TPTP Problem File: PHI007^4.p
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%------------------------------------------------------------------------------
% File : PHI007^4 : TPTP v9.0.0. Released v6.4.0.
% Domain : Philosophy
% Problem : Inconsistency of the axioms in Goedel's original manuscript
% Version : [Ben16] axioms : Biased.
% English : Scott's variant without the conjunct in the definition of essence.
% Refs : [Ben16] Benzmueller (2016), Email to Geoff Sutcliffe
% Source : [Ben16]
% Names : Sledgehammer_Inconsistency_S5U_direct_satallax_fullaxioms.p [Ben16]
% Status : Theorem
% Rating : 0.75 v9.0.0, 0.80 v8.2.0, 0.85 v8.1.0, 0.82 v7.5.0, 0.71 v7.4.0, 0.67 v7.2.0, 0.62 v7.0.0, 0.71 v6.4.0
% Syntax : Number of formulae : 52 ( 15 unt; 25 typ; 0 def)
% Number of atoms : 70 ( 15 equ; 0 cnn)
% Maximal formula atoms : 3 ( 2 avg)
% Number of connectives : 189 ( 6 ~; 0 |; 9 &; 144 @)
% ( 0 <=>; 30 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 5 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 83 ( 83 >; 0 *; 0 +; 0 <<)
% Number of symbols : 21 ( 19 usr; 1 con; 0-3 aty)
% Number of variables : 83 ( 25 ^; 52 !; 6 ?; 83 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This problem file has been generated by Sledgehammer (satallax
% translation) in default setting.
% : The $false conjecture makes this correspond to the Isabelle
% sources. Otherwise it could be omitted and the status would be
% Unsatisfiable.
%------------------------------------------------------------------------------
%----Could-be-implicit typings (6)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__QML____S5__O__092__060mu__062,type,
qML_mu: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__QML____S5__Oi,type,
qML_i: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
%----Explicit typings (19)
thf(sy_c_Equiv__Relations_Opart__equivp_001t__Nat__Onat,type,
equiv_2129914667vp_nat: ( nat > nat > $o ) > $o ).
thf(sy_c_Equiv__Relations_Opart__equivp_001t__Set__Oset_It__Nat__Onat_J,type,
equiv_235907809et_nat: ( set_nat > set_nat > $o ) > $o ).
thf(sy_c_Inconsistency__S5_OG,type,
inconsistency_G: qML_mu > qML_i > $o ).
thf(sy_c_Inconsistency__S5_ONE_001t__QML____S5__O__092__060mu__062,type,
incons1905966852QML_mu: qML_mu > qML_i > $o ).
thf(sy_c_Inconsistency__S5_OP,type,
inconsistency_P: ( qML_mu > qML_i > $o ) > qML_i > $o ).
thf(sy_c_Inconsistency__S5_Oess_001t__QML____S5__O__092__060mu__062,type,
incons1389517216QML_mu: ( qML_mu > qML_i > $o ) > qML_mu > qML_i > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le1613022364et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_QML__S5_Or,type,
qML_r: qML_i > qML_i > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Set_OBall_001t__Nat__Onat,type,
ball_nat: set_nat > ( nat > $o ) > $o ).
thf(sy_c_Set_OBall_001t__Set__Oset_It__Nat__Onat_J,type,
ball_set_nat: set_set_nat > ( set_nat > $o ) > $o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
collect_set_set_nat: ( set_set_nat > $o ) > set_set_set_nat ).
thf(sy_c_Set__Interval_Oord_OgreaterThan_001t__Nat__Onat,type,
set_greaterThan_nat: ( nat > nat > $o ) > nat > set_nat ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
%----Relevant facts (26)
thf(fact_0_A2,axiom,
! [W: qML_i,X: qML_mu > qML_i > $o,Xa: qML_mu > qML_i > $o] :
( ( ( inconsistency_P @ X @ W )
& ! [V: qML_i] :
( ( qML_r @ W @ V )
=> ! [Xb: qML_mu] :
( ( X @ Xb @ V )
=> ( Xa @ Xb @ V ) ) ) )
=> ( inconsistency_P @ Xa @ W ) ) ).
