%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP070^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n017.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 08:55:18 AM UTC 2026
% Result : Theorem 2.63s 1.59s
% Output : Refutation 2.63s
% Verified :
% SZS Type : Refutation
% Derivation depth : 12
% Number of leaves : 5
% Syntax : Number of formulae : 28 ( 19 unt; 0 typ; 0 def)
% Number of atoms : 145 ( 62 equ; 0 cnn)
% Maximal formula atoms : 10 ( 5 avg)
% Number of connectives : 1040 ( 30 ~; 12 |; 15 &; 959 @)
% ( 0 <=>; 24 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 5 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 176 ( 173 usr; 11 con; 0-6 aty)
% Number of variables : 51 ( 0 ^; 51 !; 0 ?; 51 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
multiset_a: $tType ).
thf(type_def_7,type,
tree_a: $tType ).
thf(type_def_8,type,
list_a: $tType ).
thf(type_def_9,type,
produc143150363Tree_a: $tType ).
thf(type_def_10,type,
a: $tType ).
thf(type_def_11,type,
set_a: $tType ).
thf(type_def_12,type,
produc768687417list_a: $tType ).
thf(func_def_0,type,
produc768687417list_a: $tType ).
thf(func_def_1,type,
produc143150363Tree_a: $tType ).
thf(func_def_2,type,
multiset_a: $tType ).
thf(func_def_3,type,
list_a: $tType ).
thf(func_def_4,type,
tree_a: $tType ).
thf(func_def_5,type,
set_a: $tType ).
thf(func_def_6,type,
a: $tType ).
thf(func_def_7,type,
plus_plus_multiset_a: multiset_a > multiset_a > multiset_a ).
thf(func_def_8,type,
zero_zero_multiset_a: multiset_a ).
thf(func_def_9,type,
heapIm970322378pify_a: tree_a > tree_a ).
thf(func_def_10,type,
heapIm229596386mpty_a: tree_a > $o ).
thf(func_def_11,type,
heapIm1057938560list_a: list_a > tree_a ).
thf(func_def_12,type,
heapIm1140443833left_a: tree_a > tree_a ).
thf(func_def_13,type,
heapIm1637418125tree_a: list_a > tree_a ).
thf(func_def_14,type,
heapIm837449470Leaf_a: tree_a > produc143150363Tree_a ).
thf(func_def_15,type,
heapIm1257206334ight_a: tree_a > tree_a ).
thf(func_def_16,type,
heapIm1091024090Down_a: tree_a > tree_a ).
thf(func_def_17,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_18,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_19,type,
e_a: tree_a ).
thf(func_def_20,type,
t_a: a > tree_a > tree_a > tree_a ).
thf(func_def_21,type,
in_tree_a: a > tree_a > $o ).
thf(func_def_22,type,
is_heap_a: tree_a > $o ).
thf(func_def_23,type,
multiset_a2: tree_a > multiset_a ).
thf(func_def_24,type,
val_a: tree_a > a ).
thf(func_def_25,type,
lattic146396397_Max_a: set_a > a ).
thf(func_def_26,type,
cons_a: a > list_a > list_a ).
thf(func_def_27,type,
add_mset_a: a > multiset_a > multiset_a ).
thf(func_def_28,type,
is_empty_a: multiset_a > $o ).
thf(func_def_29,type,
set_mset_a: multiset_a > set_a ).
thf(func_def_30,type,
subseteq_mset_a: multiset_a > multiset_a > $o ).
thf(func_def_31,type,
ord_less_eq_o_o_a: ( $o > $o > a ) > ( $o > $o > a ) > $o ).
thf(func_def_32,type,
ord_less_eq_o_a: ( $o > a ) > ( $o > a ) > $o ).
thf(func_def_33,type,
ord_le1199012836iset_a: multiset_a > multiset_a > $o ).
thf(func_def_34,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(func_def_35,type,
ord_less_eq_a: a > a > $o ).
thf(func_def_36,type,
order_Greatest_o_a: ( ( $o > a ) > $o ) > $o > a ).
thf(func_def_37,type,
order_Greatest_a: ( a > $o ) > a ).
thf(func_def_38,type,
produc1352981801list_a: tree_a > list_a > produc768687417list_a ).
thf(func_def_39,type,
produc686083979Tree_a: a > tree_a > produc143150363Tree_a ).
thf(func_def_40,type,
set_Tree_a_a: ( tree_a > multiset_a ) > tree_a > set_a ).
thf(func_def_41,type,
removeMax_Tree_a_a: tree_a > ( tree_a > $o ) > ( list_a > tree_a ) > ( tree_a > multiset_a ) > ( tree_a > produc143150363Tree_a ) > ( tree_a > $o ) > $o ).
thf(func_def_42,type,
ssort_Tree_a_a: ( tree_a > $o ) > ( tree_a > produc143150363Tree_a ) > tree_a > list_a > list_a ).
thf(func_def_43,type,
ssort_dom_Tree_a_a: ( tree_a > $o ) > ( tree_a > produc143150363Tree_a ) > produc768687417list_a > $o ).
thf(func_def_44,type,
remove301631099ee_a_a: ( tree_a > $o ) > ( list_a > tree_a ) > ( tree_a > multiset_a ) > ( tree_a > produc143150363Tree_a ) > ( tree_a > $o ) > $o ).
thf(func_def_45,type,
collect_a: ( a > $o ) > set_a ).
thf(func_def_46,type,
member_a: a > set_a > $o ).
thf(func_def_47,type,
l1: tree_a ).
thf(func_def_48,type,
l2: tree_a ).
thf(func_def_49,type,
r1: tree_a ).
thf(func_def_50,type,
r2: tree_a ).
thf(func_def_51,type,
v1: a ).
thf(func_def_52,type,
v2: a ).
thf(func_def_53,type,
v: a ).
thf(func_def_71,type,
sP0: a > a > a > $o ).
thf(func_def_72,type,
sP1: a > a > a > $o ).
thf(func_def_73,type,
sP2: a > a > a > $o ).
thf(func_def_74,type,
sP3: ( tree_a > $o ) > $o ).
thf(func_def_75,type,
sP4: ( tree_a > $o ) > $o ).
thf(func_def_76,type,
sP5: ( tree_a > $o ) > $o ).
thf(func_def_77,type,
sK6: multiset_a > a > multiset_a ).
thf(func_def_78,type,
sK7: ( a > a ) > a ).
thf(func_def_79,type,
sK8: ( a > a ) > a ).
thf(func_def_80,type,
sK9: ( a > a ) > a ).
thf(func_def_81,type,
sK10: ( a > a ) > a ).
thf(func_def_82,type,
sK11: multiset_a > a ).
thf(func_def_83,type,
sK12: multiset_a > multiset_a ).
thf(func_def_84,type,
sK13: ( multiset_a > $o ) > multiset_a ).
thf(func_def_85,type,
sK14: ( multiset_a > $o ) > a ).
thf(func_def_86,type,
sK15: tree_a > a ).
thf(func_def_87,type,
sK16: tree_a > tree_a ).
thf(func_def_88,type,
sK17: tree_a > tree_a ).
thf(func_def_89,type,
sK18: tree_a > a ).
thf(func_def_90,type,
sK19: tree_a > a ).
thf(func_def_91,type,
sK20: tree_a > tree_a ).
thf(func_def_92,type,
sK21: tree_a > a ).
thf(func_def_93,type,
sK22: tree_a > a ).
thf(func_def_94,type,
sK23: tree_a > a ).
thf(func_def_95,type,
sK24: tree_a > tree_a ).
thf(func_def_96,type,
sK25: tree_a > tree_a ).
thf(func_def_97,type,
sK26: tree_a > tree_a ).
thf(func_def_98,type,
sK27: tree_a > tree_a ).
thf(func_def_99,type,
sK28: tree_a > tree_a ).
thf(func_def_100,type,
sK29: tree_a > a ).
thf(func_def_101,type,
sK30: tree_a > a ).
thf(func_def_102,type,
sK31: tree_a > a ).
thf(func_def_103,type,
sK32: tree_a > a ).
thf(func_def_104,type,
sK33: tree_a > tree_a ).
thf(func_def_105,type,
sK34: tree_a > tree_a ).
thf(func_def_106,type,
sK35: tree_a > a ).
thf(func_def_107,type,
sK36: tree_a > tree_a ).
thf(func_def_108,type,
sK37: tree_a > tree_a ).
thf(func_def_109,type,
sK38: tree_a > tree_a ).
thf(func_def_110,type,
sK39: tree_a > a ).
thf(func_def_111,type,
sK40: tree_a > tree_a ).
thf(func_def_112,type,
sK41: tree_a > a ).
thf(func_def_113,type,
sK42: tree_a > a ).
thf(func_def_114,type,
sK43: tree_a > a ).
thf(func_def_115,type,
sK44: tree_a > tree_a ).
thf(func_def_116,type,
sK45: tree_a > tree_a ).
thf(func_def_117,type,
sK46: tree_a > tree_a ).
thf(func_def_118,type,
sK47: tree_a > a ).
thf(func_def_119,type,
sK48: tree_a > tree_a ).
thf(func_def_120,type,
sK49: tree_a > a ).
thf(func_def_121,type,
sK50: tree_a > tree_a ).
thf(func_def_122,type,
sK51: tree_a > a ).
thf(func_def_123,type,
sK52: tree_a > tree_a ).
thf(func_def_124,type,
sK53: tree_a > a ).
thf(func_def_125,type,
sK54: ( multiset_a > multiset_a > $o ) > multiset_a ).
thf(func_def_126,type,
sK55: ( multiset_a > multiset_a > $o ) > a ).
thf(func_def_127,type,
sK56: ( multiset_a > multiset_a > $o ) > multiset_a ).
thf(func_def_128,type,
sK57: ( multiset_a > multiset_a > $o ) > a ).
thf(func_def_129,type,
sK58: ( multiset_a > multiset_a > $o ) > multiset_a ).
thf(func_def_130,type,
sK59: ( multiset_a > multiset_a > $o ) > multiset_a ).
thf(func_def_131,type,
sK60: ( a > a ) > a ).
thf(func_def_132,type,
sK61: ( a > a ) > a ).
thf(func_def_133,type,
sK62: tree_a > tree_a ).
thf(func_def_134,type,
sK63: tree_a > tree_a ).
thf(func_def_135,type,
sK64: tree_a > a ).
thf(func_def_136,type,
sK65: tree_a > a ).
thf(func_def_137,type,
sK66: tree_a > tree_a ).
thf(func_def_138,type,
sK67: tree_a > tree_a ).
thf(func_def_139,type,
sK68: tree_a > a ).
thf(func_def_140,type,
sK69: tree_a > tree_a ).
thf(func_def_141,type,
sK70: tree_a > a ).
thf(func_def_142,type,
sK71: tree_a > tree_a ).
thf(func_def_143,type,
sK72: tree_a > a ).
thf(func_def_144,type,
sK73: tree_a > tree_a ).
thf(func_def_145,type,
sK74: tree_a > a ).
thf(func_def_146,type,
sK75: tree_a > tree_a ).
thf(func_def_147,type,
sK76: tree_a > a ).
thf(func_def_148,type,
sK77: tree_a > a ).
thf(func_def_149,type,
sK78: tree_a > tree_a ).
thf(func_def_150,type,
sK79: tree_a > tree_a ).
thf(func_def_151,type,
sK80: tree_a > a ).
thf(func_def_152,type,
sK81: ( tree_a > $o ) > tree_a ).
thf(func_def_153,type,
sK82: ( tree_a > $o ) > a ).
thf(func_def_154,type,
sK83: ( tree_a > $o ) > tree_a ).
thf(func_def_155,type,
sK84: ( multiset_a > $o ) > a ).
thf(func_def_156,type,
sK85: ( multiset_a > $o ) > multiset_a ).
thf(func_def_157,type,
sK86: multiset_a > a ).
thf(func_def_158,type,
sK87: multiset_a > multiset_a ).
thf(func_def_159,type,
sK88: multiset_a > a ).
thf(func_def_160,type,
sK89: ( a > a > $o ) > a ).
thf(func_def_161,type,
sK90: ( a > a > $o ) > a ).
thf(func_def_162,type,
sK91: ( a > a > $o ) > a ).
thf(func_def_163,type,
sK92: ( a > a > $o ) > a ).
thf(func_def_164,type,
sK93: ( multiset_a > $o ) > a ).
thf(func_def_165,type,
sK94: ( multiset_a > $o ) > multiset_a ).
thf(func_def_166,type,
sK95: ( a > a ) > a ).
thf(func_def_167,type,
sK96: ( a > a ) > a ).
thf(func_def_168,type,
sK97: multiset_a > a > multiset_a ).
thf(func_def_169,type,
sK98: tree_a > a > tree_a ).
thf(func_def_170,type,
sK99: tree_a > a > tree_a ).
thf(func_def_171,type,
sK100: tree_a > a > a ).
thf(func_def_172,type,
sK101: multiset_a > a > a > multiset_a > multiset_a ).
thf(func_def_173,type,
sK102: ( tree_a > $o ) > tree_a ).
thf(func_def_174,type,
sK103: ( tree_a > $o ) > tree_a ).
thf(func_def_175,type,
sK104: ( tree_a > $o ) > tree_a ).
thf(func_def_176,type,
sK105: ( tree_a > $o ) > a ).
thf(func_def_177,type,
sK106: ( tree_a > $o ) > a ).
thf(func_def_178,type,
sK107: ( tree_a > $o ) > tree_a ).
thf(func_def_179,type,
sK108: ( tree_a > $o ) > a ).
thf(func_def_180,type,
sK109: ( tree_a > $o ) > tree_a ).
thf(func_def_181,type,
sK110: ( tree_a > $o ) > a ).
thf(func_def_182,type,
sK111: ( tree_a > $o ) > tree_a ).
thf(func_def_183,type,
sK112: ( tree_a > $o ) > a ).
thf(func_def_184,type,
sK113: ( tree_a > $o ) > tree_a ).
thf(func_def_185,type,
sK114: ( tree_a > $o ) > tree_a ).
thf(func_def_186,type,
sK115: ( tree_a > $o ) > a ).
thf(func_def_187,type,
sK116: ( tree_a > $o ) > a ).
thf(func_def_188,type,
sK117: ( tree_a > $o ) > tree_a ).
thf(func_def_189,type,
sK118: ( tree_a > $o ) > tree_a ).
thf(func_def_190,type,
sK119: ( tree_a > $o ) > a ).
thf(func_def_191,type,
sK120: ( tree_a > $o ) > a ).
thf(func_def_192,type,
sK121: ( tree_a > $o ) > tree_a ).
thf(func_def_193,type,
sK122: ( tree_a > $o ) > a ).
thf(func_def_194,type,
sK123: ( tree_a > $o ) > tree_a ).
thf(func_def_195,type,
sK124: ( tree_a > $o ) > a ).
thf(func_def_197,type,
ph126:
!>[X0: $tType] : X0 ).
thf(f1080,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1079,f902]) ).
thf(f902,plain,
( ( ord_less_eq_a @ v2 @ v1 )
= $true ),
inference(cnf_transformation,[status(thm)],[f323]) ).
thf(f323,plain,
( ( ord_less_eq_a @ v2 @ v1 )
= $true ),
inference(fool_elimination,[status(thm)],[f322]) ).
thf(f322,plain,
ord_less_eq_a @ v2 @ v1,
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
ord_less_eq_a @ v2 @ v1,
file('/export/starexec/sandbox/tmp/tmp.geO6X9CZKJ/DTF2THF_6481.p',fact_0__092_060open_062v2_A_092_060le_062_Av1_092_060close_062) ).
thf(f1079,plain,
( ( ord_less_eq_a @ v2 @ v1 )
!= $true ),
inference(forward_demodulation,[status(thm)],[f1078,f794]) ).
thf(f794,plain,
! [X2: tree_a,X0: a,X1: tree_a] :
( ( val_a @ ( t_a @ X0 @ X1 @ X2 ) )
= X0 ),
inference(cnf_transformation,[status(thm)],[f574]) ).
thf(f574,plain,
! [X0: a,X1: tree_a,X2: tree_a] :
( ( val_a @ ( t_a @ X0 @ X1 @ X2 ) )
= X0 ),
inference(rectify,[status(thm)],[f31]) ).
thf(f31,axiom,
! [X1: a,X38: tree_a,X39: tree_a] :
( ( val_a @ ( t_a @ X1 @ X38 @ X39 ) )
= X1 ),
file('/export/starexec/sandbox/tmp/tmp.geO6X9CZKJ/DTF2THF_6481.p',fact_30_val_Osimps) ).
thf(f1078,plain,
( ( ord_less_eq_a @ v2 @ ( val_a @ ( t_a @ v1 @ l1 @ r1 ) ) )
!= $true ),
inference(forward_demodulation,[status(thm)],[f1077,f794]) ).
thf(f1077,plain,
( ( ord_less_eq_a @ ( val_a @ ( t_a @ v2 @ l2 @ r2 ) ) @ ( val_a @ ( t_a @ v1 @ l1 @ r1 ) ) )
!= $true ),
inference(subsumption_resolution,[status(thm)],[f1076,f815]) ).
thf(f815,plain,
( ( ord_less_eq_a @ v1 @ v )
= $true ),
inference(cnf_transformation,[status(thm)],[f415]) ).
thf(f415,plain,
( ( ord_less_eq_a @ v1 @ v )
= $true ),
inference(fool_elimination,[status(thm)],[f414]) ).
thf(f414,plain,
ord_less_eq_a @ v1 @ v,
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
ord_less_eq_a @ v1 @ v,
file('/export/starexec/sandbox/tmp/tmp.geO6X9CZKJ/DTF2THF_6481.p',fact_1_True) ).
thf(f1076,plain,
( ( ( ord_less_eq_a @ v1 @ v )
!= $true )
| ( ( ord_less_eq_a @ ( val_a @ ( t_a @ v2 @ l2 @ r2 ) ) @ ( val_a @ ( t_a @ v1 @ l1 @ r1 ) ) )
!= $true ) ),
inference(forward_demodulation,[status(thm)],[f1075,f794]) ).
thf(f1075,plain,
( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ v1 @ l1 @ r1 ) ) @ v )
!= $true )
| ( ( ord_less_eq_a @ ( val_a @ ( t_a @ v2 @ l2 @ r2 ) ) @ ( val_a @ ( t_a @ v1 @ l1 @ r1 ) ) )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f1069]) ).
thf(f1069,plain,
( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ v2 @ l2 @ r2 ) ) @ ( val_a @ ( t_a @ v1 @ l1 @ r1 ) ) )
!= $true )
| ( ( multiset_a2 @ ( t_a @ v @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) ) )
!= ( multiset_a2 @ ( t_a @ v @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) ) ) )
| ( ( ord_less_eq_a @ ( val_a @ ( t_a @ v1 @ l1 @ r1 ) ) @ v )
!= $true ) ),
inference(superposition,[status(thm)],[f875,f977]) ).
thf(f977,plain,
! [X2: a,X3: tree_a,X0: a,X1: tree_a,X6: tree_a,X4: a,X5: tree_a] :
( ( ( t_a @ X4 @ ( t_a @ X2 @ X5 @ X6 ) @ ( t_a @ X0 @ X3 @ X1 ) )
= ( heapIm1091024090Down_a @ ( t_a @ X4 @ ( t_a @ X2 @ X5 @ X6 ) @ ( t_a @ X0 @ X3 @ X1 ) ) ) )
| ( ( ord_less_eq_a @ ( val_a @ ( t_a @ X0 @ X3 @ X1 ) ) @ ( val_a @ ( t_a @ X2 @ X5 @ X6 ) ) )
!= $true )
| ( ( ord_less_eq_a @ ( val_a @ ( t_a @ X2 @ X5 @ X6 ) ) @ X4 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f644]) ).
thf(f644,plain,
! [X0: a,X1: tree_a,X2: a,X3: tree_a,X4: a,X5: tree_a,X6: tree_a] :
( ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ X0 @ X3 @ X1 ) ) @ ( val_a @ ( t_a @ X2 @ X5 @ X6 ) ) )
!= $true )
| ( ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ X2 @ X5 @ X6 ) ) @ X4 )
= $true )
| ( ( t_a @ ( val_a @ ( t_a @ X2 @ X5 @ X6 ) ) @ ( heapIm1091024090Down_a @ ( t_a @ X4 @ ( heapIm1140443833left_a @ ( t_a @ X2 @ X5 @ X6 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ X2 @ X5 @ X6 ) ) ) ) @ ( t_a @ X0 @ X3 @ X1 ) )
= ( heapIm1091024090Down_a @ ( t_a @ X4 @ ( t_a @ X2 @ X5 @ X6 ) @ ( t_a @ X0 @ X3 @ X1 ) ) ) ) )
& ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ X2 @ X5 @ X6 ) ) @ X4 )
!= $true )
| ( ( t_a @ X4 @ ( t_a @ X2 @ X5 @ X6 ) @ ( t_a @ X0 @ X3 @ X1 ) )
= ( heapIm1091024090Down_a @ ( t_a @ X4 @ ( t_a @ X2 @ X5 @ X6 ) @ ( t_a @ X0 @ X3 @ X1 ) ) ) ) ) ) )
& ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ X0 @ X3 @ X1 ) ) @ ( val_a @ ( t_a @ X2 @ X5 @ X6 ) ) )
= $true )
| ( ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ X0 @ X3 @ X1 ) ) @ X4 )
!= $true )
| ( ( t_a @ X4 @ ( t_a @ X2 @ X5 @ X6 ) @ ( t_a @ X0 @ X3 @ X1 ) )
= ( heapIm1091024090Down_a @ ( t_a @ X4 @ ( t_a @ X2 @ X5 @ X6 ) @ ( t_a @ X0 @ X3 @ X1 ) ) ) ) )
& ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ X0 @ X3 @ X1 ) ) @ X4 )
= $true )
| ( ( heapIm1091024090Down_a @ ( t_a @ X4 @ ( t_a @ X2 @ X5 @ X6 ) @ ( t_a @ X0 @ X3 @ X1 ) ) )
= ( t_a @ ( val_a @ ( t_a @ X0 @ X3 @ X1 ) ) @ ( t_a @ X2 @ X5 @ X6 ) @ ( heapIm1091024090Down_a @ ( t_a @ X4 @ ( heapIm1140443833left_a @ ( t_a @ X0 @ X3 @ X1 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ X0 @ X3 @ X1 ) ) ) ) ) ) ) ) ) ),
inference(ennf_transformation,[status(thm)],[f589]) ).
thf(f589,plain,
! [X0: a,X1: tree_a,X2: a,X3: tree_a,X4: a,X5: tree_a,X6: tree_a] :
( ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ X0 @ X3 @ X1 ) ) @ ( val_a @ ( t_a @ X2 @ X5 @ X6 ) ) )
!= $true )
=> ( ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ X0 @ X3 @ X1 ) ) @ X4 )
= $true )
=> ( ( t_a @ X4 @ ( t_a @ X2 @ X5 @ X6 ) @ ( t_a @ X0 @ X3 @ X1 ) )
= ( heapIm1091024090Down_a @ ( t_a @ X4 @ ( t_a @ X2 @ X5 @ X6 ) @ ( t_a @ X0 @ X3 @ X1 ) ) ) ) )
& ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ X0 @ X3 @ X1 ) ) @ X4 )
!= $true )
=> ( ( heapIm1091024090Down_a @ ( t_a @ X4 @ ( t_a @ X2 @ X5 @ X6 ) @ ( t_a @ X0 @ X3 @ X1 ) ) )
= ( t_a @ ( val_a @ ( t_a @ X0 @ X3 @ X1 ) ) @ ( t_a @ X2 @ X5 @ X6 ) @ ( heapIm1091024090Down_a @ ( t_a @ X4 @ ( heapIm1140443833left_a @ ( t_a @ X0 @ X3 @ X1 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ X0 @ X3 @ X1 ) ) ) ) ) ) ) ) )
& ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ X0 @ X3 @ X1 ) ) @ ( val_a @ ( t_a @ X2 @ X5 @ X6 ) ) )
= $true )
=> ( ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ X2 @ X5 @ X6 ) ) @ X4 )
!= $true )
=> ( ( t_a @ ( val_a @ ( t_a @ X2 @ X5 @ X6 ) ) @ ( heapIm1091024090Down_a @ ( t_a @ X4 @ ( heapIm1140443833left_a @ ( t_a @ X2 @ X5 @ X6 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ X2 @ X5 @ X6 ) ) ) ) @ ( t_a @ X0 @ X3 @ X1 ) )
= ( heapIm1091024090Down_a @ ( t_a @ X4 @ ( t_a @ X2 @ X5 @ X6 ) @ ( t_a @ X0 @ X3 @ X1 ) ) ) ) )
& ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ X2 @ X5 @ X6 ) ) @ X4 )
= $true )
=> ( ( t_a @ X4 @ ( t_a @ X2 @ X5 @ X6 ) @ ( t_a @ X0 @ X3 @ X1 ) )
= ( heapIm1091024090Down_a @ ( t_a @ X4 @ ( t_a @ X2 @ X5 @ X6 ) @ ( t_a @ X0 @ X3 @ X1 ) ) ) ) ) ) ) ),
inference(flattening,[status(thm)],[f391]) ).
