TPTP Problem File: LCL966_9.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : LCL966_9 : TPTP v8.2.0. 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)) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------