TPTP Problem File: LCL963^3.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : LCL963^3 : TPTP v8.2.0. Released v8.1.0.
% Domain : Logic Calculi
% Problem : Goedel translation of LCL451+1 (from TPTP-v5.0.0)
% Version : [BP13] axioms.
% English :
% Refs : [RO12] Raths & Otten (2012), The QMLTP Problem Library for Fi
% : [BP13] Benzmueller & Paulson (2013), Quantified Multimodal Lo
% : [Ste22] Steen (2022), An Extensible Logic Embedding Tool for L
% Source : [TPTP]
% Names : GLC451+1 [QMLTP]
% Status : CounterSatisfiable
% Rating : 0.75 v8.2.0, 1.00 v8.1.0
% Syntax : Number of formulae : 129 ( 11 unt; 52 typ; 10 def)
% Number of atoms : 911 ( 10 equ; 0 cnn)
% Maximal formula atoms : 36 ( 11 avg)
% Number of connectives : 1368 ( 1 ~; 1 |; 2 &;1361 @)
% ( 1 <=>; 2 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 11 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 103 ( 103 >; 0 *; 0 +; 0 <<)
% Number of symbols : 52 ( 51 usr; 1 con; 0-3 aty)
% Number of variables : 183 ( 177 ^; 3 !; 3 ?; 183 :)
% SPC : TH0_CSA_EQU_NAR
% Comments : This output was generated by embedproblem, version 1.7.1 (library
% version 1.3). Generated on Thu Apr 28 13:18:18 EDT 2022 using
% 'modal' embedding, version 1.5.2. Logic specification used:
% $modal == [$constants == $rigid,$quantification == $constant,
% $modalities == $modal_system_D].
%------------------------------------------------------------------------------
thf(mworld,type,
mworld: $tType ).
thf(mrel_type,type,
mrel: mworld > mworld > $o ).
thf(mactual_type,type,
mactual: mworld ).
thf(mlocal_type,type,
mlocal: ( mworld > $o ) > $o ).
thf(mlocal_def,definition,
( mlocal
= ( ^ [Phi: mworld > $o] : ( Phi @ mactual ) ) ) ).
thf(mnot_type,type,
mnot: ( mworld > $o ) > mworld > $o ).
thf(mand_type,type,
mand: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).
thf(mor_type,type,
mor: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).
thf(mimplies_type,type,
mimplies: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).
thf(mequiv_type,type,
mequiv: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).
thf(mnot_def,definition,
( mnot
= ( ^ [A: mworld > $o,W: mworld] :
~ ( A @ W ) ) ) ).
thf(mand_def,definition,
( mand
= ( ^ [A: mworld > $o,B: mworld > $o,W: mworld] :
( ( A @ W )
& ( B @ W ) ) ) ) ).
thf(mor_def,definition,
( mor
= ( ^ [A: mworld > $o,B: mworld > $o,W: mworld] :
( ( A @ W )
| ( B @ W ) ) ) ) ).
thf(mimplies_def,definition,
( mimplies
= ( ^ [A: mworld > $o,B: mworld > $o,W: mworld] :
( ( A @ W )
=> ( B @ W ) ) ) ) ).
thf(mequiv_def,definition,
( mequiv
= ( ^ [A: mworld > $o,B: mworld > $o,W: mworld] :
( ( A @ W )
<=> ( B @ W ) ) ) ) ).
thf(mbox_type,type,
mbox: ( mworld > $o ) > mworld > $o ).
thf(mbox_def,definition,
( mbox
= ( ^ [Phi: mworld > $o,W: mworld] :
! [V: mworld] :
( ( mrel @ W @ V )
=> ( Phi @ V ) ) ) ) ).
thf(mdia_type,type,
mdia: ( mworld > $o ) > mworld > $o ).
thf(mdia_def,definition,
( mdia
= ( ^ [Phi: mworld > $o,W: mworld] :
? [V: mworld] :
( ( mrel @ W @ V )
& ( Phi @ V ) ) ) ) ).
thf(mrel_serial,axiom,
! [W: mworld] :
? [V: mworld] : ( mrel @ W @ V ) ).
thf(mforall_di_type,type,
mforall_di: ( $i > mworld > $o ) > mworld > $o ).
