TPTP Problem File: GEO012-3.p
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- Solve Problem
%--------------------------------------------------------------------------
% File : GEO012-3 : TPTP v8.2.0. Bugfixed v1.2.1.
% Domain : Geometry
% Problem : Collinearity for 4 points
% Version : [Qua89] axioms : Augmented.
% English : If any three distinct points x, y, and z are collinear and
% a fourth point w is collinear with x and y, then
% w is collinear with x and z and also with x and y.
% Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a
% : [SST83] Schwabbauser et al. (1983), Metamathematische Methoden
% : [Qua89] Quaife (1989), Automated Development of Tarski's Geome
% Source : [Qua89]
% Names : T12 [Qua89]
% Status : Unsatisfiable
% Rating : 0.35 v8.2.0, 0.38 v8.1.0, 0.37 v7.5.0, 0.42 v7.4.0, 0.35 v7.3.0, 0.33 v7.1.0, 0.25 v7.0.0, 0.27 v6.4.0, 0.33 v6.3.0, 0.27 v6.2.0, 0.30 v6.1.0, 0.64 v6.0.0, 0.70 v5.5.0, 0.80 v5.4.0, 0.85 v5.3.0, 0.89 v5.2.0, 0.81 v5.1.0, 0.82 v5.0.0, 0.64 v4.1.0, 0.62 v4.0.1, 0.55 v3.7.0, 0.50 v3.5.0, 0.64 v3.4.0, 0.67 v3.3.0, 0.79 v3.2.0, 0.85 v3.1.0, 0.82 v2.7.0, 0.92 v2.6.0, 1.00 v2.0.0
% Syntax : Number of clauses : 105 ( 32 unt; 21 nHn; 79 RR)
% Number of literals : 275 ( 49 equ; 153 neg)
% Maximal clause size : 8 ( 2 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 4 ( 3 usr; 0 prp; 2-4 aty)
% Number of functors : 14 ( 14 usr; 7 con; 0-6 aty)
% Number of variables : 356 ( 18 sgn)
% SPC : CNF_UNS_RFO_SEQ_NHN
% Comments : This presentation may have alternatives/be incorrect.
% Bugfixes : v1.2.1 - Clause d12 fixed.
%--------------------------------------------------------------------------
%----Include Tarski geometry axioms
include('Axioms/GEO002-0.ax').
%----Include Tarski geometry axioms for colinearity
include('Axioms/GEO002-1.ax').
%----Include definition of reflection
include('Axioms/GEO002-2.ax').
%----Include definition of insertion
include('Axioms/GEO002-3.ax').
%--------------------------------------------------------------------------
cnf(d1,axiom,
equidistant(U,V,U,V) ).
cnf(d2,axiom,
( ~ equidistant(U,V,W,X)
| equidistant(W,X,U,V) ) ).
cnf(d3,axiom,
( ~ equidistant(U,V,W,X)
| equidistant(V,U,W,X) ) ).
cnf(d4_1,axiom,
( ~ equidistant(U,V,W,X)
| equidistant(U,V,X,W) ) ).
cnf(d4_2,axiom,
( ~ equidistant(U,V,W,X)
| equidistant(V,U,X,W) ) ).
cnf(d4_3,axiom,
( ~ equidistant(U,V,W,X)
| equidistant(W,X,V,U) ) ).
cnf(d4_4,axiom,
( ~ equidistant(U,V,W,X)
| equidistant(X,W,U,V) ) ).
cnf(d4_5,axiom,
( ~ equidistant(U,V,W,X)
| equidistant(X,W,V,U) ) ).
cnf(d5,axiom,
( ~ equidistant(U,V,W,X)
| ~ equidistant(W,X,Y,Z)
| equidistant(U,V,Y,Z) ) ).
cnf(e1,axiom,
V = extension(U,V,W,W) ).
cnf(b0,axiom,
( Y != extension(U,V,W,X)
| between(U,V,Y) ) ).
cnf(r2_1,axiom,
between(U,V,reflection(U,V)) ).
cnf(r2_2,axiom,
equidistant(V,reflection(U,V),U,V) ).
cnf(r3_1,axiom,
( U != V
| V = reflection(U,V) ) ).
cnf(r3_2,axiom,
U = reflection(U,U) ).
cnf(r4,axiom,
( V != reflection(U,V)
| U = V ) ).
cnf(d7,axiom,
equidistant(U,U,V,V) ).
cnf(d8,axiom,
( ~ equidistant(U,V,U1,V1)
| ~ equidistant(V,W,V1,W1)
| ~ between(U,V,W)
| ~ between(U1,V1,W1)
| equidistant(U,W,U1,W1) ) ).
