%------------------------------------------------------------------------------
% File : Satallax---3.5
% Problem : NUM814^5 : TPTP v9.2.0. Bugfixed v5.2.0.
% Transfm : none
% Format : tptp:raw
% Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% Computer : n016.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Oct 9 02:12:15 PM UTC 2025
% Result : Theorem 19.10s 19.24s
% Output : Proof 19.10s
% Verified :
% SZS Type : Refutation
% Derivation depth : 10
% Number of leaves : 22
% Syntax : Number of formulae : 70 ( 23 unt; 0 typ; 4 def)
% Number of atoms : 340 ( 67 equ; 0 cnn)
% Maximal formula atoms : 9 ( 4 avg)
% Number of connectives : 338 ( 107 ~; 50 |; 2 &; 119 @)
% ( 0 <=>; 54 =>; 2 <=; 0 <~>)
% Maximal formula depth : 16 ( 3 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 8 ( 8 >; 0 *; 0 +; 0 <<)
% Number of symbols : 25 ( 22 usr; 24 con; 0-2 aty)
% ( 4 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 48 ( 2 ^; 46 !; 0 ?; 48 :)
% Comments :
%------------------------------------------------------------------------------
thf(def_cEVEN1,definition,
( cEVEN1
= ( ^ [X1: $i] :
! [X2: $i > $o] :
( ~ ( ( X2 @ c0 )
=> ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ( X2 @ ( cS @ ( cS @ X3 ) ) ) ) )
=> ( X2 @ X1 ) ) ) ) ).
thf(def_cIND,definition,
( cIND
= ( ! [X1: $i > $o] :
( ~ ( ( X1 @ c0 )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ( X1 @ ( cS @ X2 ) ) ) )
=> ( !! @ X1 ) ) ) ) ).
thf(def_cODD1,definition,
( cODD1
= ( ^ [X1: $i] :
~ ( cEVEN1 @ X1 ) ) ) ).
thf(def_cPEANO1,definition,
( cPEANO1
= ( ~ ( ~ ( ! [X1: $i] :
( ( cS @ X1 )
!= c0 )
=> ~ ! [X1: $i,X2: $i] :
( ( ( cS @ X1 )
= ( cS @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ cIND ) ) ) ).
thf(cTHM404,conjecture,
( ~ ( ~ ( ! [X1: $i] :
( ( cS @ X1 )
!= c0 )
=> ~ ! [X1: $i,X2: $i] :
( ( ( cS @ X1 )
= ( cS @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: $i > $o] :
( ~ ( ( X1 @ c0 )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ( X1 @ ( cS @ X2 ) ) ) )
=> ( !! @ X1 ) ) )
=> ~ ! [X1: $i] :
~ ~ ! [X2: $i > $o] :
( ~ ( ( X2 @ c0 )
=> ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ( X2 @ ( cS @ ( cS @ X3 ) ) ) ) )
=> ( X2 @ X1 ) ) ) ).
thf(h0,negated_conjecture,
~ ( ~ ( ~ ( ! [X1: $i] :
( ( cS @ X1 )
!= c0 )
=> ~ ! [X1: $i,X2: $i] :
( ( ( cS @ X1 )
= ( cS @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: $i > $o] :
( ~ ( ( X1 @ c0 )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ( X1 @ ( cS @ X2 ) ) ) )
=> ( !! @ X1 ) ) )
=> ~ ! [X1: $i,X2: $i > $o] :
( ~ ( ( X2 @ c0 )
=> ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ( X2 @ ( cS @ ( cS @ X3 ) ) ) ) )
=> ( X2 @ X1 ) ) ),
inference(assume_negation,[status(cth)],[cTHM404]) ).
thf(ax1356,axiom,
( p1
| ~ p2 ),
file('<stdin>',ax1356) ).
thf(ax1357,axiom,
~ p1,
file('<stdin>',ax1357) ).
thf(ax1354,axiom,
( p2
| ~ p4 ),
file('<stdin>',ax1354) ).
thf(pax1121,axiom,
( p1121
=> ! [X1: $i] :
( ( fcS @ fc0 )
!= X1 ) ),
file('<stdin>',pax1121) ).
