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 : 700 ( 55 unt; 180 typ; 0 def)
% Number of atoms : 14554 ( 581 equ)
% Maximal formula atoms : 28 ( 20 avg)
% Number of connectives : 2225 ( 76 ~; 5 |; 416 &)
% ( 49 <=>;1679 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 9 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 11885 (11885 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 533 ( 177 >; 356 *; 0 +; 0 <<)
% Number of predicates : 77 ( 76 usr; 38 prp; 0-3 aty)
% Number of functors : 180 ( 180 usr; 3 con; 0-8 aty)
% Number of variables : 2320 (2262 !; 58 ?;2320 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_c_2Eiterate_2E_2E_2E,type,
c_2Eiterate_2E_2E_2E: $i ).
tff(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)))) ).
tff(stp_fo_c_2Eiterate_2E_2E_2E,type,
fo__c_2Eiterate_2E_2E_2E: ( tp__ty_2Enum_2Enum * tp__ty_2Enum_2Enum * tp__ty_2Enum_2Enum ) > tp__o ).
tff(stp_eq_fo_c_2Eiterate_2E_2E_2E,axiom,
! [X0: tp__ty_2Enum_2Enum,X1: tp__ty_2Enum_2Enum,X2: tp__ty_2Enum_2Enum] : ( inj__o(fo__c_2Eiterate_2E_2E_2E(X0,X1,X2)) = ap(ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(X0)),inj__ty_2Enum_2Enum(X1)),inj__ty_2Enum_2Enum(X2)) ) ).
tff(tp_c_2Eiterate_2EFINREC,type,
c_2Eiterate_2EFINREC: ( del * del ) > $i ).
tff(mem_c_2Eiterate_2EFINREC,axiom,
! [A_27a: del,A_27b: del] : 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)))))) ).
tff(tp_c_2Eiterate_2EITSET,type,
c_2Eiterate_2EITSET: ( del * del ) > $i ).
tff(mem_c_2Eiterate_2EITSET,axiom,
! [A_27a: del,A_27b: del] : 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)))) ).
tff(tp_c_2Eiterate_2ESum,type,
c_2Eiterate_2ESum: del > $i ).
tff(mem_c_2Eiterate_2ESum,axiom,
! [A_27a: del] : mem(c_2Eiterate_2ESum(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,ty_2Erealax_2Ereal),ty_2Erealax_2Ereal))) ).
tff(tp_c_2Eiterate_2Einf,type,
c_2Eiterate_2Einf: $i ).
tff(mem_c_2Eiterate_2Einf,axiom,
mem(c_2Eiterate_2Einf,arr(arr(ty_2Erealax_2Ereal,bool),ty_2Erealax_2Ereal)) ).
tff(tp_c_2Eiterate_2Eiterate,type,
c_2Eiterate_2Eiterate: ( del * del ) > $i ).
tff(mem_c_2Eiterate_2Eiterate,axiom,
! [A_27a: del,A_27b: del] : 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)))) ).
tff(tp_c_2Eiterate_2Emonoidal,type,
c_2Eiterate_2Emonoidal: del > $i ).
tff(mem_c_2Eiterate_2Emonoidal,axiom,
! [A_27a: del] : mem(c_2Eiterate_2Emonoidal(A_27a),arr(arr(A_27a,arr(A_27a,A_27a)),bool)) ).
tff(tp_c_2Eiterate_2Eneutral,type,
c_2Eiterate_2Eneutral: del > $i ).
tff(mem_c_2Eiterate_2Eneutral,axiom,
! [A_27a: del] : mem(c_2Eiterate_2Eneutral(A_27a),arr(arr(A_27a,arr(A_27a,A_27a)),A_27a)) ).
tff(tp_c_2Eiterate_2Ensum,type,
c_2Eiterate_2Ensum: del > $i ).
tff(mem_c_2Eiterate_2Ensum,axiom,
! [A_27a: del] : mem(c_2Eiterate_2Ensum(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,ty_2Enum_2Enum),ty_2Enum_2Enum))) ).
tff(tp_c_2Eiterate_2Epolynomial__function,type,
c_2Eiterate_2Epolynomial__function: $i ).
tff(mem_c_2Eiterate_2Epolynomial__function,axiom,
mem(c_2Eiterate_2Epolynomial__function,arr(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),bool)) ).
tff(tp_c_2Eiterate_2Esupport,type,
c_2Eiterate_2Esupport: ( del * del ) > $i ).
tff(mem_c_2Eiterate_2Esupport,axiom,
! [A_27a: del,A_27b: del] : 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))))) ).
tff(conj_thm_2Eiterate_2EREAL__LT__BETWEEN,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1b)))
<=> ? [V2x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V2x)))
& p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Erealax_2Ereal(V1b))) ) ) ).
tff(conj_thm_2Eiterate_2ESIMP__REAL__ARCH,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
? [V1n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V0x)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(V1n)))) ).
tff(conj_thm_2Eiterate_2ELOWER__BOUND__FINITE__SET__REAL,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
=> ? [V2a: tp__ty_2Erealax_2Ereal] :
! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2a)),ap(V0f,V3x))) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EUPPER__BOUND__FINITE__SET__REAL,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
=> ? [V2a: tp__ty_2Erealax_2Ereal] :
! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V3x)),inj__ty_2Erealax_2Ereal(V2a))) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__WLOG__LT,axiom,
! [V0P: $i] :
( mem(V0P,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,bool)))
=> ( ( ! [V1x: tp__ty_2Erealax_2Ereal] : p(ap(ap(V0P,inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Erealax_2Ereal(V1x)))
& ! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(V0P,inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Erealax_2Ereal(V3y)))
<=> p(ap(ap(V0P,inj__ty_2Erealax_2Ereal(V3y)),inj__ty_2Erealax_2Ereal(V2x))) )
& ! [V4x: tp__ty_2Erealax_2Ereal,V5y: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V4x)),inj__ty_2Erealax_2Ereal(V5y)))
=> p(ap(ap(V0P,inj__ty_2Erealax_2Ereal(V4x)),inj__ty_2Erealax_2Ereal(V5y))) ) )
=> ! [V6x: tp__ty_2Erealax_2Ereal,V7y: tp__ty_2Erealax_2Ereal] : p(ap(ap(V0P,inj__ty_2Erealax_2Ereal(V6x)),inj__ty_2Erealax_2Ereal(V7y))) ) ) ).
tff(ax_thm_2Eiterate_2EFINREC__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V1b: $i] :
( mem(V1b,A_27b)
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3a: $i] :
( mem(V3a,A_27b)
=> ( p(ap(ap(ap(ap(ap(c_2Eiterate_2EFINREC(A_27a,A_27b),V0f),V1b),V2s),V3a),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))
<=> ( ( V2s = c_2Epred__set_2EEMPTY(A_27a) )
& ( V3a = V1b ) ) ) ) ) ) )
& ! [V4f: $i] :
( mem(V4f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V5b: $i] :
( mem(V5b,A_27b)
=> ! [V6s: $i] :
( mem(V6s,arr(A_27a,bool))
=> ! [V7a: $i] :
( mem(V7a,A_27b)
=> ! [V8n: tp__ty_2Enum_2Enum] :
( p(ap(ap(ap(ap(ap(c_2Eiterate_2EFINREC(A_27a,A_27b),V4f),V5b),V6s),V7a),ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V8n))))
<=> ? [V9x: $i] :
( mem(V9x,A_27a)
& ? [V10c: $i] :
( 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),inj__ty_2Enum_2Enum(V8n)))
& ( V7a = ap(ap(V4f,V9x),V10c) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EFINREC__def__compute,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V1b: $i] :
( mem(V1b,A_27b)
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3a: $i] :
( mem(V3a,A_27b)
=> ( p(ap(ap(ap(ap(ap(c_2Eiterate_2EFINREC(A_27a,A_27b),V0f),V1b),V2s),V3a),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))
<=> ( ( V2s = c_2Epred__set_2EEMPTY(A_27a) )
& ( V3a = V1b ) ) ) ) ) ) )
& ! [V4f: $i] :
( mem(V4f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V5b: $i] :
( mem(V5b,A_27b)
=> ! [V6s: $i] :
( mem(V6s,arr(A_27a,bool))
=> ! [V7a: $i] :
( mem(V7a,A_27b)
=> ! [V8n: tp__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,inj__ty_2Enum_2Enum(V8n)))))
<=> ? [V9x: $i] :
( mem(V9x,A_27a)
& ? [V10c: $i] :
( 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,inj__ty_2Enum_2Enum(V8n)))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))))
& ( V7a = ap(ap(V4f,V9x),V10c) ) ) ) ) ) ) ) )
& ! [V11f: $i] :
( mem(V11f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V12b: $i] :
( mem(V12b,A_27b)
=> ! [V13s: $i] :
( mem(V13s,arr(A_27a,bool))
=> ! [V14a: $i] :
( mem(V14a,A_27b)
=> ! [V15n: tp__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,inj__ty_2Enum_2Enum(V15n)))))
<=> ? [V16x: $i] :
( mem(V16x,A_27a)
& ? [V17c: $i] :
( 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,inj__ty_2Enum_2Enum(V15n)))))
& ( V14a = ap(ap(V11f,V16x),V17c) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EFINREC__1__LEMMA,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V1b: $i] :
( mem(V1b,A_27b)
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3a: $i] :
( 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,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))
<=> ? [V4x: $i] :
( 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) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EFINREC__SUC__LEMMA,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V1b: $i] :
( mem(V1b,A_27b)
=> ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ! [V4s: $i] :
( 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: tp__ty_2Enum_2Enum,V6s: $i] :
( mem(V6s,arr(A_27a,bool))
=> ! [V7z: $i] :
( 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,inj__ty_2Enum_2Enum(V5n))))
=> ! [V8x: $i] :
( mem(V8x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V8x),V6s))
=> ? [V9w: $i] :
( 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),inj__ty_2Enum_2Enum(V5n)))
& ( V7z = ap(ap(V0f,V8x),V9w) ) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EFINREC__UNIQUE__LEMMA,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V1b: $i] :
( mem(V1b,A_27b)
=> ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ! [V4s: $i] :
( 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: tp__ty_2Enum_2Enum,V6n2: tp__ty_2Enum_2Enum,V7s: $i] :
( mem(V7s,arr(A_27a,bool))
=> ! [V8a1: $i] :
( mem(V8a1,A_27b)
=> ! [V9a2: $i] :
( mem(V9a2,A_27b)
=> ( ( p(ap(ap(ap(ap(ap(c_2Eiterate_2EFINREC(A_27a,A_27b),V0f),V1b),V7s),V8a1),inj__ty_2Enum_2Enum(V5n1)))
& p(ap(ap(ap(ap(ap(c_2Eiterate_2EFINREC(A_27a,A_27b),V0f),V1b),V7s),V9a2),inj__ty_2Enum_2Enum(V6n2))) )
=> ( ( V8a1 = V9a2 )
& ( V5n1 = V6n2 ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EFINREC__EXISTS__LEMMA,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V1b: $i] :
( mem(V1b,A_27b)
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
=> ? [V3a: $i] :
( mem(V3a,A_27b)
& ? [V4n: tp__ty_2Enum_2Enum] : p(ap(ap(ap(ap(ap(c_2Eiterate_2EFINREC(A_27a,A_27b),V0f),V1b),V2s),V3a),inj__ty_2Enum_2Enum(V4n))) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EFINREC__FUN__LEMMA,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1R: $i] :
( mem(V1R,arr(A_27a,arr(A_27b,arr(A_27c,bool))))
=> ( ( ! [V2s: $i] :
( mem(V2s,A_27a)
=> ( p(ap(V0P,V2s))
=> ? [V3a: $i] :
( mem(V3a,A_27b)
& ? [V4n: $i] :
( mem(V4n,A_27c)
& p(ap(ap(ap(V1R,V2s),V3a),V4n)) ) ) ) )
& ! [V5n1: $i] :
( mem(V5n1,A_27c)
=> ! [V6n2: $i] :
( mem(V6n2,A_27c)
=> ! [V7s: $i] :
( mem(V7s,A_27a)
=> ! [V8a1: $i] :
( mem(V8a1,A_27b)
=> ! [V9a2: $i] :
( 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: $i] :
( mem(V10f,arr(A_27a,A_27b))
& ! [V11s: $i] :
( mem(V11s,A_27a)
=> ! [V12a: $i] :
( mem(V12a,A_27b)
=> ( p(ap(V0P,V11s))
=> ( ? [V13n: $i] :
( mem(V13n,A_27c)
& p(ap(ap(ap(V1R,V11s),V12a),V13n)) )
<=> ( ap(V10f,V11s) = V12a ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EFINREC__FUN,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V1b: $i] :
( mem(V1b,A_27b)
=> ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ! [V4s: $i] :
( 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: $i] :
( mem(V5g,arr(arr(A_27a,bool),A_27b))
& ( ap(V5g,c_2Epred__set_2EEMPTY(A_27a)) = V1b )
& ! [V6s: $i] :
( mem(V6s,arr(A_27a,bool))
=> ! [V7x: $i] :
( 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))) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESET__RECURSION__LEMMA,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V1b: $i] :
( mem(V1b,A_27b)
=> ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ! [V4s: $i] :
( 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: $i] :
( mem(V5g,arr(arr(A_27a,bool),A_27b))
& ( ap(V5g,c_2Epred__set_2EEMPTY(A_27a)) = V1b )
& ! [V6x: $i] :
( mem(V6x,A_27a)
=> ! [V7s: $i] :
( 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))) ) ) ) ) ) ) ) ) ).
tff(lamtp_f3173,type,
f3173: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3173,axiom,
! [A_27a: del,A_27b: del,V4x: $i] :
( mem(V4x,A_27b)
=> ! [V3g: $i] :
( mem(V3g,arr(arr(A_27b,bool),A_27a))
=> ! [V0f: $i] :
( mem(V0f,arr(A_27b,arr(A_27a,A_27a)))
=> ! [V5s: $i] : ( 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))))) ) ) ) ) ).
tff(lamtp_f3174,type,
f3174: ( del * del * $i * $i ) > $i ).
tff(lameq_f3174,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27b,arr(A_27a,A_27a)))
=> ! [V3g: $i] :
( mem(V3g,arr(arr(A_27b,bool),A_27a))
=> ! [V4x: $i] : ( 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)) ) ) ) ).
tff(lamtp_f3175,type,
f3175: ( del * del * $i * $i ) > $i ).
tff(lameq_f3175,axiom,
! [A_27a: del,A_27b: del,V2b: $i] :
( mem(V2b,A_27a)
=> ! [V0f: $i] :
( mem(V0f,arr(A_27b,arr(A_27a,A_27a)))
=> ! [V3g: $i] : ( 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))) ) ) ) ).
tff(ax_thm_2Eiterate_2EITSET__def,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27b,arr(A_27a,A_27a)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27b,bool))
=> ! [V2b: $i] :
( 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) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EFINITE__RECURSION,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V1b: $i] :
( mem(V1b,A_27b)
=> ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ! [V4s: $i] :
( 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: $i] :
( mem(V5x,A_27a)
=> ! [V6s: $i] :
( 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))) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ECARD__UNION__EQ,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t: $i] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2u: $i] :
( 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 ) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2B,ap(c_2Epred__set_2ECARD(A_27a),V0s)),ap(c_2Epred__set_2ECARD(A_27a),V1t))) = surj__ty_2Enum_2Enum(ap(c_2Epred__set_2ECARD(A_27a),V2u)) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUBSET__RESTRICT,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1P: $i] :
( 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)) ) ) ).
tff(conj_thm_2Eiterate_2EFINITE__RESTRICT,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1P: $i] :
( 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)))) ) ) ) ).
tff(lamtp_f3176,type,
f3176: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f3176,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1x: tp__ty_2Erealax_2Ereal] : ( ap(f3176(V0a),inj__ty_2Erealax_2Ereal(V1x)) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),inj__ty_2Erealax_2Ereal(V1x)),ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1x))) ) ).
tff(lamtp_f3177,type,
f3177: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f3177,axiom,
! [V2a: tp__ty_2Erealax_2Ereal,V3x: tp__ty_2Erealax_2Ereal] : ( ap(f3177(V2a),inj__ty_2Erealax_2Ereal(V3x)) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),inj__ty_2Erealax_2Ereal(V3x)),ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3x))) ) ).
tff(lamtp_f3178,type,
f3178: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f3178,axiom,
! [V4b: tp__ty_2Erealax_2Ereal,V5x: tp__ty_2Erealax_2Ereal] : ( ap(f3178(V4b),inj__ty_2Erealax_2Ereal(V5x)) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),inj__ty_2Erealax_2Ereal(V5x)),ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V5x)),inj__ty_2Erealax_2Ereal(V4b))) ) ).
tff(lamtp_f3179,type,
f3179: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f3179,axiom,
! [V6b: tp__ty_2Erealax_2Ereal,V7x: tp__ty_2Erealax_2Ereal] : ( ap(f3179(V6b),inj__ty_2Erealax_2Ereal(V7x)) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),inj__ty_2Erealax_2Ereal(V7x)),ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V7x)),inj__ty_2Erealax_2Ereal(V6b))) ) ).
tff(lamtp_f3180,type,
f3180: ( tp__ty_2Erealax_2Ereal * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3180,axiom,
! [V9b: tp__ty_2Erealax_2Ereal,V8a: tp__ty_2Erealax_2Ereal,V10x: tp__ty_2Erealax_2Ereal] : ( ap(f3180(V9b,V8a),inj__ty_2Erealax_2Ereal(V10x)) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),inj__ty_2Erealax_2Ereal(V10x)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V8a)),inj__ty_2Erealax_2Ereal(V10x))),ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V10x)),inj__ty_2Erealax_2Ereal(V9b)))) ) ).
tff(lamtp_f3181,type,
f3181: ( tp__ty_2Erealax_2Ereal * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3181,axiom,
! [V12b: tp__ty_2Erealax_2Ereal,V11a: tp__ty_2Erealax_2Ereal,V13x: tp__ty_2Erealax_2Ereal] : ( ap(f3181(V12b,V11a),inj__ty_2Erealax_2Ereal(V13x)) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),inj__ty_2Erealax_2Ereal(V13x)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V11a)),inj__ty_2Erealax_2Ereal(V13x))),ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V13x)),inj__ty_2Erealax_2Ereal(V12b)))) ) ).
tff(lamtp_f3182,type,
f3182: ( tp__ty_2Erealax_2Ereal * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3182,axiom,
! [V15b: tp__ty_2Erealax_2Ereal,V14a: tp__ty_2Erealax_2Ereal,V16x: tp__ty_2Erealax_2Ereal] : ( ap(f3182(V15b,V14a),inj__ty_2Erealax_2Ereal(V16x)) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),inj__ty_2Erealax_2Ereal(V16x)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V14a)),inj__ty_2Erealax_2Ereal(V16x))),ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V16x)),inj__ty_2Erealax_2Ereal(V15b)))) ) ).
tff(lamtp_f3183,type,
f3183: ( tp__ty_2Erealax_2Ereal * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3183,axiom,
! [V18b: tp__ty_2Erealax_2Ereal,V17a: tp__ty_2Erealax_2Ereal,V19x: tp__ty_2Erealax_2Ereal] : ( ap(f3183(V18b,V17a),inj__ty_2Erealax_2Ereal(V19x)) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),inj__ty_2Erealax_2Ereal(V19x)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V17a)),inj__ty_2Erealax_2Ereal(V19x))),ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V19x)),inj__ty_2Erealax_2Ereal(V18b)))) ) ).
tff(conj_thm_2Eiterate_2EFINITE__REAL__INTERVAL,axiom,
( ! [V0a: tp__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: tp__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: tp__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: tp__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: tp__ty_2Erealax_2Ereal,V9b: tp__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,inj__ty_2Erealax_2Ereal(V9b)),inj__ty_2Erealax_2Ereal(V8a))) )
& ! [V11a: tp__ty_2Erealax_2Ereal,V12b: tp__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,inj__ty_2Erealax_2Ereal(V12b)),inj__ty_2Erealax_2Ereal(V11a))) )
& ! [V14a: tp__ty_2Erealax_2Ereal,V15b: tp__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,inj__ty_2Erealax_2Ereal(V15b)),inj__ty_2Erealax_2Ereal(V14a))) )
& ! [V17a: tp__ty_2Erealax_2Ereal,V18b: tp__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,inj__ty_2Erealax_2Ereal(V18b)),inj__ty_2Erealax_2Ereal(V17a))) ) ) ).
tff(conj_thm_2Eiterate_2Ereal__INFINITE,axiom,
~ p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))) ).
tff(conj_thm_2Eiterate_2ESET__PROVE__CASES,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(V0P,c_2Epred__set_2EEMPTY(A_27a)))
& ! [V1a: $i] :
( mem(V1a,A_27a)
=> ! [V2s: $i] :
( 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: $i] :
( mem(V3s,arr(A_27a,bool))
=> p(ap(V0P,V3s)) ) ) ) ).
tff(conj_thm_2Eiterate_2ECHOOSE__SUBSET__STRONG,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V0n)),ap(c_2Epred__set_2ECARD(A_27a),V1s))) )
=> ? [V2t: $i] :
( 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),inj__ty_2Enum_2Enum(V0n))) ) ) ) ).
tff(conj_thm_2Eiterate_2ECHOOSE__SUBSET,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1n)),ap(c_2Epred__set_2ECARD(A_27a),V0s)))
=> ? [V2t: $i] :
( 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),inj__ty_2Enum_2Enum(V1n))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__COMPLETE,axiom,
! [V0P: $i] :
( mem(V0P,arr(ty_2Erealax_2Ereal,bool))
=> ( ( ? [V1x: tp__ty_2Erealax_2Ereal] : p(ap(V0P,inj__ty_2Erealax_2Ereal(V1x)))
& ? [V2M: tp__ty_2Erealax_2Ereal] :
! [V3x: tp__ty_2Erealax_2Ereal] :
( p(ap(V0P,inj__ty_2Erealax_2Ereal(V3x)))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3x)),inj__ty_2Erealax_2Ereal(V2M))) ) )
=> ? [V4M: tp__ty_2Erealax_2Ereal] :
( ! [V5x: tp__ty_2Erealax_2Ereal] :
( p(ap(V0P,inj__ty_2Erealax_2Ereal(V5x)))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V5x)),inj__ty_2Erealax_2Ereal(V4M))) )
& ! [V6M_27: tp__ty_2Erealax_2Ereal] :
( ! [V7x: tp__ty_2Erealax_2Ereal] :
( p(ap(V0P,inj__ty_2Erealax_2Ereal(V7x)))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V7x)),inj__ty_2Erealax_2Ereal(V6M_27))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V4M)),inj__ty_2Erealax_2Ereal(V6M_27))) ) ) ) ) ).
