TPTP Problem File: LCL966_9.p

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%------------------------------------------------------------------------------
% File     : LCL966_9 : TPTP v9.0.0. Released v8.2.0.
% Domain   : Set Theory
% Problem  : Goedel translation of SET917+1
% Version  : [QMLTP] axioms.
% English  : 

% Refs     : [Goe69] Goedel (1969), An Interpretation of the Intuitionistic
%          : [RO12]  Raths & Otten (2012), The QMLTP Problem Library for Fi
% Source   : [QMLTP]
% Names    : GSE917+1 [QMLTP]

% Status   : CounterSatisfiable
% Rating   : 0.00 v8.2.0
% Syntax   : Number of formulae    :   32 (   2 unt;  10 typ;   0 def)
%            Number of atoms       :  187 (   0 equ)
%            Maximal formula atoms :   13 (   8 avg)
%            Number of connectives :  168 (   3   ~;   1   |;   8   &)
%                                         (   0 <=>; 156  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   19 (  13 avg)
%            Maximal term depth    :    3 (   1 avg)
%            Number of types       :    3 (   1 usr)
%            Number of type conns  :   18 (   8   >;  10   *;   0   +;   0  <<)
%            Number of predicates  :    6 (   6 usr;   0 prp; 2-3 aty)
%            Number of functors    :    3 (   3 usr;   1 con; 0-2 aty)
%            Number of variables   :  148 (; 144   !;   4   ?; 148   :)
% SPC      : TF0_CSA_NEQ_NAR

% Comments : Generated by embedproblem, version 1.7.14, rigid constant, 
%            modal_system_D, TFF embedding.
%------------------------------------------------------------------------------
tff('$ki_world_type',type,
    '$ki_world': $tType ).

tff('$ki_local_world_decl',type,
    '$ki_local_world': '$ki_world' ).

tff('$ki_accessible_decl',type,
    '$ki_accessible': ( '$ki_world' * '$ki_world' ) > $o ).

tff(mrel_serial,axiom,
    ! [W: '$ki_world'] :
    ? [V: '$ki_world'] : '$ki_accessible'(W,V) ).

tff(qmltpeq_decl,type,
    qmltpeq: ( '$ki_world' * $i * $i ) > $o ).

tff(in_decl,type,
    in: ( '$ki_world' * $i * $i ) > $o ).

tff(disjoint_decl,type,
    disjoint: ( '$ki_world' * $i * $i ) > $o ).

tff(empty_decl,type,
    empty: ( '$ki_world' * $i ) > $o ).

tff(set_intersection2_decl,type,
    set_intersection2: ( $i * $i ) > $i ).

tff(singleton_decl,type,
    singleton: $i > $i ).

tff('$ki_exists_in_world_$i_decl',type,
    '$ki_exists_in_world_$i': ( '$ki_world' * $i ) > $o ).

tff('$ki_exists_in_world_$i_nonempty',axiom,
    ! [W: '$ki_world'] :
    ? [X: $i] : '$ki_exists_in_world_$i'(W,X) ).

tff(reflexivity,axiom,
    ! [W: '$ki_world'] :
      ( '$ki_accessible'('$ki_local_world',W)
     => ! [X: $i] :
          ( '$ki_exists_in_world_$i'(W,X)
         => ! [W0: '$ki_world'] :
              ( '$ki_accessible'(W,W0)
             => qmltpeq(W0,X,X) ) ) ) ).

tff(symmetry,axiom,
    ! [W: '$ki_world'] :
      ( '$ki_accessible'('$ki_local_world',W)
     => ! [X: $i] :
          ( '$ki_exists_in_world_$i'(W,X)
         => ! [W0: '$ki_world'] :
              ( '$ki_accessible'(W,W0)
             => ! [Y: $i] :
                  ( '$ki_exists_in_world_$i'(W0,Y)
                 => ! [W1: '$ki_world'] :
                      ( '$ki_accessible'(W0,W1)
                     => ( ! [W2: '$ki_world'] :
                            ( '$ki_accessible'(W1,W2)
                           => qmltpeq(W2,X,Y) )
                       => ! [W2: '$ki_world'] :
                            ( '$ki_accessible'(W1,W2)
                           => qmltpeq(W2,Y,X) ) ) ) ) ) ) ) ).

