%--------------------------------------------------------------------------
% File     : GRP075-1 : TPTP v8.2.0. Bugfixed v2.3.0.
% Domain   : Group Theory
% Problem  : Single axiom for group theory, in double division and identity
% Version  : [McC93] (equality) axioms.
% English  : This is a single axiom for group theory, in terms of double
%            division and identity.

% Refs     : [Neu86] Neumann (1986), Yet Another Single Law for Groups
%          : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr
% Source   : [McC93]
% Names    : Axiom 2.2 [McC93]

% Status   : Unsatisfiable
% Rating   : 0.07 v8.2.0, 0.12 v8.1.0, 0.11 v7.5.0, 0.18 v7.3.0, 0.15 v7.2.0, 0.17 v7.1.0, 0.09 v7.0.0, 0.08 v6.4.0, 0.21 v6.3.0, 0.00 v6.2.0, 0.20 v6.1.0, 0.18 v6.0.0, 0.14 v5.5.0, 0.25 v5.4.0, 0.11 v5.3.0, 0.40 v5.2.0, 0.25 v5.1.0, 0.22 v5.0.0, 0.20 v4.1.0, 0.11 v4.0.1, 0.12 v4.0.0, 0.00 v3.3.0, 0.11 v3.2.0, 0.00 v2.3.0
% Syntax   : Number of clauses     :    5 (   4 unt;   0 nHn;   1 RR)
%            Number of literals    :    7 (   7 equ;   3 neg)
%            Maximal clause size   :    3 (   1 avg)
%            Maximal term depth    :    7 (   2 avg)
%            Number of predicates  :    1 (   0 usr;   0 prp; 2-2 aty)
%            Number of functors    :    9 (   9 usr;   6 con; 0-2 aty)
%            Number of variables   :    8 (   0 sgn)
% SPC      : CNF_UNS_RFO_PEQ_NUE

% Comments :
% Bugfixes : v2.3.0 - Deleted double_division, added multiply and inverse,
%            fixed prove_these_axioms.
%--------------------------------------------------------------------------
cnf(single_axiom,axiom,
    double_divide(double_divide(double_divide(X,double_divide(Y,identity)),double_divide(double_divide(Z,double_divide(U,double_divide(U,identity))),double_divide(X,identity))),Y) = Z ).

%----Definition of multiply
cnf(multiply,axiom,
    multiply(X,Y) = double_divide(double_divide(Y,X),identity) ).

%----Definition of inverse
cnf(inverse,axiom,
    inverse(X) = double_divide(X,identity) ).

%----Definition of identity
cnf(identity,axiom,
    identity = double_divide(X,inverse(X)) ).

cnf(prove_these_axioms,negated_conjecture,
    ( multiply(inverse(a1),a1) != identity
    | multiply(identity,a2) != a2
    | multiply(multiply(a3,b3),c3) != multiply(a3,multiply(b3,c3)) ) ).

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