%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP066^1 : 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 : n024.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 08:55:16 AM UTC 2026
% Result : Theorem 2.33s 1.52s
% Output : Refutation 2.33s
% Verified :
% SZS Type : Refutation
% Derivation depth : 5
% Number of leaves : 5
% Syntax : Number of formulae : 19 ( 19 unt; 0 typ; 0 def)
% Number of atoms : 19 ( 18 equ; 0 cnn)
% Maximal formula atoms : 1 ( 1 avg)
% Number of connectives : 217 ( 3 ~; 0 |; 0 &; 214 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 3 avg)
% Number of types : 33 ( 32 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 214 ( 212 usr; 19 con; 0-6 aty)
% Number of variables : 34 ( 0 ^; 34 !; 0 ?; 34 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
multiset_multiset_a: $tType ).
thf(type_def_7,type,
multis782275565iset_a: $tType ).
thf(type_def_8,type,
multis2082063201Tree_a: $tType ).
thf(type_def_9,type,
multis599418605od_a_a: $tType ).
thf(type_def_10,type,
multiset_a: $tType ).
thf(type_def_11,type,
tree_a: $tType ).
thf(type_def_12,type,
list_a: $tType ).
thf(type_def_13,type,
produc143150363Tree_a: $tType ).
thf(type_def_14,type,
a: $tType ).
thf(type_def_15,type,
product_prod_a_a: $tType ).
thf(type_def_16,type,
set_a: $tType ).
thf(type_def_17,type,
produc1127127335iset_a: $tType ).
thf(type_def_18,type,
set_Pr158363655iset_a: $tType ).
thf(type_def_19,type,
set_Pr720484615iset_a: $tType ).
thf(type_def_20,type,
set_Pr146455751iset_a: $tType ).
thf(type_def_21,type,
set_Pr1066112583iset_a: $tType ).
thf(type_def_22,type,
set_Pr220561863Tree_a: $tType ).
thf(type_def_23,type,
set_Pr295926343Tree_a: $tType ).
thf(type_def_24,type,
set_Pr1948701895od_a_a: $tType ).
thf(type_def_25,type,
set_Pr1821581383od_a_a: $tType ).
thf(type_def_26,type,
set_Product_prod_a_a: $tType ).
thf(type_def_27,type,
set_multiset_a: $tType ).
thf(type_def_28,type,
set_Pr1070816379Tree_a: $tType ).
thf(type_def_29,type,
set_Pr921883667ee_a_a: $tType ).
thf(type_def_30,type,
produc981471411ee_a_a: $tType ).
thf(type_def_31,type,
produc1293660967iset_a: $tType ).
thf(type_def_32,type,
produc741964647iset_a: $tType ).
thf(type_def_33,type,
produc1136638567Tree_a: $tType ).
thf(type_def_34,type,
produc2061001575od_a_a: $tType ).
thf(type_def_35,type,
produc558600423iset_a: $tType ).
thf(type_def_36,type,
produc1254371559Tree_a: $tType ).
thf(type_def_37,type,
produc1572603623od_a_a: $tType ).
thf(func_def_0,type,
set_Pr1066112583iset_a: $tType ).
thf(func_def_1,type,
produc741964647iset_a: $tType ).
thf(func_def_2,type,
set_Pr146455751iset_a: $tType ).
thf(func_def_3,type,
set_Pr295926343Tree_a: $tType ).
thf(func_def_4,type,
produc558600423iset_a: $tType ).
thf(func_def_5,type,
produc1136638567Tree_a: $tType ).
thf(func_def_6,type,
set_Pr1821581383od_a_a: $tType ).
thf(func_def_7,type,
set_Pr220561863Tree_a: $tType ).
thf(func_def_8,type,
produc2061001575od_a_a: $tType ).
thf(func_def_9,type,
set_Pr720484615iset_a: $tType ).
thf(func_def_10,type,
produc1254371559Tree_a: $tType ).
thf(func_def_11,type,
produc1293660967iset_a: $tType ).
thf(func_def_12,type,
set_Pr1948701895od_a_a: $tType ).
thf(func_def_13,type,
multis782275565iset_a: $tType ).
thf(func_def_14,type,
produc1572603623od_a_a: $tType ).
thf(func_def_15,type,
set_Pr158363655iset_a: $tType ).
thf(func_def_16,type,
produc1127127335iset_a: $tType ).
thf(func_def_17,type,
multis2082063201Tree_a: $tType ).
thf(func_def_18,type,
set_Pr1070816379Tree_a: $tType ).
thf(func_def_19,type,
set_Pr921883667ee_a_a: $tType ).
thf(func_def_20,type,
multis599418605od_a_a: $tType ).
thf(func_def_21,type,
produc143150363Tree_a: $tType ).
thf(func_def_22,type,
produc981471411ee_a_a: $tType ).
thf(func_def_23,type,
set_Product_prod_a_a: $tType ).
thf(func_def_24,type,
multiset_multiset_a: $tType ).
thf(func_def_25,type,
set_multiset_a: $tType ).
thf(func_def_26,type,
product_prod_a_a: $tType ).
thf(func_def_27,type,
multiset_a: $tType ).
thf(func_def_28,type,
list_a: $tType ).
thf(func_def_29,type,
tree_a: $tType ).
thf(func_def_30,type,
set_a: $tType ).
thf(func_def_31,type,
a: $tType ).
thf(func_def_32,type,
plus_p1957546689iset_a: multiset_multiset_a > multiset_multiset_a > multiset_multiset_a ).
thf(func_def_33,type,
plus_p1919096612iset_a: multis782275565iset_a > multis782275565iset_a > multis782275565iset_a ).
thf(func_def_34,type,
plus_p1065544216Tree_a: multis2082063201Tree_a > multis2082063201Tree_a > multis2082063201Tree_a ).
thf(func_def_35,type,
plus_p404985124od_a_a: multis599418605od_a_a > multis599418605od_a_a > multis599418605od_a_a ).
thf(func_def_36,type,
plus_plus_multiset_a: multiset_a > multiset_a > multiset_a ).
thf(func_def_37,type,
zero_z1580389697iset_a: multiset_multiset_a ).
thf(func_def_38,type,
zero_z1941791140iset_a: multis782275565iset_a ).
thf(func_def_39,type,
zero_z2144236696Tree_a: multis2082063201Tree_a ).
thf(func_def_40,type,
zero_z160849828od_a_a: multis599418605od_a_a ).
thf(func_def_41,type,
zero_zero_multiset_a: multiset_a ).
thf(func_def_42,type,
nO_MAT1617603563iset_a: multiset_a > multiset_a > $o ).
thf(func_def_43,type,
heapIm970322378pify_a: tree_a > tree_a ).
thf(func_def_44,type,
heapIm229596386mpty_a: tree_a > $o ).
thf(func_def_45,type,
heapIm1057938560list_a: list_a > tree_a ).
thf(func_def_46,type,
heapIm915766629_max_a: tree_a > produc143150363Tree_a ).
thf(func_def_47,type,
heapIm1140443833left_a: tree_a > tree_a ).
thf(func_def_48,type,
heapIm1637418125tree_a: list_a > tree_a ).
thf(func_def_49,type,
heapIm837449470Leaf_a: tree_a > produc143150363Tree_a ).
thf(func_def_50,type,
heapIm1257206334ight_a: tree_a > tree_a ).
thf(func_def_51,type,
heapIm1091024090Down_a: tree_a > tree_a ).
thf(func_def_52,type,
heap_Tree_a_a: tree_a > ( tree_a > $o ) > ( list_a > tree_a ) > ( tree_a > multiset_a ) > ( tree_a > tree_a ) > ( tree_a > produc143150363Tree_a ) > $o ).
thf(func_def_53,type,
heap_a_a: a > ( a > $o ) > ( list_a > a ) > ( a > multiset_a ) > ( a > tree_a ) > ( a > product_prod_a_a ) > $o ).
thf(func_def_54,type,
heap_axioms_Tree_a_a: ( tree_a > $o ) > ( list_a > tree_a ) > ( tree_a > multiset_a ) > ( tree_a > tree_a ) > ( tree_a > produc143150363Tree_a ) > $o ).
thf(func_def_55,type,
heap_axioms_a_a: ( a > $o ) > ( list_a > a ) > ( a > multiset_a ) > ( a > tree_a ) > ( a > product_prod_a_a ) > $o ).
thf(func_def_56,type,
e_a: tree_a ).
thf(func_def_57,type,
t_a: a > tree_a > tree_a > tree_a ).
thf(func_def_58,type,
in_tree_a: a > tree_a > $o ).
thf(func_def_59,type,
is_heap_a: tree_a > $o ).
thf(func_def_60,type,
multiset_a2: tree_a > multiset_a ).
thf(func_def_61,type,
val_a: tree_a > a ).
thf(func_def_62,type,
if_Pro1144176865Tree_a: $o > produc143150363Tree_a > produc143150363Tree_a > produc143150363Tree_a ).
thf(func_def_63,type,
lattic146396397_Max_a: set_a > a ).
thf(func_def_64,type,
add_mset_multiset_a: multiset_a > multiset_multiset_a > multiset_multiset_a ).
thf(func_def_65,type,
add_ms1353365509iset_a: produc1127127335iset_a > multis782275565iset_a > multis782275565iset_a ).
thf(func_def_66,type,
add_ms205941497Tree_a: produc143150363Tree_a > multis2082063201Tree_a > multis2082063201Tree_a ).
thf(func_def_67,type,
add_ms1725977093od_a_a: product_prod_a_a > multis599418605od_a_a > multis599418605od_a_a ).
thf(func_def_68,type,
add_mset_a: a > multiset_a > multiset_a ).
thf(func_def_69,type,
comm_m315775925iset_a: ( multiset_a > multiset_a > multiset_a ) > multiset_a > multiset_multiset_a > multiset_a ).
thf(func_def_70,type,
is_empty_a: multiset_a > $o ).
thf(func_def_71,type,
mult1_multiset_a: set_Pr158363655iset_a > set_Pr720484615iset_a ).
thf(func_def_72,type,
mult1_599785114iset_a: set_Pr146455751iset_a > set_Pr1066112583iset_a ).
thf(func_def_73,type,
mult1_470979598Tree_a: set_Pr220561863Tree_a > set_Pr295926343Tree_a ).
thf(func_def_74,type,
mult1_1719650714od_a_a: set_Pr1948701895od_a_a > set_Pr1821581383od_a_a ).
thf(func_def_75,type,
mult1_a: set_Product_prod_a_a > set_Pr158363655iset_a ).
thf(func_def_76,type,
mult_multiset_a: set_Pr158363655iset_a > set_Pr720484615iset_a ).
thf(func_def_77,type,
mult_a: set_Product_prod_a_a > set_Pr158363655iset_a ).
thf(func_def_78,type,
set_mset_multiset_a: multiset_multiset_a > set_multiset_a ).
thf(func_def_79,type,
set_ms1768116836iset_a: multis782275565iset_a > set_Pr158363655iset_a ).
thf(func_def_80,type,
set_ms1129970904Tree_a: multis2082063201Tree_a > set_Pr1070816379Tree_a ).
thf(func_def_81,type,
set_ms1818337124od_a_a: multis599418605od_a_a > set_Product_prod_a_a ).
thf(func_def_82,type,
set_mset_a: multiset_a > set_a ).
thf(func_def_83,type,
ord_le1748873858_a_a_o: ( tree_a > a > $o ) > ( tree_a > a > $o ) > $o ).
thf(func_def_84,type,
ord_le51495822et_a_o: ( multiset_a > multiset_a > $o ) > ( multiset_a > multiset_a > $o ) > $o ).
thf(func_def_85,type,
ord_le1327026842ee_a_o: ( a > tree_a > $o ) > ( a > tree_a > $o ) > $o ).
thf(func_def_86,type,
ord_less_eq_a_a_o: ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(func_def_87,type,
ord_le1199012836iset_a: multiset_a > multiset_a > $o ).
thf(func_def_88,type,
ord_le864074675ee_a_a: set_Pr921883667ee_a_a > set_Pr921883667ee_a_a > $o ).
thf(func_def_89,type,
ord_le1978107815iset_a: set_Pr158363655iset_a > set_Pr158363655iset_a > $o ).
thf(func_def_90,type,
ord_le1013007387Tree_a: set_Pr1070816379Tree_a > set_Pr1070816379Tree_a > $o ).
thf(func_def_91,type,
ord_le1824328871od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(func_def_92,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(func_def_93,type,
ord_less_eq_a: a > a > $o ).
thf(func_def_94,type,
produc649172771ee_a_a: tree_a > a > produc981471411ee_a_a ).
thf(func_def_95,type,
produc1444952343iset_a: multiset_multiset_a > multiset_multiset_a > produc1293660967iset_a ).
thf(func_def_96,type,
produc1226985431iset_a: multis782275565iset_a > multis782275565iset_a > produc741964647iset_a ).
thf(func_def_97,type,
produc1608222167Tree_a: multis2082063201Tree_a > multis2082063201Tree_a > produc1136638567Tree_a ).
thf(func_def_98,type,
produc1859144151od_a_a: multis599418605od_a_a > multis599418605od_a_a > produc2061001575od_a_a ).
thf(func_def_99,type,
produc2037245207iset_a: multiset_a > multiset_a > produc1127127335iset_a ).
thf(func_def_100,type,
produc1898392407iset_a: produc1127127335iset_a > produc1127127335iset_a > produc558600423iset_a ).
thf(func_def_101,type,
produc1002111575Tree_a: produc143150363Tree_a > produc143150363Tree_a > produc1254371559Tree_a ).
thf(func_def_102,type,
produc1474507607od_a_a: product_prod_a_a > product_prod_a_a > produc1572603623od_a_a ).
thf(func_def_103,type,
produc686083979Tree_a: a > tree_a > produc143150363Tree_a ).
thf(func_def_104,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(func_def_105,type,
product_fst_Tree_a_a: produc981471411ee_a_a > tree_a ).
thf(func_def_106,type,
produc653373187iset_a: produc1127127335iset_a > multiset_a ).
thf(func_def_107,type,
product_fst_a_Tree_a: produc143150363Tree_a > a ).
thf(func_def_108,type,
product_fst_a_a: product_prod_a_a > a ).
thf(func_def_109,type,
product_snd_Tree_a_a: produc981471411ee_a_a > a ).
thf(func_def_110,type,
produc148952133iset_a: produc1127127335iset_a > multiset_a ).
thf(func_def_111,type,
product_snd_a_Tree_a: produc143150363Tree_a > tree_a ).
thf(func_def_112,type,
product_snd_a_a: product_prod_a_a > a ).
thf(func_def_113,type,
produc1478296771ee_a_a: produc981471411ee_a_a > produc143150363Tree_a ).
thf(func_def_114,type,
produc506430135iset_a: produc1127127335iset_a > produc1127127335iset_a ).
thf(func_def_115,type,
produc1515207979Tree_a: produc143150363Tree_a > produc981471411ee_a_a ).
thf(func_def_116,type,
product_swap_a_a: product_prod_a_a > product_prod_a_a ).
thf(func_def_117,type,
irrefl_a: set_Product_prod_a_a > $o ).
thf(func_def_118,type,
trans_multiset_a: set_Pr158363655iset_a > $o ).
thf(func_def_119,type,
trans_a: set_Product_prod_a_a > $o ).
thf(func_def_120,type,
collection_Tree_a_a: tree_a > ( tree_a > $o ) > ( list_a > tree_a ) > ( tree_a > multiset_a ) > $o ).
thf(func_def_121,type,
collec1104713362iset_a: ( produc1127127335iset_a > $o ) > set_Pr158363655iset_a ).
thf(func_def_122,type,
collec486369286Tree_a: ( produc143150363Tree_a > $o ) > set_Pr1070816379Tree_a ).
thf(func_def_123,type,
collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
thf(func_def_124,type,
collect_a: ( a > $o ) > set_a ).
thf(func_def_125,type,
member_multiset_a: multiset_a > set_multiset_a > $o ).
thf(func_def_126,type,
member1092636252ee_a_a: produc981471411ee_a_a > set_Pr921883667ee_a_a > $o ).
thf(func_def_127,type,
member104214352iset_a: produc1293660967iset_a > set_Pr720484615iset_a > $o ).
thf(func_def_128,type,
member219455632iset_a: produc741964647iset_a > set_Pr1066112583iset_a > $o ).
thf(func_def_129,type,
member1298282640Tree_a: produc1136638567Tree_a > set_Pr295926343Tree_a > $o ).
thf(func_def_130,type,
member1349055120od_a_a: produc2061001575od_a_a > set_Pr1821581383od_a_a > $o ).
thf(func_def_131,type,
member340150864iset_a: produc1127127335iset_a > set_Pr158363655iset_a > $o ).
thf(func_def_132,type,
member567864080iset_a: produc558600423iset_a > set_Pr146455751iset_a > $o ).
thf(func_def_133,type,
member146662416Tree_a: produc1254371559Tree_a > set_Pr220561863Tree_a > $o ).
thf(func_def_134,type,
member2057358096od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_135,type,
member254315204Tree_a: produc143150363Tree_a > set_Pr1070816379Tree_a > $o ).
thf(func_def_136,type,
member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(func_def_137,type,
member_a: a > set_a > $o ).
thf(func_def_138,type,
l1: tree_a ).
thf(func_def_139,type,
l2: tree_a ).
thf(func_def_140,type,
r1: tree_a ).
thf(func_def_141,type,
r2: tree_a ).
thf(func_def_142,type,
t: tree_a ).
thf(func_def_143,type,
t2: tree_a ).
thf(func_def_144,type,
t_a2: tree_a ).
thf(func_def_145,type,
v1: a ).
thf(func_def_146,type,
v2: a ).
thf(func_def_147,type,
v: a ).
thf(func_def_148,type,
v_a: a ).
thf(func_def_149,type,
v3: a ).
thf(func_def_168,type,
sP0: ( tree_a > $o ) > $o ).
thf(func_def_169,type,
sP1: ( tree_a > $o ) > $o ).
thf(func_def_170,type,
sP2: ( tree_a > $o ) > $o ).
thf(func_def_171,type,
sK3: tree_a > a ).
thf(func_def_172,type,
sK4: tree_a > a ).
thf(func_def_173,type,
sK5: tree_a > tree_a ).
thf(func_def_174,type,
sK6: tree_a > tree_a ).
thf(func_def_175,type,
sK7: tree_a > a ).
thf(func_def_176,type,
sK8: tree_a > a ).
thf(func_def_177,type,
sK9: tree_a > tree_a ).
thf(func_def_178,type,
sK10: tree_a > a ).
thf(func_def_179,type,
sK11: tree_a > tree_a ).
thf(func_def_180,type,
sK12: tree_a > tree_a ).
thf(func_def_181,type,
sK13: tree_a > tree_a ).
thf(func_def_182,type,
sK14: tree_a > a ).
thf(func_def_183,type,
sK15: tree_a > tree_a ).
thf(func_def_184,type,
sK16: tree_a > tree_a ).
thf(func_def_185,type,
sK17: tree_a > a ).
thf(func_def_186,type,
sK18: tree_a > tree_a ).
thf(func_def_187,type,
sK19: tree_a > a ).
thf(func_def_188,type,
sK20: tree_a > tree_a ).
thf(func_def_189,type,
sK21: tree_a > a ).
thf(func_def_190,type,
sK22: tree_a > a ).
thf(func_def_191,type,
sK23: tree_a > tree_a ).
thf(func_def_192,type,
sK24: tree_a > a ).
thf(func_def_193,type,
sK25: tree_a > tree_a ).
thf(func_def_194,type,
sK26: tree_a > a ).
thf(func_def_195,type,
sK27: tree_a > a ).
thf(func_def_196,type,
sK28: tree_a > tree_a ).
thf(func_def_197,type,
sK29: tree_a > tree_a ).
thf(func_def_198,type,
sK30: tree_a > a ).
thf(func_def_199,type,
sK31: tree_a > tree_a ).
thf(func_def_200,type,
sK32: tree_a > a ).
thf(func_def_201,type,
sK33: tree_a > a ).
thf(func_def_202,type,
sK34: tree_a > tree_a ).
thf(func_def_203,type,
sK35: tree_a > tree_a ).
thf(func_def_204,type,
sK36: tree_a > tree_a ).
thf(func_def_205,type,
sK37: tree_a > a ).
thf(func_def_206,type,
sK38: tree_a > tree_a ).
thf(func_def_207,type,
sK39: tree_a > a ).
thf(func_def_208,type,
sK40: tree_a > a ).
thf(func_def_209,type,
sK41: tree_a > tree_a ).
thf(func_def_210,type,
sK42: ( tree_a > $o ) > a ).
thf(func_def_211,type,
sK43: ( tree_a > $o ) > a ).
thf(func_def_212,type,
sK44: ( tree_a > $o ) > a ).
thf(func_def_213,type,
sK45: ( tree_a > $o ) > tree_a ).
thf(func_def_214,type,
sK46: ( tree_a > $o ) > tree_a ).
thf(func_def_215,type,
sK47: ( tree_a > $o ) > tree_a ).
thf(func_def_216,type,
sK48: ( tree_a > $o ) > tree_a ).
thf(func_def_217,type,
sK49: ( tree_a > $o ) > tree_a ).
thf(func_def_218,type,
sK50: ( tree_a > $o ) > a ).
thf(func_def_219,type,
sK51: ( tree_a > $o ) > tree_a ).
thf(func_def_220,type,
sK52: ( tree_a > $o ) > a ).
thf(func_def_221,type,
sK53: ( tree_a > $o ) > tree_a ).
thf(func_def_222,type,
sK54: ( tree_a > $o ) > a ).
thf(func_def_223,type,
sK55: ( tree_a > $o ) > a ).
thf(func_def_224,type,
sK56: ( tree_a > $o ) > tree_a ).
thf(func_def_225,type,
sK57: ( tree_a > $o ) > a ).
thf(func_def_226,type,
sK58: ( tree_a > $o ) > tree_a ).
thf(func_def_227,type,
sK59: ( tree_a > $o ) > tree_a ).
thf(func_def_228,type,
sK60: ( tree_a > $o ) > a ).
thf(func_def_229,type,
sK61: ( tree_a > $o ) > tree_a ).
thf(func_def_230,type,
sK62: ( tree_a > $o ) > tree_a ).
thf(func_def_231,type,
sK63: ( tree_a > $o ) > a ).
thf(func_def_232,type,
sK64: ( tree_a > $o ) > a ).
thf(func_def_233,type,
sK65: tree_a > tree_a ).
thf(func_def_234,type,
sK66: tree_a > tree_a ).
thf(func_def_235,type,
sK67: tree_a > a ).
thf(func_def_236,type,
sK68: ( tree_a > $o ) > tree_a ).
thf(func_def_237,type,
sK69: ( tree_a > $o ) > a ).
thf(func_def_238,type,
sK70: ( tree_a > $o ) > tree_a ).
thf(func_def_239,type,
sK71: tree_a > a ).
thf(func_def_240,type,
sK72: tree_a > tree_a ).
thf(func_def_241,type,
sK73: tree_a > tree_a ).
thf(func_def_242,type,
sK74: tree_a > a ).
thf(func_def_243,type,
sK75: tree_a > a ).
thf(func_def_244,type,
sK76: tree_a > a ).
thf(func_def_245,type,
sK77: tree_a > tree_a ).
thf(func_def_246,type,
sK78: tree_a > tree_a ).
thf(func_def_247,type,
sK79: tree_a > tree_a ).
thf(func_def_248,type,
sK80: tree_a > tree_a ).
thf(func_def_249,type,
sK81: tree_a > a ).
thf(func_def_250,type,
sK82: tree_a > a ).
thf(func_def_251,type,
sK83: tree_a > a ).
thf(func_def_252,type,
sK84: tree_a > tree_a ).
thf(func_def_253,type,
sK85: tree_a > a ).
thf(func_def_254,type,
sK86: tree_a > tree_a ).
thf(func_def_255,type,
sK87: ( tree_a > a > $o ) > a ).
thf(func_def_256,type,
sK88: ( tree_a > a > $o ) > tree_a ).
thf(func_def_257,type,
sK89: a > tree_a > tree_a ).
thf(func_def_258,type,
sK90: a > tree_a > tree_a ).
thf(func_def_259,type,
sK91: a > tree_a > a ).
thf(func_def_261,type,
sK93: produc143150363Tree_a > a > tree_a ).
thf(func_def_262,type,
ph94:
!>[X0: $tType] : X0 ).
thf(f2387,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f2376,f946]) ).
thf(f946,plain,
( v_a
!= ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ v1 @ l1 @ r1 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f828]) ).
thf(f828,plain,
( v_a
!= ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ v1 @ l1 @ r1 ) ) ) ),
inference(flattening,[status(thm)],[f358]) ).
thf(f358,negated_conjecture,
( v_a
!= ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ v1 @ l1 @ r1 ) ) ) ),
inference(negated_conjecture,[status(cth)],[f357]) ).
thf(f357,conjecture,
( v_a
= ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ v1 @ l1 @ r1 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.lw13DE3eQN/DTF2THF_26921.p',conj_0) ).
thf(f2376,plain,
( v_a
= ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ v1 @ l1 @ r1 ) ) ) ),
inference(superposition,[status(thm)],[f1870,f1821]) ).
thf(f1821,plain,
( v_a
= ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ v3 @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) ) ) ) ),
inference(superposition,[status(thm)],[f1001,f1025]) ).
thf(f1025,plain,
( ( heapIm837449470Leaf_a @ ( t_a @ v3 @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) ) )
= ( produc686083979Tree_a @ v_a @ ( t_a @ v3 @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ v1 @ l1 @ r1 ) ) ) @ ( t_a @ v2 @ l2 @ r2 ) ) ) ),
inference(definition_unfolding,[status(thm)],[f993,f951]) ).
thf(f951,plain,
( t_a2
= ( t_a @ v3 @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ v1 @ l1 @ r1 ) ) ) @ ( t_a @ v2 @ l2 @ r2 ) ) ),
inference(cnf_transformation,[status(thm)],[f4]) ).
thf(f4,axiom,
( t_a2
= ( t_a @ v3 @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ v1 @ l1 @ r1 ) ) ) @ ( t_a @ v2 @ l2 @ r2 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.lw13DE3eQN/DTF2THF_26921.p',fact_3__092_060open_062t_H_A_061_AT_Av_A_Isnd_A_IremoveLeaf_A_IT_Av1_Al1_Ar1_J_J_J_A_IT_Av2_Al2_Ar2_J_092_060close_062) ).
thf(f993,plain,
( ( produc686083979Tree_a @ v_a @ t_a2 )
= ( heapIm837449470Leaf_a @ ( t_a @ v3 @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f1]) ).
thf(f1,axiom,
( ( produc686083979Tree_a @ v_a @ t_a2 )
= ( heapIm837449470Leaf_a @ ( t_a @ v3 @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.lw13DE3eQN/DTF2THF_26921.p',fact_0__C4__2_Oprems_C_I1_J) ).
thf(f1001,plain,
! [X0: a,X1: tree_a] :
( ( product_fst_a_Tree_a @ ( produc686083979Tree_a @ X0 @ X1 ) )
= X0 ),
inference(cnf_transformation,[status(thm)],[f859]) ).
thf(f859,plain,
! [X0: a,X1: tree_a] :
( ( product_fst_a_Tree_a @ ( produc686083979Tree_a @ X0 @ X1 ) )
= X0 ),
inference(rectify,[status(thm)],[f23]) ).
thf(f23,axiom,
! [X21: a,X22: tree_a] :
( ( product_fst_a_Tree_a @ ( produc686083979Tree_a @ X21 @ X22 ) )
= X21 ),
file('/export/starexec/sandbox2/tmp/tmp.lw13DE3eQN/DTF2THF_26921.p',fact_22_fst__conv) ).
thf(f1870,plain,
! [X2: tree_a,X3: a,X0: a,X1: tree_a,X6: tree_a,X4: a,X5: tree_a] :
( ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ X3 @ ( t_a @ X0 @ X1 @ X2 ) @ ( t_a @ X4 @ X5 @ X6 ) ) ) )
= ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ X0 @ X1 @ X2 ) ) ) ),
inference(superposition,[status(thm)],[f1001,f1021]) ).
thf(f1021,plain,
! [X2: tree_a,X3: a,X0: a,X1: tree_a,X6: tree_a,X4: a,X5: tree_a] :
( ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ X3 @ X6 @ X1 ) ) ) @ ( t_a @ X4 @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ X3 @ X6 @ X1 ) ) ) @ ( t_a @ X0 @ X2 @ X5 ) ) )
= ( heapIm837449470Leaf_a @ ( t_a @ X4 @ ( t_a @ X3 @ X6 @ X1 ) @ ( t_a @ X0 @ X2 @ X5 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f835]) ).
thf(f835,plain,
! [X0: a,X1: tree_a,X2: tree_a,X3: a,X4: a,X5: tree_a,X6: tree_a] :
( ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ X3 @ X6 @ X1 ) ) ) @ ( t_a @ X4 @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ X3 @ X6 @ X1 ) ) ) @ ( t_a @ X0 @ X2 @ X5 ) ) )
= ( heapIm837449470Leaf_a @ ( t_a @ X4 @ ( t_a @ X3 @ X6 @ X1 ) @ ( t_a @ X0 @ X2 @ X5 ) ) ) ),
inference(rectify,[status(thm)],[f7]) ).
thf(f7,axiom,
! [X7: a,X12: tree_a,X8: tree_a,X10: a,X6: a,X9: tree_a,X11: tree_a] :
( ( heapIm837449470Leaf_a @ ( t_a @ X6 @ ( t_a @ X10 @ X11 @ X12 ) @ ( t_a @ X7 @ X8 @ X9 ) ) )
= ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ X10 @ X11 @ X12 ) ) ) @ ( t_a @ X6 @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ X10 @ X11 @ X12 ) ) ) @ ( t_a @ X7 @ X8 @ X9 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.lw13DE3eQN/DTF2THF_26921.p',fact_6_removeLeaf_Osimps_I4_J) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.06 % Problem : ITP066^1 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.07 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.07/0.25 % Computer : n024.cluster.edu
% 0.07/0.25 % Model : x86_64 x86_64
% 0.07/0.25 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/0.25 % Memory : 8042.1875MB
% 0.07/0.25 % OS : Linux 3.10.0-693.el7.x86_64
% 0.07/0.25 % CPULimit : 300
% 0.07/0.25 % WCLimit : 300
% 0.07/0.25 % DateTime : Tue Feb 24 03:25:10 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
% 0.07/0.32 ---- Original DTF file ---
% 0.07/0.32 thf(spec,logic,$$dhol).
% 0.07/0.32 %------------------------------------------------------------------------------
% 0.07/0.32 % File : ITP066^1 : TPTP v9.2.1. Released v7.5.0.
% 0.07/0.32 % Domain : Interactive Theorem Proving
% 0.07/0.32 % Problem : Sledgehammer HeapImperative problem prob_1195__5351920_1
% 0.07/0.32 % Version : Especial.
% 0.07/0.32 % English :
% 0.07/0.32
% 0.07/0.32 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.07/0.32 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.07/0.32 % Source : [Des21]
% 0.07/0.32 % Names : HeapImperative/prob_1195__5351920_1 [Des21]
% 0.07/0.32
% 0.07/0.32 % Status : Theorem
% 0.07/0.32 % Rating : 0.33 v9.1.0, 0.25 v9.0.0, 0.40 v8.2.0, 0.31 v8.1.0, 0.27 v7.5.0
% 0.07/0.32 % Syntax : Number of formulae : 507 ( 186 unt; 150 typ; 0 def)
% 0.07/0.32 % Number of atoms : 909 ( 536 equ; 0 cnn)
% 0.07/0.32 % Maximal formula atoms : 11 ( 2 avg)
% 0.07/0.32 % Number of connectives : 3667 ( 129 ~; 18 |; 95 &;3044 @)
% 0.07/0.32 % ( 0 <=>; 381 =>; 0 <=; 0 <~>)
% 0.07/0.32 % Maximal formula depth : 23 ( 7 avg)
% 0.07/0.32 % Number of types : 33 ( 32 usr)
% 0.07/0.32 % Number of type conns : 369 ( 369 >; 0 *; 0 +; 0 <<)
% 0.07/0.32 % Number of symbols : 121 ( 118 usr; 20 con; 0-6 aty)
% 0.07/0.32 % Number of variables : 1283 ( 90 ^;1146 !; 47 ?;1283 :)
% 0.07/0.32 % SPC : TH0_THM_EQU_NAR
% 0.07/0.32
% 0.07/0.32 % Comments : This file was generated by Sledgehammer 2021-02-23 15:32:09.933
% 0.07/0.32 %------------------------------------------------------------------------------
% 0.07/0.32 % Could-be-implicit typings (32)
% 0.07/0.32 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J_J_J,type,
% 0.07/0.32 set_Pr1066112583iset_a: $tType ).
% 0.07/0.32
% 0.07/0.32 thf(ty_n_t__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J_J,type,
% 0.07/0.32 produc741964647iset_a: $tType ).
% 0.07/0.32
% 0.07/0.32 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_Mt__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J_J,type,
% 0.07/0.32 set_Pr146455751iset_a: $tType ).
% 0.07/0.32
% 0.07/0.32 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_J_J_J,type,
% 0.07/0.32 set_Pr295926343Tree_a: $tType ).
% 0.07/0.32
% 0.07/0.32 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_Mt__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J,type,
% 0.07/0.32 produc558600423iset_a: $tType ).
% 0.07/0.32
% 0.07/0.32 thf(ty_n_t__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_J_J,type,
% 0.07/0.32 produc1136638567Tree_a: $tType ).
% 0.07/0.32
% 0.07/0.32 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J,type,
% 0.07/0.32 set_Pr1821581383od_a_a: $tType ).
% 0.07/0.32
% 0.07/0.32 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_J_J,type,
% 0.07/0.32 set_Pr220561863Tree_a: $tType ).
% 0.07/0.32
% 0.07/0.32 thf(ty_n_t__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 0.07/0.32 produc2061001575od_a_a: $tType ).
% 0.07/0.32
% 0.07/0.32 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J_Mt__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J_J_J,type,
% 0.07/0.32 set_Pr720484615iset_a: $tType ).
% 0.07/0.32
% 0.07/0.32 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_J,type,
% 0.07/0.32 produc1254371559Tree_a: $tType ).
% 0.07/0.32
% 0.07/0.32 thf(ty_n_t__Product____Type__Oprod_It__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J_Mt__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J_J,type,
% 0.07/0.32 produc1293660967iset_a: $tType ).
% 0.07/0.32
% 0.07/0.32 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 0.07/0.32 set_Pr1948701895od_a_a: $tType ).
% 0.07/0.32
% 0.07/0.32 thf(ty_n_t__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J,type,
% 0.07/0.32 multis782275565iset_a: $tType ).
% 0.07/0.32
% 0.07/0.32 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.07/0.32 produc1572603623od_a_a: $tType ).
% 0.07/0.32
% 0.07/0.32 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J,type,
% 0.07/0.32 set_Pr158363655iset_a: $tType ).
% 0.07/0.32
% 0.07/0.32 thf(ty_n_t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J,type,
% 0.07/0.32 produc1127127335iset_a: $tType ).
% 0.07/0.32
% 0.07/0.32 thf(ty_n_t__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_J,type,
% 0.07/0.32 multis2082063201Tree_a: $tType ).
% 0.07/0.32
% 0.07/0.32 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_J,type,
% 0.07/0.32 set_Pr1070816379Tree_a: $tType ).
% 0.07/0.32
% 0.07/0.32 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Heap__OTree_Itf__a_J_Mtf__a_J_J,type,
% 0.07/0.32 set_Pr921883667ee_a_a: $tType ).
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% 0.07/0.32 thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_001t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J,type,
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% 0.07/0.32
% 0.07/0.32 thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.07/0.32 produc1474507607od_a_a: product_prod_a_a > product_prod_a_a > produc1572603623od_a_a ).
% 0.07/0.32
% 0.07/0.32 thf(sy_c_Product__Type_OPair_001tf__a_001t__Heap__OTree_Itf__a_J,type,
% 0.07/0.32 produc686083979Tree_a: a > tree_a > produc143150363Tree_a ).
% 0.07/0.32
% 0.07/0.32 thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
% 0.07/0.32 product_Pair_a_a: a > a > product_prod_a_a ).
% 0.07/0.32
% 0.07/0.32 thf(sy_c_Product__Type_Oprod_Ofst_001t__Heap__OTree_Itf__a_J_001tf__a,type,
% 0.07/0.32 product_fst_Tree_a_a: produc981471411ee_a_a > tree_a ).
% 0.07/0.32
% 0.07/0.32 thf(sy_c_Product__Type_Oprod_Ofst_001t__Multiset__Omultiset_Itf__a_J_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.07/0.32 produc653373187iset_a: produc1127127335iset_a > multiset_a ).
% 0.07/0.32
% 0.07/0.32 thf(sy_c_Product__Type_Oprod_Ofst_001tf__a_001t__Heap__OTree_Itf__a_J,type,
% 0.07/0.32 product_fst_a_Tree_a: produc143150363Tree_a > a ).
% 0.07/0.32
% 0.07/0.32 thf(sy_c_Product__Type_Oprod_Ofst_001tf__a_001tf__a,type,
% 0.07/0.32 product_fst_a_a: product_prod_a_a > a ).
% 0.07/0.32
% 0.07/0.32 thf(sy_c_Product__Type_Oprod_Osnd_001t__Heap__OTree_Itf__a_J_001tf__a,type,
% 0.07/0.32 product_snd_Tree_a_a: produc981471411ee_a_a > a ).
% 0.07/0.32
% 0.07/0.32 thf(sy_c_Product__Type_Oprod_Osnd_001t__Multiset__Omultiset_Itf__a_J_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.07/0.32 produc148952133iset_a: produc1127127335iset_a > multiset_a ).
% 0.07/0.32
% 0.07/0.32 thf(sy_c_Product__Type_Oprod_Osnd_001tf__a_001t__Heap__OTree_Itf__a_J,type,
% 0.07/0.32 product_snd_a_Tree_a: produc143150363Tree_a > tree_a ).
% 0.07/0.32
% 0.07/0.32 thf(sy_c_Product__Type_Oprod_Osnd_001tf__a_001tf__a,type,
% 0.07/0.32 product_snd_a_a: product_prod_a_a > a ).
% 0.07/0.32
% 0.07/0.32 thf(sy_c_Product__Type_Oprod_Oswap_001t__Heap__OTree_Itf__a_J_001tf__a,type,
% 0.07/0.32 produc1478296771ee_a_a: produc981471411ee_a_a > produc143150363Tree_a ).
% 0.07/0.32
% 0.07/0.32 thf(sy_c_Product__Type_Oprod_Oswap_001t__Multiset__Omultiset_Itf__a_J_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.07/0.32 produc506430135iset_a: produc1127127335iset_a > produc1127127335iset_a ).
% 0.07/0.32
% 0.07/0.32 thf(sy_c_Product__Type_Oprod_Oswap_001tf__a_001t__Heap__OTree_Itf__a_J,type,
% 0.07/0.32 produc1515207979Tree_a: produc143150363Tree_a > produc981471411ee_a_a ).
% 0.07/0.32
% 0.07/0.32 thf(sy_c_Product__Type_Oprod_Oswap_001tf__a_001tf__a,type,
% 0.07/0.32 product_swap_a_a: product_prod_a_a > product_prod_a_a ).
% 0.07/0.32
% 0.07/0.32 thf(sy_c_Relation_Oirrefl_001tf__a,type,
% 0.07/0.32 irrefl_a: set_Product_prod_a_a > $o ).
% 0.07/0.32
% 0.07/0.32 thf(sy_c_Relation_Otrans_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.07/0.32 trans_multiset_a: set_Pr158363655iset_a > $o ).
% 0.07/0.32
% 0.07/0.32 thf(sy_c_Relation_Otrans_001tf__a,type,
% 0.07/0.32 trans_a: set_Product_prod_a_a > $o ).
% 0.07/0.32
% 0.07/0.32 thf(sy_c_RemoveMax_OCollection_001t__Heap__OTree_Itf__a_J_001tf__a,type,
% 0.07/0.32 collection_Tree_a_a: tree_a > ( tree_a > $o ) > ( list_a > tree_a ) > ( tree_a > multiset_a ) > $o ).
% 0.07/0.32
% 0.07/0.32 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J,type,
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% 0.07/0.32
% 0.07/0.32 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J,type,
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% 0.07/0.32 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
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% 0.07/0.32
% 0.07/0.32 thf(sy_c_Set_OCollect_001tf__a,type,
% 0.07/0.32 collect_a: ( a > $o ) > set_a ).
% 0.07/0.32
% 0.07/0.32 thf(sy_c_member_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.07/0.32 member_multiset_a: multiset_a > set_multiset_a > $o ).
% 0.07/0.32
% 0.07/0.32 thf(sy_c_member_001t__Product____Type__Oprod_It__Heap__OTree_Itf__a_J_Mtf__a_J,type,
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% 0.07/0.32
% 0.07/0.32 thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J_Mt__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J_J,type,
% 0.07/0.32 member104214352iset_a: produc1293660967iset_a > set_Pr720484615iset_a > $o ).
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% 0.07/0.32 member219455632iset_a: produc741964647iset_a > set_Pr1066112583iset_a > $o ).
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% 0.07/0.32 thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_J_J,type,
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% 0.07/0.32
% 0.07/0.32 thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
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% 0.07/0.32
% 0.07/0.32 thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J,type,
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% 0.07/0.32
% 0.07/0.32 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_Mt__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J,type,
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% 0.07/0.32 thf(sy_c_member_001tf__a,type,
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% 0.07/0.32
% 0.07/0.32 thf(sy_v_l1____,type,
% 0.07/0.32 l1: tree_a ).
% 0.07/0.32
% 0.07/0.32 thf(sy_v_l2____,type,
% 0.07/0.32 l2: tree_a ).
% 0.07/0.32
% 0.07/0.32 thf(sy_v_r1____,type,
% 0.07/0.32 r1: tree_a ).
% 0.07/0.32
% 0.07/0.32 thf(sy_v_r2____,type,
% 0.07/0.32 r2: tree_a ).
% 0.07/0.32
% 0.07/0.32 thf(sy_v_t,type,
% 0.07/0.32 t: tree_a ).
% 0.07/0.32
% 0.07/0.32 thf(sy_v_t_H,type,
% 0.07/0.32 t2: tree_a ).
% 0.07/0.32
% 0.07/0.32 thf(sy_v_t_Ha____,type,
% 0.07/0.32 t_a2: tree_a ).
% 0.07/0.32
% 0.07/0.32 thf(sy_v_v1____,type,
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% 0.07/0.32
% 0.07/0.32 thf(sy_v_v2____,type,
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% 0.07/0.32
% 0.07/0.32 thf(sy_v_v_H,type,
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% 0.07/0.32
% 0.07/0.32 thf(sy_v_v_Ha____,type,
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% 0.07/0.32
% 0.07/0.32 thf(sy_v_v____,type,
% 0.07/0.32 v3: a ).
% 0.07/0.32
% 0.07/0.32 % Relevant facts (353)
% 0.07/0.32 thf(fact_0__C4__2_Oprems_C_I1_J,axiom,
% 0.07/0.32 ( ( produc686083979Tree_a @ v_a @ t_a2 )
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% 0.07/0.32
% 0.07/0.32 % "4_2.prems"(1)
% 0.07/0.32 thf(fact_1_assms_I1_J,axiom,
% 0.07/0.32 ( ( produc686083979Tree_a @ v @ t2 )
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% 0.07/0.32
% 0.07/0.32 % assms(1)
% 0.07/0.32 thf(fact_2__C4__2_Oprems_C_I2_J,axiom,
% 0.07/0.32 ( ( t_a @ v3 @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) )
% 0.07/0.32 != e_a ) ).
% 0.07/0.32
% 0.07/0.32 % "4_2.prems"(2)
% 0.07/0.32 thf(fact_3__092_060open_062t_H_A_061_AT_Av_A_Isnd_A_IremoveLeaf_A_IT_Av1_Al1_Ar1_J_J_J_A_IT_Av2_Al2_Ar2_J_092_060close_062,axiom,
% 0.07/0.32 ( t_a2
% 0.07/0.32 = ( t_a @ v3 @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ v1 @ l1 @ r1 ) ) ) @ ( t_a @ v2 @ l2 @ r2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % \<open>t' = T v (snd (removeLeaf (T v1 l1 r1))) (T v2 l2 r2)\<close>
% 0.07/0.32 thf(fact_4_Tree_Oinject,axiom,
% 0.07/0.32 ! [X21: a,X22: tree_a,X23: tree_a,Y21: a,Y22: tree_a,Y23: tree_a] :
% 0.07/0.32 ( ( ( t_a @ X21 @ X22 @ X23 )
% 0.07/0.32 = ( t_a @ Y21 @ Y22 @ Y23 ) )
% 0.07/0.32 = ( ( X21 = Y21 )
% 0.07/0.32 & ( X22 = Y22 )
% 0.07/0.32 & ( X23 = Y23 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Tree.inject
% 0.07/0.32 thf(fact_5_removeLeaf_Osimps_I5_J,axiom,
% 0.07/0.32 ! [V: a,Vd: a,Ve: tree_a,Vf: tree_a,Va: a,Vb: tree_a,Vc: tree_a] :
% 0.07/0.32 ( ( heapIm837449470Leaf_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
% 0.07/0.32 = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ Vd @ Ve @ Vf ) ) ) @ ( t_a @ V @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ Vd @ Ve @ Vf ) ) ) @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % removeLeaf.simps(5)
% 0.07/0.32 thf(fact_6_removeLeaf_Osimps_I4_J,axiom,
% 0.07/0.32 ! [V: a,Va: a,Vb: tree_a,Vc: tree_a,Vd: a,Ve: tree_a,Vf: tree_a] :
% 0.07/0.32 ( ( heapIm837449470Leaf_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 0.07/0.32 = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ Va @ Vb @ Vc ) ) ) @ ( t_a @ V @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ Va @ Vb @ Vc ) ) ) @ ( t_a @ Vd @ Ve @ Vf ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % removeLeaf.simps(4)
% 0.07/0.32 thf(fact_7_left_Osimps,axiom,
% 0.07/0.32 ! [V: a,L: tree_a,R: tree_a] :
% 0.07/0.32 ( ( heapIm1140443833left_a @ ( t_a @ V @ L @ R ) )
% 0.07/0.32 = L ) ).
% 0.07/0.32
% 0.07/0.32 % left.simps
% 0.07/0.32 thf(fact_8_right_Osimps,axiom,
% 0.07/0.32 ! [V: a,L: tree_a,R: tree_a] :
% 0.07/0.32 ( ( heapIm1257206334ight_a @ ( t_a @ V @ L @ R ) )
% 0.07/0.32 = R ) ).
% 0.07/0.32
% 0.07/0.32 % right.simps
% 0.07/0.32 thf(fact_9__C4__2_Ohyps_C_I1_J,axiom,
% 0.07/0.32 ! [V2: a,T: tree_a] :
% 0.07/0.32 ( ( ( produc686083979Tree_a @ V2 @ T )
% 0.07/0.32 = ( heapIm837449470Leaf_a @ ( t_a @ v1 @ l1 @ r1 ) ) )
% 0.07/0.32 => ( ( ( t_a @ v1 @ l1 @ r1 )
% 0.07/0.32 != e_a )
% 0.07/0.32 => ( ( plus_plus_multiset_a @ ( add_mset_a @ V2 @ zero_zero_multiset_a ) @ ( multiset_a2 @ T ) )
% 0.07/0.32 = ( multiset_a2 @ ( t_a @ v1 @ l1 @ r1 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % "4_2.hyps"(1)
% 0.07/0.32 thf(fact_10_removeLeaf_Osimps_I1_J,axiom,
% 0.07/0.32 ! [V: a] :
% 0.07/0.32 ( ( heapIm837449470Leaf_a @ ( t_a @ V @ e_a @ e_a ) )
% 0.07/0.32 = ( produc686083979Tree_a @ V @ e_a ) ) ).
% 0.07/0.32
% 0.07/0.32 % removeLeaf.simps(1)
% 0.07/0.32 thf(fact_11_fstI,axiom,
% 0.07/0.32 ! [X: produc1127127335iset_a,Y: multiset_a,Z: multiset_a] :
% 0.07/0.32 ( ( X
% 0.07/0.32 = ( produc2037245207iset_a @ Y @ Z ) )
% 0.07/0.32 => ( ( produc653373187iset_a @ X )
% 0.07/0.32 = Y ) ) ).
% 0.07/0.32
% 0.07/0.32 % fstI
% 0.07/0.32 thf(fact_12_fstI,axiom,
% 0.07/0.32 ! [X: product_prod_a_a,Y: a,Z: a] :
% 0.07/0.32 ( ( X
% 0.07/0.32 = ( product_Pair_a_a @ Y @ Z ) )
% 0.07/0.32 => ( ( product_fst_a_a @ X )
% 0.07/0.32 = Y ) ) ).
% 0.07/0.32
% 0.07/0.32 % fstI
% 0.07/0.32 thf(fact_13_fstI,axiom,
% 0.07/0.32 ! [X: produc981471411ee_a_a,Y: tree_a,Z: a] :
% 0.07/0.32 ( ( X
% 0.07/0.32 = ( produc649172771ee_a_a @ Y @ Z ) )
% 0.07/0.32 => ( ( product_fst_Tree_a_a @ X )
% 0.07/0.32 = Y ) ) ).
% 0.07/0.32
% 0.07/0.32 % fstI
% 0.07/0.32 thf(fact_14_fstI,axiom,
% 0.07/0.32 ! [X: produc143150363Tree_a,Y: a,Z: tree_a] :
% 0.07/0.32 ( ( X
% 0.07/0.32 = ( produc686083979Tree_a @ Y @ Z ) )
% 0.07/0.32 => ( ( product_fst_a_Tree_a @ X )
% 0.07/0.32 = Y ) ) ).
% 0.07/0.32
% 0.07/0.32 % fstI
% 0.07/0.32 thf(fact_15_fst__eqD,axiom,
% 0.07/0.32 ! [X: multiset_a,Y: multiset_a,A: multiset_a] :
% 0.07/0.32 ( ( ( produc653373187iset_a @ ( produc2037245207iset_a @ X @ Y ) )
% 0.07/0.32 = A )
% 0.07/0.32 => ( X = A ) ) ).
% 0.07/0.32
% 0.07/0.32 % fst_eqD
% 0.07/0.32 thf(fact_16_fst__eqD,axiom,
% 0.07/0.32 ! [X: a,Y: a,A: a] :
% 0.07/0.32 ( ( ( product_fst_a_a @ ( product_Pair_a_a @ X @ Y ) )
% 0.07/0.32 = A )
% 0.07/0.32 => ( X = A ) ) ).
% 0.07/0.32
% 0.07/0.32 % fst_eqD
% 0.07/0.32 thf(fact_17_fst__eqD,axiom,
% 0.07/0.32 ! [X: tree_a,Y: a,A: tree_a] :
% 0.07/0.32 ( ( ( product_fst_Tree_a_a @ ( produc649172771ee_a_a @ X @ Y ) )
% 0.07/0.32 = A )
% 0.07/0.32 => ( X = A ) ) ).
% 0.07/0.32
% 0.07/0.32 % fst_eqD
% 0.07/0.32 thf(fact_18_fst__eqD,axiom,
% 0.07/0.32 ! [X: a,Y: tree_a,A: a] :
% 0.07/0.32 ( ( ( product_fst_a_Tree_a @ ( produc686083979Tree_a @ X @ Y ) )
% 0.07/0.32 = A )
% 0.07/0.32 => ( X = A ) ) ).
% 0.07/0.32
% 0.07/0.32 % fst_eqD
% 0.07/0.32 thf(fact_19_fst__conv,axiom,
% 0.07/0.32 ! [X1: multiset_a,X2: multiset_a] :
% 0.07/0.32 ( ( produc653373187iset_a @ ( produc2037245207iset_a @ X1 @ X2 ) )
% 0.07/0.32 = X1 ) ).
% 0.07/0.32
% 0.07/0.32 % fst_conv
% 0.07/0.32 thf(fact_20_fst__conv,axiom,
% 0.07/0.32 ! [X1: a,X2: a] :
% 0.07/0.32 ( ( product_fst_a_a @ ( product_Pair_a_a @ X1 @ X2 ) )
% 0.07/0.32 = X1 ) ).
% 0.07/0.32
% 0.07/0.32 % fst_conv
% 0.07/0.32 thf(fact_21_fst__conv,axiom,
% 0.07/0.32 ! [X1: tree_a,X2: a] :
% 0.07/0.32 ( ( product_fst_Tree_a_a @ ( produc649172771ee_a_a @ X1 @ X2 ) )
% 0.07/0.32 = X1 ) ).
% 0.07/0.32
% 0.07/0.32 % fst_conv
% 0.07/0.32 thf(fact_22_fst__conv,axiom,
% 0.07/0.32 ! [X1: a,X2: tree_a] :
% 0.07/0.32 ( ( product_fst_a_Tree_a @ ( produc686083979Tree_a @ X1 @ X2 ) )
% 0.07/0.32 = X1 ) ).
% 0.07/0.32
% 0.07/0.32 % fst_conv
% 0.07/0.32 thf(fact_23_assms_I2_J,axiom,
% 0.07/0.32 t != e_a ).
% 0.07/0.32
% 0.07/0.32 % assms(2)
% 0.07/0.32 thf(fact_24_prod_Oinject,axiom,
% 0.07/0.32 ! [X1: tree_a,X2: a,Y1: tree_a,Y2: a] :
% 0.07/0.32 ( ( ( produc649172771ee_a_a @ X1 @ X2 )
% 0.07/0.32 = ( produc649172771ee_a_a @ Y1 @ Y2 ) )
% 0.07/0.32 = ( ( X1 = Y1 )
% 0.07/0.32 & ( X2 = Y2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % prod.inject
% 0.07/0.32 thf(fact_25_prod_Oinject,axiom,
% 0.07/0.32 ! [X1: multiset_a,X2: multiset_a,Y1: multiset_a,Y2: multiset_a] :
% 0.07/0.32 ( ( ( produc2037245207iset_a @ X1 @ X2 )
% 0.07/0.32 = ( produc2037245207iset_a @ Y1 @ Y2 ) )
% 0.07/0.32 = ( ( X1 = Y1 )
% 0.07/0.32 & ( X2 = Y2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % prod.inject
% 0.07/0.32 thf(fact_26_prod_Oinject,axiom,
% 0.07/0.32 ! [X1: a,X2: a,Y1: a,Y2: a] :
% 0.07/0.32 ( ( ( product_Pair_a_a @ X1 @ X2 )
% 0.07/0.32 = ( product_Pair_a_a @ Y1 @ Y2 ) )
% 0.07/0.32 = ( ( X1 = Y1 )
% 0.07/0.32 & ( X2 = Y2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % prod.inject
% 0.07/0.32 thf(fact_27_prod_Oinject,axiom,
% 0.07/0.32 ! [X1: a,X2: tree_a,Y1: a,Y2: tree_a] :
% 0.07/0.32 ( ( ( produc686083979Tree_a @ X1 @ X2 )
% 0.07/0.32 = ( produc686083979Tree_a @ Y1 @ Y2 ) )
% 0.07/0.32 = ( ( X1 = Y1 )
% 0.07/0.32 & ( X2 = Y2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % prod.inject
% 0.07/0.32 thf(fact_28_old_Oprod_Oinject,axiom,
% 0.07/0.32 ! [A: tree_a,B: a,A2: tree_a,B2: a] :
% 0.07/0.32 ( ( ( produc649172771ee_a_a @ A @ B )
% 0.07/0.32 = ( produc649172771ee_a_a @ A2 @ B2 ) )
% 0.07/0.32 = ( ( A = A2 )
% 0.07/0.32 & ( B = B2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % old.prod.inject
% 0.07/0.32 thf(fact_29_old_Oprod_Oinject,axiom,
% 0.07/0.32 ! [A: multiset_a,B: multiset_a,A2: multiset_a,B2: multiset_a] :
% 0.07/0.32 ( ( ( produc2037245207iset_a @ A @ B )
% 0.07/0.32 = ( produc2037245207iset_a @ A2 @ B2 ) )
% 0.07/0.32 = ( ( A = A2 )
% 0.07/0.32 & ( B = B2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % old.prod.inject
% 0.07/0.32 thf(fact_30_old_Oprod_Oinject,axiom,
% 0.07/0.32 ! [A: a,B: a,A2: a,B2: a] :
% 0.07/0.32 ( ( ( product_Pair_a_a @ A @ B )
% 0.07/0.32 = ( product_Pair_a_a @ A2 @ B2 ) )
% 0.07/0.32 = ( ( A = A2 )
% 0.07/0.32 & ( B = B2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % old.prod.inject
% 0.07/0.32 thf(fact_31_old_Oprod_Oinject,axiom,
% 0.07/0.32 ! [A: a,B: tree_a,A2: a,B2: tree_a] :
% 0.07/0.32 ( ( ( produc686083979Tree_a @ A @ B )
% 0.07/0.32 = ( produc686083979Tree_a @ A2 @ B2 ) )
% 0.07/0.32 = ( ( A = A2 )
% 0.07/0.32 & ( B = B2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % old.prod.inject
% 0.07/0.32 thf(fact_32_prod_Ocollapse,axiom,
% 0.07/0.32 ! [Prod: produc1127127335iset_a] :
% 0.07/0.32 ( ( produc2037245207iset_a @ ( produc653373187iset_a @ Prod ) @ ( produc148952133iset_a @ Prod ) )
% 0.07/0.32 = Prod ) ).
% 0.07/0.32
% 0.07/0.32 % prod.collapse
% 0.07/0.32 thf(fact_33_prod_Ocollapse,axiom,
% 0.07/0.32 ! [Prod: product_prod_a_a] :
% 0.07/0.32 ( ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) )
% 0.07/0.32 = Prod ) ).
% 0.07/0.32
% 0.07/0.32 % prod.collapse
% 0.07/0.32 thf(fact_34_prod_Ocollapse,axiom,
% 0.07/0.32 ! [Prod: produc981471411ee_a_a] :
% 0.07/0.32 ( ( produc649172771ee_a_a @ ( product_fst_Tree_a_a @ Prod ) @ ( product_snd_Tree_a_a @ Prod ) )
% 0.07/0.32 = Prod ) ).
