TPTP Problem File: LCL947_10.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : LCL947_10 : TPTP v8.2.0. Released v8.2.0.
% Domain : Set Theory
% Problem : Goedel translation of SET002+3
% 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 : GSE002+1 [QMLTP]
% Status : Theorem
% Rating : 0.00 v8.2.0
% Syntax : Number of formulae : 28 ( 2 unt; 8 typ; 0 def)
% Number of atoms : 224 ( 0 equ)
% Maximal formula atoms : 34 ( 11 avg)
% Number of connectives : 204 ( 0 ~; 2 |; 14 &)
% ( 0 <=>; 188 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 14 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 15 ( 6 >; 9 *; 0 +; 0 <<)
% Number of predicates : 5 ( 5 usr; 0 prp; 2-3 aty)
% Number of functors : 2 ( 2 usr; 1 con; 0-2 aty)
% Number of variables : 170 (; 169 !; 1 ?; 170 :)
% SPC : TF0_THM_NEQ_NAR
% Comments : Generated by embedproblem, version 1.7.14, rigid constant,
% modal_system_S4, 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' ) > $o ).
tff(mrel_reflexive,axiom,
! [W: '$ki_world'] : '$ki_accessible'(W,W) ).
tff(mrel_transitive,axiom,
! [W: '$ki_world',V: '$ki_world',U: '$ki_world'] :
( ( '$ki_accessible'(W,V)
& '$ki_accessible'(V,U) )
=> '$ki_accessible'(W,U) ) ).
tff(qmltpeq_decl,type,
qmltpeq: ( '$ki_world' * $i * $i ) > $o ).
tff(member_decl,type,
member: ( '$ki_world' * $i * $i ) > $o ).
tff(subset_decl,type,
subset: ( '$ki_world' * $i * $i ) > $o ).
tff(union_decl,type,
union: ( $i * $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'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,X,Z) ) ) ) ) ) ) ) ) ) ).
tff(union_substitution_1,axiom,
! [W: '$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'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,union(A,C),union(B,C)) ) ) ) ) ) ) ) ) ) ).
tff(union_substitution_2,axiom,
! [W: '$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'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,union(C,A),union(C,B)) ) ) ) ) ) ) ) ) ) ).
tff(member_substitution_1,axiom,
! [W: '$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'] :
( '$ki_accessible'(W2,W3)
=> member(W3,A,C) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,B,C) ) ) ) ) ) ) ) ) ) ).
tff(member_substitution_2,axiom,
! [W: '$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'] :
( '$ki_accessible'(W2,W3)
=> member(W3,C,A) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,C,B) ) ) ) ) ) ) ) ) ) ).
tff(subset_substitution_1,axiom,
! [W: '$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'] :
( '$ki_accessible'(W2,W3)
=> subset(W3,A,C) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> subset(W3,B,C) ) ) ) ) ) ) ) ) ) ).
tff(subset_substitution_2,axiom,
! [W: '$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'] :
( '$ki_accessible'(W2,W3)
=> subset(W3,C,A) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> subset(W3,C,B) ) ) ) ) ) ) ) ) ) ).
tff(subset_union,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W,B)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W0,C)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> subset(W2,B,C) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,union(B,C),C) ) ) ) ) ) ) ) ).
tff(union_defn,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W,B)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W0,C)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [D: $i] :
( '$ki_exists_in_world_$i'(W1,D)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,D,union(B,C)) )
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,D,B) )
| ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,D,C) ) ) ) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,D,B) )
| ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,D,C) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,D,union(B,C)) ) ) ) ) ) ) ) ) ) ) ).
tff(equal_defn,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W,B)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W0,C)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,B,C) )
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> subset(W2,B,C) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> subset(W2,C,B) ) ) ) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> subset(W2,B,C) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> subset(W2,C,B) ) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,B,C) ) ) ) ) ) ) ) ) ).
tff(commutativity_of_union,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W,B)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W0,C)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> qmltpeq(W1,union(B,C),union(C,B)) ) ) ) ) ) ).
tff(subset_defn,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W,B)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W0,C)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> subset(W2,B,C) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [D: $i] :
( '$ki_exists_in_world_$i'(W2,D)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,D,B) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,D,C) ) ) ) ) ) ) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [D: $i] :
( '$ki_exists_in_world_$i'(W2,D)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,D,B) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,D,C) ) ) ) ) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> subset(W2,B,C) ) ) ) ) ) ) ) ) ).
tff(reflexivity_of_subset,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W,B)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> subset(W0,B,B) ) ) ) ).
tff(equal_member_defn,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W,B)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W0,C)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,B,C) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [D: $i] :
( '$ki_exists_in_world_$i'(W2,D)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,D,B) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,D,C) ) ) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,D,C) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,D,B) ) ) ) ) ) ) ) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [D: $i] :
( '$ki_exists_in_world_$i'(W2,D)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,D,B) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,D,C) ) ) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,D,C) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,D,B) ) ) ) ) ) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,B,C) ) ) ) ) ) ) ) ) ).
tff(verify,conjecture,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W,B)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> qmltpeq(W0,union(B,B),B) ) ) ) ).
%------------------------------------------------------------------------------