ITP001 Axioms: ITP137_5.ax
%------------------------------------------------------------------------------
% File : ITP137_5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : integral_2.ax [Gau20]
% : HL4137_5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 142 ( 16 unt; 29 typ; 0 def)
% Number of atoms : 4706 ( 62 equ)
% Maximal formula atoms : 16 ( 33 avg)
% Number of connectives : 424 ( 4 ~; 2 |; 156 &)
% ( 10 <=>; 252 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 9 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 4173 (4173 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 55 ( 27 >; 28 *; 0 +; 0 <<)
% Number of predicates : 58 ( 56 usr; 39 prp; 0-2 aty)
% Number of functors : 29 ( 29 usr; 2 con; 0-4 aty)
% Number of variables : 481 ( 455 !; 26 ?; 481 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_c_2Eintegral_2Eintegrable,type,
c_2Eintegral_2Eintegrable: $i ).
tff(mem_c_2Eintegral_2Eintegrable,axiom,
mem(c_2Eintegral_2Eintegrable,arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),bool))) ).
tff(tp_c_2Eintegral_2Eintegral,type,
c_2Eintegral_2Eintegral: $i ).
tff(mem_c_2Eintegral_2Eintegral,axiom,
mem(c_2Eintegral_2Eintegral,arr(ty_2Epair_2Eprod(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),arr(arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),ty_2Erealax_2Ereal))) ).
tff(conj_thm_2Eintegral_2ESUM__SPLIT,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1f: $i] :
( mem(V1f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2n: tp__ty_2Enum_2Enum,V3p: tp__ty_2Enum_2Enum] : ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V2n))),V1f)),ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V2n))),inj__ty_2Enum_2Enum(V3p))),V1f))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V0m)),ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V2n)),inj__ty_2Enum_2Enum(V3p)))),V1f)) ) ) ).
tff(lamtp_f3106,type,
f3106: ( $i * $i ) > $i ).
tff(lameq_f3106,axiom,
! [V6d2: $i] :
( mem(V6d2,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4d1: $i] :
( mem(V4d1,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V8n: tp__ty_2Enum_2Enum] : ( ap(f3106(V6d2,V4d1),inj__ty_2Enum_2Enum(V8n)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V8n)),ap(c_2Etransc_2Edsize,V4d1))),ap(V4d1,inj__ty_2Enum_2Enum(V8n))),ap(V6d2,ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(V8n)),ap(c_2Etransc_2Edsize,V4d1)))) ) ) ) ).
tff(lamtp_f3107,type,
f3107: ( $i * $i * $i ) > $i ).
tff(lameq_f3107,axiom,
! [V5p1: $i] :
( mem(V5p1,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4d1: $i] :
( mem(V4d1,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V7p2: $i] :
( mem(V7p2,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V9n: tp__ty_2Enum_2Enum] : ( ap(f3107(V5p1,V4d1,V7p2),inj__ty_2Enum_2Enum(V9n)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V9n)),ap(c_2Etransc_2Edsize,V4d1))),ap(V5p1,inj__ty_2Enum_2Enum(V9n))),ap(V7p2,ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(V9n)),ap(c_2Etransc_2Edsize,V4d1)))) ) ) ) ) ).
tff(lamtp_f3108,type,
f3108: ( $i * $i ) > $i ).
tff(lameq_f3108,axiom,
! [V6d2: $i] :
( mem(V6d2,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4d1: $i] :
( mem(V4d1,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V10n: tp__ty_2Enum_2Enum] : ( ap(f3108(V6d2,V4d1),inj__ty_2Enum_2Enum(V10n)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V10n)),ap(c_2Etransc_2Edsize,V4d1))),ap(V4d1,inj__ty_2Enum_2Enum(V10n))),ap(V6d2,ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(V10n)),ap(c_2Etransc_2Edsize,V4d1)))) ) ) ) ).
tff(lamtp_f3109,type,
f3109: ( $i * $i * $i ) > $i ).
tff(lameq_f3109,axiom,
! [V5p1: $i] :
( mem(V5p1,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4d1: $i] :
( mem(V4d1,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V7p2: $i] :
( mem(V7p2,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V11n: tp__ty_2Enum_2Enum] : ( ap(f3109(V5p1,V4d1,V7p2),inj__ty_2Enum_2Enum(V11n)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V11n)),ap(c_2Etransc_2Edsize,V4d1))),ap(V5p1,inj__ty_2Enum_2Enum(V11n))),ap(V7p2,ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(V11n)),ap(c_2Etransc_2Edsize,V4d1)))) ) ) ) ) ).
tff(lamtp_f3110,type,
f3110: ( $i * $i ) > $i ).
tff(lameq_f3110,axiom,
! [V6d2: $i] :
( mem(V6d2,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4d1: $i] :
( mem(V4d1,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V13n: tp__ty_2Enum_2Enum] : ( ap(f3110(V6d2,V4d1),inj__ty_2Enum_2Enum(V13n)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V13n)),ap(c_2Etransc_2Edsize,V4d1))),ap(V4d1,inj__ty_2Enum_2Enum(V13n))),ap(V6d2,ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(V13n)),ap(c_2Etransc_2Edsize,V4d1)))) ) ) ) ).
tff(lamtp_f3111,type,
f3111: ( $i * $i * $i ) > $i ).
tff(lameq_f3111,axiom,
! [V5p1: $i] :
( mem(V5p1,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4d1: $i] :
( mem(V4d1,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V7p2: $i] :
( mem(V7p2,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V14n: tp__ty_2Enum_2Enum] : ( ap(f3111(V5p1,V4d1,V7p2),inj__ty_2Enum_2Enum(V14n)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V14n)),ap(c_2Etransc_2Edsize,V4d1))),ap(V5p1,inj__ty_2Enum_2Enum(V14n))),ap(V7p2,ap(ap(c_2Earithmetic_2E_2D,inj__ty_2Enum_2Enum(V14n)),ap(c_2Etransc_2Edsize,V4d1)))) ) ) ) ) ).
