ITP001 Axioms: ITP092^5.ax
%------------------------------------------------------------------------------
% File : ITP092^5 : 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 : Omega^2.ax [Gau20]
% : HL4092^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 115 ( 8 unt; 26 typ; 0 def)
% Number of atoms : 3107 ( 32 equ; 0 cnn)
% Maximal formula atoms : 125 ( 27 avg)
% Number of connectives : 4454 ( 1 ~; 5 |; 148 &;4143 @)
% ( 42 <=>; 115 =>; 0 <=; 0 <~>)
% Maximal formula depth : 31 ( 14 avg;4143 nst)
% Number of types : 4 ( 3 usr)
% Number of type conns : 14 ( 14 >; 0 *; 0 +; 0 <<)
% Number of symbols : 67 ( 64 usr; 55 con; 0-2 aty)
% Number of variables : 402 ( 4 ^ 381 !; 17 ?; 402 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2EOmega_2EMAP2,type,
c_2EOmega_2EMAP2: del > del > $i ).
thf(mem_c_2EOmega_2EMAP2,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EOmega_2EMAP2 @ A_27a @ A_27b ) @ ( arr @ A_27b @ ( arr @ ( arr @ A_27b @ ( arr @ A_27b @ A_27a ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ) ) ).
thf(stp_c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,type,
tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint: $tType ).
thf(stp_inj_c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,type,
inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint > $i ).
thf(stp_surj_c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,type,
surj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint: $i > tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint ).
thf(stp_inj_surj_c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,axiom,
! [X: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] :
( ( surj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ X ) )
= X ) ).
thf(stp_inj_mem_c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,axiom,
! [X: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] : ( mem @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ X ) @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) ) ).
thf(stp_iso_mem_c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,axiom,
! [X: $i] :
( ( mem @ X @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) )
=> ( X
= ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ ( surj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ X ) ) ) ) ).
thf(stp_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,type,
tp__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint: $tType ).
thf(stp_inj_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,type,
inj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint: tp__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint > $i ).
thf(stp_surj_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,type,
surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint: $i > tp__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint ).
thf(stp_inj_surj_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,axiom,
! [X: tp__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ ( inj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ X ) )
= X ) ).
thf(stp_inj_mem_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,axiom,
! [X: tp__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] : ( mem @ ( inj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ X ) @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) ).
thf(stp_iso_mem_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,axiom,
! [X: $i] :
( ( mem @ X @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) )
=> ( X
= ( inj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ X ) ) ) ) ).
thf(tp_c_2EOmega_2Ecalc__nightmare,type,
c_2EOmega_2Ecalc__nightmare: $i ).
thf(mem_c_2EOmega_2Ecalc__nightmare,axiom,
mem @ c_2EOmega_2Ecalc__nightmare @ ( arr @ ty_2Einteger_2Eint @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ bool ) ) ) ).
thf(tp_c_2EOmega_2Edark__shadow,type,
c_2EOmega_2Edark__shadow: $i ).
thf(mem_c_2EOmega_2Edark__shadow,axiom,
mem @ c_2EOmega_2Edark__shadow @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ bool ) ) ).
thf(tp_c_2EOmega_2Edark__shadow__cond__row,type,
c_2EOmega_2Edark__shadow__cond__row: $i ).
thf(mem_c_2EOmega_2Edark__shadow__cond__row,axiom,
mem @ c_2EOmega_2Edark__shadow__cond__row @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ bool ) ) ).
thf(tp_c_2EOmega_2Edark__shadow__condition,type,
c_2EOmega_2Edark__shadow__condition: $i ).
thf(mem_c_2EOmega_2Edark__shadow__condition,axiom,
mem @ c_2EOmega_2Edark__shadow__condition @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ bool ) ) ).
thf(tp_c_2EOmega_2Edark__shadow__row,type,
c_2EOmega_2Edark__shadow__row: $i ).
thf(mem_c_2EOmega_2Edark__shadow__row,axiom,
mem @ c_2EOmega_2Edark__shadow__row @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Einteger_2Eint @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ bool ) ) ) ).
thf(tp_c_2EOmega_2Eevallower,type,
c_2EOmega_2Eevallower: $i ).
thf(mem_c_2EOmega_2Eevallower,axiom,
mem @ c_2EOmega_2Eevallower @ ( arr @ ty_2Einteger_2Eint @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ bool ) ) ).
thf(tp_c_2EOmega_2Eevalupper,type,
c_2EOmega_2Eevalupper: $i ).
thf(mem_c_2EOmega_2Eevalupper,axiom,
mem @ c_2EOmega_2Eevalupper @ ( arr @ ty_2Einteger_2Eint @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ bool ) ) ).
thf(tp_c_2EOmega_2Efst1,type,
c_2EOmega_2Efst1: del > $i ).
thf(mem_c_2EOmega_2Efst1,axiom,
! [A_27a: del] : ( mem @ ( c_2EOmega_2Efst1 @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) @ bool ) ) ).
thf(tp_c_2EOmega_2Efst__nzero,type,
c_2EOmega_2Efst__nzero: del > $i ).
thf(mem_c_2EOmega_2Efst__nzero,axiom,
! [A_27a: del] : ( mem @ ( c_2EOmega_2Efst__nzero @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) @ bool ) ) ).
thf(tp_c_2EOmega_2Emodhat,type,
c_2EOmega_2Emodhat: $i ).
thf(mem_c_2EOmega_2Emodhat,axiom,
mem @ c_2EOmega_2Emodhat @ ( arr @ ty_2Einteger_2Eint @ ( arr @ ty_2Einteger_2Eint @ ty_2Einteger_2Eint ) ) ).
thf(stp_fo_c_2EOmega_2Emodhat,type,
fo__c_2EOmega_2Emodhat: tp__ty_2Einteger_2Eint > tp__ty_2Einteger_2Eint > tp__ty_2Einteger_2Eint ).
thf(stp_eq_fo_c_2EOmega_2Emodhat,axiom,
! [X0: tp__ty_2Einteger_2Eint,X1: tp__ty_2Einteger_2Eint] :
( ( inj__ty_2Einteger_2Eint @ ( fo__c_2EOmega_2Emodhat @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2EOmega_2Emodhat @ ( inj__ty_2Einteger_2Eint @ X0 ) ) @ ( inj__ty_2Einteger_2Eint @ X1 ) ) ) ).
thf(tp_c_2EOmega_2Enightmare,type,
c_2EOmega_2Enightmare: $i ).
thf(mem_c_2EOmega_2Enightmare,axiom,
mem @ c_2EOmega_2Enightmare @ ( arr @ ty_2Einteger_2Eint @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ bool ) ) ) ) ) ).
thf(tp_c_2EOmega_2Ereal__shadow,type,
c_2EOmega_2Ereal__shadow: $i ).
thf(mem_c_2EOmega_2Ereal__shadow,axiom,
mem @ c_2EOmega_2Ereal__shadow @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ bool ) ) ).
thf(tp_c_2EOmega_2Ershadow__row,type,
c_2EOmega_2Ershadow__row: $i ).
thf(mem_c_2EOmega_2Ershadow__row,axiom,
mem @ c_2EOmega_2Ershadow__row @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ bool ) ) ).
thf(stp_c_ty_2Elist_2Elist_ty_2Einteger_2Eint,type,
tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint: $tType ).
thf(stp_inj_c_ty_2Elist_2Elist_ty_2Einteger_2Eint,type,
inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint: tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint > $i ).
thf(stp_surj_c_ty_2Elist_2Elist_ty_2Einteger_2Eint,type,
surj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint: $i > tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint ).
thf(stp_inj_surj_c_ty_2Elist_2Elist_ty_2Einteger_2Eint,axiom,
! [X: tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint] :
( ( surj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ X ) )
= X ) ).