thf(f391,plain,
! [X0: a,X1: tree_a,X2: a,X3: tree_a,X4: a,X5: tree_a,X6: tree_a] :
( ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ X0 @ X3 @ X1 ) ) @ ( val_a @ ( t_a @ X2 @ X5 @ X6 ) ) )
!= $true )
=> ( ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ X0 @ X3 @ X1 ) ) @ X4 )
!= $true )
=> ( ( heapIm1091024090Down_a @ ( t_a @ X4 @ ( t_a @ X2 @ X5 @ X6 ) @ ( t_a @ X0 @ X3 @ X1 ) ) )
= ( t_a @ ( val_a @ ( t_a @ X0 @ X3 @ X1 ) ) @ ( t_a @ X2 @ X5 @ X6 ) @ ( heapIm1091024090Down_a @ ( t_a @ X4 @ ( heapIm1140443833left_a @ ( t_a @ X0 @ X3 @ X1 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ X0 @ X3 @ X1 ) ) ) ) ) ) )
& ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ X0 @ X3 @ X1 ) ) @ X4 )
= $true )
=> ( ( t_a @ X4 @ ( t_a @ X2 @ X5 @ X6 ) @ ( t_a @ X0 @ X3 @ X1 ) )
= ( heapIm1091024090Down_a @ ( t_a @ X4 @ ( t_a @ X2 @ X5 @ X6 ) @ ( t_a @ X0 @ X3 @ X1 ) ) ) ) ) ) )
& ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ X0 @ X3 @ X1 ) ) @ ( val_a @ ( t_a @ X2 @ X5 @ X6 ) ) )
= $true )
=> ( ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ X2 @ X5 @ X6 ) ) @ X4 )
= $true )
=> ( ( t_a @ X4 @ ( t_a @ X2 @ X5 @ X6 ) @ ( t_a @ X0 @ X3 @ X1 ) )
= ( heapIm1091024090Down_a @ ( t_a @ X4 @ ( t_a @ X2 @ X5 @ X6 ) @ ( t_a @ X0 @ X3 @ X1 ) ) ) ) )
& ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ X2 @ X5 @ X6 ) ) @ X4 )
!= $true )
=> ( ( t_a @ ( val_a @ ( t_a @ X2 @ X5 @ X6 ) ) @ ( heapIm1091024090Down_a @ ( t_a @ X4 @ ( heapIm1140443833left_a @ ( t_a @ X2 @ X5 @ X6 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ X2 @ X5 @ X6 ) ) ) ) @ ( t_a @ X0 @ X3 @ X1 ) )
= ( heapIm1091024090Down_a @ ( t_a @ X4 @ ( t_a @ X2 @ X5 @ X6 ) @ ( t_a @ X0 @ X3 @ X1 ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f390]) ).
thf(f390,plain,
! [X0: a,X1: tree_a,X2: a,X3: tree_a,X4: a,X5: tree_a,X6: tree_a] :
( ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ X0 @ X3 @ X1 ) ) @ ( val_a @ ( t_a @ X2 @ X5 @ X6 ) ) )
=> ( ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ X0 @ X3 @ X1 ) ) @ X4 )
=> ( ( heapIm1091024090Down_a @ ( t_a @ X4 @ ( t_a @ X2 @ X5 @ X6 ) @ ( t_a @ X0 @ X3 @ X1 ) ) )
= ( t_a @ ( val_a @ ( t_a @ X0 @ X3 @ X1 ) ) @ ( t_a @ X2 @ X5 @ X6 ) @ ( heapIm1091024090Down_a @ ( t_a @ X4 @ ( heapIm1140443833left_a @ ( t_a @ X0 @ X3 @ X1 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ X0 @ X3 @ X1 ) ) ) ) ) ) )
& ( ( ord_less_eq_a @ ( val_a @ ( t_a @ X0 @ X3 @ X1 ) ) @ X4 )
=> ( ( t_a @ X4 @ ( t_a @ X2 @ X5 @ X6 ) @ ( t_a @ X0 @ X3 @ X1 ) )
= ( heapIm1091024090Down_a @ ( t_a @ X4 @ ( t_a @ X2 @ X5 @ X6 ) @ ( t_a @ X0 @ X3 @ X1 ) ) ) ) ) ) )
& ( ( ord_less_eq_a @ ( val_a @ ( t_a @ X0 @ X3 @ X1 ) ) @ ( val_a @ ( t_a @ X2 @ X5 @ X6 ) ) )
=> ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ X2 @ X5 @ X6 ) ) @ X4 )
=> ( ( t_a @ X4 @ ( t_a @ X2 @ X5 @ X6 ) @ ( t_a @ X0 @ X3 @ X1 ) )
= ( heapIm1091024090Down_a @ ( t_a @ X4 @ ( t_a @ X2 @ X5 @ X6 ) @ ( t_a @ X0 @ X3 @ X1 ) ) ) ) )
& ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ X2 @ X5 @ X6 ) ) @ X4 )
=> ( ( t_a @ ( val_a @ ( t_a @ X2 @ X5 @ X6 ) ) @ ( heapIm1091024090Down_a @ ( t_a @ X4 @ ( heapIm1140443833left_a @ ( t_a @ X2 @ X5 @ X6 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ X2 @ X5 @ X6 ) ) ) ) @ ( t_a @ X0 @ X3 @ X1 ) )
= ( heapIm1091024090Down_a @ ( t_a @ X4 @ ( t_a @ X2 @ X5 @ X6 ) @ ( t_a @ X0 @ X3 @ X1 ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f33]) ).
thf(f33,axiom,
! [X18: a,X20: tree_a,X15: a,X19: tree_a,X1: a,X16: tree_a,X17: tree_a] :
( ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ X18 @ X19 @ X20 ) ) @ ( val_a @ ( t_a @ X15 @ X16 @ X17 ) ) )
=> ( ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ X18 @ X19 @ X20 ) ) @ X1 )
=> ( ( heapIm1091024090Down_a @ ( t_a @ X1 @ ( t_a @ X15 @ X16 @ X17 ) @ ( t_a @ X18 @ X19 @ X20 ) ) )
= ( t_a @ ( val_a @ ( t_a @ X18 @ X19 @ X20 ) ) @ ( t_a @ X15 @ X16 @ X17 ) @ ( heapIm1091024090Down_a @ ( t_a @ X1 @ ( heapIm1140443833left_a @ ( t_a @ X18 @ X19 @ X20 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ X18 @ X19 @ X20 ) ) ) ) ) ) )
& ( ( ord_less_eq_a @ ( val_a @ ( t_a @ X18 @ X19 @ X20 ) ) @ X1 )
=> ( ( t_a @ X1 @ ( t_a @ X15 @ X16 @ X17 ) @ ( t_a @ X18 @ X19 @ X20 ) )
= ( heapIm1091024090Down_a @ ( t_a @ X1 @ ( t_a @ X15 @ X16 @ X17 ) @ ( t_a @ X18 @ X19 @ X20 ) ) ) ) ) ) )
& ( ( ord_less_eq_a @ ( val_a @ ( t_a @ X18 @ X19 @ X20 ) ) @ ( val_a @ ( t_a @ X15 @ X16 @ X17 ) ) )
=> ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ X15 @ X16 @ X17 ) ) @ X1 )
=> ( ( t_a @ X1 @ ( t_a @ X15 @ X16 @ X17 ) @ ( t_a @ X18 @ X19 @ X20 ) )
= ( heapIm1091024090Down_a @ ( t_a @ X1 @ ( t_a @ X15 @ X16 @ X17 ) @ ( t_a @ X18 @ X19 @ X20 ) ) ) ) )
& ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ X15 @ X16 @ X17 ) ) @ X1 )
=> ( ( heapIm1091024090Down_a @ ( t_a @ X1 @ ( t_a @ X15 @ X16 @ X17 ) @ ( t_a @ X18 @ X19 @ X20 ) ) )
= ( t_a @ ( val_a @ ( t_a @ X15 @ X16 @ X17 ) ) @ ( heapIm1091024090Down_a @ ( t_a @ X1 @ ( heapIm1140443833left_a @ ( t_a @ X15 @ X16 @ X17 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ X15 @ X16 @ X17 ) ) ) ) @ ( t_a @ X18 @ X19 @ X20 ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.geO6X9CZKJ/DTF2THF_6481.p',fact_32_siftDown_Osimps_I6_J) ).
thf(f875,plain,
( ( multiset_a2 @ ( heapIm1091024090Down_a @ ( t_a @ v @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) ) ) )
!= ( multiset_a2 @ ( t_a @ v @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f558]) ).
thf(f558,plain,
( ( multiset_a2 @ ( heapIm1091024090Down_a @ ( t_a @ v @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) ) ) )
!= ( multiset_a2 @ ( t_a @ v @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) ) ) ),
inference(flattening,[status(thm)],[f216]) ).
thf(f216,negated_conjecture,
( ( multiset_a2 @ ( heapIm1091024090Down_a @ ( t_a @ v @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) ) ) )
!= ( multiset_a2 @ ( t_a @ v @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) ) ) ),
inference(negated_conjecture,[status(cth)],[f215]) ).
thf(f215,conjecture,
( ( multiset_a2 @ ( heapIm1091024090Down_a @ ( t_a @ v @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) ) ) )
= ( multiset_a2 @ ( t_a @ v @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.geO6X9CZKJ/DTF2THF_6481.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.13 % Problem : ITP070^1 : TPTP v9.2.1. Released v7.5.0.
% 0.11/0.14 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.35 % Computer : n017.cluster.edu
% 0.15/0.35 % Model : x86_64 x86_64
% 0.15/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.35 % Memory : 8042.1875MB
% 0.15/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.35 % CPULimit : 300
% 0.15/0.35 % WCLimit : 300
% 0.15/0.35 % DateTime : Tue Feb 24 08:24:11 EST 2026
% 0.15/0.35 % CPUTime :
% 0.15/0.35 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.27/0.47 ---- Original DTF file ---
% 0.27/0.47 thf(spec,logic,$$dhol).
% 0.27/0.47 %------------------------------------------------------------------------------
% 0.27/0.47 % File : ITP070^1 : TPTP v9.2.1. Released v7.5.0.
% 0.27/0.47 % Domain : Interactive Theorem Proving
% 0.27/0.47 % Problem : Sledgehammer HeapImperative problem prob_824__5349520_1
% 0.27/0.47 % Version : Especial.
% 0.27/0.47 % English :
% 0.27/0.47
% 0.27/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.27/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.27/0.47 % Source : [Des21]
% 0.27/0.47 % Names : HeapImperative/prob_824__5349520_1 [Des21]
% 0.27/0.47
% 0.27/0.47 % Status : Theorem
% 0.27/0.47 % 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
% 0.27/0.47 % Syntax : Number of formulae : 269 ( 98 unt; 54 typ; 0 def)
% 0.27/0.47 % Number of atoms : 653 ( 280 equ; 0 cnn)
% 0.27/0.47 % Maximal formula atoms : 12 ( 3 avg)
% 0.27/0.47 % Number of connectives : 2679 ( 90 ~; 7 |; 54 &;2166 @)
% 0.27/0.47 % ( 0 <=>; 362 =>; 0 <=; 0 <~>)
% 0.27/0.47 % Maximal formula depth : 25 ( 8 avg)
% 0.27/0.47 % Number of types : 8 ( 7 usr)
% 0.27/0.47 % Number of type conns : 343 ( 343 >; 0 *; 0 +; 0 <<)
% 0.27/0.47 % Number of symbols : 50 ( 47 usr; 11 con; 0-6 aty)
% 0.27/0.47 % Number of variables : 852 ( 52 ^; 787 !; 13 ?; 852 :)
% 0.27/0.47 % SPC : TH0_THM_EQU_NAR
% 0.27/0.47
% 0.27/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 15:31:22.775
% 0.27/0.47 %------------------------------------------------------------------------------
% 0.27/0.47 % Could-be-implicit typings (7)
% 0.27/0.47 thf(ty_n_t__Product____Type__Oprod_It__Heap__OTree_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
% 0.27/0.47 produc768687417list_a: $tType ).
% 0.27/0.47
% 0.27/0.47 thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J,type,
% 0.27/0.47 produc143150363Tree_a: $tType ).
% 0.27/0.47
% 0.27/0.47 thf(ty_n_t__Multiset__Omultiset_Itf__a_J,type,
% 0.27/0.47 multiset_a: $tType ).
% 0.27/0.47
% 0.27/0.47 thf(ty_n_t__List__Olist_Itf__a_J,type,
% 0.27/0.47 list_a: $tType ).
% 0.27/0.47
% 0.27/0.47 thf(ty_n_t__Heap__OTree_Itf__a_J,type,
% 0.27/0.47 tree_a: $tType ).
% 0.27/0.47
% 0.27/0.47 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 0.27/0.47 set_a: $tType ).
% 0.27/0.47
% 0.27/0.47 thf(ty_n_tf__a,type,
% 0.27/0.47 a: $tType ).
% 0.27/0.47
% 0.27/0.47 % Explicit typings (47)
% 0.27/0.47 thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.27/0.47 plus_plus_multiset_a: multiset_a > multiset_a > multiset_a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.27/0.47 zero_zero_multiset_a: multiset_a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Oheapify_001tf__a,type,
% 0.27/0.47 heapIm970322378pify_a: tree_a > tree_a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Ohs__is__empty_001tf__a,type,
% 0.27/0.47 heapIm229596386mpty_a: tree_a > $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Ohs__of__list_001tf__a,type,
% 0.27/0.47 heapIm1057938560list_a: list_a > tree_a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Oleft_001tf__a,type,
% 0.27/0.47 heapIm1140443833left_a: tree_a > tree_a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Oof__list__tree_001tf__a,type,
% 0.27/0.47 heapIm1637418125tree_a: list_a > tree_a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_OremoveLeaf_001tf__a,type,
% 0.27/0.47 heapIm837449470Leaf_a: tree_a > produc143150363Tree_a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Oright_001tf__a,type,
% 0.27/0.47 heapIm1257206334ight_a: tree_a > tree_a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_OsiftDown_001tf__a,type,
% 0.27/0.47 heapIm1091024090Down_a: tree_a > tree_a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Heap_OHeap_001t__Heap__OTree_Itf__a_J_001tf__a,type,
% 0.27/0.47 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 ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Heap_OHeap__axioms_001t__Heap__OTree_Itf__a_J_001tf__a,type,
% 0.27/0.47 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 ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Heap_OTree_OE_001tf__a,type,
% 0.27/0.47 e_a: tree_a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Heap_OTree_OT_001tf__a,type,
% 0.27/0.47 t_a: a > tree_a > tree_a > tree_a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Heap_Oin__tree_001tf__a,type,
% 0.27/0.47 in_tree_a: a > tree_a > $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Heap_Ois__heap_001tf__a,type,
% 0.27/0.47 is_heap_a: tree_a > $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Heap_Omultiset_001tf__a,type,
% 0.27/0.47 multiset_a2: tree_a > multiset_a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Heap_Oval_001tf__a,type,
% 0.27/0.47 val_a: tree_a > a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Lattices__Big_Olinorder__class_OMax_001tf__a,type,
% 0.27/0.47 lattic146396397_Max_a: set_a > a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_List_Olist_OCons_001tf__a,type,
% 0.27/0.47 cons_a: a > list_a > list_a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Multiset_Oadd__mset_001tf__a,type,
% 0.27/0.47 add_mset_a: a > multiset_a > multiset_a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Multiset_Ois__empty_001tf__a,type,
% 0.27/0.47 is_empty_a: multiset_a > $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Multiset_Oset__mset_001tf__a,type,
% 0.27/0.47 set_mset_a: multiset_a > set_a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Multiset_Osubseteq__mset_001tf__a,type,
% 0.27/0.47 subseteq_mset_a: multiset_a > multiset_a > $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_I_Eo_Mtf__a_J_J,type,
% 0.27/0.47 ord_less_eq_o_o_a: ( $o > $o > a ) > ( $o > $o > a ) > $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mtf__a_J,type,
% 0.27/0.47 ord_less_eq_o_a: ( $o > a ) > ( $o > a ) > $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.27/0.47 ord_le1199012836iset_a: multiset_a > multiset_a > $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
% 0.27/0.47 ord_less_eq_set_a: set_a > set_a > $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
% 0.27/0.47 ord_less_eq_a: a > a > $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Orderings_Oorder__class_OGreatest_001_062_I_Eo_Mtf__a_J,type,
% 0.27/0.47 order_Greatest_o_a: ( ( $o > a ) > $o ) > $o > a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Orderings_Oorder__class_OGreatest_001tf__a,type,
% 0.27/0.47 order_Greatest_a: ( a > $o ) > a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Product__Type_OPair_001t__Heap__OTree_Itf__a_J_001t__List__Olist_Itf__a_J,type,
% 0.27/0.47 produc1352981801list_a: tree_a > list_a > produc768687417list_a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Product__Type_OPair_001tf__a_001t__Heap__OTree_Itf__a_J,type,
% 0.27/0.47 produc686083979Tree_a: a > tree_a > produc143150363Tree_a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_RemoveMax_OCollection_Oset_001t__Heap__OTree_Itf__a_J_001tf__a,type,
% 0.27/0.47 set_Tree_a_a: ( tree_a > multiset_a ) > tree_a > set_a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_RemoveMax_ORemoveMax_001t__Heap__OTree_Itf__a_J_001tf__a,type,
% 0.27/0.47 removeMax_Tree_a_a: tree_a > ( tree_a > $o ) > ( list_a > tree_a ) > ( tree_a > multiset_a ) > ( tree_a > produc143150363Tree_a ) > ( tree_a > $o ) > $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_RemoveMax_ORemoveMax_Ossort_H_001t__Heap__OTree_Itf__a_J_001tf__a,type,
% 0.27/0.47 ssort_Tree_a_a: ( tree_a > $o ) > ( tree_a > produc143150363Tree_a ) > tree_a > list_a > list_a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_RemoveMax_ORemoveMax_Ossort_H__dom_001t__Heap__OTree_Itf__a_J_001tf__a,type,
% 0.27/0.47 ssort_dom_Tree_a_a: ( tree_a > $o ) > ( tree_a > produc143150363Tree_a ) > produc768687417list_a > $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_RemoveMax_ORemoveMax__axioms_001t__Heap__OTree_Itf__a_J_001tf__a,type,
% 0.27/0.47 remove301631099ee_a_a: ( tree_a > $o ) > ( list_a > tree_a ) > ( tree_a > multiset_a ) > ( tree_a > produc143150363Tree_a ) > ( tree_a > $o ) > $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_Set_OCollect_001tf__a,type,
% 0.27/0.47 collect_a: ( a > $o ) > set_a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_c_member_001tf__a,type,
% 0.27/0.47 member_a: a > set_a > $o ).
% 0.27/0.47
% 0.27/0.47 thf(sy_v_l1____,type,
% 0.27/0.47 l1: tree_a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_v_l2____,type,
% 0.27/0.47 l2: tree_a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_v_r1____,type,
% 0.27/0.47 r1: tree_a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_v_r2____,type,
% 0.27/0.47 r2: tree_a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_v_v1____,type,
% 0.27/0.47 v1: a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_v_v2____,type,
% 0.27/0.47 v2: a ).
% 0.27/0.47
% 0.27/0.47 thf(sy_v_v____,type,
% 0.27/0.47 v: a ).
% 0.27/0.47
% 0.27/0.47 % Relevant facts (214)
% 0.27/0.47 thf(fact_0__092_060open_062v2_A_092_060le_062_Av1_092_060close_062,axiom,
% 0.27/0.47 ord_less_eq_a @ v2 @ v1 ).
% 0.27/0.47
% 0.27/0.47 % \<open>v2 \<le> v1\<close>
% 0.27/0.47 thf(fact_1_True,axiom,
% 0.27/0.47 ord_less_eq_a @ v1 @ v ).
% 0.27/0.47
% 0.27/0.47 % True
% 0.27/0.47 thf(fact_2__C5__2_Ohyps_C_I2_J,axiom,
% 0.27/0.47 ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ v2 @ l2 @ r2 ) ) @ ( val_a @ ( t_a @ v1 @ l1 @ r1 ) ) )
% 0.27/0.47 => ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ v2 @ l2 @ r2 ) ) @ v )
% 0.27/0.47 => ( ( multiset_a2 @ ( heapIm1091024090Down_a @ ( t_a @ v @ ( heapIm1140443833left_a @ ( t_a @ v2 @ l2 @ r2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ v2 @ l2 @ r2 ) ) ) ) )
% 0.27/0.47 = ( multiset_a2 @ ( t_a @ v @ ( heapIm1140443833left_a @ ( t_a @ v2 @ l2 @ r2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ v2 @ l2 @ r2 ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % "5_2.hyps"(2)
% 0.27/0.47 thf(fact_3__C5__2_Ohyps_C_I1_J,axiom,
% 0.27/0.47 ( ( ord_less_eq_a @ ( val_a @ ( t_a @ v2 @ l2 @ r2 ) ) @ ( val_a @ ( t_a @ v1 @ l1 @ r1 ) ) )
% 0.27/0.47 => ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ v1 @ l1 @ r1 ) ) @ v )
% 0.27/0.47 => ( ( multiset_a2 @ ( heapIm1091024090Down_a @ ( t_a @ v @ ( heapIm1140443833left_a @ ( t_a @ v1 @ l1 @ r1 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ v1 @ l1 @ r1 ) ) ) ) )
% 0.27/0.47 = ( multiset_a2 @ ( t_a @ v @ ( heapIm1140443833left_a @ ( t_a @ v1 @ l1 @ r1 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ v1 @ l1 @ r1 ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % "5_2.hyps"(1)
% 0.27/0.47 thf(fact_4_siftDown__Node,axiom,
% 0.27/0.47 ! [T: tree_a,V: a,L: tree_a,R: tree_a] :
% 0.27/0.47 ( ( T
% 0.27/0.47 = ( t_a @ V @ L @ R ) )
% 0.27/0.47 => ? [L2: tree_a,V2: a,R2: tree_a] :
% 0.27/0.47 ( ( ( heapIm1091024090Down_a @ T )
% 0.27/0.47 = ( t_a @ V2 @ L2 @ R2 ) )
% 0.27/0.47 & ( ord_less_eq_a @ V @ V2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % siftDown_Node
% 0.27/0.47 thf(fact_5_Tree_Oinject,axiom,
% 0.27/0.47 ! [X21: a,X22: tree_a,X23: tree_a,Y21: a,Y22: tree_a,Y23: tree_a] :
% 0.27/0.47 ( ( ( t_a @ X21 @ X22 @ X23 )
% 0.27/0.47 = ( t_a @ Y21 @ Y22 @ Y23 ) )
% 0.27/0.47 = ( ( X21 = Y21 )
% 0.27/0.47 & ( X22 = Y22 )
% 0.27/0.47 & ( X23 = Y23 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Tree.inject
% 0.27/0.47 thf(fact_6_heapify_Osimps_I2_J,axiom,
% 0.27/0.47 ! [V: a,L: tree_a,R: tree_a] :
% 0.27/0.47 ( ( heapIm970322378pify_a @ ( t_a @ V @ L @ R ) )
% 0.27/0.47 = ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm970322378pify_a @ L ) @ ( heapIm970322378pify_a @ R ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % heapify.simps(2)
% 0.27/0.47 thf(fact_7_siftDown__heap__is__heap,axiom,
% 0.27/0.47 ! [L: tree_a,R: tree_a,T: tree_a,V: a] :
% 0.27/0.47 ( ( is_heap_a @ L )
% 0.27/0.47 => ( ( is_heap_a @ R )
% 0.27/0.47 => ( ( T
% 0.27/0.47 = ( t_a @ V @ L @ R ) )
% 0.27/0.47 => ( is_heap_a @ ( heapIm1091024090Down_a @ T ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % siftDown_heap_is_heap
% 0.27/0.47 thf(fact_8_siftDown__in__tree__set,axiom,
% 0.27/0.47 ( in_tree_a
% 0.27/0.47 = ( ^ [V3: a,T2: tree_a] : ( in_tree_a @ V3 @ ( heapIm1091024090Down_a @ T2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % siftDown_in_tree_set
% 0.27/0.47 thf(fact_9_left_Osimps,axiom,
% 0.27/0.47 ! [V: a,L: tree_a,R: tree_a] :
% 0.27/0.47 ( ( heapIm1140443833left_a @ ( t_a @ V @ L @ R ) )
% 0.27/0.47 = L ) ).
