ITP001 Axioms: ITP144^5.ax
%------------------------------------------------------------------------------
% File : ITP144^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : iterate^2.ax [Gau20]
% : HL4144^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 362 ( 18 unt; 11 typ; 0 def)
% Number of atoms : 13724 ( 412 equ; 0 cnn)
% Maximal formula atoms : 204 ( 37 avg)
% Number of connectives : 21151 ( 76 ~; 5 |; 416 &;19208 @)
% ( 49 <=>;1397 =>; 0 <=; 0 <~>)
% Maximal formula depth : 40 ( 15 avg;19208 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 12 ( 12 >; 0 *; 0 +; 0 <<)
% Number of symbols : 82 ( 81 usr; 73 con; 0-2 aty)
% Number of variables : 1847 ( 216 ^1573 !; 58 ?;1847 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Eiterate_2E_2E_2E,type,
c_2Eiterate_2E_2E_2E: $i ).
thf(mem_c_2Eiterate_2E_2E_2E,axiom,
mem @ c_2Eiterate_2E_2E_2E @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ).
thf(tp_c_2Eiterate_2EFINREC,type,
c_2Eiterate_2EFINREC: del > del > $i ).
thf(mem_c_2Eiterate_2EFINREC,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eiterate_2EFINREC @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) @ ( arr @ A_27b @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27b @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ) ) ) ).
thf(tp_c_2Eiterate_2EITSET,type,
c_2Eiterate_2EITSET: del > del > $i ).
thf(mem_c_2Eiterate_2EITSET,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eiterate_2EITSET @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ ( arr @ A_27b @ bool ) @ ( arr @ A_27a @ A_27a ) ) ) ) ).
thf(tp_c_2Eiterate_2ESum,type,
c_2Eiterate_2ESum: del > $i ).
thf(mem_c_2Eiterate_2ESum,axiom,
! [A_27a: del] : ( mem @ ( c_2Eiterate_2ESum @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) @ ty_2Erealax_2Ereal ) ) ) ).
thf(tp_c_2Eiterate_2Einf,type,
c_2Eiterate_2Einf: $i ).
thf(mem_c_2Eiterate_2Einf,axiom,
mem @ c_2Eiterate_2Einf @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ bool ) @ ty_2Erealax_2Ereal ) ).
thf(tp_c_2Eiterate_2Eiterate,type,
c_2Eiterate_2Eiterate: del > del > $i ).
thf(mem_c_2Eiterate_2Eiterate,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ A_27b ) ) ) ) ).
thf(tp_c_2Eiterate_2Emonoidal,type,
c_2Eiterate_2Emonoidal: del > $i ).
thf(mem_c_2Eiterate_2Emonoidal,axiom,
! [A_27a: del] : ( mem @ ( c_2Eiterate_2Emonoidal @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ bool ) ) ).
thf(tp_c_2Eiterate_2Eneutral,type,
c_2Eiterate_2Eneutral: del > $i ).
thf(mem_c_2Eiterate_2Eneutral,axiom,
! [A_27a: del] : ( mem @ ( c_2Eiterate_2Eneutral @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ A_27a ) ) ).
thf(tp_c_2Eiterate_2Ensum,type,
c_2Eiterate_2Ensum: del > $i ).
thf(mem_c_2Eiterate_2Ensum,axiom,
! [A_27a: del] : ( mem @ ( c_2Eiterate_2Ensum @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ty_2Enum_2Enum ) ) ) ).
thf(tp_c_2Eiterate_2Epolynomial__function,type,
c_2Eiterate_2Epolynomial__function: $i ).
thf(mem_c_2Eiterate_2Epolynomial__function,axiom,
mem @ c_2Eiterate_2Epolynomial__function @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ bool ) ).
thf(tp_c_2Eiterate_2Esupport,type,
c_2Eiterate_2Esupport: del > del > $i ).
thf(mem_c_2Eiterate_2Esupport,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eiterate_2Esupport @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__LT__BETWEEN,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
<=> ? [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESIMP__REAL__ARCH,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
? [V1n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Eiterate_2ELOWER__BOUND__FINITE__SET__REAL,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ? [V2a: tp__ty_2Erealax_2Ereal] :
! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( ap @ V0f @ V3x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EUPPER__BOUND__FINITE__SET__REAL,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ? [V2a: tp__ty_2Erealax_2Ereal] :
! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__WLOG__LT,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) )
=> ( ( ! [V1x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
& ! [V2x: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) )
<=> ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
& ! [V4x: tp__ty_2Erealax_2Ereal,V5y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V5y ) ) ) ) )
=> ! [V6x: tp__ty_2Erealax_2Ereal,V7y: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V6x ) ) @ ( inj__ty_2Erealax_2Ereal @ V7y ) ) ) ) ) ).
thf(ax_thm_2Eiterate_2EFINREC__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ A_27b )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2EFINREC @ A_27a @ A_27b ) @ V0f ) @ V1b ) @ V2s ) @ V3a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
<=> ( ( V2s
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ( V3a = V1b ) ) ) ) ) ) )
& ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V5b: $i] :
( ( mem @ V5b @ A_27b )
=> ! [V6s: $i] :
( ( mem @ V6s @ ( arr @ A_27a @ bool ) )
=> ! [V7a: $i] :
( ( mem @ V7a @ A_27b )
=> ! [V8n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2EFINREC @ A_27a @ A_27b ) @ V4f ) @ V5b ) @ V6s ) @ V7a ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V8n ) ) ) )
<=> ? [V9x: $i] :
( ( mem @ V9x @ A_27a )
& ? [V10c: $i] :
( ( mem @ V10c @ A_27b )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V9x ) @ V6s ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2EFINREC @ A_27a @ A_27b ) @ V4f ) @ V5b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V6s ) @ V9x ) ) @ V10c ) @ ( inj__ty_2Enum_2Enum @ V8n ) ) )
& ( V7a
= ( ap @ ( ap @ V4f @ V9x ) @ V10c ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EFINREC__def__compute,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ A_27b )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2EFINREC @ A_27a @ A_27b ) @ V0f ) @ V1b ) @ V2s ) @ V3a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
<=> ( ( V2s
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ( V3a = V1b ) ) ) ) ) ) )
& ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V5b: $i] :
( ( mem @ V5b @ A_27b )
=> ! [V6s: $i] :
( ( mem @ V6s @ ( arr @ A_27a @ bool ) )
=> ! [V7a: $i] :
( ( mem @ V7a @ A_27b )
=> ! [V8n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2EFINREC @ A_27a @ A_27b ) @ V4f ) @ V5b ) @ V6s ) @ V7a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V8n ) ) ) ) )
<=> ? [V9x: $i] :
( ( mem @ V9x @ A_27a )
& ? [V10c: $i] :
( ( mem @ V10c @ A_27b )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V9x ) @ V6s ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2EFINREC @ A_27a @ A_27b ) @ V4f ) @ V5b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V6s ) @ V9x ) ) @ V10c ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V8n ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
& ( V7a
= ( ap @ ( ap @ V4f @ V9x ) @ V10c ) ) ) ) ) ) ) ) )
& ! [V11f: $i] :
( ( mem @ V11f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V12b: $i] :
( ( mem @ V12b @ A_27b )
=> ! [V13s: $i] :
( ( mem @ V13s @ ( arr @ A_27a @ bool ) )
=> ! [V14a: $i] :
( ( mem @ V14a @ A_27b )
=> ! [V15n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2EFINREC @ A_27a @ A_27b ) @ V11f ) @ V12b ) @ V13s ) @ V14a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V15n ) ) ) ) )
<=> ? [V16x: $i] :
( ( mem @ V16x @ A_27a )
& ? [V17c: $i] :
( ( mem @ V17c @ A_27b )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V16x ) @ V13s ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2EFINREC @ A_27a @ A_27b ) @ V11f ) @ V12b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V13s ) @ V16x ) ) @ V17c ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V15n ) ) ) ) )
& ( V14a
= ( ap @ ( ap @ V11f @ V16x ) @ V17c ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EFINREC__1__LEMMA,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ A_27b )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2EFINREC @ A_27a @ A_27b ) @ V0f ) @ V1b ) @ V2s ) @ V3a ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ? [V4x: $i] :
( ( mem @ V4x @ A_27a )
& ( V2s
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V4x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& ( V3a
= ( ap @ ( ap @ V0f @ V4x ) @ V1b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EFINREC__SUC__LEMMA,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ A_27b )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ! [V4s: $i] :
( ( mem @ V4s @ A_27b )
=> ( ( V2x != V3y )
=> ( ( ap @ ( ap @ V0f @ V2x ) @ ( ap @ ( ap @ V0f @ V3y ) @ V4s ) )
= ( ap @ ( ap @ V0f @ V3y ) @ ( ap @ ( ap @ V0f @ V2x ) @ V4s ) ) ) ) ) ) )
=> ! [V5n: tp__ty_2Enum_2Enum,V6s: $i] :
( ( mem @ V6s @ ( arr @ A_27a @ bool ) )
=> ! [V7z: $i] :
( ( mem @ V7z @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2EFINREC @ A_27a @ A_27b ) @ V0f ) @ V1b ) @ V6s ) @ V7z ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) )
=> ! [V8x: $i] :
( ( mem @ V8x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V8x ) @ V6s ) )
=> ? [V9w: $i] :
( ( mem @ V9w @ A_27b )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2EFINREC @ A_27a @ A_27b ) @ V0f ) @ V1b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V6s ) @ V8x ) ) @ V9w ) @ ( inj__ty_2Enum_2Enum @ V5n ) ) )
& ( V7z
= ( ap @ ( ap @ V0f @ V8x ) @ V9w ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EFINREC__UNIQUE__LEMMA,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ A_27b )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ! [V4s: $i] :
( ( mem @ V4s @ A_27b )
=> ( ( V2x != V3y )
=> ( ( ap @ ( ap @ V0f @ V2x ) @ ( ap @ ( ap @ V0f @ V3y ) @ V4s ) )
= ( ap @ ( ap @ V0f @ V3y ) @ ( ap @ ( ap @ V0f @ V2x ) @ V4s ) ) ) ) ) ) )
=> ! [V5n1: tp__ty_2Enum_2Enum,V6n2: tp__ty_2Enum_2Enum,V7s: $i] :
( ( mem @ V7s @ ( arr @ A_27a @ bool ) )
=> ! [V8a1: $i] :
( ( mem @ V8a1 @ A_27b )
=> ! [V9a2: $i] :
( ( mem @ V9a2 @ A_27b )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2EFINREC @ A_27a @ A_27b ) @ V0f ) @ V1b ) @ V7s ) @ V8a1 ) @ ( inj__ty_2Enum_2Enum @ V5n1 ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2EFINREC @ A_27a @ A_27b ) @ V0f ) @ V1b ) @ V7s ) @ V9a2 ) @ ( inj__ty_2Enum_2Enum @ V6n2 ) ) ) )
=> ( ( V8a1 = V9a2 )
& ( V5n1 = V6n2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EFINREC__EXISTS__LEMMA,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ A_27b )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
=> ? [V3a: $i] :
( ( mem @ V3a @ A_27b )
& ? [V4n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2EFINREC @ A_27a @ A_27b ) @ V0f ) @ V1b ) @ V2s ) @ V3a ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EFINREC__FUN__LEMMA,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27c @ bool ) ) ) )
=> ( ( ! [V2s: $i] :
( ( mem @ V2s @ A_27a )
=> ( ( p @ ( ap @ V0P @ V2s ) )
=> ? [V3a: $i] :
( ( mem @ V3a @ A_27b )
& ? [V4n: $i] :
( ( mem @ V4n @ A_27c )
& ( p @ ( ap @ ( ap @ ( ap @ V1R @ V2s ) @ V3a ) @ V4n ) ) ) ) ) )
& ! [V5n1: $i] :
( ( mem @ V5n1 @ A_27c )
=> ! [V6n2: $i] :
( ( mem @ V6n2 @ A_27c )
=> ! [V7s: $i] :
( ( mem @ V7s @ A_27a )
=> ! [V8a1: $i] :
( ( mem @ V8a1 @ A_27b )
=> ! [V9a2: $i] :
( ( mem @ V9a2 @ A_27b )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ V1R @ V7s ) @ V8a1 ) @ V5n1 ) )
& ( p @ ( ap @ ( ap @ ( ap @ V1R @ V7s ) @ V9a2 ) @ V6n2 ) ) )
=> ( ( V8a1 = V9a2 )
& ( V5n1 = V6n2 ) ) ) ) ) ) ) ) )
=> ? [V10f: $i] :
( ( mem @ V10f @ ( arr @ A_27a @ A_27b ) )
& ! [V11s: $i] :
( ( mem @ V11s @ A_27a )
=> ! [V12a: $i] :
( ( mem @ V12a @ A_27b )
=> ( ( p @ ( ap @ V0P @ V11s ) )
=> ( ? [V13n: $i] :
( ( mem @ V13n @ A_27c )
& ( p @ ( ap @ ( ap @ ( ap @ V1R @ V11s ) @ V12a ) @ V13n ) ) )
<=> ( ( ap @ V10f @ V11s )
= V12a ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EFINREC__FUN,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ A_27b )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ! [V4s: $i] :
( ( mem @ V4s @ A_27b )
=> ( ( V2x != V3y )
=> ( ( ap @ ( ap @ V0f @ V2x ) @ ( ap @ ( ap @ V0f @ V3y ) @ V4s ) )
= ( ap @ ( ap @ V0f @ V3y ) @ ( ap @ ( ap @ V0f @ V2x ) @ V4s ) ) ) ) ) ) )
=> ? [V5g: $i] :
( ( mem @ V5g @ ( arr @ ( arr @ A_27a @ bool ) @ A_27b ) )
& ( ( ap @ V5g @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= V1b )
& ! [V6s: $i] :
( ( mem @ V6s @ ( arr @ A_27a @ bool ) )
=> ! [V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V6s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V7x ) @ V6s ) ) )
=> ( ( ap @ V5g @ V6s )
= ( ap @ ( ap @ V0f @ V7x ) @ ( ap @ V5g @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V6s ) @ V7x ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESET__RECURSION__LEMMA,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ A_27b )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ! [V4s: $i] :
( ( mem @ V4s @ A_27b )
=> ( ( V2x != V3y )
=> ( ( ap @ ( ap @ V0f @ V2x ) @ ( ap @ ( ap @ V0f @ V3y ) @ V4s ) )
= ( ap @ ( ap @ V0f @ V3y ) @ ( ap @ ( ap @ V0f @ V2x ) @ V4s ) ) ) ) ) ) )
=> ? [V5g: $i] :
( ( mem @ V5g @ ( arr @ ( arr @ A_27a @ bool ) @ A_27b ) )
& ( ( ap @ V5g @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= V1b )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ! [V7s: $i] :
( ( mem @ V7s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V7s ) )
=> ( ( ap @ V5g @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V6x ) @ V7s ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27b ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6x ) @ V7s ) ) @ ( ap @ V5g @ V7s ) ) @ ( ap @ ( ap @ V0f @ V6x ) @ ( ap @ V5g @ V7s ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eiterate_2EITSET__def,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2EITSET @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2b )
= ( ap
@ ( ap @ ( c_2Emin_2E_40 @ ( arr @ ( arr @ A_27b @ bool ) @ A_27a ) )
@ ( lam @ ( arr @ ( arr @ A_27b @ bool ) @ A_27a )
@ ^ [V3g: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ ( ap @ V3g @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) @ V2b ) )
@ ( ap @ ( c_2Ebool_2E_21 @ A_27b )
@ ( lam @ A_27b
@ ^ [V4x: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( arr @ A_27b @ bool ) )
@ ( lam @ ( arr @ A_27b @ bool )
@ ^ [V5s: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V5s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ ( ap @ V3g @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27b ) @ V4x ) @ V5s ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4x ) @ V5s ) ) @ ( ap @ V3g @ V5s ) ) @ ( ap @ ( ap @ V0f @ V4x ) @ ( ap @ V3g @ V5s ) ) ) ) ) ) ) ) ) ) ) )
@ V1s ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EFINITE__RECURSION,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ A_27b )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ! [V4s: $i] :
( ( mem @ V4s @ A_27b )
=> ( ( V2x != V3y )
=> ( ( ap @ ( ap @ V0f @ V2x ) @ ( ap @ ( ap @ V0f @ V3y ) @ V4s ) )
= ( ap @ ( ap @ V0f @ V3y ) @ ( ap @ ( ap @ V0f @ V2x ) @ V4s ) ) ) ) ) ) )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2EITSET @ A_27b @ A_27a ) @ V0f ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V1b )
= V1b )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ! [V6s: $i] :
( ( mem @ V6s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V6s ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2EITSET @ A_27b @ A_27a ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V5x ) @ V6s ) ) @ V1b )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27b ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V6s ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2EITSET @ A_27b @ A_27a ) @ V0f ) @ V6s ) @ V1b ) ) @ ( ap @ ( ap @ V0f @ V5x ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2EITSET @ A_27b @ A_27a ) @ V0f ) @ V6s ) @ V1b ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ECARD__UNION__EQ,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2u ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ V1t )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t )
= V2u ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1t ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V2u ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUBSET__RESTRICT,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) ) @ ( ap @ V1P @ V2x ) ) ) ) ) )
@ V0s ) ) ) ) ).
