TPTP Problem File: SET579^3.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SET579^3 : TPTP v8.2.0. Released v8.1.0.
% Domain : Set Theory
% Problem : TPTP problem SET579+3.p with axiomatized equality
% 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 : SET579+1 [QMLTP]
% Status : Theorem
% Rating : 0.20 v8.2.0, 0.31 v8.1.0
% Syntax : Number of formulae : 42 ( 11 unt; 17 typ; 10 def)
% Number of atoms : 134 ( 10 equ; 0 cnn)
% Maximal formula atoms : 13 ( 5 avg)
% Number of connectives : 210 ( 1 ~; 1 |; 2 &; 203 @)
% ( 1 <=>; 2 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 68 ( 68 >; 0 *; 0 +; 0 <<)
% Number of symbols : 17 ( 16 usr; 1 con; 0-3 aty)
% Number of variables : 66 ( 60 ^; 3 !; 3 ?; 66 :)
% SPC : TH0_THM_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].
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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(qmltpeq_decl,type,
qmltpeq: $i > $i > mworld > $o ).
thf(member_decl,type,
member: $i > $i > mworld > $o ).
thf(subset_decl,type,
subset: $i > $i > mworld > $o ).
thf(difference_decl,type,
difference: $i > $i > $i ).
thf(reflexivity,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] : ( qmltpeq @ X @ X ) ) ) ).
thf(symmetry,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] :
( mforall_di
@ ^ [Y: $i] : ( mimplies @ ( qmltpeq @ X @ Y ) @ ( qmltpeq @ Y @ X ) ) ) ) ) ).
thf(transitivity,axiom,
( mlocal
@ ( mforall_di
@ ^ [X: $i] :
( mforall_di
@ ^ [Y: $i] :
( mforall_di
@ ^ [Z: $i] : ( mimplies @ ( mand @ ( qmltpeq @ X @ Y ) @ ( qmltpeq @ Y @ Z ) ) @ ( qmltpeq @ X @ Z ) ) ) ) ) ) ).
thf(difference_substitution_1,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( difference @ A @ C ) @ ( difference @ B @ C ) ) ) ) ) ) ) ).
thf(difference_substitution_2,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( difference @ C @ A ) @ ( difference @ C @ B ) ) ) ) ) ) ) ).
thf(member_substitution_1,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( member @ A @ C ) ) @ ( member @ B @ C ) ) ) ) ) ) ).
thf(member_substitution_2,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( member @ C @ A ) ) @ ( member @ C @ B ) ) ) ) ) ) ).
thf(subset_substitution_1,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( subset @ A @ C ) ) @ ( subset @ B @ C ) ) ) ) ) ) ).
thf(subset_substitution_2,axiom,
( mlocal
@ ( mforall_di
@ ^ [A: $i] :
( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( subset @ C @ A ) ) @ ( subset @ C @ B ) ) ) ) ) ) ).
thf(difference_defn,axiom,
( mlocal
@ ( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] :
( mforall_di
@ ^ [D: $i] : ( mequiv @ ( member @ D @ ( difference @ B @ C ) ) @ ( mand @ ( member @ D @ B ) @ ( mnot @ ( member @ D @ C ) ) ) ) ) ) ) ) ).
thf(equal_defn,axiom,
( mlocal
@ ( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] : ( mequiv @ ( qmltpeq @ B @ C ) @ ( mand @ ( subset @ B @ C ) @ ( subset @ C @ B ) ) ) ) ) ) ).
thf(subset_defn,axiom,
( mlocal
@ ( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] :
( mequiv @ ( subset @ B @ C )
@ ( mforall_di
@ ^ [D: $i] : ( mimplies @ ( member @ D @ B ) @ ( member @ D @ C ) ) ) ) ) ) ) ).
thf(reflexivity_of_subset,axiom,
( mlocal
@ ( mforall_di
@ ^ [B: $i] : ( subset @ B @ B ) ) ) ).
thf(prove_th20,conjecture,
( mlocal
@ ( mforall_di
@ ^ [B: $i] :
( mforall_di
@ ^ [C: $i] :
( mforall_di
@ ^ [D: $i] :
( mimplies
@ ( mforall_di
@ ^ [E: $i] : ( mequiv @ ( member @ E @ B ) @ ( mand @ ( member @ E @ C ) @ ( mnot @ ( member @ E @ D ) ) ) ) )
@ ( qmltpeq @ B @ ( difference @ C @ D ) ) ) ) ) ) ) ).
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