% 0.27/0.47
% 0.27/0.47 % left.simps
% 0.27/0.47 thf(fact_10_right_Osimps,axiom,
% 0.27/0.47 ! [V: a,L: tree_a,R: tree_a] :
% 0.27/0.47 ( ( heapIm1257206334ight_a @ ( t_a @ V @ L @ R ) )
% 0.27/0.47 = R ) ).
% 0.27/0.47
% 0.27/0.47 % right.simps
% 0.27/0.47 thf(fact_11_siftDown_Osimps_I2_J,axiom,
% 0.27/0.47 ! [V: a] :
% 0.27/0.47 ( ( heapIm1091024090Down_a @ ( t_a @ V @ e_a @ e_a ) )
% 0.27/0.47 = ( t_a @ V @ e_a @ e_a ) ) ).
% 0.27/0.47
% 0.27/0.47 % siftDown.simps(2)
% 0.27/0.47 thf(fact_12_siftDown_Osimps_I1_J,axiom,
% 0.27/0.47 ( ( heapIm1091024090Down_a @ e_a )
% 0.27/0.47 = e_a ) ).
% 0.27/0.47
% 0.27/0.47 % siftDown.simps(1)
% 0.27/0.47 thf(fact_13_in__tree_Osimps_I1_J,axiom,
% 0.27/0.47 ! [V: a] :
% 0.27/0.47 ~ ( in_tree_a @ V @ e_a ) ).
% 0.27/0.47
% 0.27/0.47 % in_tree.simps(1)
% 0.27/0.47 thf(fact_14_is__heap_Osimps_I6_J,axiom,
% 0.27/0.47 ! [V: a,Vd: a,Ve: tree_a,Vf: tree_a,Va: a,Vb: tree_a,Vc: tree_a] :
% 0.27/0.47 ( ( is_heap_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
% 0.27/0.47 = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 0.27/0.47 & ( is_heap_a @ ( t_a @ Va @ Vb @ Vc ) )
% 0.27/0.47 & ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
% 0.27/0.47 & ( is_heap_a @ ( t_a @ Vd @ Ve @ Vf ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % is_heap.simps(6)
% 0.27/0.47 thf(fact_15_is__heap_Osimps_I5_J,axiom,
% 0.27/0.47 ! [V: a,Va: a,Vb: tree_a,Vc: tree_a,Vd: a,Ve: tree_a,Vf: tree_a] :
% 0.27/0.47 ( ( is_heap_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 0.27/0.47 = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
% 0.27/0.47 & ( is_heap_a @ ( t_a @ Vd @ Ve @ Vf ) )
% 0.27/0.47 & ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 0.27/0.47 & ( is_heap_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % is_heap.simps(5)
% 0.27/0.47 thf(fact_16_is__heap_Osimps_I4_J,axiom,
% 0.27/0.47 ! [V: a,Va: a,Vb: tree_a,Vc: tree_a] :
% 0.27/0.47 ( ( is_heap_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) )
% 0.27/0.47 = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 0.27/0.47 & ( is_heap_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % is_heap.simps(4)
% 0.27/0.47 thf(fact_17_is__heap_Osimps_I3_J,axiom,
% 0.27/0.47 ! [V: a,Va: a,Vb: tree_a,Vc: tree_a] :
% 0.27/0.47 ( ( is_heap_a @ ( t_a @ V @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) )
% 0.27/0.47 = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 0.27/0.47 & ( is_heap_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % is_heap.simps(3)
% 0.27/0.47 thf(fact_18_is__heap_Osimps_I2_J,axiom,
% 0.27/0.47 ! [V: a] : ( is_heap_a @ ( t_a @ V @ e_a @ e_a ) ) ).
% 0.27/0.47
% 0.27/0.47 % is_heap.simps(2)
% 0.27/0.47 thf(fact_19_is__heap_Osimps_I1_J,axiom,
% 0.27/0.47 is_heap_a @ e_a ).
% 0.27/0.47
% 0.27/0.47 % is_heap.simps(1)
% 0.27/0.47 thf(fact_20_heapify_Osimps_I1_J,axiom,
% 0.27/0.47 ( ( heapIm970322378pify_a @ e_a )
% 0.27/0.47 = e_a ) ).
% 0.27/0.47
% 0.27/0.47 % heapify.simps(1)
% 0.27/0.47 thf(fact_21_is__heap__max,axiom,
% 0.27/0.47 ! [V: a,T: tree_a] :
% 0.27/0.47 ( ( in_tree_a @ V @ T )
% 0.27/0.47 => ( ( is_heap_a @ T )
% 0.27/0.47 => ( ord_less_eq_a @ V @ ( val_a @ T ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % is_heap_max
% 0.27/0.47 thf(fact_22_heapify__heap__is__heap,axiom,
% 0.27/0.47 ! [T: tree_a] : ( is_heap_a @ ( heapIm970322378pify_a @ T ) ) ).
% 0.27/0.47
% 0.27/0.47 % heapify_heap_is_heap
% 0.27/0.47 thf(fact_23_siftDown__in__tree,axiom,
% 0.27/0.47 ! [T: tree_a] :
% 0.27/0.47 ( ( T != e_a )
% 0.27/0.47 => ( in_tree_a @ ( val_a @ ( heapIm1091024090Down_a @ T ) ) @ T ) ) ).
% 0.27/0.47
% 0.27/0.47 % siftDown_in_tree
% 0.27/0.47 thf(fact_24_is__heap_Ocases,axiom,
% 0.27/0.47 ! [X: tree_a] :
% 0.27/0.47 ( ( X != e_a )
% 0.27/0.47 => ( ! [V4: a] :
% 0.27/0.47 ( X
% 0.27/0.47 != ( t_a @ V4 @ e_a @ e_a ) )
% 0.27/0.47 => ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 0.27/0.47 ( X
% 0.27/0.47 != ( t_a @ V4 @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 0.27/0.47 => ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 0.27/0.47 ( X
% 0.27/0.47 != ( t_a @ V4 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
% 0.27/0.47 => ~ ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a] :
% 0.27/0.47 ( X
% 0.27/0.47 != ( t_a @ V4 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % is_heap.cases
% 0.27/0.47 thf(fact_25_Tree_Oexhaust,axiom,
% 0.27/0.47 ! [Y: tree_a] :
% 0.27/0.47 ( ( Y != e_a )
% 0.27/0.47 => ~ ! [X212: a,X222: tree_a,X232: tree_a] :
% 0.27/0.47 ( Y
% 0.27/0.47 != ( t_a @ X212 @ X222 @ X232 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Tree.exhaust
% 0.27/0.47 thf(fact_26_Tree_Oinduct,axiom,
% 0.27/0.47 ! [P: tree_a > $o,Tree: tree_a] :
% 0.27/0.47 ( ( P @ e_a )
% 0.27/0.47 => ( ! [X1: a,X2: tree_a,X3: tree_a] :
% 0.27/0.47 ( ( P @ X2 )
% 0.27/0.47 => ( ( P @ X3 )
% 0.27/0.47 => ( P @ ( t_a @ X1 @ X2 @ X3 ) ) ) )
% 0.27/0.47 => ( P @ Tree ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Tree.induct
% 0.27/0.47 thf(fact_27_Tree_Odistinct_I1_J,axiom,
% 0.27/0.47 ! [X21: a,X22: tree_a,X23: tree_a] :
% 0.27/0.47 ( e_a
% 0.27/0.47 != ( t_a @ X21 @ X22 @ X23 ) ) ).
% 0.27/0.47
% 0.27/0.47 % Tree.distinct(1)
% 0.27/0.47 thf(fact_28_siftDown_Osimps_I3_J,axiom,
% 0.27/0.47 ! [Va: a,Vb: tree_a,Vc: tree_a,V: a] :
% 0.27/0.47 ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 0.27/0.47 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) )
% 0.27/0.47 = ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) ) )
% 0.27/0.47 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 0.27/0.47 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) )
% 0.27/0.47 = ( 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 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % siftDown.simps(3)
% 0.27/0.47 thf(fact_29_siftDown_Osimps_I4_J,axiom,
% 0.27/0.47 ! [Va: a,Vb: tree_a,Vc: tree_a,V: a] :
% 0.27/0.47 ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 0.27/0.47 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) )
% 0.27/0.47 = ( t_a @ V @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) ) )
% 0.27/0.47 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 0.27/0.47 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) )
% 0.27/0.47 = ( 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 ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % siftDown.simps(4)
% 0.27/0.47 thf(fact_30_val_Osimps,axiom,
% 0.27/0.47 ! [V: a,Uu: tree_a,Uv: tree_a] :
% 0.27/0.47 ( ( val_a @ ( t_a @ V @ Uu @ Uv ) )
% 0.27/0.47 = V ) ).
% 0.27/0.47
% 0.27/0.47 % val.simps
% 0.27/0.47 thf(fact_31_siftDown_Osimps_I5_J,axiom,
% 0.27/0.47 ! [Vd: a,Ve: tree_a,Vf: tree_a,Va: a,Vb: tree_a,Vc: tree_a,V: a] :
% 0.27/0.47 ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) )
% 0.27/0.47 => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 0.27/0.47 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 0.27/0.47 = ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) ) )
% 0.27/0.47 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 0.27/0.47 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 0.27/0.47 = ( 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 ) ) ) ) ) )
% 0.27/0.47 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) )
% 0.27/0.47 => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
% 0.27/0.47 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 0.27/0.47 = ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) ) )
% 0.27/0.47 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
% 0.27/0.47 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 0.27/0.47 = ( 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 ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % siftDown.simps(5)
% 0.27/0.47 thf(fact_32_siftDown_Osimps_I6_J,axiom,
% 0.27/0.47 ! [Va: a,Vb: tree_a,Vc: tree_a,Vd: a,Ve: tree_a,Vf: tree_a,V: a] :
% 0.27/0.47 ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 0.27/0.47 => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
% 0.27/0.47 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
% 0.27/0.47 = ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) ) )
% 0.27/0.47 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
% 0.27/0.47 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
% 0.27/0.47 = ( 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 ) ) ) ) ) )
% 0.27/0.47 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 0.27/0.47 => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 0.27/0.47 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
% 0.27/0.47 = ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) ) )
% 0.27/0.47 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 0.27/0.47 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
% 0.27/0.47 = ( 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 ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % siftDown.simps(6)
% 0.27/0.47 thf(fact_33_in__tree_Osimps_I2_J,axiom,
% 0.27/0.47 ! [V: a,V5: a,L: tree_a,R: tree_a] :
% 0.27/0.47 ( ( in_tree_a @ V @ ( t_a @ V5 @ L @ R ) )
% 0.27/0.47 = ( ( V = V5 )
% 0.27/0.47 | ( in_tree_a @ V @ L )
% 0.27/0.47 | ( in_tree_a @ V @ R ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % in_tree.simps(2)
% 0.27/0.47 thf(fact_34_removeLeaf_Oinduct,axiom,
% 0.27/0.47 ! [P: tree_a > $o,A0: tree_a] :
% 0.27/0.47 ( ! [V4: a] : ( P @ ( t_a @ V4 @ e_a @ e_a ) )
% 0.27/0.47 => ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 0.27/0.47 ( ( P @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
% 0.27/0.47 => ( ( P @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
% 0.27/0.47 => ( P @ ( t_a @ V4 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) ) ) )
% 0.27/0.47 => ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 0.27/0.47 ( ( P @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
% 0.27/0.47 => ( ( P @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
% 0.27/0.47 => ( P @ ( t_a @ V4 @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) )
% 0.27/0.47 => ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a] :
% 0.27/0.47 ( ( P @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
% 0.27/0.47 => ( ( P @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
% 0.27/0.47 => ( P @ ( t_a @ V4 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) )
% 0.27/0.47 => ( ! [V4: a,Vd2: a,Ve2: tree_a,Vf2: tree_a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 0.27/0.47 ( ( P @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) )
% 0.27/0.47 => ( ( P @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) )
% 0.27/0.47 => ( P @ ( t_a @ V4 @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) )
% 0.27/0.47 => ( ( P @ e_a )
% 0.27/0.47 => ( P @ A0 ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % removeLeaf.induct
% 0.27/0.47 thf(fact_35_removeLeaf_Ocases,axiom,
% 0.27/0.47 ! [X: tree_a] :
% 0.27/0.47 ( ! [V4: a] :
% 0.27/0.47 ( X
% 0.27/0.47 != ( t_a @ V4 @ e_a @ e_a ) )
% 0.27/0.47 => ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 0.27/0.47 ( X
% 0.27/0.47 != ( t_a @ V4 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
% 0.27/0.47 => ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 0.27/0.47 ( X
% 0.27/0.47 != ( t_a @ V4 @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 0.27/0.47 => ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a] :
% 0.27/0.47 ( X
% 0.27/0.47 != ( t_a @ V4 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
% 0.27/0.47 => ( ! [V4: a,Vd2: a,Ve2: tree_a,Vf2: tree_a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 0.27/0.47 ( X
% 0.27/0.47 != ( t_a @ V4 @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 0.27/0.47 => ( X = e_a ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % removeLeaf.cases
% 0.27/0.47 thf(fact_36_siftDown_Ocases,axiom,
% 0.27/0.47 ! [X: tree_a] :
% 0.27/0.47 ( ( X != e_a )
% 0.27/0.47 => ( ! [V4: a] :
% 0.27/0.47 ( X
% 0.27/0.47 != ( t_a @ V4 @ e_a @ e_a ) )
% 0.27/0.47 => ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 0.27/0.47 ( X
% 0.27/0.47 != ( t_a @ V4 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
% 0.27/0.47 => ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 0.27/0.47 ( X
% 0.27/0.47 != ( t_a @ V4 @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 0.27/0.47 => ~ ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a] :
% 0.27/0.47 ( X
% 0.27/0.47 != ( t_a @ V4 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % siftDown.cases
% 0.27/0.47 thf(fact_37_hs__is__empty__def,axiom,
% 0.27/0.47 ( heapIm229596386mpty_a
% 0.27/0.47 = ( ^ [T2: tree_a] : ( T2 = e_a ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % hs_is_empty_def
% 0.27/0.47 thf(fact_38_order__refl,axiom,
% 0.27/0.47 ! [X: $o > a] : ( ord_less_eq_o_a @ X @ X ) ).
% 0.27/0.47
% 0.27/0.47 % order_refl
% 0.27/0.47 thf(fact_39_order__refl,axiom,
% 0.27/0.47 ! [X: a] : ( ord_less_eq_a @ X @ X ) ).
% 0.27/0.47
% 0.27/0.47 % order_refl
% 0.27/0.47 thf(fact_40_heap__top__geq,axiom,
% 0.27/0.47 ! [A: a,T: tree_a] :
% 0.27/0.47 ( ( member_a @ A @ ( set_mset_a @ ( multiset_a2 @ T ) ) )
% 0.27/0.47 => ( ( is_heap_a @ T )
% 0.27/0.47 => ( ord_less_eq_a @ A @ ( val_a @ T ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % heap_top_geq
% 0.27/0.47 thf(fact_41_hs__of__list__def,axiom,
% 0.27/0.47 ( heapIm1057938560list_a
% 0.27/0.47 = ( ^ [L3: list_a] : ( heapIm970322378pify_a @ ( heapIm1637418125tree_a @ L3 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % hs_of_list_def
% 0.27/0.47 thf(fact_42_mem__Collect__eq,axiom,
% 0.27/0.47 ! [A: a,P: a > $o] :
% 0.27/0.47 ( ( member_a @ A @ ( collect_a @ P ) )
% 0.27/0.47 = ( P @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 % mem_Collect_eq
% 0.27/0.47 thf(fact_43_Collect__mem__eq,axiom,
% 0.27/0.47 ! [A2: set_a] :
% 0.27/0.47 ( ( collect_a
% 0.27/0.47 @ ^ [X4: a] : ( member_a @ X4 @ A2 ) )
% 0.27/0.47 = A2 ) ).
% 0.27/0.47
% 0.27/0.47 % Collect_mem_eq
% 0.27/0.47 thf(fact_44_le__funD,axiom,
% 0.27/0.47 ! [F: $o > a,G: $o > a,X: $o] :
% 0.27/0.47 ( ( ord_less_eq_o_a @ F @ G )
% 0.27/0.47 => ( ord_less_eq_a @ ( F @ X ) @ ( G @ X ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % le_funD
% 0.27/0.47 thf(fact_45_le__funE,axiom,
% 0.27/0.47 ! [F: $o > a,G: $o > a,X: $o] :
% 0.27/0.47 ( ( ord_less_eq_o_a @ F @ G )
% 0.27/0.47 => ( ord_less_eq_a @ ( F @ X ) @ ( G @ X ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % le_funE
% 0.27/0.47 thf(fact_46_le__funI,axiom,
% 0.27/0.47 ! [F: $o > a,G: $o > a] :
% 0.27/0.47 ( ! [X5: $o] : ( ord_less_eq_a @ ( F @ X5 ) @ ( G @ X5 ) )
% 0.27/0.47 => ( ord_less_eq_o_a @ F @ G ) ) ).
% 0.27/0.47
% 0.27/0.47 % le_funI
% 0.27/0.47 thf(fact_47_le__fun__def,axiom,
% 0.27/0.47 ( ord_less_eq_o_a
% 0.27/0.47 = ( ^ [F2: $o > a,G2: $o > a] :
% 0.27/0.47 ! [X4: $o] : ( ord_less_eq_a @ ( F2 @ X4 ) @ ( G2 @ X4 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % le_fun_def
% 0.27/0.47 thf(fact_48_order__subst1,axiom,
% 0.27/0.47 ! [A: a,F: ( $o > a ) > a,B: $o > a,C: $o > a] :
% 0.27/0.47 ( ( ord_less_eq_a @ A @ ( F @ B ) )
% 0.27/0.47 => ( ( ord_less_eq_o_a @ B @ C )
% 0.27/0.47 => ( ! [X5: $o > a,Y2: $o > a] :
% 0.27/0.47 ( ( ord_less_eq_o_a @ X5 @ Y2 )
% 0.27/0.47 => ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.27/0.47 => ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order_subst1
% 0.27/0.47 thf(fact_49_order__subst1,axiom,
% 0.27/0.47 ! [A: $o > a,F: a > $o > a,B: a,C: a] :
% 0.27/0.47 ( ( ord_less_eq_o_a @ A @ ( F @ B ) )
% 0.27/0.47 => ( ( ord_less_eq_a @ B @ C )
% 0.27/0.47 => ( ! [X5: a,Y2: a] :
% 0.27/0.47 ( ( ord_less_eq_a @ X5 @ Y2 )
% 0.27/0.47 => ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.27/0.47 => ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order_subst1
% 0.27/0.47 thf(fact_50_order__subst1,axiom,
% 0.27/0.47 ! [A: $o > a,F: ( $o > a ) > $o > a,B: $o > a,C: $o > a] :
% 0.27/0.47 ( ( ord_less_eq_o_a @ A @ ( F @ B ) )
% 0.27/0.47 => ( ( ord_less_eq_o_a @ B @ C )
% 0.27/0.47 => ( ! [X5: $o > a,Y2: $o > a] :
% 0.27/0.47 ( ( ord_less_eq_o_a @ X5 @ Y2 )
% 0.27/0.47 => ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.27/0.47 => ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order_subst1
% 0.27/0.47 thf(fact_51_order__subst1,axiom,
% 0.27/0.47 ! [A: a,F: a > a,B: a,C: a] :
% 0.27/0.47 ( ( ord_less_eq_a @ A @ ( F @ B ) )
% 0.27/0.47 => ( ( ord_less_eq_a @ B @ C )
% 0.27/0.47 => ( ! [X5: a,Y2: a] :
% 0.27/0.47 ( ( ord_less_eq_a @ X5 @ Y2 )
% 0.27/0.47 => ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.27/0.47 => ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order_subst1
% 0.27/0.47 thf(fact_52_order__subst2,axiom,
% 0.27/0.47 ! [A: a,B: a,F: a > $o > a,C: $o > a] :
% 0.27/0.47 ( ( ord_less_eq_a @ A @ B )
% 0.27/0.47 => ( ( ord_less_eq_o_a @ ( F @ B ) @ C )
% 0.27/0.47 => ( ! [X5: a,Y2: a] :
% 0.27/0.47 ( ( ord_less_eq_a @ X5 @ Y2 )
% 0.27/0.47 => ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.27/0.47 => ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order_subst2
% 0.27/0.47 thf(fact_53_order__subst2,axiom,
% 0.27/0.47 ! [A: $o > a,B: $o > a,F: ( $o > a ) > a,C: a] :
% 0.27/0.47 ( ( ord_less_eq_o_a @ A @ B )
% 0.27/0.47 => ( ( ord_less_eq_a @ ( F @ B ) @ C )
% 0.27/0.47 => ( ! [X5: $o > a,Y2: $o > a] :
% 0.27/0.47 ( ( ord_less_eq_o_a @ X5 @ Y2 )
% 0.27/0.47 => ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.27/0.47 => ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order_subst2
% 0.27/0.47 thf(fact_54_order__subst2,axiom,
% 0.27/0.47 ! [A: $o > a,B: $o > a,F: ( $o > a ) > $o > a,C: $o > a] :
% 0.27/0.47 ( ( ord_less_eq_o_a @ A @ B )
% 0.27/0.47 => ( ( ord_less_eq_o_a @ ( F @ B ) @ C )
% 0.27/0.47 => ( ! [X5: $o > a,Y2: $o > a] :
% 0.27/0.47 ( ( ord_less_eq_o_a @ X5 @ Y2 )
% 0.27/0.47 => ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.27/0.47 => ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order_subst2
% 0.27/0.47 thf(fact_55_order__subst2,axiom,
% 0.27/0.47 ! [A: a,B: a,F: a > a,C: a] :
% 0.27/0.47 ( ( ord_less_eq_a @ A @ B )
% 0.27/0.47 => ( ( ord_less_eq_a @ ( F @ B ) @ C )
% 0.27/0.47 => ( ! [X5: a,Y2: a] :
% 0.27/0.47 ( ( ord_less_eq_a @ X5 @ Y2 )
% 0.27/0.47 => ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.27/0.47 => ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order_subst2
% 0.27/0.47 thf(fact_56_dual__order_Oantisym,axiom,
% 0.27/0.47 ! [B: $o > a,A: $o > a] :
% 0.27/0.47 ( ( ord_less_eq_o_a @ B @ A )
% 0.27/0.47 => ( ( ord_less_eq_o_a @ A @ B )
% 0.27/0.47 => ( A = B ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % dual_order.antisym
% 0.27/0.47 thf(fact_57_dual__order_Oantisym,axiom,
% 0.27/0.47 ! [B: a,A: a] :
% 0.27/0.47 ( ( ord_less_eq_a @ B @ A )
% 0.27/0.47 => ( ( ord_less_eq_a @ A @ B )
% 0.27/0.47 => ( A = B ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % dual_order.antisym
% 0.27/0.47 thf(fact_58_dual__order_Oeq__iff,axiom,
% 0.27/0.47 ( ( ^ [Y3: $o > a,Z: $o > a] : ( Y3 = Z ) )
% 0.27/0.47 = ( ^ [A3: $o > a,B2: $o > a] :
% 0.27/0.47 ( ( ord_less_eq_o_a @ B2 @ A3 )
% 0.27/0.47 & ( ord_less_eq_o_a @ A3 @ B2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % dual_order.eq_iff
% 0.27/0.47 thf(fact_59_dual__order_Oeq__iff,axiom,
% 0.27/0.47 ( ( ^ [Y3: a,Z: a] : ( Y3 = Z ) )
% 0.27/0.47 = ( ^ [A3: a,B2: a] :
% 0.27/0.47 ( ( ord_less_eq_a @ B2 @ A3 )
% 0.27/0.47 & ( ord_less_eq_a @ A3 @ B2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % dual_order.eq_iff
% 0.27/0.47 thf(fact_60_dual__order_Otrans,axiom,
% 0.27/0.47 ! [B: $o > a,A: $o > a,C: $o > a] :
% 0.27/0.47 ( ( ord_less_eq_o_a @ B @ A )
% 0.27/0.47 => ( ( ord_less_eq_o_a @ C @ B )
% 0.27/0.47 => ( ord_less_eq_o_a @ C @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % dual_order.trans
% 0.27/0.47 thf(fact_61_dual__order_Otrans,axiom,
% 0.27/0.47 ! [B: a,A: a,C: a] :
% 0.27/0.47 ( ( ord_less_eq_a @ B @ A )
% 0.27/0.47 => ( ( ord_less_eq_a @ C @ B )
% 0.27/0.47 => ( ord_less_eq_a @ C @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % dual_order.trans
% 0.27/0.47 thf(fact_62_linorder__wlog,axiom,
% 0.27/0.47 ! [P: a > a > $o,A: a,B: a] :
% 0.27/0.47 ( ! [A4: a,B3: a] :
% 0.27/0.47 ( ( ord_less_eq_a @ A4 @ B3 )
% 0.27/0.47 => ( P @ A4 @ B3 ) )
% 0.27/0.47 => ( ! [A4: a,B3: a] :
% 0.27/0.47 ( ( P @ B3 @ A4 )
% 0.27/0.47 => ( P @ A4 @ B3 ) )
% 0.27/0.47 => ( P @ A @ B ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % linorder_wlog
% 0.27/0.47 thf(fact_63_dual__order_Orefl,axiom,
% 0.27/0.47 ! [A: $o > a] : ( ord_less_eq_o_a @ A @ A ) ).