thf(conj_thm_2Eiterate_2EFINITE__RESTRICT,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) ) @ ( ap @ V1P @ V2x ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EFINITE__REAL__INTERVAL,axiom,
( ! [V0a: tp__ty_2Erealax_2Ereal] :
~ ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V1x ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V1x ) ) ) ) ) )
& ! [V2a: tp__ty_2Erealax_2Ereal] :
~ ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ V3x ) ) ) ) ) )
& ! [V4b: tp__ty_2Erealax_2Ereal] :
~ ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V5x ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ V5x ) @ ( inj__ty_2Erealax_2Ereal @ V4b ) ) ) ) ) ) )
& ! [V6b: tp__ty_2Erealax_2Ereal] :
~ ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V7x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V7x ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V7x ) @ ( inj__ty_2Erealax_2Ereal @ V6b ) ) ) ) ) ) )
& ! [V8a: tp__ty_2Erealax_2Ereal,V9b: tp__ty_2Erealax_2Ereal] :
( ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V10x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V10x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V8a ) ) @ V10x ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ V10x ) @ ( inj__ty_2Erealax_2Ereal @ V9b ) ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V9b ) ) @ ( inj__ty_2Erealax_2Ereal @ V8a ) ) ) )
& ! [V11a: tp__ty_2Erealax_2Ereal,V12b: tp__ty_2Erealax_2Ereal] :
( ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V13x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V13x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V11a ) ) @ V13x ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ V13x ) @ ( inj__ty_2Erealax_2Ereal @ V12b ) ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V12b ) ) @ ( inj__ty_2Erealax_2Ereal @ V11a ) ) ) )
& ! [V14a: tp__ty_2Erealax_2Ereal,V15b: tp__ty_2Erealax_2Ereal] :
( ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V16x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V16x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V14a ) ) @ V16x ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V16x ) @ ( inj__ty_2Erealax_2Ereal @ V15b ) ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V15b ) ) @ ( inj__ty_2Erealax_2Ereal @ V14a ) ) ) )
& ! [V17a: tp__ty_2Erealax_2Ereal,V18b: tp__ty_2Erealax_2Ereal] :
( ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V19x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V19x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V17a ) ) @ V19x ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V19x ) @ ( inj__ty_2Erealax_2Ereal @ V18b ) ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V18b ) ) @ ( inj__ty_2Erealax_2Ereal @ V17a ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2Ereal__INFINITE,axiom,
~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ) ).
thf(conj_thm_2Eiterate_2ESET__PROVE__CASES,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1a ) @ V2s ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1a ) @ V2s ) ) ) ) ) ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ V0P @ V3s ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ECHOOSE__SUBSET__STRONG,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1s ) ) ) )
=> ? [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2t ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2EHAS__SIZE @ A_27a ) @ V2t ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ECHOOSE__SUBSET,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) )
=> ? [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2t ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2EHAS__SIZE @ A_27a ) @ V2t ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__COMPLETE,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ? [V1x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) )
& ? [V2M: tp__ty_2Erealax_2Ereal] :
! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2M ) ) ) ) )
=> ? [V4M: tp__ty_2Erealax_2Ereal] :
( ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4M ) ) ) )
& ! [V6M_27: tp__ty_2Erealax_2Ereal] :
( ! [V7x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V7x ) ) @ ( inj__ty_2Erealax_2Ereal @ V6M_27 ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4M ) ) @ ( inj__ty_2Erealax_2Ereal @ V6M_27 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2Esup__alt,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Esup @ V0s ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( c_2Emin_2E_40 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1a: $i] :
( ap
@ ( ap @ c_2Ebool_2E_2F_5C
@ ( ap @ ( c_2Ebool_2E_21 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V2x ) @ V0s ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V2x ) @ V1a ) ) ) ) )
@ ( ap @ ( c_2Ebool_2E_21 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3b: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( c_2Ebool_2E_21 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V4x ) @ V0s ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V4x ) @ V3b ) ) ) ) )
@ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V1a ) @ V3b ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUP__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ! [V2b: tp__ty_2Erealax_2Ereal] :
( ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
<=> ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V1t ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Esup @ V0s ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Esup @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUP,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ? [V1b: tp__ty_2Erealax_2Ereal] :
! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) )
=> ( ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( ap @ c_2Ereal_2Esup @ V0s ) ) ) )
& ! [V4b: tp__ty_2Erealax_2Ereal] :
( ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4b ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Esup @ V0s ) ) @ ( inj__ty_2Erealax_2Ereal @ V4b ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUP__FINITE__LEMMA,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ? [V1b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ V0s ) )
& ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUP__FINITE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Ereal_2Esup @ V0s ) ) @ V0s ) )
& ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( ap @ c_2Ereal_2Esup @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__LE__SUP__FINITE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( ap @ c_2Ereal_2Esup @ V0s ) ) )
<=> ? [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__SUP__LE__FINITE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Esup @ V0s ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) )
<=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__LT__SUP__FINITE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( ap @ c_2Ereal_2Esup @ V0s ) ) )
<=> ? [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__SUP__LT__FINITE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Esup @ V0s ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) )
<=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__SUP__LE__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1y: tp__ty_2Erealax_2Ereal] :
( ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ? [V2b: tp__ty_2Erealax_2Ereal] :
! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Esup @ V0s ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) )
<=> ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__SUP__UNIQUE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1b: tp__ty_2Erealax_2Ereal] :
( ( ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
& ! [V3b_27: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V3b_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) )
=> ? [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V3b_27 ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Esup @ V0s ) )
= V1b ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__SUP__LE__S,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1b: tp__ty_2Erealax_2Ereal] :
( ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Esup @ V0s ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__SUP__LE__SUBSET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) )
& ? [V2b: tp__ty_2Erealax_2Ereal] :
! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V1t ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Esup @ V0s ) ) @ ( ap @ c_2Ereal_2Esup @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__LE__SUP,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal,V3y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3y ) ) )
& ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( ap @ c_2Ereal_2Esup @ V0s ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__SUP__BOUNDS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V0s ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( ap @ c_2Ereal_2Esup @ V0s ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Esup @ V0s ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__ABS__SUP__LE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ c_2Ereal_2Esup @ V0s ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__SUP__ASCLOSE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2e: tp__ty_2Erealax_2Ereal] :
( ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal_2Esup @ V0s ) ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUP__SING,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Esup @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) )
= V0a ) ).
thf(conj_thm_2Eiterate_2ESUP__UNIQUE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1b: tp__ty_2Erealax_2Ereal] :
( ! [V2c: tp__ty_2Erealax_2Ereal] :
( ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Esup @ V0s ) )
= V1b ) ) ) ).
thf(conj_thm_2Eiterate_2ESUP__UNION,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( V1t
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ? [V2b: tp__ty_2Erealax_2Ereal] :
! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) )
& ? [V4c: tp__ty_2Erealax_2Ereal] :
! [V5x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V1t ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4c ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Esup @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Erealax_2Ereal ) @ V0s ) @ V1t ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Emax @ ( ap @ c_2Ereal_2Esup @ V0s ) ) @ ( ap @ c_2Ereal_2Esup @ V1t ) ) ) ) ) ) ) ).
thf(ax_thm_2Eiterate_2Einf,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Eiterate_2Einf @ V0s ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( c_2Emin_2E_40 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1a: $i] :
( ap
@ ( ap @ c_2Ebool_2E_2F_5C
@ ( ap @ ( c_2Ebool_2E_21 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V2x ) @ V0s ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V1a ) @ V2x ) ) ) ) )
@ ( ap @ ( c_2Ebool_2E_21 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3b: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( c_2Ebool_2E_21 @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ V4x ) @ V0s ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V3b ) @ V4x ) ) ) ) )
@ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ V3b ) @ V1a ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EINF__EQ,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ! [V2a: tp__ty_2Erealax_2Ereal] :
( ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) )
<=> ! [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V1t ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Eiterate_2Einf @ V0s ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Eiterate_2Einf @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EINF,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ? [V1b: tp__ty_2Erealax_2Ereal] :
! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) )
=> ( ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Eiterate_2Einf @ V0s ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) )
& ! [V4b: tp__ty_2Erealax_2Ereal] :
( ! [V5x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4b ) ) @ ( inj__ty_2Erealax_2Ereal @ V5x ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V4b ) ) @ ( ap @ c_2Eiterate_2Einf @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EINF__FINITE__LEMMA,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ? [V1b: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ V0s ) )
& ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EINF__FINITE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Eiterate_2Einf @ V0s ) ) @ V0s ) )
& ! [V1x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Eiterate_2Einf @ V0s ) ) @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__LE__INF__FINITE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( ap @ c_2Eiterate_2Einf @ V0s ) ) )
<=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__INF__LE__FINITE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Eiterate_2Einf @ V0s ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) )
<=> ? [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__LT__INF__FINITE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( ap @ c_2Eiterate_2Einf @ V0s ) ) )
<=> ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__INF__LT__FINITE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Eiterate_2Einf @ V0s ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) )
<=> ? [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__INF__UNIQUE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1b: tp__ty_2Erealax_2Ereal] :
( ( ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) )
& ! [V3b_27: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b_27 ) ) )
=> ? [V4x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V3b_27 ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Eiterate_2Einf @ V0s ) )
= V1b ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__LE__INF,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1b: tp__ty_2Erealax_2Ereal] :
( ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) @ ( ap @ c_2Eiterate_2Einf @ V0s ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__LE__INF__SUBSET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( V1t
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Erealax_2Ereal ) @ V1t ) @ V0s ) )
& ? [V2b: tp__ty_2Erealax_2Ereal] :
! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Eiterate_2Einf @ V0s ) ) @ ( ap @ c_2Eiterate_2Einf @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__INF__BOUNDS,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal,V2b: tp__ty_2Erealax_2Ereal] :
( ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V0s ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) @ ( ap @ c_2Eiterate_2Einf @ V0s ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Eiterate_2Einf @ V0s ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__ABS__INF__LE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ c_2Eiterate_2Einf @ V0s ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__INF__ASCLOSE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1l: tp__ty_2Erealax_2Ereal,V2e: tp__ty_2Erealax_2Ereal] :
( ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Eiterate_2Einf @ V0s ) ) @ ( inj__ty_2Erealax_2Ereal @ V1l ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUP__UNIQUE__FINITE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V1s ) )
& ( V1s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Esup @ V1s ) )
= V0a )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V1s ) )
& ! [V2y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EINF__UNIQUE__FINITE,axiom,
! [V0a: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V1s ) )
& ( V1s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) )
=> ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Eiterate_2Einf @ V1s ) )
= V0a )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ V1s ) )
& ! [V2y: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUP__INSERT__FINITE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V1s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Esup @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V1s ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Ereal_2Emax @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal_2Esup @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EINF__INSERT__FINITE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal ) @ V1s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Eiterate_2Einf @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V1s ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ ty_2Erealax_2Ereal @ bool ) ) @ V1s ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Ereal_2Emin @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Eiterate_2Einf @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__SUP__EQ__INF,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ( ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) )
& ? [V1B: tp__ty_2Erealax_2Ereal] :
! [V2x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1B ) ) ) ) )
=> ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Esup @ V0s ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Eiterate_2Einf @ V0s ) ) )
<=> ? [V3a: tp__ty_2Erealax_2Ereal] :
( V0s
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V3a ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EINF__SING,axiom,
! [V0a: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Eiterate_2Einf @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Erealax_2Ereal ) @ ( inj__ty_2Erealax_2Ereal @ V0a ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Erealax_2Ereal ) ) ) )
= V0a ) ).
thf(ax_thm_2Eiterate_2Enumseg,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ V2x ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ V2x ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EFINITE__NUMSEG,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENUMSEG__COMBINE__R,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1p: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1p ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1p ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1p ) ) ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1p ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
= ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENUMSEG__COMBINE__L,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1p: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1p ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1p ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V1p ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1p ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
= ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENUMSEG__LREC,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
= ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENUMSEG__RREC,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
= ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENUMSEG__REC,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
=> ( ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Enum_2Enum ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EIN__NUMSEG,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum,V2p: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2p ) ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V2p ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2p ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EIN__NUMSEG__0,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENUMSEG__SING,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) ) ) ).
thf(conj_thm_2Eiterate_2ENUMSEG__EMPTY,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) )
<=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) ) ).
thf(conj_thm_2Eiterate_2ECARD__NUMSEG__LEMMA,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1d: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1d ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1d ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ECARD__NUMSEG,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) ) ).
thf(conj_thm_2Eiterate_2EHAS__SIZE__NUMSEG,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2EHAS__SIZE @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) ) ).
thf(conj_thm_2Eiterate_2ECARD__NUMSEG__1,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
= V0n ) ).
thf(conj_thm_2Eiterate_2EHAS__SIZE__NUMSEG__1,axiom,
! [V0n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2EHAS__SIZE @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ).
