%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP077^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n012.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 08:55:21 AM UTC 2026
% Result : Theorem 2.13s 1.25s
% Output : Refutation 2.13s
% Verified :
% SZS Type : Refutation
% Derivation depth : 9
% Number of leaves : 6
% Syntax : Number of formulae : 25 ( 18 unt; 0 typ; 0 def)
% Number of atoms : 46 ( 31 equ; 0 cnn)
% Maximal formula atoms : 4 ( 1 avg)
% Number of connectives : 85 ( 13 ~; 0 |; 4 &; 63 @)
% ( 0 <=>; 5 =>; 0 <=; 0 <~>)
% Maximal formula depth : 7 ( 2 avg)
% Number of types : 44 ( 43 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 151 ( 148 usr; 9 con; 0-3 aty)
% Number of variables : 9 ( 0 ^; 7 !; 2 ?; 9 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
list_a: $tType ).
thf(type_def_7,type,
set_list_a: $tType ).
thf(type_def_8,type,
a: $tType ).
thf(type_def_9,type,
set_a: $tType ).
thf(type_def_10,type,
nat: $tType ).
thf(type_def_11,type,
set_nat: $tType ).
thf(type_def_12,type,
list_multiset_list_a: $tType ).
thf(type_def_13,type,
multiset_list_a: $tType ).
thf(type_def_14,type,
list_multiset_a: $tType ).
thf(type_def_15,type,
multiset_a: $tType ).
thf(type_def_16,type,
list_nat: $tType ).
thf(type_def_17,type,
set_multiset_a: $tType ).
thf(type_def_18,type,
list_set_multiset_a: $tType ).
thf(type_def_19,type,
list_set_nat: $tType ).
thf(type_def_20,type,
list_set_a: $tType ).
thf(type_def_21,type,
list_set_list_list_a: $tType ).
thf(type_def_22,type,
set_list_list_list_a: $tType ).
thf(type_def_23,type,
list_set_list_set_a: $tType ).
thf(type_def_24,type,
set_list_list_set_a: $tType ).
thf(type_def_25,type,
list_set_list_a: $tType ).
thf(type_def_26,type,
set_list_list_a: $tType ).
thf(type_def_27,type,
set_list_multiset_a: $tType ).
thf(type_def_28,type,
set_list_nat: $tType ).
thf(type_def_29,type,
list_set_set_list_a: $tType ).
thf(type_def_30,type,
set_list_set_list_a: $tType ).
thf(type_def_31,type,
list_set_set_set_a: $tType ).
thf(type_def_32,type,
set_list_set_set_a: $tType ).
thf(type_def_33,type,
list_set_set_a: $tType ).
thf(type_def_34,type,
set_list_set_a: $tType ).
thf(type_def_35,type,
list_list_list_a: $tType ).
thf(type_def_36,type,
list_list_a: $tType ).
thf(type_def_37,type,
list_list_set_a: $tType ).
thf(type_def_38,type,
set_set_list_a: $tType ).
thf(type_def_39,type,
set_set_set_a: $tType ).
thf(type_def_40,type,
set_set_a: $tType ).
thf(type_def_41,type,
multis971982480list_a: $tType ).
thf(type_def_42,type,
multiset_multiset_a: $tType ).
thf(type_def_43,type,
multiset_nat: $tType ).
thf(type_def_44,type,
multiset_set_list_a: $tType ).
thf(type_def_45,type,
multiset_set_set_a: $tType ).
thf(type_def_46,type,
multiset_set_a: $tType ).
thf(type_def_47,type,
set_multiset_list_a: $tType ).
thf(type_def_48,type,
set_set_nat: $tType ).
thf(func_def_0,type,
set_list_list_list_a: $tType ).
thf(func_def_1,type,
multis971982480list_a: $tType ).
thf(func_def_2,type,
list_set_list_list_a: $tType ).
thf(func_def_3,type,
set_list_set_list_a: $tType ).
thf(func_def_4,type,
set_list_list_set_a: $tType ).
thf(func_def_5,type,
list_set_set_list_a: $tType ).
thf(func_def_6,type,
list_set_list_set_a: $tType ).
thf(func_def_7,type,
set_list_set_set_a: $tType ).
thf(func_def_8,type,
list_set_set_set_a: $tType ).
thf(func_def_9,type,
list_multiset_list_a: $tType ).
thf(func_def_10,type,
set_multiset_list_a: $tType ).
thf(func_def_11,type,
set_list_multiset_a: $tType ).
thf(func_def_12,type,
multiset_set_list_a: $tType ).
thf(func_def_13,type,
list_set_multiset_a: $tType ).
thf(func_def_14,type,
multiset_set_set_a: $tType ).
thf(func_def_15,type,
list_list_list_a: $tType ).
thf(func_def_16,type,
set_list_list_a: $tType ).
thf(func_def_17,type,
multiset_multiset_a: $tType ).
thf(func_def_18,type,
list_set_list_a: $tType ).
thf(func_def_19,type,
list_list_set_a: $tType ).
thf(func_def_20,type,
set_set_list_a: $tType ).
thf(func_def_21,type,
set_list_set_a: $tType ).
thf(func_def_22,type,
list_set_set_a: $tType ).
thf(func_def_23,type,
set_set_set_a: $tType ).
thf(func_def_24,type,
multiset_list_a: $tType ).
thf(func_def_25,type,
list_multiset_a: $tType ).
thf(func_def_26,type,
set_multiset_a: $tType ).
thf(func_def_27,type,
multiset_set_a: $tType ).
thf(func_def_28,type,
set_list_nat: $tType ).
thf(func_def_29,type,
list_set_nat: $tType ).
thf(func_def_30,type,
set_set_nat: $tType ).
thf(func_def_31,type,
list_list_a: $tType ).
thf(func_def_32,type,
set_list_a: $tType ).
thf(func_def_33,type,
list_set_a: $tType ).
thf(func_def_34,type,
multiset_nat: $tType ).
thf(func_def_35,type,
set_set_a: $tType ).
thf(func_def_36,type,
multiset_a: $tType ).
thf(func_def_37,type,
list_nat: $tType ).
thf(func_def_38,type,
set_nat: $tType ).
thf(func_def_39,type,
list_a: $tType ).
thf(func_def_40,type,
set_a: $tType ).
thf(func_def_41,type,
nat: $tType ).
thf(func_def_42,type,
a: $tType ).
thf(func_def_43,type,
bij_be94573046list_a: ( list_a > list_a ) > set_list_a > set_list_a > $o ).
thf(func_def_44,type,
bij_betw_list_a_a: ( list_a > a ) > set_list_a > set_a > $o ).
thf(func_def_45,type,
bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
thf(func_def_46,type,
bij_betw_a_list_a: ( a > list_a ) > set_a > set_list_a > $o ).
thf(func_def_47,type,
bij_betw_a_a: ( a > a ) > set_a > set_a > $o ).
thf(func_def_48,type,
inj_on_list_a_list_a: ( list_a > list_a ) > set_list_a > $o ).
thf(func_def_49,type,
inj_on_list_a_a: ( list_a > a ) > set_list_a > $o ).
thf(func_def_50,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(func_def_51,type,
inj_on_a_list_a: ( a > list_a ) > set_a > $o ).
thf(func_def_52,type,
inj_on_a_a: ( a > a ) > set_a > $o ).
thf(func_def_53,type,
groups1271887971list_a: list_multiset_list_a > multiset_list_a ).
thf(func_def_54,type,
groups1592617181iset_a: list_multiset_a > multiset_a ).
thf(func_def_55,type,
groups921905271st_nat: list_nat > nat ).
thf(func_def_56,type,
groups1792256535list_a: list_a > a ).
thf(func_def_57,type,
hilber921249868list_a: set_list_a > ( list_a > list_a ) > list_a > list_a ).
thf(func_def_58,type,
hilber2125729734st_a_a: set_list_a > ( list_a > a ) > a > list_a ).
thf(func_def_59,type,
hilber815131374at_nat: set_nat > ( nat > nat ) > nat > nat ).
thf(func_def_60,type,
hilber165648082list_a: set_a > ( a > list_a ) > list_a > a ).
thf(func_def_61,type,
hilbert_inv_into_a_a: set_a > ( a > a ) > a > a ).
thf(func_def_62,type,
hilber2024005914iset_a: set_multiset_a > list_set_multiset_a > $o ).
thf(func_def_63,type,
hilber1014902394mp_nat: set_nat > list_set_nat > $o ).
thf(func_def_64,type,
hilber2037636820comp_a: set_a > list_set_a > $o ).
thf(func_def_65,type,
list_ex_set_a: ( set_a > $o ) > list_set_a > $o ).
thf(func_def_66,type,
list_ex_a: ( a > $o ) > list_a > $o ).
thf(func_def_67,type,
list_update_set_a: list_set_a > nat > set_a > list_set_a ).
thf(func_def_68,type,
list_update_a: list_a > nat > a > list_a ).
thf(func_def_69,type,
listset_list_list_a: list_set_list_list_a > set_list_list_list_a ).
thf(func_def_70,type,
listset_list_set_a: list_set_list_set_a > set_list_list_set_a ).
thf(func_def_71,type,
listset_list_a: list_set_list_a > set_list_list_a ).
thf(func_def_72,type,
listset_multiset_a: list_set_multiset_a > set_list_multiset_a ).
thf(func_def_73,type,
listset_nat: list_set_nat > set_list_nat ).
thf(func_def_74,type,
listset_set_list_a: list_set_set_list_a > set_list_set_list_a ).
thf(func_def_75,type,
listset_set_set_a: list_set_set_set_a > set_list_set_set_a ).
thf(func_def_76,type,
listset_set_a: list_set_set_a > set_list_set_a ).
thf(func_def_77,type,
listset_a: list_set_a > set_list_a ).
thf(func_def_78,type,
nth_list_list_a: list_list_list_a > nat > list_list_a ).
thf(func_def_79,type,
nth_list_set_a: list_list_set_a > nat > list_set_a ).
thf(func_def_80,type,
nth_list_a: list_list_a > nat > list_a ).
thf(func_def_81,type,
nth_multiset_a: list_multiset_a > nat > multiset_a ).
thf(func_def_82,type,
nth_nat: list_nat > nat > nat ).
thf(func_def_83,type,
nth_set_list_list_a: list_set_list_list_a > nat > set_list_list_a ).
thf(func_def_84,type,
nth_set_list_set_a: list_set_list_set_a > nat > set_list_set_a ).
thf(func_def_85,type,
nth_set_list_a: list_set_list_a > nat > set_list_a ).
thf(func_def_86,type,
nth_set_multiset_a: list_set_multiset_a > nat > set_multiset_a ).
thf(func_def_87,type,
nth_set_nat: list_set_nat > nat > set_nat ).
thf(func_def_88,type,
nth_set_set_list_a: list_set_set_list_a > nat > set_set_list_a ).
thf(func_def_89,type,
nth_set_set_set_a: list_set_set_set_a > nat > set_set_set_a ).
thf(func_def_90,type,
nth_set_set_a: list_set_set_a > nat > set_set_a ).
thf(func_def_91,type,
nth_set_a: list_set_a > nat > set_a ).
thf(func_def_92,type,
nth_a: list_a > nat > a ).
thf(func_def_93,type,
rev_list_a: list_list_a > list_list_a ).
thf(func_def_94,type,
rev_nat: list_nat > list_nat ).
thf(func_def_95,type,
rev_set_list_a: list_set_list_a > list_set_list_a ).
thf(func_def_96,type,
rev_set_set_a: list_set_set_a > list_set_set_a ).
thf(func_def_97,type,
rev_set_a: list_set_a > list_set_a ).
thf(func_def_98,type,
rev_a: list_a > list_a ).
thf(func_def_99,type,
comm_m324867663list_a: multis971982480list_a > multiset_list_a ).
thf(func_def_100,type,
comm_m2145643721iset_a: multiset_multiset_a > multiset_a ).
thf(func_def_101,type,
comm_m1100186507et_nat: multiset_nat > nat ).
thf(func_def_102,type,
comm_m543484931mset_a: multiset_a > a ).
thf(func_def_103,type,
mset_list_a: list_list_a > multiset_list_a ).
thf(func_def_104,type,
mset_multiset_list_a: list_multiset_list_a > multis971982480list_a ).
thf(func_def_105,type,
mset_multiset_a: list_multiset_a > multiset_multiset_a ).
thf(func_def_106,type,
mset_nat: list_nat > multiset_nat ).
thf(func_def_107,type,
mset_set_list_a: list_set_list_a > multiset_set_list_a ).
thf(func_def_108,type,
mset_set_set_a: list_set_set_a > multiset_set_set_a ).
thf(func_def_109,type,
mset_set_a: list_set_a > multiset_set_a ).
thf(func_def_110,type,
mset_a: list_a > multiset_a ).
thf(func_def_111,type,
set_mset_list_a: multiset_list_a > set_list_a ).
thf(func_def_112,type,
set_ms1660427399list_a: multis971982480list_a > set_multiset_list_a ).
thf(func_def_113,type,
set_mset_multiset_a: multiset_multiset_a > set_multiset_a ).
thf(func_def_114,type,
set_mset_set_a: multiset_set_a > set_set_a ).
thf(func_def_115,type,
set_mset_a: multiset_a > set_a ).
thf(func_def_116,type,
size_s575106428list_a: list_list_list_a > nat ).
thf(func_def_117,type,
size_s341332310_set_a: list_list_set_a > nat ).
thf(func_def_118,type,
size_s1427607542list_a: list_list_a > nat ).
thf(func_def_119,type,
size_s1657263798iset_a: list_multiset_a > nat ).
thf(func_def_120,type,
size_size_list_nat: list_nat > nat ).
thf(func_def_121,type,
size_s240404444list_a: list_set_list_list_a > nat ).
thf(func_def_122,type,
size_s926492982_set_a: list_set_list_set_a > nat ).
thf(func_def_123,type,
size_s1635937238list_a: list_set_list_a > nat ).
thf(func_def_124,type,
size_s802269334iset_a: list_set_multiset_a > nat ).
thf(func_def_125,type,
size_s577819178et_nat: list_set_nat > nat ).
thf(func_def_126,type,
size_s469631926list_a: list_set_set_list_a > nat ).
thf(func_def_127,type,
size_s726512144_set_a: list_set_set_set_a > nat ).
thf(func_def_128,type,
size_s739728560_set_a: list_set_set_a > nat ).
thf(func_def_129,type,
size_size_list_set_a: list_set_a > nat ).
thf(func_def_130,type,
size_size_list_a: list_a > nat ).
thf(func_def_131,type,
size_s657334288_set_a: multiset_set_a > nat ).
thf(func_def_132,type,
size_size_multiset_a: multiset_a > nat ).
thf(func_def_133,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_134,type,
ord_less_set_list_a: set_list_a > set_list_a > $o ).
thf(func_def_135,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(func_def_136,type,
ord_less_set_a: set_a > set_a > $o ).
thf(func_def_137,type,
ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
thf(func_def_138,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_139,type,
ord_le1301786372list_a: set_list_a > set_list_a > $o ).
thf(func_def_140,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(func_def_141,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(func_def_142,type,
perm_set_a: list_set_a > list_set_a > $o ).
thf(func_def_143,type,
perm_a: list_a > list_a > $o ).
thf(func_def_144,type,
collect_list_list_a: ( list_list_a > $o ) > set_list_list_a ).
thf(func_def_145,type,
collect_list_set_a: ( list_set_a > $o ) > set_list_set_a ).
thf(func_def_146,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(func_def_147,type,
collect_multiset_a: ( multiset_a > $o ) > set_multiset_a ).
thf(func_def_148,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(func_def_149,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(func_def_150,type,
collect_a: ( a > $o ) > set_a ).
thf(func_def_151,type,
image_list_nat_nat: ( list_nat > nat ) > set_list_nat > set_nat ).
thf(func_def_152,type,
image_list_a_list_a: ( list_a > list_a ) > set_list_a > set_list_a ).
thf(func_def_153,type,
image_list_a_a: ( list_a > a ) > set_list_a > set_a ).
thf(func_def_154,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(func_def_155,type,
image_a_list_a: ( a > list_a ) > set_a > set_list_a ).
thf(func_def_156,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(func_def_157,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(func_def_158,type,
set_or1578766727list_a: set_list_a > set_set_list_a ).
thf(func_def_159,type,
set_or1597314339et_nat: set_nat > set_set_nat ).
thf(func_def_160,type,
set_or1164043265_set_a: set_a > set_set_a ).
thf(func_def_161,type,
member1511395513list_a: list_list_list_a > set_list_list_list_a > $o ).
thf(func_def_162,type,
member954073875_set_a: list_list_set_a > set_list_list_set_a > $o ).
thf(func_def_163,type,
member_list_list_a: list_list_a > set_list_list_a > $o ).
thf(func_def_164,type,
member518144627iset_a: list_multiset_a > set_list_multiset_a > $o ).
thf(func_def_165,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(func_def_166,type,
member101195155list_a: list_set_list_a > set_list_set_list_a > $o ).
thf(func_def_167,type,
member814963949_set_a: list_set_set_a > set_list_set_set_a > $o ).
thf(func_def_168,type,
member_list_set_a: list_set_a > set_list_set_a > $o ).
thf(func_def_169,type,
member_list_a: list_a > set_list_a > $o ).
thf(func_def_170,type,
member528267379list_a: multiset_list_a > set_multiset_list_a > $o ).
thf(func_def_171,type,
member_multiset_a: multiset_a > set_multiset_a > $o ).
thf(func_def_172,type,
member_nat: nat > set_nat > $o ).
thf(func_def_173,type,
member_set_list_a: set_list_a > set_set_list_a > $o ).
thf(func_def_174,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(func_def_175,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(func_def_176,type,
member_set_a: set_a > set_set_a > $o ).
thf(func_def_177,type,
member_a: a > set_a > $o ).
thf(func_def_178,type,
a2: set_a ).
thf(func_def_179,type,
a3: a ).
thf(func_def_180,type,
f: nat > nat ).
thf(func_def_181,type,
g: nat > nat ).
thf(func_def_182,type,
qs2: list_a ).
thf(func_def_183,type,
qs: list_a ).
thf(func_def_184,type,
ss1: list_set_a ).
thf(func_def_185,type,
ss2: list_set_a ).
thf(func_def_202,type,
sK0: list_set_a > set_a > list_a ).
thf(func_def_203,type,
sK1: list_set_a > set_a > a ).
thf(func_def_204,type,
sK2: list_a > list_set_a > set_a > list_a ).
thf(func_def_205,type,
sK3: list_a ).
thf(func_def_206,type,
sK4: list_set_a > a > list_a ).
thf(f1023,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1022,f1020]) ).
thf(f1020,plain,
( ( groups1792256535list_a @ qs )
!= ( groups1792256535list_a @ qs2 ) ),
inference(superposition,[status(thm)],[f991,f1000]) ).
thf(f1000,plain,
( ( groups1792256535list_a @ qs )
= ( groups1792256535list_a @ sK3 ) ),
inference(definition_unfolding,[status(thm)],[f982,f973]) ).
thf(f973,plain,
( a3
= ( groups1792256535list_a @ sK3 ) ),
inference(cnf_transformation,[status(thm)],[f952]) ).
thf(f952,plain,
( ( ( member_list_a @ sK3 @ ( listset_a @ ss1 ) )
= $true )
& ( a3
= ( groups1792256535list_a @ sK3 ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK3])],[f942,f951]) ).
thf(f951,plain,
( ? [X0: list_a] :
( ( ( member_list_a @ X0 @ ( listset_a @ ss1 ) )
= $true )
& ( a3
= ( groups1792256535list_a @ X0 ) ) )
=> ( ( ( member_list_a @ sK3 @ ( listset_a @ ss1 ) )
= $true )
& ( a3
= ( groups1792256535list_a @ sK3 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f942,plain,
? [X0: list_a] :
( ( ( member_list_a @ X0 @ ( listset_a @ ss1 ) )
= $true )
& ( a3
= ( groups1792256535list_a @ X0 ) ) ),
inference(ennf_transformation,[status(thm)],[f930]) ).
thf(f930,plain,
~ ! [X0: list_a] :
( ( a3
= ( groups1792256535list_a @ X0 ) )
=> ( ( member_list_a @ X0 @ ( listset_a @ ss1 ) )
!= $true ) ),
inference(flattening,[status(thm)],[f594]) ).
thf(f594,plain,
~ ! [X0: list_a] :
( ( a3
= ( groups1792256535list_a @ X0 ) )
=> ( ( member_list_a @ X0 @ ( listset_a @ ss1 ) )
!= $true ) ),
inference(fool_elimination,[status(thm)],[f593]) ).
thf(f593,plain,
~ ! [X0: list_a] :
( ( a3
= ( groups1792256535list_a @ X0 ) )
=> ~ ( member_list_a @ X0 @ ( listset_a @ ss1 ) ) ),
inference(rectify,[status(thm)],[f5]) ).
thf(f5,axiom,
~ ! [X0: list_a] :
( ( a3
= ( groups1792256535list_a @ X0 ) )
=> ~ ( member_list_a @ X0 @ ( listset_a @ ss1 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.OZNDMxpjYk/DTF2THF_9014.p',fact_4__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062qs_O_A_092_060lbrakk_062a_A_061_Asum__list_Aqs_059_Aqs_A_092_060in_062_Alistset_Ass1_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062) ).
thf(f982,plain,
( a3
= ( groups1792256535list_a @ qs ) ),
inference(cnf_transformation,[status(thm)],[f3]) ).
thf(f3,axiom,
( a3
= ( groups1792256535list_a @ qs ) ),
file('/export/starexec/sandbox2/tmp/tmp.OZNDMxpjYk/DTF2THF_9014.p',fact_2_a) ).
thf(f991,plain,
( ( groups1792256535list_a @ qs2 )
!= ( groups1792256535list_a @ sK3 ) ),
inference(definition_unfolding,[status(thm)],[f963,f973]) ).
thf(f963,plain,
( a3
!= ( groups1792256535list_a @ qs2 ) ),
inference(cnf_transformation,[status(thm)],[f929]) ).
thf(f929,plain,
( a3
!= ( groups1792256535list_a @ qs2 ) ),
inference(flattening,[status(thm)],[f355]) ).
thf(f355,negated_conjecture,
( a3
!= ( groups1792256535list_a @ qs2 ) ),
inference(negated_conjecture,[status(cth)],[f354]) ).
thf(f354,conjecture,
( a3
= ( groups1792256535list_a @ qs2 ) ),
file('/export/starexec/sandbox2/tmp/tmp.OZNDMxpjYk/DTF2THF_9014.p',conj_0) ).
thf(f1022,plain,
( ( groups1792256535list_a @ qs )
= ( groups1792256535list_a @ qs2 ) ),
inference(forward_demodulation,[status(thm)],[f1021,f979]) ).
thf(f979,plain,
! [X0: list_a] :
( ( groups1792256535list_a @ X0 )
= ( comm_m543484931mset_a @ ( mset_a @ X0 ) ) ),
inference(cnf_transformation,[status(thm)],[f931]) ).
thf(f931,plain,
! [X0: list_a] :
( ( groups1792256535list_a @ X0 )
= ( comm_m543484931mset_a @ ( mset_a @ X0 ) ) ),
inference(rectify,[status(thm)],[f25]) ).
thf(f25,axiom,
! [X9: list_a] :
( ( groups1792256535list_a @ X9 )
= ( comm_m543484931mset_a @ ( mset_a @ X9 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.OZNDMxpjYk/DTF2THF_9014.p',fact_24_sum__mset__sum__list) ).
thf(f1021,plain,
( ( groups1792256535list_a @ qs )
= ( comm_m543484931mset_a @ ( mset_a @ qs2 ) ) ),
inference(superposition,[status(thm)],[f979,f972]) ).
thf(f972,plain,
( ( mset_a @ qs2 )
= ( mset_a @ qs ) ),
inference(cnf_transformation,[status(thm)],[f1]) ).
thf(f1,axiom,
( ( mset_a @ qs2 )
= ( mset_a @ qs ) ),
file('/export/starexec/sandbox2/tmp/tmp.OZNDMxpjYk/DTF2THF_9014.p',fact_0__092_060open_062mset_Aqs2_A_061_Amset_Aqs_092_060close_062) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.06 % Problem : ITP077^1 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.06 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.07/0.24 % Computer : n012.cluster.edu
% 0.07/0.24 % Model : x86_64 x86_64
% 0.07/0.24 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/0.24 % Memory : 8042.1875MB
% 0.07/0.24 % OS : Linux 3.10.0-693.el7.x86_64
% 0.07/0.24 % CPULimit : 300
% 0.07/0.25 % WCLimit : 300
% 0.07/0.25 % DateTime : Tue Feb 24 08:25:00 EST 2026
% 0.07/0.25 % CPUTime :
% 0.07/0.25 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.07/0.31 ---- Original DTF file ---
% 0.07/0.31 thf(spec,logic,$$dhol).
% 0.07/0.31 %------------------------------------------------------------------------------
% 0.07/0.31 % File : ITP077^1 : TPTP v9.2.1. Released v7.5.0.
% 0.07/0.31 % Domain : Interactive Theorem Proving
% 0.07/0.31 % Problem : Sledgehammer Hilbert_Function problem prob_146__11622158_1
% 0.07/0.31 % Version : Especial.
% 0.07/0.31 % English :
% 0.07/0.31
% 0.07/0.31 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.07/0.31 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.07/0.31 % Source : [Des21]
% 0.07/0.31 % Names : Hilbert_Function/prob_146__11622158_1 [Des21]
% 0.07/0.31
% 0.07/0.31 % Status : Theorem
% 0.07/0.31 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% 0.07/0.31 % Syntax : Number of formulae : 540 ( 127 unt; 186 typ; 0 def)
% 0.07/0.31 % Number of atoms : 1058 ( 380 equ; 0 cnn)
% 0.07/0.31 % Maximal formula atoms : 12 ( 2 avg)
% 0.07/0.31 % Number of connectives : 3365 ( 86 ~; 11 |; 63 &;2650 @)
% 0.07/0.31 % ( 0 <=>; 555 =>; 0 <=; 0 <~>)
% 0.07/0.31 % Maximal formula depth : 17 ( 7 avg)
% 0.07/0.31 % Number of types : 44 ( 43 usr)
% 0.07/0.31 % Number of type conns : 402 ( 402 >; 0 *; 0 +; 0 <<)
% 0.07/0.31 % Number of symbols : 146 ( 143 usr; 8 con; 0-3 aty)
% 0.07/0.31 % Number of variables : 1068 ( 87 ^; 916 !; 65 ?;1068 :)
% 0.07/0.31 % SPC : TH0_THM_EQU_NAR
% 0.07/0.31
% 0.07/0.31 % Comments : This file was generated by Sledgehammer 2021-02-23 15:37:07.615
% 0.07/0.31 %------------------------------------------------------------------------------
% 0.07/0.31 % Could-be-implicit typings (43)
% 0.07/0.31 thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
% 0.07/0.31 set_list_list_list_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__Multiset__Omultiset_It__Multiset__Omultiset_It__List__Olist_Itf__a_J_J_J,type,
% 0.07/0.31 multis971982480list_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__List__Olist_It__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
% 0.07/0.31 list_set_list_list_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
% 0.07/0.31 set_list_set_list_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Set__Oset_Itf__a_J_J_J_J,type,
% 0.07/0.31 set_list_list_set_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__List__Olist_It__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
% 0.07/0.31 list_set_set_list_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__List__Olist_It__Set__Oset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J_J,type,
% 0.07/0.31 list_set_list_set_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J_J,type,
% 0.07/0.31 set_list_set_set_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__List__Olist_It__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J_J,type,
% 0.07/0.31 list_set_set_set_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__List__Olist_It__Multiset__Omultiset_It__List__Olist_Itf__a_J_J_J,type,
% 0.07/0.31 list_multiset_list_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__Set__Oset_It__Multiset__Omultiset_It__List__Olist_Itf__a_J_J_J,type,
% 0.07/0.31 set_multiset_list_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__Set__Oset_It__List__Olist_It__Multiset__Omultiset_Itf__a_J_J_J,type,
% 0.07/0.31 set_list_multiset_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
% 0.07/0.31 multiset_set_list_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__List__Olist_It__Set__Oset_It__Multiset__Omultiset_Itf__a_J_J_J,type,
% 0.07/0.31 list_set_multiset_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
% 0.07/0.31 multiset_set_set_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
% 0.07/0.31 list_list_list_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
% 0.07/0.31 set_list_list_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J,type,
% 0.07/0.31 multiset_multiset_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
% 0.07/0.31 list_set_list_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__List__Olist_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
% 0.07/0.31 list_list_set_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
% 0.07/0.31 set_set_list_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
% 0.07/0.31 set_list_set_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__List__Olist_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
% 0.07/0.31 list_set_set_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
% 0.07/0.31 set_set_set_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__Multiset__Omultiset_It__List__Olist_Itf__a_J_J,type,
% 0.07/0.31 multiset_list_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__List__Olist_It__Multiset__Omultiset_Itf__a_J_J,type,
% 0.07/0.31 list_multiset_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__Set__Oset_It__Multiset__Omultiset_Itf__a_J_J,type,
% 0.07/0.31 set_multiset_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
% 0.07/0.31 multiset_set_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
% 0.07/0.31 set_list_nat: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.07/0.31 list_set_nat: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.07/0.31 set_set_nat: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
% 0.07/0.31 list_list_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
% 0.07/0.31 set_list_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
% 0.07/0.31 list_set_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__Multiset__Omultiset_It__Nat__Onat_J,type,
% 0.07/0.31 multiset_nat: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 0.07/0.31 set_set_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__Multiset__Omultiset_Itf__a_J,type,
% 0.07/0.31 multiset_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
% 0.07/0.31 list_nat: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 0.07/0.31 set_nat: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__List__Olist_Itf__a_J,type,
% 0.07/0.31 list_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 0.07/0.31 set_a: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_t__Nat__Onat,type,
% 0.07/0.31 nat: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_n_tf__a,type,
% 0.07/0.31 a: $tType ).
% 0.07/0.31
% 0.07/0.31 % Explicit typings (143)
% 0.07/0.31 thf(sy_c_Fun_Obij__betw_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
% 0.07/0.31 bij_be94573046list_a: ( list_a > list_a ) > set_list_a > set_list_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Fun_Obij__betw_001t__List__Olist_Itf__a_J_001tf__a,type,
% 0.07/0.31 bij_betw_list_a_a: ( list_a > a ) > set_list_a > set_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.07/0.31 bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Fun_Obij__betw_001tf__a_001t__List__Olist_Itf__a_J,type,
% 0.07/0.31 bij_betw_a_list_a: ( a > list_a ) > set_a > set_list_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Fun_Obij__betw_001tf__a_001tf__a,type,
% 0.07/0.31 bij_betw_a_a: ( a > a ) > set_a > set_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
% 0.07/0.31 inj_on_list_a_list_a: ( list_a > list_a ) > set_list_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001tf__a,type,
% 0.07/0.31 inj_on_list_a_a: ( list_a > a ) > set_list_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.07/0.31 inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Fun_Oinj__on_001tf__a_001t__List__Olist_Itf__a_J,type,
% 0.07/0.31 inj_on_a_list_a: ( a > list_a ) > set_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
% 0.07/0.31 inj_on_a_a: ( a > a ) > set_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Multiset__Omultiset_It__List__Olist_Itf__a_J_J,type,
% 0.07/0.31 groups1271887971list_a: list_multiset_list_a > multiset_list_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.07/0.31 groups1592617181iset_a: list_multiset_a > multiset_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
% 0.07/0.31 groups921905271st_nat: list_nat > nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001tf__a,type,
% 0.07/0.31 groups1792256535list_a: list_a > a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Hilbert__Choice_Oinv__into_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
% 0.07/0.31 hilber921249868list_a: set_list_a > ( list_a > list_a ) > list_a > list_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Hilbert__Choice_Oinv__into_001t__List__Olist_Itf__a_J_001tf__a,type,
% 0.07/0.31 hilber2125729734st_a_a: set_list_a > ( list_a > a ) > a > list_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Hilbert__Choice_Oinv__into_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.07/0.31 hilber815131374at_nat: set_nat > ( nat > nat ) > nat > nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Hilbert__Choice_Oinv__into_001tf__a_001t__List__Olist_Itf__a_J,type,
% 0.07/0.31 hilber165648082list_a: set_a > ( a > list_a ) > list_a > a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Hilbert__Choice_Oinv__into_001tf__a_001tf__a,type,
% 0.07/0.31 hilbert_inv_into_a_a: set_a > ( a > a ) > a > a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Hilbert__Function__Mirabelle__sksumwvvhs_Odirect__decomp_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.07/0.31 hilber2024005914iset_a: set_multiset_a > list_set_multiset_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Hilbert__Function__Mirabelle__sksumwvvhs_Odirect__decomp_001t__Nat__Onat,type,
% 0.07/0.31 hilber1014902394mp_nat: set_nat > list_set_nat > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Hilbert__Function__Mirabelle__sksumwvvhs_Odirect__decomp_001tf__a,type,
% 0.07/0.31 hilber2037636820comp_a: set_a > list_set_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Olist__ex_001t__Set__Oset_Itf__a_J,type,
% 0.07/0.31 list_ex_set_a: ( set_a > $o ) > list_set_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Olist__ex_001tf__a,type,
% 0.07/0.31 list_ex_a: ( a > $o ) > list_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Olist__update_001t__Set__Oset_Itf__a_J,type,
% 0.07/0.31 list_update_set_a: list_set_a > nat > set_a > list_set_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Olist__update_001tf__a,type,
% 0.07/0.31 list_update_a: list_a > nat > a > list_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Olistset_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
% 0.07/0.31 listset_list_list_a: list_set_list_list_a > set_list_list_list_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Olistset_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
% 0.07/0.31 listset_list_set_a: list_set_list_set_a > set_list_list_set_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Olistset_001t__List__Olist_Itf__a_J,type,
% 0.07/0.31 listset_list_a: list_set_list_a > set_list_list_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Olistset_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.07/0.31 listset_multiset_a: list_set_multiset_a > set_list_multiset_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Olistset_001t__Nat__Onat,type,
% 0.07/0.31 listset_nat: list_set_nat > set_list_nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Olistset_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
% 0.07/0.31 listset_set_list_a: list_set_set_list_a > set_list_set_list_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Olistset_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Olistset_001t__Set__Oset_Itf__a_J,type,
% 0.07/0.31 listset_set_a: list_set_set_a > set_list_set_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Olistset_001tf__a,type,
% 0.07/0.31 listset_a: list_set_a > set_list_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Onth_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
% 0.07/0.31 nth_list_list_a: list_list_list_a > nat > list_list_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Onth_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
% 0.07/0.31 nth_list_set_a: list_list_set_a > nat > list_set_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Onth_001t__List__Olist_Itf__a_J,type,
% 0.07/0.31 nth_list_a: list_list_a > nat > list_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Onth_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.07/0.31 nth_multiset_a: list_multiset_a > nat > multiset_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Onth_001t__Nat__Onat,type,
% 0.07/0.31 nth_nat: list_nat > nat > nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Onth_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
% 0.07/0.31 nth_set_list_list_a: list_set_list_list_a > nat > set_list_list_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Onth_001t__Set__Oset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
% 0.07/0.31 nth_set_list_set_a: list_set_list_set_a > nat > set_list_set_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Onth_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
% 0.07/0.31 nth_set_list_a: list_set_list_a > nat > set_list_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Onth_001t__Set__Oset_It__Multiset__Omultiset_Itf__a_J_J,type,
% 0.07/0.31 nth_set_multiset_a: list_set_multiset_a > nat > set_multiset_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.07/0.31 nth_set_nat: list_set_nat > nat > set_nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Onth_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
% 0.07/0.31 nth_set_set_list_a: list_set_set_list_a > nat > set_set_list_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Onth_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
% 0.07/0.31 nth_set_set_set_a: list_set_set_set_a > nat > set_set_set_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Onth_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 0.07/0.31 nth_set_set_a: list_set_set_a > nat > set_set_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Onth_001t__Set__Oset_Itf__a_J,type,
% 0.07/0.31 nth_set_a: list_set_a > nat > set_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Onth_001tf__a,type,
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% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Orev_001t__List__Olist_Itf__a_J,type,
% 0.07/0.31 rev_list_a: list_list_a > list_list_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Orev_001t__Nat__Onat,type,
% 0.07/0.31 rev_nat: list_nat > list_nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Orev_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
% 0.07/0.31 rev_set_list_a: list_set_list_a > list_set_list_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Orev_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 0.07/0.31 rev_set_set_a: list_set_set_a > list_set_set_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Orev_001t__Set__Oset_Itf__a_J,type,
% 0.07/0.31 rev_set_a: list_set_a > list_set_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_List_Orev_001tf__a,type,
% 0.07/0.31 rev_a: list_a > list_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset_001t__Multiset__Omultiset_It__List__Olist_Itf__a_J_J,type,
% 0.07/0.31 comm_m324867663list_a: multis971982480list_a > multiset_list_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset_001t__Multiset__Omultiset_Itf__a_J,type,
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% 0.07/0.31
% 0.07/0.31 thf(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset_001t__Nat__Onat,type,
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% 0.07/0.31
% 0.07/0.31 thf(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset_001tf__a,type,
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% 0.07/0.31
% 0.07/0.31 thf(sy_c_Multiset_Omset_001t__List__Olist_Itf__a_J,type,
% 0.07/0.31 mset_list_a: list_list_a > multiset_list_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Multiset_Omset_001t__Multiset__Omultiset_It__List__Olist_Itf__a_J_J,type,
% 0.07/0.31 mset_multiset_list_a: list_multiset_list_a > multis971982480list_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Multiset_Omset_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.07/0.31 mset_multiset_a: list_multiset_a > multiset_multiset_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Multiset_Omset_001t__Nat__Onat,type,
% 0.07/0.31 mset_nat: list_nat > multiset_nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Multiset_Omset_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
% 0.07/0.31 mset_set_list_a: list_set_list_a > multiset_set_list_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Multiset_Omset_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 0.07/0.31 mset_set_set_a: list_set_set_a > multiset_set_set_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Multiset_Omset_001t__Set__Oset_Itf__a_J,type,
% 0.07/0.31 mset_set_a: list_set_a > multiset_set_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Multiset_Omset_001tf__a,type,
% 0.07/0.31 mset_a: list_a > multiset_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Multiset_Oset__mset_001t__List__Olist_Itf__a_J,type,
% 0.07/0.31 set_mset_list_a: multiset_list_a > set_list_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Multiset_Oset__mset_001t__Multiset__Omultiset_It__List__Olist_Itf__a_J_J,type,
% 0.07/0.31 set_ms1660427399list_a: multis971982480list_a > set_multiset_list_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Multiset_Oset__mset_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.07/0.31 set_mset_multiset_a: multiset_multiset_a > set_multiset_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Multiset_Oset__mset_001t__Set__Oset_Itf__a_J,type,
% 0.07/0.31 set_mset_set_a: multiset_set_a > set_set_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Multiset_Oset__mset_001tf__a,type,
% 0.07/0.31 set_mset_a: multiset_a > set_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
% 0.07/0.31 size_s575106428list_a: list_list_list_a > nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
% 0.07/0.31 size_s341332310_set_a: list_list_set_a > nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
% 0.07/0.31 size_s1427607542list_a: list_list_a > nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Multiset__Omultiset_Itf__a_J_J,type,
% 0.07/0.31 size_s1657263798iset_a: list_multiset_a > nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
% 0.07/0.31 size_size_list_nat: list_nat > nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
% 0.07/0.31 size_s240404444list_a: list_set_list_list_a > nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J_J,type,
% 0.07/0.31 size_s926492982_set_a: list_set_list_set_a > nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
% 0.07/0.31 size_s1635937238list_a: list_set_list_a > nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Multiset__Omultiset_Itf__a_J_J_J,type,
% 0.07/0.31 size_s802269334iset_a: list_set_multiset_a > nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.07/0.31 size_s577819178et_nat: list_set_nat > nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
% 0.07/0.31 size_s469631926list_a: list_set_set_list_a > nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J_J,type,
% 0.07/0.31 size_s726512144_set_a: list_set_set_set_a > nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
% 0.07/0.31 size_s739728560_set_a: list_set_set_a > nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
% 0.07/0.31 size_size_list_set_a: list_set_a > nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
% 0.07/0.31 size_size_list_a: list_a > nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
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% 0.07/0.31
% 0.07/0.31 thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.07/0.31 size_size_multiset_a: multiset_a > nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 0.07/0.31 ord_less_nat: nat > nat > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
% 0.07/0.31 ord_less_set_list_a: set_list_a > set_list_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
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% 0.07/0.31
% 0.07/0.31 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
% 0.07/0.31 ord_less_set_a: set_a > set_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J,type,
% 0.07/0.31 ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.07/0.31 ord_less_eq_nat: nat > nat > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
% 0.07/0.31 ord_le1301786372list_a: set_list_a > set_list_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.07/0.31 ord_less_eq_set_nat: set_nat > set_nat > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
% 0.07/0.31 ord_less_eq_set_a: set_a > set_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Permutation_Operm_001t__Set__Oset_Itf__a_J,type,
% 0.07/0.31 perm_set_a: list_set_a > list_set_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Permutation_Operm_001tf__a,type,
% 0.07/0.31 perm_a: list_a > list_a > $o ).
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% 0.07/0.31 thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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% 0.07/0.31 thf(sy_c_Set_OCollect_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
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% 0.07/0.31 thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
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% 0.07/0.31
% 0.07/0.31 thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
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% 0.07/0.31
% 0.07/0.31 thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
% 0.07/0.31 collect_set_a: ( set_a > $o ) > set_set_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Set_OCollect_001tf__a,type,
% 0.07/0.31 collect_a: ( a > $o ) > set_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
% 0.07/0.31 image_list_nat_nat: ( list_nat > nat ) > set_list_nat > set_nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
% 0.07/0.31 image_list_a_list_a: ( list_a > list_a ) > set_list_a > set_list_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001tf__a,type,
% 0.07/0.31 image_list_a_a: ( list_a > a ) > set_list_a > set_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.07/0.31 image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Set_Oimage_001tf__a_001t__List__Olist_Itf__a_J,type,
% 0.07/0.31 image_a_list_a: ( a > list_a ) > set_a > set_list_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
% 0.07/0.31 image_a_a: ( a > a ) > set_a > set_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
% 0.07/0.31 set_ord_lessThan_nat: nat > set_nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
% 0.07/0.31 set_or1578766727list_a: set_list_a > set_set_list_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.07/0.31 set_or1597314339et_nat: set_nat > set_set_nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_Itf__a_J,type,
% 0.07/0.31 set_or1164043265_set_a: set_a > set_set_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_member_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
% 0.07/0.31 member1511395513list_a: list_list_list_a > set_list_list_list_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_member_001t__List__Olist_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
% 0.07/0.31 member954073875_set_a: list_list_set_a > set_list_list_set_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
% 0.07/0.31 member_list_list_a: list_list_a > set_list_list_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_member_001t__List__Olist_It__Multiset__Omultiset_Itf__a_J_J,type,
% 0.07/0.31 member518144627iset_a: list_multiset_a > set_list_multiset_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
% 0.07/0.31 member_list_nat: list_nat > set_list_nat > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
% 0.07/0.31 member101195155list_a: list_set_list_a > set_list_set_list_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
% 0.07/0.31 member814963949_set_a: list_set_set_a > set_list_set_set_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_member_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
% 0.07/0.31 member_list_set_a: list_set_a > set_list_set_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
% 0.07/0.31 member_list_a: list_a > set_list_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_member_001t__Multiset__Omultiset_It__List__Olist_Itf__a_J_J,type,
% 0.07/0.31 member528267379list_a: multiset_list_a > set_multiset_list_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_member_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.07/0.31 member_multiset_a: multiset_a > set_multiset_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_member_001t__Nat__Onat,type,
% 0.07/0.31 member_nat: nat > set_nat > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_member_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
% 0.07/0.31 member_set_list_a: set_list_a > set_set_list_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.07/0.31 member_set_nat: set_nat > set_set_nat > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 0.07/0.31 member_set_set_a: set_set_a > set_set_set_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
% 0.07/0.31 member_set_a: set_a > set_set_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_member_001tf__a,type,
% 0.07/0.31 member_a: a > set_a > $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_A,type,
% 0.07/0.31 a2: set_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_a____,type,
% 0.07/0.31 a3: a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_f____,type,
% 0.07/0.31 f: nat > nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_g____,type,
% 0.07/0.31 g: nat > nat ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_qs2____,type,
% 0.07/0.31 qs2: list_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_qs____,type,
% 0.07/0.31 qs: list_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_ss1,type,
% 0.07/0.31 ss1: list_set_a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_ss2,type,
% 0.07/0.31 ss2: list_set_a ).
% 0.07/0.31
% 0.07/0.31 % Relevant facts (353)
% 0.07/0.31 thf(fact_0__092_060open_062mset_Aqs2_A_061_Amset_Aqs_092_060close_062,axiom,
% 0.07/0.31 ( ( mset_a @ qs2 )
% 0.07/0.31 = ( mset_a @ qs ) ) ).
% 0.07/0.31
% 0.07/0.31 % \<open>mset qs2 = mset qs\<close>
% 0.07/0.31 thf(fact_1__092_060open_062a_A_092_060in_062_AA_092_060close_062,axiom,
% 0.07/0.31 member_a @ a3 @ a2 ).
% 0.07/0.31
% 0.07/0.31 % \<open>a \<in> A\<close>
% 0.07/0.31 thf(fact_2_a,axiom,
% 0.07/0.31 ( a3
% 0.07/0.31 = ( groups1792256535list_a @ qs ) ) ).
% 0.07/0.31
% 0.07/0.31 % a
% 0.07/0.31 thf(fact_3_len__qs2,axiom,
% 0.07/0.31 ( ( size_size_list_a @ qs2 )
% 0.07/0.31 = ( size_size_list_a @ qs ) ) ).
% 0.07/0.31
% 0.07/0.31 % len_qs2
% 0.07/0.31 thf(fact_4__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062qs_O_A_092_060lbrakk_062a_A_061_Asum__list_Aqs_059_Aqs_A_092_060in_062_Alistset_Ass1_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.07/0.31 ~ ! [Qs: list_a] :
% 0.07/0.31 ( ( a3
% 0.07/0.31 = ( groups1792256535list_a @ Qs ) )
% 0.07/0.31 => ~ ( member_list_a @ Qs @ ( listset_a @ ss1 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % \<open>\<And>thesis. (\<And>qs. \<lbrakk>a = sum_list qs; qs \<in> listset ss1\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.07/0.31 thf(fact_5_qs2__in,axiom,
% 0.07/0.31 member_list_a @ qs2 @ ( listset_a @ ss2 ) ).
% 0.07/0.31
% 0.07/0.31 % qs2_in
% 0.07/0.31 thf(fact_6_direct__decompD_I1_J,axiom,
% 0.07/0.31 ! [A: set_multiset_a,Ss: list_set_multiset_a,Qs2: list_multiset_a] :
% 0.07/0.31 ( ( hilber2024005914iset_a @ A @ Ss )
% 0.07/0.31 => ( ( member518144627iset_a @ Qs2 @ ( listset_multiset_a @ Ss ) )
% 0.07/0.31 => ( member_multiset_a @ ( groups1592617181iset_a @ Qs2 ) @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % direct_decompD(1)
% 0.07/0.31 thf(fact_7_direct__decompD_I1_J,axiom,
% 0.07/0.31 ! [A: set_nat,Ss: list_set_nat,Qs2: list_nat] :
% 0.07/0.31 ( ( hilber1014902394mp_nat @ A @ Ss )
% 0.07/0.31 => ( ( member_list_nat @ Qs2 @ ( listset_nat @ Ss ) )
% 0.07/0.31 => ( member_nat @ ( groups921905271st_nat @ Qs2 ) @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % direct_decompD(1)
% 0.07/0.31 thf(fact_8_direct__decompD_I1_J,axiom,
% 0.07/0.31 ! [A: set_a,Ss: list_set_a,Qs2: list_a] :
% 0.07/0.31 ( ( hilber2037636820comp_a @ A @ Ss )
% 0.07/0.31 => ( ( member_list_a @ Qs2 @ ( listset_a @ Ss ) )
% 0.07/0.31 => ( member_a @ ( groups1792256535list_a @ Qs2 ) @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % direct_decompD(1)
% 0.07/0.31 thf(fact_9_direct__decompE,axiom,
% 0.07/0.31 ! [A: set_multiset_a,Ss: list_set_multiset_a,A2: multiset_a] :
% 0.07/0.31 ( ( hilber2024005914iset_a @ A @ Ss )
% 0.07/0.31 => ( ( member_multiset_a @ A2 @ A )
% 0.07/0.31 => ~ ! [Qs: list_multiset_a] :
% 0.07/0.31 ( ( member518144627iset_a @ Qs @ ( listset_multiset_a @ Ss ) )
% 0.07/0.31 => ( A2
% 0.07/0.31 != ( groups1592617181iset_a @ Qs ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % direct_decompE
% 0.07/0.31 thf(fact_10_direct__decompE,axiom,
% 0.07/0.31 ! [A: set_nat,Ss: list_set_nat,A2: nat] :
% 0.07/0.31 ( ( hilber1014902394mp_nat @ A @ Ss )
% 0.07/0.31 => ( ( member_nat @ A2 @ A )
% 0.07/0.31 => ~ ! [Qs: list_nat] :
% 0.07/0.31 ( ( member_list_nat @ Qs @ ( listset_nat @ Ss ) )
% 0.07/0.31 => ( A2
% 0.07/0.31 != ( groups921905271st_nat @ Qs ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % direct_decompE
% 0.07/0.31 thf(fact_11_direct__decompE,axiom,
% 0.07/0.31 ! [A: set_a,Ss: list_set_a,A2: a] :
% 0.07/0.31 ( ( hilber2037636820comp_a @ A @ Ss )
% 0.07/0.31 => ( ( member_a @ A2 @ A )
% 0.07/0.31 => ~ ! [Qs: list_a] :
% 0.07/0.31 ( ( member_list_a @ Qs @ ( listset_a @ Ss ) )
% 0.07/0.31 => ( A2
% 0.07/0.31 != ( groups1792256535list_a @ Qs ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % direct_decompE
% 0.07/0.31 thf(fact_12_direct__decompI__alt,axiom,
% 0.07/0.31 ! [Ss: list_set_multiset_a,A: set_multiset_a] :
% 0.07/0.31 ( ! [Qs: list_multiset_a] :
% 0.07/0.31 ( ( member518144627iset_a @ Qs @ ( listset_multiset_a @ Ss ) )
% 0.07/0.31 => ( member_multiset_a @ ( groups1592617181iset_a @ Qs ) @ A ) )
% 0.07/0.31 => ( ! [A3: multiset_a] :
% 0.07/0.31 ( ( member_multiset_a @ A3 @ A )
% 0.07/0.31 => ? [X: list_multiset_a] :
% 0.07/0.31 ( ( member518144627iset_a @ X @ ( listset_multiset_a @ Ss ) )
% 0.07/0.31 & ( A3
% 0.07/0.31 = ( groups1592617181iset_a @ X ) )
% 0.07/0.31 & ! [Y: list_multiset_a] :
% 0.07/0.31 ( ( ( member518144627iset_a @ Y @ ( listset_multiset_a @ Ss ) )
% 0.07/0.31 & ( A3
% 0.07/0.31 = ( groups1592617181iset_a @ Y ) ) )
% 0.07/0.31 => ( Y = X ) ) ) )
% 0.07/0.31 => ( hilber2024005914iset_a @ A @ Ss ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % direct_decompI_alt
% 0.07/0.31 thf(fact_13_direct__decompI__alt,axiom,
% 0.07/0.31 ! [Ss: list_set_nat,A: set_nat] :
% 0.07/0.31 ( ! [Qs: list_nat] :
% 0.07/0.31 ( ( member_list_nat @ Qs @ ( listset_nat @ Ss ) )
% 0.07/0.31 => ( member_nat @ ( groups921905271st_nat @ Qs ) @ A ) )
% 0.07/0.31 => ( ! [A3: nat] :
% 0.07/0.31 ( ( member_nat @ A3 @ A )
% 0.07/0.31 => ? [X: list_nat] :
% 0.07/0.31 ( ( member_list_nat @ X @ ( listset_nat @ Ss ) )
% 0.07/0.31 & ( A3
% 0.07/0.31 = ( groups921905271st_nat @ X ) )
% 0.07/0.31 & ! [Y: list_nat] :
% 0.07/0.31 ( ( ( member_list_nat @ Y @ ( listset_nat @ Ss ) )
% 0.07/0.31 & ( A3
% 0.07/0.31 = ( groups921905271st_nat @ Y ) ) )
% 0.07/0.31 => ( Y = X ) ) ) )
% 0.07/0.31 => ( hilber1014902394mp_nat @ A @ Ss ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % direct_decompI_alt
% 0.07/0.31 thf(fact_14_direct__decompI__alt,axiom,
% 0.07/0.31 ! [Ss: list_set_a,A: set_a] :
% 0.07/0.31 ( ! [Qs: list_a] :
% 0.07/0.31 ( ( member_list_a @ Qs @ ( listset_a @ Ss ) )
% 0.07/0.31 => ( member_a @ ( groups1792256535list_a @ Qs ) @ A ) )
% 0.07/0.31 => ( ! [A3: a] :
% 0.07/0.31 ( ( member_a @ A3 @ A )
% 0.07/0.31 => ? [X: list_a] :
% 0.07/0.31 ( ( member_list_a @ X @ ( listset_a @ Ss ) )
% 0.07/0.31 & ( A3
% 0.07/0.31 = ( groups1792256535list_a @ X ) )
% 0.07/0.31 & ! [Y: list_a] :
% 0.07/0.31 ( ( ( member_list_a @ Y @ ( listset_a @ Ss ) )
% 0.07/0.31 & ( A3
% 0.07/0.31 = ( groups1792256535list_a @ Y ) ) )
% 0.07/0.31 => ( Y = X ) ) ) )
% 0.07/0.31 => ( hilber2037636820comp_a @ A @ Ss ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % direct_decompI_alt
% 0.07/0.31 thf(fact_15_direct__decomp__unique,axiom,
% 0.07/0.31 ! [A: set_nat,Ss: list_set_nat,Qs2: list_nat,Qs3: list_nat] :
% 0.07/0.31 ( ( hilber1014902394mp_nat @ A @ Ss )
% 0.07/0.31 => ( ( member_list_nat @ Qs2 @ ( listset_nat @ Ss ) )
% 0.07/0.31 => ( ( member_list_nat @ Qs3 @ ( listset_nat @ Ss ) )
% 0.07/0.31 => ( ( ( groups921905271st_nat @ Qs2 )
% 0.07/0.31 = ( groups921905271st_nat @ Qs3 ) )
% 0.07/0.31 => ( Qs2 = Qs3 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % direct_decomp_unique
% 0.07/0.31 thf(fact_16_direct__decomp__unique,axiom,
% 0.07/0.31 ! [A: set_a,Ss: list_set_a,Qs2: list_a,Qs3: list_a] :
% 0.07/0.31 ( ( hilber2037636820comp_a @ A @ Ss )
% 0.07/0.31 => ( ( member_list_a @ Qs2 @ ( listset_a @ Ss ) )
% 0.07/0.31 => ( ( member_list_a @ Qs3 @ ( listset_a @ Ss ) )
% 0.07/0.31 => ( ( ( groups1792256535list_a @ Qs2 )
% 0.07/0.31 = ( groups1792256535list_a @ Qs3 ) )
% 0.07/0.31 => ( Qs2 = Qs3 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % direct_decomp_unique
% 0.07/0.31 thf(fact_17_sum__list_Orev,axiom,
% 0.07/0.31 ! [Xs: list_nat] :
% 0.07/0.31 ( ( groups921905271st_nat @ ( rev_nat @ Xs ) )
% 0.07/0.31 = ( groups921905271st_nat @ Xs ) ) ).
% 0.07/0.31
% 0.07/0.31 % sum_list.rev
% 0.07/0.31 thf(fact_18_sum__list_Orev,axiom,
% 0.07/0.31 ! [Xs: list_a] :
% 0.07/0.31 ( ( groups1792256535list_a @ ( rev_a @ Xs ) )
% 0.07/0.31 = ( groups1792256535list_a @ Xs ) ) ).