% 0.07/0.32
% 0.07/0.32 % prod.collapse
% 0.07/0.32 thf(fact_35_prod_Ocollapse,axiom,
% 0.07/0.32 ! [Prod: produc143150363Tree_a] :
% 0.07/0.32 ( ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ Prod ) @ ( product_snd_a_Tree_a @ Prod ) )
% 0.07/0.32 = Prod ) ).
% 0.07/0.32
% 0.07/0.32 % prod.collapse
% 0.07/0.32 thf(fact_36_multiset_Osimps_I2_J,axiom,
% 0.07/0.32 ! [V: a,L: tree_a,R: tree_a] :
% 0.07/0.32 ( ( multiset_a2 @ ( t_a @ V @ L @ R ) )
% 0.07/0.32 = ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ ( multiset_a2 @ L ) @ ( add_mset_a @ V @ zero_zero_multiset_a ) ) @ ( multiset_a2 @ R ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multiset.simps(2)
% 0.07/0.32 thf(fact_37_multiset_Osimps_I1_J,axiom,
% 0.07/0.32 ( ( multiset_a2 @ e_a )
% 0.07/0.32 = zero_zero_multiset_a ) ).
% 0.07/0.32
% 0.07/0.32 % multiset.simps(1)
% 0.07/0.32 thf(fact_38_sndI,axiom,
% 0.07/0.32 ! [X: produc1127127335iset_a,Y: multiset_a,Z: multiset_a] :
% 0.07/0.32 ( ( X
% 0.07/0.32 = ( produc2037245207iset_a @ Y @ Z ) )
% 0.07/0.32 => ( ( produc148952133iset_a @ X )
% 0.07/0.32 = Z ) ) ).
% 0.07/0.32
% 0.07/0.32 % sndI
% 0.07/0.32 thf(fact_39_sndI,axiom,
% 0.07/0.32 ! [X: product_prod_a_a,Y: a,Z: a] :
% 0.07/0.32 ( ( X
% 0.07/0.32 = ( product_Pair_a_a @ Y @ Z ) )
% 0.07/0.32 => ( ( product_snd_a_a @ X )
% 0.07/0.32 = Z ) ) ).
% 0.07/0.32
% 0.07/0.32 % sndI
% 0.07/0.32 thf(fact_40_sndI,axiom,
% 0.07/0.32 ! [X: produc981471411ee_a_a,Y: tree_a,Z: a] :
% 0.07/0.32 ( ( X
% 0.07/0.32 = ( produc649172771ee_a_a @ Y @ Z ) )
% 0.07/0.32 => ( ( product_snd_Tree_a_a @ X )
% 0.07/0.32 = Z ) ) ).
% 0.07/0.32
% 0.07/0.32 % sndI
% 0.07/0.32 thf(fact_41_sndI,axiom,
% 0.07/0.32 ! [X: produc143150363Tree_a,Y: a,Z: tree_a] :
% 0.07/0.32 ( ( X
% 0.07/0.32 = ( produc686083979Tree_a @ Y @ Z ) )
% 0.07/0.32 => ( ( product_snd_a_Tree_a @ X )
% 0.07/0.32 = Z ) ) ).
% 0.07/0.32
% 0.07/0.32 % sndI
% 0.07/0.32 thf(fact_42_snd__eqD,axiom,
% 0.07/0.32 ! [X: multiset_a,Y: multiset_a,A: multiset_a] :
% 0.07/0.32 ( ( ( produc148952133iset_a @ ( produc2037245207iset_a @ X @ Y ) )
% 0.07/0.32 = A )
% 0.07/0.32 => ( Y = A ) ) ).
% 0.07/0.32
% 0.07/0.32 % snd_eqD
% 0.07/0.32 thf(fact_43_snd__eqD,axiom,
% 0.07/0.32 ! [X: a,Y: a,A: a] :
% 0.07/0.32 ( ( ( product_snd_a_a @ ( product_Pair_a_a @ X @ Y ) )
% 0.07/0.32 = A )
% 0.07/0.32 => ( Y = A ) ) ).
% 0.07/0.32
% 0.07/0.32 % snd_eqD
% 0.07/0.32 thf(fact_44_snd__eqD,axiom,
% 0.07/0.32 ! [X: tree_a,Y: a,A: a] :
% 0.07/0.32 ( ( ( product_snd_Tree_a_a @ ( produc649172771ee_a_a @ X @ Y ) )
% 0.07/0.32 = A )
% 0.07/0.32 => ( Y = A ) ) ).
% 0.07/0.32
% 0.07/0.32 % snd_eqD
% 0.07/0.32 thf(fact_45_snd__eqD,axiom,
% 0.07/0.32 ! [X: a,Y: tree_a,A: tree_a] :
% 0.07/0.32 ( ( ( product_snd_a_Tree_a @ ( produc686083979Tree_a @ X @ Y ) )
% 0.07/0.32 = A )
% 0.07/0.32 => ( Y = A ) ) ).
% 0.07/0.32
% 0.07/0.32 % snd_eqD
% 0.07/0.32 thf(fact_46_snd__conv,axiom,
% 0.07/0.32 ! [X1: multiset_a,X2: multiset_a] :
% 0.07/0.32 ( ( produc148952133iset_a @ ( produc2037245207iset_a @ X1 @ X2 ) )
% 0.07/0.32 = X2 ) ).
% 0.07/0.32
% 0.07/0.32 % snd_conv
% 0.07/0.32 thf(fact_47_snd__conv,axiom,
% 0.07/0.32 ! [X1: a,X2: a] :
% 0.07/0.32 ( ( product_snd_a_a @ ( product_Pair_a_a @ X1 @ X2 ) )
% 0.07/0.32 = X2 ) ).
% 0.07/0.32
% 0.07/0.32 % snd_conv
% 0.07/0.32 thf(fact_48_snd__conv,axiom,
% 0.07/0.32 ! [X1: tree_a,X2: a] :
% 0.07/0.32 ( ( product_snd_Tree_a_a @ ( produc649172771ee_a_a @ X1 @ X2 ) )
% 0.07/0.32 = X2 ) ).
% 0.07/0.32
% 0.07/0.32 % snd_conv
% 0.07/0.32 thf(fact_49_snd__conv,axiom,
% 0.07/0.32 ! [X1: a,X2: tree_a] :
% 0.07/0.32 ( ( product_snd_a_Tree_a @ ( produc686083979Tree_a @ X1 @ X2 ) )
% 0.07/0.32 = X2 ) ).
% 0.07/0.32
% 0.07/0.32 % snd_conv
% 0.07/0.32 thf(fact_50_surj__pair,axiom,
% 0.07/0.32 ! [P: produc981471411ee_a_a] :
% 0.07/0.32 ? [X3: tree_a,Y3: a] :
% 0.07/0.32 ( P
% 0.07/0.32 = ( produc649172771ee_a_a @ X3 @ Y3 ) ) ).
% 0.07/0.32
% 0.07/0.32 % surj_pair
% 0.07/0.32 thf(fact_51_surj__pair,axiom,
% 0.07/0.32 ! [P: produc1127127335iset_a] :
% 0.07/0.32 ? [X3: multiset_a,Y3: multiset_a] :
% 0.07/0.32 ( P
% 0.07/0.32 = ( produc2037245207iset_a @ X3 @ Y3 ) ) ).
% 0.07/0.32
% 0.07/0.32 % surj_pair
% 0.07/0.32 thf(fact_52_surj__pair,axiom,
% 0.07/0.32 ! [P: product_prod_a_a] :
% 0.07/0.32 ? [X3: a,Y3: a] :
% 0.07/0.32 ( P
% 0.07/0.32 = ( product_Pair_a_a @ X3 @ Y3 ) ) ).
% 0.07/0.32
% 0.07/0.32 % surj_pair
% 0.07/0.32 thf(fact_53_surj__pair,axiom,
% 0.07/0.32 ! [P: produc143150363Tree_a] :
% 0.07/0.32 ? [X3: a,Y3: tree_a] :
% 0.07/0.32 ( P
% 0.07/0.32 = ( produc686083979Tree_a @ X3 @ Y3 ) ) ).
% 0.07/0.32
% 0.07/0.32 % surj_pair
% 0.07/0.32 thf(fact_54_prod__cases,axiom,
% 0.07/0.32 ! [P2: produc981471411ee_a_a > $o,P: produc981471411ee_a_a] :
% 0.07/0.32 ( ! [A3: tree_a,B3: a] : ( P2 @ ( produc649172771ee_a_a @ A3 @ B3 ) )
% 0.07/0.32 => ( P2 @ P ) ) ).
% 0.07/0.32
% 0.07/0.32 % prod_cases
% 0.07/0.32 thf(fact_55_prod__cases,axiom,
% 0.07/0.32 ! [P2: produc1127127335iset_a > $o,P: produc1127127335iset_a] :
% 0.07/0.32 ( ! [A3: multiset_a,B3: multiset_a] : ( P2 @ ( produc2037245207iset_a @ A3 @ B3 ) )
% 0.07/0.32 => ( P2 @ P ) ) ).
% 0.07/0.32
% 0.07/0.32 % prod_cases
% 0.07/0.32 thf(fact_56_prod__cases,axiom,
% 0.07/0.32 ! [P2: product_prod_a_a > $o,P: product_prod_a_a] :
% 0.07/0.32 ( ! [A3: a,B3: a] : ( P2 @ ( product_Pair_a_a @ A3 @ B3 ) )
% 0.07/0.32 => ( P2 @ P ) ) ).
% 0.07/0.32
% 0.07/0.32 % prod_cases
% 0.07/0.32 thf(fact_57_prod__cases,axiom,
% 0.07/0.32 ! [P2: produc143150363Tree_a > $o,P: produc143150363Tree_a] :
% 0.07/0.32 ( ! [A3: a,B3: tree_a] : ( P2 @ ( produc686083979Tree_a @ A3 @ B3 ) )
% 0.07/0.32 => ( P2 @ P ) ) ).
% 0.07/0.32
% 0.07/0.32 % prod_cases
% 0.07/0.32 thf(fact_58_Pair__inject,axiom,
% 0.07/0.32 ! [A: tree_a,B: a,A2: tree_a,B2: a] :
% 0.07/0.32 ( ( ( produc649172771ee_a_a @ A @ B )
% 0.07/0.32 = ( produc649172771ee_a_a @ A2 @ B2 ) )
% 0.07/0.32 => ~ ( ( A = A2 )
% 0.07/0.32 => ( B != B2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Pair_inject
% 0.07/0.32 thf(fact_59_Pair__inject,axiom,
% 0.07/0.32 ! [A: multiset_a,B: multiset_a,A2: multiset_a,B2: multiset_a] :
% 0.07/0.32 ( ( ( produc2037245207iset_a @ A @ B )
% 0.07/0.32 = ( produc2037245207iset_a @ A2 @ B2 ) )
% 0.07/0.32 => ~ ( ( A = A2 )
% 0.07/0.32 => ( B != B2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Pair_inject
% 0.07/0.32 thf(fact_60_Pair__inject,axiom,
% 0.07/0.32 ! [A: a,B: a,A2: a,B2: a] :
% 0.07/0.32 ( ( ( product_Pair_a_a @ A @ B )
% 0.07/0.32 = ( product_Pair_a_a @ A2 @ B2 ) )
% 0.07/0.32 => ~ ( ( A = A2 )
% 0.07/0.32 => ( B != B2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Pair_inject
% 0.07/0.32 thf(fact_61_Pair__inject,axiom,
% 0.07/0.32 ! [A: a,B: tree_a,A2: a,B2: tree_a] :
% 0.07/0.32 ( ( ( produc686083979Tree_a @ A @ B )
% 0.07/0.32 = ( produc686083979Tree_a @ A2 @ B2 ) )
% 0.07/0.32 => ~ ( ( A = A2 )
% 0.07/0.32 => ( B != B2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Pair_inject
% 0.07/0.32 thf(fact_62_old_Oprod_Oexhaust,axiom,
% 0.07/0.32 ! [Y: produc981471411ee_a_a] :
% 0.07/0.32 ~ ! [A3: tree_a,B3: a] :
% 0.07/0.32 ( Y
% 0.07/0.32 != ( produc649172771ee_a_a @ A3 @ B3 ) ) ).
% 0.07/0.32
% 0.07/0.32 % old.prod.exhaust
% 0.07/0.32 thf(fact_63_old_Oprod_Oexhaust,axiom,
% 0.07/0.32 ! [Y: produc1127127335iset_a] :
% 0.07/0.32 ~ ! [A3: multiset_a,B3: multiset_a] :
% 0.07/0.32 ( Y
% 0.07/0.32 != ( produc2037245207iset_a @ A3 @ B3 ) ) ).
% 0.07/0.32
% 0.07/0.32 % old.prod.exhaust
% 0.07/0.32 thf(fact_64_old_Oprod_Oexhaust,axiom,
% 0.07/0.32 ! [Y: product_prod_a_a] :
% 0.07/0.32 ~ ! [A3: a,B3: a] :
% 0.07/0.32 ( Y
% 0.07/0.32 != ( product_Pair_a_a @ A3 @ B3 ) ) ).
% 0.07/0.32
% 0.07/0.32 % old.prod.exhaust
% 0.07/0.32 thf(fact_65_old_Oprod_Oexhaust,axiom,
% 0.07/0.32 ! [Y: produc143150363Tree_a] :
% 0.07/0.32 ~ ! [A3: a,B3: tree_a] :
% 0.07/0.32 ( Y
% 0.07/0.32 != ( produc686083979Tree_a @ A3 @ B3 ) ) ).
% 0.07/0.32
% 0.07/0.32 % old.prod.exhaust
% 0.07/0.32 thf(fact_66_old_Oprod_Oinducts,axiom,
% 0.07/0.32 ! [P2: produc981471411ee_a_a > $o,Prod: produc981471411ee_a_a] :
% 0.07/0.32 ( ! [A3: tree_a,B3: a] : ( P2 @ ( produc649172771ee_a_a @ A3 @ B3 ) )
% 0.07/0.32 => ( P2 @ Prod ) ) ).
% 0.07/0.32
% 0.07/0.32 % old.prod.inducts
% 0.07/0.32 thf(fact_67_old_Oprod_Oinducts,axiom,
% 0.07/0.32 ! [P2: produc1127127335iset_a > $o,Prod: produc1127127335iset_a] :
% 0.07/0.32 ( ! [A3: multiset_a,B3: multiset_a] : ( P2 @ ( produc2037245207iset_a @ A3 @ B3 ) )
% 0.07/0.32 => ( P2 @ Prod ) ) ).
% 0.07/0.32
% 0.07/0.32 % old.prod.inducts
% 0.07/0.32 thf(fact_68_old_Oprod_Oinducts,axiom,
% 0.07/0.32 ! [P2: product_prod_a_a > $o,Prod: product_prod_a_a] :
% 0.07/0.32 ( ! [A3: a,B3: a] : ( P2 @ ( product_Pair_a_a @ A3 @ B3 ) )
% 0.07/0.32 => ( P2 @ Prod ) ) ).
% 0.07/0.32
% 0.07/0.32 % old.prod.inducts
% 0.07/0.32 thf(fact_69_old_Oprod_Oinducts,axiom,
% 0.07/0.32 ! [P2: produc143150363Tree_a > $o,Prod: produc143150363Tree_a] :
% 0.07/0.32 ( ! [A3: a,B3: tree_a] : ( P2 @ ( produc686083979Tree_a @ A3 @ B3 ) )
% 0.07/0.32 => ( P2 @ Prod ) ) ).
% 0.07/0.32
% 0.07/0.32 % old.prod.inducts
% 0.07/0.32 thf(fact_70_HS_Omultiset__empty,axiom,
% 0.07/0.32 ( ( multiset_a2 @ e_a )
% 0.07/0.32 = zero_zero_multiset_a ) ).
% 0.07/0.32
% 0.07/0.32 % HS.multiset_empty
% 0.07/0.32 thf(fact_71_surjective__pairing,axiom,
% 0.07/0.32 ! [T2: produc1127127335iset_a] :
% 0.07/0.32 ( T2
% 0.07/0.32 = ( produc2037245207iset_a @ ( produc653373187iset_a @ T2 ) @ ( produc148952133iset_a @ T2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % surjective_pairing
% 0.07/0.32 thf(fact_72_surjective__pairing,axiom,
% 0.07/0.32 ! [T2: product_prod_a_a] :
% 0.07/0.32 ( T2
% 0.07/0.32 = ( product_Pair_a_a @ ( product_fst_a_a @ T2 ) @ ( product_snd_a_a @ T2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % surjective_pairing
% 0.07/0.32 thf(fact_73_surjective__pairing,axiom,
% 0.07/0.32 ! [T2: produc981471411ee_a_a] :
% 0.07/0.32 ( T2
% 0.07/0.32 = ( produc649172771ee_a_a @ ( product_fst_Tree_a_a @ T2 ) @ ( product_snd_Tree_a_a @ T2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % surjective_pairing
% 0.07/0.32 thf(fact_74_surjective__pairing,axiom,
% 0.07/0.32 ! [T2: produc143150363Tree_a] :
% 0.07/0.32 ( T2
% 0.07/0.32 = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ T2 ) @ ( product_snd_a_Tree_a @ T2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % surjective_pairing
% 0.07/0.32 thf(fact_75_prod_Oexhaust__sel,axiom,
% 0.07/0.32 ! [Prod: produc1127127335iset_a] :
% 0.07/0.32 ( Prod
% 0.07/0.32 = ( produc2037245207iset_a @ ( produc653373187iset_a @ Prod ) @ ( produc148952133iset_a @ Prod ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % prod.exhaust_sel
% 0.07/0.32 thf(fact_76_prod_Oexhaust__sel,axiom,
% 0.07/0.32 ! [Prod: product_prod_a_a] :
% 0.07/0.32 ( Prod
% 0.07/0.32 = ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % prod.exhaust_sel
% 0.07/0.32 thf(fact_77_prod_Oexhaust__sel,axiom,
% 0.07/0.32 ! [Prod: produc981471411ee_a_a] :
% 0.07/0.32 ( Prod
% 0.07/0.32 = ( produc649172771ee_a_a @ ( product_fst_Tree_a_a @ Prod ) @ ( product_snd_Tree_a_a @ Prod ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % prod.exhaust_sel
% 0.07/0.32 thf(fact_78_prod_Oexhaust__sel,axiom,
% 0.07/0.32 ! [Prod: produc143150363Tree_a] :
% 0.07/0.32 ( Prod
% 0.07/0.32 = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ Prod ) @ ( product_snd_a_Tree_a @ Prod ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % prod.exhaust_sel
% 0.07/0.32 thf(fact_79_prod__eq__iff,axiom,
% 0.07/0.32 ( ( ^ [Y4: produc981471411ee_a_a,Z2: produc981471411ee_a_a] : ( Y4 = Z2 ) )
% 0.07/0.32 = ( ^ [S: produc981471411ee_a_a,T3: produc981471411ee_a_a] :
% 0.07/0.32 ( ( ( product_fst_Tree_a_a @ S )
% 0.07/0.32 = ( product_fst_Tree_a_a @ T3 ) )
% 0.07/0.32 & ( ( product_snd_Tree_a_a @ S )
% 0.07/0.32 = ( product_snd_Tree_a_a @ T3 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % prod_eq_iff
% 0.07/0.32 thf(fact_80_prod__eq__iff,axiom,
% 0.07/0.32 ( ( ^ [Y4: produc143150363Tree_a,Z2: produc143150363Tree_a] : ( Y4 = Z2 ) )
% 0.07/0.32 = ( ^ [S: produc143150363Tree_a,T3: produc143150363Tree_a] :
% 0.07/0.32 ( ( ( product_fst_a_Tree_a @ S )
% 0.07/0.32 = ( product_fst_a_Tree_a @ T3 ) )
% 0.07/0.32 & ( ( product_snd_a_Tree_a @ S )
% 0.07/0.32 = ( product_snd_a_Tree_a @ T3 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % prod_eq_iff
% 0.07/0.32 thf(fact_81_prod_Oexpand,axiom,
% 0.07/0.32 ! [Prod: produc981471411ee_a_a,Prod2: produc981471411ee_a_a] :
% 0.07/0.32 ( ( ( ( product_fst_Tree_a_a @ Prod )
% 0.07/0.32 = ( product_fst_Tree_a_a @ Prod2 ) )
% 0.07/0.32 & ( ( product_snd_Tree_a_a @ Prod )
% 0.07/0.32 = ( product_snd_Tree_a_a @ Prod2 ) ) )
% 0.07/0.32 => ( Prod = Prod2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % prod.expand
% 0.07/0.32 thf(fact_82_prod_Oexpand,axiom,
% 0.07/0.32 ! [Prod: produc143150363Tree_a,Prod2: produc143150363Tree_a] :
% 0.07/0.32 ( ( ( ( product_fst_a_Tree_a @ Prod )
% 0.07/0.32 = ( product_fst_a_Tree_a @ Prod2 ) )
% 0.07/0.32 & ( ( product_snd_a_Tree_a @ Prod )
% 0.07/0.32 = ( product_snd_a_Tree_a @ Prod2 ) ) )
% 0.07/0.32 => ( Prod = Prod2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % prod.expand
% 0.07/0.32 thf(fact_83_prod__eqI,axiom,
% 0.07/0.32 ! [P: produc981471411ee_a_a,Q: produc981471411ee_a_a] :
% 0.07/0.32 ( ( ( product_fst_Tree_a_a @ P )
% 0.07/0.32 = ( product_fst_Tree_a_a @ Q ) )
% 0.07/0.32 => ( ( ( product_snd_Tree_a_a @ P )
% 0.07/0.32 = ( product_snd_Tree_a_a @ Q ) )
% 0.07/0.32 => ( P = Q ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % prod_eqI
% 0.07/0.32 thf(fact_84_prod__eqI,axiom,
% 0.07/0.32 ! [P: produc143150363Tree_a,Q: produc143150363Tree_a] :
% 0.07/0.32 ( ( ( product_fst_a_Tree_a @ P )
% 0.07/0.32 = ( product_fst_a_Tree_a @ Q ) )
% 0.07/0.32 => ( ( ( product_snd_a_Tree_a @ P )
% 0.07/0.32 = ( product_snd_a_Tree_a @ Q ) )
% 0.07/0.32 => ( P = Q ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % prod_eqI
% 0.07/0.32 thf(fact_85_is__heap_Ocases,axiom,
% 0.07/0.32 ! [X: tree_a] :
% 0.07/0.32 ( ( X != e_a )
% 0.07/0.32 => ( ! [V3: a] :
% 0.07/0.32 ( X
% 0.07/0.32 != ( t_a @ V3 @ e_a @ e_a ) )
% 0.07/0.32 => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 0.07/0.32 ( X
% 0.07/0.32 != ( t_a @ V3 @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 0.07/0.32 => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 0.07/0.32 ( X
% 0.07/0.32 != ( t_a @ V3 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
% 0.07/0.32 => ~ ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a] :
% 0.07/0.32 ( X
% 0.07/0.32 != ( t_a @ V3 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % is_heap.cases
% 0.07/0.32 thf(fact_86_mem__Collect__eq,axiom,
% 0.07/0.32 ! [A: produc143150363Tree_a,P2: produc143150363Tree_a > $o] :
% 0.07/0.32 ( ( member254315204Tree_a @ A @ ( collec486369286Tree_a @ P2 ) )
% 0.07/0.32 = ( P2 @ A ) ) ).
% 0.07/0.32
% 0.07/0.32 % mem_Collect_eq
% 0.07/0.32 thf(fact_87_mem__Collect__eq,axiom,
% 0.07/0.32 ! [A: a,P2: a > $o] :
% 0.07/0.32 ( ( member_a @ A @ ( collect_a @ P2 ) )
% 0.07/0.32 = ( P2 @ A ) ) ).
% 0.07/0.32
% 0.07/0.32 % mem_Collect_eq
% 0.07/0.32 thf(fact_88_mem__Collect__eq,axiom,
% 0.07/0.32 ! [A: produc1127127335iset_a,P2: produc1127127335iset_a > $o] :
% 0.07/0.32 ( ( member340150864iset_a @ A @ ( collec1104713362iset_a @ P2 ) )
% 0.07/0.32 = ( P2 @ A ) ) ).
% 0.07/0.32
% 0.07/0.32 % mem_Collect_eq
% 0.07/0.32 thf(fact_89_mem__Collect__eq,axiom,
% 0.07/0.32 ! [A: product_prod_a_a,P2: product_prod_a_a > $o] :
% 0.07/0.32 ( ( member449909584od_a_a @ A @ ( collec645855634od_a_a @ P2 ) )
% 0.07/0.32 = ( P2 @ A ) ) ).
% 0.07/0.32
% 0.07/0.32 % mem_Collect_eq
% 0.07/0.32 thf(fact_90_Collect__mem__eq,axiom,
% 0.07/0.32 ! [A4: set_Pr1070816379Tree_a] :
% 0.07/0.32 ( ( collec486369286Tree_a
% 0.07/0.32 @ ^ [X4: produc143150363Tree_a] : ( member254315204Tree_a @ X4 @ A4 ) )
% 0.07/0.32 = A4 ) ).
% 0.07/0.32
% 0.07/0.32 % Collect_mem_eq
% 0.07/0.32 thf(fact_91_Collect__mem__eq,axiom,
% 0.07/0.32 ! [A4: set_a] :
% 0.07/0.32 ( ( collect_a
% 0.07/0.32 @ ^ [X4: a] : ( member_a @ X4 @ A4 ) )
% 0.07/0.32 = A4 ) ).
% 0.07/0.32
% 0.07/0.32 % Collect_mem_eq
% 0.07/0.32 thf(fact_92_Collect__mem__eq,axiom,
% 0.07/0.32 ! [A4: set_Pr158363655iset_a] :
% 0.07/0.32 ( ( collec1104713362iset_a
% 0.07/0.32 @ ^ [X4: produc1127127335iset_a] : ( member340150864iset_a @ X4 @ A4 ) )
% 0.07/0.32 = A4 ) ).
% 0.07/0.32
% 0.07/0.32 % Collect_mem_eq
% 0.07/0.32 thf(fact_93_Collect__mem__eq,axiom,
% 0.07/0.32 ! [A4: set_Product_prod_a_a] :
% 0.07/0.32 ( ( collec645855634od_a_a
% 0.07/0.32 @ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ A4 ) )
% 0.07/0.32 = A4 ) ).
% 0.07/0.32
% 0.07/0.32 % Collect_mem_eq
% 0.07/0.32 thf(fact_94_Tree_Oexhaust,axiom,
% 0.07/0.32 ! [Y: tree_a] :
% 0.07/0.32 ( ( Y != e_a )
% 0.07/0.32 => ~ ! [X212: a,X222: tree_a,X232: tree_a] :
% 0.07/0.32 ( Y
% 0.07/0.32 != ( t_a @ X212 @ X222 @ X232 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Tree.exhaust
% 0.07/0.32 thf(fact_95_Tree_Oinduct,axiom,
% 0.07/0.32 ! [P2: tree_a > $o,Tree: tree_a] :
% 0.07/0.32 ( ( P2 @ e_a )
% 0.07/0.32 => ( ! [X12: a,X24: tree_a,X32: tree_a] :
% 0.07/0.32 ( ( P2 @ X24 )
% 0.07/0.32 => ( ( P2 @ X32 )
% 0.07/0.32 => ( P2 @ ( t_a @ X12 @ X24 @ X32 ) ) ) )
% 0.07/0.32 => ( P2 @ Tree ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Tree.induct
% 0.07/0.32 thf(fact_96_Tree_Odistinct_I1_J,axiom,
% 0.07/0.32 ! [X21: a,X22: tree_a,X23: tree_a] :
% 0.07/0.32 ( e_a
% 0.07/0.32 != ( t_a @ X21 @ X22 @ X23 ) ) ).
% 0.07/0.32
% 0.07/0.32 % Tree.distinct(1)
% 0.07/0.32 thf(fact_97_removeLeaf_Oinduct,axiom,
% 0.07/0.32 ! [P2: tree_a > $o,A0: tree_a] :
% 0.07/0.32 ( ! [V3: a] : ( P2 @ ( t_a @ V3 @ e_a @ e_a ) )
% 0.07/0.32 => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 0.07/0.32 ( ( P2 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
% 0.07/0.32 => ( ( P2 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
% 0.07/0.32 => ( P2 @ ( t_a @ V3 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) ) ) )
% 0.07/0.32 => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 0.07/0.32 ( ( P2 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
% 0.07/0.32 => ( ( P2 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
% 0.07/0.32 => ( P2 @ ( t_a @ V3 @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) )
% 0.07/0.32 => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a] :
% 0.07/0.32 ( ( P2 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
% 0.07/0.32 => ( ( P2 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
% 0.07/0.32 => ( P2 @ ( t_a @ V3 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) )
% 0.07/0.32 => ( ! [V3: a,Vd2: a,Ve2: tree_a,Vf2: tree_a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 0.07/0.32 ( ( P2 @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) )
% 0.07/0.32 => ( ( P2 @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) )
% 0.07/0.32 => ( P2 @ ( t_a @ V3 @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) )
% 0.07/0.32 => ( ( P2 @ e_a )
% 0.07/0.32 => ( P2 @ A0 ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % removeLeaf.induct
% 0.07/0.32 thf(fact_98_removeLeaf_Ocases,axiom,
% 0.07/0.32 ! [X: tree_a] :
% 0.07/0.32 ( ! [V3: a] :
% 0.07/0.32 ( X
% 0.07/0.32 != ( t_a @ V3 @ e_a @ e_a ) )
% 0.07/0.32 => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 0.07/0.32 ( X
% 0.07/0.32 != ( t_a @ V3 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
% 0.07/0.32 => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 0.07/0.32 ( X
% 0.07/0.32 != ( t_a @ V3 @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 0.07/0.32 => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a] :
% 0.07/0.32 ( X
% 0.07/0.32 != ( t_a @ V3 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
% 0.07/0.32 => ( ! [V3: a,Vd2: a,Ve2: tree_a,Vf2: tree_a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 0.07/0.32 ( X
% 0.07/0.32 != ( t_a @ V3 @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 0.07/0.32 => ( X = e_a ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % removeLeaf.cases
% 0.07/0.32 thf(fact_99_siftDown_Ocases,axiom,
% 0.07/0.32 ! [X: tree_a] :
% 0.07/0.32 ( ( X != e_a )
% 0.07/0.32 => ( ! [V3: a] :
% 0.07/0.32 ( X
% 0.07/0.32 != ( t_a @ V3 @ e_a @ e_a ) )
% 0.07/0.32 => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 0.07/0.32 ( X
% 0.07/0.32 != ( t_a @ V3 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
% 0.07/0.32 => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 0.07/0.32 ( X
% 0.07/0.32 != ( t_a @ V3 @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 0.07/0.32 => ~ ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a] :
% 0.07/0.32 ( X
% 0.07/0.32 != ( t_a @ V3 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % siftDown.cases
% 0.07/0.32 thf(fact_100_removeLeaf_Osimps_I2_J,axiom,
% 0.07/0.32 ! [V: a,Va: a,Vb: tree_a,Vc: tree_a] :
% 0.07/0.32 ( ( heapIm837449470Leaf_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) )
% 0.07/0.32 = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ Va @ Vb @ Vc ) ) ) @ ( t_a @ V @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ Va @ Vb @ Vc ) ) ) @ e_a ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % removeLeaf.simps(2)
% 0.07/0.32 thf(fact_101_removeLeaf_Osimps_I3_J,axiom,
% 0.07/0.32 ! [V: a,Va: a,Vb: tree_a,Vc: tree_a] :
% 0.07/0.32 ( ( heapIm837449470Leaf_a @ ( t_a @ V @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) )
% 0.07/0.32 = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ Va @ Vb @ Vc ) ) ) @ ( t_a @ V @ e_a @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % removeLeaf.simps(3)
% 0.07/0.32 thf(fact_102_union__mset__add__mset__left,axiom,
% 0.07/0.32 ! [A: a,A4: multiset_a,B4: multiset_a] :
% 0.07/0.32 ( ( plus_plus_multiset_a @ ( add_mset_a @ A @ A4 ) @ B4 )
% 0.07/0.32 = ( add_mset_a @ A @ ( plus_plus_multiset_a @ A4 @ B4 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % union_mset_add_mset_left
% 0.07/0.32 thf(fact_103_union__mset__add__mset__right,axiom,
% 0.07/0.32 ! [A4: multiset_a,A: a,B4: multiset_a] :
% 0.07/0.32 ( ( plus_plus_multiset_a @ A4 @ ( add_mset_a @ A @ B4 ) )
% 0.07/0.32 = ( add_mset_a @ A @ ( plus_plus_multiset_a @ A4 @ B4 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % union_mset_add_mset_right
% 0.07/0.32 thf(fact_104_single__eq__single,axiom,
% 0.07/0.32 ! [A: a,B: a] :
% 0.07/0.32 ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
% 0.07/0.32 = ( add_mset_a @ B @ zero_zero_multiset_a ) )
% 0.07/0.32 = ( A = B ) ) ).
% 0.07/0.32
% 0.07/0.32 % single_eq_single
% 0.07/0.32 thf(fact_105_add__mset__eq__single,axiom,
% 0.07/0.32 ! [B: a,M: multiset_a,A: a] :
% 0.07/0.32 ( ( ( add_mset_a @ B @ M )
% 0.07/0.32 = ( add_mset_a @ A @ zero_zero_multiset_a ) )
% 0.07/0.32 = ( ( B = A )
% 0.07/0.32 & ( M = zero_zero_multiset_a ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % add_mset_eq_single
% 0.07/0.32 thf(fact_106_single__eq__add__mset,axiom,
% 0.07/0.32 ! [A: a,B: a,M: multiset_a] :
% 0.07/0.32 ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
% 0.07/0.32 = ( add_mset_a @ B @ M ) )
% 0.07/0.32 = ( ( B = A )
% 0.07/0.32 & ( M = zero_zero_multiset_a ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % single_eq_add_mset
% 0.07/0.32 thf(fact_107_add__mset__eq__singleton__iff,axiom,
% 0.07/0.32 ! [X: a,M: multiset_a,Y: a] :
% 0.07/0.32 ( ( ( add_mset_a @ X @ M )
% 0.07/0.32 = ( add_mset_a @ Y @ zero_zero_multiset_a ) )
% 0.07/0.32 = ( ( M = zero_zero_multiset_a )
% 0.07/0.32 & ( X = Y ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % add_mset_eq_singleton_iff
% 0.07/0.32 thf(fact_108_empty__eq__union,axiom,
% 0.07/0.32 ! [M: multiset_a,N: multiset_a] :
% 0.07/0.32 ( ( zero_zero_multiset_a
% 0.07/0.32 = ( plus_plus_multiset_a @ M @ N ) )
% 0.07/0.32 = ( ( M = zero_zero_multiset_a )
% 0.07/0.32 & ( N = zero_zero_multiset_a ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % empty_eq_union
% 0.07/0.32 thf(fact_109_union__eq__empty,axiom,
% 0.07/0.32 ! [M: multiset_a,N: multiset_a] :
% 0.07/0.32 ( ( ( plus_plus_multiset_a @ M @ N )
% 0.07/0.32 = zero_zero_multiset_a )
% 0.07/0.32 = ( ( M = zero_zero_multiset_a )
% 0.07/0.32 & ( N = zero_zero_multiset_a ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % union_eq_empty
% 0.07/0.32 thf(fact_110_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
% 0.07/0.32 ! [X: multiset_a,Y: multiset_a] :
% 0.07/0.32 ( ( ( plus_plus_multiset_a @ X @ Y )
% 0.07/0.32 = zero_zero_multiset_a )
% 0.07/0.32 = ( ( X = zero_zero_multiset_a )
% 0.07/0.32 & ( Y = zero_zero_multiset_a ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % subset_mset.add_eq_0_iff_both_eq_0
% 0.07/0.32 thf(fact_111_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
% 0.07/0.32 ! [X: multiset_a,Y: multiset_a] :
% 0.07/0.32 ( ( zero_zero_multiset_a
% 0.07/0.32 = ( plus_plus_multiset_a @ X @ Y ) )
% 0.07/0.32 = ( ( X = zero_zero_multiset_a )
% 0.07/0.32 & ( Y = zero_zero_multiset_a ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % subset_mset.zero_eq_add_iff_both_eq_0
% 0.07/0.32 thf(fact_112_add_Oleft__neutral,axiom,
% 0.07/0.32 ! [A: multiset_a] :
% 0.07/0.32 ( ( plus_plus_multiset_a @ zero_zero_multiset_a @ A )
% 0.07/0.32 = A ) ).
% 0.07/0.32
% 0.07/0.32 % add.left_neutral
% 0.07/0.32 thf(fact_113_add__right__cancel,axiom,
% 0.07/0.32 ! [B: multiset_a,A: multiset_a,C: multiset_a] :
% 0.07/0.32 ( ( ( plus_plus_multiset_a @ B @ A )
% 0.07/0.32 = ( plus_plus_multiset_a @ C @ A ) )
% 0.07/0.32 = ( B = C ) ) ).
% 0.07/0.32
% 0.07/0.32 % add_right_cancel
% 0.07/0.32 thf(fact_114_add__left__cancel,axiom,
% 0.07/0.32 ! [A: multiset_a,B: multiset_a,C: multiset_a] :
% 0.07/0.32 ( ( ( plus_plus_multiset_a @ A @ B )
% 0.07/0.32 = ( plus_plus_multiset_a @ A @ C ) )
% 0.07/0.32 = ( B = C ) ) ).
% 0.07/0.32
% 0.07/0.32 % add_left_cancel
% 0.07/0.32 thf(fact_115_multi__self__add__other__not__self,axiom,
% 0.07/0.32 ! [M: multiset_a,X: a] :
% 0.07/0.32 ( M
% 0.07/0.32 != ( add_mset_a @ X @ M ) ) ).
% 0.07/0.32
% 0.07/0.32 % multi_self_add_other_not_self
% 0.07/0.32 thf(fact_116_add__mset__add__mset__same__iff,axiom,
% 0.07/0.32 ! [A: a,A4: multiset_a,B4: multiset_a] :
% 0.07/0.32 ( ( ( add_mset_a @ A @ A4 )
% 0.07/0.32 = ( add_mset_a @ A @ B4 ) )
% 0.07/0.32 = ( A4 = B4 ) ) ).
% 0.07/0.32
% 0.07/0.32 % add_mset_add_mset_same_iff
% 0.07/0.32 thf(fact_117_add__cancel__right__right,axiom,
% 0.07/0.32 ! [A: multiset_a,B: multiset_a] :
% 0.07/0.32 ( ( A
% 0.07/0.32 = ( plus_plus_multiset_a @ A @ B ) )
% 0.07/0.32 = ( B = zero_zero_multiset_a ) ) ).
% 0.07/0.32
% 0.07/0.32 % add_cancel_right_right
% 0.07/0.32 thf(fact_118_add__cancel__right__left,axiom,
% 0.07/0.32 ! [A: multiset_a,B: multiset_a] :
% 0.07/0.32 ( ( A
% 0.07/0.32 = ( plus_plus_multiset_a @ B @ A ) )
% 0.07/0.32 = ( B = zero_zero_multiset_a ) ) ).
% 0.07/0.32
% 0.07/0.32 % add_cancel_right_left
% 0.07/0.32 thf(fact_119_add__cancel__left__right,axiom,
% 0.07/0.32 ! [A: multiset_a,B: multiset_a] :
% 0.07/0.32 ( ( ( plus_plus_multiset_a @ A @ B )
% 0.07/0.32 = A )
% 0.07/0.32 = ( B = zero_zero_multiset_a ) ) ).
% 0.07/0.32
% 0.07/0.32 % add_cancel_left_right
% 0.07/0.32 thf(fact_120_add__cancel__left__left,axiom,
% 0.07/0.32 ! [B: multiset_a,A: multiset_a] :
% 0.07/0.32 ( ( ( plus_plus_multiset_a @ B @ A )
% 0.07/0.32 = A )
% 0.07/0.32 = ( B = zero_zero_multiset_a ) ) ).
% 0.07/0.32
% 0.07/0.32 % add_cancel_left_left
% 0.07/0.32 thf(fact_121_add_Oright__neutral,axiom,
% 0.07/0.32 ! [A: multiset_a] :
% 0.07/0.32 ( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
% 0.07/0.32 = A ) ).
% 0.07/0.32
% 0.07/0.32 % add.right_neutral
% 0.07/0.32 thf(fact_122_zero__reorient,axiom,
% 0.07/0.32 ! [X: multiset_a] :
% 0.07/0.32 ( ( zero_zero_multiset_a = X )
% 0.07/0.32 = ( X = zero_zero_multiset_a ) ) ).
% 0.07/0.32
% 0.07/0.32 % zero_reorient
% 0.07/0.32 thf(fact_123_add__right__imp__eq,axiom,
% 0.07/0.32 ! [B: multiset_a,A: multiset_a,C: multiset_a] :
% 0.07/0.32 ( ( ( plus_plus_multiset_a @ B @ A )
% 0.07/0.32 = ( plus_plus_multiset_a @ C @ A ) )
% 0.07/0.32 => ( B = C ) ) ).
% 0.07/0.32
% 0.07/0.32 % add_right_imp_eq
% 0.07/0.32 thf(fact_124_add__left__imp__eq,axiom,
% 0.07/0.32 ! [A: multiset_a,B: multiset_a,C: multiset_a] :
% 0.07/0.32 ( ( ( plus_plus_multiset_a @ A @ B )
% 0.07/0.32 = ( plus_plus_multiset_a @ A @ C ) )
% 0.07/0.32 => ( B = C ) ) ).
% 0.07/0.32
% 0.07/0.32 % add_left_imp_eq
% 0.07/0.32 thf(fact_125_add_Oleft__commute,axiom,
% 0.07/0.32 ! [B: multiset_a,A: multiset_a,C: multiset_a] :
% 0.07/0.32 ( ( plus_plus_multiset_a @ B @ ( plus_plus_multiset_a @ A @ C ) )
% 0.07/0.32 = ( plus_plus_multiset_a @ A @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % add.left_commute
% 0.07/0.32 thf(fact_126_add_Ocommute,axiom,
% 0.07/0.32 ( plus_plus_multiset_a
% 0.07/0.32 = ( ^ [A5: multiset_a,B5: multiset_a] : ( plus_plus_multiset_a @ B5 @ A5 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % add.commute
% 0.07/0.32 thf(fact_127_add_Oassoc,axiom,
% 0.07/0.32 ! [A: multiset_a,B: multiset_a,C: multiset_a] :
% 0.07/0.32 ( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ A @ B ) @ C )
% 0.07/0.32 = ( plus_plus_multiset_a @ A @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % add.assoc
% 0.07/0.32 thf(fact_128_group__cancel_Oadd2,axiom,
% 0.07/0.32 ! [B4: multiset_a,K: multiset_a,B: multiset_a,A: multiset_a] :
% 0.07/0.32 ( ( B4
% 0.07/0.32 = ( plus_plus_multiset_a @ K @ B ) )
% 0.07/0.32 => ( ( plus_plus_multiset_a @ A @ B4 )
% 0.07/0.32 = ( plus_plus_multiset_a @ K @ ( plus_plus_multiset_a @ A @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % group_cancel.add2
% 0.07/0.32 thf(fact_129_group__cancel_Oadd1,axiom,
% 0.07/0.32 ! [A4: multiset_a,K: multiset_a,A: multiset_a,B: multiset_a] :
% 0.07/0.32 ( ( A4
% 0.07/0.32 = ( plus_plus_multiset_a @ K @ A ) )
% 0.07/0.32 => ( ( plus_plus_multiset_a @ A4 @ B )
% 0.07/0.32 = ( plus_plus_multiset_a @ K @ ( plus_plus_multiset_a @ A @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % group_cancel.add1
% 0.07/0.32 thf(fact_130_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.07/0.32 ! [I: multiset_a,J: multiset_a,K: multiset_a,L: multiset_a] :
% 0.07/0.32 ( ( ( I = J )
% 0.07/0.32 & ( K = L ) )
% 0.07/0.32 => ( ( plus_plus_multiset_a @ I @ K )
% 0.07/0.32 = ( plus_plus_multiset_a @ J @ L ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % add_mono_thms_linordered_semiring(4)
% 0.07/0.32 thf(fact_131_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.07/0.32 ! [A: multiset_a,B: multiset_a,C: multiset_a] :
% 0.07/0.32 ( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ A @ B ) @ C )
% 0.07/0.32 = ( plus_plus_multiset_a @ A @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % ab_semigroup_add_class.add_ac(1)
% 0.07/0.32 thf(fact_132_multi__union__self__other__eq,axiom,
% 0.07/0.32 ! [A4: multiset_a,X5: multiset_a,Y5: multiset_a] :
% 0.07/0.32 ( ( ( plus_plus_multiset_a @ A4 @ X5 )
% 0.07/0.32 = ( plus_plus_multiset_a @ A4 @ Y5 ) )
% 0.07/0.32 => ( X5 = Y5 ) ) ).
% 0.07/0.32
% 0.07/0.32 % multi_union_self_other_eq
% 0.07/0.32 thf(fact_133_union__right__cancel,axiom,
% 0.07/0.32 ! [M: multiset_a,K2: multiset_a,N: multiset_a] :
% 0.07/0.32 ( ( ( plus_plus_multiset_a @ M @ K2 )
% 0.07/0.32 = ( plus_plus_multiset_a @ N @ K2 ) )
% 0.07/0.32 = ( M = N ) ) ).
% 0.07/0.32
% 0.07/0.32 % union_right_cancel
% 0.07/0.32 thf(fact_134_union__left__cancel,axiom,
% 0.07/0.32 ! [K2: multiset_a,M: multiset_a,N: multiset_a] :
% 0.07/0.32 ( ( ( plus_plus_multiset_a @ K2 @ M )
% 0.07/0.32 = ( plus_plus_multiset_a @ K2 @ N ) )
% 0.07/0.32 = ( M = N ) ) ).
% 0.07/0.32
% 0.07/0.32 % union_left_cancel
% 0.07/0.32 thf(fact_135_union__commute,axiom,
% 0.07/0.32 ( plus_plus_multiset_a
% 0.07/0.32 = ( ^ [M2: multiset_a,N2: multiset_a] : ( plus_plus_multiset_a @ N2 @ M2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % union_commute
% 0.07/0.32 thf(fact_136_union__lcomm,axiom,
% 0.07/0.32 ! [M: multiset_a,N: multiset_a,K2: multiset_a] :
% 0.07/0.32 ( ( plus_plus_multiset_a @ M @ ( plus_plus_multiset_a @ N @ K2 ) )
% 0.07/0.32 = ( plus_plus_multiset_a @ N @ ( plus_plus_multiset_a @ M @ K2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % union_lcomm
% 0.07/0.32 thf(fact_137_union__assoc,axiom,
% 0.07/0.32 ! [M: multiset_a,N: multiset_a,K2: multiset_a] :
% 0.07/0.32 ( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ M @ N ) @ K2 )
% 0.07/0.32 = ( plus_plus_multiset_a @ M @ ( plus_plus_multiset_a @ N @ K2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % union_assoc
% 0.07/0.32 thf(fact_138_add__mset__commute,axiom,
% 0.07/0.32 ! [X: a,Y: a,M: multiset_a] :
% 0.07/0.32 ( ( add_mset_a @ X @ ( add_mset_a @ Y @ M ) )
% 0.07/0.32 = ( add_mset_a @ Y @ ( add_mset_a @ X @ M ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % add_mset_commute
% 0.07/0.32 thf(fact_139_add__eq__conv__ex,axiom,
% 0.07/0.32 ! [A: a,M: multiset_a,B: a,N: multiset_a] :
% 0.07/0.32 ( ( ( add_mset_a @ A @ M )
% 0.07/0.32 = ( add_mset_a @ B @ N ) )
% 0.07/0.32 = ( ( ( M = N )
% 0.07/0.32 & ( A = B ) )
% 0.07/0.32 | ? [K3: multiset_a] :
% 0.07/0.32 ( ( M
% 0.07/0.32 = ( add_mset_a @ B @ K3 ) )
% 0.07/0.32 & ( N
% 0.07/0.32 = ( add_mset_a @ A @ K3 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % add_eq_conv_ex
% 0.07/0.32 thf(fact_140_add_Ocomm__neutral,axiom,
% 0.07/0.32 ! [A: multiset_a] :
% 0.07/0.32 ( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
% 0.07/0.32 = A ) ).
% 0.07/0.32
% 0.07/0.32 % add.comm_neutral
% 0.07/0.32 thf(fact_141_comm__monoid__add__class_Oadd__0,axiom,
% 0.07/0.32 ! [A: multiset_a] :
% 0.07/0.32 ( ( plus_plus_multiset_a @ zero_zero_multiset_a @ A )
% 0.07/0.32 = A ) ).
% 0.07/0.32
% 0.07/0.32 % comm_monoid_add_class.add_0
% 0.07/0.32 thf(fact_142_empty__neutral_I1_J,axiom,
% 0.07/0.32 ! [X: multiset_a] :
% 0.07/0.32 ( ( plus_plus_multiset_a @ zero_zero_multiset_a @ X )
% 0.07/0.32 = X ) ).
% 0.07/0.32
% 0.07/0.32 % empty_neutral(1)
% 0.07/0.32 thf(fact_143_empty__neutral_I2_J,axiom,
% 0.07/0.32 ! [X: multiset_a] :
% 0.07/0.32 ( ( plus_plus_multiset_a @ X @ zero_zero_multiset_a )
% 0.07/0.32 = X ) ).
% 0.07/0.32
% 0.07/0.32 % empty_neutral(2)
% 0.07/0.32 thf(fact_144_multi__nonempty__split,axiom,
% 0.07/0.32 ! [M: multiset_a] :
% 0.07/0.32 ( ( M != zero_zero_multiset_a )
% 0.07/0.32 => ? [A6: multiset_a,A3: a] :
% 0.07/0.32 ( M
% 0.07/0.32 = ( add_mset_a @ A3 @ A6 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multi_nonempty_split
% 0.07/0.32 thf(fact_145_empty__not__add__mset,axiom,
% 0.07/0.32 ! [A: a,A4: multiset_a] :
% 0.07/0.32 ( zero_zero_multiset_a
% 0.07/0.32 != ( add_mset_a @ A @ A4 ) ) ).
% 0.07/0.32
% 0.07/0.32 % empty_not_add_mset
% 0.07/0.32 thf(fact_146_multiset__induct2,axiom,
% 0.07/0.32 ! [P2: multiset_a > multiset_a > $o,M: multiset_a,N: multiset_a] :
% 0.07/0.32 ( ( P2 @ zero_zero_multiset_a @ zero_zero_multiset_a )
% 0.07/0.32 => ( ! [A3: a,M3: multiset_a,N3: multiset_a] :
% 0.07/0.32 ( ( P2 @ M3 @ N3 )
% 0.07/0.32 => ( P2 @ ( add_mset_a @ A3 @ M3 ) @ N3 ) )
% 0.07/0.32 => ( ! [A3: a,M3: multiset_a,N3: multiset_a] :
% 0.07/0.32 ( ( P2 @ M3 @ N3 )
% 0.07/0.32 => ( P2 @ M3 @ ( add_mset_a @ A3 @ N3 ) ) )
% 0.07/0.32 => ( P2 @ M @ N ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multiset_induct2
% 0.07/0.32 thf(fact_147_multiset__induct,axiom,
% 0.07/0.32 ! [P2: multiset_a > $o,M: multiset_a] :
% 0.07/0.32 ( ( P2 @ zero_zero_multiset_a )
% 0.07/0.32 => ( ! [X3: a,M3: multiset_a] :
% 0.07/0.32 ( ( P2 @ M3 )
% 0.07/0.32 => ( P2 @ ( add_mset_a @ X3 @ M3 ) ) )
% 0.07/0.32 => ( P2 @ M ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multiset_induct
% 0.07/0.32 thf(fact_148_multiset__cases,axiom,
% 0.07/0.32 ! [M: multiset_a] :
% 0.07/0.32 ( ( M != zero_zero_multiset_a )
% 0.07/0.32 => ~ ! [X3: a,N3: multiset_a] :
% 0.07/0.32 ( M
% 0.07/0.32 != ( add_mset_a @ X3 @ N3 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multiset_cases
% 0.07/0.32 thf(fact_149_add__mset__add__single,axiom,
% 0.07/0.32 ( add_mset_a
% 0.07/0.32 = ( ^ [A5: a,A7: multiset_a] : ( plus_plus_multiset_a @ A7 @ ( add_mset_a @ A5 @ zero_zero_multiset_a ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % add_mset_add_single
% 0.07/0.32 thf(fact_150_union__is__single,axiom,
% 0.07/0.32 ! [M: multiset_a,N: multiset_a,A: a] :
% 0.07/0.32 ( ( ( plus_plus_multiset_a @ M @ N )
% 0.07/0.32 = ( add_mset_a @ A @ zero_zero_multiset_a ) )
% 0.07/0.32 = ( ( ( M
% 0.07/0.32 = ( add_mset_a @ A @ zero_zero_multiset_a ) )
% 0.07/0.32 & ( N = zero_zero_multiset_a ) )
% 0.07/0.32 | ( ( M = zero_zero_multiset_a )
% 0.07/0.32 & ( N
% 0.07/0.32 = ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % union_is_single
% 0.07/0.32 thf(fact_151_single__is__union,axiom,
% 0.07/0.32 ! [A: a,M: multiset_a,N: multiset_a] :
% 0.07/0.32 ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
% 0.07/0.32 = ( plus_plus_multiset_a @ M @ N ) )
% 0.07/0.32 = ( ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
% 0.07/0.32 = M )
% 0.07/0.32 & ( N = zero_zero_multiset_a ) )
% 0.07/0.32 | ( ( M = zero_zero_multiset_a )
% 0.07/0.32 & ( ( add_mset_a @ A @ zero_zero_multiset_a )
% 0.07/0.32 = N ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % single_is_union
% 0.07/0.32 thf(fact_152_exI__realizer,axiom,
% 0.07/0.32 ! [P2: multiset_a > multiset_a > $o,Y: multiset_a,X: multiset_a] :
% 0.07/0.32 ( ( P2 @ Y @ X )
% 0.07/0.32 => ( P2 @ ( produc148952133iset_a @ ( produc2037245207iset_a @ X @ Y ) ) @ ( produc653373187iset_a @ ( produc2037245207iset_a @ X @ Y ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % exI_realizer
% 0.07/0.32 thf(fact_153_exI__realizer,axiom,
% 0.07/0.32 ! [P2: a > a > $o,Y: a,X: a] :
% 0.07/0.32 ( ( P2 @ Y @ X )
% 0.07/0.32 => ( P2 @ ( product_snd_a_a @ ( product_Pair_a_a @ X @ Y ) ) @ ( product_fst_a_a @ ( product_Pair_a_a @ X @ Y ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % exI_realizer
% 0.07/0.32 thf(fact_154_exI__realizer,axiom,
% 0.07/0.32 ! [P2: a > tree_a > $o,Y: a,X: tree_a] :
% 0.07/0.32 ( ( P2 @ Y @ X )
% 0.07/0.32 => ( P2 @ ( product_snd_Tree_a_a @ ( produc649172771ee_a_a @ X @ Y ) ) @ ( product_fst_Tree_a_a @ ( produc649172771ee_a_a @ X @ Y ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % exI_realizer
% 0.07/0.32 thf(fact_155_exI__realizer,axiom,
% 0.07/0.32 ! [P2: tree_a > a > $o,Y: tree_a,X: a] :
% 0.07/0.32 ( ( P2 @ Y @ X )
% 0.07/0.32 => ( P2 @ ( product_snd_a_Tree_a @ ( produc686083979Tree_a @ X @ Y ) ) @ ( product_fst_a_Tree_a @ ( produc686083979Tree_a @ X @ Y ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % exI_realizer
% 0.07/0.32 thf(fact_156_conjI__realizer,axiom,
% 0.07/0.32 ! [P2: multiset_a > $o,P: multiset_a,Q2: multiset_a > $o,Q: multiset_a] :
% 0.07/0.32 ( ( P2 @ P )
% 0.07/0.32 => ( ( Q2 @ Q )
% 0.07/0.32 => ( ( P2 @ ( produc653373187iset_a @ ( produc2037245207iset_a @ P @ Q ) ) )
% 0.07/0.32 & ( Q2 @ ( produc148952133iset_a @ ( produc2037245207iset_a @ P @ Q ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % conjI_realizer
% 0.07/0.32 thf(fact_157_conjI__realizer,axiom,
% 0.07/0.32 ! [P2: a > $o,P: a,Q2: a > $o,Q: a] :
% 0.07/0.32 ( ( P2 @ P )
% 0.07/0.32 => ( ( Q2 @ Q )
% 0.07/0.32 => ( ( P2 @ ( product_fst_a_a @ ( product_Pair_a_a @ P @ Q ) ) )
% 0.07/0.32 & ( Q2 @ ( product_snd_a_a @ ( product_Pair_a_a @ P @ Q ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % conjI_realizer
% 0.07/0.32 thf(fact_158_conjI__realizer,axiom,
% 0.07/0.32 ! [P2: tree_a > $o,P: tree_a,Q2: a > $o,Q: a] :
% 0.07/0.32 ( ( P2 @ P )
% 0.07/0.32 => ( ( Q2 @ Q )
% 0.07/0.32 => ( ( P2 @ ( product_fst_Tree_a_a @ ( produc649172771ee_a_a @ P @ Q ) ) )
% 0.07/0.32 & ( Q2 @ ( product_snd_Tree_a_a @ ( produc649172771ee_a_a @ P @ Q ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % conjI_realizer
% 0.07/0.32 thf(fact_159_conjI__realizer,axiom,
% 0.07/0.32 ! [P2: a > $o,P: a,Q2: tree_a > $o,Q: tree_a] :
% 0.07/0.32 ( ( P2 @ P )
% 0.07/0.32 => ( ( Q2 @ Q )
% 0.07/0.32 => ( ( P2 @ ( product_fst_a_Tree_a @ ( produc686083979Tree_a @ P @ Q ) ) )
% 0.07/0.32 & ( Q2 @ ( product_snd_a_Tree_a @ ( produc686083979Tree_a @ P @ Q ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % conjI_realizer
% 0.07/0.32 thf(fact_160_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.07/0.32 ! [P2: multiset_a > multiset_a > $o,X: multiset_a,Y: multiset_a,A: produc1127127335iset_a] :
% 0.07/0.32 ( ( P2 @ X @ Y )
% 0.07/0.32 => ( ( A
% 0.07/0.32 = ( produc2037245207iset_a @ X @ Y ) )
% 0.07/0.32 => ( P2 @ ( produc653373187iset_a @ A ) @ ( produc148952133iset_a @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % BNF_Greatest_Fixpoint.subst_Pair
% 0.07/0.32 thf(fact_161_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.07/0.32 ! [P2: a > a > $o,X: a,Y: a,A: product_prod_a_a] :
% 0.07/0.32 ( ( P2 @ X @ Y )
% 0.07/0.32 => ( ( A
% 0.07/0.32 = ( product_Pair_a_a @ X @ Y ) )
% 0.07/0.32 => ( P2 @ ( product_fst_a_a @ A ) @ ( product_snd_a_a @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % BNF_Greatest_Fixpoint.subst_Pair
% 0.07/0.32 thf(fact_162_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.07/0.32 ! [P2: tree_a > a > $o,X: tree_a,Y: a,A: produc981471411ee_a_a] :
% 0.07/0.32 ( ( P2 @ X @ Y )
% 0.07/0.32 => ( ( A
% 0.07/0.32 = ( produc649172771ee_a_a @ X @ Y ) )
% 0.07/0.32 => ( P2 @ ( product_fst_Tree_a_a @ A ) @ ( product_snd_Tree_a_a @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % BNF_Greatest_Fixpoint.subst_Pair
% 0.07/0.32 thf(fact_163_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.07/0.32 ! [P2: a > tree_a > $o,X: a,Y: tree_a,A: produc143150363Tree_a] :
% 0.07/0.32 ( ( P2 @ X @ Y )
% 0.07/0.32 => ( ( A
% 0.07/0.32 = ( produc686083979Tree_a @ X @ Y ) )
% 0.07/0.32 => ( P2 @ ( product_fst_a_Tree_a @ A ) @ ( product_snd_a_Tree_a @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % BNF_Greatest_Fixpoint.subst_Pair
% 0.07/0.32 thf(fact_164_Multiset_Ois__empty__def,axiom,
% 0.07/0.32 ( is_empty_a
% 0.07/0.32 = ( ^ [A7: multiset_a] : ( A7 = zero_zero_multiset_a ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Multiset.is_empty_def
% 0.07/0.32 thf(fact_165_exE__realizer_H,axiom,
% 0.07/0.32 ! [P2: a > tree_a > $o,P: produc981471411ee_a_a] :
% 0.07/0.32 ( ( P2 @ ( product_snd_Tree_a_a @ P ) @ ( product_fst_Tree_a_a @ P ) )
% 0.07/0.32 => ~ ! [X3: tree_a,Y3: a] :
% 0.07/0.32 ~ ( P2 @ Y3 @ X3 ) ) ).
