TPTP Problem File: GRP026-3.p

%--------------------------------------------------------------------------
% File     : GRP026-3 : TPTP v9.0.0. Bugfixed v2.0.0.
% Domain   : Group Theory (Named groups)
% Problem  : All groups of order 3 are isomorphic
% Version  : [MOW76] axioms : Incomplete.
%            Theorem formulation : Does not prove full generality.
% English  : If G1 and G2 each have exactly three elements, then there
%            exists an isomorphism [a one-to-one and onto homomorphism]
%            between them.

% Refs     : [MOW76] McCharen et al. (1976), Problems and Experiments for a
% Source   : [ANL]
% Names    : order3.ver3.in [ANL]

% Status   : Satisfiable
% Rating   : 0.00 v8.1.0, 0.33 v7.5.0, 0.00 v6.2.0, 0.40 v6.1.0, 0.67 v6.0.0, 0.33 v5.5.0, 0.75 v5.4.0, 0.89 v5.3.0, 0.86 v5.0.0, 0.62 v4.1.0, 0.57 v4.0.0, 0.62 v3.5.0, 0.71 v3.4.0, 0.83 v3.2.0, 0.80 v3.1.0, 0.86 v2.7.0, 0.80 v2.6.0, 1.00 v2.1.0
% Syntax   : Number of clauses     :   60 (  38 unt;   2 nHn;  49 RR)
%            Number of literals    :  100 (   0 equ;  37 neg)
%            Maximal clause size   :    4 (   1 avg)
%            Maximal term depth    :    2 (   1 avg)
%            Number of predicates  :    3 (   3 usr;   0 prp; 2-4 aty)
%            Number of functors    :   15 (  15 usr;  11 con; 0-3 aty)
%            Number of variables   :   90 (   0 sgn)
% SPC      : CNF_SAT_RFO_NEQ

% Comments : In order to prove the theorem, the group tables and a
%            particular homomorphism are specified, and the contradiction
%            comes from the fact that this is the actual isomorphism. Not
%            only is this formulation cheating, but also it does not prove
%            the theorem in full generality.
%          : Missing an_isomorphism subsitution axioms.
% Bugfixes : v1.2.1 - Bugfix in GRP006-0.ax.
%          : v2.0.0 - Clause prove_product_holds_in_group2 fixed.
%--------------------------------------------------------------------------
%----Include the axioms for named groups
%include('Axioms/GRP006-0.ax').
%--------------------------------------------------------------------------
cnf(reflexivity,axiom,
    equalish(X,X) ).

cnf(symmetry,axiom,
    ( ~ equalish(X,Y)
    | equalish(Y,X) ) ).

cnf(transitivity,axiom,
    ( ~ equalish(X,Y)
    | ~ equalish(Y,Z)
    | equalish(X,Z) ) ).

cnf(product_substitution1,axiom,
    ( ~ equalish(Xg,Yg)
    | ~ product(Xg,X,Y,Z)
    | product(Yg,X,Y,Z) ) ).

cnf(product_substitution2,axiom,
    ( ~ equalish(X,Y)
    | ~ product(Xg,X,Z,W)
    | product(Xg,Y,Z,W) ) ).

cnf(product_substitution3,axiom,
    ( ~ equalish(X,Y)
    | ~ product(Xg,W,X,Z)
    | product(Xg,W,Y,Z) ) ).

cnf(product_substitution4,axiom,
    ( ~ equalish(X,Y)
    | ~ product(Xg,W,Z,X)
    | product(Xg,W,Z,Y) ) ).

cnf(multiply_substitution1,axiom,
    ( ~ equalish(Xg,Yg)
    | equalish(multiply(Xg,X,Y),multiply(Yg,X,Y)) ) ).

cnf(multiply_substitution2,axiom,
    ( ~ equalish(X,Y)
    | equalish(multiply(Xg,X,Z),multiply(Xg,Y,Z)) ) ).

cnf(multiply_substitution3,axiom,
    ( ~ equalish(X,Y)
    | equalish(multiply(Xg,Z,X),multiply(Xg,Z,Y)) ) ).

cnf(inverse_substitution1,axiom,
    ( ~ equalish(Xg,Yg)
    | equalish(inverse(Xg,X),inverse(Yg,X)) ) ).

cnf(inverse_substitution2,axiom,
    ( ~ equalish(X,Y)
    | equalish(inverse(Xg,X),inverse(Xg,Y)) ) ).

cnf(group_member_substitution1,axiom,
    ( ~ equalish(Xg,Yg)
    | ~ group_member(X,Xg)
    | group_member(X,Yg) ) ).

cnf(group_member_substitution2,axiom,
    ( ~ equalish(X,Y)
    | ~ group_member(X,Xg)
    | group_member(Y,Xg) ) ).

cnf(identity_substitution,axiom,
    ( ~ equalish(Xg,Yg)
    | equalish(identity_for(Xg),identity_for(Yg)) ) ).

cnf(identity_in_group,axiom,
    group_member(identity_for(Xg),Xg) ).

cnf(left_identity,axiom,
    product(Xg,identity_for(Xg),X,X) ).

cnf(right_identity,axiom,
    product(Xg,X,identity_for(Xg),X) ).

cnf(inverse_in_group,axiom,
    ( ~ group_member(X,Xg)
    | group_member(inverse(Xg,X),Xg) ) ).

cnf(left_inverse,axiom,
    product(Xg,inverse(Xg,X),X,identity_for(Xg)) ).

cnf(right_inverse,axiom,
    product(Xg,X,inverse(Xg,X),identity_for(Xg)) ).