tff(transitivity,axiom,
    ! [W: '$ki_world'] :
      ( '$ki_accessible'('$ki_local_world',W)
     => ! [X: $i] :
          ( '$ki_exists_in_world_$i'(W,X)
         => ! [W0: '$ki_world'] :
              ( '$ki_accessible'(W,W0)
             => ! [Y: $i] :
                  ( '$ki_exists_in_world_$i'(W0,Y)
                 => ! [W1: '$ki_world'] :
                      ( '$ki_accessible'(W0,W1)
                     => ! [Z: $i] :
                          ( '$ki_exists_in_world_$i'(W1,Z)
                         => ! [W2: '$ki_world'] :
                              ( '$ki_accessible'(W1,W2)
                             => ( ( ! [W3: '$ki_world'] :
                                      ( '$ki_accessible'(W2,W3)
                                     => qmltpeq(W3,X,Y) )
                                  & ! [W3: '$ki_world'] :
                                      ( '$ki_accessible'(W2,W3)
                                     => qmltpeq(W3,Y,Z) ) )
                               => ! [W3: '$ki_world'] :
                                    ( '$ki_accessible'(W2,W3)
                                   => qmltpeq(W3,X,Z) ) ) ) ) ) ) ) ) ) ).

tff(set_intersection2_substitution_1,axiom,
    ! [W: '$ki_world'] :
      ( '$ki_accessible'('$ki_local_world',W)
     => ! [A: $i] :
          ( '$ki_exists_in_world_$i'(W,A)
         => ! [W0: '$ki_world'] :
              ( '$ki_accessible'(W,W0)
             => ! [B: $i] :
                  ( '$ki_exists_in_world_$i'(W0,B)
                 => ! [W1: '$ki_world'] :
                      ( '$ki_accessible'(W0,W1)
                     => ! [C: $i] :
                          ( '$ki_exists_in_world_$i'(W1,C)
                         => ! [W2: '$ki_world'] :
                              ( '$ki_accessible'(W1,W2)
                             => ( ! [W3: '$ki_world'] :
                                    ( '$ki_accessible'(W2,W3)
                                   => qmltpeq(W3,A,B) )
                               => ! [W3: '$ki_world'] :
                                    ( '$ki_accessible'(W2,W3)
                                   => qmltpeq(W3,set_intersection2(A,C),set_intersection2(B,C)) ) ) ) ) ) ) ) ) ) ).

tff(set_intersection2_substitution_2,axiom,
    ! [W: '$ki_world'] :
      ( '$ki_accessible'('$ki_local_world',W)
     => ! [A: $i] :
          ( '$ki_exists_in_world_$i'(W,A)
         => ! [W0: '$ki_world'] :
              ( '$ki_accessible'(W,W0)
             => ! [B: $i] :
                  ( '$ki_exists_in_world_$i'(W0,B)
                 => ! [W1: '$ki_world'] :
                      ( '$ki_accessible'(W0,W1)
                     => ! [C: $i] :
                          ( '$ki_exists_in_world_$i'(W1,C)
                         => ! [W2: '$ki_world'] :
                              ( '$ki_accessible'(W1,W2)
                             => ( ! [W3: '$ki_world'] :
                                    ( '$ki_accessible'(W2,W3)
                                   => qmltpeq(W3,A,B) )
                               => ! [W3: '$ki_world'] :
                                    ( '$ki_accessible'(W2,W3)
                                   => qmltpeq(W3,set_intersection2(C,A),set_intersection2(C,B)) ) ) ) ) ) ) ) ) ) ).

tff(singleton_substitution_1,axiom,
    ! [W: '$ki_world'] :
      ( '$ki_accessible'('$ki_local_world',W)
     => ! [A: $i] :
          ( '$ki_exists_in_world_$i'(W,A)
         => ! [W0: '$ki_world'] :
              ( '$ki_accessible'(W,W0)
             => ! [B: $i] :
                  ( '$ki_exists_in_world_$i'(W0,B)
                 => ! [W1: '$ki_world'] :
                      ( '$ki_accessible'(W0,W1)
                     => ( ! [W2: '$ki_world'] :
                            ( '$ki_accessible'(W1,W2)
                           => qmltpeq(W2,A,B) )
                       => ! [W2: '$ki_world'] :
                            ( '$ki_accessible'(W1,W2)
                           => qmltpeq(W2,singleton(A),singleton(B)) ) ) ) ) ) ) ) ).