tff(conj_thm_2Eintegral_2EDIVISION__APPEND__EXPLICIT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal,V3g: $i] :
( mem(V3g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V4d1: $i] :
( mem(V4d1,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V5p1: $i] :
( mem(V5p1,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V6d2: $i] :
( mem(V6d2,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V7p2: $i] :
( mem(V7p2,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ( p(ap(ap(c_2Etransc_2Etdiv,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1b))),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V4d1),V5p1)))
& p(ap(ap(c_2Etransc_2Efine,V3g),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V4d1),V5p1)))
& p(ap(ap(c_2Etransc_2Etdiv,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1b)),inj__ty_2Erealax_2Ereal(V2c))),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V6d2),V7p2)))
& p(ap(ap(c_2Etransc_2Efine,V3g),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V6d2),V7p2))) )
=> ( p(ap(ap(c_2Etransc_2Etdiv,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V2c))),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),f3106(V6d2,V4d1)),f3107(V5p1,V4d1,V7p2))))
& p(ap(ap(c_2Etransc_2Efine,V3g),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),f3108(V6d2,V4d1)),f3109(V5p1,V4d1,V7p2))))
& ! [V12f: $i] :
( mem(V12f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Etransc_2Ersum,ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),f3110(V6d2,V4d1)),f3111(V5p1,V4d1,V7p2))),V12f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Etransc_2Ersum,ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V4d1),V5p1)),V12f)),ap(ap(c_2Etransc_2Ersum,ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V6d2),V7p2)),V12f))) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eintegral_2EDIVISION__APPEND__STRONG,axiom,
! [V0g: $i] :
( mem(V0g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal,V4D1: $i] :
( mem(V4D1,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V5p1: $i] :
( mem(V5p1,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V6D2: $i] :
( mem(V6D2,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V7p2: $i] :
( mem(V7p2,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ( p(ap(ap(c_2Etransc_2Etdiv,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V4D1),V5p1)))
& p(ap(ap(c_2Etransc_2Efine,V0g),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V4D1),V5p1)))
& p(ap(ap(c_2Etransc_2Etdiv,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2b)),inj__ty_2Erealax_2Ereal(V3c))),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V6D2),V7p2)))
& p(ap(ap(c_2Etransc_2Efine,V0g),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V6D2),V7p2))) )
=> ? [V8D: $i] :
( mem(V8D,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
& ? [V9p: $i] :
( mem(V9p,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
& p(ap(ap(c_2Etransc_2Etdiv,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V3c))),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V8D),V9p)))
& p(ap(ap(c_2Etransc_2Efine,V0g),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V8D),V9p)))
& ! [V10f: $i] :
( mem(V10f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Etransc_2Ersum,ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V8D),V9p)),V10f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Etransc_2Ersum,ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V4D1),V5p1)),V10f)),ap(ap(c_2Etransc_2Ersum,ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V6D2),V7p2)),V10f))) ) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eintegral_2EDIVISION__APPEND,axiom,
! [V0g: $i] :
( mem(V0g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal] :
( ( ? [V4D1: $i] :
( mem(V4D1,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
& ? [V5p1: $i] :
( mem(V5p1,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
& p(ap(ap(c_2Etransc_2Etdiv,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V4D1),V5p1)))
& p(ap(ap(c_2Etransc_2Efine,V0g),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V4D1),V5p1))) ) )
& ? [V6D2: $i] :
( mem(V6D2,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
& ? [V7p2: $i] :
( mem(V7p2,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
& p(ap(ap(c_2Etransc_2Etdiv,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2b)),inj__ty_2Erealax_2Ereal(V3c))),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V6D2),V7p2)))
& p(ap(ap(c_2Etransc_2Efine,V0g),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V6D2),V7p2))) ) ) )
=> ? [V8D: $i] :
( mem(V8D,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
& ? [V9p: $i] :
( mem(V9p,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
& p(ap(ap(c_2Etransc_2Etdiv,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V3c))),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V8D),V9p)))
& p(ap(ap(c_2Etransc_2Efine,V0g),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V8D),V9p))) ) ) ) ) ).
tff(ax_thm_2Eintegral_2Eintegrable__def,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2f: $i] :
( mem(V2f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1b))),V2f))
<=> ? [V3i: tp__ty_2Erealax_2Ereal] : p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1b))),V2f),inj__ty_2Erealax_2Ereal(V3i))) ) ) ).
tff(lamtp_f3112,type,
f3112: ( $i * tp__ty_2Erealax_2Ereal * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3112,axiom,
! [V2f: $i] :
( mem(V2f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1b: tp__ty_2Erealax_2Ereal,V0a: tp__ty_2Erealax_2Ereal,V3i: tp__ty_2Erealax_2Ereal] : ( ap(f3112(V2f,V1b,V0a),inj__ty_2Erealax_2Ereal(V3i)) = ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1b))),V2f),inj__ty_2Erealax_2Ereal(V3i)) ) ) ).
tff(ax_thm_2Eintegral_2Eintegral__def,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2f: $i] :
( mem(V2f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eintegral_2Eintegral,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1b))),V2f)) = surj__ty_2Erealax_2Ereal(ap(c_2Emin_2E_40(ty_2Erealax_2Ereal),f3112(V2f,V1b,V0a))) ) ) ).
tff(conj_thm_2Eintegral_2EINTEGRABLE__DINT,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),V0f))
=> p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),V0f),ap(ap(c_2Eintegral_2Eintegral,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),V0f))) ) ) ).
tff(conj_thm_2Eintegral_2EDIVISION__BOUNDS,axiom,
! [V0d: $i] :
( mem(V0d,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Etransc_2Edivision,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),V0d))
=> ! [V3n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),ap(V0d,inj__ty_2Enum_2Enum(V3n))))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0d,inj__ty_2Enum_2Enum(V3n))),inj__ty_2Erealax_2Ereal(V2b))) ) ) ) ).
tff(conj_thm_2Eintegral_2ETDIV__BOUNDS,axiom,
! [V0d: $i] :
( mem(V0d,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1p: $i] :
( mem(V1p,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Etransc_2Etdiv,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V0d),V1p)))
=> ! [V4n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2a)),ap(V0d,inj__ty_2Enum_2Enum(V4n))))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0d,inj__ty_2Enum_2Enum(V4n))),inj__ty_2Erealax_2Ereal(V3b)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2a)),ap(V1p,inj__ty_2Enum_2Enum(V4n))))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(V1p,inj__ty_2Enum_2Enum(V4n))),inj__ty_2Erealax_2Ereal(V3b))) ) ) ) ) ).
tff(conj_thm_2Eintegral_2ETDIV__LE,axiom,
! [V0d: $i] :
( mem(V0d,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1p: $i] :
( mem(V1p,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Etransc_2Etdiv,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V0d),V1p)))
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))) ) ) ) ).
tff(conj_thm_2Eintegral_2EDINT__WRONG,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2f: $i] :
( mem(V2f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V3i: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V1b)),inj__ty_2Erealax_2Ereal(V0a)))
=> p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1b))),V2f),inj__ty_2Erealax_2Ereal(V3i))) ) ) ).
tff(conj_thm_2Eintegral_2EDINT__INTEGRAL,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3i: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b)))
& p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),V0f),inj__ty_2Erealax_2Ereal(V3i))) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eintegral_2Eintegral,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),V0f)) = V3i ) ) ) ).
tff(lamtp_f3113,type,
f3113: $i > $i ).
tff(lameq_f3113,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V4x: tp__ty_2Erealax_2Ereal] : ( ap(f3113(V0f),inj__ty_2Erealax_2Ereal(V4x)) = ap(c_2Erealax_2Ereal__neg,ap(V0f,inj__ty_2Erealax_2Ereal(V4x))) ) ) ).
tff(conj_thm_2Eintegral_2EDINT__NEG,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3i: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),V0f),inj__ty_2Erealax_2Ereal(V3i)))
=> p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),f3113(V0f)),ap(c_2Erealax_2Ereal__neg,inj__ty_2Erealax_2Ereal(V3i)))) ) ) ).
tff(conj_thm_2Eintegral_2EDINT__0,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] : p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1b))),k(ty_2Erealax_2Ereal,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) ).
tff(lamtp_f3114,type,
f3114: ( $i * $i ) > $i ).
tff(lameq_f3114,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V6x: tp__ty_2Erealax_2Ereal] : ( ap(f3114(V0f,V1g),inj__ty_2Erealax_2Ereal(V6x)) = ap(ap(c_2Erealax_2Ereal__add,ap(V0f,inj__ty_2Erealax_2Ereal(V6x))),ap(V1g,inj__ty_2Erealax_2Ereal(V6x))) ) ) ) ).
tff(conj_thm_2Eintegral_2EDINT__ADD,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal,V4i: tp__ty_2Erealax_2Ereal,V5j: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V0f),inj__ty_2Erealax_2Ereal(V4i)))
& p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V1g),inj__ty_2Erealax_2Ereal(V5j))) )
=> p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),f3114(V0f,V1g)),ap(ap(c_2Erealax_2Ereal__add,inj__ty_2Erealax_2Ereal(V4i)),inj__ty_2Erealax_2Ereal(V5j)))) ) ) ) ).
tff(lamtp_f3115,type,
f3115: ( $i * $i ) > $i ).
