ITP001 Axioms: ITP147_5.ax
%------------------------------------------------------------------------------
% File : ITP147_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 : product_2.ax [Gau20]
% : HL4147_5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 129 ( 9 unt; 23 typ; 0 def)
% Number of atoms : 3472 ( 122 equ)
% Maximal formula atoms : 20 ( 26 avg)
% Number of connectives : 355 ( 2 ~; 0 |; 63 &)
% ( 4 <=>; 286 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 8 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 3013 (3013 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 51 ( 23 >; 28 *; 0 +; 0 <<)
% Number of predicates : 67 ( 66 usr; 32 prp; 0-3 aty)
% Number of functors : 23 ( 23 usr; 0 con; 1-3 aty)
% Number of variables : 397 ( 393 !; 4 ?; 397 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_c_2Eproduct_2Enproduct,type,
c_2Eproduct_2Enproduct: del > $i ).
tff(mem_c_2Eproduct_2Enproduct,axiom,
! [A_27a: del] : mem(c_2Eproduct_2Enproduct(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,ty_2Enum_2Enum),ty_2Enum_2Enum))) ).
tff(tp_c_2Eproduct_2Eproduct,type,
c_2Eproduct_2Eproduct: del > $i ).
tff(mem_c_2Eproduct_2Eproduct,axiom,
! [A_27a: del] : mem(c_2Eproduct_2Eproduct(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,ty_2Erealax_2Ereal),ty_2Erealax_2Ereal))) ).
tff(conj_thm_2Eproduct_2EREAL__ADD__AC,axiom,
! [V0m: tp__ty_2Erealax_2Ereal,V1n: tp__ty_2Erealax_2Ereal,V2p: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,inj__ty_2Erealax_2Ereal(V0m)),inj__ty_2Erealax_2Ereal(V1n))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,inj__ty_2Erealax_2Ereal(V1n)),inj__ty_2Erealax_2Ereal(V0m))) )
& ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__add,inj__ty_2Erealax_2Ereal(V0m)),inj__ty_2Erealax_2Ereal(V1n))),inj__ty_2Erealax_2Ereal(V2p))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,inj__ty_2Erealax_2Ereal(V0m)),ap(ap(c_2Erealax_2Ereal__add,inj__ty_2Erealax_2Ereal(V1n)),inj__ty_2Erealax_2Ereal(V2p)))) )
& ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,inj__ty_2Erealax_2Ereal(V0m)),ap(ap(c_2Erealax_2Ereal__add,inj__ty_2Erealax_2Ereal(V1n)),inj__ty_2Erealax_2Ereal(V2p)))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,inj__ty_2Erealax_2Ereal(V1n)),ap(ap(c_2Erealax_2Ereal__add,inj__ty_2Erealax_2Ereal(V0m)),inj__ty_2Erealax_2Ereal(V2p)))) ) ) ).
tff(conj_thm_2Eproduct_2EMULT__AC,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum,V2p: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2A,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n))) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2A,inj__ty_2Enum_2Enum(V1n)),inj__ty_2Enum_2Enum(V0m))) )
& ( surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2A,ap(ap(c_2Earithmetic_2E_2A,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n))),inj__ty_2Enum_2Enum(V2p))) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2A,inj__ty_2Enum_2Enum(V0m)),ap(ap(c_2Earithmetic_2E_2A,inj__ty_2Enum_2Enum(V1n)),inj__ty_2Enum_2Enum(V2p)))) )
& ( surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2A,inj__ty_2Enum_2Enum(V0m)),ap(ap(c_2Earithmetic_2E_2A,inj__ty_2Enum_2Enum(V1n)),inj__ty_2Enum_2Enum(V2p)))) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2A,inj__ty_2Enum_2Enum(V1n)),ap(ap(c_2Earithmetic_2E_2A,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V2p)))) ) ) ).
tff(ax_thm_2Eproduct_2Enproduct,axiom,
! [A_27a: del] : ( c_2Eproduct_2Enproduct(A_27a) = ap(c_2Eiterate_2Eiterate(A_27a,ty_2Enum_2Enum),c_2Earithmetic_2E_2A) ) ).