% 0.07/0.32
% 0.07/0.32 % exE_realizer'
% 0.07/0.32 thf(fact_166_exE__realizer_H,axiom,
% 0.07/0.32 ! [P2: tree_a > a > $o,P: produc143150363Tree_a] :
% 0.07/0.32 ( ( P2 @ ( product_snd_a_Tree_a @ P ) @ ( product_fst_a_Tree_a @ P ) )
% 0.07/0.32 => ~ ! [X3: a,Y3: tree_a] :
% 0.07/0.32 ~ ( P2 @ Y3 @ X3 ) ) ).
% 0.07/0.32
% 0.07/0.32 % exE_realizer'
% 0.07/0.32 thf(fact_167_eq__snd__iff,axiom,
% 0.07/0.32 ! [B: multiset_a,P: produc1127127335iset_a] :
% 0.07/0.32 ( ( B
% 0.07/0.32 = ( produc148952133iset_a @ P ) )
% 0.07/0.32 = ( ? [A5: multiset_a] :
% 0.07/0.32 ( P
% 0.07/0.32 = ( produc2037245207iset_a @ A5 @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % eq_snd_iff
% 0.07/0.32 thf(fact_168_eq__snd__iff,axiom,
% 0.07/0.32 ! [B: a,P: product_prod_a_a] :
% 0.07/0.32 ( ( B
% 0.07/0.32 = ( product_snd_a_a @ P ) )
% 0.07/0.32 = ( ? [A5: a] :
% 0.07/0.32 ( P
% 0.07/0.32 = ( product_Pair_a_a @ A5 @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % eq_snd_iff
% 0.07/0.32 thf(fact_169_eq__snd__iff,axiom,
% 0.07/0.32 ! [B: a,P: produc981471411ee_a_a] :
% 0.07/0.32 ( ( B
% 0.07/0.32 = ( product_snd_Tree_a_a @ P ) )
% 0.07/0.32 = ( ? [A5: tree_a] :
% 0.07/0.32 ( P
% 0.07/0.32 = ( produc649172771ee_a_a @ A5 @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % eq_snd_iff
% 0.07/0.32 thf(fact_170_eq__snd__iff,axiom,
% 0.07/0.32 ! [B: tree_a,P: produc143150363Tree_a] :
% 0.07/0.32 ( ( B
% 0.07/0.32 = ( product_snd_a_Tree_a @ P ) )
% 0.07/0.32 = ( ? [A5: a] :
% 0.07/0.32 ( P
% 0.07/0.32 = ( produc686083979Tree_a @ A5 @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % eq_snd_iff
% 0.07/0.32 thf(fact_171_eq__fst__iff,axiom,
% 0.07/0.32 ! [A: multiset_a,P: produc1127127335iset_a] :
% 0.07/0.32 ( ( A
% 0.07/0.32 = ( produc653373187iset_a @ P ) )
% 0.07/0.32 = ( ? [B5: multiset_a] :
% 0.07/0.32 ( P
% 0.07/0.32 = ( produc2037245207iset_a @ A @ B5 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % eq_fst_iff
% 0.07/0.32 thf(fact_172_eq__fst__iff,axiom,
% 0.07/0.32 ! [A: a,P: product_prod_a_a] :
% 0.07/0.32 ( ( A
% 0.07/0.32 = ( product_fst_a_a @ P ) )
% 0.07/0.32 = ( ? [B5: a] :
% 0.07/0.32 ( P
% 0.07/0.32 = ( product_Pair_a_a @ A @ B5 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % eq_fst_iff
% 0.07/0.32 thf(fact_173_eq__fst__iff,axiom,
% 0.07/0.32 ! [A: tree_a,P: produc981471411ee_a_a] :
% 0.07/0.32 ( ( A
% 0.07/0.32 = ( product_fst_Tree_a_a @ P ) )
% 0.07/0.32 = ( ? [B5: a] :
% 0.07/0.32 ( P
% 0.07/0.32 = ( produc649172771ee_a_a @ A @ B5 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % eq_fst_iff
% 0.07/0.32 thf(fact_174_eq__fst__iff,axiom,
% 0.07/0.32 ! [A: a,P: produc143150363Tree_a] :
% 0.07/0.32 ( ( A
% 0.07/0.32 = ( product_fst_a_Tree_a @ P ) )
% 0.07/0.32 = ( ? [B5: tree_a] :
% 0.07/0.32 ( P
% 0.07/0.32 = ( produc686083979Tree_a @ A @ B5 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % eq_fst_iff
% 0.07/0.32 thf(fact_175_verit__sum__simplify,axiom,
% 0.07/0.32 ! [A: multiset_a] :
% 0.07/0.32 ( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
% 0.07/0.32 = A ) ).
% 0.07/0.32
% 0.07/0.32 % verit_sum_simplify
% 0.07/0.32 thf(fact_176_HS_Ois__empty__as__list,axiom,
% 0.07/0.32 ! [E: tree_a] :
% 0.07/0.32 ( ( heapIm229596386mpty_a @ E )
% 0.07/0.32 => ( ( multiset_a2 @ E )
% 0.07/0.32 = zero_zero_multiset_a ) ) ).
% 0.07/0.32
% 0.07/0.32 % HS.is_empty_as_list
% 0.07/0.32 thf(fact_177_add__mset__replicate__mset__safe,axiom,
% 0.07/0.32 ! [M: multiset_a,A: a] :
% 0.07/0.32 ( ( nO_MAT1617603563iset_a @ zero_zero_multiset_a @ M )
% 0.07/0.32 => ( ( add_mset_a @ A @ M )
% 0.07/0.32 = ( plus_plus_multiset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) @ M ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % add_mset_replicate_mset_safe
% 0.07/0.32 thf(fact_178_HS_Ois__empty__empty,axiom,
% 0.07/0.32 heapIm229596386mpty_a @ e_a ).
% 0.07/0.32
% 0.07/0.32 % HS.is_empty_empty
% 0.07/0.32 thf(fact_179_hs__is__empty__def,axiom,
% 0.07/0.32 ( heapIm229596386mpty_a
% 0.07/0.32 = ( ^ [T3: tree_a] : ( T3 = e_a ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % hs_is_empty_def
% 0.07/0.32 thf(fact_180_HS_Ois__empty__inj,axiom,
% 0.07/0.32 ! [E: tree_a] :
% 0.07/0.32 ( ( heapIm229596386mpty_a @ E )
% 0.07/0.32 => ( E = e_a ) ) ).
% 0.07/0.32
% 0.07/0.32 % HS.is_empty_inj
% 0.07/0.32 thf(fact_181_prod_Oswap__def,axiom,
% 0.07/0.32 ( produc506430135iset_a
% 0.07/0.32 = ( ^ [P3: produc1127127335iset_a] : ( produc2037245207iset_a @ ( produc148952133iset_a @ P3 ) @ ( produc653373187iset_a @ P3 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % prod.swap_def
% 0.07/0.32 thf(fact_182_prod_Oswap__def,axiom,
% 0.07/0.32 ( product_swap_a_a
% 0.07/0.32 = ( ^ [P3: product_prod_a_a] : ( product_Pair_a_a @ ( product_snd_a_a @ P3 ) @ ( product_fst_a_a @ P3 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % prod.swap_def
% 0.07/0.32 thf(fact_183_prod_Oswap__def,axiom,
% 0.07/0.32 ( produc1478296771ee_a_a
% 0.07/0.32 = ( ^ [P3: produc981471411ee_a_a] : ( produc686083979Tree_a @ ( product_snd_Tree_a_a @ P3 ) @ ( product_fst_Tree_a_a @ P3 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % prod.swap_def
% 0.07/0.32 thf(fact_184_prod_Oswap__def,axiom,
% 0.07/0.32 ( produc1515207979Tree_a
% 0.07/0.32 = ( ^ [P3: produc143150363Tree_a] : ( produc649172771ee_a_a @ ( product_snd_a_Tree_a @ P3 ) @ ( product_fst_a_Tree_a @ P3 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % prod.swap_def
% 0.07/0.32 thf(fact_185_removeLeaf__heap__is__heap,axiom,
% 0.07/0.32 ! [T2: tree_a] :
% 0.07/0.32 ( ( is_heap_a @ T2 )
% 0.07/0.32 => ( ( T2 != e_a )
% 0.07/0.32 => ( is_heap_a @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ T2 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % removeLeaf_heap_is_heap
% 0.07/0.32 thf(fact_186_NO__MATCH__cong,axiom,
% 0.07/0.32 nO_MAT1617603563iset_a = nO_MAT1617603563iset_a ).
% 0.07/0.32
% 0.07/0.32 % NO_MATCH_cong
% 0.07/0.32 thf(fact_187_NO__MATCH__def,axiom,
% 0.07/0.32 ( nO_MAT1617603563iset_a
% 0.07/0.32 = ( ^ [Pat: multiset_a,Val: multiset_a] : $true ) ) ).
% 0.07/0.32
% 0.07/0.32 % NO_MATCH_def
% 0.07/0.32 thf(fact_188_swap__swap,axiom,
% 0.07/0.32 ! [P: produc981471411ee_a_a] :
% 0.07/0.32 ( ( produc1515207979Tree_a @ ( produc1478296771ee_a_a @ P ) )
% 0.07/0.32 = P ) ).
% 0.07/0.32
% 0.07/0.32 % swap_swap
% 0.07/0.32 thf(fact_189_swap__swap,axiom,
% 0.07/0.32 ! [P: produc143150363Tree_a] :
% 0.07/0.32 ( ( produc1478296771ee_a_a @ ( produc1515207979Tree_a @ P ) )
% 0.07/0.32 = P ) ).
% 0.07/0.32
% 0.07/0.32 % swap_swap
% 0.07/0.32 thf(fact_190_swap__simp,axiom,
% 0.07/0.32 ! [X: multiset_a,Y: multiset_a] :
% 0.07/0.32 ( ( produc506430135iset_a @ ( produc2037245207iset_a @ X @ Y ) )
% 0.07/0.32 = ( produc2037245207iset_a @ Y @ X ) ) ).
% 0.07/0.32
% 0.07/0.32 % swap_simp
% 0.07/0.32 thf(fact_191_swap__simp,axiom,
% 0.07/0.32 ! [X: a,Y: a] :
% 0.07/0.32 ( ( product_swap_a_a @ ( product_Pair_a_a @ X @ Y ) )
% 0.07/0.32 = ( product_Pair_a_a @ Y @ X ) ) ).
% 0.07/0.32
% 0.07/0.32 % swap_simp
% 0.07/0.32 thf(fact_192_swap__simp,axiom,
% 0.07/0.32 ! [X: tree_a,Y: a] :
% 0.07/0.32 ( ( produc1478296771ee_a_a @ ( produc649172771ee_a_a @ X @ Y ) )
% 0.07/0.32 = ( produc686083979Tree_a @ Y @ X ) ) ).
% 0.07/0.32
% 0.07/0.32 % swap_simp
% 0.07/0.32 thf(fact_193_swap__simp,axiom,
% 0.07/0.32 ! [X: a,Y: tree_a] :
% 0.07/0.32 ( ( produc1515207979Tree_a @ ( produc686083979Tree_a @ X @ Y ) )
% 0.07/0.32 = ( produc649172771ee_a_a @ Y @ X ) ) ).
% 0.07/0.32
% 0.07/0.32 % swap_simp
% 0.07/0.32 thf(fact_194_snd__swap,axiom,
% 0.07/0.32 ! [X: produc143150363Tree_a] :
% 0.07/0.32 ( ( product_snd_Tree_a_a @ ( produc1515207979Tree_a @ X ) )
% 0.07/0.32 = ( product_fst_a_Tree_a @ X ) ) ).
% 0.07/0.32
% 0.07/0.32 % snd_swap
% 0.07/0.32 thf(fact_195_snd__swap,axiom,
% 0.07/0.32 ! [X: produc981471411ee_a_a] :
% 0.07/0.32 ( ( product_snd_a_Tree_a @ ( produc1478296771ee_a_a @ X ) )
% 0.07/0.32 = ( product_fst_Tree_a_a @ X ) ) ).
% 0.07/0.32
% 0.07/0.32 % snd_swap
% 0.07/0.32 thf(fact_196_fst__swap,axiom,
% 0.07/0.32 ! [X: produc981471411ee_a_a] :
% 0.07/0.32 ( ( product_fst_a_Tree_a @ ( produc1478296771ee_a_a @ X ) )
% 0.07/0.32 = ( product_snd_Tree_a_a @ X ) ) ).
% 0.07/0.32
% 0.07/0.32 % fst_swap
% 0.07/0.32 thf(fact_197_fst__swap,axiom,
% 0.07/0.32 ! [X: produc143150363Tree_a] :
% 0.07/0.32 ( ( product_fst_Tree_a_a @ ( produc1515207979Tree_a @ X ) )
% 0.07/0.32 = ( product_snd_a_Tree_a @ X ) ) ).
% 0.07/0.32
% 0.07/0.32 % fst_swap
% 0.07/0.32 thf(fact_198_is__heap_Osimps_I1_J,axiom,
% 0.07/0.32 is_heap_a @ e_a ).
% 0.07/0.32
% 0.07/0.32 % is_heap.simps(1)
% 0.07/0.32 thf(fact_199_is__heap_Osimps_I2_J,axiom,
% 0.07/0.32 ! [V: a] : ( is_heap_a @ ( t_a @ V @ e_a @ e_a ) ) ).
% 0.07/0.32
% 0.07/0.32 % is_heap.simps(2)
% 0.07/0.32 thf(fact_200_removeLeaf__val__val,axiom,
% 0.07/0.32 ! [T2: tree_a] :
% 0.07/0.32 ( ( ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ T2 ) )
% 0.07/0.32 != e_a )
% 0.07/0.32 => ( ( T2 != e_a )
% 0.07/0.32 => ( ( val_a @ T2 )
% 0.07/0.32 = ( val_a @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ T2 ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % removeLeaf_val_val
% 0.07/0.32 thf(fact_201_Heap__axioms__def,axiom,
% 0.07/0.32 ( heap_axioms_a_a
% 0.07/0.32 = ( ^ [Is_empty: a > $o,Of_list: list_a > a,Multiset: a > multiset_a,As_tree: a > tree_a,Remove_max: a > product_prod_a_a] :
% 0.07/0.32 ( ! [L2: a] :
% 0.07/0.32 ( ( Multiset @ L2 )
% 0.07/0.32 = ( multiset_a2 @ ( As_tree @ L2 ) ) )
% 0.07/0.32 & ! [I2: list_a] : ( is_heap_a @ ( As_tree @ ( Of_list @ I2 ) ) )
% 0.07/0.32 & ! [T3: a] :
% 0.07/0.32 ( ( ( As_tree @ T3 )
% 0.07/0.32 = e_a )
% 0.07/0.32 = ( Is_empty @ T3 ) )
% 0.07/0.32 & ! [L2: a,M4: a,L3: a] :
% 0.07/0.32 ( ~ ( Is_empty @ L2 )
% 0.07/0.32 => ( ( ( product_Pair_a_a @ M4 @ L3 )
% 0.07/0.32 = ( Remove_max @ L2 ) )
% 0.07/0.32 => ( ( add_mset_a @ M4 @ ( Multiset @ L3 ) )
% 0.07/0.32 = ( Multiset @ L2 ) ) ) )
% 0.07/0.32 & ! [L2: a,M4: a,L3: a] :
% 0.07/0.32 ( ~ ( Is_empty @ L2 )
% 0.07/0.32 => ( ( is_heap_a @ ( As_tree @ L2 ) )
% 0.07/0.32 => ( ( ( product_Pair_a_a @ M4 @ L3 )
% 0.07/0.32 = ( Remove_max @ L2 ) )
% 0.07/0.32 => ( is_heap_a @ ( As_tree @ L3 ) ) ) ) )
% 0.07/0.32 & ! [T3: a,M4: a,T4: a] :
% 0.07/0.32 ( ~ ( Is_empty @ T3 )
% 0.07/0.32 => ( ( ( product_Pair_a_a @ M4 @ T4 )
% 0.07/0.32 = ( Remove_max @ T3 ) )
% 0.07/0.32 => ( M4
% 0.07/0.32 = ( val_a @ ( As_tree @ T3 ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Heap_axioms_def
% 0.07/0.32 thf(fact_202_Heap__axioms__def,axiom,
% 0.07/0.32 ( heap_axioms_Tree_a_a
% 0.07/0.32 = ( ^ [Is_empty: tree_a > $o,Of_list: list_a > tree_a,Multiset: tree_a > multiset_a,As_tree: tree_a > tree_a,Remove_max: tree_a > produc143150363Tree_a] :
% 0.07/0.32 ( ! [L2: tree_a] :
% 0.07/0.32 ( ( Multiset @ L2 )
% 0.07/0.32 = ( multiset_a2 @ ( As_tree @ L2 ) ) )
% 0.07/0.32 & ! [I2: list_a] : ( is_heap_a @ ( As_tree @ ( Of_list @ I2 ) ) )
% 0.07/0.32 & ! [T3: tree_a] :
% 0.07/0.32 ( ( ( As_tree @ T3 )
% 0.07/0.32 = e_a )
% 0.07/0.32 = ( Is_empty @ T3 ) )
% 0.07/0.32 & ! [L2: tree_a,M4: a,L3: tree_a] :
% 0.07/0.32 ( ~ ( Is_empty @ L2 )
% 0.07/0.32 => ( ( ( produc686083979Tree_a @ M4 @ L3 )
% 0.07/0.32 = ( Remove_max @ L2 ) )
% 0.07/0.32 => ( ( add_mset_a @ M4 @ ( Multiset @ L3 ) )
% 0.07/0.32 = ( Multiset @ L2 ) ) ) )
% 0.07/0.32 & ! [L2: tree_a,M4: a,L3: tree_a] :
% 0.07/0.32 ( ~ ( Is_empty @ L2 )
% 0.07/0.32 => ( ( is_heap_a @ ( As_tree @ L2 ) )
% 0.07/0.32 => ( ( ( produc686083979Tree_a @ M4 @ L3 )
% 0.07/0.32 = ( Remove_max @ L2 ) )
% 0.07/0.32 => ( is_heap_a @ ( As_tree @ L3 ) ) ) ) )
% 0.07/0.32 & ! [T3: tree_a,M4: a,T4: tree_a] :
% 0.07/0.32 ( ~ ( Is_empty @ T3 )
% 0.07/0.32 => ( ( ( produc686083979Tree_a @ M4 @ T4 )
% 0.07/0.32 = ( Remove_max @ T3 ) )
% 0.07/0.32 => ( M4
% 0.07/0.32 = ( val_a @ ( As_tree @ T3 ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Heap_axioms_def
% 0.07/0.32 thf(fact_203_Heap__axioms_Ointro,axiom,
% 0.07/0.32 ! [Multiset2: a > multiset_a,As_tree2: a > tree_a,Of_list2: list_a > a,Is_empty2: a > $o,Remove_max2: a > product_prod_a_a] :
% 0.07/0.32 ( ! [L4: a] :
% 0.07/0.32 ( ( Multiset2 @ L4 )
% 0.07/0.32 = ( multiset_a2 @ ( As_tree2 @ L4 ) ) )
% 0.07/0.32 => ( ! [I3: list_a] : ( is_heap_a @ ( As_tree2 @ ( Of_list2 @ I3 ) ) )
% 0.07/0.32 => ( ! [T5: a] :
% 0.07/0.32 ( ( ( As_tree2 @ T5 )
% 0.07/0.32 = e_a )
% 0.07/0.32 = ( Is_empty2 @ T5 ) )
% 0.07/0.32 => ( ! [L4: a,M5: a,L5: a] :
% 0.07/0.32 ( ~ ( Is_empty2 @ L4 )
% 0.07/0.32 => ( ( ( product_Pair_a_a @ M5 @ L5 )
% 0.07/0.32 = ( Remove_max2 @ L4 ) )
% 0.07/0.32 => ( ( add_mset_a @ M5 @ ( Multiset2 @ L5 ) )
% 0.07/0.32 = ( Multiset2 @ L4 ) ) ) )
% 0.07/0.32 => ( ! [L4: a,M5: a,L5: a] :
% 0.07/0.32 ( ~ ( Is_empty2 @ L4 )
% 0.07/0.32 => ( ( is_heap_a @ ( As_tree2 @ L4 ) )
% 0.07/0.32 => ( ( ( product_Pair_a_a @ M5 @ L5 )
% 0.07/0.32 = ( Remove_max2 @ L4 ) )
% 0.07/0.32 => ( is_heap_a @ ( As_tree2 @ L5 ) ) ) ) )
% 0.07/0.32 => ( ! [T5: a,M5: a,T6: a] :
% 0.07/0.32 ( ~ ( Is_empty2 @ T5 )
% 0.07/0.32 => ( ( ( product_Pair_a_a @ M5 @ T6 )
% 0.07/0.32 = ( Remove_max2 @ T5 ) )
% 0.07/0.32 => ( M5
% 0.07/0.32 = ( val_a @ ( As_tree2 @ T5 ) ) ) ) )
% 0.07/0.32 => ( heap_axioms_a_a @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Heap_axioms.intro
% 0.07/0.32 thf(fact_204_Heap__axioms_Ointro,axiom,
% 0.07/0.32 ! [Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Of_list2: list_a > tree_a,Is_empty2: tree_a > $o,Remove_max2: tree_a > produc143150363Tree_a] :
% 0.07/0.32 ( ! [L4: tree_a] :
% 0.07/0.32 ( ( Multiset2 @ L4 )
% 0.07/0.32 = ( multiset_a2 @ ( As_tree2 @ L4 ) ) )
% 0.07/0.32 => ( ! [I3: list_a] : ( is_heap_a @ ( As_tree2 @ ( Of_list2 @ I3 ) ) )
% 0.07/0.32 => ( ! [T5: tree_a] :
% 0.07/0.32 ( ( ( As_tree2 @ T5 )
% 0.07/0.32 = e_a )
% 0.07/0.32 = ( Is_empty2 @ T5 ) )
% 0.07/0.32 => ( ! [L4: tree_a,M5: a,L5: tree_a] :
% 0.07/0.32 ( ~ ( Is_empty2 @ L4 )
% 0.07/0.32 => ( ( ( produc686083979Tree_a @ M5 @ L5 )
% 0.07/0.32 = ( Remove_max2 @ L4 ) )
% 0.07/0.32 => ( ( add_mset_a @ M5 @ ( Multiset2 @ L5 ) )
% 0.07/0.32 = ( Multiset2 @ L4 ) ) ) )
% 0.07/0.32 => ( ! [L4: tree_a,M5: a,L5: tree_a] :
% 0.07/0.32 ( ~ ( Is_empty2 @ L4 )
% 0.07/0.32 => ( ( is_heap_a @ ( As_tree2 @ L4 ) )
% 0.07/0.32 => ( ( ( produc686083979Tree_a @ M5 @ L5 )
% 0.07/0.32 = ( Remove_max2 @ L4 ) )
% 0.07/0.32 => ( is_heap_a @ ( As_tree2 @ L5 ) ) ) ) )
% 0.07/0.32 => ( ! [T5: tree_a,M5: a,T6: tree_a] :
% 0.07/0.32 ( ~ ( Is_empty2 @ T5 )
% 0.07/0.32 => ( ( ( produc686083979Tree_a @ M5 @ T6 )
% 0.07/0.32 = ( Remove_max2 @ T5 ) )
% 0.07/0.32 => ( M5
% 0.07/0.32 = ( val_a @ ( As_tree2 @ T5 ) ) ) ) )
% 0.07/0.32 => ( heap_axioms_Tree_a_a @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Heap_axioms.intro
% 0.07/0.32 thf(fact_205_val_Osimps,axiom,
% 0.07/0.32 ! [V: a,Uu: tree_a,Uv: tree_a] :
% 0.07/0.32 ( ( val_a @ ( t_a @ V @ Uu @ Uv ) )
% 0.07/0.32 = V ) ).
% 0.07/0.32
% 0.07/0.32 % val.simps
% 0.07/0.32 thf(fact_206_is__heap_Osimps_I4_J,axiom,
% 0.07/0.32 ! [V: a,Va: a,Vb: tree_a,Vc: tree_a] :
% 0.07/0.32 ( ( is_heap_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) )
% 0.07/0.32 = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 0.07/0.32 & ( is_heap_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % is_heap.simps(4)
% 0.07/0.32 thf(fact_207_is__heap_Osimps_I3_J,axiom,
% 0.07/0.32 ! [V: a,Va: a,Vb: tree_a,Vc: tree_a] :
% 0.07/0.32 ( ( is_heap_a @ ( t_a @ V @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) )
% 0.07/0.32 = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 0.07/0.32 & ( is_heap_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % is_heap.simps(3)
% 0.07/0.32 thf(fact_208_Heap_Oremove__max__is__heap,axiom,
% 0.07/0.32 ! [Empty: a,Is_empty2: a > $o,Of_list2: list_a > a,Multiset2: a > multiset_a,As_tree2: a > tree_a,Remove_max2: a > product_prod_a_a,L: a,M6: a,L6: a] :
% 0.07/0.32 ( ( heap_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 0.07/0.32 => ( ~ ( Is_empty2 @ L )
% 0.07/0.32 => ( ( is_heap_a @ ( As_tree2 @ L ) )
% 0.07/0.32 => ( ( ( product_Pair_a_a @ M6 @ L6 )
% 0.07/0.32 = ( Remove_max2 @ L ) )
% 0.07/0.32 => ( is_heap_a @ ( As_tree2 @ L6 ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Heap.remove_max_is_heap
% 0.07/0.32 thf(fact_209_Heap_Oremove__max__is__heap,axiom,
% 0.07/0.32 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a,L: tree_a,M6: a,L6: tree_a] :
% 0.07/0.32 ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 0.07/0.32 => ( ~ ( Is_empty2 @ L )
% 0.07/0.32 => ( ( is_heap_a @ ( As_tree2 @ L ) )
% 0.07/0.32 => ( ( ( produc686083979Tree_a @ M6 @ L6 )
% 0.07/0.32 = ( Remove_max2 @ L ) )
% 0.07/0.32 => ( is_heap_a @ ( As_tree2 @ L6 ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Heap.remove_max_is_heap
% 0.07/0.32 thf(fact_210_verit__la__disequality,axiom,
% 0.07/0.32 ! [A: a,B: a] :
% 0.07/0.32 ( ( A = B )
% 0.07/0.32 | ~ ( ord_less_eq_a @ A @ B )
% 0.07/0.32 | ~ ( ord_less_eq_a @ B @ A ) ) ).
% 0.07/0.32
% 0.07/0.32 % verit_la_disequality
% 0.07/0.32 thf(fact_211_add__right__mono,axiom,
% 0.07/0.32 ! [A: multiset_a,B: multiset_a,C: multiset_a] :
% 0.07/0.32 ( ( ord_le1199012836iset_a @ A @ B )
% 0.07/0.32 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % add_right_mono
% 0.07/0.32 thf(fact_212_add__left__mono,axiom,
% 0.07/0.32 ! [A: multiset_a,B: multiset_a,C: multiset_a] :
% 0.07/0.32 ( ( ord_le1199012836iset_a @ A @ B )
% 0.07/0.32 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ C @ A ) @ ( plus_plus_multiset_a @ C @ B ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % add_left_mono
% 0.07/0.32 thf(fact_213_add__mono,axiom,
% 0.07/0.32 ! [A: multiset_a,B: multiset_a,C: multiset_a,D: multiset_a] :
% 0.07/0.32 ( ( ord_le1199012836iset_a @ A @ B )
% 0.07/0.32 => ( ( ord_le1199012836iset_a @ C @ D )
% 0.07/0.32 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ D ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % add_mono
% 0.07/0.32 thf(fact_214_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.07/0.32 ! [I: multiset_a,J: multiset_a,K: multiset_a,L: multiset_a] :
% 0.07/0.32 ( ( ( ord_le1199012836iset_a @ I @ J )
% 0.07/0.32 & ( ord_le1199012836iset_a @ K @ L ) )
% 0.07/0.32 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I @ K ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % add_mono_thms_linordered_semiring(1)
% 0.07/0.32 thf(fact_215_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.07/0.32 ! [I: multiset_a,J: multiset_a,K: multiset_a,L: multiset_a] :
% 0.07/0.32 ( ( ( I = J )
% 0.07/0.32 & ( ord_le1199012836iset_a @ K @ L ) )
% 0.07/0.32 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I @ K ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % add_mono_thms_linordered_semiring(2)
% 0.07/0.32 thf(fact_216_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.07/0.32 ! [I: multiset_a,J: multiset_a,K: multiset_a,L: multiset_a] :
% 0.07/0.32 ( ( ( ord_le1199012836iset_a @ I @ J )
% 0.07/0.32 & ( K = L ) )
% 0.07/0.32 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I @ K ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % add_mono_thms_linordered_semiring(3)
% 0.07/0.32 thf(fact_217_Heap_Oas__tree__empty,axiom,
% 0.07/0.32 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a,T2: tree_a] :
% 0.07/0.32 ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 0.07/0.32 => ( ( ( As_tree2 @ T2 )
% 0.07/0.32 = e_a )
% 0.07/0.32 = ( Is_empty2 @ T2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Heap.as_tree_empty
% 0.07/0.32 thf(fact_218_Heap_Ois__heap__of__list,axiom,
% 0.07/0.32 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a,I: list_a] :
% 0.07/0.32 ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 0.07/0.32 => ( is_heap_a @ ( As_tree2 @ ( Of_list2 @ I ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Heap.is_heap_of_list
% 0.07/0.32 thf(fact_219_Heap_Omultiset,axiom,
% 0.07/0.32 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a,L: tree_a] :
% 0.07/0.32 ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 0.07/0.32 => ( ( Multiset2 @ L )
% 0.07/0.32 = ( multiset_a2 @ ( As_tree2 @ L ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Heap.multiset
% 0.07/0.32 thf(fact_220_Heap_Oaxioms_I2_J,axiom,
% 0.07/0.32 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a] :
% 0.07/0.32 ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 0.07/0.32 => ( heap_axioms_Tree_a_a @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % Heap.axioms(2)
% 0.07/0.32 thf(fact_221_is__heap_Osimps_I6_J,axiom,
% 0.07/0.32 ! [V: a,Vd: a,Ve: tree_a,Vf: tree_a,Va: a,Vb: tree_a,Vc: tree_a] :
% 0.07/0.32 ( ( is_heap_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
% 0.07/0.32 = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 0.07/0.32 & ( is_heap_a @ ( t_a @ Va @ Vb @ Vc ) )
% 0.07/0.32 & ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
% 0.07/0.32 & ( is_heap_a @ ( t_a @ Vd @ Ve @ Vf ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % is_heap.simps(6)
% 0.07/0.32 thf(fact_222_is__heap_Osimps_I5_J,axiom,
% 0.07/0.32 ! [V: a,Va: a,Vb: tree_a,Vc: tree_a,Vd: a,Ve: tree_a,Vf: tree_a] :
% 0.07/0.32 ( ( is_heap_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 0.07/0.32 = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
% 0.07/0.32 & ( is_heap_a @ ( t_a @ Vd @ Ve @ Vf ) )
% 0.07/0.32 & ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 0.07/0.32 & ( is_heap_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % is_heap.simps(5)
% 0.07/0.32 thf(fact_223_Heap_Oremove__max__multiset_H,axiom,
% 0.07/0.32 ! [Empty: a,Is_empty2: a > $o,Of_list2: list_a > a,Multiset2: a > multiset_a,As_tree2: a > tree_a,Remove_max2: a > product_prod_a_a,L: a,M6: a,L6: a] :
% 0.07/0.32 ( ( heap_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 0.07/0.32 => ( ~ ( Is_empty2 @ L )
% 0.07/0.32 => ( ( ( product_Pair_a_a @ M6 @ L6 )
% 0.07/0.32 = ( Remove_max2 @ L ) )
% 0.07/0.32 => ( ( add_mset_a @ M6 @ ( Multiset2 @ L6 ) )
% 0.07/0.32 = ( Multiset2 @ L ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Heap.remove_max_multiset'
% 0.07/0.32 thf(fact_224_Heap_Oremove__max__multiset_H,axiom,
% 0.07/0.32 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a,L: tree_a,M6: a,L6: tree_a] :
% 0.07/0.32 ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 0.07/0.32 => ( ~ ( Is_empty2 @ L )
% 0.07/0.32 => ( ( ( produc686083979Tree_a @ M6 @ L6 )
% 0.07/0.32 = ( Remove_max2 @ L ) )
% 0.07/0.32 => ( ( add_mset_a @ M6 @ ( Multiset2 @ L6 ) )
% 0.07/0.32 = ( Multiset2 @ L ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Heap.remove_max_multiset'
% 0.07/0.32 thf(fact_225_Heap_Oremove__max__val,axiom,
% 0.07/0.32 ! [Empty: a,Is_empty2: a > $o,Of_list2: list_a > a,Multiset2: a > multiset_a,As_tree2: a > tree_a,Remove_max2: a > product_prod_a_a,T2: a,M6: a,T: a] :
% 0.07/0.32 ( ( heap_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 0.07/0.32 => ( ~ ( Is_empty2 @ T2 )
% 0.07/0.32 => ( ( ( product_Pair_a_a @ M6 @ T )
% 0.07/0.32 = ( Remove_max2 @ T2 ) )
% 0.07/0.32 => ( M6
% 0.07/0.32 = ( val_a @ ( As_tree2 @ T2 ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Heap.remove_max_val
% 0.07/0.32 thf(fact_226_Heap_Oremove__max__val,axiom,
% 0.07/0.32 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a,T2: tree_a,M6: a,T: tree_a] :
% 0.07/0.32 ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 0.07/0.32 => ( ~ ( Is_empty2 @ T2 )
% 0.07/0.32 => ( ( ( produc686083979Tree_a @ M6 @ T )
% 0.07/0.32 = ( Remove_max2 @ T2 ) )
% 0.07/0.32 => ( M6
% 0.07/0.32 = ( val_a @ ( As_tree2 @ T2 ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Heap.remove_max_val
% 0.07/0.32 thf(fact_227_siftDown_Osimps_I4_J,axiom,
% 0.07/0.32 ! [Va: a,Vb: tree_a,Vc: tree_a,V: a] :
% 0.07/0.32 ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 0.07/0.32 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) )
% 0.07/0.32 = ( t_a @ V @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) ) )
% 0.07/0.32 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 0.07/0.32 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) )
% 0.07/0.32 = ( t_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ e_a @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % siftDown.simps(4)
% 0.07/0.32 thf(fact_228_siftDown_Osimps_I3_J,axiom,
% 0.07/0.32 ! [Va: a,Vb: tree_a,Vc: tree_a,V: a] :
% 0.07/0.32 ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 0.07/0.32 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) )
% 0.07/0.32 = ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) ) )
% 0.07/0.32 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 0.07/0.32 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) )
% 0.07/0.32 = ( t_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) @ e_a ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % siftDown.simps(3)
% 0.07/0.32 thf(fact_229_siftDown_Osimps_I5_J,axiom,
% 0.07/0.32 ! [Vd: a,Ve: tree_a,Vf: tree_a,Va: a,Vb: tree_a,Vc: tree_a,V: a] :
% 0.07/0.32 ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) )
% 0.07/0.32 => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 0.07/0.32 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 0.07/0.32 = ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) ) )
% 0.07/0.32 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 0.07/0.32 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 0.07/0.32 = ( t_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) @ ( t_a @ Vd @ Ve @ Vf ) ) ) ) ) )
% 0.07/0.32 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) )
% 0.07/0.32 => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
% 0.07/0.32 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 0.07/0.32 = ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) ) )
% 0.07/0.32 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
% 0.07/0.32 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 0.07/0.32 = ( t_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ ( t_a @ Va @ Vb @ Vc ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Vd @ Ve @ Vf ) ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % siftDown.simps(5)
% 0.07/0.32 thf(fact_230_siftDown_Osimps_I1_J,axiom,
% 0.07/0.32 ( ( heapIm1091024090Down_a @ e_a )
% 0.07/0.32 = e_a ) ).
% 0.07/0.32
% 0.07/0.32 % siftDown.simps(1)
% 0.07/0.32 thf(fact_231_siftDown__multiset,axiom,
% 0.07/0.32 ! [T2: tree_a] :
% 0.07/0.32 ( ( multiset_a2 @ ( heapIm1091024090Down_a @ T2 ) )
% 0.07/0.32 = ( multiset_a2 @ T2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % siftDown_multiset
% 0.07/0.32 thf(fact_232_siftDown__Node,axiom,
% 0.07/0.32 ! [T2: tree_a,V: a,L: tree_a,R: tree_a] :
% 0.07/0.32 ( ( T2
% 0.07/0.32 = ( t_a @ V @ L @ R ) )
% 0.07/0.32 => ? [L5: tree_a,V4: a,R2: tree_a] :
% 0.07/0.32 ( ( ( heapIm1091024090Down_a @ T2 )
% 0.07/0.32 = ( t_a @ V4 @ L5 @ R2 ) )
% 0.07/0.32 & ( ord_less_eq_a @ V @ V4 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % siftDown_Node
% 0.07/0.32 thf(fact_233_siftDown_Osimps_I2_J,axiom,
% 0.07/0.32 ! [V: a] :
% 0.07/0.32 ( ( heapIm1091024090Down_a @ ( t_a @ V @ e_a @ e_a ) )
% 0.07/0.32 = ( t_a @ V @ e_a @ e_a ) ) ).
% 0.07/0.32
% 0.07/0.32 % siftDown.simps(2)
% 0.07/0.32 thf(fact_234_siftDown__heap__is__heap,axiom,
% 0.07/0.32 ! [L: tree_a,R: tree_a,T2: tree_a,V: a] :
% 0.07/0.32 ( ( is_heap_a @ L )
% 0.07/0.32 => ( ( is_heap_a @ R )
% 0.07/0.32 => ( ( T2
% 0.07/0.32 = ( t_a @ V @ L @ R ) )
% 0.07/0.32 => ( is_heap_a @ ( heapIm1091024090Down_a @ T2 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % siftDown_heap_is_heap
% 0.07/0.32 thf(fact_235_siftDown_Osimps_I6_J,axiom,
% 0.07/0.32 ! [Va: a,Vb: tree_a,Vc: tree_a,Vd: a,Ve: tree_a,Vf: tree_a,V: a] :
% 0.07/0.32 ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 0.07/0.32 => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
% 0.07/0.32 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
% 0.07/0.32 = ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) ) )
% 0.07/0.32 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
% 0.07/0.32 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
% 0.07/0.32 = ( t_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Vd @ Ve @ Vf ) ) ) ) @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ) )
% 0.07/0.32 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 0.07/0.32 => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 0.07/0.32 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
% 0.07/0.32 = ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) ) )
% 0.07/0.32 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 0.07/0.32 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
% 0.07/0.32 = ( t_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( t_a @ Vd @ Ve @ Vf ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % siftDown.simps(6)
% 0.07/0.32 thf(fact_236_siftDown__in__tree,axiom,
% 0.07/0.32 ! [T2: tree_a] :
% 0.07/0.32 ( ( T2 != e_a )
% 0.07/0.32 => ( in_tree_a @ ( val_a @ ( heapIm1091024090Down_a @ T2 ) ) @ T2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % siftDown_in_tree
% 0.07/0.32 thf(fact_237_is__heap__max,axiom,
% 0.07/0.32 ! [V: a,T2: tree_a] :
% 0.07/0.32 ( ( in_tree_a @ V @ T2 )
% 0.07/0.32 => ( ( is_heap_a @ T2 )
% 0.07/0.32 => ( ord_less_eq_a @ V @ ( val_a @ T2 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % is_heap_max
% 0.07/0.32 thf(fact_238_in__tree_Osimps_I2_J,axiom,
% 0.07/0.32 ! [V: a,V2: a,L: tree_a,R: tree_a] :
% 0.07/0.32 ( ( in_tree_a @ V @ ( t_a @ V2 @ L @ R ) )
% 0.07/0.32 = ( ( V = V2 )
% 0.07/0.32 | ( in_tree_a @ V @ L )
% 0.07/0.32 | ( in_tree_a @ V @ R ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % in_tree.simps(2)
% 0.07/0.32 thf(fact_239_in__tree_Osimps_I1_J,axiom,
% 0.07/0.32 ! [V: a] :
% 0.07/0.32 ~ ( in_tree_a @ V @ e_a ) ).
% 0.07/0.32
% 0.07/0.32 % in_tree.simps(1)
% 0.07/0.32 thf(fact_240_siftDown__in__tree__set,axiom,
% 0.07/0.32 ( in_tree_a
% 0.07/0.32 = ( ^ [V5: a,T3: tree_a] : ( in_tree_a @ V5 @ ( heapIm1091024090Down_a @ T3 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % siftDown_in_tree_set
% 0.07/0.32 thf(fact_241_hs__remove__max__def,axiom,
% 0.07/0.32 ( heapIm915766629_max_a
% 0.07/0.32 = ( ^ [T3: tree_a] :
% 0.07/0.32 ( if_Pro1144176865Tree_a
% 0.07/0.32 @ ( ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ T3 ) )
% 0.07/0.32 = e_a )
% 0.07/0.32 @ ( produc686083979Tree_a @ ( val_a @ T3 ) @ e_a )
% 0.07/0.32 @ ( produc686083979Tree_a @ ( val_a @ T3 ) @ ( heapIm1091024090Down_a @ ( t_a @ ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ T3 ) ) @ ( heapIm1140443833left_a @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ T3 ) ) ) @ ( heapIm1257206334ight_a @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ T3 ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % hs_remove_max_def
% 0.07/0.32 thf(fact_242_heapify_Osimps_I2_J,axiom,
% 0.07/0.32 ! [V: a,L: tree_a,R: tree_a] :
% 0.07/0.32 ( ( heapIm970322378pify_a @ ( t_a @ V @ L @ R ) )
% 0.07/0.32 = ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm970322378pify_a @ L ) @ ( heapIm970322378pify_a @ R ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % heapify.simps(2)
% 0.07/0.32 thf(fact_243_heapify_Osimps_I1_J,axiom,
% 0.07/0.32 ( ( heapIm970322378pify_a @ e_a )
% 0.07/0.32 = e_a ) ).
% 0.07/0.32
% 0.07/0.32 % heapify.simps(1)
% 0.07/0.32 thf(fact_244_heapify__heap__is__heap,axiom,
% 0.07/0.32 ! [T2: tree_a] : ( is_heap_a @ ( heapIm970322378pify_a @ T2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % heapify_heap_is_heap
% 0.07/0.32 thf(fact_245_multiset__heapify,axiom,
% 0.07/0.32 ! [T2: tree_a] :
% 0.07/0.32 ( ( multiset_a2 @ ( heapIm970322378pify_a @ T2 ) )
% 0.07/0.32 = ( multiset_a2 @ T2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % multiset_heapify
% 0.07/0.32 thf(fact_246_Heap__def,axiom,
% 0.07/0.32 ( heap_Tree_a_a
% 0.07/0.32 = ( ^ [Empty2: tree_a,Is_empty: tree_a > $o,Of_list: list_a > tree_a,Multiset: tree_a > multiset_a,As_tree: tree_a > tree_a,Remove_max: tree_a > produc143150363Tree_a] :
% 0.07/0.32 ( ( collection_Tree_a_a @ Empty2 @ Is_empty @ Of_list @ Multiset )
% 0.07/0.32 & ( heap_axioms_Tree_a_a @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Heap_def
% 0.07/0.32 thf(fact_247_Heap_Ointro,axiom,
% 0.07/0.32 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a] :
% 0.07/0.32 ( ( collection_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 )
% 0.07/0.32 => ( ( heap_axioms_Tree_a_a @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 0.07/0.32 => ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Heap.intro
% 0.07/0.32 thf(fact_248_Heap_Oaxioms_I1_J,axiom,
% 0.07/0.32 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a] :
% 0.07/0.32 ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 0.07/0.32 => ( collection_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % Heap.axioms(1)
% 0.07/0.32 thf(fact_249_HS_OCollection__axioms,axiom,
% 0.07/0.32 collection_Tree_a_a @ e_a @ heapIm229596386mpty_a @ heapIm1057938560list_a @ multiset_a2 ).
% 0.07/0.32
% 0.07/0.32 % HS.Collection_axioms
% 0.07/0.32 thf(fact_250_Collection_Omultiset__empty,axiom,
% 0.07/0.32 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a] :
% 0.07/0.32 ( ( collection_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 )
% 0.07/0.32 => ( ( Multiset2 @ Empty )
% 0.07/0.32 = zero_zero_multiset_a ) ) ).
% 0.07/0.32
% 0.07/0.32 % Collection.multiset_empty
% 0.07/0.32 thf(fact_251_Collection_Ois__empty__as__list,axiom,
% 0.07/0.32 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,E: tree_a] :
% 0.07/0.32 ( ( collection_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 )
% 0.07/0.32 => ( ( Is_empty2 @ E )
% 0.07/0.32 => ( ( Multiset2 @ E )
% 0.07/0.32 = zero_zero_multiset_a ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Collection.is_empty_as_list
% 0.07/0.32 thf(fact_252_hs__of__list__def,axiom,
% 0.07/0.32 ( heapIm1057938560list_a
% 0.07/0.32 = ( ^ [L2: list_a] : ( heapIm970322378pify_a @ ( heapIm1637418125tree_a @ L2 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % hs_of_list_def
% 0.07/0.32 thf(fact_253_pred__equals__eq2,axiom,
% 0.07/0.32 ! [R3: set_Pr921883667ee_a_a,S2: set_Pr921883667ee_a_a] :
% 0.07/0.32 ( ( ( ^ [X4: tree_a,Y6: a] : ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ X4 @ Y6 ) @ R3 ) )
% 0.07/0.32 = ( ^ [X4: tree_a,Y6: a] : ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ X4 @ Y6 ) @ S2 ) ) )
% 0.07/0.32 = ( R3 = S2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % pred_equals_eq2
% 0.07/0.32 thf(fact_254_pred__equals__eq2,axiom,
% 0.07/0.32 ! [R3: set_Pr158363655iset_a,S2: set_Pr158363655iset_a] :
% 0.07/0.32 ( ( ( ^ [X4: multiset_a,Y6: multiset_a] : ( member340150864iset_a @ ( produc2037245207iset_a @ X4 @ Y6 ) @ R3 ) )
% 0.07/0.32 = ( ^ [X4: multiset_a,Y6: multiset_a] : ( member340150864iset_a @ ( produc2037245207iset_a @ X4 @ Y6 ) @ S2 ) ) )
% 0.07/0.32 = ( R3 = S2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % pred_equals_eq2
% 0.07/0.32 thf(fact_255_pred__equals__eq2,axiom,
% 0.07/0.32 ! [R3: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
% 0.07/0.32 ( ( ( ^ [X4: a,Y6: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y6 ) @ R3 ) )
% 0.07/0.32 = ( ^ [X4: a,Y6: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y6 ) @ S2 ) ) )
% 0.07/0.32 = ( R3 = S2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % pred_equals_eq2
% 0.07/0.32 thf(fact_256_pred__equals__eq2,axiom,
% 0.07/0.32 ! [R3: set_Pr1070816379Tree_a,S2: set_Pr1070816379Tree_a] :
% 0.07/0.32 ( ( ( ^ [X4: a,Y6: tree_a] : ( member254315204Tree_a @ ( produc686083979Tree_a @ X4 @ Y6 ) @ R3 ) )
% 0.07/0.32 = ( ^ [X4: a,Y6: tree_a] : ( member254315204Tree_a @ ( produc686083979Tree_a @ X4 @ Y6 ) @ S2 ) ) )
% 0.07/0.32 = ( R3 = S2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % pred_equals_eq2
% 0.07/0.32 thf(fact_257_subrelI,axiom,
% 0.07/0.32 ! [R: set_Pr921883667ee_a_a,S3: set_Pr921883667ee_a_a] :
% 0.07/0.32 ( ! [X3: tree_a,Y3: a] :
% 0.07/0.32 ( ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ X3 @ Y3 ) @ R )
% 0.07/0.32 => ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ X3 @ Y3 ) @ S3 ) )
% 0.07/0.32 => ( ord_le864074675ee_a_a @ R @ S3 ) ) ).
% 0.07/0.32
% 0.07/0.32 % subrelI
% 0.07/0.32 thf(fact_258_subrelI,axiom,
% 0.07/0.32 ! [R: set_Pr158363655iset_a,S3: set_Pr158363655iset_a] :
% 0.07/0.32 ( ! [X3: multiset_a,Y3: multiset_a] :
% 0.07/0.32 ( ( member340150864iset_a @ ( produc2037245207iset_a @ X3 @ Y3 ) @ R )
% 0.07/0.32 => ( member340150864iset_a @ ( produc2037245207iset_a @ X3 @ Y3 ) @ S3 ) )
% 0.07/0.32 => ( ord_le1978107815iset_a @ R @ S3 ) ) ).
% 0.07/0.32
% 0.07/0.32 % subrelI
% 0.07/0.32 thf(fact_259_subrelI,axiom,
% 0.07/0.32 ! [R: set_Product_prod_a_a,S3: set_Product_prod_a_a] :
% 0.07/0.32 ( ! [X3: a,Y3: a] :
% 0.07/0.32 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R )
% 0.07/0.32 => ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ S3 ) )
% 0.07/0.32 => ( ord_le1824328871od_a_a @ R @ S3 ) ) ).
% 0.07/0.32
% 0.07/0.32 % subrelI
% 0.07/0.32 thf(fact_260_subrelI,axiom,
% 0.07/0.32 ! [R: set_Pr1070816379Tree_a,S3: set_Pr1070816379Tree_a] :
% 0.07/0.32 ( ! [X3: a,Y3: tree_a] :
% 0.07/0.32 ( ( member254315204Tree_a @ ( produc686083979Tree_a @ X3 @ Y3 ) @ R )
% 0.07/0.32 => ( member254315204Tree_a @ ( produc686083979Tree_a @ X3 @ Y3 ) @ S3 ) )
% 0.07/0.32 => ( ord_le1013007387Tree_a @ R @ S3 ) ) ).
