%------------------------------------------------------------------------------
% File : Lash---1.13
% Problem : ALG253^2 : TPTP v9.2.0. Bugfixed v5.2.0.
% Transfm : none
% Format : tptp:raw
% Command : lash -P picomus -M modes -p tstp -t %d %s
% Computer : n011.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 01:11:34 PM UTC 2025
% Result : Theorem 0.17s 0.53s
% Output : Proof 0.17s
% Verified :
% SZS Type : Refutation
% Derivation depth : 10
% Number of leaves : 55
% Syntax : Number of formulae : 67 ( 23 unt; 0 typ; 5 def)
% Number of atoms : 154 ( 9 equ; 0 cnn)
% Maximal formula atoms : 5 ( 2 avg)
% Number of connectives : 417 ( 30 ~; 22 |; 0 &; 245 @)
% ( 21 <=>; 98 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 4 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 9 ( 9 >; 0 *; 0 +; 0 <<)
% Number of symbols : 38 ( 35 usr; 31 con; 0-2 aty)
% ( 1 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 114 ( 37 ^; 77 !; 0 ?; 114 :)
% Comments :
%------------------------------------------------------------------------------
thf(ty_subst,type,
subst: $tType ).
thf(ty_term,type,
term: $tType ).
thf(ty_eigen__1,type,
eigen__1: term ).
thf(ty_id,type,
id: subst ).
thf(ty_lam,type,
lam: term > term ).
thf(ty_sub,type,
sub: term > subst > term ).
thf(ty_eigen__0,type,
eigen__0: term > $o ).
thf(ty_push,type,
push: term > subst > subst ).
thf(ty_var,type,
var: term > $o ).
thf(ty_eigen__6,type,
eigen__6: term ).
thf(ty_ap,type,
ap: term > term > term ).
thf(h0,assumption,
! [X1: term > $o,X2: term] :
( ( X1 @ X2 )
=> ( X1 @ ( eps__0 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(eigendef_eigen__6,definition,
( eigen__6
= ( eps__0
@ ^ [X1: term] :
~ ( ( var @ X1 )
=> ( eigen__0 @ ( sub @ X1 @ id ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__6])]) ).
thf(sP1,plain,
( sP1
<=> ! [X1: term] :
( ! [X2: term] :
( ( eigen__0 @ X2 )
=> ( eigen__0 @ ( sub @ X1 @ ( push @ X2 @ id ) ) ) )
=> ( eigen__0 @ ( lam @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ! [X1: term,X2: subst] :
( ! [X3: term] :
( ( var @ X3 )
=> ( eigen__0 @ ( sub @ X3 @ X2 ) ) )
=> ( eigen__0 @ ( sub @ X1 @ X2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ( ( var @ eigen__6 )
=> ( eigen__0 @ ( sub @ eigen__6 @ id ) ) ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ( eigen__0 @ ( sub @ eigen__6 @ id ) ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ! [X1: term] :
( ( var @ X1 )
=> ( eigen__0 @ ( sub @ X1 @ id ) ) ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(sP6,plain,
( sP6
<=> ! [X1: term > $o] :
( ! [X2: term,X3: term] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X1 @ ( ap @ X2 @ X3 ) ) ) )
=> ( ! [X2: term] :
( ! [X3: term] :
( ( X1 @ X3 )
=> ( X1 @ ( sub @ X2 @ ( push @ X3 @ id ) ) ) )
=> ( X1 @ ( lam @ X2 ) ) )
=> ! [X2: term,X3: subst] :
( ! [X4: term] :
( ( var @ X4 )
=> ( X1 @ ( sub @ X4 @ X3 ) ) )
=> ( X1 @ ( sub @ X2 @ X3 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP6])]) ).
thf(sP7,plain,
( sP7
<=> ( sP1
=> sP2 ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
thf(sP8,plain,
( sP8
<=> ( eigen__0 @ ( sub @ eigen__1 @ id ) ) ),
introduced(definition,[new_symbols(definition,[sP8])]) ).
thf(sP9,plain,
( sP9
<=> ! [X1: term] :
( ( sub @ X1 @ id )
= X1 ) ),
introduced(definition,[new_symbols(definition,[sP9])]) ).