% 0.27/0.47
% 0.27/0.47 % dual_order.refl
% 0.27/0.47 thf(fact_64_dual__order_Orefl,axiom,
% 0.27/0.47 ! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% 0.27/0.47
% 0.27/0.47 % dual_order.refl
% 0.27/0.47 thf(fact_65_order__trans,axiom,
% 0.27/0.47 ! [X: $o > a,Y: $o > a,Z2: $o > a] :
% 0.27/0.47 ( ( ord_less_eq_o_a @ X @ Y )
% 0.27/0.47 => ( ( ord_less_eq_o_a @ Y @ Z2 )
% 0.27/0.47 => ( ord_less_eq_o_a @ X @ Z2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order_trans
% 0.27/0.47 thf(fact_66_order__trans,axiom,
% 0.27/0.47 ! [X: a,Y: a,Z2: a] :
% 0.27/0.47 ( ( ord_less_eq_a @ X @ Y )
% 0.27/0.47 => ( ( ord_less_eq_a @ Y @ Z2 )
% 0.27/0.47 => ( ord_less_eq_a @ X @ Z2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order_trans
% 0.27/0.47 thf(fact_67_order__class_Oorder_Oantisym,axiom,
% 0.27/0.47 ! [A: $o > a,B: $o > a] :
% 0.27/0.47 ( ( ord_less_eq_o_a @ A @ B )
% 0.27/0.47 => ( ( ord_less_eq_o_a @ B @ A )
% 0.27/0.47 => ( A = B ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order_class.order.antisym
% 0.27/0.47 thf(fact_68_order__class_Oorder_Oantisym,axiom,
% 0.27/0.47 ! [A: a,B: a] :
% 0.27/0.47 ( ( ord_less_eq_a @ A @ B )
% 0.27/0.47 => ( ( ord_less_eq_a @ B @ A )
% 0.27/0.47 => ( A = B ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order_class.order.antisym
% 0.27/0.47 thf(fact_69_ord__le__eq__trans,axiom,
% 0.27/0.47 ! [A: $o > a,B: $o > a,C: $o > a] :
% 0.27/0.47 ( ( ord_less_eq_o_a @ A @ B )
% 0.27/0.47 => ( ( B = C )
% 0.27/0.47 => ( ord_less_eq_o_a @ A @ C ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord_le_eq_trans
% 0.27/0.47 thf(fact_70_ord__le__eq__trans,axiom,
% 0.27/0.47 ! [A: a,B: a,C: a] :
% 0.27/0.47 ( ( ord_less_eq_a @ A @ B )
% 0.27/0.47 => ( ( B = C )
% 0.27/0.47 => ( ord_less_eq_a @ A @ C ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord_le_eq_trans
% 0.27/0.47 thf(fact_71_ord__eq__le__trans,axiom,
% 0.27/0.47 ! [A: $o > a,B: $o > a,C: $o > a] :
% 0.27/0.47 ( ( A = B )
% 0.27/0.47 => ( ( ord_less_eq_o_a @ B @ C )
% 0.27/0.47 => ( ord_less_eq_o_a @ A @ C ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord_eq_le_trans
% 0.27/0.47 thf(fact_72_ord__eq__le__trans,axiom,
% 0.27/0.47 ! [A: a,B: a,C: a] :
% 0.27/0.47 ( ( A = B )
% 0.27/0.47 => ( ( ord_less_eq_a @ B @ C )
% 0.27/0.47 => ( ord_less_eq_a @ A @ C ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord_eq_le_trans
% 0.27/0.47 thf(fact_73_order__class_Oorder_Oeq__iff,axiom,
% 0.27/0.47 ( ( ^ [Y3: $o > a,Z: $o > a] : ( Y3 = Z ) )
% 0.27/0.47 = ( ^ [A3: $o > a,B2: $o > a] :
% 0.27/0.47 ( ( ord_less_eq_o_a @ A3 @ B2 )
% 0.27/0.47 & ( ord_less_eq_o_a @ B2 @ A3 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order_class.order.eq_iff
% 0.27/0.47 thf(fact_74_order__class_Oorder_Oeq__iff,axiom,
% 0.27/0.47 ( ( ^ [Y3: a,Z: a] : ( Y3 = Z ) )
% 0.27/0.47 = ( ^ [A3: a,B2: a] :
% 0.27/0.47 ( ( ord_less_eq_a @ A3 @ B2 )
% 0.27/0.47 & ( ord_less_eq_a @ B2 @ A3 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order_class.order.eq_iff
% 0.27/0.47 thf(fact_75_antisym__conv,axiom,
% 0.27/0.47 ! [Y: $o > a,X: $o > a] :
% 0.27/0.47 ( ( ord_less_eq_o_a @ Y @ X )
% 0.27/0.47 => ( ( ord_less_eq_o_a @ X @ Y )
% 0.27/0.47 = ( X = Y ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % antisym_conv
% 0.27/0.47 thf(fact_76_antisym__conv,axiom,
% 0.27/0.47 ! [Y: a,X: a] :
% 0.27/0.47 ( ( ord_less_eq_a @ Y @ X )
% 0.27/0.47 => ( ( ord_less_eq_a @ X @ Y )
% 0.27/0.47 = ( X = Y ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % antisym_conv
% 0.27/0.47 thf(fact_77_le__cases3,axiom,
% 0.27/0.47 ! [X: a,Y: a,Z2: a] :
% 0.27/0.47 ( ( ( ord_less_eq_a @ X @ Y )
% 0.27/0.47 => ~ ( ord_less_eq_a @ Y @ Z2 ) )
% 0.27/0.47 => ( ( ( ord_less_eq_a @ Y @ X )
% 0.27/0.47 => ~ ( ord_less_eq_a @ X @ Z2 ) )
% 0.27/0.47 => ( ( ( ord_less_eq_a @ X @ Z2 )
% 0.27/0.47 => ~ ( ord_less_eq_a @ Z2 @ Y ) )
% 0.27/0.47 => ( ( ( ord_less_eq_a @ Z2 @ Y )
% 0.27/0.47 => ~ ( ord_less_eq_a @ Y @ X ) )
% 0.27/0.47 => ( ( ( ord_less_eq_a @ Y @ Z2 )
% 0.27/0.47 => ~ ( ord_less_eq_a @ Z2 @ X ) )
% 0.27/0.47 => ~ ( ( ord_less_eq_a @ Z2 @ X )
% 0.27/0.47 => ~ ( ord_less_eq_a @ X @ Y ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % le_cases3
% 0.27/0.47 thf(fact_78_order_Otrans,axiom,
% 0.27/0.47 ! [A: $o > a,B: $o > a,C: $o > a] :
% 0.27/0.47 ( ( ord_less_eq_o_a @ A @ B )
% 0.27/0.47 => ( ( ord_less_eq_o_a @ B @ C )
% 0.27/0.47 => ( ord_less_eq_o_a @ A @ C ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order.trans
% 0.27/0.47 thf(fact_79_order_Otrans,axiom,
% 0.27/0.47 ! [A: a,B: a,C: a] :
% 0.27/0.47 ( ( ord_less_eq_a @ A @ B )
% 0.27/0.47 => ( ( ord_less_eq_a @ B @ C )
% 0.27/0.47 => ( ord_less_eq_a @ A @ C ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % order.trans
% 0.27/0.47 thf(fact_80_le__cases,axiom,
% 0.27/0.47 ! [X: a,Y: a] :
% 0.27/0.47 ( ~ ( ord_less_eq_a @ X @ Y )
% 0.27/0.47 => ( ord_less_eq_a @ Y @ X ) ) ).
% 0.27/0.47
% 0.27/0.47 % le_cases
% 0.27/0.47 thf(fact_81_eq__refl,axiom,
% 0.27/0.47 ! [X: $o > a,Y: $o > a] :
% 0.27/0.47 ( ( X = Y )
% 0.27/0.47 => ( ord_less_eq_o_a @ X @ Y ) ) ).
% 0.27/0.47
% 0.27/0.47 % eq_refl
% 0.27/0.47 thf(fact_82_eq__refl,axiom,
% 0.27/0.47 ! [X: a,Y: a] :
% 0.27/0.47 ( ( X = Y )
% 0.27/0.47 => ( ord_less_eq_a @ X @ Y ) ) ).
% 0.27/0.47
% 0.27/0.47 % eq_refl
% 0.27/0.47 thf(fact_83_linear,axiom,
% 0.27/0.47 ! [X: a,Y: a] :
% 0.27/0.47 ( ( ord_less_eq_a @ X @ Y )
% 0.27/0.47 | ( ord_less_eq_a @ Y @ X ) ) ).
% 0.27/0.47
% 0.27/0.47 % linear
% 0.27/0.47 thf(fact_84_antisym,axiom,
% 0.27/0.47 ! [X: $o > a,Y: $o > a] :
% 0.27/0.47 ( ( ord_less_eq_o_a @ X @ Y )
% 0.27/0.47 => ( ( ord_less_eq_o_a @ Y @ X )
% 0.27/0.47 => ( X = Y ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % antisym
% 0.27/0.47 thf(fact_85_antisym,axiom,
% 0.27/0.47 ! [X: a,Y: a] :
% 0.27/0.47 ( ( ord_less_eq_a @ X @ Y )
% 0.27/0.47 => ( ( ord_less_eq_a @ Y @ X )
% 0.27/0.47 => ( X = Y ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % antisym
% 0.27/0.47 thf(fact_86_eq__iff,axiom,
% 0.27/0.47 ( ( ^ [Y3: $o > a,Z: $o > a] : ( Y3 = Z ) )
% 0.27/0.47 = ( ^ [X4: $o > a,Y4: $o > a] :
% 0.27/0.47 ( ( ord_less_eq_o_a @ X4 @ Y4 )
% 0.27/0.47 & ( ord_less_eq_o_a @ Y4 @ X4 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % eq_iff
% 0.27/0.47 thf(fact_87_eq__iff,axiom,
% 0.27/0.47 ( ( ^ [Y3: a,Z: a] : ( Y3 = Z ) )
% 0.27/0.47 = ( ^ [X4: a,Y4: a] :
% 0.27/0.47 ( ( ord_less_eq_a @ X4 @ Y4 )
% 0.27/0.47 & ( ord_less_eq_a @ Y4 @ X4 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % eq_iff
% 0.27/0.47 thf(fact_88_ord__le__eq__subst,axiom,
% 0.27/0.47 ! [A: a,B: a,F: a > $o > a,C: $o > a] :
% 0.27/0.47 ( ( ord_less_eq_a @ A @ B )
% 0.27/0.47 => ( ( ( F @ B )
% 0.27/0.47 = C )
% 0.27/0.47 => ( ! [X5: a,Y2: a] :
% 0.27/0.47 ( ( ord_less_eq_a @ X5 @ Y2 )
% 0.27/0.47 => ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.27/0.47 => ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord_le_eq_subst
% 0.27/0.47 thf(fact_89_ord__le__eq__subst,axiom,
% 0.27/0.47 ! [A: $o > a,B: $o > a,F: ( $o > a ) > a,C: a] :
% 0.27/0.47 ( ( ord_less_eq_o_a @ A @ B )
% 0.27/0.47 => ( ( ( F @ B )
% 0.27/0.47 = C )
% 0.27/0.47 => ( ! [X5: $o > a,Y2: $o > a] :
% 0.27/0.47 ( ( ord_less_eq_o_a @ X5 @ Y2 )
% 0.27/0.47 => ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.27/0.47 => ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord_le_eq_subst
% 0.27/0.47 thf(fact_90_ord__le__eq__subst,axiom,
% 0.27/0.47 ! [A: $o > a,B: $o > a,F: ( $o > a ) > $o > a,C: $o > a] :
% 0.27/0.47 ( ( ord_less_eq_o_a @ A @ B )
% 0.27/0.47 => ( ( ( F @ B )
% 0.27/0.47 = C )
% 0.27/0.47 => ( ! [X5: $o > a,Y2: $o > a] :
% 0.27/0.47 ( ( ord_less_eq_o_a @ X5 @ Y2 )
% 0.27/0.47 => ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.27/0.47 => ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord_le_eq_subst
% 0.27/0.47 thf(fact_91_ord__le__eq__subst,axiom,
% 0.27/0.47 ! [A: a,B: a,F: a > a,C: a] :
% 0.27/0.47 ( ( ord_less_eq_a @ A @ B )
% 0.27/0.47 => ( ( ( F @ B )
% 0.27/0.47 = C )
% 0.27/0.47 => ( ! [X5: a,Y2: a] :
% 0.27/0.47 ( ( ord_less_eq_a @ X5 @ Y2 )
% 0.27/0.47 => ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.27/0.47 => ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord_le_eq_subst
% 0.27/0.47 thf(fact_92_ord__eq__le__subst,axiom,
% 0.27/0.47 ! [A: $o > a,F: a > $o > a,B: a,C: a] :
% 0.27/0.47 ( ( A
% 0.27/0.47 = ( F @ B ) )
% 0.27/0.47 => ( ( ord_less_eq_a @ B @ C )
% 0.27/0.47 => ( ! [X5: a,Y2: a] :
% 0.27/0.47 ( ( ord_less_eq_a @ X5 @ Y2 )
% 0.27/0.47 => ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.27/0.47 => ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord_eq_le_subst
% 0.27/0.47 thf(fact_93_ord__eq__le__subst,axiom,
% 0.27/0.47 ! [A: a,F: ( $o > a ) > a,B: $o > a,C: $o > a] :
% 0.27/0.47 ( ( A
% 0.27/0.47 = ( F @ B ) )
% 0.27/0.47 => ( ( ord_less_eq_o_a @ B @ C )
% 0.27/0.47 => ( ! [X5: $o > a,Y2: $o > a] :
% 0.27/0.47 ( ( ord_less_eq_o_a @ X5 @ Y2 )
% 0.27/0.47 => ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.27/0.47 => ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord_eq_le_subst
% 0.27/0.47 thf(fact_94_ord__eq__le__subst,axiom,
% 0.27/0.47 ! [A: $o > a,F: ( $o > a ) > $o > a,B: $o > a,C: $o > a] :
% 0.27/0.47 ( ( A
% 0.27/0.47 = ( F @ B ) )
% 0.27/0.47 => ( ( ord_less_eq_o_a @ B @ C )
% 0.27/0.47 => ( ! [X5: $o > a,Y2: $o > a] :
% 0.27/0.47 ( ( ord_less_eq_o_a @ X5 @ Y2 )
% 0.27/0.47 => ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.27/0.47 => ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord_eq_le_subst
% 0.27/0.47 thf(fact_95_ord__eq__le__subst,axiom,
% 0.27/0.47 ! [A: a,F: a > a,B: a,C: a] :
% 0.27/0.47 ( ( A
% 0.27/0.47 = ( F @ B ) )
% 0.27/0.47 => ( ( ord_less_eq_a @ B @ C )
% 0.27/0.47 => ( ! [X5: a,Y2: a] :
% 0.27/0.47 ( ( ord_less_eq_a @ X5 @ Y2 )
% 0.27/0.47 => ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 0.27/0.47 => ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % ord_eq_le_subst
% 0.27/0.47 thf(fact_96_heap__top__max,axiom,
% 0.27/0.47 ! [T: tree_a] :
% 0.27/0.47 ( ( T != e_a )
% 0.27/0.47 => ( ( is_heap_a @ T )
% 0.27/0.47 => ( ( val_a @ T )
% 0.27/0.47 = ( lattic146396397_Max_a @ ( set_mset_a @ ( multiset_a2 @ T ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % heap_top_max
% 0.27/0.47 thf(fact_97_of__list__tree_Osimps_I2_J,axiom,
% 0.27/0.47 ! [V: a,Tail: list_a] :
% 0.27/0.47 ( ( heapIm1637418125tree_a @ ( cons_a @ V @ Tail ) )
% 0.27/0.47 = ( t_a @ V @ ( heapIm1637418125tree_a @ Tail ) @ e_a ) ) ).
% 0.27/0.47
% 0.27/0.47 % of_list_tree.simps(2)
% 0.27/0.47 thf(fact_98_Greatest__equality,axiom,
% 0.27/0.47 ! [P: ( $o > a ) > $o,X: $o > a] :
% 0.27/0.47 ( ( P @ X )
% 0.27/0.47 => ( ! [Y2: $o > a] :
% 0.27/0.47 ( ( P @ Y2 )
% 0.27/0.47 => ( ord_less_eq_o_a @ Y2 @ X ) )
% 0.27/0.47 => ( ( order_Greatest_o_a @ P )
% 0.27/0.47 = X ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Greatest_equality
% 0.27/0.47 thf(fact_99_Greatest__equality,axiom,
% 0.27/0.47 ! [P: a > $o,X: a] :
% 0.27/0.47 ( ( P @ X )
% 0.27/0.47 => ( ! [Y2: a] :
% 0.27/0.47 ( ( P @ Y2 )
% 0.27/0.47 => ( ord_less_eq_a @ Y2 @ X ) )
% 0.27/0.47 => ( ( order_Greatest_a @ P )
% 0.27/0.47 = X ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Greatest_equality
% 0.27/0.47 thf(fact_100_GreatestI2__order,axiom,
% 0.27/0.47 ! [P: ( $o > a ) > $o,X: $o > a,Q: ( $o > a ) > $o] :
% 0.27/0.47 ( ( P @ X )
% 0.27/0.47 => ( ! [Y2: $o > a] :
% 0.27/0.47 ( ( P @ Y2 )
% 0.27/0.47 => ( ord_less_eq_o_a @ Y2 @ X ) )
% 0.27/0.47 => ( ! [X5: $o > a] :
% 0.27/0.47 ( ( P @ X5 )
% 0.27/0.47 => ( ! [Y5: $o > a] :
% 0.27/0.47 ( ( P @ Y5 )
% 0.27/0.47 => ( ord_less_eq_o_a @ Y5 @ X5 ) )
% 0.27/0.47 => ( Q @ X5 ) ) )
% 0.27/0.47 => ( Q @ ( order_Greatest_o_a @ P ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % GreatestI2_order
% 0.27/0.47 thf(fact_101_GreatestI2__order,axiom,
% 0.27/0.47 ! [P: a > $o,X: a,Q: a > $o] :
% 0.27/0.47 ( ( P @ X )
% 0.27/0.47 => ( ! [Y2: a] :
% 0.27/0.47 ( ( P @ Y2 )
% 0.27/0.47 => ( ord_less_eq_a @ Y2 @ X ) )
% 0.27/0.47 => ( ! [X5: a] :
% 0.27/0.47 ( ( P @ X5 )
% 0.27/0.47 => ( ! [Y5: a] :
% 0.27/0.47 ( ( P @ Y5 )
% 0.27/0.47 => ( ord_less_eq_a @ Y5 @ X5 ) )
% 0.27/0.47 => ( Q @ X5 ) ) )
% 0.27/0.47 => ( Q @ ( order_Greatest_a @ P ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % GreatestI2_order
% 0.27/0.47 thf(fact_102_le__rel__bool__arg__iff,axiom,
% 0.27/0.47 ( ord_less_eq_o_o_a
% 0.27/0.47 = ( ^ [X6: $o > $o > a,Y6: $o > $o > a] :
% 0.27/0.47 ( ( ord_less_eq_o_a @ ( X6 @ $false ) @ ( Y6 @ $false ) )
% 0.27/0.47 & ( ord_less_eq_o_a @ ( X6 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % le_rel_bool_arg_iff
% 0.27/0.47 thf(fact_103_le__rel__bool__arg__iff,axiom,
% 0.27/0.47 ( ord_less_eq_o_a
% 0.27/0.47 = ( ^ [X6: $o > a,Y6: $o > a] :
% 0.27/0.47 ( ( ord_less_eq_a @ ( X6 @ $false ) @ ( Y6 @ $false ) )
% 0.27/0.47 & ( ord_less_eq_a @ ( X6 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % le_rel_bool_arg_iff
% 0.27/0.47 thf(fact_104_verit__la__disequality,axiom,
% 0.27/0.47 ! [A: a,B: a] :
% 0.27/0.47 ( ( A = B )
% 0.27/0.47 | ~ ( ord_less_eq_a @ A @ B )
% 0.27/0.47 | ~ ( ord_less_eq_a @ B @ A ) ) ).
% 0.27/0.47
% 0.27/0.47 % verit_la_disequality
% 0.27/0.47 thf(fact_105_list_Oinject,axiom,
% 0.27/0.47 ! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
% 0.27/0.47 ( ( ( cons_a @ X21 @ X22 )
% 0.27/0.47 = ( cons_a @ Y21 @ Y22 ) )
% 0.27/0.47 = ( ( X21 = Y21 )
% 0.27/0.47 & ( X22 = Y22 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % list.inject
% 0.27/0.47 thf(fact_106_not__Cons__self2,axiom,
% 0.27/0.47 ! [X: a,Xs: list_a] :
% 0.27/0.47 ( ( cons_a @ X @ Xs )
% 0.27/0.47 != Xs ) ).