% 0.07/0.32
% 0.07/0.32 % subrelI
% 0.07/0.32 thf(fact_261_pred__subset__eq2,axiom,
% 0.07/0.32 ! [R3: set_Pr921883667ee_a_a,S2: set_Pr921883667ee_a_a] :
% 0.07/0.32 ( ( ord_le1748873858_a_a_o
% 0.07/0.32 @ ^ [X4: tree_a,Y6: a] : ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ X4 @ Y6 ) @ R3 )
% 0.07/0.32 @ ^ [X4: tree_a,Y6: a] : ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ X4 @ Y6 ) @ S2 ) )
% 0.07/0.32 = ( ord_le864074675ee_a_a @ R3 @ S2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % pred_subset_eq2
% 0.07/0.32 thf(fact_262_pred__subset__eq2,axiom,
% 0.07/0.32 ! [R3: set_Pr158363655iset_a,S2: set_Pr158363655iset_a] :
% 0.07/0.32 ( ( ord_le51495822et_a_o
% 0.07/0.32 @ ^ [X4: multiset_a,Y6: multiset_a] : ( member340150864iset_a @ ( produc2037245207iset_a @ X4 @ Y6 ) @ R3 )
% 0.07/0.32 @ ^ [X4: multiset_a,Y6: multiset_a] : ( member340150864iset_a @ ( produc2037245207iset_a @ X4 @ Y6 ) @ S2 ) )
% 0.07/0.32 = ( ord_le1978107815iset_a @ R3 @ S2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % pred_subset_eq2
% 0.07/0.32 thf(fact_263_pred__subset__eq2,axiom,
% 0.07/0.32 ! [R3: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
% 0.07/0.32 ( ( ord_less_eq_a_a_o
% 0.07/0.32 @ ^ [X4: a,Y6: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y6 ) @ R3 )
% 0.07/0.32 @ ^ [X4: a,Y6: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y6 ) @ S2 ) )
% 0.07/0.32 = ( ord_le1824328871od_a_a @ R3 @ S2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % pred_subset_eq2
% 0.07/0.32 thf(fact_264_pred__subset__eq2,axiom,
% 0.07/0.32 ! [R3: set_Pr1070816379Tree_a,S2: set_Pr1070816379Tree_a] :
% 0.07/0.32 ( ( ord_le1327026842ee_a_o
% 0.07/0.32 @ ^ [X4: a,Y6: tree_a] : ( member254315204Tree_a @ ( produc686083979Tree_a @ X4 @ Y6 ) @ R3 )
% 0.07/0.32 @ ^ [X4: a,Y6: tree_a] : ( member254315204Tree_a @ ( produc686083979Tree_a @ X4 @ Y6 ) @ S2 ) )
% 0.07/0.32 = ( ord_le1013007387Tree_a @ R3 @ S2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % pred_subset_eq2
% 0.07/0.32 thf(fact_265_heap__top__geq,axiom,
% 0.07/0.32 ! [A: a,T2: tree_a] :
% 0.07/0.32 ( ( member_a @ A @ ( set_mset_a @ ( multiset_a2 @ T2 ) ) )
% 0.07/0.32 => ( ( is_heap_a @ T2 )
% 0.07/0.32 => ( ord_less_eq_a @ A @ ( val_a @ T2 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % heap_top_geq
% 0.07/0.32 thf(fact_266_multi__member__last,axiom,
% 0.07/0.32 ! [X: produc143150363Tree_a] : ( member254315204Tree_a @ X @ ( set_ms1129970904Tree_a @ ( add_ms205941497Tree_a @ X @ zero_z2144236696Tree_a ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multi_member_last
% 0.07/0.32 thf(fact_267_multi__member__last,axiom,
% 0.07/0.32 ! [X: produc1127127335iset_a] : ( member340150864iset_a @ X @ ( set_ms1768116836iset_a @ ( add_ms1353365509iset_a @ X @ zero_z1941791140iset_a ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multi_member_last
% 0.07/0.32 thf(fact_268_multi__member__last,axiom,
% 0.07/0.32 ! [X: product_prod_a_a] : ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ ( add_ms1725977093od_a_a @ X @ zero_z160849828od_a_a ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multi_member_last
% 0.07/0.32 thf(fact_269_multi__member__last,axiom,
% 0.07/0.32 ! [X: multiset_a] : ( member_multiset_a @ X @ ( set_mset_multiset_a @ ( add_mset_multiset_a @ X @ zero_z1580389697iset_a ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multi_member_last
% 0.07/0.32 thf(fact_270_multi__member__last,axiom,
% 0.07/0.32 ! [X: a] : ( member_a @ X @ ( set_mset_a @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multi_member_last
% 0.07/0.32 thf(fact_271_union__single__eq__member,axiom,
% 0.07/0.32 ! [X: produc143150363Tree_a,M: multis2082063201Tree_a,N: multis2082063201Tree_a] :
% 0.07/0.32 ( ( ( add_ms205941497Tree_a @ X @ M )
% 0.07/0.32 = N )
% 0.07/0.32 => ( member254315204Tree_a @ X @ ( set_ms1129970904Tree_a @ N ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % union_single_eq_member
% 0.07/0.32 thf(fact_272_union__single__eq__member,axiom,
% 0.07/0.32 ! [X: produc1127127335iset_a,M: multis782275565iset_a,N: multis782275565iset_a] :
% 0.07/0.32 ( ( ( add_ms1353365509iset_a @ X @ M )
% 0.07/0.32 = N )
% 0.07/0.32 => ( member340150864iset_a @ X @ ( set_ms1768116836iset_a @ N ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % union_single_eq_member
% 0.07/0.32 thf(fact_273_union__single__eq__member,axiom,
% 0.07/0.32 ! [X: product_prod_a_a,M: multis599418605od_a_a,N: multis599418605od_a_a] :
% 0.07/0.32 ( ( ( add_ms1725977093od_a_a @ X @ M )
% 0.07/0.32 = N )
% 0.07/0.32 => ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ N ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % union_single_eq_member
% 0.07/0.32 thf(fact_274_union__single__eq__member,axiom,
% 0.07/0.32 ! [X: multiset_a,M: multiset_multiset_a,N: multiset_multiset_a] :
% 0.07/0.32 ( ( ( add_mset_multiset_a @ X @ M )
% 0.07/0.32 = N )
% 0.07/0.32 => ( member_multiset_a @ X @ ( set_mset_multiset_a @ N ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % union_single_eq_member
% 0.07/0.32 thf(fact_275_union__single__eq__member,axiom,
% 0.07/0.32 ! [X: a,M: multiset_a,N: multiset_a] :
% 0.07/0.32 ( ( ( add_mset_a @ X @ M )
% 0.07/0.32 = N )
% 0.07/0.32 => ( member_a @ X @ ( set_mset_a @ N ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % union_single_eq_member
% 0.07/0.32 thf(fact_276_insert__noteq__member,axiom,
% 0.07/0.32 ! [B: produc143150363Tree_a,B4: multis2082063201Tree_a,C: produc143150363Tree_a,C2: multis2082063201Tree_a] :
% 0.07/0.32 ( ( ( add_ms205941497Tree_a @ B @ B4 )
% 0.07/0.32 = ( add_ms205941497Tree_a @ C @ C2 ) )
% 0.07/0.32 => ( ( B != C )
% 0.07/0.32 => ( member254315204Tree_a @ C @ ( set_ms1129970904Tree_a @ B4 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % insert_noteq_member
% 0.07/0.32 thf(fact_277_insert__noteq__member,axiom,
% 0.07/0.32 ! [B: produc1127127335iset_a,B4: multis782275565iset_a,C: produc1127127335iset_a,C2: multis782275565iset_a] :
% 0.07/0.32 ( ( ( add_ms1353365509iset_a @ B @ B4 )
% 0.07/0.32 = ( add_ms1353365509iset_a @ C @ C2 ) )
% 0.07/0.32 => ( ( B != C )
% 0.07/0.32 => ( member340150864iset_a @ C @ ( set_ms1768116836iset_a @ B4 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % insert_noteq_member
% 0.07/0.32 thf(fact_278_insert__noteq__member,axiom,
% 0.07/0.32 ! [B: product_prod_a_a,B4: multis599418605od_a_a,C: product_prod_a_a,C2: multis599418605od_a_a] :
% 0.07/0.32 ( ( ( add_ms1725977093od_a_a @ B @ B4 )
% 0.07/0.32 = ( add_ms1725977093od_a_a @ C @ C2 ) )
% 0.07/0.32 => ( ( B != C )
% 0.07/0.32 => ( member449909584od_a_a @ C @ ( set_ms1818337124od_a_a @ B4 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % insert_noteq_member
% 0.07/0.32 thf(fact_279_insert__noteq__member,axiom,
% 0.07/0.32 ! [B: multiset_a,B4: multiset_multiset_a,C: multiset_a,C2: multiset_multiset_a] :
% 0.07/0.32 ( ( ( add_mset_multiset_a @ B @ B4 )
% 0.07/0.32 = ( add_mset_multiset_a @ C @ C2 ) )
% 0.07/0.32 => ( ( B != C )
% 0.07/0.32 => ( member_multiset_a @ C @ ( set_mset_multiset_a @ B4 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % insert_noteq_member
% 0.07/0.32 thf(fact_280_insert__noteq__member,axiom,
% 0.07/0.32 ! [B: a,B4: multiset_a,C: a,C2: multiset_a] :
% 0.07/0.32 ( ( ( add_mset_a @ B @ B4 )
% 0.07/0.32 = ( add_mset_a @ C @ C2 ) )
% 0.07/0.32 => ( ( B != C )
% 0.07/0.32 => ( member_a @ C @ ( set_mset_a @ B4 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % insert_noteq_member
% 0.07/0.32 thf(fact_281_multi__member__split,axiom,
% 0.07/0.32 ! [X: produc143150363Tree_a,M: multis2082063201Tree_a] :
% 0.07/0.32 ( ( member254315204Tree_a @ X @ ( set_ms1129970904Tree_a @ M ) )
% 0.07/0.32 => ? [A6: multis2082063201Tree_a] :
% 0.07/0.32 ( M
% 0.07/0.32 = ( add_ms205941497Tree_a @ X @ A6 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multi_member_split
% 0.07/0.32 thf(fact_282_multi__member__split,axiom,
% 0.07/0.32 ! [X: produc1127127335iset_a,M: multis782275565iset_a] :
% 0.07/0.32 ( ( member340150864iset_a @ X @ ( set_ms1768116836iset_a @ M ) )
% 0.07/0.32 => ? [A6: multis782275565iset_a] :
% 0.07/0.32 ( M
% 0.07/0.32 = ( add_ms1353365509iset_a @ X @ A6 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multi_member_split
% 0.07/0.32 thf(fact_283_multi__member__split,axiom,
% 0.07/0.32 ! [X: product_prod_a_a,M: multis599418605od_a_a] :
% 0.07/0.32 ( ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ M ) )
% 0.07/0.32 => ? [A6: multis599418605od_a_a] :
% 0.07/0.32 ( M
% 0.07/0.32 = ( add_ms1725977093od_a_a @ X @ A6 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multi_member_split
% 0.07/0.32 thf(fact_284_multi__member__split,axiom,
% 0.07/0.32 ! [X: multiset_a,M: multiset_multiset_a] :
% 0.07/0.32 ( ( member_multiset_a @ X @ ( set_mset_multiset_a @ M ) )
% 0.07/0.32 => ? [A6: multiset_multiset_a] :
% 0.07/0.32 ( M
% 0.07/0.32 = ( add_mset_multiset_a @ X @ A6 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multi_member_split
% 0.07/0.32 thf(fact_285_multi__member__split,axiom,
% 0.07/0.32 ! [X: a,M: multiset_a] :
% 0.07/0.32 ( ( member_a @ X @ ( set_mset_a @ M ) )
% 0.07/0.32 => ? [A6: multiset_a] :
% 0.07/0.32 ( M
% 0.07/0.32 = ( add_mset_a @ X @ A6 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multi_member_split
% 0.07/0.32 thf(fact_286_mset__add,axiom,
% 0.07/0.32 ! [A: produc143150363Tree_a,A4: multis2082063201Tree_a] :
% 0.07/0.32 ( ( member254315204Tree_a @ A @ ( set_ms1129970904Tree_a @ A4 ) )
% 0.07/0.32 => ~ ! [B6: multis2082063201Tree_a] :
% 0.07/0.32 ( A4
% 0.07/0.32 != ( add_ms205941497Tree_a @ A @ B6 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % mset_add
% 0.07/0.32 thf(fact_287_mset__add,axiom,
% 0.07/0.32 ! [A: produc1127127335iset_a,A4: multis782275565iset_a] :
% 0.07/0.32 ( ( member340150864iset_a @ A @ ( set_ms1768116836iset_a @ A4 ) )
% 0.07/0.32 => ~ ! [B6: multis782275565iset_a] :
% 0.07/0.32 ( A4
% 0.07/0.32 != ( add_ms1353365509iset_a @ A @ B6 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % mset_add
% 0.07/0.32 thf(fact_288_mset__add,axiom,
% 0.07/0.32 ! [A: product_prod_a_a,A4: multis599418605od_a_a] :
% 0.07/0.32 ( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ A4 ) )
% 0.07/0.32 => ~ ! [B6: multis599418605od_a_a] :
% 0.07/0.32 ( A4
% 0.07/0.32 != ( add_ms1725977093od_a_a @ A @ B6 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % mset_add
% 0.07/0.32 thf(fact_289_mset__add,axiom,
% 0.07/0.32 ! [A: multiset_a,A4: multiset_multiset_a] :
% 0.07/0.32 ( ( member_multiset_a @ A @ ( set_mset_multiset_a @ A4 ) )
% 0.07/0.32 => ~ ! [B6: multiset_multiset_a] :
% 0.07/0.32 ( A4
% 0.07/0.32 != ( add_mset_multiset_a @ A @ B6 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % mset_add
% 0.07/0.32 thf(fact_290_mset__add,axiom,
% 0.07/0.32 ! [A: a,A4: multiset_a] :
% 0.07/0.32 ( ( member_a @ A @ ( set_mset_a @ A4 ) )
% 0.07/0.32 => ~ ! [B6: multiset_a] :
% 0.07/0.32 ( A4
% 0.07/0.32 != ( add_mset_a @ A @ B6 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % mset_add
% 0.07/0.32 thf(fact_291_union__iff,axiom,
% 0.07/0.32 ! [A: produc143150363Tree_a,A4: multis2082063201Tree_a,B4: multis2082063201Tree_a] :
% 0.07/0.32 ( ( member254315204Tree_a @ A @ ( set_ms1129970904Tree_a @ ( plus_p1065544216Tree_a @ A4 @ B4 ) ) )
% 0.07/0.32 = ( ( member254315204Tree_a @ A @ ( set_ms1129970904Tree_a @ A4 ) )
% 0.07/0.32 | ( member254315204Tree_a @ A @ ( set_ms1129970904Tree_a @ B4 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % union_iff
% 0.07/0.32 thf(fact_292_union__iff,axiom,
% 0.07/0.32 ! [A: produc1127127335iset_a,A4: multis782275565iset_a,B4: multis782275565iset_a] :
% 0.07/0.32 ( ( member340150864iset_a @ A @ ( set_ms1768116836iset_a @ ( plus_p1919096612iset_a @ A4 @ B4 ) ) )
% 0.07/0.32 = ( ( member340150864iset_a @ A @ ( set_ms1768116836iset_a @ A4 ) )
% 0.07/0.32 | ( member340150864iset_a @ A @ ( set_ms1768116836iset_a @ B4 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % union_iff
% 0.07/0.32 thf(fact_293_union__iff,axiom,
% 0.07/0.32 ! [A: product_prod_a_a,A4: multis599418605od_a_a,B4: multis599418605od_a_a] :
% 0.07/0.32 ( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ ( plus_p404985124od_a_a @ A4 @ B4 ) ) )
% 0.07/0.32 = ( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ A4 ) )
% 0.07/0.32 | ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ B4 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % union_iff
% 0.07/0.32 thf(fact_294_union__iff,axiom,
% 0.07/0.32 ! [A: multiset_a,A4: multiset_multiset_a,B4: multiset_multiset_a] :
% 0.07/0.32 ( ( member_multiset_a @ A @ ( set_mset_multiset_a @ ( plus_p1957546689iset_a @ A4 @ B4 ) ) )
% 0.07/0.32 = ( ( member_multiset_a @ A @ ( set_mset_multiset_a @ A4 ) )
% 0.07/0.32 | ( member_multiset_a @ A @ ( set_mset_multiset_a @ B4 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % union_iff
% 0.07/0.32 thf(fact_295_union__iff,axiom,
% 0.07/0.32 ! [A: a,A4: multiset_a,B4: multiset_a] :
% 0.07/0.32 ( ( member_a @ A @ ( set_mset_a @ ( plus_plus_multiset_a @ A4 @ B4 ) ) )
% 0.07/0.32 = ( ( member_a @ A @ ( set_mset_a @ A4 ) )
% 0.07/0.32 | ( member_a @ A @ ( set_mset_a @ B4 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % union_iff
% 0.07/0.32 thf(fact_296_multiset__nonemptyE,axiom,
% 0.07/0.32 ! [A4: multis2082063201Tree_a] :
% 0.07/0.32 ( ( A4 != zero_z2144236696Tree_a )
% 0.07/0.32 => ~ ! [X3: produc143150363Tree_a] :
% 0.07/0.32 ~ ( member254315204Tree_a @ X3 @ ( set_ms1129970904Tree_a @ A4 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multiset_nonemptyE
% 0.07/0.32 thf(fact_297_multiset__nonemptyE,axiom,
% 0.07/0.32 ! [A4: multis782275565iset_a] :
% 0.07/0.32 ( ( A4 != zero_z1941791140iset_a )
% 0.07/0.32 => ~ ! [X3: produc1127127335iset_a] :
% 0.07/0.32 ~ ( member340150864iset_a @ X3 @ ( set_ms1768116836iset_a @ A4 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multiset_nonemptyE
% 0.07/0.32 thf(fact_298_multiset__nonemptyE,axiom,
% 0.07/0.32 ! [A4: multis599418605od_a_a] :
% 0.07/0.32 ( ( A4 != zero_z160849828od_a_a )
% 0.07/0.32 => ~ ! [X3: product_prod_a_a] :
% 0.07/0.32 ~ ( member449909584od_a_a @ X3 @ ( set_ms1818337124od_a_a @ A4 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multiset_nonemptyE
% 0.07/0.32 thf(fact_299_multiset__nonemptyE,axiom,
% 0.07/0.32 ! [A4: multiset_multiset_a] :
% 0.07/0.32 ( ( A4 != zero_z1580389697iset_a )
% 0.07/0.32 => ~ ! [X3: multiset_a] :
% 0.07/0.32 ~ ( member_multiset_a @ X3 @ ( set_mset_multiset_a @ A4 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multiset_nonemptyE
% 0.07/0.32 thf(fact_300_multiset__nonemptyE,axiom,
% 0.07/0.32 ! [A4: multiset_a] :
% 0.07/0.32 ( ( A4 != zero_zero_multiset_a )
% 0.07/0.32 => ~ ! [X3: a] :
% 0.07/0.32 ~ ( member_a @ X3 @ ( set_mset_a @ A4 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multiset_nonemptyE
% 0.07/0.32 thf(fact_301_subset__CollectI,axiom,
% 0.07/0.32 ! [B4: set_a,A4: set_a,Q2: a > $o,P2: a > $o] :
% 0.07/0.32 ( ( ord_less_eq_set_a @ B4 @ A4 )
% 0.07/0.32 => ( ! [X3: a] :
% 0.07/0.32 ( ( member_a @ X3 @ B4 )
% 0.07/0.32 => ( ( Q2 @ X3 )
% 0.07/0.32 => ( P2 @ X3 ) ) )
% 0.07/0.32 => ( ord_less_eq_set_a
% 0.07/0.32 @ ( collect_a
% 0.07/0.32 @ ^ [X4: a] :
% 0.07/0.32 ( ( member_a @ X4 @ B4 )
% 0.07/0.32 & ( Q2 @ X4 ) ) )
% 0.07/0.32 @ ( collect_a
% 0.07/0.32 @ ^ [X4: a] :
% 0.07/0.32 ( ( member_a @ X4 @ A4 )
% 0.07/0.32 & ( P2 @ X4 ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % subset_CollectI
% 0.07/0.32 thf(fact_302_subset__CollectI,axiom,
% 0.07/0.32 ! [B4: set_Pr158363655iset_a,A4: set_Pr158363655iset_a,Q2: produc1127127335iset_a > $o,P2: produc1127127335iset_a > $o] :
% 0.07/0.32 ( ( ord_le1978107815iset_a @ B4 @ A4 )
% 0.07/0.32 => ( ! [X3: produc1127127335iset_a] :
% 0.07/0.32 ( ( member340150864iset_a @ X3 @ B4 )
% 0.07/0.32 => ( ( Q2 @ X3 )
% 0.07/0.32 => ( P2 @ X3 ) ) )
% 0.07/0.32 => ( ord_le1978107815iset_a
% 0.07/0.32 @ ( collec1104713362iset_a
% 0.07/0.32 @ ^ [X4: produc1127127335iset_a] :
% 0.07/0.32 ( ( member340150864iset_a @ X4 @ B4 )
% 0.07/0.32 & ( Q2 @ X4 ) ) )
% 0.07/0.32 @ ( collec1104713362iset_a
% 0.07/0.32 @ ^ [X4: produc1127127335iset_a] :
% 0.07/0.32 ( ( member340150864iset_a @ X4 @ A4 )
% 0.07/0.32 & ( P2 @ X4 ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % subset_CollectI
% 0.07/0.32 thf(fact_303_subset__CollectI,axiom,
% 0.07/0.32 ! [B4: set_Product_prod_a_a,A4: set_Product_prod_a_a,Q2: product_prod_a_a > $o,P2: product_prod_a_a > $o] :
% 0.07/0.32 ( ( ord_le1824328871od_a_a @ B4 @ A4 )
% 0.07/0.32 => ( ! [X3: product_prod_a_a] :
% 0.07/0.32 ( ( member449909584od_a_a @ X3 @ B4 )
% 0.07/0.32 => ( ( Q2 @ X3 )
% 0.07/0.32 => ( P2 @ X3 ) ) )
% 0.07/0.32 => ( ord_le1824328871od_a_a
% 0.07/0.32 @ ( collec645855634od_a_a
% 0.07/0.32 @ ^ [X4: product_prod_a_a] :
% 0.07/0.32 ( ( member449909584od_a_a @ X4 @ B4 )
% 0.07/0.32 & ( Q2 @ X4 ) ) )
% 0.07/0.32 @ ( collec645855634od_a_a
% 0.07/0.32 @ ^ [X4: product_prod_a_a] :
% 0.07/0.32 ( ( member449909584od_a_a @ X4 @ A4 )
% 0.07/0.32 & ( P2 @ X4 ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % subset_CollectI
% 0.07/0.32 thf(fact_304_subset__CollectI,axiom,
% 0.07/0.32 ! [B4: set_Pr1070816379Tree_a,A4: set_Pr1070816379Tree_a,Q2: produc143150363Tree_a > $o,P2: produc143150363Tree_a > $o] :
% 0.07/0.32 ( ( ord_le1013007387Tree_a @ B4 @ A4 )
% 0.07/0.32 => ( ! [X3: produc143150363Tree_a] :
% 0.07/0.32 ( ( member254315204Tree_a @ X3 @ B4 )
% 0.07/0.32 => ( ( Q2 @ X3 )
% 0.07/0.32 => ( P2 @ X3 ) ) )
% 0.07/0.32 => ( ord_le1013007387Tree_a
% 0.07/0.32 @ ( collec486369286Tree_a
% 0.07/0.32 @ ^ [X4: produc143150363Tree_a] :
% 0.07/0.32 ( ( member254315204Tree_a @ X4 @ B4 )
% 0.07/0.32 & ( Q2 @ X4 ) ) )
% 0.07/0.32 @ ( collec486369286Tree_a
% 0.07/0.32 @ ^ [X4: produc143150363Tree_a] :
% 0.07/0.32 ( ( member254315204Tree_a @ X4 @ A4 )
% 0.07/0.32 & ( P2 @ X4 ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % subset_CollectI
% 0.07/0.32 thf(fact_305_subset__Collect__iff,axiom,
% 0.07/0.32 ! [B4: set_a,A4: set_a,P2: a > $o] :
% 0.07/0.32 ( ( ord_less_eq_set_a @ B4 @ A4 )
% 0.07/0.32 => ( ( ord_less_eq_set_a @ B4
% 0.07/0.32 @ ( collect_a
% 0.07/0.32 @ ^ [X4: a] :
% 0.07/0.32 ( ( member_a @ X4 @ A4 )
% 0.07/0.32 & ( P2 @ X4 ) ) ) )
% 0.07/0.32 = ( ! [X4: a] :
% 0.07/0.32 ( ( member_a @ X4 @ B4 )
% 0.07/0.32 => ( P2 @ X4 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % subset_Collect_iff
% 0.07/0.32 thf(fact_306_subset__Collect__iff,axiom,
% 0.07/0.32 ! [B4: set_Pr158363655iset_a,A4: set_Pr158363655iset_a,P2: produc1127127335iset_a > $o] :
% 0.07/0.32 ( ( ord_le1978107815iset_a @ B4 @ A4 )
% 0.07/0.32 => ( ( ord_le1978107815iset_a @ B4
% 0.07/0.32 @ ( collec1104713362iset_a
% 0.07/0.32 @ ^ [X4: produc1127127335iset_a] :
% 0.07/0.32 ( ( member340150864iset_a @ X4 @ A4 )
% 0.07/0.32 & ( P2 @ X4 ) ) ) )
% 0.07/0.32 = ( ! [X4: produc1127127335iset_a] :
% 0.07/0.32 ( ( member340150864iset_a @ X4 @ B4 )
% 0.07/0.32 => ( P2 @ X4 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % subset_Collect_iff
% 0.07/0.32 thf(fact_307_subset__Collect__iff,axiom,
% 0.07/0.32 ! [B4: set_Product_prod_a_a,A4: set_Product_prod_a_a,P2: product_prod_a_a > $o] :
% 0.07/0.32 ( ( ord_le1824328871od_a_a @ B4 @ A4 )
% 0.07/0.32 => ( ( ord_le1824328871od_a_a @ B4
% 0.07/0.32 @ ( collec645855634od_a_a
% 0.07/0.32 @ ^ [X4: product_prod_a_a] :
% 0.07/0.32 ( ( member449909584od_a_a @ X4 @ A4 )
% 0.07/0.32 & ( P2 @ X4 ) ) ) )
% 0.07/0.32 = ( ! [X4: product_prod_a_a] :
% 0.07/0.32 ( ( member449909584od_a_a @ X4 @ B4 )
% 0.07/0.32 => ( P2 @ X4 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % subset_Collect_iff
% 0.07/0.32 thf(fact_308_subset__Collect__iff,axiom,
% 0.07/0.32 ! [B4: set_Pr1070816379Tree_a,A4: set_Pr1070816379Tree_a,P2: produc143150363Tree_a > $o] :
% 0.07/0.32 ( ( ord_le1013007387Tree_a @ B4 @ A4 )
% 0.07/0.32 => ( ( ord_le1013007387Tree_a @ B4
% 0.07/0.32 @ ( collec486369286Tree_a
% 0.07/0.32 @ ^ [X4: produc143150363Tree_a] :
% 0.07/0.32 ( ( member254315204Tree_a @ X4 @ A4 )
% 0.07/0.32 & ( P2 @ X4 ) ) ) )
% 0.07/0.32 = ( ! [X4: produc143150363Tree_a] :
% 0.07/0.32 ( ( member254315204Tree_a @ X4 @ B4 )
% 0.07/0.32 => ( P2 @ X4 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % subset_Collect_iff
% 0.07/0.32 thf(fact_309_multiset__induct__min,axiom,
% 0.07/0.32 ! [P2: multiset_a > $o,M: multiset_a] :
% 0.07/0.32 ( ( P2 @ zero_zero_multiset_a )
% 0.07/0.32 => ( ! [X3: a,M3: multiset_a] :
% 0.07/0.32 ( ( P2 @ M3 )
% 0.07/0.32 => ( ! [Xa: a] :
% 0.07/0.32 ( ( member_a @ Xa @ ( set_mset_a @ M3 ) )
% 0.07/0.32 => ( ord_less_eq_a @ X3 @ Xa ) )
% 0.07/0.32 => ( P2 @ ( add_mset_a @ X3 @ M3 ) ) ) )
% 0.07/0.32 => ( P2 @ M ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multiset_induct_min
% 0.07/0.32 thf(fact_310_multiset__induct__max,axiom,
% 0.07/0.32 ! [P2: multiset_a > $o,M: multiset_a] :
% 0.07/0.32 ( ( P2 @ zero_zero_multiset_a )
% 0.07/0.32 => ( ! [X3: a,M3: multiset_a] :
% 0.07/0.32 ( ( P2 @ M3 )
% 0.07/0.32 => ( ! [Xa: a] :
% 0.07/0.32 ( ( member_a @ Xa @ ( set_mset_a @ M3 ) )
% 0.07/0.32 => ( ord_less_eq_a @ Xa @ X3 ) )
% 0.07/0.32 => ( P2 @ ( add_mset_a @ X3 @ M3 ) ) ) )
% 0.07/0.32 => ( P2 @ M ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multiset_induct_max
% 0.07/0.32 thf(fact_311_multi__member__skip,axiom,
% 0.07/0.32 ! [X: produc143150363Tree_a,XS: multis2082063201Tree_a,Y: produc143150363Tree_a] :
% 0.07/0.32 ( ( member254315204Tree_a @ X @ ( set_ms1129970904Tree_a @ XS ) )
% 0.07/0.32 => ( member254315204Tree_a @ X @ ( set_ms1129970904Tree_a @ ( plus_p1065544216Tree_a @ ( add_ms205941497Tree_a @ Y @ zero_z2144236696Tree_a ) @ XS ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multi_member_skip
% 0.07/0.32 thf(fact_312_multi__member__skip,axiom,
% 0.07/0.32 ! [X: produc1127127335iset_a,XS: multis782275565iset_a,Y: produc1127127335iset_a] :
% 0.07/0.32 ( ( member340150864iset_a @ X @ ( set_ms1768116836iset_a @ XS ) )
% 0.07/0.32 => ( member340150864iset_a @ X @ ( set_ms1768116836iset_a @ ( plus_p1919096612iset_a @ ( add_ms1353365509iset_a @ Y @ zero_z1941791140iset_a ) @ XS ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multi_member_skip
% 0.07/0.32 thf(fact_313_multi__member__skip,axiom,
% 0.07/0.32 ! [X: product_prod_a_a,XS: multis599418605od_a_a,Y: product_prod_a_a] :
% 0.07/0.32 ( ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ XS ) )
% 0.07/0.32 => ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ ( plus_p404985124od_a_a @ ( add_ms1725977093od_a_a @ Y @ zero_z160849828od_a_a ) @ XS ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multi_member_skip
% 0.07/0.32 thf(fact_314_multi__member__skip,axiom,
% 0.07/0.32 ! [X: multiset_a,XS: multiset_multiset_a,Y: multiset_a] :
% 0.07/0.32 ( ( member_multiset_a @ X @ ( set_mset_multiset_a @ XS ) )
% 0.07/0.32 => ( member_multiset_a @ X @ ( set_mset_multiset_a @ ( plus_p1957546689iset_a @ ( add_mset_multiset_a @ Y @ zero_z1580389697iset_a ) @ XS ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multi_member_skip
% 0.07/0.32 thf(fact_315_multi__member__skip,axiom,
% 0.07/0.32 ! [X: a,XS: multiset_a,Y: a] :
% 0.07/0.32 ( ( member_a @ X @ ( set_mset_a @ XS ) )
% 0.07/0.32 => ( member_a @ X @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ Y @ zero_zero_multiset_a ) @ XS ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multi_member_skip
% 0.07/0.32 thf(fact_316_multi__member__this,axiom,
% 0.07/0.32 ! [X: produc143150363Tree_a,XS: multis2082063201Tree_a] : ( member254315204Tree_a @ X @ ( set_ms1129970904Tree_a @ ( plus_p1065544216Tree_a @ ( add_ms205941497Tree_a @ X @ zero_z2144236696Tree_a ) @ XS ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multi_member_this
% 0.07/0.32 thf(fact_317_multi__member__this,axiom,
% 0.07/0.32 ! [X: produc1127127335iset_a,XS: multis782275565iset_a] : ( member340150864iset_a @ X @ ( set_ms1768116836iset_a @ ( plus_p1919096612iset_a @ ( add_ms1353365509iset_a @ X @ zero_z1941791140iset_a ) @ XS ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multi_member_this
% 0.07/0.32 thf(fact_318_multi__member__this,axiom,
% 0.07/0.32 ! [X: product_prod_a_a,XS: multis599418605od_a_a] : ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ ( plus_p404985124od_a_a @ ( add_ms1725977093od_a_a @ X @ zero_z160849828od_a_a ) @ XS ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multi_member_this
% 0.07/0.32 thf(fact_319_multi__member__this,axiom,
% 0.07/0.32 ! [X: multiset_a,XS: multiset_multiset_a] : ( member_multiset_a @ X @ ( set_mset_multiset_a @ ( plus_p1957546689iset_a @ ( add_mset_multiset_a @ X @ zero_z1580389697iset_a ) @ XS ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multi_member_this
% 0.07/0.32 thf(fact_320_multi__member__this,axiom,
% 0.07/0.32 ! [X: a,XS: multiset_a] : ( member_a @ X @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ X @ zero_zero_multiset_a ) @ XS ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % multi_member_this
% 0.07/0.32 thf(fact_321_less__add,axiom,
% 0.07/0.32 ! [N: multis2082063201Tree_a,A: produc143150363Tree_a,M0: multis2082063201Tree_a,R: set_Pr220561863Tree_a] :
% 0.07/0.32 ( ( member1298282640Tree_a @ ( produc1608222167Tree_a @ N @ ( add_ms205941497Tree_a @ A @ M0 ) ) @ ( mult1_470979598Tree_a @ R ) )
% 0.07/0.32 => ( ? [M3: multis2082063201Tree_a] :
% 0.07/0.32 ( ( member1298282640Tree_a @ ( produc1608222167Tree_a @ M3 @ M0 ) @ ( mult1_470979598Tree_a @ R ) )
% 0.07/0.32 & ( N
% 0.07/0.32 = ( add_ms205941497Tree_a @ A @ M3 ) ) )
% 0.07/0.32 | ? [K4: multis2082063201Tree_a] :
% 0.07/0.32 ( ! [B7: produc143150363Tree_a] :
% 0.07/0.32 ( ( member254315204Tree_a @ B7 @ ( set_ms1129970904Tree_a @ K4 ) )
% 0.07/0.32 => ( member146662416Tree_a @ ( produc1002111575Tree_a @ B7 @ A ) @ R ) )
% 0.07/0.32 & ( N
% 0.07/0.32 = ( plus_p1065544216Tree_a @ M0 @ K4 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % less_add
% 0.07/0.32 thf(fact_322_less__add,axiom,
% 0.07/0.32 ! [N: multis782275565iset_a,A: produc1127127335iset_a,M0: multis782275565iset_a,R: set_Pr146455751iset_a] :
% 0.07/0.32 ( ( member219455632iset_a @ ( produc1226985431iset_a @ N @ ( add_ms1353365509iset_a @ A @ M0 ) ) @ ( mult1_599785114iset_a @ R ) )
% 0.07/0.32 => ( ? [M3: multis782275565iset_a] :
% 0.07/0.32 ( ( member219455632iset_a @ ( produc1226985431iset_a @ M3 @ M0 ) @ ( mult1_599785114iset_a @ R ) )
% 0.07/0.32 & ( N
% 0.07/0.32 = ( add_ms1353365509iset_a @ A @ M3 ) ) )
% 0.07/0.32 | ? [K4: multis782275565iset_a] :
% 0.07/0.32 ( ! [B7: produc1127127335iset_a] :
% 0.07/0.32 ( ( member340150864iset_a @ B7 @ ( set_ms1768116836iset_a @ K4 ) )
% 0.07/0.32 => ( member567864080iset_a @ ( produc1898392407iset_a @ B7 @ A ) @ R ) )
% 0.07/0.32 & ( N
% 0.07/0.32 = ( plus_p1919096612iset_a @ M0 @ K4 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % less_add
% 0.07/0.32 thf(fact_323_less__add,axiom,
% 0.07/0.32 ! [N: multis599418605od_a_a,A: product_prod_a_a,M0: multis599418605od_a_a,R: set_Pr1948701895od_a_a] :
% 0.07/0.32 ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ N @ ( add_ms1725977093od_a_a @ A @ M0 ) ) @ ( mult1_1719650714od_a_a @ R ) )
% 0.07/0.32 => ( ? [M3: multis599418605od_a_a] :
% 0.07/0.32 ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ M3 @ M0 ) @ ( mult1_1719650714od_a_a @ R ) )
% 0.07/0.32 & ( N
% 0.07/0.32 = ( add_ms1725977093od_a_a @ A @ M3 ) ) )
% 0.07/0.32 | ? [K4: multis599418605od_a_a] :
% 0.07/0.32 ( ! [B7: product_prod_a_a] :
% 0.07/0.32 ( ( member449909584od_a_a @ B7 @ ( set_ms1818337124od_a_a @ K4 ) )
% 0.07/0.32 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B7 @ A ) @ R ) )
% 0.07/0.32 & ( N
% 0.07/0.32 = ( plus_p404985124od_a_a @ M0 @ K4 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % less_add
% 0.07/0.32 thf(fact_324_less__add,axiom,
% 0.07/0.32 ! [N: multiset_multiset_a,A: multiset_a,M0: multiset_multiset_a,R: set_Pr158363655iset_a] :
% 0.07/0.32 ( ( member104214352iset_a @ ( produc1444952343iset_a @ N @ ( add_mset_multiset_a @ A @ M0 ) ) @ ( mult1_multiset_a @ R ) )
% 0.07/0.32 => ( ? [M3: multiset_multiset_a] :
% 0.07/0.32 ( ( member104214352iset_a @ ( produc1444952343iset_a @ M3 @ M0 ) @ ( mult1_multiset_a @ R ) )
% 0.07/0.32 & ( N
% 0.07/0.32 = ( add_mset_multiset_a @ A @ M3 ) ) )
% 0.07/0.32 | ? [K4: multiset_multiset_a] :
% 0.07/0.32 ( ! [B7: multiset_a] :
% 0.07/0.32 ( ( member_multiset_a @ B7 @ ( set_mset_multiset_a @ K4 ) )
% 0.07/0.32 => ( member340150864iset_a @ ( produc2037245207iset_a @ B7 @ A ) @ R ) )
% 0.07/0.32 & ( N
% 0.07/0.32 = ( plus_p1957546689iset_a @ M0 @ K4 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % less_add
% 0.07/0.32 thf(fact_325_less__add,axiom,
% 0.07/0.32 ! [N: multiset_a,A: a,M0: multiset_a,R: set_Product_prod_a_a] :
% 0.07/0.32 ( ( member340150864iset_a @ ( produc2037245207iset_a @ N @ ( add_mset_a @ A @ M0 ) ) @ ( mult1_a @ R ) )
% 0.07/0.32 => ( ? [M3: multiset_a] :
% 0.07/0.32 ( ( member340150864iset_a @ ( produc2037245207iset_a @ M3 @ M0 ) @ ( mult1_a @ R ) )
% 0.07/0.32 & ( N
% 0.07/0.32 = ( add_mset_a @ A @ M3 ) ) )
% 0.07/0.32 | ? [K4: multiset_a] :
% 0.07/0.32 ( ! [B7: a] :
% 0.07/0.32 ( ( member_a @ B7 @ ( set_mset_a @ K4 ) )
% 0.07/0.32 => ( member449909584od_a_a @ ( product_Pair_a_a @ B7 @ A ) @ R ) )
% 0.07/0.32 & ( N
% 0.07/0.32 = ( plus_plus_multiset_a @ M0 @ K4 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % less_add
% 0.07/0.32 thf(fact_326_mult1I,axiom,
% 0.07/0.32 ! [M: multis2082063201Tree_a,A: produc143150363Tree_a,M0: multis2082063201Tree_a,N: multis2082063201Tree_a,K2: multis2082063201Tree_a,R: set_Pr220561863Tree_a] :
% 0.07/0.32 ( ( M
% 0.07/0.32 = ( add_ms205941497Tree_a @ A @ M0 ) )
% 0.07/0.32 => ( ( N
% 0.07/0.32 = ( plus_p1065544216Tree_a @ M0 @ K2 ) )
% 0.07/0.32 => ( ! [B3: produc143150363Tree_a] :
% 0.07/0.32 ( ( member254315204Tree_a @ B3 @ ( set_ms1129970904Tree_a @ K2 ) )
% 0.07/0.32 => ( member146662416Tree_a @ ( produc1002111575Tree_a @ B3 @ A ) @ R ) )
% 0.07/0.32 => ( member1298282640Tree_a @ ( produc1608222167Tree_a @ N @ M ) @ ( mult1_470979598Tree_a @ R ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % mult1I
% 0.07/0.32 thf(fact_327_mult1I,axiom,
% 0.07/0.32 ! [M: multis782275565iset_a,A: produc1127127335iset_a,M0: multis782275565iset_a,N: multis782275565iset_a,K2: multis782275565iset_a,R: set_Pr146455751iset_a] :
% 0.07/0.32 ( ( M
% 0.07/0.32 = ( add_ms1353365509iset_a @ A @ M0 ) )
% 0.07/0.32 => ( ( N
% 0.07/0.32 = ( plus_p1919096612iset_a @ M0 @ K2 ) )
% 0.07/0.32 => ( ! [B3: produc1127127335iset_a] :
% 0.07/0.32 ( ( member340150864iset_a @ B3 @ ( set_ms1768116836iset_a @ K2 ) )
% 0.07/0.32 => ( member567864080iset_a @ ( produc1898392407iset_a @ B3 @ A ) @ R ) )
% 0.07/0.32 => ( member219455632iset_a @ ( produc1226985431iset_a @ N @ M ) @ ( mult1_599785114iset_a @ R ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % mult1I
% 0.07/0.32 thf(fact_328_mult1I,axiom,
% 0.07/0.32 ! [M: multis599418605od_a_a,A: product_prod_a_a,M0: multis599418605od_a_a,N: multis599418605od_a_a,K2: multis599418605od_a_a,R: set_Pr1948701895od_a_a] :
% 0.07/0.32 ( ( M
% 0.07/0.32 = ( add_ms1725977093od_a_a @ A @ M0 ) )
% 0.07/0.32 => ( ( N
% 0.07/0.32 = ( plus_p404985124od_a_a @ M0 @ K2 ) )
% 0.07/0.32 => ( ! [B3: product_prod_a_a] :
% 0.07/0.32 ( ( member449909584od_a_a @ B3 @ ( set_ms1818337124od_a_a @ K2 ) )
% 0.07/0.32 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B3 @ A ) @ R ) )
% 0.07/0.32 => ( member1349055120od_a_a @ ( produc1859144151od_a_a @ N @ M ) @ ( mult1_1719650714od_a_a @ R ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % mult1I
% 0.07/0.32 thf(fact_329_mult1I,axiom,
% 0.07/0.32 ! [M: multiset_multiset_a,A: multiset_a,M0: multiset_multiset_a,N: multiset_multiset_a,K2: multiset_multiset_a,R: set_Pr158363655iset_a] :
% 0.07/0.32 ( ( M
% 0.07/0.32 = ( add_mset_multiset_a @ A @ M0 ) )
% 0.07/0.32 => ( ( N
% 0.07/0.32 = ( plus_p1957546689iset_a @ M0 @ K2 ) )
% 0.07/0.32 => ( ! [B3: multiset_a] :
% 0.07/0.32 ( ( member_multiset_a @ B3 @ ( set_mset_multiset_a @ K2 ) )
% 0.07/0.32 => ( member340150864iset_a @ ( produc2037245207iset_a @ B3 @ A ) @ R ) )
% 0.07/0.32 => ( member104214352iset_a @ ( produc1444952343iset_a @ N @ M ) @ ( mult1_multiset_a @ R ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % mult1I
% 0.07/0.32 thf(fact_330_mult1I,axiom,
% 0.07/0.32 ! [M: multiset_a,A: a,M0: multiset_a,N: multiset_a,K2: multiset_a,R: set_Product_prod_a_a] :
% 0.07/0.32 ( ( M
% 0.07/0.32 = ( add_mset_a @ A @ M0 ) )
% 0.07/0.32 => ( ( N
% 0.07/0.32 = ( plus_plus_multiset_a @ M0 @ K2 ) )
% 0.07/0.32 => ( ! [B3: a] :
% 0.07/0.32 ( ( member_a @ B3 @ ( set_mset_a @ K2 ) )
% 0.07/0.32 => ( member449909584od_a_a @ ( product_Pair_a_a @ B3 @ A ) @ R ) )
% 0.07/0.32 => ( member340150864iset_a @ ( produc2037245207iset_a @ N @ M ) @ ( mult1_a @ R ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % mult1I
% 0.07/0.32 thf(fact_331_mono__mult1,axiom,
% 0.07/0.32 ! [R: set_Product_prod_a_a,R4: set_Product_prod_a_a] :
% 0.07/0.32 ( ( ord_le1824328871od_a_a @ R @ R4 )
% 0.07/0.32 => ( ord_le1978107815iset_a @ ( mult1_a @ R ) @ ( mult1_a @ R4 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % mono_mult1
% 0.07/0.32 thf(fact_332_mult1__union,axiom,
% 0.07/0.32 ! [B4: multiset_a,D2: multiset_a,R: set_Product_prod_a_a,C2: multiset_a] :
% 0.07/0.32 ( ( member340150864iset_a @ ( produc2037245207iset_a @ B4 @ D2 ) @ ( mult1_a @ R ) )
% 0.07/0.32 => ( member340150864iset_a @ ( produc2037245207iset_a @ ( plus_plus_multiset_a @ C2 @ B4 ) @ ( plus_plus_multiset_a @ C2 @ D2 ) ) @ ( mult1_a @ R ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % mult1_union
% 0.07/0.32 thf(fact_333_not__less__empty,axiom,
% 0.07/0.32 ! [M: multiset_a,R: set_Product_prod_a_a] :
% 0.07/0.32 ~ ( member340150864iset_a @ ( produc2037245207iset_a @ M @ zero_zero_multiset_a ) @ ( mult1_a @ R ) ) ).
% 0.07/0.32
% 0.07/0.32 % not_less_empty
% 0.07/0.32 thf(fact_334_mult1E,axiom,
% 0.07/0.32 ! [N: multis2082063201Tree_a,M: multis2082063201Tree_a,R: set_Pr220561863Tree_a] :
% 0.07/0.32 ( ( member1298282640Tree_a @ ( produc1608222167Tree_a @ N @ M ) @ ( mult1_470979598Tree_a @ R ) )
% 0.07/0.32 => ~ ! [A3: produc143150363Tree_a,M02: multis2082063201Tree_a] :
% 0.07/0.32 ( ( M
% 0.07/0.32 = ( add_ms205941497Tree_a @ A3 @ M02 ) )
% 0.07/0.32 => ! [K4: multis2082063201Tree_a] :
% 0.07/0.32 ( ( N
% 0.07/0.32 = ( plus_p1065544216Tree_a @ M02 @ K4 ) )
% 0.07/0.32 => ~ ! [B7: produc143150363Tree_a] :
% 0.07/0.32 ( ( member254315204Tree_a @ B7 @ ( set_ms1129970904Tree_a @ K4 ) )
% 0.07/0.32 => ( member146662416Tree_a @ ( produc1002111575Tree_a @ B7 @ A3 ) @ R ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % mult1E
% 0.07/0.32 thf(fact_335_mult1E,axiom,
% 0.07/0.32 ! [N: multis782275565iset_a,M: multis782275565iset_a,R: set_Pr146455751iset_a] :
% 0.07/0.32 ( ( member219455632iset_a @ ( produc1226985431iset_a @ N @ M ) @ ( mult1_599785114iset_a @ R ) )
% 0.07/0.32 => ~ ! [A3: produc1127127335iset_a,M02: multis782275565iset_a] :
% 0.07/0.32 ( ( M
% 0.07/0.32 = ( add_ms1353365509iset_a @ A3 @ M02 ) )
% 0.07/0.32 => ! [K4: multis782275565iset_a] :
% 0.07/0.32 ( ( N
% 0.07/0.32 = ( plus_p1919096612iset_a @ M02 @ K4 ) )
% 0.07/0.32 => ~ ! [B7: produc1127127335iset_a] :
% 0.07/0.32 ( ( member340150864iset_a @ B7 @ ( set_ms1768116836iset_a @ K4 ) )
% 0.07/0.32 => ( member567864080iset_a @ ( produc1898392407iset_a @ B7 @ A3 ) @ R ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % mult1E
% 0.07/0.32 thf(fact_336_mult1E,axiom,
% 0.07/0.32 ! [N: multis599418605od_a_a,M: multis599418605od_a_a,R: set_Pr1948701895od_a_a] :
% 0.07/0.32 ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ N @ M ) @ ( mult1_1719650714od_a_a @ R ) )
% 0.07/0.32 => ~ ! [A3: product_prod_a_a,M02: multis599418605od_a_a] :
% 0.07/0.32 ( ( M
% 0.07/0.32 = ( add_ms1725977093od_a_a @ A3 @ M02 ) )
% 0.07/0.32 => ! [K4: multis599418605od_a_a] :
% 0.07/0.32 ( ( N
% 0.07/0.32 = ( plus_p404985124od_a_a @ M02 @ K4 ) )
% 0.07/0.32 => ~ ! [B7: product_prod_a_a] :
% 0.07/0.32 ( ( member449909584od_a_a @ B7 @ ( set_ms1818337124od_a_a @ K4 ) )
% 0.07/0.32 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B7 @ A3 ) @ R ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % mult1E
% 0.07/0.32 thf(fact_337_mult1E,axiom,
% 0.07/0.32 ! [N: multiset_multiset_a,M: multiset_multiset_a,R: set_Pr158363655iset_a] :
% 0.07/0.32 ( ( member104214352iset_a @ ( produc1444952343iset_a @ N @ M ) @ ( mult1_multiset_a @ R ) )
% 0.07/0.32 => ~ ! [A3: multiset_a,M02: multiset_multiset_a] :
% 0.07/0.32 ( ( M
% 0.07/0.32 = ( add_mset_multiset_a @ A3 @ M02 ) )
% 0.07/0.32 => ! [K4: multiset_multiset_a] :
% 0.07/0.32 ( ( N
% 0.07/0.32 = ( plus_p1957546689iset_a @ M02 @ K4 ) )
% 0.07/0.32 => ~ ! [B7: multiset_a] :
% 0.07/0.32 ( ( member_multiset_a @ B7 @ ( set_mset_multiset_a @ K4 ) )
% 0.07/0.32 => ( member340150864iset_a @ ( produc2037245207iset_a @ B7 @ A3 ) @ R ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % mult1E
% 0.07/0.32 thf(fact_338_mult1E,axiom,
% 0.07/0.32 ! [N: multiset_a,M: multiset_a,R: set_Product_prod_a_a] :
% 0.07/0.32 ( ( member340150864iset_a @ ( produc2037245207iset_a @ N @ M ) @ ( mult1_a @ R ) )
% 0.07/0.32 => ~ ! [A3: a,M02: multiset_a] :
% 0.07/0.32 ( ( M
% 0.07/0.32 = ( add_mset_a @ A3 @ M02 ) )
% 0.07/0.32 => ! [K4: multiset_a] :
% 0.07/0.32 ( ( N
% 0.07/0.32 = ( plus_plus_multiset_a @ M02 @ K4 ) )
% 0.07/0.32 => ~ ! [B7: a] :
% 0.07/0.32 ( ( member_a @ B7 @ ( set_mset_a @ K4 ) )
% 0.07/0.32 => ( member449909584od_a_a @ ( product_Pair_a_a @ B7 @ A3 ) @ R ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % mult1E
% 0.07/0.32 thf(fact_339_one__step__implies__mult,axiom,
% 0.07/0.32 ! [J2: multiset_multiset_a,K2: multiset_multiset_a,R: set_Pr158363655iset_a,I4: multiset_multiset_a] :
% 0.07/0.32 ( ( J2 != zero_z1580389697iset_a )
% 0.07/0.32 => ( ! [X3: multiset_a] :
% 0.07/0.32 ( ( member_multiset_a @ X3 @ ( set_mset_multiset_a @ K2 ) )
% 0.07/0.32 => ? [Xa: multiset_a] :
% 0.07/0.32 ( ( member_multiset_a @ Xa @ ( set_mset_multiset_a @ J2 ) )
% 0.07/0.32 & ( member340150864iset_a @ ( produc2037245207iset_a @ X3 @ Xa ) @ R ) ) )
% 0.07/0.32 => ( member104214352iset_a @ ( produc1444952343iset_a @ ( plus_p1957546689iset_a @ I4 @ K2 ) @ ( plus_p1957546689iset_a @ I4 @ J2 ) ) @ ( mult_multiset_a @ R ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % one_step_implies_mult
% 0.07/0.32 thf(fact_340_one__step__implies__mult,axiom,
% 0.07/0.32 ! [J2: multiset_a,K2: multiset_a,R: set_Product_prod_a_a,I4: multiset_a] :
% 0.07/0.32 ( ( J2 != zero_zero_multiset_a )
% 0.07/0.32 => ( ! [X3: a] :
% 0.07/0.32 ( ( member_a @ X3 @ ( set_mset_a @ K2 ) )
% 0.07/0.32 => ? [Xa: a] :
% 0.07/0.32 ( ( member_a @ Xa @ ( set_mset_a @ J2 ) )
% 0.07/0.32 & ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Xa ) @ R ) ) )
% 0.07/0.32 => ( member340150864iset_a @ ( produc2037245207iset_a @ ( plus_plus_multiset_a @ I4 @ K2 ) @ ( plus_plus_multiset_a @ I4 @ J2 ) ) @ ( mult_a @ R ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % one_step_implies_mult
% 0.07/0.32 thf(fact_341_heap__top__max,axiom,
% 0.07/0.32 ! [T2: tree_a] :
% 0.07/0.32 ( ( T2 != e_a )
% 0.07/0.32 => ( ( is_heap_a @ T2 )
% 0.07/0.32 => ( ( val_a @ T2 )
% 0.07/0.32 = ( lattic146396397_Max_a @ ( set_mset_a @ ( multiset_a2 @ T2 ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % heap_top_max
% 0.07/0.32 thf(fact_342_mono__mult,axiom,
% 0.07/0.32 ! [R: set_Product_prod_a_a,R4: set_Product_prod_a_a] :
% 0.07/0.32 ( ( ord_le1824328871od_a_a @ R @ R4 )
% 0.07/0.32 => ( ord_le1978107815iset_a @ ( mult_a @ R ) @ ( mult_a @ R4 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % mono_mult
% 0.07/0.32 thf(fact_343_mult__implies__one__step,axiom,
% 0.07/0.32 ! [R: set_Pr158363655iset_a,M: multiset_multiset_a,N: multiset_multiset_a] :
% 0.07/0.32 ( ( trans_multiset_a @ R )
% 0.07/0.32 => ( ( member104214352iset_a @ ( produc1444952343iset_a @ M @ N ) @ ( mult_multiset_a @ R ) )
% 0.07/0.32 => ? [I5: multiset_multiset_a,J3: multiset_multiset_a] :
% 0.07/0.32 ( ( N
% 0.07/0.32 = ( plus_p1957546689iset_a @ I5 @ J3 ) )
% 0.07/0.32 & ? [K4: multiset_multiset_a] :
% 0.07/0.32 ( ( M
% 0.07/0.32 = ( plus_p1957546689iset_a @ I5 @ K4 ) )
% 0.07/0.32 & ( J3 != zero_z1580389697iset_a )
% 0.07/0.32 & ! [X6: multiset_a] :
% 0.07/0.32 ( ( member_multiset_a @ X6 @ ( set_mset_multiset_a @ K4 ) )
% 0.07/0.32 => ? [Xa2: multiset_a] :
% 0.07/0.32 ( ( member_multiset_a @ Xa2 @ ( set_mset_multiset_a @ J3 ) )
% 0.07/0.32 & ( member340150864iset_a @ ( produc2037245207iset_a @ X6 @ Xa2 ) @ R ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % mult_implies_one_step
% 0.07/0.32 thf(fact_344_mult__implies__one__step,axiom,
% 0.07/0.32 ! [R: set_Product_prod_a_a,M: multiset_a,N: multiset_a] :
% 0.07/0.32 ( ( trans_a @ R )
% 0.07/0.32 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ M @ N ) @ ( mult_a @ R ) )
% 0.07/0.32 => ? [I5: multiset_a,J3: multiset_a] :
% 0.07/0.32 ( ( N
% 0.07/0.32 = ( plus_plus_multiset_a @ I5 @ J3 ) )
% 0.07/0.32 & ? [K4: multiset_a] :
% 0.07/0.32 ( ( M
% 0.07/0.32 = ( plus_plus_multiset_a @ I5 @ K4 ) )
% 0.07/0.32 & ( J3 != zero_zero_multiset_a )
% 0.07/0.32 & ! [X6: a] :
% 0.07/0.32 ( ( member_a @ X6 @ ( set_mset_a @ K4 ) )
% 0.07/0.32 => ? [Xa2: a] :
% 0.07/0.32 ( ( member_a @ Xa2 @ ( set_mset_a @ J3 ) )
% 0.07/0.32 & ( member449909584od_a_a @ ( product_Pair_a_a @ X6 @ Xa2 ) @ R ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % mult_implies_one_step
% 0.07/0.32 thf(fact_345_subset__mset_Osum__mset__0__iff,axiom,
% 0.07/0.32 ! [M: multiset_multiset_a] :
% 0.07/0.32 ( ( ( comm_m315775925iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ M )
% 0.07/0.32 = zero_zero_multiset_a )
% 0.07/0.32 = ( ! [X4: multiset_a] :
% 0.07/0.32 ( ( member_multiset_a @ X4 @ ( set_mset_multiset_a @ M ) )
% 0.07/0.32 => ( X4 = zero_zero_multiset_a ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % subset_mset.sum_mset_0_iff
% 0.07/0.32 thf(fact_346_comm__monoid__add_Osum__mset_Ocong,axiom,
% 0.07/0.32 comm_m315775925iset_a = comm_m315775925iset_a ).
% 0.07/0.32
% 0.07/0.32 % comm_monoid_add.sum_mset.cong
% 0.07/0.32 thf(fact_347_transD,axiom,
% 0.07/0.32 ! [R: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a,Z: multiset_a] :
% 0.07/0.32 ( ( trans_multiset_a @ R )
% 0.07/0.32 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ R )
% 0.07/0.32 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ Z ) @ R )
% 0.07/0.32 => ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Z ) @ R ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % transD
% 0.07/0.32 thf(fact_348_transD,axiom,
% 0.07/0.32 ! [R: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 0.07/0.32 ( ( trans_a @ R )
% 0.07/0.32 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
% 0.07/0.32 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R )
% 0.07/0.32 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ R ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % transD
% 0.07/0.32 thf(fact_349_transE,axiom,
% 0.07/0.32 ! [R: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a,Z: multiset_a] :
% 0.07/0.32 ( ( trans_multiset_a @ R )
% 0.07/0.32 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ R )
% 0.07/0.32 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ Z ) @ R )
% 0.07/0.32 => ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Z ) @ R ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % transE
% 0.07/0.32 thf(fact_350_transE,axiom,
% 0.07/0.32 ! [R: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 0.07/0.32 ( ( trans_a @ R )
% 0.07/0.32 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
% 0.07/0.32 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R )
% 0.07/0.32 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ R ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % transE
% 0.07/0.32 thf(fact_351_transI,axiom,
% 0.07/0.32 ! [R: set_Product_prod_a_a] :
% 0.07/0.32 ( ! [X3: a,Y3: a,Z3: a] :
% 0.07/0.32 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R )
% 0.07/0.32 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ Z3 ) @ R )
% 0.07/0.32 => ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Z3 ) @ R ) ) )
% 0.07/0.32 => ( trans_a @ R ) ) ).
% 0.07/0.32
% 0.07/0.32 % transI
% 0.07/0.32 thf(fact_352_mult__cancel__add__mset,axiom,
% 0.07/0.32 ! [S3: set_Product_prod_a_a,Uu: a,X5: multiset_a,Y5: multiset_a] :
% 0.07/0.32 ( ( trans_a @ S3 )
% 0.07/0.32 => ( ( irrefl_a @ S3 )
% 0.07/0.32 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ ( add_mset_a @ Uu @ X5 ) @ ( add_mset_a @ Uu @ Y5 ) ) @ ( mult_a @ S3 ) )
% 0.07/0.32 = ( member340150864iset_a @ ( produc2037245207iset_a @ X5 @ Y5 ) @ ( mult_a @ S3 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % mult_cancel_add_mset
% 0.07/0.32
% 0.07/0.32 % Helper facts (3)
% 0.07/0.32 thf(help_If_3_1_If_001t__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_T,axiom,
% 0.07/0.33 ! [P2: $o] :
% 0.07/0.33 ( ( P2 = $true )
% 0.07/0.33 | ( P2 = $false ) ) ).
% 0.07/0.33
% 0.07/0.33 thf(help_If_2_1_If_001t__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_T,axiom,
% 0.07/0.33 ! [X: produc143150363Tree_a,Y: produc143150363Tree_a] :
% 0.07/0.33 ( ( if_Pro1144176865Tree_a @ $false @ X @ Y )
% 0.07/0.33 = Y ) ).
% 0.07/0.33
% 0.07/0.33 thf(help_If_1_1_If_001t__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_T,axiom,
% 0.07/0.33 ! [X: produc143150363Tree_a,Y: produc143150363Tree_a] :
% 0.07/0.33 ( ( if_Pro1144176865Tree_a @ $true @ X @ Y )
% 0.07/0.33 = X ) ).
% 0.07/0.33
% 0.07/0.33 % Conjectures (1)
% 0.07/0.33 thf(conj_0,conjecture,
% 0.07/0.33 ( v_a
% 0.07/0.33 = ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ v1 @ l1 @ r1 ) ) ) ) ).
% 0.07/0.33
% 0.07/0.33 %------------------------------------------------------------------------------
% 0.07/0.33 ------------------------
% 2.26/1.13 ---- Embedded in THF ---
% 2.26/1.14 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.26/1.14 %%% Generated on Tue Feb 24 03:25:11 EST 2026
% 2.26/1.14 %%% using '$$dhol' embedding, version 1.3.0.
% 2.26/1.14 %%% Logic specification used:
% 2.26/1.14 %%% thf(spec, logic, $$dhol).