%----These axioms are called closure or totality in some axiomatisations
cnf(total_function1_1,axiom,
    ( ~ group_member(X,Xg)
    | ~ group_member(Y,Xg)
    | product(Xg,X,Y,multiply(Xg,X,Y)) ) ).

cnf(total_function1_2,axiom,
    ( ~ group_member(X,Xg)
    | ~ group_member(Y,Xg)
    | group_member(multiply(Xg,X,Y),Xg) ) ).

%----This axiom is called well_definedness in some axiomatisations
cnf(total_function2,axiom,
    ( ~ product(Xg,X,Y,Z)
    | ~ product(Xg,X,Y,W)
    | equalish(W,Z) ) ).

cnf(associativity1,axiom,
    ( ~ product(Xg,X,Y,Xy)
    | ~ product(Xg,Y,Z,Yz)
    | ~ product(Xg,Xy,Z,Xyz)
    | product(Xg,X,Yz,Xyz) ) ).

cnf(associativity2,axiom,
    ( ~ product(Xg,X,Y,Xy)
    | ~ product(Xg,Y,Z,Yz)
    | ~ product(Xg,X,Yz,Xyz)
    | product(Xg,Xy,Z,Xyz) ) ).

%----Definition of the two groups
cnf(a_in_group1,hypothesis,
    group_member(a,g1) ).

cnf(b_in_group1,hypothesis,
    group_member(b,g1) ).

cnf(c_in_group1,hypothesis,
    group_member(c,g1) ).

cnf(f_in_group2,hypothesis,
    group_member(f,g2) ).

cnf(g_in_group2,hypothesis,
    group_member(g,g2) ).

cnf(h_in_group2,hypothesis,
    group_member(h,g2) ).

cnf(all_of_group1,hypothesis,
    ( ~ group_member(X,g1)
    | equalish(X,a)
    | equalish(X,b)
    | equalish(X,c) ) ).

cnf(all_of_group2,hypothesis,
    ( ~ group_member(X,g2)
    | equalish(X,f)
    | equalish(X,g)
    | equalish(X,h) ) ).

cnf(a_times_a_is_a,hypothesis,
    product(g1,a,a,a) ).

cnf(a_times_b_is_b,hypothesis,
    product(g1,a,b,b) ).

cnf(b_times_a_is_b,hypothesis,
    product(g1,b,a,b) ).

cnf(a_times_c_is_c,hypothesis,
    product(g1,a,c,c) ).

cnf(c_times_a_is_c,hypothesis,
    product(g1,c,a,c) ).

cnf(b_times_b_is_c,hypothesis,
    product(g1,b,b,c) ).

cnf(b_times_c_is_a,hypothesis,
    product(g1,b,c,a) ).

cnf(c_times_b_is_a,hypothesis,
    product(g1,c,b,a) ).

cnf(c_times_c_is_b,hypothesis,
    product(g1,c,c,b) ).

cnf(f_times_f_is_f,hypothesis,
    product(g2,f,f,f) ).

cnf(f_times_g_is_g,hypothesis,
    product(g2,f,g,g) ).

cnf(g_times_f_is_g,hypothesis,
    product(g2,g,f,g) ).

cnf(f_times_h_is_h,hypothesis,
    product(g2,f,h,h) ).

cnf(h_times_f_is_h,hypothesis,
    product(g2,h,f,h) ).

cnf(g_times_g_is_h,hypothesis,
    product(g2,g,g,h) ).

cnf(g_times_h_is_f,hypothesis,
    product(g2,g,h,f) ).

cnf(h_times_g_is_f,hypothesis,
    product(g2,h,g,f) ).

cnf(h_times_h_is_g,hypothesis,
    product(g2,h,h,g) ).

%----Definition of an isomorphism
cnf(a_maps_to_f,hypothesis,
    equalish(an_isomorphism(a),f) ).

cnf(b_maps_to_g,hypothesis,
    equalish(an_isomorphism(b),g) ).

cnf(c_maps_to_h,hypothesis,
    equalish(an_isomorphism(c),h) ).

%----Denial that the isomorphism is one
cnf(d1_in_group1,hypothesis,
    group_member(d1,g1) ).

cnf(d2_in_group1,hypothesis,
    group_member(d2,g1) ).

cnf(d3_in_group1,hypothesis,
    group_member(d3,g1) ).

cnf(d1_times_d2_is_d3,hypothesis,
    product(g1,d1,d2,d3) ).

cnf(prove_product_holds_in_group2,negated_conjecture,
    ~ product(g2,an_isomorphism(d1),an_isomorphism(d2),an_isomorphism(d3)) ).

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