% A2
thf(fact_1_A5,axiom,
! [X_1: qML_i] : ( inconsistency_P @ incons1905966852QML_mu @ X_1 ) ).
% A5
thf(fact_2_G__def,axiom,
( inconsistency_G
= ( ^ [X2: qML_mu,W2: qML_i] :
! [Y: qML_mu > qML_i > $o] :
( ( inconsistency_P @ Y @ W2 )
=> ( Y @ X2 @ W2 ) ) ) ) ).
% G_def
thf(fact_3_A1a,axiom,
! [W: qML_i,X: qML_mu > qML_i > $o] :
( ( inconsistency_P
@ ^ [Y: qML_mu,Z: qML_i] :
~ ( X @ Y @ Z )
@ W )
=> ~ ( inconsistency_P @ X @ W ) ) ).
% A1a
thf(fact_4_A3,axiom,
! [X_1: qML_i] : ( inconsistency_P @ inconsistency_G @ X_1 ) ).
% A3
thf(fact_5_A1b,axiom,
! [W: qML_i,X: qML_mu > qML_i > $o] :
( ~ ( inconsistency_P @ X @ W )
=> ( inconsistency_P
@ ^ [Y: qML_mu,Z: qML_i] :
~ ( X @ Y @ Z )
@ W ) ) ).
% A1b
thf(fact_6_A4,axiom,
! [W: qML_i,X: qML_mu > qML_i > $o] :
( ( inconsistency_P @ X @ W )
=> ! [V2: qML_i] :
( ( qML_r @ W @ V2 )
=> ( inconsistency_P @ X @ V2 ) ) ) ).
% A4
thf(fact_7_QML__S5_Otrans,axiom,
! [X3: qML_i,Y2: qML_i,Z2: qML_i] :
( ( ( qML_r @ X3 @ Y2 )
& ( qML_r @ Y2 @ Z2 ) )
=> ( qML_r @ X3 @ Z2 ) ) ).
% QML_S5.trans
thf(fact_8_QML__S5_Osym,axiom,
! [X3: qML_i,Y2: qML_i] :
( ( qML_r @ X3 @ Y2 )
=> ( qML_r @ Y2 @ X3 ) ) ).
% QML_S5.sym
thf(fact_9_ref,axiom,
! [X3: qML_i] : ( qML_r @ X3 @ X3 ) ).
% ref
thf(fact_10_NE__def,axiom,
( incons1905966852QML_mu
= ( ^ [X2: qML_mu,W2: qML_i] :
! [Y: qML_mu > qML_i > $o] :
( ( incons1389517216QML_mu @ Y @ X2 @ W2 )
=> ! [V3: qML_i] :
( ( qML_r @ W2 @ V3 )
=> ? [Z: qML_mu] : ( Y @ Z @ V3 ) ) ) ) ) ).
% NE_def
thf(fact_11_ess__def,axiom,
( incons1389517216QML_mu
= ( ^ [Phi: qML_mu > qML_i > $o,X2: qML_mu,W2: qML_i] :
! [Y: qML_mu > qML_i > $o] :
( ( Y @ X2 @ W2 )
=> ! [V3: qML_i] :
( ( qML_r @ W2 @ V3 )
=> ! [Z: qML_mu] :
( ( Phi @ Z @ V3 )
=> ( Y @ Z @ V3 ) ) ) ) ) ) ).
% ess_def
thf(fact_12_mem__Collect__eq,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_13_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_14_part__equivp__typedef,axiom,
! [R: set_nat > set_nat > $o] :
( ( equiv_235907809et_nat @ R )
=> ? [D: set_set_nat] :
( member_set_set_nat @ D
@ ( collect_set_set_nat
@ ^ [C: set_set_nat] :
? [X2: set_nat] :
( ( R @ X2 @ X2 )
& ( C
= ( collect_set_nat @ ( R @ X2 ) ) ) ) ) ) ) ).