thf(conj_thm_2Eiterate_2ENUMSEG__CLAUSES,axiom,
( ! [V0m: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ ty_2Enum_2Enum @ bool ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) ) )
& ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ ty_2Enum_2Enum @ bool ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Enum_2Enum ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EFINITE__INDEX__NUMSEG,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
<=> ? [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ! [V2i: tp__ty_2Enum_2Enum,V3j: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V3j ) ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) ) )
& ( ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V2i ) )
= ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V3j ) ) ) )
=> ( V2i = V3j ) )
& ( V0s
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ A_27a ) @ V1f ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EFINITE__INDEX__NUMBERS,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
<=> ? [V1k: $i] :
( ( mem @ V1k @ ( arr @ ty_2Enum_2Enum @ bool ) )
& ? [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ! [V3i: tp__ty_2Enum_2Enum,V4j: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V3i ) ) @ V1k ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V4j ) ) @ V1k ) )
& ( ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ V3i ) )
= ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ V4j ) ) ) )
=> ( V3i = V4j ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V1k ) )
& ( V0s
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ A_27a ) @ V2f ) @ V1k ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EDISJOINT__NUMSEG,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum,V2p: tp__ty_2Enum_2Enum,V3q: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2p ) ) @ ( inj__ty_2Enum_2Enum @ V3q ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V2p ) ) )
| ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3q ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
| ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
| ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3q ) ) @ ( inj__ty_2Enum_2Enum @ V2p ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENUMSEG__ADD__SPLIT,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum,V2p: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
=> ( ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V2p ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V2p ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENUMSEG__OFFSET__IMAGE,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum,V2p: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V2p ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V2p ) ) )
= ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3i: $i] : ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V3i ) @ ( inj__ty_2Enum_2Enum @ V2p ) ) ) )
@ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUBSET__NUMSEG,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum,V2p: tp__ty_2Enum_2Enum,V3q: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2p ) ) @ ( inj__ty_2Enum_2Enum @ V3q ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
| ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2p ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V3q ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENUMSEG__LE,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ bool ) @ V1x ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ V1x ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
= ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ).
thf(conj_thm_2Eiterate_2ENUMSEG__LT,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ bool ) @ V1x ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ V1x ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ ty_2Enum_2Enum @ bool ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ETOPOLOGICAL__SORT,axiom,
! [A_27a: del,V0_3C_3C: $i] :
( ( mem @ V0_3C_3C @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V0_3C_3C @ V1x ) @ V2y ) )
& ( p @ ( ap @ ( ap @ V0_3C_3C @ V2y ) @ V1x ) ) )
=> ( V1x = V2y ) ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ! [V5z: $i] :
( ( mem @ V5z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V0_3C_3C @ V3x ) @ V4y ) )
& ( p @ ( ap @ ( ap @ V0_3C_3C @ V4y ) @ V5z ) ) )
=> ( p @ ( ap @ ( ap @ V0_3C_3C @ V3x ) @ V5z ) ) ) ) ) ) )
=> ! [V6n: tp__ty_2Enum_2Enum,V7s: $i] :
( ( mem @ V7s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2EHAS__SIZE @ A_27a ) @ V7s ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) )
=> ? [V8f: $i] :
( ( mem @ V8f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ( V7s
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ A_27a ) @ V8f ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) )
& ! [V9j: tp__ty_2Enum_2Enum,V10k: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V9j ) ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V10k ) ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V9j ) ) @ ( inj__ty_2Enum_2Enum @ V10k ) ) ) )
=> ~ ( p @ ( ap @ ( ap @ V0_3C_3C @ ( ap @ V8f @ ( inj__ty_2Enum_2Enum @ V10k ) ) ) @ ( ap @ V8f @ ( inj__ty_2Enum_2Enum @ V9j ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eiterate_2Eneutral,axiom,
! [A_27a: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ap @ ( c_2Eiterate_2Eneutral @ A_27a ) @ V0op )
= ( ap @ ( c_2Emin_2E_40 @ A_27a )
@ ( lam @ A_27a
@ ^ [V1x: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V2y: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ ( ap @ ( ap @ V0op @ V1x ) @ V2y ) ) @ V2y ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ ( ap @ ( ap @ V0op @ V2y ) @ V1x ) ) @ V2y ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eiterate_2Emonoidal,axiom,
! [A_27a: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27a ) @ V0op ) )
<=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ap @ ( ap @ V0op @ V1x ) @ V2y )
= ( ap @ ( ap @ V0op @ V2y ) @ V1x ) ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ! [V5z: $i] :
( ( mem @ V5z @ A_27a )
=> ( ( ap @ ( ap @ V0op @ V3x ) @ ( ap @ ( ap @ V0op @ V4y ) @ V5z ) )
= ( ap @ ( ap @ V0op @ ( ap @ ( ap @ V0op @ V3x ) @ V4y ) ) @ V5z ) ) ) ) )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ( ( ap @ ( ap @ V0op @ ( ap @ ( c_2Eiterate_2Eneutral @ A_27a ) @ V0op ) ) @ V6x )
= V6x ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EMONOIDAL__AC,axiom,
! [A_27a: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27a ) @ V0op ) )
=> ( ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ( ( ap @ ( ap @ V0op @ ( ap @ ( c_2Eiterate_2Eneutral @ A_27a ) @ V0op ) ) @ V1a )
= V1a ) )
& ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( ap @ ( ap @ V0op @ V2a ) @ ( ap @ ( c_2Eiterate_2Eneutral @ A_27a ) @ V0op ) )
= V2a ) )
& ! [V3a: $i] :
( ( mem @ V3a @ A_27a )
=> ! [V4b: $i] :
( ( mem @ V4b @ A_27a )
=> ( ( ap @ ( ap @ V0op @ V3a ) @ V4b )
= ( ap @ ( ap @ V0op @ V4b ) @ V3a ) ) ) )
& ! [V5a: $i] :
( ( mem @ V5a @ A_27a )
=> ! [V6b: $i] :
( ( mem @ V6b @ A_27a )
=> ! [V7c: $i] :
( ( mem @ V7c @ A_27a )
=> ( ( ap @ ( ap @ V0op @ ( ap @ ( ap @ V0op @ V5a ) @ V6b ) ) @ V7c )
= ( ap @ ( ap @ V0op @ V5a ) @ ( ap @ ( ap @ V0op @ V6b ) @ V7c ) ) ) ) ) )
& ! [V8a: $i] :
( ( mem @ V8a @ A_27a )
=> ! [V9b: $i] :
( ( mem @ V9b @ A_27a )
=> ! [V10c: $i] :
( ( mem @ V10c @ A_27a )
=> ( ( ap @ ( ap @ V0op @ V8a ) @ ( ap @ ( ap @ V0op @ V9b ) @ V10c ) )
= ( ap @ ( ap @ V0op @ V9b ) @ ( ap @ ( ap @ V0op @ V8a ) @ V10c ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eiterate_2Esupport,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27a @ A_27b ) @ V0op ) @ V1f ) @ V2s )
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ ( ap @ V1f @ V3x ) ) @ ( ap @ ( c_2Eiterate_2Eneutral @ A_27b ) @ V0op ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eiterate_2Eiterate,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ V1s ) @ V2f )
= ( ap
@ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27b ) @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27a @ A_27b ) @ V0op ) @ V2f ) @ V1s ) ) )
@ ( ap
@ ( ap
@ ( ap @ ( c_2Eiterate_2EITSET @ A_27b @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] :
( lam @ A_27b
@ ^ [V4a: $i] : ( ap @ ( ap @ V0op @ ( ap @ V2f @ V3x ) ) @ V4a ) ) ) )
@ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27a @ A_27b ) @ V0op ) @ V2f ) @ V1s ) )
@ ( ap @ ( c_2Eiterate_2Eneutral @ A_27b ) @ V0op ) ) )
@ ( ap @ ( c_2Eiterate_2Eneutral @ A_27b ) @ V0op ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EIN__SUPPORT,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ A_27a ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27b )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V2x ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27b @ A_27a ) @ V0op ) @ V1f ) @ V3s ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V2x ) @ V3s ) )
& ( ( ap @ V1f @ V2x )
!= ( ap @ ( c_2Eiterate_2Eneutral @ A_27a ) @ V0op ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUPPORT__SUPPORT,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ A_27a ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27b @ bool ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27b @ A_27a ) @ V0op ) @ V1f ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27b @ A_27a ) @ V0op ) @ V1f ) @ V2s ) )
= ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27b @ A_27a ) @ V0op ) @ V1f ) @ V2s ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUPPORT__EMPTY,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ A_27a ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27b @ bool ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3x ) @ V2s ) )
=> ( ( ap @ V1f @ V3x )
= ( ap @ ( c_2Eiterate_2Eneutral @ A_27a ) @ V0op ) ) ) )
<=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27b @ A_27a ) @ V0op ) @ V1f ) @ V2s )
= ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUPPORT__SUBSET,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ A_27a ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27b @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27b @ A_27a ) @ V0op ) @ V1f ) @ V2s ) ) @ V2s ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EFINITE__SUPPORT,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ A_27a ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2s ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27b @ A_27a ) @ V0op ) @ V1f ) @ V2s ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUPPORT__CLAUSES,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del,A_27g: del,A_27h: del,A_27i: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) )
=> ( ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27a @ A_27b ) @ V0op ) @ V1f ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27c @ A_27b ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27c )
=> ! [V4s: $i] :
( ( mem @ V4s @ ( arr @ A_27c @ bool ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27c @ A_27b ) @ V0op ) @ V2f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27c ) @ V3x ) @ V4s ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ A_27c @ bool ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ ( ap @ V2f @ V3x ) ) @ ( ap @ ( c_2Eiterate_2Eneutral @ A_27b ) @ V0op ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27c @ A_27b ) @ V0op ) @ V2f ) @ V4s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27c ) @ V3x ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27c @ A_27b ) @ V0op ) @ V2f ) @ V4s ) ) ) ) ) ) )
& ! [V5f: $i] :
( ( mem @ V5f @ ( arr @ A_27d @ A_27b ) )
=> ! [V6x: $i] :
( ( mem @ V6x @ A_27d )
=> ! [V7s: $i] :
( ( mem @ V7s @ ( arr @ A_27d @ bool ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27d @ A_27b ) @ V0op ) @ V5f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27d ) @ V7s ) @ V6x ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27d ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27d @ A_27b ) @ V0op ) @ V5f ) @ V7s ) ) @ V6x ) ) ) ) )
& ! [V8f: $i] :
( ( mem @ V8f @ ( arr @ A_27e @ A_27b ) )
=> ! [V9s: $i] :
( ( mem @ V9s @ ( arr @ A_27e @ bool ) )
=> ! [V10t: $i] :
( ( mem @ V10t @ ( arr @ A_27e @ bool ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27e @ A_27b ) @ V0op ) @ V8f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27e ) @ V9s ) @ V10t ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27e ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27e @ A_27b ) @ V0op ) @ V8f ) @ V9s ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27e @ A_27b ) @ V0op ) @ V8f ) @ V10t ) ) ) ) ) )
& ! [V11f: $i] :
( ( mem @ V11f @ ( arr @ A_27f @ A_27b ) )
=> ! [V12s: $i] :
( ( mem @ V12s @ ( arr @ A_27f @ bool ) )
=> ! [V13t: $i] :
( ( mem @ V13t @ ( arr @ A_27f @ bool ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27f @ A_27b ) @ V0op ) @ V11f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27f ) @ V12s ) @ V13t ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27f ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27f @ A_27b ) @ V0op ) @ V11f ) @ V12s ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27f @ A_27b ) @ V0op ) @ V11f ) @ V13t ) ) ) ) ) )
& ! [V14f: $i] :
( ( mem @ V14f @ ( arr @ A_27g @ A_27b ) )
=> ! [V15s: $i] :
( ( mem @ V15s @ ( arr @ A_27g @ bool ) )
=> ! [V16t: $i] :
( ( mem @ V16t @ ( arr @ A_27g @ bool ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27g @ A_27b ) @ V0op ) @ V14f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27g ) @ V15s ) @ V16t ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27g ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27g @ A_27b ) @ V0op ) @ V14f ) @ V15s ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27g @ A_27b ) @ V0op ) @ V14f ) @ V16t ) ) ) ) ) )
& ! [V17f: $i] :
( ( mem @ V17f @ ( arr @ A_27h @ A_27i ) )
=> ! [V18g: $i] :
( ( mem @ V18g @ ( arr @ A_27i @ A_27b ) )
=> ! [V19s: $i] :
( ( mem @ V19s @ ( arr @ A_27h @ bool ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27i @ A_27b ) @ V0op ) @ V18g ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27h @ A_27i ) @ V17f ) @ V19s ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27h @ A_27i ) @ V17f ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27h @ A_27b ) @ V0op ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27h @ A_27b @ A_27i ) @ V18g ) @ V17f ) ) @ V19s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUPPORT__DELTA,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27b @ A_27a ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ A_27b )
=> ( ( ap
@ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27b @ A_27a ) @ V0op )
@ ( lam @ A_27b
@ ^ [V4x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ V4x ) @ V3a ) ) @ ( ap @ V2f @ V4x ) ) @ ( ap @ ( c_2Eiterate_2Eneutral @ A_27a ) @ V0op ) ) ) )
@ V1s )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ A_27b @ bool ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3a ) @ V1s ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27b @ A_27a ) @ V0op ) @ V2f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27b ) @ V3a ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EFINITE__SUPPORT__DELTA,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27b @ bool ) )
=> ! [V1op: $i] :
( ( mem @ V1op @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27b @ A_27a ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ A_27b )
=> ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b )
@ ( ap
@ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27b @ A_27a ) @ V1op )
@ ( lam @ A_27b
@ ^ [V4x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ V4x ) @ V3a ) ) @ ( ap @ V2f @ V4x ) ) @ ( ap @ ( c_2Eiterate_2Eneutral @ A_27a ) @ V1op ) ) ) )
@ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__SUPPORT,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ A_27a ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27b @ bool ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a ) @ V0op ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27b @ A_27a ) @ V0op ) @ V1f ) @ V2s ) ) @ V1f )
= ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a ) @ V0op ) @ V2s ) @ V1f ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__EXPAND__CASES,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ A_27a ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27b @ bool ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a ) @ V0op ) @ V2s ) @ V1f )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27b @ A_27a ) @ V0op ) @ V1f ) @ V2s ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a ) @ V0op ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27b @ A_27a ) @ V0op ) @ V1f ) @ V2s ) ) @ V1f ) ) @ ( ap @ ( c_2Eiterate_2Eneutral @ A_27a ) @ V0op ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__CLAUSES__GEN,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27b ) @ V0op ) )
=> ( ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V1f )
= ( ap @ ( c_2Eiterate_2Eneutral @ A_27b ) @ V0op ) ) )
& ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4s: $i] :
( ( mem @ V4s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27b ) @ V0op ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27a @ A_27b ) @ V0op ) @ V2f ) @ V4s ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V3x ) @ V4s ) ) @ V2f )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27b ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V4s ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ V4s ) @ V2f ) ) @ ( ap @ ( ap @ V0op @ ( ap @ V2f @ V3x ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ V4s ) @ V2f ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__CLAUSES,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27a ) @ V0op ) )
=> ( ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a ) @ V0op ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) @ V1f )
= ( ap @ ( c_2Eiterate_2Eneutral @ A_27a ) @ V0op ) ) )
& ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27b @ A_27a ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27b )
=> ! [V4s: $i] :
( ( mem @ V4s @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V4s ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a ) @ V0op ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27b ) @ V3x ) @ V4s ) ) @ V2f )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3x ) @ V4s ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a ) @ V0op ) @ V4s ) @ V2f ) ) @ ( ap @ ( ap @ V0op @ ( ap @ V2f @ V3x ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a ) @ V0op ) @ V4s ) @ V2f ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__UNION,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27a ) @ V0op ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ A_27a ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27b @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27b ) @ V2s ) @ V3t ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a ) @ V0op ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27b ) @ V2s ) @ V3t ) ) @ V1f )
= ( ap @ ( ap @ V0op @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a ) @ V0op ) @ V2s ) @ V1f ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a ) @ V0op ) @ V3t ) @ V1f ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__UNION__GEN,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27b ) @ V0op ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27a @ A_27b ) @ V0op ) @ V1f ) @ V2s ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27a @ A_27b ) @ V0op ) @ V1f ) @ V3t ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27a @ A_27b ) @ V0op ) @ V1f ) @ V2s ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27a @ A_27b ) @ V0op ) @ V1f ) @ V3t ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V2s ) @ V3t ) ) @ V1f )
