ITP001 Axioms: ITP120^5.ax
%------------------------------------------------------------------------------
% File : ITP120^5 : 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 : intreal^2.ax [Gau20]
% : HL4120^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 35 ( 13 unt; 7 typ; 0 def)
% Number of atoms : 644 ( 28 equ; 0 cnn)
% Maximal formula atoms : 27 ( 18 avg)
% Number of connectives : 741 ( 0 ~; 0 |; 12 &; 720 @)
% ( 6 <=>; 3 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 6 avg; 720 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 3 ( 3 >; 0 *; 0 +; 0 <<)
% Number of symbols : 51 ( 50 usr; 47 con; 0-2 aty)
% Number of variables : 42 ( 3 ^ 39 !; 0 ?; 42 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Eintreal_2EINT__CEILING,type,
c_2Eintreal_2EINT__CEILING: $i ).
thf(mem_c_2Eintreal_2EINT__CEILING,axiom,
mem @ c_2Eintreal_2EINT__CEILING @ ( arr @ ty_2Erealax_2Ereal @ ty_2Einteger_2Eint ) ).
thf(stp_fo_c_2Eintreal_2EINT__CEILING,type,
fo__c_2Eintreal_2EINT__CEILING: tp__ty_2Erealax_2Ereal > tp__ty_2Einteger_2Eint ).
thf(stp_eq_fo_c_2Eintreal_2EINT__CEILING,axiom,
! [X0: tp__ty_2Erealax_2Ereal] :
( ( inj__ty_2Einteger_2Eint @ ( fo__c_2Eintreal_2EINT__CEILING @ X0 ) )
= ( ap @ c_2Eintreal_2EINT__CEILING @ ( inj__ty_2Erealax_2Ereal @ X0 ) ) ) ).
thf(tp_c_2Eintreal_2EINT__FLOOR,type,
c_2Eintreal_2EINT__FLOOR: $i ).
thf(mem_c_2Eintreal_2EINT__FLOOR,axiom,
mem @ c_2Eintreal_2EINT__FLOOR @ ( arr @ ty_2Erealax_2Ereal @ ty_2Einteger_2Eint ) ).
thf(stp_fo_c_2Eintreal_2EINT__FLOOR,type,
fo__c_2Eintreal_2EINT__FLOOR: tp__ty_2Erealax_2Ereal > tp__ty_2Einteger_2Eint ).
thf(stp_eq_fo_c_2Eintreal_2EINT__FLOOR,axiom,
! [X0: tp__ty_2Erealax_2Ereal] :
( ( inj__ty_2Einteger_2Eint @ ( fo__c_2Eintreal_2EINT__FLOOR @ X0 ) )
= ( ap @ c_2Eintreal_2EINT__FLOOR @ ( inj__ty_2Erealax_2Ereal @ X0 ) ) ) ).
thf(tp_c_2Eintreal_2Eis__int,type,
c_2Eintreal_2Eis__int: $i ).
thf(mem_c_2Eintreal_2Eis__int,axiom,
mem @ c_2Eintreal_2Eis__int @ ( arr @ ty_2Erealax_2Ereal @ bool ) ).
thf(tp_c_2Eintreal_2Ereal__of__int,type,
c_2Eintreal_2Ereal__of__int: $i ).
thf(mem_c_2Eintreal_2Ereal__of__int,axiom,
mem @ c_2Eintreal_2Ereal__of__int @ ( arr @ ty_2Einteger_2Eint @ ty_2Erealax_2Ereal ) ).
thf(stp_fo_c_2Eintreal_2Ereal__of__int,type,
fo__c_2Eintreal_2Ereal__of__int: tp__ty_2Einteger_2Eint > tp__ty_2Erealax_2Ereal ).
thf(stp_eq_fo_c_2Eintreal_2Ereal__of__int,axiom,
! [X0: tp__ty_2Einteger_2Eint] :
( ( inj__ty_2Erealax_2Ereal @ ( fo__c_2Eintreal_2Ereal__of__int @ X0 ) )
= ( ap @ c_2Eintreal_2Ereal__of__int @ ( inj__ty_2Einteger_2Eint @ X0 ) ) ) ).
thf(ax_thm_2Eintreal_2Ereal__of__int,axiom,
! [V0i: tp__ty_2Einteger_2Eint] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( inj__ty_2Einteger_2Eint @ V0i ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Einteger_2Eint__lt @ ( inj__ty_2Einteger_2Eint @ V0i ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Einteger_2ENum @ ( ap @ c_2Einteger_2Eint__neg @ ( inj__ty_2Einteger_2Eint @ V0i ) ) ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Einteger_2ENum @ ( inj__ty_2Einteger_2Eint @ V0i ) ) ) ) ) ) ).
thf(conj_thm_2Eintreal_2Ereal__of__int__def,axiom,
! [V0i: tp__ty_2Einteger_2Eint] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( inj__ty_2Einteger_2Eint @ V0i ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ c_2Einteger_2Eint__lt @ ( inj__ty_2Einteger_2Eint @ V0i ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Einteger_2ENum @ ( ap @ c_2Einteger_2Eint__neg @ ( inj__ty_2Einteger_2Eint @ V0i ) ) ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Einteger_2ENum @ ( inj__ty_2Einteger_2Eint @ V0i ) ) ) ) ) ) ).
