TPTP Problem File: ITP003^2.p
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%------------------------------------------------------------------------------
% File : ITP003^2 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory export of thm_2Earithmetic_2EMOD__2.p, bushy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : thm_2Earithmetic_2EMOD__2.p [Gau19]
% : HL401001^2.p [TPAP]
% Status : Theorem
% Rating : 1.00 v8.2.0, 0.85 v8.1.0, 0.82 v7.5.0
% Syntax : Number of formulae : 152 ( 23 unt; 44 typ; 0 def)
% Number of atoms : 737 ( 34 equ; 0 cnn)
% Maximal formula atoms : 21 ( 6 avg)
% Number of connectives : 1404 ( 60 ~; 53 |; 61 &;1011 @)
% ( 71 <=>; 148 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Number of types : 4 ( 2 usr)
% Number of type conns : 29 ( 29 >; 0 *; 0 +; 0 <<)
% Number of symbols : 50 ( 47 usr; 30 con; 0-2 aty)
% Number of variables : 189 ( 0 ^; 171 !; 18 ?; 189 :)
% SPC : TH0_THM_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001^2.ax').
%------------------------------------------------------------------------------
thf(tp_ty_2Enum_2Enum,type,
ty_2Enum_2Enum: del ).
thf(stp_ty_2Enum_2Enum,type,
tp__ty_2Enum_2Enum: $tType ).
thf(stp_inj_ty_2Enum_2Enum,type,
inj__ty_2Enum_2Enum: tp__ty_2Enum_2Enum > $i ).
thf(stp_surj_ty_2Enum_2Enum,type,
surj__ty_2Enum_2Enum: $i > tp__ty_2Enum_2Enum ).
thf(stp_inj_surj_ty_2Enum_2Enum,axiom,
! [X: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( inj__ty_2Enum_2Enum @ X ) )
= X ) ).
thf(stp_inj_mem_ty_2Enum_2Enum,axiom,
! [X: tp__ty_2Enum_2Enum] : ( mem @ ( inj__ty_2Enum_2Enum @ X ) @ ty_2Enum_2Enum ) ).
thf(stp_iso_mem_ty_2Enum_2Enum,axiom,
! [X: $i] :
( ( mem @ X @ ty_2Enum_2Enum )
=> ( X
= ( inj__ty_2Enum_2Enum @ ( surj__ty_2Enum_2Enum @ X ) ) ) ) ).
thf(tp_c_2Earithmetic_2EBIT1,type,
c_2Earithmetic_2EBIT1: $i ).
thf(mem_c_2Earithmetic_2EBIT1,axiom,
mem @ c_2Earithmetic_2EBIT1 @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ).
thf(stp_fo_c_2Earithmetic_2EBIT1,type,
fo__c_2Earithmetic_2EBIT1: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Earithmetic_2EBIT1,axiom,
! [X0: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Earithmetic_2EBIT1 @ X0 ) )
= ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ X0 ) ) ) ).
thf(tp_c_2Earithmetic_2EEVEN,type,
c_2Earithmetic_2EEVEN: $i ).
thf(mem_c_2Earithmetic_2EEVEN,axiom,
mem @ c_2Earithmetic_2EEVEN @ ( arr @ ty_2Enum_2Enum @ bool ) ).
thf(tp_c_2Earithmetic_2EZERO,type,
c_2Earithmetic_2EZERO: $i ).
thf(mem_c_2Earithmetic_2EZERO,axiom,
mem @ c_2Earithmetic_2EZERO @ ty_2Enum_2Enum ).
thf(stp_fo_c_2Earithmetic_2EZERO,type,
fo__c_2Earithmetic_2EZERO: tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Earithmetic_2EZERO,axiom,
( ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO )
= c_2Earithmetic_2EZERO ) ).
thf(tp_c_2Earithmetic_2EBIT2,type,
c_2Earithmetic_2EBIT2: $i ).