tff(disjoint_substitution_1,axiom,
    ! [W: '$ki_world'] :
      ( '$ki_accessible'('$ki_local_world',W)
     => ! [A: $i] :
          ( '$ki_exists_in_world_$i'(W,A)
         => ! [W0: '$ki_world'] :
              ( '$ki_accessible'(W,W0)
             => ! [B: $i] :
                  ( '$ki_exists_in_world_$i'(W0,B)
                 => ! [W1: '$ki_world'] :
                      ( '$ki_accessible'(W0,W1)
                     => ! [C: $i] :
                          ( '$ki_exists_in_world_$i'(W1,C)
                         => ! [W2: '$ki_world'] :
                              ( '$ki_accessible'(W1,W2)
                             => ( ( ! [W3: '$ki_world'] :
                                      ( '$ki_accessible'(W2,W3)
                                     => qmltpeq(W3,A,B) )
                                  & ! [W3: '$ki_world'] :
                                      ( '$ki_accessible'(W2,W3)
                                     => disjoint(W3,A,C) ) )
                               => ! [W3: '$ki_world'] :
                                    ( '$ki_accessible'(W2,W3)
                                   => disjoint(W3,B,C) ) ) ) ) ) ) ) ) ) ).

tff(disjoint_substitution_2,axiom,
    ! [W: '$ki_world'] :
      ( '$ki_accessible'('$ki_local_world',W)
     => ! [A: $i] :
          ( '$ki_exists_in_world_$i'(W,A)
         => ! [W0: '$ki_world'] :
              ( '$ki_accessible'(W,W0)
             => ! [B: $i] :
                  ( '$ki_exists_in_world_$i'(W0,B)
                 => ! [W1: '$ki_world'] :
                      ( '$ki_accessible'(W0,W1)
                     => ! [C: $i] :
                          ( '$ki_exists_in_world_$i'(W1,C)
                         => ! [W2: '$ki_world'] :
                              ( '$ki_accessible'(W1,W2)
                             => ( ( ! [W3: '$ki_world'] :
                                      ( '$ki_accessible'(W2,W3)
                                     => qmltpeq(W3,A,B) )
                                  & ! [W3: '$ki_world'] :
                                      ( '$ki_accessible'(W2,W3)
                                     => disjoint(W3,C,A) ) )
                               => ! [W3: '$ki_world'] :
                                    ( '$ki_accessible'(W2,W3)
                                   => disjoint(W3,C,B) ) ) ) ) ) ) ) ) ) ).

tff(empty_substitution_1,axiom,
    ! [W: '$ki_world'] :
      ( '$ki_accessible'('$ki_local_world',W)
     => ! [A: $i] :
          ( '$ki_exists_in_world_$i'(W,A)
         => ! [W0: '$ki_world'] :
              ( '$ki_accessible'(W,W0)
             => ! [B: $i] :
                  ( '$ki_exists_in_world_$i'(W0,B)
                 => ! [W1: '$ki_world'] :
                      ( '$ki_accessible'(W0,W1)
                     => ( ( ! [W2: '$ki_world'] :
                              ( '$ki_accessible'(W1,W2)
                             => qmltpeq(W2,A,B) )
                          & ! [W2: '$ki_world'] :
                              ( '$ki_accessible'(W1,W2)
                             => empty(W2,A) ) )
                       => ! [W2: '$ki_world'] :
                            ( '$ki_accessible'(W1,W2)
                           => empty(W2,B) ) ) ) ) ) ) ) ).