thf(pax6,axiom,
( p6
=> ! [X1: $i] :
( ( fcS @ X1 )
!= fc0 ) ),
file('<stdin>',pax6) ).
thf(ax1352,axiom,
( p4
| p6 ),
file('<stdin>',ax1352) ).
thf(ax1333,axiom,
( ~ p3
| p24 ),
file('<stdin>',ax1333) ).
thf(ax1355,axiom,
( p1
| p3 ),
file('<stdin>',ax1355) ).
thf(nax1121,axiom,
( p1121
<= ! [X1: $i] :
( ( fcS @ fc0 )
!= X1 ) ),
file('<stdin>',nax1121) ).
thf(nax506,axiom,
( p506
<= ( ~ ( ( ( fcS @ fc0 )
= fc0 )
=> ~ ! [X1: $i] :
( ( ( fcS @ fc0 )
= fc0 )
=> ( ( fcS @ fc0 )
= fc0 ) ) )
=> ( ( fcS @ f__0 )
= fc0 ) ) ),
file('<stdin>',nax506) ).
thf(ax1346,axiom,
( ~ p7
| p12 ),
file('<stdin>',ax1346) ).
thf(ax1351,axiom,
( p4
| p7 ),
file('<stdin>',ax1351) ).
thf(pax426,axiom,
( p426
=> ( ~ ( ( fc0
!= ( fcS @ fc0 ) )
=> ~ ! [X1: $i] :
( ( X1
!= ( fcS @ fc0 ) )
=> ( ( fcS @ ( fcS @ X1 ) )
!= ( fcS @ fc0 ) ) ) )
=> ( ( fcS @ fc0 )
!= ( fcS @ fc0 ) ) ) ),
file('<stdin>',pax426) ).
thf(ax850,axiom,
( ~ p24
| p426 ),
file('<stdin>',ax850) ).
thf(pax506,axiom,
( p506
=> ( ~ ( ( ( fcS @ fc0 )
= fc0 )
=> ~ ! [X1: $i] :
( ( ( fcS @ fc0 )
= fc0 )
=> ( ( fcS @ fc0 )
= fc0 ) ) )
=> ( ( fcS @ f__0 )
= fc0 ) ) ),
file('<stdin>',pax506) ).
thf(pax12,axiom,
( p12
=> ! [X1: $i] :
( ( ( fcS @ fc0 )
= ( fcS @ X1 ) )
=> ( fc0 = X1 ) ) ),
file('<stdin>',pax12) ).
thf(c_0_16,plain,
( p1
| ~ p2 ),
inference(fof_simplification,[status(thm)],[ax1356]) ).
thf(c_0_17,plain,
~ p1,
inference(fof_simplification,[status(thm)],[ax1357]) ).
thf(c_0_18,plain,
( p2
| ~ p4 ),
inference(fof_simplification,[status(thm)],[ax1354]) ).
thf(c_0_19,plain,
( p1
| ~ p2 ),
inference(split_conjunct,[status(thm)],[c_0_16]) ).
thf(c_0_20,plain,
~ p1,
inference(split_conjunct,[status(thm)],[c_0_17]) ).
thf(c_0_21,plain,
( p2
| ~ p4 ),
inference(split_conjunct,[status(thm)],[c_0_18]) ).
thf(c_0_22,plain,
~ p2,
inference(sr,[status(thm)],[c_0_19,c_0_20]) ).
thf(c_0_23,plain,
! [X132: $i] :
( ~ p1121
| ( ( fcS @ fc0 )
!= X132 ) ),
inference(shift_quantors,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[inference(fof_simplification,[status(thm)],[pax1121])])])]) ).
thf(c_0_24,plain,
! [X2604: $i] :
( ~ p6
| ( ( fcS @ X2604 )
!= fc0 ) ),
inference(shift_quantors,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[inference(fof_simplification,[status(thm)],[pax6])])])]) ).
thf(c_0_25,plain,
( p4
| p6 ),
inference(split_conjunct,[status(thm)],[ax1352]) ).
thf(c_0_26,plain,
~ p4,
inference(sr,[status(thm)],[c_0_21,c_0_22]) ).
thf(c_0_27,plain,
( ~ p3
| p24 ),
inference(fof_simplification,[status(thm)],[ax1333]) ).
