## TPTP Axioms File: FLD001-0.ax

```%--------------------------------------------------------------------------
% File     : FLD001-0 : TPTP v7.5.0. Released v2.1.0.
% Domain   : Field Theory (Ordered fields)
% Axioms   : Ordered field axioms (axiom formulation glxx)
% Version  : [Dra93] axioms : Especial.
% English  :

% Refs     : [Dra93] Draeger (1993), Anwendung des Theorembeweisers SETHEO
% Source   : [Dra93]
% Names    :

% Status   : Satisfiable
% Syntax   : Number of clauses    :   27 (   3 non-Horn;   3 unit;  27 RR)
%            Number of atoms      :   73 (   0 equality)
%            Maximal clause size  :    4 (   3 average)
%            Number of predicates :    3 (   0 propositional; 1-2 arity)
%            Number of functors   :    6 (   2 constant; 0-2 arity)
%            Number of variables  :   50 (   0 singleton)
%            Maximal term depth   :    3 (   1 average)
% SPC      :

% Comments : The missing equality axioms can be derived.
%          : Currently it is unknown if this axiomatization is complete.
%            It is definitely tuned for SETHEO.
% Bugfixes : v2.1.0 - Added different_identities clause.
%--------------------------------------------------------------------------
| ~ defined(X)
| ~ defined(Y)
| ~ defined(Z) )).

| ~ defined(X) )).

| ~ defined(X) )).

| ~ defined(X)
| ~ defined(Y) )).

cnf(associativity_multiplication,axiom,
( equalish(multiply(X,multiply(Y,Z)),multiply(multiply(X,Y),Z))
| ~ defined(X)
| ~ defined(Y)
| ~ defined(Z) )).

cnf(existence_of_identity_multiplication,axiom,
( equalish(multiply(multiplicative_identity,X),X)
| ~ defined(X) )).

cnf(existence_of_inverse_multiplication,axiom,
( equalish(multiply(X,multiplicative_inverse(X)),multiplicative_identity)
| ~ defined(X)

cnf(commutativity_multiplication,axiom,
( equalish(multiply(X,Y),multiply(Y,X))
| ~ defined(X)
| ~ defined(Y) )).

cnf(distributivity,axiom,
| ~ defined(X)
| ~ defined(Y)
| ~ defined(Z) )).

| ~ defined(X)
| ~ defined(Y) )).

| ~ defined(X) )).

cnf(well_definedness_of_multiplication,axiom,
( defined(multiply(X,Y))
| ~ defined(X)
| ~ defined(Y) )).

cnf(well_definedness_of_multiplicative_identity,axiom,
( defined(multiplicative_identity) )).

cnf(well_definedness_of_multiplicative_inverse,axiom,
( defined(multiplicative_inverse(X))
| ~ defined(X)

cnf(antisymmetry_of_order_relation,axiom,
( equalish(X,Y)
| ~ less_or_equal(X,Y)
| ~ less_or_equal(Y,X) )).

cnf(transitivity_of_order_relation,axiom,
( less_or_equal(X,Z)
| ~ less_or_equal(X,Y)
| ~ less_or_equal(Y,Z) )).

cnf(totality_of_order_relation,axiom,
( less_or_equal(X,Y)
| less_or_equal(Y,X)
| ~ defined(X)
| ~ defined(Y) )).

| ~ defined(Z)
| ~ less_or_equal(X,Y) )).

cnf(compatibility_of_order_relation_and_multiplication,axiom,

cnf(reflexivity_of_equality,axiom,
( equalish(X,X)
| ~ defined(X) )).

cnf(symmetry_of_equality,axiom,
( equalish(X,Y)
| ~ equalish(Y,X) )).

cnf(transitivity_of_equality,axiom,
( equalish(X,Z)
| ~ equalish(X,Y)
| ~ equalish(Y,Z) )).

| ~ defined(Z)
| ~ equalish(X,Y) )).

cnf(compatibility_of_equality_and_multiplication,axiom,
( equalish(multiply(X,Z),multiply(Y,Z))
| ~ defined(Z)
| ~ equalish(X,Y) )).

cnf(compatibility_of_equality_and_order_relation,axiom,
( less_or_equal(Y,Z)
| ~ less_or_equal(X,Z)
| ~ equalish(X,Y) )).

cnf(different_identities,axiom,