thf(sP10,plain,
( sP10
<=> ( ( var @ eigen__6 )
=> ( eigen__0 @ eigen__6 ) ) ),
introduced(definition,[new_symbols(definition,[sP10])]) ).
thf(sP11,plain,
( sP11
<=> ( var @ eigen__6 ) ),
introduced(definition,[new_symbols(definition,[sP11])]) ).
thf(sP12,plain,
( sP12
<=> ( sP5
=> sP8 ) ),
introduced(definition,[new_symbols(definition,[sP12])]) ).
thf(sP13,plain,
( sP13
<=> ( ! [X1: term,X2: term] :
( ( eigen__0 @ X1 )
=> ( ( eigen__0 @ X2 )
=> ( eigen__0 @ ( ap @ X1 @ X2 ) ) ) )
=> sP7 ) ),
introduced(definition,[new_symbols(definition,[sP13])]) ).
thf(sP14,plain,
( sP14
<=> ! [X1: term,X2: term] :
( ( eigen__0 @ X1 )
=> ( ( eigen__0 @ X2 )
=> ( eigen__0 @ ( ap @ X1 @ X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP14])]) ).
thf(sP15,plain,
( sP15
<=> ( eigen__0 @ eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP15])]) ).
thf(sP16,plain,
( sP16
<=> ( eigen__6
= ( sub @ eigen__6 @ id ) ) ),
introduced(definition,[new_symbols(definition,[sP16])]) ).
thf(sP17,plain,
( sP17
<=> ( ( sub @ eigen__1 @ id )
= eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP17])]) ).
thf(sP18,plain,
( sP18
<=> ! [X1: subst] :
( ! [X2: term] :
( ( var @ X2 )
=> ( eigen__0 @ ( sub @ X2 @ X1 ) ) )
=> ( eigen__0 @ ( sub @ eigen__1 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP18])]) ).
thf(sP19,plain,
( sP19
<=> ! [X1: term] :
( ( var @ X1 )
=> ( eigen__0 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP19])]) ).
thf(sP20,plain,
( sP20
<=> ( eigen__0 @ eigen__6 ) ),
introduced(definition,[new_symbols(definition,[sP20])]) ).
thf(sP21,plain,
( sP21
<=> ( ( sub @ eigen__6 @ id )
= eigen__6 ) ),
introduced(definition,[new_symbols(definition,[sP21])]) ).
thf(def_axvarid,definition,
axvarid = sP9 ).
thf(def_induction2lem,definition,
( induction2lem
= ( ! [X1: term > $o] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ! [X2: term,X3: term] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X1 @ X2 )
@ ( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X1 @ X3 )
@ ( X1 @ ( ap @ X2 @ X3 ) ) ) )
@ ( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ! [X2: term] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ! [X3: term] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X1 @ X3 )
@ ( X1 @ ( sub @ X2 @ ( push @ X3 @ id ) ) ) )
@ ( X1 @ ( lam @ X2 ) ) )
@ ! [X2: term,X3: subst] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ! [X4: term] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( var @ X4 )
@ ( X1 @ ( sub @ X4 @ X3 ) ) )
@ ( X1 @ ( sub @ X2 @ X3 ) ) ) ) ) ) ) ).
thf(def_induction2,definition,
( induction2
= ( ! [X1: term > $o] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ! [X2: term] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( var @ X2 )
@ ( X1 @ X2 ) )
@ ( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ! [X2: term,X3: term] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X1 @ X2 )
@ ( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X1 @ X3 )
@ ( X1 @ ( ap @ X2 @ X3 ) ) ) )
@ ( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ! [X2: term] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ! [X3: term] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X1 @ X3 )
@ ( X1 @ ( sub @ X2 @ ( push @ X3 @ id ) ) ) )
@ ( X1 @ ( lam @ X2 ) ) )
@ ! [X2: term] : ( X1 @ X2 ) ) ) ) ) ) ).
thf(def_induction2_lthm,definition,
( induction2_lthm
= ( ^ [X1: $o,X2: $o] :
( X1
=> X2 )
@ axvarid
@ ( ^ [X1: $o,X2: $o] :
( X1
=> X2 )
@ induction2lem
@ induction2 ) ) ) ).
