%------------------------------------------------------------------------------
% File : LEO-II---2.3.1
% Problem : SWX190+1 : TPTP v9.3.1. Released v9.3.0.
% Transfm : none
% Format : tptp:raw
% Command : leo --cores 7 --timeout 300 --proofoutput 1 --foatp e --atp e=/export/starexec/sandbox2/solver/bin/eprover /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n011.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Oct 7 11:02:31 AM UTC 2026
% Result : Theorem 8.45s 1.70s
% Output : CNFRefutation 8.45s
% Verified :
% SZS Type : Refutation
% Derivation depth : 10
% Number of leaves : 54
% Syntax : Number of formulae : 308 ( 269 unt; 0 typ; 0 def)
% Number of atoms : 997 ( 713 equ; 0 cnn)
% Maximal formula atoms : 3 ( 3 avg)
% Number of connectives : 2001 ( 86 ~; 91 |; 0 &;1790 @)
% ( 0 <=>; 34 =>; 0 <=; 0 <~>)
% Maximal formula depth : 6 ( 2 avg)
% Number of types : 1 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 27 ( 24 usr; 4 con; 0-2 aty)
% Number of variables : 576 ( 0 ^; 572 !; 4 ?; 576 :)
% Comments :
%------------------------------------------------------------------------------
thf(tp_addNat,type,
addNat: $i > $i > $i ).
thf(tp_d,type,
d: $i > $i ).
thf(tp_fail,type,
fail: $i > $i > $i ).
thf(tp_fail1,type,
fail1: $i > $i > $i ).
thf(tp_fail12,type,
fail12: $i > $i > $i ).
thf(tp_fail2,type,
fail2: $i > $i > $i ).
thf(tp_fail22,type,
fail22: $i > $i > $i ).
thf(tp_fail3,type,
fail3: $i > $i > $i ).
thf(tp_fail32,type,
fail32: $i > $i > $i ).
thf(tp_fail4,type,
fail4: $i > $i > $i ).
thf(tp_mulNat,type,
mulNat: $i > $i > $i ).
thf(tp_n,type,
n: $i > $i ).
thf(tp_opt,type,
opt: $i > $i ).
thf(tp_proj1,type,
proj1: $i > $i ).
thf(tp_proj12,type,
proj12: $i > $i ).
thf(tp_proj1N,type,
proj1N: $i > $i ).
thf(tp_proj1S,type,
proj1S: $i > $i ).
thf(tp_proj2,type,
proj2: $i > $i ).
thf(tp_proj22,type,
proj22: $i > $i ).
thf(tp_s,type,
s: $i > $i ).
thf(tp_x,type,
x: $i > $i > $i ).
thf(tp_x2,type,
x2: $i ).
thf(tp_y,type,
y: $i > $i > $i ).
thf(tp_z,type,
z: $i ).
thf(1,axiom,
! [E2: $i] :
( ( opt @ ( y @ ( n @ z ) @ E2 ) )
= ( n @ z ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_053) ).
thf(2,axiom,
! [E2: $i,X15: $i] :
( ( opt @ ( y @ ( n @ ( s @ X15 ) ) @ E2 ) )
= ( fail3 @ ( n @ ( s @ X15 ) ) @ E2 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_052) ).
thf(3,axiom,
! [X5: $i,E2: $i] :
( ( X5
!= ( n @ ( proj1N @ X5 ) ) )
=> ( ( opt @ ( y @ X5 @ E2 ) )
= ( fail3 @ X5 @ E2 ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_051) ).
thf(4,axiom,
! [E: $i] :
( ( opt @ ( x @ ( n @ z ) @ E ) )
= E ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_050) ).
thf(5,axiom,
! [E: $i,X4: $i] :
( ( opt @ ( x @ ( n @ ( s @ X4 ) ) @ E ) )
= ( fail @ ( n @ ( s @ X4 ) ) @ E ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_049) ).
thf(6,axiom,
! [Y: $i,E: $i] :
( ( Y
!= ( n @ ( proj1N @ Y ) ) )
=> ( ( opt @ ( x @ Y @ E ) )
= ( fail @ Y @ E ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_048) ).
thf(7,axiom,
! [X: $i] :
( ( X
!= ( x @ ( proj1 @ X ) @ ( proj2 @ X ) ) )
=> ( ( X
!= ( y @ ( proj12 @ X ) @ ( proj22 @ X ) ) )
=> ( ( opt @ X )
= X ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_047) ).
thf(8,axiom,
! [Y: $i] :
( ( mulNat @ z @ Y )
= z ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_046) ).
thf(9,axiom,
! [Y: $i,Z: $i] :
( ( mulNat @ ( s @ Z ) @ Y )
= ( addNat @ Y @ ( mulNat @ Z @ Y ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_045) ).
thf(10,axiom,
! [Y: $i] :
( ( addNat @ z @ Y )
= Y ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_044) ).
thf(11,axiom,
! [Y: $i,Z: $i] :
( ( addNat @ ( s @ Z ) @ Y )
= ( s @ ( addNat @ Z @ Y ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_043) ).
thf(12,axiom,
( ( d @ x2 )
= ( n @ ( s @ z ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_042) ).
thf(13,axiom,
! [H: $i,G2: $i] :
( ( d @ ( y @ H @ G2 ) )
= ( x @ ( y @ ( d @ H ) @ G2 ) @ ( y @ H @ ( d @ G2 ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_041) ).
thf(14,axiom,
! [F: $i,G: $i] :
( ( d @ ( x @ F @ G ) )
= ( x @ ( d @ F ) @ ( d @ G ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_040) ).
thf(15,axiom,
! [Y: $i] :
( ( d @ ( n @ Y ) )
= ( n @ z ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_039) ).
thf(16,axiom,
! [X5: $i] :
( ( fail3 @ X5 @ ( n @ z ) )
= ( n @ z ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_038) ).
thf(17,axiom,
! [X5: $i,X13: $i] :
( ( fail3 @ X5 @ ( n @ ( s @ X13 ) ) )
= ( fail12 @ X5 @ ( n @ ( s @ X13 ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_037) ).
thf(18,axiom,
! [X5: $i,E2: $i] :
( ( E2
!= ( n @ ( proj1N @ E2 ) ) )
=> ( ( fail3 @ X5 @ E2 )
= ( fail12 @ X5 @ E2 ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_036) ).
thf(19,axiom,
! [E2: $i] :
( ( fail12 @ ( n @ z ) @ E2 )
= ( fail22 @ ( n @ z ) @ E2 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_035) ).
thf(20,axiom,
! [E2: $i] :
( ( fail12 @ ( n @ ( s @ z ) ) @ E2 )
= E2 ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_034) ).
thf(21,axiom,
! [E2: $i,X11: $i] :
( ( fail12 @ ( n @ ( s @ ( s @ X11 ) ) ) @ E2 )
= ( fail22 @ ( n @ ( s @ ( s @ X11 ) ) ) @ E2 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_033) ).
thf(22,axiom,
! [X5: $i,E2: $i] :
( ( X5
!= ( n @ ( proj1N @ X5 ) ) )
=> ( ( fail12 @ X5 @ E2 )
= ( fail22 @ X5 @ E2 ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_032) ).
thf(23,axiom,
! [X5: $i] :
( ( fail22 @ X5 @ ( n @ z ) )
= ( fail32 @ X5 @ ( n @ z ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_031) ).
thf(24,axiom,
! [X5: $i] :
( ( fail22 @ X5 @ ( n @ ( s @ z ) ) )
= X5 ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_030) ).
thf(25,axiom,
! [X5: $i,X8: $i] :
( ( fail22 @ X5 @ ( n @ ( s @ ( s @ X8 ) ) ) )
= ( fail32 @ X5 @ ( n @ ( s @ ( s @ X8 ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_029) ).
thf(26,axiom,
! [X5: $i,E2: $i] :
( ( E2
!= ( n @ ( proj1N @ E2 ) ) )
=> ( ( fail22 @ X5 @ E2 )
= ( fail32 @ X5 @ E2 ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_028) ).
thf(27,axiom,
! [E2: $i,A3: $i,B4: $i] :
( ( fail32 @ ( y @ A3 @ B4 ) @ E2 )
= ( opt @ ( y @ A3 @ ( y @ B4 @ E2 ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_027) ).
thf(28,axiom,
! [A2: $i,B3: $i] :
( ( fail32 @ ( n @ A2 ) @ ( n @ B3 ) )
= ( n @ ( mulNat @ A2 @ B3 ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_026) ).
thf(29,axiom,
! [E2: $i,A2: $i] :
( ( E2
!= ( n @ ( proj1N @ E2 ) ) )
=> ( ( fail32 @ ( n @ A2 ) @ E2 )
= ( fail4 @ ( n @ A2 ) @ E2 ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_025) ).
thf(30,axiom,
! [X5: $i,E2: $i] :
( ( X5
!= ( n @ ( proj1N @ X5 ) ) )
=> ( ( X5
!= ( y @ ( proj12 @ X5 ) @ ( proj22 @ X5 ) ) )
=> ( ( fail32 @ X5 @ E2 )
= ( fail4 @ X5 @ E2 ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_024) ).
thf(31,axiom,
! [X5: $i,E2: $i] :
( ( fail4 @ X5 @ E2 )
= ( y @ ( opt @ X5 ) @ ( opt @ E2 ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_023) ).
thf(32,axiom,
! [Y: $i] :
( ( fail @ Y @ ( n @ z ) )
= Y ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_022) ).
thf(33,axiom,
! [Y: $i,X2: $i] :
( ( fail @ Y @ ( n @ ( s @ X2 ) ) )
= ( fail1 @ Y @ ( n @ ( s @ X2 ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_021) ).
thf(34,axiom,
! [Y: $i,E: $i] :
( ( E
!= ( n @ ( proj1N @ E ) ) )
=> ( ( fail @ Y @ E )
= ( fail1 @ Y @ E ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_020) ).
