## TPTP Problem File: GRP123-1.003.p

View Solutions - Solve Problem

```%--------------------------------------------------------------------------
% File     : GRP123-1.003 : TPTP v8.1.0. Released v1.2.0.
% Domain   : Group Theory (Quasigroups)
% Problem  : (3,2,1) conjugate orthogonality
% Version  : [Sla93] axioms.
% English  : If ab=xy and a*b = x*y then a=x and b=y, where c*b=a iff ab=c.
%            Generate the multiplication table for the specified quasi-
%            group with 3 elements.

% Refs     : [FSB93] Fujita et al. (1993), Automatic Generation of Some Res
%          : [Sla93] Slaney (1993), Email to G. Sutcliffe
%          : [Zha94] Zhang (1994), Email to G. Sutcliffe
%          : [SFS95] Slaney et al. (1995), Automated Reasoning and Exhausti
% Source   : [Sla93]
% Names    : QG1 [Sla93]
%          : QG1 [FSB93]
%          : QG1 [SFS95]
%          : Bennett QG1 [TPTP]

% Status   : Unsatisfiable
% Rating   : 0.00 v2.1.0
% Syntax   : Number of clauses     :   16 (  10 unt;   1 nHn;  15 RR)
%            Number of literals    :   34 (   0 equ;  22 neg)
%            Maximal clause size   :    5 (   2 avg)
%            Maximal term depth    :    1 (   1 avg)
%            Number of predicates  :    3 (   3 usr;   0 prp; 1-3 aty)
%            Number of functors    :    3 (   3 usr;   3 con; 0-0 aty)
%            Number of variables   :   27 (   0 sgn)
% SPC      : CNF_UNS_EPR_NEQ_NHN

% Comments : [Sla93]'s axiomatization has been modified for this.
%          : Substitution axioms are not needed, as any positive equality
%            literals should resolve on negative ones directly.
%          : [Zha94] has pointed out that either one of qg1_1
%            or qg1_2 may be used, as each implies the other in this
%            scenario, with the help of cancellation. The dependence
%            cannot be proved, so both have been left in here.
%          : tptp2X: -f tptp -s3 GRP123-1.g
%--------------------------------------------------------------------------
cnf(element_1,axiom,
group_element(e_1) ).

cnf(element_2,axiom,
group_element(e_2) ).

cnf(element_3,axiom,
group_element(e_3) ).

cnf(e_1_is_not_e_2,axiom,
~ equalish(e_1,e_2) ).

cnf(e_1_is_not_e_3,axiom,
~ equalish(e_1,e_3) ).

cnf(e_2_is_not_e_1,axiom,
~ equalish(e_2,e_1) ).

cnf(e_2_is_not_e_3,axiom,
~ equalish(e_2,e_3) ).

cnf(e_3_is_not_e_1,axiom,
~ equalish(e_3,e_1) ).

cnf(e_3_is_not_e_2,axiom,
~ equalish(e_3,e_2) ).

cnf(product_total_function1,axiom,
( ~ group_element(X)
| ~ group_element(Y)
| product(X,Y,e_1)
| product(X,Y,e_2)
| product(X,Y,e_3) ) ).

cnf(product_total_function2,axiom,
( ~ product(X,Y,W)
| ~ product(X,Y,Z)
| equalish(W,Z) ) ).

cnf(product_right_cancellation,axiom,
( ~ product(X,W,Y)
| ~ product(X,Z,Y)
| equalish(W,Z) ) ).

cnf(product_left_cancellation,axiom,
( ~ product(W,Y,X)
| ~ product(Z,Y,X)
| equalish(W,Z) ) ).

cnf(product_idempotence,axiom,
product(X,X,X) ).

cnf(qg1_1,negated_conjecture,
( ~ product(X1,Y1,Z1)
| ~ product(X2,Y2,Z1)
| ~ product(Z2,Y1,X1)
| ~ product(Z2,Y2,X2)
| equalish(X1,X2) ) ).

cnf(qg1_2,negated_conjecture,
( ~ product(X1,Y1,Z1)
| ~ product(X2,Y2,Z1)
| ~ product(Z2,Y1,X1)
| ~ product(Z2,Y2,X2)
| equalish(Y1,Y2) ) ).

%--------------------------------------------------------------------------
```