tff(lamtp_f3184,type,
f3184: ( $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3184,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] : ( ap(f3184(V0s,V1a),inj__ty_2Erealax_2Ereal(V2x)) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s)),ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Erealax_2Ereal(V1a))) ) ) ).
tff(lamtp_f3185,type,
f3185: ( $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3185,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V3b: tp__ty_2Erealax_2Ereal,V4x: tp__ty_2Erealax_2Ereal] : ( ap(f3185(V0s,V3b),inj__ty_2Erealax_2Ereal(V4x)) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),V0s)),ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V4x)),inj__ty_2Erealax_2Ereal(V3b))) ) ) ).
tff(lamtp_f3186,type,
f3186: ( $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3186,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] : ( ap(f3186(V0s,V1a),inj__ty_2Erealax_2Ereal(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,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V3b))) ) ) ).
tff(lamtp_f3187,type,
f3187: $i > $i ).
tff(lameq_f3187,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a: tp__ty_2Erealax_2Ereal] : ( ap(f3187(V0s),inj__ty_2Erealax_2Ereal(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))) ) ) ).
tff(conj_thm_2Eiterate_2Esup__alt,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ( surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Esup,V0s)) = surj__ty_2Erealax_2Ereal(ap(c_2Emin_2E_40(ty_2Erealax_2Ereal),f3187(V0s))) ) ) ).
tff(conj_thm_2Eiterate_2ESUP__EQ,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1t: $i] :
( mem(V1t,arr(ty_2Erealax_2Ereal,bool))
=> ( ! [V2b: tp__ty_2Erealax_2Ereal] :
( ! [V3x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3x)),inj__ty_2Erealax_2Ereal(V2b))) )
<=> ! [V4x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),V1t))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V4x)),inj__ty_2Erealax_2Ereal(V2b))) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Esup,V0s)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Esup,V1t)) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUP,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ( ( ( V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) )
& ? [V1b: tp__ty_2Erealax_2Ereal] :
! [V2x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Erealax_2Ereal(V1b))) ) )
=> ( ! [V3x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3x)),ap(c_2Ereal_2Esup,V0s))) )
& ! [V4b: tp__ty_2Erealax_2Ereal] :
( ! [V5x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5x)),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V5x)),inj__ty_2Erealax_2Ereal(V4b))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Esup,V0s)),inj__ty_2Erealax_2Ereal(V4b))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUP__FINITE__LEMMA,axiom,
! [V0s: $i] :
( 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: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1b)),V0s))
& ! [V2x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Erealax_2Ereal(V1b))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUP__FINITE,axiom,
! [V0s: $i] :
( 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: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1x)),ap(c_2Ereal_2Esup,V0s))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__LE__SUP__FINITE,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a: tp__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,inj__ty_2Erealax_2Ereal(V1a)),ap(c_2Ereal_2Esup,V0s)))
<=> ? [V2x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2x))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__SUP__LE__FINITE,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a: tp__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)),inj__ty_2Erealax_2Ereal(V1a)))
<=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Erealax_2Ereal(V1a))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__LT__SUP__FINITE,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a: tp__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,inj__ty_2Erealax_2Ereal(V1a)),ap(c_2Ereal_2Esup,V0s)))
<=> ? [V2x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s))
& p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2x))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__SUP__LT__FINITE,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a: tp__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)),inj__ty_2Erealax_2Ereal(V1a)))
<=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s))
=> p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Erealax_2Ereal(V1a))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__SUP__LE__EQ,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1y: tp__ty_2Erealax_2Ereal] :
( ( ( V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) )
& ? [V2b: tp__ty_2Erealax_2Ereal] :
! [V3x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3x)),inj__ty_2Erealax_2Ereal(V2b))) ) )
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Esup,V0s)),inj__ty_2Erealax_2Ereal(V1y)))
<=> ! [V4x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V4x)),inj__ty_2Erealax_2Ereal(V1y))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__SUP__UNIQUE,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1b: tp__ty_2Erealax_2Ereal] :
( ( ! [V2x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Erealax_2Ereal(V1b))) )
& ! [V3b_27: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V3b_27)),inj__ty_2Erealax_2Ereal(V1b)))
=> ? [V4x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),V0s))
& p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V3b_27)),inj__ty_2Erealax_2Ereal(V4x))) ) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Esup,V0s)) = V1b ) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__SUP__LE__S,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1b: tp__ty_2Erealax_2Ereal] :
( ( ( V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) )
& ! [V2x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Erealax_2Ereal(V1b))) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Esup,V0s)),inj__ty_2Erealax_2Ereal(V1b))) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__SUP__LE__SUBSET,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1t: $i] :
( 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: tp__ty_2Erealax_2Ereal] :
! [V3x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),V1t))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3x)),inj__ty_2Erealax_2Ereal(V2b))) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Esup,V0s)),ap(c_2Ereal_2Esup,V1t))) ) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__LE__SUP,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3y)),V0s))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V3y)))
& ! [V4x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V4x)),inj__ty_2Erealax_2Ereal(V2b))) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),ap(c_2Ereal_2Esup,V0s))) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__SUP__BOUNDS,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( ( V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),V0s))
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V3x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3x)),inj__ty_2Erealax_2Ereal(V2b))) ) ) )
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),ap(c_2Ereal_2Esup,V0s)))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Esup,V0s)),inj__ty_2Erealax_2Ereal(V2b))) ) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__ABS__SUP__LE,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( ( V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) )
& ! [V2x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,inj__ty_2Erealax_2Ereal(V2x))),inj__ty_2Erealax_2Ereal(V1a))) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(c_2Ereal_2Esup,V0s))),inj__ty_2Erealax_2Ereal(V1a))) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__SUP__ASCLOSE,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2e: tp__ty_2Erealax_2Ereal] :
( ( ( V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V3x)),inj__ty_2Erealax_2Ereal(V1l)))),inj__ty_2Erealax_2Ereal(V2e))) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(c_2Ereal_2Esup,V0s)),inj__ty_2Erealax_2Ereal(V1l)))),inj__ty_2Erealax_2Ereal(V2e))) ) ) ).
tff(conj_thm_2Eiterate_2ESUP__SING,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] : ( surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Esup,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) = V0a ) ).
tff(conj_thm_2Eiterate_2ESUP__UNIQUE,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1b: tp__ty_2Erealax_2Ereal] :
( ! [V2c: tp__ty_2Erealax_2Ereal] :
( ! [V3x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3x)),inj__ty_2Erealax_2Ereal(V2c))) )
<=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1b)),inj__ty_2Erealax_2Ereal(V2c))) )
=> ( surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Esup,V0s)) = V1b ) ) ) ).
tff(conj_thm_2Eiterate_2ESUP__UNION,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1t: $i] :
( mem(V1t,arr(ty_2Erealax_2Ereal,bool))
=> ( ( ( V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) )
& ( V1t != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) )
& ? [V2b: tp__ty_2Erealax_2Ereal] :
! [V3x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3x)),inj__ty_2Erealax_2Ereal(V2b))) )
& ? [V4c: tp__ty_2Erealax_2Ereal] :
! [V5x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5x)),V1t))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V5x)),inj__ty_2Erealax_2Ereal(V4c))) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Esup,ap(ap(c_2Epred__set_2EUNION(ty_2Erealax_2Ereal),V0s),V1t))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Emax,ap(c_2Ereal_2Esup,V0s)),ap(c_2Ereal_2Esup,V1t))) ) ) ) ) ).
tff(lamtp_f3188,type,
f3188: ( $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3188,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] : ( ap(f3188(V0s,V1a),inj__ty_2Erealax_2Ereal(V2x)) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s)),ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2x))) ) ) ).
tff(lamtp_f3189,type,
f3189: ( $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3189,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V3b: tp__ty_2Erealax_2Ereal,V4x: tp__ty_2Erealax_2Ereal] : ( ap(f3189(V0s,V3b),inj__ty_2Erealax_2Ereal(V4x)) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),V0s)),ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3b)),inj__ty_2Erealax_2Ereal(V4x))) ) ) ).
tff(lamtp_f3190,type,
f3190: ( $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3190,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] : ( ap(f3190(V0s,V1a),inj__ty_2Erealax_2Ereal(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,inj__ty_2Erealax_2Ereal(V3b)),inj__ty_2Erealax_2Ereal(V1a))) ) ) ).
tff(lamtp_f3191,type,
f3191: $i > $i ).
tff(lameq_f3191,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a: tp__ty_2Erealax_2Ereal] : ( ap(f3191(V0s),inj__ty_2Erealax_2Ereal(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))) ) ) ).
tff(ax_thm_2Eiterate_2Einf,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ( surj__ty_2Erealax_2Ereal(ap(c_2Eiterate_2Einf,V0s)) = surj__ty_2Erealax_2Ereal(ap(c_2Emin_2E_40(ty_2Erealax_2Ereal),f3191(V0s))) ) ) ).
tff(conj_thm_2Eiterate_2EINF__EQ,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1t: $i] :
( mem(V1t,arr(ty_2Erealax_2Ereal,bool))
=> ( ! [V2a: tp__ty_2Erealax_2Ereal] :
( ! [V3x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3x))) )
<=> ! [V4x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),V1t))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V4x))) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(c_2Eiterate_2Einf,V0s)) = surj__ty_2Erealax_2Ereal(ap(c_2Eiterate_2Einf,V1t)) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EINF,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ( ( ( V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) )
& ? [V1b: tp__ty_2Erealax_2Ereal] :
! [V2x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1b)),inj__ty_2Erealax_2Ereal(V2x))) ) )
=> ( ! [V3x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Eiterate_2Einf,V0s)),inj__ty_2Erealax_2Ereal(V3x))) )
& ! [V4b: tp__ty_2Erealax_2Ereal] :
( ! [V5x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V5x)),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V4b)),inj__ty_2Erealax_2Ereal(V5x))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V4b)),ap(c_2Eiterate_2Einf,V0s))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EINF__FINITE__LEMMA,axiom,
! [V0s: $i] :
( 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: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1b)),V0s))
& ! [V2x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1b)),inj__ty_2Erealax_2Ereal(V2x))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EINF__FINITE,axiom,
! [V0s: $i] :
( 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: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1x)),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Eiterate_2Einf,V0s)),inj__ty_2Erealax_2Ereal(V1x))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__LE__INF__FINITE,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a: tp__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,inj__ty_2Erealax_2Ereal(V1a)),ap(c_2Eiterate_2Einf,V0s)))
<=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2x))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__INF__LE__FINITE,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a: tp__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)),inj__ty_2Erealax_2Ereal(V1a)))
<=> ? [V2x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Erealax_2Ereal(V1a))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__LT__INF__FINITE,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a: tp__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,inj__ty_2Erealax_2Ereal(V1a)),ap(c_2Eiterate_2Einf,V0s)))
<=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s))
=> p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2x))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__INF__LT__FINITE,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a: tp__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)),inj__ty_2Erealax_2Ereal(V1a)))
<=> ? [V2x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s))
& p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Erealax_2Ereal(V1a))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__INF__UNIQUE,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1b: tp__ty_2Erealax_2Ereal] :
( ( ! [V2x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1b)),inj__ty_2Erealax_2Ereal(V2x))) )
& ! [V3b_27: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V1b)),inj__ty_2Erealax_2Ereal(V3b_27)))
=> ? [V4x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4x)),V0s))
& p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V4x)),inj__ty_2Erealax_2Ereal(V3b_27))) ) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(c_2Eiterate_2Einf,V0s)) = V1b ) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__LE__INF,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1b: tp__ty_2Erealax_2Ereal] :
( ( ( V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) )
& ! [V2x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1b)),inj__ty_2Erealax_2Ereal(V2x))) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1b)),ap(c_2Eiterate_2Einf,V0s))) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__LE__INF__SUBSET,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1t: $i] :
( 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: tp__ty_2Erealax_2Ereal] :
! [V3x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2b)),inj__ty_2Erealax_2Ereal(V3x))) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Eiterate_2Einf,V0s)),ap(c_2Eiterate_2Einf,V1t))) ) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__INF__BOUNDS,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( ( V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),V0s))
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V3x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3x)),inj__ty_2Erealax_2Ereal(V2b))) ) ) )
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),ap(c_2Eiterate_2Einf,V0s)))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Eiterate_2Einf,V0s)),inj__ty_2Erealax_2Ereal(V2b))) ) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__ABS__INF__LE,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( ( V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) )
& ! [V2x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,inj__ty_2Erealax_2Ereal(V2x))),inj__ty_2Erealax_2Ereal(V1a))) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(c_2Eiterate_2Einf,V0s))),inj__ty_2Erealax_2Ereal(V1a))) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__INF__ASCLOSE,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2e: tp__ty_2Erealax_2Ereal] :
( ( ( V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V3x)),inj__ty_2Erealax_2Ereal(V1l)))),inj__ty_2Erealax_2Ereal(V2e))) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(c_2Eiterate_2Einf,V0s)),inj__ty_2Erealax_2Ereal(V1l)))),inj__ty_2Erealax_2Ereal(V2e))) ) ) ).
tff(conj_thm_2Eiterate_2ESUP__UNIQUE__FINITE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( 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) ) )
=> ( ( surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Esup,V1s)) = V0a )
<=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),V1s))
& ! [V2y: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2y)),V1s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2y)),inj__ty_2Erealax_2Ereal(V0a))) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EINF__UNIQUE__FINITE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( 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) ) )
=> ( ( surj__ty_2Erealax_2Ereal(ap(c_2Eiterate_2Einf,V1s)) = V0a )
<=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),V1s))
& ! [V2y: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2y)),V1s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V2y))) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUP__INSERT__FINITE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1s: $i] :
( mem(V1s,arr(ty_2Erealax_2Ereal,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V1s))
=> ( surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Esup,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0x)),V1s))) = surj__ty_2Erealax_2Ereal(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))),inj__ty_2Erealax_2Ereal(V0x)),ap(ap(c_2Ereal_2Emax,inj__ty_2Erealax_2Ereal(V0x)),ap(c_2Ereal_2Esup,V1s)))) ) ) ) ).
tff(conj_thm_2Eiterate_2EINF__INSERT__FINITE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1s: $i] :
( mem(V1s,arr(ty_2Erealax_2Ereal,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V1s))
=> ( surj__ty_2Erealax_2Ereal(ap(c_2Eiterate_2Einf,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0x)),V1s))) = surj__ty_2Erealax_2Ereal(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))),inj__ty_2Erealax_2Ereal(V0x)),ap(ap(c_2Ereal_2Emin,inj__ty_2Erealax_2Ereal(V0x)),ap(c_2Eiterate_2Einf,V1s)))) ) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__SUP__EQ__INF,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ( ( ( V0s != c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal) )
& ? [V1B: tp__ty_2Erealax_2Ereal] :
! [V2x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,inj__ty_2Erealax_2Ereal(V2x))),inj__ty_2Erealax_2Ereal(V1B))) ) )
=> ( ( surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Esup,V0s)) = surj__ty_2Erealax_2Ereal(ap(c_2Eiterate_2Einf,V0s)) )
<=> ? [V3a: tp__ty_2Erealax_2Ereal] : ( V0s = ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3a)),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EINF__SING,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] : ( surj__ty_2Erealax_2Ereal(ap(c_2Eiterate_2Einf,ap(ap(c_2Epred__set_2EINSERT(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),c_2Epred__set_2EEMPTY(ty_2Erealax_2Ereal)))) = V0a ) ).
tff(lamtp_f3192,type,
f3192: ( tp__ty_2Enum_2Enum * tp__ty_2Enum_2Enum ) > $i ).
tff(lameq_f3192,axiom,
! [V1n: tp__ty_2Enum_2Enum,V0m: tp__ty_2Enum_2Enum,V2x: tp__ty_2Enum_2Enum] : ( ap(f3192(V1n,V0m),inj__ty_2Enum_2Enum(V2x)) = ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,bool),inj__ty_2Enum_2Enum(V2x)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V2x))),ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V2x)),inj__ty_2Enum_2Enum(V1n)))) ) ).
tff(ax_thm_2Eiterate_2Enumseg,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] : ( ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n)) = ap(c_2Epred__set_2EGSPEC(ty_2Enum_2Enum,ty_2Enum_2Enum),f3192(V1n,V0m)) ) ).
tff(conj_thm_2Eiterate_2EFINITE__NUMSEG,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] : p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n)))) ).
tff(conj_thm_2Eiterate_2ENUMSEG__COMBINE__R,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1p: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V0m)),ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V1p)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1p)),inj__ty_2Enum_2Enum(V2n))) )
=> ( ap(ap(c_2Epred__set_2EUNION(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1p))),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V1p)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),inj__ty_2Enum_2Enum(V2n))) = ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V2n)) ) ) ).
tff(conj_thm_2Eiterate_2ENUMSEG__COMBINE__L,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1p: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1p)))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1p)),ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V2n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))) )
=> ( ap(ap(c_2Epred__set_2EUNION(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V0m)),ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(V1p)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1p)),inj__ty_2Enum_2Enum(V2n))) = ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V2n)) ) ) ).
tff(conj_thm_2Eiterate_2ENUMSEG__LREC,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n)))
=> ( ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V0m)),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V0m)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),inj__ty_2Enum_2Enum(V1n))) = ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n)) ) ) ).
tff(conj_thm_2Eiterate_2ENUMSEG__RREC,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n)))
=> ( ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V1n)),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V0m)),ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(V1n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))) = ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n)) ) ) ).
tff(conj_thm_2Eiterate_2ENUMSEG__REC,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V0m)),ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V1n))))
=> ( ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V0m)),ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V1n))) = ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V1n))),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n))) ) ) ).
tff(conj_thm_2Eiterate_2EIN__NUMSEG,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum,V2p: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V2p)),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n))))
<=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V2p)))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V2p)),inj__ty_2Enum_2Enum(V1n))) ) ) ).
tff(conj_thm_2Eiterate_2EIN__NUMSEG__0,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V0m)),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V1n))))
<=> p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n))) ) ).
tff(conj_thm_2Eiterate_2ENUMSEG__SING,axiom,
! [V0n: tp__ty_2Enum_2Enum] : ( ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V0n)),inj__ty_2Enum_2Enum(V0n)) = ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V0n)),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)) ) ).
tff(conj_thm_2Eiterate_2ENUMSEG__EMPTY,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n)) = c_2Epred__set_2EEMPTY(ty_2Enum_2Enum) )
<=> p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V1n)),inj__ty_2Enum_2Enum(V0m))) ) ).
tff(conj_thm_2Eiterate_2ECARD__NUMSEG__LEMMA,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1d: tp__ty_2Enum_2Enum] : ( surj__ty_2Enum_2Enum(ap(c_2Epred__set_2ECARD(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V0m)),ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1d))))) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V1d)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) ) ).
tff(conj_thm_2Eiterate_2ECARD__NUMSEG,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] : ( surj__ty_2Enum_2Enum(ap(c_2Epred__set_2ECARD(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n)))) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2D,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V1n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),inj__ty_2Enum_2Enum(V0m))) ) ).
tff(conj_thm_2Eiterate_2EHAS__SIZE__NUMSEG,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Ecardinal_2EHAS__SIZE(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n))),ap(ap(c_2Earithmetic_2E_2D,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V1n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),inj__ty_2Enum_2Enum(V0m)))) ).
tff(conj_thm_2Eiterate_2ECARD__NUMSEG__1,axiom,
! [V0n: tp__ty_2Enum_2Enum] : ( surj__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,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),inj__ty_2Enum_2Enum(V0n)))) = V0n ) ).
tff(conj_thm_2Eiterate_2EHAS__SIZE__NUMSEG__1,axiom,
! [V0n: tp__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,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),inj__ty_2Enum_2Enum(V0n))),inj__ty_2Enum_2Enum(V0n))) ).
tff(conj_thm_2Eiterate_2ENUMSEG__CLAUSES,axiom,
( ! [V0m: tp__ty_2Enum_2Enum] : ( ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)) = ap(ap(ap(c_2Ebool_2ECOND(arr(ty_2Enum_2Enum,bool)),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum))),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)) )
& ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] : ( ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V2n))) = ap(ap(ap(c_2Ebool_2ECOND(arr(ty_2Enum_2Enum,bool)),ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1m)),ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V2n)))),ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V2n))),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n)))),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))) ) ) ).
tff(conj_thm_2Eiterate_2EFINITE__INDEX__NUMSEG,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
<=> ? [V1f: $i] :
( mem(V1f,arr(ty_2Enum_2Enum,A_27a))
& ! [V2i: tp__ty_2Enum_2Enum,V3j: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V2i)),ap(ap(c_2Eiterate_2E_2E_2E,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),ap(c_2Epred__set_2ECARD(A_27a),V0s))))
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V3j)),ap(ap(c_2Eiterate_2E_2E_2E,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),ap(c_2Epred__set_2ECARD(A_27a),V0s))))
& ( ap(V1f,inj__ty_2Enum_2Enum(V2i)) = ap(V1f,inj__ty_2Enum_2Enum(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,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),ap(c_2Epred__set_2ECARD(A_27a),V0s))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EFINITE__INDEX__NUMBERS,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
<=> ? [V1k: $i] :
( mem(V1k,arr(ty_2Enum_2Enum,bool))
& ? [V2f: $i] :
( mem(V2f,arr(ty_2Enum_2Enum,A_27a))
& ! [V3i: tp__ty_2Enum_2Enum,V4j: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V3i)),V1k))
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V4j)),V1k))
& ( ap(V2f,inj__ty_2Enum_2Enum(V3i)) = ap(V2f,inj__ty_2Enum_2Enum(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) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EDISJOINT__NUMSEG,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum,V2p: tp__ty_2Enum_2Enum,V3q: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Epred__set_2EDISJOINT(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n))),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2p)),inj__ty_2Enum_2Enum(V3q))))
<=> ( p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V1n)),inj__ty_2Enum_2Enum(V2p)))
| p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V3q)),inj__ty_2Enum_2Enum(V0m)))
| p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V1n)),inj__ty_2Enum_2Enum(V0m)))
| p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V3q)),inj__ty_2Enum_2Enum(V2p))) ) ) ).
tff(conj_thm_2Eiterate_2ENUMSEG__ADD__SPLIT,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum,V2p: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V0m)),ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V1n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))))
=> ( ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V0m)),ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V1n)),inj__ty_2Enum_2Enum(V2p))) = ap(ap(c_2Epred__set_2EUNION(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n))),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V1n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V1n)),inj__ty_2Enum_2Enum(V2p)))) ) ) ).
