%  problem-set/algebra/groups/ident2.ver1.clauses
%  created : 07/09/86                                              
%  revised : 08/12/88                                         
                                                            
% description:
%
% Theorem: the right identity axiom is dependant on the rest of the
% axiom set; i.e., each element has a right identity.  Note that
% this is a corollary to ident1, but also that the proof is different
% due to the introduction of a skolem function for the right
% identity of each element of the group.
%
% Note: the axioms P(x,e,x) and P(x,g(x),e) must be deleted.
                                                            
% representation:
%
% declare_predicate(1,[O]).
% declare_predicate(2,[EQUAL]).
% declare_predicate(3,[P]).
% declare_functions(1,[g,k]).
% declare_functions(2,[f,l]).
% declare_constants([e,a,b,c,d,h,j,k]).
% declare_variables([x,y,z,u,v,w]).
%
% O(x) : used for membership in a subgroup of index 2 (see index)
% P(x,y) : product of x and y is z
% g(x) : inverse of x
% k(x) : used to name some inverse of x (see ident2)
% f(x,y) : names the product of x and y
% l(x,y) : names an element of the subgroup of index 2 (see index)
% e : identity element


%  existence of identity    

P(e,x,x).
% P(x,e,x).  DELETED AS PART OF THEOREM

%  existence of inverse  

P(g(x),x,e).
% P(x,g(x),e).  DELETED AS PART OF THEOREM

%  closure  

P(x,y,f(x,y)).

%  associative property  

-P(x,y,u) | -P(y,z,v) | -P(u,z,w) | P(x,v,w).
-P(x,y,u) | -P(y,z,v) | -P(x,v,w) | P(u,z,w).

%  the operation is well defined  

-P(x,y,z) | -P(x,y,w) | EQUAL(z,w) .

%  equality axioms  

EQUAL(x,x).
-EQUAL(x,y) | EQUAL(y,x).
-EQUAL(x,y) | -EQUAL(y,z) | EQUAL(x,z).

%  equality substitution axioms  

-EQUAL(x,y) | -P(x,w,z) | P(y,w,z).
-EQUAL(x,y) | -P(w,x,z) | P(w,y,z).
-EQUAL(x,y) | -P(w,z,x) | P(w,z,y).
-EQUAL(x,y) | EQUAL(f(x,w),f(y,w)).
-EQUAL(x,y) | EQUAL(f(w,x),f(w,y)).
-EQUAL(x,y) | EQUAL(g(x),g(y)).


%  denial of the theorem 

-P(k(y),y,k(y)).