ITP001 Axioms: ITP127_7.ax
%------------------------------------------------------------------------------
% File : ITP127_7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : seq.ax [Gau19]
% : HL4127_7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 320 ( 73 unt; 133 typ; 0 def)
% Number of atoms : 445 ( 138 equ)
% Maximal formula atoms : 12 ( 1 avg)
% Number of connectives : 264 ( 6 ~; 8 |; 76 &)
% ( 33 <=>; 141 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 5 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 102 ( 64 >; 38 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 129 ( 129 usr; 46 con; 0-5 aty)
% Number of variables : 655 ( 568 !; 22 ?; 655 :)
% ( 65 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tyop_2Emetric_2Emetric,type,
tyop_2Emetric_2Emetric: $tType > $tType ).
tff(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
tff(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: ( $tType * $tType ) > $tType ).
tff(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
tff(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: ( $tType * $tType ) > $tType ).
tff(tyop_2Erealax_2Ereal,type,
tyop_2Erealax_2Ereal: $tType ).
tff(tyop_2Etopology_2Etopology,type,
tyop_2Etopology_2Etopology: $tType > $tType ).
tff(app_2E2,type,
app_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27b) * A_27a ) > A_27b ) ).
tff(p,type,
p: tyop_2Emin_2Ebool > $o ).
tff(combin_i_2E0,type,
combin_i_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,A_27a) ).
tff(combin_k_2E0,type,
combin_k_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27a)) ).
tff(combin_s_2E0,type,
combin_s_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c))) ).
tff(c_2Ebool_2E_21_2E0,type,
c_2Ebool_2E_21_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_21_2E1,type,
c_2Ebool_2E_21_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Earithmetic_2E_2A_2E0,type,
c_2Earithmetic_2E_2A_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).
tff(c_2Earithmetic_2E_2A_2E2,type,
c_2Earithmetic_2E_2A_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).
tff(c_2Earithmetic_2E_2B_2E0,type,
c_2Earithmetic_2E_2B_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).
tff(c_2Earithmetic_2E_2B_2E2,type,
c_2Earithmetic_2E_2B_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).
tff(c_2Epair_2E_2C_2E0,type,
c_2Epair_2E_2C_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b))) ).
tff(c_2Epair_2E_2C_2E2,type,
c_2Epair_2E_2C_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a * A_27b ) > tyop_2Epair_2Eprod(A_27a,A_27b) ) ).
tff(c_2Eseq_2E_2D_2D_3E_2E0,type,
c_2Eseq_2E_2D_2D_3E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Eseq_2E_2D_2D_3E_2E2,type,
c_2Eseq_2E_2D_2D_3E_2E2: ( tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) * tyop_2Erealax_2Ereal ) > tyop_2Emin_2Ebool ).
tff(c_2Ereal_2E_2F_2E0,type,
c_2Ereal_2E_2F_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Ereal_2E_2F_2E2,type,
c_2Ereal_2E_2F_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
tff(c_2Ebool_2E_2F_5C_2E0,type,
c_2Ebool_2E_2F_5C_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2E_2F_5C_2E2,type,
c_2Ebool_2E_2F_5C_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Enum_2E0_2E0,type,
c_2Enum_2E0_2E0: tyop_2Enum_2Enum ).
tff(c_2Eprim__rec_2E_3C_2E0,type,
c_2Eprim__rec_2E_3C_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).
tff(c_2Eprim__rec_2E_3C_2E2,type,
c_2Eprim__rec_2E_3C_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Emin_2Ebool ).
tff(c_2Earithmetic_2E_3C_3D_2E0,type,
c_2Earithmetic_2E_3C_3D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).
tff(c_2Earithmetic_2E_3C_3D_2E2,type,
c_2Earithmetic_2E_3C_3D_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Emin_2Ebool ).
tff(c_2Emin_2E_3D_2E0,type,
c_2Emin_2E_3D_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) ).
tff(c_2Emin_2E_3D_2E2,type,
c_2Emin_2E_3D_2E2:
!>[A_27a: $tType] : ( ( A_27a * A_27a ) > tyop_2Emin_2Ebool ) ).
tff(c_2Emin_2E_3D_3D_3E_2E0,type,
c_2Emin_2E_3D_3D_3E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Emin_2E_3D_3D_3E_2E2,type,
c_2Emin_2E_3D_3D_3E_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Earithmetic_2E_3E_3D_2E0,type,
c_2Earithmetic_2E_3E_3D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).
tff(c_2Earithmetic_2E_3E_3D_2E2,type,
c_2Earithmetic_2E_3E_3D_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Emin_2Ebool ).
tff(c_2Ebool_2E_3F_2E0,type,
c_2Ebool_2E_3F_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_3F_2E1,type,
c_2Ebool_2E_3F_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Emin_2E_40_2E0,type,
c_2Emin_2E_40_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27a) ).
tff(c_2Emin_2E_40_2E1,type,
c_2Emin_2E_40_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > A_27a ) ).
tff(c_2Epred__set_2EBIJ_2E0,type,
c_2Epred__set_2EBIJ_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))) ).
tff(c_2Epred__set_2EBIJ_2E3,type,
c_2Epred__set_2EBIJ_2E3:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27b) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Earithmetic_2EBIT1_2E0,type,
c_2Earithmetic_2EBIT1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).
tff(c_2Earithmetic_2EBIT1_2E1,type,
c_2Earithmetic_2EBIT1_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
tff(c_2Earithmetic_2EBIT2_2E0,type,
c_2Earithmetic_2EBIT2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).
tff(c_2Earithmetic_2EBIT2_2E1,type,
c_2Earithmetic_2EBIT2_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
tff(c_2Ebool_2ECOND_2E0,type,
c_2Ebool_2ECOND_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))) ).
tff(c_2Ebool_2ECOND_2E3,type,
c_2Ebool_2ECOND_2E3:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Ebool * A_27a * A_27a ) > A_27a ) ).
tff(c_2Epred__set_2ECROSS_2E0,type,
c_2Epred__set_2ECROSS_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))) ).
tff(c_2Epred__set_2ECROSS_2E2,type,
c_2Epred__set_2ECROSS_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool) ) ).
tff(c_2Ebool_2EF_2E0,type,
c_2Ebool_2EF_2E0: tyop_2Emin_2Ebool ).
tff(c_2Epred__set_2EIMAGE_2E0,type,
c_2Epred__set_2EIMAGE_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))) ).
tff(c_2Epred__set_2EIMAGE_2E2,type,
c_2Epred__set_2EIMAGE_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27b) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool) ) ).
tff(c_2Ecombin_2EK_2E0,type,
c_2Ecombin_2EK_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27a)) ).
tff(c_2Ecombin_2EK_2E1,type,
c_2Ecombin_2EK_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > tyop_2Emin_2Efun(A_27b,A_27a) ) ).
tff(c_2Ecombin_2EK_2E2,type,
c_2Ecombin_2EK_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a * A_27b ) > A_27a ) ).
tff(c_2Earithmetic_2EMAX_2E0,type,
c_2Earithmetic_2EMAX_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).
tff(c_2Earithmetic_2EMAX_2E2,type,
c_2Earithmetic_2EMAX_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).
tff(c_2Earithmetic_2ENUMERAL_2E0,type,
c_2Earithmetic_2ENUMERAL_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).
tff(c_2Earithmetic_2ENUMERAL_2E1,type,
c_2Earithmetic_2ENUMERAL_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
tff(c_2Epred__set_2ESUBSET_2E0,type,
c_2Epred__set_2ESUBSET_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)) ).
tff(c_2Epred__set_2ESUBSET_2E2,type,
c_2Epred__set_2ESUBSET_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Enum_2ESUC_2E0,type,
c_2Enum_2ESUC_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).
tff(c_2Enum_2ESUC_2E1,type,
c_2Enum_2ESUC_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
tff(c_2Ebool_2ET_2E0,type,
c_2Ebool_2ET_2E0: tyop_2Emin_2Ebool ).
tff(c_2Epair_2EUNCURRY_2E0,type,
c_2Epair_2EUNCURRY_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c)) ).
tff(c_2Epair_2EUNCURRY_2E1,type,
c_2Epair_2EUNCURRY_2E1:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)) > tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c) ) ).
tff(c_2Epred__set_2EUNIV_2E0,type,
c_2Epred__set_2EUNIV_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ).
tff(c_2Earithmetic_2EZERO_2E0,type,
c_2Earithmetic_2EZERO_2E0: tyop_2Enum_2Enum ).
tff(c_2Ebool_2E_5C_2F_2E0,type,
c_2Ebool_2E_5C_2F_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2E_5C_2F_2E2,type,
c_2Ebool_2E_5C_2F_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Ereal_2Eabs_2E0,type,
c_2Ereal_2Eabs_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Ereal_2Eabs_2E1,type,
c_2Ereal_2Eabs_2E1: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
tff(c_2Enets_2Ebounded_2E0,type,
c_2Enets_2Ebounded_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Emetric_2Emetric(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Ebool)) ).
tff(c_2Enets_2Ebounded_2E2,type,
c_2Enets_2Ebounded_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod(tyop_2Emetric_2Emetric(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))) * tyop_2Emin_2Efun(A_27b,A_27a) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Eseq_2Ecauchy_2E0,type,
c_2Eseq_2Ecauchy_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool) ).