thf(35,axiom,
! [C: $i,B2: $i] :
( ( fail1 @ ( n @ C ) @ ( n @ B2 ) )
= ( n @ ( addNat @ C @ B2 ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_019) ).
thf(36,axiom,
! [E: $i,C: $i] :
( ( E
!= ( n @ ( proj1N @ E ) ) )
=> ( ( fail1 @ ( n @ C ) @ E )
= ( fail2 @ ( n @ C ) @ E ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_018) ).
thf(37,axiom,
! [Y: $i,E: $i] :
( ( Y
!= ( n @ ( proj1N @ Y ) ) )
=> ( ( fail1 @ Y @ E )
= ( fail2 @ Y @ E ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_017) ).
thf(38,axiom,
! [E: $i,A: $i,B: $i] :
( ( ( x @ A @ B )
!= E )
=> ( ( fail2 @ ( x @ A @ B ) @ E )
= ( opt @ ( x @ A @ ( x @ B @ E ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_016) ).
thf(39,axiom,
! [Y: $i,E: $i] :
( ( Y != E )
=> ( ( Y
!= ( x @ ( proj1 @ Y ) @ ( proj2 @ Y ) ) )
=> ( ( fail2 @ Y @ E )
= ( x @ ( opt @ Y ) @ ( opt @ E ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_015) ).
thf(40,axiom,
! [Y: $i,E: $i] :
( ( Y = E )
=> ( ( fail2 @ Y @ E )
= ( y @ ( n @ ( s @ ( s @ z ) ) ) @ ( opt @ Y ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_014) ).
thf(41,axiom,
! [X: $i,X2: $i] :
( ( y @ X @ X2 )
!= x2 ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_013) ).
thf(42,axiom,
! [X: $i,X2: $i] :
( ( x @ X @ X2 )
!= x2 ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_012) ).
thf(43,axiom,
! [X: $i,X2: $i,X3: $i,X4: $i] :
( ( x @ X @ X2 )
!= ( y @ X3 @ X4 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_011) ).
thf(44,axiom,
! [X: $i] :
( ( n @ X )
!= x2 ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_010) ).
thf(45,axiom,
! [X: $i,X2: $i,X3: $i] :
( ( n @ X )
!= ( y @ X2 @ X3 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_009) ).
thf(46,axiom,
! [X: $i,X2: $i,X3: $i] :
( ( n @ X )
!= ( x @ X2 @ X3 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_008) ).
thf(47,axiom,
! [X: $i,X2: $i] :
( ( proj22 @ ( y @ X @ X2 ) )
= X2 ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_007) ).
thf(48,axiom,
! [X: $i,X2: $i] :
( ( proj12 @ ( y @ X @ X2 ) )
= X ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_006) ).
thf(49,axiom,
! [X: $i,X2: $i] :
( ( proj2 @ ( x @ X @ X2 ) )
= X2 ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_005) ).
thf(50,axiom,
! [X: $i,X2: $i] :
( ( proj1 @ ( x @ X @ X2 ) )
= X ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_004) ).
thf(51,axiom,
! [X: $i] :
( ( proj1N @ ( n @ X ) )
= X ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_003) ).
thf(52,axiom,
! [X: $i] :
( ( s @ X )
!= z ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_002) ).
thf(53,axiom,
! [X: $i] :
( ( proj1S @ ( s @ X ) )
= X ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',axiom_001) ).
thf(54,conjecture,
? [E: $i] :
( ( opt @ ( d @ E ) )
!= ( opt @ ( d @ ( opt @ E ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',goal_054) ).
thf(55,negated_conjecture,
( ( ? [E: $i] :
( ( opt @ ( d @ E ) )
!= ( opt @ ( d @ ( opt @ E ) ) ) ) )
= $false ),
inference(negate_conjecture,[status(cth)],[54]) ).
thf(56,plain,
( ( ? [E: $i] :
( ( opt @ ( d @ E ) )
!= ( opt @ ( d @ ( opt @ E ) ) ) ) )
= $false ),
inference(unfold_def,[status(thm)],[55]) ).
thf(57,plain,
( ( ! [E2: $i] :
( ( opt @ ( y @ ( n @ z ) @ E2 ) )
= ( n @ z ) ) )
= $true ),
inference(unfold_def,[status(thm)],[1]) ).
thf(58,plain,
( ( ! [E2: $i,X15: $i] :
( ( opt @ ( y @ ( n @ ( s @ X15 ) ) @ E2 ) )
= ( fail3 @ ( n @ ( s @ X15 ) ) @ E2 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[2]) ).
thf(59,plain,
( ( ! [X5: $i,E2: $i] :
( ( X5
!= ( n @ ( proj1N @ X5 ) ) )
=> ( ( opt @ ( y @ X5 @ E2 ) )
= ( fail3 @ X5 @ E2 ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[3]) ).
thf(60,plain,
( ( ! [E: $i] :
( ( opt @ ( x @ ( n @ z ) @ E ) )
= E ) )
= $true ),
inference(unfold_def,[status(thm)],[4]) ).
thf(61,plain,
( ( ! [E: $i,X4: $i] :
( ( opt @ ( x @ ( n @ ( s @ X4 ) ) @ E ) )
= ( fail @ ( n @ ( s @ X4 ) ) @ E ) ) )
= $true ),
inference(unfold_def,[status(thm)],[5]) ).
thf(62,plain,
( ( ! [Y: $i,E: $i] :
( ( Y
!= ( n @ ( proj1N @ Y ) ) )
=> ( ( opt @ ( x @ Y @ E ) )
= ( fail @ Y @ E ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[6]) ).
thf(63,plain,
( ( ! [X: $i] :
( ( X
!= ( x @ ( proj1 @ X ) @ ( proj2 @ X ) ) )
=> ( ( X
!= ( y @ ( proj12 @ X ) @ ( proj22 @ X ) ) )
=> ( ( opt @ X )
= X ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[7]) ).
thf(64,plain,
( ( ! [Y: $i] :
( ( mulNat @ z @ Y )
= z ) )
= $true ),
inference(unfold_def,[status(thm)],[8]) ).
thf(65,plain,
( ( ! [Y: $i,Z: $i] :
( ( mulNat @ ( s @ Z ) @ Y )
= ( addNat @ Y @ ( mulNat @ Z @ Y ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[9]) ).
thf(66,plain,
( ( ! [Y: $i] :
( ( addNat @ z @ Y )
= Y ) )
= $true ),
inference(unfold_def,[status(thm)],[10]) ).
thf(67,plain,
( ( ! [Y: $i,Z: $i] :
( ( addNat @ ( s @ Z ) @ Y )
= ( s @ ( addNat @ Z @ Y ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[11]) ).
thf(68,plain,
( ( ( d @ x2 )
= ( n @ ( s @ z ) ) )
= $true ),
inference(unfold_def,[status(thm)],[12]) ).
thf(69,plain,
( ( ! [H: $i,G2: $i] :
( ( d @ ( y @ H @ G2 ) )
= ( x @ ( y @ ( d @ H ) @ G2 ) @ ( y @ H @ ( d @ G2 ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[13]) ).
thf(70,plain,
( ( ! [F: $i,G: $i] :
( ( d @ ( x @ F @ G ) )
= ( x @ ( d @ F ) @ ( d @ G ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[14]) ).
thf(71,plain,
( ( ! [Y: $i] :
( ( d @ ( n @ Y ) )
= ( n @ z ) ) )
= $true ),
inference(unfold_def,[status(thm)],[15]) ).
thf(72,plain,
( ( ! [X5: $i] :
( ( fail3 @ X5 @ ( n @ z ) )
= ( n @ z ) ) )
= $true ),
inference(unfold_def,[status(thm)],[16]) ).
thf(73,plain,
( ( ! [X5: $i,X13: $i] :
( ( fail3 @ X5 @ ( n @ ( s @ X13 ) ) )
= ( fail12 @ X5 @ ( n @ ( s @ X13 ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[17]) ).
thf(74,plain,
( ( ! [X5: $i,E2: $i] :
( ( E2
!= ( n @ ( proj1N @ E2 ) ) )
=> ( ( fail3 @ X5 @ E2 )
= ( fail12 @ X5 @ E2 ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[18]) ).
thf(75,plain,
( ( ! [E2: $i] :
( ( fail12 @ ( n @ z ) @ E2 )
= ( fail22 @ ( n @ z ) @ E2 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[19]) ).
thf(76,plain,
( ( ! [E2: $i] :
( ( fail12 @ ( n @ ( s @ z ) ) @ E2 )
= E2 ) )
= $true ),
inference(unfold_def,[status(thm)],[20]) ).
thf(77,plain,
( ( ! [E2: $i,X11: $i] :
( ( fail12 @ ( n @ ( s @ ( s @ X11 ) ) ) @ E2 )
= ( fail22 @ ( n @ ( s @ ( s @ X11 ) ) ) @ E2 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[21]) ).
thf(78,plain,
( ( ! [X5: $i,E2: $i] :
( ( X5
!= ( n @ ( proj1N @ X5 ) ) )
=> ( ( fail12 @ X5 @ E2 )
= ( fail22 @ X5 @ E2 ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[22]) ).
thf(79,plain,
( ( ! [X5: $i] :
( ( fail22 @ X5 @ ( n @ z ) )
= ( fail32 @ X5 @ ( n @ z ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[23]) ).
thf(80,plain,
( ( ! [X5: $i] :
( ( fail22 @ X5 @ ( n @ ( s @ z ) ) )
= X5 ) )
= $true ),
inference(unfold_def,[status(thm)],[24]) ).
thf(81,plain,
( ( ! [X5: $i,X8: $i] :
( ( fail22 @ X5 @ ( n @ ( s @ ( s @ X8 ) ) ) )
= ( fail32 @ X5 @ ( n @ ( s @ ( s @ X8 ) ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[25]) ).
thf(82,plain,
( ( ! [X5: $i,E2: $i] :
( ( E2
!= ( n @ ( proj1N @ E2 ) ) )
=> ( ( fail22 @ X5 @ E2 )
= ( fail32 @ X5 @ E2 ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[26]) ).
