%--------------------------------------------------------------------------
% File     : GRP040-4 : TPTP v8.2.0. Released v1.0.0.
% Domain   : Group Theory (Subgroups)
% Problem  : In subgroups of order 2, inverse is an involution
% Version  : [Wos65] axioms : Augmented.
% English  :

% Refs     : [Wos65] Wos (1965), Unpublished Note
%          : [WM76]  Wilson & Minker (1976), Resolution, Refinements, and S
% Source   : [SPRFN]
% Names    : Problem 20 [Wos65]
%          : wos20 [WM76]

% Status   : Unsatisfiable
% Rating   : 0.25 v8.2.0, 0.24 v8.1.0, 0.11 v7.5.0, 0.16 v7.4.0, 0.12 v7.3.0, 0.17 v7.1.0, 0.08 v7.0.0, 0.07 v6.4.0, 0.00 v6.3.0, 0.18 v6.2.0, 0.20 v6.1.0, 0.36 v6.0.0, 0.20 v5.5.0, 0.55 v5.4.0, 0.50 v5.2.0, 0.38 v5.1.0, 0.35 v5.0.0, 0.29 v4.1.0, 0.31 v4.0.1, 0.45 v3.7.0, 0.20 v3.5.0, 0.27 v3.4.0, 0.33 v3.3.0, 0.29 v3.2.0, 0.31 v3.1.0, 0.27 v2.7.0, 0.33 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, 0.78 v2.0.0
% Syntax   : Number of clauses     :   21 (  12 unt;   2 nHn;  13 RR)
%            Number of literals    :   41 (   4 equ;  18 neg)
%            Maximal clause size   :    4 (   1 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 :
%--------------------------------------------------------------------------
%----Include group theory axioms
include('Axioms/GRP003-0.ax').
%----Include sub-group theory axioms
include('Axioms/GRP003-2.ax').
%--------------------------------------------------------------------------
%----The next four clauses are dependent
cnf(identity_is_in_subgroup,axiom,
    subgroup_member(identity) ).

cnf(closure_of_inverse,axiom,
    ( ~ subgroup_member(A)
    | subgroup_member(inverse(A)) ) ).

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 ) ).

%----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) ) ).

cnf(a_in_subgroup,hypothesis,
    ~ subgroup_member(a) ).

cnf(b_is_in_subgroup,hypothesis,
    subgroup_member(b) ).

cnf(d_in_subgroup,hypothesis,
    ~ subgroup_member(d) ).

cnf(b_times_a_inverse_is_c,hypothesis,
    product(b,inverse(a),c) ).

cnf(a_times_c_is_d,hypothesis,
    product(a,c,d) ).

cnf(prove_inverse_is_self_cancelling,negated_conjecture,
    inverse(inverse(A)) = A ).

%--------------------------------------------------------------------------