TPTP Problem File: LCL963_3.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : LCL963_3 : TPTP v8.2.0. Released v8.2.0.
% Domain : Logic Calculi
% Problem : Goedel translation of LCL451+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 : GLC451+1 [QMLTP]
% Status : CounterSatisfiable
% Rating : 0.00 v8.2.0
% Syntax : Number of formulae : 112 ( 3 unt; 43 typ; 0 def)
% Number of atoms : 722 ( 0 equ)
% Maximal formula atoms : 28 ( 10 avg)
% Number of connectives : 653 ( 0 ~; 0 |; 30 &)
% ( 0 <=>; 623 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 11 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 50 ( 41 >; 9 *; 0 +; 0 <<)
% Number of predicates : 36 ( 36 usr; 0 prp; 1-3 aty)
% Number of functors : 6 ( 6 usr; 1 con; 0-2 aty)
% Number of variables : 556 (; 554 !; 2 ?; 556 :)
% 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(r2_decl,type,
r2: '$ki_world' > $o ).
tff(r3_decl,type,
r3: '$ki_world' > $o ).
tff(r4_decl,type,
r4: '$ki_world' > $o ).
tff(r5_decl,type,
r5: '$ki_world' > $o ).
tff(equivalence_1_decl,type,
equivalence_1: '$ki_world' > $o ).
tff(equivalence_2_decl,type,
equivalence_2: '$ki_world' > $o ).
tff(cn2_decl,type,
cn2: '$ki_world' > $o ).
tff(cn1_decl,type,
cn1: '$ki_world' > $o ).
tff(cn3_decl,type,
cn3: '$ki_world' > $o ).
tff(kn2_decl,type,
kn2: '$ki_world' > $o ).
tff(kn1_decl,type,
kn1: '$ki_world' > $o ).
tff(kn3_decl,type,
kn3: '$ki_world' > $o ).
tff(equivalence_3_decl,type,
equivalence_3: '$ki_world' > $o ).
tff(op_implies_or_decl,type,
op_implies_or: '$ki_world' > $o ).
tff(op_and_decl,type,
op_and: '$ki_world' > $o ).
tff(and_2_decl,type,
and_2: '$ki_world' > $o ).
tff(and_3_decl,type,
and_3: '$ki_world' > $o ).
tff(and_1_decl,type,
and_1: '$ki_world' > $o ).
tff(implies_2_decl,type,
implies_2: '$ki_world' > $o ).
tff(implies_3_decl,type,
implies_3: '$ki_world' > $o ).
tff(op_implies_and_decl,type,
op_implies_and: '$ki_world' > $o ).
tff(implies_1_decl,type,
implies_1: '$ki_world' > $o ).
tff(substitution_of_equivalents_decl,type,
substitution_of_equivalents: '$ki_world' > $o ).
tff(op_or_decl,type,
op_or: '$ki_world' > $o ).
tff(modus_tollens_decl,type,
modus_tollens: '$ki_world' > $o ).
tff(or_1_decl,type,
or_1: '$ki_world' > $o ).
tff(or_2_decl,type,
or_2: '$ki_world' > $o ).
tff(or_3_decl,type,
or_3: '$ki_world' > $o ).
tff(op_equiv_decl,type,
op_equiv: '$ki_world' > $o ).
tff(modus_ponens_decl,type,
modus_ponens: '$ki_world' > $o ).
tff(op_implies_decl,type,
op_implies: '$ki_world' > $o ).
tff(r1_decl,type,
r1: '$ki_world' > $o ).
tff(qmltpeq_decl,type,
qmltpeq: ( '$ki_world' * $i * $i ) > $o ).
tff(is_a_theorem_decl,type,
is_a_theorem: ( '$ki_world' * $i ) > $o ).
tff(equiv_decl,type,
equiv: ( $i * $i ) > $i ).
tff(not_decl,type,
not: $i > $i ).
tff(or_decl,type,
or: ( $i * $i ) > $i ).
tff(and_decl,type,
and: ( $i * $i ) > $i ).