thf(thm,conjecture,
( sP9
=> ( sP6
=> ! [X1: term > $o] :
( ! [X2: term] :
( ( var @ X2 )
=> ( X1 @ X2 ) )
=> ( ! [X2: term,X3: term] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X1 @ ( ap @ X2 @ X3 ) ) ) )
=> ( ! [X2: term] :
( ! [X3: term] :
( ( X1 @ X3 )
=> ( X1 @ ( sub @ X2 @ ( push @ X3 @ id ) ) ) )
=> ( X1 @ ( lam @ X2 ) ) )
=> ! [X2: term] : ( X1 @ X2 ) ) ) ) ) ) ).
thf(h1,negated_conjecture,
~ ( sP9
=> ( sP6
=> ! [X1: term > $o] :
( ! [X2: term] :
( ( var @ X2 )
=> ( X1 @ X2 ) )
=> ( ! [X2: term,X3: term] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X1 @ ( ap @ X2 @ X3 ) ) ) )
=> ( ! [X2: term] :
( ! [X3: term] :
( ( X1 @ X3 )
=> ( X1 @ ( sub @ X2 @ ( push @ X3 @ id ) ) ) )
=> ( X1 @ ( lam @ X2 ) ) )
=> ! [X2: term] : ( X1 @ X2 ) ) ) ) ) ),
inference(assume_negation,[status(cth)],[thm]) ).
thf(h2,assumption,
sP9,
introduced(assumption,[]) ).
thf(h3,assumption,
~ ( sP6
=> ! [X1: term > $o] :
( ! [X2: term] :
( ( var @ X2 )
=> ( X1 @ X2 ) )
=> ( ! [X2: term,X3: term] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X1 @ ( ap @ X2 @ X3 ) ) ) )
=> ( ! [X2: term] :
( ! [X3: term] :
( ( X1 @ X3 )
=> ( X1 @ ( sub @ X2 @ ( push @ X3 @ id ) ) ) )
=> ( X1 @ ( lam @ X2 ) ) )
=> ! [X2: term] : ( X1 @ X2 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h4,assumption,
sP6,
introduced(assumption,[]) ).
thf(h5,assumption,
~ ! [X1: term > $o] :
( ! [X2: term] :
( ( var @ X2 )
=> ( X1 @ X2 ) )
=> ( ! [X2: term,X3: term] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X1 @ ( ap @ X2 @ X3 ) ) ) )
=> ( ! [X2: term] :
( ! [X3: term] :
( ( X1 @ X3 )
=> ( X1 @ ( sub @ X2 @ ( push @ X3 @ id ) ) ) )
=> ( X1 @ ( lam @ X2 ) ) )
=> ! [X2: term] : ( X1 @ X2 ) ) ) ),
introduced(assumption,[]) ).
thf(h6,assumption,
~ ( sP19
=> ( sP14
=> ( sP1
=> ( !! @ eigen__0 ) ) ) ),
introduced(assumption,[]) ).
thf(h7,assumption,
sP19,
introduced(assumption,[]) ).
thf(h8,assumption,
~ ( sP14
=> ( sP1
=> ! [X1: term] : ( eigen__0 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(h9,assumption,
sP14,
introduced(assumption,[]) ).
thf(h10,assumption,
~ ( sP1
=> ! [X1: term] : ( eigen__0 @ X1 ) ),
introduced(assumption,[]) ).
thf(h11,assumption,
sP1,
introduced(assumption,[]) ).
thf(h12,assumption,
~ ! [X1: term] : ( eigen__0 @ X1 ),
introduced(assumption,[]) ).
thf(h13,assumption,
~ sP15,
introduced(assumption,[]) ).
thf(1,plain,
( ~ sP20
| sP4
| ~ sP16 ),
inference(mating_rule,[status(thm)],[]) ).
thf(2,plain,
( ~ sP10
| ~ sP11
| sP20 ),
inference(prop_rule,[status(thm)],[]) ).
thf(3,plain,
( ~ sP21
| sP16 ),
inference(symeq,[status(thm)],[]) ).
thf(4,plain,
( ~ sP19
| sP10 ),
inference(all_rule,[status(thm)],[]) ).