% 0.27/0.47
% 0.27/0.47 % not_Cons_self2
% 0.27/0.47 thf(fact_107_Heap__axioms__def,axiom,
% 0.27/0.47 ( heap_axioms_Tree_a_a
% 0.27/0.47 = ( ^ [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] :
% 0.27/0.47 ( ! [L3: tree_a] :
% 0.27/0.47 ( ( Multiset @ L3 )
% 0.27/0.47 = ( multiset_a2 @ ( As_tree @ L3 ) ) )
% 0.27/0.47 & ! [I: list_a] : ( is_heap_a @ ( As_tree @ ( Of_list @ I ) ) )
% 0.27/0.47 & ! [T2: tree_a] :
% 0.27/0.47 ( ( ( As_tree @ T2 )
% 0.27/0.47 = e_a )
% 0.27/0.47 = ( Is_empty @ T2 ) )
% 0.27/0.47 & ! [L3: tree_a,M: a,L4: tree_a] :
% 0.27/0.47 ( ~ ( Is_empty @ L3 )
% 0.27/0.47 => ( ( ( produc686083979Tree_a @ M @ L4 )
% 0.27/0.47 = ( Remove_max @ L3 ) )
% 0.27/0.47 => ( ( add_mset_a @ M @ ( Multiset @ L4 ) )
% 0.27/0.47 = ( Multiset @ L3 ) ) ) )
% 0.27/0.47 & ! [L3: tree_a,M: a,L4: tree_a] :
% 0.27/0.47 ( ~ ( Is_empty @ L3 )
% 0.27/0.47 => ( ( is_heap_a @ ( As_tree @ L3 ) )
% 0.27/0.47 => ( ( ( produc686083979Tree_a @ M @ L4 )
% 0.27/0.47 = ( Remove_max @ L3 ) )
% 0.27/0.47 => ( is_heap_a @ ( As_tree @ L4 ) ) ) ) )
% 0.27/0.47 & ! [T2: tree_a,M: a,T3: tree_a] :
% 0.27/0.47 ( ~ ( Is_empty @ T2 )
% 0.27/0.47 => ( ( ( produc686083979Tree_a @ M @ T3 )
% 0.27/0.47 = ( Remove_max @ T2 ) )
% 0.27/0.47 => ( M
% 0.27/0.47 = ( val_a @ ( As_tree @ T2 ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Heap_axioms_def
% 0.27/0.47 thf(fact_108_Heap__axioms_Ointro,axiom,
% 0.27/0.47 ! [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] :
% 0.27/0.47 ( ! [L5: tree_a] :
% 0.27/0.47 ( ( Multiset2 @ L5 )
% 0.27/0.47 = ( multiset_a2 @ ( As_tree2 @ L5 ) ) )
% 0.27/0.47 => ( ! [I2: list_a] : ( is_heap_a @ ( As_tree2 @ ( Of_list2 @ I2 ) ) )
% 0.27/0.47 => ( ! [T4: tree_a] :
% 0.27/0.47 ( ( ( As_tree2 @ T4 )
% 0.27/0.47 = e_a )
% 0.27/0.47 = ( Is_empty2 @ T4 ) )
% 0.27/0.47 => ( ! [L5: tree_a,M2: a,L2: tree_a] :
% 0.27/0.47 ( ~ ( Is_empty2 @ L5 )
% 0.27/0.47 => ( ( ( produc686083979Tree_a @ M2 @ L2 )
% 0.27/0.47 = ( Remove_max2 @ L5 ) )
% 0.27/0.47 => ( ( add_mset_a @ M2 @ ( Multiset2 @ L2 ) )
% 0.27/0.47 = ( Multiset2 @ L5 ) ) ) )
% 0.27/0.47 => ( ! [L5: tree_a,M2: a,L2: tree_a] :
% 0.27/0.47 ( ~ ( Is_empty2 @ L5 )
% 0.27/0.47 => ( ( is_heap_a @ ( As_tree2 @ L5 ) )
% 0.27/0.47 => ( ( ( produc686083979Tree_a @ M2 @ L2 )
% 0.27/0.47 = ( Remove_max2 @ L5 ) )
% 0.27/0.47 => ( is_heap_a @ ( As_tree2 @ L2 ) ) ) ) )
% 0.27/0.47 => ( ! [T4: tree_a,M2: a,T5: tree_a] :
% 0.27/0.47 ( ~ ( Is_empty2 @ T4 )
% 0.27/0.47 => ( ( ( produc686083979Tree_a @ M2 @ T5 )
% 0.27/0.47 = ( Remove_max2 @ T4 ) )
% 0.27/0.47 => ( M2
% 0.27/0.47 = ( val_a @ ( As_tree2 @ T4 ) ) ) ) )
% 0.27/0.47 => ( heap_axioms_Tree_a_a @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Heap_axioms.intro
% 0.27/0.47 thf(fact_109_Heap_Oremove__max__multiset_H,axiom,
% 0.27/0.47 ! [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,M3: a,L6: tree_a] :
% 0.27/0.47 ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 0.27/0.47 => ( ~ ( Is_empty2 @ L )
% 0.27/0.47 => ( ( ( produc686083979Tree_a @ M3 @ L6 )
% 0.27/0.47 = ( Remove_max2 @ L ) )
% 0.27/0.47 => ( ( add_mset_a @ M3 @ ( Multiset2 @ L6 ) )
% 0.27/0.47 = ( Multiset2 @ L ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Heap.remove_max_multiset'
% 0.27/0.47 thf(fact_110_Heap_Oremove__max__val,axiom,
% 0.27/0.47 ! [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,T: tree_a,M3: a,T6: tree_a] :
% 0.27/0.47 ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 0.27/0.47 => ( ~ ( Is_empty2 @ T )
% 0.27/0.47 => ( ( ( produc686083979Tree_a @ M3 @ T6 )
% 0.27/0.47 = ( Remove_max2 @ T ) )
% 0.27/0.47 => ( M3
% 0.27/0.47 = ( val_a @ ( As_tree2 @ T ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Heap.remove_max_val
% 0.27/0.47 thf(fact_111_Heap_Oremove__max__is__heap,axiom,
% 0.27/0.47 ! [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,M3: a,L6: tree_a] :
% 0.27/0.47 ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 0.27/0.47 => ( ~ ( Is_empty2 @ L )
% 0.27/0.47 => ( ( is_heap_a @ ( As_tree2 @ L ) )
% 0.27/0.47 => ( ( ( produc686083979Tree_a @ M3 @ L6 )
% 0.27/0.47 = ( Remove_max2 @ L ) )
% 0.27/0.47 => ( is_heap_a @ ( As_tree2 @ L6 ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Heap.remove_max_is_heap
% 0.27/0.47 thf(fact_112_multi__self__add__other__not__self,axiom,
% 0.27/0.47 ! [M4: multiset_a,X: a] :
% 0.27/0.47 ( M4
% 0.27/0.47 != ( add_mset_a @ X @ M4 ) ) ).
% 0.27/0.47
% 0.27/0.47 % multi_self_add_other_not_self
% 0.27/0.47 thf(fact_113_add__mset__add__mset__same__iff,axiom,
% 0.27/0.47 ! [A: a,A2: multiset_a,B4: multiset_a] :
% 0.27/0.47 ( ( ( add_mset_a @ A @ A2 )
% 0.27/0.47 = ( add_mset_a @ A @ B4 ) )
% 0.27/0.47 = ( A2 = B4 ) ) ).
% 0.27/0.47
% 0.27/0.47 % add_mset_add_mset_same_iff
% 0.27/0.47 thf(fact_114_old_Oprod_Oinject,axiom,
% 0.27/0.47 ! [A: a,B: tree_a,A5: a,B5: tree_a] :
% 0.27/0.47 ( ( ( produc686083979Tree_a @ A @ B )
% 0.27/0.47 = ( produc686083979Tree_a @ A5 @ B5 ) )
% 0.27/0.47 = ( ( A = A5 )
% 0.27/0.47 & ( B = B5 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % old.prod.inject
% 0.27/0.47 thf(fact_115_prod_Oinject,axiom,
% 0.27/0.47 ! [X12: a,X24: tree_a,Y1: a,Y24: tree_a] :
% 0.27/0.47 ( ( ( produc686083979Tree_a @ X12 @ X24 )
% 0.27/0.47 = ( produc686083979Tree_a @ Y1 @ Y24 ) )
% 0.27/0.47 = ( ( X12 = Y1 )
% 0.27/0.47 & ( X24 = Y24 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod.inject
% 0.27/0.47 thf(fact_116_mset__add,axiom,
% 0.27/0.47 ! [A: a,A2: multiset_a] :
% 0.27/0.47 ( ( member_a @ A @ ( set_mset_a @ A2 ) )
% 0.27/0.47 => ~ ! [B6: multiset_a] :
% 0.27/0.47 ( A2
% 0.27/0.47 != ( add_mset_a @ A @ B6 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % mset_add
% 0.27/0.47 thf(fact_117_multi__member__split,axiom,
% 0.27/0.47 ! [X: a,M4: multiset_a] :
% 0.27/0.47 ( ( member_a @ X @ ( set_mset_a @ M4 ) )
% 0.27/0.47 => ? [A6: multiset_a] :
% 0.27/0.47 ( M4
% 0.27/0.47 = ( add_mset_a @ X @ A6 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % multi_member_split
% 0.27/0.47 thf(fact_118_surj__pair,axiom,
% 0.27/0.47 ! [P2: produc143150363Tree_a] :
% 0.27/0.47 ? [X5: a,Y2: tree_a] :
% 0.27/0.47 ( P2
% 0.27/0.47 = ( produc686083979Tree_a @ X5 @ Y2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % surj_pair
% 0.27/0.47 thf(fact_119_prod__cases,axiom,
% 0.27/0.47 ! [P: produc143150363Tree_a > $o,P2: produc143150363Tree_a] :
% 0.27/0.47 ( ! [A4: a,B3: tree_a] : ( P @ ( produc686083979Tree_a @ A4 @ B3 ) )
% 0.27/0.47 => ( P @ P2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % prod_cases
% 0.27/0.47 thf(fact_120_Pair__inject,axiom,
% 0.27/0.47 ! [A: a,B: tree_a,A5: a,B5: tree_a] :
% 0.27/0.47 ( ( ( produc686083979Tree_a @ A @ B )
% 0.27/0.47 = ( produc686083979Tree_a @ A5 @ B5 ) )
% 0.27/0.47 => ~ ( ( A = A5 )
% 0.27/0.47 => ( B != B5 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Pair_inject
% 0.27/0.47 thf(fact_121_old_Oprod_Oexhaust,axiom,
% 0.27/0.47 ! [Y: produc143150363Tree_a] :
% 0.27/0.47 ~ ! [A4: a,B3: tree_a] :
% 0.27/0.47 ( Y
% 0.27/0.47 != ( produc686083979Tree_a @ A4 @ B3 ) ) ).
% 0.27/0.47
% 0.27/0.47 % old.prod.exhaust
% 0.27/0.47 thf(fact_122_old_Oprod_Oinducts,axiom,
% 0.27/0.47 ! [P: produc143150363Tree_a > $o,Prod: produc143150363Tree_a] :
% 0.27/0.47 ( ! [A4: a,B3: tree_a] : ( P @ ( produc686083979Tree_a @ A4 @ B3 ) )
% 0.27/0.47 => ( P @ Prod ) ) ).
% 0.27/0.47
% 0.27/0.47 % old.prod.inducts
% 0.27/0.47 thf(fact_123_add__eq__conv__ex,axiom,
% 0.27/0.47 ! [A: a,M4: multiset_a,B: a,N: multiset_a] :
% 0.27/0.47 ( ( ( add_mset_a @ A @ M4 )
% 0.27/0.47 = ( add_mset_a @ B @ N ) )
% 0.27/0.47 = ( ( ( M4 = N )
% 0.27/0.47 & ( A = B ) )
% 0.27/0.47 | ? [K: multiset_a] :
% 0.27/0.47 ( ( M4
% 0.27/0.47 = ( add_mset_a @ B @ K ) )
% 0.27/0.47 & ( N
% 0.27/0.47 = ( add_mset_a @ A @ K ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % add_eq_conv_ex
% 0.27/0.47 thf(fact_124_add__mset__commute,axiom,
% 0.27/0.47 ! [X: a,Y: a,M4: multiset_a] :
% 0.27/0.47 ( ( add_mset_a @ X @ ( add_mset_a @ Y @ M4 ) )
% 0.27/0.47 = ( add_mset_a @ Y @ ( add_mset_a @ X @ M4 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % add_mset_commute
% 0.27/0.47 thf(fact_125_union__single__eq__member,axiom,
% 0.27/0.47 ! [X: a,M4: multiset_a,N: multiset_a] :
% 0.27/0.47 ( ( ( add_mset_a @ X @ M4 )
% 0.27/0.47 = N )
% 0.27/0.47 => ( member_a @ X @ ( set_mset_a @ N ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % union_single_eq_member
% 0.27/0.47 thf(fact_126_insert__noteq__member,axiom,
% 0.27/0.47 ! [B: a,B4: multiset_a,C: a,C2: multiset_a] :
% 0.27/0.47 ( ( ( add_mset_a @ B @ B4 )
% 0.27/0.47 = ( add_mset_a @ C @ C2 ) )
% 0.27/0.47 => ( ( B != C )
% 0.27/0.47 => ( member_a @ C @ ( set_mset_a @ B4 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % insert_noteq_member
% 0.27/0.47 thf(fact_127_removeLeaf_Osimps_I1_J,axiom,
% 0.27/0.47 ! [V: a] :
% 0.27/0.47 ( ( heapIm837449470Leaf_a @ ( t_a @ V @ e_a @ e_a ) )
% 0.27/0.47 = ( produc686083979Tree_a @ V @ e_a ) ) ).
% 0.27/0.47
% 0.27/0.47 % removeLeaf.simps(1)
% 0.27/0.47 thf(fact_128_RemoveMax__axioms__def,axiom,
% 0.27/0.47 ( remove301631099ee_a_a
% 0.27/0.47 = ( ^ [Is_empty: tree_a > $o,Of_list: list_a > tree_a,Multiset: tree_a > multiset_a,Remove_max: tree_a > produc143150363Tree_a,Inv: tree_a > $o] :
% 0.27/0.47 ( ! [X4: list_a] : ( Inv @ ( Of_list @ X4 ) )
% 0.27/0.47 & ! [L3: tree_a,M: a,L4: tree_a] :
% 0.27/0.47 ( ~ ( Is_empty @ L3 )
% 0.27/0.47 => ( ( Inv @ L3 )
% 0.27/0.47 => ( ( ( produc686083979Tree_a @ M @ L4 )
% 0.27/0.47 = ( Remove_max @ L3 ) )
% 0.27/0.47 => ( M
% 0.27/0.47 = ( lattic146396397_Max_a @ ( set_Tree_a_a @ Multiset @ L3 ) ) ) ) ) )
% 0.27/0.47 & ! [L3: tree_a,M: a,L4: tree_a] :
% 0.27/0.47 ( ~ ( Is_empty @ L3 )
% 0.27/0.47 => ( ( Inv @ L3 )
% 0.27/0.47 => ( ( ( produc686083979Tree_a @ M @ L4 )
% 0.27/0.47 = ( Remove_max @ L3 ) )
% 0.27/0.47 => ( ( add_mset_a @ M @ ( Multiset @ L4 ) )
% 0.27/0.47 = ( Multiset @ L3 ) ) ) ) )
% 0.27/0.47 & ! [L3: tree_a,M: a,L4: tree_a] :
% 0.27/0.47 ( ~ ( Is_empty @ L3 )
% 0.27/0.47 => ( ( Inv @ L3 )
% 0.27/0.47 => ( ( ( produc686083979Tree_a @ M @ L4 )
% 0.27/0.47 = ( Remove_max @ L3 ) )
% 0.27/0.47 => ( Inv @ L4 ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % RemoveMax_axioms_def
% 0.27/0.47 thf(fact_129_RemoveMax__axioms_Ointro,axiom,
% 0.27/0.47 ! [Inv2: tree_a > $o,Of_list2: list_a > tree_a,Is_empty2: tree_a > $o,Remove_max2: tree_a > produc143150363Tree_a,Multiset2: tree_a > multiset_a] :
% 0.27/0.47 ( ! [X5: list_a] : ( Inv2 @ ( Of_list2 @ X5 ) )
% 0.27/0.47 => ( ! [L5: tree_a,M2: a,L2: tree_a] :
% 0.27/0.47 ( ~ ( Is_empty2 @ L5 )
% 0.27/0.47 => ( ( Inv2 @ L5 )
% 0.27/0.47 => ( ( ( produc686083979Tree_a @ M2 @ L2 )
% 0.27/0.47 = ( Remove_max2 @ L5 ) )
% 0.27/0.47 => ( M2
% 0.27/0.47 = ( lattic146396397_Max_a @ ( set_Tree_a_a @ Multiset2 @ L5 ) ) ) ) ) )
% 0.27/0.47 => ( ! [L5: tree_a,M2: a,L2: tree_a] :
% 0.27/0.47 ( ~ ( Is_empty2 @ L5 )
% 0.27/0.47 => ( ( Inv2 @ L5 )
% 0.27/0.47 => ( ( ( produc686083979Tree_a @ M2 @ L2 )
% 0.27/0.47 = ( Remove_max2 @ L5 ) )
% 0.27/0.47 => ( ( add_mset_a @ M2 @ ( Multiset2 @ L2 ) )
% 0.27/0.47 = ( Multiset2 @ L5 ) ) ) ) )
% 0.27/0.47 => ( ! [L5: tree_a,M2: a,L2: tree_a] :
% 0.27/0.47 ( ~ ( Is_empty2 @ L5 )
% 0.27/0.47 => ( ( Inv2 @ L5 )
% 0.27/0.47 => ( ( ( produc686083979Tree_a @ M2 @ L2 )
% 0.27/0.47 = ( Remove_max2 @ L5 ) )
% 0.27/0.47 => ( Inv2 @ L2 ) ) ) )
% 0.27/0.47 => ( remove301631099ee_a_a @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv2 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % RemoveMax_axioms.intro
% 0.27/0.47 thf(fact_130_RemoveMax_Oremove__max__max,axiom,
% 0.27/0.47 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,Remove_max2: tree_a > produc143150363Tree_a,Inv2: tree_a > $o,L: tree_a,M3: a,L6: tree_a] :
% 0.27/0.47 ( ( removeMax_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv2 )
% 0.27/0.47 => ( ~ ( Is_empty2 @ L )
% 0.27/0.47 => ( ( Inv2 @ L )
% 0.27/0.47 => ( ( ( produc686083979Tree_a @ M3 @ L6 )
% 0.27/0.47 = ( Remove_max2 @ L ) )
% 0.27/0.47 => ( M3
% 0.27/0.47 = ( lattic146396397_Max_a @ ( set_Tree_a_a @ Multiset2 @ L ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % RemoveMax.remove_max_max
% 0.27/0.47 thf(fact_131_multiset__induct__max,axiom,
% 0.27/0.47 ! [P: multiset_a > $o,M4: multiset_a] :
% 0.27/0.47 ( ( P @ zero_zero_multiset_a )
% 0.27/0.47 => ( ! [X5: a,M5: multiset_a] :
% 0.27/0.47 ( ( P @ M5 )
% 0.27/0.47 => ( ! [Xa: a] :
% 0.27/0.47 ( ( member_a @ Xa @ ( set_mset_a @ M5 ) )
% 0.27/0.47 => ( ord_less_eq_a @ Xa @ X5 ) )
% 0.27/0.47 => ( P @ ( add_mset_a @ X5 @ M5 ) ) ) )
% 0.27/0.47 => ( P @ M4 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % multiset_induct_max
% 0.27/0.47 thf(fact_132_add__mset__eq__singleton__iff,axiom,
% 0.27/0.47 ! [X: a,M4: multiset_a,Y: a] :
% 0.27/0.47 ( ( ( add_mset_a @ X @ M4 )
% 0.27/0.47 = ( add_mset_a @ Y @ zero_zero_multiset_a ) )
% 0.27/0.47 = ( ( M4 = zero_zero_multiset_a )
% 0.27/0.47 & ( X = Y ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % add_mset_eq_singleton_iff
% 0.27/0.47 thf(fact_133_single__eq__add__mset,axiom,
% 0.27/0.47 ! [A: a,B: a,M4: multiset_a] :
% 0.27/0.47 ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
% 0.27/0.47 = ( add_mset_a @ B @ M4 ) )
% 0.27/0.47 = ( ( B = A )
% 0.27/0.47 & ( M4 = zero_zero_multiset_a ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % single_eq_add_mset
% 0.27/0.47 thf(fact_134_add__mset__eq__single,axiom,
% 0.27/0.47 ! [B: a,M4: multiset_a,A: a] :
% 0.27/0.47 ( ( ( add_mset_a @ B @ M4 )
% 0.27/0.47 = ( add_mset_a @ A @ zero_zero_multiset_a ) )
% 0.27/0.47 = ( ( B = A )
% 0.27/0.47 & ( M4 = zero_zero_multiset_a ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % add_mset_eq_single
% 0.27/0.47 thf(fact_135_single__eq__single,axiom,
% 0.27/0.47 ! [A: a,B: a] :
% 0.27/0.47 ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
% 0.27/0.47 = ( add_mset_a @ B @ zero_zero_multiset_a ) )
% 0.27/0.47 = ( A = B ) ) ).
% 0.27/0.47
% 0.27/0.47 % single_eq_single
% 0.27/0.47 thf(fact_136_multiset__cases,axiom,
% 0.27/0.47 ! [M4: multiset_a] :
% 0.27/0.47 ( ( M4 != zero_zero_multiset_a )
% 0.27/0.47 => ~ ! [X5: a,N2: multiset_a] :
% 0.27/0.47 ( M4
% 0.27/0.47 != ( add_mset_a @ X5 @ N2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % multiset_cases
% 0.27/0.47 thf(fact_137_multiset__induct,axiom,
% 0.27/0.47 ! [P: multiset_a > $o,M4: multiset_a] :
% 0.27/0.47 ( ( P @ zero_zero_multiset_a )
% 0.27/0.47 => ( ! [X5: a,M5: multiset_a] :
% 0.27/0.47 ( ( P @ M5 )
% 0.27/0.47 => ( P @ ( add_mset_a @ X5 @ M5 ) ) )
% 0.27/0.47 => ( P @ M4 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % multiset_induct
% 0.27/0.47 thf(fact_138_multiset__induct2,axiom,
% 0.27/0.47 ! [P: multiset_a > multiset_a > $o,M4: multiset_a,N: multiset_a] :
% 0.27/0.47 ( ( P @ zero_zero_multiset_a @ zero_zero_multiset_a )
% 0.27/0.47 => ( ! [A4: a,M5: multiset_a,N2: multiset_a] :
% 0.27/0.47 ( ( P @ M5 @ N2 )
% 0.27/0.47 => ( P @ ( add_mset_a @ A4 @ M5 ) @ N2 ) )
% 0.27/0.47 => ( ! [A4: a,M5: multiset_a,N2: multiset_a] :
% 0.27/0.47 ( ( P @ M5 @ N2 )
% 0.27/0.47 => ( P @ M5 @ ( add_mset_a @ A4 @ N2 ) ) )
% 0.27/0.47 => ( P @ M4 @ N ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % multiset_induct2
% 0.27/0.47 thf(fact_139_empty__not__add__mset,axiom,
% 0.27/0.47 ! [A: a,A2: multiset_a] :
% 0.27/0.47 ( zero_zero_multiset_a
% 0.27/0.47 != ( add_mset_a @ A @ A2 ) ) ).
% 0.27/0.47
% 0.27/0.47 % empty_not_add_mset
% 0.27/0.47 thf(fact_140_multi__nonempty__split,axiom,
% 0.27/0.47 ! [M4: multiset_a] :
% 0.27/0.47 ( ( M4 != zero_zero_multiset_a )
% 0.27/0.47 => ? [A6: multiset_a,A4: a] :
% 0.27/0.47 ( M4
% 0.27/0.47 = ( add_mset_a @ A4 @ A6 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % multi_nonempty_split
% 0.27/0.47 thf(fact_141_multiset__nonemptyE,axiom,
% 0.27/0.47 ! [A2: multiset_a] :
% 0.27/0.47 ( ( A2 != zero_zero_multiset_a )
% 0.27/0.47 => ~ ! [X5: a] :
% 0.27/0.47 ~ ( member_a @ X5 @ ( set_mset_a @ A2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % multiset_nonemptyE
% 0.27/0.47 thf(fact_142_RemoveMax_Oremove__max__inv,axiom,
% 0.27/0.47 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,Remove_max2: tree_a > produc143150363Tree_a,Inv2: tree_a > $o,L: tree_a,M3: a,L6: tree_a] :
% 0.27/0.47 ( ( removeMax_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv2 )
% 0.27/0.47 => ( ~ ( Is_empty2 @ L )
% 0.27/0.47 => ( ( Inv2 @ L )
% 0.27/0.47 => ( ( ( produc686083979Tree_a @ M3 @ L6 )
% 0.27/0.47 = ( Remove_max2 @ L ) )
% 0.27/0.47 => ( Inv2 @ L6 ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % RemoveMax.remove_max_inv
% 0.27/0.47 thf(fact_143_multi__member__last,axiom,
% 0.27/0.47 ! [X: a] : ( member_a @ X @ ( set_mset_a @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % multi_member_last
% 0.27/0.47 thf(fact_144_RemoveMax_Oremove__max__multiset,axiom,
% 0.27/0.47 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,Remove_max2: tree_a > produc143150363Tree_a,Inv2: tree_a > $o,L: tree_a,M3: a,L6: tree_a] :
% 0.27/0.47 ( ( removeMax_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv2 )
% 0.27/0.47 => ( ~ ( Is_empty2 @ L )
% 0.27/0.47 => ( ( Inv2 @ L )
% 0.27/0.47 => ( ( ( produc686083979Tree_a @ M3 @ L6 )
% 0.27/0.47 = ( Remove_max2 @ L ) )
% 0.27/0.47 => ( ( add_mset_a @ M3 @ ( Multiset2 @ L6 ) )
% 0.27/0.47 = ( Multiset2 @ L ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % RemoveMax.remove_max_multiset
% 0.27/0.47 thf(fact_145_multiset_Osimps_I1_J,axiom,
% 0.27/0.47 ( ( multiset_a2 @ e_a )
% 0.27/0.47 = zero_zero_multiset_a ) ).