tff(implies_decl,type,
implies: ( $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_$i_const',axiom,
! [W: '$ki_world',X: $i] : '$ki_exists_in_world_$i'(W,X) ).
tff('$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'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,X,Z) ) ) ) ) ) ) ) ) ) ).
tff(and_substitution_1,axiom,
! [W: '$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'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,and(A,C),and(B,C)) ) ) ) ) ) ) ) ) ) ).
tff(and_substitution_2,axiom,
! [W: '$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'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,and(C,A),and(C,B)) ) ) ) ) ) ) ) ) ) ).
tff(equiv_substitution_1,axiom,
! [W: '$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'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,equiv(A,C),equiv(B,C)) ) ) ) ) ) ) ) ) ) ).
tff(equiv_substitution_2,axiom,
! [W: '$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'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,equiv(C,A),equiv(C,B)) ) ) ) ) ) ) ) ) ) ).
tff(implies_substitution_1,axiom,
! [W: '$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'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,implies(A,C),implies(B,C)) ) ) ) ) ) ) ) ) ) ).
tff(implies_substitution_2,axiom,
! [W: '$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'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,implies(C,A),implies(C,B)) ) ) ) ) ) ) ) ) ) ).
tff(not_substitution_1,axiom,
! [W: '$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'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,not(A),not(B)) ) ) ) ) ) ) ) ).
tff(or_substitution_1,axiom,
! [W: '$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'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,or(A,C),or(B,C)) ) ) ) ) ) ) ) ) ) ).
tff(or_substitution_2,axiom,
! [W: '$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'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,or(C,A),or(C,B)) ) ) ) ) ) ) ) ) ) ).
tff(is_a_theorem_substitution_1,axiom,
! [W: '$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'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,A) ) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,B) ) ) ) ) ) ) ) ).
tff(modus_ponens_0,axiom,
( ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> modus_ponens(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> is_a_theorem(W3,X) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> is_a_theorem(W3,implies(X,Y)) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> is_a_theorem(W3,Y) ) ) ) ) ) ) ) ) )
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> is_a_theorem(W3,X) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> is_a_theorem(W3,implies(X,Y)) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> is_a_theorem(W3,Y) ) ) ) ) ) ) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> modus_ponens(W0) ) ) ) ) ).
tff(substitution_of_equivalents_0,axiom,
( ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> substitution_of_equivalents(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> is_a_theorem(W3,equiv(X,Y)) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,X,Y) ) ) ) ) ) ) ) ) )
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> is_a_theorem(W3,equiv(X,Y)) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> qmltpeq(W3,X,Y) ) ) ) ) ) ) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> substitution_of_equivalents(W0) ) ) ) ) ).
tff(modus_tollens_0,axiom,
( ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> modus_tollens(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(implies(not(Y),not(X)),implies(X,Y))) ) ) ) ) ) ) )
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(implies(not(Y),not(X)),implies(X,Y))) ) ) ) ) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> modus_tollens(W0) ) ) ) ) ).
tff(implies_1_0,axiom,
( ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> implies_1(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(X,implies(Y,X))) ) ) ) ) ) ) )
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(X,implies(Y,X))) ) ) ) ) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> implies_1(W0) ) ) ) ) ).
tff(implies_2_0,axiom,
( ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> implies_2(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(implies(X,implies(X,Y)),implies(X,Y))) ) ) ) ) ) ) )
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(implies(X,implies(X,Y)),implies(X,Y))) ) ) ) ) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> implies_2(W0) ) ) ) ) ).
tff(implies_3_0,axiom,
( ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> implies_3(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [Z: $i] :
( '$ki_exists_in_world_$i'(W2,Z)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> is_a_theorem(W3,implies(implies(X,Y),implies(implies(Y,Z),implies(X,Z)))) ) ) ) ) ) ) ) ) )
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [Z: $i] :
( '$ki_exists_in_world_$i'(W2,Z)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> is_a_theorem(W3,implies(implies(X,Y),implies(implies(Y,Z),implies(X,Z)))) ) ) ) ) ) ) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> implies_3(W0) ) ) ) ) ).
