%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : NUM662^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n028.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 : Wed Feb 25 09:24:09 AM UTC 2026
% Result : Theorem 1.52s 1.24s
% Output : Refutation 1.52s
% Verified :
% SZS Type : Refutation
% Derivation depth : 8
% Number of leaves : 6
% Syntax : Number of formulae : 23 ( 21 unt; 0 typ; 0 def)
% Number of atoms : 25 ( 24 equ; 0 cnn)
% Maximal formula atoms : 2 ( 1 avg)
% Number of connectives : 99 ( 15 ~; 0 |; 0 &; 82 @)
% ( 0 <=>; 2 =>; 0 <=; 0 <~>)
% Maximal formula depth : 5 ( 3 avg)
% Number of types : 1 ( 1 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 8 ( 6 usr; 6 con; 0-2 aty)
% Number of variables : 25 ( 0 ^; 21 !; 4 ?; 25 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
nat: $tType ).
thf(func_def_0,type,
nat: $tType ).
thf(func_def_1,type,
x: nat ).
thf(func_def_2,type,
y: nat ).
thf(func_def_3,type,
z: nat ).
thf(func_def_4,type,
pl: nat > nat > nat ).
thf(func_def_8,type,
sK0: nat ).
thf(func_def_9,type,
sK1: nat ).
thf(f27,plain,
$false,
inference(equality_resolution,[status(thm)],[f26]) ).
thf(f26,plain,
! [X0: nat] :
( ( pl @ x @ X0 )
!= ( pl @ x @ ( pl @ sK0 @ sK1 ) ) ),
inference(superposition,[status(thm)],[f25,f23]) ).
thf(f23,plain,
! [X2: nat,X0: nat,X1: nat] :
( ( pl @ ( pl @ X0 @ X1 ) @ X2 )
= ( pl @ X0 @ ( pl @ X1 @ X2 ) ) ),
inference(cnf_transformation,[status(thm)],[f19]) ).
thf(f19,plain,
! [X0: nat,X1: nat,X2: nat] :
( ( pl @ ( pl @ X0 @ X1 ) @ X2 )
= ( pl @ X0 @ ( pl @ X1 @ X2 ) ) ),
inference(rectify,[status(thm)],[f12]) ).
thf(f12,plain,
! [X0: nat,X2: nat,X1: nat] :
( ( pl @ ( pl @ X0 @ X2 ) @ X1 )
= ( pl @ X0 @ ( pl @ X2 @ X1 ) ) ),
inference(rectify,[status(thm)],[f4]) ).
thf(f4,axiom,
! [X1: nat,X4: nat,X3: nat] :
( ( pl @ ( pl @ X1 @ X3 ) @ X4 )
= ( pl @ X1 @ ( pl @ X3 @ X4 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.gXOCaPirsl/DTF2THF_860.p',satz5) ).
thf(f25,plain,
! [X0: nat] :
( ( pl @ x @ X0 )
!= ( pl @ ( pl @ x @ sK0 ) @ sK1 ) ),
inference(definition_unfolding,[status(thm)],[f22,f24]) ).
thf(f24,plain,
( z
= ( pl @ ( pl @ x @ sK0 ) @ sK1 ) ),
inference(definition_unfolding,[status(thm)],[f21,f20]) ).
thf(f20,plain,
( y
= ( pl @ x @ sK0 ) ),
inference(cnf_transformation,[status(thm)],[f16]) ).
thf(f16,plain,
( y
= ( pl @ x @ sK0 ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0])],[f13,f15]) ).
thf(f15,plain,
( ? [X0: nat] :
( y
= ( pl @ x @ X0 ) )
=> ( y
= ( pl @ x @ sK0 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f13,plain,
? [X0: nat] :
( y
= ( pl @ x @ X0 ) ),
inference(ennf_transformation,[status(thm)],[f1]) ).
thf(f1,axiom,
~ ! [X0: nat] :
( y
!= ( pl @ x @ X0 ) ),
file('/export/starexec/sandbox2/tmp/tmp.gXOCaPirsl/DTF2THF_860.p',l) ).
thf(f21,plain,
( z
= ( pl @ y @ sK1 ) ),
inference(cnf_transformation,[status(thm)],[f18]) ).
thf(f18,plain,
( z
= ( pl @ y @ sK1 ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK1])],[f14,f17]) ).