tff(conj_thm_2Eproduct_2ENPRODUCT__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_2Eproduct_2Enproduct(A_27a),c_2Epred__set_2EEMPTY(A_27a)),V0f)) = surj__ty_2Enum_2Enum(ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))) ) )
& ! [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_2Eproduct_2Enproduct(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_2Eproduct_2Enproduct(A_27b),V3s),V2f)),ap(ap(c_2Earithmetic_2E_2A,ap(V2f,V1x)),ap(ap(c_2Eproduct_2Enproduct(A_27b),V3s),V2f)))) ) ) ) ) ) ) ).
tff(conj_thm_2Eproduct_2ENPRODUCT__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_2Eproduct_2Enproduct(A_27a),ap(ap(ap(c_2Eiterate_2Esupport(A_27a,ty_2Enum_2Enum),c_2Earithmetic_2E_2A),V0f),V1s)),V0f)) = surj__ty_2Enum_2Enum(ap(ap(c_2Eproduct_2Enproduct(A_27a),V1s),V0f)) ) ) ) ).
tff(conj_thm_2Eproduct_2ENPRODUCT__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_2Eproduct_2Enproduct(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)),V0f)) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2A,ap(ap(c_2Eproduct_2Enproduct(A_27a),V1s),V0f)),ap(ap(c_2Eproduct_2Enproduct(A_27a),V2t),V0f))) ) ) ) ) ) ).
tff(conj_thm_2Eproduct_2ENPRODUCT__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_2Eproduct_2Enproduct(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V2s)),V1g)) = surj__ty_2Enum_2Enum(ap(ap(c_2Eproduct_2Enproduct(A_27a),V2s),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Enum_2Enum,A_27b),V1g),V0f))) ) ) ) ) ) ).
tff(conj_thm_2Eproduct_2ENPRODUCT__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_2Eproduct_2Enproduct(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_2A,ap(ap(c_2Eproduct_2Enproduct(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),V0f)),ap(ap(c_2Eproduct_2Enproduct(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_2Eproduct_2ENPRODUCT__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_2Eproduct_2Enproduct(A_27a),V1s),V0f))) ) ) ) ).
tff(conj_thm_2Eproduct_2ENPRODUCT__POS__LT__NUMSEG,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ! [V3x: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V3x)))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V3x)),inj__ty_2Enum_2Enum(V2n))) )
=> p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),ap(V0f,inj__ty_2Enum_2Enum(V3x)))) )
=> p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),ap(ap(c_2Eproduct_2Enproduct(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),V0f))) ) ) ).
tff(lamtp_f3396,type,
f3396: ( tp__ty_2Enum_2Enum * $i ) > $i ).
tff(lameq_f3396,axiom,
! [V3p: tp__ty_2Enum_2Enum,V1f: $i] :
( mem(V1f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V4i: tp__ty_2Enum_2Enum] : ( ap(f3396(V3p,V1f),inj__ty_2Enum_2Enum(V4i)) = ap(V1f,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V4i)),inj__ty_2Enum_2Enum(V3p))) ) ) ).
tff(conj_thm_2Eproduct_2ENPRODUCT__OFFSET,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1f: $i] :
( mem(V1f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V2m: tp__ty_2Enum_2Enum,V3p: tp__ty_2Enum_2Enum] : ( surj__ty_2Enum_2Enum(ap(ap(c_2Eproduct_2Enproduct(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(V3p))),ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V0n)),inj__ty_2Enum_2Enum(V3p)))),V1f)) = surj__ty_2Enum_2Enum(ap(ap(c_2Eproduct_2Enproduct(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V0n))),f3396(V3p,V1f))) ) ) ).
tff(conj_thm_2Eproduct_2ENPRODUCT__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_2Eproduct_2Enproduct(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(conj_thm_2Eproduct_2ENPRODUCT__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_2Eproduct_2Enproduct(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_2Eproduct_2ENPRODUCT__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_2Eproduct_2Enproduct(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))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) )
& ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] : ( surj__ty_2Enum_2Enum(ap(ap(c_2Eproduct_2Enproduct(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_2A,ap(ap(c_2Eproduct_2Enproduct(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_2Eproduct_2Enproduct(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V0f))) ) ) ) ).