tff(lamtp_f3193,type,
f3193: tp__ty_2Enum_2Enum > $i ).
tff(lameq_f3193,axiom,
! [V2p: tp__ty_2Enum_2Enum,V3i: tp__ty_2Enum_2Enum] : ( ap(f3193(V2p),inj__ty_2Enum_2Enum(V3i)) = ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V3i)),inj__ty_2Enum_2Enum(V2p)) ) ).
tff(conj_thm_2Eiterate_2ENUMSEG__OFFSET__IMAGE,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum,V2p: tp__ty_2Enum_2Enum] : ( ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V2p))),ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V1n)),inj__ty_2Enum_2Enum(V2p))) = ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Enum_2Enum),f3193(V2p)),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n))) ) ).
tff(conj_thm_2Eiterate_2ESUBSET__NUMSEG,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum,V2p: tp__ty_2Enum_2Enum,V3q: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n))),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2p)),inj__ty_2Enum_2Enum(V3q))))
<=> ( p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V1n)),inj__ty_2Enum_2Enum(V0m)))
| ( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V2p)),inj__ty_2Enum_2Enum(V0m)))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1n)),inj__ty_2Enum_2Enum(V3q))) ) ) ) ).
tff(lamtp_f3194,type,
f3194: tp__ty_2Enum_2Enum > $i ).
tff(lameq_f3194,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Enum_2Enum] : ( ap(f3194(V0n),inj__ty_2Enum_2Enum(V1x)) = ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,bool),inj__ty_2Enum_2Enum(V1x)),ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1x)),inj__ty_2Enum_2Enum(V0n))) ) ).
tff(conj_thm_2Eiterate_2ENUMSEG__LE,axiom,
! [V0n: tp__ty_2Enum_2Enum] : ( ap(c_2Epred__set_2EGSPEC(ty_2Enum_2Enum,ty_2Enum_2Enum),f3194(V0n)) = ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V0n)) ) ).
tff(lamtp_f3195,type,
f3195: tp__ty_2Enum_2Enum > $i ).
tff(lameq_f3195,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Enum_2Enum] : ( ap(f3195(V0n),inj__ty_2Enum_2Enum(V1x)) = ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,bool),inj__ty_2Enum_2Enum(V1x)),ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V1x)),inj__ty_2Enum_2Enum(V0n))) ) ).
tff(conj_thm_2Eiterate_2ENUMSEG__LT,axiom,
! [V0n: tp__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),inj__ty_2Enum_2Enum(V0n)),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(V0n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))) ) ).
tff(conj_thm_2Eiterate_2ETOPOLOGICAL__SORT,axiom,
! [A_27a: del,V0_3C_3C: $i] :
( mem(V0_3C_3C,arr(A_27a,arr(A_27a,bool)))
=> ( ( ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( ( p(ap(ap(V0_3C_3C,V1x),V2y))
& p(ap(ap(V0_3C_3C,V2y),V1x)) )
=> ( V1x = V2y ) ) ) )
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ! [V4y: $i] :
( mem(V4y,A_27a)
=> ! [V5z: $i] :
( 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: tp__ty_2Enum_2Enum,V7s: $i] :
( mem(V7s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27a),V7s),inj__ty_2Enum_2Enum(V6n)))
=> ? [V8f: $i] :
( 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,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),inj__ty_2Enum_2Enum(V6n))) )
& ! [V9j: tp__ty_2Enum_2Enum,V10k: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V9j)),ap(ap(c_2Eiterate_2E_2E_2E,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),inj__ty_2Enum_2Enum(V6n))))
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V10k)),ap(ap(c_2Eiterate_2E_2E_2E,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),inj__ty_2Enum_2Enum(V6n))))
& p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V9j)),inj__ty_2Enum_2Enum(V10k))) )
=> ~ p(ap(ap(V0_3C_3C,ap(V8f,inj__ty_2Enum_2Enum(V10k))),ap(V8f,inj__ty_2Enum_2Enum(V9j)))) ) ) ) ) ) ) ).
tff(lamtp_f3196,type,
f3196: ( del * $i * $i ) > $i ).
tff(lameq_f3196,axiom,
! [A_27a: del,V0op: $i] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] : ( 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)) ) ) ) ).
tff(lamtp_f3197,type,
f3197: ( del * $i ) > $i ).
tff(lameq_f3197,axiom,
! [A_27a: del,V0op: $i] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V1x: $i] : ( ap(f3197(A_27a,V0op),V1x) = ap(c_2Ebool_2E_21(A_27a),f3196(A_27a,V0op,V1x)) ) ) ).
tff(ax_thm_2Eiterate_2Eneutral,axiom,
! [A_27a: del,V0op: $i] :
( 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)) ) ) ).
tff(ax_thm_2Eiterate_2Emonoidal,axiom,
! [A_27a: del,V0op: $i] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V0op))
<=> ( ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( ap(ap(V0op,V1x),V2y) = ap(ap(V0op,V2y),V1x) ) ) )
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ! [V4y: $i] :
( mem(V4y,A_27a)
=> ! [V5z: $i] :
( mem(V5z,A_27a)
=> ( ap(ap(V0op,V3x),ap(ap(V0op,V4y),V5z)) = ap(ap(V0op,ap(ap(V0op,V3x),V4y)),V5z) ) ) ) )
& ! [V6x: $i] :
( mem(V6x,A_27a)
=> ( ap(ap(V0op,ap(c_2Eiterate_2Eneutral(A_27a),V0op)),V6x) = V6x ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EMONOIDAL__AC,axiom,
! [A_27a: del,V0op: $i] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V0op))
=> ( ! [V1a: $i] :
( mem(V1a,A_27a)
=> ( ap(ap(V0op,ap(c_2Eiterate_2Eneutral(A_27a),V0op)),V1a) = V1a ) )
& ! [V2a: $i] :
( mem(V2a,A_27a)
=> ( ap(ap(V0op,V2a),ap(c_2Eiterate_2Eneutral(A_27a),V0op)) = V2a ) )
& ! [V3a: $i] :
( mem(V3a,A_27a)
=> ! [V4b: $i] :
( mem(V4b,A_27a)
=> ( ap(ap(V0op,V3a),V4b) = ap(ap(V0op,V4b),V3a) ) ) )
& ! [V5a: $i] :
( mem(V5a,A_27a)
=> ! [V6b: $i] :
( mem(V6b,A_27a)
=> ! [V7c: $i] :
( mem(V7c,A_27a)
=> ( ap(ap(V0op,ap(ap(V0op,V5a),V6b)),V7c) = ap(ap(V0op,V5a),ap(ap(V0op,V6b),V7c)) ) ) ) )
& ! [V8a: $i] :
( mem(V8a,A_27a)
=> ! [V9b: $i] :
( mem(V9b,A_27a)
=> ! [V10c: $i] :
( mem(V10c,A_27a)
=> ( ap(ap(V0op,V8a),ap(ap(V0op,V9b),V10c)) = ap(ap(V0op,V9b),ap(ap(V0op,V8a),V10c)) ) ) ) ) ) ) ) ).
tff(lamtp_f3198,type,
f3198: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3198,axiom,
! [A_27b: del,A_27a: del,V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V0op: $i] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V3x: $i] : ( 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))))) ) ) ) ) ).
tff(ax_thm_2Eiterate_2Esupport,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2s: $i] :
( 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)) ) ) ) ) ).
tff(lamtp_f3199,type,
f3199: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3199,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27b))
=> ! [V3x: $i] :
( mem(V3x,A_27a)
=> ! [V4a: $i] : ( ap(f3199(A_27a,A_27b,V0op,V2f,V3x),V4a) = ap(ap(V0op,ap(V2f,V3x)),V4a) ) ) ) ) ).
tff(lamtp_f3200,type,
f3200: ( del * del * $i * $i ) > $i ).
tff(lameq_f3200,axiom,
! [A_27b: del,A_27a: del,V0op: $i] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27b))
=> ! [V3x: $i] : ( ap(f3200(A_27b,A_27a,V0op,V2f),V3x) = f3199(A_27a,A_27b,V0op,V2f,V3x) ) ) ) ).
tff(ax_thm_2Eiterate_2Eiterate,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2f: $i] :
( 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)) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EIN__SUPPORT,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27b,A_27a))
=> ! [V2x: $i] :
( mem(V2x,A_27b)
=> ! [V3s: $i] :
( 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) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUPPORT__SUPPORT,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27b,A_27a))
=> ! [V2s: $i] :
( 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) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUPPORT__EMPTY,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27b,A_27a))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27b,bool))
=> ( ! [V3x: $i] :
( 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) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUPPORT__SUBSET,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27b,A_27a))
=> ! [V2s: $i] :
( 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)) ) ) ) ).
tff(conj_thm_2Eiterate_2EFINITE__SUPPORT,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27b,A_27a))
=> ! [V2s: $i] :
( 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))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUPPORT__CLAUSES,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del,A_27g: del,A_27h: del,A_27i: del,V0op: $i] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( ! [V1f: $i] :
( 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: $i] :
( mem(V2f,arr(A_27c,A_27b))
=> ! [V3x: $i] :
( mem(V3x,A_27c)
=> ! [V4s: $i] :
( 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: $i] :
( mem(V5f,arr(A_27d,A_27b))
=> ! [V6x: $i] :
( mem(V6x,A_27d)
=> ! [V7s: $i] :
( 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: $i] :
( mem(V8f,arr(A_27e,A_27b))
=> ! [V9s: $i] :
( mem(V9s,arr(A_27e,bool))
=> ! [V10t: $i] :
( 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: $i] :
( mem(V11f,arr(A_27f,A_27b))
=> ! [V12s: $i] :
( mem(V12s,arr(A_27f,bool))
=> ! [V13t: $i] :
( 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: $i] :
( mem(V14f,arr(A_27g,A_27b))
=> ! [V15s: $i] :
( mem(V15s,arr(A_27g,bool))
=> ! [V16t: $i] :
( 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: $i] :
( mem(V17f,arr(A_27h,A_27i))
=> ! [V18g: $i] :
( mem(V18g,arr(A_27i,A_27b))
=> ! [V19s: $i] :
( 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)) ) ) ) ) ) ) ).
tff(lamtp_f3201,type,
f3201: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3201,axiom,
! [A_27a: del,A_27b: del,V2f: $i] :
( mem(V2f,arr(A_27b,A_27a))
=> ! [V3a: $i] :
( mem(V3a,A_27b)
=> ! [V0op: $i] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V4x: $i] : ( 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)) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUPPORT__DELTA,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27b,bool))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27b,A_27a))
=> ! [V3a: $i] :
( 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)) ) ) ) ) ) ).
tff(lamtp_f3202,type,
f3202: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3202,axiom,
! [A_27a: del,A_27b: del,V2f: $i] :
( mem(V2f,arr(A_27b,A_27a))
=> ! [V3a: $i] :
( mem(V3a,A_27b)
=> ! [V1op: $i] :
( mem(V1op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V4x: $i] : ( 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)) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EFINITE__SUPPORT__DELTA,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( mem(V0s,arr(A_27b,bool))
=> ! [V1op: $i] :
( mem(V1op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27b,A_27a))
=> ! [V3a: $i] :
( 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))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__SUPPORT,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27b,A_27a))
=> ! [V2s: $i] :
( 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) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__EXPAND__CASES,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27b,A_27a))
=> ! [V2s: $i] :
( 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)) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__CLAUSES__GEN,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ( ! [V1f: $i] :
( 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: $i] :
( mem(V2f,arr(A_27a,A_27b))
=> ! [V3x: $i] :
( mem(V3x,A_27a)
=> ! [V4s: $i] :
( 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))) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__CLAUSES,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V0op))
=> ( ! [V1f: $i] :
( 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: $i] :
( mem(V2f,arr(A_27b,A_27a))
=> ! [V3x: $i] :
( mem(V3x,A_27b)
=> ! [V4s: $i] :
( 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))) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__UNION,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V0op))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27b,A_27a))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27b,bool))
=> ! [V3t: $i] :
( 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)) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__UNION__GEN,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t: $i] :
( 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)) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__DIFF,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V0op))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27b,A_27a))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27b,bool))
=> ! [V3t: $i] :
( 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) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__DIFF__GEN,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t: $i] :
( 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) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__INCL__EXCL,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V0op))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27b,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27b,bool))
=> ! [V3f: $i] :
( 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)) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__CLOSED,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ! [V1P: $i] :
( mem(V1P,arr(A_27b,bool))
=> ( ( p(ap(V1P,ap(c_2Eiterate_2Eneutral(A_27b),V0op)))
& ! [V2x: $i] :
( mem(V2x,A_27b)
=> ! [V3y: $i] :
( mem(V3y,A_27b)
=> ( ( p(ap(V1P,V2x))
& p(ap(V1P,V3y)) )
=> p(ap(V1P,ap(ap(V0op,V2x),V3y))) ) ) ) )
=> ! [V4f: $i] :
( mem(V4f,arr(A_27a,A_27b))
=> ! [V5s: $i] :
( mem(V5s,arr(A_27a,bool))
=> ( ! [V6x: $i] :
( 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))) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__RELATED,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ! [V1R: $i] :
( 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: $i] :
( mem(V2x1,A_27b)
=> ! [V3y1: $i] :
( mem(V3y1,A_27b)
=> ! [V4x2: $i] :
( mem(V4x2,A_27b)
=> ! [V5y2: $i] :
( 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: $i] :
( mem(V6f,arr(A_27a,A_27b))
=> ! [V7g: $i] :
( mem(V7g,arr(A_27a,A_27b))
=> ! [V8s: $i] :
( mem(V8s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V8s))
& ! [V9x: $i] :
( 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))) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__EQ__NEUTRAL,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ! [V3x: $i] :
( 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) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__SING,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2x: $i] :
( 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) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__DELETE,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3a: $i] :
( 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) ) ) ) ) ) ) ) ).
tff(lamtp_f3203,type,
f3203: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3203,axiom,
! [A_27a: del,A_27b: del,V1f: $i] :
( mem(V1f,arr(A_27b,A_27a))
=> ! [V2a: $i] :
( mem(V2a,A_27b)
=> ! [V0op: $i] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V4x: $i] : ( 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)) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__DELTA,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V0op))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27b,A_27a))
=> ! [V2a: $i] :
( mem(V2a,A_27b)
=> ! [V3s: $i] :
( 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)) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__IMAGE,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0op: $i] :
( mem(V0op,arr(A_27c,arr(A_27c,A_27c)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27c),V0op))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2g: $i] :
( mem(V2g,arr(A_27b,A_27c))
=> ! [V3s: $i] :
( mem(V3s,arr(A_27a,bool))
=> ( ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( 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)) ) ) ) ) ) ) ) ).
tff(lamtp_f3204,type,
f3204: ( del * $i * $i * $i ) > $i ).
tff(lameq_f3204,axiom,
! [A_27a: del,V5y: $i] :
( mem(V5y,A_27a)
=> ! [V2p: $i] :
( mem(V2p,arr(A_27a,A_27a))
=> ! [V3s: $i] :
( mem(V3s,arr(A_27a,bool))
=> ! [V6x: $i] : ( 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)) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__BIJECTION,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2p: $i] :
( mem(V2p,arr(A_27a,A_27a))
=> ! [V3s: $i] :
( mem(V3s,arr(A_27a,bool))
=> ( ( ! [V4x: $i] :
( 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: $i] :
( 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)) ) ) ) ) ) ) ) ).
tff(lamtp_f3205,type,
f3205: ( del * del * del * $i * $i * $i ) > $i ).
tff(lameq_f3205,axiom,
! [A_27c: del,A_27b: del,A_27a: del,V0op: $i] :
( mem(V0op,arr(A_27c,arr(A_27c,A_27c)))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,arr(A_27b,bool)))
=> ! [V3x: $i] :
( mem(V3x,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V5i: $i] : ( 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)) ) ) ) ) ).
tff(lamtp_f3206,type,
f3206: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3206,axiom,
! [A_27a: del,A_27b: del,V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,arr(A_27b,bool)))
=> ! [V6i: $i] :
( mem(V6i,A_27a)
=> ! [V7j: $i] : ( 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)))) ) ) ) ) ).
tff(lamtp_f3207,type,
f3207: ( del * del * $i * $i ) > $i ).
tff(lameq_f3207,axiom,
! [A_27b: del,A_27a: del,V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,arr(A_27b,bool)))
=> ! [V6i: $i] : ( ap(f3207(A_27b,A_27a,V1s,V2t),V6i) = f3206(A_27a,A_27b,V1s,V2t,V6i) ) ) ) ).
tff(lamtp_f3208,type,
f3208: ( del * del * del * $i * $i ) > $i ).
tff(lameq_f3208,axiom,
! [A_27c: del,A_27a: del,A_27b: del,V8i: $i] :
( mem(V8i,A_27a)
=> ! [V3x: $i] :
( mem(V3x,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V9j: $i] : ( ap(f3208(A_27c,A_27a,A_27b,V8i,V3x),V9j) = ap(ap(V3x,V8i),V9j) ) ) ) ).
tff(lamtp_f3209,type,
f3209: ( del * del * del * $i ) > $i ).
tff(lameq_f3209,axiom,
! [A_27b: del,A_27c: del,A_27a: del,V3x: $i] :
( mem(V3x,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V8i: $i] : ( ap(f3209(A_27b,A_27c,A_27a,V3x),V8i) = f3208(A_27c,A_27a,A_27b,V8i,V3x) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__ITERATE__PRODUCT,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0op: $i] :
( mem(V0op,arr(A_27c,arr(A_27c,A_27c)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27c),V0op))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,arr(A_27b,bool)))
=> ! [V3x: $i] :
( mem(V3x,arr(A_27a,arr(A_27b,A_27c)))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& ! [V4i: $i] :
( 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))) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__EQ,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2g: $i] :
( mem(V2g,arr(A_27a,A_27b))
=> ! [V3s: $i] :
( mem(V3s,arr(A_27a,bool))
=> ( ! [V4x: $i] :
( 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) ) ) ) ) ) ) ) ).
tff(lamtp_f3210,type,
f3210: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3210,axiom,
! [A_27b: del,A_27a: del,V6y: $i] :
( mem(V6y,A_27b)
=> ! [V5h: $i] :
( mem(V5h,arr(A_27a,A_27b))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V7x: $i] : ( 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)) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__EQ__GENERAL,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0op: $i] :
( mem(V0op,arr(A_27c,arr(A_27c,A_27c)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27c),V0op))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27b,bool))
=> ! [V3f: $i] :
( mem(V3f,arr(A_27a,A_27c))
=> ! [V4g: $i] :
( mem(V4g,arr(A_27b,A_27c))
=> ! [V5h: $i] :
( mem(V5h,arr(A_27a,A_27b))
=> ( ( ! [V6y: $i] :
( 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: $i] :
( 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) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__EQ__GENERAL__INVERSES,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0op: $i] :
( mem(V0op,arr(A_27c,arr(A_27c,A_27c)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27c),V0op))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27b,bool))
=> ! [V3f: $i] :
( mem(V3f,arr(A_27a,A_27c))
=> ! [V4g: $i] :
( mem(V4g,arr(A_27b,A_27c))
=> ! [V5h: $i] :
( mem(V5h,arr(A_27a,A_27b))
=> ! [V6k: $i] :
( mem(V6k,arr(A_27b,A_27a))
=> ( ( ! [V7y: $i] :
( 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: $i] :
( 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) ) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__INJECTION,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2p: $i] :
( mem(V2p,arr(A_27a,A_27a))
=> ! [V3s: $i] :
( mem(V3s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V3s))
& ! [V4x: $i] :
( 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: $i] :
( mem(V5x,A_27a)
=> ! [V6y: $i] :
( 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) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__UNION__NONZERO,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t: $i] :
( 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: $i] :
( 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)) ) ) ) ) ) ) ) ).
tff(lamtp_f3211,type,
f3211: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3211,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27b,A_27a))
=> ! [V2g: $i] :
( mem(V2g,arr(A_27b,A_27a))
=> ! [V4x: $i] : ( ap(f3211(A_27a,A_27b,V0op,V1f,V2g),V4x) = ap(ap(V0op,ap(V1f,V4x)),ap(V2g,V4x)) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__OP,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V0op))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27b,A_27a))
=> ! [V2g: $i] :
( mem(V2g,arr(A_27b,A_27a))
=> ! [V3s: $i] :
( 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)) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__SUPERSET,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2u: $i] :
( mem(V2u,arr(A_27a,bool))
=> ! [V3v: $i] :
( mem(V3v,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V2u),V3v))
& ! [V4x: $i] :
( 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) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__IMAGE__NONZERO,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0op: $i] :
( mem(V0op,arr(A_27c,arr(A_27c,A_27c)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27c),V0op))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27b,A_27c))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27b))
=> ! [V3s: $i] :
( mem(V3s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V3s))
& ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( 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)) ) ) ) ) ) ) ) ).
tff(lamtp_f3212,type,
f3212: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3212,axiom,
! [A_27b: del,A_27a: del,V3f: $i] :
( mem(V3f,arr(A_27a,A_27b))
=> ! [V2P: $i] :
( mem(V2P,arr(A_27a,bool))
=> ! [V4g: $i] :
( mem(V4g,arr(A_27a,A_27b))
=> ! [V5x: $i] : ( 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)) ) ) ) ) ).
tff(lamtp_f3213,type,
f3213: ( del * $i * $i ) > $i ).
tff(lameq_f3213,axiom,
! [A_27a: del,V2P: $i] :
( mem(V2P,arr(A_27a,bool))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V6x: $i] : ( 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))) ) ) ) ).
tff(lamtp_f3214,type,
f3214: ( del * $i * $i ) > $i ).
tff(lameq_f3214,axiom,
! [A_27a: del,V2P: $i] :
( mem(V2P,arr(A_27a,bool))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V7x: $i] : ( 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)))) ) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__CASES,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2P: $i] :
( mem(V2P,arr(A_27a,bool))
=> ! [V3f: $i] :
( mem(V3f,arr(A_27a,A_27b))
=> ! [V4g: $i] :
( 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)) ) ) ) ) ) ) ) ) ).