thf(stp_inj_mem_c_ty_2Elist_2Elist_ty_2Einteger_2Eint,axiom,
! [X: tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint] : ( mem @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ X ) @ ( ty_2Elist_2Elist @ ty_2Einteger_2Eint ) ) ).
thf(stp_iso_mem_c_ty_2Elist_2Elist_ty_2Einteger_2Eint,axiom,
! [X: $i] :
( ( mem @ X @ ( ty_2Elist_2Elist @ ty_2Einteger_2Eint ) )
=> ( X
= ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ ( surj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ X ) ) ) ) ).
thf(tp_c_2EOmega_2Esumc,type,
c_2EOmega_2Esumc: $i ).
thf(mem_c_2EOmega_2Esumc,axiom,
mem @ c_2EOmega_2Esumc @ ( arr @ ( ty_2Elist_2Elist @ ty_2Einteger_2Eint ) @ ( arr @ ( ty_2Elist_2Elist @ ty_2Einteger_2Eint ) @ ty_2Einteger_2Eint ) ) ).
thf(stp_fo_c_2EOmega_2Esumc,type,
fo__c_2EOmega_2Esumc: tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint > tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint > tp__ty_2Einteger_2Eint ).
thf(stp_eq_fo_c_2EOmega_2Esumc,axiom,
! [X0: tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint,X1: tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint] :
( ( inj__ty_2Einteger_2Eint @ ( fo__c_2EOmega_2Esumc @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2EOmega_2Esumc @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ X0 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ X1 ) ) ) ).
thf(conj_thm_2EOmega_2EMAP2__ind,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27b @ ( arr @ ( arr @ A_27b @ ( arr @ A_27b @ A_27a ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ bool ) ) ) ) )
=> ( ( ! [V1pad: $i] :
( ( mem @ V1pad @ A_27b )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27b @ ( arr @ A_27b @ A_27a ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V1pad ) @ V2f ) @ ( c_2Elist_2ENIL @ A_27b ) ) @ ( c_2Elist_2ENIL @ A_27b ) ) ) ) )
& ! [V3pad: $i] :
( ( mem @ V3pad @ A_27b )
=> ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27b @ ( arr @ A_27b @ A_27a ) ) )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27b )
=> ! [V6ys: $i] :
( ( mem @ V6ys @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V3pad ) @ V4f ) @ ( c_2Elist_2ENIL @ A_27b ) ) @ V6ys ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V3pad ) @ V4f ) @ ( c_2Elist_2ENIL @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V5y ) @ V6ys ) ) ) ) ) ) ) )
& ! [V7pad: $i] :
( ( mem @ V7pad @ A_27b )
=> ! [V8f: $i] :
( ( mem @ V8f @ ( arr @ A_27b @ ( arr @ A_27b @ A_27a ) ) )
=> ! [V9x: $i] :
( ( mem @ V9x @ A_27b )
=> ! [V10xs: $i] :
( ( mem @ V10xs @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V7pad ) @ V8f ) @ V10xs ) @ ( c_2Elist_2ENIL @ A_27b ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V7pad ) @ V8f ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V9x ) @ V10xs ) ) @ ( c_2Elist_2ENIL @ A_27b ) ) ) ) ) ) ) )
& ! [V11pad: $i] :
( ( mem @ V11pad @ A_27b )
=> ! [V12f: $i] :
( ( mem @ V12f @ ( arr @ A_27b @ ( arr @ A_27b @ A_27a ) ) )
=> ! [V13x: $i] :
( ( mem @ V13x @ A_27b )
=> ! [V14xs: $i] :
( ( mem @ V14xs @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V15y: $i] :
( ( mem @ V15y @ A_27b )
=> ! [V16ys: $i] :
( ( mem @ V16ys @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V11pad ) @ V12f ) @ V14xs ) @ V16ys ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V11pad ) @ V12f ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V13x ) @ V14xs ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V15y ) @ V16ys ) ) ) ) ) ) ) ) ) ) )
=> ! [V17v: $i] :
( ( mem @ V17v @ A_27b )
=> ! [V18v1: $i] :
( ( mem @ V18v1 @ ( arr @ A_27b @ ( arr @ A_27b @ A_27a ) ) )
=> ! [V19v2: $i] :
( ( mem @ V19v2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V20v3: $i] :
( ( mem @ V20v3 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V17v ) @ V18v1 ) @ V19v2 ) @ V20v3 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EOmega_2EMAP2__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0pad: $i] :
( ( mem @ V0pad @ A_27b )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ ( arr @ A_27b @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EOmega_2EMAP2 @ A_27a @ A_27b ) @ V0pad ) @ V1f ) @ ( c_2Elist_2ENIL @ A_27b ) ) @ ( c_2Elist_2ENIL @ A_27b ) )
= ( c_2Elist_2ENIL @ A_27a ) ) ) )
& ! [V2ys: $i] :
( ( mem @ V2ys @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ! [V4pad: $i] :
( ( mem @ V4pad @ A_27b )
=> ! [V5f: $i] :
( ( mem @ V5f @ ( arr @ A_27b @ ( arr @ A_27b @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EOmega_2EMAP2 @ A_27a @ A_27b ) @ V4pad ) @ V5f ) @ ( c_2Elist_2ENIL @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V3y ) @ V2ys ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ ( ap @ ( ap @ V5f @ V4pad ) @ V3y ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EOmega_2EMAP2 @ A_27a @ A_27b ) @ V4pad ) @ V5f ) @ ( c_2Elist_2ENIL @ A_27b ) ) @ V2ys ) ) ) ) ) ) )
& ! [V6xs: $i] :
( ( mem @ V6xs @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V7x: $i] :
( ( mem @ V7x @ A_27b )
=> ! [V8pad: $i] :
( ( mem @ V8pad @ A_27b )
=> ! [V9f: $i] :
( ( mem @ V9f @ ( arr @ A_27b @ ( arr @ A_27b @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EOmega_2EMAP2 @ A_27a @ A_27b ) @ V8pad ) @ V9f ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V7x ) @ V6xs ) ) @ ( c_2Elist_2ENIL @ A_27b ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ ( ap @ ( ap @ V9f @ V7x ) @ V8pad ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EOmega_2EMAP2 @ A_27a @ A_27b ) @ V8pad ) @ V9f ) @ V6xs ) @ ( c_2Elist_2ENIL @ A_27b ) ) ) ) ) ) ) )
& ! [V10ys: $i] :
( ( mem @ V10ys @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V11y: $i] :
( ( mem @ V11y @ A_27b )
=> ! [V12xs: $i] :
( ( mem @ V12xs @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V13x: $i] :
( ( mem @ V13x @ A_27b )
=> ! [V14pad: $i] :
( ( mem @ V14pad @ A_27b )
=> ! [V15f: $i] :
( ( mem @ V15f @ ( arr @ A_27b @ ( arr @ A_27b @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EOmega_2EMAP2 @ A_27a @ A_27b ) @ V14pad ) @ V15f ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V13x ) @ V12xs ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V11y ) @ V10ys ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ ( ap @ ( ap @ V15f @ V13x ) @ V11y ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EOmega_2EMAP2 @ A_27a @ A_27b ) @ V14pad ) @ V15f ) @ V12xs ) @ V10ys ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EOmega_2EMAP2__zero__ADD,axiom,
! [V0xs: tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint] :
( ( ( surj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EOmega_2EMAP2 @ ty_2Einteger_2Eint @ ty_2Einteger_2Eint ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ c_2Einteger_2Eint__add ) @ ( c_2Elist_2ENIL @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V0xs ) ) )
= V0xs )
& ( ( surj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EOmega_2EMAP2 @ ty_2Einteger_2Eint @ ty_2Einteger_2Eint ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ c_2Einteger_2Eint__add ) @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V0xs ) ) @ ( c_2Elist_2ENIL @ ty_2Einteger_2Eint ) ) )
= V0xs ) ) ).