thf(mem_c_2Earithmetic_2EBIT2,axiom,
mem @ c_2Earithmetic_2EBIT2 @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ).
thf(stp_fo_c_2Earithmetic_2EBIT2,type,
fo__c_2Earithmetic_2EBIT2: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Earithmetic_2EBIT2,axiom,
! [X0: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Earithmetic_2EBIT2 @ X0 ) )
= ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ X0 ) ) ) ).
thf(tp_c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: $i ).
thf(mem_c_2Earithmetic_2ENUMERAL,axiom,
mem @ c_2Earithmetic_2ENUMERAL @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ).
thf(stp_fo_c_2Earithmetic_2ENUMERAL,type,
fo__c_2Earithmetic_2ENUMERAL: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Earithmetic_2ENUMERAL,axiom,
! [X0: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Earithmetic_2ENUMERAL @ X0 ) )
= ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ X0 ) ) ) ).
thf(tp_c_2Earithmetic_2EODD,type,
c_2Earithmetic_2EODD: $i ).
thf(mem_c_2Earithmetic_2EODD,axiom,
mem @ c_2Earithmetic_2EODD @ ( arr @ ty_2Enum_2Enum @ bool ) ).
thf(tp_c_2Earithmetic_2EMOD,type,
c_2Earithmetic_2EMOD: $i ).
thf(mem_c_2Earithmetic_2EMOD,axiom,
mem @ c_2Earithmetic_2EMOD @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) ).
thf(stp_fo_c_2Earithmetic_2EMOD,type,
fo__c_2Earithmetic_2EMOD: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Earithmetic_2EMOD,axiom,
! [X0: tp__ty_2Enum_2Enum,X1: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Earithmetic_2EMOD @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Earithmetic_2EMOD @ ( inj__ty_2Enum_2Enum @ X0 ) ) @ ( inj__ty_2Enum_2Enum @ X1 ) ) ) ).
thf(tp_c_2Earithmetic_2E_2A,type,
c_2Earithmetic_2E_2A: $i ).
thf(mem_c_2Earithmetic_2E_2A,axiom,
mem @ c_2Earithmetic_2E_2A @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) ).
thf(stp_fo_c_2Earithmetic_2E_2A,type,
fo__c_2Earithmetic_2E_2A: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Earithmetic_2E_2A,axiom,
! [X0: tp__ty_2Enum_2Enum,X1: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Earithmetic_2E_2A @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ X0 ) ) @ ( inj__ty_2Enum_2Enum @ X1 ) ) ) ).
thf(tp_c_2Earithmetic_2E_2B,type,
c_2Earithmetic_2E_2B: $i ).
thf(mem_c_2Earithmetic_2E_2B,axiom,
mem @ c_2Earithmetic_2E_2B @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) ).
thf(stp_fo_c_2Earithmetic_2E_2B,type,
fo__c_2Earithmetic_2E_2B: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Earithmetic_2E_2B,axiom,
! [X0: tp__ty_2Enum_2Enum,X1: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Earithmetic_2E_2B @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ X0 ) ) @ ( inj__ty_2Enum_2Enum @ X1 ) ) ) ).
thf(tp_c_2Ebool_2ET,type,
c_2Ebool_2ET: $i ).
thf(mem_c_2Ebool_2ET,axiom,
mem @ c_2Ebool_2ET @ bool ).
thf(ax_true_p,axiom,
p @ c_2Ebool_2ET ).
thf(tp_c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND: del > $i ).
thf(mem_c_2Ebool_2ECOND,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebool_2ECOND @ A_27a ) @ ( arr @ bool @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) ) ).
thf(tp_c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F: del > $i ).
thf(mem_c_2Ebool_2E_3F,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebool_2E_3F @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ).
thf(ax_ex_p,axiom,
! [A: del,Q: $i] :
( ( mem @ Q @ ( arr @ A @ bool ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2E_3F @ A ) @ Q ) )
<=> ? [X: $i] :
( ( mem @ X @ A )
& ( p @ ( ap @ Q @ X ) ) ) ) ) ).