% 2.26/1.14
% 2.26/1.14 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.lw13DE3eQN/DTF2DTF26921.p
% See solution above
% 2.26/1.14 ------------------------
% 2.26/1.17 ---- Cleaned THF ---
% 2.26/1.17 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J_J_J,type,
% 2.26/1.17 set_Pr1066112583iset_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J_J,type,
% 2.26/1.17 produc741964647iset_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_Mt__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J_J,type,
% 2.26/1.17 set_Pr146455751iset_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_J_J_J,type,
% 2.26/1.17 set_Pr295926343Tree_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_Mt__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J,type,
% 2.26/1.17 produc558600423iset_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_J_J,type,
% 2.26/1.17 produc1136638567Tree_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J,type,
% 2.26/1.17 set_Pr1821581383od_a_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_J_J,type,
% 2.26/1.17 set_Pr220561863Tree_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 2.26/1.17 produc2061001575od_a_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J_Mt__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J_J_J,type,
% 2.26/1.17 set_Pr720484615iset_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_J,type,
% 2.26/1.17 produc1254371559Tree_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__Product____Type__Oprod_It__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J_Mt__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J_J,type,
% 2.26/1.17 produc1293660967iset_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 2.26/1.17 set_Pr1948701895od_a_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J,type,
% 2.26/1.17 multis782275565iset_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 2.26/1.17 produc1572603623od_a_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J,type,
% 2.26/1.17 set_Pr158363655iset_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J,type,
% 2.26/1.17 produc1127127335iset_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_J,type,
% 2.26/1.17 multis2082063201Tree_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_J,type,
% 2.26/1.17 set_Pr1070816379Tree_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Heap__OTree_Itf__a_J_Mtf__a_J_J,type,
% 2.26/1.17 set_Pr921883667ee_a_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 2.26/1.17 multis599418605od_a_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J,type,
% 2.26/1.17 produc143150363Tree_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__Product____Type__Oprod_It__Heap__OTree_Itf__a_J_Mtf__a_J,type,
% 2.26/1.17 produc981471411ee_a_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 2.26/1.17 set_Product_prod_a_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J,type,
% 2.26/1.17 multiset_multiset_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__Set__Oset_It__Multiset__Omultiset_Itf__a_J_J,type,
% 2.26/1.17 set_multiset_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 2.26/1.17 product_prod_a_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__Multiset__Omultiset_Itf__a_J,type,
% 2.26/1.17 multiset_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__List__Olist_Itf__a_J,type,
% 2.26/1.17 list_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__Heap__OTree_Itf__a_J,type,
% 2.26/1.17 tree_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 2.26/1.17 set_a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(ty_n_tf__a,type,
% 2.26/1.17 a: $tType ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J,type,
% 2.26/1.17 plus_p1957546689iset_a: multiset_multiset_a > multiset_multiset_a > multiset_multiset_a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J,type,
% 2.26/1.17 plus_p1919096612iset_a: multis782275565iset_a > multis782275565iset_a > multis782275565iset_a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_J,type,
% 2.26/1.17 plus_p1065544216Tree_a: multis2082063201Tree_a > multis2082063201Tree_a > multis2082063201Tree_a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 2.26/1.17 plus_p404985124od_a_a: multis599418605od_a_a > multis599418605od_a_a > multis599418605od_a_a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.26/1.17 plus_plus_multiset_a: multiset_a > multiset_a > multiset_a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J,type,
% 2.26/1.17 zero_z1580389697iset_a: multiset_multiset_a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J,type,
% 2.26/1.17 zero_z1941791140iset_a: multis782275565iset_a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_J,type,
% 2.26/1.17 zero_z2144236696Tree_a: multis2082063201Tree_a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 2.26/1.17 zero_z160849828od_a_a: multis599418605od_a_a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.26/1.17 zero_zero_multiset_a: multiset_a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_HOL_ONO__MATCH_001t__Multiset__Omultiset_Itf__a_J_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.26/1.17 nO_MAT1617603563iset_a: multiset_a > multiset_a > $o ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Oheapify_001tf__a,type,
% 2.26/1.17 heapIm970322378pify_a: tree_a > tree_a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Ohs__is__empty_001tf__a,type,
% 2.26/1.17 heapIm229596386mpty_a: tree_a > $o ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Ohs__of__list_001tf__a,type,
% 2.26/1.17 heapIm1057938560list_a: list_a > tree_a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Ohs__remove__max_001tf__a,type,
% 2.26/1.17 heapIm915766629_max_a: tree_a > produc143150363Tree_a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Oleft_001tf__a,type,
% 2.26/1.17 heapIm1140443833left_a: tree_a > tree_a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Oof__list__tree_001tf__a,type,
% 2.26/1.17 heapIm1637418125tree_a: list_a > tree_a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_OremoveLeaf_001tf__a,type,
% 2.26/1.17 heapIm837449470Leaf_a: tree_a > produc143150363Tree_a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Oright_001tf__a,type,
% 2.26/1.17 heapIm1257206334ight_a: tree_a > tree_a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_OsiftDown_001tf__a,type,
% 2.26/1.17 heapIm1091024090Down_a: tree_a > tree_a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_Heap_OHeap_001t__Heap__OTree_Itf__a_J_001tf__a,type,
% 2.26/1.17 heap_Tree_a_a: tree_a > ( tree_a > $o ) > ( list_a > tree_a ) > ( tree_a > multiset_a ) > ( tree_a > tree_a ) > ( tree_a > produc143150363Tree_a ) > $o ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_Heap_OHeap_001tf__a_001tf__a,type,
% 2.26/1.17 heap_a_a: a > ( a > $o ) > ( list_a > a ) > ( a > multiset_a ) > ( a > tree_a ) > ( a > product_prod_a_a ) > $o ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_Heap_OHeap__axioms_001t__Heap__OTree_Itf__a_J_001tf__a,type,
% 2.26/1.17 heap_axioms_Tree_a_a: ( tree_a > $o ) > ( list_a > tree_a ) > ( tree_a > multiset_a ) > ( tree_a > tree_a ) > ( tree_a > produc143150363Tree_a ) > $o ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_Heap_OHeap__axioms_001tf__a_001tf__a,type,
% 2.26/1.17 heap_axioms_a_a: ( a > $o ) > ( list_a > a ) > ( a > multiset_a ) > ( a > tree_a ) > ( a > product_prod_a_a ) > $o ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_Heap_OTree_OE_001tf__a,type,
% 2.26/1.17 e_a: tree_a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_Heap_OTree_OT_001tf__a,type,
% 2.26/1.17 t_a: a > tree_a > tree_a > tree_a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_Heap_Oin__tree_001tf__a,type,
% 2.26/1.17 in_tree_a: a > tree_a > $o ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_Heap_Ois__heap_001tf__a,type,
% 2.26/1.17 is_heap_a: tree_a > $o ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_Heap_Omultiset_001tf__a,type,
% 2.26/1.17 multiset_a2: tree_a > multiset_a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_Heap_Oval_001tf__a,type,
% 2.26/1.17 val_a: tree_a > a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_If_001t__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J,type,
% 2.26/1.17 if_Pro1144176865Tree_a: $o > produc143150363Tree_a > produc143150363Tree_a > produc143150363Tree_a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_Lattices__Big_Olinorder__class_OMax_001tf__a,type,
% 2.26/1.17 lattic146396397_Max_a: set_a > a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_Multiset_Oadd__mset_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.26/1.17 add_mset_multiset_a: multiset_a > multiset_multiset_a > multiset_multiset_a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_Multiset_Oadd__mset_001t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J,type,
% 2.26/1.17 add_ms1353365509iset_a: produc1127127335iset_a > multis782275565iset_a > multis782275565iset_a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_Multiset_Oadd__mset_001t__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J,type,
% 2.26/1.17 add_ms205941497Tree_a: produc143150363Tree_a > multis2082063201Tree_a > multis2082063201Tree_a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_Multiset_Oadd__mset_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 2.26/1.17 add_ms1725977093od_a_a: product_prod_a_a > multis599418605od_a_a > multis599418605od_a_a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_Multiset_Oadd__mset_001tf__a,type,
% 2.26/1.17 add_mset_a: a > multiset_a > multiset_a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_Multiset_Ocomm__monoid__add_Osum__mset_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.26/1.17 comm_m315775925iset_a: ( multiset_a > multiset_a > multiset_a ) > multiset_a > multiset_multiset_a > multiset_a ).
% 2.26/1.17
% 2.26/1.17 thf(sy_c_Multiset_Ois__empty_001tf__a,type,
% 2.26/1.17 is_empty_a: multiset_a > $o ).
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% 2.26/1.17 v3: a ).
% 2.26/1.17
% 2.26/1.17 thf(fact_0__C4__2_Oprems_C_I1_J,axiom,
% 2.26/1.17 ( ( produc686083979Tree_a @ v_a @ t_a2 )
% 2.26/1.17 = ( heapIm837449470Leaf_a @ ( t_a @ v3 @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_1_assms_I1_J,axiom,
% 2.26/1.17 ( ( produc686083979Tree_a @ v @ t2 )
% 2.26/1.17 = ( heapIm837449470Leaf_a @ t ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_2__C4__2_Oprems_C_I2_J,axiom,
% 2.26/1.17 ( ( t_a @ v3 @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) )
% 2.26/1.17 != e_a ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_3__092_060open_062t_H_A_061_AT_Av_A_Isnd_A_IremoveLeaf_A_IT_Av1_Al1_Ar1_J_J_J_A_IT_Av2_Al2_Ar2_J_092_060close_062,axiom,
% 2.26/1.17 ( t_a2
% 2.26/1.17 = ( t_a @ v3 @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ v1 @ l1 @ r1 ) ) ) @ ( t_a @ v2 @ l2 @ r2 ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_4_Tree_Oinject,axiom,
% 2.26/1.17 ! [X21: a,X22: tree_a,X23: tree_a,Y21: a,Y22: tree_a,Y23: tree_a] :
% 2.26/1.17 ( ( ( t_a @ X21 @ X22 @ X23 )
% 2.26/1.17 = ( t_a @ Y21 @ Y22 @ Y23 ) )
% 2.26/1.17 = ( ( X21 = Y21 )
% 2.26/1.17 & ( X22 = Y22 )
% 2.26/1.17 & ( X23 = Y23 ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_5_removeLeaf_Osimps_I5_J,axiom,
% 2.26/1.17 ! [V: a,Vd: a,Ve: tree_a,Vf: tree_a,Va: a,Vb: tree_a,Vc: tree_a] :
% 2.26/1.17 ( ( heapIm837449470Leaf_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
% 2.26/1.17 = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ Vd @ Ve @ Vf ) ) ) @ ( t_a @ V @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ Vd @ Ve @ Vf ) ) ) @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_6_removeLeaf_Osimps_I4_J,axiom,
% 2.26/1.17 ! [V: a,Va: a,Vb: tree_a,Vc: tree_a,Vd: a,Ve: tree_a,Vf: tree_a] :
% 2.26/1.17 ( ( heapIm837449470Leaf_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 2.26/1.17 = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ Va @ Vb @ Vc ) ) ) @ ( t_a @ V @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ Va @ Vb @ Vc ) ) ) @ ( t_a @ Vd @ Ve @ Vf ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_7_left_Osimps,axiom,
% 2.26/1.17 ! [V: a,L: tree_a,R: tree_a] :
% 2.26/1.17 ( ( heapIm1140443833left_a @ ( t_a @ V @ L @ R ) )
% 2.26/1.17 = L ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_8_right_Osimps,axiom,
% 2.26/1.17 ! [V: a,L: tree_a,R: tree_a] :
% 2.26/1.17 ( ( heapIm1257206334ight_a @ ( t_a @ V @ L @ R ) )
% 2.26/1.17 = R ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_9__C4__2_Ohyps_C_I1_J,axiom,
% 2.26/1.17 ! [V2: a,T: tree_a] :
% 2.26/1.17 ( ( ( produc686083979Tree_a @ V2 @ T )
% 2.26/1.17 = ( heapIm837449470Leaf_a @ ( t_a @ v1 @ l1 @ r1 ) ) )
% 2.26/1.17 => ( ( ( t_a @ v1 @ l1 @ r1 )
% 2.26/1.17 != e_a )
% 2.26/1.17 => ( ( plus_plus_multiset_a @ ( add_mset_a @ V2 @ zero_zero_multiset_a ) @ ( multiset_a2 @ T ) )
% 2.26/1.17 = ( multiset_a2 @ ( t_a @ v1 @ l1 @ r1 ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_10_removeLeaf_Osimps_I1_J,axiom,
% 2.26/1.17 ! [V: a] :
% 2.26/1.17 ( ( heapIm837449470Leaf_a @ ( t_a @ V @ e_a @ e_a ) )
% 2.26/1.17 = ( produc686083979Tree_a @ V @ e_a ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_11_fstI,axiom,
% 2.26/1.17 ! [X: produc1127127335iset_a,Y: multiset_a,Z: multiset_a] :
% 2.26/1.17 ( ( X
% 2.26/1.17 = ( produc2037245207iset_a @ Y @ Z ) )
% 2.26/1.17 => ( ( produc653373187iset_a @ X )
% 2.26/1.17 = Y ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_12_fstI,axiom,
% 2.26/1.17 ! [X: product_prod_a_a,Y: a,Z: a] :
% 2.26/1.17 ( ( X
% 2.26/1.17 = ( product_Pair_a_a @ Y @ Z ) )
% 2.26/1.17 => ( ( product_fst_a_a @ X )
% 2.26/1.17 = Y ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_13_fstI,axiom,
% 2.26/1.17 ! [X: produc981471411ee_a_a,Y: tree_a,Z: a] :
% 2.26/1.17 ( ( X
% 2.26/1.17 = ( produc649172771ee_a_a @ Y @ Z ) )
% 2.26/1.17 => ( ( product_fst_Tree_a_a @ X )
% 2.26/1.17 = Y ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_14_fstI,axiom,
% 2.26/1.17 ! [X: produc143150363Tree_a,Y: a,Z: tree_a] :
% 2.26/1.17 ( ( X
% 2.26/1.17 = ( produc686083979Tree_a @ Y @ Z ) )
% 2.26/1.17 => ( ( product_fst_a_Tree_a @ X )
% 2.26/1.17 = Y ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_15_fst__eqD,axiom,
% 2.26/1.17 ! [X: multiset_a,Y: multiset_a,A: multiset_a] :
% 2.26/1.17 ( ( ( produc653373187iset_a @ ( produc2037245207iset_a @ X @ Y ) )
% 2.26/1.17 = A )
% 2.26/1.17 => ( X = A ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_16_fst__eqD,axiom,
% 2.26/1.17 ! [X: a,Y: a,A: a] :
% 2.26/1.17 ( ( ( product_fst_a_a @ ( product_Pair_a_a @ X @ Y ) )
% 2.26/1.17 = A )
% 2.26/1.17 => ( X = A ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_17_fst__eqD,axiom,
% 2.26/1.17 ! [X: tree_a,Y: a,A: tree_a] :
% 2.26/1.17 ( ( ( product_fst_Tree_a_a @ ( produc649172771ee_a_a @ X @ Y ) )
% 2.26/1.17 = A )
% 2.26/1.17 => ( X = A ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_18_fst__eqD,axiom,
% 2.26/1.17 ! [X: a,Y: tree_a,A: a] :
% 2.26/1.17 ( ( ( product_fst_a_Tree_a @ ( produc686083979Tree_a @ X @ Y ) )
% 2.26/1.17 = A )
% 2.26/1.17 => ( X = A ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_19_fst__conv,axiom,
% 2.26/1.17 ! [X1: multiset_a,X2: multiset_a] :
% 2.26/1.17 ( ( produc653373187iset_a @ ( produc2037245207iset_a @ X1 @ X2 ) )
% 2.26/1.17 = X1 ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_20_fst__conv,axiom,
% 2.26/1.17 ! [X1: a,X2: a] :
% 2.26/1.17 ( ( product_fst_a_a @ ( product_Pair_a_a @ X1 @ X2 ) )
% 2.26/1.17 = X1 ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_21_fst__conv,axiom,
% 2.26/1.17 ! [X1: tree_a,X2: a] :
% 2.26/1.17 ( ( product_fst_Tree_a_a @ ( produc649172771ee_a_a @ X1 @ X2 ) )
% 2.26/1.17 = X1 ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_22_fst__conv,axiom,
% 2.26/1.17 ! [X1: a,X2: tree_a] :
% 2.26/1.17 ( ( product_fst_a_Tree_a @ ( produc686083979Tree_a @ X1 @ X2 ) )
% 2.26/1.17 = X1 ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_23_assms_I2_J,axiom,
% 2.26/1.17 t != e_a ).
% 2.26/1.17
% 2.26/1.17 thf(fact_24_prod_Oinject,axiom,
% 2.26/1.17 ! [X1: tree_a,X2: a,Y1: tree_a,Y2: a] :
% 2.26/1.17 ( ( ( produc649172771ee_a_a @ X1 @ X2 )
% 2.26/1.17 = ( produc649172771ee_a_a @ Y1 @ Y2 ) )
% 2.26/1.17 = ( ( X1 = Y1 )
% 2.26/1.17 & ( X2 = Y2 ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_25_prod_Oinject,axiom,
% 2.26/1.17 ! [X1: multiset_a,X2: multiset_a,Y1: multiset_a,Y2: multiset_a] :
% 2.26/1.17 ( ( ( produc2037245207iset_a @ X1 @ X2 )
% 2.26/1.17 = ( produc2037245207iset_a @ Y1 @ Y2 ) )
% 2.26/1.17 = ( ( X1 = Y1 )
% 2.26/1.17 & ( X2 = Y2 ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_26_prod_Oinject,axiom,
% 2.26/1.17 ! [X1: a,X2: a,Y1: a,Y2: a] :
% 2.26/1.17 ( ( ( product_Pair_a_a @ X1 @ X2 )
% 2.26/1.17 = ( product_Pair_a_a @ Y1 @ Y2 ) )
% 2.26/1.17 = ( ( X1 = Y1 )
% 2.26/1.17 & ( X2 = Y2 ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_27_prod_Oinject,axiom,
% 2.26/1.17 ! [X1: a,X2: tree_a,Y1: a,Y2: tree_a] :
% 2.26/1.17 ( ( ( produc686083979Tree_a @ X1 @ X2 )
% 2.26/1.17 = ( produc686083979Tree_a @ Y1 @ Y2 ) )
% 2.26/1.17 = ( ( X1 = Y1 )
% 2.26/1.17 & ( X2 = Y2 ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_28_old_Oprod_Oinject,axiom,
% 2.26/1.17 ! [A: tree_a,B: a,A2: tree_a,B2: a] :
% 2.26/1.17 ( ( ( produc649172771ee_a_a @ A @ B )
% 2.26/1.17 = ( produc649172771ee_a_a @ A2 @ B2 ) )
% 2.26/1.17 = ( ( A = A2 )
% 2.26/1.17 & ( B = B2 ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_29_old_Oprod_Oinject,axiom,
% 2.26/1.17 ! [A: multiset_a,B: multiset_a,A2: multiset_a,B2: multiset_a] :
% 2.26/1.17 ( ( ( produc2037245207iset_a @ A @ B )
% 2.26/1.17 = ( produc2037245207iset_a @ A2 @ B2 ) )
% 2.26/1.17 = ( ( A = A2 )
% 2.26/1.17 & ( B = B2 ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_30_old_Oprod_Oinject,axiom,
% 2.26/1.17 ! [A: a,B: a,A2: a,B2: a] :
% 2.26/1.17 ( ( ( product_Pair_a_a @ A @ B )
% 2.26/1.17 = ( product_Pair_a_a @ A2 @ B2 ) )
% 2.26/1.17 = ( ( A = A2 )
% 2.26/1.17 & ( B = B2 ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_31_old_Oprod_Oinject,axiom,
% 2.26/1.17 ! [A: a,B: tree_a,A2: a,B2: tree_a] :
% 2.26/1.17 ( ( ( produc686083979Tree_a @ A @ B )
% 2.26/1.17 = ( produc686083979Tree_a @ A2 @ B2 ) )
% 2.26/1.17 = ( ( A = A2 )
% 2.26/1.17 & ( B = B2 ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_32_prod_Ocollapse,axiom,
% 2.26/1.17 ! [Prod: produc1127127335iset_a] :
% 2.26/1.17 ( ( produc2037245207iset_a @ ( produc653373187iset_a @ Prod ) @ ( produc148952133iset_a @ Prod ) )
% 2.26/1.17 = Prod ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_33_prod_Ocollapse,axiom,
% 2.26/1.17 ! [Prod: product_prod_a_a] :
% 2.26/1.17 ( ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) )
% 2.26/1.17 = Prod ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_34_prod_Ocollapse,axiom,
% 2.26/1.17 ! [Prod: produc981471411ee_a_a] :
% 2.26/1.17 ( ( produc649172771ee_a_a @ ( product_fst_Tree_a_a @ Prod ) @ ( product_snd_Tree_a_a @ Prod ) )
% 2.26/1.17 = Prod ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_35_prod_Ocollapse,axiom,
% 2.26/1.17 ! [Prod: produc143150363Tree_a] :
% 2.26/1.17 ( ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ Prod ) @ ( product_snd_a_Tree_a @ Prod ) )
% 2.26/1.17 = Prod ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_36_multiset_Osimps_I2_J,axiom,
% 2.26/1.17 ! [V: a,L: tree_a,R: tree_a] :
% 2.26/1.17 ( ( multiset_a2 @ ( t_a @ V @ L @ R ) )
% 2.26/1.17 = ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ ( multiset_a2 @ L ) @ ( add_mset_a @ V @ zero_zero_multiset_a ) ) @ ( multiset_a2 @ R ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_37_multiset_Osimps_I1_J,axiom,
% 2.26/1.17 ( ( multiset_a2 @ e_a )
% 2.26/1.17 = zero_zero_multiset_a ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_38_sndI,axiom,
% 2.26/1.17 ! [X: produc1127127335iset_a,Y: multiset_a,Z: multiset_a] :
% 2.26/1.17 ( ( X
% 2.26/1.17 = ( produc2037245207iset_a @ Y @ Z ) )
% 2.26/1.17 => ( ( produc148952133iset_a @ X )
% 2.26/1.17 = Z ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_39_sndI,axiom,
% 2.26/1.17 ! [X: product_prod_a_a,Y: a,Z: a] :
% 2.26/1.17 ( ( X
% 2.26/1.17 = ( product_Pair_a_a @ Y @ Z ) )
% 2.26/1.17 => ( ( product_snd_a_a @ X )
% 2.26/1.17 = Z ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_40_sndI,axiom,
% 2.26/1.17 ! [X: produc981471411ee_a_a,Y: tree_a,Z: a] :
% 2.26/1.17 ( ( X
% 2.26/1.17 = ( produc649172771ee_a_a @ Y @ Z ) )
% 2.26/1.17 => ( ( product_snd_Tree_a_a @ X )
% 2.26/1.17 = Z ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_41_sndI,axiom,
% 2.26/1.17 ! [X: produc143150363Tree_a,Y: a,Z: tree_a] :
% 2.26/1.17 ( ( X
% 2.26/1.17 = ( produc686083979Tree_a @ Y @ Z ) )
% 2.26/1.17 => ( ( product_snd_a_Tree_a @ X )
% 2.26/1.17 = Z ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_42_snd__eqD,axiom,
% 2.26/1.17 ! [X: multiset_a,Y: multiset_a,A: multiset_a] :
% 2.26/1.17 ( ( ( produc148952133iset_a @ ( produc2037245207iset_a @ X @ Y ) )
% 2.26/1.17 = A )
% 2.26/1.17 => ( Y = A ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_43_snd__eqD,axiom,
% 2.26/1.17 ! [X: a,Y: a,A: a] :
% 2.26/1.17 ( ( ( product_snd_a_a @ ( product_Pair_a_a @ X @ Y ) )
% 2.26/1.17 = A )
% 2.26/1.17 => ( Y = A ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_44_snd__eqD,axiom,
% 2.26/1.17 ! [X: tree_a,Y: a,A: a] :
% 2.26/1.17 ( ( ( product_snd_Tree_a_a @ ( produc649172771ee_a_a @ X @ Y ) )
% 2.26/1.17 = A )
% 2.26/1.17 => ( Y = A ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_45_snd__eqD,axiom,
% 2.26/1.17 ! [X: a,Y: tree_a,A: tree_a] :
% 2.26/1.17 ( ( ( product_snd_a_Tree_a @ ( produc686083979Tree_a @ X @ Y ) )
% 2.26/1.17 = A )
% 2.26/1.17 => ( Y = A ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_46_snd__conv,axiom,
% 2.26/1.17 ! [X1: multiset_a,X2: multiset_a] :
% 2.26/1.17 ( ( produc148952133iset_a @ ( produc2037245207iset_a @ X1 @ X2 ) )
% 2.26/1.17 = X2 ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_47_snd__conv,axiom,
% 2.26/1.17 ! [X1: a,X2: a] :
% 2.26/1.17 ( ( product_snd_a_a @ ( product_Pair_a_a @ X1 @ X2 ) )
% 2.26/1.17 = X2 ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_48_snd__conv,axiom,
% 2.26/1.17 ! [X1: tree_a,X2: a] :
% 2.26/1.17 ( ( product_snd_Tree_a_a @ ( produc649172771ee_a_a @ X1 @ X2 ) )
% 2.26/1.17 = X2 ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_49_snd__conv,axiom,
% 2.26/1.17 ! [X1: a,X2: tree_a] :
% 2.26/1.17 ( ( product_snd_a_Tree_a @ ( produc686083979Tree_a @ X1 @ X2 ) )
% 2.26/1.17 = X2 ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_50_surj__pair,axiom,
% 2.26/1.17 ! [P: produc981471411ee_a_a] :
% 2.26/1.17 ? [X3: tree_a,Y3: a] :
% 2.26/1.17 ( P
% 2.26/1.17 = ( produc649172771ee_a_a @ X3 @ Y3 ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_51_surj__pair,axiom,
% 2.26/1.17 ! [P: produc1127127335iset_a] :
% 2.26/1.17 ? [X3: multiset_a,Y3: multiset_a] :
% 2.26/1.17 ( P
% 2.26/1.17 = ( produc2037245207iset_a @ X3 @ Y3 ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_52_surj__pair,axiom,
% 2.26/1.17 ! [P: product_prod_a_a] :
% 2.26/1.17 ? [X3: a,Y3: a] :
% 2.26/1.17 ( P
% 2.26/1.17 = ( product_Pair_a_a @ X3 @ Y3 ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_53_surj__pair,axiom,
% 2.26/1.17 ! [P: produc143150363Tree_a] :
% 2.26/1.17 ? [X3: a,Y3: tree_a] :
% 2.26/1.17 ( P
% 2.26/1.17 = ( produc686083979Tree_a @ X3 @ Y3 ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_54_prod__cases,axiom,
% 2.26/1.17 ! [P2: produc981471411ee_a_a > $o,P: produc981471411ee_a_a] :
% 2.26/1.17 ( ! [A3: tree_a,B3: a] : ( P2 @ ( produc649172771ee_a_a @ A3 @ B3 ) )
% 2.26/1.17 => ( P2 @ P ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_55_prod__cases,axiom,
% 2.26/1.17 ! [P2: produc1127127335iset_a > $o,P: produc1127127335iset_a] :
% 2.26/1.17 ( ! [A3: multiset_a,B3: multiset_a] : ( P2 @ ( produc2037245207iset_a @ A3 @ B3 ) )
% 2.26/1.17 => ( P2 @ P ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_56_prod__cases,axiom,
% 2.26/1.17 ! [P2: product_prod_a_a > $o,P: product_prod_a_a] :
% 2.26/1.17 ( ! [A3: a,B3: a] : ( P2 @ ( product_Pair_a_a @ A3 @ B3 ) )
% 2.26/1.17 => ( P2 @ P ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_57_prod__cases,axiom,
% 2.26/1.17 ! [P2: produc143150363Tree_a > $o,P: produc143150363Tree_a] :
% 2.26/1.17 ( ! [A3: a,B3: tree_a] : ( P2 @ ( produc686083979Tree_a @ A3 @ B3 ) )
% 2.26/1.17 => ( P2 @ P ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_58_Pair__inject,axiom,
% 2.26/1.17 ! [A: tree_a,B: a,A2: tree_a,B2: a] :
% 2.26/1.17 ( ( ( produc649172771ee_a_a @ A @ B )
% 2.26/1.17 = ( produc649172771ee_a_a @ A2 @ B2 ) )
% 2.26/1.17 => ~ ( ( A = A2 )
% 2.26/1.17 => ( B != B2 ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_59_Pair__inject,axiom,
% 2.26/1.17 ! [A: multiset_a,B: multiset_a,A2: multiset_a,B2: multiset_a] :
% 2.26/1.17 ( ( ( produc2037245207iset_a @ A @ B )
% 2.26/1.17 = ( produc2037245207iset_a @ A2 @ B2 ) )
% 2.26/1.17 => ~ ( ( A = A2 )
% 2.26/1.17 => ( B != B2 ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_60_Pair__inject,axiom,
% 2.26/1.17 ! [A: a,B: a,A2: a,B2: a] :
% 2.26/1.17 ( ( ( product_Pair_a_a @ A @ B )
% 2.26/1.17 = ( product_Pair_a_a @ A2 @ B2 ) )
% 2.26/1.17 => ~ ( ( A = A2 )
% 2.26/1.17 => ( B != B2 ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_61_Pair__inject,axiom,
% 2.26/1.17 ! [A: a,B: tree_a,A2: a,B2: tree_a] :
% 2.26/1.17 ( ( ( produc686083979Tree_a @ A @ B )
% 2.26/1.17 = ( produc686083979Tree_a @ A2 @ B2 ) )
% 2.26/1.17 => ~ ( ( A = A2 )
% 2.26/1.17 => ( B != B2 ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_62_old_Oprod_Oexhaust,axiom,
% 2.26/1.17 ! [Y: produc981471411ee_a_a] :
% 2.26/1.17 ~ ! [A3: tree_a,B3: a] :
% 2.26/1.17 ( Y
% 2.26/1.17 != ( produc649172771ee_a_a @ A3 @ B3 ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_63_old_Oprod_Oexhaust,axiom,
% 2.26/1.17 ! [Y: produc1127127335iset_a] :
% 2.26/1.17 ~ ! [A3: multiset_a,B3: multiset_a] :
% 2.26/1.17 ( Y
% 2.26/1.17 != ( produc2037245207iset_a @ A3 @ B3 ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_64_old_Oprod_Oexhaust,axiom,
% 2.26/1.17 ! [Y: product_prod_a_a] :
% 2.26/1.17 ~ ! [A3: a,B3: a] :
% 2.26/1.17 ( Y
% 2.26/1.17 != ( product_Pair_a_a @ A3 @ B3 ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_65_old_Oprod_Oexhaust,axiom,
% 2.26/1.17 ! [Y: produc143150363Tree_a] :
% 2.26/1.17 ~ ! [A3: a,B3: tree_a] :
% 2.26/1.17 ( Y
% 2.26/1.17 != ( produc686083979Tree_a @ A3 @ B3 ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_66_old_Oprod_Oinducts,axiom,
% 2.26/1.17 ! [P2: produc981471411ee_a_a > $o,Prod: produc981471411ee_a_a] :
% 2.26/1.17 ( ! [A3: tree_a,B3: a] : ( P2 @ ( produc649172771ee_a_a @ A3 @ B3 ) )
% 2.26/1.17 => ( P2 @ Prod ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_67_old_Oprod_Oinducts,axiom,
% 2.26/1.17 ! [P2: produc1127127335iset_a > $o,Prod: produc1127127335iset_a] :
% 2.26/1.17 ( ! [A3: multiset_a,B3: multiset_a] : ( P2 @ ( produc2037245207iset_a @ A3 @ B3 ) )
% 2.26/1.17 => ( P2 @ Prod ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_68_old_Oprod_Oinducts,axiom,
% 2.26/1.17 ! [P2: product_prod_a_a > $o,Prod: product_prod_a_a] :
% 2.26/1.17 ( ! [A3: a,B3: a] : ( P2 @ ( product_Pair_a_a @ A3 @ B3 ) )
% 2.26/1.17 => ( P2 @ Prod ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_69_old_Oprod_Oinducts,axiom,
% 2.26/1.17 ! [P2: produc143150363Tree_a > $o,Prod: produc143150363Tree_a] :
% 2.26/1.17 ( ! [A3: a,B3: tree_a] : ( P2 @ ( produc686083979Tree_a @ A3 @ B3 ) )
% 2.26/1.17 => ( P2 @ Prod ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_70_HS_Omultiset__empty,axiom,
% 2.26/1.17 ( ( multiset_a2 @ e_a )
% 2.26/1.17 = zero_zero_multiset_a ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_71_surjective__pairing,axiom,
% 2.26/1.17 ! [T2: produc1127127335iset_a] :
% 2.26/1.17 ( T2
% 2.26/1.17 = ( produc2037245207iset_a @ ( produc653373187iset_a @ T2 ) @ ( produc148952133iset_a @ T2 ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_72_surjective__pairing,axiom,
% 2.26/1.17 ! [T2: product_prod_a_a] :
% 2.26/1.17 ( T2
% 2.26/1.17 = ( product_Pair_a_a @ ( product_fst_a_a @ T2 ) @ ( product_snd_a_a @ T2 ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_73_surjective__pairing,axiom,
% 2.26/1.17 ! [T2: produc981471411ee_a_a] :
% 2.26/1.17 ( T2
% 2.26/1.17 = ( produc649172771ee_a_a @ ( product_fst_Tree_a_a @ T2 ) @ ( product_snd_Tree_a_a @ T2 ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_74_surjective__pairing,axiom,
% 2.26/1.17 ! [T2: produc143150363Tree_a] :
% 2.26/1.17 ( T2
% 2.26/1.17 = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ T2 ) @ ( product_snd_a_Tree_a @ T2 ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_75_prod_Oexhaust__sel,axiom,
% 2.26/1.17 ! [Prod: produc1127127335iset_a] :
% 2.26/1.17 ( Prod
% 2.26/1.17 = ( produc2037245207iset_a @ ( produc653373187iset_a @ Prod ) @ ( produc148952133iset_a @ Prod ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_76_prod_Oexhaust__sel,axiom,
% 2.26/1.17 ! [Prod: product_prod_a_a] :
% 2.26/1.17 ( Prod
% 2.26/1.17 = ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_77_prod_Oexhaust__sel,axiom,
% 2.26/1.17 ! [Prod: produc981471411ee_a_a] :
% 2.26/1.17 ( Prod
% 2.26/1.17 = ( produc649172771ee_a_a @ ( product_fst_Tree_a_a @ Prod ) @ ( product_snd_Tree_a_a @ Prod ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_78_prod_Oexhaust__sel,axiom,
% 2.26/1.17 ! [Prod: produc143150363Tree_a] :
% 2.26/1.17 ( Prod
% 2.26/1.17 = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ Prod ) @ ( product_snd_a_Tree_a @ Prod ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_79_prod__eq__iff,axiom,
% 2.26/1.17 ( ( ^ [Y4: produc981471411ee_a_a,Z2: produc981471411ee_a_a] : ( Y4 = Z2 ) )
% 2.26/1.17 = ( ^ [S: produc981471411ee_a_a,T3: produc981471411ee_a_a] :
% 2.26/1.17 ( ( ( product_fst_Tree_a_a @ S )
% 2.26/1.17 = ( product_fst_Tree_a_a @ T3 ) )
% 2.26/1.17 & ( ( product_snd_Tree_a_a @ S )
% 2.26/1.17 = ( product_snd_Tree_a_a @ T3 ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_80_prod__eq__iff,axiom,
% 2.26/1.17 ( ( ^ [Y4: produc143150363Tree_a,Z2: produc143150363Tree_a] : ( Y4 = Z2 ) )
% 2.26/1.17 = ( ^ [S: produc143150363Tree_a,T3: produc143150363Tree_a] :
% 2.26/1.17 ( ( ( product_fst_a_Tree_a @ S )
% 2.26/1.17 = ( product_fst_a_Tree_a @ T3 ) )
% 2.26/1.17 & ( ( product_snd_a_Tree_a @ S )
% 2.26/1.17 = ( product_snd_a_Tree_a @ T3 ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_81_prod_Oexpand,axiom,
% 2.26/1.17 ! [Prod: produc981471411ee_a_a,Prod2: produc981471411ee_a_a] :
% 2.26/1.17 ( ( ( ( product_fst_Tree_a_a @ Prod )
% 2.26/1.17 = ( product_fst_Tree_a_a @ Prod2 ) )
% 2.26/1.17 & ( ( product_snd_Tree_a_a @ Prod )
% 2.26/1.17 = ( product_snd_Tree_a_a @ Prod2 ) ) )
% 2.26/1.17 => ( Prod = Prod2 ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_82_prod_Oexpand,axiom,
% 2.26/1.17 ! [Prod: produc143150363Tree_a,Prod2: produc143150363Tree_a] :
% 2.26/1.17 ( ( ( ( product_fst_a_Tree_a @ Prod )
% 2.26/1.17 = ( product_fst_a_Tree_a @ Prod2 ) )
% 2.26/1.17 & ( ( product_snd_a_Tree_a @ Prod )
% 2.26/1.17 = ( product_snd_a_Tree_a @ Prod2 ) ) )
% 2.26/1.17 => ( Prod = Prod2 ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_83_prod__eqI,axiom,
% 2.26/1.17 ! [P: produc981471411ee_a_a,Q: produc981471411ee_a_a] :
% 2.26/1.17 ( ( ( product_fst_Tree_a_a @ P )
% 2.26/1.17 = ( product_fst_Tree_a_a @ Q ) )
% 2.26/1.17 => ( ( ( product_snd_Tree_a_a @ P )
% 2.26/1.17 = ( product_snd_Tree_a_a @ Q ) )
% 2.26/1.17 => ( P = Q ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_84_prod__eqI,axiom,
% 2.26/1.17 ! [P: produc143150363Tree_a,Q: produc143150363Tree_a] :
% 2.26/1.17 ( ( ( product_fst_a_Tree_a @ P )
% 2.26/1.17 = ( product_fst_a_Tree_a @ Q ) )
% 2.26/1.17 => ( ( ( product_snd_a_Tree_a @ P )
% 2.26/1.17 = ( product_snd_a_Tree_a @ Q ) )
% 2.26/1.17 => ( P = Q ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_85_is__heap_Ocases,axiom,
% 2.26/1.17 ! [X: tree_a] :
% 2.26/1.17 ( ( X != e_a )
% 2.26/1.17 => ( ! [V3: a] :
% 2.26/1.17 ( X
% 2.26/1.17 != ( t_a @ V3 @ e_a @ e_a ) )
% 2.26/1.17 => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 2.26/1.17 ( X
% 2.26/1.17 != ( t_a @ V3 @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 2.26/1.17 => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 2.26/1.17 ( X
% 2.26/1.17 != ( t_a @ V3 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
% 2.26/1.17 => ~ ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a] :
% 2.26/1.17 ( X
% 2.26/1.17 != ( t_a @ V3 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_86_mem__Collect__eq,axiom,
% 2.26/1.17 ! [A: produc143150363Tree_a,P2: produc143150363Tree_a > $o] :
% 2.26/1.17 ( ( member254315204Tree_a @ A @ ( collec486369286Tree_a @ P2 ) )
% 2.26/1.17 = ( P2 @ A ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_87_mem__Collect__eq,axiom,
% 2.26/1.17 ! [A: a,P2: a > $o] :
% 2.26/1.17 ( ( member_a @ A @ ( collect_a @ P2 ) )
% 2.26/1.17 = ( P2 @ A ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_88_mem__Collect__eq,axiom,
% 2.26/1.17 ! [A: produc1127127335iset_a,P2: produc1127127335iset_a > $o] :
% 2.26/1.17 ( ( member340150864iset_a @ A @ ( collec1104713362iset_a @ P2 ) )
% 2.26/1.17 = ( P2 @ A ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_89_mem__Collect__eq,axiom,
% 2.26/1.17 ! [A: product_prod_a_a,P2: product_prod_a_a > $o] :
% 2.26/1.17 ( ( member449909584od_a_a @ A @ ( collec645855634od_a_a @ P2 ) )
% 2.26/1.17 = ( P2 @ A ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_90_Collect__mem__eq,axiom,
% 2.26/1.17 ! [A4: set_Pr1070816379Tree_a] :
% 2.26/1.17 ( ( collec486369286Tree_a
% 2.26/1.17 @ ^ [X4: produc143150363Tree_a] : ( member254315204Tree_a @ X4 @ A4 ) )
% 2.26/1.17 = A4 ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_91_Collect__mem__eq,axiom,
% 2.26/1.17 ! [A4: set_a] :
% 2.26/1.17 ( ( collect_a
% 2.26/1.17 @ ^ [X4: a] : ( member_a @ X4 @ A4 ) )
% 2.26/1.17 = A4 ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_92_Collect__mem__eq,axiom,
% 2.26/1.17 ! [A4: set_Pr158363655iset_a] :
% 2.26/1.17 ( ( collec1104713362iset_a
% 2.26/1.17 @ ^ [X4: produc1127127335iset_a] : ( member340150864iset_a @ X4 @ A4 ) )
% 2.26/1.17 = A4 ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_93_Collect__mem__eq,axiom,
% 2.26/1.17 ! [A4: set_Product_prod_a_a] :
% 2.26/1.17 ( ( collec645855634od_a_a
% 2.26/1.17 @ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ A4 ) )
% 2.26/1.17 = A4 ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_94_Tree_Oexhaust,axiom,
% 2.26/1.17 ! [Y: tree_a] :
% 2.26/1.17 ( ( Y != e_a )
% 2.26/1.17 => ~ ! [X212: a,X222: tree_a,X232: tree_a] :
% 2.26/1.17 ( Y
% 2.26/1.17 != ( t_a @ X212 @ X222 @ X232 ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_95_Tree_Oinduct,axiom,
% 2.26/1.17 ! [P2: tree_a > $o,Tree: tree_a] :
% 2.26/1.17 ( ( P2 @ e_a )
% 2.26/1.17 => ( ! [X12: a,X24: tree_a,X32: tree_a] :
% 2.26/1.17 ( ( P2 @ X24 )
% 2.26/1.17 => ( ( P2 @ X32 )
% 2.26/1.17 => ( P2 @ ( t_a @ X12 @ X24 @ X32 ) ) ) )
% 2.26/1.17 => ( P2 @ Tree ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_96_Tree_Odistinct_I1_J,axiom,
% 2.26/1.17 ! [X21: a,X22: tree_a,X23: tree_a] :
% 2.26/1.17 ( e_a
% 2.26/1.17 != ( t_a @ X21 @ X22 @ X23 ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_97_removeLeaf_Oinduct,axiom,
% 2.26/1.17 ! [P2: tree_a > $o,A0: tree_a] :
% 2.26/1.17 ( ! [V3: a] : ( P2 @ ( t_a @ V3 @ e_a @ e_a ) )
% 2.26/1.17 => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 2.26/1.17 ( ( P2 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
% 2.26/1.17 => ( ( P2 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
% 2.26/1.17 => ( P2 @ ( t_a @ V3 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) ) ) )
% 2.26/1.17 => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 2.26/1.17 ( ( P2 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
% 2.26/1.17 => ( ( P2 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
% 2.26/1.17 => ( P2 @ ( t_a @ V3 @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) )
% 2.26/1.17 => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a] :
% 2.26/1.17 ( ( P2 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
% 2.26/1.17 => ( ( P2 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
% 2.26/1.17 => ( P2 @ ( t_a @ V3 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) )
% 2.26/1.17 => ( ! [V3: a,Vd2: a,Ve2: tree_a,Vf2: tree_a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 2.26/1.17 ( ( P2 @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) )
% 2.26/1.17 => ( ( P2 @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) )
% 2.26/1.17 => ( P2 @ ( t_a @ V3 @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) )
% 2.26/1.17 => ( ( P2 @ e_a )
% 2.26/1.17 => ( P2 @ A0 ) ) ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_98_removeLeaf_Ocases,axiom,
% 2.26/1.17 ! [X: tree_a] :
% 2.26/1.17 ( ! [V3: a] :
% 2.26/1.17 ( X
% 2.26/1.17 != ( t_a @ V3 @ e_a @ e_a ) )
% 2.26/1.17 => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 2.26/1.17 ( X
% 2.26/1.17 != ( t_a @ V3 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
% 2.26/1.17 => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 2.26/1.17 ( X
% 2.26/1.17 != ( t_a @ V3 @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 2.26/1.17 => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a] :
% 2.26/1.17 ( X
% 2.26/1.17 != ( t_a @ V3 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
% 2.26/1.17 => ( ! [V3: a,Vd2: a,Ve2: tree_a,Vf2: tree_a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 2.26/1.17 ( X
% 2.26/1.17 != ( t_a @ V3 @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 2.26/1.17 => ( X = e_a ) ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_99_siftDown_Ocases,axiom,
% 2.26/1.17 ! [X: tree_a] :
% 2.26/1.17 ( ( X != e_a )
% 2.26/1.17 => ( ! [V3: a] :
% 2.26/1.17 ( X
% 2.26/1.17 != ( t_a @ V3 @ e_a @ e_a ) )
% 2.26/1.17 => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 2.26/1.17 ( X
% 2.26/1.17 != ( t_a @ V3 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
% 2.26/1.17 => ( ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 2.26/1.17 ( X
% 2.26/1.17 != ( t_a @ V3 @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 2.26/1.17 => ~ ! [V3: a,Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a] :
% 2.26/1.17 ( X
% 2.26/1.17 != ( t_a @ V3 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_100_removeLeaf_Osimps_I2_J,axiom,
% 2.26/1.17 ! [V: a,Va: a,Vb: tree_a,Vc: tree_a] :
% 2.26/1.17 ( ( heapIm837449470Leaf_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) )
% 2.26/1.17 = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ Va @ Vb @ Vc ) ) ) @ ( t_a @ V @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ Va @ Vb @ Vc ) ) ) @ e_a ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_101_removeLeaf_Osimps_I3_J,axiom,
% 2.26/1.17 ! [V: a,Va: a,Vb: tree_a,Vc: tree_a] :
% 2.26/1.17 ( ( heapIm837449470Leaf_a @ ( t_a @ V @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) )
% 2.26/1.17 = ( produc686083979Tree_a @ ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ Va @ Vb @ Vc ) ) ) @ ( t_a @ V @ e_a @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_102_union__mset__add__mset__left,axiom,
% 2.26/1.17 ! [A: a,A4: multiset_a,B4: multiset_a] :
% 2.26/1.17 ( ( plus_plus_multiset_a @ ( add_mset_a @ A @ A4 ) @ B4 )
% 2.26/1.17 = ( add_mset_a @ A @ ( plus_plus_multiset_a @ A4 @ B4 ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_103_union__mset__add__mset__right,axiom,
% 2.26/1.17 ! [A4: multiset_a,A: a,B4: multiset_a] :
% 2.26/1.17 ( ( plus_plus_multiset_a @ A4 @ ( add_mset_a @ A @ B4 ) )
% 2.26/1.17 = ( add_mset_a @ A @ ( plus_plus_multiset_a @ A4 @ B4 ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_104_single__eq__single,axiom,
% 2.26/1.17 ! [A: a,B: a] :
% 2.26/1.17 ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
% 2.26/1.17 = ( add_mset_a @ B @ zero_zero_multiset_a ) )
% 2.26/1.17 = ( A = B ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_105_add__mset__eq__single,axiom,
% 2.26/1.17 ! [B: a,M: multiset_a,A: a] :
% 2.26/1.17 ( ( ( add_mset_a @ B @ M )
% 2.26/1.17 = ( add_mset_a @ A @ zero_zero_multiset_a ) )
% 2.26/1.17 = ( ( B = A )
% 2.26/1.17 & ( M = zero_zero_multiset_a ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_106_single__eq__add__mset,axiom,
% 2.26/1.17 ! [A: a,B: a,M: multiset_a] :
% 2.26/1.17 ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
% 2.26/1.17 = ( add_mset_a @ B @ M ) )
% 2.26/1.17 = ( ( B = A )
% 2.26/1.17 & ( M = zero_zero_multiset_a ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_107_add__mset__eq__singleton__iff,axiom,
% 2.26/1.17 ! [X: a,M: multiset_a,Y: a] :
% 2.26/1.17 ( ( ( add_mset_a @ X @ M )
% 2.26/1.17 = ( add_mset_a @ Y @ zero_zero_multiset_a ) )
% 2.26/1.17 = ( ( M = zero_zero_multiset_a )
% 2.26/1.17 & ( X = Y ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_108_empty__eq__union,axiom,
% 2.26/1.17 ! [M: multiset_a,N: multiset_a] :
% 2.26/1.17 ( ( zero_zero_multiset_a
% 2.26/1.17 = ( plus_plus_multiset_a @ M @ N ) )
% 2.26/1.17 = ( ( M = zero_zero_multiset_a )
% 2.26/1.17 & ( N = zero_zero_multiset_a ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_109_union__eq__empty,axiom,
% 2.26/1.17 ! [M: multiset_a,N: multiset_a] :
% 2.26/1.17 ( ( ( plus_plus_multiset_a @ M @ N )
% 2.26/1.17 = zero_zero_multiset_a )
% 2.26/1.17 = ( ( M = zero_zero_multiset_a )
% 2.26/1.17 & ( N = zero_zero_multiset_a ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_110_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
% 2.26/1.17 ! [X: multiset_a,Y: multiset_a] :
% 2.26/1.17 ( ( ( plus_plus_multiset_a @ X @ Y )
% 2.26/1.17 = zero_zero_multiset_a )
% 2.26/1.17 = ( ( X = zero_zero_multiset_a )
% 2.26/1.17 & ( Y = zero_zero_multiset_a ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_111_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
% 2.26/1.17 ! [X: multiset_a,Y: multiset_a] :
% 2.26/1.17 ( ( zero_zero_multiset_a
% 2.26/1.17 = ( plus_plus_multiset_a @ X @ Y ) )
% 2.26/1.17 = ( ( X = zero_zero_multiset_a )
% 2.26/1.17 & ( Y = zero_zero_multiset_a ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_112_add_Oleft__neutral,axiom,
% 2.26/1.17 ! [A: multiset_a] :
% 2.26/1.17 ( ( plus_plus_multiset_a @ zero_zero_multiset_a @ A )
% 2.26/1.17 = A ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_113_add__right__cancel,axiom,
% 2.26/1.17 ! [B: multiset_a,A: multiset_a,C: multiset_a] :
% 2.26/1.17 ( ( ( plus_plus_multiset_a @ B @ A )
% 2.26/1.17 = ( plus_plus_multiset_a @ C @ A ) )
% 2.26/1.17 = ( B = C ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_114_add__left__cancel,axiom,
% 2.26/1.17 ! [A: multiset_a,B: multiset_a,C: multiset_a] :
% 2.26/1.17 ( ( ( plus_plus_multiset_a @ A @ B )
% 2.26/1.17 = ( plus_plus_multiset_a @ A @ C ) )
% 2.26/1.17 = ( B = C ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_115_multi__self__add__other__not__self,axiom,
% 2.26/1.17 ! [M: multiset_a,X: a] :
% 2.26/1.17 ( M
% 2.26/1.17 != ( add_mset_a @ X @ M ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_116_add__mset__add__mset__same__iff,axiom,
% 2.26/1.17 ! [A: a,A4: multiset_a,B4: multiset_a] :
% 2.26/1.17 ( ( ( add_mset_a @ A @ A4 )
% 2.26/1.17 = ( add_mset_a @ A @ B4 ) )
% 2.26/1.17 = ( A4 = B4 ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_117_add__cancel__right__right,axiom,
% 2.26/1.17 ! [A: multiset_a,B: multiset_a] :
% 2.26/1.17 ( ( A
% 2.26/1.17 = ( plus_plus_multiset_a @ A @ B ) )
% 2.26/1.17 = ( B = zero_zero_multiset_a ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_118_add__cancel__right__left,axiom,
% 2.26/1.17 ! [A: multiset_a,B: multiset_a] :
% 2.26/1.17 ( ( A
% 2.26/1.17 = ( plus_plus_multiset_a @ B @ A ) )
% 2.26/1.17 = ( B = zero_zero_multiset_a ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_119_add__cancel__left__right,axiom,
% 2.26/1.17 ! [A: multiset_a,B: multiset_a] :
% 2.26/1.17 ( ( ( plus_plus_multiset_a @ A @ B )
% 2.26/1.17 = A )
% 2.26/1.17 = ( B = zero_zero_multiset_a ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_120_add__cancel__left__left,axiom,
% 2.26/1.17 ! [B: multiset_a,A: multiset_a] :
% 2.26/1.17 ( ( ( plus_plus_multiset_a @ B @ A )
% 2.26/1.17 = A )
% 2.26/1.17 = ( B = zero_zero_multiset_a ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_121_add_Oright__neutral,axiom,
% 2.26/1.17 ! [A: multiset_a] :
% 2.26/1.17 ( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
% 2.26/1.17 = A ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_122_zero__reorient,axiom,
% 2.26/1.17 ! [X: multiset_a] :
% 2.26/1.17 ( ( zero_zero_multiset_a = X )
% 2.26/1.17 = ( X = zero_zero_multiset_a ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_123_add__right__imp__eq,axiom,
% 2.26/1.17 ! [B: multiset_a,A: multiset_a,C: multiset_a] :
% 2.26/1.17 ( ( ( plus_plus_multiset_a @ B @ A )
% 2.26/1.17 = ( plus_plus_multiset_a @ C @ A ) )
% 2.26/1.17 => ( B = C ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_124_add__left__imp__eq,axiom,
% 2.26/1.17 ! [A: multiset_a,B: multiset_a,C: multiset_a] :
% 2.26/1.17 ( ( ( plus_plus_multiset_a @ A @ B )
% 2.26/1.17 = ( plus_plus_multiset_a @ A @ C ) )
% 2.26/1.17 => ( B = C ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_125_add_Oleft__commute,axiom,
% 2.26/1.17 ! [B: multiset_a,A: multiset_a,C: multiset_a] :
% 2.26/1.17 ( ( plus_plus_multiset_a @ B @ ( plus_plus_multiset_a @ A @ C ) )
% 2.26/1.17 = ( plus_plus_multiset_a @ A @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_126_add_Ocommute,axiom,
% 2.26/1.17 ( plus_plus_multiset_a
% 2.26/1.17 = ( ^ [A5: multiset_a,B5: multiset_a] : ( plus_plus_multiset_a @ B5 @ A5 ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_127_add_Oassoc,axiom,
% 2.26/1.17 ! [A: multiset_a,B: multiset_a,C: multiset_a] :
% 2.26/1.17 ( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ A @ B ) @ C )
% 2.26/1.17 = ( plus_plus_multiset_a @ A @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_128_group__cancel_Oadd2,axiom,
% 2.26/1.17 ! [B4: multiset_a,K: multiset_a,B: multiset_a,A: multiset_a] :
% 2.26/1.17 ( ( B4
% 2.26/1.17 = ( plus_plus_multiset_a @ K @ B ) )
% 2.26/1.17 => ( ( plus_plus_multiset_a @ A @ B4 )
% 2.26/1.17 = ( plus_plus_multiset_a @ K @ ( plus_plus_multiset_a @ A @ B ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_129_group__cancel_Oadd1,axiom,
% 2.26/1.17 ! [A4: multiset_a,K: multiset_a,A: multiset_a,B: multiset_a] :
% 2.26/1.17 ( ( A4
% 2.26/1.17 = ( plus_plus_multiset_a @ K @ A ) )
% 2.26/1.17 => ( ( plus_plus_multiset_a @ A4 @ B )
% 2.26/1.17 = ( plus_plus_multiset_a @ K @ ( plus_plus_multiset_a @ A @ B ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_130_add__mono__thms__linordered__semiring_I4_J,axiom,
% 2.26/1.17 ! [I: multiset_a,J: multiset_a,K: multiset_a,L: multiset_a] :
% 2.26/1.17 ( ( ( I = J )
% 2.26/1.17 & ( K = L ) )
% 2.26/1.17 => ( ( plus_plus_multiset_a @ I @ K )
% 2.26/1.17 = ( plus_plus_multiset_a @ J @ L ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_131_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 2.26/1.17 ! [A: multiset_a,B: multiset_a,C: multiset_a] :
% 2.26/1.17 ( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ A @ B ) @ C )
% 2.26/1.17 = ( plus_plus_multiset_a @ A @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_132_multi__union__self__other__eq,axiom,
% 2.26/1.17 ! [A4: multiset_a,X5: multiset_a,Y5: multiset_a] :
% 2.26/1.17 ( ( ( plus_plus_multiset_a @ A4 @ X5 )
% 2.26/1.17 = ( plus_plus_multiset_a @ A4 @ Y5 ) )
% 2.26/1.17 => ( X5 = Y5 ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_133_union__right__cancel,axiom,
% 2.26/1.17 ! [M: multiset_a,K2: multiset_a,N: multiset_a] :
% 2.26/1.17 ( ( ( plus_plus_multiset_a @ M @ K2 )
% 2.26/1.17 = ( plus_plus_multiset_a @ N @ K2 ) )
% 2.26/1.17 = ( M = N ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_134_union__left__cancel,axiom,
% 2.26/1.17 ! [K2: multiset_a,M: multiset_a,N: multiset_a] :
% 2.26/1.17 ( ( ( plus_plus_multiset_a @ K2 @ M )
% 2.26/1.17 = ( plus_plus_multiset_a @ K2 @ N ) )
% 2.26/1.17 = ( M = N ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_135_union__commute,axiom,
% 2.26/1.17 ( plus_plus_multiset_a
% 2.26/1.17 = ( ^ [M2: multiset_a,N2: multiset_a] : ( plus_plus_multiset_a @ N2 @ M2 ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_136_union__lcomm,axiom,
% 2.26/1.17 ! [M: multiset_a,N: multiset_a,K2: multiset_a] :
% 2.26/1.17 ( ( plus_plus_multiset_a @ M @ ( plus_plus_multiset_a @ N @ K2 ) )
% 2.26/1.17 = ( plus_plus_multiset_a @ N @ ( plus_plus_multiset_a @ M @ K2 ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_137_union__assoc,axiom,
% 2.26/1.17 ! [M: multiset_a,N: multiset_a,K2: multiset_a] :
% 2.26/1.17 ( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ M @ N ) @ K2 )
% 2.26/1.17 = ( plus_plus_multiset_a @ M @ ( plus_plus_multiset_a @ N @ K2 ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_138_add__mset__commute,axiom,
% 2.26/1.17 ! [X: a,Y: a,M: multiset_a] :
% 2.26/1.17 ( ( add_mset_a @ X @ ( add_mset_a @ Y @ M ) )
% 2.26/1.17 = ( add_mset_a @ Y @ ( add_mset_a @ X @ M ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_139_add__eq__conv__ex,axiom,
% 2.26/1.17 ! [A: a,M: multiset_a,B: a,N: multiset_a] :
% 2.26/1.17 ( ( ( add_mset_a @ A @ M )
% 2.26/1.17 = ( add_mset_a @ B @ N ) )
% 2.26/1.17 = ( ( ( M = N )
% 2.26/1.17 & ( A = B ) )
% 2.26/1.17 | ? [K3: multiset_a] :
% 2.26/1.17 ( ( M
% 2.26/1.17 = ( add_mset_a @ B @ K3 ) )
% 2.26/1.17 & ( N
% 2.26/1.17 = ( add_mset_a @ A @ K3 ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_140_add_Ocomm__neutral,axiom,
% 2.26/1.17 ! [A: multiset_a] :
% 2.26/1.17 ( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
% 2.26/1.17 = A ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_141_comm__monoid__add__class_Oadd__0,axiom,
% 2.26/1.17 ! [A: multiset_a] :
% 2.26/1.17 ( ( plus_plus_multiset_a @ zero_zero_multiset_a @ A )
% 2.26/1.17 = A ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_142_empty__neutral_I1_J,axiom,
% 2.26/1.17 ! [X: multiset_a] :
% 2.26/1.17 ( ( plus_plus_multiset_a @ zero_zero_multiset_a @ X )
% 2.26/1.17 = X ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_143_empty__neutral_I2_J,axiom,
% 2.26/1.17 ! [X: multiset_a] :
% 2.26/1.17 ( ( plus_plus_multiset_a @ X @ zero_zero_multiset_a )
% 2.26/1.17 = X ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_144_multi__nonempty__split,axiom,
% 2.26/1.17 ! [M: multiset_a] :
% 2.26/1.17 ( ( M != zero_zero_multiset_a )
% 2.26/1.17 => ? [A6: multiset_a,A3: a] :
% 2.26/1.17 ( M
% 2.26/1.17 = ( add_mset_a @ A3 @ A6 ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_145_empty__not__add__mset,axiom,
% 2.26/1.17 ! [A: a,A4: multiset_a] :
% 2.26/1.17 ( zero_zero_multiset_a
% 2.26/1.17 != ( add_mset_a @ A @ A4 ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_146_multiset__induct2,axiom,
% 2.26/1.17 ! [P2: multiset_a > multiset_a > $o,M: multiset_a,N: multiset_a] :
% 2.26/1.17 ( ( P2 @ zero_zero_multiset_a @ zero_zero_multiset_a )
% 2.26/1.17 => ( ! [A3: a,M3: multiset_a,N3: multiset_a] :
% 2.26/1.17 ( ( P2 @ M3 @ N3 )
% 2.26/1.17 => ( P2 @ ( add_mset_a @ A3 @ M3 ) @ N3 ) )
% 2.26/1.17 => ( ! [A3: a,M3: multiset_a,N3: multiset_a] :
% 2.26/1.17 ( ( P2 @ M3 @ N3 )
% 2.26/1.17 => ( P2 @ M3 @ ( add_mset_a @ A3 @ N3 ) ) )
% 2.26/1.17 => ( P2 @ M @ N ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_147_multiset__induct,axiom,
% 2.26/1.17 ! [P2: multiset_a > $o,M: multiset_a] :
% 2.26/1.17 ( ( P2 @ zero_zero_multiset_a )
% 2.26/1.17 => ( ! [X3: a,M3: multiset_a] :
% 2.26/1.17 ( ( P2 @ M3 )
% 2.26/1.17 => ( P2 @ ( add_mset_a @ X3 @ M3 ) ) )
% 2.26/1.17 => ( P2 @ M ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_148_multiset__cases,axiom,
% 2.26/1.17 ! [M: multiset_a] :
% 2.26/1.17 ( ( M != zero_zero_multiset_a )
% 2.26/1.17 => ~ ! [X3: a,N3: multiset_a] :
% 2.26/1.17 ( M
% 2.26/1.17 != ( add_mset_a @ X3 @ N3 ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_149_add__mset__add__single,axiom,
% 2.26/1.17 ( add_mset_a
% 2.26/1.17 = ( ^ [A5: a,A7: multiset_a] : ( plus_plus_multiset_a @ A7 @ ( add_mset_a @ A5 @ zero_zero_multiset_a ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_150_union__is__single,axiom,
% 2.26/1.17 ! [M: multiset_a,N: multiset_a,A: a] :
% 2.26/1.17 ( ( ( plus_plus_multiset_a @ M @ N )
% 2.26/1.17 = ( add_mset_a @ A @ zero_zero_multiset_a ) )
% 2.26/1.17 = ( ( ( M
% 2.26/1.17 = ( add_mset_a @ A @ zero_zero_multiset_a ) )
% 2.26/1.17 & ( N = zero_zero_multiset_a ) )
% 2.26/1.17 | ( ( M = zero_zero_multiset_a )
% 2.26/1.17 & ( N
% 2.26/1.17 = ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_151_single__is__union,axiom,
% 2.26/1.17 ! [A: a,M: multiset_a,N: multiset_a] :
% 2.26/1.17 ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
% 2.26/1.17 = ( plus_plus_multiset_a @ M @ N ) )
% 2.26/1.17 = ( ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
% 2.26/1.17 = M )
% 2.26/1.17 & ( N = zero_zero_multiset_a ) )
% 2.26/1.17 | ( ( M = zero_zero_multiset_a )
% 2.26/1.17 & ( ( add_mset_a @ A @ zero_zero_multiset_a )
% 2.26/1.17 = N ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_152_exI__realizer,axiom,
% 2.26/1.17 ! [P2: multiset_a > multiset_a > $o,Y: multiset_a,X: multiset_a] :
% 2.26/1.17 ( ( P2 @ Y @ X )
% 2.26/1.17 => ( P2 @ ( produc148952133iset_a @ ( produc2037245207iset_a @ X @ Y ) ) @ ( produc653373187iset_a @ ( produc2037245207iset_a @ X @ Y ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_153_exI__realizer,axiom,
% 2.26/1.17 ! [P2: a > a > $o,Y: a,X: a] :
% 2.26/1.17 ( ( P2 @ Y @ X )
% 2.26/1.17 => ( P2 @ ( product_snd_a_a @ ( product_Pair_a_a @ X @ Y ) ) @ ( product_fst_a_a @ ( product_Pair_a_a @ X @ Y ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_154_exI__realizer,axiom,
% 2.26/1.17 ! [P2: a > tree_a > $o,Y: a,X: tree_a] :
% 2.26/1.17 ( ( P2 @ Y @ X )
% 2.26/1.17 => ( P2 @ ( product_snd_Tree_a_a @ ( produc649172771ee_a_a @ X @ Y ) ) @ ( product_fst_Tree_a_a @ ( produc649172771ee_a_a @ X @ Y ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_155_exI__realizer,axiom,
% 2.26/1.17 ! [P2: tree_a > a > $o,Y: tree_a,X: a] :
% 2.26/1.17 ( ( P2 @ Y @ X )
% 2.26/1.17 => ( P2 @ ( product_snd_a_Tree_a @ ( produc686083979Tree_a @ X @ Y ) ) @ ( product_fst_a_Tree_a @ ( produc686083979Tree_a @ X @ Y ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_156_conjI__realizer,axiom,
% 2.26/1.17 ! [P2: multiset_a > $o,P: multiset_a,Q2: multiset_a > $o,Q: multiset_a] :
% 2.26/1.17 ( ( P2 @ P )
% 2.26/1.17 => ( ( Q2 @ Q )
% 2.26/1.17 => ( ( P2 @ ( produc653373187iset_a @ ( produc2037245207iset_a @ P @ Q ) ) )
% 2.26/1.17 & ( Q2 @ ( produc148952133iset_a @ ( produc2037245207iset_a @ P @ Q ) ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_157_conjI__realizer,axiom,
% 2.26/1.17 ! [P2: a > $o,P: a,Q2: a > $o,Q: a] :
% 2.26/1.17 ( ( P2 @ P )
% 2.26/1.17 => ( ( Q2 @ Q )
% 2.26/1.17 => ( ( P2 @ ( product_fst_a_a @ ( product_Pair_a_a @ P @ Q ) ) )
% 2.26/1.17 & ( Q2 @ ( product_snd_a_a @ ( product_Pair_a_a @ P @ Q ) ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_158_conjI__realizer,axiom,
% 2.26/1.17 ! [P2: tree_a > $o,P: tree_a,Q2: a > $o,Q: a] :
% 2.26/1.17 ( ( P2 @ P )
% 2.26/1.17 => ( ( Q2 @ Q )
% 2.26/1.17 => ( ( P2 @ ( product_fst_Tree_a_a @ ( produc649172771ee_a_a @ P @ Q ) ) )
% 2.26/1.17 & ( Q2 @ ( product_snd_Tree_a_a @ ( produc649172771ee_a_a @ P @ Q ) ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_159_conjI__realizer,axiom,
% 2.26/1.17 ! [P2: a > $o,P: a,Q2: tree_a > $o,Q: tree_a] :
% 2.26/1.17 ( ( P2 @ P )
% 2.26/1.17 => ( ( Q2 @ Q )
% 2.26/1.17 => ( ( P2 @ ( product_fst_a_Tree_a @ ( produc686083979Tree_a @ P @ Q ) ) )
% 2.26/1.17 & ( Q2 @ ( product_snd_a_Tree_a @ ( produc686083979Tree_a @ P @ Q ) ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_160_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 2.26/1.17 ! [P2: multiset_a > multiset_a > $o,X: multiset_a,Y: multiset_a,A: produc1127127335iset_a] :
% 2.26/1.17 ( ( P2 @ X @ Y )
% 2.26/1.17 => ( ( A
% 2.26/1.17 = ( produc2037245207iset_a @ X @ Y ) )
% 2.26/1.17 => ( P2 @ ( produc653373187iset_a @ A ) @ ( produc148952133iset_a @ A ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_161_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 2.26/1.17 ! [P2: a > a > $o,X: a,Y: a,A: product_prod_a_a] :
% 2.26/1.17 ( ( P2 @ X @ Y )
% 2.26/1.17 => ( ( A
% 2.26/1.17 = ( product_Pair_a_a @ X @ Y ) )
% 2.26/1.17 => ( P2 @ ( product_fst_a_a @ A ) @ ( product_snd_a_a @ A ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_162_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 2.26/1.17 ! [P2: tree_a > a > $o,X: tree_a,Y: a,A: produc981471411ee_a_a] :
% 2.26/1.17 ( ( P2 @ X @ Y )
% 2.26/1.17 => ( ( A
% 2.26/1.17 = ( produc649172771ee_a_a @ X @ Y ) )
% 2.26/1.17 => ( P2 @ ( product_fst_Tree_a_a @ A ) @ ( product_snd_Tree_a_a @ A ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_163_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 2.26/1.17 ! [P2: a > tree_a > $o,X: a,Y: tree_a,A: produc143150363Tree_a] :
% 2.26/1.17 ( ( P2 @ X @ Y )
% 2.26/1.17 => ( ( A
% 2.26/1.17 = ( produc686083979Tree_a @ X @ Y ) )
% 2.26/1.17 => ( P2 @ ( product_fst_a_Tree_a @ A ) @ ( product_snd_a_Tree_a @ A ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_164_Multiset_Ois__empty__def,axiom,
% 2.26/1.17 ( is_empty_a
% 2.26/1.17 = ( ^ [A7: multiset_a] : ( A7 = zero_zero_multiset_a ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_165_exE__realizer_H,axiom,
% 2.26/1.17 ! [P2: a > tree_a > $o,P: produc981471411ee_a_a] :
% 2.26/1.17 ( ( P2 @ ( product_snd_Tree_a_a @ P ) @ ( product_fst_Tree_a_a @ P ) )
% 2.26/1.17 => ~ ! [X3: tree_a,Y3: a] :
% 2.26/1.17 ~ ( P2 @ Y3 @ X3 ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_166_exE__realizer_H,axiom,
% 2.26/1.17 ! [P2: tree_a > a > $o,P: produc143150363Tree_a] :
% 2.26/1.17 ( ( P2 @ ( product_snd_a_Tree_a @ P ) @ ( product_fst_a_Tree_a @ P ) )
% 2.26/1.17 => ~ ! [X3: a,Y3: tree_a] :
% 2.26/1.17 ~ ( P2 @ Y3 @ X3 ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_167_eq__snd__iff,axiom,
% 2.26/1.17 ! [B: multiset_a,P: produc1127127335iset_a] :
% 2.26/1.17 ( ( B
% 2.26/1.17 = ( produc148952133iset_a @ P ) )
% 2.26/1.17 = ( ? [A5: multiset_a] :
% 2.26/1.17 ( P
% 2.26/1.17 = ( produc2037245207iset_a @ A5 @ B ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_168_eq__snd__iff,axiom,
% 2.26/1.17 ! [B: a,P: product_prod_a_a] :
% 2.26/1.17 ( ( B
% 2.26/1.17 = ( product_snd_a_a @ P ) )
% 2.26/1.17 = ( ? [A5: a] :
% 2.26/1.17 ( P
% 2.26/1.17 = ( product_Pair_a_a @ A5 @ B ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_169_eq__snd__iff,axiom,
% 2.26/1.17 ! [B: a,P: produc981471411ee_a_a] :
% 2.26/1.17 ( ( B
% 2.26/1.17 = ( product_snd_Tree_a_a @ P ) )
% 2.26/1.17 = ( ? [A5: tree_a] :
% 2.26/1.17 ( P
% 2.26/1.17 = ( produc649172771ee_a_a @ A5 @ B ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_170_eq__snd__iff,axiom,
% 2.26/1.17 ! [B: tree_a,P: produc143150363Tree_a] :
% 2.26/1.17 ( ( B
% 2.26/1.17 = ( product_snd_a_Tree_a @ P ) )
% 2.26/1.17 = ( ? [A5: a] :
% 2.26/1.17 ( P
% 2.26/1.17 = ( produc686083979Tree_a @ A5 @ B ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_171_eq__fst__iff,axiom,
% 2.26/1.17 ! [A: multiset_a,P: produc1127127335iset_a] :
% 2.26/1.17 ( ( A
% 2.26/1.17 = ( produc653373187iset_a @ P ) )
% 2.26/1.17 = ( ? [B5: multiset_a] :
% 2.26/1.17 ( P
% 2.26/1.17 = ( produc2037245207iset_a @ A @ B5 ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_172_eq__fst__iff,axiom,
% 2.26/1.17 ! [A: a,P: product_prod_a_a] :
% 2.26/1.17 ( ( A
% 2.26/1.17 = ( product_fst_a_a @ P ) )
% 2.26/1.17 = ( ? [B5: a] :
% 2.26/1.17 ( P
% 2.26/1.17 = ( product_Pair_a_a @ A @ B5 ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_173_eq__fst__iff,axiom,
% 2.26/1.17 ! [A: tree_a,P: produc981471411ee_a_a] :
% 2.26/1.17 ( ( A
% 2.26/1.17 = ( product_fst_Tree_a_a @ P ) )
% 2.26/1.17 = ( ? [B5: a] :
% 2.26/1.17 ( P
% 2.26/1.17 = ( produc649172771ee_a_a @ A @ B5 ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_174_eq__fst__iff,axiom,
% 2.26/1.17 ! [A: a,P: produc143150363Tree_a] :
% 2.26/1.17 ( ( A
% 2.26/1.17 = ( product_fst_a_Tree_a @ P ) )
% 2.26/1.17 = ( ? [B5: tree_a] :
% 2.26/1.17 ( P
% 2.26/1.17 = ( produc686083979Tree_a @ A @ B5 ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_175_verit__sum__simplify,axiom,
% 2.26/1.17 ! [A: multiset_a] :
% 2.26/1.17 ( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
% 2.26/1.17 = A ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_176_HS_Ois__empty__as__list,axiom,
% 2.26/1.17 ! [E: tree_a] :
% 2.26/1.17 ( ( heapIm229596386mpty_a @ E )
% 2.26/1.17 => ( ( multiset_a2 @ E )
% 2.26/1.17 = zero_zero_multiset_a ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_177_add__mset__replicate__mset__safe,axiom,
% 2.26/1.17 ! [M: multiset_a,A: a] :
% 2.26/1.17 ( ( nO_MAT1617603563iset_a @ zero_zero_multiset_a @ M )
% 2.26/1.17 => ( ( add_mset_a @ A @ M )
% 2.26/1.17 = ( plus_plus_multiset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) @ M ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_178_HS_Ois__empty__empty,axiom,
% 2.26/1.17 heapIm229596386mpty_a @ e_a ).
