TPTP Axioms File: LCL015^0.ax
%------------------------------------------------------------------------------
% File : LCL015^0 : TPTP v8.2.0. Released v5.5.0.
% Domain : Logic Calculi (Quantified multimodal logic, cumulative domains)
% Axioms : Embedding of quantified multimodal logic in simple type theory
% Version : [Ben12] axioms.
% English :
% Refs : [Ben12] Benzmueller (2012), Email to Geoff Sutcliffe
% Source : [Ben12]
% Names :
% Status : Satisfiable
% Syntax : Number of formulae : 64 ( 31 unt; 33 typ; 30 def)
% Number of atoms : 91 ( 34 equ; 0 cnn)
% Maximal formula atoms : 1 ( 1 avg)
% Number of connectives : 128 ( 4 ~; 4 |; 8 &; 103 @)
% ( 0 <=>; 9 =>; 0 <=; 0 <~>)
% Maximal formula depth : 5 ( 1 avg; 103 nst)
% Number of types : 3 ( 1 usr)
% Number of type conns : 170 ( 170 >; 0 *; 0 +; 0 <<)
% Number of symbols : 34 ( 32 usr; 1 con; 0-3 aty)
% Number of variables : 83 ( 46 ^ 30 !; 7 ?; 83 :)
% SPC : THF_SAT_EQU
% Comments : LCL015^1 and this are essentially LCL013^0.ax with the following
% modifications to support cumulative domains:
% - addition of a predicate exists_in_world
% - use of this predicate in a modified definition mforall_ind
% - use of this predicate in an added axiom for cumulative domains
%------------------------------------------------------------------------------
%----Declaration of additional base type mu
thf(mu_type,type,
mu: $tType ).
%----Equality
thf(qmltpeq_type,type,
qmltpeq: mu > mu > $i > $o ).
% originale Definition
%thf(qmltpeq,definition,
% ( qmltpeq
% = ( ^ [X: mu,Y: mu,W: $i] : ( X = Y ) ) )).
% erweiterte Leibnitz-Definition
%thf(qmltpeq,definition,
% ( qmltpeq
% = ( ^ [X: mu,Y: mu,W: $i] : (![P: mu > $i > $o]: ( (P @ X @ W) <=> (P @ Y @ W) ) ) ) )).
% Leibnitz-Definition
%thf(qmltpeq,definition,
% ( qmltpeq
% = ( ^ [X: mu,Y: mu,W: $i] : (! [P: mu > $o]: ( (P @ X) <=> (P @ Y) ) ) ) )).
thf(meq_prop_type,type,
meq_prop: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(meq_prop,definition,
( meq_prop
= ( ^ [X: $i > $o,Y: $i > $o,W: $i] :
( ( X @ W )
= ( Y @ W ) ) ) ) ).
%----Modal operators not, or, box, Pi
thf(mnot_type,type,
mnot: ( $i > $o ) > $i > $o ).
thf(mnot,definition,
( mnot
= ( ^ [Phi: $i > $o,W: $i] :
~ ( Phi @ W ) ) ) ).
thf(mor_type,type,
mor: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(mor,definition,
( mor
= ( ^ [Phi: $i > $o,Psi: $i > $o,W: $i] :
( ( Phi @ W )
| ( Psi @ W ) ) ) ) ).
thf(mbox_type,type,
mbox: ( $i > $i > $o ) > ( $i > $o ) > $i > $o ).
thf(mbox,definition,
( mbox
= ( ^ [R: $i > $i > $o,Phi: $i > $o,W: $i] :
! [V: $i] :
( ~ ( R @ W @ V )
| ( Phi @ V ) ) ) ) ).
thf(mforall_prop_type,type,
mforall_prop: ( ( $i > $o ) > $i > $o ) > $i > $o ).
thf(mforall_prop,definition,
( mforall_prop
= ( ^ [Phi: ( $i > $o ) > $i > $o,W: $i] :
! [P: $i > $o] : ( Phi @ P @ W ) ) ) ).