% 0.27/0.47
% 0.27/0.47 % multiset.simps(1)
% 0.27/0.47 thf(fact_146_multiset__induct__min,axiom,
% 0.27/0.47 ! [P: multiset_a > $o,M4: multiset_a] :
% 0.27/0.47 ( ( P @ zero_zero_multiset_a )
% 0.27/0.47 => ( ! [X5: a,M5: multiset_a] :
% 0.27/0.47 ( ( P @ M5 )
% 0.27/0.47 => ( ! [Xa: a] :
% 0.27/0.47 ( ( member_a @ Xa @ ( set_mset_a @ M5 ) )
% 0.27/0.47 => ( ord_less_eq_a @ X5 @ Xa ) )
% 0.27/0.47 => ( P @ ( add_mset_a @ X5 @ M5 ) ) ) )
% 0.27/0.47 => ( P @ M4 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % multiset_induct_min
% 0.27/0.47 thf(fact_147_Multiset_Ois__empty__def,axiom,
% 0.27/0.47 ( is_empty_a
% 0.27/0.47 = ( ^ [A7: multiset_a] : ( A7 = zero_zero_multiset_a ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % Multiset.is_empty_def
% 0.27/0.47 thf(fact_148_RemoveMax_Ossort_HInduct,axiom,
% 0.27/0.47 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,Remove_max2: tree_a > produc143150363Tree_a,Inv2: tree_a > $o,L: tree_a,P: tree_a > list_a > $o,Sl: list_a] :
% 0.27/0.47 ( ( removeMax_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv2 )
% 0.27/0.47 => ( ( Inv2 @ L )
% 0.27/0.47 => ( ( P @ L @ Sl )
% 0.27/0.47 => ( ! [L5: tree_a,Sl2: list_a,M2: a,L2: tree_a] :
% 0.27/0.47 ( ~ ( Is_empty2 @ L5 )
% 0.27/0.47 => ( ( Inv2 @ L5 )
% 0.27/0.47 => ( ( ( produc686083979Tree_a @ M2 @ L2 )
% 0.27/0.47 = ( Remove_max2 @ L5 ) )
% 0.27/0.47 => ( ( P @ L5 @ Sl2 )
% 0.27/0.47 => ( P @ L2 @ ( cons_a @ M2 @ Sl2 ) ) ) ) ) )
% 0.27/0.47 => ( P @ Empty @ ( ssort_Tree_a_a @ Is_empty2 @ Remove_max2 @ L @ Sl ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % RemoveMax.ssort'Induct
% 0.27/0.47 thf(fact_149_RemoveMax_Ossort_H__dom_Ocases,axiom,
% 0.27/0.47 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,Remove_max2: tree_a > produc143150363Tree_a,Inv2: tree_a > $o,A: produc768687417list_a] :
% 0.27/0.47 ( ( removeMax_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv2 )
% 0.27/0.47 => ( ( ssort_dom_Tree_a_a @ Is_empty2 @ Remove_max2 @ A )
% 0.27/0.47 => ~ ! [L5: tree_a,Sl2: list_a] :
% 0.27/0.47 ( ( A
% 0.27/0.47 = ( produc1352981801list_a @ L5 @ Sl2 ) )
% 0.27/0.47 => ~ ( ~ ( Is_empty2 @ L5 )
% 0.27/0.47 => ! [M6: a,L7: tree_a] :
% 0.27/0.47 ( ( ( produc686083979Tree_a @ M6 @ L7 )
% 0.27/0.47 = ( Remove_max2 @ L5 ) )
% 0.27/0.47 => ( ssort_dom_Tree_a_a @ Is_empty2 @ Remove_max2 @ ( produc1352981801list_a @ L7 @ ( cons_a @ M6 @ Sl2 ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % RemoveMax.ssort'_dom.cases
% 0.27/0.47 thf(fact_150_RemoveMax_Ossort_H__dom_Osimps,axiom,
% 0.27/0.47 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,Remove_max2: tree_a > produc143150363Tree_a,Inv2: tree_a > $o,A: produc768687417list_a] :
% 0.27/0.47 ( ( removeMax_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv2 )
% 0.27/0.47 => ( ( ssort_dom_Tree_a_a @ Is_empty2 @ Remove_max2 @ A )
% 0.27/0.47 = ( ? [L3: tree_a,Sl3: list_a] :
% 0.27/0.47 ( ( A
% 0.27/0.47 = ( produc1352981801list_a @ L3 @ Sl3 ) )
% 0.27/0.47 & ! [X4: a,Y4: tree_a] :
% 0.27/0.47 ( ~ ( Is_empty2 @ L3 )
% 0.27/0.47 => ( ( ( produc686083979Tree_a @ X4 @ Y4 )
% 0.27/0.47 = ( Remove_max2 @ L3 ) )
% 0.27/0.47 => ( ssort_dom_Tree_a_a @ Is_empty2 @ Remove_max2 @ ( produc1352981801list_a @ Y4 @ ( cons_a @ X4 @ Sl3 ) ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % RemoveMax.ssort'_dom.simps
% 0.27/0.47 thf(fact_151_RemoveMax_Ossort_H__dom_Oinducts,axiom,
% 0.27/0.47 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,Remove_max2: tree_a > produc143150363Tree_a,Inv2: tree_a > $o,X: produc768687417list_a,P: produc768687417list_a > $o] :
% 0.27/0.47 ( ( removeMax_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv2 )
% 0.27/0.47 => ( ( ssort_dom_Tree_a_a @ Is_empty2 @ Remove_max2 @ X )
% 0.27/0.47 => ( ! [L5: tree_a,Sl2: list_a] :
% 0.27/0.47 ( ! [M6: a,L7: tree_a] :
% 0.27/0.47 ( ~ ( Is_empty2 @ L5 )
% 0.27/0.47 => ( ( ( produc686083979Tree_a @ M6 @ L7 )
% 0.27/0.47 = ( Remove_max2 @ L5 ) )
% 0.27/0.47 => ( ssort_dom_Tree_a_a @ Is_empty2 @ Remove_max2 @ ( produc1352981801list_a @ L7 @ ( cons_a @ M6 @ Sl2 ) ) ) ) )
% 0.27/0.47 => ( ! [M6: a,L7: tree_a] :
% 0.27/0.47 ( ~ ( Is_empty2 @ L5 )
% 0.27/0.47 => ( ( ( produc686083979Tree_a @ M6 @ L7 )
% 0.27/0.47 = ( Remove_max2 @ L5 ) )
% 0.27/0.47 => ( P @ ( produc1352981801list_a @ L7 @ ( cons_a @ M6 @ Sl2 ) ) ) ) )
% 0.27/0.47 => ( P @ ( produc1352981801list_a @ L5 @ Sl2 ) ) ) )
% 0.27/0.47 => ( P @ X ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % RemoveMax.ssort'_dom.inducts
% 0.27/0.47 thf(fact_152_RemoveMax_Ossort_H__dom_Ointros,axiom,
% 0.27/0.47 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,Remove_max2: tree_a > produc143150363Tree_a,Inv2: tree_a > $o,L: tree_a,Sl: list_a] :
% 0.27/0.47 ( ( removeMax_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv2 )
% 0.27/0.47 => ( ! [M2: a,L2: tree_a] :
% 0.27/0.47 ( ~ ( Is_empty2 @ L )
% 0.27/0.47 => ( ( ( produc686083979Tree_a @ M2 @ L2 )
% 0.27/0.47 = ( Remove_max2 @ L ) )
% 0.27/0.47 => ( ssort_dom_Tree_a_a @ Is_empty2 @ Remove_max2 @ ( produc1352981801list_a @ L2 @ ( cons_a @ M2 @ Sl ) ) ) ) )
% 0.27/0.47 => ( ssort_dom_Tree_a_a @ Is_empty2 @ Remove_max2 @ ( produc1352981801list_a @ L @ Sl ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % RemoveMax.ssort'_dom.intros
% 0.27/0.47 thf(fact_153_multiset_Osimps_I2_J,axiom,
% 0.27/0.47 ! [V: a,L: tree_a,R: tree_a] :
% 0.27/0.47 ( ( multiset_a2 @ ( t_a @ V @ L @ R ) )
% 0.27/0.47 = ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ ( multiset_a2 @ L ) @ ( add_mset_a @ V @ zero_zero_multiset_a ) ) @ ( multiset_a2 @ R ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % multiset.simps(2)
% 0.27/0.47 thf(fact_154_single__subset__iff,axiom,
% 0.27/0.47 ! [A: a,M4: multiset_a] :
% 0.27/0.47 ( ( subseteq_mset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) @ M4 )
% 0.27/0.47 = ( member_a @ A @ ( set_mset_a @ M4 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % single_subset_iff
% 0.27/0.47 thf(fact_155_subset__mset_Obot_Oextremum__unique,axiom,
% 0.27/0.47 ! [A: multiset_a] :
% 0.27/0.47 ( ( subseteq_mset_a @ A @ zero_zero_multiset_a )
% 0.27/0.47 = ( A = zero_zero_multiset_a ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_mset.bot.extremum_unique
% 0.27/0.47 thf(fact_156_subset__mset_Ole__zero__eq,axiom,
% 0.27/0.47 ! [N3: multiset_a] :
% 0.27/0.47 ( ( subseteq_mset_a @ N3 @ zero_zero_multiset_a )
% 0.27/0.47 = ( N3 = zero_zero_multiset_a ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_mset.le_zero_eq
% 0.27/0.47 thf(fact_157_subset__mset_Oadd__le__cancel__left,axiom,
% 0.27/0.47 ! [C: multiset_a,A: multiset_a,B: multiset_a] :
% 0.27/0.47 ( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C @ A ) @ ( plus_plus_multiset_a @ C @ B ) )
% 0.27/0.47 = ( subseteq_mset_a @ A @ B ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_mset.add_le_cancel_left
% 0.27/0.47 thf(fact_158_subset__mset_Oadd__le__cancel__right,axiom,
% 0.27/0.47 ! [A: multiset_a,C: multiset_a,B: multiset_a] :
% 0.27/0.47 ( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ C ) )
% 0.27/0.47 = ( subseteq_mset_a @ A @ B ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_mset.add_le_cancel_right
% 0.27/0.47 thf(fact_159_mset__subset__eq__mono__add__left__cancel,axiom,
% 0.27/0.47 ! [C2: multiset_a,A2: multiset_a,B4: multiset_a] :
% 0.27/0.47 ( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C2 @ A2 ) @ ( plus_plus_multiset_a @ C2 @ B4 ) )
% 0.27/0.47 = ( subseteq_mset_a @ A2 @ B4 ) ) ).
% 0.27/0.47
% 0.27/0.47 % mset_subset_eq_mono_add_left_cancel
% 0.27/0.47 thf(fact_160_mset__subset__eq__mono__add__right__cancel,axiom,
% 0.27/0.47 ! [A2: multiset_a,C2: multiset_a,B4: multiset_a] :
% 0.27/0.47 ( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C2 ) @ ( plus_plus_multiset_a @ B4 @ C2 ) )
% 0.27/0.47 = ( subseteq_mset_a @ A2 @ B4 ) ) ).
% 0.27/0.47
% 0.27/0.47 % mset_subset_eq_mono_add_right_cancel
% 0.27/0.47 thf(fact_161_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
% 0.27/0.47 ! [X: multiset_a,Y: multiset_a] :
% 0.27/0.47 ( ( zero_zero_multiset_a
% 0.27/0.47 = ( plus_plus_multiset_a @ X @ Y ) )
% 0.27/0.47 = ( ( X = zero_zero_multiset_a )
% 0.27/0.47 & ( Y = zero_zero_multiset_a ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_mset.zero_eq_add_iff_both_eq_0
% 0.27/0.47 thf(fact_162_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
% 0.27/0.47 ! [X: multiset_a,Y: multiset_a] :
% 0.27/0.47 ( ( ( plus_plus_multiset_a @ X @ Y )
% 0.27/0.47 = zero_zero_multiset_a )
% 0.27/0.47 = ( ( X = zero_zero_multiset_a )
% 0.27/0.47 & ( Y = zero_zero_multiset_a ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_mset.add_eq_0_iff_both_eq_0
% 0.27/0.47 thf(fact_163_union__eq__empty,axiom,
% 0.27/0.47 ! [M4: multiset_a,N: multiset_a] :
% 0.27/0.47 ( ( ( plus_plus_multiset_a @ M4 @ N )
% 0.27/0.47 = zero_zero_multiset_a )
% 0.27/0.47 = ( ( M4 = zero_zero_multiset_a )
% 0.27/0.47 & ( N = zero_zero_multiset_a ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % union_eq_empty
% 0.27/0.47 thf(fact_164_empty__eq__union,axiom,
% 0.27/0.47 ! [M4: multiset_a,N: multiset_a] :
% 0.27/0.47 ( ( zero_zero_multiset_a
% 0.27/0.47 = ( plus_plus_multiset_a @ M4 @ N ) )
% 0.27/0.47 = ( ( M4 = zero_zero_multiset_a )
% 0.27/0.47 & ( N = zero_zero_multiset_a ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % empty_eq_union
% 0.27/0.47 thf(fact_165_union__mset__add__mset__right,axiom,
% 0.27/0.47 ! [A2: multiset_a,A: a,B4: multiset_a] :
% 0.27/0.47 ( ( plus_plus_multiset_a @ A2 @ ( add_mset_a @ A @ B4 ) )
% 0.27/0.47 = ( add_mset_a @ A @ ( plus_plus_multiset_a @ A2 @ B4 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % union_mset_add_mset_right
% 0.27/0.47 thf(fact_166_union__mset__add__mset__left,axiom,
% 0.27/0.47 ! [A: a,A2: multiset_a,B4: multiset_a] :
% 0.27/0.47 ( ( plus_plus_multiset_a @ ( add_mset_a @ A @ A2 ) @ B4 )
% 0.27/0.47 = ( add_mset_a @ A @ ( plus_plus_multiset_a @ A2 @ B4 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % union_mset_add_mset_left
% 0.27/0.47 thf(fact_167_subset__mset_Oadd__le__same__cancel1,axiom,
% 0.27/0.47 ! [B: multiset_a,A: multiset_a] :
% 0.27/0.47 ( ( subseteq_mset_a @ ( plus_plus_multiset_a @ B @ A ) @ B )
% 0.27/0.47 = ( subseteq_mset_a @ A @ zero_zero_multiset_a ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_mset.add_le_same_cancel1
% 0.27/0.47 thf(fact_168_subset__mset_Oadd__le__same__cancel2,axiom,
% 0.27/0.47 ! [A: multiset_a,B: multiset_a] :
% 0.27/0.47 ( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ B ) @ B )
% 0.27/0.47 = ( subseteq_mset_a @ A @ zero_zero_multiset_a ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_mset.add_le_same_cancel2
% 0.27/0.47 thf(fact_169_subset__mset_Ole__add__same__cancel1,axiom,
% 0.27/0.47 ! [A: multiset_a,B: multiset_a] :
% 0.27/0.47 ( ( subseteq_mset_a @ A @ ( plus_plus_multiset_a @ A @ B ) )
% 0.27/0.47 = ( subseteq_mset_a @ zero_zero_multiset_a @ B ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_mset.le_add_same_cancel1
% 0.27/0.47 thf(fact_170_subset__mset_Ole__add__same__cancel2,axiom,
% 0.27/0.47 ! [A: multiset_a,B: multiset_a] :
% 0.27/0.47 ( ( subseteq_mset_a @ A @ ( plus_plus_multiset_a @ B @ A ) )
% 0.27/0.47 = ( subseteq_mset_a @ zero_zero_multiset_a @ B ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_mset.le_add_same_cancel2
% 0.27/0.47 thf(fact_171_add__mset__subseteq__single__iff,axiom,
% 0.27/0.47 ! [A: a,M4: multiset_a,B: a] :
% 0.27/0.47 ( ( subseteq_mset_a @ ( add_mset_a @ A @ M4 ) @ ( add_mset_a @ B @ zero_zero_multiset_a ) )
% 0.27/0.47 = ( ( M4 = zero_zero_multiset_a )
% 0.27/0.47 & ( A = B ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % add_mset_subseteq_single_iff
% 0.27/0.47 thf(fact_172_union__assoc,axiom,
% 0.27/0.47 ! [M4: multiset_a,N: multiset_a,K2: multiset_a] :
% 0.27/0.47 ( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ M4 @ N ) @ K2 )
% 0.27/0.47 = ( plus_plus_multiset_a @ M4 @ ( plus_plus_multiset_a @ N @ K2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % union_assoc
% 0.27/0.47 thf(fact_173_union__lcomm,axiom,
% 0.27/0.47 ! [M4: multiset_a,N: multiset_a,K2: multiset_a] :
% 0.27/0.47 ( ( plus_plus_multiset_a @ M4 @ ( plus_plus_multiset_a @ N @ K2 ) )
% 0.27/0.47 = ( plus_plus_multiset_a @ N @ ( plus_plus_multiset_a @ M4 @ K2 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % union_lcomm
% 0.27/0.47 thf(fact_174_union__commute,axiom,
% 0.27/0.47 ( plus_plus_multiset_a
% 0.27/0.47 = ( ^ [M7: multiset_a,N4: multiset_a] : ( plus_plus_multiset_a @ N4 @ M7 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % union_commute
% 0.27/0.47 thf(fact_175_union__left__cancel,axiom,
% 0.27/0.47 ! [K2: multiset_a,M4: multiset_a,N: multiset_a] :
% 0.27/0.47 ( ( ( plus_plus_multiset_a @ K2 @ M4 )
% 0.27/0.47 = ( plus_plus_multiset_a @ K2 @ N ) )
% 0.27/0.47 = ( M4 = N ) ) ).
% 0.27/0.47
% 0.27/0.47 % union_left_cancel
% 0.27/0.47 thf(fact_176_union__right__cancel,axiom,
% 0.27/0.47 ! [M4: multiset_a,K2: multiset_a,N: multiset_a] :
% 0.27/0.47 ( ( ( plus_plus_multiset_a @ M4 @ K2 )
% 0.27/0.47 = ( plus_plus_multiset_a @ N @ K2 ) )
% 0.27/0.47 = ( M4 = N ) ) ).
% 0.27/0.47
% 0.27/0.47 % union_right_cancel
% 0.27/0.47 thf(fact_177_subset__mset_Oadd__mono,axiom,
% 0.27/0.47 ! [A: multiset_a,B: multiset_a,C: multiset_a,D: multiset_a] :
% 0.27/0.47 ( ( subseteq_mset_a @ A @ B )
% 0.27/0.47 => ( ( subseteq_mset_a @ C @ D )
% 0.27/0.47 => ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ D ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_mset.add_mono
% 0.27/0.47 thf(fact_178_subset__mset_Oless__eqE,axiom,
% 0.27/0.47 ! [A: multiset_a,B: multiset_a] :
% 0.27/0.47 ( ( subseteq_mset_a @ A @ B )
% 0.27/0.47 => ~ ! [C3: multiset_a] :
% 0.27/0.47 ( B
% 0.27/0.47 != ( plus_plus_multiset_a @ A @ C3 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_mset.less_eqE
% 0.27/0.47 thf(fact_179_subset__mset_Ole__iff__add,axiom,
% 0.27/0.47 ( subseteq_mset_a
% 0.27/0.47 = ( ^ [A3: multiset_a,B2: multiset_a] :
% 0.27/0.47 ? [C4: multiset_a] :
% 0.27/0.47 ( B2
% 0.27/0.47 = ( plus_plus_multiset_a @ A3 @ C4 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_mset.le_iff_add
% 0.27/0.47 thf(fact_180_mset__subset__eq__add__left,axiom,
% 0.27/0.47 ! [A2: multiset_a,B4: multiset_a] : ( subseteq_mset_a @ A2 @ ( plus_plus_multiset_a @ A2 @ B4 ) ) ).
% 0.27/0.47
% 0.27/0.47 % mset_subset_eq_add_left
% 0.27/0.47 thf(fact_181_mset__subset__eq__mono__add,axiom,
% 0.27/0.47 ! [A2: multiset_a,B4: multiset_a,C2: multiset_a,D2: multiset_a] :
% 0.27/0.47 ( ( subseteq_mset_a @ A2 @ B4 )
% 0.27/0.47 => ( ( subseteq_mset_a @ C2 @ D2 )
% 0.27/0.47 => ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C2 ) @ ( plus_plus_multiset_a @ B4 @ D2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % mset_subset_eq_mono_add
% 0.27/0.47 thf(fact_182_mset__subset__eq__add__right,axiom,
% 0.27/0.47 ! [B4: multiset_a,A2: multiset_a] : ( subseteq_mset_a @ B4 @ ( plus_plus_multiset_a @ A2 @ B4 ) ) ).
% 0.27/0.47
% 0.27/0.47 % mset_subset_eq_add_right
% 0.27/0.47 thf(fact_183_multi__union__self__other__eq,axiom,
% 0.27/0.47 ! [A2: multiset_a,X7: multiset_a,Y7: multiset_a] :
% 0.27/0.47 ( ( ( plus_plus_multiset_a @ A2 @ X7 )
% 0.27/0.47 = ( plus_plus_multiset_a @ A2 @ Y7 ) )
% 0.27/0.47 => ( X7 = Y7 ) ) ).