thf(tp_c_2Enum_2ESUC,type,
c_2Enum_2ESUC: $i ).
thf(mem_c_2Enum_2ESUC,axiom,
mem @ c_2Enum_2ESUC @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ).
thf(stp_fo_c_2Enum_2ESUC,type,
fo__c_2Enum_2ESUC: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Enum_2ESUC,axiom,
! [X0: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Enum_2ESUC @ X0 ) )
= ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ X0 ) ) ) ).
thf(tp_c_2Enum_2E0,type,
c_2Enum_2E0: $i ).
thf(mem_c_2Enum_2E0,axiom,
mem @ c_2Enum_2E0 @ ty_2Enum_2Enum ).
thf(stp_fo_c_2Enum_2E0,type,
fo__c_2Enum_2E0: tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Enum_2E0,axiom,
( ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 )
= c_2Enum_2E0 ) ).
thf(tp_c_2Eprim__rec_2E_3C,type,
c_2Eprim__rec_2E_3C: $i ).
thf(mem_c_2Eprim__rec_2E_3C,axiom,
mem @ c_2Eprim__rec_2E_3C @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) ).
thf(tp_c_2Ebool_2EF,type,
c_2Ebool_2EF: $i ).
thf(mem_c_2Ebool_2EF,axiom,
mem @ c_2Ebool_2EF @ bool ).
thf(ax_false_p,axiom,
~ ( p @ c_2Ebool_2EF ) ).
thf(tp_c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $i ).
thf(mem_c_2Ebool_2E_2F_5C,axiom,
mem @ c_2Ebool_2E_2F_5C @ ( arr @ bool @ ( arr @ bool @ bool ) ) ).
thf(ax_and_p,axiom,
! [Q: $i] :
( ( mem @ Q @ bool )
=> ! [R: $i] :
( ( mem @ R @ bool )
=> ( ( p @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ Q ) @ R ) )
<=> ( ( p @ Q )
& ( p @ R ) ) ) ) ) ).
thf(tp_c_2Emin_2E_3D,type,
c_2Emin_2E_3D: del > $i ).
thf(mem_c_2Emin_2E_3D,axiom,
! [A_27a: del] : ( mem @ ( c_2Emin_2E_3D @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ).
thf(ax_eq_p,axiom,
! [A: del,X: $i] :
( ( mem @ X @ A )
=> ! [Y: $i] :
( ( mem @ Y @ A )
=> ( ( p @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A ) @ X ) @ Y ) )
<=> ( X = Y ) ) ) ) ).
thf(tp_c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $i ).
thf(mem_c_2Ebool_2E_5C_2F,axiom,
mem @ c_2Ebool_2E_5C_2F @ ( arr @ bool @ ( arr @ bool @ bool ) ) ).
thf(ax_or_p,axiom,
! [Q: $i] :
( ( mem @ Q @ bool )
=> ! [R: $i] :
( ( mem @ R @ bool )
=> ( ( p @ ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ Q ) @ R ) )
<=> ( ( p @ Q )
| ( p @ R ) ) ) ) ) ).
thf(tp_c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $i ).
thf(mem_c_2Ebool_2E_7E,axiom,
mem @ c_2Ebool_2E_7E @ ( arr @ bool @ bool ) ).
thf(ax_neg_p,axiom,
! [Q: $i] :
( ( mem @ Q @ bool )
=> ( ( p @ ( ap @ c_2Ebool_2E_7E @ Q ) )
<=> ~ ( p @ Q ) ) ) ).
thf(tp_c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $i ).
thf(mem_c_2Emin_2E_3D_3D_3E,axiom,
mem @ c_2Emin_2E_3D_3D_3E @ ( arr @ bool @ ( arr @ bool @ bool ) ) ).