thf(conj_thm_2EOmega_2Esumc__ind,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Elist_2Elist @ ty_2Einteger_2Eint ) @ ( arr @ ( ty_2Elist_2Elist @ ty_2Einteger_2Eint ) @ bool ) ) )
=> ( ( ! [V1v0: tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint] : ( p @ ( ap @ ( ap @ V0P @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V1v0 ) ) @ ( c_2Elist_2ENIL @ ty_2Einteger_2Eint ) ) )
& ! [V2v4: tp__ty_2Einteger_2Eint,V3v5: tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint] : ( p @ ( ap @ ( ap @ V0P @ ( c_2Elist_2ENIL @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Einteger_2Eint ) @ ( inj__ty_2Einteger_2Eint @ V2v4 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V3v5 ) ) ) )
& ! [V4c: tp__ty_2Einteger_2Eint,V5cs: tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint,V6v: tp__ty_2Einteger_2Eint,V7vs: tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ V0P @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V5cs ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V7vs ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Einteger_2Eint ) @ ( inj__ty_2Einteger_2Eint @ V4c ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V5cs ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Einteger_2Eint ) @ ( inj__ty_2Einteger_2Eint @ V6v ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V7vs ) ) ) ) ) )
=> ! [V8v: tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint,V9v1: tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint] : ( p @ ( ap @ ( ap @ V0P @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V8v ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V9v1 ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Esumc__def,axiom,
( ! [V0v0: tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint] :
( ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2EOmega_2Esumc @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V0v0 ) ) @ ( c_2Elist_2ENIL @ ty_2Einteger_2Eint ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ! [V1v5: tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint,V2v4: tp__ty_2Einteger_2Eint] :
( ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2EOmega_2Esumc @ ( c_2Elist_2ENIL @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Einteger_2Eint ) @ ( inj__ty_2Einteger_2Eint @ V2v4 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V1v5 ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ! [V3vs: tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint,V4v: tp__ty_2Einteger_2Eint,V5cs: tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint,V6c: tp__ty_2Einteger_2Eint] :
( ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2EOmega_2Esumc @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Einteger_2Eint ) @ ( inj__ty_2Einteger_2Eint @ V6c ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V5cs ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Einteger_2Eint ) @ ( inj__ty_2Einteger_2Eint @ V4v ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V3vs ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__add @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( inj__ty_2Einteger_2Eint @ V6c ) ) @ ( inj__ty_2Einteger_2Eint @ V4v ) ) ) @ ( ap @ ( ap @ c_2EOmega_2Esumc @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V5cs ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V3vs ) ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Esumc__thm,axiom,
! [V0cs: tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint,V1vs: tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint,V2c: tp__ty_2Einteger_2Eint,V3v: tp__ty_2Einteger_2Eint] :
( ( ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2EOmega_2Esumc @ ( c_2Elist_2ENIL @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V1vs ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2EOmega_2Esumc @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V0cs ) ) @ ( c_2Elist_2ENIL @ ty_2Einteger_2Eint ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ( ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2EOmega_2Esumc @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Einteger_2Eint ) @ ( inj__ty_2Einteger_2Eint @ V2c ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V0cs ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Einteger_2Eint ) @ ( inj__ty_2Einteger_2Eint @ V3v ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V1vs ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__add @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( inj__ty_2Einteger_2Eint @ V2c ) ) @ ( inj__ty_2Einteger_2Eint @ V3v ) ) ) @ ( ap @ ( ap @ c_2EOmega_2Esumc @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V0cs ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V1vs ) ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Esumc__ADD,axiom,
! [V0cs: tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint,V1vs: tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint,V2ds: tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint] :
( ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__add @ ( ap @ ( ap @ c_2EOmega_2Esumc @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V0cs ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V1vs ) ) ) @ ( ap @ ( ap @ c_2EOmega_2Esumc @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V2ds ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V1vs ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2EOmega_2Esumc @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EOmega_2EMAP2 @ ty_2Einteger_2Eint @ ty_2Einteger_2Eint ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ c_2Einteger_2Eint__add ) @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V0cs ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V2ds ) ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V1vs ) ) ) ) ).
thf(conj_thm_2EOmega_2Esumc__MULT,axiom,
! [V0cs: tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint,V1vs: tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint,V2f: tp__ty_2Einteger_2Eint] :
( ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( inj__ty_2Einteger_2Eint @ V2f ) ) @ ( ap @ ( ap @ c_2EOmega_2Esumc @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V0cs ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V1vs ) ) ) )
= ( surj__ty_2Einteger_2Eint
@ ( ap
@ ( ap @ c_2EOmega_2Esumc
@ ( ap
@ ( ap @ ( c_2Elist_2EMAP @ ty_2Einteger_2Eint @ ty_2Einteger_2Eint )
@ ( lam @ ty_2Einteger_2Eint
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( inj__ty_2Einteger_2Eint @ V2f ) ) @ V3x ) ) )
@ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V0cs ) ) )
@ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V1vs ) ) ) ) ).
thf(conj_thm_2EOmega_2Esumc__singleton,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Einteger_2Eint @ ty_2Einteger_2Eint ) )
=> ! [V1c: tp__ty_2Einteger_2Eint] :
( ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2EOmega_2Esumc @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ty_2Einteger_2Eint @ ty_2Einteger_2Eint ) @ V0f ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Einteger_2Eint ) @ ( inj__ty_2Einteger_2Eint @ V1c ) ) @ ( c_2Elist_2ENIL @ ty_2Einteger_2Eint ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Einteger_2Eint ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( c_2Elist_2ENIL @ ty_2Einteger_2Eint ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ V0f @ ( inj__ty_2Einteger_2Eint @ V1c ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Esumc__nonsingle,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Einteger_2Eint @ ty_2Einteger_2Eint ) )
=> ! [V1cs: tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint,V2c: tp__ty_2Einteger_2Eint,V3v: tp__ty_2Einteger_2Eint,V4vs: tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint] :
( ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2EOmega_2Esumc @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ty_2Einteger_2Eint @ ty_2Einteger_2Eint ) @ V0f ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Einteger_2Eint ) @ ( inj__ty_2Einteger_2Eint @ V2c ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V1cs ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Einteger_2Eint ) @ ( inj__ty_2Einteger_2Eint @ V3v ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V4vs ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__add @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ V0f @ ( inj__ty_2Einteger_2Eint @ V2c ) ) ) @ ( inj__ty_2Einteger_2Eint @ V3v ) ) ) @ ( ap @ ( ap @ c_2EOmega_2Esumc @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ty_2Einteger_2Eint @ ty_2Einteger_2Eint ) @ V0f ) @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V1cs ) ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V4vs ) ) ) ) ) ) ).
