A_27a,bool))
=> ! [V2f] :
( mem(V2f,arr(A_27a,A_27b))
=> ( ! [V3z] :
( mem(V3z,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V3z),ap(c_2Epred__set_2EGSPEC(A_27b,A_27a),f3310(A_27b,A_27a,V2f,V1P))))
=> p(ap(V0Q,V3z)) ) )
<=> ! [V5x] :
( mem(V5x,A_27a)
=> ( p(ap(V1P,V5x))
=> p(ap(V0Q,ap(V2f,V5x))) ) ) ) ) )
& ! [V6P] :
( mem(V6P,arr(A_27c,arr(A_27d,bool)))
=> ! [V7f] :
( mem(V7f,arr(A_27c,arr(A_27d,A_27b)))
=> ( ! [V8z] :
( mem(V8z,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V8z),ap(c_2Epred__set_2EGSPEC(A_27b,ty_2Epair_2Eprod(A_27c,A_27d)),ap(c_2Epair_2EUNCURRY(A_27c,A_27d,ty_2Epair_2Eprod(A_27b,bool)),f3312(A_27d,A_27b,A_27c,V7f,V6P)))))
=> p(ap(V0Q,V8z)) ) )
<=> ! [V11x] :
( mem(V11x,A_27c)
=> ! [V12y] :
( mem(V12y,A_27d)
=> ( p(ap(ap(V6P,V11x),V12y))
=> p(ap(V0Q,ap(ap(V7f,V11x),V12y))) ) ) ) ) ) )
& ! [V13P] :
( mem(V13P,arr(A_27e,arr(A_27f,arr(A_27g,bool))))
=> ! [V14f] :
( mem(V14f,arr(A_27e,arr(A_27f,arr(A_27g,A_27b))))
=> ( ! [V15z] :
( mem(V15z,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V15z),ap(c_2Epred__set_2EGSPEC(A_27b,ty_2Epair_2Eprod(A_27e,ty_2Epair_2Eprod(A_27f,A_27g))),ap(c_2Epair_2EUNCURRY(A_27e,ty_2Epair_2Eprod(A_27f,A_27g),ty_2Epair_2Eprod(A_27b,bool)),f3315(A_27f,A_27b,A_27g,A_27e,V13P,V14f)))))
=> p(ap(V0Q,V15z)) ) )
<=> ! [V19w] :
( mem(V19w,A_27e)
=> ! [V20x] :
( mem(V20x,A_27f)
=> ! [V21y] :
( mem(V21y,A_27g)
=> ( p(ap(ap(ap(V13P,V19w),V20x),V21y))
=> p(ap(V0Q,ap(ap(ap(V14f,V19w),V20x),V21y))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(lameq_f3316,axiom,
! [V0a] :
( mem(V0a,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V4i] : ap(f3316(V0a,V1x),V4i) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0a,V4i)),ap(ap(c_2Ereal_2Epow,V1x),V4i)) ) ) ).
fof(lameq_f3317,axiom,
! [V0a] :
( mem(V0a,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2y] :
( mem(V2y,ty_2Erealax_2Ereal)
=> ! [V5i] : ap(f3317(V0a,V2y),V5i) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0a,V5i)),ap(ap(c_2Ereal_2Epow,V2y),V5i)) ) ) ).
fof(lameq_f3318,axiom,
! [V2y] :
( mem(V2y,ty_2Erealax_2Ereal)
=> ! [V6j] :
( mem(V6j,ty_2Enum_2Enum)
=> ! [V0a] :
( mem(V0a,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V7i] : ap(f3318(V2y,V6j,V0a),V7i) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0a,V7i)),ap(ap(c_2Ereal_2Epow,V2y),ap(ap(c_2Earithmetic_2E_2D,ap(ap(c_2Earithmetic_2E_2D,V7i),V6j)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ) ) ) ).
fof(lameq_f3319,axiom,
! [V0a] :
( mem(V0a,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2y] :
( mem(V2y,ty_2Erealax_2Ereal)
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V6j] : ap(f3319(V0a,V2y,V3n,V1x),V6j) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,V6j),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V3n)),f3318(V2y,V6j,V0a))),ap(ap(c_2Ereal_2Epow,V1x),V6j)) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__SUB__POLYFUN,axiom,
! [V0a] :
( mem(V0a,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V2y] :
( mem(V2y,ty_2Erealax_2Ereal)
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),V3n))
=> ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),V3n)),f3316(V0a,V1x))),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),V3n)),f3317(V0a,V2y))) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2Ereal__sub,V1x),V2y)),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),ap(ap(c_2Earithmetic_2E_2D,V3n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),f3319(V0a,V2y,V3n,V1x))) ) ) ) ) ) ).