thf(ax_imp_p,axiom,
! [Q: $i] :
( ( mem @ Q @ bool )
=> ! [R: $i] :
( ( mem @ R @ bool )
=> ( ( p @ ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ Q ) @ R ) )
<=> ( ( p @ Q )
=> ( p @ R ) ) ) ) ) ).
thf(tp_c_2Ebool_2E_21,type,
c_2Ebool_2E_21: del > $i ).
thf(mem_c_2Ebool_2E_21,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebool_2E_21 @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ).
thf(ax_all_p,axiom,
! [A: del,Q: $i] :
( ( mem @ Q @ ( arr @ A @ bool ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2E_21 @ A ) @ Q ) )
<=> ! [X: $i] :
( ( mem @ X @ A )
=> ( p @ ( ap @ Q @ X ) ) ) ) ) ).
thf(conj_thm_2Earithmetic_2EONE,axiom,
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(conj_thm_2Earithmetic_2ETWO,axiom,
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Earithmetic_2EADD__0,axiom,
! [V0m: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= V0m ) ).
thf(conj_thm_2Earithmetic_2ELESS__MONO__EQ,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Earithmetic_2EADD1,axiom,
! [V0m: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Earithmetic_2EMULT__COMM,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) ) ).
thf(conj_thm_2Earithmetic_2EEVEN__ODD,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
<=> ~ ( p @ ( ap @ c_2Earithmetic_2EODD @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ).
thf(conj_thm_2Earithmetic_2EODD__EVEN,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ c_2Earithmetic_2EODD @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
<=> ~ ( p @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ).
thf(conj_thm_2Earithmetic_2EEVEN__EXISTS,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
<=> ? [V1m: tp__ty_2Enum_2Enum] :
( V0n
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) ).
thf(conj_thm_2Earithmetic_2EODD__EXISTS,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ c_2Earithmetic_2EODD @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
<=> ? [V1m: tp__ty_2Enum_2Enum] :
( V0n
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) ) ).
thf(conj_thm_2Earithmetic_2EMOD__UNIQUE,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1k: tp__ty_2Enum_2Enum,V2r: tp__ty_2Enum_2Enum] :
( ? [V3q: tp__ty_2Enum_2Enum] :
( ( V1k
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V3q ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Enum_2Enum @ V2r ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2r ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EMOD @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= V2r ) ) ).
thf(ax_thm_2Ebool_2EBOOL__CASES__AX,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( p @ V0t )
<=> $true )
| ( ( p @ V0t )
<=> $false ) ) ) ).
thf(conj_thm_2Ebool_2ETRUTH,axiom,
$true ).
thf(conj_thm_2Ebool_2EIMP__ANTISYM__AX,axiom,
! [V0t1: $i] :
( ( mem @ V0t1 @ bool )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ bool )
=> ( ( ( p @ V0t1 )
=> ( p @ V1t2 ) )
=> ( ( ( p @ V1t2 )
=> ( p @ V0t1 ) )
=> ( ( p @ V0t1 )
<=> ( p @ V1t2 ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EFALSITY,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( $false
=> ( p @ V0t ) ) ) ).
thf(conj_thm_2Ebool_2EEXCLUDED__MIDDLE,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( p @ V0t )
| ~ ( p @ V0t ) ) ) ).
thf(conj_thm_2Ebool_2EIMP__F,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( p @ V0t )
=> $false )
=> ~ ( p @ V0t ) ) ) ).
thf(conj_thm_2Ebool_2EF__IMP,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ~ ( p @ V0t )
=> ( ( p @ V0t )
=> $false ) ) ) ).
thf(conj_thm_2Ebool_2EAND__CLAUSES,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( $true
& ( p @ V0t ) )
<=> ( p @ V0t ) )
& ( ( ( p @ V0t )
& $true )
<=> ( p @ V0t ) )
& ( ( $false
& ( p @ V0t ) )
<=> $false )
& ( ( ( p @ V0t )
& $false )
<=> $false )
& ( ( ( p @ V0t )
& ( p @ V0t ) )
<=> ( p @ V0t ) ) ) ) ).