tff(conj_thm_2Eproduct_2ENPRODUCT__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_2Eproduct_2Enproduct(A_27a),V2s),V0f)) = surj__ty_2Enum_2Enum(ap(ap(c_2Eproduct_2Enproduct(A_27a),V2s),V1g)) ) ) ) ) ) ).
tff(conj_thm_2Eproduct_2ENPRODUCT__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_2Eproduct_2Enproduct(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_2Eproduct_2Enproduct(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V1g)) ) ) ) ) ).
tff(conj_thm_2Eproduct_2ENPRODUCT__EQ__0,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_2Eproduct_2Enproduct(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_2Eproduct_2ENPRODUCT__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] :
( ( surj__ty_2Enum_2Enum(ap(ap(c_2Eproduct_2Enproduct(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 )
<=> ? [V3x: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V3x)))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V3x)),inj__ty_2Enum_2Enum(V2n)))
& ( surj__ty_2Enum_2Enum(ap(V0f,inj__ty_2Enum_2Enum(V3x))) = fo__c_2Enum_2E0 ) ) ) ) ).
tff(conj_thm_2Eproduct_2ENPRODUCT__LE,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))
=> ( ( 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,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),ap(V1f,V3x)))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(V1f,V3x)),ap(V0g,V3x))) ) ) ) )
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(ap(c_2Eproduct_2Enproduct(A_27a),V2s),V1f)),ap(ap(c_2Eproduct_2Enproduct(A_27a),V2s),V0g))) ) ) ) ) ).
tff(conj_thm_2Eproduct_2ENPRODUCT__LE__NUMSEG,axiom,
! [V0g: $i] :
( mem(V0g,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1f: $i] :
( mem(V1f,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,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),ap(V1f,inj__ty_2Enum_2Enum(V4i))))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(V1f,inj__ty_2Enum_2Enum(V4i))),ap(V0g,inj__ty_2Enum_2Enum(V4i)))) ) )
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(ap(c_2Eproduct_2Enproduct(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V1f)),ap(ap(c_2Eproduct_2Enproduct(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V0g))) ) ) ) ).
tff(conj_thm_2Eproduct_2ENPRODUCT__EQ__1,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)) = surj__ty_2Enum_2Enum(ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))) ) ) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eproduct_2Enproduct(A_27a),V1s),V0f)) = surj__ty_2Enum_2Enum(ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))) ) ) ) ) ).
tff(conj_thm_2Eproduct_2ENPRODUCT__EQ__1__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))) = surj__ty_2Enum_2Enum(ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))) ) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eproduct_2Enproduct(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(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))) ) ) ) ).
tff(lamtp_f3397,type,
f3397: ( del * $i * $i ) > $i ).
tff(lameq_f3397,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(f3397(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)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))))) ) ) ) ).
tff(lamtp_f3398,type,
f3398: ( del * $i * $i ) > $i ).
tff(lameq_f3398,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(f3398(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)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))))) ) ) ) ).
tff(lamtp_f3399,type,
f3399: ( del * $i * $i ) > $i ).
tff(lameq_f3399,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(f3399(A_27a,V0f,V1g),V5x) = ap(ap(c_2Earithmetic_2E_2A,ap(V0f,V5x)),ap(V1g,V5x)) ) ) ) ).
tff(conj_thm_2Eproduct_2ENPRODUCT__MUL__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),f3397(A_27a,V0f,V2s))))
& p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3398(A_27a,V1g,V2s)))) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eproduct_2Enproduct(A_27a),V2s),f3399(A_27a,V0f,V1g))) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2A,ap(ap(c_2Eproduct_2Enproduct(A_27a),V2s),V0f)),ap(ap(c_2Eproduct_2Enproduct(A_27a),V2s),V1g))) ) ) ) ) ) ).
tff(lamtp_f3400,type,
f3400: ( del * $i * $i ) > $i ).
