TPTP Problem File: ITP009^2.p
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%------------------------------------------------------------------------------
% File : ITP009^2 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory export of thm_2Equotient_2EFUN__REL__EQ__REL.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_2Equotient_2EFUN__REL__EQ__REL.p [Gau19]
% : HL404001^2.p [TPAP]
% Status : Theorem
% Rating : 0.80 v8.2.0, 0.92 v8.1.0, 0.82 v7.5.0
% Syntax : Number of formulae : 57 ( 5 unt; 21 typ; 0 def)
% Number of atoms : 319 ( 10 equ; 0 cnn)
% Maximal formula atoms : 48 ( 8 avg)
% Number of connectives : 635 ( 4 ~; 0 |; 18 &; 531 @)
% ( 22 <=>; 60 =>; 0 <=; 0 <~>)
% Maximal formula depth : 36 ( 9 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 27 ( 27 >; 0 *; 0 +; 0 <<)
% Number of symbols : 28 ( 25 usr; 14 con; 0-4 aty)
% Number of variables : 86 ( 0 ^; 86 !; 0 ?; 86 :)
% SPC : TH0_THM_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001^2.ax').
%------------------------------------------------------------------------------
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_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_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_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_2Equotient_2E_2D_2D_3E,type,
c_2Equotient_2E_2D_2D_3E: del > del > del > del > $i ).
thf(mem_c_2Equotient_2E_2D_2D_3E,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] : ( mem @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27b @ A_27c @ A_27d ) @ ( arr @ ( arr @ A_27a @ A_27c ) @ ( arr @ ( arr @ A_27b @ A_27d ) @ ( arr @ ( arr @ A_27c @ A_27b ) @ ( arr @ A_27a @ A_27d ) ) ) ) ) ).
thf(tp_c_2Equotient_2E_3D_3D_3D_3E,type,
c_2Equotient_2E_3D_3D_3D_3E: del > del > $i ).
thf(mem_c_2Equotient_2E_3D_3D_3D_3E,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ bool ) ) ) ) ) ).
thf(tp_c_2Equotient_2EQUOTIENT,type,
c_2Equotient_2EQUOTIENT: del > del > $i ).
thf(mem_c_2Equotient_2EQUOTIENT,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ A_27a ) @ bool ) ) ) ) ).
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(tp_c_2Ecombin_2EW,type,
c_2Ecombin_2EW: del > del > $i ).
thf(mem_c_2Ecombin_2EW,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ecombin_2EW @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27b ) ) @ ( arr @ A_27a @ A_27b ) ) ) ).
thf(tp_c_2Equotient_2Erespects,type,
c_2Equotient_2Erespects: del > del > $i ).
thf(mem_c_2Equotient_2Erespects,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Equotient_2Erespects @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27b ) ) @ ( arr @ A_27a @ A_27b ) ) ) ).
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(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_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_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_2EAND__IMP__INTRO,axiom,
! [V0t1: $i] :
( ( mem @ V0t1 @ bool )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ bool )
=> ! [V2t3: $i] :
( ( mem @ V2t3 @ bool )
=> ( ( ( p @ V0t1 )
=> ( ( p @ V1t2 )
=> ( p @ V2t3 ) ) )
<=> ( ( ( p @ V0t1 )
& ( p @ V1t2 ) )
=> ( p @ V2t3 ) ) ) ) ) ) ).
thf(conj_thm_2Ecombin_2EW__THM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27b ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2EW @ A_27a @ A_27b ) @ V0f ) @ V1x )
= ( ap @ ( ap @ V0f @ V1x ) @ V1x ) ) ) ) ).
thf(conj_thm_2Equotient_2EQUOTIENT__REL,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3r: $i] :
( ( mem @ V3r @ A_27a )
=> ! [V4s: $i] :
( ( mem @ V4s @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V3r ) @ V4s ) )
<=> ( ( p @ ( ap @ ( ap @ V0R @ V3r ) @ V3r ) )
& ( p @ ( ap @ ( ap @ V0R @ V4s ) @ V4s ) )
& ( ( ap @ V1abs @ V3r )
= ( ap @ V1abs @ V4s ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EFUN__QUOTIENT,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs1: $i] :
( ( mem @ V1abs1 @ ( arr @ A_27a @ A_27c ) )
=> ! [V2rep1: $i] :
( ( mem @ V2rep1 @ ( arr @ A_27c @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c ) @ V0R1 ) @ V1abs1 ) @ V2rep1 ) )
=> ! [V3R2: $i] :
( ( mem @ V3R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V4abs2: $i] :
( ( mem @ V4abs2 @ ( arr @ A_27b @ A_27d ) )
=> ! [V5rep2: $i] :
( ( mem @ V5rep2 @ ( arr @ A_27d @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d ) @ V3R2 ) @ V4abs2 ) @ V5rep2 ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27c @ A_27d ) ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) ) @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d ) @ V2rep1 ) @ V4abs2 ) ) @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27d @ A_27c @ A_27b ) @ V1abs1 ) @ V5rep2 ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Equotient_2Erespects__def,axiom,
! [A_27a: del,A_27b: del] :
( ( c_2Equotient_2Erespects @ A_27a @ A_27b )
= ( c_2Ecombin_2EW @ A_27a @ A_27b ) ) ).
thf(conj_thm_2Equotient_2EFUN__REL__EQ__REL,conjecture,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs1: $i] :
( ( mem @ V1abs1 @ ( arr @ A_27a @ A_27c ) )
=> ! [V2rep1: $i] :
( ( mem @ V2rep1 @ ( arr @ A_27c @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c ) @ V0R1 ) @ V1abs1 ) @ V2rep1 ) )
=> ! [V3R2: $i] :
( ( mem @ V3R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V4abs2: $i] :
( ( mem @ V4abs2 @ ( arr @ A_27b @ A_27d ) )
=> ! [V5rep2: $i] :
( ( mem @ V5rep2 @ ( arr @ A_27d @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d ) @ V3R2 ) @ V4abs2 ) @ V5rep2 ) )
=> ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ A_27a @ A_27b ) )
=> ! [V7g: $i] :
( ( mem @ V7g @ ( arr @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) @ V6f ) @ V7g ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) ) @ V6f ) )
& ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) ) @ V7g ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d ) @ V2rep1 ) @ V4abs2 ) @ V6f )
= ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d ) @ V2rep1 ) @ V4abs2 ) @ V7g ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------