tff(c_2Eseq_2Ecauchy_2E1,type,
c_2Eseq_2Ecauchy_2E1: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) > tyop_2Emin_2Ebool ).
tff(c_2Eseq_2Econvergent_2E0,type,
c_2Eseq_2Econvergent_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool) ).
tff(c_2Eseq_2Econvergent_2E1,type,
c_2Eseq_2Econvergent_2E1: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) > tyop_2Emin_2Ebool ).
tff(c_2Epred__set_2Ecount_2E0,type,
c_2Epred__set_2Ecount_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).
tff(c_2Epred__set_2Ecount_2E1,type,
c_2Epred__set_2Ecount_2E1: tyop_2Enum_2Enum > tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool) ).
tff(c_2Erealax_2Einv_2E0,type,
c_2Erealax_2Einv_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Erealax_2Einv_2E1,type,
c_2Erealax_2Einv_2E1: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
tff(c_2Eseq_2Elim_2E0,type,
c_2Eseq_2Elim_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal) ).
tff(c_2Eseq_2Elim_2E1,type,
c_2Eseq_2Elim_2E1: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) > tyop_2Erealax_2Ereal ).
tff(c_2Eseq_2Emono_2E0,type,
c_2Eseq_2Emono_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool) ).
tff(c_2Eseq_2Emono_2E1,type,
c_2Eseq_2Emono_2E1: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) > tyop_2Emin_2Ebool ).
tff(c_2Eseq_2Emono__decreasing_2E0,type,
c_2Eseq_2Emono__decreasing_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool) ).
tff(c_2Eseq_2Emono__decreasing_2E1,type,
c_2Eseq_2Emono__decreasing_2E1: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) > tyop_2Emin_2Ebool ).
tff(c_2Eseq_2Emono__increasing_2E0,type,
c_2Eseq_2Emono__increasing_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool) ).
tff(c_2Eseq_2Emono__increasing_2E1,type,
c_2Eseq_2Emono__increasing_2E1: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) > tyop_2Emin_2Ebool ).
tff(c_2Emetric_2Emr1_2E0,type,
c_2Emetric_2Emr1_2E0: tyop_2Emetric_2Emetric(tyop_2Erealax_2Ereal) ).
tff(c_2Emetric_2Emtop_2E0,type,
c_2Emetric_2Emtop_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emetric_2Emetric(A_27a),tyop_2Etopology_2Etopology(A_27a)) ).
tff(c_2Emetric_2Emtop_2E1,type,
c_2Emetric_2Emtop_2E1:
!>[A_27a: $tType] : ( tyop_2Emetric_2Emetric(A_27a) > tyop_2Etopology_2Etopology(A_27a) ) ).
tff(c_2Ecombin_2Eo_2E0,type,
c_2Ecombin_2Eo_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27c),tyop_2Emin_2Efun(A_27a,A_27b))) ).
tff(c_2Ecombin_2Eo_2E2,type,
c_2Ecombin_2Eo_2E2:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Emin_2Efun(A_27c,A_27b) * tyop_2Emin_2Efun(A_27a,A_27c) ) > tyop_2Emin_2Efun(A_27a,A_27b) ) ).
tff(c_2Ereal_2Epow_2E0,type,
c_2Ereal_2Epow_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)) ).
tff(c_2Ereal_2Epow_2E2,type,
c_2Ereal_2Epow_2E2: ( tyop_2Erealax_2Ereal * tyop_2Enum_2Enum ) > tyop_2Erealax_2Ereal ).
tff(c_2Erealax_2Ereal__add_2E0,type,
c_2Erealax_2Ereal__add_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Erealax_2Ereal__add_2E2,type,
c_2Erealax_2Ereal__add_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
tff(c_2Ereal_2Ereal__ge_2E0,type,
c_2Ereal_2Ereal__ge_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Ereal_2Ereal__ge_2E2,type,
c_2Ereal_2Ereal__ge_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Emin_2Ebool ).
tff(c_2Ereal_2Ereal__gt_2E0,type,
c_2Ereal_2Ereal__gt_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Ereal_2Ereal__gt_2E2,type,
c_2Ereal_2Ereal__gt_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Emin_2Ebool ).
tff(c_2Erealax_2Ereal__lt_2E0,type,
c_2Erealax_2Ereal__lt_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Erealax_2Ereal__lt_2E2,type,
c_2Erealax_2Ereal__lt_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Emin_2Ebool ).
tff(c_2Ereal_2Ereal__lte_2E0,type,
c_2Ereal_2Ereal__lte_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Ereal_2Ereal__lte_2E2,type,
c_2Ereal_2Ereal__lte_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Emin_2Ebool ).
tff(c_2Erealax_2Ereal__mul_2E0,type,
c_2Erealax_2Ereal__mul_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Erealax_2Ereal__mul_2E2,type,
c_2Erealax_2Ereal__mul_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
tff(c_2Erealax_2Ereal__neg_2E0,type,
c_2Erealax_2Ereal__neg_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Erealax_2Ereal__neg_2E1,type,
c_2Erealax_2Ereal__neg_2E1: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
tff(c_2Ereal_2Ereal__of__num_2E0,type,
c_2Ereal_2Ereal__of__num_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) ).
tff(c_2Ereal_2Ereal__of__num_2E1,type,
c_2Ereal_2Ereal__of__num_2E1: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).
tff(c_2Ereal_2Ereal__sub_2E0,type,
c_2Ereal_2Ereal__sub_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Ereal_2Ereal__sub_2E2,type,
c_2Ereal_2Ereal__sub_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
tff(c_2Eseq_2Esubseq_2E0,type,
c_2Eseq_2Esubseq_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool) ).
tff(c_2Eseq_2Esubseq_2E1,type,
c_2Eseq_2Esubseq_2E1: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) > tyop_2Emin_2Ebool ).
tff(c_2Ereal_2Esum_2E0,type,
c_2Ereal_2Esum_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal)) ).
tff(c_2Ereal_2Esum_2E2,type,
c_2Ereal_2Esum_2E2: ( tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum) * tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) ) > tyop_2Erealax_2Ereal ).
tff(c_2Eseq_2Esuminf_2E0,type,
c_2Eseq_2Esuminf_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal) ).
tff(c_2Eseq_2Esuminf_2E1,type,
c_2Eseq_2Esuminf_2E1: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) > tyop_2Erealax_2Ereal ).
tff(c_2Eseq_2Esummable_2E0,type,
c_2Eseq_2Esummable_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool) ).
tff(c_2Eseq_2Esummable_2E1,type,
c_2Eseq_2Esummable_2E1: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) > tyop_2Emin_2Ebool ).
tff(c_2Eseq_2Esums_2E0,type,
c_2Eseq_2Esums_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Eseq_2Esums_2E2,type,
c_2Eseq_2Esums_2E2: ( tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) * tyop_2Erealax_2Ereal ) > tyop_2Emin_2Ebool ).
tff(c_2Ereal_2Esup_2E0,type,
c_2Ereal_2Esup_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal) ).
tff(c_2Ereal_2Esup_2E1,type,
c_2Ereal_2Esup_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) > tyop_2Erealax_2Ereal ).
tff(c_2Enets_2Etends_2E0,type,
c_2Enets_2Etends_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Etopology_2Etopology(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),tyop_2Emin_2Ebool))) ).
tff(c_2Enets_2Etends_2E3,type,
c_2Enets_2Etends_2E3:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27b,A_27a) * A_27a * tyop_2Epair_2Eprod(tyop_2Etopology_2Etopology(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Ebool_2E_7E_2E0,type,
c_2Ebool_2E_7E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_7E_2E1,type,
c_2Ebool_2E_7E_2E1: tyop_2Emin_2Ebool > tyop_2Emin_2Ebool ).
tff(thm_2Eextra_2Dho_2Eeq__ext,axiom,
! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] :
( ! [V2x_2E0: A_27a] : ( app_2E2(A_27a,A_27b,V0f_2E0,V2x_2E0) = app_2E2(A_27a,A_27b,V1g_2E0,V2x_2E0) )
=> ( V0f_2E0 = V1g_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Eboolext,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( ( p(V0_2E0)
<=> p(V1_2E0) )
=> ( V0_2E0 = V1_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Etruth,axiom,
p(c_2Ebool_2ET_2E0) ).
tff(thm_2Eextra_2Dho_2Enotfalse,axiom,
~ p(c_2Ebool_2EF_2E0) ).
tff(thm_2Eextra_2Dho_2Ebool__cases__ax,axiom,
! [V0t_2E0: tyop_2Emin_2Ebool] :
( ( V0t_2E0 = c_2Ebool_2ET_2E0 )
| ( V0t_2E0 = c_2Ebool_2EF_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Ei__thm,axiom,
! [A_27a: $tType,V0x_2E0: A_27a] : ( app_2E2(A_27a,A_27a,combin_i_2E0(A_27a),V0x_2E0) = V0x_2E0 ) ).
tff(thm_2Eextra_2Dho_2Ek__thm,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: A_27a,V1y_2E0: A_27b] : ( app_2E2(A_27b,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27a),combin_k_2E0(A_27a,A_27b),V0x_2E0),V1y_2E0) = V0x_2E0 ) ).