thf(ax_thm_2EOmega_2Emodhat__def,axiom,
! [V0x: tp__ty_2Einteger_2Eint,V1y: tp__ty_2Einteger_2Eint] :
( ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2EOmega_2Emodhat @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__ty_2Einteger_2Eint @ V1y ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__sub @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( inj__ty_2Einteger_2Eint @ V1y ) ) @ ( ap @ ( ap @ c_2Einteger_2Eint__div @ ( ap @ ( ap @ c_2Einteger_2Eint__add @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Einteger_2Eint @ V0x ) ) ) @ ( inj__ty_2Einteger_2Eint @ V1y ) ) ) @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Einteger_2Eint @ V1y ) ) ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Eequality__removal,axiom,
! [V0c: tp__ty_2Einteger_2Eint,V1x: tp__ty_2Einteger_2Eint,V2cs: tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint,V3vs: tp__c_ty_2Elist_2Elist_ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__lt @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Einteger_2Eint @ V0c ) ) )
=> ( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__add @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( inj__ty_2Einteger_2Eint @ V0c ) ) @ ( inj__ty_2Einteger_2Eint @ V1x ) ) ) @ ( ap @ ( ap @ c_2EOmega_2Esumc @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V2cs ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V3vs ) ) ) ) )
<=> ? [V4s: tp__ty_2Einteger_2Eint] :
( ( V1x
= ( surj__ty_2Einteger_2Eint
@ ( ap @ ( ap @ c_2Einteger_2Eint__add @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__neg @ ( ap @ ( ap @ c_2Einteger_2Eint__add @ ( inj__ty_2Einteger_2Eint @ V0c ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ ( inj__ty_2Einteger_2Eint @ V4s ) ) )
@ ( ap
@ ( ap @ c_2EOmega_2Esumc
@ ( ap
@ ( ap @ ( c_2Elist_2EMAP @ ty_2Einteger_2Eint @ ty_2Einteger_2Eint )
@ ( lam @ ty_2Einteger_2Eint
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2EOmega_2Emodhat @ V5x ) @ ( ap @ ( ap @ c_2Einteger_2Eint__add @ ( inj__ty_2Einteger_2Eint @ V0c ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
@ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V2cs ) ) )
@ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V3vs ) ) ) ) )
& ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__add @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( inj__ty_2Einteger_2Eint @ V0c ) ) @ ( inj__ty_2Einteger_2Eint @ V1x ) ) ) @ ( ap @ ( ap @ c_2EOmega_2Esumc @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V2cs ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Einteger_2Eint @ V3vs ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Eevalupper__ind,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Einteger_2Eint @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ bool ) ) )
=> ( ( ! [V1x: tp__ty_2Einteger_2Eint] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Einteger_2Eint @ V1x ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) ) )
& ! [V2x: tp__ty_2Einteger_2Eint,V3c: tp__ty_2Enum_2Enum,V4y: tp__ty_2Einteger_2Eint,V5cs: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Einteger_2Eint @ V2x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V5cs ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Einteger_2Eint @ V2x ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V3c ) ) @ ( inj__ty_2Einteger_2Eint @ V4y ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V5cs ) ) ) ) ) )
=> ! [V6v: tp__ty_2Einteger_2Eint,V7v1: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Einteger_2Eint @ V6v ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V7v1 ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Eevalupper__def,axiom,
( ! [V0x: tp__ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) ) )
<=> $true )
& ! [V1y: tp__ty_2Einteger_2Eint,V2x: tp__ty_2Einteger_2Eint,V3cs: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V4c: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V2x ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V4c ) ) @ ( inj__ty_2Einteger_2Eint @ V1y ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V3cs ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__le @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V4c ) ) ) @ ( inj__ty_2Einteger_2Eint @ V2x ) ) ) @ ( inj__ty_2Einteger_2Eint @ V1y ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V2x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V3cs ) ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Eevallower__ind,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Einteger_2Eint @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ bool ) ) )
=> ( ( ! [V1x: tp__ty_2Einteger_2Eint] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Einteger_2Eint @ V1x ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) ) )
& ! [V2x: tp__ty_2Einteger_2Eint,V3c: tp__ty_2Enum_2Enum,V4y: tp__ty_2Einteger_2Eint,V5cs: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Einteger_2Eint @ V2x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V5cs ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Einteger_2Eint @ V2x ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V3c ) ) @ ( inj__ty_2Einteger_2Eint @ V4y ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V5cs ) ) ) ) ) )
=> ! [V6v: tp__ty_2Einteger_2Eint,V7v1: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Einteger_2Eint @ V6v ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V7v1 ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Eevallower__def,axiom,
( ! [V0x: tp__ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) ) )
<=> $true )
& ! [V1y: tp__ty_2Einteger_2Eint,V2x: tp__ty_2Einteger_2Eint,V3cs: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V4c: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V2x ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V4c ) ) @ ( inj__ty_2Einteger_2Eint @ V1y ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V3cs ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__le @ ( inj__ty_2Einteger_2Eint @ V1y ) ) @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V4c ) ) ) @ ( inj__ty_2Einteger_2Eint @ V2x ) ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V2x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V3cs ) ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Esmaller__satisfies__uppers,axiom,
! [V0uppers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V1x: tp__ty_2Einteger_2Eint,V2y: tp__ty_2Einteger_2Eint] :
( ( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V1x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) )
& ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__lt @ ( inj__ty_2Einteger_2Eint @ V2y ) ) @ ( inj__ty_2Einteger_2Eint @ V1x ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V2y ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) ) ) ).
thf(conj_thm_2EOmega_2Ebigger__satisfies__lowers,axiom,
! [V0lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V1x: tp__ty_2Einteger_2Eint,V2y: tp__ty_2Einteger_2Eint] :
( ( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V1x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0lowers ) ) )
& ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__lt @ ( inj__ty_2Einteger_2Eint @ V1x ) ) @ ( inj__ty_2Einteger_2Eint @ V2y ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V2y ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0lowers ) ) ) ) ).
thf(ax_thm_2EOmega_2Efst__nzero__def,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2EOmega_2Efst__nzero @ A_27a ) @ V0x ) )
<=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( c_2Epair_2EFST @ ty_2Enum_2Enum @ A_27a ) @ V0x ) ) ) ) ) ).
thf(ax_thm_2EOmega_2Efst1__def,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2EOmega_2Efst1 @ A_27a ) @ V0x ) )
<=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epair_2EFST @ ty_2Enum_2Enum @ A_27a ) @ V0x ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Eonlylowers__satisfiable,axiom,
! [V0lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( c_2EOmega_2Efst__nzero @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0lowers ) ) )
=> ? [V1x: tp__ty_2Einteger_2Eint] : ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V1x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0lowers ) ) ) ) ).
thf(conj_thm_2EOmega_2Eonlyuppers__satisfiable,axiom,
! [V0uppers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( c_2EOmega_2Efst__nzero @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) )
=> ? [V1x: tp__ty_2Einteger_2Eint] : ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V1x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) ) ) ).
thf(conj_thm_2EOmega_2Ershadow__row__ind,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ bool ) ) )
=> ( ( ! [V1upperc: tp__ty_2Enum_2Enum,V2uppery: tp__ty_2Einteger_2Eint] : ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V1upperc ) ) @ ( inj__ty_2Einteger_2Eint @ V2uppery ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) ) )
& ! [V3upperc: tp__ty_2Enum_2Enum,V4uppery: tp__ty_2Einteger_2Eint,V5lowerc: tp__ty_2Enum_2Enum,V6lowery: tp__ty_2Einteger_2Eint,V7rs: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V3upperc ) ) @ ( inj__ty_2Einteger_2Eint @ V4uppery ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V7rs ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V3upperc ) ) @ ( inj__ty_2Einteger_2Eint @ V4uppery ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V5lowerc ) ) @ ( inj__ty_2Einteger_2Eint @ V6lowery ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V7rs ) ) ) ) ) )
=> ! [V8v: tp__ty_2Enum_2Enum,V9v1: tp__ty_2Einteger_2Eint,V10v2: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] : ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V8v ) ) @ ( inj__ty_2Einteger_2Eint @ V9v1 ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V10v2 ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Ershadow__row__def,axiom,
( ! [V0uppery: tp__ty_2Einteger_2Eint,V1upperc: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2EOmega_2Ershadow__row @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V1upperc ) ) @ ( inj__ty_2Einteger_2Eint @ V0uppery ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) ) )
<=> $true )
& ! [V2uppery: tp__ty_2Einteger_2Eint,V3upperc: tp__ty_2Enum_2Enum,V4rs: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V5lowery: tp__ty_2Einteger_2Eint,V6lowerc: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2EOmega_2Ershadow__row @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V3upperc ) ) @ ( inj__ty_2Einteger_2Eint @ V2uppery ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V6lowerc ) ) @ ( inj__ty_2Einteger_2Eint @ V5lowery ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V4rs ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__le @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V3upperc ) ) ) @ ( inj__ty_2Einteger_2Eint @ V5lowery ) ) ) @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V6lowerc ) ) ) @ ( inj__ty_2Einteger_2Eint @ V2uppery ) ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Ershadow__row @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V3upperc ) ) @ ( inj__ty_2Einteger_2Eint @ V2uppery ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V4rs ) ) ) ) ) ) ).