thf(mforall_di_def,definition,
( mforall_di
= ( ^ [A: $i > mworld > $o,W: mworld] :
! [X: $i] : ( A @ X @ W ) ) ) ).
thf(mexists_di_type,type,
mexists_di: ( $i > mworld > $o ) > mworld > $o ).
thf(mexists_di_def,definition,
( mexists_di
= ( ^ [A: $i > mworld > $o,W: mworld] :
? [X: $i] : ( A @ X @ W ) ) ) ).
thf(r2_decl,type,
r2: mworld > $o ).
thf(r3_decl,type,
r3: mworld > $o ).
thf(r4_decl,type,
r4: mworld > $o ).
thf(r5_decl,type,
r5: mworld > $o ).
thf(equivalence_1_decl,type,
equivalence_1: mworld > $o ).
thf(equivalence_2_decl,type,
equivalence_2: mworld > $o ).
thf(cn2_decl,type,
cn2: mworld > $o ).
thf(cn1_decl,type,
cn1: mworld > $o ).
thf(cn3_decl,type,
cn3: mworld > $o ).
thf(kn2_decl,type,
kn2: mworld > $o ).
thf(kn1_decl,type,
kn1: mworld > $o ).
thf(kn3_decl,type,
kn3: mworld > $o ).
thf(equivalence_3_decl,type,
equivalence_3: mworld > $o ).
thf(op_implies_or_decl,type,
op_implies_or: mworld > $o ).
thf(op_and_decl,type,
op_and: mworld > $o ).
thf(and_2_decl,type,
and_2: mworld > $o ).
thf(and_3_decl,type,
and_3: mworld > $o ).
thf(and_1_decl,type,
and_1: mworld > $o ).
thf(implies_2_decl,type,
implies_2: mworld > $o ).
thf(implies_3_decl,type,
implies_3: mworld > $o ).
thf(op_implies_and_decl,type,
op_implies_and: mworld > $o ).
thf(implies_1_decl,type,
implies_1: mworld > $o ).
thf(substitution_of_equivalents_decl,type,
substitution_of_equivalents: mworld > $o ).
thf(op_or_decl,type,
op_or: mworld > $o ).
thf(modus_tollens_decl,type,
modus_tollens: mworld > $o ).
thf(or_1_decl,type,
or_1: mworld > $o ).
thf(or_2_decl,type,
or_2: mworld > $o ).
thf(or_3_decl,type,
or_3: mworld > $o ).
thf(op_equiv_decl,type,
op_equiv: mworld > $o ).
thf(modus_ponens_decl,type,
modus_ponens: mworld > $o ).
thf(op_implies_decl,type,
op_implies: mworld > $o ).
thf(r1_decl,type,
r1: mworld > $o ).
thf(qmltpeq_decl,type,
qmltpeq: $i > $i > mworld > $o ).
thf(is_a_theorem_decl,type,
is_a_theorem: $i > mworld > $o ).
thf(equiv_decl,type,
equiv: $i > $i > $i ).
thf(not_decl,type,
not: $i > $i ).
thf(or_decl,type,
or: $i > $i > $i ).
thf(and_decl,type,
and: $i > $i > $i ).
thf(implies_decl,type,
implies: $i > $i > $i ).
thf(reflexivity,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] : ( mbox @ ( qmltpeq @ X @ X ) ) ) ) ) ).
thf(symmetry,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( mimplies @ ( mbox @ ( qmltpeq @ X @ Y ) ) @ ( mbox @ ( qmltpeq @ Y @ X ) ) ) ) ) ) ) ) ) ).
thf(transitivity,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] :
( mbox
@ ( mforall_di
@ ^ [Z: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( qmltpeq @ X @ Y ) ) @ ( mbox @ ( qmltpeq @ Y @ Z ) ) ) @ ( mbox @ ( qmltpeq @ X @ Z ) ) ) ) ) ) ) ) ) ) ) ).
