TPTP Problem File: KRS051+1.p

View Solutions - Solve Problem

%------------------------------------------------------------------------------
% File     : KRS051+1 : TPTP v9.0.0. Released v3.1.0.
% Domain   : Knowledge Representation (Semantic Web)
% Problem  : Integer multiplication in OWL DL
% Version  : Especial.
% English  : N is 2. M is 3. N times M is 6.

% Refs     : [Bec03] Bechhofer (2003), Email to G. Sutcliffe
%          : [TR+04] Tsarkov et al. (2004), Using Vampire to Reason with OW
% Source   : [Bec03]
% Names    : consistent_description-logic-Manifest905 [Bec03]

% Status   : Satisfiable
% Rating   : 0.33 v9.0.0, 0.20 v8.2.0, 0.33 v8.1.0, 0.25 v7.5.0, 0.33 v7.3.0, 0.00 v7.1.0, 0.75 v7.0.0, 0.00 v6.4.0, 0.60 v6.2.0, 0.50 v6.1.0, 0.60 v6.0.0, 0.50 v5.5.0, 0.33 v5.3.0, 0.67 v5.2.0, 0.33 v5.0.0, 0.67 v4.0.1, 0.75 v3.7.0, 0.67 v3.5.0, 0.33 v3.4.0, 0.80 v3.3.0, 0.67 v3.1.0
% Syntax   : Number of formulae    :   41 (   1 unt;   0 def)
%            Number of atoms       :  177 (  72 equ)
%            Maximal formula atoms :   50 (   4 avg)
%            Number of connectives :  157 (  21   ~;  27   |;  67   &)
%                                         (  11 <=>;  31  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   32 (   6 avg)
%            Maximal term depth    :    1 (   1 avg)
%            Number of predicates  :   14 (  13 usr;   0 prp; 1-2 aty)
%            Number of functors    :    1 (   1 usr;   1 con; 0-0 aty)
%            Number of variables   :  114 ( 100   !;  14   ?)
% SPC      : FOF_SAT_RFO_SEQ

% Comments : Sean Bechhofer says there are some errors in the encoding of
%            datatypes, so this problem may not be perfect. At least it's
%            still representative of the type of reasoning required for OWL.
%------------------------------------------------------------------------------
fof(ccardinality_N_substitution_1,axiom,
    ! [A,B] :
      ( ( A = B
        & ccardinality_N(A) )
     => ccardinality_N(B) ) ).

fof(ccardinality_N_times_M_substitution_1,axiom,
    ! [A,B] :
      ( ( A = B
        & ccardinality_N_times_M(A) )
     => ccardinality_N_times_M(B) ) ).

fof(conly_d_substitution_1,axiom,
    ! [A,B] :
      ( ( A = B
        & conly_d(A) )
     => conly_d(B) ) ).

fof(cowlNothing_substitution_1,axiom,
    ! [A,B] :
      ( ( A = B
        & cowlNothing(A) )
     => cowlNothing(B) ) ).

fof(cowlThing_substitution_1,axiom,
    ! [A,B] :
      ( ( A = B
        & cowlThing(A) )
     => cowlThing(B) ) ).

fof(rinvP_1_to_N_substitution_1,axiom,
    ! [A,B,C] :
      ( ( A = B
        & rinvP_1_to_N(A,C) )
     => rinvP_1_to_N(B,C) ) ).

fof(rinvP_1_to_N_substitution_2,axiom,
    ! [A,B,C] :
      ( ( A = B
        & rinvP_1_to_N(C,A) )
     => rinvP_1_to_N(C,B) ) ).

fof(rinvQ_1_to_M_substitution_1,axiom,
    ! [A,B,C] :
      ( ( A = B
        & rinvQ_1_to_M(A,C) )
     => rinvQ_1_to_M(B,C) ) ).

fof(rinvQ_1_to_M_substitution_2,axiom,
    ! [A,B,C] :
      ( ( A = B
        & rinvQ_1_to_M(C,A) )
     => rinvQ_1_to_M(C,B) ) ).

fof(rinvR_N_times_M_to_1_substitution_1,axiom,
    ! [A,B,C] :
      ( ( A = B
        & rinvR_N_times_M_to_1(A,C) )
     => rinvR_N_times_M_to_1(B,C) ) ).

fof(rinvR_N_times_M_to_1_substitution_2,axiom,
    ! [A,B,C] :
      ( ( A = B
        & rinvR_N_times_M_to_1(C,A) )
     => rinvR_N_times_M_to_1(C,B) ) ).