thf(ax_thm_2EOmega_2Ereal__shadow__def,axiom,
( ! [V0lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2EOmega_2Ereal__shadow @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0lowers ) ) )
<=> $true )
& ! [V1upper: tp__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V2ls: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V3lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2EOmega_2Ereal__shadow @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1upper ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2ls ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V3lowers ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2EOmega_2Ershadow__row @ ( inj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1upper ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V3lowers ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Ereal__shadow @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2ls ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V3lowers ) ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Esingleton__real__shadow,axiom,
! [V0c: tp__ty_2Enum_2Enum,V1L: tp__ty_2Einteger_2Eint,V2x: tp__ty_2Einteger_2Eint] :
( ( ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__le @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V0c ) ) ) @ ( inj__ty_2Einteger_2Eint @ V2x ) ) ) @ ( inj__ty_2Einteger_2Eint @ V1L ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0c ) ) ) )
=> ! [V3lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] :
( ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( c_2EOmega_2Efst__nzero @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V3lowers ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V2x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V3lowers ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2EOmega_2Ershadow__row @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V0c ) ) @ ( inj__ty_2Einteger_2Eint @ V1L ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V3lowers ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Ereal__shadow__revimp__uppers1,axiom,
! [V0uppers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V1lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V2L: tp__ty_2Einteger_2Eint,V3x: tp__ty_2Einteger_2Eint] :
( ( ( p @ ( ap @ ( ap @ c_2EOmega_2Ershadow__row @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Einteger_2Eint @ V2L ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V3x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V3x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( c_2EOmega_2Efst__nzero @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( c_2EOmega_2Efst1 @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) ) )
=> ? [V4x: tp__ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__le @ ( inj__ty_2Einteger_2Eint @ V4x ) ) @ ( inj__ty_2Einteger_2Eint @ V2L ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V4x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V4x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Ereal__shadow__revimp__lowers1,axiom,
! [V0uppers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V1lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V2c: tp__ty_2Enum_2Enum,V3L: tp__ty_2Einteger_2Eint,V4x: tp__ty_2Einteger_2Eint] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Ershadow__row @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) @ ( inj__ty_2Einteger_2Eint @ V3L ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V4x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V4x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( c_2EOmega_2Efst__nzero @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( c_2EOmega_2Efst1 @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) ) )
=> ? [V5x: tp__ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__le @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V2c ) ) ) @ ( inj__ty_2Einteger_2Eint @ V5x ) ) ) @ ( inj__ty_2Einteger_2Eint @ V3L ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V5x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V5x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Ereal__shadow__always__implied,axiom,
! [V0uppers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V1lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V2x: tp__ty_2Einteger_2Eint] :
( ( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V2x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V2x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( c_2EOmega_2Efst__nzero @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( c_2EOmega_2Efst__nzero @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2EOmega_2Ereal__shadow @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) ) ) ).
thf(conj_thm_2EOmega_2Eexact__shadow__case,axiom,
! [V0uppers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V1lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] :
( ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( c_2EOmega_2Efst__nzero @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( c_2EOmega_2Efst__nzero @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( c_2EOmega_2Efst1 @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) )
| ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( c_2EOmega_2Efst1 @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) ) )
=> ( ? [V2x: tp__ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V2x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V2x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2EOmega_2Ereal__shadow @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Edark__shadow__cond__row__ind,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ bool ) ) )
=> ( ( ! [V1c: tp__ty_2Enum_2Enum,V2L: tp__ty_2Einteger_2Eint] : ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V1c ) ) @ ( inj__ty_2Einteger_2Eint @ V2L ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) ) )
& ! [V3c: tp__ty_2Enum_2Enum,V4L: tp__ty_2Einteger_2Eint,V5d: tp__ty_2Enum_2Enum,V6R: tp__ty_2Einteger_2Eint,V7t: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V3c ) ) @ ( inj__ty_2Einteger_2Eint @ V4L ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V7t ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V3c ) ) @ ( inj__ty_2Einteger_2Eint @ V4L ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V5d ) ) @ ( inj__ty_2Einteger_2Eint @ V6R ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V7t ) ) ) ) ) )
=> ! [V8v: tp__ty_2Enum_2Enum,V9v1: tp__ty_2Einteger_2Eint,V10v2: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] : ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V8v ) ) @ ( inj__ty_2Einteger_2Eint @ V9v1 ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V10v2 ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Edark__shadow__cond__row__def,axiom,
( ! [V0c: tp__ty_2Enum_2Enum,V1L: tp__ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2EOmega_2Edark__shadow__cond__row @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V0c ) ) @ ( inj__ty_2Einteger_2Eint @ V1L ) ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) ) )
<=> $true )
& ! [V2t: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V3d: tp__ty_2Enum_2Enum,V4c: tp__ty_2Enum_2Enum,V5R: tp__ty_2Einteger_2Eint,V6L: tp__ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2EOmega_2Edark__shadow__cond__row @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V4c ) ) @ ( inj__ty_2Einteger_2Eint @ V6L ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V3d ) ) @ ( inj__ty_2Einteger_2Eint @ V5R ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2t ) ) ) )
<=> ( ~ ? [V7i: tp__ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__lt @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V4c ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V3d ) ) ) ) @ ( inj__ty_2Einteger_2Eint @ V7i ) ) ) @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V4c ) ) ) @ ( inj__ty_2Einteger_2Eint @ V5R ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__le @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V4c ) ) ) @ ( inj__ty_2Einteger_2Eint @ V5R ) ) ) @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V3d ) ) ) @ ( inj__ty_2Einteger_2Eint @ V6L ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__lt @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V3d ) ) ) @ ( inj__ty_2Einteger_2Eint @ V6L ) ) ) @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V4c ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V3d ) ) ) ) @ ( ap @ ( ap @ c_2Einteger_2Eint__add @ ( inj__ty_2Einteger_2Eint @ V7i ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Edark__shadow__cond__row @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V4c ) ) @ ( inj__ty_2Einteger_2Eint @ V6L ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Edark__shadow__condition__ind,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ bool ) ) )
=> ( ( ! [V1lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] : ( p @ ( ap @ ( ap @ V0P @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) )
& ! [V2c: tp__ty_2Enum_2Enum,V3L: tp__ty_2Einteger_2Eint,V4uppers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V5lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ V0P @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V4uppers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V5lowers ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) @ ( inj__ty_2Einteger_2Eint @ V3L ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V4uppers ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V5lowers ) ) ) ) )
=> ! [V6v: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V7v1: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] : ( p @ ( ap @ ( ap @ V0P @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V6v ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V7v1 ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Edark__shadow__condition__def,axiom,
( ! [V0lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2EOmega_2Edark__shadow__condition @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0lowers ) ) )
<=> $true )
& ! [V1uppers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V2lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V3c: tp__ty_2Enum_2Enum,V4L: tp__ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2EOmega_2Edark__shadow__condition @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V3c ) ) @ ( inj__ty_2Einteger_2Eint @ V4L ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1uppers ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2lowers ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2EOmega_2Edark__shadow__cond__row @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V3c ) ) @ ( inj__ty_2Einteger_2Eint @ V4L ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2lowers ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Edark__shadow__condition @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1uppers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2lowers ) ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Ebasic__shadow__equivalence,axiom,
! [V0uppers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V1lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] :
( ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( c_2EOmega_2Efst__nzero @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( c_2EOmega_2Efst__nzero @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) ) )