thf(conj_thm_2Ebool_2EOR__CLAUSES,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( $true
| ( p @ V0t ) )
<=> $true )
& ( ( ( p @ V0t )
| $true )
<=> $true )
& ( ( $false
| ( p @ V0t ) )
<=> ( p @ V0t ) )
& ( ( ( p @ V0t )
| $false )
<=> ( p @ V0t ) )
& ( ( ( p @ V0t )
| ( p @ V0t ) )
<=> ( p @ V0t ) ) ) ) ).
thf(conj_thm_2Ebool_2EIMP__CLAUSES,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( $true
=> ( p @ V0t ) )
<=> ( p @ V0t ) )
& ( ( ( p @ V0t )
=> $true )
<=> $true )
& ( ( $false
=> ( p @ V0t ) )
<=> $true )
& ( ( ( p @ V0t )
=> ( p @ V0t ) )
<=> $true )
& ( ( ( p @ V0t )
=> $false )
<=> ~ ( p @ V0t ) ) ) ) ).
thf(conj_thm_2Ebool_2ENOT__CLAUSES,axiom,
( ! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ~ ~ ( p @ V0t )
<=> ( p @ V0t ) ) )
& ( ~ $true
<=> $false )
& ( ~ $false
<=> $true ) ) ).
thf(conj_thm_2Ebool_2EREFL__CLAUSE,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( V0x = V0x )
<=> $true ) ) ).
thf(conj_thm_2Ebool_2EEQ__SYM__EQ,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ( ( V0x = V1y )
<=> ( V1y = V0x ) ) ) ) ).
thf(conj_thm_2Ebool_2EEQ__CLAUSES,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( $true
<=> ( p @ V0t ) )
<=> ( p @ V0t ) )
& ( ( ( p @ V0t )
<=> $true )
<=> ( p @ V0t ) )
& ( ( $false
<=> ( p @ V0t ) )
<=> ~ ( p @ V0t ) )
& ( ( ( p @ V0t )
<=> $false )
<=> ~ ( p @ V0t ) ) ) ) ).
thf(conj_thm_2Ebool_2ECOND__CLAUSES,axiom,
! [A_27a: del,V0t1: $i] :
( ( mem @ V0t1 @ A_27a )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ A_27a )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ c_2Ebool_2ET ) @ V0t1 ) @ V1t2 )
= V0t1 )
& ( ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ c_2Ebool_2EF ) @ V0t1 ) @ V1t2 )
= V1t2 ) ) ) ) ).
thf(conj_thm_2Ebool_2ENOT__EXISTS__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ~ ? [V1x: $i] :
( ( mem @ V1x @ A_27a )
& ( p @ ( ap @ V0P @ V1x ) ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ~ ( p @ ( ap @ V0P @ V2x ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ELEFT__AND__FORALL__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ bool )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p @ ( ap @ V0P @ V2x ) ) )
& ( p @ V1Q ) )
<=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ V0P @ V3x ) )
& ( p @ V1Q ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EEXISTS__OR__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ( ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( ( p @ ( ap @ V0P @ V2x ) )
| ( p @ ( ap @ V1Q @ V2x ) ) ) )
<=> ( ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( p @ ( ap @ V0P @ V3x ) ) )
| ? [V4x: $i] :
( ( mem @ V4x @ A_27a )
& ( p @ ( ap @ V1Q @ V4x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ELEFT__OR__EXISTS__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ bool )
=> ( ( ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ V0P @ V2x ) ) )
| ( p @ V1Q ) )
<=> ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( ( p @ ( ap @ V0P @ V3x ) )
| ( p @ V1Q ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ERIGHT__OR__EXISTS__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ V0P )
| ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ V1Q @ V2x ) ) ) )
<=> ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( ( p @ V0P )
| ( p @ ( ap @ V1Q @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ELEFT__EXISTS__AND__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ bool )
=> ( ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ V0P @ V2x ) )
& ( p @ V1Q ) )
<=> ( ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( p @ ( ap @ V0P @ V3x ) ) )