thf(83,plain,
( ( ! [E2: $i,A3: $i,B4: $i] :
( ( fail32 @ ( y @ A3 @ B4 ) @ E2 )
= ( opt @ ( y @ A3 @ ( y @ B4 @ E2 ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[27]) ).
thf(84,plain,
( ( ! [A2: $i,B3: $i] :
( ( fail32 @ ( n @ A2 ) @ ( n @ B3 ) )
= ( n @ ( mulNat @ A2 @ B3 ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[28]) ).
thf(85,plain,
( ( ! [E2: $i,A2: $i] :
( ( E2
!= ( n @ ( proj1N @ E2 ) ) )
=> ( ( fail32 @ ( n @ A2 ) @ E2 )
= ( fail4 @ ( n @ A2 ) @ E2 ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[29]) ).
thf(86,plain,
( ( ! [X5: $i,E2: $i] :
( ( X5
!= ( n @ ( proj1N @ X5 ) ) )
=> ( ( X5
!= ( y @ ( proj12 @ X5 ) @ ( proj22 @ X5 ) ) )
=> ( ( fail32 @ X5 @ E2 )
= ( fail4 @ X5 @ E2 ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[30]) ).
thf(87,plain,
( ( ! [X5: $i,E2: $i] :
( ( fail4 @ X5 @ E2 )
= ( y @ ( opt @ X5 ) @ ( opt @ E2 ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[31]) ).
thf(88,plain,
( ( ! [Y: $i] :
( ( fail @ Y @ ( n @ z ) )
= Y ) )
= $true ),
inference(unfold_def,[status(thm)],[32]) ).
thf(89,plain,
( ( ! [Y: $i,X2: $i] :
( ( fail @ Y @ ( n @ ( s @ X2 ) ) )
= ( fail1 @ Y @ ( n @ ( s @ X2 ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[33]) ).
thf(90,plain,
( ( ! [Y: $i,E: $i] :
( ( E
!= ( n @ ( proj1N @ E ) ) )
=> ( ( fail @ Y @ E )
= ( fail1 @ Y @ E ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[34]) ).
thf(91,plain,
( ( ! [C: $i,B2: $i] :
( ( fail1 @ ( n @ C ) @ ( n @ B2 ) )
= ( n @ ( addNat @ C @ B2 ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[35]) ).
thf(92,plain,
( ( ! [E: $i,C: $i] :
( ( E
!= ( n @ ( proj1N @ E ) ) )
=> ( ( fail1 @ ( n @ C ) @ E )
= ( fail2 @ ( n @ C ) @ E ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[36]) ).
thf(93,plain,
( ( ! [Y: $i,E: $i] :
( ( Y
!= ( n @ ( proj1N @ Y ) ) )
=> ( ( fail1 @ Y @ E )
= ( fail2 @ Y @ E ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[37]) ).
thf(94,plain,
( ( ! [E: $i,A: $i,B: $i] :
( ( ( x @ A @ B )
!= E )
=> ( ( fail2 @ ( x @ A @ B ) @ E )
= ( opt @ ( x @ A @ ( x @ B @ E ) ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[38]) ).
thf(95,plain,
( ( ! [Y: $i,E: $i] :
( ( Y != E )
=> ( ( Y
!= ( x @ ( proj1 @ Y ) @ ( proj2 @ Y ) ) )
=> ( ( fail2 @ Y @ E )
= ( x @ ( opt @ Y ) @ ( opt @ E ) ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[39]) ).
thf(96,plain,
( ( ! [Y: $i,E: $i] :
( ( Y = E )
=> ( ( fail2 @ Y @ E )
= ( y @ ( n @ ( s @ ( s @ z ) ) ) @ ( opt @ Y ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[40]) ).
thf(97,plain,
( ( ! [X: $i,X2: $i] :
( ( y @ X @ X2 )
!= x2 ) )
= $true ),
inference(unfold_def,[status(thm)],[41]) ).
thf(98,plain,
( ( ! [X: $i,X2: $i] :
( ( x @ X @ X2 )
!= x2 ) )
= $true ),
inference(unfold_def,[status(thm)],[42]) ).
thf(99,plain,
( ( ! [X: $i,X2: $i,X3: $i,X4: $i] :
( ( x @ X @ X2 )
!= ( y @ X3 @ X4 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[43]) ).
thf(100,plain,
( ( ! [X: $i] :
( ( n @ X )
!= x2 ) )
= $true ),
inference(unfold_def,[status(thm)],[44]) ).
thf(101,plain,
( ( ! [X: $i,X2: $i,X3: $i] :
( ( n @ X )
!= ( y @ X2 @ X3 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[45]) ).
thf(102,plain,
( ( ! [X: $i,X2: $i,X3: $i] :
( ( n @ X )
!= ( x @ X2 @ X3 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[46]) ).
thf(103,plain,
( ( ! [X: $i,X2: $i] :
( ( proj22 @ ( y @ X @ X2 ) )
= X2 ) )
= $true ),
inference(unfold_def,[status(thm)],[47]) ).
thf(104,plain,
( ( ! [X: $i,X2: $i] :
( ( proj12 @ ( y @ X @ X2 ) )
= X ) )
= $true ),
inference(unfold_def,[status(thm)],[48]) ).
thf(105,plain,
( ( ! [X: $i,X2: $i] :
( ( proj2 @ ( x @ X @ X2 ) )
= X2 ) )
= $true ),
inference(unfold_def,[status(thm)],[49]) ).
thf(106,plain,
( ( ! [X: $i,X2: $i] :
( ( proj1 @ ( x @ X @ X2 ) )
= X ) )
= $true ),
inference(unfold_def,[status(thm)],[50]) ).
thf(107,plain,
( ( ! [X: $i] :
( ( proj1N @ ( n @ X ) )
= X ) )
= $true ),
inference(unfold_def,[status(thm)],[51]) ).
thf(108,plain,
( ( ! [X: $i] :
( ( s @ X )
!= z ) )
= $true ),
inference(unfold_def,[status(thm)],[52]) ).
thf(109,plain,
( ( ! [X: $i] :
( ( proj1S @ ( s @ X ) )
= X ) )
= $true ),
inference(unfold_def,[status(thm)],[53]) ).
thf(110,plain,
( ( ~ ? [E: $i] :
( ( opt @ ( d @ E ) )
!= ( opt @ ( d @ ( opt @ E ) ) ) ) )
= $true ),
inference(polarity_switch,[status(thm)],[56]) ).
thf(111,plain,
( ( ! [E: $i] :
( ( opt @ ( d @ E ) )
= ( opt @ ( d @ ( opt @ E ) ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[110]) ).
thf(112,plain,
( ( ! [X5: $i] :
( ( X5
= ( n @ ( proj1N @ X5 ) ) )
| ! [E2: $i] :
( ( opt @ ( y @ X5 @ E2 ) )
= ( fail3 @ X5 @ E2 ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[59]) ).
thf(113,plain,
( ( ! [Y: $i] :
( ( Y
= ( n @ ( proj1N @ Y ) ) )
| ! [E: $i] :
( ( opt @ ( x @ Y @ E ) )
= ( fail @ Y @ E ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[62]) ).
thf(114,plain,
( ( ! [X: $i] :
( ( X
= ( x @ ( proj1 @ X ) @ ( proj2 @ X ) ) )
| ( X
= ( y @ ( proj12 @ X ) @ ( proj22 @ X ) ) )
| ( ( opt @ X )
= X ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[63]) ).
thf(115,plain,
( ( ! [X5: $i,E2: $i] :
( ( E2
= ( n @ ( proj1N @ E2 ) ) )
| ( ( fail3 @ X5 @ E2 )
= ( fail12 @ X5 @ E2 ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[74]) ).
thf(116,plain,
( ( ! [X5: $i] :
( ( X5
= ( n @ ( proj1N @ X5 ) ) )
| ! [E2: $i] :
( ( fail12 @ X5 @ E2 )
= ( fail22 @ X5 @ E2 ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[78]) ).
thf(117,plain,
( ( ! [X5: $i,E2: $i] :
( ( E2
= ( n @ ( proj1N @ E2 ) ) )
| ( ( fail22 @ X5 @ E2 )
= ( fail32 @ X5 @ E2 ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[82]) ).
thf(118,plain,
( ( ! [E2: $i] :
( ( E2
= ( n @ ( proj1N @ E2 ) ) )
| ! [A2: $i] :
( ( fail32 @ ( n @ A2 ) @ E2 )
= ( fail4 @ ( n @ A2 ) @ E2 ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[85]) ).
thf(119,plain,
( ( ! [X5: $i] :
( ( X5
= ( n @ ( proj1N @ X5 ) ) )
| ( X5
= ( y @ ( proj12 @ X5 ) @ ( proj22 @ X5 ) ) )
| ! [E2: $i] :
( ( fail32 @ X5 @ E2 )
= ( fail4 @ X5 @ E2 ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[86]) ).
thf(120,plain,
( ( ! [Y: $i,E: $i] :
( ( E
= ( n @ ( proj1N @ E ) ) )
| ( ( fail @ Y @ E )
= ( fail1 @ Y @ E ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[90]) ).
thf(121,plain,
( ( ! [E: $i] :
( ( E
= ( n @ ( proj1N @ E ) ) )
| ! [C: $i] :
( ( fail1 @ ( n @ C ) @ E )
= ( fail2 @ ( n @ C ) @ E ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[92]) ).
thf(122,plain,
( ( ! [Y: $i] :
( ( Y
= ( n @ ( proj1N @ Y ) ) )
| ! [E: $i] :
( ( fail1 @ Y @ E )
= ( fail2 @ Y @ E ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[93]) ).
thf(123,plain,
( ( ! [E: $i,A: $i,B: $i] :
( ( ( x @ A @ B )
= E )
| ( ( fail2 @ ( x @ A @ B ) @ E )
= ( opt @ ( x @ A @ ( x @ B @ E ) ) ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[94]) ).
