TPTP Problem File: GRP039-7.p
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- Solve Problem
%--------------------------------------------------------------------------
% File : GRP039-7 : TPTP v8.2.0. Bugfixed v1.0.1.
% Domain : Group Theory (Subgroups)
% Problem : Subgroups of index 2 are normal
% Version : [MOW76] (equality) axioms : Augmented.
% English : If O is a subgroup of G and there are exactly 2 cosets
% in G/O, then O is normal [that is, for all x in G and
% y in O, x*y*inverse(x) is back in O].
% Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a
% Source : [MOW76]
% Names : GP2 [MOW76]
% Status : Unsatisfiable
% Rating : 0.40 v8.2.0, 0.38 v8.1.0, 0.32 v7.5.0, 0.37 v7.4.0, 0.41 v7.3.0, 0.42 v7.1.0, 0.33 v7.0.0, 0.47 v6.3.0, 0.36 v6.2.0, 0.50 v6.0.0, 0.60 v5.5.0, 0.70 v5.3.0, 0.72 v5.2.0, 0.62 v5.1.0, 0.59 v5.0.0, 0.50 v4.1.0, 0.54 v4.0.1, 0.64 v4.0.0, 0.55 v3.7.0, 0.40 v3.5.0, 0.45 v3.4.0, 0.42 v3.3.0, 0.36 v3.2.0, 0.38 v3.1.0, 0.45 v2.7.0, 0.50 v2.6.0, 0.40 v2.5.0, 0.42 v2.4.0, 0.33 v2.3.0, 0.44 v2.2.1, 0.44 v2.2.0, 0.67 v2.1.0, 0.89 v2.0.0
% Syntax : Number of clauses : 16 ( 12 unt; 2 nHn; 8 RR)
% Number of literals : 24 ( 11 equ; 5 neg)
% Maximal clause size : 4 ( 1 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 8 ( 8 usr; 5 con; 0-2 aty)
% Number of variables : 16 ( 0 sgn)
% SPC : CNF_UNS_RFO_SEQ_NHN
% Comments : Used to define a subgroup of index two is a theorem which
% says that {for all x, for all y, there exists a z such that
% if x and y are both not in the subgroup O, then z is in O and
% x*z=y} if & only if {O has index 2 in G}. This z is named
% by the skolem function i(x,y). Explanation: If O is of index
% two in G, then there are exactly two cosets, namely O and
% uO for some u not in O. If both of x and y are not in O, then
% they are in uO. But then xO=yO, which implies that there
% exists some z in O such that x*z=y. If the condition holds
% that {for all x, for all y, there exists a z such that
% if x and y are both not in the subgroup O, then z is in O and
% x*z=y}, then xO=yO for all x,y not in O, which implies that
% there are at most two cosets; and there must be at least two,
% namely O and xO, since x is not in O. Therefore O must
% be of index two.
% : element_in_O2(A,B) is A^-1.B. The axioms with element_in_O2
% force index 2.
% Bugfixes : v1.0.1 - Duplicate axioms multiply_inverse_left and
% multiply_inverse_right removed.
%--------------------------------------------------------------------------
%----Include the axioms for group theory in equality form
include('Axioms/GRP004-0.ax').
%----Include the subgroup axioms in equality formulation
include('Axioms/GRP004-1.ax').
%--------------------------------------------------------------------------
%----Redundant two axioms
cnf(right_identity,axiom,
multiply(X,identity) = X ).
cnf(right_inverse,axiom,
multiply(X,inverse(X)) = identity ).
%----The next two clauses are previously established lemmas
cnf(inverse_inverse,axiom,
inverse(inverse(X)) = X ).
cnf(inverse_of_identity,axiom,
inverse(identity) = identity ).
%----This axiom is dependent
cnf(identity_in_O2,axiom,
subgroup_member(identity) ).
%----Definition of a subgroup of index 2
cnf(an_element_in_O2,axiom,
( subgroup_member(X)
| subgroup_member(Y)
| subgroup_member(element_in_O2(X,Y)) ) ).
cnf(property_of_O2,axiom,
( subgroup_member(X)
| subgroup_member(Y)
| multiply(X,element_in_O2(X,Y)) = Y ) ).
%----Denial of theorem
cnf(b_in_O2,negated_conjecture,
subgroup_member(b) ).
cnf(b_times_a_inverse_is_c,negated_conjecture,
multiply(b,inverse(a)) = c ).
cnf(a_times_c_is_d,negated_conjecture,
multiply(a,c) = d ).
cnf(prove_d_in_O2,negated_conjecture,
~ subgroup_member(d) ).
%--------------------------------------------------------------------------