## TPTP Axioms File: NUM008+0.ax

```%------------------------------------------------------------------------------
% File     : NUM008+0 : TPTP v7.5.0. Released v7.3.0.
% Domain   : Number Theory
% Axioms   : Robinson arithmetic with equality
% Version  : Especial.
% English  :

% Refs     : [BBJ03] Boolos et al. (2003), Computability and Logic
%          : [Smi07] Smith (2007), An Introduction to Goedel's Theorems
%          : [Lam18] Lampert (2018), Email to Geoff Sutcliffe
% Source   : [Lam18]
% Names    :

% Status   : Satisfiable
% Rating   : ? v7.3.0
% Syntax   : Number of formulae    :   11 (   0 unit)
%            Number of atoms       :   44 (  17 equality)
%            Maximal formula depth :    9 (   7 average)
%            Number of connectives :   47 (  14   ~;  10   |;  23   &)
%                                         (   0 <=>;   0  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of predicates  :    5 (   0 propositional; 1-3 arity)
%            Number of functors    :    0 (   0 constant; --- arity)
%            Number of variables   :   43 (   0 sgn;  23   !;  20   ?)
%            Maximal term depth    :    1 (   1 average)
% SPC      :

%------------------------------------------------------------------------------
fof(axiom_1,axiom,(
? [Y24] :
! [X19] :
( ( ~ r1(X19)
& X19 != Y24 )
| ( r1(X19)
& X19 = Y24 ) ) )).

fof(axiom_2,axiom,(
! [X11] :
? [Y21] :
! [X12] :
( ( ~ r2(X11,X12)
& X12 != Y21 )
| ( r2(X11,X12)
& X12 = Y21 ) ) )).

fof(axiom_3,axiom,(
! [X13,X14] :
? [Y22] :
! [X15] :
( ( ~ r3(X13,X14,X15)
& X15 != Y22 )
| ( r3(X13,X14,X15)
& X15 = Y22 ) ) )).

fof(axiom_4,axiom,(
! [X16,X17] :
? [Y23] :
! [X18] :
( ( ~ r4(X16,X17,X18)
& X18 != Y23 )
| ( r4(X16,X17,X18)
& X18 = Y23 ) ) )).

%Axioms of Q

fof(axiom_1a,axiom,(
! [X1,X8] :
? [Y4] :
( ? [Y5] :
( ? [Y15] :
( r2(X8,Y15)
& r3(X1,Y15,Y5) )
& Y5 = Y4 )
& ? [Y7] :
( r2(Y7,Y4)
& r3(X1,X8,Y7) ) ) )).

fof(axiom_2a,axiom,(
! [X2,X9] :
? [Y2] :
( ? [Y3] :
( ? [Y14] :
( r2(X9,Y14)
& r4(X2,Y14,Y3) )
& Y3 = Y2 )
& ? [Y6] :
( r3(Y6,X2,Y2)
& r4(X2,X9,Y6) ) ) )).

fof(axiom_3a,axiom,(
! [X3,X10] :
( ! [Y12] :
( ! [Y13] :
( ~ r2(X3,Y13)
| Y13 != Y12 )
| ~ r2(X10,Y12) )
| X3 = X10 ) )).

fof(axiom_4a,axiom,(
! [X4] :
? [Y9] :
( ? [Y16] :
( r1(Y16)
& r3(X4,Y16,Y9) )
& Y9 = X4 ) )).

fof(axiom_5a,axiom,(
! [X5] :
? [Y8] :
( ? [Y17] :
( r1(Y17)
& r4(X5,Y17,Y8) )
& ? [Y18] :
( r1(Y18)
& Y8 = Y18 ) ) )).

fof(axiom_6a,axiom,(
! [X6] :
( ? [Y19] :
( r1(Y19)
& X6 = Y19 )
| ? [Y1,Y11] :
( r2(Y1,Y11)
& X6 = Y11 ) ) )).

fof(axiom_7a,axiom,(
! [X7,Y10] :
( ! [Y20] :
( ~ r1(Y20)
| Y20 != Y10 )
| ~ r2(X7,Y10) ) )).

%------------------------------------------------------------------------------
```