% 0.07/0.31
% 0.07/0.31 % sum_list.rev
% 0.07/0.31 thf(fact_19__C1_C,axiom,
% 0.07/0.31 ! [I: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I @ ( size_size_list_a @ qs ) )
% 0.07/0.31 => ( ( nth_a @ qs2 @ I )
% 0.07/0.31 = ( nth_a @ qs @ ( g @ I ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % "1"
% 0.07/0.31 thf(fact_20_g__bij2,axiom,
% 0.07/0.31 bij_betw_nat_nat @ g @ ( set_ord_lessThan_nat @ ( size_size_list_a @ qs2 ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_a @ qs ) ) ).
% 0.07/0.31
% 0.07/0.31 % g_bij2
% 0.07/0.31 thf(fact_21_sum__mset__sum__list,axiom,
% 0.07/0.31 ! [Xs: list_nat] :
% 0.07/0.31 ( ( comm_m1100186507et_nat @ ( mset_nat @ Xs ) )
% 0.07/0.31 = ( groups921905271st_nat @ Xs ) ) ).
% 0.07/0.31
% 0.07/0.31 % sum_mset_sum_list
% 0.07/0.31 thf(fact_22_sum__mset__sum__list,axiom,
% 0.07/0.31 ! [Xs: list_multiset_list_a] :
% 0.07/0.31 ( ( comm_m324867663list_a @ ( mset_multiset_list_a @ Xs ) )
% 0.07/0.31 = ( groups1271887971list_a @ Xs ) ) ).
% 0.07/0.31
% 0.07/0.31 % sum_mset_sum_list
% 0.07/0.31 thf(fact_23_sum__mset__sum__list,axiom,
% 0.07/0.31 ! [Xs: list_multiset_a] :
% 0.07/0.31 ( ( comm_m2145643721iset_a @ ( mset_multiset_a @ Xs ) )
% 0.07/0.31 = ( groups1592617181iset_a @ Xs ) ) ).
% 0.07/0.31
% 0.07/0.31 % sum_mset_sum_list
% 0.07/0.31 thf(fact_24_sum__mset__sum__list,axiom,
% 0.07/0.31 ! [Xs: list_a] :
% 0.07/0.31 ( ( comm_m543484931mset_a @ ( mset_a @ Xs ) )
% 0.07/0.31 = ( groups1792256535list_a @ Xs ) ) ).
% 0.07/0.31
% 0.07/0.31 % sum_mset_sum_list
% 0.07/0.31 thf(fact_25_direct__decompD_I3_J,axiom,
% 0.07/0.31 ! [A: set_nat,Ss: list_set_nat] :
% 0.07/0.31 ( ( hilber1014902394mp_nat @ A @ Ss )
% 0.07/0.31 => ( ( image_list_nat_nat @ groups921905271st_nat @ ( listset_nat @ Ss ) )
% 0.07/0.31 = A ) ) ).
% 0.07/0.31
% 0.07/0.31 % direct_decompD(3)
% 0.07/0.31 thf(fact_26_direct__decompD_I3_J,axiom,
% 0.07/0.31 ! [A: set_a,Ss: list_set_a] :
% 0.07/0.31 ( ( hilber2037636820comp_a @ A @ Ss )
% 0.07/0.31 => ( ( image_list_a_a @ groups1792256535list_a @ ( listset_a @ Ss ) )
% 0.07/0.31 = A ) ) ).
% 0.07/0.31
% 0.07/0.31 % direct_decompD(3)
% 0.07/0.31 thf(fact_27_assms_I1_J,axiom,
% 0.07/0.31 hilber2037636820comp_a @ a2 @ ss1 ).
% 0.07/0.31
% 0.07/0.31 % assms(1)
% 0.07/0.31 thf(fact_28_qs__in,axiom,
% 0.07/0.31 member_list_a @ qs @ ( listset_a @ ss1 ) ).
% 0.07/0.31
% 0.07/0.31 % qs_in
% 0.07/0.31 thf(fact_29_mset__rev,axiom,
% 0.07/0.31 ! [Xs: list_set_a] :
% 0.07/0.31 ( ( mset_set_a @ ( rev_set_a @ Xs ) )
% 0.07/0.31 = ( mset_set_a @ Xs ) ) ).
% 0.07/0.31
% 0.07/0.31 % mset_rev
% 0.07/0.31 thf(fact_30_mset__rev,axiom,
% 0.07/0.31 ! [Xs: list_list_a] :
% 0.07/0.31 ( ( mset_list_a @ ( rev_list_a @ Xs ) )
% 0.07/0.31 = ( mset_list_a @ Xs ) ) ).
% 0.07/0.31
% 0.07/0.31 % mset_rev
% 0.07/0.31 thf(fact_31_mset__rev,axiom,
% 0.07/0.31 ! [Xs: list_a] :
% 0.07/0.31 ( ( mset_a @ ( rev_a @ Xs ) )
% 0.07/0.31 = ( mset_a @ Xs ) ) ).
% 0.07/0.31
% 0.07/0.31 % mset_rev
% 0.07/0.31 thf(fact_32__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062qs2_O_A_092_060lbrakk_062qs2_A_092_060in_062_Alistset_Ass2_059_Alength_Aqs2_A_061_Alength_Aqs_059_A_092_060And_062i_O_Ai_A_060_Alength_Aqs_A_092_060Longrightarrow_062_Aqs2_A_B_Ai_A_061_Aqs_A_B_Ag_Ai_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.07/0.31 ~ ! [Qs22: list_a] :
% 0.07/0.31 ( ( member_list_a @ Qs22 @ ( listset_a @ ss2 ) )
% 0.07/0.31 => ( ( ( size_size_list_a @ Qs22 )
% 0.07/0.31 = ( size_size_list_a @ qs ) )
% 0.07/0.31 => ~ ! [I2: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I2 @ ( size_size_list_a @ qs ) )
% 0.07/0.31 => ( ( nth_a @ Qs22 @ I2 )
% 0.07/0.31 = ( nth_a @ qs @ ( g @ I2 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % \<open>\<And>thesis. (\<And>qs2. \<lbrakk>qs2 \<in> listset ss2; length qs2 = length qs; \<And>i. i < length qs \<Longrightarrow> qs2 ! i = qs ! g i\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.07/0.31 thf(fact_33_assms_I2_J,axiom,
% 0.07/0.31 perm_set_a @ ss1 @ ss2 ).
% 0.07/0.31
% 0.07/0.31 % assms(2)
% 0.07/0.31 thf(fact_34_ex__mset,axiom,
% 0.07/0.31 ! [X2: multiset_set_a] :
% 0.07/0.31 ? [Xs2: list_set_a] :
% 0.07/0.31 ( ( mset_set_a @ Xs2 )
% 0.07/0.31 = X2 ) ).
% 0.07/0.31
% 0.07/0.31 % ex_mset
% 0.07/0.31 thf(fact_35_ex__mset,axiom,
% 0.07/0.31 ! [X2: multiset_list_a] :
% 0.07/0.31 ? [Xs2: list_list_a] :
% 0.07/0.31 ( ( mset_list_a @ Xs2 )
% 0.07/0.31 = X2 ) ).
% 0.07/0.31
% 0.07/0.31 % ex_mset
% 0.07/0.31 thf(fact_36_ex__mset,axiom,
% 0.07/0.31 ! [X2: multiset_a] :
% 0.07/0.31 ? [Xs2: list_a] :
% 0.07/0.31 ( ( mset_a @ Xs2 )
% 0.07/0.31 = X2 ) ).
% 0.07/0.31
% 0.07/0.31 % ex_mset
% 0.07/0.31 thf(fact_37_mset__eq__length,axiom,
% 0.07/0.31 ! [Xs: list_list_a,Ys: list_list_a] :
% 0.07/0.31 ( ( ( mset_list_a @ Xs )
% 0.07/0.31 = ( mset_list_a @ Ys ) )
% 0.07/0.31 => ( ( size_s1427607542list_a @ Xs )
% 0.07/0.31 = ( size_s1427607542list_a @ Ys ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % mset_eq_length
% 0.07/0.31 thf(fact_38_mset__eq__length,axiom,
% 0.07/0.31 ! [Xs: list_set_list_a,Ys: list_set_list_a] :
% 0.07/0.31 ( ( ( mset_set_list_a @ Xs )
% 0.07/0.31 = ( mset_set_list_a @ Ys ) )
% 0.07/0.31 => ( ( size_s1635937238list_a @ Xs )
% 0.07/0.31 = ( size_s1635937238list_a @ Ys ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % mset_eq_length
% 0.07/0.31 thf(fact_39_mset__eq__length,axiom,
% 0.07/0.31 ! [Xs: list_set_set_a,Ys: list_set_set_a] :
% 0.07/0.31 ( ( ( mset_set_set_a @ Xs )
% 0.07/0.31 = ( mset_set_set_a @ Ys ) )
% 0.07/0.31 => ( ( size_s739728560_set_a @ Xs )
% 0.07/0.31 = ( size_s739728560_set_a @ Ys ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % mset_eq_length
% 0.07/0.31 thf(fact_40_mset__eq__length,axiom,
% 0.07/0.31 ! [Xs: list_nat,Ys: list_nat] :
% 0.07/0.31 ( ( ( mset_nat @ Xs )
% 0.07/0.31 = ( mset_nat @ Ys ) )
% 0.07/0.31 => ( ( size_size_list_nat @ Xs )
% 0.07/0.31 = ( size_size_list_nat @ Ys ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % mset_eq_length
% 0.07/0.31 thf(fact_41_mset__eq__length,axiom,
% 0.07/0.31 ! [Xs: list_a,Ys: list_a] :
% 0.07/0.31 ( ( ( mset_a @ Xs )
% 0.07/0.31 = ( mset_a @ Ys ) )
% 0.07/0.31 => ( ( size_size_list_a @ Xs )
% 0.07/0.31 = ( size_size_list_a @ Ys ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % mset_eq_length
% 0.07/0.31 thf(fact_42_mset__eq__length,axiom,
% 0.07/0.31 ! [Xs: list_set_a,Ys: list_set_a] :
% 0.07/0.31 ( ( ( mset_set_a @ Xs )
% 0.07/0.31 = ( mset_set_a @ Ys ) )
% 0.07/0.31 => ( ( size_size_list_set_a @ Xs )
% 0.07/0.31 = ( size_size_list_set_a @ Ys ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % mset_eq_length
% 0.07/0.31 thf(fact_43_mset__bij,axiom,
% 0.07/0.31 ! [F: nat > nat,Xs: list_list_a,Ys: list_list_a] :
% 0.07/0.31 ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_s1427607542list_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_s1427607542list_a @ Ys ) ) )
% 0.07/0.31 => ( ! [I3: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I3 @ ( size_s1427607542list_a @ Xs ) )
% 0.07/0.31 => ( ( nth_list_a @ Xs @ I3 )
% 0.07/0.31 = ( nth_list_a @ Ys @ ( F @ I3 ) ) ) )
% 0.07/0.31 => ( ( mset_list_a @ Xs )
% 0.07/0.31 = ( mset_list_a @ Ys ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % mset_bij
% 0.07/0.31 thf(fact_44_mset__bij,axiom,
% 0.07/0.31 ! [F: nat > nat,Xs: list_set_list_a,Ys: list_set_list_a] :
% 0.07/0.31 ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_s1635937238list_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_s1635937238list_a @ Ys ) ) )
% 0.07/0.31 => ( ! [I3: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I3 @ ( size_s1635937238list_a @ Xs ) )
% 0.07/0.31 => ( ( nth_set_list_a @ Xs @ I3 )
% 0.07/0.31 = ( nth_set_list_a @ Ys @ ( F @ I3 ) ) ) )
% 0.07/0.31 => ( ( mset_set_list_a @ Xs )
% 0.07/0.31 = ( mset_set_list_a @ Ys ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % mset_bij
% 0.07/0.31 thf(fact_45_mset__bij,axiom,
% 0.07/0.31 ! [F: nat > nat,Xs: list_set_set_a,Ys: list_set_set_a] :
% 0.07/0.31 ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_s739728560_set_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_s739728560_set_a @ Ys ) ) )
% 0.07/0.31 => ( ! [I3: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I3 @ ( size_s739728560_set_a @ Xs ) )
% 0.07/0.31 => ( ( nth_set_set_a @ Xs @ I3 )
% 0.07/0.31 = ( nth_set_set_a @ Ys @ ( F @ I3 ) ) ) )
% 0.07/0.31 => ( ( mset_set_set_a @ Xs )
% 0.07/0.31 = ( mset_set_set_a @ Ys ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % mset_bij
% 0.07/0.31 thf(fact_46_mset__bij,axiom,
% 0.07/0.31 ! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
% 0.07/0.31 ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_size_list_nat @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_nat @ Ys ) ) )
% 0.07/0.31 => ( ! [I3: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
% 0.07/0.31 => ( ( nth_nat @ Xs @ I3 )
% 0.07/0.31 = ( nth_nat @ Ys @ ( F @ I3 ) ) ) )
% 0.07/0.31 => ( ( mset_nat @ Xs )
% 0.07/0.31 = ( mset_nat @ Ys ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % mset_bij
% 0.07/0.31 thf(fact_47_mset__bij,axiom,
% 0.07/0.31 ! [F: nat > nat,Xs: list_a,Ys: list_a] :
% 0.07/0.31 ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_size_list_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_a @ Ys ) ) )
% 0.07/0.31 => ( ! [I3: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
% 0.07/0.31 => ( ( nth_a @ Xs @ I3 )
% 0.07/0.31 = ( nth_a @ Ys @ ( F @ I3 ) ) ) )
% 0.07/0.31 => ( ( mset_a @ Xs )
% 0.07/0.31 = ( mset_a @ Ys ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % mset_bij
% 0.07/0.31 thf(fact_48_mset__bij,axiom,
% 0.07/0.31 ! [F: nat > nat,Xs: list_set_a,Ys: list_set_a] :
% 0.07/0.31 ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ Ys ) ) )
% 0.07/0.31 => ( ! [I3: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I3 @ ( size_size_list_set_a @ Xs ) )
% 0.07/0.31 => ( ( nth_set_a @ Xs @ I3 )
% 0.07/0.31 = ( nth_set_a @ Ys @ ( F @ I3 ) ) ) )
% 0.07/0.31 => ( ( mset_set_a @ Xs )
% 0.07/0.31 = ( mset_set_a @ Ys ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % mset_bij
% 0.07/0.31 thf(fact_49_listset__permE,axiom,
% 0.07/0.31 ! [Ys: list_set_list_a,Xs: list_set_set_list_a,F: nat > nat,Xs3: list_set_set_list_a] :
% 0.07/0.31 ( ( member101195155list_a @ Ys @ ( listset_set_list_a @ Xs ) )
% 0.07/0.31 => ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_s469631926list_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_s469631926list_a @ Xs3 ) ) )
% 0.07/0.31 => ( ! [I3: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I3 @ ( size_s469631926list_a @ Xs ) )
% 0.07/0.31 => ( ( nth_set_set_list_a @ Xs3 @ I3 )
% 0.07/0.31 = ( nth_set_set_list_a @ Xs @ ( F @ I3 ) ) ) )
% 0.07/0.31 => ~ ! [Ys2: list_set_list_a] :
% 0.07/0.31 ( ( member101195155list_a @ Ys2 @ ( listset_set_list_a @ Xs3 ) )
% 0.07/0.31 => ( ( ( size_s1635937238list_a @ Ys2 )
% 0.07/0.31 = ( size_s1635937238list_a @ Ys ) )
% 0.07/0.31 => ~ ! [I2: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I2 @ ( size_s1635937238list_a @ Ys ) )
% 0.07/0.31 => ( ( nth_set_list_a @ Ys2 @ I2 )
% 0.07/0.31 = ( nth_set_list_a @ Ys @ ( F @ I2 ) ) ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % listset_permE
% 0.07/0.31 thf(fact_50_listset__permE,axiom,
% 0.07/0.31 ! [Ys: list_set_set_a,Xs: list_set_set_set_a,F: nat > nat,Xs3: list_set_set_set_a] :
% 0.07/0.31 ( ( member814963949_set_a @ Ys @ ( listset_set_set_a @ Xs ) )
% 0.07/0.31 => ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_s726512144_set_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_s726512144_set_a @ Xs3 ) ) )
% 0.07/0.31 => ( ! [I3: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I3 @ ( size_s726512144_set_a @ Xs ) )
% 0.07/0.31 => ( ( nth_set_set_set_a @ Xs3 @ I3 )
% 0.07/0.31 = ( nth_set_set_set_a @ Xs @ ( F @ I3 ) ) ) )
% 0.07/0.31 => ~ ! [Ys2: list_set_set_a] :
% 0.07/0.31 ( ( member814963949_set_a @ Ys2 @ ( listset_set_set_a @ Xs3 ) )
% 0.07/0.31 => ( ( ( size_s739728560_set_a @ Ys2 )
% 0.07/0.31 = ( size_s739728560_set_a @ Ys ) )
% 0.07/0.31 => ~ ! [I2: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I2 @ ( size_s739728560_set_a @ Ys ) )
% 0.07/0.31 => ( ( nth_set_set_a @ Ys2 @ I2 )
% 0.07/0.31 = ( nth_set_set_a @ Ys @ ( F @ I2 ) ) ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % listset_permE
% 0.07/0.31 thf(fact_51_listset__permE,axiom,
% 0.07/0.31 ! [Ys: list_nat,Xs: list_set_nat,F: nat > nat,Xs3: list_set_nat] :
% 0.07/0.31 ( ( member_list_nat @ Ys @ ( listset_nat @ Xs ) )
% 0.07/0.31 => ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_s577819178et_nat @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_s577819178et_nat @ Xs3 ) ) )
% 0.07/0.31 => ( ! [I3: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I3 @ ( size_s577819178et_nat @ Xs ) )
% 0.07/0.31 => ( ( nth_set_nat @ Xs3 @ I3 )
% 0.07/0.31 = ( nth_set_nat @ Xs @ ( F @ I3 ) ) ) )
% 0.07/0.31 => ~ ! [Ys2: list_nat] :
% 0.07/0.31 ( ( member_list_nat @ Ys2 @ ( listset_nat @ Xs3 ) )
% 0.07/0.31 => ( ( ( size_size_list_nat @ Ys2 )
% 0.07/0.31 = ( size_size_list_nat @ Ys ) )
% 0.07/0.31 => ~ ! [I2: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
% 0.07/0.31 => ( ( nth_nat @ Ys2 @ I2 )
% 0.07/0.31 = ( nth_nat @ Ys @ ( F @ I2 ) ) ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % listset_permE
% 0.07/0.31 thf(fact_52_listset__permE,axiom,
% 0.07/0.31 ! [Ys: list_list_a,Xs: list_set_list_a,F: nat > nat,Xs3: list_set_list_a] :
% 0.07/0.31 ( ( member_list_list_a @ Ys @ ( listset_list_a @ Xs ) )
% 0.07/0.31 => ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_s1635937238list_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_s1635937238list_a @ Xs3 ) ) )
% 0.07/0.31 => ( ! [I3: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I3 @ ( size_s1635937238list_a @ Xs ) )
% 0.07/0.31 => ( ( nth_set_list_a @ Xs3 @ I3 )
% 0.07/0.31 = ( nth_set_list_a @ Xs @ ( F @ I3 ) ) ) )
% 0.07/0.31 => ~ ! [Ys2: list_list_a] :
% 0.07/0.31 ( ( member_list_list_a @ Ys2 @ ( listset_list_a @ Xs3 ) )
% 0.07/0.31 => ( ( ( size_s1427607542list_a @ Ys2 )
% 0.07/0.31 = ( size_s1427607542list_a @ Ys ) )
% 0.07/0.31 => ~ ! [I2: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I2 @ ( size_s1427607542list_a @ Ys ) )
% 0.07/0.31 => ( ( nth_list_a @ Ys2 @ I2 )
% 0.07/0.31 = ( nth_list_a @ Ys @ ( F @ I2 ) ) ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % listset_permE
% 0.07/0.31 thf(fact_53_listset__permE,axiom,
% 0.07/0.31 ! [Ys: list_set_a,Xs: list_set_set_a,F: nat > nat,Xs3: list_set_set_a] :
% 0.07/0.31 ( ( member_list_set_a @ Ys @ ( listset_set_a @ Xs ) )
% 0.07/0.31 => ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_s739728560_set_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_s739728560_set_a @ Xs3 ) ) )
% 0.07/0.31 => ( ! [I3: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I3 @ ( size_s739728560_set_a @ Xs ) )
% 0.07/0.31 => ( ( nth_set_set_a @ Xs3 @ I3 )
% 0.07/0.31 = ( nth_set_set_a @ Xs @ ( F @ I3 ) ) ) )
% 0.07/0.31 => ~ ! [Ys2: list_set_a] :
% 0.07/0.31 ( ( member_list_set_a @ Ys2 @ ( listset_set_a @ Xs3 ) )
% 0.07/0.31 => ( ( ( size_size_list_set_a @ Ys2 )
% 0.07/0.31 = ( size_size_list_set_a @ Ys ) )
% 0.07/0.31 => ~ ! [I2: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I2 @ ( size_size_list_set_a @ Ys ) )
% 0.07/0.31 => ( ( nth_set_a @ Ys2 @ I2 )
% 0.07/0.31 = ( nth_set_a @ Ys @ ( F @ I2 ) ) ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % listset_permE
% 0.07/0.31 thf(fact_54_listset__permE,axiom,
% 0.07/0.31 ! [Ys: list_a,Xs: list_set_a,F: nat > nat,Xs3: list_set_a] :
% 0.07/0.31 ( ( member_list_a @ Ys @ ( listset_a @ Xs ) )
% 0.07/0.31 => ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ Xs3 ) ) )
% 0.07/0.31 => ( ! [I3: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I3 @ ( size_size_list_set_a @ Xs ) )
% 0.07/0.31 => ( ( nth_set_a @ Xs3 @ I3 )
% 0.07/0.31 = ( nth_set_a @ Xs @ ( F @ I3 ) ) ) )
% 0.07/0.31 => ~ ! [Ys2: list_a] :
% 0.07/0.31 ( ( member_list_a @ Ys2 @ ( listset_a @ Xs3 ) )
% 0.07/0.31 => ( ( ( size_size_list_a @ Ys2 )
% 0.07/0.31 = ( size_size_list_a @ Ys ) )
% 0.07/0.31 => ~ ! [I2: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ys ) )
% 0.07/0.31 => ( ( nth_a @ Ys2 @ I2 )
% 0.07/0.31 = ( nth_a @ Ys @ ( F @ I2 ) ) ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % listset_permE
% 0.07/0.31 thf(fact_55_g__bij,axiom,
% 0.07/0.31 bij_betw_nat_nat @ g @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ ss1 ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ ss2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % g_bij
% 0.07/0.31 thf(fact_56_length__rev,axiom,
% 0.07/0.31 ! [Xs: list_list_a] :
% 0.07/0.31 ( ( size_s1427607542list_a @ ( rev_list_a @ Xs ) )
% 0.07/0.31 = ( size_s1427607542list_a @ Xs ) ) ).
% 0.07/0.31
% 0.07/0.31 % length_rev
% 0.07/0.31 thf(fact_57_length__rev,axiom,
% 0.07/0.31 ! [Xs: list_set_list_a] :
% 0.07/0.31 ( ( size_s1635937238list_a @ ( rev_set_list_a @ Xs ) )
% 0.07/0.31 = ( size_s1635937238list_a @ Xs ) ) ).
% 0.07/0.31
% 0.07/0.31 % length_rev
% 0.07/0.31 thf(fact_58_length__rev,axiom,
% 0.07/0.31 ! [Xs: list_set_set_a] :
% 0.07/0.31 ( ( size_s739728560_set_a @ ( rev_set_set_a @ Xs ) )
% 0.07/0.31 = ( size_s739728560_set_a @ Xs ) ) ).
% 0.07/0.31
% 0.07/0.31 % length_rev
% 0.07/0.31 thf(fact_59_length__rev,axiom,
% 0.07/0.31 ! [Xs: list_nat] :
% 0.07/0.31 ( ( size_size_list_nat @ ( rev_nat @ Xs ) )
% 0.07/0.31 = ( size_size_list_nat @ Xs ) ) ).
% 0.07/0.31
% 0.07/0.31 % length_rev
% 0.07/0.31 thf(fact_60_length__rev,axiom,
% 0.07/0.31 ! [Xs: list_a] :
% 0.07/0.31 ( ( size_size_list_a @ ( rev_a @ Xs ) )
% 0.07/0.31 = ( size_size_list_a @ Xs ) ) ).
% 0.07/0.31
% 0.07/0.31 % length_rev
% 0.07/0.31 thf(fact_61_length__rev,axiom,
% 0.07/0.31 ! [Xs: list_set_a] :
% 0.07/0.31 ( ( size_size_list_set_a @ ( rev_set_a @ Xs ) )
% 0.07/0.31 = ( size_size_list_set_a @ Xs ) ) ).
% 0.07/0.31
% 0.07/0.31 % length_rev
% 0.07/0.31 thf(fact_62_lessThan__iff,axiom,
% 0.07/0.31 ! [I: set_nat,K: set_nat] :
% 0.07/0.31 ( ( member_set_nat @ I @ ( set_or1597314339et_nat @ K ) )
% 0.07/0.31 = ( ord_less_set_nat @ I @ K ) ) ).
% 0.07/0.31
% 0.07/0.31 % lessThan_iff
% 0.07/0.31 thf(fact_63_lessThan__iff,axiom,
% 0.07/0.31 ! [I: set_list_a,K: set_list_a] :
% 0.07/0.31 ( ( member_set_list_a @ I @ ( set_or1578766727list_a @ K ) )
% 0.07/0.31 = ( ord_less_set_list_a @ I @ K ) ) ).
% 0.07/0.31
% 0.07/0.31 % lessThan_iff
% 0.07/0.31 thf(fact_64_lessThan__iff,axiom,
% 0.07/0.31 ! [I: set_a,K: set_a] :
% 0.07/0.31 ( ( member_set_a @ I @ ( set_or1164043265_set_a @ K ) )
% 0.07/0.31 = ( ord_less_set_a @ I @ K ) ) ).
% 0.07/0.31
% 0.07/0.31 % lessThan_iff
% 0.07/0.31 thf(fact_65_lessThan__iff,axiom,
% 0.07/0.31 ! [I: nat,K: nat] :
% 0.07/0.31 ( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
% 0.07/0.31 = ( ord_less_nat @ I @ K ) ) ).
% 0.07/0.31
% 0.07/0.31 % lessThan_iff
% 0.07/0.31 thf(fact_66_listsetI,axiom,
% 0.07/0.31 ! [Ys: list_multiset_a,Xs: list_set_multiset_a] :
% 0.07/0.31 ( ( ( size_s1657263798iset_a @ Ys )
% 0.07/0.31 = ( size_s802269334iset_a @ Xs ) )
% 0.07/0.31 => ( ! [I3: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I3 @ ( size_s802269334iset_a @ Xs ) )
% 0.07/0.31 => ( member_multiset_a @ ( nth_multiset_a @ Ys @ I3 ) @ ( nth_set_multiset_a @ Xs @ I3 ) ) )
% 0.07/0.31 => ( member518144627iset_a @ Ys @ ( listset_multiset_a @ Xs ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % listsetI
% 0.07/0.31 thf(fact_67_listsetI,axiom,
% 0.07/0.31 ! [Ys: list_list_list_a,Xs: list_set_list_list_a] :
% 0.07/0.31 ( ( ( size_s575106428list_a @ Ys )
% 0.07/0.31 = ( size_s240404444list_a @ Xs ) )
% 0.07/0.31 => ( ! [I3: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I3 @ ( size_s240404444list_a @ Xs ) )
% 0.07/0.31 => ( member_list_list_a @ ( nth_list_list_a @ Ys @ I3 ) @ ( nth_set_list_list_a @ Xs @ I3 ) ) )
% 0.07/0.31 => ( member1511395513list_a @ Ys @ ( listset_list_list_a @ Xs ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % listsetI
% 0.07/0.31 thf(fact_68_listsetI,axiom,
% 0.07/0.31 ! [Ys: list_list_set_a,Xs: list_set_list_set_a] :
% 0.07/0.31 ( ( ( size_s341332310_set_a @ Ys )
% 0.07/0.31 = ( size_s926492982_set_a @ Xs ) )
% 0.07/0.31 => ( ! [I3: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I3 @ ( size_s926492982_set_a @ Xs ) )
% 0.07/0.31 => ( member_list_set_a @ ( nth_list_set_a @ Ys @ I3 ) @ ( nth_set_list_set_a @ Xs @ I3 ) ) )
% 0.07/0.31 => ( member954073875_set_a @ Ys @ ( listset_list_set_a @ Xs ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % listsetI
% 0.07/0.31 thf(fact_69_listsetI,axiom,
% 0.07/0.31 ! [Ys: list_set_list_a,Xs: list_set_set_list_a] :
% 0.07/0.31 ( ( ( size_s1635937238list_a @ Ys )
% 0.07/0.31 = ( size_s469631926list_a @ Xs ) )
% 0.07/0.31 => ( ! [I3: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I3 @ ( size_s469631926list_a @ Xs ) )
% 0.07/0.31 => ( member_set_list_a @ ( nth_set_list_a @ Ys @ I3 ) @ ( nth_set_set_list_a @ Xs @ I3 ) ) )
% 0.07/0.31 => ( member101195155list_a @ Ys @ ( listset_set_list_a @ Xs ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % listsetI
% 0.07/0.31 thf(fact_70_listsetI,axiom,
% 0.07/0.31 ! [Ys: list_set_set_a,Xs: list_set_set_set_a] :
% 0.07/0.31 ( ( ( size_s739728560_set_a @ Ys )
% 0.07/0.31 = ( size_s726512144_set_a @ Xs ) )
% 0.07/0.31 => ( ! [I3: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I3 @ ( size_s726512144_set_a @ Xs ) )
% 0.07/0.31 => ( member_set_set_a @ ( nth_set_set_a @ Ys @ I3 ) @ ( nth_set_set_set_a @ Xs @ I3 ) ) )
% 0.07/0.31 => ( member814963949_set_a @ Ys @ ( listset_set_set_a @ Xs ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % listsetI
% 0.07/0.31 thf(fact_71_listsetI,axiom,
% 0.07/0.31 ! [Ys: list_nat,Xs: list_set_nat] :
% 0.07/0.31 ( ( ( size_size_list_nat @ Ys )
% 0.07/0.31 = ( size_s577819178et_nat @ Xs ) )
% 0.07/0.31 => ( ! [I3: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I3 @ ( size_s577819178et_nat @ Xs ) )
% 0.07/0.31 => ( member_nat @ ( nth_nat @ Ys @ I3 ) @ ( nth_set_nat @ Xs @ I3 ) ) )
% 0.07/0.31 => ( member_list_nat @ Ys @ ( listset_nat @ Xs ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % listsetI
% 0.07/0.31 thf(fact_72_listsetI,axiom,
% 0.07/0.31 ! [Ys: list_list_a,Xs: list_set_list_a] :
% 0.07/0.31 ( ( ( size_s1427607542list_a @ Ys )
% 0.07/0.31 = ( size_s1635937238list_a @ Xs ) )
% 0.07/0.31 => ( ! [I3: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I3 @ ( size_s1635937238list_a @ Xs ) )
% 0.07/0.31 => ( member_list_a @ ( nth_list_a @ Ys @ I3 ) @ ( nth_set_list_a @ Xs @ I3 ) ) )
% 0.07/0.31 => ( member_list_list_a @ Ys @ ( listset_list_a @ Xs ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % listsetI
% 0.07/0.31 thf(fact_73_listsetI,axiom,
% 0.07/0.31 ! [Ys: list_a,Xs: list_set_a] :
% 0.07/0.31 ( ( ( size_size_list_a @ Ys )
% 0.07/0.31 = ( size_size_list_set_a @ Xs ) )
% 0.07/0.31 => ( ! [I3: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I3 @ ( size_size_list_set_a @ Xs ) )
% 0.07/0.31 => ( member_a @ ( nth_a @ Ys @ I3 ) @ ( nth_set_a @ Xs @ I3 ) ) )
% 0.07/0.31 => ( member_list_a @ Ys @ ( listset_a @ Xs ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % listsetI
% 0.07/0.31 thf(fact_74_listsetI,axiom,
% 0.07/0.31 ! [Ys: list_set_a,Xs: list_set_set_a] :
% 0.07/0.31 ( ( ( size_size_list_set_a @ Ys )
% 0.07/0.31 = ( size_s739728560_set_a @ Xs ) )
% 0.07/0.31 => ( ! [I3: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I3 @ ( size_s739728560_set_a @ Xs ) )
% 0.07/0.31 => ( member_set_a @ ( nth_set_a @ Ys @ I3 ) @ ( nth_set_set_a @ Xs @ I3 ) ) )
% 0.07/0.31 => ( member_list_set_a @ Ys @ ( listset_set_a @ Xs ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % listsetI
% 0.07/0.31 thf(fact_75_len__qs,axiom,
% 0.07/0.31 ( ( size_size_list_a @ qs )
% 0.07/0.31 = ( size_size_list_set_a @ ss1 ) ) ).
% 0.07/0.31
% 0.07/0.31 % len_qs
% 0.07/0.31 thf(fact_76_f__bij,axiom,
% 0.07/0.31 bij_betw_nat_nat @ f @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ ss2 ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ ss1 ) ) ).
% 0.07/0.31
% 0.07/0.31 % f_bij
% 0.07/0.31 thf(fact_77_nth__equalityI,axiom,
% 0.07/0.31 ! [Xs: list_list_a,Ys: list_list_a] :
% 0.07/0.31 ( ( ( size_s1427607542list_a @ Xs )
% 0.07/0.31 = ( size_s1427607542list_a @ Ys ) )
% 0.07/0.31 => ( ! [I3: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I3 @ ( size_s1427607542list_a @ Xs ) )
% 0.07/0.31 => ( ( nth_list_a @ Xs @ I3 )
% 0.07/0.31 = ( nth_list_a @ Ys @ I3 ) ) )
% 0.07/0.31 => ( Xs = Ys ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % nth_equalityI
% 0.07/0.31 thf(fact_78_nth__equalityI,axiom,
% 0.07/0.31 ! [Xs: list_set_list_a,Ys: list_set_list_a] :
% 0.07/0.31 ( ( ( size_s1635937238list_a @ Xs )
% 0.07/0.31 = ( size_s1635937238list_a @ Ys ) )
% 0.07/0.31 => ( ! [I3: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I3 @ ( size_s1635937238list_a @ Xs ) )
% 0.07/0.31 => ( ( nth_set_list_a @ Xs @ I3 )
% 0.07/0.31 = ( nth_set_list_a @ Ys @ I3 ) ) )
% 0.07/0.31 => ( Xs = Ys ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % nth_equalityI
% 0.07/0.31 thf(fact_79_nth__equalityI,axiom,
% 0.07/0.31 ! [Xs: list_set_set_a,Ys: list_set_set_a] :
% 0.07/0.31 ( ( ( size_s739728560_set_a @ Xs )
% 0.07/0.31 = ( size_s739728560_set_a @ Ys ) )
% 0.07/0.31 => ( ! [I3: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I3 @ ( size_s739728560_set_a @ Xs ) )
% 0.07/0.31 => ( ( nth_set_set_a @ Xs @ I3 )
% 0.07/0.31 = ( nth_set_set_a @ Ys @ I3 ) ) )
% 0.07/0.31 => ( Xs = Ys ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % nth_equalityI
% 0.07/0.31 thf(fact_80_nth__equalityI,axiom,
% 0.07/0.31 ! [Xs: list_nat,Ys: list_nat] :
% 0.07/0.31 ( ( ( size_size_list_nat @ Xs )
% 0.07/0.31 = ( size_size_list_nat @ Ys ) )
% 0.07/0.31 => ( ! [I3: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
% 0.07/0.31 => ( ( nth_nat @ Xs @ I3 )
% 0.07/0.31 = ( nth_nat @ Ys @ I3 ) ) )
% 0.07/0.31 => ( Xs = Ys ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % nth_equalityI
% 0.07/0.31 thf(fact_81_nth__equalityI,axiom,
% 0.07/0.31 ! [Xs: list_a,Ys: list_a] :
% 0.07/0.31 ( ( ( size_size_list_a @ Xs )
% 0.07/0.31 = ( size_size_list_a @ Ys ) )
% 0.07/0.31 => ( ! [I3: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
% 0.07/0.31 => ( ( nth_a @ Xs @ I3 )
% 0.07/0.31 = ( nth_a @ Ys @ I3 ) ) )
% 0.07/0.31 => ( Xs = Ys ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % nth_equalityI
% 0.07/0.31 thf(fact_82_nth__equalityI,axiom,
% 0.07/0.31 ! [Xs: list_set_a,Ys: list_set_a] :
% 0.07/0.31 ( ( ( size_size_list_set_a @ Xs )
% 0.07/0.31 = ( size_size_list_set_a @ Ys ) )
% 0.07/0.31 => ( ! [I3: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I3 @ ( size_size_list_set_a @ Xs ) )
% 0.07/0.31 => ( ( nth_set_a @ Xs @ I3 )
% 0.07/0.31 = ( nth_set_a @ Ys @ I3 ) ) )
% 0.07/0.31 => ( Xs = Ys ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % nth_equalityI
% 0.07/0.31 thf(fact_83_Skolem__list__nth,axiom,
% 0.07/0.31 ! [K: nat,P: nat > list_a > $o] :
% 0.07/0.31 ( ( ! [I4: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I4 @ K )
% 0.07/0.31 => ? [X3: list_a] : ( P @ I4 @ X3 ) ) )
% 0.07/0.31 = ( ? [Xs4: list_list_a] :
% 0.07/0.31 ( ( ( size_s1427607542list_a @ Xs4 )
% 0.07/0.31 = K )
% 0.07/0.31 & ! [I4: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I4 @ K )
% 0.07/0.31 => ( P @ I4 @ ( nth_list_a @ Xs4 @ I4 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Skolem_list_nth
% 0.07/0.31 thf(fact_84_Skolem__list__nth,axiom,
% 0.07/0.31 ! [K: nat,P: nat > set_list_a > $o] :
% 0.07/0.31 ( ( ! [I4: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I4 @ K )
% 0.07/0.31 => ? [X3: set_list_a] : ( P @ I4 @ X3 ) ) )
% 0.07/0.31 = ( ? [Xs4: list_set_list_a] :
% 0.07/0.31 ( ( ( size_s1635937238list_a @ Xs4 )
% 0.07/0.31 = K )
% 0.07/0.31 & ! [I4: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I4 @ K )
% 0.07/0.31 => ( P @ I4 @ ( nth_set_list_a @ Xs4 @ I4 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Skolem_list_nth
% 0.07/0.31 thf(fact_85_Skolem__list__nth,axiom,
% 0.07/0.31 ! [K: nat,P: nat > set_set_a > $o] :
% 0.07/0.31 ( ( ! [I4: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I4 @ K )
% 0.07/0.31 => ? [X3: set_set_a] : ( P @ I4 @ X3 ) ) )
% 0.07/0.31 = ( ? [Xs4: list_set_set_a] :
% 0.07/0.31 ( ( ( size_s739728560_set_a @ Xs4 )
% 0.07/0.31 = K )
% 0.07/0.31 & ! [I4: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I4 @ K )
% 0.07/0.31 => ( P @ I4 @ ( nth_set_set_a @ Xs4 @ I4 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Skolem_list_nth
% 0.07/0.31 thf(fact_86_Skolem__list__nth,axiom,
% 0.07/0.31 ! [K: nat,P: nat > nat > $o] :
% 0.07/0.31 ( ( ! [I4: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I4 @ K )
% 0.07/0.31 => ? [X3: nat] : ( P @ I4 @ X3 ) ) )
% 0.07/0.31 = ( ? [Xs4: list_nat] :
% 0.07/0.31 ( ( ( size_size_list_nat @ Xs4 )
% 0.07/0.31 = K )
% 0.07/0.31 & ! [I4: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I4 @ K )
% 0.07/0.31 => ( P @ I4 @ ( nth_nat @ Xs4 @ I4 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Skolem_list_nth
% 0.07/0.31 thf(fact_87_Skolem__list__nth,axiom,
% 0.07/0.31 ! [K: nat,P: nat > a > $o] :
% 0.07/0.31 ( ( ! [I4: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I4 @ K )
% 0.07/0.31 => ? [X3: a] : ( P @ I4 @ X3 ) ) )
% 0.07/0.31 = ( ? [Xs4: list_a] :
% 0.07/0.31 ( ( ( size_size_list_a @ Xs4 )
% 0.07/0.31 = K )
% 0.07/0.31 & ! [I4: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I4 @ K )
% 0.07/0.31 => ( P @ I4 @ ( nth_a @ Xs4 @ I4 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Skolem_list_nth
% 0.07/0.31 thf(fact_88_Skolem__list__nth,axiom,
% 0.07/0.31 ! [K: nat,P: nat > set_a > $o] :
% 0.07/0.31 ( ( ! [I4: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I4 @ K )
% 0.07/0.31 => ? [X3: set_a] : ( P @ I4 @ X3 ) ) )
% 0.07/0.31 = ( ? [Xs4: list_set_a] :
% 0.07/0.31 ( ( ( size_size_list_set_a @ Xs4 )
% 0.07/0.31 = K )
% 0.07/0.31 & ! [I4: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I4 @ K )
% 0.07/0.31 => ( P @ I4 @ ( nth_set_a @ Xs4 @ I4 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Skolem_list_nth
% 0.07/0.31 thf(fact_89_list__eq__iff__nth__eq,axiom,
% 0.07/0.31 ( ( ^ [Y2: list_list_a,Z: list_list_a] : ( Y2 = Z ) )
% 0.07/0.31 = ( ^ [Xs4: list_list_a,Ys3: list_list_a] :
% 0.07/0.31 ( ( ( size_s1427607542list_a @ Xs4 )
% 0.07/0.31 = ( size_s1427607542list_a @ Ys3 ) )
% 0.07/0.31 & ! [I4: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I4 @ ( size_s1427607542list_a @ Xs4 ) )
% 0.07/0.31 => ( ( nth_list_a @ Xs4 @ I4 )
% 0.07/0.31 = ( nth_list_a @ Ys3 @ I4 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % list_eq_iff_nth_eq
% 0.07/0.31 thf(fact_90_list__eq__iff__nth__eq,axiom,
% 0.07/0.31 ( ( ^ [Y2: list_set_list_a,Z: list_set_list_a] : ( Y2 = Z ) )
% 0.07/0.31 = ( ^ [Xs4: list_set_list_a,Ys3: list_set_list_a] :
% 0.07/0.31 ( ( ( size_s1635937238list_a @ Xs4 )
% 0.07/0.31 = ( size_s1635937238list_a @ Ys3 ) )
% 0.07/0.31 & ! [I4: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I4 @ ( size_s1635937238list_a @ Xs4 ) )
% 0.07/0.31 => ( ( nth_set_list_a @ Xs4 @ I4 )
% 0.07/0.31 = ( nth_set_list_a @ Ys3 @ I4 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % list_eq_iff_nth_eq
% 0.07/0.31 thf(fact_91_list__eq__iff__nth__eq,axiom,
% 0.07/0.31 ( ( ^ [Y2: list_set_set_a,Z: list_set_set_a] : ( Y2 = Z ) )
% 0.07/0.31 = ( ^ [Xs4: list_set_set_a,Ys3: list_set_set_a] :
% 0.07/0.31 ( ( ( size_s739728560_set_a @ Xs4 )
% 0.07/0.31 = ( size_s739728560_set_a @ Ys3 ) )
% 0.07/0.31 & ! [I4: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I4 @ ( size_s739728560_set_a @ Xs4 ) )
% 0.07/0.31 => ( ( nth_set_set_a @ Xs4 @ I4 )
% 0.07/0.31 = ( nth_set_set_a @ Ys3 @ I4 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % list_eq_iff_nth_eq
% 0.07/0.31 thf(fact_92_list__eq__iff__nth__eq,axiom,
% 0.07/0.31 ( ( ^ [Y2: list_nat,Z: list_nat] : ( Y2 = Z ) )
% 0.07/0.31 = ( ^ [Xs4: list_nat,Ys3: list_nat] :
% 0.07/0.31 ( ( ( size_size_list_nat @ Xs4 )
% 0.07/0.31 = ( size_size_list_nat @ Ys3 ) )
% 0.07/0.31 & ! [I4: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs4 ) )
% 0.07/0.31 => ( ( nth_nat @ Xs4 @ I4 )
% 0.07/0.31 = ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % list_eq_iff_nth_eq
% 0.07/0.31 thf(fact_93_list__eq__iff__nth__eq,axiom,
% 0.07/0.31 ( ( ^ [Y2: list_a,Z: list_a] : ( Y2 = Z ) )
% 0.07/0.31 = ( ^ [Xs4: list_a,Ys3: list_a] :
% 0.07/0.31 ( ( ( size_size_list_a @ Xs4 )
% 0.07/0.31 = ( size_size_list_a @ Ys3 ) )
% 0.07/0.31 & ! [I4: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs4 ) )
% 0.07/0.31 => ( ( nth_a @ Xs4 @ I4 )
% 0.07/0.31 = ( nth_a @ Ys3 @ I4 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % list_eq_iff_nth_eq
% 0.07/0.31 thf(fact_94_list__eq__iff__nth__eq,axiom,
% 0.07/0.31 ( ( ^ [Y2: list_set_a,Z: list_set_a] : ( Y2 = Z ) )
% 0.07/0.31 = ( ^ [Xs4: list_set_a,Ys3: list_set_a] :
% 0.07/0.31 ( ( ( size_size_list_set_a @ Xs4 )
% 0.07/0.31 = ( size_size_list_set_a @ Ys3 ) )
% 0.07/0.31 & ! [I4: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I4 @ ( size_size_list_set_a @ Xs4 ) )
% 0.07/0.31 => ( ( nth_set_a @ Xs4 @ I4 )
% 0.07/0.31 = ( nth_set_a @ Ys3 @ I4 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % list_eq_iff_nth_eq
% 0.07/0.31 thf(fact_95_lessThan__eq__iff,axiom,
% 0.07/0.31 ! [X4: nat,Y3: nat] :
% 0.07/0.31 ( ( ( set_ord_lessThan_nat @ X4 )
% 0.07/0.31 = ( set_ord_lessThan_nat @ Y3 ) )
% 0.07/0.31 = ( X4 = Y3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % lessThan_eq_iff
% 0.07/0.31 thf(fact_96_rev__is__rev__conv,axiom,
% 0.07/0.31 ! [Xs: list_a,Ys: list_a] :
% 0.07/0.31 ( ( ( rev_a @ Xs )
% 0.07/0.31 = ( rev_a @ Ys ) )
% 0.07/0.31 = ( Xs = Ys ) ) ).
% 0.07/0.31
% 0.07/0.31 % rev_is_rev_conv
% 0.07/0.31 thf(fact_97_rev__is__rev__conv,axiom,
% 0.07/0.31 ! [Xs: list_set_a,Ys: list_set_a] :
% 0.07/0.31 ( ( ( rev_set_a @ Xs )
% 0.07/0.31 = ( rev_set_a @ Ys ) )
% 0.07/0.31 = ( Xs = Ys ) ) ).
% 0.07/0.31
% 0.07/0.31 % rev_is_rev_conv
% 0.07/0.31 thf(fact_98_rev__rev__ident,axiom,
% 0.07/0.31 ! [Xs: list_a] :
% 0.07/0.31 ( ( rev_a @ ( rev_a @ Xs ) )
% 0.07/0.31 = Xs ) ).
% 0.07/0.31
% 0.07/0.31 % rev_rev_ident
% 0.07/0.31 thf(fact_99_rev__rev__ident,axiom,
% 0.07/0.31 ! [Xs: list_set_a] :
% 0.07/0.31 ( ( rev_set_a @ ( rev_set_a @ Xs ) )
% 0.07/0.31 = Xs ) ).
% 0.07/0.31
% 0.07/0.31 % rev_rev_ident
% 0.07/0.31 thf(fact_100_len__ss1,axiom,
% 0.07/0.31 ( ( size_size_list_set_a @ ss1 )
% 0.07/0.31 = ( size_size_list_set_a @ ss2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % len_ss1
% 0.07/0.31 thf(fact_101_g,axiom,
% 0.07/0.31 ! [I: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I @ ( size_size_list_set_a @ ss1 ) )
% 0.07/0.31 => ( ( nth_set_a @ ss2 @ I )
% 0.07/0.31 = ( nth_set_a @ ss1 @ ( g @ I ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % g
% 0.07/0.31 thf(fact_102_f,axiom,
% 0.07/0.31 ! [I: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I @ ( size_size_list_set_a @ ss2 ) )
% 0.07/0.31 => ( ( nth_set_a @ ss1 @ I )
% 0.07/0.31 = ( nth_set_a @ ss2 @ ( f @ I ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % f
% 0.07/0.31 thf(fact_103__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062f_O_A_092_060lbrakk_062bij__betw_Af_A_123_O_O_060length_Ass2_125_A_123_O_O_060length_Ass1_125_059_A_092_060And_062i_O_Ai_A_060_Alength_Ass2_A_092_060Longrightarrow_062_Ass1_A_B_Ai_A_061_Ass2_A_B_Af_Ai_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.07/0.31 ~ ! [F2: nat > nat] :
% 0.07/0.31 ( ( bij_betw_nat_nat @ F2 @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ ss2 ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ ss1 ) ) )
% 0.07/0.31 => ~ ! [I2: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I2 @ ( size_size_list_set_a @ ss2 ) )
% 0.07/0.31 => ( ( nth_set_a @ ss1 @ I2 )
% 0.07/0.31 = ( nth_set_a @ ss2 @ ( F2 @ I2 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % \<open>\<And>thesis. (\<And>f. \<lbrakk>bij_betw f {..<length ss2} {..<length ss1}; \<And>i. i < length ss2 \<Longrightarrow> ss1 ! i = ss2 ! f i\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.07/0.31 thf(fact_104__092_060open_062_092_060exists_062f_O_Abij__betw_Af_A_123_O_O_060length_Ass1_125_A_123_O_O_060length_Ass2_125_A_092_060and_062_A_I_092_060forall_062i_060length_Ass1_O_Ass1_A_B_Ai_A_061_Ass2_A_B_Af_Ai_J_092_060close_062,axiom,
% 0.07/0.31 ? [F2: nat > nat] :
% 0.07/0.31 ( ( bij_betw_nat_nat @ F2 @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ ss1 ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ ss2 ) ) )
% 0.07/0.31 & ! [I2: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I2 @ ( size_size_list_set_a @ ss1 ) )
% 0.07/0.31 => ( ( nth_set_a @ ss1 @ I2 )
% 0.07/0.31 = ( nth_set_a @ ss2 @ ( F2 @ I2 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % \<open>\<exists>f. bij_betw f {..<length ss1} {..<length ss2} \<and> (\<forall>i<length ss1. ss1 ! i = ss2 ! f i)\<close>
% 0.07/0.31 thf(fact_105_f__g,axiom,
% 0.07/0.31 ! [I: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I @ ( size_size_list_set_a @ ss1 ) )
% 0.07/0.31 => ( ( f @ ( g @ I ) )
% 0.07/0.31 = I ) ) ).
% 0.07/0.31
% 0.07/0.31 % f_g
% 0.07/0.31 thf(fact_106_g__f,axiom,
% 0.07/0.31 ! [I: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I @ ( size_size_list_set_a @ ss2 ) )
% 0.07/0.31 => ( ( g @ ( f @ I ) )
% 0.07/0.31 = I ) ) ).
% 0.07/0.31
% 0.07/0.31 % g_f
% 0.07/0.31 thf(fact_107_listsetD_I2_J,axiom,
% 0.07/0.31 ! [Ys: list_set_set_a,Xs: list_set_set_set_a,I: nat] :
% 0.07/0.31 ( ( member814963949_set_a @ Ys @ ( listset_set_set_a @ Xs ) )
% 0.07/0.31 => ( ( ord_less_nat @ I @ ( size_s726512144_set_a @ Xs ) )
% 0.07/0.31 => ( member_set_set_a @ ( nth_set_set_a @ Ys @ I ) @ ( nth_set_set_set_a @ Xs @ I ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % listsetD(2)
% 0.07/0.31 thf(fact_108_listsetD_I2_J,axiom,
% 0.07/0.31 ! [Ys: list_set_list_a,Xs: list_set_set_list_a,I: nat] :
% 0.07/0.31 ( ( member101195155list_a @ Ys @ ( listset_set_list_a @ Xs ) )
% 0.07/0.31 => ( ( ord_less_nat @ I @ ( size_s469631926list_a @ Xs ) )
% 0.07/0.31 => ( member_set_list_a @ ( nth_set_list_a @ Ys @ I ) @ ( nth_set_set_list_a @ Xs @ I ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % listsetD(2)
% 0.07/0.31 thf(fact_109_listsetD_I2_J,axiom,
% 0.07/0.31 ! [Ys: list_multiset_a,Xs: list_set_multiset_a,I: nat] :
% 0.07/0.31 ( ( member518144627iset_a @ Ys @ ( listset_multiset_a @ Xs ) )
% 0.07/0.31 => ( ( ord_less_nat @ I @ ( size_s802269334iset_a @ Xs ) )
% 0.07/0.31 => ( member_multiset_a @ ( nth_multiset_a @ Ys @ I ) @ ( nth_set_multiset_a @ Xs @ I ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % listsetD(2)
% 0.07/0.31 thf(fact_110_listsetD_I2_J,axiom,
% 0.07/0.31 ! [Ys: list_list_list_a,Xs: list_set_list_list_a,I: nat] :
% 0.07/0.31 ( ( member1511395513list_a @ Ys @ ( listset_list_list_a @ Xs ) )
% 0.07/0.31 => ( ( ord_less_nat @ I @ ( size_s240404444list_a @ Xs ) )
% 0.07/0.31 => ( member_list_list_a @ ( nth_list_list_a @ Ys @ I ) @ ( nth_set_list_list_a @ Xs @ I ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % listsetD(2)
% 0.07/0.31 thf(fact_111_listsetD_I2_J,axiom,
% 0.07/0.31 ! [Ys: list_list_set_a,Xs: list_set_list_set_a,I: nat] :
% 0.07/0.31 ( ( member954073875_set_a @ Ys @ ( listset_list_set_a @ Xs ) )
% 0.07/0.31 => ( ( ord_less_nat @ I @ ( size_s926492982_set_a @ Xs ) )
% 0.07/0.31 => ( member_list_set_a @ ( nth_list_set_a @ Ys @ I ) @ ( nth_set_list_set_a @ Xs @ I ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % listsetD(2)
% 0.07/0.31 thf(fact_112_listsetD_I2_J,axiom,
% 0.07/0.31 ! [Ys: list_nat,Xs: list_set_nat,I: nat] :
% 0.07/0.31 ( ( member_list_nat @ Ys @ ( listset_nat @ Xs ) )
% 0.07/0.31 => ( ( ord_less_nat @ I @ ( size_s577819178et_nat @ Xs ) )
% 0.07/0.31 => ( member_nat @ ( nth_nat @ Ys @ I ) @ ( nth_set_nat @ Xs @ I ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % listsetD(2)
% 0.07/0.31 thf(fact_113_listsetD_I2_J,axiom,
% 0.07/0.31 ! [Ys: list_set_a,Xs: list_set_set_a,I: nat] :
% 0.07/0.31 ( ( member_list_set_a @ Ys @ ( listset_set_a @ Xs ) )
% 0.07/0.31 => ( ( ord_less_nat @ I @ ( size_s739728560_set_a @ Xs ) )
% 0.07/0.31 => ( member_set_a @ ( nth_set_a @ Ys @ I ) @ ( nth_set_set_a @ Xs @ I ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % listsetD(2)
% 0.07/0.31 thf(fact_114_listsetD_I2_J,axiom,
% 0.07/0.31 ! [Ys: list_list_a,Xs: list_set_list_a,I: nat] :
% 0.07/0.31 ( ( member_list_list_a @ Ys @ ( listset_list_a @ Xs ) )
% 0.07/0.31 => ( ( ord_less_nat @ I @ ( size_s1635937238list_a @ Xs ) )
% 0.07/0.31 => ( member_list_a @ ( nth_list_a @ Ys @ I ) @ ( nth_set_list_a @ Xs @ I ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % listsetD(2)
% 0.07/0.31 thf(fact_115_listsetD_I2_J,axiom,
% 0.07/0.31 ! [Ys: list_a,Xs: list_set_a,I: nat] :
% 0.07/0.31 ( ( member_list_a @ Ys @ ( listset_a @ Xs ) )
% 0.07/0.31 => ( ( ord_less_nat @ I @ ( size_size_list_set_a @ Xs ) )
% 0.07/0.31 => ( member_a @ ( nth_a @ Ys @ I ) @ ( nth_set_a @ Xs @ I ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % listsetD(2)
% 0.07/0.31 thf(fact_116_neq__if__length__neq,axiom,
% 0.07/0.31 ! [Xs: list_list_a,Ys: list_list_a] :
% 0.07/0.31 ( ( ( size_s1427607542list_a @ Xs )
% 0.07/0.31 != ( size_s1427607542list_a @ Ys ) )
% 0.07/0.31 => ( Xs != Ys ) ) ).