thf(and_substitution_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mbox @ ( qmltpeq @ A @ B ) ) @ ( mbox @ ( qmltpeq @ ( and @ A @ C ) @ ( and @ B @ C ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(and_substitution_2,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mbox @ ( qmltpeq @ A @ B ) ) @ ( mbox @ ( qmltpeq @ ( and @ C @ A ) @ ( and @ C @ B ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(equiv_substitution_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mbox @ ( qmltpeq @ A @ B ) ) @ ( mbox @ ( qmltpeq @ ( equiv @ A @ C ) @ ( equiv @ B @ C ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(equiv_substitution_2,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mbox @ ( qmltpeq @ A @ B ) ) @ ( mbox @ ( qmltpeq @ ( equiv @ C @ A ) @ ( equiv @ C @ B ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(implies_substitution_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mbox @ ( qmltpeq @ A @ B ) ) @ ( mbox @ ( qmltpeq @ ( implies @ A @ C ) @ ( implies @ B @ C ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(implies_substitution_2,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mbox @ ( qmltpeq @ A @ B ) ) @ ( mbox @ ( qmltpeq @ ( implies @ C @ A ) @ ( implies @ C @ B ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(not_substitution_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] : ( mbox @ ( mimplies @ ( mbox @ ( qmltpeq @ A @ B ) ) @ ( mbox @ ( qmltpeq @ ( not @ A ) @ ( not @ B ) ) ) ) ) ) ) ) ) ) ).
thf(or_substitution_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mbox @ ( qmltpeq @ A @ B ) ) @ ( mbox @ ( qmltpeq @ ( or @ A @ C ) @ ( or @ B @ C ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(or_substitution_2,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] :
( mbox
@ ( mforall_di
@ ^ [C: $i] : ( mbox @ ( mimplies @ ( mbox @ ( qmltpeq @ A @ B ) ) @ ( mbox @ ( qmltpeq @ ( or @ C @ A ) @ ( or @ C @ B ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(is_a_theorem_substitution_1,axiom,
( mlocal
@ ( mbox
@ ( mforall_di
@ ^ [A: $i] :
( mbox
@ ( mforall_di
@ ^ [B: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( qmltpeq @ A @ B ) ) @ ( mbox @ ( is_a_theorem @ A ) ) ) @ ( mbox @ ( is_a_theorem @ B ) ) ) ) ) ) ) ) ) ).
thf(modus_ponens_0,axiom,
( mlocal
@ ( mand
@ ( mbox
@ ( mimplies @ ( mbox @ modus_ponens )
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( is_a_theorem @ X ) ) @ ( mbox @ ( is_a_theorem @ ( implies @ X @ Y ) ) ) ) @ ( mbox @ ( is_a_theorem @ Y ) ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( mimplies @ ( mand @ ( mbox @ ( is_a_theorem @ X ) ) @ ( mbox @ ( is_a_theorem @ ( implies @ X @ Y ) ) ) ) @ ( mbox @ ( is_a_theorem @ Y ) ) ) ) ) ) ) )
@ ( mbox @ modus_ponens ) ) ) ) ) ).
thf(substitution_of_equivalents_0,axiom,
( mlocal
@ ( mand
@ ( mbox
@ ( mimplies @ ( mbox @ substitution_of_equivalents )
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( mimplies @ ( mbox @ ( is_a_theorem @ ( equiv @ X @ Y ) ) ) @ ( mbox @ ( qmltpeq @ X @ Y ) ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( mimplies @ ( mbox @ ( is_a_theorem @ ( equiv @ X @ Y ) ) ) @ ( mbox @ ( qmltpeq @ X @ Y ) ) ) ) ) ) ) )
@ ( mbox @ substitution_of_equivalents ) ) ) ) ) ).
thf(modus_tollens_0,axiom,
( mlocal
@ ( mand
@ ( mbox
@ ( mimplies @ ( mbox @ modus_tollens )
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( implies @ ( not @ Y ) @ ( not @ X ) ) @ ( implies @ X @ Y ) ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( implies @ ( not @ Y ) @ ( not @ X ) ) @ ( implies @ X @ Y ) ) ) ) ) ) ) )
@ ( mbox @ modus_tollens ) ) ) ) ) ).
thf(implies_1_0,axiom,
( mlocal
@ ( mand
@ ( mbox
@ ( mimplies @ ( mbox @ implies_1 )
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( is_a_theorem @ ( implies @ X @ ( implies @ Y @ X ) ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( is_a_theorem @ ( implies @ X @ ( implies @ Y @ X ) ) ) ) ) ) ) )
@ ( mbox @ implies_1 ) ) ) ) ) ).