tff(lameq_f3115,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V6x: tp__ty_2Erealax_2Ereal] : ( ap(f3115(V0f,V1g),inj__ty_2Erealax_2Ereal(V6x)) = ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,inj__ty_2Erealax_2Ereal(V6x))),ap(V1g,inj__ty_2Erealax_2Ereal(V6x))) ) ) ) ).
tff(conj_thm_2Eintegral_2EDINT__SUB,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal,V4i: tp__ty_2Erealax_2Ereal,V5j: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V0f),inj__ty_2Erealax_2Ereal(V4i)))
& p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V1g),inj__ty_2Erealax_2Ereal(V5j))) )
=> p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),f3115(V0f,V1g)),ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V4i)),inj__ty_2Erealax_2Ereal(V5j)))) ) ) ) ).
tff(lamtp_f3116,type,
f3116: $i > $i ).
tff(lameq_f3116,axiom,
! [V2D: $i] :
( mem(V2D,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3n: tp__ty_2Enum_2Enum] : ( ap(f3116(V2D),inj__ty_2Enum_2Enum(V3n)) = ap(ap(c_2Ereal_2Ereal__sub,ap(V2D,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n)))),ap(V2D,inj__ty_2Enum_2Enum(V3n))) ) ) ).
tff(conj_thm_2Eintegral_2EDSIZE__EQ,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2D: $i] :
( mem(V2D,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( p(ap(ap(c_2Etransc_2Edivision,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1b))),V2D))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),ap(c_2Etransc_2Edsize,V2D))),f3116(V2D))),ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V1b)),inj__ty_2Erealax_2Ereal(V0a)))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) ).
tff(conj_thm_2Eintegral_2EDINT__CONST,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] : p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1b))),k(ty_2Erealax_2Ereal,inj__ty_2Erealax_2Ereal(V2c))),ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V2c)),ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V1b)),inj__ty_2Erealax_2Ereal(V0a))))) ).
tff(lamtp_f3117,type,
f3117: ( $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3117,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V3c: tp__ty_2Erealax_2Ereal,V5x: tp__ty_2Erealax_2Ereal] : ( ap(f3117(V0f,V3c),inj__ty_2Erealax_2Ereal(V5x)) = ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V3c)),ap(V0f,inj__ty_2Erealax_2Ereal(V5x))) ) ) ).
tff(conj_thm_2Eintegral_2EDINT__CMUL,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal,V4i: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),V0f),inj__ty_2Erealax_2Ereal(V4i)))
=> p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),f3117(V0f,V3c)),ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V3c)),inj__ty_2Erealax_2Ereal(V4i)))) ) ) ).
tff(lamtp_f3118,type,
f3118: ( $i * tp__ty_2Erealax_2Ereal * $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3118,axiom,
! [V2f: $i] :
( mem(V2f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V0m: tp__ty_2Erealax_2Ereal,V3g: $i] :
( mem(V3g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1n: tp__ty_2Erealax_2Ereal,V8x: tp__ty_2Erealax_2Ereal] : ( ap(f3118(V2f,V0m,V3g,V1n),inj__ty_2Erealax_2Ereal(V8x)) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V0m)),ap(V2f,inj__ty_2Erealax_2Ereal(V8x)))),ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V1n)),ap(V3g,inj__ty_2Erealax_2Ereal(V8x)))) ) ) ) ).
tff(conj_thm_2Eintegral_2EDINT__LINEAR,axiom,
! [V0m: tp__ty_2Erealax_2Ereal,V1n: tp__ty_2Erealax_2Ereal,V2f: $i] :
( mem(V2f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V3g: $i] :
( mem(V3g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V4a: tp__ty_2Erealax_2Ereal,V5b: tp__ty_2Erealax_2Ereal,V6i: tp__ty_2Erealax_2Ereal,V7j: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4a)),inj__ty_2Erealax_2Ereal(V5b))),V2f),inj__ty_2Erealax_2Ereal(V6i)))
& p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4a)),inj__ty_2Erealax_2Ereal(V5b))),V3g),inj__ty_2Erealax_2Ereal(V7j))) )
=> p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4a)),inj__ty_2Erealax_2Ereal(V5b))),f3118(V2f,V0m,V3g,V1n)),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V0m)),inj__ty_2Erealax_2Ereal(V6i))),ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V1n)),inj__ty_2Erealax_2Ereal(V7j))))) ) ) ) ).
tff(conj_thm_2Eintegral_2ELT,axiom,
( ! [V0m: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))
<=> $false )
& ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V1m)),ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V2n))))
<=> ( ( V1m = V2n )
| p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))) ) ) ) ).
tff(conj_thm_2Eintegral_2ELE__0,axiom,
! [V0n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),inj__ty_2Enum_2Enum(V0n))) ).
tff(conj_thm_2Eintegral_2ELT__0,axiom,
! [V0n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V0n)))) ).
tff(conj_thm_2Eintegral_2EEQ__SUC,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum(ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V0m))) = surj__ty_2Enum_2Enum(ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V1n))) )
<=> ( V0m = V1n ) ) ).
tff(conj_thm_2Eintegral_2ELE__LT,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n)))
<=> ( p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n)))
| ( V0m = V1n ) ) ) ).
tff(conj_thm_2Eintegral_2ELT__LE,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n)))
<=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n)))
& ( V0m != V1n ) ) ) ).
tff(conj_thm_2Eintegral_2EREAL__LT__MIN,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2z: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V2z)),ap(ap(c_2Ereal_2Emin,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Erealax_2Ereal(V1y))))
<=> ( p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V2z)),inj__ty_2Erealax_2Ereal(V0x)))
& p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V2z)),inj__ty_2Erealax_2Ereal(V1y))) ) ) ).
tff(conj_thm_2Eintegral_2EREAL__LE__RMUL1,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2z: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Erealax_2Ereal(V1y)))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V2z))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Erealax_2Ereal(V2z))),ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V1y)),inj__ty_2Erealax_2Ereal(V2z)))) ) ).
tff(conj_thm_2Eintegral_2EREAL__LE__LMUL1,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2z: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V0x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1y)),inj__ty_2Erealax_2Ereal(V2z))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Erealax_2Ereal(V1y))),ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Erealax_2Ereal(V2z)))) ) ).
tff(conj_thm_2Eintegral_2EINTEGRAL__LE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal,V4i: $i] :
( mem(V4i,A_27a)
=> ! [V5j: $i] :
( mem(V5j,A_27b)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b)))
& p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V0f))
& p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V1g))
& ! [V6x: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V6x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V6x)),inj__ty_2Erealax_2Ereal(V3b))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,inj__ty_2Erealax_2Ereal(V6x))),ap(V1g,inj__ty_2Erealax_2Ereal(V6x)))) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Eintegral_2Eintegral,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V0f)),ap(ap(c_2Eintegral_2Eintegral,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V1g))) ) ) ) ) ) ).
tff(conj_thm_2Eintegral_2EDINT__LE,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal,V4i: tp__ty_2Erealax_2Ereal,V5j: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b)))
& p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V0f),inj__ty_2Erealax_2Ereal(V4i)))
& p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V1g),inj__ty_2Erealax_2Ereal(V5j)))
& ! [V6x: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V6x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V6x)),inj__ty_2Erealax_2Ereal(V3b))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0f,inj__ty_2Erealax_2Ereal(V6x))),ap(V1g,inj__ty_2Erealax_2Ereal(V6x)))) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V4i)),inj__ty_2Erealax_2Ereal(V5j))) ) ) ) ).
tff(lamtp_f3119,type,
f3119: $i > $i ).