tff(thm_2Eextra_2Dho_2Es__thm,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2x_2E0: A_27a] : ( app_2E2(A_27a,A_27c,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),combin_s_2E0(A_27a,A_27b,A_27c),V0f_2E0),V1g_2E0),V2x_2E0) = app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),V0f_2E0,V2x_2E0),app_2E2(A_27a,A_27b,V1g_2E0,V2x_2E0)) ) ).
tff(logicdef_2E_2F_5C,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_2F_5C_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
& p(V1_2E0) ) ) ).
tff(logicdef_2E_5C_2F,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_5C_2F_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
| p(V1_2E0) ) ) ).
tff(logicdef_2E_7E,axiom,
! [V0_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_7E_2E1(V0_2E0))
<=> ~ p(V0_2E0) ) ).
tff(logicdef_2E_3D_3D_3E,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Emin_2E_3D_3D_3E_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
=> p(V1_2E0) ) ) ).
tff(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0_2E0: A_27a,V1_2E0: A_27a] :
( p(c_2Emin_2E_3D_2E2(A_27a,V0_2E0,V1_2E0))
<=> ( V0_2E0 = V1_2E0 ) ) ).
tff(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2E_21_2E1(A_27a,V0f_2E0))
<=> ! [V1x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0f_2E0,V1x_2E0)) ) ).
tff(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2E_3F_2E1(A_27a,V0f_2E0))
<=> ? [V1x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0f_2E0,V1x_2E0)) ) ).
tff(arityeq1_2Ec_2Ebool_2E_21_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_21_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_21_2E0(A_27a),X0_2E0) ) ).
tff(arityeq2_2Ec_2Earithmetic_2E_2A_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2A_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2E_2A_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Earithmetic_2E_2B_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2B_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2E_2B_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Emetric_2Emetric_28tyop_2Erealax_2Ereal_29_20mono_2Etyop_2Emin_2Efun_28tyop_2Enum_2Enum_2Ctyop_2Emin_2Efun_28tyop_2Enum_2Enum_2Ctyop_2Emin_2Ebool_29_29,axiom,
! [X0_2E0: tyop_2Emetric_2Emetric(tyop_2Erealax_2Ereal),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] : ( c_2Epair_2E_2C_2E2(tyop_2Emetric_2Emetric(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),tyop_2Epair_2Eprod(tyop_2Emetric_2Emetric(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emetric_2Emetric(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),tyop_2Epair_2Eprod(tyop_2Emetric_2Emetric(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)))),c_2Epair_2E_2C_2E0(tyop_2Emetric_2Emetric(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),c_2Epair_2E_2C_2E0(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),c_2Epair_2E_2C_2E0(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Etopology_2Etopology_28tyop_2Erealax_2Ereal_29_20mono_2Etyop_2Emin_2Efun_28tyop_2Enum_2Enum_2Ctyop_2Emin_2Efun_28tyop_2Enum_2Enum_2Ctyop_2Emin_2Ebool_29_29,axiom,
! [X0_2E0: tyop_2Etopology_2Etopology(tyop_2Erealax_2Ereal),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] : ( c_2Epair_2E_2C_2E2(tyop_2Etopology_2Etopology(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),tyop_2Epair_2Eprod(tyop_2Etopology_2Etopology(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),app_2E2(tyop_2Etopology_2Etopology(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),tyop_2Epair_2Eprod(tyop_2Etopology_2Etopology(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)))),c_2Epair_2E_2C_2E0(tyop_2Etopology_2Etopology(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Eseq_2E_2D_2D_3E_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Eseq_2E_2D_2D_3E_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Eseq_2E_2D_2D_3E_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal_2E_2F_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2E_2F_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Ereal_2E_2F_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2E_2F_5C_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
& p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_2F_5C_2E0,X0_2E0),X1_2E0)) ) ).
tff(arityeq2_2Ec_2Eprim__rec_2E_3C_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Eprim__rec_2E_3C_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Earithmetic_2E_3C_3D_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Earithmetic_2E_3C_3D_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Emin_2E_3D_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: A_27a] :
( ( X0_2E0 = X1_2E0 )
<=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Emin_2E_3D_2E0(A_27a),X0_2E0),X1_2E0)) ) ).
tff(arityeq2_2Ec_2Emin_2E_3D_3D_3E_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
=> p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Emin_2E_3D_3D_3E_2E0,X0_2E0),X1_2E0)) ) ).
tff(arityeq2_2Ec_2Earithmetic_2E_3E_3D_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3E_3D_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Earithmetic_2E_3E_3D_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ebool_2E_3F_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_3F_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Emin_2E_40_2E1_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Emin_2E_40_2E1(tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal,c_2Emin_2E_40_2E0(tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq3_2Ec_2Epred__set_2EBIJ_2E3_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Epair_2Eprod_28tyop_2Enum_2Enum_2Ctyop_2Enum_2Enum_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X2_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EBIJ_2E3(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Epred__set_2EBIJ_2E0(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq1_2Ec_2Earithmetic_2EBIT1_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EBIT1_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Earithmetic_2EBIT2_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EBIT2_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2EBIT2_2E0,X0_2E0) ) ).
tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Erealax_2Ereal,X2_2E0: tyop_2Erealax_2Ereal] : ( c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),c_2Ebool_2ECOND_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2ECROSS_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2ECROSS_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool)),c_2Epred__set_2ECROSS_2E0(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EIMAGE_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Epair_2Eprod_28tyop_2Enum_2Enum_2Ctyop_2Enum_2Enum_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool)),c_2Epred__set_2EIMAGE_2E0(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2EIMAGE_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),c_2Epred__set_2EIMAGE_2E0(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ecombin_2EK_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: A_27a,X1_2E0: A_27b] : ( c_2Ecombin_2EK_2E2(A_27a,A_27b,X0_2E0,X1_2E0) = app_2E2(A_27b,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27a),c_2Ecombin_2EK_2E0(A_27a,A_27b),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ecombin_2EK_2E1_2Emono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Ecombin_2EK_2E1(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),c_2Ecombin_2EK_2E0(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum),X0_2E0) ) ).
tff(arityeq2_2Ec_2Earithmetic_2EMAX_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EMAX_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2EMAX_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Earithmetic_2ENUMERAL_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2ENUMERAL_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Epred__set_2ESUBSET_2E2_2Emono_2Etyop_2Epair_2Eprod_28tyop_2Enum_2Enum_2Ctyop_2Enum_2Enum_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2ESUBSET_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Epred__set_2ESUBSET_2E0(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Enum_2ESUC_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Enum_2ESUC_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2ESUC_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] : ( c_2Epair_2EUNCURRY_2E1(tyop_2Enum_2Enum,tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Erealax_2Ereal),c_2Epair_2EUNCURRY_2E0(tyop_2Enum_2Enum,tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2E_5C_2F_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
| p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_5C_2F_2E0,X0_2E0),X1_2E0)) ) ).
tff(arityeq1_2Ec_2Ereal_2Eabs_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Eabs_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Eabs_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Enets_2Ebounded_2E2_2Emono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Emetric_2Emetric(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Enets_2Ebounded_2E2(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,app_2E2(tyop_2Epair_2Eprod(tyop_2Emetric_2Emetric(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),c_2Enets_2Ebounded_2E0(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Eseq_2Ecauchy_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Eseq_2Ecauchy_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,c_2Eseq_2Ecauchy_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Eseq_2Econvergent_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Eseq_2Econvergent_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,c_2Eseq_2Econvergent_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Epred__set_2Ecount_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Epred__set_2Ecount_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epred__set_2Ecount_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Erealax_2Einv_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Einv_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Einv_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Eseq_2Elim_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Eseq_2Elim_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,c_2Eseq_2Elim_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Eseq_2Emono_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Eseq_2Emono_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,c_2Eseq_2Emono_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Eseq_2Emono__decreasing_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Eseq_2Emono__decreasing_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,c_2Eseq_2Emono__decreasing_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Eseq_2Emono__increasing_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Eseq_2Emono__increasing_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,c_2Eseq_2Emono__increasing_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Emetric_2Emtop_2E1_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emetric_2Emetric(tyop_2Erealax_2Ereal)] : ( c_2Emetric_2Emtop_2E1(tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emetric_2Emetric(tyop_2Erealax_2Ereal),tyop_2Etopology_2Etopology(tyop_2Erealax_2Ereal),c_2Emetric_2Emtop_2E0(tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Epair_2Eprod_28tyop_2Enum_2Enum_2Ctyop_2Enum_2Enum_29,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Erealax_2Ereal),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum))] : ( c_2Ecombin_2Eo_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),c_2Ecombin_2Eo_2E0(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal_2Epow_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Epow_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),c_2Ereal_2Epow_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Erealax_2Ereal__add_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Erealax_2Ereal__add_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal_2Ereal__ge_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__ge_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal_2Ereal__ge_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal_2Ereal__gt_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__gt_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal_2Ereal__gt_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Erealax_2Ereal__lt_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__lt_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Erealax_2Ereal__lt_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal_2Ereal__lte_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal_2Ereal__lte_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Erealax_2Ereal__mul_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Erealax_2Ereal__mul_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Erealax_2Ereal__neg_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ereal_2Ereal__of__num_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__of__num_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ereal_2Ereal__sub_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__sub_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Ereal_2Ereal__sub_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Eseq_2Esubseq_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : ( c_2Eseq_2Esubseq_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Ebool,c_2Eseq_2Esubseq_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ereal_2Esum_2E2,axiom,
! [X0_2E0: tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Ereal_2Esum_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,app_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal),c_2Ereal_2Esum_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Eseq_2Esuminf_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Eseq_2Esuminf_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,c_2Eseq_2Esuminf_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Eseq_2Esummable_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Eseq_2Esummable_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,c_2Eseq_2Esummable_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Eseq_2Esums_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Eseq_2Esums_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Eseq_2Esums_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ereal_2Esup_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal_2Esup_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal,c_2Ereal_2Esup_2E0,X0_2E0) ) ).
