ITP001 Axioms: ITP007^5.ax
%------------------------------------------------------------------------------
% File : ITP007^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : sat^2.ax [Gau20]
% : HL4007^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 24 ( 0 unt; 0 typ; 0 def)
% Number of atoms : 234 ( 0 equ; 0 cnn)
% Maximal formula atoms : 30 ( 9 avg)
% Number of connectives : 430 ( 51 ~; 38 |; 19 &; 221 @)
% ( 22 <=>; 79 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 10 avg; 221 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 7 ( 5 usr; 7 con; 0-0 aty)
% Number of variables : 47 ( 0 ^ 47 !; 0 ?; 47 :)
% SPC : TH0_SAT_NEQ_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(conj_thm_2Esat_2EAND__IMP,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_2Esat_2ENOT__NOT,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ~ ~ ( p @ V0t )
<=> ( p @ V0t ) ) ) ).
thf(conj_thm_2Esat_2EAND__INV,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ( ( ~ ( p @ V0A )
& ( p @ V0A ) )
<=> $false ) ) ).
thf(conj_thm_2Esat_2EAND__INV__IMP,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ( ( p @ V0A )
=> ( ~ ( p @ V0A )
=> $false ) ) ) ).
thf(conj_thm_2Esat_2EOR__DUAL,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_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_2ENOT__ELIM2,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ( ( ~ ( p @ V0A )
=> $false )
<=> ( p @ V0A ) ) ) ).
thf(conj_thm_2Esat_2EEQT__Imp1,axiom,
! [V0b: $i] :
( ( mem @ V0b @ bool )
=> ( ( p @ V0b )
=> ( ( p @ V0b )
<=> $true ) ) ) ).
thf(conj_thm_2Esat_2EEQF__Imp1,axiom,
! [V0b: $i] :
( ( mem @ V0b @ bool )
=> ( ~ ( p @ V0b )
=> ( ( p @ V0b )
<=> $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_2Edc__cond,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ! [V2r: $i] :
( ( mem @ V2r @ bool )
=> ! [V3s: $i] :
( ( mem @ V3s @ bool )
=> ( ( ( p @ V0p )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ V1q ) @ V2r ) @ V3s ) ) )
<=> ( ( ( p @ V0p )
| ( p @ V1q )
| ~ ( p @ V3s ) )
& ( ( p @ V0p )
| ~ ( p @ V2r )
| ~ ( p @ V1q ) )
& ( ( p @ V0p )
| ~ ( p @ V2r )
| ~ ( p @ V3s ) )
& ( ~ ( p @ V1q )
| ( p @ V2r )
| ~ ( p @ V0p ) )
& ( ( p @ V1q )
| ( p @ V3s )
| ~ ( 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__an1,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ( ( ( p @ V0p )
& ( p @ V1q ) )
=> ( p @ V0p ) ) ) ) ).
thf(conj_thm_2Esat_2Epth__an2,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 ) ) ) ).
%------------------------------------------------------------------------------