tff(lameq_f3119,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V5x: tp__ty_2Erealax_2Ereal] : ( ap(f3119(V0f),inj__ty_2Erealax_2Ereal(V5x)) = ap(c_2Ereal_2Eabs,ap(V0f,inj__ty_2Erealax_2Ereal(V5x))) ) ) ).
tff(conj_thm_2Eintegral_2EDINT__TRIANGLE,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3i: tp__ty_2Erealax_2Ereal,V4j: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b)))
& p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),V0f),inj__ty_2Erealax_2Ereal(V3i)))
& p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),f3119(V0f)),inj__ty_2Erealax_2Ereal(V4j))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,inj__ty_2Erealax_2Ereal(V3i))),inj__ty_2Erealax_2Ereal(V4j))) ) ) ).
tff(conj_thm_2Eintegral_2EDINT__EQ,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal,V4i: tp__ty_2Erealax_2Ereal,V5j: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b)))
& p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V0f),inj__ty_2Erealax_2Ereal(V4i)))
& p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V1g),inj__ty_2Erealax_2Ereal(V5j)))
& ! [V6x: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V6x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V6x)),inj__ty_2Erealax_2Ereal(V3b))) )
=> ( surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V6x))) = surj__ty_2Erealax_2Ereal(ap(V1g,inj__ty_2Erealax_2Ereal(V6x))) ) ) )
=> ( V4i = V5j ) ) ) ) ).
tff(conj_thm_2Eintegral_2EINTEGRAL__EQ,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal,V4i: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V0f),inj__ty_2Erealax_2Ereal(V4i)))
& ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V5x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V5x)),inj__ty_2Erealax_2Ereal(V3b))) )
=> ( surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V5x))) = surj__ty_2Erealax_2Ereal(ap(V1g,inj__ty_2Erealax_2Ereal(V5x))) ) ) )
=> p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V1g),inj__ty_2Erealax_2Ereal(V4i))) ) ) ) ).
tff(lamtp_f3120,type,
f3120: ( $i * $i * $i * $i ) > $i ).
tff(lameq_f3120,axiom,
! [V2f_27: $i] :
( mem(V2f_27,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V3g_27: $i] :
( mem(V3g_27,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V8x: tp__ty_2Erealax_2Ereal] : ( ap(f3120(V2f_27,V1g,V0f,V3g_27),inj__ty_2Erealax_2Ereal(V8x)) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,ap(V2f_27,inj__ty_2Erealax_2Ereal(V8x))),ap(V1g,inj__ty_2Erealax_2Ereal(V8x)))),ap(ap(c_2Erealax_2Ereal__mul,ap(V0f,inj__ty_2Erealax_2Ereal(V8x))),ap(V3g_27,inj__ty_2Erealax_2Ereal(V8x)))) ) ) ) ) ) ).
tff(conj_thm_2Eintegral_2EINTEGRATION__BY__PARTS,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2f_27: $i] :
( mem(V2f_27,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V3g_27: $i] :
( mem(V3g_27,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V4a: tp__ty_2Erealax_2Ereal,V5b: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V4a)),inj__ty_2Erealax_2Ereal(V5b)))
& ! [V6x: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V4a)),inj__ty_2Erealax_2Ereal(V6x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V6x)),inj__ty_2Erealax_2Ereal(V5b))) )
=> p(ap(ap(ap(c_2Elim_2Ediffl,V0f),ap(V2f_27,inj__ty_2Erealax_2Ereal(V6x))),inj__ty_2Erealax_2Ereal(V6x))) )
& ! [V7x: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V4a)),inj__ty_2Erealax_2Ereal(V7x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V7x)),inj__ty_2Erealax_2Ereal(V5b))) )
=> p(ap(ap(ap(c_2Elim_2Ediffl,V1g),ap(V3g_27,inj__ty_2Erealax_2Ereal(V7x))),inj__ty_2Erealax_2Ereal(V7x))) ) )
=> p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4a)),inj__ty_2Erealax_2Ereal(V5b))),f3120(V2f_27,V1g,V0f,V3g_27)),ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Erealax_2Ereal__mul,ap(V0f,inj__ty_2Erealax_2Ereal(V5b))),ap(V1g,inj__ty_2Erealax_2Ereal(V5b)))),ap(ap(c_2Erealax_2Ereal__mul,ap(V0f,inj__ty_2Erealax_2Ereal(V4a))),ap(V1g,inj__ty_2Erealax_2Ereal(V4a)))))) ) ) ) ) ) ).
tff(conj_thm_2Eintegral_2EDIVISION__LE__SUC,axiom,
! [V0d: $i] :
( mem(V0d,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Etransc_2Edivision,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),V0d))
=> ! [V3n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0d,inj__ty_2Enum_2Enum(V3n))),ap(V0d,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n))))) ) ) ).
tff(conj_thm_2Eintegral_2EDIVISION__MONO__LE,axiom,
! [V0d: $i] :
( mem(V0d,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Etransc_2Edivision,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),V0d))
=> ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V3m)),inj__ty_2Enum_2Enum(V4n)))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0d,inj__ty_2Enum_2Enum(V3m))),ap(V0d,inj__ty_2Enum_2Enum(V4n)))) ) ) ) ).
tff(conj_thm_2Eintegral_2EDIVISION__MONO__LE__SUC,axiom,
! [V0d: $i] :
( mem(V0d,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Etransc_2Edivision,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),V0d))
=> ! [V3n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0d,inj__ty_2Enum_2Enum(V3n))),ap(V0d,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n))))) ) ) ).
tff(conj_thm_2Eintegral_2EDIVISION__DSIZE__LE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2d: $i] :
( mem(V2d,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3n: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Etransc_2Edivision,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1b))),V2d))
& ( surj__ty_2Erealax_2Ereal(ap(V2d,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n)))) = surj__ty_2Erealax_2Ereal(ap(V2d,inj__ty_2Enum_2Enum(V3n))) ) )
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Etransc_2Edsize,V2d)),inj__ty_2Enum_2Enum(V3n))) ) ) ).
tff(conj_thm_2Eintegral_2EDIVISION__DSIZE__GE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2d: $i] :
( mem(V2d,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3n: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Etransc_2Edivision,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1b))),V2d))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(V2d,inj__ty_2Enum_2Enum(V3n))),ap(V2d,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n))))) )
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n))),ap(c_2Etransc_2Edsize,V2d))) ) ) ).
tff(conj_thm_2Eintegral_2EDIVISION__DSIZE__EQ,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2d: $i] :
( mem(V2d,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3n: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Etransc_2Edivision,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1b))),V2d))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(V2d,inj__ty_2Enum_2Enum(V3n))),ap(V2d,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n)))))
& ( surj__ty_2Erealax_2Ereal(ap(V2d,ap(c_2Enum_2ESUC,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n))))) = surj__ty_2Erealax_2Ereal(ap(V2d,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n)))) ) )
=> ( surj__ty_2Enum_2Enum(ap(c_2Etransc_2Edsize,V2d)) = surj__ty_2Enum_2Enum(ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n))) ) ) ) ).
tff(conj_thm_2Eintegral_2EDIVISION__DSIZE__EQ__ALT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2d: $i] :
( mem(V2d,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3n: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Etransc_2Edivision,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1b))),V2d))
& ( surj__ty_2Erealax_2Ereal(ap(V2d,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n)))) = surj__ty_2Erealax_2Ereal(ap(V2d,inj__ty_2Enum_2Enum(V3n))) )
& ! [V4i: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V4i)),inj__ty_2Enum_2Enum(V3n)))
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(V2d,inj__ty_2Enum_2Enum(V4i))),ap(V2d,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V4i))))) ) )
=> ( surj__ty_2Enum_2Enum(ap(c_2Etransc_2Edsize,V2d)) = V3n ) ) ) ).