tff(and_1_0,axiom,
( ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> and_1(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(and(X,Y),X)) ) ) ) ) ) ) )
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(and(X,Y),X)) ) ) ) ) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> and_1(W0) ) ) ) ) ).
tff(and_2_0,axiom,
( ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> and_2(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(and(X,Y),Y)) ) ) ) ) ) ) )
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(and(X,Y),Y)) ) ) ) ) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> and_2(W0) ) ) ) ) ).
tff(and_3_0,axiom,
( ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> and_3(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(X,implies(Y,and(X,Y)))) ) ) ) ) ) ) )
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(X,implies(Y,and(X,Y)))) ) ) ) ) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> and_3(W0) ) ) ) ) ).
tff(or_1_0,axiom,
( ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> or_1(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(X,or(X,Y))) ) ) ) ) ) ) )
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(X,or(X,Y))) ) ) ) ) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> or_1(W0) ) ) ) ) ).
tff(or_2_0,axiom,
( ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> or_2(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(Y,or(X,Y))) ) ) ) ) ) ) )
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(Y,or(X,Y))) ) ) ) ) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> or_2(W0) ) ) ) ) ).
tff(or_3_0,axiom,
( ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> or_3(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [Z: $i] :
( '$ki_exists_in_world_$i'(W2,Z)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> is_a_theorem(W3,implies(implies(X,Z),implies(implies(Y,Z),implies(or(X,Y),Z)))) ) ) ) ) ) ) ) ) )
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [Z: $i] :
( '$ki_exists_in_world_$i'(W2,Z)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> is_a_theorem(W3,implies(implies(X,Z),implies(implies(Y,Z),implies(or(X,Y),Z)))) ) ) ) ) ) ) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> or_3(W0) ) ) ) ) ).
tff(equivalence_1_0,axiom,
( ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> equivalence_1(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(equiv(X,Y),implies(X,Y))) ) ) ) ) ) ) )
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(equiv(X,Y),implies(X,Y))) ) ) ) ) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> equivalence_1(W0) ) ) ) ) ).
tff(equivalence_2_0,axiom,
( ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> equivalence_2(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(equiv(X,Y),implies(Y,X))) ) ) ) ) ) ) )
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(equiv(X,Y),implies(Y,X))) ) ) ) ) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> equivalence_2(W0) ) ) ) ) ).
tff(equivalence_3_0,axiom,
( ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> equivalence_3(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(implies(X,Y),implies(implies(Y,X),equiv(X,Y)))) ) ) ) ) ) ) )
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(implies(X,Y),implies(implies(Y,X),equiv(X,Y)))) ) ) ) ) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> equivalence_3(W0) ) ) ) ) ).
tff(kn1_0,axiom,
( ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> kn1(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [P: $i] :
( '$ki_exists_in_world_$i'(W0,P)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> is_a_theorem(W1,implies(P,and(P,P))) ) ) ) ) )
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [P: $i] :
( '$ki_exists_in_world_$i'(W0,P)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> is_a_theorem(W1,implies(P,and(P,P))) ) ) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> kn1(W0) ) ) ) ) ).
tff(kn2_0,axiom,
( ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> kn2(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [P: $i] :
( '$ki_exists_in_world_$i'(W0,P)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Q: $i] :
( '$ki_exists_in_world_$i'(W1,Q)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(and(P,Q),P)) ) ) ) ) ) ) )
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [P: $i] :
( '$ki_exists_in_world_$i'(W0,P)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Q: $i] :
( '$ki_exists_in_world_$i'(W1,Q)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(and(P,Q),P)) ) ) ) ) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> kn2(W0) ) ) ) ) ).