tff(lamtp_f3215,type,
f3215: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3215,axiom,
! [A_27b: del,A_27a: del,V0op: $i] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2g: $i] :
( mem(V2g,arr(A_27a,A_27b))
=> ! [V4x: $i] : ( ap(f3215(A_27b,A_27a,V0op,V1f,V2g),V4x) = ap(ap(V0op,ap(V1f,V4x)),ap(V2g,V4x)) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__OP__GEN,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( mem(V0op,arr(A_27b,arr(A_27b,A_27b)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27b),V0op))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2g: $i] :
( mem(V2g,arr(A_27a,A_27b))
=> ! [V3s: $i] :
( 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)) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__CLAUSES__NUMSEG,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,A_27a))
=> ! [V1op: $i] :
( mem(V1op,arr(A_27a,arr(A_27a,A_27a)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V1op))
=> ( ! [V2m: tp__ty_2Enum_2Enum] : ( ap(ap(ap(c_2Eiterate_2Eiterate(ty_2Enum_2Enum,A_27a),V1op),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),V0f) = ap(ap(ap(c_2Ebool_2ECOND(A_27a),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V0f,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(c_2Eiterate_2Eneutral(A_27a),V1op)) )
& ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] : ( ap(ap(ap(c_2Eiterate_2Eiterate(ty_2Enum_2Enum,A_27a),V1op),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V3m)),ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V4n)))),V0f) = ap(ap(ap(c_2Ebool_2ECOND(A_27a),ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V3m)),ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V4n)))),ap(ap(V1op,ap(ap(ap(c_2Eiterate_2Eiterate(ty_2Enum_2Enum,A_27a),V1op),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V3m)),inj__ty_2Enum_2Enum(V4n))),V0f)),ap(V0f,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V4n))))),ap(ap(ap(c_2Eiterate_2Eiterate(ty_2Enum_2Enum,A_27a),V1op),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V3m)),inj__ty_2Enum_2Enum(V4n))),V0f)) ) ) ) ) ) ).
tff(lamtp_f3216,type,
f3216: ( del * $i * $i ) > $i ).
tff(lameq_f3216,axiom,
! [A_27a: del,V0op: $i] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V1f: $i] :
( mem(V1f,arr(ty_2Enum_2Enum,A_27a))
=> ! [V4i: tp__ty_2Enum_2Enum] : ( ap(f3216(A_27a,V0op,V1f),inj__ty_2Enum_2Enum(V4i)) = ap(ap(V0op,ap(V1f,ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),inj__ty_2Enum_2Enum(V4i)))),ap(V1f,ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),inj__ty_2Enum_2Enum(V4i))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))) ) ) ) ).
tff(conj_thm_2Eiterate_2EITERATE__PAIR,axiom,
! [A_27a: del,V0op: $i] :
( mem(V0op,arr(A_27a,arr(A_27a,A_27a)))
=> ( p(ap(c_2Eiterate_2Emonoidal(A_27a),V0op))
=> ! [V1f: $i] :
( mem(V1f,arr(ty_2Enum_2Enum,A_27a))
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__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,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),inj__ty_2Enum_2Enum(V2m))),ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),inj__ty_2Enum_2Enum(V3n))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),V1f) = ap(ap(ap(c_2Eiterate_2Eiterate(ty_2Enum_2Enum,A_27a),V0op),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),f3216(A_27a,V0op,V1f)) ) ) ) ) ).
tff(ax_thm_2Eiterate_2Ensum__def,axiom,
! [A_27a: del] : ( c_2Eiterate_2Ensum(A_27a) = ap(c_2Eiterate_2Eiterate(A_27a,ty_2Enum_2Enum),c_2Earithmetic_2E_2B) ) ).
tff(conj_thm_2Eiterate_2ENEUTRAL__ADD,axiom,
surj__ty_2Enum_2Enum(ap(c_2Eiterate_2Eneutral(ty_2Enum_2Enum),c_2Earithmetic_2E_2B)) = fo__c_2Enum_2E0 ).
tff(conj_thm_2Eiterate_2ENEUTRAL__MUL,axiom,
surj__ty_2Enum_2Enum(ap(c_2Eiterate_2Eneutral(ty_2Enum_2Enum),c_2Earithmetic_2E_2A)) = surj__ty_2Enum_2Enum(ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))) ).
tff(conj_thm_2Eiterate_2EMONOIDAL__ADD,axiom,
p(ap(c_2Eiterate_2Emonoidal(ty_2Enum_2Enum),c_2Earithmetic_2E_2B)) ).
tff(conj_thm_2Eiterate_2EMONOIDAL__MUL,axiom,
p(ap(c_2Eiterate_2Emonoidal(ty_2Enum_2Enum),c_2Earithmetic_2E_2A)) ).
tff(lamtp_f3217,type,
f3217: ( del * $i * $i ) > $i ).
tff(lameq_f3217,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2x: $i] : ( 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)),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__DEGENERATE,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s: $i] :
( 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))))
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f)) = fo__c_2Enum_2E0 ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__CLAUSES,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),c_2Epred__set_2EEMPTY(A_27a)),V0f)) = fo__c_2Enum_2E0 ) )
& ! [V1x: $i] :
( mem(V1x,A_27b)
=> ! [V2f: $i] :
( mem(V2f,arr(A_27b,ty_2Enum_2Enum))
=> ! [V3s: $i] :
( mem(V3s,arr(A_27b,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27b),V3s))
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27b),ap(ap(c_2Epred__set_2EINSERT(A_27b),V1x),V3s)),V2f)) = surj__ty_2Enum_2Enum(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)))) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__UNION,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( 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)) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)),V0f)) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f)),ap(ap(c_2Eiterate_2Ensum(A_27a),V2t),V0f))) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__DIFF,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( 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)) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1s),V2t)),V0f)) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2D,ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f)),ap(ap(c_2Eiterate_2Ensum(A_27a),V2t),V0f))) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__INCL__EXCL,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t: $i] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2f: $i] :
( 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)) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(A_27a),V0s),V2f)),ap(ap(c_2Eiterate_2Ensum(A_27a),V1t),V2f))) = surj__ty_2Enum_2Enum(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))) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__SUPPORT,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( surj__ty_2Enum_2Enum(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)) = surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f)) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__ADD,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),f453(A_27a,V0f,V1g))) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V0f)),ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V1g))) ) ) ) ) ) ).
tff(lamtp_f3218,type,
f3218: ( del * $i * $i ) > $i ).
tff(lameq_f3218,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3x: $i] : ( 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)),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))) ) ) ) ).
tff(lamtp_f3219,type,
f3219: ( del * $i * $i ) > $i ).
tff(lameq_f3219,axiom,
! [A_27a: del,V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V4x: $i] : ( 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)),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))) ) ) ) ).
tff(lamtp_f3220,type,
f3220: ( del * $i * $i ) > $i ).
tff(lameq_f3220,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V5x: $i] : ( ap(f3220(A_27a,V0f,V1g),V5x) = ap(ap(c_2Earithmetic_2E_2B,ap(V0f,V5x)),ap(V1g,V5x)) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__ADD__GEN,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2s: $i] :
( 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)))) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),f3220(A_27a,V0f,V1g))) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V0f)),ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V1g))) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__EQ__0,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
=> ( surj__ty_2Enum_2Enum(ap(V0f,V2x)) = fo__c_2Enum_2E0 ) ) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f)) = fo__c_2Enum_2E0 ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__0,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V0s),k(A_27a,c_2Enum_2E0))) = fo__c_2Enum_2E0 ) ) ).
tff(lamtp_f3221,type,
f3221: ( del * $i * tp__ty_2Enum_2Enum ) > $i ).
tff(lameq_f3221,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1c: tp__ty_2Enum_2Enum,V3x: $i] : ( ap(f3221(A_27a,V0f,V1c),V3x) = ap(ap(c_2Earithmetic_2E_2A,inj__ty_2Enum_2Enum(V1c)),ap(V0f,V3x)) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__LMUL,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1c: tp__ty_2Enum_2Enum,V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),f3221(A_27a,V0f,V1c))) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2A,inj__ty_2Enum_2Enum(V1c)),ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V0f))) ) ) ) ).
tff(lamtp_f3222,type,
f3222: ( del * $i * tp__ty_2Enum_2Enum ) > $i ).
tff(lameq_f3222,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1c: tp__ty_2Enum_2Enum,V3x: $i] : ( ap(f3222(A_27a,V0f,V1c),V3x) = ap(ap(c_2Earithmetic_2E_2A,ap(V0f,V3x)),inj__ty_2Enum_2Enum(V1c)) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__RMUL,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1c: tp__ty_2Enum_2Enum,V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),f3222(A_27a,V0f,V1c))) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2A,ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V0f)),inj__ty_2Enum_2Enum(V1c))) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__LE,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& ! [V3x: $i] :
( 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))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__LT,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& ! [V3x: $i] :
( 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: $i] :
( 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))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__LT__ALL,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& ( V2s != c_2Epred__set_2EEMPTY(A_27a) )
& ! [V3x: $i] :
( 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))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__EQ,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s))
=> ( surj__ty_2Enum_2Enum(ap(V0f,V3x)) = surj__ty_2Enum_2Enum(ap(V1g,V3x)) ) ) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V0f)) = surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V1g)) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__CONST,axiom,
! [A_27a: del,V0c: tp__ty_2Enum_2Enum,V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),k(A_27a,inj__ty_2Enum_2Enum(V0c)))) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2A,ap(c_2Epred__set_2ECARD(A_27a),V1s)),inj__ty_2Enum_2Enum(V0c))) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__POS__BOUND,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1b: tp__ty_2Enum_2Enum,V2s: $i] :
( 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)),inj__ty_2Enum_2Enum(V1b))) )
=> ! [V3x: $i] :
( 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)),inj__ty_2Enum_2Enum(V1b))) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__EQ__0__IFF,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
=> ( ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f)) = fo__c_2Enum_2E0 )
<=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
=> ( surj__ty_2Enum_2Enum(ap(V0f,V2x)) = fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__POS__LT,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& ? [V2x: $i] :
( mem(V2x,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
& p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),ap(V0f,V2x))) ) )
=> p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f))) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__POS__LT__ALL,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f: $i] :
( 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: $i] :
( mem(V2i,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2i),V0s))
=> p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),ap(V1f,V2i))) ) ) )
=> p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),ap(ap(c_2Eiterate_2Ensum(A_27a),V0s),V1f))) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__DELETE,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2a: $i] :
( mem(V2a,A_27a)
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V2a),V1s)) )
=> ( surj__ty_2Enum_2Enum(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))) = surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f)) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__SING,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),c_2Epred__set_2EEMPTY(A_27a))),V0f)) = surj__ty_2Enum_2Enum(ap(V0f,V1x)) ) ) ) ).
tff(lamtp_f3223,type,
f3223: ( del * tp__ty_2Enum_2Enum * $i ) > $i ).
tff(lameq_f3223,axiom,
! [A_27a: del,V0b: tp__ty_2Enum_2Enum,V2a: $i] :
( mem(V2a,A_27a)
=> ! [V3x: $i] : ( 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)),inj__ty_2Enum_2Enum(V0b)),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__DELTA,axiom,
! [A_27a: del,V0b: tp__ty_2Enum_2Enum,V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2a: $i] :
( mem(V2a,A_27a)
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),f3223(A_27a,V0b,V2a))) = surj__ty_2Enum_2Enum(ap(ap(ap(c_2Ebool_2ECOND(ty_2Enum_2Enum),ap(ap(c_2Ebool_2EIN(A_27a),V2a),V1s)),inj__ty_2Enum_2Enum(V0b)),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) ).
tff(lamtp_f3224,type,
f3224: ( del * del * $i * $i ) > $i ).
tff(lameq_f3224,axiom,
! [A_27b: del,A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27b,ty_2Enum_2Enum)))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27b,bool))
=> ! [V3i: $i] : ( ap(f3224(A_27b,A_27a,V0f,V2t),V3i) = ap(ap(c_2Eiterate_2Ensum(A_27b),V2t),ap(V0f,V3i)) ) ) ) ).
tff(lamtp_f3225,type,
f3225: ( del * del * $i * $i ) > $i ).
tff(lameq_f3225,axiom,
! [A_27b: del,A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27b,ty_2Enum_2Enum)))
=> ! [V4j: $i] :
( mem(V4j,A_27b)
=> ! [V5i: $i] : ( ap(f3225(A_27b,A_27a,V0f,V4j),V5i) = ap(ap(V0f,V5i),V4j) ) ) ) ).
tff(lamtp_f3226,type,
f3226: ( del * del * $i * $i ) > $i ).
tff(lameq_f3226,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27b,ty_2Enum_2Enum)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V4j: $i] : ( ap(f3226(A_27a,A_27b,V0f,V1s),V4j) = ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),f3225(A_27b,A_27a,V0f,V4j)) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__SWAP,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27b,ty_2Enum_2Enum)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27b,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V2t)) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),f3224(A_27b,A_27a,V0f,V2t))) = surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27b),V2t),f3226(A_27a,A_27b,V0f,V1s))) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27b,ty_2Enum_2Enum))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ! [V3x: $i] :
( mem(V3x,A_27a)
=> ! [V4y: $i] :
( 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 ) ) ) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V2s)),V1g)) = surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Enum_2Enum,A_27b),V1g),V0f))) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__SUPERSET,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1u: $i] :
( mem(V1u,arr(A_27a,bool))
=> ! [V2v: $i] :
( mem(V2v,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1u),V2v))
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2v))
& ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1u)) )
=> ( surj__ty_2Enum_2Enum(ap(V0f,V3x)) = fo__c_2Enum_2E0 ) ) ) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V2v),V0f)) = surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V1u),V0f)) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__UNION__RZERO,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1u: $i] :
( mem(V1u,arr(A_27a,bool))
=> ! [V2v: $i] :
( mem(V2v,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1u))
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2v))
& ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1u)) )
=> ( surj__ty_2Enum_2Enum(ap(V0f,V3x)) = fo__c_2Enum_2E0 ) ) ) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1u),V2v)),V0f)) = surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V1u),V0f)) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__UNION__LZERO,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1u: $i] :
( mem(V1u,arr(A_27a,bool))
=> ! [V2v: $i] :
( mem(V2v,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2v))
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1u))
& ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2v)) )
=> ( surj__ty_2Enum_2Enum(ap(V0f,V3x)) = fo__c_2Enum_2E0 ) ) ) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1u),V2v)),V0f)) = surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V2v),V0f)) ) ) ) ) ) ).
tff(lamtp_f3227,type,
f3227: ( del * $i * $i ) > $i ).
tff(lameq_f3227,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2x: $i] : ( 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)),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__RESTRICT,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),f3227(A_27a,V0f,V1s))) = surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f)) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__BOUND,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2b: tp__ty_2Enum_2Enum] :
( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& ! [V3x: $i] :
( 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)),inj__ty_2Enum_2Enum(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)),inj__ty_2Enum_2Enum(V2b)))) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__BOUND__GEN,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2b: tp__ty_2Enum_2Enum] :
( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& ( V0s != c_2Epred__set_2EEMPTY(A_27a) )
& ! [V3x: $i] :
( 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,inj__ty_2Enum_2Enum(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)),inj__ty_2Enum_2Enum(V2b))) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__BOUND__LT,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2b: tp__ty_2Enum_2Enum] :
( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& ! [V3x: $i] :
( 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)),inj__ty_2Enum_2Enum(V2b))) ) )
& ? [V4x: $i] :
( 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)),inj__ty_2Enum_2Enum(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)),inj__ty_2Enum_2Enum(V2b)))) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__BOUND__LT__ALL,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2b: tp__ty_2Enum_2Enum] :
( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& ( V0s != c_2Epred__set_2EEMPTY(A_27a) )
& ! [V3x: $i] :
( 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)),inj__ty_2Enum_2Enum(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)),inj__ty_2Enum_2Enum(V2b)))) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__BOUND__LT__GEN,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2b: tp__ty_2Enum_2Enum] :
( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& ( V0s != c_2Epred__set_2EEMPTY(A_27a) )
& ! [V3x: $i] :
( 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,inj__ty_2Enum_2Enum(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)),inj__ty_2Enum_2Enum(V2b))) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__UNION__EQ,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ! [V3u: $i] :
( 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 ) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f)),ap(ap(c_2Eiterate_2Ensum(A_27a),V2t),V0f))) = surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V3u),V0f)) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__EQ__SUPERSET,axiom,
! [A_27a: del,V0g: $i] :
( mem(V0g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t: $i] :
( 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: $i] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V3t))
=> ( surj__ty_2Enum_2Enum(ap(V1f,V4x)) = surj__ty_2Enum_2Enum(ap(V0g,V4x)) ) ) )
& ! [V5x: $i] :
( mem(V5x,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V2s))
& ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V3t)) )
=> ( surj__ty_2Enum_2Enum(ap(V1f,V5x)) = fo__c_2Enum_2E0 ) ) ) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V1f)) = surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V3t),V0g)) ) ) ) ) ) ) ).
tff(lamtp_f3228,type,
f3228: ( del * $i * $i ) > $i ).
tff(lameq_f3228,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V3x: $i] : ( 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))) ) ) ) ).
tff(lamtp_f3229,type,
f3229: ( del * $i * $i ) > $i ).
tff(lameq_f3229,axiom,
! [A_27a: del,V2f: $i] :
( mem(V2f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V4x: $i] : ( ap(f3229(A_27a,V2f,V0P),V4x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Enum_2Enum),ap(V0P,V4x)),ap(V2f,V4x)),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__RESTRICT__SET,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,ty_2Enum_2Enum))
=> ( surj__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)) = surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),f3229(A_27a,V2f,V0P))) ) ) ) ) ).
tff(lamtp_f3230,type,
f3230: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3230,axiom,
! [A_27a: del,A_27b: del,V3t: $i] :
( mem(V3t,arr(A_27b,bool))
=> ! [V4x: $i] :
( mem(V4x,A_27a)
=> ! [V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V5y: $i] : ( 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))) ) ) ) ) ).
tff(lamtp_f3231,type,
f3231: ( del * del * $i * $i ) > $i ).
tff(lameq_f3231,axiom,
! [A_27a: del,A_27b: del,V4x: $i] :
( mem(V4x,A_27a)
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,arr(A_27b,ty_2Enum_2Enum)))
=> ! [V6y: $i] : ( ap(f3231(A_27a,A_27b,V4x,V1f),V6y) = ap(ap(V1f,V4x),V6y) ) ) ) ).
tff(lamtp_f3232,type,
f3232: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3232,axiom,
! [A_27b: del,A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V3t: $i] :
( mem(V3t,arr(A_27b,bool))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,arr(A_27b,ty_2Enum_2Enum)))
=> ! [V4x: $i] : ( 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)) ) ) ) ) ).
tff(lamtp_f3233,type,
f3233: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3233,axiom,
! [A_27b: del,A_27a: del,V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V7y: $i] :
( mem(V7y,A_27b)
=> ! [V8x: $i] : ( 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))) ) ) ) ) ).
tff(lamtp_f3234,type,
f3234: ( del * del * $i * $i ) > $i ).
tff(lameq_f3234,axiom,
! [A_27b: del,A_27a: del,V1f: $i] :
( mem(V1f,arr(A_27a,arr(A_27b,ty_2Enum_2Enum)))
=> ! [V7y: $i] :
( mem(V7y,A_27b)
=> ! [V9x: $i] : ( ap(f3234(A_27b,A_27a,V1f,V7y),V9x) = ap(ap(V1f,V9x),V7y) ) ) ) ).
tff(lamtp_f3235,type,
f3235: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3235,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,arr(A_27b,ty_2Enum_2Enum)))
=> ! [V7y: $i] : ( 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)) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__NSUM__RESTRICT,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,arr(A_27b,ty_2Enum_2Enum)))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t: $i] :
( mem(V3t,arr(A_27b,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V3t)) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),f3232(A_27b,A_27a,V0R,V3t,V1f))) = surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27b),V3t),f3235(A_27a,A_27b,V0R,V2s,V1f))) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ECARD__EQ__NSUM,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ( surj__ty_2Enum_2Enum(ap(c_2Epred__set_2ECARD(A_27a),V0s)) = surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V0s),k(A_27a,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))) ) ) ) ).
tff(lamtp_f3236,type,
f3236: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3236,axiom,
! [A_27b: del,A_27a: del,V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V4j: $i] :
( mem(V4j,A_27b)
=> ! [V5i: $i] : ( 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))) ) ) ) ) ).
tff(lamtp_f3237,type,
f3237: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3237,axiom,
! [A_27a: del,A_27b: del,V2t: $i] :
( mem(V2t,arr(A_27b,bool))
=> ! [V6i: $i] :
( mem(V6i,A_27a)
=> ! [V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V7j: $i] : ( 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))) ) ) ) ) ).
tff(lamtp_f3238,type,
f3238: ( del * del * $i * $i ) > $i ).
tff(lameq_f3238,axiom,
! [A_27b: del,A_27a: del,V2t: $i] :
( mem(V2t,arr(A_27b,bool))
=> ! [V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V6i: $i] : ( 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))) ) ) ) ).
tff(lamtp_f3239,type,
f3239: ( del * $i ) > $i ).
tff(lameq_f3239,axiom,
! [A_27b: del,V3k: $i] :
( mem(V3k,arr(A_27b,ty_2Enum_2Enum))
=> ! [V8i: $i] : ( ap(f3239(A_27b,V3k),V8i) = ap(V3k,V8i) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__MULTICOUNT__GEN,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27b,bool))
=> ! [V3k: $i] :
( 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: $i] :
( mem(V4j,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4j),V2t))
=> ( surj__ty_2Enum_2Enum(ap(c_2Epred__set_2ECARD(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3236(A_27b,A_27a,V1s,V0R,V4j)))) = surj__ty_2Enum_2Enum(ap(V3k,V4j)) ) ) ) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),f3238(A_27b,A_27a,V2t,V0R))) = surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27b),V2t),f3239(A_27b,V3k))) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__MULTICOUNT,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27b,bool))
=> ! [V3k: tp__ty_2Enum_2Enum] :
( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V2t))
& ! [V4j: $i] :
( mem(V4j,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4j),V2t))
=> ( surj__ty_2Enum_2Enum(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 ) ) ) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),f3238(A_27b,A_27a,V2t,V0R))) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2A,inj__ty_2Enum_2Enum(V3k)),ap(c_2Epred__set_2ECARD(A_27b),V2t))) ) ) ) ) ) ).
tff(lamtp_f3240,type,
f3240: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3240,axiom,
! [A_27b: del,A_27a: del,V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V3y: $i] :
( mem(V3y,A_27b)
=> ! [V4x: $i] : ( 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))) ) ) ) ) ).
tff(lamtp_f3241,type,
f3241: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3241,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V3y: $i] : ( 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) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__IMAGE__GEN,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V1g)) = surj__ty_2Enum_2Enum(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))) ) ) ) ) ) ).
tff(lamtp_f3242,type,
f3242: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3242,axiom,
! [A_27b: del,A_27a: del,V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V4y: $i] :
( mem(V4y,A_27b)
=> ! [V5x: $i] : ( 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))) ) ) ) ) ).