thf(implies_2_0,axiom,
( mlocal
@ ( mand
@ ( mbox
@ ( mimplies @ ( mbox @ implies_2 )
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( implies @ X @ ( implies @ X @ Y ) ) @ ( implies @ X @ Y ) ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( implies @ X @ ( implies @ X @ Y ) ) @ ( implies @ X @ Y ) ) ) ) ) ) ) )
@ ( mbox @ implies_2 ) ) ) ) ) ).
thf(implies_3_0,axiom,
( mlocal
@ ( mand
@ ( mbox
@ ( mimplies @ ( mbox @ implies_3 )
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] :
( mbox
@ ( mforall_di
@ ^ [Z: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( implies @ X @ Y ) @ ( implies @ ( implies @ Y @ Z ) @ ( implies @ X @ Z ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] :
( mbox
@ ( mforall_di
@ ^ [Z: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( implies @ X @ Y ) @ ( implies @ ( implies @ Y @ Z ) @ ( implies @ X @ Z ) ) ) ) ) ) ) ) ) ) )
@ ( mbox @ implies_3 ) ) ) ) ) ).
thf(and_1_0,axiom,
( mlocal
@ ( mand
@ ( mbox
@ ( mimplies @ ( mbox @ and_1 )
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( and @ X @ Y ) @ X ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( and @ X @ Y ) @ X ) ) ) ) ) ) )
@ ( mbox @ and_1 ) ) ) ) ) ).
thf(and_2_0,axiom,
( mlocal
@ ( mand
@ ( mbox
@ ( mimplies @ ( mbox @ and_2 )
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( and @ X @ Y ) @ Y ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( and @ X @ Y ) @ Y ) ) ) ) ) ) )
@ ( mbox @ and_2 ) ) ) ) ) ).
thf(and_3_0,axiom,
( mlocal
@ ( mand
@ ( mbox
@ ( mimplies @ ( mbox @ and_3 )
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( is_a_theorem @ ( implies @ X @ ( implies @ Y @ ( and @ X @ Y ) ) ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( is_a_theorem @ ( implies @ X @ ( implies @ Y @ ( and @ X @ Y ) ) ) ) ) ) ) ) )
@ ( mbox @ and_3 ) ) ) ) ) ).
thf(or_1_0,axiom,
( mlocal
@ ( mand
@ ( mbox
@ ( mimplies @ ( mbox @ or_1 )
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( is_a_theorem @ ( implies @ X @ ( or @ X @ Y ) ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( is_a_theorem @ ( implies @ X @ ( or @ X @ Y ) ) ) ) ) ) ) )
@ ( mbox @ or_1 ) ) ) ) ) ).
thf(or_2_0,axiom,
( mlocal
@ ( mand
@ ( mbox
@ ( mimplies @ ( mbox @ or_2 )
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( is_a_theorem @ ( implies @ Y @ ( or @ X @ Y ) ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( is_a_theorem @ ( implies @ Y @ ( or @ X @ Y ) ) ) ) ) ) ) )
@ ( mbox @ or_2 ) ) ) ) ) ).
thf(or_3_0,axiom,
( mlocal
@ ( mand
@ ( mbox
@ ( mimplies @ ( mbox @ or_3 )
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] :
( mbox
@ ( mforall_di
@ ^ [Z: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( implies @ X @ Z ) @ ( implies @ ( implies @ Y @ Z ) @ ( implies @ ( or @ X @ Y ) @ Z ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] :
( mbox
@ ( mforall_di
@ ^ [Z: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( implies @ X @ Z ) @ ( implies @ ( implies @ Y @ Z ) @ ( implies @ ( or @ X @ Y ) @ Z ) ) ) ) ) ) ) ) ) ) )
@ ( mbox @ or_3 ) ) ) ) ) ).
thf(equivalence_1_0,axiom,
( mlocal
@ ( mand
@ ( mbox
@ ( mimplies @ ( mbox @ equivalence_1 )
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( equiv @ X @ Y ) @ ( implies @ X @ Y ) ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( equiv @ X @ Y ) @ ( implies @ X @ Y ) ) ) ) ) ) ) )
@ ( mbox @ equivalence_1 ) ) ) ) ) ).