% part_equivp_typedef
thf(fact_15_part__equivp__typedef,axiom,
! [R: nat > nat > $o] :
( ( equiv_2129914667vp_nat @ R )
=> ? [D: set_nat] :
( member_set_nat @ D
@ ( collect_set_nat
@ ^ [C: set_nat] :
? [X2: nat] :
( ( R @ X2 @ X2 )
& ( C
= ( collect_nat @ ( R @ X2 ) ) ) ) ) ) ) ).
% part_equivp_typedef
thf(fact_16_Ball__def__raw,axiom,
( ball_nat
= ( ^ [A2: set_nat,P2: nat > $o] :
! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( P2 @ X2 ) ) ) ) ).
% Ball_def_raw
thf(fact_17_Ball__def__raw,axiom,
( ball_set_nat
= ( ^ [A2: set_set_nat,P2: set_nat > $o] :
! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( P2 @ X2 ) ) ) ) ).
% Ball_def_raw
thf(fact_18_Ball__def,axiom,
( ball_nat
= ( ^ [A2: set_nat,P2: nat > $o] :
! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( P2 @ X2 ) ) ) ) ).
% Ball_def
thf(fact_19_Ball__def,axiom,
( ball_set_nat
= ( ^ [A2: set_set_nat,P2: set_nat > $o] :
! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( P2 @ X2 ) ) ) ) ).
% Ball_def
thf(fact_20_part__equivp__def,axiom,
( equiv_2129914667vp_nat
= ( ^ [R2: nat > nat > $o] :
( ? [X2: nat] : ( R2 @ X2 @ X2 )
& ! [X2: nat,Y: nat] :
( ( R2 @ X2 @ Y )
= ( ( R2 @ X2 @ X2 )
& ( R2 @ Y @ Y )
& ( ( R2 @ X2 )
= ( R2 @ Y ) ) ) ) ) ) ) ).
% part_equivp_def
thf(fact_21_subsetI,axiom,
! [A3: set_nat,B: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ( member_nat @ X4 @ B ) )
=> ( ord_less_eq_set_nat @ A3 @ B ) ) ).
% subsetI
thf(fact_22_subsetI,axiom,
! [A3: set_set_nat,B: set_set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A3 )
=> ( member_set_nat @ X4 @ B ) )
=> ( ord_le1613022364et_nat @ A3 @ B ) ) ).
% subsetI
thf(fact_23_dvd_OgreaterThan__def,axiom,
! [L: nat] :
( ( set_greaterThan_nat
@ ^ [N: nat,M: nat] :
( ( dvd_dvd_nat @ N @ M )
& ~ ( dvd_dvd_nat @ M @ N ) )
@ L )
= ( collect_nat
@ ^ [X2: nat] :
( ( dvd_dvd_nat @ L @ X2 )
& ~ ( dvd_dvd_nat @ X2 @ L ) ) ) ) ).
% dvd.greaterThan_def
thf(fact_24_part__equivp__transp,axiom,
! [R: nat > nat > $o,X3: nat,Y2: nat,Z2: nat] :
( ( equiv_2129914667vp_nat @ R )
=> ( ( R @ X3 @ Y2 )
=> ( ( R @ Y2 @ Z2 )
=> ( R @ X3 @ Z2 ) ) ) ) ).
% part_equivp_transp
thf(fact_25_part__equivp__symp,axiom,
! [R: nat > nat > $o,X3: nat,Y2: nat] :
( ( equiv_2129914667vp_nat @ R )
=> ( ( R @ X3 @ Y2 )
=> ( R @ Y2 @ X3 ) ) ) ).
% part_equivp_symp
%----Conjectures (1)
thf(conj_0,conjecture,
$false ).
%------------------------------------------------------------------------------