% 0.07/0.31
% 0.07/0.31 % neq_if_length_neq
% 0.07/0.31 thf(fact_117_neq__if__length__neq,axiom,
% 0.07/0.31 ! [Xs: list_set_list_a,Ys: list_set_list_a] :
% 0.07/0.31 ( ( ( size_s1635937238list_a @ Xs )
% 0.07/0.31 != ( size_s1635937238list_a @ Ys ) )
% 0.07/0.31 => ( Xs != Ys ) ) ).
% 0.07/0.31
% 0.07/0.31 % neq_if_length_neq
% 0.07/0.31 thf(fact_118_neq__if__length__neq,axiom,
% 0.07/0.31 ! [Xs: list_set_set_a,Ys: list_set_set_a] :
% 0.07/0.31 ( ( ( size_s739728560_set_a @ Xs )
% 0.07/0.31 != ( size_s739728560_set_a @ Ys ) )
% 0.07/0.31 => ( Xs != Ys ) ) ).
% 0.07/0.31
% 0.07/0.31 % neq_if_length_neq
% 0.07/0.31 thf(fact_119_neq__if__length__neq,axiom,
% 0.07/0.31 ! [Xs: list_nat,Ys: list_nat] :
% 0.07/0.31 ( ( ( size_size_list_nat @ Xs )
% 0.07/0.31 != ( size_size_list_nat @ Ys ) )
% 0.07/0.31 => ( Xs != Ys ) ) ).
% 0.07/0.31
% 0.07/0.31 % neq_if_length_neq
% 0.07/0.31 thf(fact_120_neq__if__length__neq,axiom,
% 0.07/0.31 ! [Xs: list_a,Ys: list_a] :
% 0.07/0.31 ( ( ( size_size_list_a @ Xs )
% 0.07/0.31 != ( size_size_list_a @ Ys ) )
% 0.07/0.31 => ( Xs != Ys ) ) ).
% 0.07/0.31
% 0.07/0.31 % neq_if_length_neq
% 0.07/0.31 thf(fact_121_neq__if__length__neq,axiom,
% 0.07/0.31 ! [Xs: list_set_a,Ys: list_set_a] :
% 0.07/0.31 ( ( ( size_size_list_set_a @ Xs )
% 0.07/0.31 != ( size_size_list_set_a @ Ys ) )
% 0.07/0.31 => ( Xs != Ys ) ) ).
% 0.07/0.31
% 0.07/0.31 % neq_if_length_neq
% 0.07/0.31 thf(fact_122_mem__Collect__eq,axiom,
% 0.07/0.31 ! [A2: multiset_a,P: multiset_a > $o] :
% 0.07/0.31 ( ( member_multiset_a @ A2 @ ( collect_multiset_a @ P ) )
% 0.07/0.31 = ( P @ A2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % mem_Collect_eq
% 0.07/0.31 thf(fact_123_mem__Collect__eq,axiom,
% 0.07/0.31 ! [A2: list_list_a,P: list_list_a > $o] :
% 0.07/0.31 ( ( member_list_list_a @ A2 @ ( collect_list_list_a @ P ) )
% 0.07/0.31 = ( P @ A2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % mem_Collect_eq
% 0.07/0.31 thf(fact_124_mem__Collect__eq,axiom,
% 0.07/0.31 ! [A2: list_set_a,P: list_set_a > $o] :
% 0.07/0.31 ( ( member_list_set_a @ A2 @ ( collect_list_set_a @ P ) )
% 0.07/0.31 = ( P @ A2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % mem_Collect_eq
% 0.07/0.31 thf(fact_125_mem__Collect__eq,axiom,
% 0.07/0.31 ! [A2: set_a,P: set_a > $o] :
% 0.07/0.31 ( ( member_set_a @ A2 @ ( collect_set_a @ P ) )
% 0.07/0.31 = ( P @ A2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % mem_Collect_eq
% 0.07/0.31 thf(fact_126_mem__Collect__eq,axiom,
% 0.07/0.31 ! [A2: nat,P: nat > $o] :
% 0.07/0.31 ( ( member_nat @ A2 @ ( collect_nat @ P ) )
% 0.07/0.31 = ( P @ A2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % mem_Collect_eq
% 0.07/0.31 thf(fact_127_mem__Collect__eq,axiom,
% 0.07/0.31 ! [A2: a,P: a > $o] :
% 0.07/0.31 ( ( member_a @ A2 @ ( collect_a @ P ) )
% 0.07/0.31 = ( P @ A2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % mem_Collect_eq
% 0.07/0.31 thf(fact_128_mem__Collect__eq,axiom,
% 0.07/0.31 ! [A2: list_a,P: list_a > $o] :
% 0.07/0.31 ( ( member_list_a @ A2 @ ( collect_list_a @ P ) )
% 0.07/0.31 = ( P @ A2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % mem_Collect_eq
% 0.07/0.31 thf(fact_129_Collect__mem__eq,axiom,
% 0.07/0.31 ! [A: set_multiset_a] :
% 0.07/0.31 ( ( collect_multiset_a
% 0.07/0.31 @ ^ [X5: multiset_a] : ( member_multiset_a @ X5 @ A ) )
% 0.07/0.31 = A ) ).
% 0.07/0.31
% 0.07/0.31 % Collect_mem_eq
% 0.07/0.31 thf(fact_130_Collect__mem__eq,axiom,
% 0.07/0.31 ! [A: set_list_list_a] :
% 0.07/0.31 ( ( collect_list_list_a
% 0.07/0.31 @ ^ [X5: list_list_a] : ( member_list_list_a @ X5 @ A ) )
% 0.07/0.31 = A ) ).
% 0.07/0.31
% 0.07/0.31 % Collect_mem_eq
% 0.07/0.31 thf(fact_131_Collect__mem__eq,axiom,
% 0.07/0.31 ! [A: set_list_set_a] :
% 0.07/0.31 ( ( collect_list_set_a
% 0.07/0.31 @ ^ [X5: list_set_a] : ( member_list_set_a @ X5 @ A ) )
% 0.07/0.31 = A ) ).
% 0.07/0.31
% 0.07/0.31 % Collect_mem_eq
% 0.07/0.31 thf(fact_132_Collect__mem__eq,axiom,
% 0.07/0.31 ! [A: set_set_a] :
% 0.07/0.31 ( ( collect_set_a
% 0.07/0.31 @ ^ [X5: set_a] : ( member_set_a @ X5 @ A ) )
% 0.07/0.31 = A ) ).
% 0.07/0.31
% 0.07/0.31 % Collect_mem_eq
% 0.07/0.31 thf(fact_133_Collect__mem__eq,axiom,
% 0.07/0.31 ! [A: set_nat] :
% 0.07/0.31 ( ( collect_nat
% 0.07/0.31 @ ^ [X5: nat] : ( member_nat @ X5 @ A ) )
% 0.07/0.31 = A ) ).
% 0.07/0.31
% 0.07/0.31 % Collect_mem_eq
% 0.07/0.31 thf(fact_134_Collect__mem__eq,axiom,
% 0.07/0.31 ! [A: set_a] :
% 0.07/0.31 ( ( collect_a
% 0.07/0.31 @ ^ [X5: a] : ( member_a @ X5 @ A ) )
% 0.07/0.31 = A ) ).
% 0.07/0.31
% 0.07/0.31 % Collect_mem_eq
% 0.07/0.31 thf(fact_135_Collect__mem__eq,axiom,
% 0.07/0.31 ! [A: set_list_a] :
% 0.07/0.31 ( ( collect_list_a
% 0.07/0.31 @ ^ [X5: list_a] : ( member_list_a @ X5 @ A ) )
% 0.07/0.31 = A ) ).
% 0.07/0.31
% 0.07/0.31 % Collect_mem_eq
% 0.07/0.31 thf(fact_136_Collect__cong,axiom,
% 0.07/0.31 ! [P: list_a > $o,Q: list_a > $o] :
% 0.07/0.31 ( ! [X6: list_a] :
% 0.07/0.31 ( ( P @ X6 )
% 0.07/0.31 = ( Q @ X6 ) )
% 0.07/0.31 => ( ( collect_list_a @ P )
% 0.07/0.31 = ( collect_list_a @ Q ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Collect_cong
% 0.07/0.31 thf(fact_137_Collect__cong,axiom,
% 0.07/0.31 ! [P: a > $o,Q: a > $o] :
% 0.07/0.31 ( ! [X6: a] :
% 0.07/0.31 ( ( P @ X6 )
% 0.07/0.31 = ( Q @ X6 ) )
% 0.07/0.31 => ( ( collect_a @ P )
% 0.07/0.31 = ( collect_a @ Q ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Collect_cong
% 0.07/0.31 thf(fact_138_Ex__list__of__length,axiom,
% 0.07/0.31 ! [N: nat] :
% 0.07/0.31 ? [Xs2: list_list_a] :
% 0.07/0.31 ( ( size_s1427607542list_a @ Xs2 )
% 0.07/0.31 = N ) ).
% 0.07/0.31
% 0.07/0.31 % Ex_list_of_length
% 0.07/0.31 thf(fact_139_Ex__list__of__length,axiom,
% 0.07/0.31 ! [N: nat] :
% 0.07/0.31 ? [Xs2: list_set_list_a] :
% 0.07/0.31 ( ( size_s1635937238list_a @ Xs2 )
% 0.07/0.31 = N ) ).
% 0.07/0.31
% 0.07/0.31 % Ex_list_of_length
% 0.07/0.31 thf(fact_140_Ex__list__of__length,axiom,
% 0.07/0.31 ! [N: nat] :
% 0.07/0.31 ? [Xs2: list_set_set_a] :
% 0.07/0.31 ( ( size_s739728560_set_a @ Xs2 )
% 0.07/0.31 = N ) ).
% 0.07/0.31
% 0.07/0.31 % Ex_list_of_length
% 0.07/0.31 thf(fact_141_Ex__list__of__length,axiom,
% 0.07/0.31 ! [N: nat] :
% 0.07/0.31 ? [Xs2: list_nat] :
% 0.07/0.31 ( ( size_size_list_nat @ Xs2 )
% 0.07/0.31 = N ) ).
% 0.07/0.31
% 0.07/0.31 % Ex_list_of_length
% 0.07/0.31 thf(fact_142_Ex__list__of__length,axiom,
% 0.07/0.31 ! [N: nat] :
% 0.07/0.31 ? [Xs2: list_a] :
% 0.07/0.31 ( ( size_size_list_a @ Xs2 )
% 0.07/0.31 = N ) ).
% 0.07/0.31
% 0.07/0.31 % Ex_list_of_length
% 0.07/0.31 thf(fact_143_Ex__list__of__length,axiom,
% 0.07/0.31 ! [N: nat] :
% 0.07/0.31 ? [Xs2: list_set_a] :
% 0.07/0.31 ( ( size_size_list_set_a @ Xs2 )
% 0.07/0.31 = N ) ).
% 0.07/0.31
% 0.07/0.31 % Ex_list_of_length
% 0.07/0.31 thf(fact_144_rev__swap,axiom,
% 0.07/0.31 ! [Xs: list_a,Ys: list_a] :
% 0.07/0.31 ( ( ( rev_a @ Xs )
% 0.07/0.31 = Ys )
% 0.07/0.31 = ( Xs
% 0.07/0.31 = ( rev_a @ Ys ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rev_swap
% 0.07/0.31 thf(fact_145_rev__swap,axiom,
% 0.07/0.31 ! [Xs: list_set_a,Ys: list_set_a] :
% 0.07/0.31 ( ( ( rev_set_a @ Xs )
% 0.07/0.31 = Ys )
% 0.07/0.31 = ( Xs
% 0.07/0.31 = ( rev_set_a @ Ys ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rev_swap
% 0.07/0.31 thf(fact_146_lessThan__strict__subset__iff,axiom,
% 0.07/0.31 ! [M: nat,N: nat] :
% 0.07/0.31 ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
% 0.07/0.31 = ( ord_less_nat @ M @ N ) ) ).
% 0.07/0.31
% 0.07/0.31 % lessThan_strict_subset_iff
% 0.07/0.31 thf(fact_147_length__induct,axiom,
% 0.07/0.31 ! [P: list_set_list_a > $o,Xs: list_set_list_a] :
% 0.07/0.31 ( ! [Xs2: list_set_list_a] :
% 0.07/0.31 ( ! [Ys4: list_set_list_a] :
% 0.07/0.31 ( ( ord_less_nat @ ( size_s1635937238list_a @ Ys4 ) @ ( size_s1635937238list_a @ Xs2 ) )
% 0.07/0.31 => ( P @ Ys4 ) )
% 0.07/0.31 => ( P @ Xs2 ) )
% 0.07/0.31 => ( P @ Xs ) ) ).
% 0.07/0.31
% 0.07/0.31 % length_induct
% 0.07/0.31 thf(fact_148_length__induct,axiom,
% 0.07/0.31 ! [P: list_set_set_a > $o,Xs: list_set_set_a] :
% 0.07/0.31 ( ! [Xs2: list_set_set_a] :
% 0.07/0.31 ( ! [Ys4: list_set_set_a] :
% 0.07/0.31 ( ( ord_less_nat @ ( size_s739728560_set_a @ Ys4 ) @ ( size_s739728560_set_a @ Xs2 ) )
% 0.07/0.31 => ( P @ Ys4 ) )
% 0.07/0.31 => ( P @ Xs2 ) )
% 0.07/0.31 => ( P @ Xs ) ) ).
% 0.07/0.31
% 0.07/0.31 % length_induct
% 0.07/0.31 thf(fact_149_length__induct,axiom,
% 0.07/0.31 ! [P: list_nat > $o,Xs: list_nat] :
% 0.07/0.31 ( ! [Xs2: list_nat] :
% 0.07/0.31 ( ! [Ys4: list_nat] :
% 0.07/0.31 ( ( ord_less_nat @ ( size_size_list_nat @ Ys4 ) @ ( size_size_list_nat @ Xs2 ) )
% 0.07/0.31 => ( P @ Ys4 ) )
% 0.07/0.31 => ( P @ Xs2 ) )
% 0.07/0.31 => ( P @ Xs ) ) ).
% 0.07/0.31
% 0.07/0.31 % length_induct
% 0.07/0.31 thf(fact_150_length__induct,axiom,
% 0.07/0.31 ! [P: list_a > $o,Xs: list_a] :
% 0.07/0.31 ( ! [Xs2: list_a] :
% 0.07/0.31 ( ! [Ys4: list_a] :
% 0.07/0.31 ( ( ord_less_nat @ ( size_size_list_a @ Ys4 ) @ ( size_size_list_a @ Xs2 ) )
% 0.07/0.31 => ( P @ Ys4 ) )
% 0.07/0.31 => ( P @ Xs2 ) )
% 0.07/0.31 => ( P @ Xs ) ) ).
% 0.07/0.31
% 0.07/0.31 % length_induct
% 0.07/0.31 thf(fact_151_length__induct,axiom,
% 0.07/0.31 ! [P: list_set_a > $o,Xs: list_set_a] :
% 0.07/0.31 ( ! [Xs2: list_set_a] :
% 0.07/0.31 ( ! [Ys4: list_set_a] :
% 0.07/0.31 ( ( ord_less_nat @ ( size_size_list_set_a @ Ys4 ) @ ( size_size_list_set_a @ Xs2 ) )
% 0.07/0.31 => ( P @ Ys4 ) )
% 0.07/0.31 => ( P @ Xs2 ) )
% 0.07/0.31 => ( P @ Xs ) ) ).
% 0.07/0.31
% 0.07/0.31 % length_induct
% 0.07/0.31 thf(fact_152_listsetD_I1_J,axiom,
% 0.07/0.31 ! [Ys: list_a,Xs: list_set_a] :
% 0.07/0.31 ( ( member_list_a @ Ys @ ( listset_a @ Xs ) )
% 0.07/0.31 => ( ( size_size_list_a @ Ys )
% 0.07/0.31 = ( size_size_list_set_a @ Xs ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % listsetD(1)
% 0.07/0.31 thf(fact_153_listsetD_I1_J,axiom,
% 0.07/0.31 ! [Ys: list_set_a,Xs: list_set_set_a] :
% 0.07/0.31 ( ( member_list_set_a @ Ys @ ( listset_set_a @ Xs ) )
% 0.07/0.31 => ( ( size_size_list_set_a @ Ys )
% 0.07/0.31 = ( size_s739728560_set_a @ Xs ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % listsetD(1)
% 0.07/0.31 thf(fact_154_permutation__Ex__bij,axiom,
% 0.07/0.31 ! [Xs: list_a,Ys: list_a] :
% 0.07/0.31 ( ( perm_a @ Xs @ Ys )
% 0.07/0.31 => ? [F2: nat > nat] :
% 0.07/0.31 ( ( bij_betw_nat_nat @ F2 @ ( set_ord_lessThan_nat @ ( size_size_list_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_a @ Ys ) ) )
% 0.07/0.31 & ! [I2: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
% 0.07/0.31 => ( ( nth_a @ Xs @ I2 )
% 0.07/0.31 = ( nth_a @ Ys @ ( F2 @ I2 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % permutation_Ex_bij
% 0.07/0.31 thf(fact_155_permutation__Ex__bij,axiom,
% 0.07/0.31 ! [Xs: list_set_a,Ys: list_set_a] :
% 0.07/0.31 ( ( perm_set_a @ Xs @ Ys )
% 0.07/0.31 => ? [F2: nat > nat] :
% 0.07/0.31 ( ( bij_betw_nat_nat @ F2 @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ Ys ) ) )
% 0.07/0.31 & ! [I2: nat] :
% 0.07/0.31 ( ( ord_less_nat @ I2 @ ( size_size_list_set_a @ Xs ) )
% 0.07/0.31 => ( ( nth_set_a @ Xs @ I2 )
% 0.07/0.31 = ( nth_set_a @ Ys @ ( F2 @ I2 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % permutation_Ex_bij
% 0.07/0.31 thf(fact_156_g__def,axiom,
% 0.07/0.31 ( g
% 0.07/0.31 = ( hilber815131374at_nat @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ ss2 ) ) @ f ) ) ).
% 0.07/0.31
% 0.07/0.31 % g_def
% 0.07/0.31 thf(fact_157_perm__refl,axiom,
% 0.07/0.31 ! [L: list_set_a] : ( perm_set_a @ L @ L ) ).
% 0.07/0.31
% 0.07/0.31 % perm_refl
% 0.07/0.31 thf(fact_158_image__eqI,axiom,
% 0.07/0.31 ! [B: a,F: a > a,X4: a,A: set_a] :
% 0.07/0.31 ( ( B
% 0.07/0.31 = ( F @ X4 ) )
% 0.07/0.31 => ( ( member_a @ X4 @ A )
% 0.07/0.31 => ( member_a @ B @ ( image_a_a @ F @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % image_eqI
% 0.07/0.31 thf(fact_159_image__eqI,axiom,
% 0.07/0.31 ! [B: list_a,F: a > list_a,X4: a,A: set_a] :
% 0.07/0.31 ( ( B
% 0.07/0.31 = ( F @ X4 ) )
% 0.07/0.31 => ( ( member_a @ X4 @ A )
% 0.07/0.31 => ( member_list_a @ B @ ( image_a_list_a @ F @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % image_eqI
% 0.07/0.31 thf(fact_160_image__eqI,axiom,
% 0.07/0.31 ! [B: a,F: list_a > a,X4: list_a,A: set_list_a] :
% 0.07/0.31 ( ( B
% 0.07/0.31 = ( F @ X4 ) )
% 0.07/0.31 => ( ( member_list_a @ X4 @ A )
% 0.07/0.31 => ( member_a @ B @ ( image_list_a_a @ F @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % image_eqI
% 0.07/0.31 thf(fact_161_image__eqI,axiom,
% 0.07/0.31 ! [B: list_a,F: list_a > list_a,X4: list_a,A: set_list_a] :
% 0.07/0.31 ( ( B
% 0.07/0.31 = ( F @ X4 ) )
% 0.07/0.31 => ( ( member_list_a @ X4 @ A )
% 0.07/0.31 => ( member_list_a @ B @ ( image_list_a_list_a @ F @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % image_eqI
% 0.07/0.31 thf(fact_162_mset__eq__perm,axiom,
% 0.07/0.31 ! [Xs: list_a,Ys: list_a] :
% 0.07/0.31 ( ( ( mset_a @ Xs )
% 0.07/0.31 = ( mset_a @ Ys ) )
% 0.07/0.31 = ( perm_a @ Xs @ Ys ) ) ).
% 0.07/0.31
% 0.07/0.31 % mset_eq_perm
% 0.07/0.31 thf(fact_163_mset__eq__perm,axiom,
% 0.07/0.31 ! [Xs: list_set_a,Ys: list_set_a] :
% 0.07/0.31 ( ( ( mset_set_a @ Xs )
% 0.07/0.31 = ( mset_set_a @ Ys ) )
% 0.07/0.31 = ( perm_set_a @ Xs @ Ys ) ) ).
% 0.07/0.31
% 0.07/0.31 % mset_eq_perm
% 0.07/0.31 thf(fact_164_perm__rev,axiom,
% 0.07/0.31 ! [Xs: list_set_a] : ( perm_set_a @ ( rev_set_a @ Xs ) @ Xs ) ).
% 0.07/0.31
% 0.07/0.31 % perm_rev
% 0.07/0.31 thf(fact_165_perm__length,axiom,
% 0.07/0.31 ! [Xs: list_a,Ys: list_a] :
% 0.07/0.31 ( ( perm_a @ Xs @ Ys )
% 0.07/0.31 => ( ( size_size_list_a @ Xs )
% 0.07/0.31 = ( size_size_list_a @ Ys ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % perm_length
% 0.07/0.31 thf(fact_166_perm__length,axiom,
% 0.07/0.31 ! [Xs: list_set_a,Ys: list_set_a] :
% 0.07/0.31 ( ( perm_set_a @ Xs @ Ys )
% 0.07/0.31 => ( ( size_size_list_set_a @ Xs )
% 0.07/0.31 = ( size_size_list_set_a @ Ys ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % perm_length
% 0.07/0.31 thf(fact_167_bij__betw__imp__surj__on,axiom,
% 0.07/0.31 ! [F: nat > nat,A: set_nat,B2: set_nat] :
% 0.07/0.31 ( ( bij_betw_nat_nat @ F @ A @ B2 )
% 0.07/0.31 => ( ( image_nat_nat @ F @ A )
% 0.07/0.31 = B2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % bij_betw_imp_surj_on
% 0.07/0.31 thf(fact_168_mbs_Oless__not__eq,axiom,
% 0.07/0.31 ! [X4: list_a,A: set_list_a,Y3: list_a] :
% 0.07/0.31 ( ( member_list_a @ X4 @ A )
% 0.07/0.31 => ( ( ord_less_nat @ ( size_size_list_a @ X4 ) @ ( size_size_list_a @ Y3 ) )
% 0.07/0.31 => ( X4 != Y3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % mbs.less_not_eq
% 0.07/0.31 thf(fact_169_mbs_Oless__not__eq,axiom,
% 0.07/0.31 ! [X4: list_set_a,A: set_list_set_a,Y3: list_set_a] :
% 0.07/0.31 ( ( member_list_set_a @ X4 @ A )
% 0.07/0.31 => ( ( ord_less_nat @ ( size_size_list_set_a @ X4 ) @ ( size_size_list_set_a @ Y3 ) )
% 0.07/0.31 => ( X4 != Y3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % mbs.less_not_eq
% 0.07/0.31 thf(fact_170_size__mset,axiom,
% 0.07/0.31 ! [Xs: list_a] :
% 0.07/0.31 ( ( size_size_multiset_a @ ( mset_a @ Xs ) )
% 0.07/0.31 = ( size_size_list_a @ Xs ) ) ).
% 0.07/0.31
% 0.07/0.31 % size_mset
% 0.07/0.31 thf(fact_171_size__mset,axiom,
% 0.07/0.31 ! [Xs: list_set_a] :
% 0.07/0.31 ( ( size_s657334288_set_a @ ( mset_set_a @ Xs ) )
% 0.07/0.31 = ( size_size_list_set_a @ Xs ) ) ).
% 0.07/0.31
% 0.07/0.31 % size_mset
% 0.07/0.31 thf(fact_172_psubsetD,axiom,
% 0.07/0.31 ! [A: set_a,B2: set_a,C: a] :
% 0.07/0.31 ( ( ord_less_set_a @ A @ B2 )
% 0.07/0.31 => ( ( member_a @ C @ A )
% 0.07/0.31 => ( member_a @ C @ B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % psubsetD
% 0.07/0.31 thf(fact_173_psubsetD,axiom,
% 0.07/0.31 ! [A: set_list_a,B2: set_list_a,C: list_a] :
% 0.07/0.31 ( ( ord_less_set_list_a @ A @ B2 )
% 0.07/0.31 => ( ( member_list_a @ C @ A )
% 0.07/0.31 => ( member_list_a @ C @ B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % psubsetD
% 0.07/0.31 thf(fact_174_imageI,axiom,
% 0.07/0.31 ! [X4: a,A: set_a,F: a > a] :
% 0.07/0.31 ( ( member_a @ X4 @ A )
% 0.07/0.31 => ( member_a @ ( F @ X4 ) @ ( image_a_a @ F @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % imageI
% 0.07/0.31 thf(fact_175_imageI,axiom,
% 0.07/0.31 ! [X4: a,A: set_a,F: a > list_a] :
% 0.07/0.31 ( ( member_a @ X4 @ A )
% 0.07/0.31 => ( member_list_a @ ( F @ X4 ) @ ( image_a_list_a @ F @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % imageI
% 0.07/0.31 thf(fact_176_imageI,axiom,
% 0.07/0.31 ! [X4: list_a,A: set_list_a,F: list_a > a] :
% 0.07/0.31 ( ( member_list_a @ X4 @ A )
% 0.07/0.31 => ( member_a @ ( F @ X4 ) @ ( image_list_a_a @ F @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % imageI
% 0.07/0.31 thf(fact_177_imageI,axiom,
% 0.07/0.31 ! [X4: list_a,A: set_list_a,F: list_a > list_a] :
% 0.07/0.31 ( ( member_list_a @ X4 @ A )
% 0.07/0.31 => ( member_list_a @ ( F @ X4 ) @ ( image_list_a_list_a @ F @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % imageI
% 0.07/0.31 thf(fact_178_rev__image__eqI,axiom,
% 0.07/0.31 ! [X4: a,A: set_a,B: a,F: a > a] :
% 0.07/0.31 ( ( member_a @ X4 @ A )
% 0.07/0.31 => ( ( B
% 0.07/0.31 = ( F @ X4 ) )
% 0.07/0.31 => ( member_a @ B @ ( image_a_a @ F @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rev_image_eqI
% 0.07/0.31 thf(fact_179_rev__image__eqI,axiom,
% 0.07/0.31 ! [X4: a,A: set_a,B: list_a,F: a > list_a] :
% 0.07/0.31 ( ( member_a @ X4 @ A )
% 0.07/0.31 => ( ( B
% 0.07/0.31 = ( F @ X4 ) )
% 0.07/0.31 => ( member_list_a @ B @ ( image_a_list_a @ F @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rev_image_eqI
% 0.07/0.31 thf(fact_180_rev__image__eqI,axiom,
% 0.07/0.31 ! [X4: list_a,A: set_list_a,B: a,F: list_a > a] :
% 0.07/0.31 ( ( member_list_a @ X4 @ A )
% 0.07/0.31 => ( ( B
% 0.07/0.31 = ( F @ X4 ) )
% 0.07/0.31 => ( member_a @ B @ ( image_list_a_a @ F @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rev_image_eqI
% 0.07/0.31 thf(fact_181_rev__image__eqI,axiom,
% 0.07/0.31 ! [X4: list_a,A: set_list_a,B: list_a,F: list_a > list_a] :
% 0.07/0.31 ( ( member_list_a @ X4 @ A )
% 0.07/0.31 => ( ( B
% 0.07/0.31 = ( F @ X4 ) )
% 0.07/0.31 => ( member_list_a @ B @ ( image_list_a_list_a @ F @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rev_image_eqI
% 0.07/0.31 thf(fact_182_direct__decomp__def,axiom,
% 0.07/0.31 ( hilber2037636820comp_a
% 0.07/0.31 = ( ^ [A4: set_a,Ss2: list_set_a] : ( bij_betw_list_a_a @ groups1792256535list_a @ ( listset_a @ Ss2 ) @ A4 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % direct_decomp_def
% 0.07/0.31 thf(fact_183_bij__betwE,axiom,
% 0.07/0.31 ! [F: nat > nat,A: set_nat,B2: set_nat] :
% 0.07/0.31 ( ( bij_betw_nat_nat @ F @ A @ B2 )
% 0.07/0.31 => ! [X: nat] :
% 0.07/0.31 ( ( member_nat @ X @ A )
% 0.07/0.31 => ( member_nat @ ( F @ X ) @ B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % bij_betwE
% 0.07/0.31 thf(fact_184_bij__betw__inv,axiom,
% 0.07/0.31 ! [F: nat > nat,A: set_nat,B2: set_nat] :
% 0.07/0.31 ( ( bij_betw_nat_nat @ F @ A @ B2 )
% 0.07/0.31 => ? [G: nat > nat] : ( bij_betw_nat_nat @ G @ B2 @ A ) ) ).
% 0.07/0.31
% 0.07/0.31 % bij_betw_inv
% 0.07/0.31 thf(fact_185_bij__betw__cong,axiom,
% 0.07/0.31 ! [A: set_nat,F: nat > nat,G2: nat > nat,A5: set_nat] :
% 0.07/0.31 ( ! [A3: nat] :
% 0.07/0.31 ( ( member_nat @ A3 @ A )
% 0.07/0.31 => ( ( F @ A3 )
% 0.07/0.31 = ( G2 @ A3 ) ) )
% 0.07/0.31 => ( ( bij_betw_nat_nat @ F @ A @ A5 )
% 0.07/0.31 = ( bij_betw_nat_nat @ G2 @ A @ A5 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % bij_betw_cong
% 0.07/0.31 thf(fact_186_bij__betw__apply,axiom,
% 0.07/0.31 ! [F: a > a,A: set_a,B2: set_a,A2: a] :
% 0.07/0.31 ( ( bij_betw_a_a @ F @ A @ B2 )
% 0.07/0.31 => ( ( member_a @ A2 @ A )
% 0.07/0.31 => ( member_a @ ( F @ A2 ) @ B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % bij_betw_apply
% 0.07/0.31 thf(fact_187_bij__betw__apply,axiom,
% 0.07/0.31 ! [F: a > list_a,A: set_a,B2: set_list_a,A2: a] :
% 0.07/0.31 ( ( bij_betw_a_list_a @ F @ A @ B2 )
% 0.07/0.31 => ( ( member_a @ A2 @ A )
% 0.07/0.31 => ( member_list_a @ ( F @ A2 ) @ B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % bij_betw_apply
% 0.07/0.31 thf(fact_188_bij__betw__apply,axiom,
% 0.07/0.31 ! [F: list_a > a,A: set_list_a,B2: set_a,A2: list_a] :
% 0.07/0.31 ( ( bij_betw_list_a_a @ F @ A @ B2 )
% 0.07/0.31 => ( ( member_list_a @ A2 @ A )
% 0.07/0.31 => ( member_a @ ( F @ A2 ) @ B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % bij_betw_apply
% 0.07/0.31 thf(fact_189_bij__betw__apply,axiom,
% 0.07/0.31 ! [F: list_a > list_a,A: set_list_a,B2: set_list_a,A2: list_a] :
% 0.07/0.31 ( ( bij_be94573046list_a @ F @ A @ B2 )
% 0.07/0.31 => ( ( member_list_a @ A2 @ A )
% 0.07/0.31 => ( member_list_a @ ( F @ A2 ) @ B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % bij_betw_apply
% 0.07/0.31 thf(fact_190_bij__betw__apply,axiom,
% 0.07/0.31 ! [F: nat > nat,A: set_nat,B2: set_nat,A2: nat] :
% 0.07/0.31 ( ( bij_betw_nat_nat @ F @ A @ B2 )
% 0.07/0.31 => ( ( member_nat @ A2 @ A )
% 0.07/0.31 => ( member_nat @ ( F @ A2 ) @ B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % bij_betw_apply
% 0.07/0.31 thf(fact_191_bij__betw__iff__bijections,axiom,
% 0.07/0.31 ( bij_betw_a_a
% 0.07/0.31 = ( ^ [F3: a > a,A4: set_a,B3: set_a] :
% 0.07/0.31 ? [G3: a > a] :
% 0.07/0.31 ( ! [X5: a] :
% 0.07/0.31 ( ( member_a @ X5 @ A4 )
% 0.07/0.31 => ( ( member_a @ ( F3 @ X5 ) @ B3 )
% 0.07/0.31 & ( ( G3 @ ( F3 @ X5 ) )
% 0.07/0.31 = X5 ) ) )
% 0.07/0.31 & ! [X5: a] :
% 0.07/0.31 ( ( member_a @ X5 @ B3 )
% 0.07/0.31 => ( ( member_a @ ( G3 @ X5 ) @ A4 )
% 0.07/0.31 & ( ( F3 @ ( G3 @ X5 ) )
% 0.07/0.31 = X5 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % bij_betw_iff_bijections
% 0.07/0.31 thf(fact_192_bij__betw__iff__bijections,axiom,
% 0.07/0.31 ( bij_betw_list_a_a
% 0.07/0.31 = ( ^ [F3: list_a > a,A4: set_list_a,B3: set_a] :
% 0.07/0.31 ? [G3: a > list_a] :
% 0.07/0.31 ( ! [X5: list_a] :
% 0.07/0.31 ( ( member_list_a @ X5 @ A4 )
% 0.07/0.31 => ( ( member_a @ ( F3 @ X5 ) @ B3 )
% 0.07/0.31 & ( ( G3 @ ( F3 @ X5 ) )
% 0.07/0.31 = X5 ) ) )
% 0.07/0.31 & ! [X5: a] :
% 0.07/0.31 ( ( member_a @ X5 @ B3 )
% 0.07/0.31 => ( ( member_list_a @ ( G3 @ X5 ) @ A4 )
% 0.07/0.31 & ( ( F3 @ ( G3 @ X5 ) )
% 0.07/0.31 = X5 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % bij_betw_iff_bijections
% 0.07/0.31 thf(fact_193_bij__betw__iff__bijections,axiom,
% 0.07/0.31 ( bij_betw_a_list_a
% 0.07/0.31 = ( ^ [F3: a > list_a,A4: set_a,B3: set_list_a] :
% 0.07/0.31 ? [G3: list_a > a] :
% 0.07/0.31 ( ! [X5: a] :
% 0.07/0.31 ( ( member_a @ X5 @ A4 )
% 0.07/0.31 => ( ( member_list_a @ ( F3 @ X5 ) @ B3 )
% 0.07/0.31 & ( ( G3 @ ( F3 @ X5 ) )
% 0.07/0.31 = X5 ) ) )
% 0.07/0.31 & ! [X5: list_a] :
% 0.07/0.31 ( ( member_list_a @ X5 @ B3 )
% 0.07/0.31 => ( ( member_a @ ( G3 @ X5 ) @ A4 )
% 0.07/0.31 & ( ( F3 @ ( G3 @ X5 ) )
% 0.07/0.31 = X5 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % bij_betw_iff_bijections
% 0.07/0.31 thf(fact_194_bij__betw__iff__bijections,axiom,
% 0.07/0.31 ( bij_be94573046list_a
% 0.07/0.31 = ( ^ [F3: list_a > list_a,A4: set_list_a,B3: set_list_a] :
% 0.07/0.31 ? [G3: list_a > list_a] :
% 0.07/0.31 ( ! [X5: list_a] :
% 0.07/0.31 ( ( member_list_a @ X5 @ A4 )
% 0.07/0.31 => ( ( member_list_a @ ( F3 @ X5 ) @ B3 )
% 0.07/0.31 & ( ( G3 @ ( F3 @ X5 ) )
% 0.07/0.31 = X5 ) ) )
% 0.07/0.31 & ! [X5: list_a] :
% 0.07/0.31 ( ( member_list_a @ X5 @ B3 )
% 0.07/0.31 => ( ( member_list_a @ ( G3 @ X5 ) @ A4 )
% 0.07/0.31 & ( ( F3 @ ( G3 @ X5 ) )
% 0.07/0.31 = X5 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % bij_betw_iff_bijections
% 0.07/0.31 thf(fact_195_bij__betw__iff__bijections,axiom,
% 0.07/0.31 ( bij_betw_nat_nat
% 0.07/0.31 = ( ^ [F3: nat > nat,A4: set_nat,B3: set_nat] :
% 0.07/0.31 ? [G3: nat > nat] :
% 0.07/0.31 ( ! [X5: nat] :
% 0.07/0.31 ( ( member_nat @ X5 @ A4 )
% 0.07/0.31 => ( ( member_nat @ ( F3 @ X5 ) @ B3 )
% 0.07/0.31 & ( ( G3 @ ( F3 @ X5 ) )
% 0.07/0.32 = X5 ) ) )
% 0.07/0.32 & ! [X5: nat] :
% 0.07/0.32 ( ( member_nat @ X5 @ B3 )
% 0.07/0.32 => ( ( member_nat @ ( G3 @ X5 ) @ A4 )
% 0.07/0.32 & ( ( F3 @ ( G3 @ X5 ) )
% 0.07/0.32 = X5 ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % bij_betw_iff_bijections
% 0.07/0.32 thf(fact_196_perm__sym,axiom,
% 0.07/0.32 ! [Xs: list_set_a,Ys: list_set_a] :
% 0.07/0.32 ( ( perm_set_a @ Xs @ Ys )
% 0.07/0.32 => ( perm_set_a @ Ys @ Xs ) ) ).
% 0.07/0.32
% 0.07/0.32 % perm_sym
% 0.07/0.32 thf(fact_197_perm_Otrans,axiom,
% 0.07/0.32 ! [Xs: list_set_a,Ys: list_set_a,Zs: list_set_a] :
% 0.07/0.32 ( ( perm_set_a @ Xs @ Ys )
% 0.07/0.32 => ( ( perm_set_a @ Ys @ Zs )
% 0.07/0.32 => ( perm_set_a @ Xs @ Zs ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % perm.trans
% 0.07/0.32 thf(fact_198_bij__betw__inv__into__right,axiom,
% 0.07/0.32 ! [F: nat > nat,A: set_nat,A5: set_nat,A6: nat] :
% 0.07/0.32 ( ( bij_betw_nat_nat @ F @ A @ A5 )
% 0.07/0.32 => ( ( member_nat @ A6 @ A5 )
% 0.07/0.32 => ( ( F @ ( hilber815131374at_nat @ A @ F @ A6 ) )
% 0.07/0.32 = A6 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % bij_betw_inv_into_right
% 0.07/0.32 thf(fact_199_bij__betw__inv__into__left,axiom,
% 0.07/0.32 ! [F: nat > nat,A: set_nat,A5: set_nat,A2: nat] :
% 0.07/0.32 ( ( bij_betw_nat_nat @ F @ A @ A5 )
% 0.07/0.32 => ( ( member_nat @ A2 @ A )
% 0.07/0.32 => ( ( hilber815131374at_nat @ A @ F @ ( F @ A2 ) )
% 0.07/0.32 = A2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % bij_betw_inv_into_left
% 0.07/0.32 thf(fact_200_inv__into__inv__into__eq,axiom,
% 0.07/0.32 ! [F: nat > nat,A: set_nat,A5: set_nat,A2: nat] :
% 0.07/0.32 ( ( bij_betw_nat_nat @ F @ A @ A5 )
% 0.07/0.32 => ( ( member_nat @ A2 @ A )
% 0.07/0.32 => ( ( hilber815131374at_nat @ A5 @ ( hilber815131374at_nat @ A @ F ) @ A2 )
% 0.07/0.32 = ( F @ A2 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % inv_into_inv_into_eq
% 0.07/0.32 thf(fact_201_bij__betw__inv__into,axiom,
% 0.07/0.32 ! [F: nat > nat,A: set_nat,B2: set_nat] :
% 0.07/0.32 ( ( bij_betw_nat_nat @ F @ A @ B2 )
% 0.07/0.32 => ( bij_betw_nat_nat @ ( hilber815131374at_nat @ A @ F ) @ B2 @ A ) ) ).
% 0.07/0.32
% 0.07/0.32 % bij_betw_inv_into
% 0.07/0.32 thf(fact_202_inv__into__injective,axiom,
% 0.07/0.32 ! [A: set_nat,F: nat > nat,X4: nat,Y3: nat] :
% 0.07/0.32 ( ( ( hilber815131374at_nat @ A @ F @ X4 )
% 0.07/0.32 = ( hilber815131374at_nat @ A @ F @ Y3 ) )
% 0.07/0.32 => ( ( member_nat @ X4 @ ( image_nat_nat @ F @ A ) )
% 0.07/0.32 => ( ( member_nat @ Y3 @ ( image_nat_nat @ F @ A ) )
% 0.07/0.32 => ( X4 = Y3 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % inv_into_injective
% 0.07/0.32 thf(fact_203_inv__into__into,axiom,
% 0.07/0.32 ! [X4: a,F: a > a,A: set_a] :
% 0.07/0.32 ( ( member_a @ X4 @ ( image_a_a @ F @ A ) )
% 0.07/0.32 => ( member_a @ ( hilbert_inv_into_a_a @ A @ F @ X4 ) @ A ) ) ).
% 0.07/0.32
% 0.07/0.32 % inv_into_into
% 0.07/0.32 thf(fact_204_inv__into__into,axiom,
% 0.07/0.32 ! [X4: a,F: list_a > a,A: set_list_a] :
% 0.07/0.32 ( ( member_a @ X4 @ ( image_list_a_a @ F @ A ) )
% 0.07/0.32 => ( member_list_a @ ( hilber2125729734st_a_a @ A @ F @ X4 ) @ A ) ) ).
% 0.07/0.32
% 0.07/0.32 % inv_into_into
% 0.07/0.32 thf(fact_205_inv__into__into,axiom,
% 0.07/0.32 ! [X4: list_a,F: a > list_a,A: set_a] :
% 0.07/0.32 ( ( member_list_a @ X4 @ ( image_a_list_a @ F @ A ) )
% 0.07/0.32 => ( member_a @ ( hilber165648082list_a @ A @ F @ X4 ) @ A ) ) ).
% 0.07/0.32
% 0.07/0.32 % inv_into_into
% 0.07/0.32 thf(fact_206_inv__into__into,axiom,
% 0.07/0.32 ! [X4: list_a,F: list_a > list_a,A: set_list_a] :
% 0.07/0.32 ( ( member_list_a @ X4 @ ( image_list_a_list_a @ F @ A ) )
% 0.07/0.32 => ( member_list_a @ ( hilber921249868list_a @ A @ F @ X4 ) @ A ) ) ).
% 0.07/0.32
% 0.07/0.32 % inv_into_into
% 0.07/0.32 thf(fact_207_inv__into__into,axiom,
% 0.07/0.32 ! [X4: nat,F: nat > nat,A: set_nat] :
% 0.07/0.32 ( ( member_nat @ X4 @ ( image_nat_nat @ F @ A ) )
% 0.07/0.32 => ( member_nat @ ( hilber815131374at_nat @ A @ F @ X4 ) @ A ) ) ).
% 0.07/0.32
% 0.07/0.32 % inv_into_into
% 0.07/0.32 thf(fact_208_f__inv__into__f,axiom,
% 0.07/0.32 ! [Y3: nat,F: nat > nat,A: set_nat] :
% 0.07/0.32 ( ( member_nat @ Y3 @ ( image_nat_nat @ F @ A ) )
% 0.07/0.32 => ( ( F @ ( hilber815131374at_nat @ A @ F @ Y3 ) )
% 0.07/0.32 = Y3 ) ) ).
% 0.07/0.32
% 0.07/0.32 % f_inv_into_f
% 0.07/0.32 thf(fact_209_list__ex__length,axiom,
% 0.07/0.32 ( list_ex_a
% 0.07/0.32 = ( ^ [P2: a > $o,Xs4: list_a] :
% 0.07/0.32 ? [N2: nat] :
% 0.07/0.32 ( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs4 ) )
% 0.07/0.32 & ( P2 @ ( nth_a @ Xs4 @ N2 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % list_ex_length
% 0.07/0.32 thf(fact_210_list__ex__length,axiom,
% 0.07/0.32 ( list_ex_set_a
% 0.07/0.32 = ( ^ [P2: set_a > $o,Xs4: list_set_a] :
% 0.07/0.32 ? [N2: nat] :
% 0.07/0.32 ( ( ord_less_nat @ N2 @ ( size_size_list_set_a @ Xs4 ) )
% 0.07/0.32 & ( P2 @ ( nth_set_a @ Xs4 @ N2 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % list_ex_length
% 0.07/0.32 thf(fact_211_direct__decompI,axiom,
% 0.07/0.32 ! [Ss: list_set_a,A: set_a] :
% 0.07/0.32 ( ( inj_on_list_a_a @ groups1792256535list_a @ ( listset_a @ Ss ) )
% 0.07/0.32 => ( ( ( image_list_a_a @ groups1792256535list_a @ ( listset_a @ Ss ) )
% 0.07/0.32 = A )
% 0.07/0.32 => ( hilber2037636820comp_a @ A @ Ss ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % direct_decompI
% 0.07/0.32 thf(fact_212_elem__le__sum__list,axiom,
% 0.07/0.32 ! [K: nat,Ns: list_nat] :
% 0.07/0.32 ( ( ord_less_nat @ K @ ( size_size_list_nat @ Ns ) )
% 0.07/0.32 => ( ord_less_eq_nat @ ( nth_nat @ Ns @ K ) @ ( groups921905271st_nat @ Ns ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % elem_le_sum_list
% 0.07/0.32 thf(fact_213_perm__swap,axiom,
% 0.07/0.32 ! [I: nat,Xs: list_a,J: nat] :
% 0.07/0.32 ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
% 0.07/0.32 => ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
% 0.07/0.32 => ( perm_a @ ( list_update_a @ ( list_update_a @ Xs @ I @ ( nth_a @ Xs @ J ) ) @ J @ ( nth_a @ Xs @ I ) ) @ Xs ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % perm_swap
% 0.07/0.32 thf(fact_214_perm__swap,axiom,
% 0.07/0.32 ! [I: nat,Xs: list_set_a,J: nat] :
% 0.07/0.32 ( ( ord_less_nat @ I @ ( size_size_list_set_a @ Xs ) )
% 0.07/0.32 => ( ( ord_less_nat @ J @ ( size_size_list_set_a @ Xs ) )
% 0.07/0.32 => ( perm_set_a @ ( list_update_set_a @ ( list_update_set_a @ Xs @ I @ ( nth_set_a @ Xs @ J ) ) @ J @ ( nth_set_a @ Xs @ I ) ) @ Xs ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % perm_swap
% 0.07/0.32 thf(fact_215_mset__swap,axiom,
% 0.07/0.32 ! [I: nat,Ls: list_a,J: nat] :
% 0.07/0.32 ( ( ord_less_nat @ I @ ( size_size_list_a @ Ls ) )
% 0.07/0.32 => ( ( ord_less_nat @ J @ ( size_size_list_a @ Ls ) )
% 0.07/0.32 => ( ( mset_a @ ( list_update_a @ ( list_update_a @ Ls @ J @ ( nth_a @ Ls @ I ) ) @ I @ ( nth_a @ Ls @ J ) ) )
% 0.07/0.32 = ( mset_a @ Ls ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % mset_swap
% 0.07/0.32 thf(fact_216_mset__swap,axiom,
% 0.07/0.32 ! [I: nat,Ls: list_set_a,J: nat] :
% 0.07/0.32 ( ( ord_less_nat @ I @ ( size_size_list_set_a @ Ls ) )
% 0.07/0.32 => ( ( ord_less_nat @ J @ ( size_size_list_set_a @ Ls ) )
% 0.07/0.32 => ( ( mset_set_a @ ( list_update_set_a @ ( list_update_set_a @ Ls @ J @ ( nth_set_a @ Ls @ I ) ) @ I @ ( nth_set_a @ Ls @ J ) ) )
% 0.07/0.32 = ( mset_set_a @ Ls ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % mset_swap
% 0.07/0.32 thf(fact_217_nth__mem__mset,axiom,
% 0.07/0.32 ! [I: nat,Ls: list_list_a] :
% 0.07/0.32 ( ( ord_less_nat @ I @ ( size_s1427607542list_a @ Ls ) )
% 0.07/0.32 => ( member_list_a @ ( nth_list_a @ Ls @ I ) @ ( set_mset_list_a @ ( mset_list_a @ Ls ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % nth_mem_mset
% 0.07/0.32 thf(fact_218_nth__mem__mset,axiom,
% 0.07/0.32 ! [I: nat,Ls: list_a] :
% 0.07/0.32 ( ( ord_less_nat @ I @ ( size_size_list_a @ Ls ) )
% 0.07/0.32 => ( member_a @ ( nth_a @ Ls @ I ) @ ( set_mset_a @ ( mset_a @ Ls ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % nth_mem_mset
% 0.07/0.32 thf(fact_219_nth__mem__mset,axiom,
% 0.07/0.32 ! [I: nat,Ls: list_set_a] :
% 0.07/0.32 ( ( ord_less_nat @ I @ ( size_size_list_set_a @ Ls ) )
% 0.07/0.32 => ( member_set_a @ ( nth_set_a @ Ls @ I ) @ ( set_mset_set_a @ ( mset_set_a @ Ls ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % nth_mem_mset
% 0.07/0.32 thf(fact_220_in__Union__mset__iff,axiom,
% 0.07/0.32 ! [X4: a,MM: multiset_multiset_a] :
% 0.07/0.32 ( ( member_a @ X4 @ ( set_mset_a @ ( comm_m2145643721iset_a @ MM ) ) )
% 0.07/0.32 = ( ? [M2: multiset_a] :
% 0.07/0.32 ( ( member_multiset_a @ M2 @ ( set_mset_multiset_a @ MM ) )
% 0.07/0.32 & ( member_a @ X4 @ ( set_mset_a @ M2 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % in_Union_mset_iff
% 0.07/0.32 thf(fact_221_in__Union__mset__iff,axiom,
% 0.07/0.32 ! [X4: list_a,MM: multis971982480list_a] :
% 0.07/0.32 ( ( member_list_a @ X4 @ ( set_mset_list_a @ ( comm_m324867663list_a @ MM ) ) )
% 0.07/0.32 = ( ? [M2: multiset_list_a] :
% 0.07/0.32 ( ( member528267379list_a @ M2 @ ( set_ms1660427399list_a @ MM ) )
% 0.07/0.32 & ( member_list_a @ X4 @ ( set_mset_list_a @ M2 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % in_Union_mset_iff
% 0.07/0.32 thf(fact_222_lessThan__subset__iff,axiom,
% 0.07/0.32 ! [X4: nat,Y3: nat] :
% 0.07/0.32 ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X4 ) @ ( set_ord_lessThan_nat @ Y3 ) )
% 0.07/0.32 = ( ord_less_eq_nat @ X4 @ Y3 ) ) ).
% 0.07/0.32
% 0.07/0.32 % lessThan_subset_iff
% 0.07/0.32 thf(fact_223_list__update__beyond,axiom,
% 0.07/0.32 ! [Xs: list_a,I: nat,X4: a] :
% 0.07/0.32 ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ I )
% 0.07/0.32 => ( ( list_update_a @ Xs @ I @ X4 )
% 0.07/0.32 = Xs ) ) ).
% 0.07/0.32
% 0.07/0.32 % list_update_beyond
% 0.07/0.32 thf(fact_224_list__update__beyond,axiom,
% 0.07/0.32 ! [Xs: list_set_a,I: nat,X4: set_a] :
% 0.07/0.32 ( ( ord_less_eq_nat @ ( size_size_list_set_a @ Xs ) @ I )
% 0.07/0.32 => ( ( list_update_set_a @ Xs @ I @ X4 )
% 0.07/0.32 = Xs ) ) ).
% 0.07/0.32
% 0.07/0.32 % list_update_beyond
% 0.07/0.32 thf(fact_225_length__list__update,axiom,
% 0.07/0.32 ! [Xs: list_a,I: nat,X4: a] :
% 0.07/0.32 ( ( size_size_list_a @ ( list_update_a @ Xs @ I @ X4 ) )
% 0.07/0.32 = ( size_size_list_a @ Xs ) ) ).
% 0.07/0.32
% 0.07/0.32 % length_list_update
% 0.07/0.32 thf(fact_226_length__list__update,axiom,
% 0.07/0.32 ! [Xs: list_set_a,I: nat,X4: set_a] :
% 0.07/0.32 ( ( size_size_list_set_a @ ( list_update_set_a @ Xs @ I @ X4 ) )
% 0.07/0.32 = ( size_size_list_set_a @ Xs ) ) ).
% 0.07/0.32
% 0.07/0.32 % length_list_update
% 0.07/0.32 thf(fact_227_inv__into__f__f,axiom,
% 0.07/0.32 ! [F: nat > nat,A: set_nat,X4: nat] :
% 0.07/0.32 ( ( inj_on_nat_nat @ F @ A )
% 0.07/0.32 => ( ( member_nat @ X4 @ A )
% 0.07/0.32 => ( ( hilber815131374at_nat @ A @ F @ ( F @ X4 ) )
% 0.07/0.32 = X4 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % inv_into_f_f
% 0.07/0.32 thf(fact_228_list__update__id,axiom,
% 0.07/0.32 ! [Xs: list_a,I: nat] :
% 0.07/0.32 ( ( list_update_a @ Xs @ I @ ( nth_a @ Xs @ I ) )
% 0.07/0.32 = Xs ) ).
% 0.07/0.32
% 0.07/0.32 % list_update_id
% 0.07/0.32 thf(fact_229_list__update__id,axiom,
% 0.07/0.32 ! [Xs: list_set_a,I: nat] :
% 0.07/0.32 ( ( list_update_set_a @ Xs @ I @ ( nth_set_a @ Xs @ I ) )
% 0.07/0.32 = Xs ) ).
% 0.07/0.32
% 0.07/0.32 % list_update_id
% 0.07/0.32 thf(fact_230_nth__list__update__neq,axiom,
% 0.07/0.32 ! [I: nat,J: nat,Xs: list_a,X4: a] :
% 0.07/0.32 ( ( I != J )
% 0.07/0.32 => ( ( nth_a @ ( list_update_a @ Xs @ I @ X4 ) @ J )
% 0.07/0.32 = ( nth_a @ Xs @ J ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % nth_list_update_neq
% 0.07/0.32 thf(fact_231_nth__list__update__neq,axiom,
% 0.07/0.32 ! [I: nat,J: nat,Xs: list_set_a,X4: set_a] :
% 0.07/0.32 ( ( I != J )
% 0.07/0.32 => ( ( nth_set_a @ ( list_update_set_a @ Xs @ I @ X4 ) @ J )
% 0.07/0.32 = ( nth_set_a @ Xs @ J ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % nth_list_update_neq
% 0.07/0.32 thf(fact_232_inv__into__image__cancel,axiom,
% 0.07/0.32 ! [F: nat > nat,A: set_nat,S: set_nat] :
% 0.07/0.32 ( ( inj_on_nat_nat @ F @ A )
% 0.07/0.32 => ( ( ord_less_eq_set_nat @ S @ A )
% 0.07/0.32 => ( ( image_nat_nat @ ( hilber815131374at_nat @ A @ F ) @ ( image_nat_nat @ F @ S ) )
% 0.07/0.32 = S ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % inv_into_image_cancel
% 0.07/0.32 thf(fact_233_nth__list__update__eq,axiom,
% 0.07/0.32 ! [I: nat,Xs: list_a,X4: a] :
% 0.07/0.32 ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
% 0.07/0.32 => ( ( nth_a @ ( list_update_a @ Xs @ I @ X4 ) @ I )
% 0.07/0.32 = X4 ) ) ).