%----Further modal operators
thf(mtrue_type,type,
mtrue: $i > $o ).
thf(mtrue,definition,
( mtrue
= ( ^ [W: $i] : $true ) ) ).
thf(mfalse_type,type,
mfalse: $i > $o ).
thf(mfalse,definition,
( mfalse
= ( mnot @ mtrue ) ) ).
thf(mand_type,type,
mand: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(mand,definition,
( mand
= ( ^ [Phi: $i > $o,Psi: $i > $o] : ( mnot @ ( mor @ ( mnot @ Phi ) @ ( mnot @ Psi ) ) ) ) ) ).
thf(mimplies_type,type,
mimplies: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(mimplies,definition,
( mimplies
= ( ^ [Phi: $i > $o,Psi: $i > $o] : ( mor @ ( mnot @ Phi ) @ Psi ) ) ) ).
thf(mimplied_type,type,
mimplied: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(mimplied,definition,
( mimplied
= ( ^ [Phi: $i > $o,Psi: $i > $o] : ( mor @ ( mnot @ Psi ) @ Phi ) ) ) ).
thf(mequiv_type,type,
mequiv: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(mequiv,definition,
( mequiv
= ( ^ [Phi: $i > $o,Psi: $i > $o] : ( mand @ ( mimplies @ Phi @ Psi ) @ ( mimplies @ Psi @ Phi ) ) ) ) ).
thf(mxor_type,type,
mxor: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(mxor,definition,
( mxor
= ( ^ [Phi: $i > $o,Psi: $i > $o] : ( mnot @ ( mequiv @ Phi @ Psi ) ) ) ) ).
thf(mdia_type,type,
mdia: ( $i > $i > $o ) > ( $i > $o ) > $i > $o ).
thf(mdia,definition,
( mdia
= ( ^ [R: $i > $i > $o,Phi: $i > $o] : ( mnot @ ( mbox @ R @ ( mnot @ Phi ) ) ) ) ) ).
%--- (new for cumulative)
%---Declaration of existence predicate for simulating cumulative domain
thf(exists_in_world_type,type,
exists_in_world: mu > $i > $o ).
%----The domains are non-empty
thf(nonempty_ax,axiom,
! [V: $i] :
? [X: mu] : ( exists_in_world @ X @ V ) ).
thf(mforall_ind_type,type,
mforall_ind: ( mu > $i > $o ) > $i > $o ).
%--- (new for cumulative)
thf(mforall_ind,definition,
( mforall_ind
= ( ^ [Phi: mu > $i > $o,W: $i] :
! [X: mu] :
( ( exists_in_world @ X @ W )
=> ( Phi @ X @ W ) ) ) ) ).
%thf(mforall_ind,definition,
% ( mforall_ind
% = ( ^ [Phi: mu > $i > $o,W: $i] :
% ! [X: mu] :
% ( Phi @ X @ W ) ) )).
thf(mexists_ind_type,type,
mexists_ind: ( mu > $i > $o ) > $i > $o ).
thf(mexists_ind,definition,
( mexists_ind
= ( ^ [Phi: mu > $i > $o] :
( mnot
@ ( mforall_ind
@ ^ [X: mu] : ( mnot @ ( Phi @ X ) ) ) ) ) ) ).
thf(mexists_prop_type,type,
mexists_prop: ( ( $i > $o ) > $i > $o ) > $i > $o ).
thf(mexists_prop,definition,
( mexists_prop
= ( ^ [Phi: ( $i > $o ) > $i > $o] :
( mnot
@ ( mforall_prop
@ ^ [P: $i > $o] : ( mnot @ ( Phi @ P ) ) ) ) ) ) ).
%----Definition of properties of accessibility relations
thf(mreflexive_type,type,
mreflexive: ( $i > $i > $o ) > $o ).
thf(mreflexive,definition,
( mreflexive
= ( ^ [R: $i > $i > $o] :
! [S: $i] : ( R @ S @ S ) ) ) ).
thf(msymmetric_type,type,
msymmetric: ( $i > $i > $o ) > $o ).
thf(msymmetric,definition,
( msymmetric
= ( ^ [R: $i > $i > $o] :
! [S: $i,T: $i] :
( ( R @ S @ T )
=> ( R @ T @ S ) ) ) ) ).
thf(mserial_type,type,
mserial: ( $i > $i > $o ) > $o ).