fof(lameq_f3320,axiom,
! [V6j] :
( mem(V6j,ty_2Enum_2Enum)
=> ! [V0a] :
( mem(V0a,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2y] :
( mem(V2y,ty_2Erealax_2Ereal)
=> ! [V7k] : ap(f3320(V6j,V0a,V2y),V7k) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0a,ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Earithmetic_2E_2B,V6j),V7k)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),ap(ap(c_2Ereal_2Epow,V2y),V7k)) ) ) ) ).
fof(lameq_f3321,axiom,
! [V2y] :
( mem(V2y,ty_2Erealax_2Ereal)
=> ! [V0a] :
( mem(V0a,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V6j] : ap(f3321(V2y,V0a,V3n,V1x),V6j) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),ap(ap(c_2Earithmetic_2E_2D,ap(ap(c_2Earithmetic_2E_2D,V3n),V6j)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),f3320(V6j,V0a,V2y))),ap(ap(c_2Ereal_2Epow,V1x),V6j)) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__SUB__POLYFUN__ALT,axiom,
! [V0a] :
( mem(V0a,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ! [V2y] :
( mem(V2y,ty_2Erealax_2Ereal)
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),V3n))
=> ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),V3n)),f3316(V0a,V1x))),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),V3n)),f3317(V0a,V2y))) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2Ereal__sub,V1x),V2y)),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),ap(ap(c_2Earithmetic_2E_2D,V3n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))),f3321(V2y,V0a,V3n,V1x))) ) ) ) ) ) ).
fof(lameq_f3322,axiom,
! [V1c] :
( mem(V1c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ! [V4i] : ap(f3322(V1c,V3x),V4i) = ap(ap(c_2Erealax_2Ereal__mul,ap(V1c,V4i)),ap(ap(c_2Ereal_2Epow,V3x),V4i)) ) ) ).
fof(lameq_f3323,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1c] :
( mem(V1c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3x] : ap(f3323(V0n,V1c),V3x) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),V3x),ap(ap(c_2Emin_2E_3D(ty_2Erealax_2Ereal),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),V0n)),f3322(V1c,V3x))),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ).
fof(lameq_f3324,axiom,
! [V1c] :
( mem(V1c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V5x] :
( mem(V5x,ty_2Erealax_2Ereal)
=> ! [V6i] : ap(f3324(V1c,V5x),V6i) = ap(ap(c_2Erealax_2Ereal__mul,ap(V1c,V6i)),ap(ap(c_2Ereal_2Epow,V5x),V6i)) ) ) ).
fof(lameq_f3325,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1c] :
( mem(V1c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V5x] : ap(f3325(V0n,V1c),V5x) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),V5x),ap(ap(c_2Emin_2E_3D(ty_2Erealax_2Ereal),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),V0n)),f3324(V1c,V5x))),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__POLYFUN__ROOTBOUND,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1c] :
( mem(V1c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ~ ! [V2i] :
( mem(V2i,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V2i),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),V0n)))
=> ap(V1c,V2i) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) )
=> ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3323(V0n,V1c))))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Epred__set_2ECARD(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3325(V0n,V1c)))),V0n)) ) ) ) ) ).
fof(lameq_f3326,axiom,
! [V1c] :
( mem(V1c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ! [V3i] : ap(f3326(V1c,V2x),V3i) = ap(ap(c_2Erealax_2Ereal__mul,ap(V1c,V3i)),ap(ap(c_2Ereal_2Epow,V2x),V3i)) ) ) ).
fof(lameq_f3327,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1c] :
( mem(V1c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2x] : ap(f3327(V0n,V1c),V2x) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),V2x),ap(ap(c_2Emin_2E_3D(ty_2Erealax_2Ereal),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),V0n)),f3326(V1c,V2x))),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0))) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__POLYFUN__FINITE__ROOTS,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1c] :
( mem(V1c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3327(V0n,V1c))))
<=> ? [V4i] :
( mem(V4i,ty_2Enum_2Enum)
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V4i),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),V0n)))
& ap(V1c,V4i) != ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__POLYFUN__EQ__0,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1c] :
( mem(V1c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ! [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
=> ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),V0n)),f3326(V1c,V2x)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) )
<=> ! [V4i] :
( mem(V4i,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V4i),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),V0n)))
=> ap(V1c,V4i) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EREAL__POLYFUN__EQ__CONST,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1c] :
( mem(V1c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2k] :
( mem(V2k,ty_2Erealax_2Ereal)
=> ( ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),V0n)),f3322(V1c,V3x)) = V2k )
<=> ( ap(V1c,c_2Enum_2E0) = V2k
& ! [V5i] :
( mem(V5i,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V5i),ap(ap(c_2Eiterate_2E_2E_2E,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),V0n)))
=> ap(V1c,V5i) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ) ) ) ) ).