tff(in_substitution_1,axiom,
    ! [W: '$ki_world'] :
      ( '$ki_accessible'('$ki_local_world',W)
     => ! [A: $i] :
          ( '$ki_exists_in_world_$i'(W,A)
         => ! [W0: '$ki_world'] :
              ( '$ki_accessible'(W,W0)
             => ! [B: $i] :
                  ( '$ki_exists_in_world_$i'(W0,B)
                 => ! [W1: '$ki_world'] :
                      ( '$ki_accessible'(W0,W1)
                     => ! [C: $i] :
                          ( '$ki_exists_in_world_$i'(W1,C)
                         => ! [W2: '$ki_world'] :
                              ( '$ki_accessible'(W1,W2)
                             => ( ( ! [W3: '$ki_world'] :
                                      ( '$ki_accessible'(W2,W3)
                                     => qmltpeq(W3,A,B) )
                                  & ! [W3: '$ki_world'] :
                                      ( '$ki_accessible'(W2,W3)
                                     => in(W3,A,C) ) )
                               => ! [W3: '$ki_world'] :
                                    ( '$ki_accessible'(W2,W3)
                                   => in(W3,B,C) ) ) ) ) ) ) ) ) ) ).

tff(in_substitution_2,axiom,
    ! [W: '$ki_world'] :
      ( '$ki_accessible'('$ki_local_world',W)
     => ! [A: $i] :
          ( '$ki_exists_in_world_$i'(W,A)
         => ! [W0: '$ki_world'] :
              ( '$ki_accessible'(W,W0)
             => ! [B: $i] :
                  ( '$ki_exists_in_world_$i'(W0,B)
                 => ! [W1: '$ki_world'] :
                      ( '$ki_accessible'(W0,W1)
                     => ! [C: $i] :
                          ( '$ki_exists_in_world_$i'(W1,C)
                         => ! [W2: '$ki_world'] :
                              ( '$ki_accessible'(W1,W2)
                             => ( ( ! [W3: '$ki_world'] :
                                      ( '$ki_accessible'(W2,W3)
                                     => qmltpeq(W3,A,B) )
                                  & ! [W3: '$ki_world'] :
                                      ( '$ki_accessible'(W2,W3)
                                     => in(W3,C,A) ) )
                               => ! [W3: '$ki_world'] :
                                    ( '$ki_accessible'(W2,W3)
                                   => in(W3,C,B) ) ) ) ) ) ) ) ) ) ).

tff(antisymmetry_r2_hidden,axiom,
    ! [W: '$ki_world'] :
      ( '$ki_accessible'('$ki_local_world',W)
     => ! [A: $i] :
          ( '$ki_exists_in_world_$i'(W,A)
         => ! [W0: '$ki_world'] :
              ( '$ki_accessible'(W,W0)
             => ! [B: $i] :
                  ( '$ki_exists_in_world_$i'(W0,B)
                 => ! [W1: '$ki_world'] :
                      ( '$ki_accessible'(W0,W1)
                     => ( ! [W2: '$ki_world'] :
                            ( '$ki_accessible'(W1,W2)
                           => in(W2,A,B) )
                       => ! [W2: '$ki_world'] :
                            ( '$ki_accessible'(W1,W2)
                           => ~ ! [W3: '$ki_world'] :
                                  ( '$ki_accessible'(W2,W3)
                                 => in(W3,B,A) ) ) ) ) ) ) ) ) ).

tff(commutativity_k3_xboole_0,axiom,
    ! [W: '$ki_world'] :
      ( '$ki_accessible'('$ki_local_world',W)
     => ! [A: $i] :
          ( '$ki_exists_in_world_$i'(W,A)
         => ! [W0: '$ki_world'] :
              ( '$ki_accessible'(W,W0)
             => ! [B: $i] :
                  ( '$ki_exists_in_world_$i'(W0,B)
                 => ! [W1: '$ki_world'] :
                      ( '$ki_accessible'(W0,W1)
                     => qmltpeq(W1,set_intersection2(A,B),set_intersection2(B,A)) ) ) ) ) ) ).

tff(idempotence_k3_xboole_0,axiom,
    ! [W: '$ki_world'] :
      ( '$ki_accessible'('$ki_local_world',W)
     => ! [A: $i] :
          ( '$ki_exists_in_world_$i'(W,A)
         => ! [W0: '$ki_world'] :
              ( '$ki_accessible'(W,W0)
             => ! [B: $i] :
                  ( '$ki_exists_in_world_$i'(W0,B)
                 => ! [W1: '$ki_world'] :
                      ( '$ki_accessible'(W0,W1)
                     => qmltpeq(W1,set_intersection2(A,A),A) ) ) ) ) ) ).