tff(lamtp_f3243,type,
f3243: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3243,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V4y: $i] : ( 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) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__GROUP,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t: $i] :
( 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)) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27b),V3t),f3243(A_27a,A_27b,V0f,V2s,V1g))) = surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V1g)) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__SUBSET,axiom,
! [A_27a: del,V0u: $i] :
( mem(V0u,arr(A_27a,bool))
=> ! [V1v: $i] :
( mem(V1v,arr(A_27a,bool))
=> ! [V2f: $i] :
( 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: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0u),V1v)))
=> ( surj__ty_2Enum_2Enum(ap(V2f,V3x)) = fo__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))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__SUBSET__SIMPLE,axiom,
! [A_27a: del,V0u: $i] :
( mem(V0u,arr(A_27a,bool))
=> ! [V1v: $i] :
( mem(V1v,arr(A_27a,bool))
=> ! [V2f: $i] :
( 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))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__LE__GEN,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ( ! [V3x: $i] :
( 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))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__IMAGE__NONZERO,axiom,
! [A_27a: del,A_27b: del,V0d: $i] :
( mem(V0d,arr(A_27b,ty_2Enum_2Enum))
=> ! [V1i: $i] :
( mem(V1i,arr(A_27a,A_27b))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ! [V4y: $i] :
( 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) ) )
=> ( surj__ty_2Enum_2Enum(ap(V0d,ap(V1i,V3x))) = fo__c_2Enum_2E0 ) ) ) ) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1i),V2s)),V0d)) = surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Enum_2Enum,A_27b),V0d),V1i))) ) ) ) ) ) ).
tff(lamtp_f3244,type,
f3244: ( del * $i * $i * $i ) > $i ).
tff(lameq_f3244,axiom,
! [A_27a: del,V4y: $i] :
( mem(V4y,A_27a)
=> ! [V1p: $i] :
( mem(V1p,arr(A_27a,A_27a))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V5x: $i] : ( 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)) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__BIJECTION,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1p: $i] :
( mem(V1p,arr(A_27a,A_27a))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ( ! [V3x: $i] :
( 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: $i] :
( 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))) ) ) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V0f)) = surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Enum_2Enum,A_27a),V0f),V1p))) ) ) ) ) ) ).
tff(lamtp_f3245,type,
f3245: ( del * del * $i * $i ) > $i ).
tff(lameq_f3245,axiom,
! [A_27b: del,A_27a: del,V1t: $i] :
( mem(V1t,arr(A_27a,arr(A_27b,bool)))
=> ! [V2x: $i] :
( mem(V2x,arr(A_27a,arr(A_27b,ty_2Enum_2Enum)))
=> ! [V4i: $i] : ( ap(f3245(A_27b,A_27a,V1t,V2x),V4i) = ap(ap(c_2Eiterate_2Ensum(A_27b),ap(V1t,V4i)),ap(V2x,V4i)) ) ) ) ).
tff(lamtp_f3246,type,
f3246: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3246,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t: $i] :
( mem(V1t,arr(A_27a,arr(A_27b,bool)))
=> ! [V5i: $i] :
( mem(V5i,A_27a)
=> ! [V6j: $i] : ( 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)))) ) ) ) ) ).
tff(lamtp_f3247,type,
f3247: ( del * del * $i * $i ) > $i ).
tff(lameq_f3247,axiom,
! [A_27b: del,A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t: $i] :
( mem(V1t,arr(A_27a,arr(A_27b,bool)))
=> ! [V5i: $i] : ( ap(f3247(A_27b,A_27a,V0s,V1t),V5i) = f3246(A_27a,A_27b,V0s,V1t,V5i) ) ) ) ).
tff(lamtp_f3248,type,
f3248: ( del * del * $i * $i ) > $i ).
tff(lameq_f3248,axiom,
! [A_27a: del,A_27b: del,V7i: $i] :
( mem(V7i,A_27a)
=> ! [V2x: $i] :
( mem(V2x,arr(A_27a,arr(A_27b,ty_2Enum_2Enum)))
=> ! [V8j: $i] : ( ap(f3248(A_27a,A_27b,V7i,V2x),V8j) = ap(ap(V2x,V7i),V8j) ) ) ) ).
tff(lamtp_f3249,type,
f3249: ( del * del * $i ) > $i ).
tff(lameq_f3249,axiom,
! [A_27b: del,A_27a: del,V2x: $i] :
( mem(V2x,arr(A_27a,arr(A_27b,ty_2Enum_2Enum)))
=> ! [V7i: $i] : ( ap(f3249(A_27b,A_27a,V2x),V7i) = f3248(A_27a,A_27b,V7i,V2x) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__NSUM__PRODUCT,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t: $i] :
( mem(V1t,arr(A_27a,arr(A_27b,bool)))
=> ! [V2x: $i] :
( mem(V2x,arr(A_27a,arr(A_27b,ty_2Enum_2Enum)))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& ! [V3i: $i] :
( 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))) ) ) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V0s),f3245(A_27b,A_27a,V1t,V2x))) = surj__ty_2Enum_2Enum(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)))) ) ) ) ) ) ).
tff(lamtp_f3250,type,
f3250: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3250,axiom,
! [A_27b: del,A_27a: del,V5y: $i] :
( mem(V5y,A_27b)
=> ! [V4h: $i] :
( mem(V4h,arr(A_27a,A_27b))
=> ! [V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V6x: $i] : ( 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)) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__EQ__GENERAL,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t: $i] :
( mem(V1t,arr(A_27b,bool))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V3g: $i] :
( mem(V3g,arr(A_27b,ty_2Enum_2Enum))
=> ! [V4h: $i] :
( mem(V4h,arr(A_27a,A_27b))
=> ( ( ! [V5y: $i] :
( 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: $i] :
( 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))
& ( surj__ty_2Enum_2Enum(ap(V3g,ap(V4h,V7x))) = surj__ty_2Enum_2Enum(ap(V2f,V7x)) ) ) ) ) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V0s),V2f)) = surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27b),V1t),V3g)) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__EQ__GENERAL__INVERSES,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t: $i] :
( mem(V1t,arr(A_27b,bool))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V3g: $i] :
( mem(V3g,arr(A_27b,ty_2Enum_2Enum))
=> ! [V4h: $i] :
( mem(V4h,arr(A_27a,A_27b))
=> ! [V5k: $i] :
( mem(V5k,arr(A_27b,A_27a))
=> ( ( ! [V6y: $i] :
( 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: $i] :
( 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 )
& ( surj__ty_2Enum_2Enum(ap(V3g,ap(V4h,V7x))) = surj__ty_2Enum_2Enum(ap(V2f,V7x)) ) ) ) ) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V0s),V2f)) = surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27b),V1t),V3g)) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__INJECTION,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1p: $i] :
( mem(V1p,arr(A_27a,A_27a))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& ! [V3x: $i] :
( 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: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( 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 ) ) ) ) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Enum_2Enum,A_27a),V0f),V1p))) = surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V0f)) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__UNION__NONZERO,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( 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: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t)))
=> ( surj__ty_2Enum_2Enum(ap(V0f,V3x)) = fo__c_2Enum_2E0 ) ) ) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)),V0f)) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f)),ap(ap(c_2Eiterate_2Ensum(A_27a),V2t),V0f))) ) ) ) ) ) ).
tff(lamtp_f3251,type,
f3251: ( del * $i ) > $i ).
tff(lameq_f3251,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V6t: $i] : ( ap(f3251(A_27a,V0f),V6t) = ap(ap(c_2Eiterate_2Ensum(A_27a),V6t),V0f) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__BIGUNION__NONZERO,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s: $i] :
( mem(V1s,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(arr(A_27a,bool)),V1s))
& ! [V2t: $i] :
( 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: $i] :
( mem(V3t1,arr(A_27a,bool))
=> ! [V4t2: $i] :
( mem(V4t2,arr(A_27a,bool))
=> ! [V5x: $i] :
( 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)) )
=> ( surj__ty_2Enum_2Enum(ap(V0f,V5x)) = fo__c_2Enum_2E0 ) ) ) ) ) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),V1s)),V0f)) = surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(arr(A_27a,bool)),V1s),f3251(A_27a,V0f))) ) ) ) ) ).
tff(lamtp_f3252,type,
f3252: ( del * $i * $i * $i ) > $i ).
tff(lameq_f3252,axiom,
! [A_27a: del,V2f: $i] :
( mem(V2f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1P: $i] :
( mem(V1P,arr(A_27a,bool))
=> ! [V3g: $i] :
( mem(V3g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V4x: $i] : ( 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)) ) ) ) ) ).
tff(lamtp_f3253,type,
f3253: ( del * $i * $i ) > $i ).
tff(lameq_f3253,axiom,
! [A_27a: del,V1P: $i] :
( mem(V1P,arr(A_27a,bool))
=> ! [V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V5x: $i] : ( 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))) ) ) ) ).
tff(lamtp_f3254,type,
f3254: ( del * $i * $i ) > $i ).
tff(lameq_f3254,axiom,
! [A_27a: del,V1P: $i] :
( mem(V1P,arr(A_27a,bool))
=> ! [V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V6x: $i] : ( 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)))) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__CASES,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1P: $i] :
( mem(V1P,arr(A_27a,bool))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V3g: $i] :
( mem(V3g,arr(A_27a,ty_2Enum_2Enum))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(A_27a),V0s),f3252(A_27a,V2f,V1P,V3g))) = surj__ty_2Enum_2Enum(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))) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__CLOSED,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(ty_2Enum_2Enum,bool))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(V0P,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))
& ! [V3x: tp__ty_2Enum_2Enum,V4y: tp__ty_2Enum_2Enum] :
( ( p(ap(V0P,inj__ty_2Enum_2Enum(V3x)))
& p(ap(V0P,inj__ty_2Enum_2Enum(V4y))) )
=> p(ap(V0P,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V3x)),inj__ty_2Enum_2Enum(V4y)))) )
& ! [V5a: $i] :
( 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))) ) ) ) ) ).
tff(lamtp_f3255,type,
f3255: ( $i * $i ) > $i ).
tff(lameq_f3255,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V4i: tp__ty_2Enum_2Enum] : ( ap(f3255(V0f,V1g),inj__ty_2Enum_2Enum(V4i)) = ap(ap(c_2Earithmetic_2E_2B,ap(V0f,inj__ty_2Enum_2Enum(V4i))),ap(V1g,inj__ty_2Enum_2Enum(V4i))) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__ADD__NUMSEG,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] : ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),f3255(V0f,V1g))) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V0f)),ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V1g))) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__LE__NUMSEG,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ! [V4i: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V4i)))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V4i)),inj__ty_2Enum_2Enum(V3n))) )
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(V0f,inj__ty_2Enum_2Enum(V4i))),ap(V1g,inj__ty_2Enum_2Enum(V4i)))) )
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V0f)),ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V1g))) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__EQ__NUMSEG,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ! [V4i: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V4i)))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V4i)),inj__ty_2Enum_2Enum(V3n))) )
=> ( surj__ty_2Enum_2Enum(ap(V0f,inj__ty_2Enum_2Enum(V4i))) = surj__ty_2Enum_2Enum(ap(V1g,inj__ty_2Enum_2Enum(V4i))) ) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V0f)) = surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V1g)) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__CONST__NUMSEG,axiom,
! [V0c: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] : ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),k(ty_2Enum_2Enum,inj__ty_2Enum_2Enum(V0c)))) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2A,ap(ap(c_2Earithmetic_2E_2D,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V2n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),inj__ty_2Enum_2Enum(V1m))),inj__ty_2Enum_2Enum(V0c))) ) ).
tff(conj_thm_2Eiterate_2ENSUM__EQ__0__NUMSEG,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ! [V3i: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V3i)))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V3i)),inj__ty_2Enum_2Enum(V2n))) )
=> ( surj__ty_2Enum_2Enum(ap(V0f,inj__ty_2Enum_2Enum(V3i))) = fo__c_2Enum_2E0 ) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),V0f)) = fo__c_2Enum_2E0 ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__EQ__0__IFF__NUMSEG,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),V0f)) = fo__c_2Enum_2E0 )
<=> ! [V3i: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V3i)))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V3i)),inj__ty_2Enum_2Enum(V2n))) )
=> ( surj__ty_2Enum_2Enum(ap(V0f,inj__ty_2Enum_2Enum(V3i))) = fo__c_2Enum_2E0 ) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__TRIV__NUMSEG,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V2n)),inj__ty_2Enum_2Enum(V1m)))
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),V0f)) = fo__c_2Enum_2E0 ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__SING__NUMSEG,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1n: tp__ty_2Enum_2Enum] : ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1n)),inj__ty_2Enum_2Enum(V1n))),V0f)) = surj__ty_2Enum_2Enum(ap(V0f,inj__ty_2Enum_2Enum(V1n))) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__CLAUSES__NUMSEG,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ( ! [V1m: tp__ty_2Enum_2Enum] : ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),V0f)) = surj__ty_2Enum_2Enum(ap(ap(ap(c_2Ebool_2ECOND(ty_2Enum_2Enum),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V0f,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) )
& ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] : ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n)))),V0f)) = surj__ty_2Enum_2Enum(ap(ap(ap(c_2Ebool_2ECOND(ty_2Enum_2Enum),ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V2m)),ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n)))),ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V0f)),ap(V0f,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n))))),ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V0f))) ) ) ) ).
tff(lamtp_f3256,type,
f3256: ( tp__ty_2Enum_2Enum * tp__ty_2Enum_2Enum * $i ) > $i ).
tff(lameq_f3256,axiom,
! [V2c: tp__ty_2Enum_2Enum,V3d: tp__ty_2Enum_2Enum,V4f: $i] :
( mem(V4f,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)))
=> ! [V5i: tp__ty_2Enum_2Enum] : ( ap(f3256(V2c,V3d,V4f),inj__ty_2Enum_2Enum(V5i)) = ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2c)),inj__ty_2Enum_2Enum(V3d))),ap(V4f,inj__ty_2Enum_2Enum(V5i))) ) ) ).
tff(lamtp_f3257,type,
f3257: ( $i * tp__ty_2Enum_2Enum ) > $i ).
tff(lameq_f3257,axiom,
! [V4f: $i] :
( mem(V4f,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)))
=> ! [V6j: tp__ty_2Enum_2Enum,V7i: tp__ty_2Enum_2Enum] : ( ap(f3257(V4f,V6j),inj__ty_2Enum_2Enum(V7i)) = ap(ap(V4f,inj__ty_2Enum_2Enum(V7i)),inj__ty_2Enum_2Enum(V6j)) ) ) ).
tff(lamtp_f3258,type,
f3258: ( tp__ty_2Enum_2Enum * tp__ty_2Enum_2Enum * $i ) > $i ).
tff(lameq_f3258,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum,V4f: $i] :
( mem(V4f,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)))
=> ! [V6j: tp__ty_2Enum_2Enum] : ( ap(f3258(V0a,V1b,V4f),inj__ty_2Enum_2Enum(V6j)) = ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V0a)),inj__ty_2Enum_2Enum(V1b))),f3257(V4f,V6j)) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__SWAP__NUMSEG,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum,V2c: tp__ty_2Enum_2Enum,V3d: tp__ty_2Enum_2Enum,V4f: $i] :
( mem(V4f,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)))
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V0a)),inj__ty_2Enum_2Enum(V1b))),f3256(V2c,V3d,V4f))) = surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2c)),inj__ty_2Enum_2Enum(V3d))),f3258(V0a,V1b,V4f))) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__ADD__SPLIT,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum,V3p: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1m)),ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V2n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))))
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V2n)),inj__ty_2Enum_2Enum(V3p)))),V0f)) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),V0f)),ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V2n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V2n)),inj__ty_2Enum_2Enum(V3p)))),V0f))) ) ) ) ).
tff(lamtp_f3259,type,
f3259: ( tp__ty_2Enum_2Enum * $i ) > $i ).
tff(lameq_f3259,axiom,
! [V0p: tp__ty_2Enum_2Enum,V1f: $i] :
( mem(V1f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V4i: tp__ty_2Enum_2Enum] : ( ap(f3259(V0p,V1f),inj__ty_2Enum_2Enum(V4i)) = ap(V1f,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V4i)),inj__ty_2Enum_2Enum(V0p))) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__OFFSET,axiom,
! [V0p: tp__ty_2Enum_2Enum,V1f: $i] :
( mem(V1f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] : ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V0p))),ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V3n)),inj__ty_2Enum_2Enum(V0p)))),V1f)) = surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),f3259(V0p,V1f))) ) ) ).
tff(lamtp_f3260,type,
f3260: ( tp__ty_2Enum_2Enum * $i ) > $i ).
tff(lameq_f3260,axiom,
! [V1m: tp__ty_2Enum_2Enum,V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V3i: tp__ty_2Enum_2Enum] : ( ap(f3260(V1m,V0f),inj__ty_2Enum_2Enum(V3i)) = ap(V0f,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V3i)),inj__ty_2Enum_2Enum(V1m))) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__OFFSET__0,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n)))
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),V0f)) = surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(V2n)),inj__ty_2Enum_2Enum(V1m)))),f3260(V1m,V0f))) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__CLAUSES__LEFT,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n)))
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),V0f)) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2B,ap(V0f,inj__ty_2Enum_2Enum(V1m))),ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V1m)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),inj__ty_2Enum_2Enum(V2n))),V0f))) ) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__CLAUSES__RIGHT,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V2n)))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),V0f)) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(V2n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),V0f)),ap(V0f,inj__ty_2Enum_2Enum(V2n)))) ) ) ) ).
tff(lamtp_f3261,type,
f3261: $i > $i ).
tff(lameq_f3261,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V3i: tp__ty_2Enum_2Enum] : ( ap(f3261(V0f),inj__ty_2Enum_2Enum(V3i)) = ap(ap(c_2Earithmetic_2E_2B,ap(V0f,ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),inj__ty_2Enum_2Enum(V3i)))),ap(V0f,ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),inj__ty_2Enum_2Enum(V3i))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))) ) ) ).
tff(conj_thm_2Eiterate_2ENSUM__PAIR,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] : ( surj__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,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),inj__ty_2Enum_2Enum(V1m))),ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),inj__ty_2Enum_2Enum(V2n))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),V0f)) = surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),f3261(V0f))) ) ) ).
tff(lamtp_f3262,type,
f3262: ( del * $i * tp__ty_2Enum_2Enum ) > $i ).
tff(lameq_f3262,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1n: tp__ty_2Enum_2Enum,V3i: $i] : ( ap(f3262(A_27a,V0f,V1n),V3i) = ap(ap(c_2Earithmetic_2EMOD,ap(V0f,V3i)),inj__ty_2Enum_2Enum(V1n)) ) ) ).
tff(conj_thm_2Eiterate_2EMOD__NSUM__MOD,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1n: tp__ty_2Enum_2Enum,V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& ( V1n != fo__c_2Enum_2E0 ) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2EMOD,ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),V0f)),inj__ty_2Enum_2Enum(V1n))) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2EMOD,ap(ap(c_2Eiterate_2Ensum(A_27a),V2s),f3262(A_27a,V0f,V1n))),inj__ty_2Enum_2Enum(V1n))) ) ) ) ) ).
tff(lamtp_f3263,type,
f3263: ( $i * tp__ty_2Enum_2Enum ) > $i ).
tff(lameq_f3263,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V3n: tp__ty_2Enum_2Enum,V4i: tp__ty_2Enum_2Enum] : ( ap(f3263(V0f,V3n),inj__ty_2Enum_2Enum(V4i)) = ap(ap(c_2Earithmetic_2EMOD,ap(V0f,inj__ty_2Enum_2Enum(V4i))),inj__ty_2Enum_2Enum(V3n)) ) ) ).
tff(conj_thm_2Eiterate_2EMOD__NSUM__MOD__NUMSEG,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1a: tp__ty_2Enum_2Enum,V2b: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( V3n != fo__c_2Enum_2E0 )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2EMOD,ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1a)),inj__ty_2Enum_2Enum(V2b))),V0f)),inj__ty_2Enum_2Enum(V3n))) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2EMOD,ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1a)),inj__ty_2Enum_2Enum(V2b))),f3263(V0f,V3n))),inj__ty_2Enum_2Enum(V3n))) ) ) ) ).
tff(conj_thm_2Eiterate_2ECARD__BIGUNION,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(arr(A_27a,bool)),V0s))
& ! [V1t: $i] :
( 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: $i] :
( mem(V2t,arr(A_27a,bool))
=> ! [V3u: $i] :
( 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) ) ) ) ) )
=> ( surj__ty_2Enum_2Enum(ap(c_2Epred__set_2ECARD(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),V0s))) = surj__ty_2Enum_2Enum(ap(ap(c_2Eiterate_2Ensum(arr(A_27a,bool)),V0s),c_2Epred__set_2ECARD(A_27a))) ) ) ) ).
tff(ax_thm_2Eiterate_2Esum__def,axiom,
! [A_27a: del] : ( c_2Eiterate_2ESum(A_27a) = ap(c_2Eiterate_2Eiterate(A_27a,ty_2Erealax_2Ereal),c_2Erealax_2Ereal__add) ) ).
tff(conj_thm_2Eiterate_2ENEUTRAL__REAL__ADD,axiom,
surj__ty_2Erealax_2Ereal(ap(c_2Eiterate_2Eneutral(ty_2Erealax_2Ereal),c_2Erealax_2Ereal__add)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ).
tff(conj_thm_2Eiterate_2ENEUTRAL__REAL__MUL,axiom,
surj__ty_2Erealax_2Ereal(ap(c_2Eiterate_2Eneutral(ty_2Erealax_2Ereal),c_2Erealax_2Ereal__mul)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) ).
tff(conj_thm_2Eiterate_2EMONOIDAL__REAL__ADD,axiom,
p(ap(c_2Eiterate_2Emonoidal(ty_2Erealax_2Ereal),c_2Erealax_2Ereal__add)) ).
tff(conj_thm_2Eiterate_2EMONOIDAL__REAL__MUL,axiom,
p(ap(c_2Eiterate_2Emonoidal(ty_2Erealax_2Ereal),c_2Erealax_2Ereal__mul)) ).
tff(lamtp_f3264,type,
f3264: ( del * $i * $i ) > $i ).