tff(lameq_f3400,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V3x: $i] : ( ap(f3400(A_27a,V0f,V1g),V3x) = ap(ap(c_2Earithmetic_2E_2A,ap(V0f,V3x)),ap(V1g,V3x)) ) ) ) ).
tff(conj_thm_2Eproduct_2ENPRODUCT__MUL,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_2Eproduct_2Enproduct(A_27a),V2s),f3400(A_27a,V0f,V1g))) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2A,ap(ap(c_2Eproduct_2Enproduct(A_27a),V2s),V0f)),ap(ap(c_2Eproduct_2Enproduct(A_27a),V2s),V1g))) ) ) ) ) ) ).
tff(lamtp_f3401,type,
f3401: ( $i * $i ) > $i ).
tff(lameq_f3401,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V4x: tp__ty_2Enum_2Enum] : ( ap(f3401(V0f,V1g),inj__ty_2Enum_2Enum(V4x)) = ap(ap(c_2Earithmetic_2E_2A,ap(V0f,inj__ty_2Enum_2Enum(V4x))),ap(V1g,inj__ty_2Enum_2Enum(V4x))) ) ) ) ).
tff(conj_thm_2Eproduct_2ENPRODUCT__MUL__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_2Eproduct_2Enproduct(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),f3401(V0f,V1g))) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2A,ap(ap(c_2Eproduct_2Enproduct(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V0f)),ap(ap(c_2Eproduct_2Enproduct(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V1g))) ) ) ) ).
tff(conj_thm_2Eproduct_2ENPRODUCT__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_2Eproduct_2Enproduct(A_27a),V1s),k(A_27a,inj__ty_2Enum_2Enum(V0c)))) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2EEXP,inj__ty_2Enum_2Enum(V0c)),ap(c_2Epred__set_2ECARD(A_27a),V1s))) ) ) ) ).
tff(conj_thm_2Eproduct_2ENPRODUCT__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_2Eproduct_2Enproduct(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_2EEXP,inj__ty_2Enum_2Enum(V0c)),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)))) ) ).
tff(conj_thm_2Eproduct_2ENPRODUCT__CONST__NUMSEG__1,axiom,
! [V0c: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] : ( surj__ty_2Enum_2Enum(ap(ap(c_2Eproduct_2Enproduct(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(V1n))),k(ty_2Enum_2Enum,inj__ty_2Enum_2Enum(V0c)))) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2EEXP,inj__ty_2Enum_2Enum(V0c)),inj__ty_2Enum_2Enum(V1n))) ) ).
tff(conj_thm_2Eproduct_2ENPRODUCT__ONE,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eproduct_2Enproduct(A_27a),V0s),k(A_27a,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))) = surj__ty_2Enum_2Enum(ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))) ) ) ).
tff(conj_thm_2Eproduct_2ENPRODUCT__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,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))
& ! [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_2A,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_2Eproduct_2Enproduct(A_27a),V2s),V1f))) ) ) ) ) ).
tff(conj_thm_2Eproduct_2ENPRODUCT__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_2Eproduct_2Enproduct(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_2A,ap(V0f,inj__ty_2Enum_2Enum(V1m))),ap(ap(c_2Eproduct_2Enproduct(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_2Eproduct_2ENPRODUCT__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_2Eproduct_2Enproduct(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_2A,ap(ap(c_2Eproduct_2Enproduct(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(conj_thm_2Eproduct_2ENPRODUCT__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)) = surj__ty_2Enum_2Enum(ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))) ) ) ) )
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Eproduct_2Enproduct(A_27a),V2v),V0f)) = surj__ty_2Enum_2Enum(ap(ap(c_2Eproduct_2Enproduct(A_27a),V1u),V0f)) ) ) ) ) ) ).
tff(lamtp_f3402,type,
f3402: $i > $i ).
tff(lameq_f3402,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V3i: tp__ty_2Enum_2Enum] : ( ap(f3402(V0f),inj__ty_2Enum_2Enum(V3i)) = ap(ap(c_2Earithmetic_2E_2A,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_2Eproduct_2ENPRODUCT__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_2Eproduct_2Enproduct(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_2Eproduct_2Enproduct(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),f3402(V0f))) ) ) ).