tff(l28_zfmisc_1,axiom,
    ! [W: '$ki_world'] :
      ( '$ki_accessible'('$ki_local_world',W)
     => ! [A: $i] :
          ( '$ki_exists_in_world_$i'(W,A)
         => ! [W0: '$ki_world'] :
              ( '$ki_accessible'(W,W0)
             => ! [B: $i] :
                  ( '$ki_exists_in_world_$i'(W0,B)
                 => ! [W1: '$ki_world'] :
                      ( '$ki_accessible'(W0,W1)
                     => ( ! [W2: '$ki_world'] :
                            ( '$ki_accessible'(W1,W2)
                           => ~ ! [W3: '$ki_world'] :
                                  ( '$ki_accessible'(W2,W3)
                                 => in(W3,A,B) ) )
                       => ! [W2: '$ki_world'] :
                            ( '$ki_accessible'(W1,W2)
                           => disjoint(W2,singleton(A),B) ) ) ) ) ) ) ) ).

tff(l32_zfmisc_1,axiom,
    ! [W: '$ki_world'] :
      ( '$ki_accessible'('$ki_local_world',W)
     => ! [A: $i] :
          ( '$ki_exists_in_world_$i'(W,A)
         => ! [W0: '$ki_world'] :
              ( '$ki_accessible'(W,W0)
             => ! [B: $i] :
                  ( '$ki_exists_in_world_$i'(W0,B)
                 => ! [W1: '$ki_world'] :
                      ( '$ki_accessible'(W0,W1)
                     => ( ! [W2: '$ki_world'] :
                            ( '$ki_accessible'(W1,W2)
                           => in(W2,A,B) )
                       => ! [W2: '$ki_world'] :
                            ( '$ki_accessible'(W1,W2)
                           => qmltpeq(W2,set_intersection2(B,singleton(A)),singleton(A)) ) ) ) ) ) ) ) ).

tff(rc1_xboole_0,axiom,
    ? [A: $i] :
      ( '$ki_exists_in_world_$i'('$ki_local_world',A)
      & ! [W: '$ki_world'] :
          ( '$ki_accessible'('$ki_local_world',W)
         => empty(W,A) ) ) ).

tff(rc2_xboole_0,axiom,
    ? [A: $i] :
      ( '$ki_exists_in_world_$i'('$ki_local_world',A)
      & ! [W: '$ki_world'] :
          ( '$ki_accessible'('$ki_local_world',W)
         => ~ ! [W0: '$ki_world'] :
                ( '$ki_accessible'(W,W0)
               => empty(W0,A) ) ) ) ).

tff(symmetry_r1_xboole_0,axiom,
    ! [W: '$ki_world'] :
      ( '$ki_accessible'('$ki_local_world',W)
     => ! [A: $i] :
          ( '$ki_exists_in_world_$i'(W,A)
         => ! [W0: '$ki_world'] :
              ( '$ki_accessible'(W,W0)
             => ! [B: $i] :
                  ( '$ki_exists_in_world_$i'(W0,B)
                 => ! [W1: '$ki_world'] :
                      ( '$ki_accessible'(W0,W1)
                     => ( ! [W2: '$ki_world'] :
                            ( '$ki_accessible'(W1,W2)
                           => disjoint(W2,A,B) )
                       => ! [W2: '$ki_world'] :
                            ( '$ki_accessible'(W1,W2)
                           => disjoint(W2,B,A) ) ) ) ) ) ) ) ).

tff(verify,conjecture,
    ! [W: '$ki_world'] :
      ( '$ki_accessible'('$ki_local_world',W)
     => ! [A: $i] :
          ( '$ki_exists_in_world_$i'(W,A)
         => ! [W0: '$ki_world'] :
              ( '$ki_accessible'(W,W0)
             => ! [B: $i] :
                  ( '$ki_exists_in_world_$i'(W0,B)
                 => ( ! [W1: '$ki_world'] :
                        ( '$ki_accessible'(W0,W1)
                       => disjoint(W1,singleton(A),B) )
                    | ! [W1: '$ki_world'] :
                        ( '$ki_accessible'(W0,W1)
                       => qmltpeq(W1,set_intersection2(singleton(A),B),singleton(A)) ) ) ) ) ) ) ).

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