% problem-set/algebra/rings/Stickel.eq.clauses
% created : 07/11/86                 
% revised : 07/13/88                 
                                        
% description : 
%
% The following clauses are given by Stickel as a complete set 
% of reductions for ring theory. 

% representation :
%
% declare_predicate(2,EQUAL).
% declare_functions(2,[j,f]).
% declare_function(1,g).
% declare_constants([0]).
% declare_variables([x,y,z]).
%
% j(x,y) is the sum of x and y, f(x,y) is the product of x and y; 
% g(x) is the additive inverse of x; 0 is the additive identity element


% existence of left identity for addition 

EQUAL(j(0,x),x).

% existence of left additive inverse  

EQUAL(j(g(x),x),0).

% distributive property of product over sum    

EQUAL(f(x,j(y,z)),j(f(x,y),f(x,z))).
EQUAL(f(j(x,y),z),j(f(x,z),f(y,z))).

% inverse of identity is identity 

EQUAL(g(0),0).

% inverse of inverse of x is x itself 

EQUAL(g(g(x)),x).

% behavior of 0 and the multiplication operation  

EQUAL(f(x,0),0).
EQUAL(f(0,x),0).

% inverse of (x + y) is inverse(x) + inverse(y)  

EQUAL(g(j(x,y)),j(g(x),g(y))).

% x * inverse(y) = inverse (x * y)

EQUAL(f(x,g(y)),g(f(x,y))).
EQUAL(f(g(x),y),g(f(x,y))).