= ( ap @ ( ap @ V0op @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ V2s ) @ V1f ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ V3t ) @ V1f ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__DIFF,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27a ) @ V0op ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ A_27a ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27b @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ V3t ) @ V2s ) ) )
=> ( ( ap @ ( ap @ V0op @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a ) @ V0op ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27b ) @ V2s ) @ V3t ) ) @ V1f ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a ) @ V0op ) @ V3t ) @ V1f ) )
= ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a ) @ V0op ) @ V2s ) @ V1f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__DIFF__GEN,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27b ) @ V0op ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27a @ A_27b ) @ V0op ) @ V1f ) @ V2s ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27a @ A_27b ) @ V0op ) @ V1f ) @ V3t ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27a @ A_27b ) @ V0op ) @ V1f ) @ V2s ) ) ) )
=> ( ( ap @ ( ap @ V0op @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V2s ) @ V3t ) ) @ V1f ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ V3t ) @ V1f ) )
= ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ V2s ) @ V1f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__INCL__EXCL,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27a ) @ V0op ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27b @ A_27a ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2t ) ) )
=> ( ( ap @ ( ap @ V0op @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a ) @ V0op ) @ V1s ) @ V3f ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a ) @ V0op ) @ V2t ) @ V3f ) )
= ( ap @ ( ap @ V0op @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a ) @ V0op ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27b ) @ V1s ) @ V2t ) ) @ V3f ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a ) @ V0op ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27b ) @ V1s ) @ V2t ) ) @ V3f ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__CLOSED,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27b ) @ V0op ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ V1P @ ( ap @ ( c_2Eiterate_2Eneutral @ A_27b ) @ V0op ) ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27b )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( ( p @ ( ap @ V1P @ V2x ) )
& ( p @ ( ap @ V1P @ V3y ) ) )
=> ( p @ ( ap @ V1P @ ( ap @ ( ap @ V0op @ V2x ) @ V3y ) ) ) ) ) ) )
=> ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27a @ A_27b ) )
=> ! [V5s: $i] :
( ( mem @ V5s @ ( arr @ A_27a @ bool ) )
=> ( ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6x ) @ V5s ) )
& ( ( ap @ V4f @ V6x )
!= ( ap @ ( c_2Eiterate_2Eneutral @ A_27b ) @ V0op ) ) )
=> ( p @ ( ap @ V1P @ ( ap @ V4f @ V6x ) ) ) ) )
=> ( p @ ( ap @ V1P @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ V5s ) @ V4f ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__RELATED,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27b ) @ V0op ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ V1R @ ( ap @ ( c_2Eiterate_2Eneutral @ A_27b ) @ V0op ) ) @ ( ap @ ( c_2Eiterate_2Eneutral @ A_27b ) @ V0op ) ) )
& ! [V2x1: $i] :
( ( mem @ V2x1 @ A_27b )
=> ! [V3y1: $i] :
( ( mem @ V3y1 @ A_27b )
=> ! [V4x2: $i] :
( ( mem @ V4x2 @ A_27b )
=> ! [V5y2: $i] :
( ( mem @ V5y2 @ A_27b )
=> ( ( ( p @ ( ap @ ( ap @ V1R @ V2x1 ) @ V4x2 ) )
& ( p @ ( ap @ ( ap @ V1R @ V3y1 ) @ V5y2 ) ) )
=> ( p @ ( ap @ ( ap @ V1R @ ( ap @ ( ap @ V0op @ V2x1 ) @ V3y1 ) ) @ ( ap @ ( ap @ V0op @ V4x2 ) @ V5y2 ) ) ) ) ) ) ) ) )
=> ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ A_27a @ A_27b ) )
=> ! [V7g: $i] :
( ( mem @ V7g @ ( arr @ A_27a @ A_27b ) )
=> ! [V8s: $i] :
( ( mem @ V8s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V8s ) )
& ! [V9x: $i] :
( ( mem @ V9x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V9x ) @ V8s ) )
=> ( p @ ( ap @ ( ap @ V1R @ ( ap @ V6f @ V9x ) ) @ ( ap @ V7g @ V9x ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ V1R @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ V8s ) @ V6f ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ V8s ) @ V7g ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__EQ__NEUTRAL,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27b ) @ V0op ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
=> ( ( ap @ V1f @ V3x )
= ( ap @ ( c_2Eiterate_2Eneutral @ A_27b ) @ V0op ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ V2s ) @ V1f )
= ( ap @ ( c_2Eiterate_2Eneutral @ A_27b ) @ V0op ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__SING,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27b ) @ V0op ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) @ V1f )
= ( ap @ V1f @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__DELETE,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27b ) @ V0op ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ A_27a )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3a ) @ V2s ) ) )
=> ( ( ap @ ( ap @ V0op @ ( ap @ V1f @ V3a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V2s ) @ V3a ) ) @ V1f ) )
= ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ V2s ) @ V1f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__DELTA,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27a ) @ V0op ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ A_27a ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27b )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27b @ bool ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a ) @ V0op ) @ V3s )
@ ( lam @ A_27b
@ ^ [V4x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ V4x ) @ V2a ) ) @ ( ap @ V1f @ V4x ) ) @ ( ap @ ( c_2Eiterate_2Eneutral @ A_27a ) @ V0op ) ) ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V2a ) @ V3s ) ) @ ( ap @ V1f @ V2a ) ) @ ( ap @ ( c_2Eiterate_2Eneutral @ A_27a ) @ V0op ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__IMAGE,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27c @ ( arr @ A_27c @ A_27c ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27c ) @ V0op ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27b @ A_27c ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ( ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V3s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5y ) @ V3s ) )
& ( ( ap @ V1f @ V4x )
= ( ap @ V1f @ V5y ) ) )
=> ( V4x = V5y ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27c ) @ V0op ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V1f ) @ V3s ) ) @ V2g )
= ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27c ) @ V0op ) @ V3s ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b ) @ V2g ) @ V1f ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__BIJECTION,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27b ) @ V0op ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2p: $i] :
( ( mem @ V2p @ ( arr @ A_27a @ A_27a ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ( ( ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V3s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ V2p @ V4x ) ) @ V3s ) ) ) )
& ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5y ) @ V3s ) )
=> ( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V6x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6x ) @ V3s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ ( ap @ V2p @ V6x ) ) @ V5y ) ) ) ) ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ V3s ) @ V1f )
= ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ V3s ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27b @ A_27a ) @ V1f ) @ V2p ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__ITERATE__PRODUCT,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27c @ ( arr @ A_27c @ A_27c ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27c ) @ V0op ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ! [V4i: $i] :
( ( mem @ V4i @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4i ) @ V1s ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ ( ap @ V2t @ V4i ) ) ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27c ) @ V0op ) @ V1s )
@ ( lam @ A_27a
@ ^ [V5i: $i] : ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27c ) @ V0op ) @ ( ap @ V2t @ V5i ) ) @ ( ap @ V3x @ V5i ) ) ) )
= ( ap
@ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c ) @ V0op )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
@ ( lam @ A_27a
@ ^ [V6i: $i] :
( lam @ A_27b
@ ^ [V7j: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V6i ) @ V7j ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6i ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V7j ) @ ( ap @ V2t @ V6i ) ) ) ) ) ) ) ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c )
@ ( lam @ A_27a
@ ^ [V8i: $i] :
( lam @ A_27b
@ ^ [V9j: $i] : ( ap @ ( ap @ V3x @ V8i ) @ V9j ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__EQ,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27b ) @ V0op ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ A_27b ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ( ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V3s ) )
=> ( ( ap @ V1f @ V4x )
= ( ap @ V2g @ V4x ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ V3s ) @ V1f )
= ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ V3s ) @ V2g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__EQ__GENERAL,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27c @ ( arr @ A_27c @ A_27c ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27c ) @ V0op ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ A_27c ) )
=> ! [V4g: $i] :
( ( mem @ V4g @ ( arr @ A_27b @ A_27c ) )
=> ! [V5h: $i] :
( ( mem @ V5h @ ( arr @ A_27a @ A_27b ) )
=> ( ( ! [V6y: $i] :
( ( mem @ V6y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V6y ) @ V2t ) )
=> ( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V7x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V7x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ ( ap @ V5h @ V7x ) ) @ V6y ) ) ) ) ) ) )
& ! [V8x: $i] :
( ( mem @ V8x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V8x ) @ V1s ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V5h @ V8x ) ) @ V2t ) )
& ( ( ap @ V4g @ ( ap @ V5h @ V8x ) )
= ( ap @ V3f @ V8x ) ) ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27c ) @ V0op ) @ V1s ) @ V3f )
= ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27c ) @ V0op ) @ V2t ) @ V4g ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__EQ__GENERAL__INVERSES,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27c @ ( arr @ A_27c @ A_27c ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27c ) @ V0op ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ A_27c ) )
=> ! [V4g: $i] :
( ( mem @ V4g @ ( arr @ A_27b @ A_27c ) )
=> ! [V5h: $i] :
( ( mem @ V5h @ ( arr @ A_27a @ A_27b ) )
=> ! [V6k: $i] :
( ( mem @ V6k @ ( arr @ A_27b @ A_27a ) )
=> ( ( ! [V7y: $i] :
( ( mem @ V7y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V7y ) @ V2t ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ V6k @ V7y ) ) @ V1s ) )
& ( ( ap @ V5h @ ( ap @ V6k @ V7y ) )
= V7y ) ) ) )
& ! [V8x: $i] :
( ( mem @ V8x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V8x ) @ V1s ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V5h @ V8x ) ) @ V2t ) )
& ( ( ap @ V6k @ ( ap @ V5h @ V8x ) )
= V8x )
& ( ( ap @ V4g @ ( ap @ V5h @ V8x ) )
= ( ap @ V3f @ V8x ) ) ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27c ) @ V0op ) @ V1s ) @ V3f )
= ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27c ) @ V0op ) @ V2t ) @ V4g ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__INJECTION,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27b ) @ V0op ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2p: $i] :
( ( mem @ V2p @ ( arr @ A_27a @ A_27a ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V3s ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V3s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ V2p @ V4x ) ) @ V3s ) ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ! [V6y: $i] :
( ( mem @ V6y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V3s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6y ) @ V3s ) )
& ( ( ap @ V2p @ V5x )
= ( ap @ V2p @ V6y ) ) )
=> ( V5x = V6y ) ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ V3s ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27b @ A_27a ) @ V1f ) @ V2p ) )
= ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ V3s ) @ V1f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__UNION__NONZERO,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27b ) @ V0op ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V3t ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V2s ) @ V3t ) ) )
=> ( ( ap @ V1f @ V4x )
= ( ap @ ( c_2Eiterate_2Eneutral @ A_27b ) @ V0op ) ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V2s ) @ V3t ) ) @ V1f )
= ( ap @ ( ap @ V0op @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ V2s ) @ V1f ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ V3t ) @ V1f ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__OP,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27a ) @ V0op ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ A_27a ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27b @ A_27a ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V3s ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a ) @ V0op ) @ V3s )
@ ( lam @ A_27b
@ ^ [V4x: $i] : ( ap @ ( ap @ V0op @ ( ap @ V1f @ V4x ) ) @ ( ap @ V2g @ V4x ) ) ) )
= ( ap @ ( ap @ V0op @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a ) @ V0op ) @ V3s ) @ V1f ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27a ) @ V0op ) @ V3s ) @ V2g ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__SUPERSET,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27b ) @ V0op ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27a @ bool ) )
=> ! [V3v: $i] :
( ( mem @ V3v @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2u ) @ V3v ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V3v ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V2u ) ) )
=> ( ( ap @ V1f @ V4x )
= ( ap @ ( c_2Eiterate_2Eneutral @ A_27b ) @ V0op ) ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ V3v ) @ V1f )
= ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ V2u ) @ V1f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__IMAGE__NONZERO,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27c @ ( arr @ A_27c @ A_27c ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27c ) @ V0op ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ A_27c ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V3s ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V3s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5y ) @ V3s ) )
& ( V4x != V5y )
& ( ( ap @ V2f @ V4x )
= ( ap @ V2f @ V5y ) ) )
=> ( ( ap @ V1g @ ( ap @ V2f @ V4x ) )
= ( ap @ ( c_2Eiterate_2Eneutral @ A_27c ) @ V0op ) ) ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27b @ A_27c ) @ V0op ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V2f ) @ V3s ) ) @ V1g )
= ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27c ) @ V0op ) @ V3s ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b ) @ V1g ) @ V2f ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__CASES,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27b ) @ V0op ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2P: $i] :
( ( mem @ V2P @ ( arr @ A_27a @ bool ) )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ A_27b ) )
=> ! [V4g: $i] :
( ( mem @ V4g @ ( arr @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ V1s )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27b ) @ ( ap @ V2P @ V5x ) ) @ ( ap @ V3f @ V5x ) ) @ ( ap @ V4g @ V5x ) ) ) )
= ( ap
@ ( ap @ V0op
@ ( ap
@ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V6x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V6x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6x ) @ V1s ) ) @ ( ap @ V2P @ V6x ) ) ) ) ) )
@ V3f ) )
@ ( ap
@ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V7x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V7x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V7x ) @ V1s ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ V2P @ V7x ) ) ) ) ) ) )
@ V4g ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__OP__GEN,axiom,
! [A_27a: del,A_27b: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27b ) @ V0op ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ A_27b ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27a @ A_27b ) @ V0op ) @ V1f ) @ V3s ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27a @ A_27b ) @ V0op ) @ V2g ) @ V3s ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ V3s )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ V0op @ ( ap @ V1f @ V4x ) ) @ ( ap @ V2g @ V4x ) ) ) )
= ( ap @ ( ap @ V0op @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ V3s ) @ V1f ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ A_27b ) @ V0op ) @ V3s ) @ V2g ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__CLAUSES__NUMSEG,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V1op: $i] :
( ( mem @ V1op @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27a ) @ V1op ) )
=> ( ! [V2m: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ ty_2Enum_2Enum @ A_27a ) @ V1op ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0f )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( c_2Eiterate_2Eneutral @ A_27a ) @ V1op ) ) )
& ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ ty_2Enum_2Enum @ A_27a ) @ V1op ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) @ V0f )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) @ ( ap @ ( ap @ V1op @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ ty_2Enum_2Enum @ A_27a ) @ V1op ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ V0f ) ) @ ( ap @ V0f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ ty_2Enum_2Enum @ A_27a ) @ V1op ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ V0f ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EITERATE__PAIR,axiom,
! [A_27a: del,V0op: $i] :
( ( mem @ V0op @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( p @ ( ap @ ( c_2Eiterate_2Emonoidal @ A_27a ) @ V0op ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ ty_2Enum_2Enum @ A_27a ) @ V0op ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ V1f )
= ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Eiterate @ ty_2Enum_2Enum @ A_27a ) @ V0op ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4i: $i] : ( ap @ ( ap @ V0op @ ( ap @ V1f @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ V4i ) ) ) @ ( ap @ V1f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ V4i ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eiterate_2Ensum__def,axiom,
! [A_27a: del] :
( ( c_2Eiterate_2Ensum @ A_27a )
= ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ ty_2Enum_2Enum ) @ c_2Earithmetic_2E_2B ) ) ).