% 0.07/0.32
% 0.07/0.32 % nth_list_update_eq
% 0.07/0.32 thf(fact_234_nth__list__update__eq,axiom,
% 0.07/0.32 ! [I: nat,Xs: list_set_a,X4: set_a] :
% 0.07/0.32 ( ( ord_less_nat @ I @ ( size_size_list_set_a @ Xs ) )
% 0.07/0.32 => ( ( nth_set_a @ ( list_update_set_a @ Xs @ I @ X4 ) @ I )
% 0.07/0.32 = X4 ) ) ).
% 0.07/0.32
% 0.07/0.32 % nth_list_update_eq
% 0.07/0.32 thf(fact_235_inj__on__inv__into,axiom,
% 0.07/0.32 ! [B2: set_nat,F: nat > nat,A: set_nat] :
% 0.07/0.32 ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
% 0.07/0.32 => ( inj_on_nat_nat @ ( hilber815131374at_nat @ A @ F ) @ B2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % inj_on_inv_into
% 0.07/0.32 thf(fact_236_inj__on__image__mem__iff,axiom,
% 0.07/0.32 ! [F: a > a,B2: set_a,A2: a,A: set_a] :
% 0.07/0.32 ( ( inj_on_a_a @ F @ B2 )
% 0.07/0.32 => ( ( member_a @ A2 @ B2 )
% 0.07/0.32 => ( ( ord_less_eq_set_a @ A @ B2 )
% 0.07/0.32 => ( ( member_a @ ( F @ A2 ) @ ( image_a_a @ F @ A ) )
% 0.07/0.32 = ( member_a @ A2 @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % inj_on_image_mem_iff
% 0.07/0.32 thf(fact_237_inj__on__image__mem__iff,axiom,
% 0.07/0.32 ! [F: a > list_a,B2: set_a,A2: a,A: set_a] :
% 0.07/0.32 ( ( inj_on_a_list_a @ F @ B2 )
% 0.07/0.32 => ( ( member_a @ A2 @ B2 )
% 0.07/0.32 => ( ( ord_less_eq_set_a @ A @ B2 )
% 0.07/0.32 => ( ( member_list_a @ ( F @ A2 ) @ ( image_a_list_a @ F @ A ) )
% 0.07/0.32 = ( member_a @ A2 @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % inj_on_image_mem_iff
% 0.07/0.32 thf(fact_238_inj__on__image__mem__iff,axiom,
% 0.07/0.32 ! [F: list_a > a,B2: set_list_a,A2: list_a,A: set_list_a] :
% 0.07/0.32 ( ( inj_on_list_a_a @ F @ B2 )
% 0.07/0.32 => ( ( member_list_a @ A2 @ B2 )
% 0.07/0.32 => ( ( ord_le1301786372list_a @ A @ B2 )
% 0.07/0.32 => ( ( member_a @ ( F @ A2 ) @ ( image_list_a_a @ F @ A ) )
% 0.07/0.32 = ( member_list_a @ A2 @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % inj_on_image_mem_iff
% 0.07/0.32 thf(fact_239_inj__on__image__mem__iff,axiom,
% 0.07/0.32 ! [F: list_a > list_a,B2: set_list_a,A2: list_a,A: set_list_a] :
% 0.07/0.32 ( ( inj_on_list_a_list_a @ F @ B2 )
% 0.07/0.32 => ( ( member_list_a @ A2 @ B2 )
% 0.07/0.32 => ( ( ord_le1301786372list_a @ A @ B2 )
% 0.07/0.32 => ( ( member_list_a @ ( F @ A2 ) @ ( image_list_a_list_a @ F @ A ) )
% 0.07/0.32 = ( member_list_a @ A2 @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % inj_on_image_mem_iff
% 0.07/0.32 thf(fact_240_bij__betw__imp__inj__on,axiom,
% 0.07/0.32 ! [F: nat > nat,A: set_nat,B2: set_nat] :
% 0.07/0.32 ( ( bij_betw_nat_nat @ F @ A @ B2 )
% 0.07/0.32 => ( inj_on_nat_nat @ F @ A ) ) ).
% 0.07/0.32
% 0.07/0.32 % bij_betw_imp_inj_on
% 0.07/0.32 thf(fact_241_inv__into__f__eq,axiom,
% 0.07/0.32 ! [F: nat > nat,A: set_nat,X4: nat,Y3: nat] :
% 0.07/0.32 ( ( inj_on_nat_nat @ F @ A )
% 0.07/0.32 => ( ( member_nat @ X4 @ A )
% 0.07/0.32 => ( ( ( F @ X4 )
% 0.07/0.32 = Y3 )
% 0.07/0.32 => ( ( hilber815131374at_nat @ A @ F @ Y3 )
% 0.07/0.32 = X4 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % inv_into_f_eq
% 0.07/0.32 thf(fact_242_image__subsetI,axiom,
% 0.07/0.32 ! [A: set_a,F: a > a,B2: set_a] :
% 0.07/0.32 ( ! [X6: a] :
% 0.07/0.32 ( ( member_a @ X6 @ A )
% 0.07/0.32 => ( member_a @ ( F @ X6 ) @ B2 ) )
% 0.07/0.32 => ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % image_subsetI
% 0.07/0.32 thf(fact_243_image__subsetI,axiom,
% 0.07/0.32 ! [A: set_a,F: a > list_a,B2: set_list_a] :
% 0.07/0.32 ( ! [X6: a] :
% 0.07/0.32 ( ( member_a @ X6 @ A )
% 0.07/0.32 => ( member_list_a @ ( F @ X6 ) @ B2 ) )
% 0.07/0.32 => ( ord_le1301786372list_a @ ( image_a_list_a @ F @ A ) @ B2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % image_subsetI
% 0.07/0.32 thf(fact_244_image__subsetI,axiom,
% 0.07/0.32 ! [A: set_list_a,F: list_a > a,B2: set_a] :
% 0.07/0.32 ( ! [X6: list_a] :
% 0.07/0.32 ( ( member_list_a @ X6 @ A )
% 0.07/0.32 => ( member_a @ ( F @ X6 ) @ B2 ) )
% 0.07/0.32 => ( ord_less_eq_set_a @ ( image_list_a_a @ F @ A ) @ B2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % image_subsetI
% 0.07/0.32 thf(fact_245_image__subsetI,axiom,
% 0.07/0.32 ! [A: set_list_a,F: list_a > list_a,B2: set_list_a] :
% 0.07/0.32 ( ! [X6: list_a] :
% 0.07/0.32 ( ( member_list_a @ X6 @ A )
% 0.07/0.32 => ( member_list_a @ ( F @ X6 ) @ B2 ) )
% 0.07/0.32 => ( ord_le1301786372list_a @ ( image_list_a_list_a @ F @ A ) @ B2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % image_subsetI
% 0.07/0.32 thf(fact_246_bij__betw__def,axiom,
% 0.07/0.32 ( bij_betw_nat_nat
% 0.07/0.32 = ( ^ [F3: nat > nat,A4: set_nat,B3: set_nat] :
% 0.07/0.32 ( ( inj_on_nat_nat @ F3 @ A4 )
% 0.07/0.32 & ( ( image_nat_nat @ F3 @ A4 )
% 0.07/0.32 = B3 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % bij_betw_def
% 0.07/0.32 thf(fact_247_bij__betw__imageI,axiom,
% 0.07/0.32 ! [F: nat > nat,A: set_nat,B2: set_nat] :
% 0.07/0.32 ( ( inj_on_nat_nat @ F @ A )
% 0.07/0.32 => ( ( ( image_nat_nat @ F @ A )
% 0.07/0.32 = B2 )
% 0.07/0.32 => ( bij_betw_nat_nat @ F @ A @ B2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % bij_betw_imageI
% 0.07/0.32 thf(fact_248_inj__on__imp__bij__betw,axiom,
% 0.07/0.32 ! [F: nat > nat,A: set_nat] :
% 0.07/0.32 ( ( inj_on_nat_nat @ F @ A )
% 0.07/0.32 => ( bij_betw_nat_nat @ F @ A @ ( image_nat_nat @ F @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % inj_on_imp_bij_betw
% 0.07/0.32 thf(fact_249_mbs_Ominimal,axiom,
% 0.07/0.32 ! [X4: list_a,A: set_list_a,P: list_a > $o] :
% 0.07/0.32 ( ( member_list_a @ X4 @ A )
% 0.07/0.32 => ( ( P @ X4 )
% 0.07/0.32 => ? [X6: list_a] :
% 0.07/0.32 ( ( member_list_a @ X6 @ A )
% 0.07/0.32 & ( ord_less_eq_nat @ ( size_size_list_a @ X6 ) @ ( size_size_list_a @ X4 ) )
% 0.07/0.32 & ( P @ X6 )
% 0.07/0.32 & ! [Xa: list_a] :
% 0.07/0.32 ( ( member_list_a @ Xa @ A )
% 0.07/0.32 => ( ( ord_less_nat @ ( size_size_list_a @ Xa ) @ ( size_size_list_a @ X6 ) )
% 0.07/0.32 => ~ ( P @ Xa ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % mbs.minimal
% 0.07/0.32 thf(fact_250_mbs_Ominimal,axiom,
% 0.07/0.32 ! [X4: list_set_a,A: set_list_set_a,P: list_set_a > $o] :
% 0.07/0.32 ( ( member_list_set_a @ X4 @ A )
% 0.07/0.32 => ( ( P @ X4 )
% 0.07/0.32 => ? [X6: list_set_a] :
% 0.07/0.32 ( ( member_list_set_a @ X6 @ A )
% 0.07/0.32 & ( ord_less_eq_nat @ ( size_size_list_set_a @ X6 ) @ ( size_size_list_set_a @ X4 ) )
% 0.07/0.32 & ( P @ X6 )
% 0.07/0.32 & ! [Xa: list_set_a] :
% 0.07/0.32 ( ( member_list_set_a @ Xa @ A )
% 0.07/0.32 => ( ( ord_less_nat @ ( size_size_list_set_a @ Xa ) @ ( size_size_list_set_a @ X6 ) )
% 0.07/0.32 => ~ ( P @ Xa ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % mbs.minimal
% 0.07/0.32 thf(fact_251_bij__betw__subset,axiom,
% 0.07/0.32 ! [F: nat > nat,A: set_nat,A5: set_nat,B2: set_nat,B4: set_nat] :
% 0.07/0.32 ( ( bij_betw_nat_nat @ F @ A @ A5 )
% 0.07/0.32 => ( ( ord_less_eq_set_nat @ B2 @ A )
% 0.07/0.32 => ( ( ( image_nat_nat @ F @ B2 )
% 0.07/0.32 = B4 )
% 0.07/0.32 => ( bij_betw_nat_nat @ F @ B2 @ B4 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % bij_betw_subset
% 0.07/0.32 thf(fact_252_bij__betw__byWitness,axiom,
% 0.07/0.32 ! [A: set_nat,F4: nat > nat,F: nat > nat,A5: set_nat] :
% 0.07/0.32 ( ! [X6: nat] :
% 0.07/0.32 ( ( member_nat @ X6 @ A )
% 0.07/0.32 => ( ( F4 @ ( F @ X6 ) )
% 0.07/0.32 = X6 ) )
% 0.07/0.32 => ( ! [X6: nat] :
% 0.07/0.32 ( ( member_nat @ X6 @ A5 )
% 0.07/0.32 => ( ( F @ ( F4 @ X6 ) )
% 0.07/0.32 = X6 ) )
% 0.07/0.32 => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ A5 )
% 0.07/0.32 => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F4 @ A5 ) @ A )
% 0.07/0.32 => ( bij_betw_nat_nat @ F @ A @ A5 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % bij_betw_byWitness
% 0.07/0.32 thf(fact_253_image__inv__into__cancel,axiom,
% 0.07/0.32 ! [F: nat > nat,A: set_nat,A5: set_nat,B4: set_nat] :
% 0.07/0.32 ( ( ( image_nat_nat @ F @ A )
% 0.07/0.32 = A5 )
% 0.07/0.32 => ( ( ord_less_eq_set_nat @ B4 @ A5 )
% 0.07/0.32 => ( ( image_nat_nat @ F @ ( image_nat_nat @ ( hilber815131374at_nat @ A @ F ) @ B4 ) )
% 0.07/0.32 = B4 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % image_inv_into_cancel
% 0.07/0.32 thf(fact_254_listset__mono,axiom,
% 0.07/0.32 ! [Xs: list_set_a,Ys: list_set_a] :
% 0.07/0.32 ( ( ( size_size_list_set_a @ Xs )
% 0.07/0.32 = ( size_size_list_set_a @ Ys ) )
% 0.07/0.32 => ( ! [I3: nat] :
% 0.07/0.32 ( ( ord_less_nat @ I3 @ ( size_size_list_set_a @ Ys ) )
% 0.07/0.32 => ( ord_less_eq_set_a @ ( nth_set_a @ Xs @ I3 ) @ ( nth_set_a @ Ys @ I3 ) ) )
% 0.07/0.32 => ( ord_le1301786372list_a @ ( listset_a @ Xs ) @ ( listset_a @ Ys ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % listset_mono
% 0.07/0.32 thf(fact_255_nth__list__update,axiom,
% 0.07/0.32 ! [I: nat,Xs: list_a,J: nat,X4: a] :
% 0.07/0.32 ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
% 0.07/0.32 => ( ( ( I = J )
% 0.07/0.32 => ( ( nth_a @ ( list_update_a @ Xs @ I @ X4 ) @ J )
% 0.07/0.32 = X4 ) )
% 0.07/0.32 & ( ( I != J )
% 0.07/0.32 => ( ( nth_a @ ( list_update_a @ Xs @ I @ X4 ) @ J )
% 0.07/0.32 = ( nth_a @ Xs @ J ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % nth_list_update
% 0.07/0.32 thf(fact_256_nth__list__update,axiom,
% 0.07/0.32 ! [I: nat,Xs: list_set_a,J: nat,X4: set_a] :
% 0.07/0.32 ( ( ord_less_nat @ I @ ( size_size_list_set_a @ Xs ) )
% 0.07/0.32 => ( ( ( I = J )
% 0.07/0.32 => ( ( nth_set_a @ ( list_update_set_a @ Xs @ I @ X4 ) @ J )
% 0.07/0.32 = X4 ) )
% 0.07/0.32 & ( ( I != J )
% 0.07/0.32 => ( ( nth_set_a @ ( list_update_set_a @ Xs @ I @ X4 ) @ J )
% 0.07/0.32 = ( nth_set_a @ Xs @ J ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % nth_list_update
% 0.07/0.32 thf(fact_257_list__update__same__conv,axiom,
% 0.07/0.32 ! [I: nat,Xs: list_a,X4: a] :
% 0.07/0.32 ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
% 0.07/0.32 => ( ( ( list_update_a @ Xs @ I @ X4 )
% 0.07/0.32 = Xs )
% 0.07/0.32 = ( ( nth_a @ Xs @ I )
% 0.07/0.32 = X4 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % list_update_same_conv
% 0.07/0.32 thf(fact_258_list__update__same__conv,axiom,
% 0.07/0.32 ! [I: nat,Xs: list_set_a,X4: set_a] :
% 0.07/0.32 ( ( ord_less_nat @ I @ ( size_size_list_set_a @ Xs ) )
% 0.07/0.32 => ( ( ( list_update_set_a @ Xs @ I @ X4 )
% 0.07/0.32 = Xs )
% 0.07/0.32 = ( ( nth_set_a @ Xs @ I )
% 0.07/0.32 = X4 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % list_update_same_conv
% 0.07/0.32 thf(fact_259_bij__betw__inv__into__subset,axiom,
% 0.07/0.32 ! [F: nat > nat,A: set_nat,A5: set_nat,B2: set_nat,B4: set_nat] :
% 0.07/0.32 ( ( bij_betw_nat_nat @ F @ A @ A5 )
% 0.07/0.32 => ( ( ord_less_eq_set_nat @ B2 @ A )
% 0.07/0.32 => ( ( ( image_nat_nat @ F @ B2 )
% 0.07/0.32 = B4 )
% 0.07/0.32 => ( bij_betw_nat_nat @ ( hilber815131374at_nat @ A @ F ) @ B4 @ B2 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % bij_betw_inv_into_subset
% 0.07/0.32 thf(fact_260_direct__decompD_I2_J,axiom,
% 0.07/0.32 ! [A: set_a,Ss: list_set_a] :
% 0.07/0.32 ( ( hilber2037636820comp_a @ A @ Ss )
% 0.07/0.32 => ( inj_on_list_a_a @ groups1792256535list_a @ ( listset_a @ Ss ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % direct_decompD(2)
% 0.07/0.32 thf(fact_261_Schroeder__Bernstein,axiom,
% 0.07/0.32 ! [F: nat > nat,A: set_nat,B2: set_nat,G2: nat > nat] :
% 0.07/0.32 ( ( inj_on_nat_nat @ F @ A )
% 0.07/0.32 => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 )
% 0.07/0.32 => ( ( inj_on_nat_nat @ G2 @ B2 )
% 0.07/0.32 => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G2 @ B2 ) @ A )
% 0.07/0.32 => ? [H: nat > nat] : ( bij_betw_nat_nat @ H @ A @ B2 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Schroeder_Bernstein
% 0.07/0.32 thf(fact_262_order__refl,axiom,
% 0.07/0.32 ! [X4: nat] : ( ord_less_eq_nat @ X4 @ X4 ) ).
% 0.07/0.32
% 0.07/0.32 % order_refl
% 0.07/0.32 thf(fact_263_minf_I8_J,axiom,
% 0.07/0.32 ! [T: nat] :
% 0.07/0.32 ? [Z2: nat] :
% 0.07/0.32 ! [X: nat] :
% 0.07/0.32 ( ( ord_less_nat @ X @ Z2 )
% 0.07/0.32 => ~ ( ord_less_eq_nat @ T @ X ) ) ).
% 0.07/0.32
% 0.07/0.32 % minf(8)
% 0.07/0.32 thf(fact_264_minf_I6_J,axiom,
% 0.07/0.32 ! [T: nat] :
% 0.07/0.32 ? [Z2: nat] :
% 0.07/0.32 ! [X: nat] :
% 0.07/0.32 ( ( ord_less_nat @ X @ Z2 )
% 0.07/0.32 => ( ord_less_eq_nat @ X @ T ) ) ).
% 0.07/0.32
% 0.07/0.32 % minf(6)
% 0.07/0.32 thf(fact_265_subsetI,axiom,
% 0.07/0.32 ! [A: set_a,B2: set_a] :
% 0.07/0.32 ( ! [X6: a] :
% 0.07/0.32 ( ( member_a @ X6 @ A )
% 0.07/0.32 => ( member_a @ X6 @ B2 ) )
% 0.07/0.32 => ( ord_less_eq_set_a @ A @ B2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % subsetI
% 0.07/0.32 thf(fact_266_subsetI,axiom,
% 0.07/0.32 ! [A: set_list_a,B2: set_list_a] :
% 0.07/0.32 ( ! [X6: list_a] :
% 0.07/0.32 ( ( member_list_a @ X6 @ A )
% 0.07/0.32 => ( member_list_a @ X6 @ B2 ) )
% 0.07/0.32 => ( ord_le1301786372list_a @ A @ B2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % subsetI
% 0.07/0.32 thf(fact_267_bounded__Max__nat,axiom,
% 0.07/0.32 ! [P: nat > $o,X4: nat,M3: nat] :
% 0.07/0.32 ( ( P @ X4 )
% 0.07/0.32 => ( ! [X6: nat] :
% 0.07/0.32 ( ( P @ X6 )
% 0.07/0.32 => ( ord_less_eq_nat @ X6 @ M3 ) )
% 0.07/0.32 => ~ ! [M4: nat] :
% 0.07/0.32 ( ( P @ M4 )
% 0.07/0.32 => ~ ! [X: nat] :
% 0.07/0.32 ( ( P @ X )
% 0.07/0.32 => ( ord_less_eq_nat @ X @ M4 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % bounded_Max_nat
% 0.07/0.32 thf(fact_268_subset__iff,axiom,
% 0.07/0.32 ( ord_less_eq_set_a
% 0.07/0.32 = ( ^ [A4: set_a,B3: set_a] :
% 0.07/0.32 ! [T2: a] :
% 0.07/0.32 ( ( member_a @ T2 @ A4 )
% 0.07/0.32 => ( member_a @ T2 @ B3 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % subset_iff
% 0.07/0.32 thf(fact_269_subset__iff,axiom,
% 0.07/0.32 ( ord_le1301786372list_a
% 0.07/0.32 = ( ^ [A4: set_list_a,B3: set_list_a] :
% 0.07/0.32 ! [T2: list_a] :
% 0.07/0.32 ( ( member_list_a @ T2 @ A4 )
% 0.07/0.32 => ( member_list_a @ T2 @ B3 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % subset_iff
% 0.07/0.32 thf(fact_270_subset__eq,axiom,
% 0.07/0.32 ( ord_less_eq_set_a
% 0.07/0.32 = ( ^ [A4: set_a,B3: set_a] :
% 0.07/0.32 ! [X5: a] :
% 0.07/0.32 ( ( member_a @ X5 @ A4 )
% 0.07/0.32 => ( member_a @ X5 @ B3 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % subset_eq
% 0.07/0.32 thf(fact_271_subset__eq,axiom,
% 0.07/0.32 ( ord_le1301786372list_a
% 0.07/0.32 = ( ^ [A4: set_list_a,B3: set_list_a] :
% 0.07/0.32 ! [X5: list_a] :
% 0.07/0.32 ( ( member_list_a @ X5 @ A4 )
% 0.07/0.32 => ( member_list_a @ X5 @ B3 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % subset_eq
% 0.07/0.32 thf(fact_272_subsetD,axiom,
% 0.07/0.32 ! [A: set_a,B2: set_a,C: a] :
% 0.07/0.32 ( ( ord_less_eq_set_a @ A @ B2 )
% 0.07/0.32 => ( ( member_a @ C @ A )
% 0.07/0.32 => ( member_a @ C @ B2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % subsetD
% 0.07/0.32 thf(fact_273_subsetD,axiom,
% 0.07/0.32 ! [A: set_list_a,B2: set_list_a,C: list_a] :
% 0.07/0.32 ( ( ord_le1301786372list_a @ A @ B2 )
% 0.07/0.32 => ( ( member_list_a @ C @ A )
% 0.07/0.32 => ( member_list_a @ C @ B2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % subsetD
% 0.07/0.32 thf(fact_274_in__mono,axiom,
% 0.07/0.32 ! [A: set_a,B2: set_a,X4: a] :
% 0.07/0.32 ( ( ord_less_eq_set_a @ A @ B2 )
% 0.07/0.32 => ( ( member_a @ X4 @ A )
% 0.07/0.32 => ( member_a @ X4 @ B2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % in_mono
% 0.07/0.32 thf(fact_275_in__mono,axiom,
% 0.07/0.32 ! [A: set_list_a,B2: set_list_a,X4: list_a] :
% 0.07/0.32 ( ( ord_le1301786372list_a @ A @ B2 )
% 0.07/0.32 => ( ( member_list_a @ X4 @ A )
% 0.07/0.32 => ( member_list_a @ X4 @ B2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % in_mono
% 0.07/0.32 thf(fact_276_dual__order_Oantisym,axiom,
% 0.07/0.32 ! [B: nat,A2: nat] :
% 0.07/0.32 ( ( ord_less_eq_nat @ B @ A2 )
% 0.07/0.32 => ( ( ord_less_eq_nat @ A2 @ B )
% 0.07/0.32 => ( A2 = B ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % dual_order.antisym
% 0.07/0.32 thf(fact_277_dual__order_Oeq__iff,axiom,
% 0.07/0.32 ( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
% 0.07/0.32 = ( ^ [A7: nat,B5: nat] :
% 0.07/0.32 ( ( ord_less_eq_nat @ B5 @ A7 )
% 0.07/0.32 & ( ord_less_eq_nat @ A7 @ B5 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % dual_order.eq_iff
% 0.07/0.32 thf(fact_278_dual__order_Otrans,axiom,
% 0.07/0.32 ! [B: nat,A2: nat,C: nat] :
% 0.07/0.32 ( ( ord_less_eq_nat @ B @ A2 )
% 0.07/0.32 => ( ( ord_less_eq_nat @ C @ B )
% 0.07/0.32 => ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % dual_order.trans
% 0.07/0.32 thf(fact_279_linorder__wlog,axiom,
% 0.07/0.32 ! [P: nat > nat > $o,A2: nat,B: nat] :
% 0.07/0.32 ( ! [A3: nat,B6: nat] :
% 0.07/0.32 ( ( ord_less_eq_nat @ A3 @ B6 )
% 0.07/0.32 => ( P @ A3 @ B6 ) )
% 0.07/0.32 => ( ! [A3: nat,B6: nat] :
% 0.07/0.32 ( ( P @ B6 @ A3 )
% 0.07/0.32 => ( P @ A3 @ B6 ) )
% 0.07/0.32 => ( P @ A2 @ B ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % linorder_wlog
% 0.07/0.32 thf(fact_280_dual__order_Orefl,axiom,
% 0.07/0.32 ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% 0.07/0.32
% 0.07/0.32 % dual_order.refl
% 0.07/0.32 thf(fact_281_order__trans,axiom,
% 0.07/0.32 ! [X4: nat,Y3: nat,Z3: nat] :
% 0.07/0.32 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.07/0.32 => ( ( ord_less_eq_nat @ Y3 @ Z3 )
% 0.07/0.32 => ( ord_less_eq_nat @ X4 @ Z3 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % order_trans
% 0.07/0.32 thf(fact_282_order__class_Oorder_Oantisym,axiom,
% 0.07/0.32 ! [A2: nat,B: nat] :
% 0.07/0.32 ( ( ord_less_eq_nat @ A2 @ B )
% 0.07/0.32 => ( ( ord_less_eq_nat @ B @ A2 )
% 0.07/0.32 => ( A2 = B ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % order_class.order.antisym
% 0.07/0.32 thf(fact_283_ord__le__eq__trans,axiom,
% 0.07/0.32 ! [A2: nat,B: nat,C: nat] :
% 0.07/0.32 ( ( ord_less_eq_nat @ A2 @ B )
% 0.07/0.32 => ( ( B = C )
% 0.07/0.32 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % ord_le_eq_trans
% 0.07/0.32 thf(fact_284_ord__eq__le__trans,axiom,
% 0.07/0.32 ! [A2: nat,B: nat,C: nat] :
% 0.07/0.32 ( ( A2 = B )
% 0.07/0.32 => ( ( ord_less_eq_nat @ B @ C )
% 0.07/0.32 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % ord_eq_le_trans
% 0.07/0.32 thf(fact_285_order__class_Oorder_Oeq__iff,axiom,
% 0.07/0.32 ( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
% 0.07/0.32 = ( ^ [A7: nat,B5: nat] :
% 0.07/0.32 ( ( ord_less_eq_nat @ A7 @ B5 )
% 0.07/0.32 & ( ord_less_eq_nat @ B5 @ A7 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % order_class.order.eq_iff
% 0.07/0.32 thf(fact_286_antisym__conv,axiom,
% 0.07/0.32 ! [Y3: nat,X4: nat] :
% 0.07/0.32 ( ( ord_less_eq_nat @ Y3 @ X4 )
% 0.07/0.32 => ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.07/0.32 = ( X4 = Y3 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % antisym_conv
% 0.07/0.32 thf(fact_287_le__cases3,axiom,
% 0.07/0.32 ! [X4: nat,Y3: nat,Z3: nat] :
% 0.07/0.32 ( ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.07/0.32 => ~ ( ord_less_eq_nat @ Y3 @ Z3 ) )
% 0.07/0.32 => ( ( ( ord_less_eq_nat @ Y3 @ X4 )
% 0.07/0.32 => ~ ( ord_less_eq_nat @ X4 @ Z3 ) )
% 0.07/0.32 => ( ( ( ord_less_eq_nat @ X4 @ Z3 )
% 0.07/0.32 => ~ ( ord_less_eq_nat @ Z3 @ Y3 ) )
% 0.07/0.32 => ( ( ( ord_less_eq_nat @ Z3 @ Y3 )
% 0.07/0.32 => ~ ( ord_less_eq_nat @ Y3 @ X4 ) )
% 0.07/0.32 => ( ( ( ord_less_eq_nat @ Y3 @ Z3 )
% 0.07/0.32 => ~ ( ord_less_eq_nat @ Z3 @ X4 ) )
% 0.07/0.32 => ~ ( ( ord_less_eq_nat @ Z3 @ X4 )
% 0.07/0.32 => ~ ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % le_cases3
% 0.07/0.32 thf(fact_288_order_Otrans,axiom,
% 0.07/0.32 ! [A2: nat,B: nat,C: nat] :
% 0.07/0.32 ( ( ord_less_eq_nat @ A2 @ B )
% 0.07/0.32 => ( ( ord_less_eq_nat @ B @ C )
% 0.07/0.32 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % order.trans
% 0.07/0.32 thf(fact_289_le__cases,axiom,
% 0.07/0.32 ! [X4: nat,Y3: nat] :
% 0.07/0.32 ( ~ ( ord_less_eq_nat @ X4 @ Y3 )
% 0.07/0.32 => ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% 0.07/0.32
% 0.07/0.32 % le_cases
% 0.07/0.32 thf(fact_290_eq__refl,axiom,
% 0.07/0.32 ! [X4: nat,Y3: nat] :
% 0.07/0.32 ( ( X4 = Y3 )
% 0.07/0.32 => ( ord_less_eq_nat @ X4 @ Y3 ) ) ).
% 0.07/0.32
% 0.07/0.32 % eq_refl
% 0.07/0.32 thf(fact_291_linear,axiom,
% 0.07/0.32 ! [X4: nat,Y3: nat] :
% 0.07/0.32 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.07/0.32 | ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% 0.07/0.32
% 0.07/0.32 % linear
% 0.07/0.32 thf(fact_292_antisym,axiom,
% 0.07/0.32 ! [X4: nat,Y3: nat] :
% 0.07/0.32 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.07/0.32 => ( ( ord_less_eq_nat @ Y3 @ X4 )
% 0.07/0.32 => ( X4 = Y3 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % antisym
% 0.07/0.32 thf(fact_293_le__rel__bool__arg__iff,axiom,
% 0.07/0.32 ( ord_less_eq_o_nat
% 0.07/0.32 = ( ^ [X7: $o > nat,Y4: $o > nat] :
% 0.07/0.32 ( ( ord_less_eq_nat @ ( X7 @ $false ) @ ( Y4 @ $false ) )
% 0.07/0.32 & ( ord_less_eq_nat @ ( X7 @ $true ) @ ( Y4 @ $true ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % le_rel_bool_arg_iff
% 0.07/0.32 thf(fact_294_eq__iff,axiom,
% 0.07/0.32 ( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
% 0.07/0.32 = ( ^ [X5: nat,Y5: nat] :
% 0.07/0.32 ( ( ord_less_eq_nat @ X5 @ Y5 )
% 0.07/0.32 & ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % eq_iff
% 0.07/0.32 thf(fact_295_ord__le__eq__subst,axiom,
% 0.07/0.32 ! [A2: nat,B: nat,F: nat > nat,C: nat] :
% 0.07/0.32 ( ( ord_less_eq_nat @ A2 @ B )
% 0.07/0.32 => ( ( ( F @ B )
% 0.07/0.32 = C )
% 0.07/0.32 => ( ! [X6: nat,Y: nat] :
% 0.07/0.32 ( ( ord_less_eq_nat @ X6 @ Y )
% 0.07/0.32 => ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y ) ) )
% 0.07/0.32 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % ord_le_eq_subst
% 0.07/0.32 thf(fact_296_ord__eq__le__subst,axiom,
% 0.07/0.32 ! [A2: nat,F: nat > nat,B: nat,C: nat] :
% 0.07/0.32 ( ( A2
% 0.07/0.32 = ( F @ B ) )
% 0.07/0.32 => ( ( ord_less_eq_nat @ B @ C )
% 0.07/0.32 => ( ! [X6: nat,Y: nat] :
% 0.07/0.32 ( ( ord_less_eq_nat @ X6 @ Y )
% 0.07/0.32 => ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y ) ) )
% 0.07/0.32 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % ord_eq_le_subst
% 0.07/0.32 thf(fact_297_order__subst2,axiom,
% 0.07/0.32 ! [A2: nat,B: nat,F: nat > nat,C: nat] :
% 0.07/0.32 ( ( ord_less_eq_nat @ A2 @ B )
% 0.07/0.32 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.07/0.32 => ( ! [X6: nat,Y: nat] :
% 0.07/0.32 ( ( ord_less_eq_nat @ X6 @ Y )
% 0.07/0.32 => ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y ) ) )
% 0.07/0.32 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % order_subst2
% 0.07/0.32 thf(fact_298_order__subst1,axiom,
% 0.07/0.32 ! [A2: nat,F: nat > nat,B: nat,C: nat] :
% 0.07/0.32 ( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
% 0.07/0.32 => ( ( ord_less_eq_nat @ B @ C )
% 0.07/0.32 => ( ! [X6: nat,Y: nat] :
% 0.07/0.32 ( ( ord_less_eq_nat @ X6 @ Y )
% 0.07/0.32 => ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y ) ) )
% 0.07/0.32 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % order_subst1
% 0.07/0.32 thf(fact_299_dual__order_Ostrict__implies__not__eq,axiom,
% 0.07/0.32 ! [B: nat,A2: nat] :
% 0.07/0.32 ( ( ord_less_nat @ B @ A2 )
% 0.07/0.32 => ( A2 != B ) ) ).
% 0.07/0.32
% 0.07/0.32 % dual_order.strict_implies_not_eq
% 0.07/0.32 thf(fact_300_order_Ostrict__implies__not__eq,axiom,
% 0.07/0.32 ! [A2: nat,B: nat] :
% 0.07/0.32 ( ( ord_less_nat @ A2 @ B )
% 0.07/0.32 => ( A2 != B ) ) ).
% 0.07/0.32
% 0.07/0.32 % order.strict_implies_not_eq
% 0.07/0.32 thf(fact_301_not__less__iff__gr__or__eq,axiom,
% 0.07/0.32 ! [X4: nat,Y3: nat] :
% 0.07/0.32 ( ( ~ ( ord_less_nat @ X4 @ Y3 ) )
% 0.07/0.32 = ( ( ord_less_nat @ Y3 @ X4 )
% 0.07/0.32 | ( X4 = Y3 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % not_less_iff_gr_or_eq
% 0.07/0.32 thf(fact_302_dual__order_Ostrict__trans,axiom,
% 0.07/0.32 ! [B: nat,A2: nat,C: nat] :
% 0.07/0.32 ( ( ord_less_nat @ B @ A2 )
% 0.07/0.32 => ( ( ord_less_nat @ C @ B )
% 0.07/0.32 => ( ord_less_nat @ C @ A2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % dual_order.strict_trans
% 0.07/0.32 thf(fact_303_linorder__less__wlog,axiom,
% 0.07/0.32 ! [P: nat > nat > $o,A2: nat,B: nat] :
% 0.07/0.32 ( ! [A3: nat,B6: nat] :
% 0.07/0.32 ( ( ord_less_nat @ A3 @ B6 )
% 0.07/0.32 => ( P @ A3 @ B6 ) )
% 0.07/0.32 => ( ! [A3: nat] : ( P @ A3 @ A3 )
% 0.07/0.32 => ( ! [A3: nat,B6: nat] :
% 0.07/0.32 ( ( P @ B6 @ A3 )
% 0.07/0.32 => ( P @ A3 @ B6 ) )
% 0.07/0.32 => ( P @ A2 @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % linorder_less_wlog
% 0.07/0.32 thf(fact_304_exists__least__iff,axiom,
% 0.07/0.32 ( ( ^ [P3: nat > $o] :
% 0.07/0.32 ? [X3: nat] : ( P3 @ X3 ) )
% 0.07/0.32 = ( ^ [P2: nat > $o] :
% 0.07/0.32 ? [N2: nat] :
% 0.07/0.32 ( ( P2 @ N2 )
% 0.07/0.32 & ! [M5: nat] :
% 0.07/0.32 ( ( ord_less_nat @ M5 @ N2 )
% 0.07/0.32 => ~ ( P2 @ M5 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % exists_least_iff
% 0.07/0.32 thf(fact_305_less__imp__not__less,axiom,
% 0.07/0.32 ! [X4: nat,Y3: nat] :
% 0.07/0.32 ( ( ord_less_nat @ X4 @ Y3 )
% 0.07/0.32 => ~ ( ord_less_nat @ Y3 @ X4 ) ) ).
% 0.07/0.32
% 0.07/0.32 % less_imp_not_less
% 0.07/0.32 thf(fact_306_order_Ostrict__trans,axiom,
% 0.07/0.32 ! [A2: nat,B: nat,C: nat] :
% 0.07/0.32 ( ( ord_less_nat @ A2 @ B )
% 0.07/0.32 => ( ( ord_less_nat @ B @ C )
% 0.07/0.32 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % order.strict_trans
% 0.07/0.32 thf(fact_307_dual__order_Oirrefl,axiom,
% 0.07/0.32 ! [A2: nat] :
% 0.07/0.32 ~ ( ord_less_nat @ A2 @ A2 ) ).
% 0.07/0.32
% 0.07/0.32 % dual_order.irrefl
% 0.07/0.32 thf(fact_308_linorder__cases,axiom,
% 0.07/0.32 ! [X4: nat,Y3: nat] :
% 0.07/0.32 ( ~ ( ord_less_nat @ X4 @ Y3 )
% 0.07/0.32 => ( ( X4 != Y3 )
% 0.07/0.32 => ( ord_less_nat @ Y3 @ X4 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % linorder_cases
% 0.07/0.32 thf(fact_309_less__imp__triv,axiom,
% 0.07/0.32 ! [X4: nat,Y3: nat,P: $o] :
% 0.07/0.32 ( ( ord_less_nat @ X4 @ Y3 )
% 0.07/0.32 => ( ( ord_less_nat @ Y3 @ X4 )
% 0.07/0.32 => P ) ) ).
% 0.07/0.32
% 0.07/0.32 % less_imp_triv
% 0.07/0.32 thf(fact_310_less__imp__not__eq2,axiom,
% 0.07/0.32 ! [X4: nat,Y3: nat] :
% 0.07/0.32 ( ( ord_less_nat @ X4 @ Y3 )
% 0.07/0.32 => ( Y3 != X4 ) ) ).
% 0.07/0.32
% 0.07/0.32 % less_imp_not_eq2
% 0.07/0.32 thf(fact_311_antisym__conv3,axiom,
% 0.07/0.32 ! [Y3: nat,X4: nat] :
% 0.07/0.32 ( ~ ( ord_less_nat @ Y3 @ X4 )
% 0.07/0.32 => ( ( ~ ( ord_less_nat @ X4 @ Y3 ) )
% 0.07/0.32 = ( X4 = Y3 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % antisym_conv3
% 0.07/0.32 thf(fact_312_less__induct,axiom,
% 0.07/0.32 ! [P: nat > $o,A2: nat] :
% 0.07/0.32 ( ! [X6: nat] :
% 0.07/0.32 ( ! [Y6: nat] :
% 0.07/0.32 ( ( ord_less_nat @ Y6 @ X6 )
% 0.07/0.32 => ( P @ Y6 ) )
% 0.07/0.32 => ( P @ X6 ) )
% 0.07/0.32 => ( P @ A2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % less_induct
% 0.07/0.32 thf(fact_313_less__not__sym,axiom,
% 0.07/0.32 ! [X4: nat,Y3: nat] :
% 0.07/0.32 ( ( ord_less_nat @ X4 @ Y3 )
% 0.07/0.32 => ~ ( ord_less_nat @ Y3 @ X4 ) ) ).
% 0.07/0.32
% 0.07/0.32 % less_not_sym
% 0.07/0.32 thf(fact_314_less__imp__not__eq,axiom,
% 0.07/0.32 ! [X4: nat,Y3: nat] :
% 0.07/0.32 ( ( ord_less_nat @ X4 @ Y3 )
% 0.07/0.32 => ( X4 != Y3 ) ) ).
% 0.07/0.32
% 0.07/0.32 % less_imp_not_eq
% 0.07/0.32 thf(fact_315_dual__order_Oasym,axiom,
% 0.07/0.32 ! [B: nat,A2: nat] :
% 0.07/0.32 ( ( ord_less_nat @ B @ A2 )
% 0.07/0.32 => ~ ( ord_less_nat @ A2 @ B ) ) ).
% 0.07/0.32
% 0.07/0.32 % dual_order.asym
% 0.07/0.32 thf(fact_316_ord__less__eq__trans,axiom,
% 0.07/0.32 ! [A2: nat,B: nat,C: nat] :
% 0.07/0.32 ( ( ord_less_nat @ A2 @ B )
% 0.07/0.32 => ( ( B = C )
% 0.07/0.32 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % ord_less_eq_trans
% 0.07/0.32 thf(fact_317_ord__eq__less__trans,axiom,
% 0.07/0.32 ! [A2: nat,B: nat,C: nat] :
% 0.07/0.32 ( ( A2 = B )
% 0.07/0.32 => ( ( ord_less_nat @ B @ C )
% 0.07/0.32 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % ord_eq_less_trans
% 0.07/0.32 thf(fact_318_less__irrefl,axiom,
% 0.07/0.32 ! [X4: nat] :
% 0.07/0.32 ~ ( ord_less_nat @ X4 @ X4 ) ).
% 0.07/0.32
% 0.07/0.32 % less_irrefl
% 0.07/0.32 thf(fact_319_less__linear,axiom,
% 0.07/0.32 ! [X4: nat,Y3: nat] :
% 0.07/0.32 ( ( ord_less_nat @ X4 @ Y3 )
% 0.07/0.32 | ( X4 = Y3 )
% 0.07/0.32 | ( ord_less_nat @ Y3 @ X4 ) ) ).
% 0.07/0.32
% 0.07/0.32 % less_linear
% 0.07/0.32 thf(fact_320_less__trans,axiom,
% 0.07/0.32 ! [X4: nat,Y3: nat,Z3: nat] :
% 0.07/0.32 ( ( ord_less_nat @ X4 @ Y3 )
% 0.07/0.32 => ( ( ord_less_nat @ Y3 @ Z3 )
% 0.07/0.32 => ( ord_less_nat @ X4 @ Z3 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % less_trans
% 0.07/0.32 thf(fact_321_less__asym_H,axiom,
% 0.07/0.32 ! [A2: nat,B: nat] :
% 0.07/0.32 ( ( ord_less_nat @ A2 @ B )
% 0.07/0.32 => ~ ( ord_less_nat @ B @ A2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % less_asym'
% 0.07/0.32 thf(fact_322_less__asym,axiom,
% 0.07/0.32 ! [X4: nat,Y3: nat] :
% 0.07/0.32 ( ( ord_less_nat @ X4 @ Y3 )
% 0.07/0.32 => ~ ( ord_less_nat @ Y3 @ X4 ) ) ).
% 0.07/0.32
% 0.07/0.32 % less_asym
% 0.07/0.32 thf(fact_323_less__imp__neq,axiom,
% 0.07/0.32 ! [X4: nat,Y3: nat] :
% 0.07/0.32 ( ( ord_less_nat @ X4 @ Y3 )
% 0.07/0.32 => ( X4 != Y3 ) ) ).
% 0.07/0.32
% 0.07/0.32 % less_imp_neq
% 0.07/0.32 thf(fact_324_order_Oasym,axiom,
% 0.07/0.32 ! [A2: nat,B: nat] :
% 0.07/0.32 ( ( ord_less_nat @ A2 @ B )
% 0.07/0.32 => ~ ( ord_less_nat @ B @ A2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % order.asym
% 0.07/0.32 thf(fact_325_neq__iff,axiom,
% 0.07/0.32 ! [X4: nat,Y3: nat] :
% 0.07/0.32 ( ( X4 != Y3 )
% 0.07/0.32 = ( ( ord_less_nat @ X4 @ Y3 )
% 0.07/0.32 | ( ord_less_nat @ Y3 @ X4 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % neq_iff
% 0.07/0.32 thf(fact_326_neqE,axiom,
% 0.07/0.32 ! [X4: nat,Y3: nat] :
% 0.07/0.32 ( ( X4 != Y3 )
% 0.07/0.32 => ( ~ ( ord_less_nat @ X4 @ Y3 )
% 0.07/0.32 => ( ord_less_nat @ Y3 @ X4 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % neqE
% 0.07/0.32 thf(fact_327_gt__ex,axiom,
% 0.07/0.32 ! [X4: nat] :
% 0.07/0.32 ? [X_1: nat] : ( ord_less_nat @ X4 @ X_1 ) ).
% 0.07/0.32
% 0.07/0.32 % gt_ex
% 0.07/0.32 thf(fact_328_order__less__subst2,axiom,
% 0.07/0.32 ! [A2: nat,B: nat,F: nat > nat,C: nat] :
% 0.07/0.32 ( ( ord_less_nat @ A2 @ B )
% 0.07/0.32 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.07/0.32 => ( ! [X6: nat,Y: nat] :
% 0.07/0.32 ( ( ord_less_nat @ X6 @ Y )
% 0.07/0.32 => ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y ) ) )
% 0.07/0.32 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % order_less_subst2
% 0.07/0.32 thf(fact_329_order__less__subst1,axiom,
% 0.07/0.32 ! [A2: nat,F: nat > nat,B: nat,C: nat] :
% 0.07/0.32 ( ( ord_less_nat @ A2 @ ( F @ B ) )
% 0.07/0.32 => ( ( ord_less_nat @ B @ C )
% 0.07/0.32 => ( ! [X6: nat,Y: nat] :
% 0.07/0.32 ( ( ord_less_nat @ X6 @ Y )
% 0.07/0.32 => ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y ) ) )
% 0.07/0.32 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % order_less_subst1
% 0.07/0.32 thf(fact_330_ord__less__eq__subst,axiom,
% 0.07/0.32 ! [A2: nat,B: nat,F: nat > nat,C: nat] :
% 0.07/0.32 ( ( ord_less_nat @ A2 @ B )
% 0.07/0.32 => ( ( ( F @ B )
% 0.07/0.32 = C )
% 0.07/0.32 => ( ! [X6: nat,Y: nat] :
% 0.07/0.32 ( ( ord_less_nat @ X6 @ Y )
% 0.07/0.32 => ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y ) ) )
% 0.07/0.32 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % ord_less_eq_subst
% 0.07/0.32 thf(fact_331_ord__eq__less__subst,axiom,
% 0.07/0.32 ! [A2: nat,F: nat > nat,B: nat,C: nat] :
% 0.07/0.32 ( ( A2
% 0.07/0.32 = ( F @ B ) )
% 0.07/0.32 => ( ( ord_less_nat @ B @ C )
% 0.07/0.32 => ( ! [X6: nat,Y: nat] :
% 0.07/0.32 ( ( ord_less_nat @ X6 @ Y )
% 0.07/0.32 => ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y ) ) )
% 0.07/0.32 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % ord_eq_less_subst
% 0.07/0.32 thf(fact_332_pinf_I1_J,axiom,
% 0.07/0.32 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.07/0.32 ( ? [Z4: nat] :
% 0.07/0.32 ! [X6: nat] :
% 0.07/0.32 ( ( ord_less_nat @ Z4 @ X6 )
% 0.07/0.32 => ( ( P @ X6 )
% 0.07/0.32 = ( P4 @ X6 ) ) )
% 0.07/0.32 => ( ? [Z4: nat] :
% 0.07/0.32 ! [X6: nat] :
% 0.07/0.32 ( ( ord_less_nat @ Z4 @ X6 )
% 0.07/0.32 => ( ( Q @ X6 )
% 0.07/0.32 = ( Q2 @ X6 ) ) )
% 0.07/0.32 => ? [Z2: nat] :
% 0.07/0.32 ! [X: nat] :
% 0.07/0.32 ( ( ord_less_nat @ Z2 @ X )
% 0.07/0.32 => ( ( ( P @ X )
% 0.07/0.32 & ( Q @ X ) )
% 0.07/0.32 = ( ( P4 @ X )
% 0.07/0.32 & ( Q2 @ X ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % pinf(1)
% 0.07/0.32 thf(fact_333_pinf_I2_J,axiom,
% 0.07/0.32 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.07/0.32 ( ? [Z4: nat] :
% 0.07/0.32 ! [X6: nat] :
% 0.07/0.32 ( ( ord_less_nat @ Z4 @ X6 )
% 0.07/0.32 => ( ( P @ X6 )
% 0.07/0.32 = ( P4 @ X6 ) ) )
% 0.07/0.32 => ( ? [Z4: nat] :
% 0.07/0.32 ! [X6: nat] :
% 0.07/0.32 ( ( ord_less_nat @ Z4 @ X6 )
% 0.07/0.32 => ( ( Q @ X6 )
% 0.07/0.32 = ( Q2 @ X6 ) ) )
% 0.07/0.32 => ? [Z2: nat] :
% 0.07/0.32 ! [X: nat] :
% 0.07/0.32 ( ( ord_less_nat @ Z2 @ X )
% 0.07/0.32 => ( ( ( P @ X )
% 0.07/0.32 | ( Q @ X ) )
% 0.07/0.32 = ( ( P4 @ X )
% 0.07/0.32 | ( Q2 @ X ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % pinf(2)
% 0.07/0.32 thf(fact_334_pinf_I3_J,axiom,
% 0.07/0.32 ! [T: nat] :
% 0.07/0.32 ? [Z2: nat] :
% 0.07/0.32 ! [X: nat] :
% 0.07/0.32 ( ( ord_less_nat @ Z2 @ X )
% 0.07/0.32 => ( X != T ) ) ).
% 0.07/0.32
% 0.07/0.32 % pinf(3)
% 0.07/0.32 thf(fact_335_pinf_I4_J,axiom,
% 0.07/0.32 ! [T: nat] :
% 0.07/0.32 ? [Z2: nat] :
% 0.07/0.32 ! [X: nat] :
% 0.07/0.32 ( ( ord_less_nat @ Z2 @ X )
% 0.07/0.32 => ( X != T ) ) ).
% 0.07/0.32
% 0.07/0.32 % pinf(4)
% 0.07/0.32 thf(fact_336_pinf_I5_J,axiom,
% 0.07/0.32 ! [T: nat] :
% 0.07/0.32 ? [Z2: nat] :
% 0.07/0.32 ! [X: nat] :
% 0.07/0.32 ( ( ord_less_nat @ Z2 @ X )
% 0.07/0.32 => ~ ( ord_less_nat @ X @ T ) ) ).
% 0.07/0.32
% 0.07/0.32 % pinf(5)
% 0.07/0.32 thf(fact_337_pinf_I7_J,axiom,
% 0.07/0.32 ! [T: nat] :
% 0.07/0.32 ? [Z2: nat] :
% 0.07/0.32 ! [X: nat] :
% 0.07/0.32 ( ( ord_less_nat @ Z2 @ X )
% 0.07/0.32 => ( ord_less_nat @ T @ X ) ) ).
% 0.07/0.32
% 0.07/0.32 % pinf(7)
% 0.07/0.32 thf(fact_338_minf_I1_J,axiom,
% 0.07/0.32 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.07/0.32 ( ? [Z4: nat] :
% 0.07/0.32 ! [X6: nat] :
% 0.07/0.32 ( ( ord_less_nat @ X6 @ Z4 )
% 0.07/0.32 => ( ( P @ X6 )
% 0.07/0.32 = ( P4 @ X6 ) ) )
% 0.07/0.32 => ( ? [Z4: nat] :
% 0.07/0.32 ! [X6: nat] :
% 0.07/0.32 ( ( ord_less_nat @ X6 @ Z4 )
% 0.07/0.32 => ( ( Q @ X6 )
% 0.07/0.32 = ( Q2 @ X6 ) ) )
% 0.07/0.32 => ? [Z2: nat] :
% 0.07/0.32 ! [X: nat] :
% 0.07/0.32 ( ( ord_less_nat @ X @ Z2 )
% 0.07/0.32 => ( ( ( P @ X )
% 0.07/0.32 & ( Q @ X ) )
% 0.07/0.32 = ( ( P4 @ X )
% 0.07/0.32 & ( Q2 @ X ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % minf(1)
% 0.07/0.32 thf(fact_339_minf_I2_J,axiom,
% 0.07/0.32 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.07/0.32 ( ? [Z4: nat] :
% 0.07/0.32 ! [X6: nat] :
% 0.07/0.32 ( ( ord_less_nat @ X6 @ Z4 )
% 0.07/0.32 => ( ( P @ X6 )
% 0.07/0.32 = ( P4 @ X6 ) ) )
% 0.07/0.32 => ( ? [Z4: nat] :
% 0.07/0.32 ! [X6: nat] :
% 0.07/0.32 ( ( ord_less_nat @ X6 @ Z4 )
% 0.07/0.32 => ( ( Q @ X6 )
% 0.07/0.32 = ( Q2 @ X6 ) ) )
% 0.07/0.32 => ? [Z2: nat] :
% 0.07/0.32 ! [X: nat] :
% 0.07/0.32 ( ( ord_less_nat @ X @ Z2 )
% 0.07/0.32 => ( ( ( P @ X )
% 0.07/0.32 | ( Q @ X ) )
% 0.07/0.32 = ( ( P4 @ X )
% 0.07/0.32 | ( Q2 @ X ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % minf(2)
% 0.07/0.32 thf(fact_340_minf_I3_J,axiom,
% 0.07/0.32 ! [T: nat] :
% 0.07/0.32 ? [Z2: nat] :
% 0.07/0.32 ! [X: nat] :
% 0.07/0.32 ( ( ord_less_nat @ X @ Z2 )
% 0.07/0.32 => ( X != T ) ) ).
% 0.07/0.32
% 0.07/0.32 % minf(3)
% 0.07/0.32 thf(fact_341_minf_I4_J,axiom,
% 0.07/0.32 ! [T: nat] :
% 0.07/0.32 ? [Z2: nat] :
% 0.07/0.32 ! [X: nat] :
% 0.07/0.32 ( ( ord_less_nat @ X @ Z2 )
% 0.07/0.32 => ( X != T ) ) ).
% 0.07/0.32
% 0.07/0.32 % minf(4)
% 0.07/0.32 thf(fact_342_minf_I5_J,axiom,
% 0.07/0.32 ! [T: nat] :
% 0.07/0.32 ? [Z2: nat] :
% 0.07/0.32 ! [X: nat] :
% 0.07/0.32 ( ( ord_less_nat @ X @ Z2 )
% 0.07/0.32 => ( ord_less_nat @ X @ T ) ) ).
% 0.07/0.32
% 0.07/0.32 % minf(5)
% 0.07/0.32 thf(fact_343_minf_I7_J,axiom,
% 0.07/0.32 ! [T: nat] :
% 0.07/0.32 ? [Z2: nat] :
% 0.07/0.32 ! [X: nat] :
% 0.07/0.32 ( ( ord_less_nat @ X @ Z2 )
% 0.07/0.32 => ~ ( ord_less_nat @ T @ X ) ) ).
% 0.07/0.32
% 0.07/0.32 % minf(7)
% 0.07/0.32 thf(fact_344_order_Onot__eq__order__implies__strict,axiom,
% 0.07/0.32 ! [A2: nat,B: nat] :
% 0.07/0.32 ( ( A2 != B )
% 0.07/0.32 => ( ( ord_less_eq_nat @ A2 @ B )
% 0.07/0.32 => ( ord_less_nat @ A2 @ B ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % order.not_eq_order_implies_strict
% 0.07/0.32 thf(fact_345_dual__order_Ostrict__implies__order,axiom,
% 0.07/0.32 ! [B: nat,A2: nat] :
% 0.07/0.32 ( ( ord_less_nat @ B @ A2 )
% 0.07/0.32 => ( ord_less_eq_nat @ B @ A2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % dual_order.strict_implies_order
% 0.07/0.32 thf(fact_346_dual__order_Ostrict__iff__order,axiom,
% 0.07/0.32 ( ord_less_nat
% 0.07/0.32 = ( ^ [B5: nat,A7: nat] :
% 0.07/0.32 ( ( ord_less_eq_nat @ B5 @ A7 )
% 0.07/0.32 & ( A7 != B5 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % dual_order.strict_iff_order
% 0.07/0.32 thf(fact_347_dual__order_Oorder__iff__strict,axiom,
% 0.07/0.32 ( ord_less_eq_nat
% 0.07/0.32 = ( ^ [B5: nat,A7: nat] :
% 0.07/0.32 ( ( ord_less_nat @ B5 @ A7 )
% 0.07/0.32 | ( A7 = B5 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % dual_order.order_iff_strict
% 0.07/0.32 thf(fact_348_order_Ostrict__implies__order,axiom,
% 0.07/0.32 ! [A2: nat,B: nat] :
% 0.07/0.32 ( ( ord_less_nat @ A2 @ B )
% 0.07/0.32 => ( ord_less_eq_nat @ A2 @ B ) ) ).