tff(arityeq3_2Ec_2Enets_2Etends_2E3_2Emono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Erealax_2Ereal,X2_2E0: tyop_2Epair_2Eprod(tyop_2Etopology_2Etopology(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)))] : ( c_2Enets_2Etends_2E3(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Etopology_2Etopology(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Etopology_2Etopology(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Etopology_2Etopology(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),tyop_2Emin_2Ebool)),c_2Enets_2Etends_2E0(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq1_2Ec_2Ebool_2E_7E_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Ebool] :
( ~ p(X0_2E0)
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,c_2Ebool_2E_7E_2E0,X0_2E0)) ) ).
tff(thm_2Eseq_2Etends__num__real,axiom,
! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1x0_2E0: tyop_2Erealax_2Ereal] : ( c_2Eseq_2E_2D_2D_3E_2E2(V0x_2E0,V1x0_2E0) = c_2Enets_2Etends_2E3(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,V0x_2E0,V1x0_2E0,c_2Epair_2E_2C_2E2(tyop_2Etopology_2Etopology(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Emetric_2Emtop_2E1(tyop_2Erealax_2Ereal,c_2Emetric_2Emr1_2E0),c_2Earithmetic_2E_3E_3D_2E0)) ) ).
tff(thm_2Eseq_2Econvergent,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Eseq_2Econvergent_2E1(V0f_2E0))
<=> ? [V1l_2E0: tyop_2Erealax_2Ereal] : p(c_2Eseq_2E_2D_2D_3E_2E2(V0f_2E0,V1l_2E0)) ) ).
tff(thm_2Eseq_2Ecauchy,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Eseq_2Ecauchy_2E1(V0f_2E0))
<=> ! [V1e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1e_2E0))
=> ? [V2N_2E0: tyop_2Enum_2Enum] :
! [V3m_2E0: tyop_2Enum_2Enum,V4n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Earithmetic_2E_3E_3D_2E2(V3m_2E0,V2N_2E0))
& p(c_2Earithmetic_2E_3E_3D_2E2(V4n_2E0,V2N_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4n_2E0))),V1e_2E0)) ) ) ) ).
tff(thm_2Eseq_2Elim,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0f_2E0),V1l_2E0) = c_2Eseq_2E_2D_2D_3E_2E2(V0f_2E0,V1l_2E0) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Eseq_2Elim_2E1(V0f_2E0) = c_2Emin_2E_40_2E1(tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0f_2E0)) ) ) ).
tff(thm_2Eseq_2Esubseq,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
( p(c_2Eseq_2Esubseq_2E1(V0f_2E0))
<=> ! [V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V1m_2E0,V2n_2E0))
=> p(c_2Eprim__rec_2E_3C_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,V1m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,V2n_2E0))) ) ) ).
tff(thm_2Eseq_2Emono,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Eseq_2Emono_2E1(V0f_2E0))
<=> ( ! [V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V2n_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2n_2E0))) )
| ! [V3m_2E0: tyop_2Enum_2Enum,V4n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V3m_2E0,V4n_2E0))
=> p(c_2Ereal_2Ereal__ge_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4n_2E0))) ) ) ) ).
tff(thm_2Eseq_2Esums,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V2n_2E0) = c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,V2n_2E0),V0f_2E0) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Erealax_2Ereal] : ( c_2Eseq_2Esums_2E2(V0f_2E0,V1s_2E0) = c_2Eseq_2E_2D_2D_3E_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V1s_2E0) ) ) ).
tff(thm_2Eseq_2Esummable,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Eseq_2Esummable_2E1(V0f_2E0))
<=> ? [V1s_2E0: tyop_2Erealax_2Ereal] : p(c_2Eseq_2Esums_2E2(V0f_2E0,V1s_2E0)) ) ).
tff(thm_2Eseq_2Esuminf,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0f_2E0),V1s_2E0) = c_2Eseq_2Esums_2E2(V0f_2E0,V1s_2E0) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Eseq_2Esuminf_2E1(V0f_2E0) = c_2Emin_2E_40_2E1(tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0f_2E0)) ) ) ).
tff(thm_2Eseq_2Emono__increasing__def,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Eseq_2Emono__increasing_2E1(V0f_2E0))
<=> ! [V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V2n_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2n_2E0))) ) ) ).
tff(thm_2Eseq_2Emono__decreasing__def,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Eseq_2Emono__decreasing_2E1(V0f_2E0))
<=> ! [V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V2n_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1m_2E0))) ) ) ).
tff(thm_2Eseq_2ESEQ,axiom,
! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1x0_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Eseq_2E_2D_2D_3E_2E2(V0x_2E0,V1x0_2E0))
<=> ! [V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
=> ? [V3N_2E0: tyop_2Enum_2Enum] :
! [V4n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3E_3D_2E2(V4n_2E0,V3N_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V4n_2E0),V1x0_2E0)),V2e_2E0)) ) ) ) ).
tff(thm_2Eseq_2ESEQ__CONST,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0k_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0k_2E0),V1x_2E0) = V0k_2E0 )
=> ! [V0k_2E0: tyop_2Erealax_2Ereal] : p(c_2Eseq_2E_2D_2D_3E_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0k_2E0),V0k_2E0)) ) ).
tff(thm_2Eseq_2ESEQ__ADD,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2y_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0x_2E0),V2y_2E0),V4n_2E0) = c_2Erealax_2Ereal__add_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V4n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2y_2E0,V4n_2E0)) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1x0_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3y0_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Eseq_2E_2D_2D_3E_2E2(V0x_2E0,V1x0_2E0))
& p(c_2Eseq_2E_2D_2D_3E_2E2(V2y_2E0,V3y0_2E0)) )
=> p(c_2Eseq_2E_2D_2D_3E_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0x_2E0),V2y_2E0),c_2Erealax_2Ereal__add_2E2(V1x0_2E0,V3y0_2E0))) ) ) ).
tff(thm_2Eseq_2ESEQ__MUL,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2y_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0x_2E0),V2y_2E0),V4n_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V4n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2y_2E0,V4n_2E0)) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1x0_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3y0_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Eseq_2E_2D_2D_3E_2E2(V0x_2E0,V1x0_2E0))
& p(c_2Eseq_2E_2D_2D_3E_2E2(V2y_2E0,V3y0_2E0)) )
=> p(c_2Eseq_2E_2D_2D_3E_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0x_2E0),V2y_2E0),c_2Erealax_2Ereal__mul_2E2(V1x0_2E0,V3y0_2E0))) ) ) ).
tff(thm_2Eseq_2ESEQ__NEG,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V2n_2E0) = c_2Erealax_2Ereal__neg_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V2n_2E0)) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1x0_2E0: tyop_2Erealax_2Ereal] : ( c_2Eseq_2E_2D_2D_3E_2E2(V0x_2E0,V1x0_2E0) = c_2Eseq_2E_2D_2D_3E_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),c_2Erealax_2Ereal__neg_2E1(V1x0_2E0)) ) ) ).
tff(thm_2Eseq_2ESEQ__INV,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V2n_2E0) = c_2Erealax_2Einv_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V2n_2E0)) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1x0_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Eseq_2E_2D_2D_3E_2E2(V0x_2E0,V1x0_2E0))
& ( V1x0_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> p(c_2Eseq_2E_2D_2D_3E_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),c_2Erealax_2Einv_2E1(V1x0_2E0))) ) ) ).
tff(thm_2Eseq_2ESEQ__SUB,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2y_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0x_2E0),V2y_2E0),V4n_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V4n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2y_2E0,V4n_2E0)) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1x0_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3y0_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Eseq_2E_2D_2D_3E_2E2(V0x_2E0,V1x0_2E0))
& p(c_2Eseq_2E_2D_2D_3E_2E2(V2y_2E0,V3y0_2E0)) )
=> p(c_2Eseq_2E_2D_2D_3E_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0x_2E0),V2y_2E0),c_2Ereal_2Ereal__sub_2E2(V1x0_2E0,V3y0_2E0))) ) ) ).
tff(thm_2Eseq_2ESEQ__DIV,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2y_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0x_2E0),V2y_2E0),V4n_2E0) = c_2Ereal_2E_2F_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V4n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2y_2E0,V4n_2E0)) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1x0_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3y0_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Eseq_2E_2D_2D_3E_2E2(V0x_2E0,V1x0_2E0))
& p(c_2Eseq_2E_2D_2D_3E_2E2(V2y_2E0,V3y0_2E0))
& ( V3y0_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> p(c_2Eseq_2E_2D_2D_3E_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0x_2E0),V2y_2E0),c_2Ereal_2E_2F_2E2(V1x0_2E0,V3y0_2E0))) ) ) ).