thf(124,plain,
( ( ! [Y: $i,E: $i] :
( ( Y = E )
| ( Y
= ( x @ ( proj1 @ Y ) @ ( proj2 @ Y ) ) )
| ( ( fail2 @ Y @ E )
= ( x @ ( opt @ Y ) @ ( opt @ E ) ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[95]) ).
thf(125,plain,
( ( ! [Y: $i,E: $i] :
( ( Y != E )
| ( ( fail2 @ Y @ E )
= ( y @ ( n @ ( s @ ( s @ z ) ) ) @ ( opt @ Y ) ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[96]) ).
thf(126,plain,
( ( ! [X: $i,X2: $i] :
( ( y @ X @ X2 )
!= x2 ) )
= $true ),
inference(extcnf_combined,[status(esa)],[97]) ).
thf(127,plain,
( ( ! [X: $i,X2: $i] :
( ( x @ X @ X2 )
!= x2 ) )
= $true ),
inference(extcnf_combined,[status(esa)],[98]) ).
thf(128,plain,
( ( ! [X: $i,X2: $i,X3: $i,X4: $i] :
( ( x @ X @ X2 )
!= ( y @ X3 @ X4 ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[99]) ).
thf(129,plain,
( ( ! [X: $i] :
( ( n @ X )
!= x2 ) )
= $true ),
inference(extcnf_combined,[status(esa)],[100]) ).
thf(130,plain,
( ( ! [X: $i,X2: $i,X3: $i] :
( ( n @ X )
!= ( y @ X2 @ X3 ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[101]) ).
thf(131,plain,
( ( ! [X: $i,X2: $i,X3: $i] :
( ( n @ X )
!= ( x @ X2 @ X3 ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[102]) ).
thf(132,plain,
( ( ! [X: $i] :
( ( s @ X )
!= z ) )
= $true ),
inference(extcnf_combined,[status(esa)],[108]) ).
thf(133,plain,
( ( ! [X: $i] :
( ( proj1S @ ( s @ X ) )
= X ) )
= $true ),
inference(copy,[status(thm)],[109]) ).
thf(134,plain,
( ( ! [X: $i] :
( ( s @ X )
!= z ) )
= $true ),
inference(copy,[status(thm)],[132]) ).
thf(135,plain,
( ( ! [X: $i] :
( ( proj1N @ ( n @ X ) )
= X ) )
= $true ),
inference(copy,[status(thm)],[107]) ).
thf(136,plain,
( ( ! [X: $i,X2: $i] :
( ( proj1 @ ( x @ X @ X2 ) )
= X ) )
= $true ),
inference(copy,[status(thm)],[106]) ).
thf(137,plain,
( ( ! [X: $i,X2: $i] :
( ( proj2 @ ( x @ X @ X2 ) )
= X2 ) )
= $true ),
inference(copy,[status(thm)],[105]) ).
thf(138,plain,
( ( ! [X: $i,X2: $i] :
( ( proj12 @ ( y @ X @ X2 ) )
= X ) )
= $true ),
inference(copy,[status(thm)],[104]) ).
thf(139,plain,
( ( ! [X: $i,X2: $i] :
( ( proj22 @ ( y @ X @ X2 ) )
= X2 ) )
= $true ),
inference(copy,[status(thm)],[103]) ).
thf(140,plain,
( ( ! [X: $i,X2: $i,X3: $i] :
( ( n @ X )
!= ( x @ X2 @ X3 ) ) )
= $true ),
inference(copy,[status(thm)],[131]) ).
thf(141,plain,
( ( ! [X: $i,X2: $i,X3: $i] :
( ( n @ X )
!= ( y @ X2 @ X3 ) ) )
= $true ),
inference(copy,[status(thm)],[130]) ).
thf(142,plain,
( ( ! [X: $i] :
( ( n @ X )
!= x2 ) )
= $true ),
inference(copy,[status(thm)],[129]) ).
thf(143,plain,
( ( ! [X: $i,X2: $i,X3: $i,X4: $i] :
( ( x @ X @ X2 )
!= ( y @ X3 @ X4 ) ) )
= $true ),
inference(copy,[status(thm)],[128]) ).
thf(144,plain,
( ( ! [X: $i,X2: $i] :
( ( x @ X @ X2 )
!= x2 ) )
= $true ),
inference(copy,[status(thm)],[127]) ).
thf(145,plain,
( ( ! [X: $i,X2: $i] :
( ( y @ X @ X2 )
!= x2 ) )
= $true ),
inference(copy,[status(thm)],[126]) ).
thf(146,plain,
( ( ! [Y: $i,E: $i] :
( ( Y != E )
| ( ( fail2 @ Y @ E )
= ( y @ ( n @ ( s @ ( s @ z ) ) ) @ ( opt @ Y ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[125]) ).
thf(147,plain,
( ( ! [Y: $i,E: $i] :
( ( Y = E )
| ( Y
= ( x @ ( proj1 @ Y ) @ ( proj2 @ Y ) ) )
| ( ( fail2 @ Y @ E )
= ( x @ ( opt @ Y ) @ ( opt @ E ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[124]) ).
thf(148,plain,
( ( ! [E: $i,A: $i,B: $i] :
( ( ( x @ A @ B )
= E )
| ( ( fail2 @ ( x @ A @ B ) @ E )
= ( opt @ ( x @ A @ ( x @ B @ E ) ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[123]) ).
thf(149,plain,
( ( ! [Y: $i] :
( ( Y
= ( n @ ( proj1N @ Y ) ) )
| ! [E: $i] :
( ( fail1 @ Y @ E )
= ( fail2 @ Y @ E ) ) ) )
= $true ),
inference(copy,[status(thm)],[122]) ).
thf(150,plain,
( ( ! [E: $i] :
( ( E
= ( n @ ( proj1N @ E ) ) )
| ! [C: $i] :
( ( fail1 @ ( n @ C ) @ E )
= ( fail2 @ ( n @ C ) @ E ) ) ) )
= $true ),
inference(copy,[status(thm)],[121]) ).
thf(151,plain,
( ( ! [C: $i,B2: $i] :
( ( fail1 @ ( n @ C ) @ ( n @ B2 ) )
= ( n @ ( addNat @ C @ B2 ) ) ) )
= $true ),
inference(copy,[status(thm)],[91]) ).
thf(152,plain,
( ( ! [Y: $i,E: $i] :
( ( E
= ( n @ ( proj1N @ E ) ) )
| ( ( fail @ Y @ E )
= ( fail1 @ Y @ E ) ) ) )
= $true ),
inference(copy,[status(thm)],[120]) ).
thf(153,plain,
( ( ! [Y: $i,X2: $i] :
( ( fail @ Y @ ( n @ ( s @ X2 ) ) )
= ( fail1 @ Y @ ( n @ ( s @ X2 ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[89]) ).
thf(154,plain,
( ( ! [Y: $i] :
( ( fail @ Y @ ( n @ z ) )
= Y ) )
= $true ),
inference(copy,[status(thm)],[88]) ).
thf(155,plain,
( ( ! [X5: $i,E2: $i] :
( ( fail4 @ X5 @ E2 )
= ( y @ ( opt @ X5 ) @ ( opt @ E2 ) ) ) )
= $true ),
inference(copy,[status(thm)],[87]) ).
thf(156,plain,
( ( ! [X5: $i] :
( ( X5
= ( n @ ( proj1N @ X5 ) ) )
| ( X5
= ( y @ ( proj12 @ X5 ) @ ( proj22 @ X5 ) ) )
| ! [E2: $i] :
( ( fail32 @ X5 @ E2 )
= ( fail4 @ X5 @ E2 ) ) ) )
= $true ),
inference(copy,[status(thm)],[119]) ).
thf(157,plain,
( ( ! [E2: $i] :
( ( E2
= ( n @ ( proj1N @ E2 ) ) )
| ! [A2: $i] :
( ( fail32 @ ( n @ A2 ) @ E2 )
= ( fail4 @ ( n @ A2 ) @ E2 ) ) ) )
= $true ),
inference(copy,[status(thm)],[118]) ).
thf(158,plain,
( ( ! [A2: $i,B3: $i] :
( ( fail32 @ ( n @ A2 ) @ ( n @ B3 ) )
= ( n @ ( mulNat @ A2 @ B3 ) ) ) )
= $true ),
inference(copy,[status(thm)],[84]) ).
thf(159,plain,
( ( ! [E2: $i,A3: $i,B4: $i] :
( ( fail32 @ ( y @ A3 @ B4 ) @ E2 )
= ( opt @ ( y @ A3 @ ( y @ B4 @ E2 ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[83]) ).
thf(160,plain,
( ( ! [X5: $i,E2: $i] :
( ( E2
= ( n @ ( proj1N @ E2 ) ) )
| ( ( fail22 @ X5 @ E2 )
= ( fail32 @ X5 @ E2 ) ) ) )
= $true ),
inference(copy,[status(thm)],[117]) ).
thf(161,plain,
( ( ! [X5: $i,X8: $i] :
( ( fail22 @ X5 @ ( n @ ( s @ ( s @ X8 ) ) ) )
= ( fail32 @ X5 @ ( n @ ( s @ ( s @ X8 ) ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[81]) ).
thf(162,plain,
( ( ! [X5: $i] :
( ( fail22 @ X5 @ ( n @ ( s @ z ) ) )
= X5 ) )
= $true ),
inference(copy,[status(thm)],[80]) ).
thf(163,plain,
( ( ! [X5: $i] :
( ( fail22 @ X5 @ ( n @ z ) )
= ( fail32 @ X5 @ ( n @ z ) ) ) )
= $true ),
inference(copy,[status(thm)],[79]) ).
thf(164,plain,
( ( ! [X5: $i] :
( ( X5
= ( n @ ( proj1N @ X5 ) ) )
| ! [E2: $i] :
( ( fail12 @ X5 @ E2 )
= ( fail22 @ X5 @ E2 ) ) ) )
= $true ),
inference(copy,[status(thm)],[116]) ).