tff(conj_thm_2Eintegral_2Enum__MAX,axiom,
! [V0P: $i] :
( mem(V0P,arr(ty_2Enum_2Enum,bool))
=> ( ( ? [V1x: tp__ty_2Enum_2Enum] : p(ap(V0P,inj__ty_2Enum_2Enum(V1x)))
& ? [V2M: tp__ty_2Enum_2Enum] :
! [V3x: tp__ty_2Enum_2Enum] :
( p(ap(V0P,inj__ty_2Enum_2Enum(V3x)))
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V3x)),inj__ty_2Enum_2Enum(V2M))) ) )
<=> ? [V4m: tp__ty_2Enum_2Enum] :
( p(ap(V0P,inj__ty_2Enum_2Enum(V4m)))
& ! [V5x: tp__ty_2Enum_2Enum] :
( p(ap(V0P,inj__ty_2Enum_2Enum(V5x)))
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V5x)),inj__ty_2Enum_2Enum(V4m))) ) ) ) ) ).
tff(conj_thm_2Eintegral_2EDIVISION__INTERMEDIATE,axiom,
! [V0d: $i] :
( mem(V0d,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Etransc_2Edivision,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),V0d))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V3c)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3c)),inj__ty_2Erealax_2Ereal(V2b))) )
=> ? [V4n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V4n)),ap(c_2Etransc_2Edsize,V0d)))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0d,inj__ty_2Enum_2Enum(V4n))),inj__ty_2Erealax_2Ereal(V3c)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3c)),ap(V0d,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V4n))))) ) ) ) ).
tff(conj_thm_2Eintegral_2EDINT__COMBINE,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal,V4i: tp__ty_2Erealax_2Ereal,V5j: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2b)),inj__ty_2Erealax_2Ereal(V3c)))
& p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),V0f),inj__ty_2Erealax_2Ereal(V4i)))
& p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2b)),inj__ty_2Erealax_2Ereal(V3c))),V0f),inj__ty_2Erealax_2Ereal(V5j))) )
=> p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V3c))),V0f),ap(ap(c_2Erealax_2Ereal__add,inj__ty_2Erealax_2Ereal(V4i)),inj__ty_2Erealax_2Ereal(V5j)))) ) ) ).
tff(conj_thm_2Eintegral_2ESUM__EQ__0,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum,V2f: $i] :
( mem(V2f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ! [V3r: tp__ty_2Enum_2Enum] :
( ( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V3r)))
& p(ap(ap(c_2Eprim__rec_2E_3C,inj__ty_2Enum_2Enum(V3r)),ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n)))) )
=> ( surj__ty_2Erealax_2Ereal(ap(V2f,inj__ty_2Enum_2Enum(V3r))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n))),V2f)) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ) ).
tff(lamtp_f3121,type,
f3121: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f3121,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] : ( ap(f3121(V0a),inj__ty_2Erealax_2Ereal(V2x)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Emin_2E_3D(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Erealax_2Ereal(V0a))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ).
tff(conj_thm_2Eintegral_2EDINT__DELTA__LEFT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] : p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1b))),f3121(V0a)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) ).
tff(lamtp_f3122,type,
f3122: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f3122,axiom,
! [V1b: tp__ty_2Erealax_2Ereal,V2x: tp__ty_2Erealax_2Ereal] : ( ap(f3122(V1b),inj__ty_2Erealax_2Ereal(V2x)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Emin_2E_3D(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Erealax_2Ereal(V1b))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ).
tff(conj_thm_2Eintegral_2EDINT__DELTA__RIGHT,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] : p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1b))),f3122(V1b)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) ).
tff(lamtp_f3123,type,
f3123: tp__ty_2Erealax_2Ereal > $i ).
tff(lameq_f3123,axiom,
! [V2c: tp__ty_2Erealax_2Ereal,V3x: tp__ty_2Erealax_2Ereal] : ( ap(f3123(V2c),inj__ty_2Erealax_2Ereal(V3x)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Erealax_2Ereal),ap(ap(c_2Emin_2E_3D(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3x)),inj__ty_2Erealax_2Ereal(V2c))),ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ).
tff(conj_thm_2Eintegral_2EDINT__DELTA,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] : p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1b))),f3123(V2c)),ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) ).
tff(conj_thm_2Eintegral_2EDINT__POINT__SPIKE,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal,V4c: tp__ty_2Erealax_2Ereal,V5i: tp__ty_2Erealax_2Ereal] :
( ( ! [V6x: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V6x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V6x)),inj__ty_2Erealax_2Ereal(V3b)))
& ( V6x != V4c ) )
=> ( surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V6x))) = surj__ty_2Erealax_2Ereal(ap(V1g,inj__ty_2Erealax_2Ereal(V6x))) ) )
& p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V0f),inj__ty_2Erealax_2Ereal(V5i))) )
=> p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V1g),inj__ty_2Erealax_2Ereal(V5i))) ) ) ) ).
tff(conj_thm_2Eintegral_2EDINT__FINITE__SPIKE,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal,V4s: $i] :
( mem(V4s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V5i: tp__ty_2Erealax_2Ereal] :
( ( p(ap(c_2Epred__set_2EFINITE(ty_2Erealax_2Ereal),V4s))
& ! [V6x: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V6x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V6x)),inj__ty_2Erealax_2Ereal(V3b)))
& ~ p(ap(ap(c_2Ebool_2EIN(ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V6x)),V4s)) )
=> ( surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V6x))) = surj__ty_2Erealax_2Ereal(ap(V1g,inj__ty_2Erealax_2Ereal(V6x))) ) )
& p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V0f),inj__ty_2Erealax_2Ereal(V5i))) )
=> p(ap(ap(ap(c_2Etransc_2EDint,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V1g),inj__ty_2Erealax_2Ereal(V5i))) ) ) ) ) ).
tff(conj_thm_2Eintegral_2EREAL__POW__LBOUND,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V0x)))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(V1n))),inj__ty_2Erealax_2Ereal(V0x)))),ap(ap(c_2Ereal_2Epow,ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),inj__ty_2Erealax_2Ereal(V0x))),inj__ty_2Enum_2Enum(V1n)))) ) ).
tff(conj_thm_2Eintegral_2EREAL__ARCH__POW,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),inj__ty_2Erealax_2Ereal(V0x)))
=> ? [V2n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V1y)),ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Enum_2Enum(V2n)))) ) ).
tff(conj_thm_2Eintegral_2EREAL__ARCH__POW2,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
? [V1n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V0x)),ap(ap(c_2Ereal_2Epow,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),inj__ty_2Enum_2Enum(V1n)))) ).
tff(conj_thm_2Eintegral_2EREAL__POW__LE__1,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),inj__ty_2Erealax_2Ereal(V1x)))
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V1x)),inj__ty_2Enum_2Enum(V0n)))) ) ).
tff(conj_thm_2Eintegral_2EREAL__POW__MONO,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum,V2x: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))),inj__ty_2Erealax_2Ereal(V2x)))
& p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V0m)),inj__ty_2Enum_2Enum(V1n))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Enum_2Enum(V0m))),ap(ap(c_2Ereal_2Epow,inj__ty_2Erealax_2Ereal(V2x)),inj__ty_2Enum_2Enum(V1n)))) ) ).
tff(conj_thm_2Eintegral_2EREAL__LE__INV2,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V0x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V0x)),inj__ty_2Erealax_2Ereal(V1y))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Erealax_2Einv,inj__ty_2Erealax_2Ereal(V1y))),ap(c_2Erealax_2Einv,inj__ty_2Erealax_2Ereal(V0x)))) ) ).