tff(thm_2Eseq_2ESEQ__UNIQ,axiom,
! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1x1_2E0: tyop_2Erealax_2Ereal,V2x2_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Eseq_2E_2D_2D_3E_2E2(V0x_2E0,V1x1_2E0))
& p(c_2Eseq_2E_2D_2D_3E_2E2(V0x_2E0,V2x2_2E0)) )
=> ( V1x1_2E0 = V2x2_2E0 ) ) ).
tff(thm_2Eseq_2ESEQ__LIM,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Eseq_2Econvergent_2E1(V0f_2E0) = c_2Eseq_2E_2D_2D_3E_2E2(V0f_2E0,c_2Eseq_2Elim_2E1(V0f_2E0)) ) ).
tff(thm_2Eseq_2ESUBSEQ__SUC,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
( p(c_2Eseq_2Esubseq_2E1(V0f_2E0))
<=> ! [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Eprim__rec_2E_3C_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,V1n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,c_2Enum_2ESUC_2E1(V1n_2E0)))) ) ).
tff(thm_2Eseq_2EMONO__SUC,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Eseq_2Emono_2E1(V0f_2E0))
<=> ( ! [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__ge_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Enum_2ESUC_2E1(V1n_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1n_2E0)))
| ! [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Enum_2ESUC_2E1(V2n_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2n_2E0))) ) ) ).
tff(thm_2Eseq_2EMAX__LEMMA,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1N_2E0: tyop_2Enum_2Enum] :
? [V2k_2E0: tyop_2Erealax_2Ereal] :
! [V3n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V3n_2E0,V1N_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,V3n_2E0)),V2k_2E0)) ) ).
tff(thm_2Eseq_2ESEQ__BOUNDED,axiom,
! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Enets_2Ebounded_2E2(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,c_2Epair_2E_2C_2E2(tyop_2Emetric_2Emetric(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Emetric_2Emr1_2E0,c_2Earithmetic_2E_3E_3D_2E0),V0s_2E0))
<=> ? [V1k_2E0: tyop_2Erealax_2Ereal] :
! [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,V2n_2E0)),V1k_2E0)) ) ).
tff(thm_2Eseq_2ESEQ__BOUNDED__2,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1k_2E0: tyop_2Erealax_2Ereal,V2k_27_2E0: tyop_2Erealax_2Ereal] :
( ! [V3n_2E0: tyop_2Enum_2Enum] :
( p(c_2Ereal_2Ereal__lte_2E2(V1k_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3n_2E0)))
& p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3n_2E0),V2k_27_2E0)) )
=> p(c_2Enets_2Ebounded_2E2(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,c_2Epair_2E_2C_2E2(tyop_2Emetric_2Emetric(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Emetric_2Emr1_2E0,c_2Earithmetic_2E_3E_3D_2E0),V0f_2E0)) ) ).
tff(thm_2Eseq_2ESEQ__CBOUNDED,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Eseq_2Ecauchy_2E1(V0f_2E0))
=> p(c_2Enets_2Ebounded_2E2(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,c_2Epair_2E_2C_2E2(tyop_2Emetric_2Emetric(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Emetric_2Emr1_2E0,c_2Earithmetic_2E_3E_3D_2E0),V0f_2E0)) ) ).
tff(thm_2Eseq_2ESEQ__ICONV,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ( p(c_2Enets_2Ebounded_2E2(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,c_2Epair_2E_2C_2E2(tyop_2Emetric_2Emetric(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Emetric_2Emr1_2E0,c_2Earithmetic_2E_3E_3D_2E0),V0f_2E0))
& ! [V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3E_3D_2E2(V1m_2E0,V2n_2E0))
=> p(c_2Ereal_2Ereal__ge_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2n_2E0))) ) )
=> p(c_2Eseq_2Econvergent_2E1(V0f_2E0)) ) ).
tff(thm_2Eseq_2ESEQ__NEG__CONV,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V1n_2E0) = c_2Erealax_2Ereal__neg_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Eseq_2Econvergent_2E1(V0f_2E0) = c_2Eseq_2Econvergent_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0)) ) ) ).
tff(thm_2Eseq_2ESEQ__NEG__BOUNDED,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V1n_2E0) = c_2Erealax_2Ereal__neg_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Enets_2Ebounded_2E2(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,c_2Epair_2E_2C_2E2(tyop_2Emetric_2Emetric(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Emetric_2Emr1_2E0,c_2Earithmetic_2E_3E_3D_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0)) = c_2Enets_2Ebounded_2E2(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,c_2Epair_2E_2C_2E2(tyop_2Emetric_2Emetric(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Emetric_2Emr1_2E0,c_2Earithmetic_2E_3E_3D_2E0),V0f_2E0) ) ) ).
tff(thm_2Eseq_2ESEQ__BCONV,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ( p(c_2Enets_2Ebounded_2E2(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,c_2Epair_2E_2C_2E2(tyop_2Emetric_2Emetric(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Emetric_2Emr1_2E0,c_2Earithmetic_2E_3E_3D_2E0),V0f_2E0))
& p(c_2Eseq_2Emono_2E1(V0f_2E0)) )
=> p(c_2Eseq_2Econvergent_2E1(V0f_2E0)) ) ).
tff(thm_2Eseq_2ESEQ__MONOSUB,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0s_2E0),V1f_2E0),V2n_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1f_2E0,V2n_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
? [V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
( p(c_2Eseq_2Esubseq_2E1(V1f_2E0))
& p(c_2Eseq_2Emono_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0s_2E0),V1f_2E0))) ) ) ).
tff(thm_2Eseq_2ESEQ__SBOUNDED,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0s_2E0),V1f_2E0),V2n_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0s_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1f_2E0,V2n_2E0)) )
=> ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
( p(c_2Enets_2Ebounded_2E2(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,c_2Epair_2E_2C_2E2(tyop_2Emetric_2Emetric(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Emetric_2Emr1_2E0,c_2Earithmetic_2E_3E_3D_2E0),V0s_2E0))
=> p(c_2Enets_2Ebounded_2E2(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,c_2Epair_2E_2C_2E2(tyop_2Emetric_2Emetric(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Emetric_2Emr1_2E0,c_2Earithmetic_2E_3E_3D_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0s_2E0),V1f_2E0))) ) ) ).
tff(thm_2Eseq_2ESEQ__SUBLE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
( p(c_2Eseq_2Esubseq_2E1(V0f_2E0))
=> ! [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,V1n_2E0))) ) ).
tff(thm_2Eseq_2ESEQ__DIRECT,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
( p(c_2Eseq_2Esubseq_2E1(V0f_2E0))
=> ! [V1N1_2E0: tyop_2Enum_2Enum,V2N2_2E0: tyop_2Enum_2Enum] :
? [V3n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3E_3D_2E2(V3n_2E0,V1N1_2E0))
& p(c_2Earithmetic_2E_3E_3D_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,V3n_2E0),V2N2_2E0)) ) ) ).
tff(thm_2Eseq_2ESEQ__CAUCHY,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Eseq_2Ecauchy_2E1(V0f_2E0) = c_2Eseq_2Econvergent_2E1(V0f_2E0) ) ).
tff(thm_2Eseq_2ESEQ__LE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2l_2E0: tyop_2Erealax_2Ereal,V3m_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Eseq_2E_2D_2D_3E_2E2(V0f_2E0,V2l_2E0))
& p(c_2Eseq_2E_2D_2D_3E_2E2(V1g_2E0,V3m_2E0))
& ? [V4N_2E0: tyop_2Enum_2Enum] :
! [V5n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3E_3D_2E2(V5n_2E0,V4N_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V5n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V5n_2E0))) ) )
=> p(c_2Ereal_2Ereal__lte_2E2(V2l_2E0,V3m_2E0)) ) ).
tff(thm_2Eseq_2ESEQ__SUC,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V2n_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Enum_2ESUC_2E1(V2n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal] : ( c_2Eseq_2E_2D_2D_3E_2E2(V0f_2E0,V1l_2E0) = c_2Eseq_2E_2D_2D_3E_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V1l_2E0) ) ) ).
tff(thm_2Eseq_2ESEQ__ABS,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V1n_2E0) = c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Eseq_2E_2D_2D_3E_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) = c_2Eseq_2E_2D_2D_3E_2E2(V0f_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) ) ) ).
tff(thm_2Eseq_2ESEQ__ABS__IMP,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V2n_2E0) = c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Eseq_2E_2D_2D_3E_2E2(V0f_2E0,V1l_2E0))
=> p(c_2Eseq_2E_2D_2D_3E_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),c_2Ereal_2Eabs_2E1(V1l_2E0))) ) ) ).
tff(thm_2Eseq_2ESEQ__INV0,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V4n_2E0) = c_2Erealax_2Einv_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ! [V1y_2E0: tyop_2Erealax_2Ereal] :
? [V2N_2E0: tyop_2Enum_2Enum] :
! [V3n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3E_3D_2E2(V3n_2E0,V2N_2E0))
=> p(c_2Ereal_2Ereal__gt_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3n_2E0),V1y_2E0)) )
=> p(c_2Eseq_2E_2D_2D_3E_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) ) ).
tff(thm_2Eseq_2ESEQ__POWER__ABS,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0c_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0c_2E0),V1n_2E0) = c_2Ereal_2Epow_2E2(c_2Ereal_2Eabs_2E1(V0c_2E0),V1n_2E0) )
=> ! [V0c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(V0c_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))
=> p(c_2Eseq_2E_2D_2D_3E_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0c_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) ) ).