tff(kn3_0,axiom,
( ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> kn3(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [P: $i] :
( '$ki_exists_in_world_$i'(W0,P)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Q: $i] :
( '$ki_exists_in_world_$i'(W1,Q)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [R: $i] :
( '$ki_exists_in_world_$i'(W2,R)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> is_a_theorem(W3,implies(implies(P,Q),implies(not(and(Q,R)),not(and(R,P))))) ) ) ) ) ) ) ) ) )
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [P: $i] :
( '$ki_exists_in_world_$i'(W0,P)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Q: $i] :
( '$ki_exists_in_world_$i'(W1,Q)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [R: $i] :
( '$ki_exists_in_world_$i'(W2,R)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> is_a_theorem(W3,implies(implies(P,Q),implies(not(and(Q,R)),not(and(R,P))))) ) ) ) ) ) ) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> kn3(W0) ) ) ) ) ).
tff(cn1_0,axiom,
( ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> cn1(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [P: $i] :
( '$ki_exists_in_world_$i'(W0,P)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Q: $i] :
( '$ki_exists_in_world_$i'(W1,Q)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [R: $i] :
( '$ki_exists_in_world_$i'(W2,R)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> is_a_theorem(W3,implies(implies(P,Q),implies(implies(Q,R),implies(P,R)))) ) ) ) ) ) ) ) ) )
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [P: $i] :
( '$ki_exists_in_world_$i'(W0,P)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Q: $i] :
( '$ki_exists_in_world_$i'(W1,Q)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [R: $i] :
( '$ki_exists_in_world_$i'(W2,R)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> is_a_theorem(W3,implies(implies(P,Q),implies(implies(Q,R),implies(P,R)))) ) ) ) ) ) ) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> cn1(W0) ) ) ) ) ).
tff(cn2_0,axiom,
( ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> cn2(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [P: $i] :
( '$ki_exists_in_world_$i'(W0,P)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Q: $i] :
( '$ki_exists_in_world_$i'(W1,Q)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(P,implies(not(P),Q))) ) ) ) ) ) ) )
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [P: $i] :
( '$ki_exists_in_world_$i'(W0,P)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Q: $i] :
( '$ki_exists_in_world_$i'(W1,Q)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(P,implies(not(P),Q))) ) ) ) ) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> cn2(W0) ) ) ) ) ).
tff(cn3_0,axiom,
( ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> cn3(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [P: $i] :
( '$ki_exists_in_world_$i'(W0,P)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> is_a_theorem(W1,implies(implies(not(P),P),P)) ) ) ) ) )
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [P: $i] :
( '$ki_exists_in_world_$i'(W0,P)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> is_a_theorem(W1,implies(implies(not(P),P),P)) ) ) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> cn3(W0) ) ) ) ) ).
tff(r1_0,axiom,
( ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> r1(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [P: $i] :
( '$ki_exists_in_world_$i'(W0,P)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> is_a_theorem(W1,implies(or(P,P),P)) ) ) ) ) )
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [P: $i] :
( '$ki_exists_in_world_$i'(W0,P)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> is_a_theorem(W1,implies(or(P,P),P)) ) ) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> r1(W0) ) ) ) ) ).
tff(r2_0,axiom,
( ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> r2(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [P: $i] :
( '$ki_exists_in_world_$i'(W0,P)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Q: $i] :
( '$ki_exists_in_world_$i'(W1,Q)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(Q,or(P,Q))) ) ) ) ) ) ) )
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [P: $i] :
( '$ki_exists_in_world_$i'(W0,P)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Q: $i] :
( '$ki_exists_in_world_$i'(W1,Q)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(Q,or(P,Q))) ) ) ) ) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> r2(W0) ) ) ) ) ).
tff(r3_0,axiom,
( ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> r3(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [P: $i] :
( '$ki_exists_in_world_$i'(W0,P)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Q: $i] :
( '$ki_exists_in_world_$i'(W1,Q)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(or(P,Q),or(Q,P))) ) ) ) ) ) ) )
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [P: $i] :
( '$ki_exists_in_world_$i'(W0,P)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Q: $i] :
( '$ki_exists_in_world_$i'(W1,Q)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> is_a_theorem(W2,implies(or(P,Q),or(Q,P))) ) ) ) ) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> r3(W0) ) ) ) ) ).
