## TPTP Problem File: GRP123-9.004.p

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```%--------------------------------------------------------------------------
% File     : GRP123-9.004 : TPTP v8.1.0. Bugfixed v1.2.1.
% Domain   : Group Theory (Quasigroups)
% Problem  : (3,2,1) conjugate orthogonality
% Version  : [Sla93] axioms : Augmented.
%            Theorem formulation : Uses a second group.
% 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 4 elements.

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

% Status   : Satisfiable
% Rating   : 0.00 v7.3.0, 0.25 v7.0.0, 0.00 v6.2.0, 0.17 v6.1.0, 0.20 v6.0.0, 0.00 v5.0.0, 0.14 v4.1.0, 0.12 v4.0.1, 0.00 v3.4.0, 0.20 v3.3.0, 0.00 v3.2.0, 0.33 v3.1.0, 0.17 v2.7.0, 0.25 v2.6.0, 0.17 v2.5.0, 0.20 v2.4.0, 0.00 v2.2.1, 0.33 v2.2.0, 1.00 v2.1.0
% Syntax   : Number of clauses     :   29 (  18 unt;   4 nHn;  27 RR)
%            Number of literals    :   63 (   0 equ;  34 neg)
%            Maximal clause size   :    6 (   2 avg)
%            Maximal term depth    :    1 (   1 avg)
%            Number of predicates  :    5 (   5 usr;   0 prp; 1-3 aty)
%            Number of functors    :    4 (   4 usr;   4 con; 0-0 aty)
%            Number of variables   :   38 (   0 sgn)
% SPC      : CNF_SAT_EPR_NEQ

% Comments : [SFS93]'s axiomatization has been modified for this.
%          : Substitution axioms are not needed, as any positive equality
%            literals should resolve on negative ones directly.
%          : Version 9 has surjectivity and rotation
%          : tptp2X: -f tptp -s4 GRP123-9.g
% Bugfixes : v1.2.1 - Clauses row_surjectivity and column_surjectivity fixed.
%--------------------------------------------------------------------------
cnf(row_surjectivity,axiom,
( ~ group_element(X)
| ~ group_element(Y)
| product(e_1,X,Y)
| product(e_2,X,Y)
| product(e_3,X,Y)
| product(e_4,X,Y) ) ).

cnf(column_surjectivity,axiom,
( ~ group_element(X)
| ~ group_element(Y)
| product(X,e_1,Y)
| product(X,e_2,Y)
| product(X,e_3,Y)
| product(X,e_4,Y) ) ).

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(element_4,axiom,
group_element(e_4) ).

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_1_is_not_e_4,axiom,
~ equalish(e_1,e_4) ).

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_2_is_not_e_4,axiom,
~ equalish(e_2,e_4) ).

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(e_3_is_not_e_4,axiom,
~ equalish(e_3,e_4) ).

cnf(e_4_is_not_e_1,axiom,
~ equalish(e_4,e_1) ).

cnf(e_4_is_not_e_2,axiom,
~ equalish(e_4,e_2) ).

cnf(e_4_is_not_e_3,axiom,
~ equalish(e_4,e_3) ).

cnf(product1_total_function1,axiom,
( ~ group_element(X)
| ~ group_element(Y)
| product1(X,Y,e_1)
| product1(X,Y,e_2)
| product1(X,Y,e_3)
| product1(X,Y,e_4) ) ).

cnf(product1_total_function2,axiom,
( ~ product1(X,Y,W)
| ~ product1(X,Y,Z)
| equalish(W,Z) ) ).

cnf(product1_right_cancellation,axiom,
( ~ product1(X,W,Y)
| ~ product1(X,Z,Y)
| equalish(W,Z) ) ).

cnf(product1_left_cancellation,axiom,
( ~ product1(W,Y,X)
| ~ product1(Z,Y,X)
| equalish(W,Z) ) ).

cnf(product1_idempotence,axiom,
product1(X,X,X) ).

cnf(product2_total_function1,axiom,
( ~ group_element(X)
| ~ group_element(Y)
| product2(X,Y,e_1)
| product2(X,Y,e_2)
| product2(X,Y,e_3)
| product2(X,Y,e_4) ) ).

cnf(product2_total_function2,axiom,
( ~ product2(X,Y,W)
| ~ product2(X,Y,Z)
| equalish(W,Z) ) ).

cnf(product2_right_cancellation,axiom,
( ~ product2(X,W,Y)
| ~ product2(X,Z,Y)
| equalish(W,Z) ) ).

cnf(product2_left_cancellation,axiom,
( ~ product2(W,Y,X)
| ~ product2(Z,Y,X)
| equalish(W,Z) ) ).

cnf(product2_idempotence,axiom,
product2(X,X,X) ).

cnf(qg1a,negated_conjecture,
( ~ product1(X,Y,Z1)
| ~ product1(Z1,Y,Z2)
| product2(Z2,X,Y) ) ).

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