cnf(d9,axiom,
( ~ between(U,V,W)
| ~ between(U,V,X)
| ~ equidistant(V,W,V,X)
| U = V
| W = X ) ).
cnf(d10_1,axiom,
( ~ between(U,V,W)
| U = V
| W = extension(U,V,V,W) ) ).
cnf(d10_2,axiom,
( ~ equidistant(W,X,Y,Z)
| extension(U,V,W,X) = extension(U,V,Y,Z)
| U = V ) ).
cnf(d10_3,axiom,
( extension(U,V,U,V) = extension(U,V,V,U)
| U = V ) ).
cnf(r5,axiom,
equidistant(V,U,V,reflection(reflection(U,V),V)) ).
cnf(r6,axiom,
U = reflection(reflection(U,V),V) ).
cnf(t3,axiom,
between(U,V,V) ).
cnf(b1,axiom,
( ~ between(U,W,X)
| U != X
| between(V,W,X) ) ).
cnf(t1,axiom,
( ~ between(U,V,W)
| between(W,V,U) ) ).
cnf(t2,axiom,
between(U,U,V) ).
cnf(b2,axiom,
( ~ between(U,V,W)
| ~ between(V,U,W)
| U = V ) ).
cnf(b3,axiom,
( ~ between(U,V,W)
| ~ between(U,W,V)
| V = W ) ).
cnf(t6_1,axiom,
( ~ between(U,V,W)
| ~ between(V,U,W)
| U = V
| V = W ) ).
cnf(t6_2,axiom,
( ~ between(U,V,W)
| ~ between(U,W,V)
| U = V
| V = W ) ).
cnf(b4,axiom,
( ~ between(U,V,W)
| ~ between(V,W,X)
| between(U,V,W) ) ).
cnf(b5,axiom,
( ~ between(U,V,W)
| ~ between(U,W,X)
| between(V,W,X) ) ).
cnf(b6,axiom,
( ~ between(U,V,W)
| ~ between(V,W,X)
| between(U,W,X)
| V = W ) ).
cnf(b7,axiom,
( ~ between(U,V,W)
| ~ between(V,W,X)
| between(U,V,X)
| V = W ) ).
cnf(b8,axiom,
( ~ between(U,V,X)
| ~ between(V,W,X)
| between(U,W,X) ) ).
cnf(b9,axiom,
( ~ between(U,V,W)
| ~ between(U,W,X)
| between(U,V,X) ) ).
cnf(e2_1,axiom,
lower_dimension_point_1 != lower_dimension_point_2 ).
cnf(e2_2,axiom,
lower_dimension_point_2 != lower_dimension_point_3 ).
cnf(e2_3,axiom,
lower_dimension_point_1 != lower_dimension_point_3 ).
cnf(e3_1,axiom,
V != extension(U,V,lower_dimension_point_1,lower_dimension_point_2) ).
cnf(e3_2,axiom,
equidistant(V,extension(U,V,lower_dimension_point_1,lower_dimension_point_2),X,extension(W,X,lower_dimension_point_1,lower_dimension_point_2)) ).
cnf(e3_3,axiom,
between(U,V,extension(U,V,lower_dimension_point_1,lower_dimension_point_2)) ).
cnf(b10,axiom,
( ~ between(U,V,W)
| ~ between(U1,V1,W)
| ~ between(U,X,U1)
| between(X,inner_pasch(V1,inner_pasch(U,X,U1,V1,W),U,V,W),W)
| between(V,inner_pasch(V1,inner_pasch(U,X,U1,V1,W),U,V,W),V1) ) ).
cnf(d11,axiom,
( ~ between(U,V,W)
| ~ equidistant(U,W,U,W1)
| ~ equidistant(V,W,V,W1)
| U = V
| W = W1 ) ).
cnf(d12,axiom,
( ~ equidistant(U,V,U1,V1)
| ~ equidistant(U,W,U1,W1)
| ~ equidistant(U,X,U1,X1)
| ~ equidistant(W,X,W1,X1)
| ~ between(U,V,W)
| ~ between(U1,V1,W1)
| equidistant(V,X,V1,X1) ) ).
cnf(d13,axiom,
( ~ between(U,V,W)
| ~ between(U1,V1,W1)
| ~ equidistant(U,V,U1,V1)
| ~ equidistant(U,W,U1,W1)
| equidistant(V,W,V1,W1) ) ).