tff(r4_0,axiom,
( ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> r4(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [P: $i] :
( '$ki_exists_in_world_$i'(W0,P)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Q: $i] :
( '$ki_exists_in_world_$i'(W1,Q)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [R: $i] :
( '$ki_exists_in_world_$i'(W2,R)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> is_a_theorem(W3,implies(or(P,or(Q,R)),or(Q,or(P,R)))) ) ) ) ) ) ) ) ) )
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [P: $i] :
( '$ki_exists_in_world_$i'(W0,P)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Q: $i] :
( '$ki_exists_in_world_$i'(W1,Q)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [R: $i] :
( '$ki_exists_in_world_$i'(W2,R)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> is_a_theorem(W3,implies(or(P,or(Q,R)),or(Q,or(P,R)))) ) ) ) ) ) ) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> r4(W0) ) ) ) ) ).
tff(r5_0,axiom,
( ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> r5(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [P: $i] :
( '$ki_exists_in_world_$i'(W0,P)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Q: $i] :
( '$ki_exists_in_world_$i'(W1,Q)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [R: $i] :
( '$ki_exists_in_world_$i'(W2,R)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> is_a_theorem(W3,implies(implies(Q,R),implies(or(P,Q),or(P,R)))) ) ) ) ) ) ) ) ) )
& ! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [P: $i] :
( '$ki_exists_in_world_$i'(W0,P)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Q: $i] :
( '$ki_exists_in_world_$i'(W1,Q)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [R: $i] :
( '$ki_exists_in_world_$i'(W2,R)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> is_a_theorem(W3,implies(implies(Q,R),implies(or(P,Q),or(P,R)))) ) ) ) ) ) ) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> r5(W0) ) ) ) ) ).
tff(op_or_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> op_or(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,or(X,Y),not(and(not(X),not(Y)))) ) ) ) ) ) ) ) ).
tff(op_and_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> op_and(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,and(X,Y),not(or(not(X),not(Y)))) ) ) ) ) ) ) ) ).
tff(op_implies_and_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> op_implies_and(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,implies(X,Y),not(and(X,not(Y)))) ) ) ) ) ) ) ) ).
tff(op_implies_or_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> op_implies_or(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,implies(X,Y),or(not(X),Y)) ) ) ) ) ) ) ) ).
tff(op_equiv_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> op_equiv(W0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W0,X)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W1,Y)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> qmltpeq(W2,equiv(X,Y),and(implies(X,Y),implies(Y,X))) ) ) ) ) ) ) ) ).
tff(hilbert_op_or,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> op_or(W) ) ).
tff(hilbert_op_implies_and,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> op_implies_and(W) ) ).
tff(hilbert_op_equiv,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> op_equiv(W) ) ).
tff(hilbert_modus_ponens,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> modus_ponens(W) ) ).
tff(hilbert_modus_tollens,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> modus_tollens(W) ) ).
tff(hilbert_implies_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> implies_1(W) ) ).
tff(hilbert_implies_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> implies_2(W) ) ).
tff(hilbert_implies_3,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> implies_3(W) ) ).
tff(hilbert_and_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> and_1(W) ) ).
tff(hilbert_and_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> and_2(W) ) ).
tff(hilbert_and_3,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> and_3(W) ) ).
tff(hilbert_or_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> or_1(W) ) ).
tff(hilbert_or_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> or_2(W) ) ).
tff(hilbert_or_3,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> or_3(W) ) ).
tff(hilbert_equivalence_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> equivalence_1(W) ) ).
tff(hilbert_equivalence_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> equivalence_2(W) ) ).
tff(hilbert_equivalence_3,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> equivalence_3(W) ) ).
tff(substitution_of_equivalents_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> substitution_of_equivalents(W) ) ).
tff(luka_op_or,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> op_or(W) ) ).
tff(luka_op_implies,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> op_implies(W) ) ).
tff(luka_op_equiv,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> op_equiv(W) ) ).
tff(verify,conjecture,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> cn1(W) ) ).
%------------------------------------------------------------------------------