% 2.26/1.17
% 2.26/1.17 thf(fact_179_hs__is__empty__def,axiom,
% 2.26/1.17 ( heapIm229596386mpty_a
% 2.26/1.17 = ( ^ [T3: tree_a] : ( T3 = e_a ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_180_HS_Ois__empty__inj,axiom,
% 2.26/1.17 ! [E: tree_a] :
% 2.26/1.17 ( ( heapIm229596386mpty_a @ E )
% 2.26/1.17 => ( E = e_a ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_181_prod_Oswap__def,axiom,
% 2.26/1.17 ( produc506430135iset_a
% 2.26/1.17 = ( ^ [P3: produc1127127335iset_a] : ( produc2037245207iset_a @ ( produc148952133iset_a @ P3 ) @ ( produc653373187iset_a @ P3 ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_182_prod_Oswap__def,axiom,
% 2.26/1.17 ( product_swap_a_a
% 2.26/1.17 = ( ^ [P3: product_prod_a_a] : ( product_Pair_a_a @ ( product_snd_a_a @ P3 ) @ ( product_fst_a_a @ P3 ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_183_prod_Oswap__def,axiom,
% 2.26/1.17 ( produc1478296771ee_a_a
% 2.26/1.17 = ( ^ [P3: produc981471411ee_a_a] : ( produc686083979Tree_a @ ( product_snd_Tree_a_a @ P3 ) @ ( product_fst_Tree_a_a @ P3 ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_184_prod_Oswap__def,axiom,
% 2.26/1.17 ( produc1515207979Tree_a
% 2.26/1.17 = ( ^ [P3: produc143150363Tree_a] : ( produc649172771ee_a_a @ ( product_snd_a_Tree_a @ P3 ) @ ( product_fst_a_Tree_a @ P3 ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_185_removeLeaf__heap__is__heap,axiom,
% 2.26/1.17 ! [T2: tree_a] :
% 2.26/1.17 ( ( is_heap_a @ T2 )
% 2.26/1.17 => ( ( T2 != e_a )
% 2.26/1.17 => ( is_heap_a @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ T2 ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_186_NO__MATCH__cong,axiom,
% 2.26/1.17 nO_MAT1617603563iset_a = nO_MAT1617603563iset_a ).
% 2.26/1.17
% 2.26/1.17 thf(fact_187_NO__MATCH__def,axiom,
% 2.26/1.17 ( nO_MAT1617603563iset_a
% 2.26/1.17 = ( ^ [Pat: multiset_a,Val: multiset_a] : $true ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_188_swap__swap,axiom,
% 2.26/1.17 ! [P: produc981471411ee_a_a] :
% 2.26/1.17 ( ( produc1515207979Tree_a @ ( produc1478296771ee_a_a @ P ) )
% 2.26/1.17 = P ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_189_swap__swap,axiom,
% 2.26/1.17 ! [P: produc143150363Tree_a] :
% 2.26/1.17 ( ( produc1478296771ee_a_a @ ( produc1515207979Tree_a @ P ) )
% 2.26/1.17 = P ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_190_swap__simp,axiom,
% 2.26/1.17 ! [X: multiset_a,Y: multiset_a] :
% 2.26/1.17 ( ( produc506430135iset_a @ ( produc2037245207iset_a @ X @ Y ) )
% 2.26/1.17 = ( produc2037245207iset_a @ Y @ X ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_191_swap__simp,axiom,
% 2.26/1.17 ! [X: a,Y: a] :
% 2.26/1.17 ( ( product_swap_a_a @ ( product_Pair_a_a @ X @ Y ) )
% 2.26/1.17 = ( product_Pair_a_a @ Y @ X ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_192_swap__simp,axiom,
% 2.26/1.17 ! [X: tree_a,Y: a] :
% 2.26/1.17 ( ( produc1478296771ee_a_a @ ( produc649172771ee_a_a @ X @ Y ) )
% 2.26/1.17 = ( produc686083979Tree_a @ Y @ X ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_193_swap__simp,axiom,
% 2.26/1.17 ! [X: a,Y: tree_a] :
% 2.26/1.17 ( ( produc1515207979Tree_a @ ( produc686083979Tree_a @ X @ Y ) )
% 2.26/1.17 = ( produc649172771ee_a_a @ Y @ X ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_194_snd__swap,axiom,
% 2.26/1.17 ! [X: produc143150363Tree_a] :
% 2.26/1.17 ( ( product_snd_Tree_a_a @ ( produc1515207979Tree_a @ X ) )
% 2.26/1.17 = ( product_fst_a_Tree_a @ X ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_195_snd__swap,axiom,
% 2.26/1.17 ! [X: produc981471411ee_a_a] :
% 2.26/1.17 ( ( product_snd_a_Tree_a @ ( produc1478296771ee_a_a @ X ) )
% 2.26/1.17 = ( product_fst_Tree_a_a @ X ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_196_fst__swap,axiom,
% 2.26/1.17 ! [X: produc981471411ee_a_a] :
% 2.26/1.17 ( ( product_fst_a_Tree_a @ ( produc1478296771ee_a_a @ X ) )
% 2.26/1.17 = ( product_snd_Tree_a_a @ X ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_197_fst__swap,axiom,
% 2.26/1.17 ! [X: produc143150363Tree_a] :
% 2.26/1.17 ( ( product_fst_Tree_a_a @ ( produc1515207979Tree_a @ X ) )
% 2.26/1.17 = ( product_snd_a_Tree_a @ X ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_198_is__heap_Osimps_I1_J,axiom,
% 2.26/1.17 is_heap_a @ e_a ).
% 2.26/1.17
% 2.26/1.17 thf(fact_199_is__heap_Osimps_I2_J,axiom,
% 2.26/1.17 ! [V: a] : ( is_heap_a @ ( t_a @ V @ e_a @ e_a ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_200_removeLeaf__val__val,axiom,
% 2.26/1.17 ! [T2: tree_a] :
% 2.26/1.17 ( ( ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ T2 ) )
% 2.26/1.17 != e_a )
% 2.26/1.17 => ( ( T2 != e_a )
% 2.26/1.17 => ( ( val_a @ T2 )
% 2.26/1.17 = ( val_a @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ T2 ) ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_201_Heap__axioms__def,axiom,
% 2.26/1.17 ( heap_axioms_a_a
% 2.26/1.17 = ( ^ [Is_empty: a > $o,Of_list: list_a > a,Multiset: a > multiset_a,As_tree: a > tree_a,Remove_max: a > product_prod_a_a] :
% 2.26/1.17 ( ! [L2: a] :
% 2.26/1.17 ( ( Multiset @ L2 )
% 2.26/1.17 = ( multiset_a2 @ ( As_tree @ L2 ) ) )
% 2.26/1.17 & ! [I2: list_a] : ( is_heap_a @ ( As_tree @ ( Of_list @ I2 ) ) )
% 2.26/1.17 & ! [T3: a] :
% 2.26/1.17 ( ( ( As_tree @ T3 )
% 2.26/1.17 = e_a )
% 2.26/1.17 = ( Is_empty @ T3 ) )
% 2.26/1.17 & ! [L2: a,M4: a,L3: a] :
% 2.26/1.17 ( ~ ( Is_empty @ L2 )
% 2.26/1.17 => ( ( ( product_Pair_a_a @ M4 @ L3 )
% 2.26/1.17 = ( Remove_max @ L2 ) )
% 2.26/1.17 => ( ( add_mset_a @ M4 @ ( Multiset @ L3 ) )
% 2.26/1.17 = ( Multiset @ L2 ) ) ) )
% 2.26/1.17 & ! [L2: a,M4: a,L3: a] :
% 2.26/1.17 ( ~ ( Is_empty @ L2 )
% 2.26/1.17 => ( ( is_heap_a @ ( As_tree @ L2 ) )
% 2.26/1.17 => ( ( ( product_Pair_a_a @ M4 @ L3 )
% 2.26/1.17 = ( Remove_max @ L2 ) )
% 2.26/1.17 => ( is_heap_a @ ( As_tree @ L3 ) ) ) ) )
% 2.26/1.17 & ! [T3: a,M4: a,T4: a] :
% 2.26/1.17 ( ~ ( Is_empty @ T3 )
% 2.26/1.17 => ( ( ( product_Pair_a_a @ M4 @ T4 )
% 2.26/1.17 = ( Remove_max @ T3 ) )
% 2.26/1.17 => ( M4
% 2.26/1.17 = ( val_a @ ( As_tree @ T3 ) ) ) ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_202_Heap__axioms__def,axiom,
% 2.26/1.17 ( heap_axioms_Tree_a_a
% 2.26/1.17 = ( ^ [Is_empty: tree_a > $o,Of_list: list_a > tree_a,Multiset: tree_a > multiset_a,As_tree: tree_a > tree_a,Remove_max: tree_a > produc143150363Tree_a] :
% 2.26/1.17 ( ! [L2: tree_a] :
% 2.26/1.17 ( ( Multiset @ L2 )
% 2.26/1.17 = ( multiset_a2 @ ( As_tree @ L2 ) ) )
% 2.26/1.17 & ! [I2: list_a] : ( is_heap_a @ ( As_tree @ ( Of_list @ I2 ) ) )
% 2.26/1.17 & ! [T3: tree_a] :
% 2.26/1.17 ( ( ( As_tree @ T3 )
% 2.26/1.17 = e_a )
% 2.26/1.17 = ( Is_empty @ T3 ) )
% 2.26/1.17 & ! [L2: tree_a,M4: a,L3: tree_a] :
% 2.26/1.17 ( ~ ( Is_empty @ L2 )
% 2.26/1.17 => ( ( ( produc686083979Tree_a @ M4 @ L3 )
% 2.26/1.17 = ( Remove_max @ L2 ) )
% 2.26/1.17 => ( ( add_mset_a @ M4 @ ( Multiset @ L3 ) )
% 2.26/1.17 = ( Multiset @ L2 ) ) ) )
% 2.26/1.17 & ! [L2: tree_a,M4: a,L3: tree_a] :
% 2.26/1.17 ( ~ ( Is_empty @ L2 )
% 2.26/1.17 => ( ( is_heap_a @ ( As_tree @ L2 ) )
% 2.26/1.17 => ( ( ( produc686083979Tree_a @ M4 @ L3 )
% 2.26/1.17 = ( Remove_max @ L2 ) )
% 2.26/1.17 => ( is_heap_a @ ( As_tree @ L3 ) ) ) ) )
% 2.26/1.17 & ! [T3: tree_a,M4: a,T4: tree_a] :
% 2.26/1.17 ( ~ ( Is_empty @ T3 )
% 2.26/1.17 => ( ( ( produc686083979Tree_a @ M4 @ T4 )
% 2.26/1.17 = ( Remove_max @ T3 ) )
% 2.26/1.17 => ( M4
% 2.26/1.17 = ( val_a @ ( As_tree @ T3 ) ) ) ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_203_Heap__axioms_Ointro,axiom,
% 2.26/1.17 ! [Multiset2: a > multiset_a,As_tree2: a > tree_a,Of_list2: list_a > a,Is_empty2: a > $o,Remove_max2: a > product_prod_a_a] :
% 2.26/1.17 ( ! [L4: a] :
% 2.26/1.17 ( ( Multiset2 @ L4 )
% 2.26/1.17 = ( multiset_a2 @ ( As_tree2 @ L4 ) ) )
% 2.26/1.17 => ( ! [I3: list_a] : ( is_heap_a @ ( As_tree2 @ ( Of_list2 @ I3 ) ) )
% 2.26/1.17 => ( ! [T5: a] :
% 2.26/1.17 ( ( ( As_tree2 @ T5 )
% 2.26/1.17 = e_a )
% 2.26/1.17 = ( Is_empty2 @ T5 ) )
% 2.26/1.17 => ( ! [L4: a,M5: a,L5: a] :
% 2.26/1.17 ( ~ ( Is_empty2 @ L4 )
% 2.26/1.17 => ( ( ( product_Pair_a_a @ M5 @ L5 )
% 2.26/1.17 = ( Remove_max2 @ L4 ) )
% 2.26/1.17 => ( ( add_mset_a @ M5 @ ( Multiset2 @ L5 ) )
% 2.26/1.17 = ( Multiset2 @ L4 ) ) ) )
% 2.26/1.17 => ( ! [L4: a,M5: a,L5: a] :
% 2.26/1.17 ( ~ ( Is_empty2 @ L4 )
% 2.26/1.17 => ( ( is_heap_a @ ( As_tree2 @ L4 ) )
% 2.26/1.17 => ( ( ( product_Pair_a_a @ M5 @ L5 )
% 2.26/1.17 = ( Remove_max2 @ L4 ) )
% 2.26/1.17 => ( is_heap_a @ ( As_tree2 @ L5 ) ) ) ) )
% 2.26/1.17 => ( ! [T5: a,M5: a,T6: a] :
% 2.26/1.17 ( ~ ( Is_empty2 @ T5 )
% 2.26/1.17 => ( ( ( product_Pair_a_a @ M5 @ T6 )
% 2.26/1.17 = ( Remove_max2 @ T5 ) )
% 2.26/1.17 => ( M5
% 2.26/1.17 = ( val_a @ ( As_tree2 @ T5 ) ) ) ) )
% 2.26/1.17 => ( heap_axioms_a_a @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 ) ) ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_204_Heap__axioms_Ointro,axiom,
% 2.26/1.17 ! [Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Of_list2: list_a > tree_a,Is_empty2: tree_a > $o,Remove_max2: tree_a > produc143150363Tree_a] :
% 2.26/1.17 ( ! [L4: tree_a] :
% 2.26/1.17 ( ( Multiset2 @ L4 )
% 2.26/1.17 = ( multiset_a2 @ ( As_tree2 @ L4 ) ) )
% 2.26/1.17 => ( ! [I3: list_a] : ( is_heap_a @ ( As_tree2 @ ( Of_list2 @ I3 ) ) )
% 2.26/1.17 => ( ! [T5: tree_a] :
% 2.26/1.17 ( ( ( As_tree2 @ T5 )
% 2.26/1.17 = e_a )
% 2.26/1.17 = ( Is_empty2 @ T5 ) )
% 2.26/1.17 => ( ! [L4: tree_a,M5: a,L5: tree_a] :
% 2.26/1.17 ( ~ ( Is_empty2 @ L4 )
% 2.26/1.17 => ( ( ( produc686083979Tree_a @ M5 @ L5 )
% 2.26/1.17 = ( Remove_max2 @ L4 ) )
% 2.26/1.17 => ( ( add_mset_a @ M5 @ ( Multiset2 @ L5 ) )
% 2.26/1.17 = ( Multiset2 @ L4 ) ) ) )
% 2.26/1.17 => ( ! [L4: tree_a,M5: a,L5: tree_a] :
% 2.26/1.17 ( ~ ( Is_empty2 @ L4 )
% 2.26/1.17 => ( ( is_heap_a @ ( As_tree2 @ L4 ) )
% 2.26/1.17 => ( ( ( produc686083979Tree_a @ M5 @ L5 )
% 2.26/1.17 = ( Remove_max2 @ L4 ) )
% 2.26/1.17 => ( is_heap_a @ ( As_tree2 @ L5 ) ) ) ) )
% 2.26/1.17 => ( ! [T5: tree_a,M5: a,T6: tree_a] :
% 2.26/1.17 ( ~ ( Is_empty2 @ T5 )
% 2.26/1.17 => ( ( ( produc686083979Tree_a @ M5 @ T6 )
% 2.26/1.17 = ( Remove_max2 @ T5 ) )
% 2.26/1.17 => ( M5
% 2.26/1.17 = ( val_a @ ( As_tree2 @ T5 ) ) ) ) )
% 2.26/1.17 => ( heap_axioms_Tree_a_a @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 ) ) ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_205_val_Osimps,axiom,
% 2.26/1.17 ! [V: a,Uu: tree_a,Uv: tree_a] :
% 2.26/1.17 ( ( val_a @ ( t_a @ V @ Uu @ Uv ) )
% 2.26/1.17 = V ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_206_is__heap_Osimps_I4_J,axiom,
% 2.26/1.17 ! [V: a,Va: a,Vb: tree_a,Vc: tree_a] :
% 2.26/1.17 ( ( is_heap_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) )
% 2.26/1.17 = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 2.26/1.17 & ( is_heap_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_207_is__heap_Osimps_I3_J,axiom,
% 2.26/1.17 ! [V: a,Va: a,Vb: tree_a,Vc: tree_a] :
% 2.26/1.17 ( ( is_heap_a @ ( t_a @ V @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) )
% 2.26/1.17 = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 2.26/1.17 & ( is_heap_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_208_Heap_Oremove__max__is__heap,axiom,
% 2.26/1.17 ! [Empty: a,Is_empty2: a > $o,Of_list2: list_a > a,Multiset2: a > multiset_a,As_tree2: a > tree_a,Remove_max2: a > product_prod_a_a,L: a,M6: a,L6: a] :
% 2.26/1.17 ( ( heap_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 2.26/1.17 => ( ~ ( Is_empty2 @ L )
% 2.26/1.17 => ( ( is_heap_a @ ( As_tree2 @ L ) )
% 2.26/1.17 => ( ( ( product_Pair_a_a @ M6 @ L6 )
% 2.26/1.17 = ( Remove_max2 @ L ) )
% 2.26/1.17 => ( is_heap_a @ ( As_tree2 @ L6 ) ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_209_Heap_Oremove__max__is__heap,axiom,
% 2.26/1.17 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a,L: tree_a,M6: a,L6: tree_a] :
% 2.26/1.17 ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 2.26/1.17 => ( ~ ( Is_empty2 @ L )
% 2.26/1.17 => ( ( is_heap_a @ ( As_tree2 @ L ) )
% 2.26/1.17 => ( ( ( produc686083979Tree_a @ M6 @ L6 )
% 2.26/1.17 = ( Remove_max2 @ L ) )
% 2.26/1.17 => ( is_heap_a @ ( As_tree2 @ L6 ) ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_210_verit__la__disequality,axiom,
% 2.26/1.17 ! [A: a,B: a] :
% 2.26/1.17 ( ( A = B )
% 2.26/1.17 | ~ ( ord_less_eq_a @ A @ B )
% 2.26/1.17 | ~ ( ord_less_eq_a @ B @ A ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_211_add__right__mono,axiom,
% 2.26/1.17 ! [A: multiset_a,B: multiset_a,C: multiset_a] :
% 2.26/1.17 ( ( ord_le1199012836iset_a @ A @ B )
% 2.26/1.17 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_212_add__left__mono,axiom,
% 2.26/1.17 ! [A: multiset_a,B: multiset_a,C: multiset_a] :
% 2.26/1.17 ( ( ord_le1199012836iset_a @ A @ B )
% 2.26/1.17 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ C @ A ) @ ( plus_plus_multiset_a @ C @ B ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_213_add__mono,axiom,
% 2.26/1.17 ! [A: multiset_a,B: multiset_a,C: multiset_a,D: multiset_a] :
% 2.26/1.17 ( ( ord_le1199012836iset_a @ A @ B )
% 2.26/1.17 => ( ( ord_le1199012836iset_a @ C @ D )
% 2.26/1.17 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ D ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_214_add__mono__thms__linordered__semiring_I1_J,axiom,
% 2.26/1.17 ! [I: multiset_a,J: multiset_a,K: multiset_a,L: multiset_a] :
% 2.26/1.17 ( ( ( ord_le1199012836iset_a @ I @ J )
% 2.26/1.17 & ( ord_le1199012836iset_a @ K @ L ) )
% 2.26/1.17 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I @ K ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_215_add__mono__thms__linordered__semiring_I2_J,axiom,
% 2.26/1.17 ! [I: multiset_a,J: multiset_a,K: multiset_a,L: multiset_a] :
% 2.26/1.17 ( ( ( I = J )
% 2.26/1.17 & ( ord_le1199012836iset_a @ K @ L ) )
% 2.26/1.17 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I @ K ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_216_add__mono__thms__linordered__semiring_I3_J,axiom,
% 2.26/1.17 ! [I: multiset_a,J: multiset_a,K: multiset_a,L: multiset_a] :
% 2.26/1.17 ( ( ( ord_le1199012836iset_a @ I @ J )
% 2.26/1.17 & ( K = L ) )
% 2.26/1.17 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I @ K ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_217_Heap_Oas__tree__empty,axiom,
% 2.26/1.17 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a,T2: tree_a] :
% 2.26/1.17 ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 2.26/1.17 => ( ( ( As_tree2 @ T2 )
% 2.26/1.17 = e_a )
% 2.26/1.17 = ( Is_empty2 @ T2 ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_218_Heap_Ois__heap__of__list,axiom,
% 2.26/1.17 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a,I: list_a] :
% 2.26/1.17 ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 2.26/1.17 => ( is_heap_a @ ( As_tree2 @ ( Of_list2 @ I ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_219_Heap_Omultiset,axiom,
% 2.26/1.17 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a,L: tree_a] :
% 2.26/1.17 ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 2.26/1.17 => ( ( Multiset2 @ L )
% 2.26/1.17 = ( multiset_a2 @ ( As_tree2 @ L ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_220_Heap_Oaxioms_I2_J,axiom,
% 2.26/1.17 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a] :
% 2.26/1.17 ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 2.26/1.17 => ( heap_axioms_Tree_a_a @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_221_is__heap_Osimps_I6_J,axiom,
% 2.26/1.17 ! [V: a,Vd: a,Ve: tree_a,Vf: tree_a,Va: a,Vb: tree_a,Vc: tree_a] :
% 2.26/1.17 ( ( is_heap_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
% 2.26/1.17 = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 2.26/1.17 & ( is_heap_a @ ( t_a @ Va @ Vb @ Vc ) )
% 2.26/1.17 & ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
% 2.26/1.17 & ( is_heap_a @ ( t_a @ Vd @ Ve @ Vf ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_222_is__heap_Osimps_I5_J,axiom,
% 2.26/1.17 ! [V: a,Va: a,Vb: tree_a,Vc: tree_a,Vd: a,Ve: tree_a,Vf: tree_a] :
% 2.26/1.17 ( ( is_heap_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 2.26/1.17 = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
% 2.26/1.17 & ( is_heap_a @ ( t_a @ Vd @ Ve @ Vf ) )
% 2.26/1.17 & ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 2.26/1.17 & ( is_heap_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_223_Heap_Oremove__max__multiset_H,axiom,
% 2.26/1.17 ! [Empty: a,Is_empty2: a > $o,Of_list2: list_a > a,Multiset2: a > multiset_a,As_tree2: a > tree_a,Remove_max2: a > product_prod_a_a,L: a,M6: a,L6: a] :
% 2.26/1.17 ( ( heap_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 2.26/1.17 => ( ~ ( Is_empty2 @ L )
% 2.26/1.17 => ( ( ( product_Pair_a_a @ M6 @ L6 )
% 2.26/1.17 = ( Remove_max2 @ L ) )
% 2.26/1.17 => ( ( add_mset_a @ M6 @ ( Multiset2 @ L6 ) )
% 2.26/1.17 = ( Multiset2 @ L ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_224_Heap_Oremove__max__multiset_H,axiom,
% 2.26/1.17 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a,L: tree_a,M6: a,L6: tree_a] :
% 2.26/1.17 ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 2.26/1.17 => ( ~ ( Is_empty2 @ L )
% 2.26/1.17 => ( ( ( produc686083979Tree_a @ M6 @ L6 )
% 2.26/1.17 = ( Remove_max2 @ L ) )
% 2.26/1.17 => ( ( add_mset_a @ M6 @ ( Multiset2 @ L6 ) )
% 2.26/1.17 = ( Multiset2 @ L ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_225_Heap_Oremove__max__val,axiom,
% 2.26/1.17 ! [Empty: a,Is_empty2: a > $o,Of_list2: list_a > a,Multiset2: a > multiset_a,As_tree2: a > tree_a,Remove_max2: a > product_prod_a_a,T2: a,M6: a,T: a] :
% 2.26/1.17 ( ( heap_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 2.26/1.17 => ( ~ ( Is_empty2 @ T2 )
% 2.26/1.17 => ( ( ( product_Pair_a_a @ M6 @ T )
% 2.26/1.17 = ( Remove_max2 @ T2 ) )
% 2.26/1.17 => ( M6
% 2.26/1.17 = ( val_a @ ( As_tree2 @ T2 ) ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_226_Heap_Oremove__max__val,axiom,
% 2.26/1.17 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a,T2: tree_a,M6: a,T: tree_a] :
% 2.26/1.17 ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 2.26/1.17 => ( ~ ( Is_empty2 @ T2 )
% 2.26/1.17 => ( ( ( produc686083979Tree_a @ M6 @ T )
% 2.26/1.17 = ( Remove_max2 @ T2 ) )
% 2.26/1.17 => ( M6
% 2.26/1.17 = ( val_a @ ( As_tree2 @ T2 ) ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_227_siftDown_Osimps_I4_J,axiom,
% 2.26/1.17 ! [Va: a,Vb: tree_a,Vc: tree_a,V: a] :
% 2.26/1.17 ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 2.26/1.17 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) )
% 2.26/1.17 = ( t_a @ V @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) ) )
% 2.26/1.17 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 2.26/1.17 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) )
% 2.26/1.17 = ( t_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ e_a @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_228_siftDown_Osimps_I3_J,axiom,
% 2.26/1.17 ! [Va: a,Vb: tree_a,Vc: tree_a,V: a] :
% 2.26/1.17 ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 2.26/1.17 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) )
% 2.26/1.17 = ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) ) )
% 2.26/1.17 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 2.26/1.17 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) )
% 2.26/1.17 = ( t_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) @ e_a ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_229_siftDown_Osimps_I5_J,axiom,
% 2.26/1.17 ! [Vd: a,Ve: tree_a,Vf: tree_a,Va: a,Vb: tree_a,Vc: tree_a,V: a] :
% 2.26/1.17 ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) )
% 2.26/1.17 => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 2.26/1.17 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 2.26/1.17 = ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) ) )
% 2.26/1.17 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 2.26/1.17 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 2.26/1.17 = ( t_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) @ ( t_a @ Vd @ Ve @ Vf ) ) ) ) ) )
% 2.26/1.17 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) )
% 2.26/1.17 => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
% 2.26/1.17 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 2.26/1.17 = ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) ) )
% 2.26/1.17 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
% 2.26/1.17 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 2.26/1.17 = ( t_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ ( t_a @ Va @ Vb @ Vc ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Vd @ Ve @ Vf ) ) ) ) ) ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_230_siftDown_Osimps_I1_J,axiom,
% 2.26/1.17 ( ( heapIm1091024090Down_a @ e_a )
% 2.26/1.17 = e_a ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_231_siftDown__multiset,axiom,
% 2.26/1.17 ! [T2: tree_a] :
% 2.26/1.17 ( ( multiset_a2 @ ( heapIm1091024090Down_a @ T2 ) )
% 2.26/1.17 = ( multiset_a2 @ T2 ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_232_siftDown__Node,axiom,
% 2.26/1.17 ! [T2: tree_a,V: a,L: tree_a,R: tree_a] :
% 2.26/1.17 ( ( T2
% 2.26/1.17 = ( t_a @ V @ L @ R ) )
% 2.26/1.17 => ? [L5: tree_a,V4: a,R2: tree_a] :
% 2.26/1.17 ( ( ( heapIm1091024090Down_a @ T2 )
% 2.26/1.17 = ( t_a @ V4 @ L5 @ R2 ) )
% 2.26/1.17 & ( ord_less_eq_a @ V @ V4 ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_233_siftDown_Osimps_I2_J,axiom,
% 2.26/1.17 ! [V: a] :
% 2.26/1.17 ( ( heapIm1091024090Down_a @ ( t_a @ V @ e_a @ e_a ) )
% 2.26/1.17 = ( t_a @ V @ e_a @ e_a ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_234_siftDown__heap__is__heap,axiom,
% 2.26/1.17 ! [L: tree_a,R: tree_a,T2: tree_a,V: a] :
% 2.26/1.17 ( ( is_heap_a @ L )
% 2.26/1.17 => ( ( is_heap_a @ R )
% 2.26/1.17 => ( ( T2
% 2.26/1.17 = ( t_a @ V @ L @ R ) )
% 2.26/1.17 => ( is_heap_a @ ( heapIm1091024090Down_a @ T2 ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_235_siftDown_Osimps_I6_J,axiom,
% 2.26/1.17 ! [Va: a,Vb: tree_a,Vc: tree_a,Vd: a,Ve: tree_a,Vf: tree_a,V: a] :
% 2.26/1.17 ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 2.26/1.17 => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
% 2.26/1.17 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
% 2.26/1.17 = ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) ) )
% 2.26/1.17 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
% 2.26/1.17 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
% 2.26/1.17 = ( t_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Vd @ Ve @ Vf ) ) ) ) @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ) )
% 2.26/1.17 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 2.26/1.17 => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 2.26/1.17 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
% 2.26/1.17 = ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) ) )
% 2.26/1.17 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 2.26/1.17 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
% 2.26/1.17 = ( t_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( t_a @ Vd @ Ve @ Vf ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ) ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_236_siftDown__in__tree,axiom,
% 2.26/1.17 ! [T2: tree_a] :
% 2.26/1.17 ( ( T2 != e_a )
% 2.26/1.17 => ( in_tree_a @ ( val_a @ ( heapIm1091024090Down_a @ T2 ) ) @ T2 ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_237_is__heap__max,axiom,
% 2.26/1.17 ! [V: a,T2: tree_a] :
% 2.26/1.17 ( ( in_tree_a @ V @ T2 )
% 2.26/1.17 => ( ( is_heap_a @ T2 )
% 2.26/1.17 => ( ord_less_eq_a @ V @ ( val_a @ T2 ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_238_in__tree_Osimps_I2_J,axiom,
% 2.26/1.17 ! [V: a,V2: a,L: tree_a,R: tree_a] :
% 2.26/1.17 ( ( in_tree_a @ V @ ( t_a @ V2 @ L @ R ) )
% 2.26/1.17 = ( ( V = V2 )
% 2.26/1.17 | ( in_tree_a @ V @ L )
% 2.26/1.17 | ( in_tree_a @ V @ R ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_239_in__tree_Osimps_I1_J,axiom,
% 2.26/1.17 ! [V: a] :
% 2.26/1.17 ~ ( in_tree_a @ V @ e_a ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_240_siftDown__in__tree__set,axiom,
% 2.26/1.17 ( in_tree_a
% 2.26/1.17 = ( ^ [V5: a,T3: tree_a] : ( in_tree_a @ V5 @ ( heapIm1091024090Down_a @ T3 ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_241_hs__remove__max__def,axiom,
% 2.26/1.17 ( heapIm915766629_max_a
% 2.26/1.17 = ( ^ [T3: tree_a] :
% 2.26/1.17 ( if_Pro1144176865Tree_a
% 2.26/1.17 @ ( ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ T3 ) )
% 2.26/1.17 = e_a )
% 2.26/1.17 @ ( produc686083979Tree_a @ ( val_a @ T3 ) @ e_a )
% 2.26/1.17 @ ( produc686083979Tree_a @ ( val_a @ T3 ) @ ( heapIm1091024090Down_a @ ( t_a @ ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ T3 ) ) @ ( heapIm1140443833left_a @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ T3 ) ) ) @ ( heapIm1257206334ight_a @ ( product_snd_a_Tree_a @ ( heapIm837449470Leaf_a @ T3 ) ) ) ) ) ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_242_heapify_Osimps_I2_J,axiom,
% 2.26/1.17 ! [V: a,L: tree_a,R: tree_a] :
% 2.26/1.17 ( ( heapIm970322378pify_a @ ( t_a @ V @ L @ R ) )
% 2.26/1.17 = ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm970322378pify_a @ L ) @ ( heapIm970322378pify_a @ R ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_243_heapify_Osimps_I1_J,axiom,
% 2.26/1.17 ( ( heapIm970322378pify_a @ e_a )
% 2.26/1.17 = e_a ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_244_heapify__heap__is__heap,axiom,
% 2.26/1.17 ! [T2: tree_a] : ( is_heap_a @ ( heapIm970322378pify_a @ T2 ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_245_multiset__heapify,axiom,
% 2.26/1.17 ! [T2: tree_a] :
% 2.26/1.17 ( ( multiset_a2 @ ( heapIm970322378pify_a @ T2 ) )
% 2.26/1.17 = ( multiset_a2 @ T2 ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_246_Heap__def,axiom,
% 2.26/1.17 ( heap_Tree_a_a
% 2.26/1.17 = ( ^ [Empty2: tree_a,Is_empty: tree_a > $o,Of_list: list_a > tree_a,Multiset: tree_a > multiset_a,As_tree: tree_a > tree_a,Remove_max: tree_a > produc143150363Tree_a] :
% 2.26/1.17 ( ( collection_Tree_a_a @ Empty2 @ Is_empty @ Of_list @ Multiset )
% 2.26/1.17 & ( heap_axioms_Tree_a_a @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max ) ) ) ) ).
% 2.26/1.17
% 2.26/1.17 thf(fact_247_Heap_Ointro,axiom,
% 2.26/1.18 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a] :
% 2.26/1.18 ( ( collection_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 )
% 2.26/1.18 => ( ( heap_axioms_Tree_a_a @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 2.26/1.18 => ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_248_Heap_Oaxioms_I1_J,axiom,
% 2.26/1.18 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a] :
% 2.26/1.18 ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 2.26/1.18 => ( collection_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_249_HS_OCollection__axioms,axiom,
% 2.26/1.18 collection_Tree_a_a @ e_a @ heapIm229596386mpty_a @ heapIm1057938560list_a @ multiset_a2 ).