tff(conj_thm_2Eproduct_2ENPRODUCT__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_2A,ap(V0f,V2a)),ap(ap(c_2Eproduct_2Enproduct(A_27a),ap(ap(c_2Epred__set_2EDELETE(A_27a),V1s),V2a)),V0f))) = surj__ty_2Enum_2Enum(ap(ap(c_2Eproduct_2Enproduct(A_27a),V1s),V0f)) ) ) ) ) ) ).
tff(conj_thm_2Eproduct_2ENPRODUCT__FACT,axiom,
! [V0n: tp__ty_2Enum_2Enum] : ( surj__ty_2Enum_2Enum(ap(ap(c_2Eproduct_2Enproduct(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))),i(ty_2Enum_2Enum))) = surj__ty_2Enum_2Enum(ap(c_2Earithmetic_2EFACT,inj__ty_2Enum_2Enum(V0n))) ) ).
tff(lamtp_f3403,type,
f3403: ( del * tp__ty_2Enum_2Enum * $i ) > $i ).
tff(lameq_f3403,axiom,
! [A_27a: del,V0b: tp__ty_2Enum_2Enum,V2a: $i] :
( mem(V2a,A_27a)
=> ! [V3x: $i] : ( ap(f3403(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)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))) ) ) ).
tff(conj_thm_2Eproduct_2ENPRODUCT__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_2Eproduct_2Enproduct(A_27a),V1s),f3403(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)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) ) ) ) ).
tff(ax_thm_2Eproduct_2Eproduct,axiom,
! [A_27a: del] : ( c_2Eproduct_2Eproduct(A_27a) = ap(c_2Eiterate_2Eiterate(A_27a,ty_2Erealax_2Ereal),c_2Erealax_2Ereal__mul) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__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_2Eproduct_2Eproduct(A_27a),c_2Epred__set_2EEMPTY(A_27a)),V0f)) = 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))))) ) )
& ! [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_2Eproduct_2Eproduct(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_2Eproduct_2Eproduct(A_27b),V3s),V2f)),ap(ap(c_2Erealax_2Ereal__mul,ap(V2f,V1x)),ap(ap(c_2Eproduct_2Eproduct(A_27b),V3s),V2f)))) ) ) ) ) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__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_2Eproduct_2Eproduct(A_27a),ap(ap(ap(c_2Eiterate_2Esupport(A_27a,ty_2Erealax_2Ereal),c_2Erealax_2Ereal__mul),V0f),V1s)),V0f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eproduct_2Eproduct(A_27a),V1s),V0f)) ) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__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_2Eproduct_2Eproduct(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)),V0f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Eproduct_2Eproduct(A_27a),V1s),V0f)),ap(ap(c_2Eproduct_2Eproduct(A_27a),V2t),V0f))) ) ) ) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__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_2Eproduct_2Eproduct(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V2s)),V1g)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eproduct_2Eproduct(A_27a),V2s),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Erealax_2Ereal,A_27b),V1g),V0f))) ) ) ) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__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_2Eproduct_2Eproduct(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__mul,ap(ap(c_2Eproduct_2Eproduct(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),V0f)),ap(ap(c_2Eproduct_2Eproduct(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_2Eproduct_2EPRODUCT__POS__LE,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))) ) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Eproduct_2Eproduct(A_27a),V1s),V0f))) ) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__POS__LE__NUMSEG,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ! [V3x: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V3x)))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V3x)),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(V3x)))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Eproduct_2Eproduct(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),V0f))) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__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_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V0f,V2x))) ) ) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(ap(c_2Eproduct_2Eproduct(A_27a),V1s),V0f))) ) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__POS__LT__NUMSEG,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ! [V3x: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V3x)))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V3x)),inj__ty_2Enum_2Enum(V2n))) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V0f,inj__ty_2Enum_2Enum(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_2Eproduct_2Eproduct(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),V0f))) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__OFFSET,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1f: $i] :
( mem(V1f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2m: tp__ty_2Enum_2Enum,V3p: tp__ty_2Enum_2Enum] : ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eproduct_2Eproduct(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(V3p))),ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V0n)),inj__ty_2Enum_2Enum(V3p)))),V1f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eproduct_2Eproduct(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V0n))),f3396(V3p,V1f))) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__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_2Eproduct_2Eproduct(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(conj_thm_2Eproduct_2EPRODUCT__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_2Eproduct_2Eproduct(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_2Eproduct_2EPRODUCT__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_2Eproduct_2Eproduct(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,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))) )
& ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] : ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eproduct_2Eproduct(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__mul,ap(ap(c_2Eproduct_2Eproduct(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_2Eproduct_2Eproduct(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V0f))) ) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__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_2Eproduct_2Eproduct(A_27a),V2s),V0f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eproduct_2Eproduct(A_27a),V2s),V1g)) ) ) ) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__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_2Eproduct_2Eproduct(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_2Eproduct_2Eproduct(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V1g)) ) ) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__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))
=> ( ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eproduct_2Eproduct(A_27a),V1s),V0f)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) )
<=> ? [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))) ) ) ) ) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__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] :
( ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eproduct_2Eproduct(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))) )
<=> ? [V3x: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V3x)))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V3x)),inj__ty_2Enum_2Enum(V2n)))
& ( surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Enum_2Enum(V3x))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__LE,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))
=> ( ( 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(V1f,V3x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(V1f,V3x)),ap(V0g,V3x))) ) ) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eproduct_2Eproduct(A_27a),V2s),V1f)),ap(ap(c_2Eproduct_2Eproduct(A_27a),V2s),V0g))) ) ) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__LE__NUMSEG,axiom,
! [V0g: $i] :
( mem(V0g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1f: $i] :
( mem(V1f,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(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V1f,inj__ty_2Enum_2Enum(V4i))))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(V1f,inj__ty_2Enum_2Enum(V4i))),ap(V0g,inj__ty_2Enum_2Enum(V4i)))) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eproduct_2Eproduct(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V1f)),ap(ap(c_2Eproduct_2Eproduct(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V0g))) ) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__EQ__1,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,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) ) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eproduct_2Eproduct(A_27a),V1s),V0f)) = 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_2Eproduct_2EPRODUCT__EQ__1__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,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eproduct_2Eproduct(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,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) ) ) ) ).
tff(lamtp_f3404,type,
f3404: ( del * $i * $i ) > $i ).
tff(lameq_f3404,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(f3404(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,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))))) ) ) ) ).
tff(lamtp_f3405,type,
f3405: ( del * $i * $i ) > $i ).
tff(lameq_f3405,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(f3405(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,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))))) ) ) ) ).
tff(lamtp_f3406,type,
f3406: ( del * $i * $i ) > $i ).
tff(lameq_f3406,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(f3406(A_27a,V0f,V1g),V5x) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0f,V5x)),ap(V1g,V5x)) ) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__MUL__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),f3404(A_27a,V0f,V2s))))
& p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f3405(A_27a,V1g,V2s)))) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eproduct_2Eproduct(A_27a),V2s),f3406(A_27a,V0f,V1g))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Eproduct_2Eproduct(A_27a),V2s),V0f)),ap(ap(c_2Eproduct_2Eproduct(A_27a),V2s),V1g))) ) ) ) ) ) ).
tff(lamtp_f3407,type,
f3407: ( del * $i * $i ) > $i ).
tff(lameq_f3407,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(f3407(A_27a,V0f,V1g),V3x) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0f,V3x)),ap(V1g,V3x)) ) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__MUL,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_2Eproduct_2Eproduct(A_27a),V2s),f3407(A_27a,V0f,V1g))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Eproduct_2Eproduct(A_27a),V2s),V0f)),ap(ap(c_2Eproduct_2Eproduct(A_27a),V2s),V1g))) ) ) ) ) ) ).
tff(lamtp_f3408,type,
f3408: ( $i * $i ) > $i ).