thf(c_0_28,plain,
( p1
| p3 ),
inference(split_conjunct,[status(thm)],[ax1355]) ).
thf(c_0_29,plain,
( ( ( fcS @ fc0 )
= esk65_0 )
| p1121 ),
inference(skolemize,[status(esa)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[inference(fof_simplification,[status(thm)],[nax1121])])])]) ).
thf(c_0_30,plain,
! [X1: $i] :
( ~ p1121
| ( ( fcS @ fc0 )
!= X1 ) ),
inference(split_conjunct,[status(thm)],[c_0_23]) ).
thf(c_0_31,plain,
( ( ( ( fcS @ fc0 )
= fc0 )
| p506 )
& ( ( ( fcS @ f__0 )
!= fc0 )
| p506 ) ),
inference(distribute,[status(thm)],[inference(fof_nnf,[status(thm)],[inference(fof_simplification,[status(thm)],[nax506])])]) ).
thf(c_0_32,plain,
! [X1: $i] :
( ~ p6
| ( ( fcS @ X1 )
!= fc0 ) ),
inference(split_conjunct,[status(thm)],[c_0_24]) ).
thf(c_0_33,plain,
p6,
inference(sr,[status(thm)],[c_0_25,c_0_26]) ).
thf(c_0_34,plain,
( ~ p7
| p12 ),
inference(fof_simplification,[status(thm)],[ax1346]) ).
thf(c_0_35,plain,
( p4
| p7 ),
inference(split_conjunct,[status(thm)],[ax1351]) ).
thf(c_0_36,plain,
( ( ( esk1008_0
!= ( fcS @ fc0 ) )
| ( fc0
= ( fcS @ fc0 ) )
| ( ( fcS @ fc0 )
!= ( fcS @ fc0 ) )
| ~ p426 )
& ( ( ( fcS @ ( fcS @ esk1008_0 ) )
= ( fcS @ fc0 ) )
| ( fc0
= ( fcS @ fc0 ) )
| ( ( fcS @ fc0 )
!= ( fcS @ fc0 ) )
| ~ p426 ) ),
inference(distribute,[status(thm)],[inference(skolemize,[status(esa)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[inference(fof_simplification,[status(thm)],[pax426])])])])]) ).
thf(c_0_37,plain,
( ~ p24
| p426 ),
inference(fof_simplification,[status(thm)],[ax850]) ).
thf(c_0_38,plain,
( p24
| ~ p3 ),
inference(split_conjunct,[status(thm)],[c_0_27]) ).
thf(c_0_39,plain,
p3,
inference(sr,[status(thm)],[c_0_28,c_0_20]) ).
thf(c_0_40,plain,
( ~ p506
| ( ( fcS @ fc0 )
!= fc0 )
| ( ( fcS @ f__0 )
= fc0 ) ),
inference(fof_nnf,[status(thm)],[inference(fof_simplification,[status(thm)],[pax506])]) ).
thf(c_0_41,plain,
( ( ( fcS @ fc0 )
= esk65_0 )
| p1121 ),
inference(split_conjunct,[status(thm)],[c_0_29]) ).
thf(c_0_42,plain,
~ p1121,
inference(er,[status(thm)],[c_0_30]) ).
thf(c_0_43,plain,
( ( ( fcS @ fc0 )
= fc0 )
| p506 ),
inference(split_conjunct,[status(thm)],[c_0_31]) ).
thf(c_0_44,plain,
! [X1: $i] :
( ( fcS @ X1 )
!= fc0 ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[c_0_32,c_0_33])]) ).
thf(c_0_45,plain,
! [X2598: $i] :
( ~ p12
| ( ( fcS @ fc0 )
!= ( fcS @ X2598 ) )
| ( fc0 = X2598 ) ),
inference(shift_quantors,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[pax12])])]) ).
thf(c_0_46,plain,
( p12
| ~ p7 ),
inference(split_conjunct,[status(thm)],[c_0_34]) ).
thf(c_0_47,plain,
p7,
inference(sr,[status(thm)],[c_0_35,c_0_26]) ).
