TPTP Problem File: GRP039-3.p
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- Solve Problem
%--------------------------------------------------------------------------
% File : GRP039-3 : TPTP v8.2.0. Released v1.0.0.
% Domain : Group Theory (Subgroups)
% Problem : Subgroups of index 2 are normal
% Version : [Wos65] 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% Refs : [Wos65] Wos (1965), Unpublished Note
% : [WM76] Wilson & Minker (1976), Resolution, Refinements, and S
% Source : [SPRFN]
% Names : wos19 [WM76]
% Status : Unsatisfiable
% Rating : 0.20 v8.2.0, 0.14 v8.1.0, 0.16 v7.5.0, 0.21 v7.4.0, 0.24 v7.3.0, 0.17 v7.1.0, 0.08 v7.0.0, 0.27 v6.4.0, 0.07 v6.3.0, 0.18 v6.2.0, 0.20 v6.1.0, 0.36 v6.0.0, 0.30 v5.5.0, 0.55 v5.3.0, 0.50 v5.2.0, 0.44 v5.1.0, 0.47 v5.0.0, 0.43 v4.1.0, 0.38 v4.0.1, 0.45 v4.0.0, 0.55 v3.7.0, 0.40 v3.5.0, 0.45 v3.4.0, 0.50 v3.3.0, 0.43 v3.2.0, 0.46 v3.1.0, 0.36 v2.7.0, 0.50 v2.6.0, 0.40 v2.5.0, 0.50 v2.4.0, 0.44 v2.2.1, 0.56 v2.2.0, 0.67 v2.1.0, 0.78 v2.0.0
% Syntax : Number of clauses : 20 ( 11 unt; 2 nHn; 12 RR)
% Number of literals : 40 ( 4 equ; 17 neg)
% Maximal clause size : 4 ( 2 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 3 ( 2 usr; 0 prp; 1-3 aty)
% Number of functors : 8 ( 8 usr; 5 con; 0-2 aty)
% Number of variables : 39 ( 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.
%--------------------------------------------------------------------------
%----Include group theory axioms
include('Axioms/GRP003-0.ax').
%----Include sub-group theory axioms
include('Axioms/GRP003-2.ax').
%--------------------------------------------------------------------------
%----The next five clauses are dependent lemmas
cnf(product_right_cancellation,axiom,
( ~ product(A,B,C)
| ~ product(A,D,C)
| D = B ) ).
cnf(product_left_cancellation,axiom,
( ~ product(A,B,C)
| ~ product(D,B,C)
| D = A ) ).
cnf(inverse_is_self_cancelling,axiom,
inverse(inverse(A)) = A ).
cnf(identity_is_in_subgroup,axiom,
subgroup_member(identity) ).
cnf(subgroup_member_inverse_are_in_subgroup,axiom,
( ~ subgroup_member(A)
| subgroup_member(inverse(A)) ) ).
%----Definition of subgroup of index 2
cnf(an_element_in_O2,axiom,
( subgroup_member(element_in_O2(A,B))
| subgroup_member(B)
| subgroup_member(A) ) ).
cnf(property_of_O2,axiom,
( product(A,element_in_O2(A,B),B)
| subgroup_member(B)
| subgroup_member(A) ) ).
%----Denial of theorem
cnf(b_is_in_subgroup,negated_conjecture,
subgroup_member(b) ).
cnf(b_times_a_inverse_is_c,negated_conjecture,
product(b,inverse(a),c) ).
cnf(a_times_c_is_d,negated_conjecture,
product(a,c,d) ).
cnf(prove_d_is_in_subgroup,negated_conjecture,
~ subgroup_member(d) ).
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