cnf(d14,axiom,
( ~ equidistant(U,V,U1,V1)
| ~ equidistant(V,W,V1,W1)
| ~ equidistant(U,X,U1,X1)
| ~ equidistant(W,X,W1,X1)
| ~ between(U,V,W)
| ~ between(U1,V1,W1)
| equidistant(V,X,V1,X1) ) ).
cnf(d15,axiom,
( ~ between(U,V,W)
| ~ equidistant(U,V,U,X)
| ~ equidistant(W,V,W,X)
| V = X ) ).
cnf(i2_1,axiom,
equidistant(U,V,U1,insertion(U1,W1,U,V)) ).
cnf(i2_2,axiom,
( ~ between(U,V,W)
| ~ equidistant(U,W,U1,W1)
| between(U1,insertion(U1,W1,U,V),W1) ) ).
cnf(i2_3,axiom,
( ~ between(U,V,W)
| ~ equidistant(U,W,U1,W1)
| equidistant(V,W,insertion(U1,W1,U,V),W1) ) ).
cnf(i3,axiom,
( ~ between(U,V,W)
| V = insertion(U,W,U,V) ) ).
cnf(i4,axiom,
( ~ equidistant(W,X,Y,Z)
| insertion(U,V,W,X) = insertion(U,V,Y,Z) ) ).
cnf(b11,axiom,
( ~ equidistant(U,V,U1,V1)
| ~ equidistant(V,W,V1,W1)
| ~ equidistant(U,W,U1,W1)
| ~ between(U,V,W)
| between(U1,V1,W1) ) ).
cnf(b12,axiom,
( ~ between(U,V,W)
| ~ between(U,V,X)
| U = V
| between(U,W,X)
| between(U,X,W) ) ).
cnf(b13,axiom,
( ~ between(U,V,W)
| ~ between(U,V,X)
| U = V
| between(V,W,X)
| between(V,X,W) ) ).
cnf(t7,axiom,
( ~ between(U,W,X)
| ~ between(V,W,X)
| W = X
| between(U,V,W)
| between(V,U,W) ) ).
cnf(t9,axiom,
( ~ between(U,V,X)
| ~ between(U,W,X)
| between(U,V,W)
| between(U,W,V) ) ).
cnf(b14,axiom,
( ~ between(U,V,X)
| ~ between(U,W,X)
| between(V,W,X)
| between(W,V,X) ) ).
cnf(t8,axiom,
( ~ between(U,V,Y)
| ~ between(V,W,X)
| ~ between(U,X,Y)
| between(U,W,Y) ) ).
cnf(b15,axiom,
( ~ between(U,V,W)
| ~ equidistant(U,V,U,W)
| V = W ) ).
cnf(c2_1,axiom,
( ~ between(W,V,U)
| colinear(U,V,W) ) ).
cnf(c2_2,axiom,
( ~ between(U,W,V)
| colinear(U,V,W) ) ).
cnf(c2_3,axiom,
( ~ between(V,U,W)
| colinear(U,V,W) ) ).
cnf(t10_1,axiom,
( ~ colinear(U,V,W)
| colinear(W,V,U) ) ).
cnf(t10_2,axiom,
( ~ colinear(U,V,W)
| colinear(V,W,U) ) ).
cnf(t10_3,axiom,
( ~ colinear(U,V,W)
| colinear(U,W,V) ) ).
cnf(t10_4,axiom,
( ~ colinear(U,V,W)
| colinear(W,U,V) ) ).
cnf(t10_5,axiom,
( ~ colinear(U,V,W)
| colinear(V,U,W) ) ).
cnf(t11,axiom,
~ colinear(lower_dimension_point_1,lower_dimension_point_2,lower_dimension_point_3) ).
cnf(c3_1,axiom,
colinear(X,X,Y) ).
cnf(c3_2,axiom,
colinear(X,Y,X) ).
cnf(c3_3,axiom,
colinear(Y,X,X) ).
cnf(c3_4,axiom,
( X != Y
| colinear(X,Z,Y) ) ).
cnf(c4,axiom,
( ~ equidistant(U,V,U1,V1)
| ~ equidistant(V,W,V1,W1)
| ~ equidistant(U,W,U1,W1)
| ~ colinear(U,V,W)
| colinear(U1,V1,W1) ) ).
cnf(a_not_b,hypothesis,
a != b ).
cnf(abc_colinear,hypothesis,
colinear(a,b,c) ).
cnf(abd_colinear,hypothesis,
colinear(a,b,d) ).
cnf(prove_colinearity,negated_conjecture,
( ~ colinear(a,c,d)
| ~ colinear(b,c,d) ) ).
%--------------------------------------------------------------------------