TPTP Problem File: ITP005^3.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP005^3 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 syntactic 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]
% : HL402001^3.p [TPAP]
% Status : Theorem
% Rating : 1.00 v7.5.0
% Syntax : Number of formulae : 43 ( 13 unt; 20 typ; 0 def)
% Number of atoms : 49 ( 24 equ; 3 cnn)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 174 ( 3 ~; 1 |; 7 &; 139 @)
% ( 19 <=>; 5 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 6 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 52 ( 52 >; 0 *; 0 +; 0 <<)
% Number of symbols : 21 ( 19 usr; 3 con; 0-5 aty)
% Number of variables : 94 ( 4 ^; 68 !; 3 ?; 94 :)
% ( 19 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2EF,type,
c_2Ebool_2EF: $o ).
thf(c_2Epair_2EFST,type,
c_2Epair_2EFST:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27a ) ).
thf(c_2Epred__set_2EGSPEC,type,
c_2Epred__set_2EGSPEC:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > ( tyop_2Epair_2Eprod @ A_27a @ $o ) ) > A_27a > $o ) ).
thf(c_2Ebool_2EIN,type,
c_2Ebool_2EIN:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).
thf(c_2Epair_2ESND,type,
c_2Epair_2ESND:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27b ) ).
thf(c_2Ebool_2ET,type,
c_2Ebool_2ET: $o ).
thf(c_2Epair_2EUNCURRY,type,
c_2Epair_2EUNCURRY:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27c ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c ) ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Eset__relation_2Erel__to__reln,type,
c_2Eset__relation_2Erel__to__reln:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) ).
thf(c_2Eset__relation_2Ereln__to__rel,type,
c_2Eset__relation_2Ereln__to__rel:
!>[A_27a: $tType,A_27b: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) > A_27a > A_27b > $o ) ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Ebool_2ETRUTH,axiom,
c_2Ebool_2ET ).
thf(thm_2Ebool_2EIMP__ANTISYM__AX,axiom,
! [V0t1: $o,V1t2: $o] :
( ( V0t1
=> V1t2 )
=> ( ( V1t2
=> V0t1 )
=> ( V0t1 = V1t2 ) ) ) ).
thf(thm_2Ebool_2EFORALL__SIMP,axiom,
! [A_27a: $tType,V0t: $o] :
( ! [V1x: A_27a] : V0t
<=> V0t ) ).
thf(thm_2Ebool_2EREFL__CLAUSE,axiom,
! [A_27a: $tType,V0x: A_27a] :
( ( V0x = V0x )
<=> c_2Ebool_2ET ) ).
thf(thm_2Ebool_2EEQ__SYM__EQ,axiom,
! [A_27a: $tType,V0x: A_27a,V1y: A_27a] :
( ( V0x = V1y )
<=> ( V1y = V0x ) ) ).
thf(thm_2Ebool_2EFUN__EQ__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1g: A_27a > A_27b] :
( ( V0f = V1g )
<=> ! [V2x: A_27a] :
( ( V0f @ V2x )
= ( V1g @ V2x ) ) ) ).
thf(thm_2Ebool_2EEQ__CLAUSES,axiom,
! [V0t: $o] :
( ( ( c_2Ebool_2ET = V0t )
<=> V0t )
& ( ( V0t = c_2Ebool_2ET )
<=> V0t )
& ( ( c_2Ebool_2EF = V0t )
<=> ( (~) @ V0t ) )
& ( ( V0t = c_2Ebool_2EF )
<=> ( (~) @ V0t ) ) ) ).
thf(thm_2Ebool_2EUNWIND__THM2,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1a: A_27a] :
( ? [V2x: A_27a] :
( ( V2x = V1a )
& ( V0P @ V2x ) )
<=> ( V0P @ V1a ) ) ).
thf(thm_2Epair_2EPAIR__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0y: A_27b,V1x: A_27a,V2b: A_27b,V3a: A_27a] :
( ( ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1x @ V0y )
= ( c_2Epair_2E_2C @ A_27a @ A_27b @ V3a @ V2b ) )
<=> ( ( V1x = V3a )
& ( V0y = V2b ) ) ) ).
thf(thm_2Epair_2ECLOSED__PAIR__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1y: A_27b,V2a: A_27a,V3b: A_27b] :
( ( ( c_2Epair_2E_2C @ A_27a @ A_27b @ V0x @ V1y )
= ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2a @ V3b ) )
<=> ( ( V0x = V2a )
& ( V1y = V3b ) ) ) ).
thf(thm_2Epair_2EPAIR,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( ( c_2Epair_2E_2C @ A_27a @ A_27b @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V0x ) @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V0x ) )
= V0x ) ).
thf(thm_2Epair_2EUNCURRY__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b > A_27c,V1x: A_27a,V2y: A_27b] :
( ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c @ V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1x @ V2y ) )
= ( V0f @ V1x @ V2y ) ) ).
thf(thm_2Epred__set_2EGSPECIFICATION,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27b > ( tyop_2Epair_2Eprod @ A_27a @ $o ),V1v: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V1v @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27b @ V0f ) )
<=> ? [V2x: A_27b] :
( ( c_2Epair_2E_2C @ A_27a @ $o @ V1v @ c_2Ebool_2ET )
= ( V0f @ V2x ) ) ) ).
thf(thm_2Eset__relation_2Ereln__to__rel__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o] :
( ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b @ V0r )
= ( ^ [V1x: A_27a,V2y: A_27b] : ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1x @ V2y ) @ V0r ) ) ) ).
thf(thm_2Eset__relation_2Erel__to__reln__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o] :
( ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b @ V0R )
= ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( tyop_2Epair_2Eprod @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ $o )
@ ^ [V1x: A_27a,V2y: A_27b] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ $o @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1x @ V2y ) @ ( V0R @ V1x @ V2y ) ) ) ) ) ).
thf(thm_2Eset__relation_2Erel__to__reln__inv,conjecture,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o] :
( ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b @ V0R ) )
= V0R ) ).
%------------------------------------------------------------------------------