thf(165,plain,
( ( ! [E2: $i,X11: $i] :
( ( fail12 @ ( n @ ( s @ ( s @ X11 ) ) ) @ E2 )
= ( fail22 @ ( n @ ( s @ ( s @ X11 ) ) ) @ E2 ) ) )
= $true ),
inference(copy,[status(thm)],[77]) ).
thf(166,plain,
( ( ! [E2: $i] :
( ( fail12 @ ( n @ ( s @ z ) ) @ E2 )
= E2 ) )
= $true ),
inference(copy,[status(thm)],[76]) ).
thf(167,plain,
( ( ! [E2: $i] :
( ( fail12 @ ( n @ z ) @ E2 )
= ( fail22 @ ( n @ z ) @ E2 ) ) )
= $true ),
inference(copy,[status(thm)],[75]) ).
thf(168,plain,
( ( ! [X5: $i,E2: $i] :
( ( E2
= ( n @ ( proj1N @ E2 ) ) )
| ( ( fail3 @ X5 @ E2 )
= ( fail12 @ X5 @ E2 ) ) ) )
= $true ),
inference(copy,[status(thm)],[115]) ).
thf(169,plain,
( ( ! [X5: $i,X13: $i] :
( ( fail3 @ X5 @ ( n @ ( s @ X13 ) ) )
= ( fail12 @ X5 @ ( n @ ( s @ X13 ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[73]) ).
thf(170,plain,
( ( ! [X5: $i] :
( ( fail3 @ X5 @ ( n @ z ) )
= ( n @ z ) ) )
= $true ),
inference(copy,[status(thm)],[72]) ).
thf(171,plain,
( ( ! [Y: $i] :
( ( d @ ( n @ Y ) )
= ( n @ z ) ) )
= $true ),
inference(copy,[status(thm)],[71]) ).
thf(172,plain,
( ( ! [F: $i,G: $i] :
( ( d @ ( x @ F @ G ) )
= ( x @ ( d @ F ) @ ( d @ G ) ) ) )
= $true ),
inference(copy,[status(thm)],[70]) ).
thf(173,plain,
( ( ! [H: $i,G2: $i] :
( ( d @ ( y @ H @ G2 ) )
= ( x @ ( y @ ( d @ H ) @ G2 ) @ ( y @ H @ ( d @ G2 ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[69]) ).
thf(174,plain,
( ( ( d @ x2 )
= ( n @ ( s @ z ) ) )
= $true ),
inference(copy,[status(thm)],[68]) ).
thf(175,plain,
( ( ! [Y: $i,Z: $i] :
( ( addNat @ ( s @ Z ) @ Y )
= ( s @ ( addNat @ Z @ Y ) ) ) )
= $true ),
inference(copy,[status(thm)],[67]) ).
thf(176,plain,
( ( ! [Y: $i] :
( ( addNat @ z @ Y )
= Y ) )
= $true ),
inference(copy,[status(thm)],[66]) ).
thf(177,plain,
( ( ! [Y: $i,Z: $i] :
( ( mulNat @ ( s @ Z ) @ Y )
= ( addNat @ Y @ ( mulNat @ Z @ Y ) ) ) )
= $true ),
inference(copy,[status(thm)],[65]) ).
thf(178,plain,
( ( ! [Y: $i] :
( ( mulNat @ z @ Y )
= z ) )
= $true ),
inference(copy,[status(thm)],[64]) ).
thf(179,plain,
( ( ! [X: $i] :
( ( X
= ( x @ ( proj1 @ X ) @ ( proj2 @ X ) ) )
| ( X
= ( y @ ( proj12 @ X ) @ ( proj22 @ X ) ) )
| ( ( opt @ X )
= X ) ) )
= $true ),
inference(copy,[status(thm)],[114]) ).
thf(180,plain,
( ( ! [Y: $i] :
( ( Y
= ( n @ ( proj1N @ Y ) ) )
| ! [E: $i] :
( ( opt @ ( x @ Y @ E ) )
= ( fail @ Y @ E ) ) ) )
= $true ),
inference(copy,[status(thm)],[113]) ).
thf(181,plain,
( ( ! [E: $i,X4: $i] :
( ( opt @ ( x @ ( n @ ( s @ X4 ) ) @ E ) )
= ( fail @ ( n @ ( s @ X4 ) ) @ E ) ) )
= $true ),
inference(copy,[status(thm)],[61]) ).
thf(182,plain,
( ( ! [E: $i] :
( ( opt @ ( x @ ( n @ z ) @ E ) )
= E ) )
= $true ),
inference(copy,[status(thm)],[60]) ).
thf(183,plain,
( ( ! [X5: $i] :
( ( X5
= ( n @ ( proj1N @ X5 ) ) )
| ! [E2: $i] :
( ( opt @ ( y @ X5 @ E2 ) )
= ( fail3 @ X5 @ E2 ) ) ) )
= $true ),
inference(copy,[status(thm)],[112]) ).
thf(184,plain,
( ( ! [E2: $i,X15: $i] :
( ( opt @ ( y @ ( n @ ( s @ X15 ) ) @ E2 ) )
= ( fail3 @ ( n @ ( s @ X15 ) ) @ E2 ) ) )
= $true ),
inference(copy,[status(thm)],[58]) ).
thf(185,plain,
( ( ! [E2: $i] :
( ( opt @ ( y @ ( n @ z ) @ E2 ) )
= ( n @ z ) ) )
= $true ),
inference(copy,[status(thm)],[57]) ).
thf(186,plain,
( ( ! [E: $i] :
( ( opt @ ( d @ E ) )
= ( opt @ ( d @ ( opt @ E ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[111]) ).
thf(187,plain,
! [SV1: $i] :
( ( ( proj1S @ ( s @ SV1 ) )
= SV1 )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[133]) ).
thf(188,plain,
! [SV2: $i] :
( ( ( s @ SV2 )
!= z )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[134]) ).
thf(189,plain,
! [SV3: $i] :
( ( ( proj1N @ ( n @ SV3 ) )
= SV3 )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[135]) ).
thf(190,plain,
! [SV4: $i] :
( ( ! [SY95: $i] :
( ( proj1 @ ( x @ SV4 @ SY95 ) )
= SV4 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[136]) ).
thf(191,plain,
! [SV5: $i] :
( ( ! [SY96: $i] :
( ( proj2 @ ( x @ SV5 @ SY96 ) )
= SY96 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[137]) ).
thf(192,plain,
! [SV6: $i] :
( ( ! [SY97: $i] :
( ( proj12 @ ( y @ SV6 @ SY97 ) )
= SV6 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[138]) ).
thf(193,plain,
! [SV7: $i] :
( ( ! [SY98: $i] :
( ( proj22 @ ( y @ SV7 @ SY98 ) )
= SY98 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[139]) ).
thf(194,plain,
! [SV8: $i] :
( ( ! [SY99: $i,SY100: $i] :
( ( n @ SV8 )
!= ( x @ SY99 @ SY100 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[140]) ).
thf(195,plain,
! [SV9: $i] :
( ( ! [SY101: $i,SY102: $i] :
( ( n @ SV9 )
!= ( y @ SY101 @ SY102 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[141]) ).
thf(196,plain,
! [SV10: $i] :
( ( ( n @ SV10 )
!= x2 )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[142]) ).
thf(197,plain,
! [SV11: $i] :
( ( ! [SY103: $i,SY104: $i,SY105: $i] :
( ( x @ SV11 @ SY103 )
!= ( y @ SY104 @ SY105 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[143]) ).
thf(198,plain,
! [SV12: $i] :
( ( ! [SY106: $i] :
( ( x @ SV12 @ SY106 )
!= x2 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[144]) ).
thf(199,plain,
! [SV13: $i] :
( ( ! [SY107: $i] :
( ( y @ SV13 @ SY107 )
!= x2 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[145]) ).
thf(200,plain,
! [SV14: $i] :
( ( ! [SY108: $i] :
( ( SV14 != SY108 )
| ( ( fail2 @ SV14 @ SY108 )
= ( y @ ( n @ ( s @ ( s @ z ) ) ) @ ( opt @ SV14 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[146]) ).
thf(201,plain,
! [SV15: $i] :
( ( ! [SY109: $i] :
( ( SV15 = SY109 )
| ( SV15
= ( x @ ( proj1 @ SV15 ) @ ( proj2 @ SV15 ) ) )
| ( ( fail2 @ SV15 @ SY109 )
= ( x @ ( opt @ SV15 ) @ ( opt @ SY109 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[147]) ).
thf(202,plain,
! [SV16: $i] :
( ( ! [SY110: $i,SY111: $i] :
( ( ( x @ SY110 @ SY111 )
= SV16 )
| ( ( fail2 @ ( x @ SY110 @ SY111 ) @ SV16 )
= ( opt @ ( x @ SY110 @ ( x @ SY111 @ SV16 ) ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[148]) ).
thf(203,plain,
! [SV17: $i] :
( ( ( SV17
= ( n @ ( proj1N @ SV17 ) ) )
| ! [SY112: $i] :
( ( fail1 @ SV17 @ SY112 )
= ( fail2 @ SV17 @ SY112 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[149]) ).
thf(204,plain,
! [SV18: $i] :
( ( ( SV18
= ( n @ ( proj1N @ SV18 ) ) )
| ! [SY113: $i] :
( ( fail1 @ ( n @ SY113 ) @ SV18 )
= ( fail2 @ ( n @ SY113 ) @ SV18 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[150]) ).
thf(205,plain,
! [SV19: $i] :
( ( ! [SY114: $i] :
( ( fail1 @ ( n @ SV19 ) @ ( n @ SY114 ) )
= ( n @ ( addNat @ SV19 @ SY114 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[151]) ).