thf(ax_thm_2Eintreal_2EINT__FLOOR__def,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Eintreal_2EINT__FLOOR @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= ( surj__ty_2Einteger_2Eint
@ ( ap @ c_2Einteger_2ELEAST__INT
@ ( lam @ ty_2Einteger_2Eint
@ ^ [V1i: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( ap @ ( ap @ c_2Einteger_2Eint__add @ V1i ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eintreal_2EINT__CEILING__def,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Eintreal_2EINT__CEILING @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
= ( surj__ty_2Einteger_2Eint
@ ( ap @ c_2Einteger_2ELEAST__INT
@ ( lam @ ty_2Einteger_2Eint
@ ^ [V1i: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ c_2Eintreal_2Ereal__of__int @ V1i ) ) ) ) ) ) ).
thf(ax_thm_2Eintreal_2Eis__int__def,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ c_2Eintreal_2Eis__int @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) )
<=> ( V0x
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( ap @ c_2Eintreal_2EINT__FLOOR @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ) ).
thf(conj_thm_2Eintreal_2Ereal__of__int__monotonic,axiom,
! [V0i: tp__ty_2Einteger_2Eint,V1j: tp__ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__lt @ ( inj__ty_2Einteger_2Eint @ V0i ) ) @ ( inj__ty_2Einteger_2Eint @ V1j ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( inj__ty_2Einteger_2Eint @ V0i ) ) ) @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( inj__ty_2Einteger_2Eint @ V1j ) ) ) ) ) ).
thf(conj_thm_2Eintreal_2EINT__FLOOR__BOUNDS,axiom,
! [V0r: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( ap @ c_2Eintreal_2EINT__FLOOR @ ( inj__ty_2Erealax_2Ereal @ V0r ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0r ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0r ) ) @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( ap @ ( ap @ c_2Einteger_2Eint__add @ ( ap @ c_2Eintreal_2EINT__FLOOR @ ( inj__ty_2Erealax_2Ereal @ V0r ) ) ) @ ( 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_2Eintreal_2EINT__FLOOR,axiom,
! [V0r: tp__ty_2Erealax_2Ereal,V1i: tp__ty_2Einteger_2Eint] :
( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Eintreal_2EINT__FLOOR @ ( inj__ty_2Erealax_2Ereal @ V0r ) ) )
= V1i )
<=> ( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( inj__ty_2Einteger_2Eint @ V1i ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0r ) ) )
& ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0r ) ) @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( ap @ ( ap @ c_2Einteger_2Eint__add @ ( inj__ty_2Einteger_2Eint @ V1i ) ) @ ( 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_2Eintreal_2EINT__CEILING__INT__FLOOR,axiom,
! [V0r: tp__ty_2Erealax_2Ereal] :
( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Eintreal_2EINT__CEILING @ ( inj__ty_2Erealax_2Ereal @ V0r ) ) )
= ( surj__ty_2Einteger_2Eint
@ ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ty_2Einteger_2Eint @ ty_2Einteger_2Eint )
@ ( lam @ ty_2Einteger_2Eint
@ ^ [V1i: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Einteger_2Eint ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ c_2Eintreal_2Ereal__of__int @ V1i ) ) @ ( inj__ty_2Erealax_2Ereal @ V0r ) ) ) @ V1i ) @ ( ap @ ( ap @ c_2Einteger_2Eint__add @ V1i ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) )
@ ( ap @ c_2Eintreal_2EINT__FLOOR @ ( inj__ty_2Erealax_2Ereal @ V0r ) ) ) ) ) ).
thf(conj_thm_2Eintreal_2EINT__CEILING__BOUNDS,axiom,
! [V0r: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( ap @ ( ap @ c_2Einteger_2Eint__sub @ ( ap @ c_2Eintreal_2EINT__CEILING @ ( inj__ty_2Erealax_2Ereal @ V0r ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0r ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0r ) ) @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( ap @ c_2Eintreal_2EINT__CEILING @ ( inj__ty_2Erealax_2Ereal @ V0r ) ) ) ) ) ) ).
thf(conj_thm_2Eintreal_2EINT__CEILING,axiom,
! [V0r: tp__ty_2Erealax_2Ereal,V1i: tp__ty_2Einteger_2Eint] :
( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Eintreal_2EINT__CEILING @ ( inj__ty_2Erealax_2Ereal @ V0r ) ) )
= V1i )
<=> ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( ap @ ( ap @ c_2Einteger_2Eint__sub @ ( inj__ty_2Einteger_2Eint @ V1i ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V0r ) ) )
& ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( inj__ty_2Erealax_2Ereal @ V0r ) ) @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( inj__ty_2Einteger_2Eint @ V1i ) ) ) ) ) ) ).