=> ( ? [V2x: tp__ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V2x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V2x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2EOmega_2Ereal__shadow @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Edark__shadow__condition @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Edark__shadow__row__ind,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Einteger_2Eint @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ bool ) ) ) )
=> ( ( ! [V1c: tp__ty_2Enum_2Enum,V2L: tp__ty_2Einteger_2Eint] : ( p @ ( ap @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V1c ) ) @ ( inj__ty_2Einteger_2Eint @ V2L ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) ) )
& ! [V3c: tp__ty_2Enum_2Enum,V4L: tp__ty_2Einteger_2Eint,V5d: tp__ty_2Enum_2Enum,V6R: tp__ty_2Einteger_2Eint,V7rs: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V3c ) ) @ ( inj__ty_2Einteger_2Eint @ V4L ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V7rs ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V3c ) ) @ ( inj__ty_2Einteger_2Eint @ V4L ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V5d ) ) @ ( inj__ty_2Einteger_2Eint @ V6R ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V7rs ) ) ) ) ) )
=> ! [V8v: tp__ty_2Enum_2Enum,V9v1: tp__ty_2Einteger_2Eint,V10v2: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] : ( p @ ( ap @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V8v ) ) @ ( inj__ty_2Einteger_2Eint @ V9v1 ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V10v2 ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Edark__shadow__row__def,axiom,
( ! [V0c: tp__ty_2Enum_2Enum,V1L: tp__ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ ( ap @ c_2EOmega_2Edark__shadow__row @ ( inj__ty_2Enum_2Enum @ V0c ) ) @ ( inj__ty_2Einteger_2Eint @ V1L ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) ) )
<=> $true )
& ! [V2rs: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V3d: tp__ty_2Enum_2Enum,V4c: tp__ty_2Enum_2Enum,V5R: tp__ty_2Einteger_2Eint,V6L: tp__ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ ( ap @ c_2EOmega_2Edark__shadow__row @ ( inj__ty_2Enum_2Enum @ V4c ) ) @ ( inj__ty_2Einteger_2Eint @ V6L ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V3d ) ) @ ( inj__ty_2Einteger_2Eint @ V5R ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2rs ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__ge @ ( ap @ ( ap @ c_2Einteger_2Eint__sub @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V3d ) ) ) @ ( inj__ty_2Einteger_2Eint @ V6L ) ) ) @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V4c ) ) ) @ ( inj__ty_2Einteger_2Eint @ V5R ) ) ) ) @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ ( ap @ c_2Einteger_2Eint__sub @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V4c ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Einteger_2Eint__sub @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V3d ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2EOmega_2Edark__shadow__row @ ( inj__ty_2Enum_2Enum @ V4c ) ) @ ( inj__ty_2Einteger_2Eint @ V6L ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2rs ) ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Edark__shadow__ind,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ bool ) ) )
=> ( ( ! [V1lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] : ( p @ ( ap @ ( ap @ V0P @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) )
& ! [V2c: tp__ty_2Enum_2Enum,V3L: tp__ty_2Einteger_2Eint,V4uppers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V5lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ V0P @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V4uppers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V5lowers ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) @ ( inj__ty_2Einteger_2Eint @ V3L ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V4uppers ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V5lowers ) ) ) ) )
=> ! [V6v: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V7v1: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] : ( p @ ( ap @ ( ap @ V0P @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V6v ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V7v1 ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Edark__shadow__def,axiom,
( ! [V0lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2EOmega_2Edark__shadow @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0lowers ) ) )
<=> $true )
& ! [V1uppers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V2lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V3c: tp__ty_2Enum_2Enum,V4L: tp__ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2EOmega_2Edark__shadow @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V3c ) ) @ ( inj__ty_2Einteger_2Eint @ V4L ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1uppers ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2lowers ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( ap @ c_2EOmega_2Edark__shadow__row @ ( inj__ty_2Enum_2Enum @ V3c ) ) @ ( inj__ty_2Einteger_2Eint @ V4L ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2lowers ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Edark__shadow @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1uppers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2lowers ) ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Enightmare__ind,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Einteger_2Eint @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ bool ) ) ) ) ) )
=> ( ( ! [V1x: tp__ty_2Einteger_2Eint,V2c: tp__ty_2Enum_2Enum,V3uppers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V4lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] : ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ ( inj__ty_2Einteger_2Eint @ V1x ) ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V3uppers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V4lowers ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) ) )
& ! [V5x: tp__ty_2Einteger_2Eint,V6c: tp__ty_2Enum_2Enum,V7uppers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V8lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V9d: tp__ty_2Enum_2Enum,V10R: tp__ty_2Einteger_2Eint,V11rs: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ ( inj__ty_2Einteger_2Eint @ V5x ) ) @ ( inj__ty_2Enum_2Enum @ V6c ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V7uppers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V8lowers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V11rs ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ ( inj__ty_2Einteger_2Eint @ V5x ) ) @ ( inj__ty_2Enum_2Enum @ V6c ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V7uppers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V8lowers ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V9d ) ) @ ( inj__ty_2Einteger_2Eint @ V10R ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V11rs ) ) ) ) ) )
=> ! [V12v: tp__ty_2Einteger_2Eint,V13v1: tp__ty_2Enum_2Enum,V14v2: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V15v3: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V16v4: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] : ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ ( inj__ty_2Einteger_2Eint @ V12v ) ) @ ( inj__ty_2Enum_2Enum @ V13v1 ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V14v2 ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V15v3 ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V16v4 ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Enightmare__def,axiom,
( ! [V0x: tp__ty_2Einteger_2Eint,V1uppers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V2lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V3c: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ c_2EOmega_2Enightmare @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V3c ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1uppers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2lowers ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) ) )
<=> $false )
& ! [V4x: tp__ty_2Einteger_2Eint,V5uppers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V6rs: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V7lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V8d: tp__ty_2Enum_2Enum,V9c: tp__ty_2Enum_2Enum,V10R: tp__ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ c_2EOmega_2Enightmare @ ( inj__ty_2Einteger_2Eint @ V4x ) ) @ ( inj__ty_2Enum_2Enum @ V9c ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V5uppers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V7lowers ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V8d ) ) @ ( inj__ty_2Einteger_2Eint @ V10R ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V6rs ) ) ) )
<=> ( ? [V11i: tp__ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__le @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Einteger_2Eint @ V11i ) ) )
& ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__le @ ( inj__ty_2Einteger_2Eint @ V11i ) ) @ ( ap @ ( ap @ c_2Einteger_2Eint__div @ ( ap @ ( ap @ c_2Einteger_2Eint__sub @ ( ap @ ( ap @ c_2Einteger_2Eint__sub @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V9c ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V8d ) ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V9c ) ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V8d ) ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V9c ) ) ) ) )
& ( ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V8d ) ) ) @ ( inj__ty_2Einteger_2Eint @ V4x ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__add @ ( inj__ty_2Einteger_2Eint @ V10R ) ) @ ( inj__ty_2Einteger_2Eint @ V11i ) ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V4x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V5uppers ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V4x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V7lowers ) ) ) )
| ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ c_2EOmega_2Enightmare @ ( inj__ty_2Einteger_2Eint @ V4x ) ) @ ( inj__ty_2Enum_2Enum @ V9c ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V5uppers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V7lowers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V6rs ) ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Enightmare__implies__LHS,axiom,
! [V0rs: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V1x: tp__ty_2Einteger_2Eint,V2uppers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V3lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V4c: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ c_2EOmega_2Enightmare @ ( inj__ty_2Einteger_2Eint @ V1x ) ) @ ( inj__ty_2Enum_2Enum @ V4c ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2uppers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V3lowers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0rs ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V1x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2uppers ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V1x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V3lowers ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Edark__shadow__FORALL,axiom,
! [V0uppers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V1lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2EOmega_2Edark__shadow @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) )