tff(conj_thm_2Eintegral_2EGAUGE__MIN__FINITE,axiom,
! [V0s: $i] :
( mem(V0s,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1gs: $i] :
( mem(V1gs,arr(ty_2Enum_2Enum,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)))
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ! [V3m: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V3m)),inj__ty_2Enum_2Enum(V2n)))
=> p(ap(ap(c_2Etransc_2Egauge,V0s),ap(V1gs,inj__ty_2Enum_2Enum(V3m)))) )
=> ? [V4g: $i] :
( mem(V4g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
& p(ap(ap(c_2Etransc_2Egauge,V0s),V4g))
& ! [V5d: $i] :
( mem(V5d,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V6p: $i] :
( mem(V6p,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( p(ap(ap(c_2Etransc_2Efine,V4g),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V5d),V6p)))
=> ! [V7m: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V7m)),inj__ty_2Enum_2Enum(V2n)))
=> p(ap(ap(c_2Etransc_2Efine,ap(V1gs,inj__ty_2Enum_2Enum(V7m))),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V5d),V6p))) ) ) ) ) ) ) ) ) ).
tff(lamtp_f3124,type,
f3124: ( tp__ty_2Erealax_2Ereal * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3124,axiom,
! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V5x: tp__ty_2Erealax_2Ereal] : ( ap(f3124(V1a,V2b),inj__ty_2Erealax_2Ereal(V5x)) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V5x))),ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V5x)),inj__ty_2Erealax_2Ereal(V2b))) ) ).
tff(conj_thm_2Eintegral_2EINTEGRABLE__CAUCHY,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),V0f))
<=> ! [V3e: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V3e)))
=> ? [V4g: $i] :
( mem(V4g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
& p(ap(ap(c_2Etransc_2Egauge,f3124(V1a,V2b)),V4g))
& ! [V6d1: $i] :
( mem(V6d1,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V7p1: $i] :
( mem(V7p1,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V8d2: $i] :
( mem(V8d2,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V9p2: $i] :
( mem(V9p2,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ( p(ap(ap(c_2Etransc_2Etdiv,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V6d1),V7p1)))
& p(ap(ap(c_2Etransc_2Efine,V4g),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V6d1),V7p1)))
& p(ap(ap(c_2Etransc_2Etdiv,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V8d2),V9p2)))
& p(ap(ap(c_2Etransc_2Efine,V4g),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V8d2),V9p2))) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Etransc_2Ersum,ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V6d1),V7p1)),V0f)),ap(ap(c_2Etransc_2Ersum,ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V8d2),V9p2)),V0f)))),inj__ty_2Erealax_2Ereal(V3e))) ) ) ) ) ) ) ) ) ) ).
tff(lamtp_f3125,type,
f3125: $i > $i ).
tff(lameq_f3125,axiom,
! [V0d: $i] :
( mem(V0d,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3i: tp__ty_2Enum_2Enum] : ( ap(f3125(V0d),inj__ty_2Enum_2Enum(V3i)) = ap(ap(c_2Ereal_2Ereal__sub,ap(V0d,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3i)))),ap(V0d,inj__ty_2Enum_2Enum(V3i))) ) ) ).
tff(conj_thm_2Eintegral_2ESUM__DIFFS,axiom,
! [V0d: $i] :
( mem(V0d,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] : ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Esum,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n))),f3125(V0d))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Ereal__sub,ap(V0d,ap(ap(c_2Earithmetic_2E_2B,inj__ty_2Enum_2Enum(V1m)),inj__ty_2Enum_2Enum(V2n)))),ap(V0d,inj__ty_2Enum_2Enum(V1m)))) ) ) ).
tff(conj_thm_2Eintegral_2ERSUM__BOUND,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2d: $i] :
( mem(V2d,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3p: $i] :
( mem(V3p,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4e: tp__ty_2Erealax_2Ereal,V5f: $i] :
( mem(V5f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ( p(ap(ap(c_2Etransc_2Etdiv,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1b))),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V2d),V3p)))
& ! [V6x: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V6x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V6x)),inj__ty_2Erealax_2Ereal(V1b))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(V5f,inj__ty_2Erealax_2Ereal(V6x)))),inj__ty_2Erealax_2Ereal(V4e))) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Etransc_2Ersum,ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V2d),V3p)),V5f))),ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V4e)),ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V1b)),inj__ty_2Erealax_2Ereal(V0a))))) ) ) ) ) ).
tff(conj_thm_2Eintegral_2ERSUM__DIFF__BOUND,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2d: $i] :
( mem(V2d,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3p: $i] :
( mem(V3p,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V4e: tp__ty_2Erealax_2Ereal,V5f: $i] :
( mem(V5f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V6g: $i] :
( mem(V6g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ( ( p(ap(ap(c_2Etransc_2Etdiv,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1b))),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V2d),V3p)))
& ! [V7x: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V7x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V7x)),inj__ty_2Erealax_2Ereal(V1b))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(V5f,inj__ty_2Erealax_2Ereal(V7x))),ap(V6g,inj__ty_2Erealax_2Ereal(V7x))))),inj__ty_2Erealax_2Ereal(V4e))) ) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Etransc_2Ersum,ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V2d),V3p)),V5f)),ap(ap(c_2Etransc_2Ersum,ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V2d),V3p)),V6g)))),ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V4e)),ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V1b)),inj__ty_2Erealax_2Ereal(V0a))))) ) ) ) ) ) ).
tff(conj_thm_2Eintegral_2EINTEGRABLE__LIMIT,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ! [V3e: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V3e)))
=> ? [V4g: $i] :
( mem(V4g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
& ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V5x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V5x)),inj__ty_2Erealax_2Ereal(V2b))) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,inj__ty_2Erealax_2Ereal(V5x))),ap(V4g,inj__ty_2Erealax_2Ereal(V5x))))),inj__ty_2Erealax_2Ereal(V3e))) )
& p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),V4g)) ) )
=> p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),V0f)) ) ) ).
tff(conj_thm_2Eintegral_2EINTEGRABLE__CONST,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] : p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1b))),k(ty_2Erealax_2Ereal,inj__ty_2Erealax_2Ereal(V2c)))) ).
tff(lamtp_f3126,type,
f3126: ( $i * $i ) > $i ).
tff(lameq_f3126,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V4x: tp__ty_2Erealax_2Ereal] : ( ap(f3126(V0f,V1g),inj__ty_2Erealax_2Ereal(V4x)) = ap(ap(c_2Erealax_2Ereal__add,ap(V0f,inj__ty_2Erealax_2Ereal(V4x))),ap(V1g,inj__ty_2Erealax_2Ereal(V4x))) ) ) ) ).
tff(conj_thm_2Eintegral_2EINTEGRABLE__ADD,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b)))
& p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V0f))
& p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V1g)) )
=> p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),f3126(V0f,V1g))) ) ) ) ).
tff(lamtp_f3127,type,
f3127: ( $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3127,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V3c: tp__ty_2Erealax_2Ereal,V4x: tp__ty_2Erealax_2Ereal] : ( ap(f3127(V0f,V3c),inj__ty_2Erealax_2Ereal(V4x)) = ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V3c)),ap(V0f,inj__ty_2Erealax_2Ereal(V4x))) ) ) ).
tff(conj_thm_2Eintegral_2EINTEGRABLE__CMUL,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b)))
& p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),V0f)) )
=> p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),f3127(V0f,V3c))) ) ) ).
tff(conj_thm_2Eintegral_2EINTEGRABLE__COMBINE,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2b)),inj__ty_2Erealax_2Ereal(V3c)))
& p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),V0f))
& p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2b)),inj__ty_2Erealax_2Ereal(V3c))),V0f)) )
=> p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V3c))),V0f)) ) ) ).