fof(lameq_f3328,axiom,
! [V2c] :
( mem(V2c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ! [V4i] : ap(f3328(V2c,V3x),V4i) = ap(ap(c_2Erealax_2Ereal__mul,ap(V2c,V4i)),ap(ap(c_2Ereal_2Epow,V3x),V4i)) ) ) ).
fof(ax_thm_2Eiterate_2Epolynomial__function,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Eiterate_2Epolynomial__function,V0p))
<=> ? [V1m] :
( mem(V1m,ty_2Enum_2Enum)
& ? [V2c] :
( mem(V2c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
& ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ap(V0p,V3x) = ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,c_2Enum_2E0),V1m)),f3328(V2c,V3x)) ) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__CONST,axiom,
! [V0c] :
( mem(V0c,ty_2Erealax_2Ereal)
=> p(ap(c_2Eiterate_2Epolynomial__function,k(ty_2Erealax_2Ereal,V0c))) ) ).
fof(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__ID,axiom,
p(ap(c_2Eiterate_2Epolynomial__function,i(ty_2Erealax_2Ereal))) ).
fof(lameq_f3329,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1q] :
( mem(V1q,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2x] : ap(f3329(V0p,V1q),V2x) = ap(ap(c_2Erealax_2Ereal__add,ap(V0p,V2x)),ap(V1q,V2x)) ) ) ).
fof(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__ADD,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1q] :
( mem(V1q,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Eiterate_2Epolynomial__function,V0p))
& p(ap(c_2Eiterate_2Epolynomial__function,V1q)) )
=> p(ap(c_2Eiterate_2Epolynomial__function,f3329(V0p,V1q))) ) ) ) ).
fof(lameq_f3330,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1c] :
( mem(V1c,ty_2Erealax_2Ereal)
=> ! [V2x] : ap(f3330(V0p,V1c),V2x) = ap(ap(c_2Erealax_2Ereal__mul,V1c),ap(V0p,V2x)) ) ) ).
fof(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__LMUL,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1c] :
( mem(V1c,ty_2Erealax_2Ereal)
=> ( p(ap(c_2Eiterate_2Epolynomial__function,V0p))
=> p(ap(c_2Eiterate_2Epolynomial__function,f3330(V0p,V1c))) ) ) ) ).
fof(lameq_f3331,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1c] :
( mem(V1c,ty_2Erealax_2Ereal)
=> ! [V2x] : ap(f3331(V0p,V1c),V2x) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0p,V2x)),V1c) ) ) ).
fof(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__RMUL,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1c] :
( mem(V1c,ty_2Erealax_2Ereal)
=> ( p(ap(c_2Eiterate_2Epolynomial__function,V0p))
=> p(ap(c_2Eiterate_2Epolynomial__function,f3331(V0p,V1c))) ) ) ) ).
fof(lameq_f3332,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1x] : ap(f3332(V0p),V1x) = ap(c_2Erealax_2Ereal__neg,ap(V0p,V1x)) ) ).
fof(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__NEG,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Eiterate_2Epolynomial__function,f3332(V0p)))
<=> p(ap(c_2Eiterate_2Epolynomial__function,V0p)) ) ) ).
fof(lameq_f3333,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1q] :
( mem(V1q,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2x] : ap(f3333(V0p,V1q),V2x) = ap(ap(c_2Ereal_2Ereal__sub,ap(V0p,V2x)),ap(V1q,V2x)) ) ) ).
fof(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__SUB,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1q] :
( mem(V1q,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Eiterate_2Epolynomial__function,V0p))
& p(ap(c_2Eiterate_2Epolynomial__function,V1q)) )
=> p(ap(c_2Eiterate_2Epolynomial__function,f3333(V0p,V1q))) ) ) ) ).
fof(lameq_f3334,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1q] :
( mem(V1q,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2x] : ap(f3334(V0p,V1q),V2x) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0p,V2x)),ap(V1q,V2x)) ) ) ).
fof(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__MUL,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1q] :
( mem(V1q,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Eiterate_2Epolynomial__function,V0p))
& p(ap(c_2Eiterate_2Epolynomial__function,V1q)) )
=> p(ap(c_2Eiterate_2Epolynomial__function,f3334(V0p,V1q))) ) ) ) ).