<=> ! [V2c: tp__ty_2Enum_2Enum,V3d: tp__ty_2Enum_2Enum,V4L: tp__ty_2Einteger_2Eint,V5R: tp__ty_2Einteger_2Eint] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) @ ( inj__ty_2Einteger_2Eint @ V4L ) ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V3d ) ) @ ( inj__ty_2Einteger_2Eint @ V5R ) ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__ge @ ( ap @ ( ap @ c_2Einteger_2Eint__sub @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V3d ) ) ) @ ( inj__ty_2Einteger_2Eint @ V4L ) ) ) @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V2c ) ) ) @ ( inj__ty_2Einteger_2Eint @ V5R ) ) ) ) @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ ( ap @ c_2Einteger_2Eint__sub @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V2c ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ c_2Einteger_2Eint__sub @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V3d ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Ereal__shadow__FORALL,axiom,
! [V0uppers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V1lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2EOmega_2Ereal__shadow @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) )
<=> ! [V2c: tp__ty_2Enum_2Enum,V3d: tp__ty_2Enum_2Enum,V4L: tp__ty_2Einteger_2Eint,V5R: tp__ty_2Einteger_2Eint] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) @ ( inj__ty_2Einteger_2Eint @ V4L ) ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V3d ) ) @ ( inj__ty_2Einteger_2Eint @ V5R ) ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__le @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V2c ) ) ) @ ( inj__ty_2Einteger_2Eint @ V5R ) ) ) @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V3d ) ) ) @ ( inj__ty_2Einteger_2Eint @ V4L ) ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Eevalupper__FORALL,axiom,
! [V0uppers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V1x: tp__ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V1x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) )
<=> ! [V2c: tp__ty_2Enum_2Enum,V3L: tp__ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) @ ( inj__ty_2Einteger_2Eint @ V3L ) ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__le @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V2c ) ) ) @ ( inj__ty_2Einteger_2Eint @ V1x ) ) ) @ ( inj__ty_2Einteger_2Eint @ V3L ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Eevallower__FORALL,axiom,
! [V0lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V1x: tp__ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V1x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0lowers ) ) )
<=> ! [V2d: tp__ty_2Enum_2Enum,V3R: tp__ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V2d ) ) @ ( inj__ty_2Einteger_2Eint @ V3R ) ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0lowers ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__le @ ( inj__ty_2Einteger_2Eint @ V3R ) ) @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V2d ) ) ) @ ( inj__ty_2Einteger_2Eint @ V1x ) ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Enightmare__EXISTS,axiom,
! [V0rs: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V1x: tp__ty_2Einteger_2Eint,V2c: tp__ty_2Enum_2Enum,V3uppers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V4lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ c_2EOmega_2Enightmare @ ( inj__ty_2Einteger_2Eint @ V1x ) ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V3uppers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V4lowers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0rs ) ) )
<=> ? [V5i: tp__ty_2Einteger_2Eint,V6d: tp__ty_2Enum_2Enum,V7R: tp__ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__le @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Einteger_2Eint @ V5i ) ) )
& ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__le @ ( inj__ty_2Einteger_2Eint @ V5i ) ) @ ( ap @ ( ap @ c_2Einteger_2Eint__div @ ( ap @ ( ap @ c_2Einteger_2Eint__sub @ ( ap @ ( ap @ c_2Einteger_2Eint__sub @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V6d ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V2c ) ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V2c ) ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V6d ) ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V2c ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V6d ) ) @ ( inj__ty_2Einteger_2Eint @ V7R ) ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0rs ) ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V1x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V3uppers ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V1x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V4lowers ) ) )
& ( ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V6d ) ) ) @ ( inj__ty_2Einteger_2Eint @ V1x ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__add @ ( inj__ty_2Einteger_2Eint @ V7R ) ) @ ( inj__ty_2Einteger_2Eint @ V5i ) ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Efinal__equivalence,axiom,
! [V0uppers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V1lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V2m: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( c_2EOmega_2Efst__nzero @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( c_2EOmega_2Efst__nzero @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) )
@ ( lam @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint )
@ ^ [V3p: $i] : ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Epair_2EFST @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ V3p ) ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) )
@ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) ) )
=> ( ? [V4x: tp__ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V4x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V4x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2EOmega_2Ereal__shadow @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) )
& ( ( p @ ( ap @ ( ap @ c_2EOmega_2Edark__shadow @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) )
| ? [V5x: tp__ty_2Einteger_2Eint] : ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ c_2EOmega_2Enightmare @ ( inj__ty_2Einteger_2Eint @ V5x ) ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Edarkrow__implies__realrow,axiom,
! [V0lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V1c: tp__ty_2Enum_2Enum,V2L: tp__ty_2Einteger_2Eint] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V1c ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( c_2EOmega_2Efst__nzero @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0lowers ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ c_2EOmega_2Edark__shadow__row @ ( inj__ty_2Enum_2Enum @ V1c ) ) @ ( inj__ty_2Einteger_2Eint @ V2L ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0lowers ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2EOmega_2Ershadow__row @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V1c ) ) @ ( inj__ty_2Einteger_2Eint @ V2L ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0lowers ) ) ) ) ).
thf(conj_thm_2EOmega_2Edark__implies__real,axiom,
! [V0uppers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V1lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] :
( ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( c_2EOmega_2Efst__nzero @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( c_2EOmega_2Efst__nzero @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Edark__shadow @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2EOmega_2Ereal__shadow @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) ) ) ).
thf(conj_thm_2EOmega_2Ealternative__equivalence,axiom,
! [V0uppers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V1lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V2m: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( c_2EOmega_2Efst__nzero @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( c_2EOmega_2Efst__nzero @ ty_2Einteger_2Eint ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Elist_2EEVERY @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) )
@ ( lam @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint )
@ ^ [V3p: $i] : ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Epair_2EFST @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ V3p ) ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) )
@ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) ) )
=> ( ? [V4x: tp__ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V4x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V4x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2EOmega_2Edark__shadow @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) )
| ? [V5x: tp__ty_2Einteger_2Eint] : ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ c_2EOmega_2Enightmare @ ( inj__ty_2Einteger_2Eint @ V5x ) ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V0uppers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1lowers ) ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Eeval__base,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1x: tp__ty_2Einteger_2Eint] :
( ( p @ V0p )
<=> ( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V1x ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V1x ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) ) )
& $true
& ( p @ V0p ) ) ) ) ).
thf(conj_thm_2EOmega_2Eeval__step__upper1,axiom,
! [V0x: tp__ty_2Einteger_2Eint,V1ups: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V2lows: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V3ex: $i] :
( ( mem @ V3ex @ bool )
=> ! [V4c: tp__ty_2Enum_2Enum,V5r: tp__ty_2Einteger_2Eint] :
( ( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1ups ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2lows ) ) )
& ( p @ V3ex )
& ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__le @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V4c ) ) ) @ ( inj__ty_2Einteger_2Eint @ V0x ) ) ) @ ( inj__ty_2Einteger_2Eint @ V5r ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V4c ) ) @ ( inj__ty_2Einteger_2Eint @ V5r ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1ups ) ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2lows ) ) )
& ( p @ V3ex ) ) ) ) ).
thf(conj_thm_2EOmega_2Eeval__step__upper2,axiom,
! [V0x: tp__ty_2Einteger_2Eint,V1ups: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V2lows: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V3ex: $i] :
( ( mem @ V3ex @ bool )
=> ! [V4c: tp__ty_2Enum_2Enum,V5r: tp__ty_2Einteger_2Eint,V6p: $i] :
( ( mem @ V6p @ bool )
=> ( ( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1ups ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2lows ) ) )
& ( p @ V3ex )
& ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__le @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V4c ) ) ) @ ( inj__ty_2Einteger_2Eint @ V0x ) ) ) @ ( inj__ty_2Einteger_2Eint @ V5r ) ) )
& ( p @ V6p ) )
<=> ( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V4c ) ) @ ( inj__ty_2Einteger_2Eint @ V5r ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1ups ) ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2lows ) ) )
& ( p @ V3ex )
& ( p @ V6p ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Eeval__step__lower1,axiom,
! [V0x: tp__ty_2Einteger_2Eint,V1ups: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V2lows: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V3ex: $i] :
( ( mem @ V3ex @ bool )
=> ! [V4r: tp__ty_2Einteger_2Eint,V5c: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1ups ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2lows ) ) )
& ( p @ V3ex )
& ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__le @ ( inj__ty_2Einteger_2Eint @ V4r ) ) @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V5c ) ) ) @ ( inj__ty_2Einteger_2Eint @ V0x ) ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1ups ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V5c ) ) @ ( inj__ty_2Einteger_2Eint @ V4r ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2lows ) ) ) )
& ( p @ V3ex ) ) ) ) ).