% 0.07/0.32
% 0.07/0.32 % order.strict_implies_order
% 0.07/0.32 thf(fact_349_dual__order_Ostrict__trans2,axiom,
% 0.07/0.32 ! [B: nat,A2: nat,C: nat] :
% 0.07/0.32 ( ( ord_less_nat @ B @ A2 )
% 0.07/0.32 => ( ( ord_less_eq_nat @ C @ B )
% 0.07/0.32 => ( ord_less_nat @ C @ A2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % dual_order.strict_trans2
% 0.07/0.32 thf(fact_350_dual__order_Ostrict__trans1,axiom,
% 0.07/0.32 ! [B: nat,A2: nat,C: nat] :
% 0.07/0.32 ( ( ord_less_eq_nat @ B @ A2 )
% 0.07/0.32 => ( ( ord_less_nat @ C @ B )
% 0.07/0.32 => ( ord_less_nat @ C @ A2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % dual_order.strict_trans1
% 0.07/0.32 thf(fact_351_order_Ostrict__iff__order,axiom,
% 0.07/0.32 ( ord_less_nat
% 0.07/0.32 = ( ^ [A7: nat,B5: nat] :
% 0.07/0.32 ( ( ord_less_eq_nat @ A7 @ B5 )
% 0.07/0.32 & ( A7 != B5 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % order.strict_iff_order
% 0.07/0.32 thf(fact_352_order_Oorder__iff__strict,axiom,
% 0.07/0.32 ( ord_less_eq_nat
% 0.07/0.32 = ( ^ [A7: nat,B5: nat] :
% 0.07/0.32 ( ( ord_less_nat @ A7 @ B5 )
% 0.07/0.32 | ( A7 = B5 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % order.order_iff_strict
% 0.07/0.32
% 0.07/0.32 % Conjectures (1)
% 0.07/0.32 thf(conj_0,conjecture,
% 0.07/0.32 ( a3
% 0.07/0.32 = ( groups1792256535list_a @ qs2 ) ) ).
% 0.07/0.32
% 0.07/0.32 %------------------------------------------------------------------------------
% 0.07/0.32 ------------------------
% 1.81/1.09 ---- Embedded in THF ---
% 1.81/1.09 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.81/1.09 %%% Generated on Tue Feb 24 08:25:01 EST 2026
% 1.81/1.09 %%% using '$$dhol' embedding, version 1.3.0.
% 1.81/1.09 %%% Logic specification used:
% 1.81/1.09 %%% thf(spec, logic, $$dhol).
% 1.81/1.09
% 1.81/1.09 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.OZNDMxpjYk/DTF2DTF9014.p
% See solution above
% 1.81/1.10 ------------------------
% 2.09/1.13 ---- Cleaned THF ---
% 2.09/1.13 thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
% 2.09/1.13 set_list_list_list_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__Multiset__Omultiset_It__Multiset__Omultiset_It__List__Olist_Itf__a_J_J_J,type,
% 2.09/1.13 multis971982480list_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__List__Olist_It__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
% 2.09/1.13 list_set_list_list_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
% 2.09/1.13 set_list_set_list_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Set__Oset_Itf__a_J_J_J_J,type,
% 2.09/1.13 set_list_list_set_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__List__Olist_It__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
% 2.09/1.13 list_set_set_list_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__List__Olist_It__Set__Oset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J_J,type,
% 2.09/1.13 list_set_list_set_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J_J,type,
% 2.09/1.13 set_list_set_set_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__List__Olist_It__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J_J,type,
% 2.09/1.13 list_set_set_set_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__List__Olist_It__Multiset__Omultiset_It__List__Olist_Itf__a_J_J_J,type,
% 2.09/1.13 list_multiset_list_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__Set__Oset_It__Multiset__Omultiset_It__List__Olist_Itf__a_J_J_J,type,
% 2.09/1.13 set_multiset_list_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__Set__Oset_It__List__Olist_It__Multiset__Omultiset_Itf__a_J_J_J,type,
% 2.09/1.13 set_list_multiset_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
% 2.09/1.13 multiset_set_list_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__List__Olist_It__Set__Oset_It__Multiset__Omultiset_Itf__a_J_J_J,type,
% 2.09/1.13 list_set_multiset_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
% 2.09/1.13 multiset_set_set_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
% 2.09/1.13 list_list_list_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
% 2.09/1.13 set_list_list_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J,type,
% 2.09/1.13 multiset_multiset_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
% 2.09/1.13 list_set_list_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__List__Olist_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
% 2.09/1.13 list_list_set_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
% 2.09/1.13 set_set_list_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
% 2.09/1.13 set_list_set_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__List__Olist_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
% 2.09/1.13 list_set_set_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
% 2.09/1.13 set_set_set_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__Multiset__Omultiset_It__List__Olist_Itf__a_J_J,type,
% 2.09/1.13 multiset_list_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__List__Olist_It__Multiset__Omultiset_Itf__a_J_J,type,
% 2.09/1.13 list_multiset_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__Set__Oset_It__Multiset__Omultiset_Itf__a_J_J,type,
% 2.09/1.13 set_multiset_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
% 2.09/1.13 multiset_set_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
% 2.09/1.13 set_list_nat: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
% 2.09/1.13 list_set_nat: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 2.09/1.13 set_set_nat: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
% 2.09/1.13 list_list_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
% 2.09/1.13 set_list_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
% 2.09/1.13 list_set_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__Multiset__Omultiset_It__Nat__Onat_J,type,
% 2.09/1.13 multiset_nat: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 2.09/1.13 set_set_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__Multiset__Omultiset_Itf__a_J,type,
% 2.09/1.13 multiset_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
% 2.09/1.13 list_nat: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 2.09/1.13 set_nat: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__List__Olist_Itf__a_J,type,
% 2.09/1.13 list_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 2.09/1.13 set_a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_t__Nat__Onat,type,
% 2.09/1.13 nat: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(ty_n_tf__a,type,
% 2.09/1.13 a: $tType ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Fun_Obij__betw_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
% 2.09/1.13 bij_be94573046list_a: ( list_a > list_a ) > set_list_a > set_list_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Fun_Obij__betw_001t__List__Olist_Itf__a_J_001tf__a,type,
% 2.09/1.13 bij_betw_list_a_a: ( list_a > a ) > set_list_a > set_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
% 2.09/1.13 bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Fun_Obij__betw_001tf__a_001t__List__Olist_Itf__a_J,type,
% 2.09/1.13 bij_betw_a_list_a: ( a > list_a ) > set_a > set_list_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Fun_Obij__betw_001tf__a_001tf__a,type,
% 2.09/1.13 bij_betw_a_a: ( a > a ) > set_a > set_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
% 2.09/1.13 inj_on_list_a_list_a: ( list_a > list_a ) > set_list_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001tf__a,type,
% 2.09/1.13 inj_on_list_a_a: ( list_a > a ) > set_list_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
% 2.09/1.13 inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Fun_Oinj__on_001tf__a_001t__List__Olist_Itf__a_J,type,
% 2.09/1.13 inj_on_a_list_a: ( a > list_a ) > set_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
% 2.09/1.13 inj_on_a_a: ( a > a ) > set_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Multiset__Omultiset_It__List__Olist_Itf__a_J_J,type,
% 2.09/1.13 groups1271887971list_a: list_multiset_list_a > multiset_list_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.09/1.13 groups1592617181iset_a: list_multiset_a > multiset_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
% 2.09/1.13 groups921905271st_nat: list_nat > nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001tf__a,type,
% 2.09/1.13 groups1792256535list_a: list_a > a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Hilbert__Choice_Oinv__into_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
% 2.09/1.13 hilber921249868list_a: set_list_a > ( list_a > list_a ) > list_a > list_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Hilbert__Choice_Oinv__into_001t__List__Olist_Itf__a_J_001tf__a,type,
% 2.09/1.13 hilber2125729734st_a_a: set_list_a > ( list_a > a ) > a > list_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Hilbert__Choice_Oinv__into_001t__Nat__Onat_001t__Nat__Onat,type,
% 2.09/1.13 hilber815131374at_nat: set_nat > ( nat > nat ) > nat > nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Hilbert__Choice_Oinv__into_001tf__a_001t__List__Olist_Itf__a_J,type,
% 2.09/1.13 hilber165648082list_a: set_a > ( a > list_a ) > list_a > a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Hilbert__Choice_Oinv__into_001tf__a_001tf__a,type,
% 2.09/1.13 hilbert_inv_into_a_a: set_a > ( a > a ) > a > a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Hilbert__Function__Mirabelle__sksumwvvhs_Odirect__decomp_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.09/1.13 hilber2024005914iset_a: set_multiset_a > list_set_multiset_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Hilbert__Function__Mirabelle__sksumwvvhs_Odirect__decomp_001t__Nat__Onat,type,
% 2.09/1.13 hilber1014902394mp_nat: set_nat > list_set_nat > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Hilbert__Function__Mirabelle__sksumwvvhs_Odirect__decomp_001tf__a,type,
% 2.09/1.13 hilber2037636820comp_a: set_a > list_set_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Olist__ex_001t__Set__Oset_Itf__a_J,type,
% 2.09/1.13 list_ex_set_a: ( set_a > $o ) > list_set_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Olist__ex_001tf__a,type,
% 2.09/1.13 list_ex_a: ( a > $o ) > list_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Olist__update_001t__Set__Oset_Itf__a_J,type,
% 2.09/1.13 list_update_set_a: list_set_a > nat > set_a > list_set_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Olist__update_001tf__a,type,
% 2.09/1.13 list_update_a: list_a > nat > a > list_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Olistset_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
% 2.09/1.13 listset_list_list_a: list_set_list_list_a > set_list_list_list_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Olistset_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
% 2.09/1.13 listset_list_set_a: list_set_list_set_a > set_list_list_set_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Olistset_001t__List__Olist_Itf__a_J,type,
% 2.09/1.13 listset_list_a: list_set_list_a > set_list_list_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Olistset_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.09/1.13 listset_multiset_a: list_set_multiset_a > set_list_multiset_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Olistset_001t__Nat__Onat,type,
% 2.09/1.13 listset_nat: list_set_nat > set_list_nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Olistset_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
% 2.09/1.13 listset_set_list_a: list_set_set_list_a > set_list_set_list_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Olistset_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 2.09/1.13 listset_set_set_a: list_set_set_set_a > set_list_set_set_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Olistset_001t__Set__Oset_Itf__a_J,type,
% 2.09/1.13 listset_set_a: list_set_set_a > set_list_set_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Olistset_001tf__a,type,
% 2.09/1.13 listset_a: list_set_a > set_list_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Onth_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
% 2.09/1.13 nth_list_list_a: list_list_list_a > nat > list_list_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Onth_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
% 2.09/1.13 nth_list_set_a: list_list_set_a > nat > list_set_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Onth_001t__List__Olist_Itf__a_J,type,
% 2.09/1.13 nth_list_a: list_list_a > nat > list_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Onth_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.09/1.13 nth_multiset_a: list_multiset_a > nat > multiset_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Onth_001t__Nat__Onat,type,
% 2.09/1.13 nth_nat: list_nat > nat > nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Onth_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
% 2.09/1.13 nth_set_list_list_a: list_set_list_list_a > nat > set_list_list_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Onth_001t__Set__Oset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
% 2.09/1.13 nth_set_list_set_a: list_set_list_set_a > nat > set_list_set_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Onth_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
% 2.09/1.13 nth_set_list_a: list_set_list_a > nat > set_list_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Onth_001t__Set__Oset_It__Multiset__Omultiset_Itf__a_J_J,type,
% 2.09/1.13 nth_set_multiset_a: list_set_multiset_a > nat > set_multiset_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
% 2.09/1.13 nth_set_nat: list_set_nat > nat > set_nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Onth_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
% 2.09/1.13 nth_set_set_list_a: list_set_set_list_a > nat > set_set_list_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Onth_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
% 2.09/1.13 nth_set_set_set_a: list_set_set_set_a > nat > set_set_set_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Onth_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 2.09/1.13 nth_set_set_a: list_set_set_a > nat > set_set_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Onth_001t__Set__Oset_Itf__a_J,type,
% 2.09/1.13 nth_set_a: list_set_a > nat > set_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Onth_001tf__a,type,
% 2.09/1.13 nth_a: list_a > nat > a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Orev_001t__List__Olist_Itf__a_J,type,
% 2.09/1.13 rev_list_a: list_list_a > list_list_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Orev_001t__Nat__Onat,type,
% 2.09/1.13 rev_nat: list_nat > list_nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Orev_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
% 2.09/1.13 rev_set_list_a: list_set_list_a > list_set_list_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Orev_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 2.09/1.13 rev_set_set_a: list_set_set_a > list_set_set_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Orev_001t__Set__Oset_Itf__a_J,type,
% 2.09/1.13 rev_set_a: list_set_a > list_set_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_List_Orev_001tf__a,type,
% 2.09/1.13 rev_a: list_a > list_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset_001t__Multiset__Omultiset_It__List__Olist_Itf__a_J_J,type,
% 2.09/1.13 comm_m324867663list_a: multis971982480list_a > multiset_list_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.09/1.13 comm_m2145643721iset_a: multiset_multiset_a > multiset_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset_001t__Nat__Onat,type,
% 2.09/1.13 comm_m1100186507et_nat: multiset_nat > nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset_001tf__a,type,
% 2.09/1.13 comm_m543484931mset_a: multiset_a > a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Multiset_Omset_001t__List__Olist_Itf__a_J,type,
% 2.09/1.13 mset_list_a: list_list_a > multiset_list_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Multiset_Omset_001t__Multiset__Omultiset_It__List__Olist_Itf__a_J_J,type,
% 2.09/1.13 mset_multiset_list_a: list_multiset_list_a > multis971982480list_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Multiset_Omset_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.09/1.13 mset_multiset_a: list_multiset_a > multiset_multiset_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Multiset_Omset_001t__Nat__Onat,type,
% 2.09/1.13 mset_nat: list_nat > multiset_nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Multiset_Omset_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
% 2.09/1.13 mset_set_list_a: list_set_list_a > multiset_set_list_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Multiset_Omset_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 2.09/1.13 mset_set_set_a: list_set_set_a > multiset_set_set_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Multiset_Omset_001t__Set__Oset_Itf__a_J,type,
% 2.09/1.13 mset_set_a: list_set_a > multiset_set_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Multiset_Omset_001tf__a,type,
% 2.09/1.13 mset_a: list_a > multiset_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Multiset_Oset__mset_001t__List__Olist_Itf__a_J,type,
% 2.09/1.13 set_mset_list_a: multiset_list_a > set_list_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Multiset_Oset__mset_001t__Multiset__Omultiset_It__List__Olist_Itf__a_J_J,type,
% 2.09/1.13 set_ms1660427399list_a: multis971982480list_a > set_multiset_list_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Multiset_Oset__mset_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.09/1.13 set_mset_multiset_a: multiset_multiset_a > set_multiset_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Multiset_Oset__mset_001t__Set__Oset_Itf__a_J,type,
% 2.09/1.13 set_mset_set_a: multiset_set_a > set_set_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Multiset_Oset__mset_001tf__a,type,
% 2.09/1.13 set_mset_a: multiset_a > set_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
% 2.09/1.13 size_s575106428list_a: list_list_list_a > nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
% 2.09/1.13 size_s341332310_set_a: list_list_set_a > nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
% 2.09/1.13 size_s1427607542list_a: list_list_a > nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Multiset__Omultiset_Itf__a_J_J,type,
% 2.09/1.13 size_s1657263798iset_a: list_multiset_a > nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
% 2.09/1.13 size_size_list_nat: list_nat > nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
% 2.09/1.13 size_s240404444list_a: list_set_list_list_a > nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J_J,type,
% 2.09/1.13 size_s926492982_set_a: list_set_list_set_a > nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
% 2.09/1.13 size_s1635937238list_a: list_set_list_a > nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Multiset__Omultiset_Itf__a_J_J_J,type,
% 2.09/1.13 size_s802269334iset_a: list_set_multiset_a > nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
% 2.09/1.13 size_s577819178et_nat: list_set_nat > nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
% 2.09/1.13 size_s469631926list_a: list_set_set_list_a > nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J_J,type,
% 2.09/1.13 size_s726512144_set_a: list_set_set_set_a > nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
% 2.09/1.13 size_s739728560_set_a: list_set_set_a > nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
% 2.09/1.13 size_size_list_set_a: list_set_a > nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
% 2.09/1.13 size_size_list_a: list_a > nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
% 2.09/1.13 size_s657334288_set_a: multiset_set_a > nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.09/1.13 size_size_multiset_a: multiset_a > nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 2.09/1.13 ord_less_nat: nat > nat > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
% 2.09/1.13 ord_less_set_list_a: set_list_a > set_list_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
% 2.09/1.13 ord_less_set_nat: set_nat > set_nat > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
% 2.09/1.13 ord_less_set_a: set_a > set_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J,type,
% 2.09/1.13 ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 2.09/1.13 ord_less_eq_nat: nat > nat > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
% 2.09/1.13 ord_le1301786372list_a: set_list_a > set_list_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
% 2.09/1.13 ord_less_eq_set_nat: set_nat > set_nat > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
% 2.09/1.13 ord_less_eq_set_a: set_a > set_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Permutation_Operm_001t__Set__Oset_Itf__a_J,type,
% 2.09/1.13 perm_set_a: list_set_a > list_set_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Permutation_Operm_001tf__a,type,
% 2.09/1.13 perm_a: list_a > list_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
% 2.09/1.13 collect_list_list_a: ( list_list_a > $o ) > set_list_list_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Set_OCollect_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
% 2.09/1.13 collect_list_set_a: ( list_set_a > $o ) > set_list_set_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
% 2.09/1.13 collect_list_a: ( list_a > $o ) > set_list_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Set_OCollect_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.09/1.13 collect_multiset_a: ( multiset_a > $o ) > set_multiset_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 2.09/1.13 collect_nat: ( nat > $o ) > set_nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
% 2.09/1.13 collect_set_a: ( set_a > $o ) > set_set_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Set_OCollect_001tf__a,type,
% 2.09/1.13 collect_a: ( a > $o ) > set_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
% 2.09/1.13 image_list_nat_nat: ( list_nat > nat ) > set_list_nat > set_nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
% 2.09/1.13 image_list_a_list_a: ( list_a > list_a ) > set_list_a > set_list_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001tf__a,type,
% 2.09/1.13 image_list_a_a: ( list_a > a ) > set_list_a > set_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
% 2.09/1.13 image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Set_Oimage_001tf__a_001t__List__Olist_Itf__a_J,type,
% 2.09/1.13 image_a_list_a: ( a > list_a ) > set_a > set_list_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
% 2.09/1.13 image_a_a: ( a > a ) > set_a > set_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
% 2.09/1.13 set_ord_lessThan_nat: nat > set_nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
% 2.09/1.13 set_or1578766727list_a: set_list_a > set_set_list_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
% 2.09/1.13 set_or1597314339et_nat: set_nat > set_set_nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_Itf__a_J,type,
% 2.09/1.13 set_or1164043265_set_a: set_a > set_set_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_member_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
% 2.09/1.13 member1511395513list_a: list_list_list_a > set_list_list_list_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_member_001t__List__Olist_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
% 2.09/1.13 member954073875_set_a: list_list_set_a > set_list_list_set_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
% 2.09/1.13 member_list_list_a: list_list_a > set_list_list_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_member_001t__List__Olist_It__Multiset__Omultiset_Itf__a_J_J,type,
% 2.09/1.13 member518144627iset_a: list_multiset_a > set_list_multiset_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
% 2.09/1.13 member_list_nat: list_nat > set_list_nat > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
% 2.09/1.13 member101195155list_a: list_set_list_a > set_list_set_list_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
% 2.09/1.13 member814963949_set_a: list_set_set_a > set_list_set_set_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_member_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
% 2.09/1.13 member_list_set_a: list_set_a > set_list_set_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
% 2.09/1.13 member_list_a: list_a > set_list_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_member_001t__Multiset__Omultiset_It__List__Olist_Itf__a_J_J,type,
% 2.09/1.13 member528267379list_a: multiset_list_a > set_multiset_list_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_member_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.09/1.13 member_multiset_a: multiset_a > set_multiset_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_member_001t__Nat__Onat,type,
% 2.09/1.13 member_nat: nat > set_nat > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_member_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
% 2.09/1.13 member_set_list_a: set_list_a > set_set_list_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
% 2.09/1.13 member_set_nat: set_nat > set_set_nat > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 2.09/1.13 member_set_set_a: set_set_a > set_set_set_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
% 2.09/1.13 member_set_a: set_a > set_set_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_c_member_001tf__a,type,
% 2.09/1.13 member_a: a > set_a > $o ).
% 2.09/1.13
% 2.09/1.13 thf(sy_v_A,type,
% 2.09/1.13 a2: set_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_v_a____,type,
% 2.09/1.13 a3: a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_v_f____,type,
% 2.09/1.13 f: nat > nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_v_g____,type,
% 2.09/1.13 g: nat > nat ).
% 2.09/1.13
% 2.09/1.13 thf(sy_v_qs2____,type,
% 2.09/1.13 qs2: list_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_v_qs____,type,
% 2.09/1.13 qs: list_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_v_ss1,type,
% 2.09/1.13 ss1: list_set_a ).
% 2.09/1.13
% 2.09/1.13 thf(sy_v_ss2,type,
% 2.09/1.13 ss2: list_set_a ).
% 2.09/1.13
% 2.09/1.13 thf(fact_0__092_060open_062mset_Aqs2_A_061_Amset_Aqs_092_060close_062,axiom,
% 2.09/1.13 ( ( mset_a @ qs2 )
% 2.09/1.13 = ( mset_a @ qs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_1__092_060open_062a_A_092_060in_062_AA_092_060close_062,axiom,
% 2.09/1.13 member_a @ a3 @ a2 ).
% 2.09/1.13
% 2.09/1.13 thf(fact_2_a,axiom,
% 2.09/1.13 ( a3
% 2.09/1.13 = ( groups1792256535list_a @ qs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_3_len__qs2,axiom,
% 2.09/1.13 ( ( size_size_list_a @ qs2 )
% 2.09/1.13 = ( size_size_list_a @ qs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_4__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062qs_O_A_092_060lbrakk_062a_A_061_Asum__list_Aqs_059_Aqs_A_092_060in_062_Alistset_Ass1_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 2.09/1.13 ~ ! [Qs: list_a] :
% 2.09/1.13 ( ( a3
% 2.09/1.13 = ( groups1792256535list_a @ Qs ) )
% 2.09/1.13 => ~ ( member_list_a @ Qs @ ( listset_a @ ss1 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_5_qs2__in,axiom,
% 2.09/1.13 member_list_a @ qs2 @ ( listset_a @ ss2 ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_6_direct__decompD_I1_J,axiom,
% 2.09/1.13 ! [A: set_multiset_a,Ss: list_set_multiset_a,Qs2: list_multiset_a] :
% 2.09/1.13 ( ( hilber2024005914iset_a @ A @ Ss )
% 2.09/1.13 => ( ( member518144627iset_a @ Qs2 @ ( listset_multiset_a @ Ss ) )
% 2.09/1.13 => ( member_multiset_a @ ( groups1592617181iset_a @ Qs2 ) @ A ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_7_direct__decompD_I1_J,axiom,
% 2.09/1.13 ! [A: set_nat,Ss: list_set_nat,Qs2: list_nat] :
% 2.09/1.13 ( ( hilber1014902394mp_nat @ A @ Ss )
% 2.09/1.13 => ( ( member_list_nat @ Qs2 @ ( listset_nat @ Ss ) )
% 2.09/1.13 => ( member_nat @ ( groups921905271st_nat @ Qs2 ) @ A ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_8_direct__decompD_I1_J,axiom,
% 2.09/1.13 ! [A: set_a,Ss: list_set_a,Qs2: list_a] :
% 2.09/1.13 ( ( hilber2037636820comp_a @ A @ Ss )
% 2.09/1.13 => ( ( member_list_a @ Qs2 @ ( listset_a @ Ss ) )
% 2.09/1.13 => ( member_a @ ( groups1792256535list_a @ Qs2 ) @ A ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_9_direct__decompE,axiom,
% 2.09/1.13 ! [A: set_multiset_a,Ss: list_set_multiset_a,A2: multiset_a] :
% 2.09/1.13 ( ( hilber2024005914iset_a @ A @ Ss )
% 2.09/1.13 => ( ( member_multiset_a @ A2 @ A )
% 2.09/1.13 => ~ ! [Qs: list_multiset_a] :
% 2.09/1.13 ( ( member518144627iset_a @ Qs @ ( listset_multiset_a @ Ss ) )
% 2.09/1.13 => ( A2
% 2.09/1.13 != ( groups1592617181iset_a @ Qs ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_10_direct__decompE,axiom,
% 2.09/1.13 ! [A: set_nat,Ss: list_set_nat,A2: nat] :
% 2.09/1.13 ( ( hilber1014902394mp_nat @ A @ Ss )
% 2.09/1.13 => ( ( member_nat @ A2 @ A )
% 2.09/1.13 => ~ ! [Qs: list_nat] :
% 2.09/1.13 ( ( member_list_nat @ Qs @ ( listset_nat @ Ss ) )
% 2.09/1.13 => ( A2
% 2.09/1.13 != ( groups921905271st_nat @ Qs ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_11_direct__decompE,axiom,
% 2.09/1.13 ! [A: set_a,Ss: list_set_a,A2: a] :
% 2.09/1.13 ( ( hilber2037636820comp_a @ A @ Ss )
% 2.09/1.13 => ( ( member_a @ A2 @ A )
% 2.09/1.13 => ~ ! [Qs: list_a] :
% 2.09/1.13 ( ( member_list_a @ Qs @ ( listset_a @ Ss ) )
% 2.09/1.13 => ( A2
% 2.09/1.13 != ( groups1792256535list_a @ Qs ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_12_direct__decompI__alt,axiom,
% 2.09/1.13 ! [Ss: list_set_multiset_a,A: set_multiset_a] :
% 2.09/1.13 ( ! [Qs: list_multiset_a] :
% 2.09/1.13 ( ( member518144627iset_a @ Qs @ ( listset_multiset_a @ Ss ) )
% 2.09/1.13 => ( member_multiset_a @ ( groups1592617181iset_a @ Qs ) @ A ) )
% 2.09/1.13 => ( ! [A3: multiset_a] :
% 2.09/1.13 ( ( member_multiset_a @ A3 @ A )
% 2.09/1.13 => ? [X: list_multiset_a] :
% 2.09/1.13 ( ( member518144627iset_a @ X @ ( listset_multiset_a @ Ss ) )
% 2.09/1.13 & ( A3
% 2.09/1.13 = ( groups1592617181iset_a @ X ) )
% 2.09/1.13 & ! [Y: list_multiset_a] :
% 2.09/1.13 ( ( ( member518144627iset_a @ Y @ ( listset_multiset_a @ Ss ) )
% 2.09/1.13 & ( A3
% 2.09/1.13 = ( groups1592617181iset_a @ Y ) ) )
% 2.09/1.13 => ( Y = X ) ) ) )
% 2.09/1.13 => ( hilber2024005914iset_a @ A @ Ss ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_13_direct__decompI__alt,axiom,
% 2.09/1.13 ! [Ss: list_set_nat,A: set_nat] :
% 2.09/1.13 ( ! [Qs: list_nat] :
% 2.09/1.13 ( ( member_list_nat @ Qs @ ( listset_nat @ Ss ) )
% 2.09/1.13 => ( member_nat @ ( groups921905271st_nat @ Qs ) @ A ) )
% 2.09/1.13 => ( ! [A3: nat] :
% 2.09/1.13 ( ( member_nat @ A3 @ A )
% 2.09/1.13 => ? [X: list_nat] :
% 2.09/1.13 ( ( member_list_nat @ X @ ( listset_nat @ Ss ) )
% 2.09/1.13 & ( A3
% 2.09/1.13 = ( groups921905271st_nat @ X ) )
% 2.09/1.13 & ! [Y: list_nat] :
% 2.09/1.13 ( ( ( member_list_nat @ Y @ ( listset_nat @ Ss ) )
% 2.09/1.13 & ( A3
% 2.09/1.13 = ( groups921905271st_nat @ Y ) ) )
% 2.09/1.13 => ( Y = X ) ) ) )
% 2.09/1.13 => ( hilber1014902394mp_nat @ A @ Ss ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_14_direct__decompI__alt,axiom,
% 2.09/1.13 ! [Ss: list_set_a,A: set_a] :
% 2.09/1.13 ( ! [Qs: list_a] :
% 2.09/1.13 ( ( member_list_a @ Qs @ ( listset_a @ Ss ) )
% 2.09/1.13 => ( member_a @ ( groups1792256535list_a @ Qs ) @ A ) )
% 2.09/1.13 => ( ! [A3: a] :
% 2.09/1.13 ( ( member_a @ A3 @ A )
% 2.09/1.13 => ? [X: list_a] :
% 2.09/1.13 ( ( member_list_a @ X @ ( listset_a @ Ss ) )
% 2.09/1.13 & ( A3
% 2.09/1.13 = ( groups1792256535list_a @ X ) )
% 2.09/1.13 & ! [Y: list_a] :
% 2.09/1.13 ( ( ( member_list_a @ Y @ ( listset_a @ Ss ) )
% 2.09/1.13 & ( A3
% 2.09/1.13 = ( groups1792256535list_a @ Y ) ) )
% 2.09/1.13 => ( Y = X ) ) ) )
% 2.09/1.13 => ( hilber2037636820comp_a @ A @ Ss ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_15_direct__decomp__unique,axiom,
% 2.09/1.13 ! [A: set_nat,Ss: list_set_nat,Qs2: list_nat,Qs3: list_nat] :
% 2.09/1.13 ( ( hilber1014902394mp_nat @ A @ Ss )
% 2.09/1.13 => ( ( member_list_nat @ Qs2 @ ( listset_nat @ Ss ) )
% 2.09/1.13 => ( ( member_list_nat @ Qs3 @ ( listset_nat @ Ss ) )
% 2.09/1.13 => ( ( ( groups921905271st_nat @ Qs2 )
% 2.09/1.13 = ( groups921905271st_nat @ Qs3 ) )
% 2.09/1.13 => ( Qs2 = Qs3 ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_16_direct__decomp__unique,axiom,
% 2.09/1.13 ! [A: set_a,Ss: list_set_a,Qs2: list_a,Qs3: list_a] :
% 2.09/1.13 ( ( hilber2037636820comp_a @ A @ Ss )
% 2.09/1.13 => ( ( member_list_a @ Qs2 @ ( listset_a @ Ss ) )
% 2.09/1.13 => ( ( member_list_a @ Qs3 @ ( listset_a @ Ss ) )
% 2.09/1.13 => ( ( ( groups1792256535list_a @ Qs2 )
% 2.09/1.13 = ( groups1792256535list_a @ Qs3 ) )
% 2.09/1.13 => ( Qs2 = Qs3 ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_17_sum__list_Orev,axiom,
% 2.09/1.13 ! [Xs: list_nat] :
% 2.09/1.13 ( ( groups921905271st_nat @ ( rev_nat @ Xs ) )
% 2.09/1.13 = ( groups921905271st_nat @ Xs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_18_sum__list_Orev,axiom,
% 2.09/1.13 ! [Xs: list_a] :
% 2.09/1.13 ( ( groups1792256535list_a @ ( rev_a @ Xs ) )
% 2.09/1.13 = ( groups1792256535list_a @ Xs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_19__C1_C,axiom,
% 2.09/1.13 ! [I: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I @ ( size_size_list_a @ qs ) )
% 2.09/1.13 => ( ( nth_a @ qs2 @ I )
% 2.09/1.13 = ( nth_a @ qs @ ( g @ I ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_20_g__bij2,axiom,
% 2.09/1.13 bij_betw_nat_nat @ g @ ( set_ord_lessThan_nat @ ( size_size_list_a @ qs2 ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_a @ qs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_21_sum__mset__sum__list,axiom,
% 2.09/1.13 ! [Xs: list_nat] :
% 2.09/1.13 ( ( comm_m1100186507et_nat @ ( mset_nat @ Xs ) )
% 2.09/1.13 = ( groups921905271st_nat @ Xs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_22_sum__mset__sum__list,axiom,
% 2.09/1.13 ! [Xs: list_multiset_list_a] :
% 2.09/1.13 ( ( comm_m324867663list_a @ ( mset_multiset_list_a @ Xs ) )
% 2.09/1.13 = ( groups1271887971list_a @ Xs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_23_sum__mset__sum__list,axiom,
% 2.09/1.13 ! [Xs: list_multiset_a] :
% 2.09/1.13 ( ( comm_m2145643721iset_a @ ( mset_multiset_a @ Xs ) )
% 2.09/1.13 = ( groups1592617181iset_a @ Xs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_24_sum__mset__sum__list,axiom,
% 2.09/1.13 ! [Xs: list_a] :
% 2.09/1.13 ( ( comm_m543484931mset_a @ ( mset_a @ Xs ) )
% 2.09/1.13 = ( groups1792256535list_a @ Xs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_25_direct__decompD_I3_J,axiom,
% 2.09/1.13 ! [A: set_nat,Ss: list_set_nat] :
% 2.09/1.13 ( ( hilber1014902394mp_nat @ A @ Ss )
% 2.09/1.13 => ( ( image_list_nat_nat @ groups921905271st_nat @ ( listset_nat @ Ss ) )
% 2.09/1.13 = A ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_26_direct__decompD_I3_J,axiom,
% 2.09/1.13 ! [A: set_a,Ss: list_set_a] :
% 2.09/1.13 ( ( hilber2037636820comp_a @ A @ Ss )
% 2.09/1.13 => ( ( image_list_a_a @ groups1792256535list_a @ ( listset_a @ Ss ) )
% 2.09/1.13 = A ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_27_assms_I1_J,axiom,
% 2.09/1.13 hilber2037636820comp_a @ a2 @ ss1 ).
% 2.09/1.13
% 2.09/1.13 thf(fact_28_qs__in,axiom,
% 2.09/1.13 member_list_a @ qs @ ( listset_a @ ss1 ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_29_mset__rev,axiom,
% 2.09/1.13 ! [Xs: list_set_a] :
% 2.09/1.13 ( ( mset_set_a @ ( rev_set_a @ Xs ) )
% 2.09/1.13 = ( mset_set_a @ Xs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_30_mset__rev,axiom,
% 2.09/1.13 ! [Xs: list_list_a] :
% 2.09/1.13 ( ( mset_list_a @ ( rev_list_a @ Xs ) )
% 2.09/1.13 = ( mset_list_a @ Xs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_31_mset__rev,axiom,
% 2.09/1.13 ! [Xs: list_a] :
% 2.09/1.13 ( ( mset_a @ ( rev_a @ Xs ) )
% 2.09/1.13 = ( mset_a @ Xs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_32__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062qs2_O_A_092_060lbrakk_062qs2_A_092_060in_062_Alistset_Ass2_059_Alength_Aqs2_A_061_Alength_Aqs_059_A_092_060And_062i_O_Ai_A_060_Alength_Aqs_A_092_060Longrightarrow_062_Aqs2_A_B_Ai_A_061_Aqs_A_B_Ag_Ai_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 2.09/1.13 ~ ! [Qs22: list_a] :
% 2.09/1.13 ( ( member_list_a @ Qs22 @ ( listset_a @ ss2 ) )
% 2.09/1.13 => ( ( ( size_size_list_a @ Qs22 )
% 2.09/1.13 = ( size_size_list_a @ qs ) )
% 2.09/1.13 => ~ ! [I2: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I2 @ ( size_size_list_a @ qs ) )
% 2.09/1.13 => ( ( nth_a @ Qs22 @ I2 )
% 2.09/1.13 = ( nth_a @ qs @ ( g @ I2 ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_33_assms_I2_J,axiom,
% 2.09/1.13 perm_set_a @ ss1 @ ss2 ).
% 2.09/1.13
% 2.09/1.13 thf(fact_34_ex__mset,axiom,
% 2.09/1.13 ! [X2: multiset_set_a] :
% 2.09/1.13 ? [Xs2: list_set_a] :
% 2.09/1.13 ( ( mset_set_a @ Xs2 )
% 2.09/1.13 = X2 ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_35_ex__mset,axiom,
% 2.09/1.13 ! [X2: multiset_list_a] :
% 2.09/1.13 ? [Xs2: list_list_a] :
% 2.09/1.13 ( ( mset_list_a @ Xs2 )
% 2.09/1.13 = X2 ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_36_ex__mset,axiom,
% 2.09/1.13 ! [X2: multiset_a] :
% 2.09/1.13 ? [Xs2: list_a] :
% 2.09/1.13 ( ( mset_a @ Xs2 )
% 2.09/1.13 = X2 ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_37_mset__eq__length,axiom,
% 2.09/1.13 ! [Xs: list_list_a,Ys: list_list_a] :
% 2.09/1.13 ( ( ( mset_list_a @ Xs )
% 2.09/1.13 = ( mset_list_a @ Ys ) )
% 2.09/1.13 => ( ( size_s1427607542list_a @ Xs )
% 2.09/1.13 = ( size_s1427607542list_a @ Ys ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_38_mset__eq__length,axiom,
% 2.09/1.13 ! [Xs: list_set_list_a,Ys: list_set_list_a] :
% 2.09/1.13 ( ( ( mset_set_list_a @ Xs )
% 2.09/1.13 = ( mset_set_list_a @ Ys ) )
% 2.09/1.13 => ( ( size_s1635937238list_a @ Xs )
% 2.09/1.13 = ( size_s1635937238list_a @ Ys ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_39_mset__eq__length,axiom,
% 2.09/1.13 ! [Xs: list_set_set_a,Ys: list_set_set_a] :
% 2.09/1.13 ( ( ( mset_set_set_a @ Xs )
% 2.09/1.13 = ( mset_set_set_a @ Ys ) )
% 2.09/1.13 => ( ( size_s739728560_set_a @ Xs )
% 2.09/1.13 = ( size_s739728560_set_a @ Ys ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_40_mset__eq__length,axiom,
% 2.09/1.13 ! [Xs: list_nat,Ys: list_nat] :
% 2.09/1.13 ( ( ( mset_nat @ Xs )
% 2.09/1.13 = ( mset_nat @ Ys ) )
% 2.09/1.13 => ( ( size_size_list_nat @ Xs )
% 2.09/1.13 = ( size_size_list_nat @ Ys ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_41_mset__eq__length,axiom,
% 2.09/1.13 ! [Xs: list_a,Ys: list_a] :
% 2.09/1.13 ( ( ( mset_a @ Xs )
% 2.09/1.13 = ( mset_a @ Ys ) )
% 2.09/1.13 => ( ( size_size_list_a @ Xs )
% 2.09/1.13 = ( size_size_list_a @ Ys ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_42_mset__eq__length,axiom,
% 2.09/1.13 ! [Xs: list_set_a,Ys: list_set_a] :
% 2.09/1.13 ( ( ( mset_set_a @ Xs )
% 2.09/1.13 = ( mset_set_a @ Ys ) )
% 2.09/1.13 => ( ( size_size_list_set_a @ Xs )
% 2.09/1.13 = ( size_size_list_set_a @ Ys ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_43_mset__bij,axiom,
% 2.09/1.13 ! [F: nat > nat,Xs: list_list_a,Ys: list_list_a] :
% 2.09/1.13 ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_s1427607542list_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_s1427607542list_a @ Ys ) ) )
% 2.09/1.13 => ( ! [I3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I3 @ ( size_s1427607542list_a @ Xs ) )
% 2.09/1.13 => ( ( nth_list_a @ Xs @ I3 )
% 2.09/1.13 = ( nth_list_a @ Ys @ ( F @ I3 ) ) ) )
% 2.09/1.13 => ( ( mset_list_a @ Xs )
% 2.09/1.13 = ( mset_list_a @ Ys ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_44_mset__bij,axiom,
% 2.09/1.13 ! [F: nat > nat,Xs: list_set_list_a,Ys: list_set_list_a] :
% 2.09/1.13 ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_s1635937238list_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_s1635937238list_a @ Ys ) ) )
% 2.09/1.13 => ( ! [I3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I3 @ ( size_s1635937238list_a @ Xs ) )
% 2.09/1.13 => ( ( nth_set_list_a @ Xs @ I3 )
% 2.09/1.13 = ( nth_set_list_a @ Ys @ ( F @ I3 ) ) ) )
% 2.09/1.13 => ( ( mset_set_list_a @ Xs )
% 2.09/1.13 = ( mset_set_list_a @ Ys ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_45_mset__bij,axiom,
% 2.09/1.13 ! [F: nat > nat,Xs: list_set_set_a,Ys: list_set_set_a] :
% 2.09/1.13 ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_s739728560_set_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_s739728560_set_a @ Ys ) ) )
% 2.09/1.13 => ( ! [I3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I3 @ ( size_s739728560_set_a @ Xs ) )
% 2.09/1.13 => ( ( nth_set_set_a @ Xs @ I3 )
% 2.09/1.13 = ( nth_set_set_a @ Ys @ ( F @ I3 ) ) ) )
% 2.09/1.13 => ( ( mset_set_set_a @ Xs )
% 2.09/1.13 = ( mset_set_set_a @ Ys ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_46_mset__bij,axiom,
% 2.09/1.13 ! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
% 2.09/1.13 ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_size_list_nat @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_nat @ Ys ) ) )
% 2.09/1.13 => ( ! [I3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
% 2.09/1.13 => ( ( nth_nat @ Xs @ I3 )
% 2.09/1.13 = ( nth_nat @ Ys @ ( F @ I3 ) ) ) )
% 2.09/1.13 => ( ( mset_nat @ Xs )
% 2.09/1.13 = ( mset_nat @ Ys ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_47_mset__bij,axiom,
% 2.09/1.13 ! [F: nat > nat,Xs: list_a,Ys: list_a] :
% 2.09/1.13 ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_size_list_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_a @ Ys ) ) )
% 2.09/1.13 => ( ! [I3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
% 2.09/1.13 => ( ( nth_a @ Xs @ I3 )
% 2.09/1.13 = ( nth_a @ Ys @ ( F @ I3 ) ) ) )
% 2.09/1.13 => ( ( mset_a @ Xs )
% 2.09/1.13 = ( mset_a @ Ys ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_48_mset__bij,axiom,
% 2.09/1.13 ! [F: nat > nat,Xs: list_set_a,Ys: list_set_a] :
% 2.09/1.13 ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ Ys ) ) )
% 2.09/1.13 => ( ! [I3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I3 @ ( size_size_list_set_a @ Xs ) )
% 2.09/1.13 => ( ( nth_set_a @ Xs @ I3 )
% 2.09/1.13 = ( nth_set_a @ Ys @ ( F @ I3 ) ) ) )
% 2.09/1.13 => ( ( mset_set_a @ Xs )
% 2.09/1.13 = ( mset_set_a @ Ys ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_49_listset__permE,axiom,
% 2.09/1.13 ! [Ys: list_set_list_a,Xs: list_set_set_list_a,F: nat > nat,Xs3: list_set_set_list_a] :
% 2.09/1.13 ( ( member101195155list_a @ Ys @ ( listset_set_list_a @ Xs ) )
% 2.09/1.13 => ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_s469631926list_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_s469631926list_a @ Xs3 ) ) )
% 2.09/1.13 => ( ! [I3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I3 @ ( size_s469631926list_a @ Xs ) )
% 2.09/1.13 => ( ( nth_set_set_list_a @ Xs3 @ I3 )
% 2.09/1.13 = ( nth_set_set_list_a @ Xs @ ( F @ I3 ) ) ) )
% 2.09/1.13 => ~ ! [Ys2: list_set_list_a] :
% 2.09/1.13 ( ( member101195155list_a @ Ys2 @ ( listset_set_list_a @ Xs3 ) )
% 2.09/1.13 => ( ( ( size_s1635937238list_a @ Ys2 )
% 2.09/1.13 = ( size_s1635937238list_a @ Ys ) )
% 2.09/1.13 => ~ ! [I2: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I2 @ ( size_s1635937238list_a @ Ys ) )
% 2.09/1.13 => ( ( nth_set_list_a @ Ys2 @ I2 )
% 2.09/1.13 = ( nth_set_list_a @ Ys @ ( F @ I2 ) ) ) ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_50_listset__permE,axiom,
% 2.09/1.13 ! [Ys: list_set_set_a,Xs: list_set_set_set_a,F: nat > nat,Xs3: list_set_set_set_a] :
% 2.09/1.13 ( ( member814963949_set_a @ Ys @ ( listset_set_set_a @ Xs ) )
% 2.09/1.13 => ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_s726512144_set_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_s726512144_set_a @ Xs3 ) ) )
% 2.09/1.13 => ( ! [I3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I3 @ ( size_s726512144_set_a @ Xs ) )
% 2.09/1.13 => ( ( nth_set_set_set_a @ Xs3 @ I3 )
% 2.09/1.13 = ( nth_set_set_set_a @ Xs @ ( F @ I3 ) ) ) )
% 2.09/1.13 => ~ ! [Ys2: list_set_set_a] :
% 2.09/1.13 ( ( member814963949_set_a @ Ys2 @ ( listset_set_set_a @ Xs3 ) )
% 2.09/1.13 => ( ( ( size_s739728560_set_a @ Ys2 )
% 2.09/1.13 = ( size_s739728560_set_a @ Ys ) )
% 2.09/1.13 => ~ ! [I2: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I2 @ ( size_s739728560_set_a @ Ys ) )
% 2.09/1.13 => ( ( nth_set_set_a @ Ys2 @ I2 )
% 2.09/1.13 = ( nth_set_set_a @ Ys @ ( F @ I2 ) ) ) ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_51_listset__permE,axiom,
% 2.09/1.13 ! [Ys: list_nat,Xs: list_set_nat,F: nat > nat,Xs3: list_set_nat] :
% 2.09/1.13 ( ( member_list_nat @ Ys @ ( listset_nat @ Xs ) )
% 2.09/1.13 => ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_s577819178et_nat @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_s577819178et_nat @ Xs3 ) ) )
% 2.09/1.13 => ( ! [I3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I3 @ ( size_s577819178et_nat @ Xs ) )
% 2.09/1.13 => ( ( nth_set_nat @ Xs3 @ I3 )
% 2.09/1.13 = ( nth_set_nat @ Xs @ ( F @ I3 ) ) ) )
% 2.09/1.13 => ~ ! [Ys2: list_nat] :
% 2.09/1.13 ( ( member_list_nat @ Ys2 @ ( listset_nat @ Xs3 ) )
% 2.09/1.13 => ( ( ( size_size_list_nat @ Ys2 )
% 2.09/1.13 = ( size_size_list_nat @ Ys ) )
% 2.09/1.13 => ~ ! [I2: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
% 2.09/1.13 => ( ( nth_nat @ Ys2 @ I2 )
% 2.09/1.13 = ( nth_nat @ Ys @ ( F @ I2 ) ) ) ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_52_listset__permE,axiom,
% 2.09/1.13 ! [Ys: list_list_a,Xs: list_set_list_a,F: nat > nat,Xs3: list_set_list_a] :
% 2.09/1.13 ( ( member_list_list_a @ Ys @ ( listset_list_a @ Xs ) )
% 2.09/1.13 => ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_s1635937238list_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_s1635937238list_a @ Xs3 ) ) )
% 2.09/1.13 => ( ! [I3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I3 @ ( size_s1635937238list_a @ Xs ) )
% 2.09/1.13 => ( ( nth_set_list_a @ Xs3 @ I3 )
% 2.09/1.13 = ( nth_set_list_a @ Xs @ ( F @ I3 ) ) ) )
% 2.09/1.13 => ~ ! [Ys2: list_list_a] :
% 2.09/1.13 ( ( member_list_list_a @ Ys2 @ ( listset_list_a @ Xs3 ) )
% 2.09/1.13 => ( ( ( size_s1427607542list_a @ Ys2 )
% 2.09/1.13 = ( size_s1427607542list_a @ Ys ) )
% 2.09/1.13 => ~ ! [I2: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I2 @ ( size_s1427607542list_a @ Ys ) )
% 2.09/1.13 => ( ( nth_list_a @ Ys2 @ I2 )
% 2.09/1.13 = ( nth_list_a @ Ys @ ( F @ I2 ) ) ) ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_53_listset__permE,axiom,
% 2.09/1.13 ! [Ys: list_set_a,Xs: list_set_set_a,F: nat > nat,Xs3: list_set_set_a] :