thf(206,plain,
! [SV20: $i] :
( ( ! [SY115: $i] :
( ( SY115
= ( n @ ( proj1N @ SY115 ) ) )
| ( ( fail @ SV20 @ SY115 )
= ( fail1 @ SV20 @ SY115 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[152]) ).
thf(207,plain,
! [SV21: $i] :
( ( ! [SY116: $i] :
( ( fail @ SV21 @ ( n @ ( s @ SY116 ) ) )
= ( fail1 @ SV21 @ ( n @ ( s @ SY116 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[153]) ).
thf(208,plain,
! [SV22: $i] :
( ( ( fail @ SV22 @ ( n @ z ) )
= SV22 )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[154]) ).
thf(209,plain,
! [SV23: $i] :
( ( ! [SY117: $i] :
( ( fail4 @ SV23 @ SY117 )
= ( y @ ( opt @ SV23 ) @ ( opt @ SY117 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[155]) ).
thf(210,plain,
! [SV24: $i] :
( ( ( SV24
= ( n @ ( proj1N @ SV24 ) ) )
| ( SV24
= ( y @ ( proj12 @ SV24 ) @ ( proj22 @ SV24 ) ) )
| ! [SY118: $i] :
( ( fail32 @ SV24 @ SY118 )
= ( fail4 @ SV24 @ SY118 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[156]) ).
thf(211,plain,
! [SV25: $i] :
( ( ( SV25
= ( n @ ( proj1N @ SV25 ) ) )
| ! [SY119: $i] :
( ( fail32 @ ( n @ SY119 ) @ SV25 )
= ( fail4 @ ( n @ SY119 ) @ SV25 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[157]) ).
thf(212,plain,
! [SV26: $i] :
( ( ! [SY120: $i] :
( ( fail32 @ ( n @ SV26 ) @ ( n @ SY120 ) )
= ( n @ ( mulNat @ SV26 @ SY120 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[158]) ).
thf(213,plain,
! [SV27: $i] :
( ( ! [SY121: $i,SY122: $i] :
( ( fail32 @ ( y @ SY121 @ SY122 ) @ SV27 )
= ( opt @ ( y @ SY121 @ ( y @ SY122 @ SV27 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[159]) ).
thf(214,plain,
! [SV28: $i] :
( ( ! [SY123: $i] :
( ( SY123
= ( n @ ( proj1N @ SY123 ) ) )
| ( ( fail22 @ SV28 @ SY123 )
= ( fail32 @ SV28 @ SY123 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[160]) ).
thf(215,plain,
! [SV29: $i] :
( ( ! [SY124: $i] :
( ( fail22 @ SV29 @ ( n @ ( s @ ( s @ SY124 ) ) ) )
= ( fail32 @ SV29 @ ( n @ ( s @ ( s @ SY124 ) ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[161]) ).
thf(216,plain,
! [SV30: $i] :
( ( ( fail22 @ SV30 @ ( n @ ( s @ z ) ) )
= SV30 )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[162]) ).
thf(217,plain,
! [SV31: $i] :
( ( ( fail22 @ SV31 @ ( n @ z ) )
= ( fail32 @ SV31 @ ( n @ z ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[163]) ).
thf(218,plain,
! [SV32: $i] :
( ( ( SV32
= ( n @ ( proj1N @ SV32 ) ) )
| ! [SY125: $i] :
( ( fail12 @ SV32 @ SY125 )
= ( fail22 @ SV32 @ SY125 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[164]) ).
thf(219,plain,
! [SV33: $i] :
( ( ! [SY126: $i] :
( ( fail12 @ ( n @ ( s @ ( s @ SY126 ) ) ) @ SV33 )
= ( fail22 @ ( n @ ( s @ ( s @ SY126 ) ) ) @ SV33 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[165]) ).
thf(220,plain,
! [SV34: $i] :
( ( ( fail12 @ ( n @ ( s @ z ) ) @ SV34 )
= SV34 )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[166]) ).
thf(221,plain,
! [SV35: $i] :
( ( ( fail12 @ ( n @ z ) @ SV35 )
= ( fail22 @ ( n @ z ) @ SV35 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[167]) ).
thf(222,plain,
! [SV36: $i] :
( ( ! [SY127: $i] :
( ( SY127
= ( n @ ( proj1N @ SY127 ) ) )
| ( ( fail3 @ SV36 @ SY127 )
= ( fail12 @ SV36 @ SY127 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[168]) ).
thf(223,plain,
! [SV37: $i] :
( ( ! [SY128: $i] :
( ( fail3 @ SV37 @ ( n @ ( s @ SY128 ) ) )
= ( fail12 @ SV37 @ ( n @ ( s @ SY128 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[169]) ).
thf(224,plain,
! [SV38: $i] :
( ( ( fail3 @ SV38 @ ( n @ z ) )
= ( n @ z ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[170]) ).
thf(225,plain,
! [SV39: $i] :
( ( ( d @ ( n @ SV39 ) )
= ( n @ z ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[171]) ).
thf(226,plain,
! [SV40: $i] :
( ( ! [SY129: $i] :
( ( d @ ( x @ SV40 @ SY129 ) )
= ( x @ ( d @ SV40 ) @ ( d @ SY129 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[172]) ).
thf(227,plain,
! [SV41: $i] :
( ( ! [SY130: $i] :
( ( d @ ( y @ SV41 @ SY130 ) )
= ( x @ ( y @ ( d @ SV41 ) @ SY130 ) @ ( y @ SV41 @ ( d @ SY130 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[173]) ).
thf(228,plain,
! [SV42: $i] :
( ( ! [SY131: $i] :
( ( addNat @ ( s @ SY131 ) @ SV42 )
= ( s @ ( addNat @ SY131 @ SV42 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[175]) ).
thf(229,plain,
! [SV43: $i] :
( ( ( addNat @ z @ SV43 )
= SV43 )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[176]) ).
thf(230,plain,
! [SV44: $i] :
( ( ! [SY132: $i] :
( ( mulNat @ ( s @ SY132 ) @ SV44 )
= ( addNat @ SV44 @ ( mulNat @ SY132 @ SV44 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[177]) ).
thf(231,plain,
! [SV45: $i] :
( ( ( mulNat @ z @ SV45 )
= z )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[178]) ).
thf(232,plain,
! [SV46: $i] :
( ( ( SV46
= ( x @ ( proj1 @ SV46 ) @ ( proj2 @ SV46 ) ) )
| ( SV46
= ( y @ ( proj12 @ SV46 ) @ ( proj22 @ SV46 ) ) )
| ( ( opt @ SV46 )
= SV46 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[179]) ).
thf(233,plain,
! [SV47: $i] :
( ( ( SV47
= ( n @ ( proj1N @ SV47 ) ) )
| ! [SY133: $i] :
( ( opt @ ( x @ SV47 @ SY133 ) )
= ( fail @ SV47 @ SY133 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[180]) ).
thf(234,plain,
! [SV48: $i] :
( ( ! [SY134: $i] :
( ( opt @ ( x @ ( n @ ( s @ SY134 ) ) @ SV48 ) )
= ( fail @ ( n @ ( s @ SY134 ) ) @ SV48 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[181]) ).
thf(235,plain,
! [SV49: $i] :
( ( ( opt @ ( x @ ( n @ z ) @ SV49 ) )
= SV49 )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[182]) ).
thf(236,plain,
! [SV50: $i] :
( ( ( SV50
= ( n @ ( proj1N @ SV50 ) ) )
| ! [SY135: $i] :
( ( opt @ ( y @ SV50 @ SY135 ) )
= ( fail3 @ SV50 @ SY135 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[183]) ).
thf(237,plain,
! [SV51: $i] :
( ( ! [SY136: $i] :
( ( opt @ ( y @ ( n @ ( s @ SY136 ) ) @ SV51 ) )
= ( fail3 @ ( n @ ( s @ SY136 ) ) @ SV51 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[184]) ).
thf(238,plain,
! [SV52: $i] :
( ( ( opt @ ( y @ ( n @ z ) @ SV52 ) )
= ( n @ z ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[185]) ).
thf(239,plain,
! [SV53: $i] :
( ( ( opt @ ( d @ SV53 ) )
= ( opt @ ( d @ ( opt @ SV53 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[186]) ).
thf(240,plain,
! [SV2: $i] :
( ( ( s @ SV2 )
= z )
= $false ),
inference(extcnf_not_pos,[status(thm)],[188]) ).
thf(241,plain,
! [SV54: $i,SV4: $i] :
( ( ( proj1 @ ( x @ SV4 @ SV54 ) )
= SV4 )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[190]) ).
thf(242,plain,
! [SV55: $i,SV5: $i] :
( ( ( proj2 @ ( x @ SV5 @ SV55 ) )
= SV55 )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[191]) ).
thf(243,plain,
! [SV56: $i,SV6: $i] :
( ( ( proj12 @ ( y @ SV6 @ SV56 ) )
= SV6 )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[192]) ).
thf(244,plain,
! [SV57: $i,SV7: $i] :
( ( ( proj22 @ ( y @ SV7 @ SV57 ) )
= SV57 )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[193]) ).
thf(245,plain,
! [SV58: $i,SV8: $i] :
( ( ! [SY137: $i] :
( ( n @ SV8 )
!= ( x @ SV58 @ SY137 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[194]) ).
thf(246,plain,
! [SV59: $i,SV9: $i] :
( ( ! [SY138: $i] :
( ( n @ SV9 )
!= ( y @ SV59 @ SY138 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[195]) ).
thf(247,plain,
! [SV10: $i] :
( ( ( n @ SV10 )
= x2 )
= $false ),
inference(extcnf_not_pos,[status(thm)],[196]) ).
thf(248,plain,
! [SV60: $i,SV11: $i] :
( ( ! [SY139: $i,SY140: $i] :
( ( x @ SV11 @ SV60 )
!= ( y @ SY139 @ SY140 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[197]) ).
thf(249,plain,
! [SV61: $i,SV12: $i] :
( ( ( x @ SV12 @ SV61 )
!= x2 )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[198]) ).