tff(thm_2Eseq_2ESEQ__POWER,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0c_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0c_2E0),V1n_2E0) = c_2Ereal_2Epow_2E2(V0c_2E0,V1n_2E0) )
=> ! [V0c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(V0c_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))
=> p(c_2Eseq_2E_2D_2D_3E_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0c_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) ) ).
tff(thm_2Eseq_2ENEST__LEMMA,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ( ! [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__ge_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Enum_2ESUC_2E1(V2n_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2n_2E0)))
& ! [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,c_2Enum_2ESUC_2E1(V3n_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V3n_2E0)))
& ! [V4n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V4n_2E0))) )
=> ? [V5l_2E0: tyop_2Erealax_2Ereal,V6m_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V5l_2E0,V6m_2E0))
& ! [V7n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V7n_2E0),V5l_2E0))
& p(c_2Eseq_2E_2D_2D_3E_2E2(V0f_2E0,V5l_2E0))
& ! [V8n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(V6m_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V8n_2E0)))
& p(c_2Eseq_2E_2D_2D_3E_2E2(V1g_2E0,V6m_2E0)) ) ) ).
tff(thm_2Eseq_2ENEST__LEMMA__UNIQ,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V5n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V5n_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V5n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V5n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ( ! [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__ge_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Enum_2ESUC_2E1(V2n_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2n_2E0)))
& ! [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,c_2Enum_2ESUC_2E1(V3n_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V3n_2E0)))
& ! [V4n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V4n_2E0)))
& p(c_2Eseq_2E_2D_2D_3E_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) )
=> ? [V6l_2E0: tyop_2Erealax_2Ereal] :
( ! [V7n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V7n_2E0),V6l_2E0))
& p(c_2Eseq_2E_2D_2D_3E_2E2(V0f_2E0,V6l_2E0))
& ! [V8n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(V6l_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V8n_2E0)))
& p(c_2Eseq_2E_2D_2D_3E_2E2(V1g_2E0,V6l_2E0)) ) ) ) ).
tff(thm_2Eseq_2EBOLZANO__LEMMA,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool)] :
( ( ! [V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal,V3c_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V2b_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V2b_2E0,V3c_2E0))
& p(app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,V0P_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V2b_2E0)))
& p(app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,V0P_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V2b_2E0,V3c_2E0))) )
=> p(app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,V0P_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1a_2E0,V3c_2E0))) )
& ! [V4x_2E0: tyop_2Erealax_2Ereal] :
? [V5d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V5d_2E0))
& ! [V6a_2E0: tyop_2Erealax_2Ereal,V7b_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Ereal_2Ereal__lte_2E2(V6a_2E0,V4x_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(V4x_2E0,V7b_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__sub_2E2(V7b_2E0,V6a_2E0),V5d_2E0)) )
=> p(app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,V0P_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V6a_2E0,V7b_2E0))) ) ) )
=> ! [V8a_2E0: tyop_2Erealax_2Ereal,V9b_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V8a_2E0,V9b_2E0))
=> p(app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,V0P_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V8a_2E0,V9b_2E0))) ) ) ).
tff(thm_2Eseq_2ESUM__SUMMABLE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Eseq_2Esums_2E2(V0f_2E0,V1l_2E0))
=> p(c_2Eseq_2Esummable_2E1(V0f_2E0)) ) ).
tff(thm_2Eseq_2ESUMMABLE__SUM,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Eseq_2Esummable_2E1(V0f_2E0))
=> p(c_2Eseq_2Esums_2E2(V0f_2E0,c_2Eseq_2Esuminf_2E1(V0f_2E0))) ) ).
tff(thm_2Eseq_2ESUM__UNIQ,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Eseq_2Esums_2E2(V0f_2E0,V1x_2E0))
=> ( V1x_2E0 = c_2Eseq_2Esuminf_2E1(V0f_2E0) ) ) ).
tff(thm_2Eseq_2ESER__0,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1n_2E0: tyop_2Enum_2Enum] :
( ! [V2m_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V2m_2E0))
=> ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2m_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
=> p(c_2Eseq_2Esums_2E2(V0f_2E0,c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,V1n_2E0),V0f_2E0))) ) ).
tff(thm_2Eseq_2ESER__POS__LE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eseq_2Esummable_2E1(V0f_2E0))
& ! [V2m_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V2m_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2m_2E0))) ) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,V1n_2E0),V0f_2E0),c_2Eseq_2Esuminf_2E1(V0f_2E0))) ) ).
tff(thm_2Eseq_2ESER__POS__LT,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eseq_2Esummable_2E1(V0f_2E0))
& ! [V2m_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V2m_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2m_2E0))) ) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,V1n_2E0),V0f_2E0),c_2Eseq_2Esuminf_2E1(V0f_2E0))) ) ).
tff(thm_2Eseq_2ESER__GROUP,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V1k_2E0: tyop_2Enum_2Enum,V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1k_2E0),V0f_2E0),V2n_2E0) = c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2E_2A_2E2(V2n_2E0,V1k_2E0),V1k_2E0),V0f_2E0) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1k_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eseq_2Esummable_2E1(V0f_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V1k_2E0)) )
=> p(c_2Eseq_2Esums_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1k_2E0),V0f_2E0),c_2Eseq_2Esuminf_2E1(V0f_2E0))) ) ) ).
tff(thm_2Eseq_2ESER__PAIR,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V1n_2E0) = c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V1n_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),V0f_2E0) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Eseq_2Esummable_2E1(V0f_2E0))
=> p(c_2Eseq_2Esums_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),c_2Eseq_2Esuminf_2E1(V0f_2E0))) ) ) ).
tff(thm_2Eseq_2ESER__OFFSET,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1k_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1k_2E0),V2n_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2B_2E2(V2n_2E0,V1k_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Eseq_2Esummable_2E1(V0f_2E0))
=> ! [V1k_2E0: tyop_2Enum_2Enum] : p(c_2Eseq_2Esums_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1k_2E0),c_2Ereal_2Ereal__sub_2E2(c_2Eseq_2Esuminf_2E1(V0f_2E0),c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,V1k_2E0),V0f_2E0)))) ) ) ).
tff(thm_2Eseq_2ESER__POS__LT__PAIR,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eseq_2Esummable_2E1(V0f_2E0))
& ! [V2d_2E0: tyop_2Enum_2Enum] : p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Erealax_2Ereal__add_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2B_2E2(V1n_2E0,c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V2d_2E0))),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2B_2E2(V1n_2E0,c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V2d_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))))) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,V1n_2E0),V0f_2E0),c_2Eseq_2Esuminf_2E1(V0f_2E0))) ) ).
tff(thm_2Eseq_2ESER__ADD,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2y_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0x_2E0),V2y_2E0),V4n_2E0) = c_2Erealax_2Ereal__add_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V4n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2y_2E0,V4n_2E0)) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1x0_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3y0_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Eseq_2Esums_2E2(V0x_2E0,V1x0_2E0))
& p(c_2Eseq_2Esums_2E2(V2y_2E0,V3y0_2E0)) )
=> p(c_2Eseq_2Esums_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0x_2E0),V2y_2E0),c_2Erealax_2Ereal__add_2E2(V1x0_2E0,V3y0_2E0))) ) ) ).
tff(thm_2Eseq_2ESER__CMUL,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V2c_2E0: tyop_2Erealax_2Ereal,V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V2c_2E0),V0x_2E0),V3n_2E0) = c_2Erealax_2Ereal__mul_2E2(V2c_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V3n_2E0)) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1x0_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Eseq_2Esums_2E2(V0x_2E0,V1x0_2E0))
=> p(c_2Eseq_2Esums_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V2c_2E0),V0x_2E0),c_2Erealax_2Ereal__mul_2E2(V2c_2E0,V1x0_2E0))) ) ) ).
tff(thm_2Eseq_2ESER__NEG,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V2n_2E0) = c_2Erealax_2Ereal__neg_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V2n_2E0)) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1x0_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Eseq_2Esums_2E2(V0x_2E0,V1x0_2E0))
=> p(c_2Eseq_2Esums_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),c_2Erealax_2Ereal__neg_2E1(V1x0_2E0))) ) ) ).
tff(thm_2Eseq_2ESER__SUB,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2y_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0x_2E0),V2y_2E0),V4n_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V4n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2y_2E0,V4n_2E0)) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1x0_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3y0_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Eseq_2Esums_2E2(V0x_2E0,V1x0_2E0))
& p(c_2Eseq_2Esums_2E2(V2y_2E0,V3y0_2E0)) )
=> p(c_2Eseq_2Esums_2E2(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0x_2E0),V2y_2E0),c_2Ereal_2Ereal__sub_2E2(V1x0_2E0,V3y0_2E0))) ) ) ).
tff(thm_2Eseq_2ESER__CDIV,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2c_2E0: tyop_2Erealax_2Ereal,V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0x_2E0),V2c_2E0),V3n_2E0) = c_2Ereal_2E_2F_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0x_2E0,V3n_2E0),V2c_2E0) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1x0_2E0: tyop_2Erealax_2Ereal,V2c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Eseq_2Esums_2E2(V0x_2E0,V1x0_2E0))
=> p(c_2Eseq_2Esums_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0x_2E0),V2c_2E0),c_2Ereal_2E_2F_2E2(V1x0_2E0,V2c_2E0))) ) ) ).