tff(conj_thm_2Eintegral_2EINTEGRABLE__POINT__SPIKE,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal,V4c: tp__ty_2Erealax_2Ereal] :
( ( ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V5x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V5x)),inj__ty_2Erealax_2Ereal(V3b)))
& ( V5x != V4c ) )
=> ( surj__ty_2Erealax_2Ereal(ap(V0f,inj__ty_2Erealax_2Ereal(V5x))) = surj__ty_2Erealax_2Ereal(ap(V1g,inj__ty_2Erealax_2Ereal(V5x))) ) )
& p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V0f)) )
=> p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V1g)) ) ) ) ).
tff(conj_thm_2Eintegral_2ESUP__INTERVAL,axiom,
! [V0P: $i] :
( mem(V0P,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ? [V3x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V3x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3x)),inj__ty_2Erealax_2Ereal(V2b)))
& p(ap(V0P,inj__ty_2Erealax_2Ereal(V3x))) )
=> ? [V4s: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V4s)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V4s)),inj__ty_2Erealax_2Ereal(V2b)))
& ! [V5y: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V5y)),inj__ty_2Erealax_2Ereal(V4s)))
<=> ? [V6x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V6x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V6x)),inj__ty_2Erealax_2Ereal(V2b)))
& p(ap(V0P,inj__ty_2Erealax_2Ereal(V6x)))
& p(ap(ap(c_2Erealax_2Ereal__lt,inj__ty_2Erealax_2Ereal(V5y)),inj__ty_2Erealax_2Ereal(V6x))) ) ) ) ) ) ).
tff(conj_thm_2Eintegral_2EBOLZANO__LEMMA__ALT,axiom,
! [V0P: $i] :
( mem(V0P,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,bool)))
=> ( ( ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2b)),inj__ty_2Erealax_2Ereal(V3c)))
& p(ap(ap(V0P,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b)))
& p(ap(ap(V0P,inj__ty_2Erealax_2Ereal(V2b)),inj__ty_2Erealax_2Ereal(V3c))) )
=> p(ap(ap(V0P,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V3c))) )
& ! [V4x: tp__ty_2Erealax_2Ereal] :
? [V5d: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V5d)))
& ! [V6a: tp__ty_2Erealax_2Ereal,V7b: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V6a)),inj__ty_2Erealax_2Ereal(V4x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V4x)),inj__ty_2Erealax_2Ereal(V7b)))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V7b)),inj__ty_2Erealax_2Ereal(V6a))),inj__ty_2Erealax_2Ereal(V5d))) )
=> p(ap(ap(V0P,inj__ty_2Erealax_2Ereal(V6a)),inj__ty_2Erealax_2Ereal(V7b))) ) ) )
=> ! [V8a: tp__ty_2Erealax_2Ereal,V9b: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V8a)),inj__ty_2Erealax_2Ereal(V9b)))
=> p(ap(ap(V0P,inj__ty_2Erealax_2Ereal(V8a)),inj__ty_2Erealax_2Ereal(V9b))) ) ) ) ).
tff(conj_thm_2Eintegral_2ECONT__UNIFORM,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b)))
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V3x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3x)),inj__ty_2Erealax_2Ereal(V2b))) )
=> p(ap(ap(c_2Elim_2Econtl,V0f),inj__ty_2Erealax_2Ereal(V3x))) ) )
=> ! [V4e: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V4e)))
=> ? [V5d: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V5d)))
& ! [V6x: tp__ty_2Erealax_2Ereal,V7y: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V6x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V6x)),inj__ty_2Erealax_2Ereal(V2b)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V7y)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V7y)),inj__ty_2Erealax_2Ereal(V2b)))
& p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V6x)),inj__ty_2Erealax_2Ereal(V7y)))),inj__ty_2Erealax_2Ereal(V5d))) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(V0f,inj__ty_2Erealax_2Ereal(V6x))),ap(V0f,inj__ty_2Erealax_2Ereal(V7y))))),inj__ty_2Erealax_2Ereal(V4e))) ) ) ) ) ) ).
tff(conj_thm_2Eintegral_2EINTEGRABLE__CONTINUOUS,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V3x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3x)),inj__ty_2Erealax_2Ereal(V2b))) )
=> p(ap(ap(c_2Elim_2Econtl,V0f),inj__ty_2Erealax_2Ereal(V3x))) )
=> p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),V0f)) ) ) ).
tff(lamtp_f3128,type,
f3128: ( tp__ty_2Erealax_2Ereal * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3128,axiom,
! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V7x: tp__ty_2Erealax_2Ereal] : ( ap(f3128(V1a,V2b),inj__ty_2Erealax_2Ereal(V7x)) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V7x))),ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V7x)),inj__ty_2Erealax_2Ereal(V2b))) ) ).
tff(conj_thm_2Eintegral_2EINTEGRABLE__SPLIT__SIDES,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V3c)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3c)),inj__ty_2Erealax_2Ereal(V2b)))
& p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),V0f)) )
=> ? [V4i: tp__ty_2Erealax_2Ereal] :
! [V5e: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),inj__ty_2Erealax_2Ereal(V5e)))
=> ? [V6g: $i] :
( mem(V6g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
& p(ap(ap(c_2Etransc_2Egauge,f3128(V1a,V2b)),V6g))
& ! [V8d1: $i] :
( mem(V8d1,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V9p1: $i] :
( mem(V9p1,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V10d2: $i] :
( mem(V10d2,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V11p2: $i] :
( mem(V11p2,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ( p(ap(ap(c_2Etransc_2Etdiv,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V3c))),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V8d1),V9p1)))
& p(ap(ap(c_2Etransc_2Efine,V6g),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V8d1),V9p1)))
& p(ap(ap(c_2Etransc_2Etdiv,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3c)),inj__ty_2Erealax_2Ereal(V2b))),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V10d2),V11p2)))
& p(ap(ap(c_2Etransc_2Efine,V6g),ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V10d2),V11p2))) )
=> p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Etransc_2Ersum,ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V8d1),V9p1)),V0f)),ap(ap(c_2Etransc_2Ersum,ap(ap(c_2Epair_2E_2C(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)),V10d2),V11p2)),V0f))),inj__ty_2Erealax_2Ereal(V4i)))),inj__ty_2Erealax_2Ereal(V5e))) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eintegral_2EINTEGRABLE__SUBINTERVAL__LEFT,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V3c)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3c)),inj__ty_2Erealax_2Ereal(V2b)))
& p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),V0f)) )
=> p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V3c))),V0f)) ) ) ).
tff(conj_thm_2Eintegral_2EINTEGRABLE__SUBINTERVAL__RIGHT,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V3c)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3c)),inj__ty_2Erealax_2Ereal(V2b)))
& p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),V0f)) )
=> p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3c)),inj__ty_2Erealax_2Ereal(V2b))),V0f)) ) ) ).
tff(conj_thm_2Eintegral_2EINTEGRABLE__SUBINTERVAL,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal,V4d: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V3c)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3c)),inj__ty_2Erealax_2Ereal(V4d)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V4d)),inj__ty_2Erealax_2Ereal(V2b)))
& p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),V0f)) )
=> p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V3c)),inj__ty_2Erealax_2Ereal(V4d))),V0f)) ) ) ).
tff(conj_thm_2Eintegral_2EINTEGRAL__0,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1b)))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eintegral_2Eintegral,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1b))),k(ty_2Erealax_2Ereal,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))) = surj__ty_2Erealax_2Ereal(ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ).