thf(conj_thm_2Eintreal_2EINT__FLOOR__EQNS,axiom,
( ! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Eintreal_2EINT__FLOOR @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
& ! [V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Eintreal_2EINT__FLOOR @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__neg @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) )
& ! [V2n: tp__ty_2Enum_2Enum,V3m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) )
=> ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Eintreal_2EINT__FLOOR @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__div @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) ) ) )
& ! [V4n: tp__ty_2Enum_2Enum,V5m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V5m ) ) )
=> ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Eintreal_2EINT__FLOOR @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V5m ) ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__div @ ( ap @ c_2Einteger_2Eint__neg @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V5m ) ) ) ) ) ) ) ).
thf(conj_thm_2Eintreal_2EINT__FLOOR__compute,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Eintreal_2EINT__FLOOR @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
& ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Eintreal_2EINT__FLOOR @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Einteger_2Eint__neg @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
& ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Eintreal_2EINT__FLOOR @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__div @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) ) )
& ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Eintreal_2EINT__FLOOR @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__div @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) ) )
& ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Eintreal_2EINT__FLOOR @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__div @ ( ap @ c_2Einteger_2Eint__neg @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) ) )
& ( ( surj__ty_2Einteger_2Eint @ ( ap @ c_2Eintreal_2EINT__FLOOR @ ( ap @ ( ap @ c_2Ereal_2E_2F @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) ) )
= ( surj__ty_2Einteger_2Eint @ ( ap @ ( ap @ c_2Einteger_2Eint__div @ ( ap @ c_2Einteger_2Eint__neg @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eintreal_2Ereal__of__int__num,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ).
thf(conj_thm_2Eintreal_2Ereal__of__int__add,axiom,
! [V0m: tp__ty_2Einteger_2Eint,V1n: tp__ty_2Einteger_2Eint] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( ap @ ( ap @ c_2Einteger_2Eint__add @ ( inj__ty_2Einteger_2Eint @ V0m ) ) @ ( inj__ty_2Einteger_2Eint @ V1n ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( inj__ty_2Einteger_2Eint @ V0m ) ) ) @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( inj__ty_2Einteger_2Eint @ V1n ) ) ) ) ) ).
thf(conj_thm_2Eintreal_2Ereal__of__int__neg,axiom,
! [V0m: tp__ty_2Einteger_2Eint] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( ap @ c_2Einteger_2Eint__neg @ ( inj__ty_2Einteger_2Eint @ V0m ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( inj__ty_2Einteger_2Eint @ V0m ) ) ) ) ) ).
thf(conj_thm_2Eintreal_2Ereal__of__int__sub,axiom,
! [V0m: tp__ty_2Einteger_2Eint,V1n: tp__ty_2Einteger_2Eint] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( ap @ ( ap @ c_2Einteger_2Eint__sub @ ( inj__ty_2Einteger_2Eint @ V0m ) ) @ ( inj__ty_2Einteger_2Eint @ V1n ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( inj__ty_2Einteger_2Eint @ V0m ) ) ) @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( inj__ty_2Einteger_2Eint @ V1n ) ) ) ) ) ).
thf(conj_thm_2Eintreal_2Ereal__of__int__mul,axiom,
! [V0m: tp__ty_2Einteger_2Eint,V1n: tp__ty_2Einteger_2Eint] :
( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( ap @ ( ap @ c_2Einteger_2Eint__mul @ ( inj__ty_2Einteger_2Eint @ V0m ) ) @ ( inj__ty_2Einteger_2Eint @ V1n ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( inj__ty_2Einteger_2Eint @ V0m ) ) ) @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( inj__ty_2Einteger_2Eint @ V1n ) ) ) ) ) ).
thf(conj_thm_2Eintreal_2Ereal__of__int__lt,axiom,
! [V0m: tp__ty_2Einteger_2Eint,V1n: tp__ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( inj__ty_2Einteger_2Eint @ V0m ) ) ) @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( inj__ty_2Einteger_2Eint @ V1n ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__lt @ ( inj__ty_2Einteger_2Eint @ V0m ) ) @ ( inj__ty_2Einteger_2Eint @ V1n ) ) ) ) ).
thf(conj_thm_2Eintreal_2Ereal__of__int__11,axiom,
! [V0m: tp__ty_2Einteger_2Eint,V1n: tp__ty_2Einteger_2Eint] :
( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( inj__ty_2Einteger_2Eint @ V0m ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( inj__ty_2Einteger_2Eint @ V1n ) ) ) )
<=> ( V0m = V1n ) ) ).
thf(conj_thm_2Eintreal_2Ereal__of__int__le,axiom,
! [V0m: tp__ty_2Einteger_2Eint,V1n: tp__ty_2Einteger_2Eint] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( inj__ty_2Einteger_2Eint @ V0m ) ) ) @ ( ap @ c_2Eintreal_2Ereal__of__int @ ( inj__ty_2Einteger_2Eint @ V1n ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Einteger_2Eint__le @ ( inj__ty_2Einteger_2Eint @ V0m ) ) @ ( inj__ty_2Einteger_2Eint @ V1n ) ) ) ) ).
%------------------------------------------------------------------------------