thf(mserial,definition,
( mserial
= ( ^ [R: $i > $i > $o] :
! [S: $i] :
? [T: $i] : ( R @ S @ T ) ) ) ).
thf(mtransitive_type,type,
mtransitive: ( $i > $i > $o ) > $o ).
thf(mtransitive,definition,
( mtransitive
= ( ^ [R: $i > $i > $o] :
! [S: $i,T: $i,U: $i] :
( ( ( R @ S @ T )
& ( R @ T @ U ) )
=> ( R @ S @ U ) ) ) ) ).
thf(meuclidean_type,type,
meuclidean: ( $i > $i > $o ) > $o ).
thf(meuclidean,definition,
( meuclidean
= ( ^ [R: $i > $i > $o] :
! [S: $i,T: $i,U: $i] :
( ( ( R @ S @ T )
& ( R @ S @ U ) )
=> ( R @ T @ U ) ) ) ) ).
thf(mpartially_functional_type,type,
mpartially_functional: ( $i > $i > $o ) > $o ).
thf(mpartially_functional,definition,
( mpartially_functional
= ( ^ [R: $i > $i > $o] :
! [S: $i,T: $i,U: $i] :
( ( ( R @ S @ T )
& ( R @ S @ U ) )
=> ( T = U ) ) ) ) ).
thf(mfunctional_type,type,
mfunctional: ( $i > $i > $o ) > $o ).
thf(mfunctional,definition,
( mfunctional
= ( ^ [R: $i > $i > $o] :
! [S: $i] :
? [T: $i] :
( ( R @ S @ T )
& ! [U: $i] :
( ( R @ S @ U )
=> ( T = U ) ) ) ) ) ).
thf(mweakly_dense_type,type,
mweakly_dense: ( $i > $i > $o ) > $o ).
thf(mweakly_dense,definition,
( mweakly_dense
= ( ^ [R: $i > $i > $o] :
! [S: $i,T: $i,U: $i] :
( ( R @ S @ T )
=> ? [U: $i] :
( ( R @ S @ U )
& ( R @ U @ T ) ) ) ) ) ).
thf(mweakly_connected_type,type,
mweakly_connected: ( $i > $i > $o ) > $o ).
thf(mweakly_connected,definition,
( mweakly_connected
= ( ^ [R: $i > $i > $o] :
! [S: $i,T: $i,U: $i] :
( ( ( R @ S @ T )
& ( R @ S @ U ) )
=> ( ( R @ T @ U )
| ( T = U )
| ( R @ U @ T ) ) ) ) ) ).
thf(mweakly_directed_type,type,
mweakly_directed: ( $i > $i > $o ) > $o ).
thf(mweakly_directed,definition,
( mweakly_directed
= ( ^ [R: $i > $i > $o] :
! [S: $i,T: $i,U: $i] :
( ( ( R @ S @ T )
& ( R @ S @ U ) )
=> ? [V: $i] :
( ( R @ T @ V )
& ( R @ U @ V ) ) ) ) ) ).
%----Definition of validity
thf(mvalid_type,type,
mvalid: ( $i > $o ) > $o ).
thf(mvalid,definition,
( mvalid
= ( ^ [Phi: $i > $o] :
! [W: $i] : ( Phi @ W ) ) ) ).
%----Definition of satisfiability
thf(msatisfiable_type,type,
msatisfiable: ( $i > $o ) > $o ).
thf(msatisfiable,definition,
( msatisfiable
= ( ^ [Phi: $i > $o] :
? [W: $i] : ( Phi @ W ) ) ) ).
%----Definition of countersatisfiability
thf(mcountersatisfiable_type,type,
mcountersatisfiable: ( $i > $o ) > $o ).
thf(mcountersatisfiable,definition,
( mcountersatisfiable
= ( ^ [Phi: $i > $o] :
? [W: $i] :
~ ( Phi @ W ) ) ) ).
%----Definition of invalidity
thf(minvalid_type,type,
minvalid: ( $i > $o ) > $o ).
thf(minvalid,definition,
( minvalid
= ( ^ [Phi: $i > $o] :
! [W: $i] :
~ ( Phi @ W ) ) ) ).
%------------------------------------------------------------------------------