tff(lameq_f3408,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4x: tp__ty_2Enum_2Enum] : ( ap(f3408(V0f,V1g),inj__ty_2Enum_2Enum(V4x)) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0f,inj__ty_2Enum_2Enum(V4x))),ap(V1g,inj__ty_2Enum_2Enum(V4x))) ) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__MUL__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_2Eproduct_2Eproduct(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),f3408(V0f,V1g))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Eproduct_2Eproduct(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V0f)),ap(ap(c_2Eproduct_2Eproduct(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V1g))) ) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__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_2Eproduct_2Eproduct(A_27a),V1s),k(A_27a,inj__ty_2Erealax_2Ereal(V0c)))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V0c)),ap(c_2Epred__set_2ECARD(A_27a),V1s))) ) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__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_2Eproduct_2Eproduct(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_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V0c)),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)))) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__CONST__NUMSEG__1,axiom,
! [V0c: tp__ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] : ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eproduct_2Eproduct(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(V1n))),k(ty_2Enum_2Enum,inj__ty_2Erealax_2Ereal(V0c)))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V0c)),inj__ty_2Enum_2Enum(V1n))) ) ).
tff(lamtp_f3409,type,
f3409: ( del * $i ) > $i ).
tff(lameq_f3409,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2i: $i] : ( ap(f3409(A_27a,V0f),V2i) = ap(c_2Erealax_2Ereal__neg,ap(V0f,V2i)) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__NEG,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_2Eproduct_2Eproduct(A_27a),V1s),f3409(A_27a,V0f))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2Epow,ap(c_2Erealax_2Ereal__neg,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),ap(c_2Epred__set_2ECARD(A_27a),V1s))),ap(ap(c_2Eproduct_2Eproduct(A_27a),V1s),V0f))) ) ) ) ) ).
tff(lamtp_f3410,type,
f3410: $i > $i ).
tff(lameq_f3410,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3i: tp__ty_2Enum_2Enum] : ( ap(f3410(V0f),inj__ty_2Enum_2Enum(V3i)) = ap(c_2Erealax_2Ereal__neg,ap(V0f,inj__ty_2Enum_2Enum(V3i))) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__NEG__NUMSEG,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_2Eproduct_2Eproduct(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),f3410(V0f))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2Epow,ap(c_2Erealax_2Ereal__neg,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_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)))),ap(ap(c_2Eproduct_2Eproduct(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),V0f))) ) ) ).
tff(lamtp_f3411,type,
f3411: $i > $i ).
tff(lameq_f3411,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2i: tp__ty_2Enum_2Enum] : ( ap(f3411(V0f),inj__ty_2Enum_2Enum(V2i)) = ap(c_2Erealax_2Ereal__neg,ap(V0f,inj__ty_2Enum_2Enum(V2i))) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__NEG__NUMSEG__1,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1n: tp__ty_2Enum_2Enum] : ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eproduct_2Eproduct(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(V1n))),f3411(V0f))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2Epow,ap(c_2Erealax_2Ereal__neg,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),inj__ty_2Enum_2Enum(V1n))),ap(ap(c_2Eproduct_2Eproduct(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(V1n))),V0f))) ) ) ).
tff(lamtp_f3412,type,
f3412: ( del * $i ) > $i ).
tff(lameq_f3412,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2x: $i] : ( ap(f3412(A_27a,V0f),V2x) = ap(c_2Erealax_2Einv,ap(V0f,V2x)) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__INV,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_2Eproduct_2Eproduct(A_27a),V1s),f3412(A_27a,V0f))) = surj__ty_2Erealax_2Ereal(ap(c_2Erealax_2Einv,ap(ap(c_2Eproduct_2Eproduct(A_27a),V1s),V0f))) ) ) ) ) ).
tff(lamtp_f3413,type,
f3413: ( del * $i * $i ) > $i ).
tff(lameq_f3413,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(f3413(A_27a,V0f,V1g),V3x) = ap(ap(c_2Ereal_2E_2F,ap(V0f,V3x)),ap(V1g,V3x)) ) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__DIV,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_2Eproduct_2Eproduct(A_27a),V2s),f3413(A_27a,V0f,V1g))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2E_2F,ap(ap(c_2Eproduct_2Eproduct(A_27a),V2s),V0f)),ap(ap(c_2Eproduct_2Eproduct(A_27a),V2s),V1g))) ) ) ) ) ) ).