tff(thm_2Eseq_2ESER__CAUCHY,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Eseq_2Esummable_2E1(V0f_2E0))
<=> ! [V1e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1e_2E0))
=> ? [V2N_2E0: tyop_2Enum_2Enum] :
! [V3m_2E0: tyop_2Enum_2Enum,V4n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3E_3D_2E2(V3m_2E0,V2N_2E0))
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V3m_2E0,V4n_2E0),V0f_2E0)),V1e_2E0)) ) ) ) ).
tff(thm_2Eseq_2ESER__ZERO,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Eseq_2Esummable_2E1(V0f_2E0))
=> p(c_2Eseq_2E_2D_2D_3E_2E2(V0f_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) ).
tff(thm_2Eseq_2ESER__COMPAR,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ( ? [V2N_2E0: tyop_2Enum_2Enum] :
! [V3n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3E_3D_2E2(V3n_2E0,V2N_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3n_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V3n_2E0))) )
& p(c_2Eseq_2Esummable_2E1(V1g_2E0)) )
=> p(c_2Eseq_2Esummable_2E1(V0f_2E0)) ) ).
tff(thm_2Eseq_2ESER__COMPARA,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4k_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V4k_2E0) = c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4k_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ( ? [V2N_2E0: tyop_2Enum_2Enum] :
! [V3n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3E_3D_2E2(V3n_2E0,V2N_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3n_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V3n_2E0))) )
& p(c_2Eseq_2Esummable_2E1(V1g_2E0)) )
=> p(c_2Eseq_2Esummable_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0))) ) ) ).
tff(thm_2Eseq_2ESER__LE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ( ! [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V2n_2E0)))
& p(c_2Eseq_2Esummable_2E1(V0f_2E0))
& p(c_2Eseq_2Esummable_2E1(V1g_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Eseq_2Esuminf_2E1(V0f_2E0),c_2Eseq_2Esuminf_2E1(V1g_2E0))) ) ).
tff(thm_2Eseq_2ESER__LE2,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ( ! [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2n_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V2n_2E0)))
& p(c_2Eseq_2Esummable_2E1(V1g_2E0)) )
=> ( p(c_2Eseq_2Esummable_2E1(V0f_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Eseq_2Esuminf_2E1(V0f_2E0),c_2Eseq_2Esuminf_2E1(V1g_2E0))) ) ) ).
tff(thm_2Eseq_2ESER__ACONV,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V1n_2E0) = c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Eseq_2Esummable_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0)))
=> p(c_2Eseq_2Esummable_2E1(V0f_2E0)) ) ) ).
tff(thm_2Eseq_2ESER__ABS,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F1_2E0,V0f_2E0),V2n_2E0) = c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2n_2E0)) )
=> ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V1n_2E0) = c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Eseq_2Esummable_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0)))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Eseq_2Esuminf_2E1(V0f_2E0)),c_2Eseq_2Esuminf_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F1_2E0,V0f_2E0)))) ) ) ) ).
tff(thm_2Eseq_2EGP__FINITE,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal,V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V2n_2E0) = c_2Ereal_2Epow_2E2(V0x_2E0,V2n_2E0) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal] :
( ( V0x_2E0 != c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) )
=> ! [V1n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,V1n_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0)) = c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Epow_2E2(V0x_2E0,V1n_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Ereal_2Ereal__sub_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ) ) ).
tff(thm_2Eseq_2EGP,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V1n_2E0) = c_2Ereal_2Epow_2E2(V0x_2E0,V1n_2E0) )
=> ! [V0x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Eabs_2E1(V0x_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))
=> p(c_2Eseq_2Esums_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),c_2Erealax_2Einv_2E1(c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0x_2E0)))) ) ) ).
tff(thm_2Eseq_2EABS__NEG__LEMMA,axiom,
! [V0c_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(V0c_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
=> ! [V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(V1x_2E0),c_2Erealax_2Ereal__mul_2E2(V0c_2E0,c_2Ereal_2Eabs_2E1(V2y_2E0))))
=> ( V1x_2E0 = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ) ).
tff(thm_2Eseq_2ESER__RATIO,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1c_2E0: tyop_2Erealax_2Ereal,V2N_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Erealax_2Ereal__lt_2E2(V1c_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))
& ! [V3n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3E_3D_2E2(V3n_2E0,V2N_2E0))
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Enum_2ESUC_2E1(V3n_2E0))),c_2Erealax_2Ereal__mul_2E2(V1c_2E0,c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3n_2E0))))) ) )
=> p(c_2Eseq_2Esummable_2E1(V0f_2E0)) ) ).
tff(thm_2Eseq_2ELE__SEQ__IMP__LE__LIM,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ( ! [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2f_2E0,V3n_2E0)))
& p(c_2Eseq_2E_2D_2D_3E_2E2(V2f_2E0,V1y_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,V1y_2E0)) ) ).
tff(thm_2Eseq_2ESEQ__LE__IMP__LIM__LE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ( ! [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2f_2E0,V3n_2E0),V0x_2E0))
& p(c_2Eseq_2E_2D_2D_3E_2E2(V2f_2E0,V1y_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(V1y_2E0,V0x_2E0)) ) ).
tff(thm_2Eseq_2ESEQ__MONO__LE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal,V2n_2E0: tyop_2Enum_2Enum] :
( ( ! [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2B_2E2(V3n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))))
& p(c_2Eseq_2E_2D_2D_3E_2E2(V0f_2E0,V1x_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2n_2E0),V1x_2E0)) ) ).
tff(thm_2Eseq_2ESEQ__LE__MONO,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal,V2n_2E0: tyop_2Enum_2Enum] :
( ( ! [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2B_2E2(V3n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3n_2E0)))
& p(c_2Eseq_2E_2D_2D_3E_2E2(V0f_2E0,V1x_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(V1x_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2n_2E0))) ) ).
tff(thm_2Eseq_2Emono__increasing__suc,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Eseq_2Emono__increasing_2E1(V0f_2E0))
<=> ! [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Enum_2ESUC_2E1(V1n_2E0)))) ) ).
tff(thm_2Eseq_2Emono__decreasing__suc,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( p(c_2Eseq_2Emono__decreasing_2E1(V0f_2E0))
<=> ! [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Enum_2ESUC_2E1(V1n_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1n_2E0))) ) ).
tff(thm_2Eseq_2Emono__increasing__converges__to__sup,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1r_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Eseq_2Emono__increasing_2E1(V0f_2E0))
& p(c_2Eseq_2E_2D_2D_3E_2E2(V0f_2E0,V1r_2E0)) )
=> ( V1r_2E0 = c_2Ereal_2Esup_2E1(c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))) ) ) ).
tff(thm_2Eseq_2EINCREASING__SEQ,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1l_2E0: tyop_2Erealax_2Ereal] :
( ( ! [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Enum_2ESUC_2E1(V2n_2E0))))
& ! [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3n_2E0),V1l_2E0))
& ! [V4e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V4e_2E0))
=> ? [V5n_2E0: tyop_2Enum_2Enum] : p(c_2Erealax_2Ereal__lt_2E2(V1l_2E0,c_2Erealax_2Ereal__add_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V5n_2E0),V4e_2E0))) ) )
=> p(c_2Eseq_2E_2D_2D_3E_2E2(V0f_2E0,V1l_2E0)) ) ).
tff(thm_2Eseq_2EMAX__LE__X,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2k_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2EMAX_2E2(V0m_2E0,V1n_2E0),V2k_2E0))
<=> ( p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V2k_2E0))
& p(c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V2k_2E0)) ) ) ).
tff(thm_2Eseq_2EX__LE__MAX,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2k_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V2k_2E0,c_2Earithmetic_2EMAX_2E2(V0m_2E0,V1n_2E0)))
<=> ( p(c_2Earithmetic_2E_3C_3D_2E2(V2k_2E0,V0m_2E0))
| p(c_2Earithmetic_2E_3C_3D_2E2(V2k_2E0,V1n_2E0)) ) ) ).
tff(thm_2Eseq_2ETRANSFORM__2D__NUM,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
( ( ! [V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,V1m_2E0),V2n_2E0))
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,V2n_2E0),V1m_2E0)) )
& ! [V3m_2E0: tyop_2Enum_2Enum,V4n_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,V3m_2E0),c_2Earithmetic_2E_2B_2E2(V3m_2E0,V4n_2E0))) )
=> ! [V5m_2E0: tyop_2Enum_2Enum,V6n_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,V5m_2E0),V6n_2E0)) ) ).
tff(thm_2Eseq_2ETRIANGLE__2D__NUM,axiom,
! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
( ! [V1d_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,V2n_2E0),c_2Earithmetic_2E_2B_2E2(V1d_2E0,V2n_2E0)))
=> ! [V3m_2E0: tyop_2Enum_2Enum,V4n_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V3m_2E0,V4n_2E0))
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0P_2E0,V3m_2E0),V4n_2E0)) ) ) ).
tff(thm_2Eseq_2ESEQ__SANDWICH,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2h_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3l_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Eseq_2E_2D_2D_3E_2E2(V0f_2E0,V3l_2E0))
& p(c_2Eseq_2E_2D_2D_3E_2E2(V2h_2E0,V3l_2E0))
& ! [V4n_2E0: tyop_2Enum_2Enum] :
( p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V4n_2E0)))
& p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V4n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2h_2E0,V4n_2E0))) ) )
=> p(c_2Eseq_2E_2D_2D_3E_2E2(V1g_2E0,V3l_2E0)) ) ).