fof(rp_N_to_1_substitution_1,axiom,
    ! [A,B,C] :
      ( ( A = B
        & rp_N_to_1(A,C) )
     => rp_N_to_1(B,C) ) ).

fof(rp_N_to_1_substitution_2,axiom,
    ! [A,B,C] :
      ( ( A = B
        & rp_N_to_1(C,A) )
     => rp_N_to_1(C,B) ) ).

fof(rq_M_to_1_substitution_1,axiom,
    ! [A,B,C] :
      ( ( A = B
        & rq_M_to_1(A,C) )
     => rq_M_to_1(B,C) ) ).

fof(rq_M_to_1_substitution_2,axiom,
    ! [A,B,C] :
      ( ( A = B
        & rq_M_to_1(C,A) )
     => rq_M_to_1(C,B) ) ).

fof(rr_N_times_M_to_1_substitution_1,axiom,
    ! [A,B,C] :
      ( ( A = B
        & rr_N_times_M_to_1(A,C) )
     => rr_N_times_M_to_1(B,C) ) ).

fof(rr_N_times_M_to_1_substitution_2,axiom,
    ! [A,B,C] :
      ( ( A = B
        & rr_N_times_M_to_1(C,A) )
     => rr_N_times_M_to_1(C,B) ) ).

fof(xsd_integer_substitution_1,axiom,
    ! [A,B] :
      ( ( A = B
        & xsd_integer(A) )
     => xsd_integer(B) ) ).

fof(xsd_string_substitution_1,axiom,
    ! [A,B] :
      ( ( A = B
        & xsd_string(A) )
     => xsd_string(B) ) ).

%----Thing and Nothing
fof(axiom_0,axiom,
    ! [X] :
      ( cowlThing(X)
      & ~ cowlNothing(X) ) ).

%----String and Integer disjoint
fof(axiom_1,axiom,
    ! [X] :
      ( xsd_string(X)
    <=> ~ xsd_integer(X) ) ).

%----Equality ccardinality_N
fof(axiom_2,axiom,
    ! [X] :
      ( ccardinality_N(X)
    <=> ( ? [Y0,Y1,Y2] :
            ( rinvQ_1_to_M(X,Y0)
            & rinvQ_1_to_M(X,Y1)
            & rinvQ_1_to_M(X,Y2)
            & Y0 != Y1
            & Y0 != Y2
            & Y1 != Y2 )
        & ! [Y0,Y1,Y2,Y3] :
            ( ( rinvQ_1_to_M(X,Y0)
              & rinvQ_1_to_M(X,Y1)
              & rinvQ_1_to_M(X,Y2)
              & rinvQ_1_to_M(X,Y3) )
           => ( Y0 = Y1
              | Y0 = Y2
              | Y0 = Y3
              | Y1 = Y2
              | Y1 = Y3
              | Y2 = Y3 ) ) ) ) ).

%----Equality ccardinality_N
fof(axiom_3,axiom,
    ! [X] :
      ( ccardinality_N(X)
    <=> ? [Y] :
          ( rp_N_to_1(X,Y)
          & conly_d(Y) ) ) ).

%----Equality ccardinality_N_times_M
fof(axiom_4,axiom,
    ! [X] :
      ( ccardinality_N_times_M(X)
    <=> ? [Y] :
          ( rr_N_times_M_to_1(X,Y)
          & conly_d(Y) ) ) ).

%----Equality ccardinality_N_times_M
fof(axiom_5,axiom,
    ! [X] :
      ( ccardinality_N_times_M(X)
    <=> ? [Y] :
          ( rq_M_to_1(X,Y)
          & ccardinality_N(Y) ) ) ).