tff(lamtp_f3414,type,
f3414: ( $i * $i ) > $i ).
tff(lameq_f3414,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4x: tp__ty_2Enum_2Enum] : ( ap(f3414(V0f,V1g),inj__ty_2Enum_2Enum(V4x)) = ap(ap(c_2Ereal_2E_2F,ap(V0f,inj__ty_2Enum_2Enum(V4x))),ap(V1g,inj__ty_2Enum_2Enum(V4x))) ) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__DIV__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_2Eproduct_2Eproduct(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),f3414(V0f,V1g))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2E_2F,ap(ap(c_2Eproduct_2Eproduct(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V0f)),ap(ap(c_2Eproduct_2Eproduct(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n))),V1g))) ) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__ONE,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eproduct_2Eproduct(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))))))) = 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_2Eproduct_2EPRODUCT__LE__1,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)))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,V2x)),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))) ) ) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eproduct_2Eproduct(A_27a),V1s),V0f)),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_2Eproduct_2EPRODUCT__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))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eproduct_2Eproduct(A_27a),V1s),f3273(A_27a,V0f))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Eabs,ap(ap(c_2Eproduct_2Eproduct(A_27a),V1s),V0f))) ) ) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__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,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))))
& ! [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__mul,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_2Eproduct_2Eproduct(A_27a),V2s),V1f))) ) ) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__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_2Eproduct_2Eproduct(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__mul,ap(V0f,inj__ty_2Enum_2Enum(V1m))),ap(ap(c_2Eproduct_2Eproduct(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_2Eproduct_2EPRODUCT__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_2Eproduct_2Eproduct(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__mul,ap(ap(c_2Eproduct_2Eproduct(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(conj_thm_2Eproduct_2EREAL__OF__NUM__NPRODUCT,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_2Eproduct_2Enproduct(A_27a),V1s),V0f))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eproduct_2Eproduct(A_27a),V1s),f3285(A_27a,V0f))) ) ) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__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,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) ) ) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eproduct_2Eproduct(A_27a),V2v),V0f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eproduct_2Eproduct(A_27a),V1u),V0f)) ) ) ) ) ) ).
tff(lamtp_f3415,type,
f3415: $i > $i ).
tff(lameq_f3415,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3i: tp__ty_2Enum_2Enum] : ( ap(f3415(V0f),inj__ty_2Enum_2Enum(V3i)) = ap(ap(c_2Erealax_2Ereal__mul,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_2Eproduct_2EPRODUCT__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_2Eproduct_2Eproduct(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_2Eproduct_2Eproduct(ty_2Enum_2Enum),ap(ap(c_2Eiterate_2E_2E_2E,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),f3415(V0f))) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__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_2Erealax_2Ereal__mul,ap(V0f,V2a)),ap(ap(c_2Eproduct_2Eproduct(A_27a),ap(ap(c_2Epred__set_2EDELETE(A_27a),V1s),V2a)),V0f))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eproduct_2Eproduct(A_27a),V1s),V0f)) ) ) ) ) ) ).
tff(lamtp_f3416,type,
f3416: ( del * tp__ty_2Erealax_2Ereal * $i ) > $i ).
tff(lameq_f3416,axiom,
! [A_27a: del,V0b: tp__ty_2Erealax_2Ereal,V2a: $i] :
( mem(V2a,A_27a)
=> ! [V3x: $i] : ( ap(f3416(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,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))) ) ) ).
tff(conj_thm_2Eproduct_2EPRODUCT__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_2Eproduct_2Eproduct(A_27a),V1s),f3416(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,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))) ) ) ) ).
tff(conj_thm_2Eproduct_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_2Eproduct_2Eproduct(A_27a),V2s),f3303(A_27a,V0f))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Eproduct_2Eproduct(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_2Eproduct_2Eproduct(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_2Eproduct_2Eproduct(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_2Eproduct_2Eproduct(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_2Eproduct_2Eproduct(A_27b),ap(c_2Epred__set_2EGSPEC(A_27b,A_27b),f3307(A_27b,V13p))),V12g)) ) ) ) ) ) ) ).
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