% 0.27/0.47
% 0.27/0.47 % multi_union_self_other_eq
% 0.27/0.47 thf(fact_184_subset__mset_Oadd__left__mono,axiom,
% 0.27/0.47 ! [A: multiset_a,B: multiset_a,C: multiset_a] :
% 0.27/0.47 ( ( subseteq_mset_a @ A @ B )
% 0.27/0.47 => ( subseteq_mset_a @ ( plus_plus_multiset_a @ C @ A ) @ ( plus_plus_multiset_a @ C @ B ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_mset.add_left_mono
% 0.27/0.47 thf(fact_185_mset__subset__eq__exists__conv,axiom,
% 0.27/0.47 ( subseteq_mset_a
% 0.27/0.47 = ( ^ [A7: multiset_a,B7: multiset_a] :
% 0.27/0.47 ? [C5: multiset_a] :
% 0.27/0.47 ( B7
% 0.27/0.47 = ( plus_plus_multiset_a @ A7 @ C5 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % mset_subset_eq_exists_conv
% 0.27/0.47 thf(fact_186_subset__mset_Oadd__right__mono,axiom,
% 0.27/0.47 ! [A: multiset_a,B: multiset_a,C: multiset_a] :
% 0.27/0.47 ( ( subseteq_mset_a @ A @ B )
% 0.27/0.47 => ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_mset.add_right_mono
% 0.27/0.47 thf(fact_187_subset__mset_Oadd__le__imp__le__left,axiom,
% 0.27/0.47 ! [C: multiset_a,A: multiset_a,B: multiset_a] :
% 0.27/0.47 ( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C @ A ) @ ( plus_plus_multiset_a @ C @ B ) )
% 0.27/0.47 => ( subseteq_mset_a @ A @ B ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_mset.add_le_imp_le_left
% 0.27/0.47 thf(fact_188_subset__mset_Oadd__le__imp__le__right,axiom,
% 0.27/0.47 ! [A: multiset_a,C: multiset_a,B: multiset_a] :
% 0.27/0.47 ( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ C ) )
% 0.27/0.47 => ( subseteq_mset_a @ A @ B ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_mset.add_le_imp_le_right
% 0.27/0.47 thf(fact_189_union__iff,axiom,
% 0.27/0.47 ! [A: a,A2: multiset_a,B4: multiset_a] :
% 0.27/0.47 ( ( member_a @ A @ ( set_mset_a @ ( plus_plus_multiset_a @ A2 @ B4 ) ) )
% 0.27/0.47 = ( ( member_a @ A @ ( set_mset_a @ A2 ) )
% 0.27/0.47 | ( member_a @ A @ ( set_mset_a @ B4 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % union_iff
% 0.27/0.47 thf(fact_190_mset__subset__eqD,axiom,
% 0.27/0.47 ! [A2: multiset_a,B4: multiset_a,X: a] :
% 0.27/0.47 ( ( subseteq_mset_a @ A2 @ B4 )
% 0.27/0.47 => ( ( member_a @ X @ ( set_mset_a @ A2 ) )
% 0.27/0.47 => ( member_a @ X @ ( set_mset_a @ B4 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % mset_subset_eqD
% 0.27/0.47 thf(fact_191_set__mset__mono,axiom,
% 0.27/0.47 ! [A2: multiset_a,B4: multiset_a] :
% 0.27/0.47 ( ( subseteq_mset_a @ A2 @ B4 )
% 0.27/0.47 => ( ord_less_eq_set_a @ ( set_mset_a @ A2 ) @ ( set_mset_a @ B4 ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % set_mset_mono
% 0.27/0.47 thf(fact_192_mset__subset__eq__add__mset__cancel,axiom,
% 0.27/0.47 ! [A: a,A2: multiset_a,B4: multiset_a] :
% 0.27/0.47 ( ( subseteq_mset_a @ ( add_mset_a @ A @ A2 ) @ ( add_mset_a @ A @ B4 ) )
% 0.27/0.47 = ( subseteq_mset_a @ A2 @ B4 ) ) ).
% 0.27/0.47
% 0.27/0.47 % mset_subset_eq_add_mset_cancel
% 0.27/0.47 thf(fact_193_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.27/0.47 ! [I3: multiset_a,J: multiset_a,K3: multiset_a,L: multiset_a] :
% 0.27/0.47 ( ( ( ord_le1199012836iset_a @ I3 @ J )
% 0.27/0.47 & ( K3 = L ) )
% 0.27/0.47 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I3 @ K3 ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % add_mono_thms_linordered_semiring(3)
% 0.27/0.47 thf(fact_194_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.27/0.47 ! [I3: multiset_a,J: multiset_a,K3: multiset_a,L: multiset_a] :
% 0.27/0.47 ( ( ( I3 = J )
% 0.27/0.47 & ( ord_le1199012836iset_a @ K3 @ L ) )
% 0.27/0.47 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I3 @ K3 ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % add_mono_thms_linordered_semiring(2)
% 0.27/0.47 thf(fact_195_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.27/0.47 ! [I3: multiset_a,J: multiset_a,K3: multiset_a,L: multiset_a] :
% 0.27/0.47 ( ( ( ord_le1199012836iset_a @ I3 @ J )
% 0.27/0.47 & ( ord_le1199012836iset_a @ K3 @ L ) )
% 0.27/0.47 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I3 @ K3 ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % add_mono_thms_linordered_semiring(1)
% 0.27/0.47 thf(fact_196_add__mono,axiom,
% 0.27/0.47 ! [A: multiset_a,B: multiset_a,C: multiset_a,D: multiset_a] :
% 0.27/0.47 ( ( ord_le1199012836iset_a @ A @ B )
% 0.27/0.47 => ( ( ord_le1199012836iset_a @ C @ D )
% 0.27/0.47 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ D ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % add_mono
% 0.27/0.47 thf(fact_197_add__left__mono,axiom,
% 0.27/0.47 ! [A: multiset_a,B: multiset_a,C: multiset_a] :
% 0.27/0.47 ( ( ord_le1199012836iset_a @ A @ B )
% 0.27/0.47 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ C @ A ) @ ( plus_plus_multiset_a @ C @ B ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % add_left_mono
% 0.27/0.47 thf(fact_198_add__right__mono,axiom,
% 0.27/0.47 ! [A: multiset_a,B: multiset_a,C: multiset_a] :
% 0.27/0.47 ( ( ord_le1199012836iset_a @ A @ B )
% 0.27/0.47 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % add_right_mono
% 0.27/0.47 thf(fact_199_empty__neutral_I2_J,axiom,
% 0.27/0.47 ! [X: multiset_a] :
% 0.27/0.47 ( ( plus_plus_multiset_a @ X @ zero_zero_multiset_a )
% 0.27/0.47 = X ) ).
% 0.27/0.47
% 0.27/0.47 % empty_neutral(2)
% 0.27/0.47 thf(fact_200_empty__neutral_I1_J,axiom,
% 0.27/0.47 ! [X: multiset_a] :
% 0.27/0.47 ( ( plus_plus_multiset_a @ zero_zero_multiset_a @ X )
% 0.27/0.47 = X ) ).
% 0.27/0.47
% 0.27/0.47 % empty_neutral(1)
% 0.27/0.47 thf(fact_201_empty__le,axiom,
% 0.27/0.47 ! [A2: multiset_a] : ( subseteq_mset_a @ zero_zero_multiset_a @ A2 ) ).
% 0.27/0.47
% 0.27/0.47 % empty_le
% 0.27/0.47 thf(fact_202_subset__mset_Ozero__le,axiom,
% 0.27/0.47 ! [X: multiset_a] : ( subseteq_mset_a @ zero_zero_multiset_a @ X ) ).
% 0.27/0.47
% 0.27/0.47 % subset_mset.zero_le
% 0.27/0.47 thf(fact_203_subset__mset_Obot_Oextremum,axiom,
% 0.27/0.47 ! [A: multiset_a] : ( subseteq_mset_a @ zero_zero_multiset_a @ A ) ).
% 0.27/0.47
% 0.27/0.47 % subset_mset.bot.extremum
% 0.27/0.47 thf(fact_204_subset__mset_Oadd__decreasing,axiom,
% 0.27/0.47 ! [A: multiset_a,C: multiset_a,B: multiset_a] :
% 0.27/0.47 ( ( subseteq_mset_a @ A @ zero_zero_multiset_a )
% 0.27/0.47 => ( ( subseteq_mset_a @ C @ B )
% 0.27/0.47 => ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C ) @ B ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_mset.add_decreasing
% 0.27/0.47 thf(fact_205_subset__mset_Oadd__increasing,axiom,
% 0.27/0.47 ! [A: multiset_a,B: multiset_a,C: multiset_a] :
% 0.27/0.47 ( ( subseteq_mset_a @ zero_zero_multiset_a @ A )
% 0.27/0.47 => ( ( subseteq_mset_a @ B @ C )
% 0.27/0.47 => ( subseteq_mset_a @ B @ ( plus_plus_multiset_a @ A @ C ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_mset.add_increasing
% 0.27/0.47 thf(fact_206_subset__mset_Oadd__decreasing2,axiom,
% 0.27/0.47 ! [C: multiset_a,A: multiset_a,B: multiset_a] :
% 0.27/0.47 ( ( subseteq_mset_a @ C @ zero_zero_multiset_a )
% 0.27/0.47 => ( ( subseteq_mset_a @ A @ B )
% 0.27/0.47 => ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C ) @ B ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_mset.add_decreasing2
% 0.27/0.47 thf(fact_207_subset__mset_Oadd__increasing2,axiom,
% 0.27/0.47 ! [C: multiset_a,B: multiset_a,A: multiset_a] :
% 0.27/0.47 ( ( subseteq_mset_a @ zero_zero_multiset_a @ C )
% 0.27/0.47 => ( ( subseteq_mset_a @ B @ A )
% 0.27/0.47 => ( subseteq_mset_a @ B @ ( plus_plus_multiset_a @ A @ C ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_mset.add_increasing2
% 0.27/0.47 thf(fact_208_subset__mset_Oadd__nonneg__nonneg,axiom,
% 0.27/0.47 ! [A: multiset_a,B: multiset_a] :
% 0.27/0.47 ( ( subseteq_mset_a @ zero_zero_multiset_a @ A )
% 0.27/0.47 => ( ( subseteq_mset_a @ zero_zero_multiset_a @ B )
% 0.27/0.47 => ( subseteq_mset_a @ zero_zero_multiset_a @ ( plus_plus_multiset_a @ A @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_mset.add_nonneg_nonneg
% 0.27/0.47 thf(fact_209_subset__mset_Oadd__nonpos__nonpos,axiom,
% 0.27/0.47 ! [A: multiset_a,B: multiset_a] :
% 0.27/0.47 ( ( subseteq_mset_a @ A @ zero_zero_multiset_a )
% 0.27/0.47 => ( ( subseteq_mset_a @ B @ zero_zero_multiset_a )
% 0.27/0.47 => ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ B ) @ zero_zero_multiset_a ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_mset.add_nonpos_nonpos
% 0.27/0.47 thf(fact_210_subset__mset_Oadd__nonneg__eq__0__iff,axiom,
% 0.27/0.47 ! [X: multiset_a,Y: multiset_a] :
% 0.27/0.47 ( ( subseteq_mset_a @ zero_zero_multiset_a @ X )
% 0.27/0.47 => ( ( subseteq_mset_a @ zero_zero_multiset_a @ Y )
% 0.27/0.47 => ( ( ( plus_plus_multiset_a @ X @ Y )
% 0.27/0.47 = zero_zero_multiset_a )
% 0.27/0.47 = ( ( X = zero_zero_multiset_a )
% 0.27/0.47 & ( Y = zero_zero_multiset_a ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_mset.add_nonneg_eq_0_iff
% 0.27/0.47 thf(fact_211_subset__mset_Oadd__nonpos__eq__0__iff,axiom,
% 0.27/0.47 ! [X: multiset_a,Y: multiset_a] :
% 0.27/0.47 ( ( subseteq_mset_a @ X @ zero_zero_multiset_a )
% 0.27/0.47 => ( ( subseteq_mset_a @ Y @ zero_zero_multiset_a )
% 0.27/0.47 => ( ( ( plus_plus_multiset_a @ X @ Y )
% 0.27/0.47 = zero_zero_multiset_a )
% 0.27/0.47 = ( ( X = zero_zero_multiset_a )
% 0.27/0.47 & ( Y = zero_zero_multiset_a ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_mset.add_nonpos_eq_0_iff
% 0.27/0.47 thf(fact_212_subset__mset_Obot_Oextremum__uniqueI,axiom,
% 0.27/0.47 ! [A: multiset_a] :
% 0.27/0.47 ( ( subseteq_mset_a @ A @ zero_zero_multiset_a )
% 0.27/0.47 => ( A = zero_zero_multiset_a ) ) ).
% 0.27/0.47
% 0.27/0.47 % subset_mset.bot.extremum_uniqueI
% 0.27/0.47 thf(fact_213_verit__sum__simplify,axiom,
% 0.27/0.47 ! [A: multiset_a] :
% 0.27/0.47 ( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
% 0.27/0.47 = A ) ).
% 0.27/0.47
% 0.27/0.47 % verit_sum_simplify
% 0.27/0.47
% 0.27/0.47 % Conjectures (1)
% 0.27/0.47 thf(conj_0,conjecture,
% 0.27/0.47 ( ( multiset_a2 @ ( heapIm1091024090Down_a @ ( t_a @ v @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) ) ) )
% 0.27/0.47 = ( multiset_a2 @ ( t_a @ v @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 %------------------------------------------------------------------------------
% 0.27/0.47 ------------------------
% 2.41/1.47 ---- Embedded in THF ---
% 2.41/1.47 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.41/1.47 %%% Generated on Tue Feb 24 08:24:12 EST 2026
% 2.41/1.47 %%% using '$$dhol' embedding, version 1.3.0.
% 2.41/1.47 %%% Logic specification used:
% 2.41/1.47 %%% thf(spec, logic, $$dhol).
% 2.41/1.47
% 2.41/1.47 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.geO6X9CZKJ/DTF2DTF6481.p
% See solution above
% 2.41/1.47 ------------------------
% 2.41/1.50 ---- Cleaned THF ---
% 2.41/1.50 thf(ty_n_t__Product____Type__Oprod_It__Heap__OTree_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
% 2.41/1.50 produc768687417list_a: $tType ).
% 2.41/1.50
% 2.41/1.50 thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J,type,
% 2.41/1.50 produc143150363Tree_a: $tType ).
% 2.41/1.50
% 2.41/1.50 thf(ty_n_t__Multiset__Omultiset_Itf__a_J,type,
% 2.41/1.50 multiset_a: $tType ).
% 2.41/1.50
% 2.41/1.50 thf(ty_n_t__List__Olist_Itf__a_J,type,
% 2.41/1.50 list_a: $tType ).
% 2.41/1.50
% 2.41/1.50 thf(ty_n_t__Heap__OTree_Itf__a_J,type,
% 2.41/1.50 tree_a: $tType ).
% 2.41/1.50
% 2.41/1.50 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 2.41/1.50 set_a: $tType ).
% 2.41/1.50
% 2.41/1.50 thf(ty_n_tf__a,type,
% 2.41/1.50 a: $tType ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.41/1.50 plus_plus_multiset_a: multiset_a > multiset_a > multiset_a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.41/1.50 zero_zero_multiset_a: multiset_a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Oheapify_001tf__a,type,
% 2.41/1.50 heapIm970322378pify_a: tree_a > tree_a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Ohs__is__empty_001tf__a,type,
% 2.41/1.50 heapIm229596386mpty_a: tree_a > $o ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Ohs__of__list_001tf__a,type,
% 2.41/1.50 heapIm1057938560list_a: list_a > tree_a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Oleft_001tf__a,type,
% 2.41/1.50 heapIm1140443833left_a: tree_a > tree_a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Oof__list__tree_001tf__a,type,
% 2.41/1.50 heapIm1637418125tree_a: list_a > tree_a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_OremoveLeaf_001tf__a,type,
% 2.41/1.50 heapIm837449470Leaf_a: tree_a > produc143150363Tree_a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Oright_001tf__a,type,
% 2.41/1.50 heapIm1257206334ight_a: tree_a > tree_a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_OsiftDown_001tf__a,type,
% 2.41/1.50 heapIm1091024090Down_a: tree_a > tree_a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_Heap_OHeap_001t__Heap__OTree_Itf__a_J_001tf__a,type,
% 2.41/1.50 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 ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_Heap_OHeap__axioms_001t__Heap__OTree_Itf__a_J_001tf__a,type,
% 2.41/1.50 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 ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_Heap_OTree_OE_001tf__a,type,
% 2.41/1.50 e_a: tree_a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_Heap_OTree_OT_001tf__a,type,
% 2.41/1.50 t_a: a > tree_a > tree_a > tree_a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_Heap_Oin__tree_001tf__a,type,
% 2.41/1.50 in_tree_a: a > tree_a > $o ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_Heap_Ois__heap_001tf__a,type,
% 2.41/1.50 is_heap_a: tree_a > $o ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_Heap_Omultiset_001tf__a,type,
% 2.41/1.50 multiset_a2: tree_a > multiset_a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_Heap_Oval_001tf__a,type,
% 2.41/1.50 val_a: tree_a > a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_Lattices__Big_Olinorder__class_OMax_001tf__a,type,
% 2.41/1.50 lattic146396397_Max_a: set_a > a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_List_Olist_OCons_001tf__a,type,
% 2.41/1.50 cons_a: a > list_a > list_a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_Multiset_Oadd__mset_001tf__a,type,
% 2.41/1.50 add_mset_a: a > multiset_a > multiset_a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_Multiset_Ois__empty_001tf__a,type,
% 2.41/1.50 is_empty_a: multiset_a > $o ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_Multiset_Oset__mset_001tf__a,type,
% 2.41/1.50 set_mset_a: multiset_a > set_a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_Multiset_Osubseteq__mset_001tf__a,type,
% 2.41/1.50 subseteq_mset_a: multiset_a > multiset_a > $o ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_I_Eo_Mtf__a_J_J,type,
% 2.41/1.50 ord_less_eq_o_o_a: ( $o > $o > a ) > ( $o > $o > a ) > $o ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mtf__a_J,type,
% 2.41/1.50 ord_less_eq_o_a: ( $o > a ) > ( $o > a ) > $o ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.41/1.50 ord_le1199012836iset_a: multiset_a > multiset_a > $o ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
% 2.41/1.50 ord_less_eq_set_a: set_a > set_a > $o ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
% 2.41/1.50 ord_less_eq_a: a > a > $o ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_Orderings_Oorder__class_OGreatest_001_062_I_Eo_Mtf__a_J,type,
% 2.41/1.50 order_Greatest_o_a: ( ( $o > a ) > $o ) > $o > a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_Orderings_Oorder__class_OGreatest_001tf__a,type,
% 2.41/1.50 order_Greatest_a: ( a > $o ) > a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_Product__Type_OPair_001t__Heap__OTree_Itf__a_J_001t__List__Olist_Itf__a_J,type,
% 2.41/1.50 produc1352981801list_a: tree_a > list_a > produc768687417list_a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_Product__Type_OPair_001tf__a_001t__Heap__OTree_Itf__a_J,type,
% 2.41/1.50 produc686083979Tree_a: a > tree_a > produc143150363Tree_a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_RemoveMax_OCollection_Oset_001t__Heap__OTree_Itf__a_J_001tf__a,type,
% 2.41/1.50 set_Tree_a_a: ( tree_a > multiset_a ) > tree_a > set_a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_RemoveMax_ORemoveMax_001t__Heap__OTree_Itf__a_J_001tf__a,type,
% 2.41/1.50 removeMax_Tree_a_a: tree_a > ( tree_a > $o ) > ( list_a > tree_a ) > ( tree_a > multiset_a ) > ( tree_a > produc143150363Tree_a ) > ( tree_a > $o ) > $o ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_RemoveMax_ORemoveMax_Ossort_H_001t__Heap__OTree_Itf__a_J_001tf__a,type,
% 2.41/1.50 ssort_Tree_a_a: ( tree_a > $o ) > ( tree_a > produc143150363Tree_a ) > tree_a > list_a > list_a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_RemoveMax_ORemoveMax_Ossort_H__dom_001t__Heap__OTree_Itf__a_J_001tf__a,type,
% 2.41/1.50 ssort_dom_Tree_a_a: ( tree_a > $o ) > ( tree_a > produc143150363Tree_a ) > produc768687417list_a > $o ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_RemoveMax_ORemoveMax__axioms_001t__Heap__OTree_Itf__a_J_001tf__a,type,
% 2.41/1.50 remove301631099ee_a_a: ( tree_a > $o ) > ( list_a > tree_a ) > ( tree_a > multiset_a ) > ( tree_a > produc143150363Tree_a ) > ( tree_a > $o ) > $o ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_Set_OCollect_001tf__a,type,
% 2.41/1.50 collect_a: ( a > $o ) > set_a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_c_member_001tf__a,type,
% 2.41/1.50 member_a: a > set_a > $o ).
% 2.41/1.50
% 2.41/1.50 thf(sy_v_l1____,type,
% 2.41/1.50 l1: tree_a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_v_l2____,type,
% 2.41/1.50 l2: tree_a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_v_r1____,type,
% 2.41/1.50 r1: tree_a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_v_r2____,type,
% 2.41/1.50 r2: tree_a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_v_v1____,type,
% 2.41/1.50 v1: a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_v_v2____,type,
% 2.41/1.50 v2: a ).
% 2.41/1.50
% 2.41/1.50 thf(sy_v_v____,type,
% 2.41/1.50 v: a ).
% 2.41/1.50
% 2.41/1.50 thf(fact_0__092_060open_062v2_A_092_060le_062_Av1_092_060close_062,axiom,
% 2.41/1.50 ord_less_eq_a @ v2 @ v1 ).
% 2.41/1.50
% 2.41/1.50 thf(fact_1_True,axiom,
% 2.41/1.50 ord_less_eq_a @ v1 @ v ).
% 2.41/1.50
% 2.41/1.50 thf(fact_2__C5__2_Ohyps_C_I2_J,axiom,
% 2.41/1.50 ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ v2 @ l2 @ r2 ) ) @ ( val_a @ ( t_a @ v1 @ l1 @ r1 ) ) )
% 2.41/1.50 => ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ v2 @ l2 @ r2 ) ) @ v )
% 2.41/1.50 => ( ( multiset_a2 @ ( heapIm1091024090Down_a @ ( t_a @ v @ ( heapIm1140443833left_a @ ( t_a @ v2 @ l2 @ r2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ v2 @ l2 @ r2 ) ) ) ) )
% 2.41/1.50 = ( multiset_a2 @ ( t_a @ v @ ( heapIm1140443833left_a @ ( t_a @ v2 @ l2 @ r2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ v2 @ l2 @ r2 ) ) ) ) ) ) ) ).
% 2.41/1.50
% 2.41/1.50 thf(fact_3__C5__2_Ohyps_C_I1_J,axiom,
% 2.41/1.50 ( ( ord_less_eq_a @ ( val_a @ ( t_a @ v2 @ l2 @ r2 ) ) @ ( val_a @ ( t_a @ v1 @ l1 @ r1 ) ) )
% 2.41/1.50 => ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ v1 @ l1 @ r1 ) ) @ v )
% 2.41/1.50 => ( ( multiset_a2 @ ( heapIm1091024090Down_a @ ( t_a @ v @ ( heapIm1140443833left_a @ ( t_a @ v1 @ l1 @ r1 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ v1 @ l1 @ r1 ) ) ) ) )
% 2.41/1.50 = ( multiset_a2 @ ( t_a @ v @ ( heapIm1140443833left_a @ ( t_a @ v1 @ l1 @ r1 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ v1 @ l1 @ r1 ) ) ) ) ) ) ) ).
