TPTP Problem File: GRP028-3.p
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%--------------------------------------------------------------------------
% File : GRP028-3 : TPTP v9.0.0. Released v1.1.0.
% Domain : Group Theory (Semigroups)
% Problem : In semigroups, left and right solutions => right id exists
% Version : [MOW76] axioms : Reduced > Incomplete.
% English : If there are left and right solutions, then there is a right
% identity element.
% Refs : [Luc68] Luckham (1968), Some Tree-paring Strategies for Theore
% : [MOW76] McCharen et al. (1976), Problems and Experiments for a
% Source : [Luc68]
% Names : Example 1 [Luc68]
% Status : Unsatisfiable
% Rating : 0.00 v5.4.0, 0.06 v5.3.0, 0.10 v5.2.0, 0.00 v2.1.0, 0.00 v2.0.0
% Syntax : Number of clauses : 6 ( 4 unt; 0 nHn; 3 RR)
% Number of literals : 12 ( 0 equ; 7 neg)
% Maximal clause size : 4 ( 2 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of predicates : 1 ( 1 usr; 0 prp; 3-3 aty)
% Number of functors : 4 ( 4 usr; 0 con; 1-2 aty)
% Number of variables : 19 ( 0 sgn)
% SPC : CNF_UNS_RFO_NEQ_HRN
% Comments : [Luc68] uses less axioms than [MOW76].
%--------------------------------------------------------------------------
%----Include Semigroup axioms
% include('Axioms/GRP002-0.ax').
%--------------------------------------------------------------------------
%----This axiom is called closure or totality in some axiomatisations
cnf(total_function1,axiom,
product(X,Y,multiply(X,Y)) ).
%----This axiom is called well_definedness in some axiomatisations
%input_clause(total_function2,axiom,
% [--product(X,Y,Z),
% --product(X,Y,W),
% ++equal(Z,W)]).
cnf(associativity1,axiom,
( ~ product(X,Y,U)
| ~ product(Y,Z,V)
| ~ product(U,Z,W)
| product(X,V,W) ) ).
cnf(associativity2,axiom,
( ~ product(X,Y,U)
| ~ product(Y,Z,V)
| ~ product(X,V,W)
| product(U,Z,W) ) ).
cnf(left_soln,hypothesis,
product(left_solution(X,Y),X,Y) ).
cnf(right_soln,hypothesis,
product(X,right_solution(X,Y),Y) ).
%----There is an element for which no X is identity
cnf(prove_there_is_a_right_identity,negated_conjecture,
~ product(not_identity(X),X,not_identity(X)) ).
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