% 2.26/1.18
% 2.26/1.18 thf(fact_250_Collection_Omultiset__empty,axiom,
% 2.26/1.18 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a] :
% 2.26/1.18 ( ( collection_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 )
% 2.26/1.18 => ( ( Multiset2 @ Empty )
% 2.26/1.18 = zero_zero_multiset_a ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_251_Collection_Ois__empty__as__list,axiom,
% 2.26/1.18 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,E: tree_a] :
% 2.26/1.18 ( ( collection_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 )
% 2.26/1.18 => ( ( Is_empty2 @ E )
% 2.26/1.18 => ( ( Multiset2 @ E )
% 2.26/1.18 = zero_zero_multiset_a ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_252_hs__of__list__def,axiom,
% 2.26/1.18 ( heapIm1057938560list_a
% 2.26/1.18 = ( ^ [L2: list_a] : ( heapIm970322378pify_a @ ( heapIm1637418125tree_a @ L2 ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_253_pred__equals__eq2,axiom,
% 2.26/1.18 ! [R3: set_Pr921883667ee_a_a,S2: set_Pr921883667ee_a_a] :
% 2.26/1.18 ( ( ( ^ [X4: tree_a,Y6: a] : ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ X4 @ Y6 ) @ R3 ) )
% 2.26/1.18 = ( ^ [X4: tree_a,Y6: a] : ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ X4 @ Y6 ) @ S2 ) ) )
% 2.26/1.18 = ( R3 = S2 ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_254_pred__equals__eq2,axiom,
% 2.26/1.18 ! [R3: set_Pr158363655iset_a,S2: set_Pr158363655iset_a] :
% 2.26/1.18 ( ( ( ^ [X4: multiset_a,Y6: multiset_a] : ( member340150864iset_a @ ( produc2037245207iset_a @ X4 @ Y6 ) @ R3 ) )
% 2.26/1.18 = ( ^ [X4: multiset_a,Y6: multiset_a] : ( member340150864iset_a @ ( produc2037245207iset_a @ X4 @ Y6 ) @ S2 ) ) )
% 2.26/1.18 = ( R3 = S2 ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_255_pred__equals__eq2,axiom,
% 2.26/1.18 ! [R3: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
% 2.26/1.18 ( ( ( ^ [X4: a,Y6: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y6 ) @ R3 ) )
% 2.26/1.18 = ( ^ [X4: a,Y6: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y6 ) @ S2 ) ) )
% 2.26/1.18 = ( R3 = S2 ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_256_pred__equals__eq2,axiom,
% 2.26/1.18 ! [R3: set_Pr1070816379Tree_a,S2: set_Pr1070816379Tree_a] :
% 2.26/1.18 ( ( ( ^ [X4: a,Y6: tree_a] : ( member254315204Tree_a @ ( produc686083979Tree_a @ X4 @ Y6 ) @ R3 ) )
% 2.26/1.18 = ( ^ [X4: a,Y6: tree_a] : ( member254315204Tree_a @ ( produc686083979Tree_a @ X4 @ Y6 ) @ S2 ) ) )
% 2.26/1.18 = ( R3 = S2 ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_257_subrelI,axiom,
% 2.26/1.18 ! [R: set_Pr921883667ee_a_a,S3: set_Pr921883667ee_a_a] :
% 2.26/1.18 ( ! [X3: tree_a,Y3: a] :
% 2.26/1.18 ( ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ X3 @ Y3 ) @ R )
% 2.26/1.18 => ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ X3 @ Y3 ) @ S3 ) )
% 2.26/1.18 => ( ord_le864074675ee_a_a @ R @ S3 ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_258_subrelI,axiom,
% 2.26/1.18 ! [R: set_Pr158363655iset_a,S3: set_Pr158363655iset_a] :
% 2.26/1.18 ( ! [X3: multiset_a,Y3: multiset_a] :
% 2.26/1.18 ( ( member340150864iset_a @ ( produc2037245207iset_a @ X3 @ Y3 ) @ R )
% 2.26/1.18 => ( member340150864iset_a @ ( produc2037245207iset_a @ X3 @ Y3 ) @ S3 ) )
% 2.26/1.18 => ( ord_le1978107815iset_a @ R @ S3 ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_259_subrelI,axiom,
% 2.26/1.18 ! [R: set_Product_prod_a_a,S3: set_Product_prod_a_a] :
% 2.26/1.18 ( ! [X3: a,Y3: a] :
% 2.26/1.18 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R )
% 2.26/1.18 => ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ S3 ) )
% 2.26/1.18 => ( ord_le1824328871od_a_a @ R @ S3 ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_260_subrelI,axiom,
% 2.26/1.18 ! [R: set_Pr1070816379Tree_a,S3: set_Pr1070816379Tree_a] :
% 2.26/1.18 ( ! [X3: a,Y3: tree_a] :
% 2.26/1.18 ( ( member254315204Tree_a @ ( produc686083979Tree_a @ X3 @ Y3 ) @ R )
% 2.26/1.18 => ( member254315204Tree_a @ ( produc686083979Tree_a @ X3 @ Y3 ) @ S3 ) )
% 2.26/1.18 => ( ord_le1013007387Tree_a @ R @ S3 ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_261_pred__subset__eq2,axiom,
% 2.26/1.18 ! [R3: set_Pr921883667ee_a_a,S2: set_Pr921883667ee_a_a] :
% 2.26/1.18 ( ( ord_le1748873858_a_a_o
% 2.26/1.18 @ ^ [X4: tree_a,Y6: a] : ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ X4 @ Y6 ) @ R3 )
% 2.26/1.18 @ ^ [X4: tree_a,Y6: a] : ( member1092636252ee_a_a @ ( produc649172771ee_a_a @ X4 @ Y6 ) @ S2 ) )
% 2.26/1.18 = ( ord_le864074675ee_a_a @ R3 @ S2 ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_262_pred__subset__eq2,axiom,
% 2.26/1.18 ! [R3: set_Pr158363655iset_a,S2: set_Pr158363655iset_a] :
% 2.26/1.18 ( ( ord_le51495822et_a_o
% 2.26/1.18 @ ^ [X4: multiset_a,Y6: multiset_a] : ( member340150864iset_a @ ( produc2037245207iset_a @ X4 @ Y6 ) @ R3 )
% 2.26/1.18 @ ^ [X4: multiset_a,Y6: multiset_a] : ( member340150864iset_a @ ( produc2037245207iset_a @ X4 @ Y6 ) @ S2 ) )
% 2.26/1.18 = ( ord_le1978107815iset_a @ R3 @ S2 ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_263_pred__subset__eq2,axiom,
% 2.26/1.18 ! [R3: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
% 2.26/1.18 ( ( ord_less_eq_a_a_o
% 2.26/1.18 @ ^ [X4: a,Y6: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y6 ) @ R3 )
% 2.26/1.18 @ ^ [X4: a,Y6: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y6 ) @ S2 ) )
% 2.26/1.18 = ( ord_le1824328871od_a_a @ R3 @ S2 ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_264_pred__subset__eq2,axiom,
% 2.26/1.18 ! [R3: set_Pr1070816379Tree_a,S2: set_Pr1070816379Tree_a] :
% 2.26/1.18 ( ( ord_le1327026842ee_a_o
% 2.26/1.18 @ ^ [X4: a,Y6: tree_a] : ( member254315204Tree_a @ ( produc686083979Tree_a @ X4 @ Y6 ) @ R3 )
% 2.26/1.18 @ ^ [X4: a,Y6: tree_a] : ( member254315204Tree_a @ ( produc686083979Tree_a @ X4 @ Y6 ) @ S2 ) )
% 2.26/1.18 = ( ord_le1013007387Tree_a @ R3 @ S2 ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_265_heap__top__geq,axiom,
% 2.26/1.18 ! [A: a,T2: tree_a] :
% 2.26/1.18 ( ( member_a @ A @ ( set_mset_a @ ( multiset_a2 @ T2 ) ) )
% 2.26/1.18 => ( ( is_heap_a @ T2 )
% 2.26/1.18 => ( ord_less_eq_a @ A @ ( val_a @ T2 ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_266_multi__member__last,axiom,
% 2.26/1.18 ! [X: produc143150363Tree_a] : ( member254315204Tree_a @ X @ ( set_ms1129970904Tree_a @ ( add_ms205941497Tree_a @ X @ zero_z2144236696Tree_a ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_267_multi__member__last,axiom,
% 2.26/1.18 ! [X: produc1127127335iset_a] : ( member340150864iset_a @ X @ ( set_ms1768116836iset_a @ ( add_ms1353365509iset_a @ X @ zero_z1941791140iset_a ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_268_multi__member__last,axiom,
% 2.26/1.18 ! [X: product_prod_a_a] : ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ ( add_ms1725977093od_a_a @ X @ zero_z160849828od_a_a ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_269_multi__member__last,axiom,
% 2.26/1.18 ! [X: multiset_a] : ( member_multiset_a @ X @ ( set_mset_multiset_a @ ( add_mset_multiset_a @ X @ zero_z1580389697iset_a ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_270_multi__member__last,axiom,
% 2.26/1.18 ! [X: a] : ( member_a @ X @ ( set_mset_a @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_271_union__single__eq__member,axiom,
% 2.26/1.18 ! [X: produc143150363Tree_a,M: multis2082063201Tree_a,N: multis2082063201Tree_a] :
% 2.26/1.18 ( ( ( add_ms205941497Tree_a @ X @ M )
% 2.26/1.18 = N )
% 2.26/1.18 => ( member254315204Tree_a @ X @ ( set_ms1129970904Tree_a @ N ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_272_union__single__eq__member,axiom,
% 2.26/1.18 ! [X: produc1127127335iset_a,M: multis782275565iset_a,N: multis782275565iset_a] :
% 2.26/1.18 ( ( ( add_ms1353365509iset_a @ X @ M )
% 2.26/1.18 = N )
% 2.26/1.18 => ( member340150864iset_a @ X @ ( set_ms1768116836iset_a @ N ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_273_union__single__eq__member,axiom,
% 2.26/1.18 ! [X: product_prod_a_a,M: multis599418605od_a_a,N: multis599418605od_a_a] :
% 2.26/1.18 ( ( ( add_ms1725977093od_a_a @ X @ M )
% 2.26/1.18 = N )
% 2.26/1.18 => ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ N ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_274_union__single__eq__member,axiom,
% 2.26/1.18 ! [X: multiset_a,M: multiset_multiset_a,N: multiset_multiset_a] :
% 2.26/1.18 ( ( ( add_mset_multiset_a @ X @ M )
% 2.26/1.18 = N )
% 2.26/1.18 => ( member_multiset_a @ X @ ( set_mset_multiset_a @ N ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_275_union__single__eq__member,axiom,
% 2.26/1.18 ! [X: a,M: multiset_a,N: multiset_a] :
% 2.26/1.18 ( ( ( add_mset_a @ X @ M )
% 2.26/1.18 = N )
% 2.26/1.18 => ( member_a @ X @ ( set_mset_a @ N ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_276_insert__noteq__member,axiom,
% 2.26/1.18 ! [B: produc143150363Tree_a,B4: multis2082063201Tree_a,C: produc143150363Tree_a,C2: multis2082063201Tree_a] :
% 2.26/1.18 ( ( ( add_ms205941497Tree_a @ B @ B4 )
% 2.26/1.18 = ( add_ms205941497Tree_a @ C @ C2 ) )
% 2.26/1.18 => ( ( B != C )
% 2.26/1.18 => ( member254315204Tree_a @ C @ ( set_ms1129970904Tree_a @ B4 ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_277_insert__noteq__member,axiom,
% 2.26/1.18 ! [B: produc1127127335iset_a,B4: multis782275565iset_a,C: produc1127127335iset_a,C2: multis782275565iset_a] :
% 2.26/1.18 ( ( ( add_ms1353365509iset_a @ B @ B4 )
% 2.26/1.18 = ( add_ms1353365509iset_a @ C @ C2 ) )
% 2.26/1.18 => ( ( B != C )
% 2.26/1.18 => ( member340150864iset_a @ C @ ( set_ms1768116836iset_a @ B4 ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_278_insert__noteq__member,axiom,
% 2.26/1.18 ! [B: product_prod_a_a,B4: multis599418605od_a_a,C: product_prod_a_a,C2: multis599418605od_a_a] :
% 2.26/1.18 ( ( ( add_ms1725977093od_a_a @ B @ B4 )
% 2.26/1.18 = ( add_ms1725977093od_a_a @ C @ C2 ) )
% 2.26/1.18 => ( ( B != C )
% 2.26/1.18 => ( member449909584od_a_a @ C @ ( set_ms1818337124od_a_a @ B4 ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_279_insert__noteq__member,axiom,
% 2.26/1.18 ! [B: multiset_a,B4: multiset_multiset_a,C: multiset_a,C2: multiset_multiset_a] :
% 2.26/1.18 ( ( ( add_mset_multiset_a @ B @ B4 )
% 2.26/1.18 = ( add_mset_multiset_a @ C @ C2 ) )
% 2.26/1.18 => ( ( B != C )
% 2.26/1.18 => ( member_multiset_a @ C @ ( set_mset_multiset_a @ B4 ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_280_insert__noteq__member,axiom,
% 2.26/1.18 ! [B: a,B4: multiset_a,C: a,C2: multiset_a] :
% 2.26/1.18 ( ( ( add_mset_a @ B @ B4 )
% 2.26/1.18 = ( add_mset_a @ C @ C2 ) )
% 2.26/1.18 => ( ( B != C )
% 2.26/1.18 => ( member_a @ C @ ( set_mset_a @ B4 ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_281_multi__member__split,axiom,
% 2.26/1.18 ! [X: produc143150363Tree_a,M: multis2082063201Tree_a] :
% 2.26/1.18 ( ( member254315204Tree_a @ X @ ( set_ms1129970904Tree_a @ M ) )
% 2.26/1.18 => ? [A6: multis2082063201Tree_a] :
% 2.26/1.18 ( M
% 2.26/1.18 = ( add_ms205941497Tree_a @ X @ A6 ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_282_multi__member__split,axiom,
% 2.26/1.18 ! [X: produc1127127335iset_a,M: multis782275565iset_a] :
% 2.26/1.18 ( ( member340150864iset_a @ X @ ( set_ms1768116836iset_a @ M ) )
% 2.26/1.18 => ? [A6: multis782275565iset_a] :
% 2.26/1.18 ( M
% 2.26/1.18 = ( add_ms1353365509iset_a @ X @ A6 ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_283_multi__member__split,axiom,
% 2.26/1.18 ! [X: product_prod_a_a,M: multis599418605od_a_a] :
% 2.26/1.18 ( ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ M ) )
% 2.26/1.18 => ? [A6: multis599418605od_a_a] :
% 2.26/1.18 ( M
% 2.26/1.18 = ( add_ms1725977093od_a_a @ X @ A6 ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_284_multi__member__split,axiom,
% 2.26/1.18 ! [X: multiset_a,M: multiset_multiset_a] :
% 2.26/1.18 ( ( member_multiset_a @ X @ ( set_mset_multiset_a @ M ) )
% 2.26/1.18 => ? [A6: multiset_multiset_a] :
% 2.26/1.18 ( M
% 2.26/1.18 = ( add_mset_multiset_a @ X @ A6 ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_285_multi__member__split,axiom,
% 2.26/1.18 ! [X: a,M: multiset_a] :
% 2.26/1.18 ( ( member_a @ X @ ( set_mset_a @ M ) )
% 2.26/1.18 => ? [A6: multiset_a] :
% 2.26/1.18 ( M
% 2.26/1.18 = ( add_mset_a @ X @ A6 ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_286_mset__add,axiom,
% 2.26/1.18 ! [A: produc143150363Tree_a,A4: multis2082063201Tree_a] :
% 2.26/1.18 ( ( member254315204Tree_a @ A @ ( set_ms1129970904Tree_a @ A4 ) )
% 2.26/1.18 => ~ ! [B6: multis2082063201Tree_a] :
% 2.26/1.18 ( A4
% 2.26/1.18 != ( add_ms205941497Tree_a @ A @ B6 ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_287_mset__add,axiom,
% 2.26/1.18 ! [A: produc1127127335iset_a,A4: multis782275565iset_a] :
% 2.26/1.18 ( ( member340150864iset_a @ A @ ( set_ms1768116836iset_a @ A4 ) )
% 2.26/1.18 => ~ ! [B6: multis782275565iset_a] :
% 2.26/1.18 ( A4
% 2.26/1.18 != ( add_ms1353365509iset_a @ A @ B6 ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_288_mset__add,axiom,
% 2.26/1.18 ! [A: product_prod_a_a,A4: multis599418605od_a_a] :
% 2.26/1.18 ( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ A4 ) )
% 2.26/1.18 => ~ ! [B6: multis599418605od_a_a] :
% 2.26/1.18 ( A4
% 2.26/1.18 != ( add_ms1725977093od_a_a @ A @ B6 ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_289_mset__add,axiom,
% 2.26/1.18 ! [A: multiset_a,A4: multiset_multiset_a] :
% 2.26/1.18 ( ( member_multiset_a @ A @ ( set_mset_multiset_a @ A4 ) )
% 2.26/1.18 => ~ ! [B6: multiset_multiset_a] :
% 2.26/1.18 ( A4
% 2.26/1.18 != ( add_mset_multiset_a @ A @ B6 ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_290_mset__add,axiom,
% 2.26/1.18 ! [A: a,A4: multiset_a] :
% 2.26/1.18 ( ( member_a @ A @ ( set_mset_a @ A4 ) )
% 2.26/1.18 => ~ ! [B6: multiset_a] :
% 2.26/1.18 ( A4
% 2.26/1.18 != ( add_mset_a @ A @ B6 ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_291_union__iff,axiom,
% 2.26/1.18 ! [A: produc143150363Tree_a,A4: multis2082063201Tree_a,B4: multis2082063201Tree_a] :
% 2.26/1.18 ( ( member254315204Tree_a @ A @ ( set_ms1129970904Tree_a @ ( plus_p1065544216Tree_a @ A4 @ B4 ) ) )
% 2.26/1.18 = ( ( member254315204Tree_a @ A @ ( set_ms1129970904Tree_a @ A4 ) )
% 2.26/1.18 | ( member254315204Tree_a @ A @ ( set_ms1129970904Tree_a @ B4 ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_292_union__iff,axiom,
% 2.26/1.18 ! [A: produc1127127335iset_a,A4: multis782275565iset_a,B4: multis782275565iset_a] :
% 2.26/1.18 ( ( member340150864iset_a @ A @ ( set_ms1768116836iset_a @ ( plus_p1919096612iset_a @ A4 @ B4 ) ) )
% 2.26/1.18 = ( ( member340150864iset_a @ A @ ( set_ms1768116836iset_a @ A4 ) )
% 2.26/1.18 | ( member340150864iset_a @ A @ ( set_ms1768116836iset_a @ B4 ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_293_union__iff,axiom,
% 2.26/1.18 ! [A: product_prod_a_a,A4: multis599418605od_a_a,B4: multis599418605od_a_a] :
% 2.26/1.18 ( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ ( plus_p404985124od_a_a @ A4 @ B4 ) ) )
% 2.26/1.18 = ( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ A4 ) )
% 2.26/1.18 | ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ B4 ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_294_union__iff,axiom,
% 2.26/1.18 ! [A: multiset_a,A4: multiset_multiset_a,B4: multiset_multiset_a] :
% 2.26/1.18 ( ( member_multiset_a @ A @ ( set_mset_multiset_a @ ( plus_p1957546689iset_a @ A4 @ B4 ) ) )
% 2.26/1.18 = ( ( member_multiset_a @ A @ ( set_mset_multiset_a @ A4 ) )
% 2.26/1.18 | ( member_multiset_a @ A @ ( set_mset_multiset_a @ B4 ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_295_union__iff,axiom,
% 2.26/1.18 ! [A: a,A4: multiset_a,B4: multiset_a] :
% 2.26/1.18 ( ( member_a @ A @ ( set_mset_a @ ( plus_plus_multiset_a @ A4 @ B4 ) ) )
% 2.26/1.18 = ( ( member_a @ A @ ( set_mset_a @ A4 ) )
% 2.26/1.18 | ( member_a @ A @ ( set_mset_a @ B4 ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_296_multiset__nonemptyE,axiom,
% 2.26/1.18 ! [A4: multis2082063201Tree_a] :
% 2.26/1.18 ( ( A4 != zero_z2144236696Tree_a )
% 2.26/1.18 => ~ ! [X3: produc143150363Tree_a] :
% 2.26/1.18 ~ ( member254315204Tree_a @ X3 @ ( set_ms1129970904Tree_a @ A4 ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_297_multiset__nonemptyE,axiom,
% 2.26/1.18 ! [A4: multis782275565iset_a] :
% 2.26/1.18 ( ( A4 != zero_z1941791140iset_a )
% 2.26/1.18 => ~ ! [X3: produc1127127335iset_a] :
% 2.26/1.18 ~ ( member340150864iset_a @ X3 @ ( set_ms1768116836iset_a @ A4 ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_298_multiset__nonemptyE,axiom,
% 2.26/1.18 ! [A4: multis599418605od_a_a] :
% 2.26/1.18 ( ( A4 != zero_z160849828od_a_a )
% 2.26/1.18 => ~ ! [X3: product_prod_a_a] :
% 2.26/1.18 ~ ( member449909584od_a_a @ X3 @ ( set_ms1818337124od_a_a @ A4 ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_299_multiset__nonemptyE,axiom,
% 2.26/1.18 ! [A4: multiset_multiset_a] :
% 2.26/1.18 ( ( A4 != zero_z1580389697iset_a )
% 2.26/1.18 => ~ ! [X3: multiset_a] :
% 2.26/1.18 ~ ( member_multiset_a @ X3 @ ( set_mset_multiset_a @ A4 ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_300_multiset__nonemptyE,axiom,
% 2.26/1.18 ! [A4: multiset_a] :
% 2.26/1.18 ( ( A4 != zero_zero_multiset_a )
% 2.26/1.18 => ~ ! [X3: a] :
% 2.26/1.18 ~ ( member_a @ X3 @ ( set_mset_a @ A4 ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_301_subset__CollectI,axiom,
% 2.26/1.18 ! [B4: set_a,A4: set_a,Q2: a > $o,P2: a > $o] :
% 2.26/1.18 ( ( ord_less_eq_set_a @ B4 @ A4 )
% 2.26/1.18 => ( ! [X3: a] :
% 2.26/1.18 ( ( member_a @ X3 @ B4 )
% 2.26/1.18 => ( ( Q2 @ X3 )
% 2.26/1.18 => ( P2 @ X3 ) ) )
% 2.26/1.18 => ( ord_less_eq_set_a
% 2.26/1.18 @ ( collect_a
% 2.26/1.18 @ ^ [X4: a] :
% 2.26/1.18 ( ( member_a @ X4 @ B4 )
% 2.26/1.18 & ( Q2 @ X4 ) ) )
% 2.26/1.18 @ ( collect_a
% 2.26/1.18 @ ^ [X4: a] :
% 2.26/1.18 ( ( member_a @ X4 @ A4 )
% 2.26/1.18 & ( P2 @ X4 ) ) ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_302_subset__CollectI,axiom,
% 2.26/1.18 ! [B4: set_Pr158363655iset_a,A4: set_Pr158363655iset_a,Q2: produc1127127335iset_a > $o,P2: produc1127127335iset_a > $o] :
% 2.26/1.18 ( ( ord_le1978107815iset_a @ B4 @ A4 )
% 2.26/1.18 => ( ! [X3: produc1127127335iset_a] :
% 2.26/1.18 ( ( member340150864iset_a @ X3 @ B4 )
% 2.26/1.18 => ( ( Q2 @ X3 )
% 2.26/1.18 => ( P2 @ X3 ) ) )
% 2.26/1.18 => ( ord_le1978107815iset_a
% 2.26/1.18 @ ( collec1104713362iset_a
% 2.26/1.18 @ ^ [X4: produc1127127335iset_a] :
% 2.26/1.18 ( ( member340150864iset_a @ X4 @ B4 )
% 2.26/1.18 & ( Q2 @ X4 ) ) )
% 2.26/1.18 @ ( collec1104713362iset_a
% 2.26/1.18 @ ^ [X4: produc1127127335iset_a] :
% 2.26/1.18 ( ( member340150864iset_a @ X4 @ A4 )
% 2.26/1.18 & ( P2 @ X4 ) ) ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_303_subset__CollectI,axiom,
% 2.26/1.18 ! [B4: set_Product_prod_a_a,A4: set_Product_prod_a_a,Q2: product_prod_a_a > $o,P2: product_prod_a_a > $o] :
% 2.26/1.18 ( ( ord_le1824328871od_a_a @ B4 @ A4 )
% 2.26/1.18 => ( ! [X3: product_prod_a_a] :
% 2.26/1.18 ( ( member449909584od_a_a @ X3 @ B4 )
% 2.26/1.18 => ( ( Q2 @ X3 )
% 2.26/1.18 => ( P2 @ X3 ) ) )
% 2.26/1.18 => ( ord_le1824328871od_a_a
% 2.26/1.18 @ ( collec645855634od_a_a
% 2.26/1.18 @ ^ [X4: product_prod_a_a] :
% 2.26/1.18 ( ( member449909584od_a_a @ X4 @ B4 )
% 2.26/1.18 & ( Q2 @ X4 ) ) )
% 2.26/1.18 @ ( collec645855634od_a_a
% 2.26/1.18 @ ^ [X4: product_prod_a_a] :
% 2.26/1.18 ( ( member449909584od_a_a @ X4 @ A4 )
% 2.26/1.18 & ( P2 @ X4 ) ) ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_304_subset__CollectI,axiom,
% 2.26/1.18 ! [B4: set_Pr1070816379Tree_a,A4: set_Pr1070816379Tree_a,Q2: produc143150363Tree_a > $o,P2: produc143150363Tree_a > $o] :
% 2.26/1.18 ( ( ord_le1013007387Tree_a @ B4 @ A4 )
% 2.26/1.18 => ( ! [X3: produc143150363Tree_a] :
% 2.26/1.18 ( ( member254315204Tree_a @ X3 @ B4 )
% 2.26/1.18 => ( ( Q2 @ X3 )
% 2.26/1.18 => ( P2 @ X3 ) ) )
% 2.26/1.18 => ( ord_le1013007387Tree_a
% 2.26/1.18 @ ( collec486369286Tree_a
% 2.26/1.18 @ ^ [X4: produc143150363Tree_a] :
% 2.26/1.18 ( ( member254315204Tree_a @ X4 @ B4 )
% 2.26/1.18 & ( Q2 @ X4 ) ) )
% 2.26/1.18 @ ( collec486369286Tree_a
% 2.26/1.18 @ ^ [X4: produc143150363Tree_a] :
% 2.26/1.18 ( ( member254315204Tree_a @ X4 @ A4 )
% 2.26/1.18 & ( P2 @ X4 ) ) ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_305_subset__Collect__iff,axiom,
% 2.26/1.18 ! [B4: set_a,A4: set_a,P2: a > $o] :
% 2.26/1.18 ( ( ord_less_eq_set_a @ B4 @ A4 )
% 2.26/1.18 => ( ( ord_less_eq_set_a @ B4
% 2.26/1.18 @ ( collect_a
% 2.26/1.18 @ ^ [X4: a] :
% 2.26/1.18 ( ( member_a @ X4 @ A4 )
% 2.26/1.18 & ( P2 @ X4 ) ) ) )
% 2.26/1.18 = ( ! [X4: a] :
% 2.26/1.18 ( ( member_a @ X4 @ B4 )
% 2.26/1.18 => ( P2 @ X4 ) ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_306_subset__Collect__iff,axiom,
% 2.26/1.18 ! [B4: set_Pr158363655iset_a,A4: set_Pr158363655iset_a,P2: produc1127127335iset_a > $o] :
% 2.26/1.18 ( ( ord_le1978107815iset_a @ B4 @ A4 )
% 2.26/1.18 => ( ( ord_le1978107815iset_a @ B4
% 2.26/1.18 @ ( collec1104713362iset_a
% 2.26/1.18 @ ^ [X4: produc1127127335iset_a] :
% 2.26/1.18 ( ( member340150864iset_a @ X4 @ A4 )
% 2.26/1.18 & ( P2 @ X4 ) ) ) )
% 2.26/1.18 = ( ! [X4: produc1127127335iset_a] :
% 2.26/1.18 ( ( member340150864iset_a @ X4 @ B4 )
% 2.26/1.18 => ( P2 @ X4 ) ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_307_subset__Collect__iff,axiom,
% 2.26/1.18 ! [B4: set_Product_prod_a_a,A4: set_Product_prod_a_a,P2: product_prod_a_a > $o] :
% 2.26/1.18 ( ( ord_le1824328871od_a_a @ B4 @ A4 )
% 2.26/1.18 => ( ( ord_le1824328871od_a_a @ B4
% 2.26/1.18 @ ( collec645855634od_a_a
% 2.26/1.18 @ ^ [X4: product_prod_a_a] :
% 2.26/1.18 ( ( member449909584od_a_a @ X4 @ A4 )
% 2.26/1.18 & ( P2 @ X4 ) ) ) )
% 2.26/1.18 = ( ! [X4: product_prod_a_a] :
% 2.26/1.18 ( ( member449909584od_a_a @ X4 @ B4 )
% 2.26/1.18 => ( P2 @ X4 ) ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_308_subset__Collect__iff,axiom,
% 2.26/1.18 ! [B4: set_Pr1070816379Tree_a,A4: set_Pr1070816379Tree_a,P2: produc143150363Tree_a > $o] :
% 2.26/1.18 ( ( ord_le1013007387Tree_a @ B4 @ A4 )
% 2.26/1.18 => ( ( ord_le1013007387Tree_a @ B4
% 2.26/1.18 @ ( collec486369286Tree_a
% 2.26/1.18 @ ^ [X4: produc143150363Tree_a] :
% 2.26/1.18 ( ( member254315204Tree_a @ X4 @ A4 )
% 2.26/1.18 & ( P2 @ X4 ) ) ) )
% 2.26/1.18 = ( ! [X4: produc143150363Tree_a] :
% 2.26/1.18 ( ( member254315204Tree_a @ X4 @ B4 )
% 2.26/1.18 => ( P2 @ X4 ) ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_309_multiset__induct__min,axiom,
% 2.26/1.18 ! [P2: multiset_a > $o,M: multiset_a] :
% 2.26/1.18 ( ( P2 @ zero_zero_multiset_a )
% 2.26/1.18 => ( ! [X3: a,M3: multiset_a] :
% 2.26/1.18 ( ( P2 @ M3 )
% 2.26/1.18 => ( ! [Xa: a] :
% 2.26/1.18 ( ( member_a @ Xa @ ( set_mset_a @ M3 ) )
% 2.26/1.18 => ( ord_less_eq_a @ X3 @ Xa ) )
% 2.26/1.18 => ( P2 @ ( add_mset_a @ X3 @ M3 ) ) ) )
% 2.26/1.18 => ( P2 @ M ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_310_multiset__induct__max,axiom,
% 2.26/1.18 ! [P2: multiset_a > $o,M: multiset_a] :
% 2.26/1.18 ( ( P2 @ zero_zero_multiset_a )
% 2.26/1.18 => ( ! [X3: a,M3: multiset_a] :
% 2.26/1.18 ( ( P2 @ M3 )
% 2.26/1.18 => ( ! [Xa: a] :
% 2.26/1.18 ( ( member_a @ Xa @ ( set_mset_a @ M3 ) )
% 2.26/1.18 => ( ord_less_eq_a @ Xa @ X3 ) )
% 2.26/1.18 => ( P2 @ ( add_mset_a @ X3 @ M3 ) ) ) )
% 2.26/1.18 => ( P2 @ M ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_311_multi__member__skip,axiom,
% 2.26/1.18 ! [X: produc143150363Tree_a,XS: multis2082063201Tree_a,Y: produc143150363Tree_a] :
% 2.26/1.18 ( ( member254315204Tree_a @ X @ ( set_ms1129970904Tree_a @ XS ) )
% 2.26/1.18 => ( member254315204Tree_a @ X @ ( set_ms1129970904Tree_a @ ( plus_p1065544216Tree_a @ ( add_ms205941497Tree_a @ Y @ zero_z2144236696Tree_a ) @ XS ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_312_multi__member__skip,axiom,
% 2.26/1.18 ! [X: produc1127127335iset_a,XS: multis782275565iset_a,Y: produc1127127335iset_a] :
% 2.26/1.18 ( ( member340150864iset_a @ X @ ( set_ms1768116836iset_a @ XS ) )
% 2.26/1.18 => ( member340150864iset_a @ X @ ( set_ms1768116836iset_a @ ( plus_p1919096612iset_a @ ( add_ms1353365509iset_a @ Y @ zero_z1941791140iset_a ) @ XS ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_313_multi__member__skip,axiom,
% 2.26/1.18 ! [X: product_prod_a_a,XS: multis599418605od_a_a,Y: product_prod_a_a] :
% 2.26/1.18 ( ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ XS ) )
% 2.26/1.18 => ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ ( plus_p404985124od_a_a @ ( add_ms1725977093od_a_a @ Y @ zero_z160849828od_a_a ) @ XS ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_314_multi__member__skip,axiom,
% 2.26/1.18 ! [X: multiset_a,XS: multiset_multiset_a,Y: multiset_a] :
% 2.26/1.18 ( ( member_multiset_a @ X @ ( set_mset_multiset_a @ XS ) )
% 2.26/1.18 => ( member_multiset_a @ X @ ( set_mset_multiset_a @ ( plus_p1957546689iset_a @ ( add_mset_multiset_a @ Y @ zero_z1580389697iset_a ) @ XS ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_315_multi__member__skip,axiom,
% 2.26/1.18 ! [X: a,XS: multiset_a,Y: a] :
% 2.26/1.18 ( ( member_a @ X @ ( set_mset_a @ XS ) )
% 2.26/1.18 => ( member_a @ X @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ Y @ zero_zero_multiset_a ) @ XS ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_316_multi__member__this,axiom,
% 2.26/1.18 ! [X: produc143150363Tree_a,XS: multis2082063201Tree_a] : ( member254315204Tree_a @ X @ ( set_ms1129970904Tree_a @ ( plus_p1065544216Tree_a @ ( add_ms205941497Tree_a @ X @ zero_z2144236696Tree_a ) @ XS ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_317_multi__member__this,axiom,
% 2.26/1.18 ! [X: produc1127127335iset_a,XS: multis782275565iset_a] : ( member340150864iset_a @ X @ ( set_ms1768116836iset_a @ ( plus_p1919096612iset_a @ ( add_ms1353365509iset_a @ X @ zero_z1941791140iset_a ) @ XS ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_318_multi__member__this,axiom,
% 2.26/1.18 ! [X: product_prod_a_a,XS: multis599418605od_a_a] : ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ ( plus_p404985124od_a_a @ ( add_ms1725977093od_a_a @ X @ zero_z160849828od_a_a ) @ XS ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_319_multi__member__this,axiom,
% 2.26/1.18 ! [X: multiset_a,XS: multiset_multiset_a] : ( member_multiset_a @ X @ ( set_mset_multiset_a @ ( plus_p1957546689iset_a @ ( add_mset_multiset_a @ X @ zero_z1580389697iset_a ) @ XS ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_320_multi__member__this,axiom,
% 2.26/1.18 ! [X: a,XS: multiset_a] : ( member_a @ X @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ X @ zero_zero_multiset_a ) @ XS ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_321_less__add,axiom,
% 2.26/1.18 ! [N: multis2082063201Tree_a,A: produc143150363Tree_a,M0: multis2082063201Tree_a,R: set_Pr220561863Tree_a] :
% 2.26/1.18 ( ( member1298282640Tree_a @ ( produc1608222167Tree_a @ N @ ( add_ms205941497Tree_a @ A @ M0 ) ) @ ( mult1_470979598Tree_a @ R ) )
% 2.26/1.18 => ( ? [M3: multis2082063201Tree_a] :
% 2.26/1.18 ( ( member1298282640Tree_a @ ( produc1608222167Tree_a @ M3 @ M0 ) @ ( mult1_470979598Tree_a @ R ) )
% 2.26/1.18 & ( N
% 2.26/1.18 = ( add_ms205941497Tree_a @ A @ M3 ) ) )
% 2.26/1.18 | ? [K4: multis2082063201Tree_a] :
% 2.26/1.18 ( ! [B7: produc143150363Tree_a] :
% 2.26/1.18 ( ( member254315204Tree_a @ B7 @ ( set_ms1129970904Tree_a @ K4 ) )
% 2.26/1.18 => ( member146662416Tree_a @ ( produc1002111575Tree_a @ B7 @ A ) @ R ) )
% 2.26/1.18 & ( N
% 2.26/1.18 = ( plus_p1065544216Tree_a @ M0 @ K4 ) ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_322_less__add,axiom,
% 2.26/1.18 ! [N: multis782275565iset_a,A: produc1127127335iset_a,M0: multis782275565iset_a,R: set_Pr146455751iset_a] :
% 2.26/1.18 ( ( member219455632iset_a @ ( produc1226985431iset_a @ N @ ( add_ms1353365509iset_a @ A @ M0 ) ) @ ( mult1_599785114iset_a @ R ) )
% 2.26/1.18 => ( ? [M3: multis782275565iset_a] :
% 2.26/1.18 ( ( member219455632iset_a @ ( produc1226985431iset_a @ M3 @ M0 ) @ ( mult1_599785114iset_a @ R ) )
% 2.26/1.18 & ( N
% 2.26/1.18 = ( add_ms1353365509iset_a @ A @ M3 ) ) )
% 2.26/1.18 | ? [K4: multis782275565iset_a] :
% 2.26/1.18 ( ! [B7: produc1127127335iset_a] :
% 2.26/1.18 ( ( member340150864iset_a @ B7 @ ( set_ms1768116836iset_a @ K4 ) )
% 2.26/1.18 => ( member567864080iset_a @ ( produc1898392407iset_a @ B7 @ A ) @ R ) )
% 2.26/1.18 & ( N
% 2.26/1.18 = ( plus_p1919096612iset_a @ M0 @ K4 ) ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_323_less__add,axiom,
% 2.26/1.18 ! [N: multis599418605od_a_a,A: product_prod_a_a,M0: multis599418605od_a_a,R: set_Pr1948701895od_a_a] :
% 2.26/1.18 ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ N @ ( add_ms1725977093od_a_a @ A @ M0 ) ) @ ( mult1_1719650714od_a_a @ R ) )
% 2.26/1.18 => ( ? [M3: multis599418605od_a_a] :
% 2.26/1.18 ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ M3 @ M0 ) @ ( mult1_1719650714od_a_a @ R ) )
% 2.26/1.18 & ( N
% 2.26/1.18 = ( add_ms1725977093od_a_a @ A @ M3 ) ) )
% 2.26/1.18 | ? [K4: multis599418605od_a_a] :
% 2.26/1.18 ( ! [B7: product_prod_a_a] :
% 2.26/1.18 ( ( member449909584od_a_a @ B7 @ ( set_ms1818337124od_a_a @ K4 ) )
% 2.26/1.18 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B7 @ A ) @ R ) )
% 2.26/1.18 & ( N
% 2.26/1.18 = ( plus_p404985124od_a_a @ M0 @ K4 ) ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_324_less__add,axiom,
% 2.26/1.18 ! [N: multiset_multiset_a,A: multiset_a,M0: multiset_multiset_a,R: set_Pr158363655iset_a] :
% 2.26/1.18 ( ( member104214352iset_a @ ( produc1444952343iset_a @ N @ ( add_mset_multiset_a @ A @ M0 ) ) @ ( mult1_multiset_a @ R ) )
% 2.26/1.18 => ( ? [M3: multiset_multiset_a] :
% 2.26/1.18 ( ( member104214352iset_a @ ( produc1444952343iset_a @ M3 @ M0 ) @ ( mult1_multiset_a @ R ) )
% 2.26/1.18 & ( N
% 2.26/1.18 = ( add_mset_multiset_a @ A @ M3 ) ) )
% 2.26/1.18 | ? [K4: multiset_multiset_a] :
% 2.26/1.18 ( ! [B7: multiset_a] :
% 2.26/1.18 ( ( member_multiset_a @ B7 @ ( set_mset_multiset_a @ K4 ) )
% 2.26/1.18 => ( member340150864iset_a @ ( produc2037245207iset_a @ B7 @ A ) @ R ) )
% 2.26/1.18 & ( N
% 2.26/1.18 = ( plus_p1957546689iset_a @ M0 @ K4 ) ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_325_less__add,axiom,
% 2.26/1.18 ! [N: multiset_a,A: a,M0: multiset_a,R: set_Product_prod_a_a] :
% 2.26/1.18 ( ( member340150864iset_a @ ( produc2037245207iset_a @ N @ ( add_mset_a @ A @ M0 ) ) @ ( mult1_a @ R ) )
% 2.26/1.18 => ( ? [M3: multiset_a] :
% 2.26/1.18 ( ( member340150864iset_a @ ( produc2037245207iset_a @ M3 @ M0 ) @ ( mult1_a @ R ) )
% 2.26/1.18 & ( N
% 2.26/1.18 = ( add_mset_a @ A @ M3 ) ) )
% 2.26/1.18 | ? [K4: multiset_a] :
% 2.26/1.18 ( ! [B7: a] :
% 2.26/1.18 ( ( member_a @ B7 @ ( set_mset_a @ K4 ) )
% 2.26/1.18 => ( member449909584od_a_a @ ( product_Pair_a_a @ B7 @ A ) @ R ) )
% 2.26/1.18 & ( N
% 2.26/1.18 = ( plus_plus_multiset_a @ M0 @ K4 ) ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_326_mult1I,axiom,
% 2.26/1.18 ! [M: multis2082063201Tree_a,A: produc143150363Tree_a,M0: multis2082063201Tree_a,N: multis2082063201Tree_a,K2: multis2082063201Tree_a,R: set_Pr220561863Tree_a] :
% 2.26/1.18 ( ( M
% 2.26/1.18 = ( add_ms205941497Tree_a @ A @ M0 ) )
% 2.26/1.18 => ( ( N
% 2.26/1.18 = ( plus_p1065544216Tree_a @ M0 @ K2 ) )
% 2.26/1.18 => ( ! [B3: produc143150363Tree_a] :
% 2.26/1.18 ( ( member254315204Tree_a @ B3 @ ( set_ms1129970904Tree_a @ K2 ) )
% 2.26/1.18 => ( member146662416Tree_a @ ( produc1002111575Tree_a @ B3 @ A ) @ R ) )
% 2.26/1.18 => ( member1298282640Tree_a @ ( produc1608222167Tree_a @ N @ M ) @ ( mult1_470979598Tree_a @ R ) ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_327_mult1I,axiom,
% 2.26/1.18 ! [M: multis782275565iset_a,A: produc1127127335iset_a,M0: multis782275565iset_a,N: multis782275565iset_a,K2: multis782275565iset_a,R: set_Pr146455751iset_a] :
% 2.26/1.18 ( ( M
% 2.26/1.18 = ( add_ms1353365509iset_a @ A @ M0 ) )
% 2.26/1.18 => ( ( N
% 2.26/1.18 = ( plus_p1919096612iset_a @ M0 @ K2 ) )
% 2.26/1.18 => ( ! [B3: produc1127127335iset_a] :
% 2.26/1.18 ( ( member340150864iset_a @ B3 @ ( set_ms1768116836iset_a @ K2 ) )
% 2.26/1.18 => ( member567864080iset_a @ ( produc1898392407iset_a @ B3 @ A ) @ R ) )
% 2.26/1.18 => ( member219455632iset_a @ ( produc1226985431iset_a @ N @ M ) @ ( mult1_599785114iset_a @ R ) ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_328_mult1I,axiom,
% 2.26/1.18 ! [M: multis599418605od_a_a,A: product_prod_a_a,M0: multis599418605od_a_a,N: multis599418605od_a_a,K2: multis599418605od_a_a,R: set_Pr1948701895od_a_a] :
% 2.26/1.18 ( ( M
% 2.26/1.18 = ( add_ms1725977093od_a_a @ A @ M0 ) )
% 2.26/1.18 => ( ( N
% 2.26/1.18 = ( plus_p404985124od_a_a @ M0 @ K2 ) )
% 2.26/1.18 => ( ! [B3: product_prod_a_a] :
% 2.26/1.18 ( ( member449909584od_a_a @ B3 @ ( set_ms1818337124od_a_a @ K2 ) )
% 2.26/1.18 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B3 @ A ) @ R ) )
% 2.26/1.18 => ( member1349055120od_a_a @ ( produc1859144151od_a_a @ N @ M ) @ ( mult1_1719650714od_a_a @ R ) ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_329_mult1I,axiom,
% 2.26/1.18 ! [M: multiset_multiset_a,A: multiset_a,M0: multiset_multiset_a,N: multiset_multiset_a,K2: multiset_multiset_a,R: set_Pr158363655iset_a] :
% 2.26/1.18 ( ( M
% 2.26/1.18 = ( add_mset_multiset_a @ A @ M0 ) )
% 2.26/1.18 => ( ( N
% 2.26/1.18 = ( plus_p1957546689iset_a @ M0 @ K2 ) )
% 2.26/1.18 => ( ! [B3: multiset_a] :
% 2.26/1.18 ( ( member_multiset_a @ B3 @ ( set_mset_multiset_a @ K2 ) )
% 2.26/1.18 => ( member340150864iset_a @ ( produc2037245207iset_a @ B3 @ A ) @ R ) )
% 2.26/1.18 => ( member104214352iset_a @ ( produc1444952343iset_a @ N @ M ) @ ( mult1_multiset_a @ R ) ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_330_mult1I,axiom,
% 2.26/1.18 ! [M: multiset_a,A: a,M0: multiset_a,N: multiset_a,K2: multiset_a,R: set_Product_prod_a_a] :
% 2.26/1.18 ( ( M
% 2.26/1.18 = ( add_mset_a @ A @ M0 ) )
% 2.26/1.18 => ( ( N
% 2.26/1.18 = ( plus_plus_multiset_a @ M0 @ K2 ) )
% 2.26/1.18 => ( ! [B3: a] :
% 2.26/1.18 ( ( member_a @ B3 @ ( set_mset_a @ K2 ) )
% 2.26/1.18 => ( member449909584od_a_a @ ( product_Pair_a_a @ B3 @ A ) @ R ) )
% 2.26/1.18 => ( member340150864iset_a @ ( produc2037245207iset_a @ N @ M ) @ ( mult1_a @ R ) ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_331_mono__mult1,axiom,
% 2.26/1.18 ! [R: set_Product_prod_a_a,R4: set_Product_prod_a_a] :
% 2.26/1.18 ( ( ord_le1824328871od_a_a @ R @ R4 )
% 2.26/1.18 => ( ord_le1978107815iset_a @ ( mult1_a @ R ) @ ( mult1_a @ R4 ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_332_mult1__union,axiom,
% 2.26/1.18 ! [B4: multiset_a,D2: multiset_a,R: set_Product_prod_a_a,C2: multiset_a] :
% 2.26/1.18 ( ( member340150864iset_a @ ( produc2037245207iset_a @ B4 @ D2 ) @ ( mult1_a @ R ) )
% 2.26/1.18 => ( member340150864iset_a @ ( produc2037245207iset_a @ ( plus_plus_multiset_a @ C2 @ B4 ) @ ( plus_plus_multiset_a @ C2 @ D2 ) ) @ ( mult1_a @ R ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_333_not__less__empty,axiom,
% 2.26/1.18 ! [M: multiset_a,R: set_Product_prod_a_a] :
% 2.26/1.18 ~ ( member340150864iset_a @ ( produc2037245207iset_a @ M @ zero_zero_multiset_a ) @ ( mult1_a @ R ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_334_mult1E,axiom,
% 2.26/1.18 ! [N: multis2082063201Tree_a,M: multis2082063201Tree_a,R: set_Pr220561863Tree_a] :
% 2.26/1.18 ( ( member1298282640Tree_a @ ( produc1608222167Tree_a @ N @ M ) @ ( mult1_470979598Tree_a @ R ) )
% 2.26/1.18 => ~ ! [A3: produc143150363Tree_a,M02: multis2082063201Tree_a] :
% 2.26/1.18 ( ( M
% 2.26/1.18 = ( add_ms205941497Tree_a @ A3 @ M02 ) )
% 2.26/1.18 => ! [K4: multis2082063201Tree_a] :
% 2.26/1.18 ( ( N
% 2.26/1.18 = ( plus_p1065544216Tree_a @ M02 @ K4 ) )
% 2.26/1.18 => ~ ! [B7: produc143150363Tree_a] :
% 2.26/1.18 ( ( member254315204Tree_a @ B7 @ ( set_ms1129970904Tree_a @ K4 ) )
% 2.26/1.18 => ( member146662416Tree_a @ ( produc1002111575Tree_a @ B7 @ A3 ) @ R ) ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_335_mult1E,axiom,
% 2.26/1.18 ! [N: multis782275565iset_a,M: multis782275565iset_a,R: set_Pr146455751iset_a] :
% 2.26/1.18 ( ( member219455632iset_a @ ( produc1226985431iset_a @ N @ M ) @ ( mult1_599785114iset_a @ R ) )
% 2.26/1.18 => ~ ! [A3: produc1127127335iset_a,M02: multis782275565iset_a] :
% 2.26/1.18 ( ( M
% 2.26/1.18 = ( add_ms1353365509iset_a @ A3 @ M02 ) )
% 2.26/1.18 => ! [K4: multis782275565iset_a] :
% 2.26/1.18 ( ( N
% 2.26/1.18 = ( plus_p1919096612iset_a @ M02 @ K4 ) )
% 2.26/1.18 => ~ ! [B7: produc1127127335iset_a] :
% 2.26/1.18 ( ( member340150864iset_a @ B7 @ ( set_ms1768116836iset_a @ K4 ) )
% 2.26/1.18 => ( member567864080iset_a @ ( produc1898392407iset_a @ B7 @ A3 ) @ R ) ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_336_mult1E,axiom,
% 2.26/1.18 ! [N: multis599418605od_a_a,M: multis599418605od_a_a,R: set_Pr1948701895od_a_a] :
% 2.26/1.18 ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ N @ M ) @ ( mult1_1719650714od_a_a @ R ) )
% 2.26/1.18 => ~ ! [A3: product_prod_a_a,M02: multis599418605od_a_a] :
% 2.26/1.18 ( ( M
% 2.26/1.18 = ( add_ms1725977093od_a_a @ A3 @ M02 ) )
% 2.26/1.18 => ! [K4: multis599418605od_a_a] :
% 2.26/1.18 ( ( N
% 2.26/1.18 = ( plus_p404985124od_a_a @ M02 @ K4 ) )
% 2.26/1.18 => ~ ! [B7: product_prod_a_a] :
% 2.26/1.18 ( ( member449909584od_a_a @ B7 @ ( set_ms1818337124od_a_a @ K4 ) )
% 2.26/1.18 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B7 @ A3 ) @ R ) ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_337_mult1E,axiom,
% 2.26/1.18 ! [N: multiset_multiset_a,M: multiset_multiset_a,R: set_Pr158363655iset_a] :
% 2.26/1.18 ( ( member104214352iset_a @ ( produc1444952343iset_a @ N @ M ) @ ( mult1_multiset_a @ R ) )
% 2.26/1.18 => ~ ! [A3: multiset_a,M02: multiset_multiset_a] :
% 2.26/1.18 ( ( M
% 2.26/1.18 = ( add_mset_multiset_a @ A3 @ M02 ) )
% 2.26/1.18 => ! [K4: multiset_multiset_a] :
% 2.26/1.18 ( ( N
% 2.26/1.18 = ( plus_p1957546689iset_a @ M02 @ K4 ) )
% 2.26/1.18 => ~ ! [B7: multiset_a] :
% 2.26/1.18 ( ( member_multiset_a @ B7 @ ( set_mset_multiset_a @ K4 ) )
% 2.26/1.18 => ( member340150864iset_a @ ( produc2037245207iset_a @ B7 @ A3 ) @ R ) ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_338_mult1E,axiom,
% 2.26/1.18 ! [N: multiset_a,M: multiset_a,R: set_Product_prod_a_a] :
% 2.26/1.18 ( ( member340150864iset_a @ ( produc2037245207iset_a @ N @ M ) @ ( mult1_a @ R ) )
% 2.26/1.18 => ~ ! [A3: a,M02: multiset_a] :
% 2.26/1.18 ( ( M
% 2.26/1.18 = ( add_mset_a @ A3 @ M02 ) )
% 2.26/1.18 => ! [K4: multiset_a] :
% 2.26/1.18 ( ( N
% 2.26/1.18 = ( plus_plus_multiset_a @ M02 @ K4 ) )
% 2.26/1.18 => ~ ! [B7: a] :
% 2.26/1.18 ( ( member_a @ B7 @ ( set_mset_a @ K4 ) )
% 2.26/1.18 => ( member449909584od_a_a @ ( product_Pair_a_a @ B7 @ A3 ) @ R ) ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_339_one__step__implies__mult,axiom,
% 2.26/1.18 ! [J2: multiset_multiset_a,K2: multiset_multiset_a,R: set_Pr158363655iset_a,I4: multiset_multiset_a] :
% 2.26/1.18 ( ( J2 != zero_z1580389697iset_a )
% 2.26/1.18 => ( ! [X3: multiset_a] :
% 2.26/1.18 ( ( member_multiset_a @ X3 @ ( set_mset_multiset_a @ K2 ) )
% 2.26/1.18 => ? [Xa: multiset_a] :
% 2.26/1.18 ( ( member_multiset_a @ Xa @ ( set_mset_multiset_a @ J2 ) )
% 2.26/1.18 & ( member340150864iset_a @ ( produc2037245207iset_a @ X3 @ Xa ) @ R ) ) )
% 2.26/1.18 => ( member104214352iset_a @ ( produc1444952343iset_a @ ( plus_p1957546689iset_a @ I4 @ K2 ) @ ( plus_p1957546689iset_a @ I4 @ J2 ) ) @ ( mult_multiset_a @ R ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_340_one__step__implies__mult,axiom,
% 2.26/1.18 ! [J2: multiset_a,K2: multiset_a,R: set_Product_prod_a_a,I4: multiset_a] :
% 2.26/1.18 ( ( J2 != zero_zero_multiset_a )
% 2.26/1.18 => ( ! [X3: a] :
% 2.26/1.18 ( ( member_a @ X3 @ ( set_mset_a @ K2 ) )
% 2.26/1.18 => ? [Xa: a] :
% 2.26/1.18 ( ( member_a @ Xa @ ( set_mset_a @ J2 ) )
% 2.26/1.18 & ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Xa ) @ R ) ) )
% 2.26/1.18 => ( member340150864iset_a @ ( produc2037245207iset_a @ ( plus_plus_multiset_a @ I4 @ K2 ) @ ( plus_plus_multiset_a @ I4 @ J2 ) ) @ ( mult_a @ R ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_341_heap__top__max,axiom,
% 2.26/1.18 ! [T2: tree_a] :
% 2.26/1.18 ( ( T2 != e_a )
% 2.26/1.18 => ( ( is_heap_a @ T2 )
% 2.26/1.18 => ( ( val_a @ T2 )
% 2.26/1.18 = ( lattic146396397_Max_a @ ( set_mset_a @ ( multiset_a2 @ T2 ) ) ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_342_mono__mult,axiom,
% 2.26/1.18 ! [R: set_Product_prod_a_a,R4: set_Product_prod_a_a] :
% 2.26/1.18 ( ( ord_le1824328871od_a_a @ R @ R4 )
% 2.26/1.18 => ( ord_le1978107815iset_a @ ( mult_a @ R ) @ ( mult_a @ R4 ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_343_mult__implies__one__step,axiom,
% 2.26/1.18 ! [R: set_Pr158363655iset_a,M: multiset_multiset_a,N: multiset_multiset_a] :
% 2.26/1.18 ( ( trans_multiset_a @ R )
% 2.26/1.18 => ( ( member104214352iset_a @ ( produc1444952343iset_a @ M @ N ) @ ( mult_multiset_a @ R ) )
% 2.26/1.18 => ? [I5: multiset_multiset_a,J3: multiset_multiset_a] :
% 2.26/1.18 ( ( N
% 2.26/1.18 = ( plus_p1957546689iset_a @ I5 @ J3 ) )
% 2.26/1.18 & ? [K4: multiset_multiset_a] :
% 2.26/1.18 ( ( M
% 2.26/1.18 = ( plus_p1957546689iset_a @ I5 @ K4 ) )
% 2.26/1.18 & ( J3 != zero_z1580389697iset_a )
% 2.26/1.18 & ! [X6: multiset_a] :
% 2.26/1.18 ( ( member_multiset_a @ X6 @ ( set_mset_multiset_a @ K4 ) )
% 2.26/1.18 => ? [Xa2: multiset_a] :
% 2.26/1.18 ( ( member_multiset_a @ Xa2 @ ( set_mset_multiset_a @ J3 ) )
% 2.26/1.18 & ( member340150864iset_a @ ( produc2037245207iset_a @ X6 @ Xa2 ) @ R ) ) ) ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_344_mult__implies__one__step,axiom,
% 2.26/1.18 ! [R: set_Product_prod_a_a,M: multiset_a,N: multiset_a] :
% 2.26/1.18 ( ( trans_a @ R )
% 2.26/1.18 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ M @ N ) @ ( mult_a @ R ) )
% 2.26/1.18 => ? [I5: multiset_a,J3: multiset_a] :
% 2.26/1.18 ( ( N
% 2.26/1.18 = ( plus_plus_multiset_a @ I5 @ J3 ) )
% 2.26/1.18 & ? [K4: multiset_a] :
% 2.26/1.18 ( ( M
% 2.26/1.18 = ( plus_plus_multiset_a @ I5 @ K4 ) )
% 2.26/1.18 & ( J3 != zero_zero_multiset_a )
% 2.26/1.18 & ! [X6: a] :
% 2.26/1.18 ( ( member_a @ X6 @ ( set_mset_a @ K4 ) )
% 2.26/1.18 => ? [Xa2: a] :
% 2.26/1.18 ( ( member_a @ Xa2 @ ( set_mset_a @ J3 ) )
% 2.26/1.18 & ( member449909584od_a_a @ ( product_Pair_a_a @ X6 @ Xa2 ) @ R ) ) ) ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_345_subset__mset_Osum__mset__0__iff,axiom,
% 2.26/1.18 ! [M: multiset_multiset_a] :
% 2.26/1.18 ( ( ( comm_m315775925iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ M )
% 2.26/1.18 = zero_zero_multiset_a )
% 2.26/1.18 = ( ! [X4: multiset_a] :
% 2.26/1.18 ( ( member_multiset_a @ X4 @ ( set_mset_multiset_a @ M ) )
% 2.26/1.18 => ( X4 = zero_zero_multiset_a ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_346_comm__monoid__add_Osum__mset_Ocong,axiom,
% 2.26/1.18 comm_m315775925iset_a = comm_m315775925iset_a ).
% 2.26/1.18
% 2.26/1.18 thf(fact_347_transD,axiom,
% 2.26/1.18 ! [R: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a,Z: multiset_a] :
% 2.26/1.18 ( ( trans_multiset_a @ R )
% 2.26/1.18 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ R )
% 2.26/1.18 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ Z ) @ R )
% 2.26/1.18 => ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Z ) @ R ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_348_transD,axiom,
% 2.26/1.18 ! [R: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 2.26/1.18 ( ( trans_a @ R )
% 2.26/1.18 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
% 2.26/1.18 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R )
% 2.26/1.18 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ R ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_349_transE,axiom,
% 2.26/1.18 ! [R: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a,Z: multiset_a] :
% 2.26/1.18 ( ( trans_multiset_a @ R )
% 2.26/1.18 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ R )
% 2.26/1.18 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ Z ) @ R )
% 2.26/1.18 => ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Z ) @ R ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_350_transE,axiom,
% 2.26/1.18 ! [R: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 2.26/1.18 ( ( trans_a @ R )
% 2.26/1.18 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
% 2.26/1.18 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R )
% 2.26/1.18 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ R ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_351_transI,axiom,
% 2.26/1.18 ! [R: set_Product_prod_a_a] :
% 2.26/1.18 ( ! [X3: a,Y3: a,Z3: a] :
% 2.26/1.18 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R )
% 2.26/1.18 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ Z3 ) @ R )
% 2.26/1.18 => ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Z3 ) @ R ) ) )
% 2.26/1.18 => ( trans_a @ R ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(fact_352_mult__cancel__add__mset,axiom,
% 2.26/1.18 ! [S3: set_Product_prod_a_a,Uu: a,X5: multiset_a,Y5: multiset_a] :
% 2.26/1.18 ( ( trans_a @ S3 )
% 2.26/1.18 => ( ( irrefl_a @ S3 )
% 2.26/1.18 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ ( add_mset_a @ Uu @ X5 ) @ ( add_mset_a @ Uu @ Y5 ) ) @ ( mult_a @ S3 ) )
% 2.26/1.18 = ( member340150864iset_a @ ( produc2037245207iset_a @ X5 @ Y5 ) @ ( mult_a @ S3 ) ) ) ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(help_If_3_1_If_001t__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_T,axiom,
% 2.26/1.18 ! [P2: $o] :
% 2.26/1.18 ( ( P2 = $true )
% 2.26/1.18 | ( P2 = $false ) ) ).
% 2.26/1.18
% 2.26/1.18 thf(help_If_2_1_If_001t__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_T,axiom,
% 2.26/1.18 ! [X: produc143150363Tree_a,Y: produc143150363Tree_a] :
% 2.26/1.18 ( ( if_Pro1144176865Tree_a @ $false @ X @ Y )
% 2.26/1.18 = Y ) ).
% 2.26/1.18
% 2.26/1.18 thf(help_If_1_1_If_001t__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J_T,axiom,
% 2.26/1.18 ! [X: produc143150363Tree_a,Y: produc143150363Tree_a] :
% 2.26/1.18 ( ( if_Pro1144176865Tree_a @ $true @ X @ Y )
% 2.26/1.18 = X ) ).
% 2.26/1.18
% 2.26/1.18 thf(conj_0,conjecture,
% 2.26/1.18 ( v_a
% 2.26/1.18 = ( product_fst_a_Tree_a @ ( heapIm837449470Leaf_a @ ( t_a @ v1 @ l1 @ r1 ) ) ) ) ).
% 2.26/1.18 ------------------------
% 2.33/1.20 % (27121)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_26921 for (2999ds/27Mi)
% 2.33/1.20 % (27124)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_26921 for (2999ds/275Mi)
% 2.33/1.20 % (27122)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_26921 for (2999ds/2Mi)
% 2.33/1.20 % (27123)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_26921 for (2999ds/2Mi)
% 2.33/1.20 % (27119)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_26921 for (2999ds/183Mi)
% 2.33/1.20 % (27120)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_26921 for (2999ds/4Mi)
% 2.33/1.20 % (27122)Instruction limit reached!
% 2.33/1.20 % (27122)------------------------------
% 2.33/1.20 % (27122)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.20 % (27122)Termination reason: Unknown
% 2.33/1.20 % (27122)Termination phase: shuffling
% 2.33/1.20
% 2.33/1.20 % (27122)Memory used [KB]: 1535
% 2.33/1.20 % (27123)Instruction limit reached!
% 2.33/1.20 % (27123)------------------------------
% 2.33/1.20 % (27123)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.20 % (27122)Time elapsed: 0.002 s
% 2.33/1.20 % (27122)Instructions burned: 2 (million)
% 2.33/1.20 % (27122)------------------------------
% 2.33/1.20 % (27122)------------------------------
% 2.33/1.20 % (27123)Termination reason: Unknown
% 2.33/1.20 % (27123)Termination phase: shuffling
% 2.33/1.20
% 2.33/1.20 % (27123)Memory used [KB]: 1535
% 2.33/1.20 % (27123)Time elapsed: 0.002 s
% 2.33/1.20 % (27123)Instructions burned: 2 (million)
% 2.33/1.20 % (27123)------------------------------
% 2.33/1.20 % (27123)------------------------------
% 2.33/1.20 % (27125)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_26921 for (2999ds/18Mi)
% 2.33/1.20 % (27120)Instruction limit reached!
% 2.33/1.20 % (27120)------------------------------
% 2.33/1.20 % (27120)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.20 % (27120)Termination reason: Unknown
% 2.33/1.20 % (27120)Termination phase: shuffling
% 2.33/1.20
% 2.33/1.20 % (27120)Memory used [KB]: 1663
% 2.33/1.20 % (27120)Time elapsed: 0.003 s
% 2.33/1.20 % (27120)Instructions burned: 4 (million)
% 2.33/1.20 % (27120)------------------------------
% 2.33/1.20 % (27120)------------------------------
% 2.33/1.21 % (27121)Instruction limit reached!
% 2.33/1.21 % (27121)------------------------------
% 2.33/1.21 % (27121)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.21 % (27121)Termination reason: Unknown
% 2.33/1.21 % (27121)Termination phase: shuffling
% 2.33/1.21
% 2.33/1.21 % (27121)Memory used [KB]: 2046
% 2.33/1.21 % (27121)Time elapsed: 0.010 s
% 2.33/1.21 % (27121)Instructions burned: 27 (million)
% 2.33/1.21 % (27121)------------------------------
% 2.33/1.21 % (27121)------------------------------
% 2.33/1.21 % (27125)Instruction limit reached!
% 2.33/1.21 % (27125)------------------------------
% 2.33/1.21 % (27125)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.21 % (27125)Termination reason: Unknown
% 2.33/1.21 % (27125)Termination phase: shuffling
% 2.33/1.21
% 2.33/1.21 % (27125)Memory used [KB]: 1918
% 2.33/1.21 % (27125)Time elapsed: 0.009 s
% 2.33/1.21 % (27125)Instructions burned: 19 (million)
% 2.33/1.21 % (27125)------------------------------
% 2.33/1.21 % (27125)------------------------------
% 2.33/1.21 % (27128)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_26921 for (2999ds/15Mi)
% 2.33/1.22 % (27126)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_26921 for (2999ds/3Mi)
% 2.33/1.22 % (27127)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_26921 for (2999ds/37Mi)
% 2.33/1.22 % (27126)Instruction limit reached!
% 2.33/1.22 % (27126)------------------------------
% 2.33/1.22 % (27126)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.22 % (27126)Termination reason: Unknown
% 2.33/1.22 % (27126)Termination phase: shuffling
% 2.33/1.22
% 2.33/1.22 % (27126)Memory used [KB]: 1663
% 2.33/1.22 % (27126)Time elapsed: 0.004 s
% 2.33/1.22 % (27126)Instructions burned: 4 (million)
% 2.33/1.22 % (27126)------------------------------
% 2.33/1.22 % (27126)------------------------------
% 2.33/1.22 % (27136)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_26921 for (2999ds/3Mi)
% 2.33/1.22 % (27136)Instruction limit reached!
% 2.33/1.22 % (27136)------------------------------
% 2.33/1.22 % (27136)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.22 % (27136)Termination reason: Unknown
% 2.33/1.22 % (27136)Termination phase: shuffling
% 2.33/1.22
% 2.33/1.22 % (27136)Memory used [KB]: 1663
% 2.33/1.22 % (27136)Time elapsed: 0.002 s
% 2.33/1.22 % (27136)Instructions burned: 3 (million)
% 2.33/1.22 % (27136)------------------------------
% 2.33/1.22 % (27136)------------------------------
% 2.33/1.22 % (27128)Instruction limit reached!
% 2.33/1.22 % (27128)------------------------------
% 2.33/1.22 % (27128)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.22 % (27128)Termination reason: Unknown
% 2.33/1.22 % (27128)Termination phase: shuffling
% 2.33/1.22
% 2.33/1.22 % (27128)Memory used [KB]: 1918
% 2.33/1.22 % (27128)Time elapsed: 0.009 s
% 2.33/1.22 % (27128)Instructions burned: 15 (million)
% 2.33/1.22 % (27128)------------------------------
% 2.33/1.22 % (27128)------------------------------
% 2.33/1.23 % (27137)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_26921 for (2999ds/1041Mi)
% 2.33/1.23 % (27138)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_26921 for (2999ds/7Mi)
% 2.33/1.23 % (27127)Instruction limit reached!
% 2.33/1.23 % (27127)------------------------------
% 2.33/1.23 % (27127)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.23 % (27127)Termination reason: Unknown
% 2.33/1.23 % (27127)Termination phase: shuffling
% 2.33/1.23
% 2.33/1.23 % (27127)Memory used [KB]: 2302
% 2.33/1.23 % (27127)Time elapsed: 0.016 s
% 2.33/1.23 % (27127)Instructions burned: 37 (million)
% 2.33/1.23 % (27127)------------------------------
% 2.33/1.23 % (27127)------------------------------
% 2.33/1.23 % (27138)Instruction limit reached!