%----Equality conly_d
fof(axiom_6,axiom,
    ! [X] :
      ( conly_d(X)
    <=> ( ? [Y0,Y1,Y2,Y3,Y4,Y5] :
            ( rinvR_N_times_M_to_1(X,Y0)
            & rinvR_N_times_M_to_1(X,Y1)
            & rinvR_N_times_M_to_1(X,Y2)
            & rinvR_N_times_M_to_1(X,Y3)
            & rinvR_N_times_M_to_1(X,Y4)
            & rinvR_N_times_M_to_1(X,Y5)
            & Y0 != Y1
            & Y0 != Y2
            & Y0 != Y3
            & Y0 != Y4
            & Y0 != Y5
            & Y1 != Y2
            & Y1 != Y3
            & Y1 != Y4
            & Y1 != Y5
            & Y2 != Y3
            & Y2 != Y4
            & Y2 != Y5
            & Y3 != Y4
            & Y3 != Y5
            & Y4 != Y5 )
        & ! [Y0,Y1,Y2,Y3,Y4,Y5,Y6] :
            ( ( rinvR_N_times_M_to_1(X,Y0)
              & rinvR_N_times_M_to_1(X,Y1)
              & rinvR_N_times_M_to_1(X,Y2)
              & rinvR_N_times_M_to_1(X,Y3)
              & rinvR_N_times_M_to_1(X,Y4)
              & rinvR_N_times_M_to_1(X,Y5)
              & rinvR_N_times_M_to_1(X,Y6) )
           => ( Y0 = Y1
              | Y0 = Y2
              | Y0 = Y3
              | Y0 = Y4
              | Y0 = Y5
              | Y0 = Y6
              | Y1 = Y2
              | Y1 = Y3
              | Y1 = Y4
              | Y1 = Y5
              | Y1 = Y6
              | Y2 = Y3
              | Y2 = Y4
              | Y2 = Y5
              | Y2 = Y6
              | Y3 = Y4
              | Y3 = Y5
              | Y3 = Y6
              | Y4 = Y5
              | Y4 = Y6
              | Y5 = Y6 ) ) ) ) ).

%----Equality conly_d
fof(axiom_7,axiom,
    ! [X] :
      ( conly_d(X)
    <=> ( ? [Y0,Y1] :
            ( rinvP_1_to_N(X,Y0)
            & rinvP_1_to_N(X,Y1)
            & Y0 != Y1 )
        & ! [Y0,Y1,Y2] :
            ( ( rinvP_1_to_N(X,Y0)
              & rinvP_1_to_N(X,Y1)
              & rinvP_1_to_N(X,Y2) )
           => ( Y0 = Y1
              | Y0 = Y2
              | Y1 = Y2 ) ) ) ) ).

%----Enumeration conly_d
fof(axiom_8,axiom,
    ! [X] :
      ( conly_d(X)
    <=> X = id ) ).

%----Functional: rp_N_to_1
fof(axiom_9,axiom,
    ! [X,Y,Z] :
      ( ( rp_N_to_1(X,Y)
        & rp_N_to_1(X,Z) )
     => Y = Z ) ).

%----Domain: rp_N_to_1
fof(axiom_10,axiom,
    ! [X,Y] :
      ( rp_N_to_1(X,Y)
     => ccardinality_N(X) ) ).

%----Range: rp_N_to_1
fof(axiom_11,axiom,
    ! [X,Y] :
      ( rp_N_to_1(X,Y)
     => conly_d(Y) ) ).

%----Inverse: rp_N_to_1
fof(axiom_12,axiom,
    ! [X,Y] :
      ( rp_N_to_1(X,Y)
    <=> rinvP_1_to_N(Y,X) ) ).

%----Functional: rq_M_to_1
fof(axiom_13,axiom,
    ! [X,Y,Z] :
      ( ( rq_M_to_1(X,Y)
        & rq_M_to_1(X,Z) )
     => Y = Z ) ).

%----Domain: rq_M_to_1
fof(axiom_14,axiom,
    ! [X,Y] :
      ( rq_M_to_1(X,Y)
     => ccardinality_N_times_M(X) ) ).

%----Range: rq_M_to_1
fof(axiom_15,axiom,
    ! [X,Y] :
      ( rq_M_to_1(X,Y)
     => ccardinality_N(Y) ) ).

%----Inverse: rq_M_to_1
fof(axiom_16,axiom,
    ! [X,Y] :
      ( rq_M_to_1(X,Y)
    <=> rinvQ_1_to_M(Y,X) ) ).

%----Functional: rr_N_times_M_to_1
fof(axiom_17,axiom,
    ! [X,Y,Z] :
      ( ( rr_N_times_M_to_1(X,Y)
        & rr_N_times_M_to_1(X,Z) )
     => Y = Z ) ).

%----Domain: rr_N_times_M_to_1
fof(axiom_18,axiom,
    ! [X,Y] :
      ( rr_N_times_M_to_1(X,Y)
     => ccardinality_N_times_M(X) ) ).

%----Range: rr_N_times_M_to_1
fof(axiom_19,axiom,
    ! [X,Y] :
      ( rr_N_times_M_to_1(X,Y)
     => conly_d(Y) ) ).

%----Inverse: rr_N_times_M_to_1
fof(axiom_20,axiom,
    ! [X,Y] :
      ( rr_N_times_M_to_1(X,Y)
    <=> rinvR_N_times_M_to_1(Y,X) ) ).

%----id
fof(axiom_21,axiom,
    cowlThing(id) ).

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