## TPTP Axioms File: BOO002-0.ax

```%--------------------------------------------------------------------------
% File     : BOO002-0 : TPTP v7.5.0. Released v1.0.0.
% Domain   : Boolean Algebra
% Axioms   : Boolean algebra axioms
% Version  : [MOW76] axioms.
% English  :

% Refs     : [Whi61] Whitesitt (1961), Boolean Algebra and Its Applications
%          : [MOW76] McCharen et al. (1976), Problems and Experiments for a
% Source   : [MOW76]
% Names    :

% Status   : Satisfiable
% Syntax   : Number of clauses    :   22 (   0 non-Horn;  10 unit;  12 RR)
%            Number of atoms      :   60 (   2 equality)
%            Maximal clause size  :    5 (   3 average)
%            Number of predicates :    3 (   0 propositional; 2-3 arity)
%            Number of functors   :    5 (   2 constant; 0-2 arity)
%            Number of variables  :   82 (   0 singleton)
%            Maximal term depth   :    2 (   1 average)
% SPC      :

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

cnf(closure_of_multiplication,axiom,
( product(X,Y,multiply(X,Y)) )).

( ~ sum(X,Y,Z)
| sum(Y,X,Z) )).

cnf(commutativity_of_multiplication,axiom,
( ~ product(X,Y,Z)
| product(Y,X,Z) )).

cnf(multiplicative_identity1,axiom,
( product(multiplicative_identity,X,X) )).

cnf(multiplicative_identity2,axiom,
( product(X,multiplicative_identity,X) )).

cnf(distributivity1,axiom,
( ~ product(X,Y,V1)
| ~ product(X,Z,V2)
| ~ sum(Y,Z,V3)
| ~ product(X,V3,V4)
| sum(V1,V2,V4) )).

cnf(distributivity2,axiom,
( ~ product(X,Y,V1)
| ~ product(X,Z,V2)
| ~ sum(Y,Z,V3)
| ~ sum(V1,V2,V4)
| product(X,V3,V4) )).

cnf(distributivity3,axiom,
( ~ product(Y,X,V1)
| ~ product(Z,X,V2)
| ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| sum(V1,V2,V4) )).

cnf(distributivity4,axiom,
( ~ product(Y,X,V1)
| ~ product(Z,X,V2)
| ~ sum(Y,Z,V3)
| ~ sum(V1,V2,V4)
| product(V3,X,V4) )).

cnf(distributivity5,axiom,
( ~ sum(X,Y,V1)
| ~ sum(X,Z,V2)
| ~ product(Y,Z,V3)
| ~ sum(X,V3,V4)
| product(V1,V2,V4) )).

cnf(distributivity6,axiom,
( ~ sum(X,Y,V1)
| ~ sum(X,Z,V2)
| ~ product(Y,Z,V3)
| ~ product(V1,V2,V4)
| sum(X,V3,V4) )).

cnf(distributivity7,axiom,
( ~ sum(Y,X,V1)
| ~ sum(Z,X,V2)
| ~ product(Y,Z,V3)
| ~ sum(V3,X,V4)
| product(V1,V2,V4) )).

cnf(distributivity8,axiom,
( ~ sum(Y,X,V1)
| ~ sum(Z,X,V2)
| ~ product(Y,Z,V3)
| ~ product(V1,V2,V4)
| sum(V3,X,V4) )).

( sum(inverse(X),X,multiplicative_identity) )).

( sum(X,inverse(X),multiplicative_identity) )).

cnf(multiplicative_inverse1,axiom,

cnf(multiplicative_inverse2,axiom,

%-----Well definedness of the operations