thf(conj_thm_2Eiterate_2ENEUTRAL__ADD,axiom,
( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Eiterate_2Eneutral @ ty_2Enum_2Enum ) @ c_2Earithmetic_2E_2B ) )
= fo__c_2Enum_2E0 ) ).
thf(conj_thm_2Eiterate_2ENEUTRAL__MUL,axiom,
( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Eiterate_2Eneutral @ ty_2Enum_2Enum ) @ c_2Earithmetic_2E_2A ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EMONOIDAL__ADD,axiom,
p @ ( ap @ ( c_2Eiterate_2Emonoidal @ ty_2Enum_2Enum ) @ c_2Earithmetic_2E_2B ) ).
thf(conj_thm_2Eiterate_2EMONOIDAL__MUL,axiom,
p @ ( ap @ ( c_2Eiterate_2Emonoidal @ ty_2Enum_2Enum ) @ c_2Earithmetic_2E_2A ) ).
thf(conj_thm_2Eiterate_2ENSUM__DEGENERATE,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( ap @ V0f @ V2x ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V1s ) @ V0f ) )
= fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__CLAUSES,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V0f ) )
= fo__c_2Enum_2E0 ) )
& ! [V1x: $i] :
( ( mem @ V1x @ A_27b )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27b @ ty_2Enum_2Enum ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V3s ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27b ) @ V1x ) @ V3s ) ) @ V2f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V1x ) @ V3s ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27b ) @ V3s ) @ V2f ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ V2f @ V1x ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27b ) @ V3s ) @ V2f ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__UNION,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V1s ) @ V2t ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V1s ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2t ) @ V0f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__DIFF,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2t ) @ V1s ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V1s ) @ V2t ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V1s ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2t ) @ V0f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__INCL__EXCL,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1t ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V0s ) @ V2f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V1t ) @ V2f ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t ) ) @ V2f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ V1t ) ) @ V2f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__SUPPORT,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27a @ ty_2Enum_2Enum ) @ c_2Earithmetic_2E_2B ) @ V0f ) @ V1s ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V1s ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__ADD,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
=> ( ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s ) @ V1g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__ADD__GEN,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( ap @ V0f @ V3x ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) )
& ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V2s ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( ap @ V1g @ V4x ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ V0f @ V5x ) ) @ ( ap @ V1g @ V5x ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s ) @ V1g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__EQ__0,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ V0f @ V2x ) )
= fo__c_2Enum_2E0 ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V1s ) @ V0f ) )
= fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__0,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V0s )
@ ( lam @ A_27a
@ ^ [V1n: $i] : ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
= fo__c_2Enum_2E0 ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__LMUL,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1c: tp__ty_2Enum_2Enum,V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V1c ) ) @ ( ap @ V0f @ V3x ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V1c ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__RMUL,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1c: tp__ty_2Enum_2Enum,V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ V0f @ V3x ) ) @ ( inj__ty_2Enum_2Enum @ V1c ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s ) @ V0f ) ) @ ( inj__ty_2Enum_2Enum @ V1c ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__LE,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s ) @ V1g ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__LT,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) ) )
& ? [V4x: $i] :
( ( mem @ V4x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V2s ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ V0f @ V4x ) ) @ ( ap @ V1g @ V4x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s ) @ V1g ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__LT__ALL,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
& ( V2s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s ) @ V1g ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__EQ,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ V0f @ V3x ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ V1g @ V3x ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s ) @ V1g ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__CONST,axiom,
! [A_27a: del,V0c: tp__ty_2Enum_2Enum,V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ( ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V2n: $i] : ( inj__ty_2Enum_2Enum @ V0c ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1s ) ) @ ( inj__ty_2Enum_2Enum @ V0c ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__POS__BOUND,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1b: tp__ty_2Enum_2Enum,V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s ) @ V0f ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ V0f @ V3x ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__EQ__0__IFF,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V1s ) @ V0f ) )
= fo__c_2Enum_2E0 )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ V0f @ V2x ) )
= fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__POS__LT,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ V0f @ V2x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V1s ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__POS__LT__ALL,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ! [V2i: $i] :
( ( mem @ V2i @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2i ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ V1f @ V2i ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V0s ) @ V1f ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__DELETE,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2a ) @ V1s ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ V0f @ V2a ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V1s ) @ V2a ) ) @ V0f ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V1s ) @ V0f ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__SING,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ V0f @ V1x ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__DELTA,axiom,
! [A_27a: del,V0b: tp__ty_2Enum_2Enum,V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V3x ) @ V2a ) ) @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2a ) @ V1s ) ) @ ( inj__ty_2Enum_2Enum @ V0b ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__SWAP,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2Enum_2Enum ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2t ) ) )
=> ( ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V3i: $i] : ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27b ) @ V2t ) @ ( ap @ V0f @ V3i ) ) ) ) )
= ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27b ) @ V2t )
@ ( lam @ A_27b
@ ^ [V4j: $i] :
( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V5i: $i] : ( ap @ ( ap @ V0f @ V5i ) @ V4j ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ ty_2Enum_2Enum ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4y ) @ V2s ) )
& ( ( ap @ V0f @ V3x )
= ( ap @ V0f @ V4y ) ) )
=> ( V3x = V4y ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2s ) ) @ V1g ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ty_2Enum_2Enum @ A_27b ) @ V1g ) @ V0f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__SUPERSET,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1u: $i] :
( ( mem @ V1u @ ( arr @ A_27a @ bool ) )
=> ! [V2v: $i] :
( ( mem @ V2v @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1u ) @ V2v ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2v ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1u ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ V0f @ V3x ) )
= fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2v ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V1u ) @ V0f ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__UNION__RZERO,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1u: $i] :
( ( mem @ V1u @ ( arr @ A_27a @ bool ) )
=> ! [V2v: $i] :
( ( mem @ V2v @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1u ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2v ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1u ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ V0f @ V3x ) )
= fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1u ) @ V2v ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V1u ) @ V0f ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__UNION__LZERO,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1u: $i] :
( ( mem @ V1u @ ( arr @ A_27a @ bool ) )
=> ! [V2v: $i] :
( ( mem @ V2v @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2v ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1u ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2v ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ V0f @ V3x ) )
= fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1u ) @ V2v ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2v ) @ V0f ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__RESTRICT,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ( ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) ) @ ( ap @ V0f @ V2x ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V1s ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__BOUND,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ V1f @ V3x ) ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V0s ) @ V1f ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__BOUND__GEN,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ V1f @ V3x ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( inj__ty_2Enum_2Enum @ V2b ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V0s ) @ V1f ) ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__BOUND__LT,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ V1f @ V3x ) ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) ) ) )
& ? [V4x: $i] :
( ( mem @ V4x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ V1f @ V4x ) ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V0s ) @ V1f ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__BOUND__LT__ALL,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ V1f @ V3x ) ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V0s ) @ V1f ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__BOUND__LT__GEN,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2b: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ V1f @ V3x ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( inj__ty_2Enum_2Enum @ V2b ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V0s ) @ V1f ) ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__UNION__EQ,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V3u ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1s ) @ V2t )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t )
= V3u ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V1s ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2t ) @ V0f ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V3u ) @ V0f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__EQ__SUPERSET,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V3t ) @ V2s ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V3t ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ V1f @ V4x ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ V0g @ V4x ) ) ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V2s ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V3t ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ V1f @ V5x ) )
= fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s ) @ V1f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V3t ) @ V0g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__RESTRICT__SET,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( surj__ty_2Enum_2Enum
@ ( ap
@ ( ap @ ( c_2Eiterate_2Ensum @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) ) @ ( ap @ V0P @ V3x ) ) ) ) ) )
@ V2f ) )
= ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ V0P @ V4x ) ) @ ( ap @ V2f @ V4x ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__NSUM__RESTRICT,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2Enum_2Enum ) ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V3t ) ) )
=> ( ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s )
@ ( lam @ A_27a
@ ^ [V4x: $i] :
( ap
@ ( ap @ ( c_2Eiterate_2Ensum @ A_27b )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b )
@ ( lam @ A_27b
@ ^ [V5y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ V5y ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V5y ) @ V3t ) ) @ ( ap @ ( ap @ V0R @ V4x ) @ V5y ) ) ) ) ) )
@ ( lam @ A_27b
@ ^ [V6y: $i] : ( ap @ ( ap @ V1f @ V4x ) @ V6y ) ) ) ) ) )
= ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27b ) @ V3t )
@ ( lam @ A_27b
@ ^ [V7y: $i] :
( ap
@ ( ap @ ( c_2Eiterate_2Ensum @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V8x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V8x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V8x ) @ V2s ) ) @ ( ap @ ( ap @ V0R @ V8x ) @ V7y ) ) ) ) ) )
@ ( lam @ A_27a
@ ^ [V9x: $i] : ( ap @ ( ap @ V1f @ V9x ) @ V7y ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ECARD__EQ__NSUM,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) )
= ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V0s )
@ ( lam @ A_27a
@ ^ [V1x: $i] : ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__MULTICOUNT__GEN,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ! [V3k: $i] :
( ( mem @ V3k @ ( arr @ A_27b @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2t ) )
& ! [V4j: $i] :
( ( mem @ V4j @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4j ) @ V2t ) )
=> ( ( surj__ty_2Enum_2Enum
@ ( ap @ ( c_2Epred__set_2ECARD @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V5i: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V5i ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5i ) @ V1s ) ) @ ( ap @ ( ap @ V0R @ V5i ) @ V4j ) ) ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ V3k @ V4j ) ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V6i: $i] :
( ap @ ( c_2Epred__set_2ECARD @ A_27b )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b )
@ ( lam @ A_27b
@ ^ [V7j: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ V7j ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V7j ) @ V2t ) ) @ ( ap @ ( ap @ V0R @ V6i ) @ V7j ) ) ) ) ) ) ) ) )
= ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27b ) @ V2t )
@ ( lam @ A_27b
@ ^ [V8i: $i] : ( ap @ V3k @ V8i ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__MULTICOUNT,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ! [V3k: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2t ) )
& ! [V4j: $i] :
( ( mem @ V4j @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4j ) @ V2t ) )
=> ( ( surj__ty_2Enum_2Enum
@ ( ap @ ( c_2Epred__set_2ECARD @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V5i: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V5i ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5i ) @ V1s ) ) @ ( ap @ ( ap @ V0R @ V5i ) @ V4j ) ) ) ) ) ) )
= V3k ) ) ) )
=> ( ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V6i: $i] :
( ap @ ( c_2Epred__set_2ECARD @ A_27b )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b )
@ ( lam @ A_27b
@ ^ [V7j: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ V7j ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V7j ) @ V2t ) ) @ ( ap @ ( ap @ V0R @ V6i ) @ V7j ) ) ) ) ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V3k ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ V2t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__IMAGE__GEN,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s ) @ V1g ) )
= ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2s ) )
@ ( lam @ A_27b
@ ^ [V3y: $i] :
( ap
@ ( ap @ ( c_2Eiterate_2Ensum @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ ( ap @ V0f @ V4x ) ) @ V3y ) ) ) ) ) )
@ V1g ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__GROUP,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2s ) ) @ V3t ) ) )
=> ( ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27b ) @ V3t )
@ ( lam @ A_27b
@ ^ [V4y: $i] :
( ap
@ ( ap @ ( c_2Eiterate_2Ensum @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V5x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ ( ap @ V0f @ V5x ) ) @ V4y ) ) ) ) ) )
@ V1g ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s ) @ V1g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__SUBSET,axiom,
! [A_27a: del,V0u: $i] :
( ( mem @ V0u @ ( arr @ A_27a @ bool ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ ( arr @ A_27a @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0u ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1v ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0u ) @ V1v ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ V2f @ V3x ) )
= fo__c_2Enum_2E0 ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V0u ) @ V2f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V1v ) @ V2f ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__SUBSET__SIMPLE,axiom,
! [A_27a: del,V0u: $i] :
( ( mem @ V0u @ ( arr @ A_27a @ bool ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ ( arr @ A_27a @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1v ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0u ) @ V1v ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V0u ) @ V2f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V1v ) @ V2f ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__LE__GEN,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) ) )
& ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V2s ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( ap @ V1g @ V4x ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s ) @ V1g ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__IMAGE__NONZERO,axiom,
! [A_27a: del,A_27b: del,V0d: $i] :
( ( mem @ V0d @ ( arr @ A_27b @ ty_2Enum_2Enum ) )
=> ! [V1i: $i] :
( ( mem @ V1i @ ( arr @ A_27a @ A_27b ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4y ) @ V2s ) )
& ( V3x != V4y )
& ( ( ap @ V1i @ V3x )
= ( ap @ V1i @ V4y ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ V0d @ ( ap @ V1i @ V3x ) ) )
= fo__c_2Enum_2E0 ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V1i ) @ V2s ) ) @ V0d ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ty_2Enum_2Enum @ A_27b ) @ V0d ) @ V1i ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__BIJECTION,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( arr @ A_27a @ A_27a ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ V1p @ V3x ) ) @ V2s ) ) ) )
& ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4y ) @ V2s ) )
=> ( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ ( ap @ V1p @ V5x ) ) @ V4y ) ) ) ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ty_2Enum_2Enum @ A_27a ) @ V0f ) @ V1p ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__NSUM__PRODUCT,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2Enum_2Enum ) ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ! [V3i: $i] :
( ( mem @ V3i @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3i ) @ V0s ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ ( ap @ V1t @ V3i ) ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V0s )
@ ( lam @ A_27a
@ ^ [V4i: $i] : ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27b ) @ ( ap @ V1t @ V4i ) ) @ ( ap @ V2x @ V4i ) ) ) ) )
= ( surj__ty_2Enum_2Enum
@ ( ap
@ ( ap @ ( c_2Eiterate_2Ensum @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
@ ( lam @ A_27a
@ ^ [V5i: $i] :
( lam @ A_27b
@ ^ [V6j: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V5i ) @ V6j ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5i ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V6j ) @ ( ap @ V1t @ V5i ) ) ) ) ) ) ) ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ty_2Enum_2Enum )
@ ( lam @ A_27a
@ ^ [V7i: $i] :
( lam @ A_27b
@ ^ [V8j: $i] : ( ap @ ( ap @ V2x @ V7i ) @ V8j ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__EQ__GENERAL,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27b @ ty_2Enum_2Enum ) )
=> ! [V4h: $i] :
( ( mem @ V4h @ ( arr @ A_27a @ A_27b ) )
=> ( ( ! [V5y: $i] :
( ( mem @ V5y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V5y ) @ V1t ) )
=> ( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V6x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6x ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ ( ap @ V4h @ V6x ) ) @ V5y ) ) ) ) ) ) )
& ! [V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V7x ) @ V0s ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V4h @ V7x ) ) @ V1t ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ V3g @ ( ap @ V4h @ V7x ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ V2f @ V7x ) ) ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V0s ) @ V2f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27b ) @ V1t ) @ V3g ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__EQ__GENERAL__INVERSES,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27b @ ty_2Enum_2Enum ) )