thf(250,plain,
! [SV62: $i,SV13: $i] :
( ( ( y @ SV13 @ SV62 )
!= x2 )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[199]) ).
thf(251,plain,
! [SV63: $i,SV14: $i] :
( ( ( SV14 != SV63 )
| ( ( fail2 @ SV14 @ SV63 )
= ( y @ ( n @ ( s @ ( s @ z ) ) ) @ ( opt @ SV14 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[200]) ).
thf(252,plain,
! [SV64: $i,SV15: $i] :
( ( ( SV15 = SV64 )
| ( SV15
= ( x @ ( proj1 @ SV15 ) @ ( proj2 @ SV15 ) ) )
| ( ( fail2 @ SV15 @ SV64 )
= ( x @ ( opt @ SV15 ) @ ( opt @ SV64 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[201]) ).
thf(253,plain,
! [SV16: $i,SV65: $i] :
( ( ! [SY141: $i] :
( ( ( x @ SV65 @ SY141 )
= SV16 )
| ( ( fail2 @ ( x @ SV65 @ SY141 ) @ SV16 )
= ( opt @ ( x @ SV65 @ ( x @ SY141 @ SV16 ) ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[202]) ).
thf(254,plain,
! [SV17: $i] :
( ( ( SV17
= ( n @ ( proj1N @ SV17 ) ) )
= $true )
| ( ( ! [SY112: $i] :
( ( fail1 @ SV17 @ SY112 )
= ( fail2 @ SV17 @ SY112 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[203]) ).
thf(255,plain,
! [SV18: $i] :
( ( ( SV18
= ( n @ ( proj1N @ SV18 ) ) )
= $true )
| ( ( ! [SY113: $i] :
( ( fail1 @ ( n @ SY113 ) @ SV18 )
= ( fail2 @ ( n @ SY113 ) @ SV18 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[204]) ).
thf(256,plain,
! [SV66: $i,SV19: $i] :
( ( ( fail1 @ ( n @ SV19 ) @ ( n @ SV66 ) )
= ( n @ ( addNat @ SV19 @ SV66 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[205]) ).
thf(257,plain,
! [SV20: $i,SV67: $i] :
( ( ( SV67
= ( n @ ( proj1N @ SV67 ) ) )
| ( ( fail @ SV20 @ SV67 )
= ( fail1 @ SV20 @ SV67 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[206]) ).
thf(258,plain,
! [SV68: $i,SV21: $i] :
( ( ( fail @ SV21 @ ( n @ ( s @ SV68 ) ) )
= ( fail1 @ SV21 @ ( n @ ( s @ SV68 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[207]) ).
thf(259,plain,
! [SV69: $i,SV23: $i] :
( ( ( fail4 @ SV23 @ SV69 )
= ( y @ ( opt @ SV23 ) @ ( opt @ SV69 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[209]) ).
thf(260,plain,
! [SV24: $i] :
( ( ( SV24
= ( n @ ( proj1N @ SV24 ) ) )
= $true )
| ( ( ( SV24
= ( y @ ( proj12 @ SV24 ) @ ( proj22 @ SV24 ) ) )
| ! [SY118: $i] :
( ( fail32 @ SV24 @ SY118 )
= ( fail4 @ SV24 @ SY118 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[210]) ).
thf(261,plain,
! [SV25: $i] :
( ( ( SV25
= ( n @ ( proj1N @ SV25 ) ) )
= $true )
| ( ( ! [SY119: $i] :
( ( fail32 @ ( n @ SY119 ) @ SV25 )
= ( fail4 @ ( n @ SY119 ) @ SV25 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[211]) ).
thf(262,plain,
! [SV70: $i,SV26: $i] :
( ( ( fail32 @ ( n @ SV26 ) @ ( n @ SV70 ) )
= ( n @ ( mulNat @ SV26 @ SV70 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[212]) ).
thf(263,plain,
! [SV27: $i,SV71: $i] :
( ( ! [SY142: $i] :
( ( fail32 @ ( y @ SV71 @ SY142 ) @ SV27 )
= ( opt @ ( y @ SV71 @ ( y @ SY142 @ SV27 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[213]) ).
thf(264,plain,
! [SV28: $i,SV72: $i] :
( ( ( SV72
= ( n @ ( proj1N @ SV72 ) ) )
| ( ( fail22 @ SV28 @ SV72 )
= ( fail32 @ SV28 @ SV72 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[214]) ).
thf(265,plain,
! [SV73: $i,SV29: $i] :
( ( ( fail22 @ SV29 @ ( n @ ( s @ ( s @ SV73 ) ) ) )
= ( fail32 @ SV29 @ ( n @ ( s @ ( s @ SV73 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[215]) ).
thf(266,plain,
! [SV32: $i] :
( ( ( SV32
= ( n @ ( proj1N @ SV32 ) ) )
= $true )
| ( ( ! [SY125: $i] :
( ( fail12 @ SV32 @ SY125 )
= ( fail22 @ SV32 @ SY125 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[218]) ).
thf(267,plain,
! [SV33: $i,SV74: $i] :
( ( ( fail12 @ ( n @ ( s @ ( s @ SV74 ) ) ) @ SV33 )
= ( fail22 @ ( n @ ( s @ ( s @ SV74 ) ) ) @ SV33 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[219]) ).
thf(268,plain,
! [SV36: $i,SV75: $i] :
( ( ( SV75
= ( n @ ( proj1N @ SV75 ) ) )
| ( ( fail3 @ SV36 @ SV75 )
= ( fail12 @ SV36 @ SV75 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[222]) ).
thf(269,plain,
! [SV76: $i,SV37: $i] :
( ( ( fail3 @ SV37 @ ( n @ ( s @ SV76 ) ) )
= ( fail12 @ SV37 @ ( n @ ( s @ SV76 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[223]) ).
thf(270,plain,
! [SV77: $i,SV40: $i] :
( ( ( d @ ( x @ SV40 @ SV77 ) )
= ( x @ ( d @ SV40 ) @ ( d @ SV77 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[226]) ).
thf(271,plain,
! [SV78: $i,SV41: $i] :
( ( ( d @ ( y @ SV41 @ SV78 ) )
= ( x @ ( y @ ( d @ SV41 ) @ SV78 ) @ ( y @ SV41 @ ( d @ SV78 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[227]) ).
thf(272,plain,
! [SV42: $i,SV79: $i] :
( ( ( addNat @ ( s @ SV79 ) @ SV42 )
= ( s @ ( addNat @ SV79 @ SV42 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[228]) ).
thf(273,plain,
! [SV44: $i,SV80: $i] :
( ( ( mulNat @ ( s @ SV80 ) @ SV44 )
= ( addNat @ SV44 @ ( mulNat @ SV80 @ SV44 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[230]) ).
thf(274,plain,
! [SV46: $i] :
( ( ( SV46
= ( x @ ( proj1 @ SV46 ) @ ( proj2 @ SV46 ) ) )
= $true )
| ( ( ( SV46
= ( y @ ( proj12 @ SV46 ) @ ( proj22 @ SV46 ) ) )
| ( ( opt @ SV46 )
= SV46 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[232]) ).
thf(275,plain,
! [SV47: $i] :
( ( ( SV47
= ( n @ ( proj1N @ SV47 ) ) )
= $true )
| ( ( ! [SY133: $i] :
( ( opt @ ( x @ SV47 @ SY133 ) )
= ( fail @ SV47 @ SY133 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[233]) ).
thf(276,plain,
! [SV48: $i,SV81: $i] :
( ( ( opt @ ( x @ ( n @ ( s @ SV81 ) ) @ SV48 ) )
= ( fail @ ( n @ ( s @ SV81 ) ) @ SV48 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[234]) ).
thf(277,plain,
! [SV50: $i] :
( ( ( SV50
= ( n @ ( proj1N @ SV50 ) ) )
= $true )
| ( ( ! [SY135: $i] :
( ( opt @ ( y @ SV50 @ SY135 ) )
= ( fail3 @ SV50 @ SY135 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[236]) ).
thf(278,plain,
! [SV51: $i,SV82: $i] :
( ( ( opt @ ( y @ ( n @ ( s @ SV82 ) ) @ SV51 ) )
= ( fail3 @ ( n @ ( s @ SV82 ) ) @ SV51 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[237]) ).
thf(279,plain,
! [SV83: $i,SV58: $i,SV8: $i] :
( ( ( n @ SV8 )
!= ( x @ SV58 @ SV83 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[245]) ).
thf(280,plain,
! [SV84: $i,SV59: $i,SV9: $i] :
( ( ( n @ SV9 )
!= ( y @ SV59 @ SV84 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[246]) ).
thf(281,plain,
! [SV85: $i,SV60: $i,SV11: $i] :
( ( ! [SY143: $i] :
( ( x @ SV11 @ SV60 )
!= ( y @ SV85 @ SY143 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[248]) ).
thf(282,plain,
! [SV61: $i,SV12: $i] :
( ( ( x @ SV12 @ SV61 )
= x2 )
= $false ),
inference(extcnf_not_pos,[status(thm)],[249]) ).
thf(283,plain,
! [SV62: $i,SV13: $i] :
( ( ( y @ SV13 @ SV62 )
= x2 )
= $false ),
inference(extcnf_not_pos,[status(thm)],[250]) ).
thf(284,plain,
! [SV63: $i,SV14: $i] :
( ( ( SV14 != SV63 )
= $true )
| ( ( ( fail2 @ SV14 @ SV63 )
= ( y @ ( n @ ( s @ ( s @ z ) ) ) @ ( opt @ SV14 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[251]) ).
thf(285,plain,
! [SV64: $i,SV15: $i] :
( ( ( SV15 = SV64 )
= $true )
| ( ( ( SV15
= ( x @ ( proj1 @ SV15 ) @ ( proj2 @ SV15 ) ) )
| ( ( fail2 @ SV15 @ SV64 )
= ( x @ ( opt @ SV15 ) @ ( opt @ SV64 ) ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[252]) ).
