## TPTP Axioms File: NUM009+0.ax

```%------------------------------------------------------------------------------
% File     : NUM009+0 : TPTP v7.5.0. Released v7.3.0.
% Domain   : Number Theory
% Axioms   : Robinson arithmetic without 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    :   18 (   1 unit)
%            Number of atoms       :   75 (   0 equality)
%            Maximal formula depth :   13 (   8 average)
%            Number of connectives :   91 (  34   ~;  26   |;  31   &)
%                                         (   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   :   67 (   0 sgn;  47   !;  20   ?)
%            Maximal term depth    :    1 (   1 average)
% SPC      :

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

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

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

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

fof(axiom_5,axiom,(
! [X20] : id(X20,X20) )).

fof(axiom_6,axiom,(
! [X21,X22] :
( ~ id(X21,X22)
| id(X22,X21) ) )).

fof(axiom_7,axiom,(
! [X23,X24,X25] :
( ~ id(X23,X24)
| id(X23,X25)
| ~ id(X24,X25) ) )).

fof(axiom_8,axiom,(
! [X26,X27] :
( ~ id(X26,X27)
| ( ~ r1(X26)
& ~ r1(X27) )
| ( r1(X26)
& r1(X27) ) ) )).

fof(axiom_9,axiom,(
! [X28,X29,X30,X31] :
( ~ id(X28,X30)
| ~ id(X29,X31)
| ( ~ r2(X28,X29)
& ~ r2(X30,X31) )
| ( r2(X28,X29)
& r2(X30,X31) ) ) )).

fof(axiom_10,axiom,(
! [X32,X33,X34,X35,X36,X37] :
( ~ id(X32,X35)
| ~ id(X33,X36)
| ~ id(X34,X37)
| ( ~ r3(X32,X33,X34)
& ~ r3(X35,X36,X37) )
| ( r3(X32,X33,X34)
& r3(X35,X36,X37) ) ) )).

fof(axiom_11,axiom,(
! [X38,X39,X40,X41,X42,X43] :
( ~ id(X38,X41)
| ~ id(X39,X42)
| ~ id(X40,X43)
| ( ~ r4(X38,X39,X40)
& ~ r4(X41,X42,X43) )
| ( r4(X38,X39,X40)
& r4(X41,X42,X43) ) ) )).

%----Axioms of Q
fof(axiom_1a,axiom,(
! [X1,X8] :
? [Y4] :
( ? [Y5] :
( id(Y5,Y4)
& ? [Y15] :
( r2(X8,Y15)
& r3(X1,Y15,Y5) ) )
& ? [Y7] :
( r2(Y7,Y4)
& r3(X1,X8,Y7) ) ) )).

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

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

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

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

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

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

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