TPTP Problem File: ITP012^3.p
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%------------------------------------------------------------------------------
% File : ITP012^3 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 syntactic export of thm_2Einteger_2EINT__DIVIDES__RSUB.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_2Einteger_2EINT__DIVIDES__RSUB.p [Gau19]
% : HL405501^3.p [TPAP]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 49 ( 8 unt; 16 typ; 0 def)
% Number of atoms : 102 ( 12 equ; 46 cnn)
% Maximal formula atoms : 8 ( 3 avg)
% Number of connectives : 274 ( 46 ~; 34 |; 27 &; 99 @)
% ( 38 <=>; 30 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 6 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 27 ( 27 >; 0 *; 0 +; 0 <<)
% Number of symbols : 16 ( 14 usr; 3 con; 0-3 aty)
% Number of variables : 74 ( 0 ^; 69 !; 2 ?; 74 :)
% ( 3 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
thf(tyop_2Einteger_2Eint,type,
tyop_2Einteger_2Eint: $tType ).
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
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_2Ebool_2ET,type,
c_2Ebool_2ET: $o ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Einteger_2Eint__add,type,
c_2Einteger_2Eint__add: tyop_2Einteger_2Eint > tyop_2Einteger_2Eint > tyop_2Einteger_2Eint ).
thf(c_2Einteger_2Eint__divides,type,
c_2Einteger_2Eint__divides: tyop_2Einteger_2Eint > tyop_2Einteger_2Eint > $o ).
thf(c_2Einteger_2Eint__neg,type,
c_2Einteger_2Eint__neg: tyop_2Einteger_2Eint > tyop_2Einteger_2Eint ).
thf(c_2Einteger_2Eint__sub,type,
c_2Einteger_2Eint__sub: tyop_2Einteger_2Eint > tyop_2Einteger_2Eint > tyop_2Einteger_2Eint ).
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_2EIMP__F,axiom,
! [V0t: $o] :
( ( V0t
=> c_2Ebool_2EF )
=> ( (~) @ V0t ) ) ).
thf(thm_2Ebool_2EF__IMP,axiom,
! [V0t: $o] :
( ( (~) @ V0t )
=> ( V0t
=> c_2Ebool_2EF ) ) ).
thf(thm_2Ebool_2EIMP__CLAUSES,axiom,
! [V0t: $o] :
( ( ( c_2Ebool_2ET
=> V0t )
<=> V0t )
& ( ( V0t
=> c_2Ebool_2ET )
<=> c_2Ebool_2ET )
& ( ( c_2Ebool_2EF
=> V0t )
<=> c_2Ebool_2ET )
& ( ( V0t
=> V0t )
<=> c_2Ebool_2ET )
& ( ( V0t
=> c_2Ebool_2EF )
<=> ( (~) @ V0t ) ) ) ).
thf(thm_2Ebool_2ENOT__CLAUSES,axiom,
( ! [V0t: $o] :
( ( (~) @ ( (~) @ V0t ) )
<=> V0t )
& ( ( (~) @ c_2Ebool_2ET )
<=> c_2Ebool_2EF )
& ( ( (~) @ c_2Ebool_2EF )
<=> c_2Ebool_2ET ) ) ).
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_2ENOT__FORALL__THM,axiom,
! [A_27a: $tType,V0P: A_27a > $o] :
( ( (~)
@ ! [V1x: A_27a] : ( V0P @ V1x ) )
<=> ? [V2x: A_27a] : ( (~) @ ( V0P @ V2x ) ) ) ).
thf(thm_2Ebool_2EFORALL__AND__THM,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: A_27a > $o] :
( ! [V2x: A_27a] :
( ( V0P @ V2x )
& ( V1Q @ V2x ) )
<=> ( ! [V3x: A_27a] : ( V0P @ V3x )
& ! [V4x: A_27a] : ( V1Q @ V4x ) ) ) ).
thf(thm_2Ebool_2EDISJ__ASSOC,axiom,
! [V0A: $o,V1B: $o,V2C: $o] :
( ( V0A
| V1B
| V2C )
<=> ( V0A
| V1B
| V2C ) ) ).
thf(thm_2Ebool_2EDISJ__SYM,axiom,
! [V0A: $o,V1B: $o] :
( ( V0A
| V1B )
<=> ( V1B
| V0A ) ) ).
thf(thm_2Ebool_2EDE__MORGAN__THM,axiom,
! [V0A: $o,V1B: $o] :
( ( ( (~)
@ ( V0A
& V1B ) )
<=> ( ( (~) @ V0A )
| ( (~) @ V1B ) ) )
& ( ( (~)
@ ( V0A
| V1B ) )
<=> ( ( (~) @ V0A )
& ( (~) @ V1B ) ) ) ) ).
thf(thm_2Einteger_2Eint__sub,axiom,
! [V0x: tyop_2Einteger_2Eint,V1y: tyop_2Einteger_2Eint] :
( ( c_2Einteger_2Eint__sub @ V0x @ V1y )
= ( c_2Einteger_2Eint__add @ V0x @ ( c_2Einteger_2Eint__neg @ V1y ) ) ) ).