thf(286,plain,
! [SV16: $i,SV86: $i,SV65: $i] :
( ( ( ( x @ SV65 @ SV86 )
= SV16 )
| ( ( fail2 @ ( x @ SV65 @ SV86 ) @ SV16 )
= ( opt @ ( x @ SV65 @ ( x @ SV86 @ SV16 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[253]) ).
thf(287,plain,
! [SV87: $i,SV17: $i] :
( ( ( ( fail1 @ SV17 @ SV87 )
= ( fail2 @ SV17 @ SV87 ) )
= $true )
| ( ( SV17
= ( n @ ( proj1N @ SV17 ) ) )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[254]) ).
thf(288,plain,
! [SV18: $i,SV88: $i] :
( ( ( ( fail1 @ ( n @ SV88 ) @ SV18 )
= ( fail2 @ ( n @ SV88 ) @ SV18 ) )
= $true )
| ( ( SV18
= ( n @ ( proj1N @ SV18 ) ) )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[255]) ).
thf(289,plain,
! [SV20: $i,SV67: $i] :
( ( ( SV67
= ( n @ ( proj1N @ SV67 ) ) )
= $true )
| ( ( ( fail @ SV20 @ SV67 )
= ( fail1 @ SV20 @ SV67 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[257]) ).
thf(290,plain,
! [SV24: $i] :
( ( ( SV24
= ( y @ ( proj12 @ SV24 ) @ ( proj22 @ SV24 ) ) )
= $true )
| ( ( ! [SY118: $i] :
( ( fail32 @ SV24 @ SY118 )
= ( fail4 @ SV24 @ SY118 ) ) )
= $true )
| ( ( SV24
= ( n @ ( proj1N @ SV24 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[260]) ).
thf(291,plain,
! [SV25: $i,SV89: $i] :
( ( ( ( fail32 @ ( n @ SV89 ) @ SV25 )
= ( fail4 @ ( n @ SV89 ) @ SV25 ) )
= $true )
| ( ( SV25
= ( n @ ( proj1N @ SV25 ) ) )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[261]) ).
thf(292,plain,
! [SV27: $i,SV90: $i,SV71: $i] :
( ( ( fail32 @ ( y @ SV71 @ SV90 ) @ SV27 )
= ( opt @ ( y @ SV71 @ ( y @ SV90 @ SV27 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[263]) ).
thf(293,plain,
! [SV28: $i,SV72: $i] :
( ( ( SV72
= ( n @ ( proj1N @ SV72 ) ) )
= $true )
| ( ( ( fail22 @ SV28 @ SV72 )
= ( fail32 @ SV28 @ SV72 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[264]) ).
thf(294,plain,
! [SV91: $i,SV32: $i] :
( ( ( ( fail12 @ SV32 @ SV91 )
= ( fail22 @ SV32 @ SV91 ) )
= $true )
| ( ( SV32
= ( n @ ( proj1N @ SV32 ) ) )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[266]) ).
thf(295,plain,
! [SV36: $i,SV75: $i] :
( ( ( SV75
= ( n @ ( proj1N @ SV75 ) ) )
= $true )
| ( ( ( fail3 @ SV36 @ SV75 )
= ( fail12 @ SV36 @ SV75 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[268]) ).
thf(296,plain,
! [SV46: $i] :
( ( ( SV46
= ( y @ ( proj12 @ SV46 ) @ ( proj22 @ SV46 ) ) )
= $true )
| ( ( ( opt @ SV46 )
= SV46 )
= $true )
| ( ( SV46
= ( x @ ( proj1 @ SV46 ) @ ( proj2 @ SV46 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[274]) ).
thf(297,plain,
! [SV92: $i,SV47: $i] :
( ( ( ( opt @ ( x @ SV47 @ SV92 ) )
= ( fail @ SV47 @ SV92 ) )
= $true )
| ( ( SV47
= ( n @ ( proj1N @ SV47 ) ) )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[275]) ).
thf(298,plain,
! [SV93: $i,SV50: $i] :
( ( ( ( opt @ ( y @ SV50 @ SV93 ) )
= ( fail3 @ SV50 @ SV93 ) )
= $true )
| ( ( SV50
= ( n @ ( proj1N @ SV50 ) ) )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[277]) ).
thf(299,plain,
! [SV83: $i,SV58: $i,SV8: $i] :
( ( ( n @ SV8 )
= ( x @ SV58 @ SV83 ) )
= $false ),
inference(extcnf_not_pos,[status(thm)],[279]) ).
thf(300,plain,
! [SV84: $i,SV59: $i,SV9: $i] :
( ( ( n @ SV9 )
= ( y @ SV59 @ SV84 ) )
= $false ),
inference(extcnf_not_pos,[status(thm)],[280]) ).
thf(301,plain,
! [SV94: $i,SV85: $i,SV60: $i,SV11: $i] :
( ( ( x @ SV11 @ SV60 )
!= ( y @ SV85 @ SV94 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[281]) ).
thf(302,plain,
! [SV63: $i,SV14: $i] :
( ( ( SV14 = SV63 )
= $false )
| ( ( ( fail2 @ SV14 @ SV63 )
= ( y @ ( n @ ( s @ ( s @ z ) ) ) @ ( opt @ SV14 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[284]) ).
thf(303,plain,
! [SV64: $i,SV15: $i] :
( ( ( SV15
= ( x @ ( proj1 @ SV15 ) @ ( proj2 @ SV15 ) ) )
= $true )
| ( ( ( fail2 @ SV15 @ SV64 )
= ( x @ ( opt @ SV15 ) @ ( opt @ SV64 ) ) )
= $true )
| ( ( SV15 = SV64 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[285]) ).
thf(304,plain,
! [SV16: $i,SV86: $i,SV65: $i] :
( ( ( ( x @ SV65 @ SV86 )
= SV16 )
= $true )
| ( ( ( fail2 @ ( x @ SV65 @ SV86 ) @ SV16 )
= ( opt @ ( x @ SV65 @ ( x @ SV86 @ SV16 ) ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[286]) ).
thf(305,plain,
! [SV95: $i,SV24: $i] :
( ( ( ( fail32 @ SV24 @ SV95 )
= ( fail4 @ SV24 @ SV95 ) )
= $true )
| ( ( SV24
= ( y @ ( proj12 @ SV24 ) @ ( proj22 @ SV24 ) ) )
= $true )
| ( ( SV24
= ( n @ ( proj1N @ SV24 ) ) )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[290]) ).
thf(306,plain,
! [SV94: $i,SV85: $i,SV60: $i,SV11: $i] :
( ( ( x @ SV11 @ SV60 )
= ( y @ SV85 @ SV94 ) )
= $false ),
inference(extcnf_not_pos,[status(thm)],[301]) ).
thf(307,plain,
$false = $true,
inference(fo_atp_e,[status(thm)],[174,306,305,304,303,302,300,299,298,297,296,295,294,293,292,291,289,288,287,283,282,278,276,273,272,271,270,269,267,265,262,259,258,256,247,244,243,242,241,240,239,238,235,231,229,225,224,221,220,217,216,208,189,187]) ).
thf(308,plain,
$false,
inference(solved_all_splits,[solved_all_splits(join,[])],[307]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SWX190+1 : TPTP v9.3.1. Released v9.3.0.
% 0.00/0.04 % Command : leo --cores 7 --timeout 300 --proofoutput 1 --foatp e --atp e=/export/starexec/sandbox2/solver/bin/eprover /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.07/0.17 % Computer : n011.cluster.edu
% 0.07/0.17 % Model : x86_64 x86_64
% 0.07/0.17 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/0.17 % Memory : 8046.5625MB
% 0.07/0.17 % OS : Linux 6.8.0-71-generic
% 0.07/0.17 % CPULimit : 300
% 0.07/0.17 % WCLimit : 300
% 0.07/0.17 % DateTime : Tue Oct 6 17:11:21 UTC 2026
% 0.07/0.18 % CPUTime :
% 0.07/0.18 Running leo --cores 7 --timeout 300 --proofoutput 1 --foatp e --atp e=/export/starexec/sandbox2/solver/bin/eprover /export/starexec/sandbox2/benchmark/theBenchmark.p
% 8.45/1.70
% 8.45/1.70 No.of.Axioms: 53
% 8.45/1.70
% 8.45/1.70 Length.of.Defs: 0
% 8.45/1.70
% 8.45/1.70 Contains.Choice.Funs: false
% 8.45/1.70 .........
% 8.45/1.70
% 8.45/1.70 ********************************
% 8.45/1.70 * All subproblems solved! *
% 8.45/1.70 ********************************
% 8.45/1.70 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : (rf:0,axioms:53,ps:3,u:6,ude:true,rLeibEQ:true,rAndEQ:true,use_choice:true,use_extuni:true,use_extcnf_combined:true,expand_extuni:false,foatp:e,atp_timeout:74,atp_calls_frequency:5,ordering:none,proof_output:1,protocol_output:false,clause_count:307,loop_count:0,foatp_calls:1,translation:fof_experiment)
% 8.45/1.70
% 8.45/1.70 %**** Beginning of derivation protocol ****
% 8.45/1.70 % SZS output start CNFRefutation
% See solution above
% 8.45/1.70
% 8.45/1.70 %**** End of derivation protocol ****
% 8.45/1.70 %**** no. of clauses in derivation: 308 ****
% 8.45/1.70 %**** clause counter: 307 ****
% 8.45/1.70
% 8.45/1.70 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : (rf:0,axioms:53,ps:3,u:6,ude:true,rLeibEQ:true,rAndEQ:true,use_choice:true,use_extuni:true,use_extcnf_combined:true,expand_extuni:false,foatp:e,atp_timeout:74,atp_calls_frequency:5,ordering:none,proof_output:1,protocol_output:false,clause_count:307,loop_count:0,foatp_calls:1,translation:fof_experiment)
%------------------------------------------------------------------------------