%--------------------------------------------------------------------------
% File     : RNG005-0 : TPTP v7.2.0. Released v1.0.0.
% Domain   : Ring Theory
% Axioms   : Ring theory (equality) axioms
% Version  : [LW92] (equality) axioms.
% English  :

% Refs     : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr
%          : [LW92]  Lusk & Wos (1992), Benchmark Problems in Which Equalit
% Source   : [LW92]
% Names    :

% Status   : Satisfiable
% Syntax   : Number of clauses    :    9 (   0 non-Horn;   9 unit;   0 RR)
%            Number of atoms      :    9 (   9 equality)
%            Maximal clause size  :    1 (   1 average)
%            Number of predicates :    1 (   0 propositional; 2-2 arity)
%            Number of functors   :    4 (   1 constant; 0-2 arity)
%            Number of variables  :   18 (   0 singleton)
%            Maximal term depth   :    3 (   2 average)
% SPC      : 

% Comments : These axioms are used in [Wos88] p.203.
%--------------------------------------------------------------------------
%----There exists an additive identity element
cnf(left_additive_identity,axiom,
    ( add(additive_identity,X) = X )).

cnf(right_additive_identity,axiom,
    ( add(X,additive_identity) = X )).

%----Existence of left additive additive_inverse
cnf(left_additive_inverse,axiom,
    ( add(additive_inverse(X),X) = additive_identity )).

cnf(right_additive_inverse,axiom,
    ( add(X,additive_inverse(X)) = additive_identity )).

%----Associativity for addition
cnf(associativity_for_addition,axiom,
    ( add(X,add(Y,Z)) = add(add(X,Y),Z) )).

%----Commutativity for addition
cnf(commutativity_for_addition,axiom,
    ( add(X,Y) = add(Y,X) )).

%----Associativity for multiplication
cnf(associativity_for_multiplication,axiom,
    ( multiply(X,multiply(Y,Z)) = multiply(multiply(X,Y),Z) )).

%----Distributive property of product over sum
cnf(distribute1,axiom,
    ( multiply(X,add(Y,Z)) = add(multiply(X,Y),multiply(X,Z)) )).

cnf(distribute2,axiom,
    ( multiply(add(X,Y),Z) = add(multiply(X,Z),multiply(Y,Z)) )).

%--------------------------------------------------------------------------