thf(equivalence_2_0,axiom,
( mlocal
@ ( mand
@ ( mbox
@ ( mimplies @ ( mbox @ equivalence_2 )
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( equiv @ X @ Y ) @ ( implies @ Y @ X ) ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( equiv @ X @ Y ) @ ( implies @ Y @ X ) ) ) ) ) ) ) )
@ ( mbox @ equivalence_2 ) ) ) ) ) ).
thf(equivalence_3_0,axiom,
( mlocal
@ ( mand
@ ( mbox
@ ( mimplies @ ( mbox @ equivalence_3 )
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( implies @ X @ Y ) @ ( implies @ ( implies @ Y @ X ) @ ( equiv @ X @ Y ) ) ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( implies @ X @ Y ) @ ( implies @ ( implies @ Y @ X ) @ ( equiv @ X @ Y ) ) ) ) ) ) ) ) )
@ ( mbox @ equivalence_3 ) ) ) ) ) ).
thf(kn1_0,axiom,
( mlocal
@ ( mand
@ ( mbox
@ ( mimplies @ ( mbox @ kn1 )
@ ( mbox
@ ( mforall_di
@ ^ [P: $i] : ( mbox @ ( is_a_theorem @ ( implies @ P @ ( and @ P @ P ) ) ) ) ) ) ) )
@ ( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [P: $i] : ( mbox @ ( is_a_theorem @ ( implies @ P @ ( and @ P @ P ) ) ) ) ) )
@ ( mbox @ kn1 ) ) ) ) ) ).
thf(kn2_0,axiom,
( mlocal
@ ( mand
@ ( mbox
@ ( mimplies @ ( mbox @ kn2 )
@ ( mbox
@ ( mforall_di
@ ^ [P: $i] :
( mbox
@ ( mforall_di
@ ^ [Q: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( and @ P @ Q ) @ P ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [P: $i] :
( mbox
@ ( mforall_di
@ ^ [Q: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( and @ P @ Q ) @ P ) ) ) ) ) ) )
@ ( mbox @ kn2 ) ) ) ) ) ).
thf(kn3_0,axiom,
( mlocal
@ ( mand
@ ( mbox
@ ( mimplies @ ( mbox @ kn3 )
@ ( mbox
@ ( mforall_di
@ ^ [P: $i] :
( mbox
@ ( mforall_di
@ ^ [Q: $i] :
( mbox
@ ( mforall_di
@ ^ [R: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( implies @ P @ Q ) @ ( implies @ ( not @ ( and @ Q @ R ) ) @ ( not @ ( and @ R @ P ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [P: $i] :
( mbox
@ ( mforall_di
@ ^ [Q: $i] :
( mbox
@ ( mforall_di
@ ^ [R: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( implies @ P @ Q ) @ ( implies @ ( not @ ( and @ Q @ R ) ) @ ( not @ ( and @ R @ P ) ) ) ) ) ) ) ) ) ) ) )
@ ( mbox @ kn3 ) ) ) ) ) ).
thf(cn1_0,axiom,
( mlocal
@ ( mand
@ ( mbox
@ ( mimplies @ ( mbox @ cn1 )
@ ( mbox
@ ( mforall_di
@ ^ [P: $i] :
( mbox
@ ( mforall_di
@ ^ [Q: $i] :
( mbox
@ ( mforall_di
@ ^ [R: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( implies @ P @ Q ) @ ( implies @ ( implies @ Q @ R ) @ ( implies @ P @ R ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [P: $i] :
( mbox
@ ( mforall_di
@ ^ [Q: $i] :
( mbox
@ ( mforall_di
@ ^ [R: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( implies @ P @ Q ) @ ( implies @ ( implies @ Q @ R ) @ ( implies @ P @ R ) ) ) ) ) ) ) ) ) ) )
@ ( mbox @ cn1 ) ) ) ) ) ).