thf(thm_2Einteger_2EINT__DIVIDES__RADD,axiom,
! [V0p: tyop_2Einteger_2Eint,V1q: tyop_2Einteger_2Eint,V2r: tyop_2Einteger_2Eint] :
( ( c_2Einteger_2Eint__divides @ V0p @ V1q )
=> ( ( c_2Einteger_2Eint__divides @ V0p @ ( c_2Einteger_2Eint__add @ V2r @ V1q ) )
= ( c_2Einteger_2Eint__divides @ V0p @ V2r ) ) ) ).
thf(thm_2Einteger_2EINT__DIVIDES__NEG,axiom,
! [V0p: tyop_2Einteger_2Eint,V1q: tyop_2Einteger_2Eint] :
( ( ( c_2Einteger_2Eint__divides @ V0p @ ( c_2Einteger_2Eint__neg @ V1q ) )
= ( c_2Einteger_2Eint__divides @ V0p @ V1q ) )
& ( ( c_2Einteger_2Eint__divides @ ( c_2Einteger_2Eint__neg @ V0p ) @ V1q )
= ( c_2Einteger_2Eint__divides @ V0p @ V1q ) ) ) ).
thf(thm_2Esat_2ENOT__NOT,axiom,
! [V0t: $o] :
( ( (~) @ ( (~) @ V0t ) )
<=> V0t ) ).
thf(thm_2Esat_2EAND__INV__IMP,axiom,
! [V0A: $o] :
( V0A
=> ( ( (~) @ V0A )
=> c_2Ebool_2EF ) ) ).
thf(thm_2Esat_2EOR__DUAL2,axiom,
! [V0B: $o,V1A: $o] :
( ( ( (~)
@ ( V1A
| V0B ) )
=> c_2Ebool_2EF )
<=> ( ( V1A
=> c_2Ebool_2EF )
=> ( ( (~) @ V0B )
=> c_2Ebool_2EF ) ) ) ).
thf(thm_2Esat_2EOR__DUAL3,axiom,
! [V0B: $o,V1A: $o] :
( ( ( (~)
@ ( ( (~) @ V1A )
| V0B ) )
=> c_2Ebool_2EF )
<=> ( V1A
=> ( ( (~) @ V0B )
=> c_2Ebool_2EF ) ) ) ).
thf(thm_2Esat_2EAND__INV2,axiom,
! [V0A: $o] :
( ( ( (~) @ V0A )
=> c_2Ebool_2EF )
=> ( ( V0A
=> c_2Ebool_2EF )
=> c_2Ebool_2EF ) ) ).
thf(thm_2Esat_2Edc__eq,axiom,
! [V0r: $o,V1q: $o,V2p: $o] :
( ( V2p
<=> ( V1q = V0r ) )
<=> ( ( V2p
| V1q
| V0r )
& ( V2p
| ( (~) @ V0r )
| ( (~) @ V1q ) )
& ( V1q
| ( (~) @ V0r )
| ( (~) @ V2p ) )
& ( V0r
| ( (~) @ V1q )
| ( (~) @ V2p ) ) ) ) ).
thf(thm_2Esat_2Edc__conj,axiom,
! [V0r: $o,V1q: $o,V2p: $o] :
( ( V2p
<=> ( V1q
& V0r ) )
<=> ( ( V2p
| ( (~) @ V1q )
| ( (~) @ V0r ) )
& ( V1q
| ( (~) @ V2p ) )
& ( V0r
| ( (~) @ V2p ) ) ) ) ).
thf(thm_2Esat_2Edc__disj,axiom,
! [V0r: $o,V1q: $o,V2p: $o] :
( ( V2p
<=> ( V1q
| V0r ) )
<=> ( ( V2p
| ( (~) @ V1q ) )
& ( V2p
| ( (~) @ V0r ) )
& ( V1q
| V0r
| ( (~) @ V2p ) ) ) ) ).
thf(thm_2Esat_2Edc__imp,axiom,
! [V0r: $o,V1q: $o,V2p: $o] :
( ( V2p
<=> ( V1q
=> V0r ) )
<=> ( ( V2p
| V1q )
& ( V2p
| ( (~) @ V0r ) )
& ( ( (~) @ V1q )
| V0r
| ( (~) @ V2p ) ) ) ) ).
thf(thm_2Esat_2Edc__neg,axiom,
! [V0q: $o,V1p: $o] :
( ( V1p
<=> ( (~) @ V0q ) )
<=> ( ( V1p
| V0q )
& ( ( (~) @ V0q )
| ( (~) @ V1p ) ) ) ) ).
thf(thm_2Einteger_2EINT__DIVIDES__RSUB,conjecture,
! [V0p: tyop_2Einteger_2Eint,V1q: tyop_2Einteger_2Eint,V2r: tyop_2Einteger_2Eint] :
( ( c_2Einteger_2Eint__divides @ V0p @ V1q )
=> ( ( c_2Einteger_2Eint__divides @ V0p @ ( c_2Einteger_2Eint__sub @ V2r @ V1q ) )
= ( c_2Einteger_2Eint__divides @ V0p @ V2r ) ) ) ).
%------------------------------------------------------------------------------