% 2.09/1.13 ( ( member_list_set_a @ Ys @ ( listset_set_a @ Xs ) )
% 2.09/1.13 => ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_s739728560_set_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_s739728560_set_a @ Xs3 ) ) )
% 2.09/1.13 => ( ! [I3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I3 @ ( size_s739728560_set_a @ Xs ) )
% 2.09/1.13 => ( ( nth_set_set_a @ Xs3 @ I3 )
% 2.09/1.13 = ( nth_set_set_a @ Xs @ ( F @ I3 ) ) ) )
% 2.09/1.13 => ~ ! [Ys2: list_set_a] :
% 2.09/1.13 ( ( member_list_set_a @ Ys2 @ ( listset_set_a @ Xs3 ) )
% 2.09/1.13 => ( ( ( size_size_list_set_a @ Ys2 )
% 2.09/1.13 = ( size_size_list_set_a @ Ys ) )
% 2.09/1.13 => ~ ! [I2: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I2 @ ( size_size_list_set_a @ Ys ) )
% 2.09/1.13 => ( ( nth_set_a @ Ys2 @ I2 )
% 2.09/1.13 = ( nth_set_a @ Ys @ ( F @ I2 ) ) ) ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_54_listset__permE,axiom,
% 2.09/1.13 ! [Ys: list_a,Xs: list_set_a,F: nat > nat,Xs3: list_set_a] :
% 2.09/1.13 ( ( member_list_a @ Ys @ ( listset_a @ Xs ) )
% 2.09/1.13 => ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ Xs3 ) ) )
% 2.09/1.13 => ( ! [I3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I3 @ ( size_size_list_set_a @ Xs ) )
% 2.09/1.13 => ( ( nth_set_a @ Xs3 @ I3 )
% 2.09/1.13 = ( nth_set_a @ Xs @ ( F @ I3 ) ) ) )
% 2.09/1.13 => ~ ! [Ys2: list_a] :
% 2.09/1.13 ( ( member_list_a @ Ys2 @ ( listset_a @ Xs3 ) )
% 2.09/1.13 => ( ( ( size_size_list_a @ Ys2 )
% 2.09/1.13 = ( size_size_list_a @ Ys ) )
% 2.09/1.13 => ~ ! [I2: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ys ) )
% 2.09/1.13 => ( ( nth_a @ Ys2 @ I2 )
% 2.09/1.13 = ( nth_a @ Ys @ ( F @ I2 ) ) ) ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_55_g__bij,axiom,
% 2.09/1.13 bij_betw_nat_nat @ g @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ ss1 ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ ss2 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_56_length__rev,axiom,
% 2.09/1.13 ! [Xs: list_list_a] :
% 2.09/1.13 ( ( size_s1427607542list_a @ ( rev_list_a @ Xs ) )
% 2.09/1.13 = ( size_s1427607542list_a @ Xs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_57_length__rev,axiom,
% 2.09/1.13 ! [Xs: list_set_list_a] :
% 2.09/1.13 ( ( size_s1635937238list_a @ ( rev_set_list_a @ Xs ) )
% 2.09/1.13 = ( size_s1635937238list_a @ Xs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_58_length__rev,axiom,
% 2.09/1.13 ! [Xs: list_set_set_a] :
% 2.09/1.13 ( ( size_s739728560_set_a @ ( rev_set_set_a @ Xs ) )
% 2.09/1.13 = ( size_s739728560_set_a @ Xs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_59_length__rev,axiom,
% 2.09/1.13 ! [Xs: list_nat] :
% 2.09/1.13 ( ( size_size_list_nat @ ( rev_nat @ Xs ) )
% 2.09/1.13 = ( size_size_list_nat @ Xs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_60_length__rev,axiom,
% 2.09/1.13 ! [Xs: list_a] :
% 2.09/1.13 ( ( size_size_list_a @ ( rev_a @ Xs ) )
% 2.09/1.13 = ( size_size_list_a @ Xs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_61_length__rev,axiom,
% 2.09/1.13 ! [Xs: list_set_a] :
% 2.09/1.13 ( ( size_size_list_set_a @ ( rev_set_a @ Xs ) )
% 2.09/1.13 = ( size_size_list_set_a @ Xs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_62_lessThan__iff,axiom,
% 2.09/1.13 ! [I: set_nat,K: set_nat] :
% 2.09/1.13 ( ( member_set_nat @ I @ ( set_or1597314339et_nat @ K ) )
% 2.09/1.13 = ( ord_less_set_nat @ I @ K ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_63_lessThan__iff,axiom,
% 2.09/1.13 ! [I: set_list_a,K: set_list_a] :
% 2.09/1.13 ( ( member_set_list_a @ I @ ( set_or1578766727list_a @ K ) )
% 2.09/1.13 = ( ord_less_set_list_a @ I @ K ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_64_lessThan__iff,axiom,
% 2.09/1.13 ! [I: set_a,K: set_a] :
% 2.09/1.13 ( ( member_set_a @ I @ ( set_or1164043265_set_a @ K ) )
% 2.09/1.13 = ( ord_less_set_a @ I @ K ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_65_lessThan__iff,axiom,
% 2.09/1.13 ! [I: nat,K: nat] :
% 2.09/1.13 ( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
% 2.09/1.13 = ( ord_less_nat @ I @ K ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_66_listsetI,axiom,
% 2.09/1.13 ! [Ys: list_multiset_a,Xs: list_set_multiset_a] :
% 2.09/1.13 ( ( ( size_s1657263798iset_a @ Ys )
% 2.09/1.13 = ( size_s802269334iset_a @ Xs ) )
% 2.09/1.13 => ( ! [I3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I3 @ ( size_s802269334iset_a @ Xs ) )
% 2.09/1.13 => ( member_multiset_a @ ( nth_multiset_a @ Ys @ I3 ) @ ( nth_set_multiset_a @ Xs @ I3 ) ) )
% 2.09/1.13 => ( member518144627iset_a @ Ys @ ( listset_multiset_a @ Xs ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_67_listsetI,axiom,
% 2.09/1.13 ! [Ys: list_list_list_a,Xs: list_set_list_list_a] :
% 2.09/1.13 ( ( ( size_s575106428list_a @ Ys )
% 2.09/1.13 = ( size_s240404444list_a @ Xs ) )
% 2.09/1.13 => ( ! [I3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I3 @ ( size_s240404444list_a @ Xs ) )
% 2.09/1.13 => ( member_list_list_a @ ( nth_list_list_a @ Ys @ I3 ) @ ( nth_set_list_list_a @ Xs @ I3 ) ) )
% 2.09/1.13 => ( member1511395513list_a @ Ys @ ( listset_list_list_a @ Xs ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_68_listsetI,axiom,
% 2.09/1.13 ! [Ys: list_list_set_a,Xs: list_set_list_set_a] :
% 2.09/1.13 ( ( ( size_s341332310_set_a @ Ys )
% 2.09/1.13 = ( size_s926492982_set_a @ Xs ) )
% 2.09/1.13 => ( ! [I3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I3 @ ( size_s926492982_set_a @ Xs ) )
% 2.09/1.13 => ( member_list_set_a @ ( nth_list_set_a @ Ys @ I3 ) @ ( nth_set_list_set_a @ Xs @ I3 ) ) )
% 2.09/1.13 => ( member954073875_set_a @ Ys @ ( listset_list_set_a @ Xs ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_69_listsetI,axiom,
% 2.09/1.13 ! [Ys: list_set_list_a,Xs: list_set_set_list_a] :
% 2.09/1.13 ( ( ( size_s1635937238list_a @ Ys )
% 2.09/1.13 = ( size_s469631926list_a @ Xs ) )
% 2.09/1.13 => ( ! [I3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I3 @ ( size_s469631926list_a @ Xs ) )
% 2.09/1.13 => ( member_set_list_a @ ( nth_set_list_a @ Ys @ I3 ) @ ( nth_set_set_list_a @ Xs @ I3 ) ) )
% 2.09/1.13 => ( member101195155list_a @ Ys @ ( listset_set_list_a @ Xs ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_70_listsetI,axiom,
% 2.09/1.13 ! [Ys: list_set_set_a,Xs: list_set_set_set_a] :
% 2.09/1.13 ( ( ( size_s739728560_set_a @ Ys )
% 2.09/1.13 = ( size_s726512144_set_a @ Xs ) )
% 2.09/1.13 => ( ! [I3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I3 @ ( size_s726512144_set_a @ Xs ) )
% 2.09/1.13 => ( member_set_set_a @ ( nth_set_set_a @ Ys @ I3 ) @ ( nth_set_set_set_a @ Xs @ I3 ) ) )
% 2.09/1.13 => ( member814963949_set_a @ Ys @ ( listset_set_set_a @ Xs ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_71_listsetI,axiom,
% 2.09/1.13 ! [Ys: list_nat,Xs: list_set_nat] :
% 2.09/1.13 ( ( ( size_size_list_nat @ Ys )
% 2.09/1.13 = ( size_s577819178et_nat @ Xs ) )
% 2.09/1.13 => ( ! [I3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I3 @ ( size_s577819178et_nat @ Xs ) )
% 2.09/1.13 => ( member_nat @ ( nth_nat @ Ys @ I3 ) @ ( nth_set_nat @ Xs @ I3 ) ) )
% 2.09/1.13 => ( member_list_nat @ Ys @ ( listset_nat @ Xs ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_72_listsetI,axiom,
% 2.09/1.13 ! [Ys: list_list_a,Xs: list_set_list_a] :
% 2.09/1.13 ( ( ( size_s1427607542list_a @ Ys )
% 2.09/1.13 = ( size_s1635937238list_a @ Xs ) )
% 2.09/1.13 => ( ! [I3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I3 @ ( size_s1635937238list_a @ Xs ) )
% 2.09/1.13 => ( member_list_a @ ( nth_list_a @ Ys @ I3 ) @ ( nth_set_list_a @ Xs @ I3 ) ) )
% 2.09/1.13 => ( member_list_list_a @ Ys @ ( listset_list_a @ Xs ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_73_listsetI,axiom,
% 2.09/1.13 ! [Ys: list_a,Xs: list_set_a] :
% 2.09/1.13 ( ( ( size_size_list_a @ Ys )
% 2.09/1.13 = ( size_size_list_set_a @ Xs ) )
% 2.09/1.13 => ( ! [I3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I3 @ ( size_size_list_set_a @ Xs ) )
% 2.09/1.13 => ( member_a @ ( nth_a @ Ys @ I3 ) @ ( nth_set_a @ Xs @ I3 ) ) )
% 2.09/1.13 => ( member_list_a @ Ys @ ( listset_a @ Xs ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_74_listsetI,axiom,
% 2.09/1.13 ! [Ys: list_set_a,Xs: list_set_set_a] :
% 2.09/1.13 ( ( ( size_size_list_set_a @ Ys )
% 2.09/1.13 = ( size_s739728560_set_a @ Xs ) )
% 2.09/1.13 => ( ! [I3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I3 @ ( size_s739728560_set_a @ Xs ) )
% 2.09/1.13 => ( member_set_a @ ( nth_set_a @ Ys @ I3 ) @ ( nth_set_set_a @ Xs @ I3 ) ) )
% 2.09/1.13 => ( member_list_set_a @ Ys @ ( listset_set_a @ Xs ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_75_len__qs,axiom,
% 2.09/1.13 ( ( size_size_list_a @ qs )
% 2.09/1.13 = ( size_size_list_set_a @ ss1 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_76_f__bij,axiom,
% 2.09/1.13 bij_betw_nat_nat @ f @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ ss2 ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ ss1 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_77_nth__equalityI,axiom,
% 2.09/1.13 ! [Xs: list_list_a,Ys: list_list_a] :
% 2.09/1.13 ( ( ( size_s1427607542list_a @ Xs )
% 2.09/1.13 = ( size_s1427607542list_a @ Ys ) )
% 2.09/1.13 => ( ! [I3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I3 @ ( size_s1427607542list_a @ Xs ) )
% 2.09/1.13 => ( ( nth_list_a @ Xs @ I3 )
% 2.09/1.13 = ( nth_list_a @ Ys @ I3 ) ) )
% 2.09/1.13 => ( Xs = Ys ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_78_nth__equalityI,axiom,
% 2.09/1.13 ! [Xs: list_set_list_a,Ys: list_set_list_a] :
% 2.09/1.13 ( ( ( size_s1635937238list_a @ Xs )
% 2.09/1.13 = ( size_s1635937238list_a @ Ys ) )
% 2.09/1.13 => ( ! [I3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I3 @ ( size_s1635937238list_a @ Xs ) )
% 2.09/1.13 => ( ( nth_set_list_a @ Xs @ I3 )
% 2.09/1.13 = ( nth_set_list_a @ Ys @ I3 ) ) )
% 2.09/1.13 => ( Xs = Ys ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_79_nth__equalityI,axiom,
% 2.09/1.13 ! [Xs: list_set_set_a,Ys: list_set_set_a] :
% 2.09/1.13 ( ( ( size_s739728560_set_a @ Xs )
% 2.09/1.13 = ( size_s739728560_set_a @ Ys ) )
% 2.09/1.13 => ( ! [I3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I3 @ ( size_s739728560_set_a @ Xs ) )
% 2.09/1.13 => ( ( nth_set_set_a @ Xs @ I3 )
% 2.09/1.13 = ( nth_set_set_a @ Ys @ I3 ) ) )
% 2.09/1.13 => ( Xs = Ys ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_80_nth__equalityI,axiom,
% 2.09/1.13 ! [Xs: list_nat,Ys: list_nat] :
% 2.09/1.13 ( ( ( size_size_list_nat @ Xs )
% 2.09/1.13 = ( size_size_list_nat @ Ys ) )
% 2.09/1.13 => ( ! [I3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
% 2.09/1.13 => ( ( nth_nat @ Xs @ I3 )
% 2.09/1.13 = ( nth_nat @ Ys @ I3 ) ) )
% 2.09/1.13 => ( Xs = Ys ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_81_nth__equalityI,axiom,
% 2.09/1.13 ! [Xs: list_a,Ys: list_a] :
% 2.09/1.13 ( ( ( size_size_list_a @ Xs )
% 2.09/1.13 = ( size_size_list_a @ Ys ) )
% 2.09/1.13 => ( ! [I3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
% 2.09/1.13 => ( ( nth_a @ Xs @ I3 )
% 2.09/1.13 = ( nth_a @ Ys @ I3 ) ) )
% 2.09/1.13 => ( Xs = Ys ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_82_nth__equalityI,axiom,
% 2.09/1.13 ! [Xs: list_set_a,Ys: list_set_a] :
% 2.09/1.13 ( ( ( size_size_list_set_a @ Xs )
% 2.09/1.13 = ( size_size_list_set_a @ Ys ) )
% 2.09/1.13 => ( ! [I3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I3 @ ( size_size_list_set_a @ Xs ) )
% 2.09/1.13 => ( ( nth_set_a @ Xs @ I3 )
% 2.09/1.13 = ( nth_set_a @ Ys @ I3 ) ) )
% 2.09/1.13 => ( Xs = Ys ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_83_Skolem__list__nth,axiom,
% 2.09/1.13 ! [K: nat,P: nat > list_a > $o] :
% 2.09/1.13 ( ( ! [I4: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I4 @ K )
% 2.09/1.13 => ? [X3: list_a] : ( P @ I4 @ X3 ) ) )
% 2.09/1.13 = ( ? [Xs4: list_list_a] :
% 2.09/1.13 ( ( ( size_s1427607542list_a @ Xs4 )
% 2.09/1.13 = K )
% 2.09/1.13 & ! [I4: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I4 @ K )
% 2.09/1.13 => ( P @ I4 @ ( nth_list_a @ Xs4 @ I4 ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_84_Skolem__list__nth,axiom,
% 2.09/1.13 ! [K: nat,P: nat > set_list_a > $o] :
% 2.09/1.13 ( ( ! [I4: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I4 @ K )
% 2.09/1.13 => ? [X3: set_list_a] : ( P @ I4 @ X3 ) ) )
% 2.09/1.13 = ( ? [Xs4: list_set_list_a] :
% 2.09/1.13 ( ( ( size_s1635937238list_a @ Xs4 )
% 2.09/1.13 = K )
% 2.09/1.13 & ! [I4: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I4 @ K )
% 2.09/1.13 => ( P @ I4 @ ( nth_set_list_a @ Xs4 @ I4 ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_85_Skolem__list__nth,axiom,
% 2.09/1.13 ! [K: nat,P: nat > set_set_a > $o] :
% 2.09/1.13 ( ( ! [I4: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I4 @ K )
% 2.09/1.13 => ? [X3: set_set_a] : ( P @ I4 @ X3 ) ) )
% 2.09/1.13 = ( ? [Xs4: list_set_set_a] :
% 2.09/1.13 ( ( ( size_s739728560_set_a @ Xs4 )
% 2.09/1.13 = K )
% 2.09/1.13 & ! [I4: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I4 @ K )
% 2.09/1.13 => ( P @ I4 @ ( nth_set_set_a @ Xs4 @ I4 ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_86_Skolem__list__nth,axiom,
% 2.09/1.13 ! [K: nat,P: nat > nat > $o] :
% 2.09/1.13 ( ( ! [I4: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I4 @ K )
% 2.09/1.13 => ? [X3: nat] : ( P @ I4 @ X3 ) ) )
% 2.09/1.13 = ( ? [Xs4: list_nat] :
% 2.09/1.13 ( ( ( size_size_list_nat @ Xs4 )
% 2.09/1.13 = K )
% 2.09/1.13 & ! [I4: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I4 @ K )
% 2.09/1.13 => ( P @ I4 @ ( nth_nat @ Xs4 @ I4 ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_87_Skolem__list__nth,axiom,
% 2.09/1.13 ! [K: nat,P: nat > a > $o] :
% 2.09/1.13 ( ( ! [I4: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I4 @ K )
% 2.09/1.13 => ? [X3: a] : ( P @ I4 @ X3 ) ) )
% 2.09/1.13 = ( ? [Xs4: list_a] :
% 2.09/1.13 ( ( ( size_size_list_a @ Xs4 )
% 2.09/1.13 = K )
% 2.09/1.13 & ! [I4: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I4 @ K )
% 2.09/1.13 => ( P @ I4 @ ( nth_a @ Xs4 @ I4 ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_88_Skolem__list__nth,axiom,
% 2.09/1.13 ! [K: nat,P: nat > set_a > $o] :
% 2.09/1.13 ( ( ! [I4: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I4 @ K )
% 2.09/1.13 => ? [X3: set_a] : ( P @ I4 @ X3 ) ) )
% 2.09/1.13 = ( ? [Xs4: list_set_a] :
% 2.09/1.13 ( ( ( size_size_list_set_a @ Xs4 )
% 2.09/1.13 = K )
% 2.09/1.13 & ! [I4: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I4 @ K )
% 2.09/1.13 => ( P @ I4 @ ( nth_set_a @ Xs4 @ I4 ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_89_list__eq__iff__nth__eq,axiom,
% 2.09/1.13 ( ( ^ [Y2: list_list_a,Z: list_list_a] : ( Y2 = Z ) )
% 2.09/1.13 = ( ^ [Xs4: list_list_a,Ys3: list_list_a] :
% 2.09/1.13 ( ( ( size_s1427607542list_a @ Xs4 )
% 2.09/1.13 = ( size_s1427607542list_a @ Ys3 ) )
% 2.09/1.13 & ! [I4: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I4 @ ( size_s1427607542list_a @ Xs4 ) )
% 2.09/1.13 => ( ( nth_list_a @ Xs4 @ I4 )
% 2.09/1.13 = ( nth_list_a @ Ys3 @ I4 ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_90_list__eq__iff__nth__eq,axiom,
% 2.09/1.13 ( ( ^ [Y2: list_set_list_a,Z: list_set_list_a] : ( Y2 = Z ) )
% 2.09/1.13 = ( ^ [Xs4: list_set_list_a,Ys3: list_set_list_a] :
% 2.09/1.13 ( ( ( size_s1635937238list_a @ Xs4 )
% 2.09/1.13 = ( size_s1635937238list_a @ Ys3 ) )
% 2.09/1.13 & ! [I4: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I4 @ ( size_s1635937238list_a @ Xs4 ) )
% 2.09/1.13 => ( ( nth_set_list_a @ Xs4 @ I4 )
% 2.09/1.13 = ( nth_set_list_a @ Ys3 @ I4 ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_91_list__eq__iff__nth__eq,axiom,
% 2.09/1.13 ( ( ^ [Y2: list_set_set_a,Z: list_set_set_a] : ( Y2 = Z ) )
% 2.09/1.13 = ( ^ [Xs4: list_set_set_a,Ys3: list_set_set_a] :
% 2.09/1.13 ( ( ( size_s739728560_set_a @ Xs4 )
% 2.09/1.13 = ( size_s739728560_set_a @ Ys3 ) )
% 2.09/1.13 & ! [I4: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I4 @ ( size_s739728560_set_a @ Xs4 ) )
% 2.09/1.13 => ( ( nth_set_set_a @ Xs4 @ I4 )
% 2.09/1.13 = ( nth_set_set_a @ Ys3 @ I4 ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_92_list__eq__iff__nth__eq,axiom,
% 2.09/1.13 ( ( ^ [Y2: list_nat,Z: list_nat] : ( Y2 = Z ) )
% 2.09/1.13 = ( ^ [Xs4: list_nat,Ys3: list_nat] :
% 2.09/1.13 ( ( ( size_size_list_nat @ Xs4 )
% 2.09/1.13 = ( size_size_list_nat @ Ys3 ) )
% 2.09/1.13 & ! [I4: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs4 ) )
% 2.09/1.13 => ( ( nth_nat @ Xs4 @ I4 )
% 2.09/1.13 = ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_93_list__eq__iff__nth__eq,axiom,
% 2.09/1.13 ( ( ^ [Y2: list_a,Z: list_a] : ( Y2 = Z ) )
% 2.09/1.13 = ( ^ [Xs4: list_a,Ys3: list_a] :
% 2.09/1.13 ( ( ( size_size_list_a @ Xs4 )
% 2.09/1.13 = ( size_size_list_a @ Ys3 ) )
% 2.09/1.13 & ! [I4: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs4 ) )
% 2.09/1.13 => ( ( nth_a @ Xs4 @ I4 )
% 2.09/1.13 = ( nth_a @ Ys3 @ I4 ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_94_list__eq__iff__nth__eq,axiom,
% 2.09/1.13 ( ( ^ [Y2: list_set_a,Z: list_set_a] : ( Y2 = Z ) )
% 2.09/1.13 = ( ^ [Xs4: list_set_a,Ys3: list_set_a] :
% 2.09/1.13 ( ( ( size_size_list_set_a @ Xs4 )
% 2.09/1.13 = ( size_size_list_set_a @ Ys3 ) )
% 2.09/1.13 & ! [I4: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I4 @ ( size_size_list_set_a @ Xs4 ) )
% 2.09/1.13 => ( ( nth_set_a @ Xs4 @ I4 )
% 2.09/1.13 = ( nth_set_a @ Ys3 @ I4 ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_95_lessThan__eq__iff,axiom,
% 2.09/1.13 ! [X4: nat,Y3: nat] :
% 2.09/1.13 ( ( ( set_ord_lessThan_nat @ X4 )
% 2.09/1.13 = ( set_ord_lessThan_nat @ Y3 ) )
% 2.09/1.13 = ( X4 = Y3 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_96_rev__is__rev__conv,axiom,
% 2.09/1.13 ! [Xs: list_a,Ys: list_a] :
% 2.09/1.13 ( ( ( rev_a @ Xs )
% 2.09/1.13 = ( rev_a @ Ys ) )
% 2.09/1.13 = ( Xs = Ys ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_97_rev__is__rev__conv,axiom,
% 2.09/1.13 ! [Xs: list_set_a,Ys: list_set_a] :
% 2.09/1.13 ( ( ( rev_set_a @ Xs )
% 2.09/1.13 = ( rev_set_a @ Ys ) )
% 2.09/1.13 = ( Xs = Ys ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_98_rev__rev__ident,axiom,
% 2.09/1.13 ! [Xs: list_a] :
% 2.09/1.13 ( ( rev_a @ ( rev_a @ Xs ) )
% 2.09/1.13 = Xs ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_99_rev__rev__ident,axiom,
% 2.09/1.13 ! [Xs: list_set_a] :
% 2.09/1.13 ( ( rev_set_a @ ( rev_set_a @ Xs ) )
% 2.09/1.13 = Xs ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_100_len__ss1,axiom,
% 2.09/1.13 ( ( size_size_list_set_a @ ss1 )
% 2.09/1.13 = ( size_size_list_set_a @ ss2 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_101_g,axiom,
% 2.09/1.13 ! [I: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I @ ( size_size_list_set_a @ ss1 ) )
% 2.09/1.13 => ( ( nth_set_a @ ss2 @ I )
% 2.09/1.13 = ( nth_set_a @ ss1 @ ( g @ I ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_102_f,axiom,
% 2.09/1.13 ! [I: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I @ ( size_size_list_set_a @ ss2 ) )
% 2.09/1.13 => ( ( nth_set_a @ ss1 @ I )
% 2.09/1.13 = ( nth_set_a @ ss2 @ ( f @ I ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_103__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062f_O_A_092_060lbrakk_062bij__betw_Af_A_123_O_O_060length_Ass2_125_A_123_O_O_060length_Ass1_125_059_A_092_060And_062i_O_Ai_A_060_Alength_Ass2_A_092_060Longrightarrow_062_Ass1_A_B_Ai_A_061_Ass2_A_B_Af_Ai_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 2.09/1.13 ~ ! [F2: nat > nat] :
% 2.09/1.13 ( ( bij_betw_nat_nat @ F2 @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ ss2 ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ ss1 ) ) )
% 2.09/1.13 => ~ ! [I2: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I2 @ ( size_size_list_set_a @ ss2 ) )
% 2.09/1.13 => ( ( nth_set_a @ ss1 @ I2 )
% 2.09/1.13 = ( nth_set_a @ ss2 @ ( F2 @ I2 ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_104__092_060open_062_092_060exists_062f_O_Abij__betw_Af_A_123_O_O_060length_Ass1_125_A_123_O_O_060length_Ass2_125_A_092_060and_062_A_I_092_060forall_062i_060length_Ass1_O_Ass1_A_B_Ai_A_061_Ass2_A_B_Af_Ai_J_092_060close_062,axiom,
% 2.09/1.13 ? [F2: nat > nat] :
% 2.09/1.13 ( ( bij_betw_nat_nat @ F2 @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ ss1 ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ ss2 ) ) )
% 2.09/1.13 & ! [I2: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I2 @ ( size_size_list_set_a @ ss1 ) )
% 2.09/1.13 => ( ( nth_set_a @ ss1 @ I2 )
% 2.09/1.13 = ( nth_set_a @ ss2 @ ( F2 @ I2 ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_105_f__g,axiom,
% 2.09/1.13 ! [I: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I @ ( size_size_list_set_a @ ss1 ) )
% 2.09/1.13 => ( ( f @ ( g @ I ) )
% 2.09/1.13 = I ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_106_g__f,axiom,
% 2.09/1.13 ! [I: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I @ ( size_size_list_set_a @ ss2 ) )
% 2.09/1.13 => ( ( g @ ( f @ I ) )
% 2.09/1.13 = I ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_107_listsetD_I2_J,axiom,
% 2.09/1.13 ! [Ys: list_set_set_a,Xs: list_set_set_set_a,I: nat] :
% 2.09/1.13 ( ( member814963949_set_a @ Ys @ ( listset_set_set_a @ Xs ) )
% 2.09/1.13 => ( ( ord_less_nat @ I @ ( size_s726512144_set_a @ Xs ) )
% 2.09/1.13 => ( member_set_set_a @ ( nth_set_set_a @ Ys @ I ) @ ( nth_set_set_set_a @ Xs @ I ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_108_listsetD_I2_J,axiom,
% 2.09/1.13 ! [Ys: list_set_list_a,Xs: list_set_set_list_a,I: nat] :
% 2.09/1.13 ( ( member101195155list_a @ Ys @ ( listset_set_list_a @ Xs ) )
% 2.09/1.13 => ( ( ord_less_nat @ I @ ( size_s469631926list_a @ Xs ) )
% 2.09/1.13 => ( member_set_list_a @ ( nth_set_list_a @ Ys @ I ) @ ( nth_set_set_list_a @ Xs @ I ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_109_listsetD_I2_J,axiom,
% 2.09/1.13 ! [Ys: list_multiset_a,Xs: list_set_multiset_a,I: nat] :
% 2.09/1.13 ( ( member518144627iset_a @ Ys @ ( listset_multiset_a @ Xs ) )
% 2.09/1.13 => ( ( ord_less_nat @ I @ ( size_s802269334iset_a @ Xs ) )
% 2.09/1.13 => ( member_multiset_a @ ( nth_multiset_a @ Ys @ I ) @ ( nth_set_multiset_a @ Xs @ I ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_110_listsetD_I2_J,axiom,
% 2.09/1.13 ! [Ys: list_list_list_a,Xs: list_set_list_list_a,I: nat] :
% 2.09/1.13 ( ( member1511395513list_a @ Ys @ ( listset_list_list_a @ Xs ) )
% 2.09/1.13 => ( ( ord_less_nat @ I @ ( size_s240404444list_a @ Xs ) )
% 2.09/1.13 => ( member_list_list_a @ ( nth_list_list_a @ Ys @ I ) @ ( nth_set_list_list_a @ Xs @ I ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_111_listsetD_I2_J,axiom,
% 2.09/1.13 ! [Ys: list_list_set_a,Xs: list_set_list_set_a,I: nat] :
% 2.09/1.13 ( ( member954073875_set_a @ Ys @ ( listset_list_set_a @ Xs ) )
% 2.09/1.13 => ( ( ord_less_nat @ I @ ( size_s926492982_set_a @ Xs ) )
% 2.09/1.13 => ( member_list_set_a @ ( nth_list_set_a @ Ys @ I ) @ ( nth_set_list_set_a @ Xs @ I ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_112_listsetD_I2_J,axiom,
% 2.09/1.13 ! [Ys: list_nat,Xs: list_set_nat,I: nat] :
% 2.09/1.13 ( ( member_list_nat @ Ys @ ( listset_nat @ Xs ) )
% 2.09/1.13 => ( ( ord_less_nat @ I @ ( size_s577819178et_nat @ Xs ) )
% 2.09/1.13 => ( member_nat @ ( nth_nat @ Ys @ I ) @ ( nth_set_nat @ Xs @ I ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_113_listsetD_I2_J,axiom,
% 2.09/1.13 ! [Ys: list_set_a,Xs: list_set_set_a,I: nat] :
% 2.09/1.13 ( ( member_list_set_a @ Ys @ ( listset_set_a @ Xs ) )
% 2.09/1.13 => ( ( ord_less_nat @ I @ ( size_s739728560_set_a @ Xs ) )
% 2.09/1.13 => ( member_set_a @ ( nth_set_a @ Ys @ I ) @ ( nth_set_set_a @ Xs @ I ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_114_listsetD_I2_J,axiom,
% 2.09/1.13 ! [Ys: list_list_a,Xs: list_set_list_a,I: nat] :
% 2.09/1.13 ( ( member_list_list_a @ Ys @ ( listset_list_a @ Xs ) )
% 2.09/1.13 => ( ( ord_less_nat @ I @ ( size_s1635937238list_a @ Xs ) )
% 2.09/1.13 => ( member_list_a @ ( nth_list_a @ Ys @ I ) @ ( nth_set_list_a @ Xs @ I ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_115_listsetD_I2_J,axiom,
% 2.09/1.13 ! [Ys: list_a,Xs: list_set_a,I: nat] :
% 2.09/1.13 ( ( member_list_a @ Ys @ ( listset_a @ Xs ) )
% 2.09/1.13 => ( ( ord_less_nat @ I @ ( size_size_list_set_a @ Xs ) )
% 2.09/1.13 => ( member_a @ ( nth_a @ Ys @ I ) @ ( nth_set_a @ Xs @ I ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_116_neq__if__length__neq,axiom,
% 2.09/1.13 ! [Xs: list_list_a,Ys: list_list_a] :
% 2.09/1.13 ( ( ( size_s1427607542list_a @ Xs )
% 2.09/1.13 != ( size_s1427607542list_a @ Ys ) )
% 2.09/1.13 => ( Xs != Ys ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_117_neq__if__length__neq,axiom,
% 2.09/1.13 ! [Xs: list_set_list_a,Ys: list_set_list_a] :
% 2.09/1.13 ( ( ( size_s1635937238list_a @ Xs )
% 2.09/1.13 != ( size_s1635937238list_a @ Ys ) )
% 2.09/1.13 => ( Xs != Ys ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_118_neq__if__length__neq,axiom,
% 2.09/1.13 ! [Xs: list_set_set_a,Ys: list_set_set_a] :
% 2.09/1.13 ( ( ( size_s739728560_set_a @ Xs )
% 2.09/1.13 != ( size_s739728560_set_a @ Ys ) )
% 2.09/1.13 => ( Xs != Ys ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_119_neq__if__length__neq,axiom,
% 2.09/1.13 ! [Xs: list_nat,Ys: list_nat] :
% 2.09/1.13 ( ( ( size_size_list_nat @ Xs )
% 2.09/1.13 != ( size_size_list_nat @ Ys ) )
% 2.09/1.13 => ( Xs != Ys ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_120_neq__if__length__neq,axiom,
% 2.09/1.13 ! [Xs: list_a,Ys: list_a] :
% 2.09/1.13 ( ( ( size_size_list_a @ Xs )
% 2.09/1.13 != ( size_size_list_a @ Ys ) )
% 2.09/1.13 => ( Xs != Ys ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_121_neq__if__length__neq,axiom,
% 2.09/1.13 ! [Xs: list_set_a,Ys: list_set_a] :
% 2.09/1.13 ( ( ( size_size_list_set_a @ Xs )
% 2.09/1.13 != ( size_size_list_set_a @ Ys ) )
% 2.09/1.13 => ( Xs != Ys ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_122_mem__Collect__eq,axiom,
% 2.09/1.13 ! [A2: multiset_a,P: multiset_a > $o] :
% 2.09/1.13 ( ( member_multiset_a @ A2 @ ( collect_multiset_a @ P ) )
% 2.09/1.13 = ( P @ A2 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_123_mem__Collect__eq,axiom,
% 2.09/1.13 ! [A2: list_list_a,P: list_list_a > $o] :
% 2.09/1.13 ( ( member_list_list_a @ A2 @ ( collect_list_list_a @ P ) )
% 2.09/1.13 = ( P @ A2 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_124_mem__Collect__eq,axiom,
% 2.09/1.13 ! [A2: list_set_a,P: list_set_a > $o] :
% 2.09/1.13 ( ( member_list_set_a @ A2 @ ( collect_list_set_a @ P ) )
% 2.09/1.13 = ( P @ A2 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_125_mem__Collect__eq,axiom,
% 2.09/1.13 ! [A2: set_a,P: set_a > $o] :
% 2.09/1.13 ( ( member_set_a @ A2 @ ( collect_set_a @ P ) )
% 2.09/1.13 = ( P @ A2 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_126_mem__Collect__eq,axiom,
% 2.09/1.13 ! [A2: nat,P: nat > $o] :
% 2.09/1.13 ( ( member_nat @ A2 @ ( collect_nat @ P ) )
% 2.09/1.13 = ( P @ A2 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_127_mem__Collect__eq,axiom,
% 2.09/1.13 ! [A2: a,P: a > $o] :
% 2.09/1.13 ( ( member_a @ A2 @ ( collect_a @ P ) )
% 2.09/1.13 = ( P @ A2 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_128_mem__Collect__eq,axiom,
% 2.09/1.13 ! [A2: list_a,P: list_a > $o] :
% 2.09/1.13 ( ( member_list_a @ A2 @ ( collect_list_a @ P ) )
% 2.09/1.13 = ( P @ A2 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_129_Collect__mem__eq,axiom,
% 2.09/1.13 ! [A: set_multiset_a] :
% 2.09/1.13 ( ( collect_multiset_a
% 2.09/1.13 @ ^ [X5: multiset_a] : ( member_multiset_a @ X5 @ A ) )
% 2.09/1.13 = A ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_130_Collect__mem__eq,axiom,
% 2.09/1.13 ! [A: set_list_list_a] :
% 2.09/1.13 ( ( collect_list_list_a
% 2.09/1.13 @ ^ [X5: list_list_a] : ( member_list_list_a @ X5 @ A ) )
% 2.09/1.13 = A ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_131_Collect__mem__eq,axiom,
% 2.09/1.13 ! [A: set_list_set_a] :
% 2.09/1.13 ( ( collect_list_set_a
% 2.09/1.13 @ ^ [X5: list_set_a] : ( member_list_set_a @ X5 @ A ) )
% 2.09/1.13 = A ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_132_Collect__mem__eq,axiom,
% 2.09/1.13 ! [A: set_set_a] :
% 2.09/1.13 ( ( collect_set_a
% 2.09/1.13 @ ^ [X5: set_a] : ( member_set_a @ X5 @ A ) )
% 2.09/1.13 = A ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_133_Collect__mem__eq,axiom,
% 2.09/1.13 ! [A: set_nat] :
% 2.09/1.13 ( ( collect_nat
% 2.09/1.13 @ ^ [X5: nat] : ( member_nat @ X5 @ A ) )
% 2.09/1.13 = A ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_134_Collect__mem__eq,axiom,
% 2.09/1.13 ! [A: set_a] :
% 2.09/1.13 ( ( collect_a
% 2.09/1.13 @ ^ [X5: a] : ( member_a @ X5 @ A ) )
% 2.09/1.13 = A ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_135_Collect__mem__eq,axiom,
% 2.09/1.13 ! [A: set_list_a] :
% 2.09/1.13 ( ( collect_list_a
% 2.09/1.13 @ ^ [X5: list_a] : ( member_list_a @ X5 @ A ) )
% 2.09/1.13 = A ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_136_Collect__cong,axiom,
% 2.09/1.13 ! [P: list_a > $o,Q: list_a > $o] :
% 2.09/1.13 ( ! [X6: list_a] :
% 2.09/1.13 ( ( P @ X6 )
% 2.09/1.13 = ( Q @ X6 ) )
% 2.09/1.13 => ( ( collect_list_a @ P )
% 2.09/1.13 = ( collect_list_a @ Q ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_137_Collect__cong,axiom,
% 2.09/1.13 ! [P: a > $o,Q: a > $o] :
% 2.09/1.13 ( ! [X6: a] :
% 2.09/1.13 ( ( P @ X6 )
% 2.09/1.13 = ( Q @ X6 ) )
% 2.09/1.13 => ( ( collect_a @ P )
% 2.09/1.13 = ( collect_a @ Q ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_138_Ex__list__of__length,axiom,
% 2.09/1.13 ! [N: nat] :
% 2.09/1.13 ? [Xs2: list_list_a] :
% 2.09/1.13 ( ( size_s1427607542list_a @ Xs2 )
% 2.09/1.13 = N ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_139_Ex__list__of__length,axiom,
% 2.09/1.13 ! [N: nat] :
% 2.09/1.13 ? [Xs2: list_set_list_a] :
% 2.09/1.13 ( ( size_s1635937238list_a @ Xs2 )
% 2.09/1.13 = N ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_140_Ex__list__of__length,axiom,
% 2.09/1.13 ! [N: nat] :
% 2.09/1.13 ? [Xs2: list_set_set_a] :
% 2.09/1.13 ( ( size_s739728560_set_a @ Xs2 )
% 2.09/1.13 = N ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_141_Ex__list__of__length,axiom,
% 2.09/1.13 ! [N: nat] :
% 2.09/1.13 ? [Xs2: list_nat] :
% 2.09/1.13 ( ( size_size_list_nat @ Xs2 )
% 2.09/1.13 = N ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_142_Ex__list__of__length,axiom,
% 2.09/1.13 ! [N: nat] :
% 2.09/1.13 ? [Xs2: list_a] :
% 2.09/1.13 ( ( size_size_list_a @ Xs2 )
% 2.09/1.13 = N ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_143_Ex__list__of__length,axiom,
% 2.09/1.13 ! [N: nat] :
% 2.09/1.13 ? [Xs2: list_set_a] :
% 2.09/1.13 ( ( size_size_list_set_a @ Xs2 )
% 2.09/1.13 = N ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_144_rev__swap,axiom,
% 2.09/1.13 ! [Xs: list_a,Ys: list_a] :
% 2.09/1.13 ( ( ( rev_a @ Xs )
% 2.09/1.13 = Ys )
% 2.09/1.13 = ( Xs
% 2.09/1.13 = ( rev_a @ Ys ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_145_rev__swap,axiom,
% 2.09/1.13 ! [Xs: list_set_a,Ys: list_set_a] :
% 2.09/1.13 ( ( ( rev_set_a @ Xs )
% 2.09/1.13 = Ys )
% 2.09/1.13 = ( Xs
% 2.09/1.13 = ( rev_set_a @ Ys ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_146_lessThan__strict__subset__iff,axiom,
% 2.09/1.13 ! [M: nat,N: nat] :
% 2.09/1.13 ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
% 2.09/1.13 = ( ord_less_nat @ M @ N ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_147_length__induct,axiom,
% 2.09/1.13 ! [P: list_set_list_a > $o,Xs: list_set_list_a] :
% 2.09/1.13 ( ! [Xs2: list_set_list_a] :
% 2.09/1.13 ( ! [Ys4: list_set_list_a] :
% 2.09/1.13 ( ( ord_less_nat @ ( size_s1635937238list_a @ Ys4 ) @ ( size_s1635937238list_a @ Xs2 ) )
% 2.09/1.13 => ( P @ Ys4 ) )
% 2.09/1.13 => ( P @ Xs2 ) )
% 2.09/1.13 => ( P @ Xs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_148_length__induct,axiom,
% 2.09/1.13 ! [P: list_set_set_a > $o,Xs: list_set_set_a] :
% 2.09/1.13 ( ! [Xs2: list_set_set_a] :
% 2.09/1.13 ( ! [Ys4: list_set_set_a] :
% 2.09/1.13 ( ( ord_less_nat @ ( size_s739728560_set_a @ Ys4 ) @ ( size_s739728560_set_a @ Xs2 ) )
% 2.09/1.13 => ( P @ Ys4 ) )
% 2.09/1.13 => ( P @ Xs2 ) )
% 2.09/1.13 => ( P @ Xs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_149_length__induct,axiom,
% 2.09/1.13 ! [P: list_nat > $o,Xs: list_nat] :
% 2.09/1.13 ( ! [Xs2: list_nat] :
% 2.09/1.13 ( ! [Ys4: list_nat] :
% 2.09/1.13 ( ( ord_less_nat @ ( size_size_list_nat @ Ys4 ) @ ( size_size_list_nat @ Xs2 ) )
% 2.09/1.13 => ( P @ Ys4 ) )
% 2.09/1.13 => ( P @ Xs2 ) )
% 2.09/1.13 => ( P @ Xs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_150_length__induct,axiom,
% 2.09/1.13 ! [P: list_a > $o,Xs: list_a] :
% 2.09/1.13 ( ! [Xs2: list_a] :
% 2.09/1.13 ( ! [Ys4: list_a] :
% 2.09/1.13 ( ( ord_less_nat @ ( size_size_list_a @ Ys4 ) @ ( size_size_list_a @ Xs2 ) )
% 2.09/1.13 => ( P @ Ys4 ) )
% 2.09/1.13 => ( P @ Xs2 ) )
% 2.09/1.13 => ( P @ Xs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_151_length__induct,axiom,
% 2.09/1.13 ! [P: list_set_a > $o,Xs: list_set_a] :
% 2.09/1.13 ( ! [Xs2: list_set_a] :
% 2.09/1.13 ( ! [Ys4: list_set_a] :
% 2.09/1.13 ( ( ord_less_nat @ ( size_size_list_set_a @ Ys4 ) @ ( size_size_list_set_a @ Xs2 ) )
% 2.09/1.13 => ( P @ Ys4 ) )
% 2.09/1.13 => ( P @ Xs2 ) )
% 2.09/1.13 => ( P @ Xs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_152_listsetD_I1_J,axiom,
% 2.09/1.13 ! [Ys: list_a,Xs: list_set_a] :
% 2.09/1.13 ( ( member_list_a @ Ys @ ( listset_a @ Xs ) )
% 2.09/1.13 => ( ( size_size_list_a @ Ys )
% 2.09/1.13 = ( size_size_list_set_a @ Xs ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_153_listsetD_I1_J,axiom,
% 2.09/1.13 ! [Ys: list_set_a,Xs: list_set_set_a] :
% 2.09/1.13 ( ( member_list_set_a @ Ys @ ( listset_set_a @ Xs ) )
% 2.09/1.13 => ( ( size_size_list_set_a @ Ys )
% 2.09/1.13 = ( size_s739728560_set_a @ Xs ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_154_permutation__Ex__bij,axiom,
% 2.09/1.13 ! [Xs: list_a,Ys: list_a] :
% 2.09/1.13 ( ( perm_a @ Xs @ Ys )
% 2.09/1.13 => ? [F2: nat > nat] :
% 2.09/1.13 ( ( bij_betw_nat_nat @ F2 @ ( set_ord_lessThan_nat @ ( size_size_list_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_a @ Ys ) ) )
% 2.09/1.13 & ! [I2: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
% 2.09/1.13 => ( ( nth_a @ Xs @ I2 )
% 2.09/1.13 = ( nth_a @ Ys @ ( F2 @ I2 ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_155_permutation__Ex__bij,axiom,
% 2.09/1.13 ! [Xs: list_set_a,Ys: list_set_a] :
% 2.09/1.13 ( ( perm_set_a @ Xs @ Ys )
% 2.09/1.13 => ? [F2: nat > nat] :
% 2.09/1.13 ( ( bij_betw_nat_nat @ F2 @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ Ys ) ) )
% 2.09/1.13 & ! [I2: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I2 @ ( size_size_list_set_a @ Xs ) )
% 2.09/1.13 => ( ( nth_set_a @ Xs @ I2 )
% 2.09/1.13 = ( nth_set_a @ Ys @ ( F2 @ I2 ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_156_g__def,axiom,
% 2.09/1.13 ( g
% 2.09/1.13 = ( hilber815131374at_nat @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ ss2 ) ) @ f ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_157_perm__refl,axiom,
% 2.09/1.13 ! [L: list_set_a] : ( perm_set_a @ L @ L ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_158_image__eqI,axiom,
% 2.09/1.13 ! [B: a,F: a > a,X4: a,A: set_a] :
% 2.09/1.13 ( ( B
% 2.09/1.13 = ( F @ X4 ) )
% 2.09/1.13 => ( ( member_a @ X4 @ A )
% 2.09/1.13 => ( member_a @ B @ ( image_a_a @ F @ A ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_159_image__eqI,axiom,
% 2.09/1.13 ! [B: list_a,F: a > list_a,X4: a,A: set_a] :
% 2.09/1.13 ( ( B
% 2.09/1.13 = ( F @ X4 ) )
% 2.09/1.13 => ( ( member_a @ X4 @ A )
% 2.09/1.13 => ( member_list_a @ B @ ( image_a_list_a @ F @ A ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_160_image__eqI,axiom,
% 2.09/1.13 ! [B: a,F: list_a > a,X4: list_a,A: set_list_a] :
% 2.09/1.13 ( ( B
% 2.09/1.13 = ( F @ X4 ) )
% 2.09/1.13 => ( ( member_list_a @ X4 @ A )
% 2.09/1.13 => ( member_a @ B @ ( image_list_a_a @ F @ A ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_161_image__eqI,axiom,
% 2.09/1.13 ! [B: list_a,F: list_a > list_a,X4: list_a,A: set_list_a] :
% 2.09/1.13 ( ( B
% 2.09/1.13 = ( F @ X4 ) )
% 2.09/1.13 => ( ( member_list_a @ X4 @ A )
% 2.09/1.13 => ( member_list_a @ B @ ( image_list_a_list_a @ F @ A ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_162_mset__eq__perm,axiom,
% 2.09/1.13 ! [Xs: list_a,Ys: list_a] :
% 2.09/1.13 ( ( ( mset_a @ Xs )
% 2.09/1.13 = ( mset_a @ Ys ) )
% 2.09/1.13 = ( perm_a @ Xs @ Ys ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_163_mset__eq__perm,axiom,
% 2.09/1.13 ! [Xs: list_set_a,Ys: list_set_a] :
% 2.09/1.13 ( ( ( mset_set_a @ Xs )
% 2.09/1.13 = ( mset_set_a @ Ys ) )
% 2.09/1.13 = ( perm_set_a @ Xs @ Ys ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_164_perm__rev,axiom,
% 2.09/1.13 ! [Xs: list_set_a] : ( perm_set_a @ ( rev_set_a @ Xs ) @ Xs ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_165_perm__length,axiom,
% 2.09/1.13 ! [Xs: list_a,Ys: list_a] :
% 2.09/1.13 ( ( perm_a @ Xs @ Ys )
% 2.09/1.13 => ( ( size_size_list_a @ Xs )
% 2.09/1.13 = ( size_size_list_a @ Ys ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_166_perm__length,axiom,
% 2.09/1.13 ! [Xs: list_set_a,Ys: list_set_a] :
% 2.09/1.13 ( ( perm_set_a @ Xs @ Ys )
% 2.09/1.13 => ( ( size_size_list_set_a @ Xs )
% 2.09/1.13 = ( size_size_list_set_a @ Ys ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_167_bij__betw__imp__surj__on,axiom,
% 2.09/1.13 ! [F: nat > nat,A: set_nat,B2: set_nat] :
% 2.09/1.13 ( ( bij_betw_nat_nat @ F @ A @ B2 )
% 2.09/1.13 => ( ( image_nat_nat @ F @ A )
% 2.09/1.13 = B2 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_168_mbs_Oless__not__eq,axiom,
% 2.09/1.13 ! [X4: list_a,A: set_list_a,Y3: list_a] :
% 2.09/1.13 ( ( member_list_a @ X4 @ A )
% 2.09/1.13 => ( ( ord_less_nat @ ( size_size_list_a @ X4 ) @ ( size_size_list_a @ Y3 ) )
% 2.09/1.13 => ( X4 != Y3 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_169_mbs_Oless__not__eq,axiom,
% 2.09/1.13 ! [X4: list_set_a,A: set_list_set_a,Y3: list_set_a] :
% 2.09/1.13 ( ( member_list_set_a @ X4 @ A )
% 2.09/1.13 => ( ( ord_less_nat @ ( size_size_list_set_a @ X4 ) @ ( size_size_list_set_a @ Y3 ) )
% 2.09/1.13 => ( X4 != Y3 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_170_size__mset,axiom,
% 2.09/1.13 ! [Xs: list_a] :
% 2.09/1.13 ( ( size_size_multiset_a @ ( mset_a @ Xs ) )
% 2.09/1.13 = ( size_size_list_a @ Xs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_171_size__mset,axiom,
% 2.09/1.13 ! [Xs: list_set_a] :
% 2.09/1.13 ( ( size_s657334288_set_a @ ( mset_set_a @ Xs ) )
% 2.09/1.13 = ( size_size_list_set_a @ Xs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_172_psubsetD,axiom,
% 2.09/1.13 ! [A: set_a,B2: set_a,C: a] :
% 2.09/1.13 ( ( ord_less_set_a @ A @ B2 )
% 2.09/1.13 => ( ( member_a @ C @ A )
% 2.09/1.13 => ( member_a @ C @ B2 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_173_psubsetD,axiom,
% 2.09/1.13 ! [A: set_list_a,B2: set_list_a,C: list_a] :
% 2.09/1.13 ( ( ord_less_set_list_a @ A @ B2 )
% 2.09/1.13 => ( ( member_list_a @ C @ A )
% 2.09/1.13 => ( member_list_a @ C @ B2 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_174_imageI,axiom,
% 2.09/1.13 ! [X4: a,A: set_a,F: a > a] :
% 2.09/1.13 ( ( member_a @ X4 @ A )
% 2.09/1.13 => ( member_a @ ( F @ X4 ) @ ( image_a_a @ F @ A ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_175_imageI,axiom,
% 2.09/1.13 ! [X4: a,A: set_a,F: a > list_a] :
% 2.09/1.13 ( ( member_a @ X4 @ A )
% 2.09/1.13 => ( member_list_a @ ( F @ X4 ) @ ( image_a_list_a @ F @ A ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_176_imageI,axiom,
% 2.09/1.13 ! [X4: list_a,A: set_list_a,F: list_a > a] :
% 2.09/1.13 ( ( member_list_a @ X4 @ A )
% 2.09/1.13 => ( member_a @ ( F @ X4 ) @ ( image_list_a_a @ F @ A ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_177_imageI,axiom,
% 2.09/1.13 ! [X4: list_a,A: set_list_a,F: list_a > list_a] :
% 2.09/1.13 ( ( member_list_a @ X4 @ A )
% 2.09/1.13 => ( member_list_a @ ( F @ X4 ) @ ( image_list_a_list_a @ F @ A ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_178_rev__image__eqI,axiom,
% 2.09/1.13 ! [X4: a,A: set_a,B: a,F: a > a] :
% 2.09/1.13 ( ( member_a @ X4 @ A )
% 2.09/1.13 => ( ( B
% 2.09/1.13 = ( F @ X4 ) )
% 2.09/1.13 => ( member_a @ B @ ( image_a_a @ F @ A ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_179_rev__image__eqI,axiom,
% 2.09/1.13 ! [X4: a,A: set_a,B: list_a,F: a > list_a] :
% 2.09/1.13 ( ( member_a @ X4 @ A )
% 2.09/1.13 => ( ( B
% 2.09/1.13 = ( F @ X4 ) )
% 2.09/1.13 => ( member_list_a @ B @ ( image_a_list_a @ F @ A ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_180_rev__image__eqI,axiom,
% 2.09/1.13 ! [X4: list_a,A: set_list_a,B: a,F: list_a > a] :
% 2.09/1.13 ( ( member_list_a @ X4 @ A )
% 2.09/1.13 => ( ( B
% 2.09/1.13 = ( F @ X4 ) )
% 2.09/1.13 => ( member_a @ B @ ( image_list_a_a @ F @ A ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_181_rev__image__eqI,axiom,
% 2.09/1.13 ! [X4: list_a,A: set_list_a,B: list_a,F: list_a > list_a] :
% 2.09/1.13 ( ( member_list_a @ X4 @ A )
% 2.09/1.13 => ( ( B
% 2.09/1.13 = ( F @ X4 ) )
% 2.09/1.13 => ( member_list_a @ B @ ( image_list_a_list_a @ F @ A ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_182_direct__decomp__def,axiom,
% 2.09/1.13 ( hilber2037636820comp_a
% 2.09/1.13 = ( ^ [A4: set_a,Ss2: list_set_a] : ( bij_betw_list_a_a @ groups1792256535list_a @ ( listset_a @ Ss2 ) @ A4 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_183_bij__betwE,axiom,
% 2.09/1.13 ! [F: nat > nat,A: set_nat,B2: set_nat] :
% 2.09/1.13 ( ( bij_betw_nat_nat @ F @ A @ B2 )
% 2.09/1.13 => ! [X: nat] :
% 2.09/1.13 ( ( member_nat @ X @ A )
% 2.09/1.13 => ( member_nat @ ( F @ X ) @ B2 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_184_bij__betw__inv,axiom,
% 2.09/1.13 ! [F: nat > nat,A: set_nat,B2: set_nat] :
% 2.09/1.13 ( ( bij_betw_nat_nat @ F @ A @ B2 )
% 2.09/1.13 => ? [G: nat > nat] : ( bij_betw_nat_nat @ G @ B2 @ A ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_185_bij__betw__cong,axiom,
% 2.09/1.13 ! [A: set_nat,F: nat > nat,G2: nat > nat,A5: set_nat] :
% 2.09/1.13 ( ! [A3: nat] :
% 2.09/1.13 ( ( member_nat @ A3 @ A )
% 2.09/1.13 => ( ( F @ A3 )
% 2.09/1.13 = ( G2 @ A3 ) ) )
% 2.09/1.13 => ( ( bij_betw_nat_nat @ F @ A @ A5 )
% 2.09/1.13 = ( bij_betw_nat_nat @ G2 @ A @ A5 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_186_bij__betw__apply,axiom,
% 2.09/1.13 ! [F: a > a,A: set_a,B2: set_a,A2: a] :
% 2.09/1.13 ( ( bij_betw_a_a @ F @ A @ B2 )
% 2.09/1.13 => ( ( member_a @ A2 @ A )
% 2.09/1.13 => ( member_a @ ( F @ A2 ) @ B2 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_187_bij__betw__apply,axiom,
% 2.09/1.13 ! [F: a > list_a,A: set_a,B2: set_list_a,A2: a] :
% 2.09/1.13 ( ( bij_betw_a_list_a @ F @ A @ B2 )
% 2.09/1.13 => ( ( member_a @ A2 @ A )
% 2.09/1.13 => ( member_list_a @ ( F @ A2 ) @ B2 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_188_bij__betw__apply,axiom,
% 2.09/1.13 ! [F: list_a > a,A: set_list_a,B2: set_a,A2: list_a] :
% 2.09/1.13 ( ( bij_betw_list_a_a @ F @ A @ B2 )
% 2.09/1.13 => ( ( member_list_a @ A2 @ A )
% 2.09/1.13 => ( member_a @ ( F @ A2 ) @ B2 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_189_bij__betw__apply,axiom,
% 2.09/1.13 ! [F: list_a > list_a,A: set_list_a,B2: set_list_a,A2: list_a] :
% 2.09/1.13 ( ( bij_be94573046list_a @ F @ A @ B2 )
% 2.09/1.13 => ( ( member_list_a @ A2 @ A )
% 2.09/1.13 => ( member_list_a @ ( F @ A2 ) @ B2 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_190_bij__betw__apply,axiom,
% 2.09/1.13 ! [F: nat > nat,A: set_nat,B2: set_nat,A2: nat] :
% 2.09/1.13 ( ( bij_betw_nat_nat @ F @ A @ B2 )
% 2.09/1.13 => ( ( member_nat @ A2 @ A )
% 2.09/1.13 => ( member_nat @ ( F @ A2 ) @ B2 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_191_bij__betw__iff__bijections,axiom,
% 2.09/1.13 ( bij_betw_a_a
% 2.09/1.13 = ( ^ [F3: a > a,A4: set_a,B3: set_a] :
% 2.09/1.13 ? [G3: a > a] :
% 2.09/1.13 ( ! [X5: a] :
% 2.09/1.13 ( ( member_a @ X5 @ A4 )
% 2.09/1.13 => ( ( member_a @ ( F3 @ X5 ) @ B3 )
% 2.09/1.13 & ( ( G3 @ ( F3 @ X5 ) )
% 2.09/1.13 = X5 ) ) )
% 2.09/1.13 & ! [X5: a] :
% 2.09/1.13 ( ( member_a @ X5 @ B3 )
% 2.09/1.13 => ( ( member_a @ ( G3 @ X5 ) @ A4 )
% 2.09/1.13 & ( ( F3 @ ( G3 @ X5 ) )
% 2.09/1.13 = X5 ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_192_bij__betw__iff__bijections,axiom,
% 2.09/1.13 ( bij_betw_list_a_a
% 2.09/1.13 = ( ^ [F3: list_a > a,A4: set_list_a,B3: set_a] :
% 2.09/1.13 ? [G3: a > list_a] :
% 2.09/1.13 ( ! [X5: list_a] :
% 2.09/1.13 ( ( member_list_a @ X5 @ A4 )
% 2.09/1.13 => ( ( member_a @ ( F3 @ X5 ) @ B3 )
% 2.09/1.13 & ( ( G3 @ ( F3 @ X5 ) )
% 2.09/1.13 = X5 ) ) )
% 2.09/1.13 & ! [X5: a] :
% 2.09/1.13 ( ( member_a @ X5 @ B3 )
% 2.09/1.13 => ( ( member_list_a @ ( G3 @ X5 ) @ A4 )
% 2.09/1.13 & ( ( F3 @ ( G3 @ X5 ) )
% 2.09/1.13 = X5 ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_193_bij__betw__iff__bijections,axiom,
% 2.09/1.13 ( bij_betw_a_list_a
% 2.09/1.13 = ( ^ [F3: a > list_a,A4: set_a,B3: set_list_a] :
% 2.09/1.13 ? [G3: list_a > a] :
% 2.09/1.13 ( ! [X5: a] :
% 2.09/1.13 ( ( member_a @ X5 @ A4 )
% 2.09/1.13 => ( ( member_list_a @ ( F3 @ X5 ) @ B3 )
% 2.09/1.13 & ( ( G3 @ ( F3 @ X5 ) )
% 2.09/1.13 = X5 ) ) )
% 2.09/1.13 & ! [X5: list_a] :
% 2.09/1.13 ( ( member_list_a @ X5 @ B3 )
% 2.09/1.13 => ( ( member_a @ ( G3 @ X5 ) @ A4 )
% 2.09/1.13 & ( ( F3 @ ( G3 @ X5 ) )
% 2.09/1.13 = X5 ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_194_bij__betw__iff__bijections,axiom,
% 2.09/1.13 ( bij_be94573046list_a
% 2.09/1.13 = ( ^ [F3: list_a > list_a,A4: set_list_a,B3: set_list_a] :
% 2.09/1.13 ? [G3: list_a > list_a] :
% 2.09/1.13 ( ! [X5: list_a] :
% 2.09/1.13 ( ( member_list_a @ X5 @ A4 )
% 2.09/1.13 => ( ( member_list_a @ ( F3 @ X5 ) @ B3 )
% 2.09/1.13 & ( ( G3 @ ( F3 @ X5 ) )
% 2.09/1.13 = X5 ) ) )
% 2.09/1.13 & ! [X5: list_a] :
% 2.09/1.13 ( ( member_list_a @ X5 @ B3 )
% 2.09/1.13 => ( ( member_list_a @ ( G3 @ X5 ) @ A4 )
% 2.09/1.13 & ( ( F3 @ ( G3 @ X5 ) )
% 2.09/1.13 = X5 ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_195_bij__betw__iff__bijections,axiom,
% 2.09/1.13 ( bij_betw_nat_nat
% 2.09/1.13 = ( ^ [F3: nat > nat,A4: set_nat,B3: set_nat] :
% 2.09/1.13 ? [G3: nat > nat] :
% 2.09/1.13 ( ! [X5: nat] :
% 2.09/1.13 ( ( member_nat @ X5 @ A4 )
% 2.09/1.13 => ( ( member_nat @ ( F3 @ X5 ) @ B3 )
% 2.09/1.13 & ( ( G3 @ ( F3 @ X5 ) )
% 2.09/1.13 = X5 ) ) )
% 2.09/1.13 & ! [X5: nat] :
% 2.09/1.13 ( ( member_nat @ X5 @ B3 )
% 2.09/1.13 => ( ( member_nat @ ( G3 @ X5 ) @ A4 )
