%--------------------------------------------------------------------------
% File     : RNG008-5 : TPTP v9.0.0. Released v1.0.0.
% Domain   : Ring Theory
% Problem  : Boolean rings are commutative
% Version  : [MOW76] axioms : Augmented.
%            Theorem formulation : Equality.
% English  : Given a ring in which for all x, x * x = x, prove that for
%            all x and y, x * y = y * x.

% Refs     : [MOW76] McCharen et al. (1976), Problems and Experiments for a
%          : [PS81]  Peterson & Stickel (1981), Complete Sets of Reductions
% Source   : [ANL]
% Names    : commute.ver1.in [ANL]

% Status   : Unsatisfiable
% Rating   : 0.08 v9.0.0, 0.06 v8.2.0, 0.00 v8.1.0, 0.11 v7.5.0, 0.10 v7.4.0, 0.22 v7.2.0, 0.25 v7.1.0, 0.14 v6.3.0, 0.00 v6.1.0, 0.20 v6.0.0, 0.33 v5.5.0, 0.69 v5.4.0, 0.67 v5.3.0, 0.75 v5.2.0, 0.38 v5.1.0, 0.43 v5.0.0, 0.29 v4.1.0, 0.33 v4.0.1, 0.17 v4.0.0, 0.33 v3.7.0, 0.17 v3.5.0, 0.00 v3.2.0, 0.14 v3.1.0, 0.33 v2.7.0, 0.17 v2.6.0, 0.43 v2.5.0, 0.20 v2.4.0, 0.33 v2.3.0, 0.17 v2.2.1, 0.67 v2.2.0, 0.71 v2.1.0, 0.50 v2.0.0
% Syntax   : Number of clauses     :   26 (  15 unt;   0 nHn;  14 RR)
%            Number of literals    :   59 (   2 equ;  34 neg)
%            Maximal clause size   :    5 (   2 avg)
%            Maximal term depth    :    3 (   1 avg)
%            Number of predicates  :    3 (   2 usr;   0 prp; 2-3 aty)
%            Number of functors    :    7 (   7 usr;   4 con; 0-2 aty)
%            Number of variables   :   79 (   2 sgn)
% SPC      : CNF_UNS_RFO_SEQ_HRN

% Comments : Extra lemmas based on [PS81] equality axioms.
%--------------------------------------------------------------------------
%----Include ring theory axioms
include('Axioms/RNG001-0.ax').
%--------------------------------------------------------------------------
%----The next six clauses are the extra lemmas.
%----Inverse of identity is identity
cnf(additive_inverse_identity,axiom,
    sum(additive_inverse(additive_identity),additive_identity,additive_identity) ).

%----Inverse of inverse of X is X
cnf(additive_inverse_additive_inverse,axiom,
    sum(additive_inverse(additive_inverse(X)),additive_identity,X) ).

%----Behavior of additive_identity and the multiplication operation
cnf(multiply_additive_id1,axiom,
    product(X,additive_identity,additive_identity) ).

cnf(multiply_additive_id2,axiom,
    product(additive_identity,X,additive_identity) ).

%----Inverse of (x + y) is additive_inverse(x) + additive_inverse(y),
cnf(distribute_additive_inverse,axiom,
    sum(additive_inverse(X),additive_inverse(Y),additive_inverse(add(X,Y))) ).

%----x * additive_inverse(y) = additive_inverse (x * y),
cnf(multiply_additive_inverse,axiom,
    product(X,additive_inverse(Y),additive_inverse(multiply(X,Y))) ).

%----Clauses for the theorem
cnf(x_squared_is_x,hypothesis,
    product(X,X,X) ).

cnf(a_times_b_is_c,hypothesis,
    product(a,b,c) ).

cnf(prove_b_times_a_is_c,negated_conjecture,
    ~ product(b,a,c) ).

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