thf(f17,plain,
( ? [X0: nat] :
( z
= ( pl @ y @ X0 ) )
=> ( z
= ( pl @ y @ sK1 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f14,plain,
? [X0: nat] :
( z
= ( pl @ y @ X0 ) ),
inference(ennf_transformation,[status(thm)],[f11]) ).
thf(f11,plain,
~ ! [X0: nat] :
( z
!= ( pl @ y @ X0 ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
~ ! [X1: nat] :
( z
!= ( pl @ y @ X1 ) ),
file('/export/starexec/sandbox2/tmp/tmp.gXOCaPirsl/DTF2THF_860.p',k) ).
thf(f22,plain,
! [X0: nat] :
( ( pl @ x @ X0 )
!= z ),
inference(cnf_transformation,[status(thm)],[f10]) ).
thf(f10,plain,
! [X0: nat] :
( ( pl @ x @ X0 )
!= z ),
inference(flattening,[status(thm)],[f6]) ).
thf(f6,negated_conjecture,
~ ~ ! [X0: nat] :
( ( pl @ x @ X0 )
!= z ),
inference(negated_conjecture,[status(cth)],[f5]) ).
thf(f5,conjecture,
~ ! [X0: nat] :
( ( pl @ x @ X0 )
!= z ),
file('/export/starexec/sandbox2/tmp/tmp.gXOCaPirsl/DTF2THF_860.p',satz15) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : NUM662^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n028.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 11:09:57 EST 2026
% 0.16/0.33 % CPUTime :
% 0.16/0.33 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.19/0.44 ---- Original DTF file ---
% 0.19/0.44 thf(spec,logic,$$dhol).
% 0.19/0.44 %------------------------------------------------------------------------------
% 0.19/0.44 % File : NUM662^1 : TPTP v9.2.1. Released v3.7.0.
% 0.19/0.44 % Domain : Number Theory
% 0.19/0.44 % Problem : Landau theorem 15
% 0.19/0.44 % Version : Especial.
% 0.19/0.44 % English : ~(forall x_0:nat.~(z = pl x x_0))
% 0.19/0.44
% 0.19/0.44 % Refs : [Lan30] Landau (1930), Grundlagen der Analysis
% 0.19/0.44 % : [vBJ79] van Benthem Jutting (1979), Checking Landau's "Grundla
% 0.19/0.44 % : [Bro09] Brown (2009), Email to Geoff Sutcliffe
% 0.19/0.44 % Source : [Bro09]
% 0.19/0.44 % Names : satz15 [Lan30]
% 0.19/0.44
% 0.19/0.44 % Status : Theorem
% 0.19/0.44 % : Without extensionality : Theorem
% 0.19/0.44 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.30 v8.2.0, 0.31 v8.1.0, 0.09 v7.5.0, 0.00 v7.4.0, 0.22 v7.2.0, 0.00 v7.1.0, 0.12 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.14 v5.5.0, 0.33 v5.4.0, 0.20 v5.3.0, 0.40 v4.1.0, 0.00 v4.0.0, 0.33 v3.7.0
% 0.19/0.44 % Syntax : Number of formulae : 10 ( 4 unt; 5 typ; 0 def)
% 0.19/0.44 % Number of atoms : 4 ( 4 equ; 0 cnn)
% 0.19/0.44 % Maximal formula atoms : 1 ( 0 avg)
% 0.19/0.44 % Number of connectives : 23 ( 8 ~; 0 |; 0 &; 14 @)
% 0.19/0.44 % ( 0 <=>; 1 =>; 0 <=; 0 <~>)
% 0.19/0.44 % Maximal formula depth : 5 ( 4 avg)
% 0.19/0.44 % Number of types : 2 ( 1 usr)
% 0.19/0.44 % Number of type conns : 2 ( 2 >; 0 *; 0 +; 0 <<)
% 0.19/0.44 % Number of symbols : 5 ( 4 usr; 3 con; 0-2 aty)
% 0.19/0.44 % Number of variables : 7 ( 0 ^; 7 !; 0 ?; 7 :)
% 0.19/0.44 % SPC : TH0_THM_EQU_NAR
% 0.19/0.44
% 0.19/0.44 % Comments :
% 0.19/0.44 %------------------------------------------------------------------------------
% 0.19/0.44 thf(nat_type,type,
% 0.19/0.44 nat: $tType ).