% 2.09/1.13 & ( ( F3 @ ( G3 @ X5 ) )
% 2.09/1.13 = X5 ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_196_perm__sym,axiom,
% 2.09/1.13 ! [Xs: list_set_a,Ys: list_set_a] :
% 2.09/1.13 ( ( perm_set_a @ Xs @ Ys )
% 2.09/1.13 => ( perm_set_a @ Ys @ Xs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_197_perm_Otrans,axiom,
% 2.09/1.13 ! [Xs: list_set_a,Ys: list_set_a,Zs: list_set_a] :
% 2.09/1.13 ( ( perm_set_a @ Xs @ Ys )
% 2.09/1.13 => ( ( perm_set_a @ Ys @ Zs )
% 2.09/1.13 => ( perm_set_a @ Xs @ Zs ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_198_bij__betw__inv__into__right,axiom,
% 2.09/1.13 ! [F: nat > nat,A: set_nat,A5: set_nat,A6: nat] :
% 2.09/1.13 ( ( bij_betw_nat_nat @ F @ A @ A5 )
% 2.09/1.13 => ( ( member_nat @ A6 @ A5 )
% 2.09/1.13 => ( ( F @ ( hilber815131374at_nat @ A @ F @ A6 ) )
% 2.09/1.13 = A6 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_199_bij__betw__inv__into__left,axiom,
% 2.09/1.13 ! [F: nat > nat,A: set_nat,A5: set_nat,A2: nat] :
% 2.09/1.13 ( ( bij_betw_nat_nat @ F @ A @ A5 )
% 2.09/1.13 => ( ( member_nat @ A2 @ A )
% 2.09/1.13 => ( ( hilber815131374at_nat @ A @ F @ ( F @ A2 ) )
% 2.09/1.13 = A2 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_200_inv__into__inv__into__eq,axiom,
% 2.09/1.13 ! [F: nat > nat,A: set_nat,A5: set_nat,A2: nat] :
% 2.09/1.13 ( ( bij_betw_nat_nat @ F @ A @ A5 )
% 2.09/1.13 => ( ( member_nat @ A2 @ A )
% 2.09/1.13 => ( ( hilber815131374at_nat @ A5 @ ( hilber815131374at_nat @ A @ F ) @ A2 )
% 2.09/1.13 = ( F @ A2 ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_201_bij__betw__inv__into,axiom,
% 2.09/1.13 ! [F: nat > nat,A: set_nat,B2: set_nat] :
% 2.09/1.13 ( ( bij_betw_nat_nat @ F @ A @ B2 )
% 2.09/1.13 => ( bij_betw_nat_nat @ ( hilber815131374at_nat @ A @ F ) @ B2 @ A ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_202_inv__into__injective,axiom,
% 2.09/1.13 ! [A: set_nat,F: nat > nat,X4: nat,Y3: nat] :
% 2.09/1.13 ( ( ( hilber815131374at_nat @ A @ F @ X4 )
% 2.09/1.13 = ( hilber815131374at_nat @ A @ F @ Y3 ) )
% 2.09/1.13 => ( ( member_nat @ X4 @ ( image_nat_nat @ F @ A ) )
% 2.09/1.13 => ( ( member_nat @ Y3 @ ( image_nat_nat @ F @ A ) )
% 2.09/1.13 => ( X4 = Y3 ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_203_inv__into__into,axiom,
% 2.09/1.13 ! [X4: a,F: a > a,A: set_a] :
% 2.09/1.13 ( ( member_a @ X4 @ ( image_a_a @ F @ A ) )
% 2.09/1.13 => ( member_a @ ( hilbert_inv_into_a_a @ A @ F @ X4 ) @ A ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_204_inv__into__into,axiom,
% 2.09/1.13 ! [X4: a,F: list_a > a,A: set_list_a] :
% 2.09/1.13 ( ( member_a @ X4 @ ( image_list_a_a @ F @ A ) )
% 2.09/1.13 => ( member_list_a @ ( hilber2125729734st_a_a @ A @ F @ X4 ) @ A ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_205_inv__into__into,axiom,
% 2.09/1.13 ! [X4: list_a,F: a > list_a,A: set_a] :
% 2.09/1.13 ( ( member_list_a @ X4 @ ( image_a_list_a @ F @ A ) )
% 2.09/1.13 => ( member_a @ ( hilber165648082list_a @ A @ F @ X4 ) @ A ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_206_inv__into__into,axiom,
% 2.09/1.13 ! [X4: list_a,F: list_a > list_a,A: set_list_a] :
% 2.09/1.13 ( ( member_list_a @ X4 @ ( image_list_a_list_a @ F @ A ) )
% 2.09/1.13 => ( member_list_a @ ( hilber921249868list_a @ A @ F @ X4 ) @ A ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_207_inv__into__into,axiom,
% 2.09/1.13 ! [X4: nat,F: nat > nat,A: set_nat] :
% 2.09/1.13 ( ( member_nat @ X4 @ ( image_nat_nat @ F @ A ) )
% 2.09/1.13 => ( member_nat @ ( hilber815131374at_nat @ A @ F @ X4 ) @ A ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_208_f__inv__into__f,axiom,
% 2.09/1.13 ! [Y3: nat,F: nat > nat,A: set_nat] :
% 2.09/1.13 ( ( member_nat @ Y3 @ ( image_nat_nat @ F @ A ) )
% 2.09/1.13 => ( ( F @ ( hilber815131374at_nat @ A @ F @ Y3 ) )
% 2.09/1.13 = Y3 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_209_list__ex__length,axiom,
% 2.09/1.13 ( list_ex_a
% 2.09/1.13 = ( ^ [P2: a > $o,Xs4: list_a] :
% 2.09/1.13 ? [N2: nat] :
% 2.09/1.13 ( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs4 ) )
% 2.09/1.13 & ( P2 @ ( nth_a @ Xs4 @ N2 ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_210_list__ex__length,axiom,
% 2.09/1.13 ( list_ex_set_a
% 2.09/1.13 = ( ^ [P2: set_a > $o,Xs4: list_set_a] :
% 2.09/1.13 ? [N2: nat] :
% 2.09/1.13 ( ( ord_less_nat @ N2 @ ( size_size_list_set_a @ Xs4 ) )
% 2.09/1.13 & ( P2 @ ( nth_set_a @ Xs4 @ N2 ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_211_direct__decompI,axiom,
% 2.09/1.13 ! [Ss: list_set_a,A: set_a] :
% 2.09/1.13 ( ( inj_on_list_a_a @ groups1792256535list_a @ ( listset_a @ Ss ) )
% 2.09/1.13 => ( ( ( image_list_a_a @ groups1792256535list_a @ ( listset_a @ Ss ) )
% 2.09/1.13 = A )
% 2.09/1.13 => ( hilber2037636820comp_a @ A @ Ss ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_212_elem__le__sum__list,axiom,
% 2.09/1.13 ! [K: nat,Ns: list_nat] :
% 2.09/1.13 ( ( ord_less_nat @ K @ ( size_size_list_nat @ Ns ) )
% 2.09/1.13 => ( ord_less_eq_nat @ ( nth_nat @ Ns @ K ) @ ( groups921905271st_nat @ Ns ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_213_perm__swap,axiom,
% 2.09/1.13 ! [I: nat,Xs: list_a,J: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
% 2.09/1.13 => ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
% 2.09/1.13 => ( perm_a @ ( list_update_a @ ( list_update_a @ Xs @ I @ ( nth_a @ Xs @ J ) ) @ J @ ( nth_a @ Xs @ I ) ) @ Xs ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_214_perm__swap,axiom,
% 2.09/1.13 ! [I: nat,Xs: list_set_a,J: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I @ ( size_size_list_set_a @ Xs ) )
% 2.09/1.13 => ( ( ord_less_nat @ J @ ( size_size_list_set_a @ Xs ) )
% 2.09/1.13 => ( perm_set_a @ ( list_update_set_a @ ( list_update_set_a @ Xs @ I @ ( nth_set_a @ Xs @ J ) ) @ J @ ( nth_set_a @ Xs @ I ) ) @ Xs ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_215_mset__swap,axiom,
% 2.09/1.13 ! [I: nat,Ls: list_a,J: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I @ ( size_size_list_a @ Ls ) )
% 2.09/1.13 => ( ( ord_less_nat @ J @ ( size_size_list_a @ Ls ) )
% 2.09/1.13 => ( ( mset_a @ ( list_update_a @ ( list_update_a @ Ls @ J @ ( nth_a @ Ls @ I ) ) @ I @ ( nth_a @ Ls @ J ) ) )
% 2.09/1.13 = ( mset_a @ Ls ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_216_mset__swap,axiom,
% 2.09/1.13 ! [I: nat,Ls: list_set_a,J: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I @ ( size_size_list_set_a @ Ls ) )
% 2.09/1.13 => ( ( ord_less_nat @ J @ ( size_size_list_set_a @ Ls ) )
% 2.09/1.13 => ( ( mset_set_a @ ( list_update_set_a @ ( list_update_set_a @ Ls @ J @ ( nth_set_a @ Ls @ I ) ) @ I @ ( nth_set_a @ Ls @ J ) ) )
% 2.09/1.13 = ( mset_set_a @ Ls ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_217_nth__mem__mset,axiom,
% 2.09/1.13 ! [I: nat,Ls: list_list_a] :
% 2.09/1.13 ( ( ord_less_nat @ I @ ( size_s1427607542list_a @ Ls ) )
% 2.09/1.13 => ( member_list_a @ ( nth_list_a @ Ls @ I ) @ ( set_mset_list_a @ ( mset_list_a @ Ls ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_218_nth__mem__mset,axiom,
% 2.09/1.13 ! [I: nat,Ls: list_a] :
% 2.09/1.13 ( ( ord_less_nat @ I @ ( size_size_list_a @ Ls ) )
% 2.09/1.13 => ( member_a @ ( nth_a @ Ls @ I ) @ ( set_mset_a @ ( mset_a @ Ls ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_219_nth__mem__mset,axiom,
% 2.09/1.13 ! [I: nat,Ls: list_set_a] :
% 2.09/1.13 ( ( ord_less_nat @ I @ ( size_size_list_set_a @ Ls ) )
% 2.09/1.13 => ( member_set_a @ ( nth_set_a @ Ls @ I ) @ ( set_mset_set_a @ ( mset_set_a @ Ls ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_220_in__Union__mset__iff,axiom,
% 2.09/1.13 ! [X4: a,MM: multiset_multiset_a] :
% 2.09/1.13 ( ( member_a @ X4 @ ( set_mset_a @ ( comm_m2145643721iset_a @ MM ) ) )
% 2.09/1.13 = ( ? [M2: multiset_a] :
% 2.09/1.13 ( ( member_multiset_a @ M2 @ ( set_mset_multiset_a @ MM ) )
% 2.09/1.13 & ( member_a @ X4 @ ( set_mset_a @ M2 ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_221_in__Union__mset__iff,axiom,
% 2.09/1.13 ! [X4: list_a,MM: multis971982480list_a] :
% 2.09/1.13 ( ( member_list_a @ X4 @ ( set_mset_list_a @ ( comm_m324867663list_a @ MM ) ) )
% 2.09/1.13 = ( ? [M2: multiset_list_a] :
% 2.09/1.13 ( ( member528267379list_a @ M2 @ ( set_ms1660427399list_a @ MM ) )
% 2.09/1.13 & ( member_list_a @ X4 @ ( set_mset_list_a @ M2 ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_222_lessThan__subset__iff,axiom,
% 2.09/1.13 ! [X4: nat,Y3: nat] :
% 2.09/1.13 ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X4 ) @ ( set_ord_lessThan_nat @ Y3 ) )
% 2.09/1.13 = ( ord_less_eq_nat @ X4 @ Y3 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_223_list__update__beyond,axiom,
% 2.09/1.13 ! [Xs: list_a,I: nat,X4: a] :
% 2.09/1.13 ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ I )
% 2.09/1.13 => ( ( list_update_a @ Xs @ I @ X4 )
% 2.09/1.13 = Xs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_224_list__update__beyond,axiom,
% 2.09/1.13 ! [Xs: list_set_a,I: nat,X4: set_a] :
% 2.09/1.13 ( ( ord_less_eq_nat @ ( size_size_list_set_a @ Xs ) @ I )
% 2.09/1.13 => ( ( list_update_set_a @ Xs @ I @ X4 )
% 2.09/1.13 = Xs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_225_length__list__update,axiom,
% 2.09/1.13 ! [Xs: list_a,I: nat,X4: a] :
% 2.09/1.13 ( ( size_size_list_a @ ( list_update_a @ Xs @ I @ X4 ) )
% 2.09/1.13 = ( size_size_list_a @ Xs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_226_length__list__update,axiom,
% 2.09/1.13 ! [Xs: list_set_a,I: nat,X4: set_a] :
% 2.09/1.13 ( ( size_size_list_set_a @ ( list_update_set_a @ Xs @ I @ X4 ) )
% 2.09/1.13 = ( size_size_list_set_a @ Xs ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_227_inv__into__f__f,axiom,
% 2.09/1.13 ! [F: nat > nat,A: set_nat,X4: nat] :
% 2.09/1.13 ( ( inj_on_nat_nat @ F @ A )
% 2.09/1.13 => ( ( member_nat @ X4 @ A )
% 2.09/1.13 => ( ( hilber815131374at_nat @ A @ F @ ( F @ X4 ) )
% 2.09/1.13 = X4 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_228_list__update__id,axiom,
% 2.09/1.13 ! [Xs: list_a,I: nat] :
% 2.09/1.13 ( ( list_update_a @ Xs @ I @ ( nth_a @ Xs @ I ) )
% 2.09/1.13 = Xs ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_229_list__update__id,axiom,
% 2.09/1.13 ! [Xs: list_set_a,I: nat] :
% 2.09/1.13 ( ( list_update_set_a @ Xs @ I @ ( nth_set_a @ Xs @ I ) )
% 2.09/1.13 = Xs ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_230_nth__list__update__neq,axiom,
% 2.09/1.13 ! [I: nat,J: nat,Xs: list_a,X4: a] :
% 2.09/1.13 ( ( I != J )
% 2.09/1.13 => ( ( nth_a @ ( list_update_a @ Xs @ I @ X4 ) @ J )
% 2.09/1.13 = ( nth_a @ Xs @ J ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_231_nth__list__update__neq,axiom,
% 2.09/1.13 ! [I: nat,J: nat,Xs: list_set_a,X4: set_a] :
% 2.09/1.13 ( ( I != J )
% 2.09/1.13 => ( ( nth_set_a @ ( list_update_set_a @ Xs @ I @ X4 ) @ J )
% 2.09/1.13 = ( nth_set_a @ Xs @ J ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_232_inv__into__image__cancel,axiom,
% 2.09/1.13 ! [F: nat > nat,A: set_nat,S: set_nat] :
% 2.09/1.13 ( ( inj_on_nat_nat @ F @ A )
% 2.09/1.13 => ( ( ord_less_eq_set_nat @ S @ A )
% 2.09/1.13 => ( ( image_nat_nat @ ( hilber815131374at_nat @ A @ F ) @ ( image_nat_nat @ F @ S ) )
% 2.09/1.13 = S ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_233_nth__list__update__eq,axiom,
% 2.09/1.13 ! [I: nat,Xs: list_a,X4: a] :
% 2.09/1.13 ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
% 2.09/1.13 => ( ( nth_a @ ( list_update_a @ Xs @ I @ X4 ) @ I )
% 2.09/1.13 = X4 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_234_nth__list__update__eq,axiom,
% 2.09/1.13 ! [I: nat,Xs: list_set_a,X4: set_a] :
% 2.09/1.13 ( ( ord_less_nat @ I @ ( size_size_list_set_a @ Xs ) )
% 2.09/1.13 => ( ( nth_set_a @ ( list_update_set_a @ Xs @ I @ X4 ) @ I )
% 2.09/1.13 = X4 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_235_inj__on__inv__into,axiom,
% 2.09/1.13 ! [B2: set_nat,F: nat > nat,A: set_nat] :
% 2.09/1.13 ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
% 2.09/1.13 => ( inj_on_nat_nat @ ( hilber815131374at_nat @ A @ F ) @ B2 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_236_inj__on__image__mem__iff,axiom,
% 2.09/1.13 ! [F: a > a,B2: set_a,A2: a,A: set_a] :
% 2.09/1.13 ( ( inj_on_a_a @ F @ B2 )
% 2.09/1.13 => ( ( member_a @ A2 @ B2 )
% 2.09/1.13 => ( ( ord_less_eq_set_a @ A @ B2 )
% 2.09/1.13 => ( ( member_a @ ( F @ A2 ) @ ( image_a_a @ F @ A ) )
% 2.09/1.13 = ( member_a @ A2 @ A ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_237_inj__on__image__mem__iff,axiom,
% 2.09/1.13 ! [F: a > list_a,B2: set_a,A2: a,A: set_a] :
% 2.09/1.13 ( ( inj_on_a_list_a @ F @ B2 )
% 2.09/1.13 => ( ( member_a @ A2 @ B2 )
% 2.09/1.13 => ( ( ord_less_eq_set_a @ A @ B2 )
% 2.09/1.13 => ( ( member_list_a @ ( F @ A2 ) @ ( image_a_list_a @ F @ A ) )
% 2.09/1.13 = ( member_a @ A2 @ A ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_238_inj__on__image__mem__iff,axiom,
% 2.09/1.13 ! [F: list_a > a,B2: set_list_a,A2: list_a,A: set_list_a] :
% 2.09/1.13 ( ( inj_on_list_a_a @ F @ B2 )
% 2.09/1.13 => ( ( member_list_a @ A2 @ B2 )
% 2.09/1.13 => ( ( ord_le1301786372list_a @ A @ B2 )
% 2.09/1.13 => ( ( member_a @ ( F @ A2 ) @ ( image_list_a_a @ F @ A ) )
% 2.09/1.13 = ( member_list_a @ A2 @ A ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_239_inj__on__image__mem__iff,axiom,
% 2.09/1.13 ! [F: list_a > list_a,B2: set_list_a,A2: list_a,A: set_list_a] :
% 2.09/1.13 ( ( inj_on_list_a_list_a @ F @ B2 )
% 2.09/1.13 => ( ( member_list_a @ A2 @ B2 )
% 2.09/1.13 => ( ( ord_le1301786372list_a @ A @ B2 )
% 2.09/1.13 => ( ( member_list_a @ ( F @ A2 ) @ ( image_list_a_list_a @ F @ A ) )
% 2.09/1.13 = ( member_list_a @ A2 @ A ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_240_bij__betw__imp__inj__on,axiom,
% 2.09/1.13 ! [F: nat > nat,A: set_nat,B2: set_nat] :
% 2.09/1.13 ( ( bij_betw_nat_nat @ F @ A @ B2 )
% 2.09/1.13 => ( inj_on_nat_nat @ F @ A ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_241_inv__into__f__eq,axiom,
% 2.09/1.13 ! [F: nat > nat,A: set_nat,X4: nat,Y3: nat] :
% 2.09/1.13 ( ( inj_on_nat_nat @ F @ A )
% 2.09/1.13 => ( ( member_nat @ X4 @ A )
% 2.09/1.13 => ( ( ( F @ X4 )
% 2.09/1.13 = Y3 )
% 2.09/1.13 => ( ( hilber815131374at_nat @ A @ F @ Y3 )
% 2.09/1.13 = X4 ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_242_image__subsetI,axiom,
% 2.09/1.13 ! [A: set_a,F: a > a,B2: set_a] :
% 2.09/1.13 ( ! [X6: a] :
% 2.09/1.13 ( ( member_a @ X6 @ A )
% 2.09/1.13 => ( member_a @ ( F @ X6 ) @ B2 ) )
% 2.09/1.13 => ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B2 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_243_image__subsetI,axiom,
% 2.09/1.13 ! [A: set_a,F: a > list_a,B2: set_list_a] :
% 2.09/1.13 ( ! [X6: a] :
% 2.09/1.13 ( ( member_a @ X6 @ A )
% 2.09/1.13 => ( member_list_a @ ( F @ X6 ) @ B2 ) )
% 2.09/1.13 => ( ord_le1301786372list_a @ ( image_a_list_a @ F @ A ) @ B2 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_244_image__subsetI,axiom,
% 2.09/1.13 ! [A: set_list_a,F: list_a > a,B2: set_a] :
% 2.09/1.13 ( ! [X6: list_a] :
% 2.09/1.13 ( ( member_list_a @ X6 @ A )
% 2.09/1.13 => ( member_a @ ( F @ X6 ) @ B2 ) )
% 2.09/1.13 => ( ord_less_eq_set_a @ ( image_list_a_a @ F @ A ) @ B2 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_245_image__subsetI,axiom,
% 2.09/1.13 ! [A: set_list_a,F: list_a > list_a,B2: set_list_a] :
% 2.09/1.13 ( ! [X6: list_a] :
% 2.09/1.13 ( ( member_list_a @ X6 @ A )
% 2.09/1.13 => ( member_list_a @ ( F @ X6 ) @ B2 ) )
% 2.09/1.13 => ( ord_le1301786372list_a @ ( image_list_a_list_a @ F @ A ) @ B2 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_246_bij__betw__def,axiom,
% 2.09/1.13 ( bij_betw_nat_nat
% 2.09/1.13 = ( ^ [F3: nat > nat,A4: set_nat,B3: set_nat] :
% 2.09/1.13 ( ( inj_on_nat_nat @ F3 @ A4 )
% 2.09/1.13 & ( ( image_nat_nat @ F3 @ A4 )
% 2.09/1.13 = B3 ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_247_bij__betw__imageI,axiom,
% 2.09/1.13 ! [F: nat > nat,A: set_nat,B2: set_nat] :
% 2.09/1.13 ( ( inj_on_nat_nat @ F @ A )
% 2.09/1.13 => ( ( ( image_nat_nat @ F @ A )
% 2.09/1.13 = B2 )
% 2.09/1.13 => ( bij_betw_nat_nat @ F @ A @ B2 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_248_inj__on__imp__bij__betw,axiom,
% 2.09/1.13 ! [F: nat > nat,A: set_nat] :
% 2.09/1.13 ( ( inj_on_nat_nat @ F @ A )
% 2.09/1.13 => ( bij_betw_nat_nat @ F @ A @ ( image_nat_nat @ F @ A ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_249_mbs_Ominimal,axiom,
% 2.09/1.13 ! [X4: list_a,A: set_list_a,P: list_a > $o] :
% 2.09/1.13 ( ( member_list_a @ X4 @ A )
% 2.09/1.13 => ( ( P @ X4 )
% 2.09/1.13 => ? [X6: list_a] :
% 2.09/1.13 ( ( member_list_a @ X6 @ A )
% 2.09/1.13 & ( ord_less_eq_nat @ ( size_size_list_a @ X6 ) @ ( size_size_list_a @ X4 ) )
% 2.09/1.13 & ( P @ X6 )
% 2.09/1.13 & ! [Xa: list_a] :
% 2.09/1.13 ( ( member_list_a @ Xa @ A )
% 2.09/1.13 => ( ( ord_less_nat @ ( size_size_list_a @ Xa ) @ ( size_size_list_a @ X6 ) )
% 2.09/1.13 => ~ ( P @ Xa ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_250_mbs_Ominimal,axiom,
% 2.09/1.13 ! [X4: list_set_a,A: set_list_set_a,P: list_set_a > $o] :
% 2.09/1.13 ( ( member_list_set_a @ X4 @ A )
% 2.09/1.13 => ( ( P @ X4 )
% 2.09/1.13 => ? [X6: list_set_a] :
% 2.09/1.13 ( ( member_list_set_a @ X6 @ A )
% 2.09/1.13 & ( ord_less_eq_nat @ ( size_size_list_set_a @ X6 ) @ ( size_size_list_set_a @ X4 ) )
% 2.09/1.13 & ( P @ X6 )
% 2.09/1.13 & ! [Xa: list_set_a] :
% 2.09/1.13 ( ( member_list_set_a @ Xa @ A )
% 2.09/1.13 => ( ( ord_less_nat @ ( size_size_list_set_a @ Xa ) @ ( size_size_list_set_a @ X6 ) )
% 2.09/1.13 => ~ ( P @ Xa ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_251_bij__betw__subset,axiom,
% 2.09/1.13 ! [F: nat > nat,A: set_nat,A5: set_nat,B2: set_nat,B4: set_nat] :
% 2.09/1.13 ( ( bij_betw_nat_nat @ F @ A @ A5 )
% 2.09/1.13 => ( ( ord_less_eq_set_nat @ B2 @ A )
% 2.09/1.13 => ( ( ( image_nat_nat @ F @ B2 )
% 2.09/1.13 = B4 )
% 2.09/1.13 => ( bij_betw_nat_nat @ F @ B2 @ B4 ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_252_bij__betw__byWitness,axiom,
% 2.09/1.13 ! [A: set_nat,F4: nat > nat,F: nat > nat,A5: set_nat] :
% 2.09/1.13 ( ! [X6: nat] :
% 2.09/1.13 ( ( member_nat @ X6 @ A )
% 2.09/1.13 => ( ( F4 @ ( F @ X6 ) )
% 2.09/1.13 = X6 ) )
% 2.09/1.13 => ( ! [X6: nat] :
% 2.09/1.13 ( ( member_nat @ X6 @ A5 )
% 2.09/1.13 => ( ( F @ ( F4 @ X6 ) )
% 2.09/1.13 = X6 ) )
% 2.09/1.13 => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ A5 )
% 2.09/1.13 => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F4 @ A5 ) @ A )
% 2.09/1.13 => ( bij_betw_nat_nat @ F @ A @ A5 ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_253_image__inv__into__cancel,axiom,
% 2.09/1.13 ! [F: nat > nat,A: set_nat,A5: set_nat,B4: set_nat] :
% 2.09/1.13 ( ( ( image_nat_nat @ F @ A )
% 2.09/1.13 = A5 )
% 2.09/1.13 => ( ( ord_less_eq_set_nat @ B4 @ A5 )
% 2.09/1.13 => ( ( image_nat_nat @ F @ ( image_nat_nat @ ( hilber815131374at_nat @ A @ F ) @ B4 ) )
% 2.09/1.13 = B4 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_254_listset__mono,axiom,
% 2.09/1.13 ! [Xs: list_set_a,Ys: list_set_a] :
% 2.09/1.13 ( ( ( size_size_list_set_a @ Xs )
% 2.09/1.13 = ( size_size_list_set_a @ Ys ) )
% 2.09/1.13 => ( ! [I3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ I3 @ ( size_size_list_set_a @ Ys ) )
% 2.09/1.13 => ( ord_less_eq_set_a @ ( nth_set_a @ Xs @ I3 ) @ ( nth_set_a @ Ys @ I3 ) ) )
% 2.09/1.13 => ( ord_le1301786372list_a @ ( listset_a @ Xs ) @ ( listset_a @ Ys ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_255_nth__list__update,axiom,
% 2.09/1.13 ! [I: nat,Xs: list_a,J: nat,X4: a] :
% 2.09/1.13 ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
% 2.09/1.13 => ( ( ( I = J )
% 2.09/1.13 => ( ( nth_a @ ( list_update_a @ Xs @ I @ X4 ) @ J )
% 2.09/1.13 = X4 ) )
% 2.09/1.13 & ( ( I != J )
% 2.09/1.13 => ( ( nth_a @ ( list_update_a @ Xs @ I @ X4 ) @ J )
% 2.09/1.13 = ( nth_a @ Xs @ J ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_256_nth__list__update,axiom,
% 2.09/1.13 ! [I: nat,Xs: list_set_a,J: nat,X4: set_a] :
% 2.09/1.13 ( ( ord_less_nat @ I @ ( size_size_list_set_a @ Xs ) )
% 2.09/1.13 => ( ( ( I = J )
% 2.09/1.13 => ( ( nth_set_a @ ( list_update_set_a @ Xs @ I @ X4 ) @ J )
% 2.09/1.13 = X4 ) )
% 2.09/1.13 & ( ( I != J )
% 2.09/1.13 => ( ( nth_set_a @ ( list_update_set_a @ Xs @ I @ X4 ) @ J )
% 2.09/1.13 = ( nth_set_a @ Xs @ J ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_257_list__update__same__conv,axiom,
% 2.09/1.13 ! [I: nat,Xs: list_a,X4: a] :
% 2.09/1.13 ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
% 2.09/1.13 => ( ( ( list_update_a @ Xs @ I @ X4 )
% 2.09/1.13 = Xs )
% 2.09/1.13 = ( ( nth_a @ Xs @ I )
% 2.09/1.13 = X4 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_258_list__update__same__conv,axiom,
% 2.09/1.13 ! [I: nat,Xs: list_set_a,X4: set_a] :
% 2.09/1.13 ( ( ord_less_nat @ I @ ( size_size_list_set_a @ Xs ) )
% 2.09/1.13 => ( ( ( list_update_set_a @ Xs @ I @ X4 )
% 2.09/1.13 = Xs )
% 2.09/1.13 = ( ( nth_set_a @ Xs @ I )
% 2.09/1.13 = X4 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_259_bij__betw__inv__into__subset,axiom,
% 2.09/1.13 ! [F: nat > nat,A: set_nat,A5: set_nat,B2: set_nat,B4: set_nat] :
% 2.09/1.13 ( ( bij_betw_nat_nat @ F @ A @ A5 )
% 2.09/1.13 => ( ( ord_less_eq_set_nat @ B2 @ A )
% 2.09/1.13 => ( ( ( image_nat_nat @ F @ B2 )
% 2.09/1.13 = B4 )
% 2.09/1.13 => ( bij_betw_nat_nat @ ( hilber815131374at_nat @ A @ F ) @ B4 @ B2 ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_260_direct__decompD_I2_J,axiom,
% 2.09/1.13 ! [A: set_a,Ss: list_set_a] :
% 2.09/1.13 ( ( hilber2037636820comp_a @ A @ Ss )
% 2.09/1.13 => ( inj_on_list_a_a @ groups1792256535list_a @ ( listset_a @ Ss ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_261_Schroeder__Bernstein,axiom,
% 2.09/1.13 ! [F: nat > nat,A: set_nat,B2: set_nat,G2: nat > nat] :
% 2.09/1.13 ( ( inj_on_nat_nat @ F @ A )
% 2.09/1.13 => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 )
% 2.09/1.13 => ( ( inj_on_nat_nat @ G2 @ B2 )
% 2.09/1.13 => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G2 @ B2 ) @ A )
% 2.09/1.13 => ? [H: nat > nat] : ( bij_betw_nat_nat @ H @ A @ B2 ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_262_order__refl,axiom,
% 2.09/1.13 ! [X4: nat] : ( ord_less_eq_nat @ X4 @ X4 ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_263_minf_I8_J,axiom,
% 2.09/1.13 ! [T: nat] :
% 2.09/1.13 ? [Z2: nat] :
% 2.09/1.13 ! [X: nat] :
% 2.09/1.13 ( ( ord_less_nat @ X @ Z2 )
% 2.09/1.13 => ~ ( ord_less_eq_nat @ T @ X ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_264_minf_I6_J,axiom,
% 2.09/1.13 ! [T: nat] :
% 2.09/1.13 ? [Z2: nat] :
% 2.09/1.13 ! [X: nat] :
% 2.09/1.13 ( ( ord_less_nat @ X @ Z2 )
% 2.09/1.13 => ( ord_less_eq_nat @ X @ T ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_265_subsetI,axiom,
% 2.09/1.13 ! [A: set_a,B2: set_a] :
% 2.09/1.13 ( ! [X6: a] :
% 2.09/1.13 ( ( member_a @ X6 @ A )
% 2.09/1.13 => ( member_a @ X6 @ B2 ) )
% 2.09/1.13 => ( ord_less_eq_set_a @ A @ B2 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_266_subsetI,axiom,
% 2.09/1.13 ! [A: set_list_a,B2: set_list_a] :
% 2.09/1.13 ( ! [X6: list_a] :
% 2.09/1.13 ( ( member_list_a @ X6 @ A )
% 2.09/1.13 => ( member_list_a @ X6 @ B2 ) )
% 2.09/1.13 => ( ord_le1301786372list_a @ A @ B2 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_267_bounded__Max__nat,axiom,
% 2.09/1.13 ! [P: nat > $o,X4: nat,M3: nat] :
% 2.09/1.13 ( ( P @ X4 )
% 2.09/1.13 => ( ! [X6: nat] :
% 2.09/1.13 ( ( P @ X6 )
% 2.09/1.13 => ( ord_less_eq_nat @ X6 @ M3 ) )
% 2.09/1.13 => ~ ! [M4: nat] :
% 2.09/1.13 ( ( P @ M4 )
% 2.09/1.13 => ~ ! [X: nat] :
% 2.09/1.13 ( ( P @ X )
% 2.09/1.13 => ( ord_less_eq_nat @ X @ M4 ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_268_subset__iff,axiom,
% 2.09/1.13 ( ord_less_eq_set_a
% 2.09/1.13 = ( ^ [A4: set_a,B3: set_a] :
% 2.09/1.13 ! [T2: a] :
% 2.09/1.13 ( ( member_a @ T2 @ A4 )
% 2.09/1.13 => ( member_a @ T2 @ B3 ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_269_subset__iff,axiom,
% 2.09/1.13 ( ord_le1301786372list_a
% 2.09/1.13 = ( ^ [A4: set_list_a,B3: set_list_a] :
% 2.09/1.13 ! [T2: list_a] :
% 2.09/1.13 ( ( member_list_a @ T2 @ A4 )
% 2.09/1.13 => ( member_list_a @ T2 @ B3 ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_270_subset__eq,axiom,
% 2.09/1.13 ( ord_less_eq_set_a
% 2.09/1.13 = ( ^ [A4: set_a,B3: set_a] :
% 2.09/1.13 ! [X5: a] :
% 2.09/1.13 ( ( member_a @ X5 @ A4 )
% 2.09/1.13 => ( member_a @ X5 @ B3 ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_271_subset__eq,axiom,
% 2.09/1.13 ( ord_le1301786372list_a
% 2.09/1.13 = ( ^ [A4: set_list_a,B3: set_list_a] :
% 2.09/1.13 ! [X5: list_a] :
% 2.09/1.13 ( ( member_list_a @ X5 @ A4 )
% 2.09/1.13 => ( member_list_a @ X5 @ B3 ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_272_subsetD,axiom,
% 2.09/1.13 ! [A: set_a,B2: set_a,C: a] :
% 2.09/1.13 ( ( ord_less_eq_set_a @ A @ B2 )
% 2.09/1.13 => ( ( member_a @ C @ A )
% 2.09/1.13 => ( member_a @ C @ B2 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_273_subsetD,axiom,
% 2.09/1.13 ! [A: set_list_a,B2: set_list_a,C: list_a] :
% 2.09/1.13 ( ( ord_le1301786372list_a @ A @ B2 )
% 2.09/1.13 => ( ( member_list_a @ C @ A )
% 2.09/1.13 => ( member_list_a @ C @ B2 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_274_in__mono,axiom,
% 2.09/1.13 ! [A: set_a,B2: set_a,X4: a] :
% 2.09/1.13 ( ( ord_less_eq_set_a @ A @ B2 )
% 2.09/1.13 => ( ( member_a @ X4 @ A )
% 2.09/1.13 => ( member_a @ X4 @ B2 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_275_in__mono,axiom,
% 2.09/1.13 ! [A: set_list_a,B2: set_list_a,X4: list_a] :
% 2.09/1.13 ( ( ord_le1301786372list_a @ A @ B2 )
% 2.09/1.13 => ( ( member_list_a @ X4 @ A )
% 2.09/1.13 => ( member_list_a @ X4 @ B2 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_276_dual__order_Oantisym,axiom,
% 2.09/1.13 ! [B: nat,A2: nat] :
% 2.09/1.13 ( ( ord_less_eq_nat @ B @ A2 )
% 2.09/1.13 => ( ( ord_less_eq_nat @ A2 @ B )
% 2.09/1.13 => ( A2 = B ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_277_dual__order_Oeq__iff,axiom,
% 2.09/1.13 ( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
% 2.09/1.13 = ( ^ [A7: nat,B5: nat] :
% 2.09/1.13 ( ( ord_less_eq_nat @ B5 @ A7 )
% 2.09/1.13 & ( ord_less_eq_nat @ A7 @ B5 ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_278_dual__order_Otrans,axiom,
% 2.09/1.13 ! [B: nat,A2: nat,C: nat] :
% 2.09/1.13 ( ( ord_less_eq_nat @ B @ A2 )
% 2.09/1.13 => ( ( ord_less_eq_nat @ C @ B )
% 2.09/1.13 => ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_279_linorder__wlog,axiom,
% 2.09/1.13 ! [P: nat > nat > $o,A2: nat,B: nat] :
% 2.09/1.13 ( ! [A3: nat,B6: nat] :
% 2.09/1.13 ( ( ord_less_eq_nat @ A3 @ B6 )
% 2.09/1.13 => ( P @ A3 @ B6 ) )
% 2.09/1.13 => ( ! [A3: nat,B6: nat] :
% 2.09/1.13 ( ( P @ B6 @ A3 )
% 2.09/1.13 => ( P @ A3 @ B6 ) )
% 2.09/1.13 => ( P @ A2 @ B ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_280_dual__order_Orefl,axiom,
% 2.09/1.13 ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_281_order__trans,axiom,
% 2.09/1.13 ! [X4: nat,Y3: nat,Z3: nat] :
% 2.09/1.13 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 2.09/1.13 => ( ( ord_less_eq_nat @ Y3 @ Z3 )
% 2.09/1.13 => ( ord_less_eq_nat @ X4 @ Z3 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_282_order__class_Oorder_Oantisym,axiom,
% 2.09/1.13 ! [A2: nat,B: nat] :
% 2.09/1.13 ( ( ord_less_eq_nat @ A2 @ B )
% 2.09/1.13 => ( ( ord_less_eq_nat @ B @ A2 )
% 2.09/1.13 => ( A2 = B ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_283_ord__le__eq__trans,axiom,
% 2.09/1.13 ! [A2: nat,B: nat,C: nat] :
% 2.09/1.13 ( ( ord_less_eq_nat @ A2 @ B )
% 2.09/1.13 => ( ( B = C )
% 2.09/1.13 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_284_ord__eq__le__trans,axiom,
% 2.09/1.13 ! [A2: nat,B: nat,C: nat] :
% 2.09/1.13 ( ( A2 = B )
% 2.09/1.13 => ( ( ord_less_eq_nat @ B @ C )
% 2.09/1.13 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_285_order__class_Oorder_Oeq__iff,axiom,
% 2.09/1.13 ( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
% 2.09/1.13 = ( ^ [A7: nat,B5: nat] :
% 2.09/1.13 ( ( ord_less_eq_nat @ A7 @ B5 )
% 2.09/1.13 & ( ord_less_eq_nat @ B5 @ A7 ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_286_antisym__conv,axiom,
% 2.09/1.13 ! [Y3: nat,X4: nat] :
% 2.09/1.13 ( ( ord_less_eq_nat @ Y3 @ X4 )
% 2.09/1.13 => ( ( ord_less_eq_nat @ X4 @ Y3 )
% 2.09/1.13 = ( X4 = Y3 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_287_le__cases3,axiom,
% 2.09/1.13 ! [X4: nat,Y3: nat,Z3: nat] :
% 2.09/1.13 ( ( ( ord_less_eq_nat @ X4 @ Y3 )
% 2.09/1.13 => ~ ( ord_less_eq_nat @ Y3 @ Z3 ) )
% 2.09/1.13 => ( ( ( ord_less_eq_nat @ Y3 @ X4 )
% 2.09/1.13 => ~ ( ord_less_eq_nat @ X4 @ Z3 ) )
% 2.09/1.13 => ( ( ( ord_less_eq_nat @ X4 @ Z3 )
% 2.09/1.13 => ~ ( ord_less_eq_nat @ Z3 @ Y3 ) )
% 2.09/1.13 => ( ( ( ord_less_eq_nat @ Z3 @ Y3 )
% 2.09/1.13 => ~ ( ord_less_eq_nat @ Y3 @ X4 ) )
% 2.09/1.13 => ( ( ( ord_less_eq_nat @ Y3 @ Z3 )
% 2.09/1.13 => ~ ( ord_less_eq_nat @ Z3 @ X4 ) )
% 2.09/1.13 => ~ ( ( ord_less_eq_nat @ Z3 @ X4 )
% 2.09/1.13 => ~ ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_288_order_Otrans,axiom,
% 2.09/1.13 ! [A2: nat,B: nat,C: nat] :
% 2.09/1.13 ( ( ord_less_eq_nat @ A2 @ B )
% 2.09/1.13 => ( ( ord_less_eq_nat @ B @ C )
% 2.09/1.13 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_289_le__cases,axiom,
% 2.09/1.13 ! [X4: nat,Y3: nat] :
% 2.09/1.13 ( ~ ( ord_less_eq_nat @ X4 @ Y3 )
% 2.09/1.13 => ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_290_eq__refl,axiom,
% 2.09/1.13 ! [X4: nat,Y3: nat] :
% 2.09/1.13 ( ( X4 = Y3 )
% 2.09/1.13 => ( ord_less_eq_nat @ X4 @ Y3 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_291_linear,axiom,
% 2.09/1.13 ! [X4: nat,Y3: nat] :
% 2.09/1.13 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 2.09/1.13 | ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_292_antisym,axiom,
% 2.09/1.13 ! [X4: nat,Y3: nat] :
% 2.09/1.13 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 2.09/1.13 => ( ( ord_less_eq_nat @ Y3 @ X4 )
% 2.09/1.13 => ( X4 = Y3 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_293_le__rel__bool__arg__iff,axiom,
% 2.09/1.13 ( ord_less_eq_o_nat
% 2.09/1.13 = ( ^ [X7: $o > nat,Y4: $o > nat] :
% 2.09/1.13 ( ( ord_less_eq_nat @ ( X7 @ $false ) @ ( Y4 @ $false ) )
% 2.09/1.13 & ( ord_less_eq_nat @ ( X7 @ $true ) @ ( Y4 @ $true ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_294_eq__iff,axiom,
% 2.09/1.13 ( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
% 2.09/1.13 = ( ^ [X5: nat,Y5: nat] :
% 2.09/1.13 ( ( ord_less_eq_nat @ X5 @ Y5 )
% 2.09/1.13 & ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_295_ord__le__eq__subst,axiom,
% 2.09/1.13 ! [A2: nat,B: nat,F: nat > nat,C: nat] :
% 2.09/1.13 ( ( ord_less_eq_nat @ A2 @ B )
% 2.09/1.13 => ( ( ( F @ B )
% 2.09/1.13 = C )
% 2.09/1.13 => ( ! [X6: nat,Y: nat] :
% 2.09/1.13 ( ( ord_less_eq_nat @ X6 @ Y )
% 2.09/1.13 => ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y ) ) )
% 2.09/1.13 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_296_ord__eq__le__subst,axiom,
% 2.09/1.13 ! [A2: nat,F: nat > nat,B: nat,C: nat] :
% 2.09/1.13 ( ( A2
% 2.09/1.13 = ( F @ B ) )
% 2.09/1.13 => ( ( ord_less_eq_nat @ B @ C )
% 2.09/1.13 => ( ! [X6: nat,Y: nat] :
% 2.09/1.13 ( ( ord_less_eq_nat @ X6 @ Y )
% 2.09/1.13 => ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y ) ) )
% 2.09/1.13 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_297_order__subst2,axiom,
% 2.09/1.13 ! [A2: nat,B: nat,F: nat > nat,C: nat] :
% 2.09/1.13 ( ( ord_less_eq_nat @ A2 @ B )
% 2.09/1.13 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 2.09/1.13 => ( ! [X6: nat,Y: nat] :
% 2.09/1.13 ( ( ord_less_eq_nat @ X6 @ Y )
% 2.09/1.13 => ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y ) ) )
% 2.09/1.13 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_298_order__subst1,axiom,
% 2.09/1.13 ! [A2: nat,F: nat > nat,B: nat,C: nat] :
% 2.09/1.13 ( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
% 2.09/1.13 => ( ( ord_less_eq_nat @ B @ C )
% 2.09/1.13 => ( ! [X6: nat,Y: nat] :
% 2.09/1.13 ( ( ord_less_eq_nat @ X6 @ Y )
% 2.09/1.13 => ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y ) ) )
% 2.09/1.13 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_299_dual__order_Ostrict__implies__not__eq,axiom,
% 2.09/1.13 ! [B: nat,A2: nat] :
% 2.09/1.13 ( ( ord_less_nat @ B @ A2 )
% 2.09/1.13 => ( A2 != B ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_300_order_Ostrict__implies__not__eq,axiom,
% 2.09/1.13 ! [A2: nat,B: nat] :
% 2.09/1.13 ( ( ord_less_nat @ A2 @ B )
% 2.09/1.13 => ( A2 != B ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_301_not__less__iff__gr__or__eq,axiom,
% 2.09/1.13 ! [X4: nat,Y3: nat] :
% 2.09/1.13 ( ( ~ ( ord_less_nat @ X4 @ Y3 ) )
% 2.09/1.13 = ( ( ord_less_nat @ Y3 @ X4 )
% 2.09/1.13 | ( X4 = Y3 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_302_dual__order_Ostrict__trans,axiom,
% 2.09/1.13 ! [B: nat,A2: nat,C: nat] :
% 2.09/1.13 ( ( ord_less_nat @ B @ A2 )
% 2.09/1.13 => ( ( ord_less_nat @ C @ B )
% 2.09/1.13 => ( ord_less_nat @ C @ A2 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_303_linorder__less__wlog,axiom,
% 2.09/1.13 ! [P: nat > nat > $o,A2: nat,B: nat] :
% 2.09/1.13 ( ! [A3: nat,B6: nat] :
% 2.09/1.13 ( ( ord_less_nat @ A3 @ B6 )
% 2.09/1.13 => ( P @ A3 @ B6 ) )
% 2.09/1.13 => ( ! [A3: nat] : ( P @ A3 @ A3 )
% 2.09/1.13 => ( ! [A3: nat,B6: nat] :
% 2.09/1.13 ( ( P @ B6 @ A3 )
% 2.09/1.13 => ( P @ A3 @ B6 ) )
% 2.09/1.13 => ( P @ A2 @ B ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_304_exists__least__iff,axiom,
% 2.09/1.13 ( ( ^ [P3: nat > $o] :
% 2.09/1.13 ? [X3: nat] : ( P3 @ X3 ) )
% 2.09/1.13 = ( ^ [P2: nat > $o] :
% 2.09/1.13 ? [N2: nat] :
% 2.09/1.13 ( ( P2 @ N2 )
% 2.09/1.13 & ! [M5: nat] :
% 2.09/1.13 ( ( ord_less_nat @ M5 @ N2 )
% 2.09/1.13 => ~ ( P2 @ M5 ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_305_less__imp__not__less,axiom,
% 2.09/1.13 ! [X4: nat,Y3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ X4 @ Y3 )
% 2.09/1.13 => ~ ( ord_less_nat @ Y3 @ X4 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_306_order_Ostrict__trans,axiom,
% 2.09/1.13 ! [A2: nat,B: nat,C: nat] :
% 2.09/1.13 ( ( ord_less_nat @ A2 @ B )
% 2.09/1.13 => ( ( ord_less_nat @ B @ C )
% 2.09/1.13 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_307_dual__order_Oirrefl,axiom,
% 2.09/1.13 ! [A2: nat] :
% 2.09/1.13 ~ ( ord_less_nat @ A2 @ A2 ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_308_linorder__cases,axiom,
% 2.09/1.13 ! [X4: nat,Y3: nat] :
% 2.09/1.13 ( ~ ( ord_less_nat @ X4 @ Y3 )
% 2.09/1.13 => ( ( X4 != Y3 )
% 2.09/1.13 => ( ord_less_nat @ Y3 @ X4 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_309_less__imp__triv,axiom,
% 2.09/1.13 ! [X4: nat,Y3: nat,P: $o] :
% 2.09/1.13 ( ( ord_less_nat @ X4 @ Y3 )
% 2.09/1.13 => ( ( ord_less_nat @ Y3 @ X4 )
% 2.09/1.13 => P ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_310_less__imp__not__eq2,axiom,
% 2.09/1.13 ! [X4: nat,Y3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ X4 @ Y3 )
% 2.09/1.13 => ( Y3 != X4 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_311_antisym__conv3,axiom,
% 2.09/1.13 ! [Y3: nat,X4: nat] :
% 2.09/1.13 ( ~ ( ord_less_nat @ Y3 @ X4 )
% 2.09/1.13 => ( ( ~ ( ord_less_nat @ X4 @ Y3 ) )
% 2.09/1.13 = ( X4 = Y3 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_312_less__induct,axiom,
% 2.09/1.13 ! [P: nat > $o,A2: nat] :
% 2.09/1.13 ( ! [X6: nat] :
% 2.09/1.13 ( ! [Y6: nat] :
% 2.09/1.13 ( ( ord_less_nat @ Y6 @ X6 )
% 2.09/1.13 => ( P @ Y6 ) )
% 2.09/1.13 => ( P @ X6 ) )
% 2.09/1.13 => ( P @ A2 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_313_less__not__sym,axiom,
% 2.09/1.13 ! [X4: nat,Y3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ X4 @ Y3 )
% 2.09/1.13 => ~ ( ord_less_nat @ Y3 @ X4 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_314_less__imp__not__eq,axiom,
% 2.09/1.13 ! [X4: nat,Y3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ X4 @ Y3 )
% 2.09/1.13 => ( X4 != Y3 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_315_dual__order_Oasym,axiom,
% 2.09/1.13 ! [B: nat,A2: nat] :
% 2.09/1.13 ( ( ord_less_nat @ B @ A2 )
% 2.09/1.13 => ~ ( ord_less_nat @ A2 @ B ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_316_ord__less__eq__trans,axiom,
% 2.09/1.13 ! [A2: nat,B: nat,C: nat] :
% 2.09/1.13 ( ( ord_less_nat @ A2 @ B )
% 2.09/1.13 => ( ( B = C )
% 2.09/1.13 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_317_ord__eq__less__trans,axiom,
% 2.09/1.13 ! [A2: nat,B: nat,C: nat] :
% 2.09/1.13 ( ( A2 = B )
% 2.09/1.13 => ( ( ord_less_nat @ B @ C )
% 2.09/1.13 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_318_less__irrefl,axiom,
% 2.09/1.13 ! [X4: nat] :
% 2.09/1.13 ~ ( ord_less_nat @ X4 @ X4 ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_319_less__linear,axiom,
% 2.09/1.13 ! [X4: nat,Y3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ X4 @ Y3 )
% 2.09/1.13 | ( X4 = Y3 )
% 2.09/1.13 | ( ord_less_nat @ Y3 @ X4 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_320_less__trans,axiom,
% 2.09/1.13 ! [X4: nat,Y3: nat,Z3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ X4 @ Y3 )
% 2.09/1.13 => ( ( ord_less_nat @ Y3 @ Z3 )
% 2.09/1.13 => ( ord_less_nat @ X4 @ Z3 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_321_less__asym_H,axiom,
% 2.09/1.13 ! [A2: nat,B: nat] :
% 2.09/1.13 ( ( ord_less_nat @ A2 @ B )
% 2.09/1.13 => ~ ( ord_less_nat @ B @ A2 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_322_less__asym,axiom,
% 2.09/1.13 ! [X4: nat,Y3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ X4 @ Y3 )
% 2.09/1.13 => ~ ( ord_less_nat @ Y3 @ X4 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_323_less__imp__neq,axiom,
% 2.09/1.13 ! [X4: nat,Y3: nat] :
% 2.09/1.13 ( ( ord_less_nat @ X4 @ Y3 )
% 2.09/1.13 => ( X4 != Y3 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_324_order_Oasym,axiom,
% 2.09/1.13 ! [A2: nat,B: nat] :
% 2.09/1.13 ( ( ord_less_nat @ A2 @ B )
% 2.09/1.13 => ~ ( ord_less_nat @ B @ A2 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_325_neq__iff,axiom,
% 2.09/1.13 ! [X4: nat,Y3: nat] :
% 2.09/1.13 ( ( X4 != Y3 )
% 2.09/1.13 = ( ( ord_less_nat @ X4 @ Y3 )
% 2.09/1.13 | ( ord_less_nat @ Y3 @ X4 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_326_neqE,axiom,
% 2.09/1.13 ! [X4: nat,Y3: nat] :
% 2.09/1.13 ( ( X4 != Y3 )
% 2.09/1.13 => ( ~ ( ord_less_nat @ X4 @ Y3 )
% 2.09/1.13 => ( ord_less_nat @ Y3 @ X4 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_327_gt__ex,axiom,
% 2.09/1.13 ! [X4: nat] :
% 2.09/1.13 ? [X_1: nat] : ( ord_less_nat @ X4 @ X_1 ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_328_order__less__subst2,axiom,
% 2.09/1.13 ! [A2: nat,B: nat,F: nat > nat,C: nat] :
% 2.09/1.13 ( ( ord_less_nat @ A2 @ B )
% 2.09/1.13 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 2.09/1.13 => ( ! [X6: nat,Y: nat] :
% 2.09/1.13 ( ( ord_less_nat @ X6 @ Y )
% 2.09/1.13 => ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y ) ) )
% 2.09/1.13 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_329_order__less__subst1,axiom,
% 2.09/1.13 ! [A2: nat,F: nat > nat,B: nat,C: nat] :
% 2.09/1.13 ( ( ord_less_nat @ A2 @ ( F @ B ) )
% 2.09/1.13 => ( ( ord_less_nat @ B @ C )
% 2.09/1.13 => ( ! [X6: nat,Y: nat] :
% 2.09/1.13 ( ( ord_less_nat @ X6 @ Y )
% 2.09/1.13 => ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y ) ) )
% 2.09/1.13 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_330_ord__less__eq__subst,axiom,
% 2.09/1.13 ! [A2: nat,B: nat,F: nat > nat,C: nat] :
% 2.09/1.13 ( ( ord_less_nat @ A2 @ B )
% 2.09/1.13 => ( ( ( F @ B )
% 2.09/1.13 = C )
% 2.09/1.13 => ( ! [X6: nat,Y: nat] :
% 2.09/1.13 ( ( ord_less_nat @ X6 @ Y )
% 2.09/1.13 => ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y ) ) )
% 2.09/1.13 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_331_ord__eq__less__subst,axiom,
% 2.09/1.13 ! [A2: nat,F: nat > nat,B: nat,C: nat] :
% 2.09/1.13 ( ( A2
% 2.09/1.13 = ( F @ B ) )
% 2.09/1.13 => ( ( ord_less_nat @ B @ C )
% 2.09/1.13 => ( ! [X6: nat,Y: nat] :
% 2.09/1.13 ( ( ord_less_nat @ X6 @ Y )
% 2.09/1.13 => ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y ) ) )
% 2.09/1.13 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_332_pinf_I1_J,axiom,
% 2.09/1.13 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 2.09/1.13 ( ? [Z4: nat] :
% 2.09/1.13 ! [X6: nat] :
% 2.09/1.13 ( ( ord_less_nat @ Z4 @ X6 )
% 2.09/1.13 => ( ( P @ X6 )
% 2.09/1.13 = ( P4 @ X6 ) ) )
% 2.09/1.13 => ( ? [Z4: nat] :
% 2.09/1.13 ! [X6: nat] :
% 2.09/1.13 ( ( ord_less_nat @ Z4 @ X6 )
% 2.09/1.13 => ( ( Q @ X6 )
% 2.09/1.13 = ( Q2 @ X6 ) ) )
% 2.09/1.13 => ? [Z2: nat] :
% 2.09/1.13 ! [X: nat] :
% 2.09/1.13 ( ( ord_less_nat @ Z2 @ X )
% 2.09/1.13 => ( ( ( P @ X )
% 2.09/1.13 & ( Q @ X ) )
% 2.09/1.13 = ( ( P4 @ X )
% 2.09/1.13 & ( Q2 @ X ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_333_pinf_I2_J,axiom,
% 2.09/1.13 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 2.09/1.13 ( ? [Z4: nat] :
% 2.09/1.13 ! [X6: nat] :
% 2.09/1.13 ( ( ord_less_nat @ Z4 @ X6 )
% 2.09/1.13 => ( ( P @ X6 )
% 2.09/1.13 = ( P4 @ X6 ) ) )
% 2.09/1.13 => ( ? [Z4: nat] :
% 2.09/1.13 ! [X6: nat] :
% 2.09/1.13 ( ( ord_less_nat @ Z4 @ X6 )
% 2.09/1.13 => ( ( Q @ X6 )
% 2.09/1.13 = ( Q2 @ X6 ) ) )
% 2.09/1.13 => ? [Z2: nat] :
% 2.09/1.13 ! [X: nat] :
% 2.09/1.13 ( ( ord_less_nat @ Z2 @ X )
% 2.09/1.13 => ( ( ( P @ X )
% 2.09/1.13 | ( Q @ X ) )
% 2.09/1.13 = ( ( P4 @ X )
% 2.09/1.13 | ( Q2 @ X ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_334_pinf_I3_J,axiom,
% 2.09/1.13 ! [T: nat] :
% 2.09/1.13 ? [Z2: nat] :
% 2.09/1.13 ! [X: nat] :
% 2.09/1.13 ( ( ord_less_nat @ Z2 @ X )
% 2.09/1.13 => ( X != T ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_335_pinf_I4_J,axiom,
% 2.09/1.13 ! [T: nat] :
% 2.09/1.13 ? [Z2: nat] :
% 2.09/1.13 ! [X: nat] :
% 2.09/1.13 ( ( ord_less_nat @ Z2 @ X )
% 2.09/1.13 => ( X != T ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_336_pinf_I5_J,axiom,
% 2.09/1.13 ! [T: nat] :
% 2.09/1.13 ? [Z2: nat] :
% 2.09/1.13 ! [X: nat] :
% 2.09/1.13 ( ( ord_less_nat @ Z2 @ X )
% 2.09/1.13 => ~ ( ord_less_nat @ X @ T ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_337_pinf_I7_J,axiom,
% 2.09/1.13 ! [T: nat] :
% 2.09/1.13 ? [Z2: nat] :
% 2.09/1.13 ! [X: nat] :
% 2.09/1.13 ( ( ord_less_nat @ Z2 @ X )
% 2.09/1.13 => ( ord_less_nat @ T @ X ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_338_minf_I1_J,axiom,
% 2.09/1.13 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 2.09/1.13 ( ? [Z4: nat] :
% 2.09/1.13 ! [X6: nat] :
% 2.09/1.13 ( ( ord_less_nat @ X6 @ Z4 )
% 2.09/1.13 => ( ( P @ X6 )
% 2.09/1.13 = ( P4 @ X6 ) ) )
% 2.09/1.13 => ( ? [Z4: nat] :
% 2.09/1.13 ! [X6: nat] :
% 2.09/1.13 ( ( ord_less_nat @ X6 @ Z4 )
% 2.09/1.13 => ( ( Q @ X6 )
% 2.09/1.13 = ( Q2 @ X6 ) ) )
% 2.09/1.13 => ? [Z2: nat] :
% 2.09/1.13 ! [X: nat] :
% 2.09/1.13 ( ( ord_less_nat @ X @ Z2 )
% 2.09/1.13 => ( ( ( P @ X )
% 2.09/1.13 & ( Q @ X ) )
% 2.09/1.13 = ( ( P4 @ X )
% 2.09/1.13 & ( Q2 @ X ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_339_minf_I2_J,axiom,
% 2.09/1.13 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 2.09/1.13 ( ? [Z4: nat] :
% 2.09/1.13 ! [X6: nat] :
% 2.09/1.13 ( ( ord_less_nat @ X6 @ Z4 )
% 2.09/1.13 => ( ( P @ X6 )
% 2.09/1.13 = ( P4 @ X6 ) ) )
% 2.09/1.13 => ( ? [Z4: nat] :
% 2.09/1.13 ! [X6: nat] :
% 2.09/1.13 ( ( ord_less_nat @ X6 @ Z4 )
% 2.09/1.13 => ( ( Q @ X6 )
% 2.09/1.13 = ( Q2 @ X6 ) ) )
% 2.09/1.13 => ? [Z2: nat] :
% 2.09/1.13 ! [X: nat] :
% 2.09/1.13 ( ( ord_less_nat @ X @ Z2 )
% 2.09/1.13 => ( ( ( P @ X )
% 2.09/1.13 | ( Q @ X ) )
% 2.09/1.13 = ( ( P4 @ X )
% 2.09/1.13 | ( Q2 @ X ) ) ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_340_minf_I3_J,axiom,
% 2.09/1.13 ! [T: nat] :
% 2.09/1.13 ? [Z2: nat] :
% 2.09/1.13 ! [X: nat] :
% 2.09/1.13 ( ( ord_less_nat @ X @ Z2 )
% 2.09/1.13 => ( X != T ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_341_minf_I4_J,axiom,
% 2.09/1.13 ! [T: nat] :
% 2.09/1.13 ? [Z2: nat] :
% 2.09/1.13 ! [X: nat] :
% 2.09/1.13 ( ( ord_less_nat @ X @ Z2 )
% 2.09/1.13 => ( X != T ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_342_minf_I5_J,axiom,
% 2.09/1.13 ! [T: nat] :
% 2.09/1.13 ? [Z2: nat] :
% 2.09/1.13 ! [X: nat] :
% 2.09/1.13 ( ( ord_less_nat @ X @ Z2 )
% 2.09/1.13 => ( ord_less_nat @ X @ T ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_343_minf_I7_J,axiom,
% 2.09/1.13 ! [T: nat] :
% 2.09/1.13 ? [Z2: nat] :
% 2.09/1.13 ! [X: nat] :
% 2.09/1.13 ( ( ord_less_nat @ X @ Z2 )
% 2.09/1.13 => ~ ( ord_less_nat @ T @ X ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_344_order_Onot__eq__order__implies__strict,axiom,
% 2.09/1.13 ! [A2: nat,B: nat] :
% 2.09/1.13 ( ( A2 != B )
% 2.09/1.13 => ( ( ord_less_eq_nat @ A2 @ B )
% 2.09/1.13 => ( ord_less_nat @ A2 @ B ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_345_dual__order_Ostrict__implies__order,axiom,
% 2.09/1.13 ! [B: nat,A2: nat] :
% 2.09/1.13 ( ( ord_less_nat @ B @ A2 )
% 2.09/1.13 => ( ord_less_eq_nat @ B @ A2 ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_346_dual__order_Ostrict__iff__order,axiom,
% 2.09/1.13 ( ord_less_nat
% 2.09/1.13 = ( ^ [B5: nat,A7: nat] :
% 2.09/1.13 ( ( ord_less_eq_nat @ B5 @ A7 )
% 2.09/1.13 & ( A7 != B5 ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_347_dual__order_Oorder__iff__strict,axiom,
% 2.09/1.13 ( ord_less_eq_nat
% 2.09/1.13 = ( ^ [B5: nat,A7: nat] :
% 2.09/1.13 ( ( ord_less_nat @ B5 @ A7 )
% 2.09/1.13 | ( A7 = B5 ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_348_order_Ostrict__implies__order,axiom,
% 2.09/1.13 ! [A2: nat,B: nat] :
% 2.09/1.13 ( ( ord_less_nat @ A2 @ B )
% 2.09/1.13 => ( ord_less_eq_nat @ A2 @ B ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_349_dual__order_Ostrict__trans2,axiom,
% 2.09/1.13 ! [B: nat,A2: nat,C: nat] :
% 2.09/1.13 ( ( ord_less_nat @ B @ A2 )
% 2.09/1.13 => ( ( ord_less_eq_nat @ C @ B )
% 2.09/1.13 => ( ord_less_nat @ C @ A2 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_350_dual__order_Ostrict__trans1,axiom,
% 2.09/1.13 ! [B: nat,A2: nat,C: nat] :
% 2.09/1.13 ( ( ord_less_eq_nat @ B @ A2 )
% 2.09/1.13 => ( ( ord_less_nat @ C @ B )
% 2.09/1.13 => ( ord_less_nat @ C @ A2 ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_351_order_Ostrict__iff__order,axiom,
% 2.09/1.13 ( ord_less_nat
% 2.09/1.13 = ( ^ [A7: nat,B5: nat] :
% 2.09/1.13 ( ( ord_less_eq_nat @ A7 @ B5 )
% 2.09/1.13 & ( A7 != B5 ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(fact_352_order_Oorder__iff__strict,axiom,
% 2.09/1.13 ( ord_less_eq_nat
% 2.09/1.13 = ( ^ [A7: nat,B5: nat] :
% 2.09/1.13 ( ( ord_less_nat @ A7 @ B5 )
% 2.09/1.13 | ( A7 = B5 ) ) ) ) ).