thf(conj_thm_2EOmega_2Eeval__step__lower2,axiom,
! [V0x: tp__ty_2Einteger_2Eint,V1ups: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V2lows: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V3ex: $i] :
( ( mem @ V3ex @ bool )
=> ! [V4r: tp__ty_2Einteger_2Eint,V5c: tp__ty_2Enum_2Enum,V6p: $i] :
( ( mem @ V6p @ bool )
=> ( ( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1ups ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2lows ) ) )
& ( p @ V3ex )
& ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__le @ ( inj__ty_2Einteger_2Eint @ V4r ) ) @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V5c ) ) ) @ ( inj__ty_2Einteger_2Eint @ V0x ) ) ) )
& ( p @ V6p ) )
<=> ( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1ups ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V5c ) ) @ ( inj__ty_2Einteger_2Eint @ V4r ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2lows ) ) ) )
& ( p @ V3ex )
& ( p @ V6p ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Eeval__step__extra1,axiom,
! [V0x: tp__ty_2Einteger_2Eint,V1ups: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V2lows: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V3ex_27: $i] :
( ( mem @ V3ex_27 @ bool )
=> ( ( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1ups ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2lows ) ) )
& $true
& ( p @ V3ex_27 ) )
<=> ( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1ups ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2lows ) ) )
& ( p @ V3ex_27 ) ) ) ) ).
thf(conj_thm_2EOmega_2Eeval__step__extra2,axiom,
! [V0x: tp__ty_2Einteger_2Eint,V1ups: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V2lows: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V3ex: $i] :
( ( mem @ V3ex @ bool )
=> ! [V4ex_27: $i] :
( ( mem @ V4ex_27 @ bool )
=> ( ( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1ups ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2lows ) ) )
& ( p @ V3ex )
& ( p @ V4ex_27 ) )
<=> ( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1ups ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2lows ) ) )
& ( p @ V3ex )
& ( p @ V4ex_27 ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Eeval__step__extra3,axiom,
! [V0x: tp__ty_2Einteger_2Eint,V1ups: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V2lows: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V3ex_27: $i] :
( ( mem @ V3ex_27 @ bool )
=> ! [V4p: $i] :
( ( mem @ V4p @ bool )
=> ( ( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1ups ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2lows ) ) )
& $true
& ( p @ V3ex_27 )
& ( p @ V4p ) )
<=> ( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1ups ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2lows ) ) )
& ( p @ V3ex_27 )
& ( p @ V4p ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Eeval__step__extra4,axiom,
! [V0x: tp__ty_2Einteger_2Eint,V1ups: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V2lows: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V3ex: $i] :
( ( mem @ V3ex @ bool )
=> ! [V4ex_27: $i] :
( ( mem @ V4ex_27 @ bool )
=> ! [V5p: $i] :
( ( mem @ V5p @ bool )
=> ( ( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1ups ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2lows ) ) )
& ( p @ V3ex )
& ( p @ V4ex_27 )
& ( p @ V5p ) )
<=> ( ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V1ups ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2lows ) ) )
& ( p @ V3ex )
& ( p @ V4ex_27 )
& ( p @ V5p ) ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Ecalc__nightmare__ind,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Einteger_2Eint @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ bool ) ) ) )
=> ( ( ! [V1x: tp__ty_2Einteger_2Eint,V2c: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( ap @ V0P @ ( inj__ty_2Einteger_2Eint @ V1x ) ) @ ( inj__ty_2Enum_2Enum @ V2c ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) ) )
& ! [V3x: tp__ty_2Einteger_2Eint,V4c: tp__ty_2Enum_2Enum,V5d: tp__ty_2Enum_2Enum,V6R: tp__ty_2Einteger_2Eint,V7rs: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ ( ap @ V0P @ ( inj__ty_2Einteger_2Eint @ V3x ) ) @ ( inj__ty_2Enum_2Enum @ V4c ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V7rs ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ ( inj__ty_2Einteger_2Eint @ V3x ) ) @ ( inj__ty_2Enum_2Enum @ V4c ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V5d ) ) @ ( inj__ty_2Einteger_2Eint @ V6R ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V7rs ) ) ) ) ) )
=> ! [V8v: tp__ty_2Einteger_2Eint,V9v1: tp__ty_2Enum_2Enum,V10v2: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] : ( p @ ( ap @ ( ap @ ( ap @ V0P @ ( inj__ty_2Einteger_2Eint @ V8v ) ) @ ( inj__ty_2Enum_2Enum @ V9v1 ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V10v2 ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Ecalc__nightmare__def,axiom,
( ! [V0x: tp__ty_2Einteger_2Eint,V1c: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( ap @ c_2EOmega_2Ecalc__nightmare @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1c ) ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) ) )
<=> $false )
& ! [V2x: tp__ty_2Einteger_2Eint,V3rs: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V4d: tp__ty_2Enum_2Enum,V5c: tp__ty_2Enum_2Enum,V6R: tp__ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ ( ap @ c_2EOmega_2Ecalc__nightmare @ ( inj__ty_2Einteger_2Eint @ V2x ) ) @ ( inj__ty_2Enum_2Enum @ V5c ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Einteger_2Eint ) @ ( inj__ty_2Enum_2Enum @ V4d ) ) @ ( inj__ty_2Einteger_2Eint @ V6R ) ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V3rs ) ) ) )
<=> ( ? [V7i: tp__ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__le @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Einteger_2Eint @ V7i ) ) )
& ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__le @ ( inj__ty_2Einteger_2Eint @ V7i ) ) @ ( ap @ ( ap @ c_2Einteger_2Eint__div @ ( ap @ ( ap @ c_2Einteger_2Eint__sub @ ( ap @ ( ap @ c_2Einteger_2Eint__sub @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V5c ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V4d ) ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V5c ) ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V4d ) ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V5c ) ) ) ) )
& ( ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V4d ) ) ) @ ( inj__ty_2Einteger_2Eint @ V2x ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__add @ ( inj__ty_2Einteger_2Eint @ V6R ) ) @ ( inj__ty_2Einteger_2Eint @ V7i ) ) ) ) )
| ( p @ ( ap @ ( ap @ ( ap @ c_2EOmega_2Ecalc__nightmare @ ( inj__ty_2Einteger_2Eint @ V2x ) ) @ ( inj__ty_2Enum_2Enum @ V5c ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V3rs ) ) ) ) ) ) ).
thf(conj_thm_2EOmega_2Ecalculational__nightmare,axiom,
! [V0x: tp__ty_2Einteger_2Eint,V1c: tp__ty_2Enum_2Enum,V2uppers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V3lowers: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint,V4rs: tp__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ c_2EOmega_2Enightmare @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1c ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2uppers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V3lowers ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V4rs ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( ap @ c_2EOmega_2Ecalc__nightmare @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1c ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V4rs ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevalupper @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V2uppers ) ) )
& ( p @ ( ap @ ( ap @ c_2EOmega_2Eevallower @ ( inj__ty_2Einteger_2Eint @ V0x ) ) @ ( inj__c_ty_2Elist_2Elist_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Einteger_2Eint @ V3lowers ) ) ) ) ) ).
%------------------------------------------------------------------------------