% 0.19/0.44
% 0.19/0.44 thf(x,type,
% 0.19/0.44 x: nat ).
% 0.19/0.44
% 0.19/0.44 thf(y,type,
% 0.19/0.44 y: nat ).
% 0.19/0.44
% 0.19/0.44 thf(z,type,
% 0.19/0.44 z: nat ).
% 0.19/0.44
% 0.19/0.44 thf(pl,type,
% 0.19/0.44 pl: nat > nat > nat ).
% 0.19/0.44
% 0.19/0.44 thf(l,axiom,
% 0.19/0.44 ~ ! [Xx_0: nat] :
% 0.19/0.44 ( y
% 0.19/0.44 != ( pl @ x @ Xx_0 ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(k,axiom,
% 0.19/0.44 ~ ! [Xx: nat] :
% 0.19/0.44 ( z
% 0.19/0.44 != ( pl @ y @ Xx ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(et,axiom,
% 0.19/0.44 ! [Xa: $o] :
% 0.19/0.44 ( ~ ~ Xa
% 0.19/0.44 => Xa ) ).
% 0.19/0.44
% 0.19/0.44 thf(satz5,axiom,
% 0.19/0.44 ! [Xx: nat,Xy: nat,Xz: nat] :
% 0.19/0.44 ( ( pl @ ( pl @ Xx @ Xy ) @ Xz )
% 0.19/0.44 = ( pl @ Xx @ ( pl @ Xy @ Xz ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(satz15,conjecture,
% 0.19/0.44 ~ ! [Xx_0: nat] :
% 0.19/0.44 ( z
% 0.19/0.44 != ( pl @ x @ Xx_0 ) ) ).
% 0.19/0.44
% 0.19/0.44 %------------------------------------------------------------------------------
% 0.19/0.44 ------------------------
% 1.52/1.21 ---- Embedded in THF ---
% 1.52/1.21 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.52/1.21 %%% Generated on Tue Feb 24 11:09:57 EST 2026
% 1.52/1.21 %%% using '$$dhol' embedding, version 1.3.0.
% 1.52/1.21 %%% Logic specification used:
% 1.52/1.21 %%% thf(spec, logic, $$dhol).
% 1.52/1.21
% 1.52/1.21 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.gXOCaPirsl/DTF2DTF860.p
% See solution above
% 1.52/1.21 ------------------------
% 1.52/1.22 ---- Cleaned THF ---
% 1.52/1.22 thf(nat_type,type,
% 1.52/1.22 nat: $tType ).
% 1.52/1.22
% 1.52/1.22 thf(x,type,
% 1.52/1.22 x: nat ).
% 1.52/1.22
% 1.52/1.22 thf(y,type,
% 1.52/1.22 y: nat ).
% 1.52/1.22
% 1.52/1.22 thf(z,type,
% 1.52/1.22 z: nat ).
% 1.52/1.22
% 1.52/1.22 thf(pl,type,
% 1.52/1.22 pl: nat > nat > nat ).
% 1.52/1.22
% 1.52/1.22 thf(l,axiom,
% 1.52/1.22 ~ ! [Xx_0: nat] :
% 1.52/1.22 ( y
% 1.52/1.22 != ( pl @ x @ Xx_0 ) ) ).
% 1.52/1.22
% 1.52/1.22 thf(k,axiom,
% 1.52/1.22 ~ ! [Xx: nat] :
% 1.52/1.22 ( z
% 1.52/1.22 != ( pl @ y @ Xx ) ) ).
% 1.52/1.22
% 1.52/1.22 thf(et,axiom,
% 1.52/1.22 ! [Xa: $o] :
% 1.52/1.22 ( ~ ~ Xa
% 1.52/1.22 => Xa ) ).
% 1.52/1.22
% 1.52/1.22 thf(satz5,axiom,
% 1.52/1.22 ! [Xx: nat,Xy: nat,Xz: nat] :
% 1.52/1.22 ( ( pl @ ( pl @ Xx @ Xy ) @ Xz )
% 1.52/1.22 = ( pl @ Xx @ ( pl @ Xy @ Xz ) ) ) ).
% 1.52/1.22
% 1.52/1.22 thf(satz15,conjecture,
% 1.52/1.22 ~ ! [Xx_0: nat] :
% 1.52/1.22 ( z
% 1.52/1.22 != ( pl @ x @ Xx_0 ) ) ).
