TPTP Problem File: ITP005^2.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP005^2 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory export of thm_2Eset__relation_2Erel__to__reln__inv.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_2Eset__relation_2Erel__to__reln__inv.p [Gau19]
% : HL402001^2.p [TPAP]
% Status : Theorem
% Rating : 1.00 v7.5.0
% Syntax : Number of formulae : 73 ( 4 unt; 26 typ; 0 def)
% Number of atoms : 296 ( 24 equ; 0 cnn)
% Maximal formula atoms : 14 ( 6 avg)
% Number of connectives : 603 ( 4 ~; 0 |; 10 &; 511 @)
% ( 23 <=>; 55 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 8 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 34 ( 34 >; 0 *; 0 +; 0 <<)
% Number of symbols : 34 ( 31 usr; 15 con; 0-3 aty)
% Number of variables : 111 ( 4 ^; 104 !; 3 ?; 111 :)
% SPC : TH0_THM_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001^2.ax').
%------------------------------------------------------------------------------
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_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_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_ty_2Epair_2Eprod,type,
ty_2Epair_2Eprod: del > del > del ).
thf(tp_c_2Epair_2ESND,type,
c_2Epair_2ESND: del > del > $i ).
thf(mem_c_2Epair_2ESND,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27b ) ) ).
thf(tp_c_2Epair_2EFST,type,
c_2Epair_2EFST: del > del > $i ).
thf(mem_c_2Epair_2EFST,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) ) ).
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_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_2Ebool_2EIN,type,
c_2Ebool_2EIN: del > $i ).
thf(mem_c_2Ebool_2EIN,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebool_2EIN @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Eset__relation_2Ereln__to__rel,type,
c_2Eset__relation_2Ereln__to__rel: del > del > $i ).
thf(mem_c_2Eset__relation_2Ereln__to__rel,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) ) ).
thf(tp_c_2Epair_2E_2C,type,
c_2Epair_2E_2C: del > del > $i ).
thf(mem_c_2Epair_2E_2C,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ ( arr @ A_27a @ ( arr @ A_27b @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ) ).
thf(tp_c_2Epair_2EUNCURRY,type,
c_2Epair_2EUNCURRY: del > del > del > $i ).
thf(mem_c_2Epair_2EUNCURRY,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c ) ) ) ).
thf(tp_c_2Epred__set_2EGSPEC,type,
c_2Epred__set_2EGSPEC: del > del > $i ).
thf(mem_c_2Epred__set_2EGSPEC,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ ( ty_2Epair_2Eprod @ A_27a @ bool ) ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2Eset__relation_2Erel__to__reln,type,
c_2Eset__relation_2Erel__to__reln: del > del > $i ).
thf(mem_c_2Eset__relation_2Erel__to__reln,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) ) ) ).
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_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_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_2EFORALL__SIMP,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( p @ V0t ) )
<=> ( p @ V0t ) ) ) ).
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_2EFUN__EQ__THM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ A_27b ) )
=> ( ( V0f = V1g )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ap @ V0f @ V2x )
= ( ap @ V1g @ V2x ) ) ) ) ) ) ).
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_2EUNWIND__THM2,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ( ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( V2x = V1a )
& ( p @ ( ap @ V0P @ V2x ) ) )
<=> ( p @ ( ap @ V0P @ V1a ) ) ) ) ) ).
thf(conj_thm_2Epair_2EPAIR__EQ,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27b )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ! [V3b: $i] :
( ( mem @ V3b @ A_27b )
=> ( ( ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V0x ) @ V1y )
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2a ) @ V3b ) )
<=> ( ( V0x = V2a )
& ( V1y = V3b ) ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2ECLOSED__PAIR__EQ,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27b )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ! [V3b: $i] :
( ( mem @ V3b @ A_27b )
=> ( ( ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V0x ) @ V1y )
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2a ) @ V3b ) )
<=> ( ( V0x = V2a )
& ( V1y = V3b ) ) ) ) ) ) ) ).
thf(ax_thm_2Epair_2EPAIR,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0x ) ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V0x ) )
= V0x ) ) ).
thf(conj_thm_2Epair_2EUNCURRY__DEF,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1x ) @ V2y ) )
= ( ap @ ( ap @ V0f @ V1x ) @ V2y ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2EGSPECIFICATION,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ ( ty_2Epair_2Eprod @ A_27a @ bool ) ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1v ) @ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27b ) @ V0f ) ) )
<=> ? [V2x: $i] :
( ( mem @ V2x @ A_27b )
& ( ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V1v ) @ c_2Ebool_2ET )
= ( ap @ V0f @ V2x ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Ereln__to__rel__def,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ( ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ V0r )
= ( lam @ A_27a
@ ^ [V1x: $i] :
( lam @ A_27b
@ ^ [V2y: $i] : ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1x ) @ V2y ) ) @ V0r ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Erel__to__reln__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ V0R )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
@ ( lam @ A_27a
@ ^ [V1x: $i] :
( lam @ A_27b
@ ^ [V2y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1x ) @ V2y ) ) @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Erel__to__reln__inv,conjecture,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ V0R ) )
= V0R ) ) ).
%------------------------------------------------------------------------------