tff(lameq_f3264,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2x: $i] : ( 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,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))))) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__DEGENERATE,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s: $i] :
( 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))))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__CLAUSES,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),c_2Epred__set_2EEMPTY(A_27a)),V0f)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) )
& ! [V1x: $i] :
( mem(V1x,A_27b)
=> ! [V2f: $i] :
( mem(V2f,arr(A_27b,ty_2Erealax_2Ereal))
=> ! [V3s: $i] :
( mem(V3s,arr(A_27b,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27b),V3s))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27b),ap(ap(c_2Epred__set_2EINSERT(A_27b),V1x),V3s)),V2f)) = surj__ty_2Erealax_2Ereal(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)))) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__UNION,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( 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)) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)),V0f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f)),ap(ap(c_2Eiterate_2ESum(A_27a),V2t),V0f))) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__DIFF,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( 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)) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1s),V2t)),V0f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f)),ap(ap(c_2Eiterate_2ESum(A_27a),V2t),V0f))) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__INCL__EXCL,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t: $i] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2f: $i] :
( 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)) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(A_27a),V0s),V2f)),ap(ap(c_2Eiterate_2ESum(A_27a),V1t),V2f))) = surj__ty_2Erealax_2Ereal(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))) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__SUPPORT,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( surj__ty_2Erealax_2Ereal(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)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f)) ) ) ) ).
tff(lamtp_f3265,type,
f3265: ( del * $i * $i ) > $i ).
tff(lameq_f3265,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V3x: $i] : ( ap(f3265(A_27a,V0f,V1g),V3x) = ap(ap(c_2Erealax_2Ereal__add,ap(V0f,V3x)),ap(V1g,V3x)) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__ADD,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V2s),f3265(A_27a,V0f,V1g))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f)),ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1g))) ) ) ) ) ) ).
tff(lamtp_f3266,type,
f3266: ( del * $i * $i ) > $i ).
tff(lameq_f3266,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3x: $i] : ( 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,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))))) ) ) ) ).
tff(lamtp_f3267,type,
f3267: ( del * $i * $i ) > $i ).
tff(lameq_f3267,axiom,
! [A_27a: del,V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V4x: $i] : ( 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,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))))) ) ) ) ).
tff(lamtp_f3268,type,
f3268: ( del * $i * $i ) > $i ).
tff(lameq_f3268,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V5x: $i] : ( ap(f3268(A_27a,V0f,V1g),V5x) = ap(ap(c_2Erealax_2Ereal__add,ap(V0f,V5x)),ap(V1g,V5x)) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__ADD__GEN,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s: $i] :
( 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)))) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V2s),f3268(A_27a,V0f,V1g))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f)),ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1g))) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__EQ__0,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
=> ( surj__ty_2Erealax_2Ereal(ap(V0f,V2x)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__0,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V0s),k(A_27a,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ).
tff(lamtp_f3269,type,
f3269: ( del * $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3269,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1c: tp__ty_2Erealax_2Ereal,V3x: $i] : ( ap(f3269(A_27a,V0f,V1c),V3x) = ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V1c)),ap(V0f,V3x)) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__LMUL,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1c: tp__ty_2Erealax_2Ereal,V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V2s),f3269(A_27a,V0f,V1c))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V1c)),ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f))) ) ) ) ).
tff(lamtp_f3270,type,
f3270: ( del * $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3270,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1c: tp__ty_2Erealax_2Ereal,V3x: $i] : ( ap(f3270(A_27a,V0f,V1c),V3x) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0f,V3x)),inj__ty_2Erealax_2Ereal(V1c)) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__RMUL,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1c: tp__ty_2Erealax_2Ereal,V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V2s),f3270(A_27a,V0f,V1c))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f)),inj__ty_2Erealax_2Ereal(V1c))) ) ) ) ).
tff(lamtp_f3271,type,
f3271: ( del * $i ) > $i ).
tff(lameq_f3271,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2x: $i] : ( ap(f3271(A_27a,V0f),V2x) = ap(c_2Erealax_2Ereal__neg,ap(V0f,V2x)) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__NEG,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3271(A_27a,V0f))) = surj__ty_2Erealax_2Ereal(ap(c_2Erealax_2Ereal__neg,ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f))) ) ) ) ).
tff(lamtp_f3272,type,
f3272: ( del * $i * $i ) > $i ).
tff(lameq_f3272,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V3x: $i] : ( ap(f3272(A_27a,V0f,V1g),V3x) = ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,V3x)),ap(V1g,V3x)) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__SUB,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V2s),f3272(A_27a,V0f,V1g))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f)),ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1g))) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__LE,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& ! [V3x: $i] :
( 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))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__LT,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& ! [V3x: $i] :
( 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: $i] :
( 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))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__LT__ALL,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& ( V2s != c_2Epred__set_2EEMPTY(A_27a) )
& ! [V3x: $i] :
( 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))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__POS__LT,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& ! [V2x: $i] :
( 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,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V0f,V2x))) ) )
& ? [V3x: $i] :
( 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,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V0f,V3x))) ) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f))) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__POS__LT__ALL,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f: $i] :
( 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: $i] :
( 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,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V1f,V2i))) ) ) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Eiterate_2ESum(A_27a),V0s),V1f))) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__EQ,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s))
=> ( surj__ty_2Erealax_2Ereal(ap(V0f,V3x)) = surj__ty_2Erealax_2Ereal(ap(V1g,V3x)) ) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1g)) ) ) ) ) ) ).
tff(lamtp_f3273,type,
f3273: ( del * $i ) > $i ).
tff(lameq_f3273,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2x: $i] : ( ap(f3273(A_27a,V0f),V2x) = ap(c_2Ereal_2Eabs,ap(V0f,V2x)) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__ABS,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s: $i] :
( 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)))) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__ABS__LE,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& ! [V3x: $i] :
( 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))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__CONST,axiom,
! [A_27a: del,V0c: tp__ty_2Erealax_2Ereal,V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V1s),k(A_27a,inj__ty_2Erealax_2Ereal(V0c)))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V1s))),inj__ty_2Erealax_2Ereal(V0c))) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__POS__LE,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ! [V2x: $i] :
( 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,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V0f,V2x))) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f))) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__POS__BOUND,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1b: tp__ty_2Erealax_2Ereal,V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& ! [V3x: $i] :
( 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,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V0f,V3x))) ) )
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f)),inj__ty_2Erealax_2Ereal(V1b))) )
=> ! [V4x: $i] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V2s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V4x)),inj__ty_2Erealax_2Ereal(V1b))) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__POS__EQ__0,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& ! [V2x: $i] :
( 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,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V0f,V2x))) ) )
& ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) )
=> ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s))
=> ( surj__ty_2Erealax_2Ereal(ap(V0f,V3x)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__ZERO__EXISTS,axiom,
! [A_27a: del,V0u: $i] :
( mem(V0u,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0u)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) )
=> ( ! [V2i: $i] :
( mem(V2i,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2i),V1s))
=> ( surj__ty_2Erealax_2Ereal(ap(V0u,V2i)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) )
| ? [V3j: $i] :
( mem(V3j,A_27a)
& ? [V4k: $i] :
( 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,inj__ty_2Enum_2Enum(fo__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,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__DELETE,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2a: $i] :
( mem(V2a,A_27a)
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V2a),V1s)) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),ap(ap(c_2Epred__set_2EDELETE(A_27a),V1s),V2a)),V0f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f)),ap(V0f,V2a))) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__DELETE__CASES,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2a: $i] :
( mem(V2a,A_27a)
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),ap(ap(c_2Epred__set_2EDELETE(A_27a),V1s),V2a)),V0f)) = surj__ty_2Erealax_2Ereal(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))) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__SING,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),c_2Epred__set_2EEMPTY(A_27a))),V0f)) = surj__ty_2Erealax_2Ereal(ap(V0f,V1x)) ) ) ) ).
tff(lamtp_f3274,type,
f3274: ( del * tp__ty_2Erealax_2Ereal * $i ) > $i ).
tff(lameq_f3274,axiom,
! [A_27a: del,V0b: tp__ty_2Erealax_2Ereal,V2a: $i] :
( mem(V2a,A_27a)
=> ! [V3x: $i] : ( 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)),inj__ty_2Erealax_2Ereal(V0b)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__DELTA,axiom,
! [A_27a: del,V0b: tp__ty_2Erealax_2Ereal,V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2a: $i] :
( mem(V2a,A_27a)
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3274(A_27a,V0b,V2a))) = surj__ty_2Erealax_2Ereal(ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Ebool_2EIN(A_27a),V2a),V1s)),inj__ty_2Erealax_2Ereal(V0b)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) ) ) ) ).
tff(lamtp_f3275,type,
f3275: ( del * del * $i * $i ) > $i ).
tff(lameq_f3275,axiom,
! [A_27b: del,A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27b,ty_2Erealax_2Ereal)))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27b,bool))
=> ! [V3i: $i] : ( ap(f3275(A_27b,A_27a,V0f,V2t),V3i) = ap(ap(c_2Eiterate_2ESum(A_27b),V2t),ap(V0f,V3i)) ) ) ) ).
tff(lamtp_f3276,type,
f3276: ( del * del * $i * $i ) > $i ).
tff(lameq_f3276,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27b,ty_2Erealax_2Ereal)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V4j: $i] : ( ap(f3276(A_27a,A_27b,V0f,V1s),V4j) = ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3225(A_27b,A_27a,V0f,V4j)) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__SWAP,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27b,ty_2Erealax_2Ereal)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27b,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V2t)) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3275(A_27b,A_27a,V0f,V2t))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27b),V2t),f3276(A_27a,A_27b,V0f,V1s))) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27b,ty_2Erealax_2Ereal))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ! [V3x: $i] :
( mem(V3x,A_27a)
=> ! [V4y: $i] :
( 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 ) ) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V2s)),V1g)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V2s),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Erealax_2Ereal,A_27b),V1g),V0f))) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__SUPERSET,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1u: $i] :
( mem(V1u,arr(A_27a,bool))
=> ! [V2v: $i] :
( mem(V2v,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1u),V2v))
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2v))
& ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1u)) )
=> ( surj__ty_2Erealax_2Ereal(ap(V0f,V3x)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V2v),V0f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V1u),V0f)) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__UNION__RZERO,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1u: $i] :
( mem(V1u,arr(A_27a,bool))
=> ! [V2v: $i] :
( mem(V2v,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1u))
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2v))
& ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1u)) )
=> ( surj__ty_2Erealax_2Ereal(ap(V0f,V3x)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1u),V2v)),V0f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V1u),V0f)) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__UNION__LZERO,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1u: $i] :
( mem(V1u,arr(A_27a,bool))
=> ! [V2v: $i] :
( mem(V2v,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2v))
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1u))
& ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2v)) )
=> ( surj__ty_2Erealax_2Ereal(ap(V0f,V3x)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1u),V2v)),V0f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V2v),V0f)) ) ) ) ) ) ).
tff(lamtp_f3277,type,
f3277: ( del * $i * $i ) > $i ).
tff(lameq_f3277,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2x: $i] : ( 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,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__RESTRICT,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3277(A_27a,V0f,V1s))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f)) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__BOUND,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2b: tp__ty_2Erealax_2Ereal] :
( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V1f,V3x)),inj__ty_2Erealax_2Ereal(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))),inj__ty_2Erealax_2Ereal(V2b)))) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__BOUND__GEN,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2b: tp__ty_2Erealax_2Ereal] :
( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& ( V0s != c_2Epred__set_2EEMPTY(A_27a) )
& ! [V3x: $i] :
( 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,inj__ty_2Erealax_2Ereal(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)),inj__ty_2Erealax_2Ereal(V2b))) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__ABS__BOUND,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2b: tp__ty_2Erealax_2Ereal] :
( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& ! [V3x: $i] :
( 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))),inj__ty_2Erealax_2Ereal(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))),inj__ty_2Erealax_2Ereal(V2b)))) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__BOUND__LT,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2b: tp__ty_2Erealax_2Ereal] :
( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V1f,V3x)),inj__ty_2Erealax_2Ereal(V2b))) ) )
& ? [V4x: $i] :
( mem(V4x,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V0s))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(V1f,V4x)),inj__ty_2Erealax_2Ereal(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))),inj__ty_2Erealax_2Ereal(V2b)))) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__BOUND__LT__ALL,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2b: tp__ty_2Erealax_2Ereal] :
( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& ( V0s != c_2Epred__set_2EEMPTY(A_27a) )
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s))
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(V1f,V3x)),inj__ty_2Erealax_2Ereal(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))),inj__ty_2Erealax_2Ereal(V2b)))) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__BOUND__LT__GEN,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2b: tp__ty_2Erealax_2Ereal] :
( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& ( V0s != c_2Epred__set_2EEMPTY(A_27a) )
& ! [V3x: $i] :
( 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,inj__ty_2Erealax_2Ereal(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)),inj__ty_2Erealax_2Ereal(V2b))) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__UNION__EQ,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ! [V3u: $i] :
( 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 ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f)),ap(ap(c_2Eiterate_2ESum(A_27a),V2t),V0f))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V3u),V0f)) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__EQ__SUPERSET,axiom,
! [A_27a: del,V0g: $i] :
( mem(V0g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t: $i] :
( 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: $i] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V3t))
=> ( surj__ty_2Erealax_2Ereal(ap(V1f,V4x)) = surj__ty_2Erealax_2Ereal(ap(V0g,V4x)) ) ) )
& ! [V5x: $i] :
( mem(V5x,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V2s))
& ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V3t)) )
=> ( surj__ty_2Erealax_2Ereal(ap(V1f,V5x)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V3t),V0g)) ) ) ) ) ) ) ).
tff(lamtp_f3278,type,
f3278: ( del * $i * $i ) > $i ).
tff(lameq_f3278,axiom,
! [A_27a: del,V2f: $i] :
( mem(V2f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V4x: $i] : ( 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,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__RESTRICT__SET,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,ty_2Erealax_2Ereal))
=> ( surj__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)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3278(A_27a,V2f,V0P))) ) ) ) ) ).
tff(lamtp_f3279,type,
f3279: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3279,axiom,
! [A_27b: del,A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V3t: $i] :
( mem(V3t,arr(A_27b,bool))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,arr(A_27b,ty_2Erealax_2Ereal)))
=> ! [V4x: $i] : ( 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)) ) ) ) ) ).
tff(lamtp_f3280,type,
f3280: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3280,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,arr(A_27b,ty_2Erealax_2Ereal)))
=> ! [V7y: $i] : ( 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)) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__SUM__RESTRICT,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,arr(A_27b,ty_2Erealax_2Ereal)))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t: $i] :
( mem(V3t,arr(A_27b,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V3t)) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V2s),f3279(A_27b,A_27a,V0R,V3t,V1f))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27b),V3t),f3280(A_27a,A_27b,V0R,V2s,V1f))) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ECARD__EQ__SUM,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ( surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V0s))) = surj__ty_2Erealax_2Ereal(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,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))))) ) ) ) ).
tff(lamtp_f3281,type,
f3281: ( del * del * $i * $i ) > $i ).
tff(lameq_f3281,axiom,
! [A_27b: del,A_27a: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27b,bool))
=> ! [V6i: $i] : ( 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)))) ) ) ) ).
tff(lamtp_f3282,type,
f3282: ( del * $i ) > $i ).
tff(lameq_f3282,axiom,
! [A_27b: del,V3k: $i] :
( mem(V3k,arr(A_27b,ty_2Enum_2Enum))
=> ! [V8i: $i] : ( ap(f3282(A_27b,V3k),V8i) = ap(c_2Ereal_2Ereal__of__num,ap(V3k,V8i)) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__MULTICOUNT__GEN,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27b,bool))
=> ! [V3k: $i] :
( 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: $i] :
( mem(V4j,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4j),V2t))
=> ( surj__ty_2Enum_2Enum(ap(c_2Epred__set_2ECARD(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3236(A_27b,A_27a,V1s,V0R,V4j)))) = surj__ty_2Enum_2Enum(ap(V3k,V4j)) ) ) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3281(A_27b,A_27a,V0R,V2t))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27b),V2t),f3282(A_27b,V3k))) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__MULTICOUNT,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27b,bool))
=> ! [V3k: tp__ty_2Enum_2Enum] :
( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V2t))
& ! [V4j: $i] :
( mem(V4j,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4j),V2t))
=> ( surj__ty_2Enum_2Enum(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 ) ) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3281(A_27b,A_27a,V0R,V2t))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,ap(ap(c_2Earithmetic_2E_2A,inj__ty_2Enum_2Enum(V3k)),ap(c_2Epred__set_2ECARD(A_27b),V2t)))) ) ) ) ) ) ).
tff(lamtp_f3283,type,
f3283: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3283,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V3y: $i] : ( 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) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__IMAGE__GEN,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1g)) = surj__ty_2Erealax_2Ereal(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))) ) ) ) ) ) ).
tff(lamtp_f3284,type,
f3284: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f3284,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V4y: $i] : ( 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) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__GROUP,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t: $i] :
( 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)) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27b),V3t),f3284(A_27a,A_27b,V0f,V2s,V1g))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1g)) ) ) ) ) ) ) ).
tff(lamtp_f3285,type,
f3285: ( del * $i ) > $i ).
tff(lameq_f3285,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2x: $i] : ( ap(f3285(A_27a,V0f),V2x) = ap(c_2Ereal_2Ereal__of__num,ap(V0f,V2x)) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__OF__NUM__SUM,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
=> ( surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,ap(ap(c_2Eiterate_2Ensum(A_27a),V1s),V0f))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V1s),f3285(A_27a,V0f))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__SUBSET,axiom,
! [A_27a: del,V0u: $i] :
( mem(V0u,arr(A_27a,bool))
=> ! [V1v: $i] :
( mem(V1v,arr(A_27a,bool))
=> ! [V2f: $i] :
( 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: $i] :
( 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,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) ) )
& ! [V4x: $i] :
( 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,inj__ty_2Enum_2Enum(fo__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))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__SUBSET__SIMPLE,axiom,
! [A_27a: del,V0u: $i] :
( mem(V0u,arr(A_27a,bool))
=> ! [V1v: $i] :
( mem(V1v,arr(A_27a,bool))
=> ! [V2f: $i] :
( 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: $i] :
( 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,inj__ty_2Enum_2Enum(fo__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))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__IMAGE__NONZERO,axiom,
! [A_27a: del,A_27b: del,V0d: $i] :
( mem(V0d,arr(A_27b,ty_2Erealax_2Ereal))
=> ! [V1i: $i] :
( mem(V1i,arr(A_27a,A_27b))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ! [V4y: $i] :
( 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) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(V0d,ap(V1i,V3x))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1i),V2s)),V0d)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V2s),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Erealax_2Ereal,A_27b),V0d),V1i))) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__BIJECTION,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1p: $i] :
( mem(V1p,arr(A_27a,A_27a))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ( ! [V3x: $i] :
( 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: $i] :
( 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))) ) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V2s),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Erealax_2Ereal,A_27a),V0f),V1p))) ) ) ) ) ) ).
tff(lamtp_f3286,type,
f3286: ( del * del * $i * $i ) > $i ).
tff(lameq_f3286,axiom,
! [A_27b: del,A_27a: del,V1t: $i] :
( mem(V1t,arr(A_27a,arr(A_27b,bool)))
=> ! [V2x: $i] :
( mem(V2x,arr(A_27a,arr(A_27b,ty_2Erealax_2Ereal)))
=> ! [V4i: $i] : ( ap(f3286(A_27b,A_27a,V1t,V2x),V4i) = ap(ap(c_2Eiterate_2ESum(A_27b),ap(V1t,V4i)),ap(V2x,V4i)) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__SUM__PRODUCT,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t: $i] :
( mem(V1t,arr(A_27a,arr(A_27b,bool)))
=> ! [V2x: $i] :
( mem(V2x,arr(A_27a,arr(A_27b,ty_2Erealax_2Ereal)))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& ! [V3i: $i] :
( 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))) ) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V0s),f3286(A_27b,A_27a,V1t,V2x))) = surj__ty_2Erealax_2Ereal(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)))) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__EQ__GENERAL,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t: $i] :
( mem(V1t,arr(A_27b,bool))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V3g: $i] :
( mem(V3g,arr(A_27b,ty_2Erealax_2Ereal))
=> ! [V4h: $i] :
( mem(V4h,arr(A_27a,A_27b))
=> ( ( ! [V5y: $i] :
( 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: $i] :
( 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))
& ( surj__ty_2Erealax_2Ereal(ap(V3g,ap(V4h,V7x))) = surj__ty_2Erealax_2Ereal(ap(V2f,V7x)) ) ) ) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V0s),V2f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27b),V1t),V3g)) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__EQ__GENERAL__INVERSES,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t: $i] :
( mem(V1t,arr(A_27b,bool))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V3g: $i] :
( mem(V3g,arr(A_27b,ty_2Erealax_2Ereal))
=> ! [V4h: $i] :
( mem(V4h,arr(A_27a,A_27b))
=> ! [V5k: $i] :
( mem(V5k,arr(A_27b,A_27a))
=> ( ( ! [V6y: $i] :
( 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: $i] :
( 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 )
& ( surj__ty_2Erealax_2Ereal(ap(V3g,ap(V4h,V7x))) = surj__ty_2Erealax_2Ereal(ap(V2f,V7x)) ) ) ) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V0s),V2f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27b),V1t),V3g)) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__INJECTION,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1p: $i] :
( mem(V1p,arr(A_27a,A_27a))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& ! [V3x: $i] :
( 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: $i] :
( mem(V4x,A_27a)
=> ! [V5y: $i] :
( 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 ) ) ) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V2s),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Erealax_2Ereal,A_27a),V0f),V1p))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V0f)) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__UNION__NONZERO,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( 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: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),V2t)))
=> ( surj__ty_2Erealax_2Ereal(ap(V0f,V3x)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)),V0f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(A_27a),V1s),V0f)),ap(ap(c_2Eiterate_2ESum(A_27a),V2t),V0f))) ) ) ) ) ) ).
tff(lamtp_f3287,type,
f3287: ( del * $i ) > $i ).
tff(lameq_f3287,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V6t: $i] : ( ap(f3287(A_27a,V0f),V6t) = ap(ap(c_2Eiterate_2ESum(A_27a),V6t),V0f) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__BIGUNION__NONZERO,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s: $i] :
( mem(V1s,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(arr(A_27a,bool)),V1s))
& ! [V2t: $i] :
( 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: $i] :
( mem(V3t1,arr(A_27a,bool))
=> ! [V4t2: $i] :
( mem(V4t2,arr(A_27a,bool))
=> ! [V5x: $i] :
( 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)) )
=> ( surj__ty_2Erealax_2Ereal(ap(V0f,V5x)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) ) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),V1s)),V0f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(arr(A_27a,bool)),V1s),f3287(A_27a,V0f))) ) ) ) ) ).
tff(lamtp_f3288,type,
f3288: ( del * $i * $i * $i ) > $i ).