% 2.33/1.23 % (27138)------------------------------
% 2.33/1.23 % (27138)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.23 % (27138)Termination reason: Unknown
% 2.33/1.23 % (27138)Termination phase: shuffling
% 2.33/1.23
% 2.33/1.23 % (27138)Memory used [KB]: 1791
% 2.33/1.23 % (27138)Time elapsed: 0.004 s
% 2.33/1.23 % (27138)Instructions burned: 8 (million)
% 2.33/1.23 % (27138)------------------------------
% 2.33/1.23 % (27138)------------------------------
% 2.33/1.23 % (27139)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_26921 for (2999ds/16Mi)
% 2.33/1.24 % (27142)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_26921 for (2999ds/3Mi)
% 2.33/1.24 % (27142)Instruction limit reached!
% 2.33/1.24 % (27142)------------------------------
% 2.33/1.24 % (27142)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.24 % (27142)Termination reason: Unknown
% 2.33/1.24 % (27142)Termination phase: shuffling
% 2.33/1.24
% 2.33/1.24 % (27142)Memory used [KB]: 1663
% 2.33/1.24 % (27142)Time elapsed: 0.003 s
% 2.33/1.24 % (27142)Instructions burned: 3 (million)
% 2.33/1.24 % (27142)------------------------------
% 2.33/1.24 % (27142)------------------------------
% 2.33/1.24 % (27146)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_26921 for (2999ds/7Mi)
% 2.33/1.24 % (27139)Instruction limit reached!
% 2.33/1.24 % (27139)------------------------------
% 2.33/1.24 % (27139)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.24 % (27139)Termination reason: Unknown
% 2.33/1.24 % (27139)Termination phase: shuffling
% 2.33/1.24
% 2.33/1.24 % (27139)Memory used [KB]: 1918
% 2.33/1.24 % (27139)Time elapsed: 0.010 s
% 2.33/1.24 % (27139)Instructions burned: 17 (million)
% 2.33/1.24 % (27139)------------------------------
% 2.33/1.24 % (27139)------------------------------
% 2.33/1.24 % (27124)Refutation not found, incomplete strategy
% 2.33/1.24 % (27124)------------------------------
% 2.33/1.24 % (27124)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.24 % (27124)Termination reason: Refutation not found, incomplete strategy
% 2.33/1.24
% 2.33/1.24
% 2.33/1.24 % (27124)Memory used [KB]: 7291
% 2.33/1.24 % (27124)Time elapsed: 0.045 s
% 2.33/1.24 % (27124)Instructions burned: 94 (million)
% 2.33/1.24 % (27124)------------------------------
% 2.33/1.24 % (27124)------------------------------
% 2.33/1.24 % (27146)Instruction limit reached!
% 2.33/1.24 % (27146)------------------------------
% 2.33/1.24 % (27146)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.24 % (27146)Termination reason: Unknown
% 2.33/1.24 % (27146)Termination phase: shuffling
% 2.33/1.24
% 2.33/1.24 % (27146)Memory used [KB]: 1791
% 2.33/1.24 % (27146)Time elapsed: 0.004 s
% 2.33/1.24 % (27146)Instructions burned: 9 (million)
% 2.33/1.24 % (27146)------------------------------
% 2.33/1.24 % (27146)------------------------------
% 2.33/1.25 % (27145)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_26921 for (2999ds/3Mi)
% 2.33/1.25 % (27145)Instruction limit reached!
% 2.33/1.25 % (27145)------------------------------
% 2.33/1.25 % (27145)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.25 % (27145)Termination reason: Unknown
% 2.33/1.25 % (27145)Termination phase: shuffling
% 2.33/1.25
% 2.33/1.25 % (27145)Memory used [KB]: 1663
% 2.33/1.25 % (27145)Time elapsed: 0.003 s
% 2.33/1.25 % (27145)Instructions burned: 5 (million)
% 2.33/1.25 % (27145)------------------------------
% 2.33/1.25 % (27145)------------------------------
% 2.33/1.26 % (27150)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_26921 for (2999ds/3Mi)
% 2.33/1.26 % (27154)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_26921 for (2999ds/710Mi)
% 2.33/1.26 % (27151)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_26921 for (2999ds/4Mi)
% 2.33/1.26 % (27150)Instruction limit reached!
% 2.33/1.26 % (27150)------------------------------
% 2.33/1.26 % (27150)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.26 % (27150)Termination reason: Unknown
% 2.33/1.26 % (27150)Termination phase: shuffling
% 2.33/1.26
% 2.33/1.26 % (27150)Memory used [KB]: 1663
% 2.33/1.26 % (27150)Time elapsed: 0.003 s
% 2.33/1.26 % (27150)Instructions burned: 3 (million)
% 2.33/1.26 % (27150)------------------------------
% 2.33/1.26 % (27150)------------------------------
% 2.33/1.26 % (27152)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_26921 for (2999ds/18Mi)
% 2.33/1.26 % (27155)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_26921 for (2999ds/6Mi)
% 2.33/1.26 % (27151)Instruction limit reached!
% 2.33/1.26 % (27151)------------------------------
% 2.33/1.26 % (27151)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.26 % (27151)Termination reason: Unknown
% 2.33/1.26 % (27151)Termination phase: shuffling
% 2.33/1.26
% 2.33/1.26 % (27151)Memory used [KB]: 1663
% 2.33/1.26 % (27151)Time elapsed: 0.004 s
% 2.33/1.26 % (27151)Instructions burned: 5 (million)
% 2.33/1.26 % (27151)------------------------------
% 2.33/1.26 % (27151)------------------------------
% 2.33/1.26 % (27155)Instruction limit reached!
% 2.33/1.26 % (27155)------------------------------
% 2.33/1.26 % (27155)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.26 % (27155)Termination reason: Unknown
% 2.33/1.26 % (27155)Termination phase: shuffling
% 2.33/1.26
% 2.33/1.26 % (27155)Memory used [KB]: 1791
% 2.33/1.26 % (27155)Time elapsed: 0.005 s
% 2.33/1.26 % (27155)Instructions burned: 7 (million)
% 2.33/1.26 % (27155)------------------------------
% 2.33/1.26 % (27155)------------------------------
% 2.33/1.27 % (27152)Instruction limit reached!
% 2.33/1.27 % (27152)------------------------------
% 2.33/1.27 % (27152)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.27 % (27152)Termination reason: Unknown
% 2.33/1.27 % (27152)Termination phase: shuffling
% 2.33/1.27
% 2.33/1.27 % (27152)Memory used [KB]: 1918
% 2.33/1.27 % (27152)Time elapsed: 0.010 s
% 2.33/1.27 % (27152)Instructions burned: 18 (million)
% 2.33/1.27 % (27152)------------------------------
% 2.33/1.27 % (27152)------------------------------
% 2.33/1.28 % (27158)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_26921 for (2999ds/902Mi)
% 2.33/1.28 % (27163)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_26921 for (2999ds/6Mi)
% 2.33/1.28 % (27160)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_26921 for (2999ds/21Mi)
% 2.33/1.28 % (27163)Instruction limit reached!
% 2.33/1.28 % (27163)------------------------------
% 2.33/1.28 % (27163)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.28 % (27163)Termination reason: Unknown
% 2.33/1.28 % (27163)Termination phase: shuffling
% 2.33/1.28
% 2.33/1.28 % (27163)Memory used [KB]: 1791
% 2.33/1.28 % (27163)Time elapsed: 0.003 s
% 2.33/1.28 % (27163)Instructions burned: 6 (million)
% 2.33/1.28 % (27163)------------------------------
% 2.33/1.28 % (27163)------------------------------
% 2.33/1.28 % (27161)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_26921 for (2999ds/5Mi)
% 2.33/1.28 % (27161)Instruction limit reached!
% 2.33/1.28 % (27161)------------------------------
% 2.33/1.28 % (27161)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.28 % (27161)Termination reason: Unknown
% 2.33/1.28 % (27161)Termination phase: shuffling
% 2.33/1.28
% 2.33/1.28 % (27161)Memory used [KB]: 1791
% 2.33/1.28 % (27161)Time elapsed: 0.004 s
% 2.33/1.28 % (27161)Instructions burned: 6 (million)
% 2.33/1.28 % (27161)------------------------------
% 2.33/1.28 % (27161)------------------------------
% 2.33/1.28 % (27119)Instruction limit reached!
% 2.33/1.28 % (27119)------------------------------
% 2.33/1.28 % (27119)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.28 % (27119)Termination reason: Unknown
% 2.33/1.28 % (27119)Termination phase: Saturation
% 2.33/1.28
% 2.33/1.28 % (27119)Memory used [KB]: 7547
% 2.33/1.28 % (27119)Time elapsed: 0.084 s
% 2.33/1.28 % (27119)Instructions burned: 184 (million)
% 2.33/1.28 % (27119)------------------------------
% 2.33/1.28 % (27119)------------------------------
% 2.33/1.29 % (27167)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_26921 for (2999ds/377Mi)
% 2.33/1.29 % (27160)Instruction limit reached!
% 2.33/1.29 % (27160)------------------------------
% 2.33/1.29 % (27160)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.29 % (27160)Termination reason: Unknown
% 2.33/1.29 % (27160)Termination phase: shuffling
% 2.33/1.29
% 2.33/1.29 % (27160)Memory used [KB]: 2046
% 2.33/1.29 % (27160)Time elapsed: 0.012 s
% 2.33/1.29 % (27160)Instructions burned: 22 (million)
% 2.33/1.29 % (27160)------------------------------
% 2.33/1.29 % (27160)------------------------------
% 2.33/1.30 % (27169)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_26921 for (2998ds/19Mi)
% 2.33/1.30 % (27168)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_26921 for (2998ds/779Mi)
% 2.33/1.31 % (27172)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_26921 for (2998ds/879Mi)
% 2.33/1.31 % (27169)Instruction limit reached!
% 2.33/1.31 % (27169)------------------------------
% 2.33/1.31 % (27169)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.31 % (27169)Termination reason: Unknown
% 2.33/1.31 % (27169)Termination phase: shuffling
% 2.33/1.31
% 2.33/1.31 % (27169)Memory used [KB]: 2046
% 2.33/1.31 % (27169)Time elapsed: 0.010 s
% 2.33/1.31 % (27169)Instructions burned: 19 (million)
% 2.33/1.31 % (27169)------------------------------
% 2.33/1.31 % (27169)------------------------------
% 2.33/1.33 % (27176)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_26921 for (2998ds/17Mi)
% 2.33/1.33 % (27176)Instruction limit reached!
% 2.33/1.33 % (27176)------------------------------
% 2.33/1.33 % (27176)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.33 % (27176)Termination reason: Unknown
% 2.33/1.33 % (27176)Termination phase: shuffling
% 2.33/1.33
% 2.33/1.33 % (27176)Memory used [KB]: 1918
% 2.33/1.33 % (27176)Time elapsed: 0.011 s
% 2.33/1.33 % (27176)Instructions burned: 18 (million)
% 2.33/1.33 % (27176)------------------------------
% 2.33/1.33 % (27176)------------------------------
% 2.33/1.35 % (27179)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_26921 for (2998ds/3Mi)
% 2.33/1.36 % (27179)Instruction limit reached!
% 2.33/1.36 % (27179)------------------------------
% 2.33/1.36 % (27179)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.36 % (27179)Termination reason: Unknown
% 2.33/1.36 % (27179)Termination phase: shuffling
% 2.33/1.36
% 2.33/1.36 % (27179)Memory used [KB]: 1663
% 2.33/1.36 % (27179)Time elapsed: 0.004 s
% 2.33/1.36 % (27179)Instructions burned: 4 (million)
% 2.33/1.36 % (27179)------------------------------
% 2.33/1.36 % (27179)------------------------------
% 2.33/1.36 % (27158)Refutation not found, incomplete strategy
% 2.33/1.36 % (27158)------------------------------
% 2.33/1.36 % (27158)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.36 % (27158)Termination reason: Refutation not found, incomplete strategy
% 2.33/1.36
% 2.33/1.36
% 2.33/1.36 % (27158)Memory used [KB]: 8315
% 2.33/1.36 % (27158)Time elapsed: 0.090 s
% 2.33/1.36 % (27158)Instructions burned: 186 (million)
% 2.33/1.36 % (27158)------------------------------
% 2.33/1.36 % (27158)------------------------------
% 2.33/1.37 % (27182)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_26921 for (2998ds/30Mi)
% 2.33/1.38 % (27183)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_26921 for (2998ds/127Mi)
% 2.33/1.39 % (27182)Instruction limit reached!
% 2.33/1.39 % (27182)------------------------------
% 2.33/1.39 % (27182)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.39 % (27182)Termination reason: Unknown
% 2.33/1.39 % (27182)Termination phase: shuffling
% 2.33/1.39
% 2.33/1.39 % (27182)Memory used [KB]: 2302
% 2.33/1.39 % (27182)Time elapsed: 0.017 s
% 2.33/1.39 % (27182)Instructions burned: 31 (million)
% 2.33/1.39 % (27182)------------------------------
% 2.33/1.39 % (27182)------------------------------
% 2.33/1.40 % (27186)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_26921 for (2997ds/100Mi)
% 2.33/1.42 % (27167)Instruction limit reached!
% 2.33/1.42 % (27167)------------------------------
% 2.33/1.42 % (27167)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.42 % (27167)Termination reason: Unknown
% 2.33/1.42 % (27167)Termination phase: Saturation
% 2.33/1.42
% 2.33/1.42 % (27167)Memory used [KB]: 8187
% 2.33/1.42 % (27167)Time elapsed: 0.132 s
% 2.33/1.42 % (27167)Instructions burned: 378 (million)
% 2.33/1.42 % (27167)------------------------------
% 2.33/1.42 % (27167)------------------------------
% 2.33/1.42 % (27183)Instruction limit reached!
% 2.33/1.42 % (27183)------------------------------
% 2.33/1.42 % (27183)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.42 % (27183)Termination reason: Unknown
% 2.33/1.42 % (27183)Termination phase: Property scanning
% 2.33/1.42
% 2.33/1.42 % (27183)Memory used [KB]: 2942
% 2.33/1.42 % (27183)Time elapsed: 0.045 s
% 2.33/1.42 % (27183)Instructions burned: 127 (million)
% 2.33/1.42 % (27183)------------------------------
% 2.33/1.42 % (27183)------------------------------
% 2.33/1.43 % (27189)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_26921 for (2997ds/3Mi)
% 2.33/1.43 % (27186)Instruction limit reached!
% 2.33/1.43 % (27186)------------------------------
% 2.33/1.43 % (27186)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.43 % (27186)Termination reason: Unknown
% 2.33/1.43 % (27186)Termination phase: Property scanning
% 2.33/1.43
% 2.33/1.43 % (27186)Memory used [KB]: 3070
% 2.33/1.43 % (27186)Time elapsed: 0.028 s
% 2.33/1.43 % (27186)Instructions burned: 100 (million)
% 2.33/1.43 % (27186)------------------------------
% 2.33/1.43 % (27186)------------------------------
% 2.33/1.43 % (27189)Instruction limit reached!
% 2.33/1.43 % (27189)------------------------------
% 2.33/1.43 % (27189)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.43 % (27189)Termination reason: Unknown
% 2.33/1.43 % (27189)Termination phase: shuffling
% 2.33/1.43
% 2.33/1.43 % (27189)Memory used [KB]: 1663
% 2.33/1.43 % (27189)Time elapsed: 0.003 s
% 2.33/1.43 % (27189)Instructions burned: 5 (million)
% 2.33/1.43 % (27189)------------------------------
% 2.33/1.43 % (27189)------------------------------
% 2.33/1.43 % (27190)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_26921 for (2997ds/20Mi)
% 2.33/1.44 % (27191)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_26921 for (2997ds/86Mi)
% 2.33/1.44 % (27192)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_26921 for (2997ds/107Mi)
% 2.33/1.44 % (27190)Instruction limit reached!
% 2.33/1.44 % (27190)------------------------------
% 2.33/1.44 % (27190)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.44 % (27190)Termination reason: Unknown
% 2.33/1.44 % (27190)Termination phase: shuffling
% 2.33/1.44
% 2.33/1.44 % (27190)Memory used [KB]: 2046
% 2.33/1.44 % (27190)Time elapsed: 0.008 s
% 2.33/1.44 % (27190)Instructions burned: 22 (million)
% 2.33/1.44 % (27190)------------------------------
% 2.33/1.44 % (27190)------------------------------
% 2.33/1.45 % (27193)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_26921 for (2997ds/39Mi)
% 2.33/1.46 % (27193)Instruction limit reached!
% 2.33/1.46 % (27193)------------------------------
% 2.33/1.46 % (27193)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.46 % (27193)Termination reason: Unknown
% 2.33/1.46 % (27193)Termination phase: shuffling
% 2.33/1.46
% 2.33/1.46 % (27193)Memory used [KB]: 2430
% 2.33/1.46 % (27193)Time elapsed: 0.013 s
% 2.33/1.46 % (27193)Instructions burned: 41 (million)
% 2.33/1.46 % (27193)------------------------------
% 2.33/1.46 % (27193)------------------------------
% 2.33/1.47 % (27191)Instruction limit reached!
% 2.33/1.47 % (27191)------------------------------
% 2.33/1.47 % (27191)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.47 % (27191)Termination reason: Unknown
% 2.33/1.47 % (27191)Termination phase: Property scanning
% 2.33/1.47
% 2.33/1.47 % (27191)Memory used [KB]: 2942
% 2.33/1.47 % (27191)Time elapsed: 0.027 s
% 2.33/1.47 % (27191)Instructions burned: 88 (million)
% 2.33/1.47 % (27191)------------------------------
% 2.33/1.47 % (27191)------------------------------
% 2.33/1.47 % (27192)Instruction limit reached!
% 2.33/1.47 % (27192)------------------------------
% 2.33/1.47 % (27192)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.47 % (27192)Termination reason: Unknown
% 2.33/1.47 % (27192)Termination phase: Saturation
% 2.33/1.47
% 2.33/1.47 % (27192)Memory used [KB]: 2686
% 2.33/1.47 % (27192)Time elapsed: 0.032 s
% 2.33/1.47 % (27192)Instructions burned: 109 (million)
% 2.33/1.47 % (27192)------------------------------
% 2.33/1.47 % (27192)------------------------------
% 2.33/1.47 % (27194)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_26921 for (2997ds/448Mi)
% 2.33/1.48 % (27195)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_26921 for (2997ds/46Mi)
% 2.33/1.48 % (27196)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_26921 for (2997ds/98Mi)
% 2.33/1.49 % (27195)Instruction limit reached!
% 2.33/1.49 % (27195)------------------------------
% 2.33/1.49 % (27195)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.49 % (27195)Termination reason: Unknown
% 2.33/1.49 % (27195)Termination phase: Property scanning
% 2.33/1.49
% 2.33/1.49 % (27195)Memory used [KB]: 2302
% 2.33/1.49 % (27195)Time elapsed: 0.016 s
% 2.33/1.49 % (27195)Instructions burned: 49 (million)
% 2.33/1.49 % (27195)------------------------------
% 2.33/1.49 % (27195)------------------------------
% 2.33/1.50 % (27197)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_26921 for (2996ds/507Mi)
% 2.33/1.51 % (27196)Instruction limit reached!
% 2.33/1.51 % (27196)------------------------------
% 2.33/1.51 % (27196)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.51 % (27196)Termination reason: Unknown
% 2.33/1.51 % (27196)Termination phase: Property scanning
% 2.33/1.51
% 2.33/1.51 % (27196)Memory used [KB]: 2942
% 2.33/1.51 % (27196)Time elapsed: 0.028 s
% 2.33/1.51 % (27196)Instructions burned: 101 (million)
% 2.33/1.51 % (27196)------------------------------
% 2.33/1.51 % (27196)------------------------------
% 2.33/1.51 % (27172)First to succeed.
% 2.33/1.52 % (27172)Refutation found. Thanks to Tanya!
% 2.33/1.52 % SZS status Theorem for DTF2THF_26921
% 2.33/1.52 % SZS output start Proof for DTF2THF_26921
% 2.33/1.52 thf(type_def_5, type, multiset_multiset_a: $tType).
% 2.33/1.52 thf(type_def_7, type, multis782275565iset_a: $tType).
% 2.33/1.52 thf(type_def_8, type, multis2082063201Tree_a: $tType).
% 2.33/1.52 thf(type_def_9, type, multis599418605od_a_a: $tType).
% 2.33/1.52 thf(type_def_10, type, multiset_a: $tType).
% 2.33/1.52 thf(type_def_11, type, tree_a: $tType).
% 2.33/1.52 thf(type_def_12, type, list_a: $tType).
% 2.33/1.52 thf(type_def_13, type, produc143150363Tree_a: $tType).
% 2.33/1.52 thf(type_def_14, type, a: $tType).
% 2.33/1.52 thf(type_def_15, type, product_prod_a_a: $tType).
% 2.33/1.52 thf(type_def_16, type, set_a: $tType).
% 2.33/1.52 thf(type_def_17, type, produc1127127335iset_a: $tType).
% 2.33/1.52 thf(type_def_18, type, set_Pr158363655iset_a: $tType).
% 2.33/1.52 thf(type_def_19, type, set_Pr720484615iset_a: $tType).
% 2.33/1.52 thf(type_def_20, type, set_Pr146455751iset_a: $tType).
% 2.33/1.52 thf(type_def_21, type, set_Pr1066112583iset_a: $tType).
% 2.33/1.52 thf(type_def_22, type, set_Pr220561863Tree_a: $tType).
% 2.33/1.52 thf(type_def_23, type, set_Pr295926343Tree_a: $tType).
% 2.33/1.52 thf(type_def_24, type, set_Pr1948701895od_a_a: $tType).
% 2.33/1.52 thf(type_def_25, type, set_Pr1821581383od_a_a: $tType).
% 2.33/1.52 thf(type_def_26, type, set_Product_prod_a_a: $tType).
% 2.33/1.52 thf(type_def_27, type, set_multiset_a: $tType).
% 2.33/1.52 thf(type_def_28, type, set_Pr1070816379Tree_a: $tType).
% 2.33/1.52 thf(type_def_29, type, set_Pr921883667ee_a_a: $tType).
% 2.33/1.52 thf(type_def_30, type, produc981471411ee_a_a: $tType).
% 2.33/1.52 thf(type_def_31, type, produc1293660967iset_a: $tType).
% 2.33/1.52 thf(type_def_32, type, produc741964647iset_a: $tType).
% 2.33/1.52 thf(type_def_33, type, produc1136638567Tree_a: $tType).
% 2.33/1.52 thf(type_def_34, type, produc2061001575od_a_a: $tType).
% 2.33/1.52 thf(type_def_35, type, produc558600423iset_a: $tType).
% 2.33/1.52 thf(type_def_36, type, produc1254371559Tree_a: $tType).
% 2.33/1.52 thf(type_def_37, type, produc1572603623od_a_a: $tType).
% 2.33/1.52 thf(func_def_0, type, set_Pr1066112583iset_a: $tType).
% 2.33/1.52 thf(func_def_1, type, produc741964647iset_a: $tType).
% 2.33/1.52 thf(func_def_2, type, set_Pr146455751iset_a: $tType).
% 2.33/1.52 thf(func_def_3, type, set_Pr295926343Tree_a: $tType).
% 2.33/1.52 thf(func_def_4, type, produc558600423iset_a: $tType).
% 2.33/1.52 thf(func_def_5, type, produc1136638567Tree_a: $tType).
% 2.33/1.52 thf(func_def_6, type, set_Pr1821581383od_a_a: $tType).
% 2.33/1.52 thf(func_def_7, type, set_Pr220561863Tree_a: $tType).
% 2.33/1.52 thf(func_def_8, type, produc2061001575od_a_a: $tType).
% 2.33/1.52 thf(func_def_9, type, set_Pr720484615iset_a: $tType).
% 2.33/1.52 thf(func_def_10, type, produc1254371559Tree_a: $tType).
% 2.33/1.52 thf(func_def_11, type, produc1293660967iset_a: $tType).
% 2.33/1.52 thf(func_def_12, type, set_Pr1948701895od_a_a: $tType).
% 2.33/1.52 thf(func_def_13, type, multis782275565iset_a: $tType).
% 2.33/1.52 thf(func_def_14, type, produc1572603623od_a_a: $tType).
% 2.33/1.52 thf(func_def_15, type, set_Pr158363655iset_a: $tType).
% 2.33/1.52 thf(func_def_16, type, produc1127127335iset_a: $tType).
% 2.33/1.52 thf(func_def_17, type, multis2082063201Tree_a: $tType).
% 2.33/1.52 thf(func_def_18, type, set_Pr1070816379Tree_a: $tType).
% 2.33/1.52 thf(func_def_19, type, set_Pr921883667ee_a_a: $tType).
% 2.33/1.52 thf(func_def_20, type, multis599418605od_a_a: $tType).
% 2.33/1.52 thf(func_def_21, type, produc143150363Tree_a: $tType).
% 2.33/1.52 thf(func_def_22, type, produc981471411ee_a_a: $tType).
% 2.33/1.52 thf(func_def_23, type, set_Product_prod_a_a: $tType).
% 2.33/1.52 thf(func_def_24, type, multiset_multiset_a: $tType).
% 2.33/1.52 thf(func_def_25, type, set_multiset_a: $tType).
% 2.33/1.52 thf(func_def_26, type, product_prod_a_a: $tType).
% 2.33/1.52 thf(func_def_27, type, multiset_a: $tType).
% 2.33/1.52 thf(func_def_28, type, list_a: $tType).
% 2.33/1.52 thf(func_def_29, type, tree_a: $tType).
% 2.33/1.52 thf(func_def_30, type, set_a: $tType).
% 2.33/1.52 thf(func_def_31, type, a: $tType).
% 2.33/1.52 thf(func_def_32, type, plus_p1957546689iset_a: multiset_multiset_a > multiset_multiset_a > multiset_multiset_a).
% 2.33/1.52 thf(func_def_33, type, plus_p1919096612iset_a: multis782275565iset_a > multis782275565iset_a > multis782275565iset_a).
% 2.33/1.52 thf(func_def_34, type, plus_p1065544216Tree_a: multis2082063201Tree_a > multis2082063201Tree_a > multis2082063201Tree_a).
% 2.33/1.52 thf(func_def_35, type, plus_p404985124od_a_a: multis599418605od_a_a > multis599418605od_a_a > multis599418605od_a_a).
% 2.33/1.52 thf(func_def_36, type, plus_plus_multiset_a: multiset_a > multiset_a > multiset_a).
% 2.33/1.52 thf(func_def_37, type, zero_z1580389697iset_a: multiset_multiset_a).
% 2.33/1.52 thf(func_def_38, type, zero_z1941791140iset_a: multis782275565iset_a).
% 2.33/1.52 thf(func_def_39, type, zero_z2144236696Tree_a: multis2082063201Tree_a).
% 2.33/1.52 thf(func_def_40, type, zero_z160849828od_a_a: multis599418605od_a_a).
% 2.33/1.52 thf(func_def_41, type, zero_zero_multiset_a: multiset_a).
% 2.33/1.52 thf(func_def_42, type, nO_MAT1617603563iset_a: multiset_a > multiset_a > $o).
% 2.33/1.52 thf(func_def_43, type, heapIm970322378pify_a: tree_a > tree_a).
% 2.33/1.52 thf(func_def_44, type, heapIm229596386mpty_a: tree_a > $o).
% 2.33/1.52 thf(func_def_45, type, heapIm1057938560list_a: list_a > tree_a).
% 2.33/1.52 thf(func_def_46, type, heapIm915766629_max_a: tree_a > produc143150363Tree_a).
% 2.33/1.52 thf(func_def_47, type, heapIm1140443833left_a: tree_a > tree_a).
% 2.33/1.52 thf(func_def_48, type, heapIm1637418125tree_a: list_a > tree_a).
% 2.33/1.52 thf(func_def_49, type, heapIm837449470Leaf_a: tree_a > produc143150363Tree_a).
% 2.33/1.52 thf(func_def_50, type, heapIm1257206334ight_a: tree_a > tree_a).
% 2.33/1.52 thf(func_def_51, type, heapIm1091024090Down_a: tree_a > tree_a).
% 2.33/1.52 thf(func_def_52, type, heap_Tree_a_a: tree_a > (tree_a > $o) > (list_a > tree_a) > (tree_a > multiset_a) > (tree_a > tree_a) > (tree_a > produc143150363Tree_a) > $o).
% 2.33/1.52 thf(func_def_53, type, heap_a_a: a > (a > $o) > (list_a > a) > (a > multiset_a) > (a > tree_a) > (a > product_prod_a_a) > $o).
% 2.33/1.52 thf(func_def_54, type, heap_axioms_Tree_a_a: (tree_a > $o) > (list_a > tree_a) > (tree_a > multiset_a) > (tree_a > tree_a) > (tree_a > produc143150363Tree_a) > $o).
% 2.33/1.52 thf(func_def_55, type, heap_axioms_a_a: (a > $o) > (list_a > a) > (a > multiset_a) > (a > tree_a) > (a > product_prod_a_a) > $o).
% 2.33/1.52 thf(func_def_56, type, e_a: tree_a).
% 2.33/1.52 thf(func_def_57, type, t_a: a > tree_a > tree_a > tree_a).
% 2.33/1.52 thf(func_def_58, type, in_tree_a: a > tree_a > $o).
% 2.33/1.52 thf(func_def_59, type, is_heap_a: tree_a > $o).
% 2.33/1.52 thf(func_def_60, type, multiset_a2: tree_a > multiset_a).
% 2.33/1.52 thf(func_def_61, type, val_a: tree_a > a).
% 2.33/1.52 thf(func_def_62, type, if_Pro1144176865Tree_a: $o > produc143150363Tree_a > produc143150363Tree_a > produc143150363Tree_a).
% 2.33/1.52 thf(func_def_63, type, lattic146396397_Max_a: set_a > a).
% 2.33/1.52 thf(func_def_64, type, add_mset_multiset_a: multiset_a > multiset_multiset_a > multiset_multiset_a).
% 2.33/1.52 thf(func_def_65, type, add_ms1353365509iset_a: produc1127127335iset_a > multis782275565iset_a > multis782275565iset_a).
% 2.33/1.52 thf(func_def_66, type, add_ms205941497Tree_a: produc143150363Tree_a > multis2082063201Tree_a > multis2082063201Tree_a).
% 2.33/1.52 thf(func_def_67, type, add_ms1725977093od_a_a: product_prod_a_a > multis599418605od_a_a > multis599418605od_a_a).
% 2.33/1.52 thf(func_def_68, type, add_mset_a: a > multiset_a > multiset_a).
% 2.33/1.52 thf(func_def_69, type, comm_m315775925iset_a: (multiset_a > multiset_a > multiset_a) > multiset_a > multiset_multiset_a > multiset_a).
% 2.33/1.52 thf(func_def_70, type, is_empty_a: multiset_a > $o).
% 2.33/1.52 thf(func_def_71, type, mult1_multiset_a: set_Pr158363655iset_a > set_Pr720484615iset_a).
% 2.33/1.52 thf(func_def_72, type, mult1_599785114iset_a: set_Pr146455751iset_a > set_Pr1066112583iset_a).
% 2.33/1.52 thf(func_def_73, type, mult1_470979598Tree_a: set_Pr220561863Tree_a > set_Pr295926343Tree_a).
% 2.33/1.52 thf(func_def_74, type, mult1_1719650714od_a_a: set_Pr1948701895od_a_a > set_Pr1821581383od_a_a).
% 2.33/1.52 thf(func_def_75, type, mult1_a: set_Product_prod_a_a > set_Pr158363655iset_a).
% 2.33/1.52 thf(func_def_76, type, mult_multiset_a: set_Pr158363655iset_a > set_Pr720484615iset_a).
% 2.33/1.52 thf(func_def_77, type, mult_a: set_Product_prod_a_a > set_Pr158363655iset_a).
% 2.33/1.52 thf(func_def_78, type, set_mset_multiset_a: multiset_multiset_a > set_multiset_a).
% 2.33/1.52 thf(func_def_79, type, set_ms1768116836iset_a: multis782275565iset_a > set_Pr158363655iset_a).
% 2.33/1.52 thf(func_def_80, type, set_ms1129970904Tree_a: multis2082063201Tree_a > set_Pr1070816379Tree_a).
% 2.33/1.52 thf(func_def_81, type, set_ms1818337124od_a_a: multis599418605od_a_a > set_Product_prod_a_a).
% 2.33/1.52 thf(func_def_82, type, set_mset_a: multiset_a > set_a).
% 2.33/1.52 thf(func_def_83, type, ord_le1748873858_a_a_o: (tree_a > a > $o) > (tree_a > a > $o) > $o).
% 2.33/1.52 thf(func_def_84, type, ord_le51495822et_a_o: (multiset_a > multiset_a > $o) > (multiset_a > multiset_a > $o) > $o).
% 2.33/1.52 thf(func_def_85, type, ord_le1327026842ee_a_o: (a > tree_a > $o) > (a > tree_a > $o) > $o).
% 2.33/1.52 thf(func_def_86, type, ord_less_eq_a_a_o: (a > a > $o) > (a > a > $o) > $o).
% 2.33/1.52 thf(func_def_87, type, ord_le1199012836iset_a: multiset_a > multiset_a > $o).
% 2.33/1.52 thf(func_def_88, type, ord_le864074675ee_a_a: set_Pr921883667ee_a_a > set_Pr921883667ee_a_a > $o).
% 2.33/1.52 thf(func_def_89, type, ord_le1978107815iset_a: set_Pr158363655iset_a > set_Pr158363655iset_a > $o).
% 2.33/1.52 thf(func_def_90, type, ord_le1013007387Tree_a: set_Pr1070816379Tree_a > set_Pr1070816379Tree_a > $o).
% 2.33/1.52 thf(func_def_91, type, ord_le1824328871od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o).
% 2.33/1.52 thf(func_def_92, type, ord_less_eq_set_a: set_a > set_a > $o).
% 2.33/1.52 thf(func_def_93, type, ord_less_eq_a: a > a > $o).
% 2.33/1.52 thf(func_def_94, type, produc649172771ee_a_a: tree_a > a > produc981471411ee_a_a).
% 2.33/1.52 thf(func_def_95, type, produc1444952343iset_a: multiset_multiset_a > multiset_multiset_a > produc1293660967iset_a).
% 2.33/1.52 thf(func_def_96, type, produc1226985431iset_a: multis782275565iset_a > multis782275565iset_a > produc741964647iset_a).
% 2.33/1.52 thf(func_def_97, type, produc1608222167Tree_a: multis2082063201Tree_a > multis2082063201Tree_a > produc1136638567Tree_a).
% 2.33/1.52 thf(func_def_98, type, produc1859144151od_a_a: multis599418605od_a_a > multis599418605od_a_a > produc2061001575od_a_a).
% 2.33/1.52 thf(func_def_99, type, produc2037245207iset_a: multiset_a > multiset_a > produc1127127335iset_a).
% 2.33/1.52 thf(func_def_100, type, produc1898392407iset_a: produc1127127335iset_a > produc1127127335iset_a > produc558600423iset_a).
% 2.33/1.52 thf(func_def_101, type, produc1002111575Tree_a: produc143150363Tree_a > produc143150363Tree_a > produc1254371559Tree_a).
% 2.33/1.52 thf(func_def_102, type, produc1474507607od_a_a: product_prod_a_a > product_prod_a_a > produc1572603623od_a_a).
% 2.33/1.52 thf(func_def_103, type, produc686083979Tree_a: a > tree_a > produc143150363Tree_a).
% 2.33/1.52 thf(func_def_104, type, product_Pair_a_a: a > a > product_prod_a_a).
% 2.33/1.52 thf(func_def_105, type, product_fst_Tree_a_a: produc981471411ee_a_a > tree_a).
% 2.33/1.52 thf(func_def_106, type, produc653373187iset_a: produc1127127335iset_a > multiset_a).
% 2.33/1.52 thf(func_def_107, type, product_fst_a_Tree_a: produc143150363Tree_a > a).
% 2.33/1.52 thf(func_def_108, type, product_fst_a_a: product_prod_a_a > a).
% 2.33/1.52 thf(func_def_109, type, product_snd_Tree_a_a: produc981471411ee_a_a > a).
% 2.33/1.52 thf(func_def_110, type, produc148952133iset_a: produc1127127335iset_a > multiset_a).
% 2.33/1.52 thf(func_def_111, type, product_snd_a_Tree_a: produc143150363Tree_a > tree_a).
% 2.33/1.52 thf(func_def_112, type, product_snd_a_a: product_prod_a_a > a).
% 2.33/1.52 thf(func_def_113, type, produc1478296771ee_a_a: produc981471411ee_a_a > produc143150363Tree_a).
% 2.33/1.52 thf(func_def_114, type, produc506430135iset_a: produc1127127335iset_a > produc1127127335iset_a).
% 2.33/1.52 thf(func_def_115, type, produc1515207979Tree_a: produc143150363Tree_a > produc981471411ee_a_a).
% 2.33/1.52 thf(func_def_116, type, product_swap_a_a: product_prod_a_a > product_prod_a_a).
% 2.33/1.52 thf(func_def_117, type, irrefl_a: set_Product_prod_a_a > $o).
% 2.33/1.52 thf(func_def_118, type, trans_multiset_a: set_Pr158363655iset_a > $o).
% 2.33/1.52 thf(func_def_119, type, trans_a: set_Product_prod_a_a > $o).
% 2.33/1.52 thf(func_def_120, type, collection_Tree_a_a: tree_a > (tree_a > $o) > (list_a > tree_a) > (tree_a > multiset_a) > $o).
% 2.33/1.52 thf(func_def_121, type, collec1104713362iset_a: (produc1127127335iset_a > $o) > set_Pr158363655iset_a).
% 2.33/1.52 thf(func_def_122, type, collec486369286Tree_a: (produc143150363Tree_a > $o) > set_Pr1070816379Tree_a).
% 2.33/1.52 thf(func_def_123, type, collec645855634od_a_a: (product_prod_a_a > $o) > set_Product_prod_a_a).
% 2.33/1.52 thf(func_def_124, type, collect_a: (a > $o) > set_a).
% 2.33/1.52 thf(func_def_125, type, member_multiset_a: multiset_a > set_multiset_a > $o).
% 2.33/1.52 thf(func_def_126, type, member1092636252ee_a_a: produc981471411ee_a_a > set_Pr921883667ee_a_a > $o).
% 2.33/1.52 thf(func_def_127, type, member104214352iset_a: produc1293660967iset_a > set_Pr720484615iset_a > $o).
% 2.33/1.52 thf(func_def_128, type, member219455632iset_a: produc741964647iset_a > set_Pr1066112583iset_a > $o).
% 2.33/1.52 thf(func_def_129, type, member1298282640Tree_a: produc1136638567Tree_a > set_Pr295926343Tree_a > $o).
% 2.33/1.52 thf(func_def_130, type, member1349055120od_a_a: produc2061001575od_a_a > set_Pr1821581383od_a_a > $o).
% 2.33/1.52 thf(func_def_131, type, member340150864iset_a: produc1127127335iset_a > set_Pr158363655iset_a > $o).
% 2.33/1.52 thf(func_def_132, type, member567864080iset_a: produc558600423iset_a > set_Pr146455751iset_a > $o).
% 2.33/1.52 thf(func_def_133, type, member146662416Tree_a: produc1254371559Tree_a > set_Pr220561863Tree_a > $o).
% 2.33/1.52 thf(func_def_134, type, member2057358096od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > $o).
% 2.33/1.52 thf(func_def_135, type, member254315204Tree_a: produc143150363Tree_a > set_Pr1070816379Tree_a > $o).
% 2.33/1.52 thf(func_def_136, type, member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o).
% 2.33/1.52 thf(func_def_137, type, member_a: a > set_a > $o).
% 2.33/1.52 thf(func_def_138, type, l1: tree_a).
% 2.33/1.52 thf(func_def_139, type, l2: tree_a).
% 2.33/1.52 thf(func_def_140, type, r1: tree_a).
% 2.33/1.52 thf(func_def_141, type, r2: tree_a).
% 2.33/1.52 thf(func_def_142, type, t: tree_a).
% 2.33/1.52 thf(func_def_143, type, t2: tree_a).
% 2.33/1.52 thf(func_def_144, type, t_a2: tree_a).
% 2.33/1.52 thf(func_def_145, type, v1: a).
% 2.33/1.52 thf(func_def_146, type, v2: a).
% 2.33/1.52 thf(func_def_147, type, v: a).
% 2.33/1.52 thf(func_def_148, type, v_a: a).
% 2.33/1.52 thf(func_def_149, type, v3: a).
% 2.33/1.52 thf(func_def_168, type, sP0: (tree_a > $o) > $o).
% 2.33/1.52 thf(func_def_169, type, sP1: (tree_a > $o) > $o).
% 2.33/1.52 thf(func_def_170, type, sP2: (tree_a > $o) > $o).
% 2.33/1.52 thf(func_def_171, type, sK3: tree_a > a).
% 2.33/1.52 thf(func_def_172, type, sK4: tree_a > a).
% 2.33/1.52 thf(func_def_173, type, sK5: tree_a > tree_a).
% 2.33/1.52 thf(func_def_174, type, sK6: tree_a > tree_a).
% 2.33/1.52 thf(func_def_175, type, sK7: tree_a > a).
% 2.33/1.52 thf(func_def_176, type, sK8: tree_a > a).
% 2.33/1.52 thf(func_def_177, type, sK9: tree_a > tree_a).
% 2.33/1.52 thf(func_def_178, type, sK10: tree_a > a).
% 2.33/1.52 thf(func_def_179, type, sK11: tree_a > tree_a).
% 2.33/1.52 thf(func_def_180, type, sK12: tree_a > tree_a).
% 2.33/1.52 thf(func_def_181, type, sK13: tree_a > tree_a).
% 2.33/1.52 thf(func_def_182, type, sK14: tree_a > a).
% 2.33/1.52 thf(func_def_183, type, sK15: tree_a > tree_a).
% 2.33/1.52 thf(func_def_184, type, sK16: tree_a > tree_a).
% 2.33/1.52 thf(func_def_185, type, sK17: tree_a > a).
% 2.33/1.52 thf(func_def_186, type, sK18: tree_a > tree_a).
% 2.33/1.52 thf(func_def_187, type, sK19: tree_a > a).
% 2.33/1.52 thf(func_def_188, type, sK20: tree_a > tree_a).
% 2.33/1.52 thf(func_def_189, type, sK21: tree_a > a).
% 2.33/1.52 thf(func_def_190, type, sK22: tree_a > a).
% 2.33/1.52 thf(func_def_191, type, sK23: tree_a > tree_a).
% 2.33/1.52 thf(func_def_192, type, sK24: tree_a > a).
% 2.33/1.52 thf(func_def_193, type, sK25: tree_a > tree_a).
% 2.33/1.52 thf(func_def_194, type, sK26: tree_a > a).
% 2.33/1.52 thf(func_def_195, type, sK27: tree_a > a).
% 2.33/1.52 thf(func_def_196, type, sK28: tree_a > tree_a).
% 2.33/1.52 thf(func_def_197, type, sK29: tree_a > tree_a).
% 2.33/1.52 thf(func_def_198, type, sK30: tree_a > a).
% 2.33/1.52 thf(func_def_199, type, sK31: tree_a > tree_a).
% 2.33/1.52 thf(func_def_200, type, sK32: tree_a > a).
% 2.33/1.52 thf(func_def_201, type, sK33: tree_a > a).
% 2.33/1.52 thf(func_def_202, type, sK34: tree_a > tree_a).
% 2.33/1.52 thf(func_def_203, type, sK35: tree_a > tree_a).
% 2.33/1.52 thf(func_def_204, type, sK36: tree_a > tree_a).
% 2.33/1.52 thf(func_def_205, type, sK37: tree_a > a).
% 2.33/1.52 thf(func_def_206, type, sK38: tree_a > tree_a).
% 2.33/1.52 thf(func_def_207, type, sK39: tree_a > a).
% 2.33/1.52 thf(func_def_208, type, sK40: tree_a > a).
% 2.33/1.52 thf(func_def_209, type, sK41: tree_a > tree_a).
% 2.33/1.52 thf(func_def_210, type, sK42: (tree_a > $o) > a).
% 2.33/1.52 thf(func_def_211, type, sK43: (tree_a > $o) > a).
% 2.33/1.52 thf(func_def_212, type, sK44: (tree_a > $o) > a).
% 2.33/1.52 thf(func_def_213, type, sK45: (tree_a > $o) > tree_a).
% 2.33/1.52 thf(func_def_214, type, sK46: (tree_a > $o) > tree_a).
% 2.33/1.52 thf(func_def_215, type, sK47: (tree_a > $o) > tree_a).
% 2.33/1.52 thf(func_def_216, type, sK48: (tree_a > $o) > tree_a).
% 2.33/1.52 thf(func_def_217, type, sK49: (tree_a > $o) > tree_a).
% 2.33/1.52 thf(func_def_218, type, sK50: (tree_a > $o) > a).
% 2.33/1.52 thf(func_def_219, type, sK51: (tree_a > $o) > tree_a).
% 2.33/1.52 thf(func_def_220, type, sK52: (tree_a > $o) > a).
% 2.33/1.52 thf(func_def_221, type, sK53: (tree_a > $o) > tree_a).
% 2.33/1.52 thf(func_def_222, type, sK54: (tree_a > $o) > a).
% 2.33/1.52 thf(func_def_223, type, sK55: (tree_a > $o) > a).
% 2.33/1.52 thf(func_def_224, type, sK56: (tree_a > $o) > tree_a).
% 2.33/1.52 thf(func_def_225, type, sK57: (tree_a > $o) > a).
% 2.33/1.52 thf(func_def_226, type, sK58: (tree_a > $o) > tree_a).
% 2.33/1.52 thf(func_def_227, type, sK59: (tree_a > $o) > tree_a).
% 2.33/1.52 thf(func_def_228, type, sK60: (tree_a > $o) > a).
% 2.33/1.52 thf(func_def_229, type, sK61: (tree_a > $o) > tree_a).
% 2.33/1.52 thf(func_def_230, type, sK62: (tree_a > $o) > tree_a).
% 2.33/1.52 thf(func_def_231, type, sK63: (tree_a > $o) > a).
% 2.33/1.52 thf(func_def_232, type, sK64: (tree_a > $o) > a).
% 2.33/1.52 thf(func_def_233, type, sK65: tree_a > tree_a).
% 2.33/1.52 thf(func_def_234, type, sK66: tree_a > tree_a).
% 2.33/1.52 thf(func_def_235, type, sK67: tree_a > a).
% 2.33/1.52 thf(func_def_236, type, sK68: (tree_a > $o) > tree_a).
% 2.33/1.52 thf(func_def_237, type, sK69: (tree_a > $o) > a).
% 2.33/1.52 thf(func_def_238, type, sK70: (tree_a > $o) > tree_a).
% 2.33/1.52 thf(func_def_239, type, sK71: tree_a > a).
% 2.33/1.52 thf(func_def_240, type, sK72: tree_a > tree_a).
% 2.33/1.52 thf(func_def_241, type, sK73: tree_a > tree_a).
% 2.33/1.52 thf(func_def_242, type, sK74: tree_a > a).
% 2.33/1.52 thf(func_def_243, type, sK75: tree_a > a).
% 2.33/1.52 thf(func_def_244, type, sK76: tree_a > a).
% 2.33/1.52 thf(func_def_245, type, sK77: tree_a > tree_a).
% 2.33/1.52 thf(func_def_246, type, sK78: tree_a > tree_a).
% 2.33/1.52 thf(func_def_247, type, sK79: tree_a > tree_a).
% 2.33/1.52 thf(func_def_248, type, sK80: tree_a > tree_a).
% 2.33/1.52 thf(func_def_249, type, sK81: tree_a > a).
% 2.33/1.52 thf(func_def_250, type, sK82: tree_a > a).
% 2.33/1.52 thf(func_def_251, type, sK83: tree_a > a).
% 2.33/1.52 thf(func_def_252, type, sK84: tree_a > tree_a).
% 2.33/1.52 thf(func_def_253, type, sK85: tree_a > a).
% 2.33/1.52 thf(func_def_254, type, sK86: tree_a > tree_a).
% 2.33/1.52 thf(func_def_255, type, sK87: (tree_a > a > $o) > a).
% 2.33/1.52 thf(func_def_256, type, sK88: (tree_a > a > $o) > tree_a).
% 2.33/1.52 thf(func_def_257, type, sK89: a > tree_a > tree_a).
% 2.33/1.52 thf(func_def_258, type, sK90: a > tree_a > tree_a).
% 2.33/1.52 thf(func_def_259, type, sK91: a > tree_a > a).
% 2.33/1.52 thf(func_def_261, type, sK93: produc143150363Tree_a > a > tree_a).
% 2.33/1.52 thf(func_def_262, type, ph94: !>[X0: $tType]:(X0)).
% 2.33/1.52 thf(f2387,plain,(
% 2.33/1.52 $false),
% 2.33/1.52 inference(subsumption_resolution,[status(thm)],[f2376,f946])).
% 2.33/1.52 thf(f946,plain,(
% 2.33/1.52 (v_a != (product_fst_a_Tree_a @ (heapIm837449470Leaf_a @ (t_a @ v1 @ l1 @ r1))))),
% 2.33/1.52 inference(cnf_transformation,[status(thm)],[f828])).
% 2.33/1.52 thf(f828,plain,(
% 2.33/1.52 (v_a != (product_fst_a_Tree_a @ (heapIm837449470Leaf_a @ (t_a @ v1 @ l1 @ r1))))),
% 2.33/1.52 inference(flattening,[status(thm)],[f358])).
% 2.33/1.52 thf(f358,negated_conjecture,(
% 2.33/1.52 ~(v_a = (product_fst_a_Tree_a @ (heapIm837449470Leaf_a @ (t_a @ v1 @ l1 @ r1))))),
% 2.33/1.52 inference(negated_conjecture,[status(cth)],[f357])).
% 2.33/1.52 thf(f357,conjecture,(
% 2.33/1.52 (v_a = (product_fst_a_Tree_a @ (heapIm837449470Leaf_a @ (t_a @ v1 @ l1 @ r1))))),
% 2.33/1.52 file('/export/starexec/sandbox2/tmp/tmp.lw13DE3eQN/DTF2THF_26921.p',conj_0)).
% 2.33/1.52 thf(f2376,plain,(
% 2.33/1.52 (v_a = (product_fst_a_Tree_a @ (heapIm837449470Leaf_a @ (t_a @ v1 @ l1 @ r1))))),
% 2.33/1.52 inference(superposition,[status(thm)],[f1870,f1821])).
% 2.33/1.52 thf(f1821,plain,(
% 2.33/1.52 (v_a = (product_fst_a_Tree_a @ (heapIm837449470Leaf_a @ (t_a @ v3 @ (t_a @ v1 @ l1 @ r1) @ (t_a @ v2 @ l2 @ r2)))))),
% 2.33/1.52 inference(superposition,[status(thm)],[f1001,f1025])).
% 2.33/1.52 thf(f1025,plain,(
% 2.33/1.52 ((heapIm837449470Leaf_a @ (t_a @ v3 @ (t_a @ v1 @ l1 @ r1) @ (t_a @ v2 @ l2 @ r2))) = (produc686083979Tree_a @ v_a @ (t_a @ v3 @ (product_snd_a_Tree_a @ (heapIm837449470Leaf_a @ (t_a @ v1 @ l1 @ r1))) @ (t_a @ v2 @ l2 @ r2))))),
% 2.33/1.52 inference(definition_unfolding,[status(thm)],[f993,f951])).
% 2.33/1.52 thf(f951,plain,(
% 2.33/1.52 (t_a2 = (t_a @ v3 @ (product_snd_a_Tree_a @ (heapIm837449470Leaf_a @ (t_a @ v1 @ l1 @ r1))) @ (t_a @ v2 @ l2 @ r2)))),
% 2.33/1.52 inference(cnf_transformation,[status(thm)],[f4])).
% 2.33/1.52 thf(f4,axiom,(
% 2.33/1.52 (t_a2 = (t_a @ v3 @ (product_snd_a_Tree_a @ (heapIm837449470Leaf_a @ (t_a @ v1 @ l1 @ r1))) @ (t_a @ v2 @ l2 @ r2)))),
% 2.33/1.52 file('/export/starexec/sandbox2/tmp/tmp.lw13DE3eQN/DTF2THF_26921.p',fact_3__092_060open_062t_H_A_061_AT_Av_A_Isnd_A_IremoveLeaf_A_IT_Av1_Al1_Ar1_J_J_J_A_IT_Av2_Al2_Ar2_J_092_060close_062)).
% 2.33/1.52 thf(f993,plain,(
% 2.33/1.52 ((produc686083979Tree_a @ v_a @ t_a2) = (heapIm837449470Leaf_a @ (t_a @ v3 @ (t_a @ v1 @ l1 @ r1) @ (t_a @ v2 @ l2 @ r2))))),
% 2.33/1.52 inference(cnf_transformation,[status(thm)],[f1])).
% 2.33/1.52 thf(f1,axiom,(
% 2.33/1.52 ((produc686083979Tree_a @ v_a @ t_a2) = (heapIm837449470Leaf_a @ (t_a @ v3 @ (t_a @ v1 @ l1 @ r1) @ (t_a @ v2 @ l2 @ r2))))),
% 2.33/1.52 file('/export/starexec/sandbox2/tmp/tmp.lw13DE3eQN/DTF2THF_26921.p',fact_0__C4__2_Oprems_C_I1_J)).
% 2.33/1.52 thf(f1001,plain,(
% 2.33/1.52 ( ! [X0 : a,X1 : tree_a] : (((product_fst_a_Tree_a @ (produc686083979Tree_a @ X0 @ X1)) = X0)) )),
% 2.33/1.52 inference(cnf_transformation,[status(thm)],[f859])).
% 2.33/1.52 thf(f859,plain,(
% 2.33/1.52 ! [X0 : a,X1 : tree_a] : ((product_fst_a_Tree_a @ (produc686083979Tree_a @ X0 @ X1)) = X0)),
% 2.33/1.52 inference(rectify,[status(thm)],[f23])).
% 2.33/1.52 thf(f23,axiom,(
% 2.33/1.52 ! [X21 : a,X22 : tree_a] : ((product_fst_a_Tree_a @ (produc686083979Tree_a @ X21 @ X22)) = X21)),
% 2.33/1.52 file('/export/starexec/sandbox2/tmp/tmp.lw13DE3eQN/DTF2THF_26921.p',fact_22_fst__conv)).
% 2.33/1.52 thf(f1870,plain,(
% 2.33/1.52 ( ! [X2 : tree_a,X3 : a,X0 : a,X1 : tree_a,X6 : tree_a,X4 : a,X5 : tree_a] : (((product_fst_a_Tree_a @ (heapIm837449470Leaf_a @ (t_a @ X3 @ (t_a @ X0 @ X1 @ X2) @ (t_a @ X4 @ X5 @ X6)))) = (product_fst_a_Tree_a @ (heapIm837449470Leaf_a @ (t_a @ X0 @ X1 @ X2))))) )),
% 2.33/1.52 inference(superposition,[status(thm)],[f1001,f1021])).
% 2.33/1.52 thf(f1021,plain,(
% 2.33/1.52 ( ! [X2 : tree_a,X3 : a,X0 : a,X1 : tree_a,X6 : tree_a,X4 : a,X5 : tree_a] : (((produc686083979Tree_a @ (product_fst_a_Tree_a @ (heapIm837449470Leaf_a @ (t_a @ X3 @ X6 @ X1))) @ (t_a @ X4 @ (product_snd_a_Tree_a @ (heapIm837449470Leaf_a @ (t_a @ X3 @ X6 @ X1))) @ (t_a @ X0 @ X2 @ X5))) = (heapIm837449470Leaf_a @ (t_a @ X4 @ (t_a @ X3 @ X6 @ X1) @ (t_a @ X0 @ X2 @ X5))))) )),
% 2.33/1.52 inference(cnf_transformation,[status(thm)],[f835])).
% 2.33/1.52 thf(f835,plain,(
% 2.33/1.52 ! [X0 : a,X1 : tree_a,X2 : tree_a,X3 : a,X4 : a,X5 : tree_a,X6 : tree_a] : ((produc686083979Tree_a @ (product_fst_a_Tree_a @ (heapIm837449470Leaf_a @ (t_a @ X3 @ X6 @ X1))) @ (t_a @ X4 @ (product_snd_a_Tree_a @ (heapIm837449470Leaf_a @ (t_a @ X3 @ X6 @ X1))) @ (t_a @ X0 @ X2 @ X5))) = (heapIm837449470Leaf_a @ (t_a @ X4 @ (t_a @ X3 @ X6 @ X1) @ (t_a @ X0 @ X2 @ X5))))),
% 2.33/1.52 inference(rectify,[status(thm)],[f7])).
% 2.33/1.52 thf(f7,axiom,(
% 2.33/1.52 ! [X7 : a,X12 : tree_a,X8 : tree_a,X10 : a,X6 : a,X9 : tree_a,X11 : tree_a] : ((heapIm837449470Leaf_a @ (t_a @ X6 @ (t_a @ X10 @ X11 @ X12) @ (t_a @ X7 @ X8 @ X9))) = (produc686083979Tree_a @ (product_fst_a_Tree_a @ (heapIm837449470Leaf_a @ (t_a @ X10 @ X11 @ X12))) @ (t_a @ X6 @ (product_snd_a_Tree_a @ (heapIm837449470Leaf_a @ (t_a @ X10 @ X11 @ X12))) @ (t_a @ X7 @ X8 @ X9))))),
% 2.33/1.52 file('/export/starexec/sandbox2/tmp/tmp.lw13DE3eQN/DTF2THF_26921.p',fact_6_removeLeaf_Osimps_I4_J)).
% 2.33/1.52 % SZS output end Proof for DTF2THF_26921
% 2.33/1.52 % (27172)------------------------------
% 2.33/1.52 % (27172)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.52 % (27172)Termination reason: Refutation
% 2.33/1.52
% 2.33/1.52 % (27172)Memory used [KB]: 8059
% 2.33/1.52 % (27172)Time elapsed: 0.210 s
% 2.33/1.52 % (27172)Instructions burned: 623 (million)
% 2.33/1.52 % (27172)------------------------------
% 2.33/1.52 % (27172)------------------------------
% 2.33/1.52 % (27116)Success in time 0.334 s
%------------------------------------------------------------------------------