thf(cn2_0,axiom,
( mlocal
@ ( mand
@ ( mbox
@ ( mimplies @ ( mbox @ cn2 )
@ ( mbox
@ ( mforall_di
@ ^ [P: $i] :
( mbox
@ ( mforall_di
@ ^ [Q: $i] : ( mbox @ ( is_a_theorem @ ( implies @ P @ ( implies @ ( not @ P ) @ Q ) ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [P: $i] :
( mbox
@ ( mforall_di
@ ^ [Q: $i] : ( mbox @ ( is_a_theorem @ ( implies @ P @ ( implies @ ( not @ P ) @ Q ) ) ) ) ) ) ) )
@ ( mbox @ cn2 ) ) ) ) ) ).
thf(cn3_0,axiom,
( mlocal
@ ( mand
@ ( mbox
@ ( mimplies @ ( mbox @ cn3 )
@ ( mbox
@ ( mforall_di
@ ^ [P: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( implies @ ( not @ P ) @ P ) @ P ) ) ) ) ) ) )
@ ( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [P: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( implies @ ( not @ P ) @ P ) @ P ) ) ) ) )
@ ( mbox @ cn3 ) ) ) ) ) ).
thf(r1_0,axiom,
( mlocal
@ ( mand
@ ( mbox
@ ( mimplies @ ( mbox @ r1 )
@ ( mbox
@ ( mforall_di
@ ^ [P: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( or @ P @ P ) @ P ) ) ) ) ) ) )
@ ( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [P: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( or @ P @ P ) @ P ) ) ) ) )
@ ( mbox @ r1 ) ) ) ) ) ).
thf(r2_0,axiom,
( mlocal
@ ( mand
@ ( mbox
@ ( mimplies @ ( mbox @ r2 )
@ ( mbox
@ ( mforall_di
@ ^ [P: $i] :
( mbox
@ ( mforall_di
@ ^ [Q: $i] : ( mbox @ ( is_a_theorem @ ( implies @ Q @ ( or @ P @ Q ) ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [P: $i] :
( mbox
@ ( mforall_di
@ ^ [Q: $i] : ( mbox @ ( is_a_theorem @ ( implies @ Q @ ( or @ P @ Q ) ) ) ) ) ) ) )
@ ( mbox @ r2 ) ) ) ) ) ).
thf(r3_0,axiom,
( mlocal
@ ( mand
@ ( mbox
@ ( mimplies @ ( mbox @ r3 )
@ ( mbox
@ ( mforall_di
@ ^ [P: $i] :
( mbox
@ ( mforall_di
@ ^ [Q: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( or @ P @ Q ) @ ( or @ Q @ P ) ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [P: $i] :
( mbox
@ ( mforall_di
@ ^ [Q: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( or @ P @ Q ) @ ( or @ Q @ P ) ) ) ) ) ) ) )
@ ( mbox @ r3 ) ) ) ) ) ).
thf(r4_0,axiom,
( mlocal
@ ( mand
@ ( mbox
@ ( mimplies @ ( mbox @ r4 )
@ ( mbox
@ ( mforall_di
@ ^ [P: $i] :
( mbox
@ ( mforall_di
@ ^ [Q: $i] :
( mbox
@ ( mforall_di
@ ^ [R: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( or @ P @ ( or @ Q @ R ) ) @ ( or @ Q @ ( or @ P @ R ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [P: $i] :
( mbox
@ ( mforall_di
@ ^ [Q: $i] :
( mbox
@ ( mforall_di
@ ^ [R: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( or @ P @ ( or @ Q @ R ) ) @ ( or @ Q @ ( or @ P @ R ) ) ) ) ) ) ) ) ) ) )
@ ( mbox @ r4 ) ) ) ) ) ).
thf(r5_0,axiom,
( mlocal
@ ( mand
@ ( mbox
@ ( mimplies @ ( mbox @ r5 )
@ ( mbox
@ ( mforall_di
@ ^ [P: $i] :
( mbox
@ ( mforall_di
@ ^ [Q: $i] :
( mbox
@ ( mforall_di
@ ^ [R: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( implies @ Q @ R ) @ ( implies @ ( or @ P @ Q ) @ ( or @ P @ R ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( mbox
@ ( mimplies
@ ( mbox
@ ( mforall_di
@ ^ [P: $i] :
( mbox
@ ( mforall_di
@ ^ [Q: $i] :
( mbox
@ ( mforall_di
@ ^ [R: $i] : ( mbox @ ( is_a_theorem @ ( implies @ ( implies @ Q @ R ) @ ( implies @ ( or @ P @ Q ) @ ( or @ P @ R ) ) ) ) ) ) ) ) ) ) )
@ ( mbox @ r5 ) ) ) ) ) ).