tff(lameq_f3288,axiom,
! [A_27a: del,V2f: $i] :
( mem(V2f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1P: $i] :
( mem(V1P,arr(A_27a,bool))
=> ! [V3g: $i] :
( mem(V3g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V4x: $i] : ( 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)) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__CASES,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1P: $i] :
( mem(V1P,arr(A_27a,bool))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V3g: $i] :
( mem(V3g,arr(A_27a,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V0s),f3288(A_27a,V2f,V1P,V3g))) = surj__ty_2Erealax_2Ereal(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))) ) ) ) ) ) ) ).
tff(lamtp_f3289,type,
f3289: ( del * tp__ty_2Erealax_2Ereal * $i * $i ) > $i ).
tff(lameq_f3289,axiom,
! [A_27a: del,V0y: tp__ty_2Erealax_2Ereal,V3a: $i] :
( mem(V3a,A_27a)
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V4x: $i] : ( 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)),inj__ty_2Erealax_2Ereal(V0y)),ap(V1f,V4x)) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__CASES__1,axiom,
! [A_27a: del,V0y: tp__ty_2Erealax_2Ereal,V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3a: $i] :
( mem(V3a,A_27a)
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V3a),V2s)) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V2s),f3289(A_27a,V0y,V3a,V1f))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1f)),ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V0y)),ap(V1f,V3a)))) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__LE__INCLUDED,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27b,ty_2Erealax_2Ereal))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t: $i] :
( mem(V3t,arr(A_27b,bool))
=> ! [V4i: $i] :
( 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: $i] :
( 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,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V1g,V5y))) ) )
& ! [V6x: $i] :
( mem(V6x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V6x),V2s))
=> ? [V7y: $i] :
( 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))) ) ) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__IMAGE__LE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27b,ty_2Erealax_2Ereal))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
& ! [V3x: $i] :
( 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,inj__ty_2Enum_2Enum(fo__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)))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__CLOSED,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(V0P,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))
& ! [V3x: tp__ty_2Erealax_2Ereal,V4y: tp__ty_2Erealax_2Ereal] :
( ( p(ap(V0P,inj__ty_2Erealax_2Ereal(V3x)))
& p(ap(V0P,inj__ty_2Erealax_2Ereal(V4y))) )
=> p(ap(V0P,ap(ap(c_2Erealax_2Ereal__add,inj__ty_2Erealax_2Ereal(V3x)),inj__ty_2Erealax_2Ereal(V4y)))) )
& ! [V5a: $i] :
( 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))) ) ) ) ) ).
tff(lamtp_f3290,type,
f3290: ( $i * $i ) > $i ).
tff(lameq_f3290,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4i: tp__ty_2Enum_2Enum] : ( ap(f3290(V0f,V1g),inj__ty_2Enum_2Enum(V4i)) = ap(ap(c_2Erealax_2Ereal__add,ap(V0f,inj__ty_2Enum_2Enum(V4i))),ap(V1g,inj__ty_2Enum_2Enum(V4i))) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__ADD__NUMSEG,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] : ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),f3290(V0f,V1g))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V0f)),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V1g))) ) ) ) ).
tff(lamtp_f3291,type,
f3291: ( $i * $i ) > $i ).
tff(lameq_f3291,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4i: tp__ty_2Enum_2Enum] : ( ap(f3291(V0f,V1g),inj__ty_2Enum_2Enum(V4i)) = ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,inj__ty_2Enum_2Enum(V4i))),ap(V1g,inj__ty_2Enum_2Enum(V4i))) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__SUB__NUMSEG,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] : ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),f3291(V0f,V1g))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V0f)),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V1g))) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__LE__NUMSEG,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ! [V4i: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V4i)))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V4i)),inj__ty_2Enum_2Enum(V3n))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,inj__ty_2Enum_2Enum(V4i))),ap(V1g,inj__ty_2Enum_2Enum(V4i)))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V0f)),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V1g))) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__EQ__NUMSEG,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ! [V4i: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V4i)))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V4i)),inj__ty_2Enum_2Enum(V3n))) )
=> ( surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Enum_2Enum(V4i))) = surj__ty_2Erealax_2Ereal(ap(V1g,inj__ty_2Enum_2Enum(V4i))) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V0f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V1g)) ) ) ) ) ).
tff(lamtp_f3292,type,
f3292: $i > $i ).
tff(lameq_f3292,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3i: tp__ty_2Enum_2Enum] : ( ap(f3292(V0f),inj__ty_2Enum_2Enum(V3i)) = ap(c_2Ereal_2Eabs,ap(V0f,inj__ty_2Enum_2Enum(V3i))) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__ABS__NUMSEG,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__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,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),V0f))),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),f3292(V0f)))) ) ).
tff(conj_thm_2Eiterate_2ESUM__CONST__NUMSEG,axiom,
! [V0c: tp__ty_2Erealax_2Ereal,V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] : ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),k(ty_2Enum_2Enum,inj__ty_2Erealax_2Ereal(V0c)))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,ap(ap(c_2Earithmetic_2E_2D,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V2n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),inj__ty_2Enum_2Enum(V1m)))),inj__ty_2Erealax_2Ereal(V0c))) ) ).
tff(conj_thm_2Eiterate_2ESUM__EQ__0__NUMSEG,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ! [V3i: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V3i)))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V3i)),inj__ty_2Enum_2Enum(V2n))) )
=> ( surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Enum_2Enum(V3i))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),V0f)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__TRIV__NUMSEG,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V2n)),inj__ty_2Enum_2Enum(V1m)))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),V0f)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__POS__LE__NUMSEG,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum,V2f: $i] :
( mem(V2f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ! [V3p: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V3p)))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V3p)),inj__ty_2Enum_2Enum(V1n))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V2f,inj__ty_2Enum_2Enum(V3p)))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n))),V2f))) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__POS__EQ__0__NUMSEG,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( ! [V3p: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V3p)))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V3p)),inj__ty_2Enum_2Enum(V2n))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V0f,inj__ty_2Enum_2Enum(V3p)))) )
& ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),V0f)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) )
=> ! [V4p: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V4p)))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V4p)),inj__ty_2Enum_2Enum(V2n))) )
=> ( surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Enum_2Enum(V4p))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__SING__NUMSEG,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1n: tp__ty_2Enum_2Enum] : ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1n)),inj__ty_2Enum_2Enum(V1n))),V0f)) = surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Enum_2Enum(V1n))) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__CLAUSES__NUMSEG,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ! [V1m: tp__ty_2Enum_2Enum] : ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),V0f)) = surj__ty_2Erealax_2Ereal(ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V0f,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) )
& ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] : ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n)))),V0f)) = surj__ty_2Erealax_2Ereal(ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V2m)),ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n)))),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V0f)),ap(V0f,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n))))),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V0f))) ) ) ) ).
tff(lamtp_f3293,type,
f3293: ( tp__ty_2Enum_2Enum * tp__ty_2Enum_2Enum * $i ) > $i ).
tff(lameq_f3293,axiom,
! [V2c: tp__ty_2Enum_2Enum,V3d: tp__ty_2Enum_2Enum,V4f: $i] :
( mem(V4f,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)))
=> ! [V5i: tp__ty_2Enum_2Enum] : ( ap(f3293(V2c,V3d,V4f),inj__ty_2Enum_2Enum(V5i)) = ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2c)),inj__ty_2Enum_2Enum(V3d))),ap(V4f,inj__ty_2Enum_2Enum(V5i))) ) ) ).
tff(lamtp_f3294,type,
f3294: ( tp__ty_2Enum_2Enum * tp__ty_2Enum_2Enum * $i ) > $i ).
tff(lameq_f3294,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum,V4f: $i] :
( mem(V4f,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)))
=> ! [V6j: tp__ty_2Enum_2Enum] : ( ap(f3294(V0a,V1b,V4f),inj__ty_2Enum_2Enum(V6j)) = ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V0a)),inj__ty_2Enum_2Enum(V1b))),f3257(V4f,V6j)) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__SWAP__NUMSEG,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum,V2c: tp__ty_2Enum_2Enum,V3d: tp__ty_2Enum_2Enum,V4f: $i] :
( mem(V4f,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V0a)),inj__ty_2Enum_2Enum(V1b))),f3293(V2c,V3d,V4f))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2c)),inj__ty_2Enum_2Enum(V3d))),f3294(V0a,V1b,V4f))) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__ADD__SPLIT,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum,V3p: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1m)),ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V2n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V2n)),inj__ty_2Enum_2Enum(V3p)))),V0f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),V0f)),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V2n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V2n)),inj__ty_2Enum_2Enum(V3p)))),V0f))) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__OFFSET,axiom,
! [V0p: tp__ty_2Enum_2Enum,V1f: $i] :
( mem(V1f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] : ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V0p))),ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V3n)),inj__ty_2Enum_2Enum(V0p)))),V1f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),f3259(V0p,V1f))) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__OFFSET__0,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n)))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),V0f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(V2n)),inj__ty_2Enum_2Enum(V1m)))),f3260(V1m,V0f))) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__CLAUSES__LEFT,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n)))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),V0f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(V0f,inj__ty_2Enum_2Enum(V1m))),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V1m)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),inj__ty_2Enum_2Enum(V2n))),V0f))) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__CLAUSES__RIGHT,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V2n)))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),V0f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(V2n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),V0f)),ap(V0f,inj__ty_2Enum_2Enum(V2n)))) ) ) ) ).
tff(lamtp_f3295,type,
f3295: $i > $i ).
tff(lameq_f3295,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3i: tp__ty_2Enum_2Enum] : ( ap(f3295(V0f),inj__ty_2Enum_2Enum(V3i)) = ap(ap(c_2Erealax_2Ereal__add,ap(V0f,ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),inj__ty_2Enum_2Enum(V3i)))),ap(V0f,ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),inj__ty_2Enum_2Enum(V3i))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__PAIR,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] : ( surj__ty_2Erealax_2Ereal(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,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),inj__ty_2Enum_2Enum(V1m))),ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Earithmetic_2E_2A,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),inj__ty_2Enum_2Enum(V2n))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),V0f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),f3295(V0f))) ) ) ).
tff(lamtp_f3296,type,
f3296: $i > $i ).
tff(lameq_f3296,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V3i: tp__ty_2Enum_2Enum] : ( ap(f3296(V0f),inj__ty_2Enum_2Enum(V3i)) = ap(c_2Ereal_2Ereal__of__num,ap(V0f,inj__ty_2Enum_2Enum(V3i))) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__OF__NUM__SUM__NUMSEG,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] : ( surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,ap(ap(c_2Eiterate_2Ensum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),V0f))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),f3296(V0f))) ) ) ).
tff(lamtp_f3297,type,
f3297: ( $i * $i ) > $i ).
tff(lameq_f3297,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4k: tp__ty_2Enum_2Enum] : ( ap(f3297(V0f,V1g),inj__ty_2Enum_2Enum(V4k)) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0f,inj__ty_2Enum_2Enum(V4k))),ap(ap(c_2Ereal_2Ereal__sub,ap(V1g,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V4k)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),ap(V1g,inj__ty_2Enum_2Enum(V4k)))) ) ) ) ).
tff(lamtp_f3298,type,
f3298: ( $i * $i ) > $i ).
tff(lameq_f3298,axiom,
! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V5k: tp__ty_2Enum_2Enum] : ( ap(f3298(V1g,V0f),inj__ty_2Enum_2Enum(V5k)) = ap(ap(c_2Erealax_2Ereal__mul,ap(V1g,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V5k)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V5k)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),ap(V0f,inj__ty_2Enum_2Enum(V5k)))) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__PARTIAL__SUC,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] : ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),f3297(V0f,V1g))) = surj__ty_2Erealax_2Ereal(ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(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,inj__ty_2Enum_2Enum(V3n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),ap(V1g,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V3n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))))),ap(ap(c_2Erealax_2Ereal__mul,ap(V0f,inj__ty_2Enum_2Enum(V2m))),ap(V1g,inj__ty_2Enum_2Enum(V2m))))),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),f3298(V1g,V0f)))),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) ) ) ) ).
tff(lamtp_f3299,type,
f3299: ( $i * $i ) > $i ).
tff(lameq_f3299,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4k: tp__ty_2Enum_2Enum] : ( ap(f3299(V0f,V1g),inj__ty_2Enum_2Enum(V4k)) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0f,inj__ty_2Enum_2Enum(V4k))),ap(ap(c_2Ereal_2Ereal__sub,ap(V1g,inj__ty_2Enum_2Enum(V4k))),ap(V1g,ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(V4k)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))))) ) ) ) ).
tff(lamtp_f3300,type,
f3300: ( $i * $i ) > $i ).
tff(lameq_f3300,axiom,
! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V5k: tp__ty_2Enum_2Enum] : ( ap(f3300(V1g,V0f),inj__ty_2Enum_2Enum(V5k)) = ap(ap(c_2Erealax_2Ereal__mul,ap(V1g,inj__ty_2Enum_2Enum(V5k))),ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V5k)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),ap(V0f,inj__ty_2Enum_2Enum(V5k)))) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__PARTIAL__PRE,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] : ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),f3299(V0f,V1g))) = surj__ty_2Erealax_2Ereal(ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(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,inj__ty_2Enum_2Enum(V3n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),ap(V1g,inj__ty_2Enum_2Enum(V3n)))),ap(ap(c_2Erealax_2Ereal__mul,ap(V0f,inj__ty_2Enum_2Enum(V2m))),ap(V1g,ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(V2m)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))))),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),f3300(V1g,V0f)))),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) ) ) ) ).
tff(lamtp_f3301,type,
f3301: $i > $i ).
tff(lameq_f3301,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3k: tp__ty_2Enum_2Enum] : ( ap(f3301(V0f),inj__ty_2Enum_2Enum(V3k)) = ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,inj__ty_2Enum_2Enum(V3k))),ap(V0f,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V3k)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__DIFFS,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] : ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),f3301(V0f))) = surj__ty_2Erealax_2Ereal(ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,inj__ty_2Enum_2Enum(V1m))),ap(V0f,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V2n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))))),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) ) ) ).
tff(lamtp_f3302,type,
f3302: $i > $i ).
tff(lameq_f3302,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3k: tp__ty_2Enum_2Enum] : ( ap(f3302(V0f),inj__ty_2Enum_2Enum(V3k)) = ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V3k)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),ap(V0f,inj__ty_2Enum_2Enum(V3k))) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__DIFFS__ALT,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] : ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),f3302(V0f))) = surj__ty_2Erealax_2Ereal(ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V2n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),ap(V0f,inj__ty_2Enum_2Enum(V1m)))),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__COMBINE__R,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum,V3p: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1m)),ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V2n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V2n)),inj__ty_2Enum_2Enum(V3p))) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),V0f)),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V2n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),inj__ty_2Enum_2Enum(V3p))),V0f))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V3p))),V0f)) ) ) ) ).
tff(conj_thm_2Eiterate_2ESUM__COMBINE__L,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum,V3p: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V2n)))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n)))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V2n)),ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V3p)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(V2n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),V0f)),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2n)),inj__ty_2Enum_2Enum(V3p))),V0f))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V3p))),V0f)) ) ) ) ).
tff(lamtp_f3303,type,
f3303: ( del * $i ) > $i ).
tff(lameq_f3303,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V4i: $i] : ( ap(f3303(A_27a,V0f),V4i) = ap(V0f,V4i) ) ) ).
tff(lamtp_f3304,type,
f3304: $i > $i ).
tff(lameq_f3304,axiom,
! [V5f: $i] :
( mem(V5f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V10i: tp__ty_2Enum_2Enum] : ( ap(f3304(V5f),inj__ty_2Enum_2Enum(V10i)) = ap(V5f,inj__ty_2Enum_2Enum(V10i)) ) ) ).
tff(lamtp_f3305,type,
f3305: ( del * $i ) > $i ).
tff(lameq_f3305,axiom,
! [A_27b: del,V13p: $i] :
( mem(V13p,arr(A_27b,bool))
=> ! [V15y: $i] : ( ap(f3305(A_27b,V13p),V15y) = ap(ap(c_2Epair_2E_2C(A_27b,bool),V15y),ap(V13p,V15y)) ) ) ).
tff(lamtp_f3306,type,
f3306: ( del * $i ) > $i ).
tff(lameq_f3306,axiom,
! [A_27b: del,V11f: $i] :
( mem(V11f,arr(A_27b,ty_2Erealax_2Ereal))
=> ! [V16i: $i] : ( ap(f3306(A_27b,V11f),V16i) = ap(V11f,V16i) ) ) ).
tff(lamtp_f3307,type,
f3307: ( del * $i ) > $i ).
tff(lameq_f3307,axiom,
! [A_27b: del,V13p: $i] :
( mem(V13p,arr(A_27b,bool))
=> ! [V17y: $i] : ( ap(f3307(A_27b,V13p),V17y) = ap(ap(c_2Epair_2E_2C(A_27b,bool),V17y),ap(V13p,V17y)) ) ) ).
tff(conj_thm_2Eiterate_2ETH,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2s: $i] :
( mem(V2s,arr(A_27a,bool))
=> ( ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s))
=> ( surj__ty_2Erealax_2Ereal(ap(V0f,V3x)) = surj__ty_2Erealax_2Ereal(ap(V1g,V3x)) ) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V2s),f3303(A_27a,V0f))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27a),V2s),V1g)) ) ) ) ) )
& ! [V5f: $i] :
( mem(V5f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V6g: $i] :
( mem(V6g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V7a: tp__ty_2Enum_2Enum,V8b: tp__ty_2Enum_2Enum] :
( ! [V9i: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V7a)),inj__ty_2Enum_2Enum(V9i)))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V9i)),inj__ty_2Enum_2Enum(V8b))) )
=> ( surj__ty_2Erealax_2Ereal(ap(V5f,inj__ty_2Enum_2Enum(V9i))) = surj__ty_2Erealax_2Ereal(ap(V6g,inj__ty_2Enum_2Enum(V9i))) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V7a)),inj__ty_2Enum_2Enum(V8b))),f3304(V5f))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V7a)),inj__ty_2Enum_2Enum(V8b))),V6g)) ) ) ) )
& ! [V11f: $i] :
( mem(V11f,arr(A_27b,ty_2Erealax_2Ereal))
=> ! [V12g: $i] :
( mem(V12g,arr(A_27b,ty_2Erealax_2Ereal))
=> ! [V13p: $i] :
( mem(V13p,arr(A_27b,bool))
=> ( ! [V14x: $i] :
( mem(V14x,A_27b)
=> ( p(ap(V13p,V14x))
=> ( surj__ty_2Erealax_2Ereal(ap(V11f,V14x)) = surj__ty_2Erealax_2Ereal(ap(V12g,V14x)) ) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27b),ap(c_2Epred__set_2EGSPEC(A_27b,A_27b),f3305(A_27b,V13p))),f3306(A_27b,V11f))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(A_27b),ap(c_2Epred__set_2EGSPEC(A_27b,A_27b),f3307(A_27b,V13p))),V12g)) ) ) ) ) ) ) ).
tff(lamtp_f3308,type,
f3308: ( tp__ty_2Erealax_2Ereal * tp__ty_2Enum_2Enum * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3308,axiom,
! [V1y: tp__ty_2Erealax_2Ereal,V2n: tp__ty_2Enum_2Enum,V0x: tp__ty_2Erealax_2Ereal,V3i: tp__ty_2Enum_2Enum] : ( ap(f3308(V1y,V2n,V0x),inj__ty_2Enum_2Enum(V3i)) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Enum_2Enum(V3i))),ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V1y)),ap(ap(c_2Earithmetic_2E_2D,ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(V2n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),inj__ty_2Enum_2Enum(V3i)))) ) ).
tff(conj_thm_2Eiterate_2EREAL__SUB__POW,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),inj__ty_2Enum_2Enum(V2n)))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Enum_2Enum(V2n))),ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V1y)),inj__ty_2Enum_2Enum(V2n)))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Erealax_2Ereal(V1y))),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(V2n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),f3308(V1y,V2n,V0x)))) ) ) ).
tff(lamtp_f3309,type,
f3309: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f3309,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V2i: tp__ty_2Enum_2Enum] : ( ap(f3309(V0x),inj__ty_2Enum_2Enum(V2i)) = ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Enum_2Enum(V2i)) ) ).
tff(conj_thm_2Eiterate_2EREAL__SUB__POW__R1,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),inj__ty_2Enum_2Enum(V1n)))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Enum_2Enum(V1n))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V0x)),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(V1n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),f3309(V0x)))) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__SUB__POW__L1,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),inj__ty_2Enum_2Enum(V1n)))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Ereal__sub,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Enum_2Enum(V1n)))) = surj__ty_2Erealax_2Ereal(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,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),inj__ty_2Erealax_2Ereal(V0x))),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(V1n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),f3309(V0x)))) ) ) ).
tff(lamtp_f3310,type,
f3310: ( del * del * $i * $i ) > $i ).
tff(lameq_f3310,axiom,
! [A_27b: del,A_27a: del,V2f: $i] :
( mem(V2f,arr(A_27a,A_27b))
=> ! [V1P: $i] :
( mem(V1P,arr(A_27a,bool))
=> ! [V4x: $i] : ( ap(f3310(A_27b,A_27a,V2f,V1P),V4x) = ap(ap(c_2Epair_2E_2C(A_27b,bool),ap(V2f,V4x)),ap(V1P,V4x)) ) ) ) ).
tff(lamtp_f3311,type,
f3311: ( del * del * del * $i * $i * $i ) > $i ).
tff(lameq_f3311,axiom,
! [A_27c: del,A_27b: del,A_27d: del,V7f: $i] :
( mem(V7f,arr(A_27c,arr(A_27d,A_27b)))
=> ! [V9x: $i] :
( mem(V9x,A_27c)
=> ! [V6P: $i] :
( mem(V6P,arr(A_27c,arr(A_27d,bool)))
=> ! [V10y: $i] : ( 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)) ) ) ) ) ).
tff(lamtp_f3312,type,
f3312: ( del * del * del * $i * $i ) > $i ).
tff(lameq_f3312,axiom,
! [A_27d: del,A_27b: del,A_27c: del,V7f: $i] :
( mem(V7f,arr(A_27c,arr(A_27d,A_27b)))
=> ! [V6P: $i] :
( mem(V6P,arr(A_27c,arr(A_27d,bool)))
=> ! [V9x: $i] : ( ap(f3312(A_27d,A_27b,A_27c,V7f,V6P),V9x) = f3311(A_27c,A_27b,A_27d,V7f,V9x,V6P) ) ) ) ).
tff(lamtp_f3313,type,
f3313: ( del * del * del * del * $i * $i * $i * $i ) > $i ).