& ( p @ V1Q ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ERIGHT__EXISTS__AND__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ( ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ V0P )
& ( p @ ( ap @ V1Q @ V2x ) ) )
<=> ( ( p @ V0P )
& ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( p @ ( ap @ V1Q @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ERIGHT__FORALL__OR__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ V0P )
| ( p @ ( ap @ V1Q @ V2x ) ) ) )
<=> ( ( p @ V0P )
| ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( p @ ( ap @ V1Q @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EDISJ__ASSOC,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ! [V2C: $i] :
( ( mem @ V2C @ bool )
=> ( ( ( p @ V0A )
| ( p @ V1B )
| ( p @ V2C ) )
<=> ( ( p @ V0A )
| ( p @ V1B )
| ( p @ V2C ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EDISJ__SYM,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ( ( ( p @ V0A )
| ( p @ V1B ) )
<=> ( ( p @ V1B )
| ( p @ V0A ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EDE__MORGAN__THM,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ( ( ~ ( ( p @ V0A )
& ( p @ V1B ) )
<=> ( ~ ( p @ V0A )
| ~ ( p @ V1B ) ) )
& ( ~ ( ( p @ V0A )
| ( p @ V1B ) )
<=> ( ~ ( p @ V0A )
& ~ ( p @ V1B ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ECOND__RATOR,axiom,
! [A_27a: del,A_27b: del,V0b: $i] :
( ( mem @ V0b @ bool )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ A_27b ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ A_27a @ A_27b ) ) @ V0b ) @ V1f ) @ V2g ) @ V3x )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27b ) @ V0b ) @ ( ap @ V1f @ V3x ) ) @ ( ap @ V2g @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ECOND__RAND,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ bool )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ap @ V0f @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ V1b ) @ V2x ) @ V3y ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27b ) @ V1b ) @ ( ap @ V0f @ V2x ) ) @ ( ap @ V0f @ V3y ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ECOND__EXPAND,axiom,
! [V0b: $i] :
( ( mem @ V0b @ bool )
=> ! [V1t1: $i] :
( ( mem @ V1t1 @ bool )
=> ! [V2t2: $i] :
( ( mem @ V2t2 @ bool )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ V0b ) @ V1t1 ) @ V2t2 ) )
<=> ( ( ~ ( p @ V0b )
| ( p @ V1t1 ) )
& ( ( p @ V0b )
| ( p @ V2t2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ESKOLEM__THM,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ? [V2y: $i] :
( ( mem @ V2y @ A_27b )
& ( p @ ( ap @ ( ap @ V0P @ V1x ) @ V2y ) ) ) )
<=> ? [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ A_27b ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( p @ ( ap @ ( ap @ V0P @ V4x ) @ ( ap @ V3f @ V4x ) ) ) ) ) ) ) ).
thf(conj_thm_2Eprim__rec_2ELESS__0,axiom,
! [V0n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ).
thf(conj_thm_2Esat_2ENOT__NOT,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ~ ~ ( p @ V0t )
<=> ( p @ V0t ) ) ) ).
thf(conj_thm_2Esat_2EAND__INV__IMP,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ( ( p @ V0A )
=> ( ~ ( p @ V0A )
=> $false ) ) ) ).
thf(conj_thm_2Esat_2EOR__DUAL2,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ( ( ~ ( ( p @ V0A )
| ( p @ V1B ) )
=> $false )
<=> ( ( ( p @ V0A )
=> $false )
=> ( ~ ( p @ V1B )
=> $false ) ) ) ) ) ).
thf(conj_thm_2Esat_2EOR__DUAL3,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ( ( ~ ( ~ ( p @ V0A )
| ( p @ V1B ) )
=> $false )
<=> ( ( p @ V0A )
=> ( ~ ( p @ V1B )
=> $false ) ) ) ) ) ).
