TPTP Problem File: GRP025-3.p
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%--------------------------------------------------------------------------
% File : GRP025-3 : TPTP v8.2.0. Bugfixed v1.2.1.
% Domain : Group Theory (Named groups)
% Problem : All groups of order 2 are isomorphic
% Version : [MOW76] axioms : Incomplete.
% Theorem formulation : Does not prove full generality.
% English : If G1 has exactly two elements and G2 has exactly two
% 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 : order2.ver3.in [ANL]
% Status : Satisfiable
% Rating : 0.00 v8.1.0, 0.33 v7.5.0, 0.00 v6.2.0, 0.20 v6.1.0, 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 v2.0.0
% Syntax : Number of clauses : 47 ( 25 unt; 2 nHn; 36 RR)
% Number of literals : 85 ( 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 : 13 ( 13 usr; 9 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.
%--------------------------------------------------------------------------
%----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_member_of_group1,hypothesis,
group_member(a,g1) ).
cnf(b_member_of_group1,hypothesis,
group_member(b,g1) ).
cnf(c_member_of_group2,hypothesis,
group_member(c,g2) ).
cnf(d_member_of_group2,hypothesis,
group_member(d,g2) ).
cnf(a_and_b_only_members_of_group1,hypothesis,
( ~ group_member(X,g1)
| equalish(X,a)
| equalish(X,b) ) ).
cnf(c_and_d_only_members_of_group2,hypothesis,
( ~ group_member(X,g2)
| equalish(X,c)
| equalish(X,d) ) ).
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(b_times_b_is_a,hypothesis,
product(g1,b,b,a) ).
cnf(c_times_c_is_c,hypothesis,
product(g2,c,c,c) ).
cnf(c_times_d_is_d,hypothesis,
product(g2,c,d,d) ).
cnf(d_times_c_is_d,hypothesis,
product(g2,d,c,d) ).
cnf(d_times_d_is_c,hypothesis,
product(g2,d,d,c) ).
%----Definition of the isomorphism
cnf(a_maps_to_c,hypothesis,
equalish(an_isomorphism(a),c) ).
cnf(b_maps_to_d,hypothesis,
equalish(an_isomorphism(b),d) ).
%----Denial that the isomorphism is indeed one
cnf(d1_member_of_group1,hypothesis,
group_member(d1,g1) ).
cnf(d2_member_of_group1,hypothesis,
group_member(d2,g1) ).
cnf(d3_member_of_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)) ).
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