TPTP Problem File: GRP027-1.p
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- Solve Problem
%--------------------------------------------------------------------------
% File : GRP027-1 : TPTP v9.0.0. Bugfixed v1.2.1.
% Domain : Group Theory (Named groups)
% Problem : All groups of order 5 are cyclic
% Version : [MOW76] axioms.
% English : There exists an element in G that generates all other
% elements by taking powers of that element.
% Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a
% Source : [TPTP]
% Names :
% Status : Satisfiable
% Rating : 0.56 v9.0.0, 0.50 v8.2.0, 0.70 v8.1.0, 0.62 v7.5.0, 0.67 v7.4.0, 0.64 v7.3.0, 0.67 v7.1.0, 0.62 v7.0.0, 0.57 v6.4.0, 0.29 v6.3.0, 0.25 v6.2.0, 0.30 v6.1.0, 0.44 v6.0.0, 0.43 v5.5.0, 0.62 v5.4.0, 0.80 v5.3.0, 0.78 v5.2.0, 0.80 v5.0.0, 0.78 v4.1.0, 0.71 v4.0.1, 0.80 v4.0.0, 0.50 v3.7.0, 0.33 v3.4.0, 0.50 v3.3.0, 0.33 v3.2.0, 0.80 v3.1.0, 0.67 v2.7.0, 0.33 v2.6.0, 0.86 v2.5.0, 0.67 v2.3.0, 1.00 v2.1.0
% Syntax : Number of clauses : 24 ( 17 unt; 1 nHn; 18 RR)
% Number of literals : 42 ( 9 equ; 19 neg)
% Maximal clause size : 6 ( 1 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 3 ( 2 usr; 0 prp; 2-4 aty)
% Number of functors : 10 ( 10 usr; 6 con; 0-3 aty)
% Number of variables : 43 ( 0 sgn)
% SPC : CNF_SAT_RFO_EQU_NUE
% Comments : This theorem is proven via the fact that any element raised
% to the power of the group gives the identity element back,
% and that therefore only powers up to 5 in this case need
% be investigated.
% Bugfixes : v1.2.1 - Bugfix in GRP006-0.ax.
%--------------------------------------------------------------------------
%----Include the axioms for named groups
include('Axioms/GRP006-0.ax').
%--------------------------------------------------------------------------
%----Elements of the group of order 5
cnf(a_in_group,hypothesis,
group_member(a,g) ).
cnf(b_in_group,hypothesis,
group_member(b,g) ).
cnf(c_in_group,hypothesis,
group_member(c,g) ).
cnf(d_in_group,hypothesis,
group_member(d,g) ).
cnf(i_in_group,hypothesis,
group_member(i,g) ).
cnf(i_is_identity,hypothesis,
identity_for(g) = i ).
cnf(all_of_group,hypothesis,
( ~ group_member(X,g)
| X = a
| X = b
| X = c
| X = d
| X = i ) ).
cnf(multiplication_to_identity,hypothesis,
multiply(g,X,multiply(g,X,multiply(g,X,multiply(g,X,X)))) = i ).
cnf(all_multiply_to_identity,hypothesis,
not_power_of(g,X) != X ).
%----Denial of theorem : For all X in g, there exists a Y such that X<>Y,
%----and X^2<>Y, and X^3<>Y, and X^4<>Y, and X^5<>Y.
cnf(x2_is_not_power,negated_conjecture,
~ product(g,X,X,not_power_of(g,X)) ).
cnf(x3_is_not_power,negated_conjecture,
~ product(g,X,multiply(g,X,X),not_power_of(g,X)) ).
cnf(x4_is_not_power,negated_conjecture,
~ product(g,X,multiply(g,X,multiply(g,X,X)),not_power_of(g,X)) ).
cnf(x5_is_not_power,negated_conjecture,
~ product(g,X,multiply(g,X,multiply(g,X,multiply(g,X,X))),not_power_of(g,X)) ).
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