thf(c_0_48,plain,
( ( ( fcS @ ( fcS @ esk1008_0 ) )
= ( fcS @ fc0 ) )
| ( fc0
= ( fcS @ fc0 ) )
| ( ( fcS @ fc0 )
!= ( fcS @ fc0 ) )
| ~ p426 ),
inference(split_conjunct,[status(thm)],[c_0_36]) ).
thf(c_0_49,plain,
( p426
| ~ p24 ),
inference(split_conjunct,[status(thm)],[c_0_37]) ).
thf(c_0_50,plain,
p24,
inference(cn,[status(thm)],[inference(rw,[status(thm)],[c_0_38,c_0_39])]) ).
thf(c_0_51,plain,
( ( ( fcS @ f__0 )
= fc0 )
| ~ p506
| ( ( fcS @ fc0 )
!= fc0 ) ),
inference(split_conjunct,[status(thm)],[c_0_40]) ).
thf(c_0_52,plain,
( ( fcS @ fc0 )
= esk65_0 ),
inference(sr,[status(thm)],[c_0_41,c_0_42]) ).
thf(c_0_53,plain,
p506,
inference(sr,[status(thm)],[c_0_43,c_0_44]) ).
thf(c_0_54,plain,
! [X1: $i] :
( ( fc0 = X1 )
| ~ p12
| ( ( fcS @ fc0 )
!= ( fcS @ X1 ) ) ),
inference(split_conjunct,[status(thm)],[c_0_45]) ).
thf(c_0_55,plain,
p12,
inference(cn,[status(thm)],[inference(rw,[status(thm)],[c_0_46,c_0_47])]) ).
thf(c_0_56,plain,
( ( fc0
= ( fcS @ fc0 ) )
| ( ( fcS @ ( fcS @ esk1008_0 ) )
= ( fcS @ fc0 ) )
| ~ p426 ),
inference(cn,[status(thm)],[c_0_48]) ).
thf(c_0_57,plain,
p426,
inference(cn,[status(thm)],[inference(rw,[status(thm)],[c_0_49,c_0_50])]) ).
thf(c_0_58,plain,
esk65_0 != fc0,
inference(sr,[status(thm)],[inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[c_0_51,c_0_52]),c_0_53])]),c_0_44]) ).
thf(c_0_59,plain,
! [X1: $i] :
( ( fc0 = X1 )
| ( ( fcS @ X1 )
!= esk65_0 ) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[c_0_54,c_0_52]),c_0_55])]) ).
thf(c_0_60,plain,
( ( fcS @ ( fcS @ esk1008_0 ) )
= esk65_0 ),
inference(sr,[status(thm)],[inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[c_0_56,c_0_52]),c_0_52]),c_0_57])]),c_0_58]) ).
thf(c_0_61,plain,
$false,
inference(sr,[status(thm)],[inference(spm,[status(thm)],[c_0_59,c_0_60]),c_0_44]),
[proof] ).
thf(1,plain,
$false,
inference(eprover,[status(thm),assumptions([h0])],[]) ).
thf(0,theorem,
( ~ ( ~ ( ! [X1: $i] :
( ( cS @ X1 )
!= c0 )
=> ~ ! [X1: $i,X2: $i] :
( ( ( cS @ X1 )
= ( cS @ X2 ) )
=> ( X1 = X2 ) ) )
=> ~ ! [X1: $i > $o] :
( ~ ( ( X1 @ c0 )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ( X1 @ ( cS @ X2 ) ) ) )
=> ( !! @ X1 ) ) )
=> ~ ! [X1: $i] :
~ ~ ! [X2: $i > $o] :
( ~ ( ( X2 @ c0 )
=> ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ( X2 @ ( cS @ ( cS @ X3 ) ) ) ) )
=> ( X2 @ X1 ) ) ),
inference(contra,[status(thm),contra(discharge,[h0])],[1,h0]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : NUM814^5 : TPTP v9.2.0. Bugfixed v5.2.0.
% 0.11/0.13 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% 0.12/0.33 % Computer : n016.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Wed Oct 8 22:40:53 EDT 2025
% 0.12/0.33 % CPUTime :
% 19.10/19.24 % SZS status Theorem
% 19.10/19.24 % Mode: mode485
% 19.10/19.24 % Inferences: 33
% 19.10/19.24 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------