% 1.52/1.22 ------------------------
% 1.52/1.24 % (1128)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_860 for (2999ds/275Mi)
% 1.52/1.24 % (1127)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_860 for (2999ds/2Mi)
% 1.52/1.24 % (1128)First to succeed.
% 1.52/1.24 % (1127)Instruction limit reached!
% 1.52/1.24 % (1127)------------------------------
% 1.52/1.24 % (1127)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.52/1.24 % (1127)Termination reason: Unknown
% 1.52/1.24 % (1127)Termination phase: Saturation
% 1.52/1.24
% 1.52/1.24 % (1127)Memory used [KB]: 1023
% 1.52/1.24 % (1127)Time elapsed: 0.002 s
% 1.52/1.24 % (1127)Instructions burned: 2 (million)
% 1.52/1.24 % (1127)------------------------------
% 1.52/1.24 % (1127)------------------------------
% 1.52/1.24 % (1128)Refutation found. Thanks to Tanya!
% 1.52/1.24 % SZS status Theorem for DTF2THF_860
% 1.52/1.24 % SZS output start Proof for DTF2THF_860
% 1.52/1.24 thf(type_def_5, type, nat: $tType).
% 1.52/1.24 thf(func_def_0, type, nat: $tType).
% 1.52/1.24 thf(func_def_1, type, x: nat).
% 1.52/1.24 thf(func_def_2, type, y: nat).
% 1.52/1.24 thf(func_def_3, type, z: nat).
% 1.52/1.24 thf(func_def_4, type, pl: nat > nat > nat).
% 1.52/1.24 thf(func_def_8, type, sK0: nat).
% 1.52/1.24 thf(func_def_9, type, sK1: nat).
% 1.52/1.24 thf(f27,plain,(
% 1.52/1.24 $false),
% 1.52/1.24 inference(equality_resolution,[status(thm)],[f26])).
% 1.52/1.24 thf(f26,plain,(
% 1.52/1.24 ( ! [X0 : nat] : (((pl @ x @ X0) != (pl @ x @ (pl @ sK0 @ sK1)))) )),
% 1.52/1.24 inference(superposition,[status(thm)],[f25,f23])).
% 1.52/1.24 thf(f23,plain,(
% 1.52/1.24 ( ! [X2 : nat,X0 : nat,X1 : nat] : (((pl @ (pl @ X0 @ X1) @ X2) = (pl @ X0 @ (pl @ X1 @ X2)))) )),
% 1.52/1.24 inference(cnf_transformation,[status(thm)],[f19])).
% 1.52/1.24 thf(f19,plain,(
% 1.52/1.24 ! [X0 : nat,X1 : nat,X2 : nat] : ((pl @ (pl @ X0 @ X1) @ X2) = (pl @ X0 @ (pl @ X1 @ X2)))),
% 1.52/1.24 inference(rectify,[status(thm)],[f12])).
% 1.52/1.24 thf(f12,plain,(
% 1.52/1.24 ! [X0 : nat,X2 : nat,X1 : nat] : ((pl @ (pl @ X0 @ X2) @ X1) = (pl @ X0 @ (pl @ X2 @ X1)))),
% 1.52/1.24 inference(rectify,[status(thm)],[f4])).
% 1.52/1.24 thf(f4,axiom,(
% 1.52/1.24 ! [X1 : nat,X4 : nat,X3 : nat] : ((pl @ (pl @ X1 @ X3) @ X4) = (pl @ X1 @ (pl @ X3 @ X4)))),
% 1.52/1.24 file('/export/starexec/sandbox2/tmp/tmp.gXOCaPirsl/DTF2THF_860.p',satz5)).
% 1.52/1.24 thf(f25,plain,(
% 1.52/1.24 ( ! [X0 : nat] : (((pl @ x @ X0) != (pl @ (pl @ x @ sK0) @ sK1))) )),
% 1.52/1.24 inference(definition_unfolding,[status(thm)],[f22,f24])).
% 1.52/1.24 thf(f24,plain,(
% 1.52/1.24 (z = (pl @ (pl @ x @ sK0) @ sK1))),
% 1.52/1.24 inference(definition_unfolding,[status(thm)],[f21,f20])).