thf(conj_thm_2Esat_2EAND__INV2,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ( ( ~ ( p @ V0A )
=> $false )
=> ( ( ( p @ V0A )
=> $false )
=> $false ) ) ) ).
thf(conj_thm_2Esat_2Edc__eq,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ! [V2r: $i] :
( ( mem @ V2r @ bool )
=> ( ( ( p @ V0p )
<=> ( ( p @ V1q )
<=> ( p @ V2r ) ) )
<=> ( ( ( p @ V0p )
| ( p @ V1q )
| ( p @ V2r ) )
& ( ( p @ V0p )
| ~ ( p @ V2r )
| ~ ( p @ V1q ) )
& ( ( p @ V1q )
| ~ ( p @ V2r )
| ~ ( p @ V0p ) )
& ( ( p @ V2r )
| ~ ( p @ V1q )
| ~ ( p @ V0p ) ) ) ) ) ) ) ).
thf(conj_thm_2Esat_2Edc__conj,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ! [V2r: $i] :
( ( mem @ V2r @ bool )
=> ( ( ( p @ V0p )
<=> ( ( p @ V1q )
& ( p @ V2r ) ) )
<=> ( ( ( p @ V0p )
| ~ ( p @ V1q )
| ~ ( p @ V2r ) )
& ( ( p @ V1q )
| ~ ( p @ V0p ) )
& ( ( p @ V2r )
| ~ ( p @ V0p ) ) ) ) ) ) ) ).
thf(conj_thm_2Esat_2Edc__disj,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ! [V2r: $i] :
( ( mem @ V2r @ bool )
=> ( ( ( p @ V0p )
<=> ( ( p @ V1q )
| ( p @ V2r ) ) )
<=> ( ( ( p @ V0p )
| ~ ( p @ V1q ) )
& ( ( p @ V0p )
| ~ ( p @ V2r ) )
& ( ( p @ V1q )
| ( p @ V2r )
| ~ ( p @ V0p ) ) ) ) ) ) ) ).
thf(conj_thm_2Esat_2Edc__imp,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ! [V2r: $i] :
( ( mem @ V2r @ bool )
=> ( ( ( p @ V0p )
<=> ( ( p @ V1q )
=> ( p @ V2r ) ) )
<=> ( ( ( p @ V0p )
| ( p @ V1q ) )
& ( ( p @ V0p )
| ~ ( p @ V2r ) )
& ( ~ ( p @ V1q )
| ( p @ V2r )
| ~ ( p @ V0p ) ) ) ) ) ) ) ).
thf(conj_thm_2Esat_2Edc__neg,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ( ( ( p @ V0p )
<=> ~ ( p @ V1q ) )
<=> ( ( ( p @ V0p )
| ( p @ V1q ) )
& ( ~ ( p @ V1q )
| ~ ( p @ V0p ) ) ) ) ) ) ).
thf(conj_thm_2Esat_2Epth__ni1,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ( ~ ( ( p @ V0p )
=> ( p @ V1q ) )
=> ( p @ V0p ) ) ) ) ).
thf(conj_thm_2Esat_2Epth__ni2,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ( ~ ( ( p @ V0p )
=> ( p @ V1q ) )
=> ~ ( p @ V1q ) ) ) ) ).
thf(conj_thm_2Esat_2Epth__no1,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ( ~ ( ( p @ V0p )
| ( p @ V1q ) )
=> ~ ( p @ V0p ) ) ) ) ).
thf(conj_thm_2Esat_2Epth__no2,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ( ~ ( ( p @ V0p )
| ( p @ V1q ) )
=> ~ ( p @ V1q ) ) ) ) ).
thf(conj_thm_2Esat_2Epth__nn,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ( ~ ~ ( p @ V0p )
=> ( p @ V0p ) ) ) ).
thf(conj_thm_2Earithmetic_2EMOD__2,conjecture,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EMOD @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
%------------------------------------------------------------------------------