tff(lameq_f3313,axiom,
! [A_27e: del,A_27f: del,A_27b: del,A_27g: del,V14f: $i] :
( mem(V14f,arr(A_27e,arr(A_27f,arr(A_27g,A_27b))))
=> ! [V17x: $i] :
( mem(V17x,A_27f)
=> ! [V13P: $i] :
( mem(V13P,arr(A_27e,arr(A_27f,arr(A_27g,bool))))
=> ! [V16w: $i] :
( mem(V16w,A_27e)
=> ! [V18y: $i] : ( 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)) ) ) ) ) ) ).
tff(lamtp_f3314,type,
f3314: ( del * del * del * del * $i * $i * $i ) > $i ).
tff(lameq_f3314,axiom,
! [A_27g: del,A_27b: del,A_27e: del,A_27f: del,V14f: $i] :
( mem(V14f,arr(A_27e,arr(A_27f,arr(A_27g,A_27b))))
=> ! [V13P: $i] :
( mem(V13P,arr(A_27e,arr(A_27f,arr(A_27g,bool))))
=> ! [V16w: $i] :
( mem(V16w,A_27e)
=> ! [V17x: $i] : ( 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) ) ) ) ) ).
tff(lamtp_f3315,type,
f3315: ( del * del * del * del * $i * $i ) > $i ).
tff(lameq_f3315,axiom,
! [A_27f: del,A_27b: del,A_27g: del,A_27e: del,V13P: $i] :
( mem(V13P,arr(A_27e,arr(A_27f,arr(A_27g,bool))))
=> ! [V14f: $i] :
( mem(V14f,arr(A_27e,arr(A_27f,arr(A_27g,A_27b))))
=> ! [V16w: $i] : ( 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)) ) ) ) ).
tff(conj_thm_2Eiterate_2EFORALL__IN__GSPEC,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del,A_27g: del,V0Q: $i] :
( mem(V0Q,arr(A_27b,bool))
=> ( ! [V1P: $i] :
( mem(V1P,arr(A_27a,bool))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27b))
=> ( ! [V3z: $i] :
( 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: $i] :
( mem(V5x,A_27a)
=> ( p(ap(V1P,V5x))
=> p(ap(V0Q,ap(V2f,V5x))) ) ) ) ) )
& ! [V6P: $i] :
( mem(V6P,arr(A_27c,arr(A_27d,bool)))
=> ! [V7f: $i] :
( mem(V7f,arr(A_27c,arr(A_27d,A_27b)))
=> ( ! [V8z: $i] :
( 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: $i] :
( mem(V11x,A_27c)
=> ! [V12y: $i] :
( mem(V12y,A_27d)
=> ( p(ap(ap(V6P,V11x),V12y))
=> p(ap(V0Q,ap(ap(V7f,V11x),V12y))) ) ) ) ) ) )
& ! [V13P: $i] :
( mem(V13P,arr(A_27e,arr(A_27f,arr(A_27g,bool))))
=> ! [V14f: $i] :
( mem(V14f,arr(A_27e,arr(A_27f,arr(A_27g,A_27b))))
=> ( ! [V15z: $i] :
( 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: $i] :
( mem(V19w,A_27e)
=> ! [V20x: $i] :
( mem(V20x,A_27f)
=> ! [V21y: $i] :
( mem(V21y,A_27g)
=> ( p(ap(ap(ap(V13P,V19w),V20x),V21y))
=> p(ap(V0Q,ap(ap(ap(V14f,V19w),V20x),V21y))) ) ) ) ) ) ) ) ) ) ).
tff(lamtp_f3316,type,
f3316: ( $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3316,axiom,
! [V0a: $i] :
( mem(V0a,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x: tp__ty_2Erealax_2Ereal,V4i: tp__ty_2Enum_2Enum] : ( ap(f3316(V0a,V1x),inj__ty_2Enum_2Enum(V4i)) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0a,inj__ty_2Enum_2Enum(V4i))),ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Enum_2Enum(V4i))) ) ) ).
tff(lamtp_f3317,type,
f3317: ( $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3317,axiom,
! [V0a: $i] :
( mem(V0a,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2y: tp__ty_2Erealax_2Ereal,V5i: tp__ty_2Enum_2Enum] : ( ap(f3317(V0a,V2y),inj__ty_2Enum_2Enum(V5i)) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0a,inj__ty_2Enum_2Enum(V5i))),ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V2y)),inj__ty_2Enum_2Enum(V5i))) ) ) ).
tff(lamtp_f3318,type,
f3318: ( tp__ty_2Erealax_2Ereal * tp__ty_2Enum_2Enum * $i ) > $i ).
tff(lameq_f3318,axiom,
! [V2y: tp__ty_2Erealax_2Ereal,V6j: tp__ty_2Enum_2Enum,V0a: $i] :
( mem(V0a,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V7i: tp__ty_2Enum_2Enum] : ( ap(f3318(V2y,V6j,V0a),inj__ty_2Enum_2Enum(V7i)) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0a,inj__ty_2Enum_2Enum(V7i))),ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V2y)),ap(ap(c_2Earithmetic_2E_2D,ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(V7i)),inj__ty_2Enum_2Enum(V6j))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))) ) ) ).
tff(lamtp_f3319,type,
f3319: ( $i * tp__ty_2Erealax_2Ereal * tp__ty_2Enum_2Enum * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3319,axiom,
! [V0a: $i] :
( mem(V0a,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2y: tp__ty_2Erealax_2Ereal,V3n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal,V6j: tp__ty_2Enum_2Enum] : ( ap(f3319(V0a,V2y,V3n,V1x),inj__ty_2Enum_2Enum(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,inj__ty_2Enum_2Enum(V6j)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),inj__ty_2Enum_2Enum(V3n))),f3318(V2y,V6j,V0a))),ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Enum_2Enum(V6j))) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__SUB__POLYFUN,axiom,
! [V0a: $i] :
( mem(V0a,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal,V3n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),inj__ty_2Enum_2Enum(V3n)))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V3n))),f3316(V0a,V1x))),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V3n))),f3317(V0a,V2y)))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Erealax_2Ereal(V2y))),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(V3n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),f3319(V0a,V2y,V3n,V1x)))) ) ) ) ).
tff(lamtp_f3320,type,
f3320: ( tp__ty_2Enum_2Enum * $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3320,axiom,
! [V6j: tp__ty_2Enum_2Enum,V0a: $i] :
( mem(V0a,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2y: tp__ty_2Erealax_2Ereal,V7k: tp__ty_2Enum_2Enum] : ( ap(f3320(V6j,V0a,V2y),inj__ty_2Enum_2Enum(V7k)) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0a,ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V6j)),inj__ty_2Enum_2Enum(V7k))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V2y)),inj__ty_2Enum_2Enum(V7k))) ) ) ).
tff(lamtp_f3321,type,
f3321: ( tp__ty_2Erealax_2Ereal * $i * tp__ty_2Enum_2Enum * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3321,axiom,
! [V2y: tp__ty_2Erealax_2Ereal,V0a: $i] :
( mem(V0a,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal,V6j: tp__ty_2Enum_2Enum] : ( ap(f3321(V2y,V0a,V3n,V1x),inj__ty_2Enum_2Enum(V6j)) = ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),ap(ap(c_2Earithmetic_2E_2D,ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(V3n)),inj__ty_2Enum_2Enum(V6j))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),f3320(V6j,V0a,V2y))),ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Enum_2Enum(V6j))) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__SUB__POLYFUN__ALT,axiom,
! [V0a: $i] :
( mem(V0a,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal,V3n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),inj__ty_2Enum_2Enum(V3n)))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V3n))),f3316(V0a,V1x))),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V3n))),f3317(V0a,V2y)))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Erealax_2Ereal(V2y))),ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(V3n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),f3321(V2y,V0a,V3n,V1x)))) ) ) ) ).
tff(lamtp_f3322,type,
f3322: ( $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3322,axiom,
! [V1c: $i] :
( mem(V1c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3x: tp__ty_2Erealax_2Ereal,V4i: tp__ty_2Enum_2Enum] : ( ap(f3322(V1c,V3x),inj__ty_2Enum_2Enum(V4i)) = ap(ap(c_2Erealax_2Ereal__mul,ap(V1c,inj__ty_2Enum_2Enum(V4i))),ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V3x)),inj__ty_2Enum_2Enum(V4i))) ) ) ).
tff(lamtp_f3323,type,
f3323: ( tp__ty_2Enum_2Enum * $i ) > $i ).
tff(lameq_f3323,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1c: $i] :
( mem(V1c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3x: tp__ty_2Erealax_2Ereal] : ( ap(f3323(V0n,V1c),inj__ty_2Erealax_2Ereal(V3x)) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),inj__ty_2Erealax_2Ereal(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,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V0n))),f3322(V1c,V3x))),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) ) ) ).
tff(lamtp_f3324,type,
f3324: ( $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3324,axiom,
! [V1c: $i] :
( mem(V1c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V5x: tp__ty_2Erealax_2Ereal,V6i: tp__ty_2Enum_2Enum] : ( ap(f3324(V1c,V5x),inj__ty_2Enum_2Enum(V6i)) = ap(ap(c_2Erealax_2Ereal__mul,ap(V1c,inj__ty_2Enum_2Enum(V6i))),ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V5x)),inj__ty_2Enum_2Enum(V6i))) ) ) ).
tff(lamtp_f3325,type,
f3325: ( tp__ty_2Enum_2Enum * $i ) > $i ).
tff(lameq_f3325,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1c: $i] :
( mem(V1c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V5x: tp__ty_2Erealax_2Ereal] : ( ap(f3325(V0n,V1c),inj__ty_2Erealax_2Ereal(V5x)) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),inj__ty_2Erealax_2Ereal(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,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V0n))),f3324(V1c,V5x))),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__POLYFUN__ROOTBOUND,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1c: $i] :
( mem(V1c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ~ ! [V2i: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V2i)),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V0n))))
=> ( surj__ty_2Erealax_2Ereal(ap(V1c,inj__ty_2Enum_2Enum(V2i))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__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)))),inj__ty_2Enum_2Enum(V0n))) ) ) ) ).
tff(lamtp_f3326,type,
f3326: ( $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3326,axiom,
! [V1c: $i] :
( mem(V1c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2x: tp__ty_2Erealax_2Ereal,V3i: tp__ty_2Enum_2Enum] : ( ap(f3326(V1c,V2x),inj__ty_2Enum_2Enum(V3i)) = ap(ap(c_2Erealax_2Ereal__mul,ap(V1c,inj__ty_2Enum_2Enum(V3i))),ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Enum_2Enum(V3i))) ) ) ).
tff(lamtp_f3327,type,
f3327: ( tp__ty_2Enum_2Enum * $i ) > $i ).
tff(lameq_f3327,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1c: $i] :
( mem(V1c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2x: tp__ty_2Erealax_2Ereal] : ( ap(f3327(V0n,V1c),inj__ty_2Erealax_2Ereal(V2x)) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),inj__ty_2Erealax_2Ereal(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,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V0n))),f3326(V1c,V2x))),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__POLYFUN__FINITE__ROOTS,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1c: $i] :
( 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: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V4i)),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V0n))))
& ( surj__ty_2Erealax_2Ereal(ap(V1c,inj__ty_2Enum_2Enum(V4i))) != surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__POLYFUN__EQ__0,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1c: $i] :
( mem(V1c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ! [V2x: tp__ty_2Erealax_2Ereal] : ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V0n))),f3326(V1c,V2x))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) )
<=> ! [V4i: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V4i)),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V0n))))
=> ( surj__ty_2Erealax_2Ereal(ap(V1c,inj__ty_2Enum_2Enum(V4i))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EREAL__POLYFUN__EQ__CONST,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1c: $i] :
( mem(V1c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2k: tp__ty_2Erealax_2Ereal] :
( ! [V3x: tp__ty_2Erealax_2Ereal] : ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V0n))),f3322(V1c,V3x))) = V2k )
<=> ( ( surj__ty_2Erealax_2Ereal(ap(V1c,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) = V2k )
& ! [V5i: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V5i)),ap(ap(c_2Eiterate_2E_2E_2E,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),inj__ty_2Enum_2Enum(V0n))))
=> ( surj__ty_2Erealax_2Ereal(ap(V1c,inj__ty_2Enum_2Enum(V5i))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) ) ) ).
tff(lamtp_f3328,type,
f3328: ( $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3328,axiom,
! [V2c: $i] :
( mem(V2c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3x: tp__ty_2Erealax_2Ereal,V4i: tp__ty_2Enum_2Enum] : ( ap(f3328(V2c,V3x),inj__ty_2Enum_2Enum(V4i)) = ap(ap(c_2Erealax_2Ereal__mul,ap(V2c,inj__ty_2Enum_2Enum(V4i))),ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V3x)),inj__ty_2Enum_2Enum(V4i))) ) ) ).
tff(ax_thm_2Eiterate_2Epolynomial__function,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Eiterate_2Epolynomial__function,V0p))
<=> ? [V1m: tp__ty_2Enum_2Enum,V2c: $i] :
( mem(V2c,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
& ! [V3x: tp__ty_2Erealax_2Ereal] : ( surj__ty_2Erealax_2Ereal(ap(V0p,inj__ty_2Erealax_2Ereal(V3x))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eiterate_2ESum(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V1m))),f3328(V2c,V3x))) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__CONST,axiom,
! [V0c: tp__ty_2Erealax_2Ereal] : p(ap(c_2Eiterate_2Epolynomial__function,k(ty_2Erealax_2Ereal,inj__ty_2Erealax_2Ereal(V0c)))) ).
tff(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__ID,axiom,
p(ap(c_2Eiterate_2Epolynomial__function,i(ty_2Erealax_2Ereal))) ).
tff(lamtp_f3329,type,
f3329: ( $i * $i ) > $i ).
tff(lameq_f3329,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1q: $i] :
( mem(V1q,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2x: tp__ty_2Erealax_2Ereal] : ( ap(f3329(V0p,V1q),inj__ty_2Erealax_2Ereal(V2x)) = ap(ap(c_2Erealax_2Ereal__add,ap(V0p,inj__ty_2Erealax_2Ereal(V2x))),ap(V1q,inj__ty_2Erealax_2Ereal(V2x))) ) ) ) ).
tff(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__ADD,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1q: $i] :
( 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))) ) ) ) ).
tff(lamtp_f3330,type,
f3330: ( $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3330,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1c: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] : ( ap(f3330(V0p,V1c),inj__ty_2Erealax_2Ereal(V2x)) = ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V1c)),ap(V0p,inj__ty_2Erealax_2Ereal(V2x))) ) ) ).
tff(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__LMUL,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1c: tp__ty_2Erealax_2Ereal] :
( p(ap(c_2Eiterate_2Epolynomial__function,V0p))
=> p(ap(c_2Eiterate_2Epolynomial__function,f3330(V0p,V1c))) ) ) ).
tff(lamtp_f3331,type,
f3331: ( $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3331,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1c: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] : ( ap(f3331(V0p,V1c),inj__ty_2Erealax_2Ereal(V2x)) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0p,inj__ty_2Erealax_2Ereal(V2x))),inj__ty_2Erealax_2Ereal(V1c)) ) ) ).
tff(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__RMUL,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1c: tp__ty_2Erealax_2Ereal] :
( p(ap(c_2Eiterate_2Epolynomial__function,V0p))
=> p(ap(c_2Eiterate_2Epolynomial__function,f3331(V0p,V1c))) ) ) ).
tff(lamtp_f3332,type,
f3332: $i > $i ).
tff(lameq_f3332,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1x: tp__ty_2Erealax_2Ereal] : ( ap(f3332(V0p),inj__ty_2Erealax_2Ereal(V1x)) = ap(c_2Erealax_2Ereal__neg,ap(V0p,inj__ty_2Erealax_2Ereal(V1x))) ) ) ).
tff(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__NEG,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Eiterate_2Epolynomial__function,f3332(V0p)))
<=> p(ap(c_2Eiterate_2Epolynomial__function,V0p)) ) ) ).
tff(lamtp_f3333,type,
f3333: ( $i * $i ) > $i ).
tff(lameq_f3333,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1q: $i] :
( mem(V1q,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2x: tp__ty_2Erealax_2Ereal] : ( ap(f3333(V0p,V1q),inj__ty_2Erealax_2Ereal(V2x)) = ap(ap(c_2Ereal_2Ereal__sub,ap(V0p,inj__ty_2Erealax_2Ereal(V2x))),ap(V1q,inj__ty_2Erealax_2Ereal(V2x))) ) ) ) ).
tff(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__SUB,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1q: $i] :
( 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))) ) ) ) ).
tff(lamtp_f3334,type,
f3334: ( $i * $i ) > $i ).
tff(lameq_f3334,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1q: $i] :
( mem(V1q,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2x: tp__ty_2Erealax_2Ereal] : ( ap(f3334(V0p,V1q),inj__ty_2Erealax_2Ereal(V2x)) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0p,inj__ty_2Erealax_2Ereal(V2x))),ap(V1q,inj__ty_2Erealax_2Ereal(V2x))) ) ) ) ).
tff(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__MUL,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1q: $i] :
( 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))) ) ) ) ).
tff(lamtp_f3335,type,
f3335: ( del * $i * $i ) > $i ).
tff(lameq_f3335,axiom,
! [A_27a: del,V1p: $i] :
( mem(V1p,arr(ty_2Erealax_2Ereal,arr(A_27a,ty_2Erealax_2Ereal)))
=> ! [V2i: $i] :
( mem(V2i,A_27a)
=> ! [V3x: tp__ty_2Erealax_2Ereal] : ( ap(f3335(A_27a,V1p,V2i),inj__ty_2Erealax_2Ereal(V3x)) = ap(ap(V1p,inj__ty_2Erealax_2Ereal(V3x)),V2i) ) ) ) ).
tff(lamtp_f3336,type,
f3336: ( del * $i * $i ) > $i ).
tff(lameq_f3336,axiom,
! [A_27a: del,V1p: $i] :
( mem(V1p,arr(ty_2Erealax_2Ereal,arr(A_27a,ty_2Erealax_2Ereal)))
=> ! [V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V4x: tp__ty_2Erealax_2Ereal] : ( ap(f3336(A_27a,V1p,V0s),inj__ty_2Erealax_2Ereal(V4x)) = ap(ap(c_2Eiterate_2ESum(A_27a),V0s),ap(V1p,inj__ty_2Erealax_2Ereal(V4x))) ) ) ) ).
tff(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__SUM,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1p: $i] :
( mem(V1p,arr(ty_2Erealax_2Ereal,arr(A_27a,ty_2Erealax_2Ereal)))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& ! [V2i: $i] :
( 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))) ) ) ) ).
tff(lamtp_f3337,type,
f3337: ( $i * tp__ty_2Enum_2Enum ) > $i ).
tff(lameq_f3337,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1n: tp__ty_2Enum_2Enum,V2x: tp__ty_2Erealax_2Ereal] : ( ap(f3337(V0p,V1n),inj__ty_2Erealax_2Ereal(V2x)) = ap(ap(c_2Ereal_2Epow,ap(V0p,inj__ty_2Erealax_2Ereal(V2x))),inj__ty_2Enum_2Enum(V1n)) ) ) ).
tff(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__POW,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1n: tp__ty_2Enum_2Enum] :
( p(ap(c_2Eiterate_2Epolynomial__function,V0p))
=> p(ap(c_2Eiterate_2Epolynomial__function,f3337(V0p,V1n))) ) ) ).
tff(lamtp_f3338,type,
f3338: ( $i * $i ) > $i ).
tff(lameq_f3338,axiom,
! [V4p: $i] :
( mem(V4p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V5q: $i] :
( mem(V5q,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V6x: tp__ty_2Erealax_2Ereal] : ( ap(f3338(V4p,V5q),inj__ty_2Erealax_2Ereal(V6x)) = ap(ap(c_2Erealax_2Ereal__add,ap(V4p,inj__ty_2Erealax_2Ereal(V6x))),ap(V5q,inj__ty_2Erealax_2Ereal(V6x))) ) ) ) ).
tff(lamtp_f3339,type,
f3339: ( $i * $i ) > $i ).
tff(lameq_f3339,axiom,
! [V7p: $i] :
( mem(V7p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V8q: $i] :
( mem(V8q,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V9x: tp__ty_2Erealax_2Ereal] : ( ap(f3339(V7p,V8q),inj__ty_2Erealax_2Ereal(V9x)) = ap(ap(c_2Erealax_2Ereal__mul,ap(V7p,inj__ty_2Erealax_2Ereal(V9x))),ap(V8q,inj__ty_2Erealax_2Ereal(V9x))) ) ) ) ).
tff(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__INDUCT,axiom,
! [V0P: $i] :
( mem(V0P,arr(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),bool))
=> ( ( p(ap(V0P,i(ty_2Erealax_2Ereal)))
& ! [V2c: tp__ty_2Erealax_2Ereal] : p(ap(V0P,k(ty_2Erealax_2Ereal,inj__ty_2Erealax_2Ereal(V2c))))
& ! [V4p: $i] :
( mem(V4p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V5q: $i] :
( mem(V5q,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ( p(ap(V0P,V4p))
& p(ap(V0P,V5q)) )
=> p(ap(V0P,f3338(V4p,V5q))) ) ) )
& ! [V7p: $i] :
( mem(V7p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V8q: $i] :
( mem(V8q,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ( p(ap(V0P,V7p))
& p(ap(V0P,V8q)) )
=> p(ap(V0P,f3339(V7p,V8q))) ) ) ) )
=> ! [V10p: $i] :
( mem(V10p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( p(ap(c_2Eiterate_2Epolynomial__function,V10p))
=> p(ap(V0P,V10p)) ) ) ) ) ).
tff(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__o,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1q: $i] :
( 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))) ) ) ) ).
tff(lamtp_f3340,type,
f3340: ( tp__ty_2Erealax_2Ereal * $i ) > $i ).
tff(lameq_f3340,axiom,
! [V1a: tp__ty_2Erealax_2Ereal,V0p: $i] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2x: tp__ty_2Erealax_2Ereal] : ( ap(f3340(V1a,V0p),inj__ty_2Erealax_2Ereal(V2x)) = ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,bool),inj__ty_2Erealax_2Ereal(V2x)),ap(ap(c_2Emin_2E_3D(ty_2Erealax_2Ereal),ap(V0p,inj__ty_2Erealax_2Ereal(V2x))),inj__ty_2Erealax_2Ereal(V1a))) ) ) ).
tff(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__FINITE__ROOTS,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a: tp__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: tp__ty_2Erealax_2Ereal] : ( surj__ty_2Erealax_2Ereal(ap(V0p,inj__ty_2Erealax_2Ereal(V3x))) = V1a ) ) ) ) ).
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