TPTP Problem File: LCL948_10.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : LCL948_10 : TPTP v8.2.0. Released v8.2.0.
% Domain : Set Theory
% Problem : Goedel translation of SET002+4
% 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 : CounterSatisfiable
% Rating : 0.00 v8.2.0
% Syntax : Number of formulae : 53 ( 2 unt; 17 typ; 0 def)
% Number of atoms : 438 ( 0 equ)
% Maximal formula atoms : 24 ( 12 avg)
% Number of connectives : 405 ( 3 ~; 4 |; 28 &)
% ( 0 <=>; 370 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 16 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 28 ( 14 >; 14 *; 0 +; 0 <<)
% Number of predicates : 6 ( 6 usr; 0 prp; 2-3 aty)
% Number of functors : 10 ( 10 usr; 2 con; 0-2 aty)
% Number of variables : 333 (; 330 !; 3 ?; 333 :)
% SPC : TF0_CSA_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(empty_set_decl,type,
empty_set: $i ).
tff(qmltpeq_decl,type,
qmltpeq: ( '$ki_world' * $i * $i ) > $o ).
tff(equal_set_decl,type,
equal_set: ( '$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(singleton_decl,type,
singleton: $i > $i ).
tff(product_decl,type,
product: $i > $i ).
tff(unordered_pair_decl,type,
unordered_pair: ( $i * $i ) > $i ).
tff(intersection_decl,type,
intersection: ( $i * $i ) > $i ).
tff(difference_decl,type,
difference: ( $i * $i ) > $i ).
tff(sum_decl,type,
sum: $i > $i ).
tff(union_decl,type,
union: ( $i * $i ) > $i ).
tff(power_set_decl,type,
power_set: $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'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,X,Z) ) ) ) ) ) ) ) ) ) ).
tff(difference_substitution_1,axiom,
! [W: '$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'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,difference(A,C),difference(B,C)) ) ) ) ) ) ) ) ) ) ).
tff(difference_substitution_2,axiom,
! [W: '$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'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,difference(C,A),difference(C,B)) ) ) ) ) ) ) ) ) ) ).
tff(intersection_substitution_1,axiom,
! [W: '$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'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,intersection(A,C),intersection(B,C)) ) ) ) ) ) ) ) ) ) ).
tff(intersection_substitution_2,axiom,
! [W: '$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'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,intersection(C,A),intersection(C,B)) ) ) ) ) ) ) ) ) ) ).
tff(power_set_substitution_1,axiom,
! [W: '$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'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,power_set(A),power_set(B)) ) ) ) ) ) ) ) ).
tff(product_substitution_1,axiom,
! [W: '$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'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,product(A),product(B)) ) ) ) ) ) ) ) ).
tff(singleton_substitution_1,axiom,
! [W: '$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'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,singleton(A),singleton(B)) ) ) ) ) ) ) ) ).
tff(sum_substitution_1,axiom,
! [W: '$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'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,sum(A),sum(B)) ) ) ) ) ) ) ) ).
tff(union_substitution_1,axiom,
! [W: '$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'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,union(C,A),union(C,B)) ) ) ) ) ) ) ) ) ) ).
tff(unordered_pair_substitution_1,axiom,
! [W: '$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'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,unordered_pair(A,C),unordered_pair(B,C)) ) ) ) ) ) ) ) ) ) ).
tff(unordered_pair_substitution_2,axiom,
! [W: '$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'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,unordered_pair(C,A),unordered_pair(C,B)) ) ) ) ) ) ) ) ) ) ).
tff(equal_set_substitution_1,axiom,
! [W: '$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'] :
( '$ki_accessible'(W2,W3)
=> equal_set(W3,A,C) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> equal_set(W3,B,C) ) ) ) ) ) ) ) ) ) ).
tff(equal_set_substitution_2,axiom,
! [W: '$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'] :
( '$ki_accessible'(W2,W3)
=> equal_set(W3,C,A) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> equal_set(W3,C,B) ) ) ) ) ) ) ) ) ) ).
tff(member_substitution_1,axiom,
! [W: '$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'] :
( '$ki_accessible'(W2,W3)
=> subset(W3,C,B) ) ) ) ) ) ) ) ) ) ).
tff(subset_0,axiom,
! [W: '$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'] :
( '$ki_accessible'(W1,W2)
=> subset(W2,A,B) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W2,X)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,X,A) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,X,B) ) ) ) ) ) ) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W2,X)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,X,A) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,X,B) ) ) ) ) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> subset(W2,A,B) ) ) ) ) ) ) ) ) ).
tff(equal_set_0,axiom,
! [W: '$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'] :
( '$ki_accessible'(W1,W2)
=> equal_set(W2,A,B) )
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> subset(W2,A,B) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> subset(W2,B,A) ) ) ) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> subset(W2,A,B) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> subset(W2,B,A) ) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> equal_set(W2,A,B) ) ) ) ) ) ) ) ) ).
tff(power_set_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W0,A)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,X,power_set(A)) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> subset(W2,X,A) ) ) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> subset(W2,X,A) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,X,power_set(A)) ) ) ) ) ) ) ) ) ).
tff(intersection_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W0,A)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W1,B)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,X,intersection(A,B)) )
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,X,A) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,X,B) ) ) ) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,X,A) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,X,B) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,X,intersection(A,B)) ) ) ) ) ) ) ) ) ) ) ).
tff(union_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W0,A)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W1,B)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,X,union(A,B)) )
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,X,A) )
| ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,X,B) ) ) ) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,X,A) )
| ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,X,B) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,X,union(A,B)) ) ) ) ) ) ) ) ) ) ) ).
tff(empty_set_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ~ ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> member(W1,X,empty_set) ) ) ) ) ).
tff(difference_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W,B)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W0,A)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [E: $i] :
( '$ki_exists_in_world_$i'(W1,E)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,B,difference(E,A)) )
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,B,E) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ~ ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,B,A) ) ) ) ) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,B,E) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ~ ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,B,A) ) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,B,difference(E,A)) ) ) ) ) ) ) ) ) ) ) ).
tff(singleton_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W0,A)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,X,singleton(A)) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,X,A) ) ) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,X,A) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,X,singleton(A)) ) ) ) ) ) ) ) ) ).
tff(unordered_pair_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W0,A)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W1,B)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,X,unordered_pair(A,B)) )
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,X,A) )
| ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,X,B) ) ) ) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,X,A) )
| ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,X,B) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> member(W3,X,unordered_pair(A,B)) ) ) ) ) ) ) ) ) ) ) ).
tff(sum_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W0,A)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,X,sum(A)) )
=> ? [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,Y,A) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,X,Y) ) ) ) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ? [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,Y,A) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,X,Y) ) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,X,sum(A)) ) ) ) ) ) ) ) ) ).
tff(product_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W0,A)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,X,product(A)) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W2,Y)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,Y,A) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,X,Y) ) ) ) ) ) ) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W2,Y)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,Y,A) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> member(W4,X,Y) ) ) ) ) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> member(W2,X,product(A)) ) ) ) ) ) ) ) ) ).
tff(verify,conjecture,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> equal_set(W0,union(A,A),A) ) ) ) ).
%------------------------------------------------------------------------------