tff(thm_2Eseq_2ESER__POS,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ( p(c_2Eseq_2Esummable_2E1(V0f_2E0))
& ! [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1n_2E0))) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eseq_2Esuminf_2E1(V0f_2E0))) ) ).
tff(thm_2Eseq_2ESER__POS__MONO,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V2n_2E0) = c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,V2n_2E0),V0f_2E0) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ! [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1n_2E0)))
=> p(c_2Eseq_2Emono_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0))) ) ) ).
tff(thm_2Eseq_2EPOS__SUMMABLE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ( ! [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1n_2E0)))
& ? [V2x_2E0: tyop_2Erealax_2Ereal] :
! [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,V3n_2E0),V0f_2E0),V2x_2E0)) )
=> p(c_2Eseq_2Esummable_2E1(V0f_2E0)) ) ).
tff(thm_2Eseq_2ESUMMABLE__LE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Eseq_2Esummable_2E1(V0f_2E0))
& ! [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,V2n_2E0),V0f_2E0),V1x_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Eseq_2Esuminf_2E1(V0f_2E0),V1x_2E0)) ) ).
tff(thm_2Eseq_2ESUMS__EQ,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Eseq_2Esums_2E2(V0f_2E0,V1x_2E0))
<=> ( p(c_2Eseq_2Esummable_2E1(V0f_2E0))
& ( c_2Eseq_2Esuminf_2E1(V0f_2E0) = V1x_2E0 ) ) ) ).
tff(thm_2Eseq_2ESUMINF__POS,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ( ! [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1n_2E0)))
& p(c_2Eseq_2Esummable_2E1(V0f_2E0)) )
=> p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eseq_2Esuminf_2E1(V0f_2E0))) ) ).
tff(thm_2Eseq_2ESUM__PICK,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V3m_2E0: tyop_2Enum_2Enum,V1k_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F1_2E0,V3m_2E0),V1k_2E0))
<=> ( V3m_2E0 = V1k_2E0 ) )
=> ( ! [V1k_2E0: tyop_2Enum_2Enum,V2x_2E0: tyop_2Erealax_2Ereal,V3m_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1k_2E0),V2x_2E0),V3m_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F1_2E0,V3m_2E0),V1k_2E0),V2x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) )
=> ! [V0n_2E0: tyop_2Enum_2Enum,V1k_2E0: tyop_2Enum_2Enum,V2x_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,V0n_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1k_2E0),V2x_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Eprim__rec_2E_3C_2E2(V1k_2E0,V0n_2E0),V2x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) ) ) ) ).
tff(thm_2Eseq_2ESUM__LT,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] :
( ( ! [V4r_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Earithmetic_2E_3C_3D_2E2(V2m_2E0,V4r_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(V4r_2E0,c_2Earithmetic_2E_2B_2E2(V3n_2E0,V2m_2E0))) )
=> p(c_2Erealax_2Ereal__lt_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4r_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V4r_2E0))) )
& p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V3n_2E0)) )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2m_2E0,V3n_2E0),V0f_2E0),c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V2m_2E0,V3n_2E0),V1g_2E0))) ) ).
tff(thm_2Eseq_2ESUM__CONST__R,axiom,
! [V0n_2E0: tyop_2Enum_2Enum,V1r_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Esum_2E2(c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2E0_2E0,V0n_2E0),c_2Ecombin_2EK_2E1(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,V1r_2E0)) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(V0n_2E0),V1r_2E0) ) ).
tff(thm_2Eseq_2ESUMS__ZERO,axiom,
p(c_2Eseq_2Esums_2E2(c_2Ecombin_2EK_2E1(tyop_2Erealax_2Ereal,tyop_2Enum_2Enum,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ).
tff(thm_2Eseq_2ELT__SUC,axiom,
! [V0a_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V0a_2E0,c_2Enum_2ESUC_2E1(V1b_2E0)))
<=> ( p(c_2Eprim__rec_2E_3C_2E2(V0a_2E0,V1b_2E0))
| ( V0a_2E0 = V1b_2E0 ) ) ) ).
tff(thm_2Eseq_2ELE__SUC,axiom,
! [V0a_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum] :
( p(c_2Earithmetic_2E_3C_3D_2E2(V0a_2E0,c_2Enum_2ESUC_2E1(V1b_2E0)))
<=> ( p(c_2Earithmetic_2E_3C_3D_2E2(V0a_2E0,V1b_2E0))
| ( V0a_2E0 = c_2Enum_2ESUC_2E1(V1b_2E0) ) ) ) ).
tff(thm_2Eseq_2EK__PARTIAL,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: A_27a,V1x_27_2E0: A_27b] : ( c_2Ecombin_2EK_2E2(A_27a,A_27b,V0x_2E0,V1x_27_2E0) = V0x_2E0 ) ).
tff(thm_2Eseq_2EHALF__POS,axiom,
p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))))) ).
tff(thm_2Eseq_2EHALF__LT__1,axiom,
p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ).
tff(thm_2Eseq_2EHALF__CANCEL,axiom,
c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ).
tff(thm_2Eseq_2EX__HALF__HALF,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),V0x_2E0),c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),V0x_2E0)) = V0x_2E0 ) ).
tff(thm_2Eseq_2EONE__MINUS__HALF,axiom,
c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0))))) = c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))) ).
tff(thm_2Eseq_2ENUM__2D__BIJ__BIG__SQUARE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),V1N_2E0: tyop_2Enum_2Enum] :
( p(c_2Epred__set_2EBIJ_2E3(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V0f_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum),c_2Epred__set_2ECROSS_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum),c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))))
=> ? [V2k_2E0: tyop_2Enum_2Enum] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V0f_2E0,c_2Epred__set_2Ecount_2E1(V1N_2E0)),c_2Epred__set_2ECROSS_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Epred__set_2Ecount_2E1(V2k_2E0),c_2Epred__set_2Ecount_2E1(V2k_2E0)))) ) ).
tff(thm_2Eseq_2ENUM__2D__BIJ__SMALL__SQUARE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),V1k_2E0: tyop_2Enum_2Enum] :
( p(c_2Epred__set_2EBIJ_2E3(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V0f_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum),c_2Epred__set_2ECROSS_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum),c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum))))
=> ? [V2N_2E0: tyop_2Enum_2Enum] : p(c_2Epred__set_2ESUBSET_2E2(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Epred__set_2ECROSS_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Epred__set_2Ecount_2E1(V1k_2E0),c_2Epred__set_2Ecount_2E1(V1k_2E0)),c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V0f_2E0,c_2Epred__set_2Ecount_2E1(V2N_2E0)))) ) ).
tff(thm_2Eseq_2ESUMINF__ADD,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F1_2E0,V0f_2E0),V1g_2E0),V3n_2E0) = c_2Erealax_2Ereal__add_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V3n_2E0)) )
=> ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V2n_2E0) = c_2Erealax_2Ereal__add_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V2n_2E0)) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ( p(c_2Eseq_2Esummable_2E1(V0f_2E0))
& p(c_2Eseq_2Esummable_2E1(V1g_2E0)) )
=> ( p(c_2Eseq_2Esummable_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0)))
& ( c_2Erealax_2Ereal__add_2E2(c_2Eseq_2Esuminf_2E1(V0f_2E0),c_2Eseq_2Esuminf_2E1(V1g_2E0)) = c_2Eseq_2Esuminf_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F1_2E0,V0f_2E0),V1g_2E0)) ) ) ) ) ) ).
tff(thm_2Eseq_2ESUMINF__2D,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2h_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum))] :
( ( ! [V3m_2E0: tyop_2Enum_2Enum,V4n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V0f_2E0,V3m_2E0),V4n_2E0)))
& ! [V5n_2E0: tyop_2Enum_2Enum] : p(c_2Eseq_2Esums_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V0f_2E0,V5n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V5n_2E0)))
& p(c_2Eseq_2Esummable_2E1(V1g_2E0))
& p(c_2Epred__set_2EBIJ_2E3(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V2h_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum),c_2Epred__set_2ECROSS_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum),c_2Epred__set_2EUNIV_2E0(tyop_2Enum_2Enum)))) )
=> p(c_2Eseq_2Esums_2E2(c_2Ecombin_2Eo_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Epair_2EUNCURRY_2E1(tyop_2Enum_2Enum,tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0),V2h_2E0),c_2Eseq_2Esuminf_2E1(V1g_2E0))) ) ).
tff(thm_2Eseq_2EPOW__HALF__SER,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ! [V0n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,F0_2E0,V0n_2E0) = c_2Ereal_2Epow_2E2(c_2Ereal_2E_2F_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Earithmetic_2E_2B_2E2(V0n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) )
=> p(c_2Eseq_2Esums_2E2(F0_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ).
tff(thm_2Eseq_2ESER__POS__COMPARE,axiom,
! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
( ( ! [V2n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2n_2E0)))
& p(c_2Eseq_2Esummable_2E1(V1g_2E0))
& ! [V3n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V3n_2E0))) )
=> ( p(c_2Eseq_2Esummable_2E1(V0f_2E0))
& p(c_2Ereal_2Ereal__lte_2E2(c_2Eseq_2Esuminf_2E1(V0f_2E0),c_2Eseq_2Esuminf_2E1(V1g_2E0))) ) ) ).
%------------------------------------------------------------------------------