% 2.09/1.13
% 2.09/1.13 thf(conj_0,conjecture,
% 2.09/1.13 ( a3
% 2.09/1.13 = ( groups1792256535list_a @ qs2 ) ) ).
% 2.09/1.13 ------------------------
% 2.09/1.15 % (9218)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_9014 for (2999ds/183Mi)
% 2.09/1.15 % (9219)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_9014 for (2999ds/4Mi)
% 2.09/1.15 % (9220)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_9014 for (2999ds/27Mi)
% 2.09/1.15 % (9221)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_9014 for (2999ds/2Mi)
% 2.09/1.15 % (9222)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_9014 for (2999ds/2Mi)
% 2.09/1.15 % (9223)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_9014 for (2999ds/275Mi)
% 2.09/1.15 % (9224)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_9014 for (2999ds/18Mi)
% 2.09/1.16 % (9221)Instruction limit reached!
% 2.09/1.16 % (9221)------------------------------
% 2.09/1.16 % (9221)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.16 % (9221)Termination reason: Unknown
% 2.09/1.16 % (9221)Termination phase: shuffling
% 2.09/1.16
% 2.09/1.16 % (9221)Memory used [KB]: 1535
% 2.09/1.16 % (9221)Time elapsed: 0.002 s
% 2.09/1.16 % (9221)Instructions burned: 2 (million)
% 2.09/1.16 % (9221)------------------------------
% 2.09/1.16 % (9221)------------------------------
% 2.09/1.16 % (9222)Instruction limit reached!
% 2.09/1.16 % (9222)------------------------------
% 2.09/1.16 % (9222)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.16 % (9222)Termination reason: Unknown
% 2.09/1.16 % (9222)Termination phase: shuffling
% 2.09/1.16
% 2.09/1.16 % (9222)Memory used [KB]: 1535
% 2.09/1.16 % (9222)Time elapsed: 0.002 s
% 2.09/1.16 % (9222)Instructions burned: 2 (million)
% 2.09/1.16 % (9222)------------------------------
% 2.09/1.16 % (9222)------------------------------
% 2.09/1.16 % (9219)Instruction limit reached!
% 2.09/1.16 % (9219)------------------------------
% 2.09/1.16 % (9219)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.16 % (9219)Termination reason: Unknown
% 2.09/1.16 % (9219)Termination phase: shuffling
% 2.09/1.16
% 2.09/1.16 % (9219)Memory used [KB]: 1663
% 2.09/1.16 % (9219)Time elapsed: 0.003 s
% 2.09/1.16 % (9219)Instructions burned: 6 (million)
% 2.09/1.16 % (9219)------------------------------
% 2.09/1.16 % (9219)------------------------------
% 2.09/1.16 % (9224)Instruction limit reached!
% 2.09/1.16 % (9224)------------------------------
% 2.09/1.16 % (9224)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.16 % (9224)Termination reason: Unknown
% 2.09/1.16 % (9224)Termination phase: shuffling
% 2.09/1.16
% 2.09/1.16 % (9224)Memory used [KB]: 1791
% 2.09/1.16 % (9224)Time elapsed: 0.007 s
% 2.09/1.16 % (9224)Instructions burned: 18 (million)
% 2.09/1.16 % (9224)------------------------------
% 2.09/1.16 % (9224)------------------------------
% 2.09/1.16 % (9220)Instruction limit reached!
% 2.09/1.16 % (9220)------------------------------
% 2.09/1.16 % (9220)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.16 % (9220)Termination reason: Unknown
% 2.09/1.16 % (9220)Termination phase: shuffling
% 2.09/1.16
% 2.09/1.16 % (9220)Memory used [KB]: 2046
% 2.09/1.16 % (9220)Time elapsed: 0.010 s
% 2.09/1.16 % (9220)Instructions burned: 28 (million)
% 2.09/1.16 % (9220)------------------------------
% 2.09/1.16 % (9220)------------------------------
% 2.09/1.16 % (9225)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_9014 for (2999ds/3Mi)
% 2.09/1.16 % (9226)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_9014 for (2999ds/37Mi)
% 2.09/1.17 % (9227)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_9014 for (2999ds/15Mi)
% 2.09/1.17 % (9225)Instruction limit reached!
% 2.09/1.17 % (9225)------------------------------
% 2.09/1.17 % (9225)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.17 % (9225)Termination reason: Unknown
% 2.09/1.17 % (9225)Termination phase: shuffling
% 2.09/1.17
% 2.09/1.17 % (9225)Memory used [KB]: 1535
% 2.09/1.17 % (9225)Time elapsed: 0.002 s
% 2.09/1.17 % (9225)Instructions burned: 3 (million)
% 2.09/1.17 % (9225)------------------------------
% 2.09/1.17 % (9225)------------------------------
% 2.09/1.17 % (9228)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_9014 for (2999ds/3Mi)
% 2.09/1.17 % (9227)Instruction limit reached!
% 2.09/1.17 % (9227)------------------------------
% 2.09/1.17 % (9227)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.17 % (9227)Termination reason: Unknown
% 2.09/1.17 % (9227)Termination phase: shuffling
% 2.09/1.17
% 2.09/1.17 % (9227)Memory used [KB]: 1791
% 2.09/1.17 % (9227)Time elapsed: 0.006 s
% 2.09/1.17 % (9227)Instructions burned: 15 (million)
% 2.09/1.17 % (9227)------------------------------
% 2.09/1.17 % (9227)------------------------------
% 2.09/1.17 % (9228)Instruction limit reached!
% 2.09/1.17 % (9228)------------------------------
% 2.09/1.17 % (9228)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.17 % (9228)Termination reason: Unknown
% 2.09/1.17 % (9228)Termination phase: shuffling
% 2.09/1.17
% 2.09/1.17 % (9228)Memory used [KB]: 1663
% 2.09/1.17 % (9228)Time elapsed: 0.003 s
% 2.09/1.17 % (9228)Instructions burned: 5 (million)
% 2.09/1.17 % (9228)------------------------------
% 2.09/1.17 % (9228)------------------------------
% 2.09/1.17 % (9229)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_9014 for (2999ds/1041Mi)
% 2.09/1.18 % (9230)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_9014 for (2999ds/7Mi)
% 2.09/1.18 % (9226)Instruction limit reached!
% 2.09/1.18 % (9226)------------------------------
% 2.09/1.18 % (9226)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.18 % (9226)Termination reason: Unknown
% 2.09/1.18 % (9226)Termination phase: shuffling
% 2.09/1.18
% 2.09/1.18 % (9226)Memory used [KB]: 2302
% 2.09/1.18 % (9226)Time elapsed: 0.012 s
% 2.09/1.18 % (9226)Instructions burned: 40 (million)
% 2.09/1.18 % (9226)------------------------------
% 2.09/1.18 % (9226)------------------------------
% 2.09/1.18 % (9230)Instruction limit reached!
% 2.09/1.18 % (9230)------------------------------
% 2.09/1.18 % (9230)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.18 % (9230)Termination reason: Unknown
% 2.09/1.18 % (9230)Termination phase: shuffling
% 2.09/1.18
% 2.09/1.18 % (9230)Memory used [KB]: 1663
% 2.09/1.18 % (9230)Time elapsed: 0.004 s
% 2.09/1.18 % (9230)Instructions burned: 8 (million)
% 2.09/1.18 % (9230)------------------------------
% 2.09/1.18 % (9230)------------------------------
% 2.09/1.18 % (9231)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_9014 for (2999ds/16Mi)
% 2.09/1.18 % (9223)Refutation not found, incomplete strategy
% 2.09/1.18 % (9223)------------------------------
% 2.09/1.18 % (9223)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.18 % (9223)Termination reason: Refutation not found, incomplete strategy
% 2.09/1.18
% 2.09/1.18
% 2.09/1.18 % (9223)Memory used [KB]: 7036
% 2.09/1.18 % (9223)Time elapsed: 0.027 s
% 2.09/1.18 % (9223)Instructions burned: 86 (million)
% 2.09/1.18 % (9223)------------------------------
% 2.09/1.18 % (9223)------------------------------
% 2.09/1.18 % (9232)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_9014 for (2999ds/3Mi)
% 2.09/1.18 % (9232)Instruction limit reached!
% 2.09/1.18 % (9232)------------------------------
% 2.09/1.18 % (9232)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.18 % (9232)Termination reason: Unknown
% 2.09/1.18 % (9232)Termination phase: shuffling
% 2.09/1.18
% 2.09/1.18 % (9232)Memory used [KB]: 1663
% 2.09/1.18 % (9232)Time elapsed: 0.003 s
% 2.09/1.18 % (9232)Instructions burned: 6 (million)
% 2.09/1.18 % (9232)------------------------------
% 2.09/1.18 % (9232)------------------------------
% 2.09/1.19 % (9233)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_9014 for (2999ds/3Mi)
% 2.09/1.19 % (9231)Instruction limit reached!
% 2.09/1.19 % (9231)------------------------------
% 2.09/1.19 % (9231)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.19 % (9231)Termination reason: Unknown
% 2.09/1.19 % (9231)Termination phase: shuffling
% 2.09/1.19
% 2.09/1.19 % (9231)Memory used [KB]: 1791
% 2.09/1.19 % (9231)Time elapsed: 0.007 s
% 2.09/1.19 % (9231)Instructions burned: 18 (million)
% 2.09/1.19 % (9231)------------------------------
% 2.09/1.19 % (9231)------------------------------
% 2.09/1.19 % (9233)Instruction limit reached!
% 2.09/1.19 % (9233)------------------------------
% 2.09/1.19 % (9233)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.19 % (9233)Termination reason: Unknown
% 2.09/1.19 % (9233)Termination phase: shuffling
% 2.09/1.19
% 2.09/1.19 % (9233)Memory used [KB]: 1535
% 2.09/1.19 % (9233)Time elapsed: 0.002 s
% 2.09/1.19 % (9233)Instructions burned: 3 (million)
% 2.09/1.19 % (9233)------------------------------
% 2.09/1.19 % (9233)------------------------------
% 2.09/1.19 % (9234)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_9014 for (2999ds/7Mi)
% 2.09/1.19 % (9234)Instruction limit reached!
% 2.09/1.19 % (9234)------------------------------
% 2.09/1.19 % (9234)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.19 % (9234)Termination reason: Unknown
% 2.09/1.19 % (9234)Termination phase: shuffling
% 2.09/1.19
% 2.09/1.19 % (9234)Memory used [KB]: 1663
% 2.09/1.19 % (9234)Time elapsed: 0.004 s
% 2.09/1.19 % (9234)Instructions burned: 9 (million)
% 2.09/1.19 % (9234)------------------------------
% 2.09/1.19 % (9234)------------------------------
% 2.09/1.19 % (9235)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_9014 for (2999ds/3Mi)
% 2.09/1.19 % (9235)Instruction limit reached!
% 2.09/1.19 % (9235)------------------------------
% 2.09/1.19 % (9235)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.19 % (9235)Termination reason: Unknown
% 2.09/1.19 % (9235)Termination phase: shuffling
% 2.09/1.19
% 2.09/1.19 % (9235)Memory used [KB]: 1535
% 2.09/1.19 % (9235)Time elapsed: 0.002 s
% 2.09/1.19 % (9235)Instructions burned: 3 (million)
% 2.09/1.19 % (9235)------------------------------
% 2.09/1.19 % (9235)------------------------------
% 2.09/1.19 % (9236)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_9014 for (2999ds/4Mi)
% 2.09/1.19 % (9236)Instruction limit reached!
% 2.09/1.19 % (9236)------------------------------
% 2.09/1.19 % (9236)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.19 % (9236)Termination reason: Unknown
% 2.09/1.19 % (9236)Termination phase: shuffling
% 2.09/1.19
% 2.09/1.19 % (9236)Memory used [KB]: 1663
% 2.09/1.19 % (9236)Time elapsed: 0.003 s
% 2.09/1.19 % (9236)Instructions burned: 6 (million)
% 2.09/1.19 % (9236)------------------------------
% 2.09/1.19 % (9236)------------------------------
% 2.09/1.20 % (9237)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_9014 for (2999ds/18Mi)
% 2.09/1.20 % (9238)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_9014 for (2999ds/710Mi)
% 2.13/1.20 % (9239)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_9014 for (2999ds/6Mi)
% 2.13/1.20 % (9240)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_9014 for (2999ds/902Mi)
% 2.13/1.20 % (9237)Instruction limit reached!
% 2.13/1.20 % (9237)------------------------------
% 2.13/1.20 % (9237)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.13/1.20 % (9237)Termination reason: Unknown
% 2.13/1.20 % (9237)Termination phase: shuffling
% 2.13/1.20
% 2.13/1.20 % (9237)Memory used [KB]: 1791
% 2.13/1.20 % (9237)Time elapsed: 0.007 s
% 2.13/1.20 % (9237)Instructions burned: 19 (million)
% 2.13/1.20 % (9237)------------------------------
% 2.13/1.20 % (9237)------------------------------
% 2.13/1.20 % (9239)Instruction limit reached!
% 2.13/1.20 % (9239)------------------------------
% 2.13/1.20 % (9239)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.13/1.20 % (9239)Termination reason: Unknown
% 2.13/1.20 % (9239)Termination phase: shuffling
% 2.13/1.20
% 2.13/1.20 % (9239)Memory used [KB]: 1663
% 2.13/1.20 % (9239)Time elapsed: 0.003 s
% 2.13/1.20 % (9239)Instructions burned: 6 (million)
% 2.13/1.20 % (9239)------------------------------
% 2.13/1.20 % (9239)------------------------------
% 2.13/1.20 % (9241)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_9014 for (2999ds/21Mi)
% 2.13/1.21 % (9218)Instruction limit reached!
% 2.13/1.21 % (9218)------------------------------
% 2.13/1.21 % (9218)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.13/1.21 % (9218)Termination reason: Unknown
% 2.13/1.21 % (9218)Termination phase: Saturation
% 2.13/1.21
% 2.13/1.21 % (9218)Memory used [KB]: 7419
% 2.13/1.21 % (9218)Time elapsed: 0.052 s
% 2.13/1.21 % (9218)Instructions burned: 186 (million)
% 2.13/1.21 % (9218)------------------------------
% 2.13/1.21 % (9218)------------------------------
% 2.13/1.21 % (9242)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_9014 for (2999ds/5Mi)
% 2.13/1.21 % (9241)Instruction limit reached!
% 2.13/1.21 % (9241)------------------------------
% 2.13/1.21 % (9241)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.13/1.21 % (9241)Termination reason: Unknown
% 2.13/1.21 % (9241)Termination phase: shuffling
% 2.13/1.21
% 2.13/1.21 % (9241)Memory used [KB]: 1918
% 2.13/1.21 % (9241)Time elapsed: 0.008 s
% 2.13/1.21 % (9241)Instructions burned: 22 (million)
% 2.13/1.21 % (9241)------------------------------
% 2.13/1.21 % (9241)------------------------------
% 2.13/1.21 % (9243)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_9014 for (2999ds/6Mi)
% 2.13/1.21 % (9242)Instruction limit reached!
% 2.13/1.21 % (9242)------------------------------
% 2.13/1.21 % (9242)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.13/1.21 % (9242)Termination reason: Unknown
% 2.13/1.21 % (9242)Termination phase: shuffling
% 2.13/1.21
% 2.13/1.21 % (9242)Memory used [KB]: 1663
% 2.13/1.21 % (9242)Time elapsed: 0.003 s
% 2.13/1.21 % (9242)Instructions burned: 6 (million)
% 2.13/1.21 % (9242)------------------------------
% 2.13/1.21 % (9242)------------------------------
% 2.13/1.21 % (9243)Instruction limit reached!
% 2.13/1.21 % (9243)------------------------------
% 2.13/1.21 % (9243)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.13/1.21 % (9243)Termination reason: Unknown
% 2.13/1.21 % (9243)Termination phase: shuffling
% 2.13/1.21
% 2.13/1.21 % (9243)Memory used [KB]: 1663
% 2.13/1.21 % (9243)Time elapsed: 0.003 s
% 2.13/1.21 % (9243)Instructions burned: 6 (million)
% 2.13/1.21 % (9243)------------------------------
% 2.13/1.21 % (9243)------------------------------
% 2.13/1.21 % (9238)Refutation not found, incomplete strategy
% 2.13/1.21 % (9238)------------------------------
% 2.13/1.21 % (9238)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.13/1.21 % (9238)Termination reason: Refutation not found, incomplete strategy
% 2.13/1.21
% 2.13/1.21
% 2.13/1.21 % (9238)Memory used [KB]: 6652
% 2.13/1.21 % (9238)Time elapsed: 0.018 s
% 2.13/1.21 % (9238)Instructions burned: 56 (million)
% 2.13/1.21 % (9238)------------------------------
% 2.13/1.21 % (9238)------------------------------
% 2.13/1.21 % (9244)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_9014 for (2999ds/377Mi)
% 2.13/1.22 % (9245)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_9014 for (2999ds/779Mi)
% 2.13/1.22 % (9246)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_9014 for (2999ds/19Mi)
% 2.13/1.22 % (9247)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_9014 for (2999ds/879Mi)
% 2.13/1.22 % (9248)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_9014 for (2999ds/17Mi)
% 2.13/1.23 % (9246)Instruction limit reached!
% 2.13/1.23 % (9246)------------------------------
% 2.13/1.23 % (9246)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.13/1.23 % (9246)Termination reason: Unknown
% 2.13/1.23 % (9246)Termination phase: shuffling
% 2.13/1.23
% 2.13/1.23 % (9246)Memory used [KB]: 1918
% 2.13/1.23 % (9246)Time elapsed: 0.007 s
% 2.13/1.23 % (9246)Instructions burned: 19 (million)
% 2.13/1.23 % (9246)------------------------------
% 2.13/1.23 % (9246)------------------------------
% 2.13/1.23 % (9248)Instruction limit reached!
% 2.13/1.23 % (9248)------------------------------
% 2.13/1.23 % (9248)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.13/1.23 % (9248)Termination reason: Unknown
% 2.13/1.23 % (9248)Termination phase: shuffling
% 2.13/1.23
% 2.13/1.23 % (9248)Memory used [KB]: 1791
% 2.13/1.23 % (9248)Time elapsed: 0.007 s
% 2.13/1.23 % (9248)Instructions burned: 17 (million)
% 2.13/1.23 % (9248)------------------------------
% 2.13/1.23 % (9248)------------------------------
% 2.13/1.24 % (9249)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_9014 for (2998ds/3Mi)
% 2.13/1.24 % (9249)Instruction limit reached!
% 2.13/1.24 % (9249)------------------------------
% 2.13/1.24 % (9249)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.13/1.24 % (9249)Termination reason: Unknown
% 2.13/1.24 % (9249)Termination phase: shuffling
% 2.13/1.24
% 2.13/1.24 % (9249)Memory used [KB]: 1535
% 2.13/1.24 % (9249)Time elapsed: 0.002 s
% 2.13/1.24 % (9249)Instructions burned: 3 (million)
% 2.13/1.24 % (9249)------------------------------
% 2.13/1.24 % (9249)------------------------------
% 2.13/1.24 % (9250)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_9014 for (2998ds/30Mi)
% 2.13/1.25 % (9247)First to succeed.
% 2.13/1.25 % (9251)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_9014 for (2998ds/127Mi)
% 2.13/1.25 % (9247)Refutation found. Thanks to Tanya!
% 2.13/1.25 % SZS status Theorem for DTF2THF_9014
% 2.13/1.25 % SZS output start Proof for DTF2THF_9014
% 2.13/1.25 thf(type_def_5, type, list_a: $tType).
% 2.13/1.25 thf(type_def_7, type, set_list_a: $tType).
% 2.13/1.25 thf(type_def_8, type, a: $tType).
% 2.13/1.25 thf(type_def_9, type, set_a: $tType).
% 2.13/1.25 thf(type_def_10, type, nat: $tType).
% 2.13/1.25 thf(type_def_11, type, set_nat: $tType).
% 2.13/1.25 thf(type_def_12, type, list_multiset_list_a: $tType).
% 2.13/1.25 thf(type_def_13, type, multiset_list_a: $tType).
% 2.13/1.25 thf(type_def_14, type, list_multiset_a: $tType).
% 2.13/1.25 thf(type_def_15, type, multiset_a: $tType).
% 2.13/1.25 thf(type_def_16, type, list_nat: $tType).
% 2.13/1.25 thf(type_def_17, type, set_multiset_a: $tType).
% 2.13/1.25 thf(type_def_18, type, list_set_multiset_a: $tType).
% 2.13/1.25 thf(type_def_19, type, list_set_nat: $tType).
% 2.13/1.25 thf(type_def_20, type, list_set_a: $tType).
% 2.13/1.25 thf(type_def_21, type, list_set_list_list_a: $tType).
% 2.13/1.25 thf(type_def_22, type, set_list_list_list_a: $tType).
% 2.13/1.25 thf(type_def_23, type, list_set_list_set_a: $tType).
% 2.13/1.25 thf(type_def_24, type, set_list_list_set_a: $tType).
% 2.13/1.25 thf(type_def_25, type, list_set_list_a: $tType).
% 2.13/1.25 thf(type_def_26, type, set_list_list_a: $tType).
% 2.13/1.25 thf(type_def_27, type, set_list_multiset_a: $tType).
% 2.13/1.25 thf(type_def_28, type, set_list_nat: $tType).
% 2.13/1.25 thf(type_def_29, type, list_set_set_list_a: $tType).
% 2.13/1.25 thf(type_def_30, type, set_list_set_list_a: $tType).
% 2.13/1.25 thf(type_def_31, type, list_set_set_set_a: $tType).
% 2.13/1.25 thf(type_def_32, type, set_list_set_set_a: $tType).
% 2.13/1.25 thf(type_def_33, type, list_set_set_a: $tType).
% 2.13/1.25 thf(type_def_34, type, set_list_set_a: $tType).
% 2.13/1.25 thf(type_def_35, type, list_list_list_a: $tType).
% 2.13/1.25 thf(type_def_36, type, list_list_a: $tType).
% 2.13/1.25 thf(type_def_37, type, list_list_set_a: $tType).
% 2.13/1.25 thf(type_def_38, type, set_set_list_a: $tType).
% 2.13/1.25 thf(type_def_39, type, set_set_set_a: $tType).
% 2.13/1.25 thf(type_def_40, type, set_set_a: $tType).
% 2.13/1.25 thf(type_def_41, type, multis971982480list_a: $tType).
% 2.13/1.25 thf(type_def_42, type, multiset_multiset_a: $tType).
% 2.13/1.25 thf(type_def_43, type, multiset_nat: $tType).
% 2.13/1.25 thf(type_def_44, type, multiset_set_list_a: $tType).
% 2.13/1.25 thf(type_def_45, type, multiset_set_set_a: $tType).
% 2.13/1.25 thf(type_def_46, type, multiset_set_a: $tType).
% 2.13/1.25 thf(type_def_47, type, set_multiset_list_a: $tType).
% 2.13/1.25 thf(type_def_48, type, set_set_nat: $tType).
% 2.13/1.25 thf(func_def_0, type, set_list_list_list_a: $tType).
% 2.13/1.25 thf(func_def_1, type, multis971982480list_a: $tType).
% 2.13/1.25 thf(func_def_2, type, list_set_list_list_a: $tType).
% 2.13/1.25 thf(func_def_3, type, set_list_set_list_a: $tType).
% 2.13/1.25 thf(func_def_4, type, set_list_list_set_a: $tType).
% 2.13/1.25 thf(func_def_5, type, list_set_set_list_a: $tType).
% 2.13/1.25 thf(func_def_6, type, list_set_list_set_a: $tType).
% 2.13/1.25 thf(func_def_7, type, set_list_set_set_a: $tType).
% 2.13/1.25 thf(func_def_8, type, list_set_set_set_a: $tType).
% 2.13/1.25 thf(func_def_9, type, list_multiset_list_a: $tType).
% 2.13/1.25 thf(func_def_10, type, set_multiset_list_a: $tType).
% 2.13/1.25 thf(func_def_11, type, set_list_multiset_a: $tType).
% 2.13/1.25 thf(func_def_12, type, multiset_set_list_a: $tType).
% 2.13/1.25 thf(func_def_13, type, list_set_multiset_a: $tType).
% 2.13/1.25 thf(func_def_14, type, multiset_set_set_a: $tType).
% 2.13/1.25 thf(func_def_15, type, list_list_list_a: $tType).
% 2.13/1.25 thf(func_def_16, type, set_list_list_a: $tType).
% 2.13/1.25 thf(func_def_17, type, multiset_multiset_a: $tType).
% 2.13/1.25 thf(func_def_18, type, list_set_list_a: $tType).
% 2.13/1.25 thf(func_def_19, type, list_list_set_a: $tType).
% 2.13/1.25 thf(func_def_20, type, set_set_list_a: $tType).
% 2.13/1.25 thf(func_def_21, type, set_list_set_a: $tType).
% 2.13/1.25 thf(func_def_22, type, list_set_set_a: $tType).
% 2.13/1.25 thf(func_def_23, type, set_set_set_a: $tType).
% 2.13/1.25 thf(func_def_24, type, multiset_list_a: $tType).
% 2.13/1.25 thf(func_def_25, type, list_multiset_a: $tType).
% 2.13/1.25 thf(func_def_26, type, set_multiset_a: $tType).
% 2.13/1.25 thf(func_def_27, type, multiset_set_a: $tType).
% 2.13/1.25 thf(func_def_28, type, set_list_nat: $tType).
% 2.13/1.25 thf(func_def_29, type, list_set_nat: $tType).
% 2.13/1.25 thf(func_def_30, type, set_set_nat: $tType).
% 2.13/1.25 thf(func_def_31, type, list_list_a: $tType).
% 2.13/1.25 thf(func_def_32, type, set_list_a: $tType).
% 2.13/1.25 thf(func_def_33, type, list_set_a: $tType).
% 2.13/1.25 thf(func_def_34, type, multiset_nat: $tType).
% 2.13/1.25 thf(func_def_35, type, set_set_a: $tType).
% 2.13/1.25 thf(func_def_36, type, multiset_a: $tType).
% 2.13/1.25 thf(func_def_37, type, list_nat: $tType).
% 2.13/1.25 thf(func_def_38, type, set_nat: $tType).
% 2.13/1.25 thf(func_def_39, type, list_a: $tType).
% 2.13/1.25 thf(func_def_40, type, set_a: $tType).
% 2.13/1.25 thf(func_def_41, type, nat: $tType).
% 2.13/1.25 thf(func_def_42, type, a: $tType).
% 2.13/1.25 thf(func_def_43, type, bij_be94573046list_a: (list_a > list_a) > set_list_a > set_list_a > $o).
% 2.13/1.25 thf(func_def_44, type, bij_betw_list_a_a: (list_a > a) > set_list_a > set_a > $o).
% 2.13/1.25 thf(func_def_45, type, bij_betw_nat_nat: (nat > nat) > set_nat > set_nat > $o).
% 2.13/1.25 thf(func_def_46, type, bij_betw_a_list_a: (a > list_a) > set_a > set_list_a > $o).
% 2.13/1.25 thf(func_def_47, type, bij_betw_a_a: (a > a) > set_a > set_a > $o).
% 2.13/1.25 thf(func_def_48, type, inj_on_list_a_list_a: (list_a > list_a) > set_list_a > $o).
% 2.13/1.25 thf(func_def_49, type, inj_on_list_a_a: (list_a > a) > set_list_a > $o).
% 2.13/1.25 thf(func_def_50, type, inj_on_nat_nat: (nat > nat) > set_nat > $o).
% 2.13/1.25 thf(func_def_51, type, inj_on_a_list_a: (a > list_a) > set_a > $o).
% 2.13/1.25 thf(func_def_52, type, inj_on_a_a: (a > a) > set_a > $o).
% 2.13/1.25 thf(func_def_53, type, groups1271887971list_a: list_multiset_list_a > multiset_list_a).
% 2.13/1.25 thf(func_def_54, type, groups1592617181iset_a: list_multiset_a > multiset_a).
% 2.13/1.25 thf(func_def_55, type, groups921905271st_nat: list_nat > nat).
% 2.13/1.25 thf(func_def_56, type, groups1792256535list_a: list_a > a).
% 2.13/1.25 thf(func_def_57, type, hilber921249868list_a: set_list_a > (list_a > list_a) > list_a > list_a).
% 2.13/1.25 thf(func_def_58, type, hilber2125729734st_a_a: set_list_a > (list_a > a) > a > list_a).
% 2.13/1.25 thf(func_def_59, type, hilber815131374at_nat: set_nat > (nat > nat) > nat > nat).
% 2.13/1.25 thf(func_def_60, type, hilber165648082list_a: set_a > (a > list_a) > list_a > a).
% 2.13/1.25 thf(func_def_61, type, hilbert_inv_into_a_a: set_a > (a > a) > a > a).
% 2.13/1.25 thf(func_def_62, type, hilber2024005914iset_a: set_multiset_a > list_set_multiset_a > $o).
% 2.13/1.25 thf(func_def_63, type, hilber1014902394mp_nat: set_nat > list_set_nat > $o).
% 2.13/1.25 thf(func_def_64, type, hilber2037636820comp_a: set_a > list_set_a > $o).
% 2.13/1.25 thf(func_def_65, type, list_ex_set_a: (set_a > $o) > list_set_a > $o).
% 2.13/1.25 thf(func_def_66, type, list_ex_a: (a > $o) > list_a > $o).
% 2.13/1.25 thf(func_def_67, type, list_update_set_a: list_set_a > nat > set_a > list_set_a).
% 2.13/1.25 thf(func_def_68, type, list_update_a: list_a > nat > a > list_a).
% 2.13/1.25 thf(func_def_69, type, listset_list_list_a: list_set_list_list_a > set_list_list_list_a).
% 2.13/1.25 thf(func_def_70, type, listset_list_set_a: list_set_list_set_a > set_list_list_set_a).
% 2.13/1.25 thf(func_def_71, type, listset_list_a: list_set_list_a > set_list_list_a).
% 2.13/1.25 thf(func_def_72, type, listset_multiset_a: list_set_multiset_a > set_list_multiset_a).
% 2.13/1.25 thf(func_def_73, type, listset_nat: list_set_nat > set_list_nat).
% 2.13/1.25 thf(func_def_74, type, listset_set_list_a: list_set_set_list_a > set_list_set_list_a).
% 2.13/1.25 thf(func_def_75, type, listset_set_set_a: list_set_set_set_a > set_list_set_set_a).
% 2.13/1.25 thf(func_def_76, type, listset_set_a: list_set_set_a > set_list_set_a).
% 2.13/1.25 thf(func_def_77, type, listset_a: list_set_a > set_list_a).
% 2.13/1.25 thf(func_def_78, type, nth_list_list_a: list_list_list_a > nat > list_list_a).
% 2.13/1.25 thf(func_def_79, type, nth_list_set_a: list_list_set_a > nat > list_set_a).
% 2.13/1.25 thf(func_def_80, type, nth_list_a: list_list_a > nat > list_a).
% 2.13/1.25 thf(func_def_81, type, nth_multiset_a: list_multiset_a > nat > multiset_a).
% 2.13/1.25 thf(func_def_82, type, nth_nat: list_nat > nat > nat).
% 2.13/1.25 thf(func_def_83, type, nth_set_list_list_a: list_set_list_list_a > nat > set_list_list_a).
% 2.13/1.25 thf(func_def_84, type, nth_set_list_set_a: list_set_list_set_a > nat > set_list_set_a).
% 2.13/1.25 thf(func_def_85, type, nth_set_list_a: list_set_list_a > nat > set_list_a).
% 2.13/1.25 thf(func_def_86, type, nth_set_multiset_a: list_set_multiset_a > nat > set_multiset_a).
% 2.13/1.25 thf(func_def_87, type, nth_set_nat: list_set_nat > nat > set_nat).
% 2.13/1.25 thf(func_def_88, type, nth_set_set_list_a: list_set_set_list_a > nat > set_set_list_a).
% 2.13/1.25 thf(func_def_89, type, nth_set_set_set_a: list_set_set_set_a > nat > set_set_set_a).
% 2.13/1.25 thf(func_def_90, type, nth_set_set_a: list_set_set_a > nat > set_set_a).
% 2.13/1.25 thf(func_def_91, type, nth_set_a: list_set_a > nat > set_a).
% 2.13/1.25 thf(func_def_92, type, nth_a: list_a > nat > a).
% 2.13/1.25 thf(func_def_93, type, rev_list_a: list_list_a > list_list_a).
% 2.13/1.25 thf(func_def_94, type, rev_nat: list_nat > list_nat).
% 2.13/1.25 thf(func_def_95, type, rev_set_list_a: list_set_list_a > list_set_list_a).
% 2.13/1.25 thf(func_def_96, type, rev_set_set_a: list_set_set_a > list_set_set_a).
% 2.13/1.25 thf(func_def_97, type, rev_set_a: list_set_a > list_set_a).
% 2.13/1.25 thf(func_def_98, type, rev_a: list_a > list_a).
% 2.13/1.25 thf(func_def_99, type, comm_m324867663list_a: multis971982480list_a > multiset_list_a).
% 2.13/1.25 thf(func_def_100, type, comm_m2145643721iset_a: multiset_multiset_a > multiset_a).
% 2.13/1.25 thf(func_def_101, type, comm_m1100186507et_nat: multiset_nat > nat).
% 2.13/1.25 thf(func_def_102, type, comm_m543484931mset_a: multiset_a > a).
% 2.13/1.25 thf(func_def_103, type, mset_list_a: list_list_a > multiset_list_a).
% 2.13/1.25 thf(func_def_104, type, mset_multiset_list_a: list_multiset_list_a > multis971982480list_a).
% 2.13/1.25 thf(func_def_105, type, mset_multiset_a: list_multiset_a > multiset_multiset_a).
% 2.13/1.25 thf(func_def_106, type, mset_nat: list_nat > multiset_nat).
% 2.13/1.25 thf(func_def_107, type, mset_set_list_a: list_set_list_a > multiset_set_list_a).
% 2.13/1.25 thf(func_def_108, type, mset_set_set_a: list_set_set_a > multiset_set_set_a).
% 2.13/1.25 thf(func_def_109, type, mset_set_a: list_set_a > multiset_set_a).
% 2.13/1.25 thf(func_def_110, type, mset_a: list_a > multiset_a).
% 2.13/1.25 thf(func_def_111, type, set_mset_list_a: multiset_list_a > set_list_a).
% 2.13/1.25 thf(func_def_112, type, set_ms1660427399list_a: multis971982480list_a > set_multiset_list_a).
% 2.13/1.25 thf(func_def_113, type, set_mset_multiset_a: multiset_multiset_a > set_multiset_a).
% 2.13/1.25 thf(func_def_114, type, set_mset_set_a: multiset_set_a > set_set_a).
% 2.13/1.25 thf(func_def_115, type, set_mset_a: multiset_a > set_a).
% 2.13/1.25 thf(func_def_116, type, size_s575106428list_a: list_list_list_a > nat).
% 2.13/1.25 thf(func_def_117, type, size_s341332310_set_a: list_list_set_a > nat).
% 2.13/1.25 thf(func_def_118, type, size_s1427607542list_a: list_list_a > nat).
% 2.13/1.25 thf(func_def_119, type, size_s1657263798iset_a: list_multiset_a > nat).
% 2.13/1.25 thf(func_def_120, type, size_size_list_nat: list_nat > nat).
% 2.13/1.25 thf(func_def_121, type, size_s240404444list_a: list_set_list_list_a > nat).
% 2.13/1.25 thf(func_def_122, type, size_s926492982_set_a: list_set_list_set_a > nat).
% 2.13/1.25 thf(func_def_123, type, size_s1635937238list_a: list_set_list_a > nat).
% 2.13/1.25 thf(func_def_124, type, size_s802269334iset_a: list_set_multiset_a > nat).
% 2.13/1.25 thf(func_def_125, type, size_s577819178et_nat: list_set_nat > nat).
% 2.13/1.25 thf(func_def_126, type, size_s469631926list_a: list_set_set_list_a > nat).
% 2.13/1.25 thf(func_def_127, type, size_s726512144_set_a: list_set_set_set_a > nat).
% 2.13/1.25 thf(func_def_128, type, size_s739728560_set_a: list_set_set_a > nat).
% 2.13/1.25 thf(func_def_129, type, size_size_list_set_a: list_set_a > nat).
% 2.13/1.25 thf(func_def_130, type, size_size_list_a: list_a > nat).
% 2.13/1.25 thf(func_def_131, type, size_s657334288_set_a: multiset_set_a > nat).
% 2.13/1.25 thf(func_def_132, type, size_size_multiset_a: multiset_a > nat).
% 2.13/1.25 thf(func_def_133, type, ord_less_nat: nat > nat > $o).
% 2.13/1.25 thf(func_def_134, type, ord_less_set_list_a: set_list_a > set_list_a > $o).
% 2.13/1.25 thf(func_def_135, type, ord_less_set_nat: set_nat > set_nat > $o).
% 2.13/1.25 thf(func_def_136, type, ord_less_set_a: set_a > set_a > $o).
% 2.13/1.25 thf(func_def_137, type, ord_less_eq_o_nat: ($o > nat) > ($o > nat) > $o).
% 2.13/1.25 thf(func_def_138, type, ord_less_eq_nat: nat > nat > $o).
% 2.13/1.25 thf(func_def_139, type, ord_le1301786372list_a: set_list_a > set_list_a > $o).
% 2.13/1.25 thf(func_def_140, type, ord_less_eq_set_nat: set_nat > set_nat > $o).
% 2.13/1.25 thf(func_def_141, type, ord_less_eq_set_a: set_a > set_a > $o).
% 2.13/1.25 thf(func_def_142, type, perm_set_a: list_set_a > list_set_a > $o).
% 2.13/1.25 thf(func_def_143, type, perm_a: list_a > list_a > $o).
% 2.13/1.25 thf(func_def_144, type, collect_list_list_a: (list_list_a > $o) > set_list_list_a).
% 2.13/1.25 thf(func_def_145, type, collect_list_set_a: (list_set_a > $o) > set_list_set_a).
% 2.13/1.25 thf(func_def_146, type, collect_list_a: (list_a > $o) > set_list_a).
% 2.13/1.25 thf(func_def_147, type, collect_multiset_a: (multiset_a > $o) > set_multiset_a).
% 2.13/1.25 thf(func_def_148, type, collect_nat: (nat > $o) > set_nat).
% 2.13/1.25 thf(func_def_149, type, collect_set_a: (set_a > $o) > set_set_a).
% 2.13/1.25 thf(func_def_150, type, collect_a: (a > $o) > set_a).
% 2.13/1.25 thf(func_def_151, type, image_list_nat_nat: (list_nat > nat) > set_list_nat > set_nat).
% 2.13/1.25 thf(func_def_152, type, image_list_a_list_a: (list_a > list_a) > set_list_a > set_list_a).
% 2.13/1.25 thf(func_def_153, type, image_list_a_a: (list_a > a) > set_list_a > set_a).
% 2.13/1.25 thf(func_def_154, type, image_nat_nat: (nat > nat) > set_nat > set_nat).
% 2.13/1.25 thf(func_def_155, type, image_a_list_a: (a > list_a) > set_a > set_list_a).
% 2.13/1.25 thf(func_def_156, type, image_a_a: (a > a) > set_a > set_a).
% 2.13/1.25 thf(func_def_157, type, set_ord_lessThan_nat: nat > set_nat).
% 2.13/1.25 thf(func_def_158, type, set_or1578766727list_a: set_list_a > set_set_list_a).
% 2.13/1.25 thf(func_def_159, type, set_or1597314339et_nat: set_nat > set_set_nat).
% 2.13/1.25 thf(func_def_160, type, set_or1164043265_set_a: set_a > set_set_a).
% 2.13/1.25 thf(func_def_161, type, member1511395513list_a: list_list_list_a > set_list_list_list_a > $o).
% 2.13/1.25 thf(func_def_162, type, member954073875_set_a: list_list_set_a > set_list_list_set_a > $o).
% 2.13/1.25 thf(func_def_163, type, member_list_list_a: list_list_a > set_list_list_a > $o).
% 2.13/1.25 thf(func_def_164, type, member518144627iset_a: list_multiset_a > set_list_multiset_a > $o).
% 2.13/1.25 thf(func_def_165, type, member_list_nat: list_nat > set_list_nat > $o).
% 2.13/1.25 thf(func_def_166, type, member101195155list_a: list_set_list_a > set_list_set_list_a > $o).
% 2.13/1.25 thf(func_def_167, type, member814963949_set_a: list_set_set_a > set_list_set_set_a > $o).
% 2.13/1.25 thf(func_def_168, type, member_list_set_a: list_set_a > set_list_set_a > $o).
% 2.13/1.25 thf(func_def_169, type, member_list_a: list_a > set_list_a > $o).
% 2.13/1.25 thf(func_def_170, type, member528267379list_a: multiset_list_a > set_multiset_list_a > $o).
% 2.13/1.25 thf(func_def_171, type, member_multiset_a: multiset_a > set_multiset_a > $o).
% 2.13/1.25 thf(func_def_172, type, member_nat: nat > set_nat > $o).
% 2.13/1.25 thf(func_def_173, type, member_set_list_a: set_list_a > set_set_list_a > $o).
% 2.13/1.25 thf(func_def_174, type, member_set_nat: set_nat > set_set_nat > $o).
% 2.13/1.25 thf(func_def_175, type, member_set_set_a: set_set_a > set_set_set_a > $o).
% 2.13/1.25 thf(func_def_176, type, member_set_a: set_a > set_set_a > $o).
% 2.13/1.25 thf(func_def_177, type, member_a: a > set_a > $o).
% 2.13/1.25 thf(func_def_178, type, a2: set_a).
% 2.13/1.25 thf(func_def_179, type, a3: a).
% 2.13/1.25 thf(func_def_180, type, f: nat > nat).
% 2.13/1.25 thf(func_def_181, type, g: nat > nat).
% 2.13/1.25 thf(func_def_182, type, qs2: list_a).
% 2.13/1.25 thf(func_def_183, type, qs: list_a).
% 2.13/1.25 thf(func_def_184, type, ss1: list_set_a).
% 2.13/1.25 thf(func_def_185, type, ss2: list_set_a).
% 2.13/1.25 thf(func_def_202, type, sK0: list_set_a > set_a > list_a).
% 2.13/1.25 thf(func_def_203, type, sK1: list_set_a > set_a > a).
% 2.13/1.25 thf(func_def_204, type, sK2: list_a > list_set_a > set_a > list_a).
% 2.13/1.25 thf(func_def_205, type, sK3: list_a).
% 2.13/1.25 thf(func_def_206, type, sK4: list_set_a > a > list_a).
% 2.13/1.25 thf(f1023,plain,(
% 2.13/1.25 $false),
% 2.13/1.25 inference(subsumption_resolution,[status(thm)],[f1022,f1020])).
% 2.13/1.25 thf(f1020,plain,(
% 2.13/1.25 ((groups1792256535list_a @ qs) != (groups1792256535list_a @ qs2))),
% 2.13/1.25 inference(superposition,[status(thm)],[f991,f1000])).
% 2.13/1.25 thf(f1000,plain,(
% 2.13/1.25 ((groups1792256535list_a @ qs) = (groups1792256535list_a @ sK3))),
% 2.13/1.25 inference(definition_unfolding,[status(thm)],[f982,f973])).
% 2.13/1.25 thf(f973,plain,(
% 2.13/1.25 (a3 = (groups1792256535list_a @ sK3))),
% 2.13/1.25 inference(cnf_transformation,[status(thm)],[f952])).
% 2.13/1.25 thf(f952,plain,(
% 2.13/1.25 ((member_list_a @ sK3 @ (listset_a @ ss1)) = $true) & (a3 = (groups1792256535list_a @ sK3))),
% 2.13/1.25 inference(skolemisation,[status(esa),new_symbols(skolem,[sK3])],[f942,f951])).
% 2.13/1.25 thf(f951,plain,(
% 2.13/1.25 ? [X0 : list_a] : (((member_list_a @ X0 @ (listset_a @ ss1)) = $true) & (a3 = (groups1792256535list_a @ X0))) => (((member_list_a @ sK3 @ (listset_a @ ss1)) = $true) & (a3 = (groups1792256535list_a @ sK3)))),
% 2.13/1.25 introduced(definition,[],[choice_axiom])).
% 2.13/1.25 thf(f942,plain,(
% 2.13/1.25 ? [X0 : list_a] : (((member_list_a @ X0 @ (listset_a @ ss1)) = $true) & (a3 = (groups1792256535list_a @ X0)))),
% 2.13/1.25 inference(ennf_transformation,[status(thm)],[f930])).
% 2.13/1.25 thf(f930,plain,(
% 2.13/1.25 ~! [X0 : list_a] : ((a3 = (groups1792256535list_a @ X0)) => ((member_list_a @ X0 @ (listset_a @ ss1)) != $true))),
% 2.13/1.25 inference(flattening,[status(thm)],[f594])).
% 2.13/1.25 thf(f594,plain,(
% 2.13/1.25 ~! [X0 : list_a] : ((a3 = (groups1792256535list_a @ X0)) => ~((member_list_a @ X0 @ (listset_a @ ss1)) = $true))),
% 2.13/1.25 inference(fool_elimination,[status(thm)],[f593])).
% 2.13/1.25 thf(f593,plain,(
% 2.13/1.25 ~! [X0 : list_a] : ((a3 = (groups1792256535list_a @ X0)) => ~(member_list_a @ X0 @ (listset_a @ ss1)))),
% 2.13/1.25 inference(rectify,[status(thm)],[f5])).
% 2.13/1.25 thf(f5,axiom,(
% 2.13/1.25 ~! [X0 : list_a] : ((a3 = (groups1792256535list_a @ X0)) => ~(member_list_a @ X0 @ (listset_a @ ss1)))),
% 2.13/1.25 file('/export/starexec/sandbox2/tmp/tmp.OZNDMxpjYk/DTF2THF_9014.p',fact_4__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062qs_O_A_092_060lbrakk_062a_A_061_Asum__list_Aqs_059_Aqs_A_092_060in_062_Alistset_Ass1_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062)).
% 2.13/1.25 thf(f982,plain,(
% 2.13/1.25 (a3 = (groups1792256535list_a @ qs))),
% 2.13/1.25 inference(cnf_transformation,[status(thm)],[f3])).
% 2.13/1.25 thf(f3,axiom,(
% 2.13/1.25 (a3 = (groups1792256535list_a @ qs))),
% 2.13/1.25 file('/export/starexec/sandbox2/tmp/tmp.OZNDMxpjYk/DTF2THF_9014.p',fact_2_a)).
% 2.13/1.25 thf(f991,plain,(
% 2.13/1.25 ((groups1792256535list_a @ qs2) != (groups1792256535list_a @ sK3))),
% 2.13/1.25 inference(definition_unfolding,[status(thm)],[f963,f973])).
% 2.13/1.25 thf(f963,plain,(
% 2.13/1.25 (a3 != (groups1792256535list_a @ qs2))),
% 2.13/1.25 inference(cnf_transformation,[status(thm)],[f929])).
% 2.13/1.25 thf(f929,plain,(
% 2.13/1.25 (a3 != (groups1792256535list_a @ qs2))),
% 2.13/1.25 inference(flattening,[status(thm)],[f355])).
% 2.13/1.25 thf(f355,negated_conjecture,(
% 2.13/1.25 ~(a3 = (groups1792256535list_a @ qs2))),
% 2.13/1.25 inference(negated_conjecture,[status(cth)],[f354])).
% 2.13/1.25 thf(f354,conjecture,(
% 2.13/1.25 (a3 = (groups1792256535list_a @ qs2))),
% 2.13/1.25 file('/export/starexec/sandbox2/tmp/tmp.OZNDMxpjYk/DTF2THF_9014.p',conj_0)).
% 2.13/1.25 thf(f1022,plain,(
% 2.13/1.25 ((groups1792256535list_a @ qs) = (groups1792256535list_a @ qs2))),
% 2.13/1.25 inference(forward_demodulation,[status(thm)],[f1021,f979])).
% 2.13/1.25 thf(f979,plain,(
% 2.13/1.25 ( ! [X0 : list_a] : (((groups1792256535list_a @ X0) = (comm_m543484931mset_a @ (mset_a @ X0)))) )),
% 2.13/1.25 inference(cnf_transformation,[status(thm)],[f931])).
% 2.13/1.25 thf(f931,plain,(
% 2.13/1.25 ! [X0 : list_a] : ((groups1792256535list_a @ X0) = (comm_m543484931mset_a @ (mset_a @ X0)))),
% 2.13/1.25 inference(rectify,[status(thm)],[f25])).
% 2.13/1.25 thf(f25,axiom,(
% 2.13/1.25 ! [X9 : list_a] : ((groups1792256535list_a @ X9) = (comm_m543484931mset_a @ (mset_a @ X9)))),
% 2.13/1.25 file('/export/starexec/sandbox2/tmp/tmp.OZNDMxpjYk/DTF2THF_9014.p',fact_24_sum__mset__sum__list)).
% 2.13/1.25 thf(f1021,plain,(
% 2.13/1.25 ((groups1792256535list_a @ qs) = (comm_m543484931mset_a @ (mset_a @ qs2)))),
% 2.13/1.25 inference(superposition,[status(thm)],[f979,f972])).
% 2.13/1.25 thf(f972,plain,(
% 2.13/1.25 ((mset_a @ qs2) = (mset_a @ qs))),
% 2.13/1.25 inference(cnf_transformation,[status(thm)],[f1])).
% 2.13/1.25 thf(f1,axiom,(
% 2.13/1.25 ((mset_a @ qs2) = (mset_a @ qs))),
% 2.13/1.25 file('/export/starexec/sandbox2/tmp/tmp.OZNDMxpjYk/DTF2THF_9014.p',fact_0__092_060open_062mset_Aqs2_A_061_Amset_Aqs_092_060close_062)).
% 2.13/1.25 % SZS output end Proof for DTF2THF_9014
% 2.13/1.25 % (9247)------------------------------
% 2.13/1.25 % (9247)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.13/1.25 % (9247)Termination reason: Refutation
% 2.13/1.25
% 2.13/1.25 % (9247)Memory used [KB]: 6908
% 2.13/1.25 % (9247)Time elapsed: 0.025 s
% 2.13/1.25 % (9247)Instructions burned: 77 (million)
% 2.13/1.25 % (9247)------------------------------
% 2.13/1.25 % (9247)------------------------------
% 2.13/1.25 % (9217)Success in time 0.123 s
%------------------------------------------------------------------------------