=> ! [V4h: $i] :
( ( mem @ V4h @ ( arr @ A_27a @ A_27b ) )
=> ! [V5k: $i] :
( ( mem @ V5k @ ( arr @ A_27b @ A_27a ) )
=> ( ( ! [V6y: $i] :
( ( mem @ V6y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V6y ) @ V1t ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ V5k @ V6y ) ) @ V0s ) )
& ( ( ap @ V4h @ ( ap @ V5k @ V6y ) )
= V6y ) ) ) )
& ! [V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V7x ) @ V0s ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V4h @ V7x ) ) @ V1t ) )
& ( ( ap @ V5k @ ( ap @ V4h @ V7x ) )
= V7x )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ V3g @ ( ap @ V4h @ V7x ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ V2f @ V7x ) ) ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V0s ) @ V2f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27b ) @ V1t ) @ V3g ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__INJECTION,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( arr @ A_27a @ A_27a ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ V1p @ V3x ) ) @ V2s ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5y ) @ V2s ) )
& ( ( ap @ V1p @ V4x )
= ( ap @ V1p @ V5y ) ) )
=> ( V4x = V5y ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ty_2Enum_2Enum @ A_27a ) @ V0f ) @ V1p ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s ) @ V0f ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__UNION__NONZERO,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2t ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1s ) @ V2t ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ V0f @ V3x ) )
= fo__c_2Enum_2E0 ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V1s ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2t ) @ V0f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__BIGUNION__NONZERO,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ A_27a @ bool ) ) @ V1s ) )
& ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2t ) @ V1s ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2t ) ) ) )
& ! [V3t1: $i] :
( ( mem @ V3t1 @ ( arr @ A_27a @ bool ) )
=> ! [V4t2: $i] :
( ( mem @ V4t2 @ ( arr @ A_27a @ bool ) )
=> ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3t1 ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V4t2 ) @ V1s ) )
& ( V3t1 != V4t2 )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V3t1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V4t2 ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ V0f @ V5x ) )
= fo__c_2Enum_2E0 ) ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V1s ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ( arr @ A_27a @ bool ) ) @ V1s )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V6t: $i] : ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V6t ) @ V0f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__CASES,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V0s )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ V1P @ V4x ) ) @ ( ap @ V2f @ V4x ) ) @ ( ap @ V3g @ V4x ) ) ) ) )
= ( surj__ty_2Enum_2Enum
@ ( ap
@ ( ap @ c_2Earithmetic_2E_2B
@ ( ap
@ ( ap @ ( c_2Eiterate_2Ensum @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V5x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V0s ) ) @ ( ap @ V1P @ V5x ) ) ) ) ) )
@ V2f ) )
@ ( ap
@ ( ap @ ( c_2Eiterate_2Ensum @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V6x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V6x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6x ) @ V0s ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ V1P @ V6x ) ) ) ) ) ) )
@ V3g ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__CLOSED,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
& ! [V3x: tp__ty_2Enum_2Enum,V4y: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V3x ) ) )
& ( p @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V4y ) ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V3x ) ) @ ( inj__ty_2Enum_2Enum @ V4y ) ) ) ) )
& ! [V5a: $i] :
( ( mem @ V5a @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5a ) @ V2s ) )
=> ( p @ ( ap @ V0P @ ( ap @ V1f @ V5a ) ) ) ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s ) @ V1f ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__ADD__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4i: $i] : ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ V0f @ V4i ) ) @ ( ap @ V1g @ V4i ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V1g ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__LE__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ! [V4i: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V4i ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V4i ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V4i ) ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V4i ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V1g ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__EQ__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ! [V4i: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V4i ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V4i ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V4i ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V4i ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V1g ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__CONST__NUMSEG,axiom,
! [V0c: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( inj__ty_2Enum_2Enum @ V0c ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) @ ( inj__ty_2Enum_2Enum @ V0c ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__EQ__0__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ! [V3i: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V3i ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3i ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3i ) ) )
= fo__c_2Enum_2E0 ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) )
= fo__c_2Enum_2E0 ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__EQ__0__IFF__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) )
= fo__c_2Enum_2E0 )
<=> ! [V3i: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V3i ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3i ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3i ) ) )
= fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__TRIV__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) )
= fo__c_2Enum_2E0 ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__SING__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__CLAUSES__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ( ! [V1m: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V0f ) ) @ ( ap @ V0f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__SWAP__NUMSEG,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum,V2c: tp__ty_2Enum_2Enum,V3d: tp__ty_2Enum_2Enum,V4f: $i] :
( ( mem @ V4f @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) )
=> ( ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5i: $i] : ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2c ) ) @ ( inj__ty_2Enum_2Enum @ V3d ) ) ) @ ( ap @ V4f @ V5i ) ) ) ) )
= ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2c ) ) @ ( inj__ty_2Enum_2Enum @ V3d ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V6j: $i] :
( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7i: $i] : ( ap @ ( ap @ V4f @ V7i ) @ V6j ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__ADD__SPLIT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum,V3p: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V3p ) ) ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V3p ) ) ) ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__OFFSET,axiom,
! [V0p: tp__ty_2Enum_2Enum,V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V0p ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( inj__ty_2Enum_2Enum @ V0p ) ) ) ) @ V1f ) )
= ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4i: $i] : ( ap @ V1f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V4i ) @ ( inj__ty_2Enum_2Enum @ V0p ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__OFFSET__0,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3i: $i] : ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V3i ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__CLAUSES__LEFT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__CLAUSES__RIGHT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ V0f ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENSUM__PAIR,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ V0f ) )
= ( surj__ty_2Enum_2Enum
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3i: $i] : ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ V3i ) ) ) @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ V3i ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EMOD__NSUM__MOD,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1n: tp__ty_2Enum_2Enum,V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
& ( V1n != fo__c_2Enum_2E0 ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EMOD @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s ) @ V0f ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
= ( surj__ty_2Enum_2Enum
@ ( ap
@ ( ap @ c_2Earithmetic_2EMOD
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V2s )
@ ( lam @ A_27a
@ ^ [V3i: $i] : ( ap @ ( ap @ c_2Earithmetic_2EMOD @ ( ap @ V0f @ V3i ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) )
@ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EMOD__NSUM__MOD__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1a: tp__ty_2Enum_2Enum,V2b: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( V3n != fo__c_2Enum_2E0 )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EMOD @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) ) @ V0f ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
= ( surj__ty_2Enum_2Enum
@ ( ap
@ ( ap @ c_2Earithmetic_2EMOD
@ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4i: $i] : ( ap @ ( ap @ c_2Earithmetic_2EMOD @ ( ap @ V0f @ V4i ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) )
@ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ECARD__BIGUNION,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ A_27a @ bool ) ) @ V0s ) )
& ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1t ) @ V0s ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1t ) ) ) )
& ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2t ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3u ) @ V0s ) )
& ( V2t != V3u ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V2t ) @ V3u )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V0s ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ( arr @ A_27a @ bool ) ) @ V0s ) @ ( c_2Epred__set_2ECARD @ A_27a ) ) ) ) ) ) ).
thf(ax_thm_2Eiterate_2Esum__def,axiom,
! [A_27a: del] :
( ( c_2Eiterate_2ESum @ A_27a )
= ( ap @ ( c_2Eiterate_2Eiterate @ A_27a @ ty_2Erealax_2Ereal ) @ c_2Erealax_2Ereal__add ) ) ).
thf(conj_thm_2Eiterate_2ENEUTRAL__REAL__ADD,axiom,
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( c_2Eiterate_2Eneutral @ ty_2Erealax_2Ereal ) @ c_2Erealax_2Ereal__add ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Eiterate_2ENEUTRAL__REAL__MUL,axiom,
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( c_2Eiterate_2Eneutral @ ty_2Erealax_2Ereal ) @ c_2Erealax_2Ereal__mul ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EMONOIDAL__REAL__ADD,axiom,
p @ ( ap @ ( c_2Eiterate_2Emonoidal @ ty_2Erealax_2Ereal ) @ c_2Erealax_2Ereal__add ) ).
thf(conj_thm_2Eiterate_2EMONOIDAL__REAL__MUL,axiom,
p @ ( ap @ ( c_2Eiterate_2Emonoidal @ ty_2Erealax_2Ereal ) @ c_2Erealax_2Ereal__mul ) ).
thf(conj_thm_2Eiterate_2ESUM__DEGENERATE,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V2x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__CLAUSES,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
& ! [V1x: $i] :
( ( mem @ V1x @ A_27b )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V3s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27b ) @ V1x ) @ V3s ) ) @ V2f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V1x ) @ V3s ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27b ) @ V3s ) @ V2f ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V2f @ V1x ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27b ) @ V3s ) @ V2f ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__UNION,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V1s ) @ V2t ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2t ) @ V0f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__DIFF,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2t ) @ V1s ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V1s ) @ V2t ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2t ) @ V0f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__INCL__EXCL,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1t ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V0s ) @ V2f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1t ) @ V2f ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t ) ) @ V2f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ V1t ) ) @ V2f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__SUPPORT,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Eiterate_2Esupport @ A_27a @ ty_2Erealax_2Ereal ) @ c_2Erealax_2Ereal__add ) @ V0f ) @ V1s ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__ADD,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ V1g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__ADD__GEN,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V0f @ V3x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ) )
& ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V2s ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V1g @ V4x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0f @ V5x ) ) @ ( ap @ V1g @ V5x ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ V1g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__EQ__0,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ V2x ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__0,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V0s )
@ ( lam @ A_27a
@ ^ [V1n: $i] : ( 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 ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__LMUL,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1c: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( ap @ V0f @ V3x ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__RMUL,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1c: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ V0f ) ) @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__NEG,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ V0f @ V2x ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__SUB,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ V1g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__LE,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ V1g ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__LT,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) ) )
& ? [V4x: $i] :
( ( mem @ V4x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V2s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ V0f @ V4x ) ) @ ( ap @ V1g @ V4x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ V1g ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__LT__ALL,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
& ( V2s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ V1g ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__POS__LT,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0f @ V2x ) ) ) ) )
& ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0f @ V3x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__POS__LT__ALL,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ! [V2i: $i] :
( ( mem @ V2i @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2i ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V2i ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V0s ) @ V1f ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__EQ,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ V3x ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ V3x ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ V1g ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__ABS,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ( p
@ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s ) @ V0f ) ) )
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ V2x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__ABS__LE,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ V3x ) ) ) @ ( ap @ V1g @ V3x ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ V0f ) ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ V1g ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__CONST,axiom,
! [A_27a: del,V0c: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V2n: $i] : ( inj__ty_2Erealax_2Ereal @ V0c ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1s ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__POS__LE,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0f @ V2x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__POS__BOUND,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1b: tp__ty_2Erealax_2Ereal,V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0f @ V3x ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ V0f ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__POS__EQ__0,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0f @ V2x ) ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ V3x ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__ZERO__EXISTS,axiom,
! [A_27a: del,V0u: $i] :
( ( mem @ V0u @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s ) @ V0u ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( ! [V2i: $i] :
( ( mem @ V2i @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2i ) @ V1s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0u @ V2i ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
| ? [V3j: $i] :
( ( mem @ V3j @ A_27a )
& ? [V4k: $i] :
( ( mem @ V4k @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3j ) @ V1s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ V0u @ V3j ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4k ) @ V1s ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__gt @ ( ap @ V0u @ V4k ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__DELETE,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2a ) @ V1s ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V1s ) @ V2a ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s ) @ V0f ) ) @ ( ap @ V0f @ V2a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__DELETE__CASES,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V1s ) @ V2a ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2a ) @ V1s ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s ) @ V0f ) ) @ ( ap @ V0f @ V2a ) ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s ) @ V0f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__SING,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ V1x ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__DELTA,axiom,
! [A_27a: del,V0b: tp__ty_2Erealax_2Ereal,V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V3x ) @ V2a ) ) @ ( inj__ty_2Erealax_2Ereal @ V0b ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2a ) @ V1s ) ) @ ( inj__ty_2Erealax_2Ereal @ V0b ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__SWAP,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2t ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V3i: $i] : ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27b ) @ V2t ) @ ( ap @ V0f @ V3i ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27b ) @ V2t )
@ ( lam @ A_27b
@ ^ [V4j: $i] :
( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V5i: $i] : ( ap @ ( ap @ V0f @ V5i ) @ V4j ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4y ) @ V2s ) )
& ( ( ap @ V0f @ V3x )
= ( ap @ V0f @ V4y ) ) )
=> ( V3x = V4y ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2s ) ) @ V1g ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ty_2Erealax_2Ereal @ A_27b ) @ V1g ) @ V0f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__SUPERSET,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1u: $i] :
( ( mem @ V1u @ ( arr @ A_27a @ bool ) )
=> ! [V2v: $i] :
( ( mem @ V2v @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1u ) @ V2v ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2v ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1u ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ V3x ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2v ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1u ) @ V0f ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__UNION__RZERO,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1u: $i] :
( ( mem @ V1u @ ( arr @ A_27a @ bool ) )
=> ! [V2v: $i] :
( ( mem @ V2v @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1u ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2v ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1u ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ V3x ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1u ) @ V2v ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1u ) @ V0f ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__UNION__LZERO,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1u: $i] :
( ( mem @ V1u @ ( arr @ A_27a @ bool ) )
=> ! [V2v: $i] :
( ( mem @ V2v @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2v ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1u ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2v ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ V3x ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1u ) @ V2v ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2v ) @ V0f ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__RESTRICT,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) ) @ ( ap @ V0f @ V2x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__BOUND,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V1f @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V0s ) @ V1f ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__BOUND__GEN,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V1f @ V3x ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V0s ) @ V1f ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__ABS__BOUND,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ V1f @ V3x ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V0s ) @ V1f ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__BOUND__LT,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V1f @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) )