thf(5,plain,
( ~ sP9
| sP21 ),
inference(all_rule,[status(thm)],[]) ).
thf(6,plain,
( ~ sP8
| sP15
| ~ sP17 ),
inference(mating_rule,[status(thm)],[]) ).
thf(7,plain,
( sP3
| ~ sP4 ),
inference(prop_rule,[status(thm)],[]) ).
thf(8,plain,
( sP3
| sP11 ),
inference(prop_rule,[status(thm)],[]) ).
thf(9,plain,
( sP5
| ~ sP3 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__6]) ).
thf(10,plain,
( ~ sP12
| ~ sP5
| sP8 ),
inference(prop_rule,[status(thm)],[]) ).
thf(11,plain,
( ~ sP18
| sP12 ),
inference(all_rule,[status(thm)],[]) ).
thf(12,plain,
( ~ sP2
| sP18 ),
inference(all_rule,[status(thm)],[]) ).
thf(13,plain,
( ~ sP7
| ~ sP1
| sP2 ),
inference(prop_rule,[status(thm)],[]) ).
thf(14,plain,
( ~ sP13
| ~ sP14
| sP7 ),
inference(prop_rule,[status(thm)],[]) ).
thf(15,plain,
( ~ sP6
| sP13 ),
inference(all_rule,[status(thm)],[]) ).
thf(16,plain,
( ~ sP9
| sP17 ),
inference(all_rule,[status(thm)],[]) ).
thf(17,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h13,h11,h12,h9,h10,h7,h8,h6,h4,h5,h2,h3,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,h2,h4,h7,h9,h11,h13]) ).
thf(18,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h11,h12,h9,h10,h7,h8,h6,h4,h5,h2,h3,h1,h0]),tab_negall(discharge,[h13]),tab_negall(eigenvar,eigen__1)],[h12,17,h13]) ).
thf(19,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h9,h10,h7,h8,h6,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h11,h12])],[h10,18,h11,h12]) ).
thf(20,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h7,h8,h6,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h9,h10])],[h8,19,h9,h10]) ).
thf(21,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h6,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h7,h8])],[h6,20,h7,h8]) ).
thf(22,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h4,h5,h2,h3,h1,h0]),tab_negall(discharge,[h6]),tab_negall(eigenvar,eigen__0)],[h5,21,h6]) ).
thf(23,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h2,h3,h1,h0]),tab_negimp(discharge,[h4,h5])],[h3,22,h4,h5]) ).
thf(24,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h1,h0]),tab_negimp(discharge,[h2,h3])],[h1,23,h2,h3]) ).
thf(25,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h1]),eigenvar_choice(discharge,[h0])],[24,h0]) ).
thf(0,theorem,
( sP9
=> ( sP6
=> ! [X1: term > $o] :
( ! [X2: term] :
( ( var @ X2 )
=> ( X1 @ X2 ) )
=> ( ! [X2: term,X3: term] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X1 @ ( ap @ X2 @ X3 ) ) ) )
=> ( ! [X2: term] :
( ! [X3: term] :
( ( X1 @ X3 )
=> ( X1 @ ( sub @ X2 @ ( push @ X3 @ id ) ) ) )
=> ( X1 @ ( lam @ X2 ) ) )
=> ! [X2: term] : ( X1 @ X2 ) ) ) ) ) ),
inference(contra,[status(thm),contra(discharge,[h1])],[24,h1]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : ALG253^2 : TPTP v9.2.0. Bugfixed v5.2.0.
% 0.11/0.12 % Command : lash -P picomus -M modes -p tstp -t %d %s
% 0.13/0.32 % Computer : n011.cluster.edu
% 0.13/0.32 % Model : x86_64 x86_64
% 0.13/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.32 % Memory : 8042.1875MB
% 0.13/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.32 % CPULimit : 300
% 0.13/0.32 % WCLimit : 300
% 0.13/0.32 % DateTime : Wed Oct 8 23:35:38 EDT 2025
% 0.13/0.32 % CPUTime :
% 0.17/0.53 % SZS status Theorem
% 0.17/0.53 % Mode: cade22sinegrackle2x6978
% 0.17/0.53 % Steps: 1879
% 0.17/0.53 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------