thf(op_or_0,axiom,
( mlocal
@ ( mbox
@ ( mimplies @ ( mbox @ op_or )
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( qmltpeq @ ( or @ X @ Y ) @ ( not @ ( and @ ( not @ X ) @ ( not @ Y ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(op_and_0,axiom,
( mlocal
@ ( mbox
@ ( mimplies @ ( mbox @ op_and )
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( qmltpeq @ ( and @ X @ Y ) @ ( not @ ( or @ ( not @ X ) @ ( not @ Y ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(op_implies_and_0,axiom,
( mlocal
@ ( mbox
@ ( mimplies @ ( mbox @ op_implies_and )
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( qmltpeq @ ( implies @ X @ Y ) @ ( not @ ( and @ X @ ( not @ Y ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(op_implies_or_0,axiom,
( mlocal
@ ( mbox
@ ( mimplies @ ( mbox @ op_implies_or )
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( qmltpeq @ ( implies @ X @ Y ) @ ( or @ ( not @ X ) @ Y ) ) ) ) ) ) ) ) ) ) ).
thf(op_equiv_0,axiom,
( mlocal
@ ( mbox
@ ( mimplies @ ( mbox @ op_equiv )
@ ( mbox
@ ( mforall_di
@ ^ [X: $i] :
( mbox
@ ( mforall_di
@ ^ [Y: $i] : ( mbox @ ( qmltpeq @ ( equiv @ X @ Y ) @ ( and @ ( implies @ X @ Y ) @ ( implies @ Y @ X ) ) ) ) ) ) ) ) ) ) ) ).
thf(hilbert_op_or,axiom,
mlocal @ ( mbox @ op_or ) ).
thf(hilbert_op_implies_and,axiom,
mlocal @ ( mbox @ op_implies_and ) ).
thf(hilbert_op_equiv,axiom,
mlocal @ ( mbox @ op_equiv ) ).
thf(hilbert_modus_ponens,axiom,
mlocal @ ( mbox @ modus_ponens ) ).
thf(hilbert_modus_tollens,axiom,
mlocal @ ( mbox @ modus_tollens ) ).
thf(hilbert_implies_1,axiom,
mlocal @ ( mbox @ implies_1 ) ).
thf(hilbert_implies_2,axiom,
mlocal @ ( mbox @ implies_2 ) ).
thf(hilbert_implies_3,axiom,
mlocal @ ( mbox @ implies_3 ) ).
thf(hilbert_and_1,axiom,
mlocal @ ( mbox @ and_1 ) ).
thf(hilbert_and_2,axiom,
mlocal @ ( mbox @ and_2 ) ).
thf(hilbert_and_3,axiom,
mlocal @ ( mbox @ and_3 ) ).
thf(hilbert_or_1,axiom,
mlocal @ ( mbox @ or_1 ) ).
thf(hilbert_or_2,axiom,
mlocal @ ( mbox @ or_2 ) ).
thf(hilbert_or_3,axiom,
mlocal @ ( mbox @ or_3 ) ).
thf(hilbert_equivalence_1,axiom,
mlocal @ ( mbox @ equivalence_1 ) ).
thf(hilbert_equivalence_2,axiom,
mlocal @ ( mbox @ equivalence_2 ) ).
thf(hilbert_equivalence_3,axiom,
mlocal @ ( mbox @ equivalence_3 ) ).
thf(substitution_of_equivalents_1,axiom,
mlocal @ ( mbox @ substitution_of_equivalents ) ).
thf(luka_op_or,axiom,
mlocal @ ( mbox @ op_or ) ).
thf(luka_op_implies,axiom,
mlocal @ ( mbox @ op_implies ) ).
thf(luka_op_equiv,axiom,
mlocal @ ( mbox @ op_equiv ) ).
thf(luka_cn1,conjecture,
mlocal @ ( mbox @ cn1 ) ).
%------------------------------------------------------------------------------