& ? [V4x: $i] :
( ( mem @ V4x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V0s ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ V1f @ V4x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V0s ) @ V1f ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__BOUND__LT__ALL,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ V1f @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V0s ) @ V1f ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__BOUND__LT__GEN,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2b: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ V1f @ V3x ) ) @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V0s ) @ V1f ) ) @ ( inj__ty_2Erealax_2Ereal @ V2b ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__UNION__EQ,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ! [V3u: $i] :
( ( mem @ V3u @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V3u ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1s ) @ V2t )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t )
= V3u ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2t ) @ V0f ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V3u ) @ V0f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__EQ__SUPERSET,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V3t ) @ V2s ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V3t ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ V4x ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0g @ V4x ) ) ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V2s ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V3t ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ V5x ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ V1f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V3t ) @ V0g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__RESTRICT__SET,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ ( c_2Eiterate_2ESum @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V3x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) ) @ ( ap @ V0P @ V3x ) ) ) ) ) )
@ V2f ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ V0P @ V4x ) ) @ ( ap @ V2f @ V4x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__SUM__RESTRICT,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V3t ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s )
@ ( lam @ A_27a
@ ^ [V4x: $i] :
( ap
@ ( ap @ ( c_2Eiterate_2ESum @ A_27b )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b )
@ ( lam @ A_27b
@ ^ [V5y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ V5y ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V5y ) @ V3t ) ) @ ( ap @ ( ap @ V0R @ V4x ) @ V5y ) ) ) ) ) )
@ ( lam @ A_27b
@ ^ [V6y: $i] : ( ap @ ( ap @ V1f @ V4x ) @ V6y ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27b ) @ V3t )
@ ( lam @ A_27b
@ ^ [V7y: $i] :
( ap
@ ( ap @ ( c_2Eiterate_2ESum @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V8x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V8x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V8x ) @ V2s ) ) @ ( ap @ ( ap @ V0R @ V8x ) @ V7y ) ) ) ) ) )
@ ( lam @ A_27a
@ ^ [V9x: $i] : ( ap @ ( ap @ V1f @ V9x ) @ V7y ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ECARD__EQ__SUM,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V0s )
@ ( lam @ A_27a
@ ^ [V1x: $i] : ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__MULTICOUNT__GEN,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ! [V3k: $i] :
( ( mem @ V3k @ ( arr @ A_27b @ ty_2Enum_2Enum ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2t ) )
& ! [V4j: $i] :
( ( mem @ V4j @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4j ) @ V2t ) )
=> ( ( surj__ty_2Enum_2Enum
@ ( ap @ ( c_2Epred__set_2ECARD @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V5i: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V5i ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5i ) @ V1s ) ) @ ( ap @ ( ap @ V0R @ V5i ) @ V4j ) ) ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ V3k @ V4j ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V6i: $i] :
( ap @ c_2Ereal_2Ereal__of__num
@ ( ap @ ( c_2Epred__set_2ECARD @ A_27b )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b )
@ ( lam @ A_27b
@ ^ [V7j: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ V7j ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V7j ) @ V2t ) ) @ ( ap @ ( ap @ V0R @ V6i ) @ V7j ) ) ) ) ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27b ) @ V2t )
@ ( lam @ A_27b
@ ^ [V8i: $i] : ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ V3k @ V8i ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__MULTICOUNT,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ! [V3k: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2t ) )
& ! [V4j: $i] :
( ( mem @ V4j @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4j ) @ V2t ) )
=> ( ( surj__ty_2Enum_2Enum
@ ( ap @ ( c_2Epred__set_2ECARD @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V5i: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V5i ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5i ) @ V1s ) ) @ ( ap @ ( ap @ V0R @ V5i ) @ V4j ) ) ) ) ) ) )
= V3k ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V6i: $i] :
( ap @ c_2Ereal_2Ereal__of__num
@ ( ap @ ( c_2Epred__set_2ECARD @ A_27b )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b )
@ ( lam @ A_27b
@ ^ [V7j: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ V7j ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V7j ) @ V2t ) ) @ ( ap @ ( ap @ V0R @ V6i ) @ V7j ) ) ) ) ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V3k ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ V2t ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__IMAGE__GEN,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ V1g ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2s ) )
@ ( lam @ A_27b
@ ^ [V3y: $i] :
( ap
@ ( ap @ ( c_2Eiterate_2ESum @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V4x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ ( ap @ V0f @ V4x ) ) @ V3y ) ) ) ) ) )
@ V1g ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__GROUP,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2s ) ) @ V3t ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27b ) @ V3t )
@ ( lam @ A_27b
@ ^ [V4y: $i] :
( ap
@ ( ap @ ( c_2Eiterate_2ESum @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V5x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ ( ap @ V0f @ V5x ) ) @ V4y ) ) ) ) ) )
@ V1g ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ V1g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__OF__NUM__SUM,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ A_27a ) @ V1s ) @ V0f ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ V0f @ V2x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__SUBSET,axiom,
! [A_27a: del,V0u: $i] :
( ( mem @ V0u @ ( arr @ A_27a @ bool ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ ( arr @ A_27a @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0u ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1v ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0u ) @ V1v ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V2f @ V3x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V1v ) @ V0u ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V2f @ V4x ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V0u ) @ V2f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1v ) @ V2f ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__SUBSET__SIMPLE,axiom,
! [A_27a: del,V0u: $i] :
( ( mem @ V0u @ ( arr @ A_27a @ bool ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ ( arr @ A_27a @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1v ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0u ) @ V1v ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V1v ) @ V0u ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V2f @ V3x ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V0u ) @ V2f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1v ) @ V2f ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__IMAGE__NONZERO,axiom,
! [A_27a: del,A_27b: del,V0d: $i] :
( ( mem @ V0d @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) )
=> ! [V1i: $i] :
( ( mem @ V1i @ ( arr @ A_27a @ A_27b ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4y ) @ V2s ) )
& ( V3x != V4y )
& ( ( ap @ V1i @ V3x )
= ( ap @ V1i @ V4y ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0d @ ( ap @ V1i @ V3x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V1i ) @ V2s ) ) @ V0d ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ty_2Erealax_2Ereal @ A_27b ) @ V0d ) @ V1i ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__BIJECTION,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( arr @ A_27a @ A_27a ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ V1p @ V3x ) ) @ V2s ) ) ) )
& ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4y ) @ V2s ) )
=> ( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ ( ap @ V1p @ V5x ) ) @ V4y ) ) ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ty_2Erealax_2Ereal @ A_27a ) @ V0f ) @ V1p ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__SUM__PRODUCT,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ! [V3i: $i] :
( ( mem @ V3i @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3i ) @ V0s ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ ( ap @ V1t @ V3i ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V0s )
@ ( lam @ A_27a
@ ^ [V4i: $i] : ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27b ) @ ( ap @ V1t @ V4i ) ) @ ( ap @ V2x @ V4i ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ ( c_2Eiterate_2ESum @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
@ ( lam @ A_27a
@ ^ [V5i: $i] :
( lam @ A_27b
@ ^ [V6j: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V5i ) @ V6j ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5i ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V6j ) @ ( ap @ V1t @ V5i ) ) ) ) ) ) ) ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ty_2Erealax_2Ereal )
@ ( lam @ A_27a
@ ^ [V7i: $i] :
( lam @ A_27b
@ ^ [V8j: $i] : ( ap @ ( ap @ V2x @ V7i ) @ V8j ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__EQ__GENERAL,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) )
=> ! [V4h: $i] :
( ( mem @ V4h @ ( arr @ A_27a @ A_27b ) )
=> ( ( ! [V5y: $i] :
( ( mem @ V5y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V5y ) @ V1t ) )
=> ( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V6x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6x ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ ( ap @ V4h @ V6x ) ) @ V5y ) ) ) ) ) ) )
& ! [V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V7x ) @ V0s ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V4h @ V7x ) ) @ V1t ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V3g @ ( ap @ V4h @ V7x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V2f @ V7x ) ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V0s ) @ V2f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27b ) @ V1t ) @ V3g ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__EQ__GENERAL__INVERSES,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) )
=> ! [V4h: $i] :
( ( mem @ V4h @ ( arr @ A_27a @ A_27b ) )
=> ! [V5k: $i] :
( ( mem @ V5k @ ( arr @ A_27b @ A_27a ) )
=> ( ( ! [V6y: $i] :
( ( mem @ V6y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V6y ) @ V1t ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ V5k @ V6y ) ) @ V0s ) )
& ( ( ap @ V4h @ ( ap @ V5k @ V6y ) )
= V6y ) ) ) )
& ! [V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V7x ) @ V0s ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V4h @ V7x ) ) @ V1t ) )
& ( ( ap @ V5k @ ( ap @ V4h @ V7x ) )
= V7x )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V3g @ ( ap @ V4h @ V7x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V2f @ V7x ) ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V0s ) @ V2f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27b ) @ V1t ) @ V3g ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__INJECTION,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( arr @ A_27a @ A_27a ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ V1p @ V3x ) ) @ V2s ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5y ) @ V2s ) )
& ( ( ap @ V1p @ V4x )
= ( ap @ V1p @ V5y ) ) )
=> ( V4x = V5y ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ty_2Erealax_2Ereal @ A_27a ) @ V0f ) @ V1p ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ V0f ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__UNION__NONZERO,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2t ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1s ) @ V2t ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ V3x ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V1s ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2t ) @ V0f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__BIGUNION__NONZERO,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ A_27a @ bool ) ) @ V1s ) )
& ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2t ) @ V1s ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2t ) ) ) )
& ! [V3t1: $i] :
( ( mem @ V3t1 @ ( arr @ A_27a @ bool ) )
=> ! [V4t2: $i] :
( ( mem @ V4t2 @ ( arr @ A_27a @ bool ) )
=> ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3t1 ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V4t2 ) @ V1s ) )
& ( V3t1 != V4t2 )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V3t1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V4t2 ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ V5x ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V1s ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ( arr @ A_27a @ bool ) ) @ V1s )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V6t: $i] : ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V6t ) @ V0f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__CASES,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V0s )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ V1P @ V4x ) ) @ ( ap @ V2f @ V4x ) ) @ ( ap @ V3g @ V4x ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ c_2Erealax_2Ereal__add
@ ( ap
@ ( ap @ ( c_2Eiterate_2ESum @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V5x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V0s ) ) @ ( ap @ V1P @ V5x ) ) ) ) ) )
@ V2f ) )
@ ( ap
@ ( ap @ ( c_2Eiterate_2ESum @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V6x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V6x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6x ) @ V0s ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ V1P @ V6x ) ) ) ) ) ) )
@ V3g ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__CASES__1,axiom,
! [A_27a: del,V0y: tp__ty_2Erealax_2Ereal,V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ A_27a )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3a ) @ V2s ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V4x ) @ V3a ) ) @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) @ ( ap @ V1f @ V4x ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ V1f ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0y ) ) @ ( ap @ V1f @ V3a ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__LE__INCLUDED,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27b @ bool ) )
=> ! [V4i: $i] :
( ( mem @ V4i @ ( arr @ A_27b @ A_27a ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V3t ) )
& ! [V5y: $i] :
( ( mem @ V5y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V5y ) @ V3t ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1g @ V5y ) ) ) ) )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6x ) @ V2s ) )
=> ? [V7y: $i] :
( ( mem @ V7y @ A_27b )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V7y ) @ V3t ) )
& ( ( ap @ V4i @ V7y )
= V6x )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ V6x ) ) @ ( ap @ V1g @ V7y ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27b ) @ V3t ) @ V1g ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__IMAGE__LE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1g @ ( ap @ V0f @ V3x ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2s ) ) @ V1g ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ty_2Erealax_2Ereal @ A_27b ) @ V1g ) @ V0f ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__CLOSED,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Erealax_2Ereal @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ! [V3x: tp__ty_2Erealax_2Ereal,V4y: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
& ( p @ ( ap @ V0P @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V4y ) ) ) ) )
& ! [V5a: $i] :
( ( mem @ V5a @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5a ) @ V2s ) )
=> ( p @ ( ap @ V0P @ ( ap @ V1f @ V5a ) ) ) ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ V1f ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__ADD__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4i: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0f @ V4i ) ) @ ( ap @ V1g @ V4i ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V1g ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__SUB__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4i: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ V4i ) ) @ ( ap @ V1g @ V4i ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V1g ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__LE__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ! [V4i: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V4i ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V4i ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V4i ) ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V4i ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V1g ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__EQ__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ! [V4i: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V4i ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V4i ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V4i ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V4i ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V1g ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__ABS__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( p
@ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) ) )
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3i: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ V0f @ V3i ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__CONST__NUMSEG,axiom,
! [V0c: tp__ty_2Erealax_2Ereal,V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( inj__ty_2Erealax_2Ereal @ V0c ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0c ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__EQ__0__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ! [V3i: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V3i ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3i ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3i ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__TRIV__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__POS__LE__NUMSEG,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum,V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ! [V3p: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V3p ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3p ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ V3p ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ V2f ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__POS__EQ__0__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( ! [V3p: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V3p ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3p ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3p ) ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ! [V4p: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V4p ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V4p ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V4p ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__SING__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__CLAUSES__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ! [V1m: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
& ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V0f ) ) @ ( ap @ V0f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__SWAP__NUMSEG,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum,V2c: tp__ty_2Enum_2Enum,V3d: tp__ty_2Enum_2Enum,V4f: $i] :
( ( mem @ V4f @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5i: $i] : ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2c ) ) @ ( inj__ty_2Enum_2Enum @ V3d ) ) ) @ ( ap @ V4f @ V5i ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2c ) ) @ ( inj__ty_2Enum_2Enum @ V3d ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V6j: $i] :
( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7i: $i] : ( ap @ ( ap @ V4f @ V7i ) @ V6j ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__ADD__SPLIT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum,V3p: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V3p ) ) ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V3p ) ) ) ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__OFFSET,axiom,