% 1.52/1.24 thf(f20,plain,(
% 1.52/1.24 (y = (pl @ x @ sK0))),
% 1.52/1.24 inference(cnf_transformation,[status(thm)],[f16])).
% 1.52/1.24 thf(f16,plain,(
% 1.52/1.24 (y = (pl @ x @ sK0))),
% 1.52/1.24 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0])],[f13,f15])).
% 1.52/1.24 thf(f15,plain,(
% 1.52/1.24 ? [X0 : nat] : (y = (pl @ x @ X0)) => (y = (pl @ x @ sK0))),
% 1.52/1.24 introduced(definition,[],[choice_axiom])).
% 1.52/1.24 thf(f13,plain,(
% 1.52/1.24 ? [X0 : nat] : (y = (pl @ x @ X0))),
% 1.52/1.24 inference(ennf_transformation,[status(thm)],[f1])).
% 1.52/1.24 thf(f1,axiom,(
% 1.52/1.24 ~! [X0 : nat] : (y != (pl @ x @ X0))),
% 1.52/1.24 file('/export/starexec/sandbox2/tmp/tmp.gXOCaPirsl/DTF2THF_860.p',l)).
% 1.52/1.24 thf(f21,plain,(
% 1.52/1.24 (z = (pl @ y @ sK1))),
% 1.52/1.24 inference(cnf_transformation,[status(thm)],[f18])).
% 1.52/1.24 thf(f18,plain,(
% 1.52/1.24 (z = (pl @ y @ sK1))),
% 1.52/1.24 inference(skolemisation,[status(esa),new_symbols(skolem,[sK1])],[f14,f17])).
% 1.52/1.24 thf(f17,plain,(
% 1.52/1.24 ? [X0 : nat] : (z = (pl @ y @ X0)) => (z = (pl @ y @ sK1))),
% 1.52/1.24 introduced(definition,[],[choice_axiom])).
% 1.52/1.24 thf(f14,plain,(
% 1.52/1.24 ? [X0 : nat] : (z = (pl @ y @ X0))),
% 1.52/1.24 inference(ennf_transformation,[status(thm)],[f11])).
% 1.52/1.24 thf(f11,plain,(
% 1.52/1.24 ~! [X0 : nat] : (z != (pl @ y @ X0))),
% 1.52/1.24 inference(rectify,[status(thm)],[f2])).
% 1.52/1.24 thf(f2,axiom,(
% 1.52/1.24 ~! [X1 : nat] : (z != (pl @ y @ X1))),
% 1.52/1.24 file('/export/starexec/sandbox2/tmp/tmp.gXOCaPirsl/DTF2THF_860.p',k)).
% 1.52/1.24 thf(f22,plain,(
% 1.52/1.24 ( ! [X0 : nat] : (((pl @ x @ X0) != z)) )),
% 1.52/1.24 inference(cnf_transformation,[status(thm)],[f10])).
% 1.52/1.24 thf(f10,plain,(
% 1.52/1.24 ! [X0 : nat] : ((pl @ x @ X0) != z)),
% 1.52/1.24 inference(flattening,[status(thm)],[f6])).
% 1.52/1.24 thf(f6,negated_conjecture,(
% 1.52/1.24 ~~! [X0 : nat] : ((pl @ x @ X0) != z)),
% 1.52/1.24 inference(negated_conjecture,[status(cth)],[f5])).
% 1.52/1.24 thf(f5,conjecture,(
% 1.52/1.24 ~! [X0 : nat] : ((pl @ x @ X0) != z)),
% 1.52/1.24 file('/export/starexec/sandbox2/tmp/tmp.gXOCaPirsl/DTF2THF_860.p',satz15)).
% 1.52/1.24 % SZS output end Proof for DTF2THF_860
% 1.52/1.24 % (1128)------------------------------
% 1.52/1.24 % (1128)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.52/1.24 % (1128)Termination reason: Refutation
% 1.52/1.24
% 1.52/1.24 % (1128)Memory used [KB]: 5500
% 1.52/1.24 % (1128)Time elapsed: 0.003 s
% 1.52/1.24 % (1128)Instructions burned: 1 (million)
% 1.52/1.24 % (1128)------------------------------
% 1.52/1.24 % (1128)------------------------------
% 1.52/1.24 % (1122)Success in time 0.012 s
%------------------------------------------------------------------------------