tff(conj_thm_2Eintegral_2EINTEGRAL__CONST,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal,V2c: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1b)))
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eintegral_2Eintegral,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V0a)),inj__ty_2Erealax_2Ereal(V1b))),k(ty_2Erealax_2Ereal,inj__ty_2Erealax_2Ereal(V2c)))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V2c)),ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V1b)),inj__ty_2Erealax_2Ereal(V0a)))) ) ) ).
tff(conj_thm_2Eintegral_2EINTEGRAL__CMUL,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1c: tp__ty_2Erealax_2Ereal,V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b)))
& p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V0f)) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eintegral_2Eintegral,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),f2824(V0f,V1c))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__mul,inj__ty_2Erealax_2Ereal(V1c)),ap(ap(c_2Eintegral_2Eintegral,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V0f))) ) ) ) ).
tff(conj_thm_2Eintegral_2EINTEGRAL__ADD,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b)))
& p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V0f))
& p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V1g)) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eintegral_2Eintegral,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),f3126(V0f,V1g))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eintegral_2Eintegral,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V0f)),ap(ap(c_2Eintegral_2Eintegral,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V1g))) ) ) ) ) ).
tff(conj_thm_2Eintegral_2EINTEGRAL__SUB,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2a: tp__ty_2Erealax_2Ereal,V3b: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b)))
& p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V0f))
& p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V1g)) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eintegral_2Eintegral,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),f2814(V0f,V1g))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Eintegral_2Eintegral,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V0f)),ap(ap(c_2Eintegral_2Eintegral,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2a)),inj__ty_2Erealax_2Ereal(V3b))),V1g))) ) ) ) ) ).
tff(lamtp_f3129,type,
f3129: ( $i * $i ) > $i ).
tff(lameq_f3129,axiom,
! [V2f_27: $i] :
( mem(V2f_27,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V8x: tp__ty_2Erealax_2Ereal] : ( ap(f3129(V2f_27,V1g),inj__ty_2Erealax_2Ereal(V8x)) = ap(ap(c_2Erealax_2Ereal__mul,ap(V2f_27,inj__ty_2Erealax_2Ereal(V8x))),ap(V1g,inj__ty_2Erealax_2Ereal(V8x))) ) ) ) ).
tff(lamtp_f3130,type,
f3130: ( $i * $i ) > $i ).
tff(lameq_f3130,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V3g_27: $i] :
( mem(V3g_27,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V9x: tp__ty_2Erealax_2Ereal] : ( ap(f3130(V0f,V3g_27),inj__ty_2Erealax_2Ereal(V9x)) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0f,inj__ty_2Erealax_2Ereal(V9x))),ap(V3g_27,inj__ty_2Erealax_2Ereal(V9x))) ) ) ) ).
tff(lamtp_f3131,type,
f3131: ( $i * $i ) > $i ).
tff(lameq_f3131,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V3g_27: $i] :
( mem(V3g_27,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V10x: tp__ty_2Erealax_2Ereal] : ( ap(f3131(V0f,V3g_27),inj__ty_2Erealax_2Ereal(V10x)) = ap(ap(c_2Erealax_2Ereal__mul,ap(V0f,inj__ty_2Erealax_2Ereal(V10x))),ap(V3g_27,inj__ty_2Erealax_2Ereal(V10x))) ) ) ) ).
tff(lamtp_f3132,type,
f3132: ( $i * $i ) > $i ).
tff(lameq_f3132,axiom,
! [V2f_27: $i] :
( mem(V2f_27,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V11x: tp__ty_2Erealax_2Ereal] : ( ap(f3132(V2f_27,V1g),inj__ty_2Erealax_2Ereal(V11x)) = ap(ap(c_2Erealax_2Ereal__mul,ap(V2f_27,inj__ty_2Erealax_2Ereal(V11x))),ap(V1g,inj__ty_2Erealax_2Ereal(V11x))) ) ) ) ).
tff(conj_thm_2Eintegral_2EINTEGRAL__BY__PARTS,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V2f_27: $i] :
( mem(V2f_27,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V3g_27: $i] :
( mem(V3g_27,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V4a: tp__ty_2Erealax_2Ereal,V5b: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V4a)),inj__ty_2Erealax_2Ereal(V5b)))
& ! [V6x: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V4a)),inj__ty_2Erealax_2Ereal(V6x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V6x)),inj__ty_2Erealax_2Ereal(V5b))) )
=> p(ap(ap(ap(c_2Elim_2Ediffl,V0f),ap(V2f_27,inj__ty_2Erealax_2Ereal(V6x))),inj__ty_2Erealax_2Ereal(V6x))) )
& ! [V7x: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V4a)),inj__ty_2Erealax_2Ereal(V7x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V7x)),inj__ty_2Erealax_2Ereal(V5b))) )
=> p(ap(ap(ap(c_2Elim_2Ediffl,V1g),ap(V3g_27,inj__ty_2Erealax_2Ereal(V7x))),inj__ty_2Erealax_2Ereal(V7x))) )
& p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4a)),inj__ty_2Erealax_2Ereal(V5b))),f3129(V2f_27,V1g)))
& p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4a)),inj__ty_2Erealax_2Ereal(V5b))),f3130(V0f,V3g_27))) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eintegral_2Eintegral,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4a)),inj__ty_2Erealax_2Ereal(V5b))),f3131(V0f,V3g_27))) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Ereal_2Ereal__sub,ap(ap(c_2Erealax_2Ereal__mul,ap(V0f,inj__ty_2Erealax_2Ereal(V5b))),ap(V1g,inj__ty_2Erealax_2Ereal(V5b)))),ap(ap(c_2Erealax_2Ereal__mul,ap(V0f,inj__ty_2Erealax_2Ereal(V4a))),ap(V1g,inj__ty_2Erealax_2Ereal(V4a))))),ap(ap(c_2Eintegral_2Eintegral,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V4a)),inj__ty_2Erealax_2Ereal(V5b))),f3132(V2f_27,V1g)))) ) ) ) ) ) ) ).
tff(conj_thm_2Eintegral_2EINTEGRAL__COMBINE,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3c: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V2b)),inj__ty_2Erealax_2Ereal(V3c)))
& p(ap(ap(c_2Eintegral_2Eintegrable,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V3c))),V0f)) )
=> ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eintegral_2Eintegral,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V3c))),V0f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Eintegral_2Eintegral,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),V0f)),ap(ap(c_2Eintegral_2Eintegral,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V2b)),inj__ty_2Erealax_2Ereal(V3c))),V0f))) ) ) ) ).
tff(conj_thm_2Eintegral_2EINTEGRAL__MVT1,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b)))
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V3x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V3x)),inj__ty_2Erealax_2Ereal(V2b))) )
=> p(ap(ap(c_2Elim_2Econtl,V0f),inj__ty_2Erealax_2Ereal(V3x))) ) )
=> ? [V4x: tp__ty_2Erealax_2Ereal] :
( p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V4x)))
& p(ap(ap(c_2Ereal_2Ereal__lte,inj__ty_2Erealax_2Ereal(V4x)),inj__ty_2Erealax_2Ereal(V2b)))
& ( surj__ty_2Erealax_2Ereal(ap(ap(c_2Eintegral_2Eintegral,ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),inj__ty_2Erealax_2Ereal(V1a)),inj__ty_2Erealax_2Ereal(V2b))),V0f)) = surj__ty_2Erealax_2Ereal(ap(ap(c_2Erealax_2Ereal__mul,ap(V0f,inj__ty_2Erealax_2Ereal(V4x))),ap(ap(c_2Ereal_2Ereal__sub,inj__ty_2Erealax_2Ereal(V2b)),inj__ty_2Erealax_2Ereal(V1a)))) ) ) ) ) ).
%------------------------------------------------------------------------------