! [V0p: tp__ty_2Enum_2Enum,V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V0p ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( inj__ty_2Enum_2Enum @ V0p ) ) ) ) @ V1f ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4i: $i] : ( ap @ V1f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V4i ) @ ( inj__ty_2Enum_2Enum @ V0p ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__OFFSET__0,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3i: $i] : ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V3i ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__CLAUSES__LEFT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__CLAUSES__RIGHT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ V0f ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__PAIR,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ V0f ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3i: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ V3i ) ) ) @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ V3i ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__OF__NUM__SUM__NUMSEG,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ ( ap @ ( c_2Eiterate_2Ensum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3i: $i] : ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ V0f @ V3i ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__PARTIAL__SUC,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4k: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ V4k ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V1g @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V4k ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ V1g @ V4k ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
@ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ V1g @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5k: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V1g @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V5k ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V5k ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ V0f @ V5k ) ) ) ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__PARTIAL__PRE,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4k: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ V4k ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V1g @ V4k ) ) @ ( ap @ V1g @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V4k ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
@ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) @ ( ap @ V1g @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5k: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V1g @ V5k ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V5k ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ V0f @ V5k ) ) ) ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__DIFFS,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3k: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ V3k ) ) @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V3k ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__DIFFS__ALT,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3k: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V3k ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ V0f @ V3k ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__COMBINE__R,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum,V3p: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V3p ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V3p ) ) ) @ V0f ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V3p ) ) ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ESUM__COMBINE__L,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum,V3p: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V3p ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V3p ) ) ) @ V0f ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V3p ) ) ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2ETH,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ V3x ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V1g @ V3x ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s )
@ ( lam @ A_27a
@ ^ [V4i: $i] : ( ap @ V0f @ V4i ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V2s ) @ V1g ) ) ) ) ) ) )
& ! [V5f: $i] :
( ( mem @ V5f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V6g: $i] :
( ( mem @ V6g @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V7a: tp__ty_2Enum_2Enum,V8b: tp__ty_2Enum_2Enum] :
( ! [V9i: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V7a ) ) @ ( inj__ty_2Enum_2Enum @ V9i ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V9i ) ) @ ( inj__ty_2Enum_2Enum @ V8b ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V5f @ ( inj__ty_2Enum_2Enum @ V9i ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V6g @ ( inj__ty_2Enum_2Enum @ V9i ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V7a ) ) @ ( inj__ty_2Enum_2Enum @ V8b ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V10i: $i] : ( ap @ V5f @ V10i ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ V7a ) ) @ ( inj__ty_2Enum_2Enum @ V8b ) ) ) @ V6g ) ) ) ) ) )
& ! [V11f: $i] :
( ( mem @ V11f @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) )
=> ! [V12g: $i] :
( ( mem @ V12g @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) )
=> ! [V13p: $i] :
( ( mem @ V13p @ ( arr @ A_27b @ bool ) )
=> ( ! [V14x: $i] :
( ( mem @ V14x @ A_27b )
=> ( ( p @ ( ap @ V13p @ V14x ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V11f @ V14x ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ V12g @ V14x ) ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ ( c_2Eiterate_2ESum @ A_27b )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b )
@ ( lam @ A_27b
@ ^ [V15y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ V15y ) @ ( ap @ V13p @ V15y ) ) ) ) )
@ ( lam @ A_27b
@ ^ [V16i: $i] : ( ap @ V11f @ V16i ) ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ ( c_2Eiterate_2ESum @ A_27b )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b )
@ ( lam @ A_27b
@ ^ [V17y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ V17y ) @ ( ap @ V13p @ V17y ) ) ) ) )
@ V12g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__SUB__POW,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) )
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3i: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V3i ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V3i ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__SUB__POW__R1,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2i: $i] : ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V2i ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__SUB__POW__L1,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2i: $i] : ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ V2i ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EFORALL__IN__GSPEC,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del,A_27g: del,V0Q: $i] :
( ( mem @ V0Q @ ( arr @ A_27b @ bool ) )
=> ( ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ( ! [V3z: $i] :
( ( mem @ V3z @ A_27b )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3z )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ ( ap @ V2f @ V4x ) ) @ ( ap @ V1P @ V4x ) ) ) ) ) )
=> ( p @ ( ap @ V0Q @ V3z ) ) ) )
<=> ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( p @ ( ap @ V1P @ V5x ) )
=> ( p @ ( ap @ V0Q @ ( ap @ V2f @ V5x ) ) ) ) ) ) ) )
& ! [V6P: $i] :
( ( mem @ V6P @ ( arr @ A_27c @ ( arr @ A_27d @ bool ) ) )
=> ! [V7f: $i] :
( ( mem @ V7f @ ( arr @ A_27c @ ( arr @ A_27d @ A_27b ) ) )
=> ( ! [V8z: $i] :
( ( mem @ V8z @ A_27b )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V8z )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ ( ty_2Epair_2Eprod @ A_27c @ A_27d ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27c @ A_27d @ ( ty_2Epair_2Eprod @ A_27b @ bool ) )
@ ( lam @ A_27c
@ ^ [V9x: $i] :
( lam @ A_27d
@ ^ [V10y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ ( ap @ ( ap @ V7f @ V9x ) @ V10y ) ) @ ( ap @ ( ap @ V6P @ V9x ) @ V10y ) ) ) ) ) ) ) )
=> ( p @ ( ap @ V0Q @ V8z ) ) ) )
<=> ! [V11x: $i] :
( ( mem @ V11x @ A_27c )
=> ! [V12y: $i] :
( ( mem @ V12y @ A_27d )
=> ( ( p @ ( ap @ ( ap @ V6P @ V11x ) @ V12y ) )
=> ( p @ ( ap @ V0Q @ ( ap @ ( ap @ V7f @ V11x ) @ V12y ) ) ) ) ) ) ) ) )
& ! [V13P: $i] :
( ( mem @ V13P @ ( arr @ A_27e @ ( arr @ A_27f @ ( arr @ A_27g @ bool ) ) ) )
=> ! [V14f: $i] :
( ( mem @ V14f @ ( arr @ A_27e @ ( arr @ A_27f @ ( arr @ A_27g @ A_27b ) ) ) )
=> ( ! [V15z: $i] :
( ( mem @ V15z @ A_27b )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V15z )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ ( ty_2Epair_2Eprod @ A_27e @ ( ty_2Epair_2Eprod @ A_27f @ A_27g ) ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27e @ ( ty_2Epair_2Eprod @ A_27f @ A_27g ) @ ( ty_2Epair_2Eprod @ A_27b @ bool ) )
@ ( lam @ A_27e
@ ^ [V16w: $i] :
( ap @ ( c_2Epair_2EUNCURRY @ A_27f @ A_27g @ ( ty_2Epair_2Eprod @ A_27b @ bool ) )
@ ( lam @ A_27f
@ ^ [V17x: $i] :
( lam @ A_27g
@ ^ [V18y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ ( ap @ ( ap @ ( ap @ V14f @ V16w ) @ V17x ) @ V18y ) ) @ ( ap @ ( ap @ ( ap @ V13P @ V16w ) @ V17x ) @ V18y ) ) ) ) ) ) ) ) ) )
=> ( p @ ( ap @ V0Q @ V15z ) ) ) )
<=> ! [V19w: $i] :
( ( mem @ V19w @ A_27e )
=> ! [V20x: $i] :
( ( mem @ V20x @ A_27f )
=> ! [V21y: $i] :
( ( mem @ V21y @ A_27g )
=> ( ( p @ ( ap @ ( ap @ ( ap @ V13P @ V19w ) @ V20x ) @ V21y ) )
=> ( p @ ( ap @ V0Q @ ( ap @ ( ap @ ( ap @ V14f @ V19w ) @ V20x ) @ V21y ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__SUB__POLYFUN,axiom,
! [V0a: $i] :
( ( mem @ V0a @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal,V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ c_2Ereal_2Ereal__sub
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4i: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0a @ V4i ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V4i ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5i: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0a @ V5i ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ V5i ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) )
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V6j: $i] :
( ap
@ ( ap @ c_2Erealax_2Ereal__mul
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V6j ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7i: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0a @ V7i ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V7i ) @ V6j ) ) @ ( 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 ) ) @ V6j ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__SUB__POLYFUN__ALT,axiom,
! [V0a: $i] :
( ( mem @ V0a @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V1x: tp__ty_2Erealax_2Ereal,V2y: tp__ty_2Erealax_2Ereal,V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
=> ( ( surj__ty_2Erealax_2Ereal
@ ( ap
@ ( ap @ c_2Ereal_2Ereal__sub
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4i: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0a @ V4i ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V4i ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5i: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0a @ V5i ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ V5i ) ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) )
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V6j: $i] :
( ap
@ ( ap @ c_2Erealax_2Ereal__mul
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ V6j ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7k: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0a @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V6j ) @ V7k ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2y ) ) @ V7k ) ) ) ) )
@ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V1x ) ) @ V6j ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__POLYFUN__ROOTBOUND,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1c: $i] :
( ( mem @ V1c @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ~ ! [V2i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1c @ ( inj__ty_2Enum_2Enum @ V2i ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
=> ( ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V3x )
@ ( ap
@ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal )
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4i: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V1c @ V4i ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ V3x ) @ V4i ) ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) )
& ( p
@ ( ap
@ ( ap @ c_2Earithmetic_2E_3C_3D
@ ( ap @ ( c_2Epred__set_2ECARD @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V5x: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V5x )
@ ( ap
@ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal )
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V6i: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V1c @ V6i ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ V5x ) @ V6i ) ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) )
@ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__POLYFUN__FINITE__ROOTS,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1c: $i] :
( ( mem @ V1c @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2x )
@ ( ap
@ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal )
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3i: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V1c @ V3i ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ V2x ) @ V3i ) ) ) ) )
@ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) )
<=> ? [V4i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V4i ) ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
& ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1c @ ( inj__ty_2Enum_2Enum @ V4i ) ) )
!= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__POLYFUN__EQ__0,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1c: $i] :
( ( mem @ V1c @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ( ! [V2x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3i: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V1c @ V3i ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) @ V3i ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ! [V4i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V4i ) ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1c @ ( inj__ty_2Enum_2Enum @ V4i ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EREAL__POLYFUN__EQ__CONST,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1c: $i] :
( ( mem @ V1c @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
=> ! [V2k: tp__ty_2Erealax_2Ereal] :
( ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4i: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V1c @ V4i ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V4i ) ) ) ) )
= V2k )
<=> ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1c @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= V2k )
& ! [V5i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V5i ) ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1c @ ( inj__ty_2Enum_2Enum @ V5i ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eiterate_2Epolynomial__function,axiom,
! [V0p: $i] :
( ( mem @ V0p @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Eiterate_2Epolynomial__function @ V0p ) )
<=> ? [V1m: tp__ty_2Enum_2Enum,V2c: $i] :
( ( mem @ V2c @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
& ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0p @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
= ( surj__ty_2Erealax_2Ereal
@ ( ap @ ( ap @ ( c_2Eiterate_2ESum @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eiterate_2E_2E_2E @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4i: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V2c @ V4i ) ) @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) @ V4i ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__CONST,axiom,
! [V0c: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap @ c_2Eiterate_2Epolynomial__function
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( inj__ty_2Erealax_2Ereal @ V0c ) ) ) ) ).
thf(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__ID,axiom,
( p
@ ( ap @ c_2Eiterate_2Epolynomial__function
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V0x: $i] : V0x ) ) ) ).
thf(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__ADD,axiom,
! [V0p: $i] :
( ( mem @ V0p @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1q: $i] :
( ( mem @ V1q @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ c_2Eiterate_2Epolynomial__function @ V0p ) )
& ( p @ ( ap @ c_2Eiterate_2Epolynomial__function @ V1q ) ) )
=> ( p
@ ( ap @ c_2Eiterate_2Epolynomial__function
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0p @ V2x ) ) @ ( ap @ V1q @ V2x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__LMUL,axiom,
! [V0p: $i] :
( ( mem @ V0p @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Eiterate_2Epolynomial__function @ V0p ) )
=> ( p
@ ( ap @ c_2Eiterate_2Epolynomial__function
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) @ ( ap @ V0p @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__RMUL,axiom,
! [V0p: $i] :
( ( mem @ V0p @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1c: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Eiterate_2Epolynomial__function @ V0p ) )
=> ( p
@ ( ap @ c_2Eiterate_2Epolynomial__function
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0p @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1c ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__NEG,axiom,
! [V0p: $i] :
( ( mem @ V0p @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( p
@ ( ap @ c_2Eiterate_2Epolynomial__function
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ V0p @ V1x ) ) ) ) )
<=> ( p @ ( ap @ c_2Eiterate_2Epolynomial__function @ V0p ) ) ) ) ).
thf(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__SUB,axiom,
! [V0p: $i] :
( ( mem @ V0p @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1q: $i] :
( ( mem @ V1q @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ c_2Eiterate_2Epolynomial__function @ V0p ) )
& ( p @ ( ap @ c_2Eiterate_2Epolynomial__function @ V1q ) ) )
=> ( p
@ ( ap @ c_2Eiterate_2Epolynomial__function
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V0p @ V2x ) ) @ ( ap @ V1q @ V2x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__MUL,axiom,
! [V0p: $i] :
( ( mem @ V0p @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1q: $i] :
( ( mem @ V1q @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ c_2Eiterate_2Epolynomial__function @ V0p ) )
& ( p @ ( ap @ c_2Eiterate_2Epolynomial__function @ V1q ) ) )
=> ( p
@ ( ap @ c_2Eiterate_2Epolynomial__function
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0p @ V2x ) ) @ ( ap @ V1q @ V2x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__SUM,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( arr @ ty_2Erealax_2Ereal @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ! [V2i: $i] :
( ( mem @ V2i @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2i ) @ V0s ) )
=> ( p
@ ( ap @ c_2Eiterate_2Epolynomial__function
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( ap @ ( ap @ V1p @ V3x ) @ V2i ) ) ) ) ) ) )
=> ( p
@ ( ap @ c_2Eiterate_2Epolynomial__function
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Eiterate_2ESum @ A_27a ) @ V0s ) @ ( ap @ V1p @ V4x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__POW,axiom,
! [V0p: $i] :
( ( mem @ V0p @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ c_2Eiterate_2Epolynomial__function @ V0p ) )
=> ( p
@ ( ap @ c_2Eiterate_2Epolynomial__function
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ V0p @ V2x ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__INDUCT,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ bool ) )
=> ( ( ( p
@ ( ap @ V0P
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V1x: $i] : V1x ) ) )
& ! [V2c: tp__ty_2Erealax_2Ereal] :
( p
@ ( ap @ V0P
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3x: $i] : ( inj__ty_2Erealax_2Ereal @ V2c ) ) ) )
& ! [V4p: $i] :
( ( mem @ V4p @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V5q: $i] :
( ( mem @ V5q @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ V0P @ V4p ) )
& ( p @ ( ap @ V0P @ V5q ) ) )
=> ( p
@ ( ap @ V0P
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V6x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V4p @ V6x ) ) @ ( ap @ V5q @ V6x ) ) ) ) ) ) ) )
& ! [V7p: $i] :
( ( mem @ V7p @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V8q: $i] :
( ( mem @ V8q @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ V0P @ V7p ) )
& ( p @ ( ap @ V0P @ V8q ) ) )
=> ( p
@ ( ap @ V0P
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V9x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V7p @ V9x ) ) @ ( ap @ V8q @ V9x ) ) ) ) ) ) ) ) )
=> ! [V10p: $i] :
( ( mem @ V10p @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ c_2Eiterate_2Epolynomial__function @ V10p ) )
=> ( p @ ( ap @ V0P @ V10p ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__o,axiom,
! [V0p: $i] :
( ( mem @ V0p @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1q: $i] :
( ( mem @ V1q @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ( ( ( p @ ( ap @ c_2Eiterate_2Epolynomial__function @ V0p ) )
& ( p @ ( ap @ c_2Eiterate_2Epolynomial__function @ V1q ) ) )
=> ( p @ ( ap @ c_2Eiterate_2Epolynomial__function @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) @ V0p ) @ V1q ) ) ) ) ) ) ).
thf(conj_thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__FINITE__ROOTS,axiom,
! [V0p: $i] :
( ( mem @ V0p @ ( arr @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal ) )
=> ! [V1a: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Eiterate_2Epolynomial__function @ V0p ) )
=> ( ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Erealax_2Ereal @ bool ) @ V2x ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V0p @ V2x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1a ) ) ) ) ) ) )
<=> ~ ! [V3x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0p @ ( inj__ty_2Erealax_2Ereal @ V3x ) ) )
= V1a ) ) ) ) ).
%------------------------------------------------------------------------------