fof(lameq_f3335,axiom,
! [A_27a,V1p] :
( mem(V1p,arr(ty_2Erealax_2Ereal,arr(A_27a,ty_2Erealax_2Ereal)))
=> ! [V2i] :
( mem(V2i,A_27a)
=> ! [V3x] : ap(f3335(A_27a,V1p,V2i),V3x) = ap(ap(V1p,V3x),V2i) ) ) ).
fof(lameq_f3336,axiom,
! [A_27a,V1p] :
( mem(V1p,arr(ty_2Erealax_2Ereal,arr(A_27a,ty_2Erealax_2Ereal)))
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V4x] : ap(f3336(A_27a,V1p,V0s),V4x) = ap(ap(c_2Eiterate_2ESum(A_27a),V0s),ap(V1p,V4x)) ) ) ).
fof(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__SUM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1p] :
( mem(V1p,arr(ty_2Erealax_2Ereal,arr(A_27a,ty_2Erealax_2Ereal)))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& ! [V2i] :
( mem(V2i,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2i),V0s))
=> p(ap(c_2Eiterate_2Epolynomial__function,f3335(A_27a,V1p,V2i))) ) ) )
=> p(ap(c_2Eiterate_2Epolynomial__function,f3336(A_27a,V1p,V0s))) ) ) ) ) ).
fof(lameq_f3337,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ! [V2x] : ap(f3337(V0p,V1n),V2x) = ap(ap(c_2Ereal_2Epow,ap(V0p,V2x)),V1n) ) ) ).
fof(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__POW,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( p(ap(c_2Eiterate_2Epolynomial__function,V0p))
=> p(ap(c_2Eiterate_2Epolynomial__function,f3337(V0p,V1n))) ) ) ) ).
fof(lameq_f3338,axiom,
! [V4p] :
( mem(V4p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V5q] :
( mem(V5q,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V6x] : ap(f3338(V4p,V5q),V6x) = ap(ap(c_2Erealax_2Ereal__add,ap(V4p,V6x)),ap(V5q,V6x)) ) ) ).
fof(lameq_f3339,axiom,
! [V7p] :
( mem(V7p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V8q] :
( mem(V8q,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V9x] : ap(f3339(V7p,V8q),V9x) = ap(ap(c_2Erealax_2Ereal__mul,ap(V7p,V9x)),ap(V8q,V9x)) ) ) ).
fof(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__INDUCT,axiom,
! [V0P] :
( mem(V0P,arr(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),bool))
=> ( ( p(ap(V0P,i(ty_2Erealax_2Ereal)))
& ! [V2c] :
( mem(V2c,ty_2Erealax_2Ereal)
=> p(ap(V0P,k(ty_2Erealax_2Ereal,V2c))) )
& ! [V4p] :
( mem(V4p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V5q] :
( mem(V5q,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ( p(ap(V0P,V4p))
& p(ap(V0P,V5q)) )
=> p(ap(V0P,f3338(V4p,V5q))) ) ) )
& ! [V7p] :
( mem(V7p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V8q] :
( mem(V8q,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ( p(ap(V0P,V7p))
& p(ap(V0P,V8q)) )
=> p(ap(V0P,f3339(V7p,V8q))) ) ) ) )
=> ! [V10p] :
( mem(V10p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Eiterate_2Epolynomial__function,V10p))
=> p(ap(V0P,V10p)) ) ) ) ) ).
fof(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__o,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1q] :
( mem(V1q,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ( p(ap(c_2Eiterate_2Epolynomial__function,V0p))
& p(ap(c_2Eiterate_2Epolynomial__function,V1q)) )
=> p(ap(c_2Eiterate_2Epolynomial__function,ap(ap(c_2Ecombin_2Eo(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0p),V1q))) ) ) ) ).
fof(lameq_f3340,axiom,
! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ! [V0p] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2x] : ap(f3340(V1a,V0p),V2x) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),V2x),ap(ap(c_2Emin_2E_3D(ty_2Erealax_2Ereal),ap(V0p,V2x)),V1a)) ) ) ).
fof(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__FINITE__ROOTS,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a] :
( mem(V1a,ty_2Erealax_2Ereal)
=> ( p(ap(c_2Eiterate_2Epolynomial__function,V0p))
=> ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),ap(c_2Epred__set_2EGSPEC(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f3340(V1a,V0p))))
<=> ~ ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ap(V0p,V3x) = V1a ) ) ) ) ) ).
%------------------------------------------------------------------------------