%------------------------------------------------------------------------------
% File : Lash---1.13
% Problem : ITP134^1 : TPTP v9.2.0. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : lash -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 01:45:12 PM UTC 2025
% Result : Theorem 20.29s 20.54s
% Output : Proof 20.29s
% Verified :
% SZS Type : ERROR: Analysing output (MakeTreeStats fails)
% Comments :
%------------------------------------------------------------------------------
thf(ty_nat,type,
nat: $tType ).
thf(ty_set_nat_nat,type,
set_nat_nat: $tType ).
thf(ty_collect_nat_nat,type,
collect_nat_nat: ( ( nat > nat ) > $o ) > set_nat_nat ).
thf(ty_one_one_nat,type,
one_one_nat: nat ).
thf(ty_number1551313001itions,type,
number1551313001itions: ( nat > nat ) > nat > $o ).
thf(ty_zero_zero_nat,type,
zero_zero_nat: nat ).
thf(ty_ord_le1415039317at_nat,type,
ord_le1415039317at_nat: set_nat_nat > set_nat_nat > $o ).
thf(ty_plus_plus_nat,type,
plus_plus_nat: nat > nat > nat ).
thf(ty_finite570312790at_nat,type,
finite570312790at_nat: set_nat_nat > $o ).
thf(ty_ord_less_eq_nat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(ty_n,type,
n: nat ).
thf(sP1,plain,
( sP1
<=> ( ord_le1415039317at_nat
@ ( collect_nat_nat
@ ^ [X1: nat > nat] : ( number1551313001itions @ X1 @ n ) )
@ ( collect_nat_nat
@ ^ [X1: nat > nat] :
~ ( ! [X2: nat] : ( ord_less_eq_nat @ ( X1 @ X2 ) @ n )
=> ~ ! [X2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ n @ one_one_nat ) @ X2 )
=> ( ( X1 @ X2 )
= zero_zero_nat ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ( sP1
=> ( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [X1: nat > nat] : ( number1551313001itions @ X1 @ n ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ! [X1: set_nat_nat,X2: set_nat_nat] :
( ( finite570312790at_nat @ X1 )
=> ( ( ord_le1415039317at_nat @ X2 @ X1 )
=> ( finite570312790at_nat @ X2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ( ( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [X1: nat > nat] :
~ ( ! [X2: nat] : ( ord_less_eq_nat @ ( X1 @ X2 ) @ n )
=> ~ ! [X2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ n @ one_one_nat ) @ X2 )
=> ( ( X1 @ X2 )
= zero_zero_nat ) ) ) ) )
=> sP2 ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [X1: nat > nat] : ( number1551313001itions @ X1 @ n ) ) ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(sP6,plain,
( sP6
<=> ( finite570312790at_nat
@ ( collect_nat_nat
@ ^ [X1: nat > nat] :
~ ( ! [X2: nat] : ( ord_less_eq_nat @ ( X1 @ X2 ) @ n )
=> ~ ! [X2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ n @ one_one_nat ) @ X2 )
=> ( ( X1 @ X2 )
= zero_zero_nat ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP6])]) ).
thf(sP7,plain,
( sP7
<=> ! [X1: set_nat_nat] :
( sP6
=> ( ( ord_le1415039317at_nat @ X1
@ ( collect_nat_nat
@ ^ [X2: nat > nat] :
~ ( ! [X3: nat] : ( ord_less_eq_nat @ ( X2 @ X3 ) @ n )
=> ~ ! [X3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ n @ one_one_nat ) @ X3 )
=> ( ( X2 @ X3 )
= zero_zero_nat ) ) ) ) )
=> ( finite570312790at_nat @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
thf(conj_0,conjecture,
sP5 ).
thf(h0,negated_conjecture,
~ sP5,
inference(assume_negation,[status(cth)],[conj_0]) ).
thf(1,plain,
( ~ sP2
| ~ sP1
| sP5 ),
inference(prop_rule,[status(thm)],[]) ).
thf(2,plain,
( ~ sP4
| ~ sP6
| sP2 ),
inference(prop_rule,[status(thm)],[]) ).
thf(3,plain,
( ~ sP7
| sP4 ),
inference(all_rule,[status(thm)],[]) ).
thf(4,plain,
( ~ sP3
| sP7 ),
inference(all_rule,[status(thm)],[]) ).
thf(fact_62_rev__finite__subset,axiom,
sP3 ).
thf(fact_1__092_060open_062_123p_O_Ap_Apartitions_An_125_A_092_060subseteq_062_A_123f_O_A_I_092_060forall_062i_O_Af_Ai_A_092_060le_062_An_J_A_092_060and_062_A_I_092_060forall_062i_092_060ge_062n_A_L_A1_O_Af_Ai_A_061_A0_J_125_092_060close_062,axiom,
sP1 ).
thf(fact_0__092_060open_062finite_A_123f_O_A_I_092_060forall_062i_O_Af_Ai_A_092_060le_062_An_J_A_092_060and_062_A_I_092_060forall_062i_092_060ge_062n_A_L_A1_O_Af_Ai_A_061_A0_J_125_092_060close_062,axiom,
sP6 ).
thf(5,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h0])],[1,2,3,4,h0,fact_62_rev__finite__subset,fact_1__092_060open_062_123p_O_Ap_Apartitions_An_125_A_092_060subseteq_062_A_123f_O_A_I_092_060forall_062i_O_Af_Ai_A_092_060le_062_An_J_A_092_060and_062_A_I_092_060forall_062i_092_060ge_062n_A_L_A1_O_Af_Ai_A_061_A0_J_125_092_060close_062,fact_0__092_060open_062finite_A_123f_O_A_I_092_060forall_062i_O_Af_Ai_A_092_060le_062_An_J_A_092_060and_062_A_I_092_060forall_062i_092_060ge_062n_A_L_A1_O_Af_Ai_A_061_A0_J_125_092_060close_062]) ).
thf(0,theorem,
sP5,
inference(contra,[status(thm),contra(discharge,[h0])],[5,h0]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.13 % Problem : ITP134^1 : TPTP v9.2.0. Released v7.5.0.
% 0.12/0.14 % Command : lash -P picomus -M modes -p tstp -t %d %s
% 0.14/0.35 % Computer : n016.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Wed Oct 8 18:30:38 EDT 2025
% 0.14/0.36 % CPUTime :
% 20.29/20.54 % SZS status Theorem
% 20.29/20.54 % Mode: cade22sinegrackle2xfaf3
% 20.29/20.54 % Steps: 865
% 20.29/20.54 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------