% 2.41/1.50
% 2.41/1.50 thf(fact_4_siftDown__Node,axiom,
% 2.41/1.50 ! [T: tree_a,V: a,L: tree_a,R: tree_a] :
% 2.41/1.50 ( ( T
% 2.41/1.50 = ( t_a @ V @ L @ R ) )
% 2.41/1.50 => ? [L2: tree_a,V2: a,R2: tree_a] :
% 2.41/1.50 ( ( ( heapIm1091024090Down_a @ T )
% 2.63/1.50 = ( t_a @ V2 @ L2 @ R2 ) )
% 2.63/1.50 & ( ord_less_eq_a @ V @ V2 ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_5_Tree_Oinject,axiom,
% 2.63/1.50 ! [X21: a,X22: tree_a,X23: tree_a,Y21: a,Y22: tree_a,Y23: tree_a] :
% 2.63/1.50 ( ( ( t_a @ X21 @ X22 @ X23 )
% 2.63/1.50 = ( t_a @ Y21 @ Y22 @ Y23 ) )
% 2.63/1.50 = ( ( X21 = Y21 )
% 2.63/1.50 & ( X22 = Y22 )
% 2.63/1.50 & ( X23 = Y23 ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_6_heapify_Osimps_I2_J,axiom,
% 2.63/1.50 ! [V: a,L: tree_a,R: tree_a] :
% 2.63/1.50 ( ( heapIm970322378pify_a @ ( t_a @ V @ L @ R ) )
% 2.63/1.50 = ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm970322378pify_a @ L ) @ ( heapIm970322378pify_a @ R ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_7_siftDown__heap__is__heap,axiom,
% 2.63/1.50 ! [L: tree_a,R: tree_a,T: tree_a,V: a] :
% 2.63/1.50 ( ( is_heap_a @ L )
% 2.63/1.50 => ( ( is_heap_a @ R )
% 2.63/1.50 => ( ( T
% 2.63/1.50 = ( t_a @ V @ L @ R ) )
% 2.63/1.50 => ( is_heap_a @ ( heapIm1091024090Down_a @ T ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_8_siftDown__in__tree__set,axiom,
% 2.63/1.50 ( in_tree_a
% 2.63/1.50 = ( ^ [V3: a,T2: tree_a] : ( in_tree_a @ V3 @ ( heapIm1091024090Down_a @ T2 ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_9_left_Osimps,axiom,
% 2.63/1.50 ! [V: a,L: tree_a,R: tree_a] :
% 2.63/1.50 ( ( heapIm1140443833left_a @ ( t_a @ V @ L @ R ) )
% 2.63/1.50 = L ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_10_right_Osimps,axiom,
% 2.63/1.50 ! [V: a,L: tree_a,R: tree_a] :
% 2.63/1.50 ( ( heapIm1257206334ight_a @ ( t_a @ V @ L @ R ) )
% 2.63/1.50 = R ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_11_siftDown_Osimps_I2_J,axiom,
% 2.63/1.50 ! [V: a] :
% 2.63/1.50 ( ( heapIm1091024090Down_a @ ( t_a @ V @ e_a @ e_a ) )
% 2.63/1.50 = ( t_a @ V @ e_a @ e_a ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_12_siftDown_Osimps_I1_J,axiom,
% 2.63/1.50 ( ( heapIm1091024090Down_a @ e_a )
% 2.63/1.50 = e_a ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_13_in__tree_Osimps_I1_J,axiom,
% 2.63/1.50 ! [V: a] :
% 2.63/1.50 ~ ( in_tree_a @ V @ e_a ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_14_is__heap_Osimps_I6_J,axiom,
% 2.63/1.50 ! [V: a,Vd: a,Ve: tree_a,Vf: tree_a,Va: a,Vb: tree_a,Vc: tree_a] :
% 2.63/1.50 ( ( is_heap_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
% 2.63/1.50 = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 2.63/1.50 & ( is_heap_a @ ( t_a @ Va @ Vb @ Vc ) )
% 2.63/1.50 & ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
% 2.63/1.50 & ( is_heap_a @ ( t_a @ Vd @ Ve @ Vf ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_15_is__heap_Osimps_I5_J,axiom,
% 2.63/1.50 ! [V: a,Va: a,Vb: tree_a,Vc: tree_a,Vd: a,Ve: tree_a,Vf: tree_a] :
% 2.63/1.50 ( ( is_heap_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 2.63/1.50 = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
% 2.63/1.50 & ( is_heap_a @ ( t_a @ Vd @ Ve @ Vf ) )
% 2.63/1.50 & ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 2.63/1.50 & ( is_heap_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_16_is__heap_Osimps_I4_J,axiom,
% 2.63/1.50 ! [V: a,Va: a,Vb: tree_a,Vc: tree_a] :
% 2.63/1.50 ( ( is_heap_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) )
% 2.63/1.50 = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 2.63/1.50 & ( is_heap_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_17_is__heap_Osimps_I3_J,axiom,
% 2.63/1.50 ! [V: a,Va: a,Vb: tree_a,Vc: tree_a] :
% 2.63/1.50 ( ( is_heap_a @ ( t_a @ V @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) )
% 2.63/1.50 = ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 2.63/1.50 & ( is_heap_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_18_is__heap_Osimps_I2_J,axiom,
% 2.63/1.50 ! [V: a] : ( is_heap_a @ ( t_a @ V @ e_a @ e_a ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_19_is__heap_Osimps_I1_J,axiom,
% 2.63/1.50 is_heap_a @ e_a ).
% 2.63/1.50
% 2.63/1.50 thf(fact_20_heapify_Osimps_I1_J,axiom,
% 2.63/1.50 ( ( heapIm970322378pify_a @ e_a )
% 2.63/1.50 = e_a ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_21_is__heap__max,axiom,
% 2.63/1.50 ! [V: a,T: tree_a] :
% 2.63/1.50 ( ( in_tree_a @ V @ T )
% 2.63/1.50 => ( ( is_heap_a @ T )
% 2.63/1.50 => ( ord_less_eq_a @ V @ ( val_a @ T ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_22_heapify__heap__is__heap,axiom,
% 2.63/1.50 ! [T: tree_a] : ( is_heap_a @ ( heapIm970322378pify_a @ T ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_23_siftDown__in__tree,axiom,
% 2.63/1.50 ! [T: tree_a] :
% 2.63/1.50 ( ( T != e_a )
% 2.63/1.50 => ( in_tree_a @ ( val_a @ ( heapIm1091024090Down_a @ T ) ) @ T ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_24_is__heap_Ocases,axiom,
% 2.63/1.50 ! [X: tree_a] :
% 2.63/1.50 ( ( X != e_a )
% 2.63/1.50 => ( ! [V4: a] :
% 2.63/1.50 ( X
% 2.63/1.50 != ( t_a @ V4 @ e_a @ e_a ) )
% 2.63/1.50 => ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 2.63/1.50 ( X
% 2.63/1.50 != ( t_a @ V4 @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 2.63/1.50 => ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 2.63/1.50 ( X
% 2.63/1.50 != ( t_a @ V4 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
% 2.63/1.50 => ~ ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a] :
% 2.63/1.50 ( X
% 2.63/1.50 != ( t_a @ V4 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_25_Tree_Oexhaust,axiom,
% 2.63/1.50 ! [Y: tree_a] :
% 2.63/1.50 ( ( Y != e_a )
% 2.63/1.50 => ~ ! [X212: a,X222: tree_a,X232: tree_a] :
% 2.63/1.50 ( Y
% 2.63/1.50 != ( t_a @ X212 @ X222 @ X232 ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_26_Tree_Oinduct,axiom,
% 2.63/1.50 ! [P: tree_a > $o,Tree: tree_a] :
% 2.63/1.50 ( ( P @ e_a )
% 2.63/1.50 => ( ! [X1: a,X2: tree_a,X3: tree_a] :
% 2.63/1.50 ( ( P @ X2 )
% 2.63/1.50 => ( ( P @ X3 )
% 2.63/1.50 => ( P @ ( t_a @ X1 @ X2 @ X3 ) ) ) )
% 2.63/1.50 => ( P @ Tree ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_27_Tree_Odistinct_I1_J,axiom,
% 2.63/1.50 ! [X21: a,X22: tree_a,X23: tree_a] :
% 2.63/1.50 ( e_a
% 2.63/1.50 != ( t_a @ X21 @ X22 @ X23 ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_28_siftDown_Osimps_I3_J,axiom,
% 2.63/1.50 ! [Va: a,Vb: tree_a,Vc: tree_a,V: a] :
% 2.63/1.50 ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 2.63/1.50 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) )
% 2.63/1.50 = ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) ) )
% 2.63/1.50 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 2.63/1.50 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) )
% 2.63/1.50 = ( 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 ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_29_siftDown_Osimps_I4_J,axiom,
% 2.63/1.50 ! [Va: a,Vb: tree_a,Vc: tree_a,V: a] :
% 2.63/1.50 ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 2.63/1.50 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) )
% 2.63/1.50 = ( t_a @ V @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) ) )
% 2.63/1.50 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 2.63/1.50 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) )
% 2.63/1.50 = ( 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 ) ) ) ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_30_val_Osimps,axiom,
% 2.63/1.50 ! [V: a,Uu: tree_a,Uv: tree_a] :
% 2.63/1.50 ( ( val_a @ ( t_a @ V @ Uu @ Uv ) )
% 2.63/1.50 = V ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_31_siftDown_Osimps_I5_J,axiom,
% 2.63/1.50 ! [Vd: a,Ve: tree_a,Vf: tree_a,Va: a,Vb: tree_a,Vc: tree_a,V: a] :
% 2.63/1.50 ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) )
% 2.63/1.50 => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 2.63/1.50 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 2.63/1.50 = ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) ) )
% 2.63/1.50 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 2.63/1.50 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 2.63/1.50 = ( 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 ) ) ) ) ) )
% 2.63/1.50 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) )
% 2.63/1.50 => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
% 2.63/1.50 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 2.63/1.50 = ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) ) )
% 2.63/1.50 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
% 2.63/1.50 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 2.63/1.50 = ( 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 ) ) ) ) ) ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_32_siftDown_Osimps_I6_J,axiom,
% 2.63/1.50 ! [Va: a,Vb: tree_a,Vc: tree_a,Vd: a,Ve: tree_a,Vf: tree_a,V: a] :
% 2.63/1.50 ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 2.63/1.50 => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
% 2.63/1.50 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
% 2.63/1.50 = ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) ) )
% 2.63/1.50 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
% 2.63/1.50 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
% 2.63/1.50 = ( 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 ) ) ) ) ) )
% 2.63/1.50 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) )
% 2.63/1.50 => ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 2.63/1.50 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
% 2.63/1.50 = ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) ) )
% 2.63/1.50 & ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
% 2.63/1.50 => ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
% 2.63/1.50 = ( 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 ) ) ) ) ) ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_33_in__tree_Osimps_I2_J,axiom,
% 2.63/1.50 ! [V: a,V5: a,L: tree_a,R: tree_a] :
% 2.63/1.50 ( ( in_tree_a @ V @ ( t_a @ V5 @ L @ R ) )
% 2.63/1.50 = ( ( V = V5 )
% 2.63/1.50 | ( in_tree_a @ V @ L )
% 2.63/1.50 | ( in_tree_a @ V @ R ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_34_removeLeaf_Oinduct,axiom,
% 2.63/1.50 ! [P: tree_a > $o,A0: tree_a] :
% 2.63/1.50 ( ! [V4: a] : ( P @ ( t_a @ V4 @ e_a @ e_a ) )
% 2.63/1.50 => ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 2.63/1.50 ( ( P @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
% 2.63/1.50 => ( ( P @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
% 2.63/1.50 => ( P @ ( t_a @ V4 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) ) ) )
% 2.63/1.50 => ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 2.63/1.50 ( ( P @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
% 2.63/1.50 => ( ( P @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
% 2.63/1.50 => ( P @ ( t_a @ V4 @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) )
% 2.63/1.50 => ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a] :
% 2.63/1.50 ( ( P @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
% 2.63/1.50 => ( ( P @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
% 2.63/1.50 => ( P @ ( t_a @ V4 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) )
% 2.63/1.50 => ( ! [V4: a,Vd2: a,Ve2: tree_a,Vf2: tree_a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 2.63/1.50 ( ( P @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) )
% 2.63/1.50 => ( ( P @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) )
% 2.63/1.50 => ( P @ ( t_a @ V4 @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) )
% 2.63/1.50 => ( ( P @ e_a )
% 2.63/1.50 => ( P @ A0 ) ) ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_35_removeLeaf_Ocases,axiom,
% 2.63/1.50 ! [X: tree_a] :
% 2.63/1.50 ( ! [V4: a] :
% 2.63/1.50 ( X
% 2.63/1.50 != ( t_a @ V4 @ e_a @ e_a ) )
% 2.63/1.50 => ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 2.63/1.50 ( X
% 2.63/1.50 != ( t_a @ V4 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
% 2.63/1.50 => ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 2.63/1.50 ( X
% 2.63/1.50 != ( t_a @ V4 @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 2.63/1.50 => ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a] :
% 2.63/1.50 ( X
% 2.63/1.50 != ( t_a @ V4 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
% 2.63/1.50 => ( ! [V4: a,Vd2: a,Ve2: tree_a,Vf2: tree_a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 2.63/1.50 ( X
% 2.63/1.50 != ( t_a @ V4 @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 2.63/1.50 => ( X = e_a ) ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_36_siftDown_Ocases,axiom,
% 2.63/1.50 ! [X: tree_a] :
% 2.63/1.50 ( ( X != e_a )
% 2.63/1.50 => ( ! [V4: a] :
% 2.63/1.50 ( X
% 2.63/1.50 != ( t_a @ V4 @ e_a @ e_a ) )
% 2.63/1.50 => ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 2.63/1.50 ( X
% 2.63/1.50 != ( t_a @ V4 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
% 2.63/1.50 => ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
% 2.63/1.50 ( X
% 2.63/1.50 != ( t_a @ V4 @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
% 2.63/1.50 => ~ ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a] :
% 2.63/1.50 ( X
% 2.63/1.50 != ( t_a @ V4 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_37_hs__is__empty__def,axiom,
% 2.63/1.50 ( heapIm229596386mpty_a
% 2.63/1.50 = ( ^ [T2: tree_a] : ( T2 = e_a ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_38_order__refl,axiom,
% 2.63/1.50 ! [X: $o > a] : ( ord_less_eq_o_a @ X @ X ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_39_order__refl,axiom,
% 2.63/1.50 ! [X: a] : ( ord_less_eq_a @ X @ X ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_40_heap__top__geq,axiom,
% 2.63/1.50 ! [A: a,T: tree_a] :
% 2.63/1.50 ( ( member_a @ A @ ( set_mset_a @ ( multiset_a2 @ T ) ) )
% 2.63/1.50 => ( ( is_heap_a @ T )
% 2.63/1.50 => ( ord_less_eq_a @ A @ ( val_a @ T ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_41_hs__of__list__def,axiom,
% 2.63/1.50 ( heapIm1057938560list_a
% 2.63/1.50 = ( ^ [L3: list_a] : ( heapIm970322378pify_a @ ( heapIm1637418125tree_a @ L3 ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_42_mem__Collect__eq,axiom,
% 2.63/1.50 ! [A: a,P: a > $o] :
% 2.63/1.50 ( ( member_a @ A @ ( collect_a @ P ) )
% 2.63/1.50 = ( P @ A ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_43_Collect__mem__eq,axiom,
% 2.63/1.50 ! [A2: set_a] :
% 2.63/1.50 ( ( collect_a
% 2.63/1.50 @ ^ [X4: a] : ( member_a @ X4 @ A2 ) )
% 2.63/1.50 = A2 ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_44_le__funD,axiom,
% 2.63/1.50 ! [F: $o > a,G: $o > a,X: $o] :
% 2.63/1.50 ( ( ord_less_eq_o_a @ F @ G )
% 2.63/1.50 => ( ord_less_eq_a @ ( F @ X ) @ ( G @ X ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_45_le__funE,axiom,
% 2.63/1.50 ! [F: $o > a,G: $o > a,X: $o] :
% 2.63/1.50 ( ( ord_less_eq_o_a @ F @ G )
% 2.63/1.50 => ( ord_less_eq_a @ ( F @ X ) @ ( G @ X ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_46_le__funI,axiom,
% 2.63/1.50 ! [F: $o > a,G: $o > a] :
% 2.63/1.50 ( ! [X5: $o] : ( ord_less_eq_a @ ( F @ X5 ) @ ( G @ X5 ) )
% 2.63/1.50 => ( ord_less_eq_o_a @ F @ G ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_47_le__fun__def,axiom,
% 2.63/1.50 ( ord_less_eq_o_a
% 2.63/1.50 = ( ^ [F2: $o > a,G2: $o > a] :
% 2.63/1.50 ! [X4: $o] : ( ord_less_eq_a @ ( F2 @ X4 ) @ ( G2 @ X4 ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_48_order__subst1,axiom,
% 2.63/1.50 ! [A: a,F: ( $o > a ) > a,B: $o > a,C: $o > a] :
% 2.63/1.50 ( ( ord_less_eq_a @ A @ ( F @ B ) )
% 2.63/1.50 => ( ( ord_less_eq_o_a @ B @ C )
% 2.63/1.50 => ( ! [X5: $o > a,Y2: $o > a] :
% 2.63/1.50 ( ( ord_less_eq_o_a @ X5 @ Y2 )
% 2.63/1.50 => ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 2.63/1.50 => ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_49_order__subst1,axiom,
% 2.63/1.50 ! [A: $o > a,F: a > $o > a,B: a,C: a] :
% 2.63/1.50 ( ( ord_less_eq_o_a @ A @ ( F @ B ) )
% 2.63/1.50 => ( ( ord_less_eq_a @ B @ C )
% 2.63/1.50 => ( ! [X5: a,Y2: a] :
% 2.63/1.50 ( ( ord_less_eq_a @ X5 @ Y2 )
% 2.63/1.50 => ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 2.63/1.50 => ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_50_order__subst1,axiom,
% 2.63/1.50 ! [A: $o > a,F: ( $o > a ) > $o > a,B: $o > a,C: $o > a] :
% 2.63/1.50 ( ( ord_less_eq_o_a @ A @ ( F @ B ) )
% 2.63/1.50 => ( ( ord_less_eq_o_a @ B @ C )
% 2.63/1.50 => ( ! [X5: $o > a,Y2: $o > a] :
% 2.63/1.50 ( ( ord_less_eq_o_a @ X5 @ Y2 )
% 2.63/1.50 => ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 2.63/1.50 => ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_51_order__subst1,axiom,
% 2.63/1.50 ! [A: a,F: a > a,B: a,C: a] :
% 2.63/1.50 ( ( ord_less_eq_a @ A @ ( F @ B ) )
% 2.63/1.50 => ( ( ord_less_eq_a @ B @ C )
% 2.63/1.50 => ( ! [X5: a,Y2: a] :
% 2.63/1.50 ( ( ord_less_eq_a @ X5 @ Y2 )
% 2.63/1.50 => ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 2.63/1.50 => ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_52_order__subst2,axiom,
% 2.63/1.50 ! [A: a,B: a,F: a > $o > a,C: $o > a] :
% 2.63/1.50 ( ( ord_less_eq_a @ A @ B )
% 2.63/1.50 => ( ( ord_less_eq_o_a @ ( F @ B ) @ C )
% 2.63/1.50 => ( ! [X5: a,Y2: a] :
% 2.63/1.50 ( ( ord_less_eq_a @ X5 @ Y2 )
% 2.63/1.50 => ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 2.63/1.50 => ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_53_order__subst2,axiom,
% 2.63/1.50 ! [A: $o > a,B: $o > a,F: ( $o > a ) > a,C: a] :
% 2.63/1.50 ( ( ord_less_eq_o_a @ A @ B )
% 2.63/1.50 => ( ( ord_less_eq_a @ ( F @ B ) @ C )
% 2.63/1.50 => ( ! [X5: $o > a,Y2: $o > a] :
% 2.63/1.50 ( ( ord_less_eq_o_a @ X5 @ Y2 )
% 2.63/1.50 => ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 2.63/1.50 => ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_54_order__subst2,axiom,
% 2.63/1.50 ! [A: $o > a,B: $o > a,F: ( $o > a ) > $o > a,C: $o > a] :
% 2.63/1.50 ( ( ord_less_eq_o_a @ A @ B )
% 2.63/1.50 => ( ( ord_less_eq_o_a @ ( F @ B ) @ C )
% 2.63/1.50 => ( ! [X5: $o > a,Y2: $o > a] :
% 2.63/1.50 ( ( ord_less_eq_o_a @ X5 @ Y2 )
% 2.63/1.50 => ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 2.63/1.50 => ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_55_order__subst2,axiom,
% 2.63/1.50 ! [A: a,B: a,F: a > a,C: a] :
% 2.63/1.50 ( ( ord_less_eq_a @ A @ B )
% 2.63/1.50 => ( ( ord_less_eq_a @ ( F @ B ) @ C )
% 2.63/1.50 => ( ! [X5: a,Y2: a] :
% 2.63/1.50 ( ( ord_less_eq_a @ X5 @ Y2 )
% 2.63/1.50 => ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 2.63/1.50 => ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_56_dual__order_Oantisym,axiom,
% 2.63/1.50 ! [B: $o > a,A: $o > a] :
% 2.63/1.50 ( ( ord_less_eq_o_a @ B @ A )
% 2.63/1.50 => ( ( ord_less_eq_o_a @ A @ B )
% 2.63/1.50 => ( A = B ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_57_dual__order_Oantisym,axiom,
% 2.63/1.50 ! [B: a,A: a] :
% 2.63/1.50 ( ( ord_less_eq_a @ B @ A )
% 2.63/1.50 => ( ( ord_less_eq_a @ A @ B )
% 2.63/1.50 => ( A = B ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_58_dual__order_Oeq__iff,axiom,
% 2.63/1.50 ( ( ^ [Y3: $o > a,Z: $o > a] : ( Y3 = Z ) )
% 2.63/1.50 = ( ^ [A3: $o > a,B2: $o > a] :
% 2.63/1.50 ( ( ord_less_eq_o_a @ B2 @ A3 )
% 2.63/1.50 & ( ord_less_eq_o_a @ A3 @ B2 ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_59_dual__order_Oeq__iff,axiom,
% 2.63/1.50 ( ( ^ [Y3: a,Z: a] : ( Y3 = Z ) )
% 2.63/1.50 = ( ^ [A3: a,B2: a] :
% 2.63/1.50 ( ( ord_less_eq_a @ B2 @ A3 )
% 2.63/1.50 & ( ord_less_eq_a @ A3 @ B2 ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_60_dual__order_Otrans,axiom,
% 2.63/1.50 ! [B: $o > a,A: $o > a,C: $o > a] :
% 2.63/1.50 ( ( ord_less_eq_o_a @ B @ A )
% 2.63/1.50 => ( ( ord_less_eq_o_a @ C @ B )
% 2.63/1.50 => ( ord_less_eq_o_a @ C @ A ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_61_dual__order_Otrans,axiom,
% 2.63/1.50 ! [B: a,A: a,C: a] :
% 2.63/1.50 ( ( ord_less_eq_a @ B @ A )
% 2.63/1.50 => ( ( ord_less_eq_a @ C @ B )
% 2.63/1.50 => ( ord_less_eq_a @ C @ A ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_62_linorder__wlog,axiom,
% 2.63/1.50 ! [P: a > a > $o,A: a,B: a] :
% 2.63/1.50 ( ! [A4: a,B3: a] :
% 2.63/1.50 ( ( ord_less_eq_a @ A4 @ B3 )
% 2.63/1.50 => ( P @ A4 @ B3 ) )
% 2.63/1.50 => ( ! [A4: a,B3: a] :
% 2.63/1.50 ( ( P @ B3 @ A4 )
% 2.63/1.50 => ( P @ A4 @ B3 ) )
% 2.63/1.50 => ( P @ A @ B ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_63_dual__order_Orefl,axiom,
% 2.63/1.50 ! [A: $o > a] : ( ord_less_eq_o_a @ A @ A ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_64_dual__order_Orefl,axiom,
% 2.63/1.50 ! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_65_order__trans,axiom,
% 2.63/1.50 ! [X: $o > a,Y: $o > a,Z2: $o > a] :
% 2.63/1.50 ( ( ord_less_eq_o_a @ X @ Y )
% 2.63/1.50 => ( ( ord_less_eq_o_a @ Y @ Z2 )
% 2.63/1.50 => ( ord_less_eq_o_a @ X @ Z2 ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_66_order__trans,axiom,
% 2.63/1.50 ! [X: a,Y: a,Z2: a] :
% 2.63/1.50 ( ( ord_less_eq_a @ X @ Y )
% 2.63/1.50 => ( ( ord_less_eq_a @ Y @ Z2 )
% 2.63/1.50 => ( ord_less_eq_a @ X @ Z2 ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_67_order__class_Oorder_Oantisym,axiom,
% 2.63/1.50 ! [A: $o > a,B: $o > a] :
% 2.63/1.50 ( ( ord_less_eq_o_a @ A @ B )
% 2.63/1.50 => ( ( ord_less_eq_o_a @ B @ A )
% 2.63/1.50 => ( A = B ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_68_order__class_Oorder_Oantisym,axiom,
% 2.63/1.50 ! [A: a,B: a] :
% 2.63/1.50 ( ( ord_less_eq_a @ A @ B )
% 2.63/1.50 => ( ( ord_less_eq_a @ B @ A )
% 2.63/1.50 => ( A = B ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_69_ord__le__eq__trans,axiom,
% 2.63/1.50 ! [A: $o > a,B: $o > a,C: $o > a] :
% 2.63/1.50 ( ( ord_less_eq_o_a @ A @ B )
% 2.63/1.50 => ( ( B = C )
% 2.63/1.50 => ( ord_less_eq_o_a @ A @ C ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_70_ord__le__eq__trans,axiom,
% 2.63/1.50 ! [A: a,B: a,C: a] :
% 2.63/1.50 ( ( ord_less_eq_a @ A @ B )
% 2.63/1.50 => ( ( B = C )
% 2.63/1.50 => ( ord_less_eq_a @ A @ C ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_71_ord__eq__le__trans,axiom,
% 2.63/1.50 ! [A: $o > a,B: $o > a,C: $o > a] :
% 2.63/1.50 ( ( A = B )
% 2.63/1.50 => ( ( ord_less_eq_o_a @ B @ C )
% 2.63/1.50 => ( ord_less_eq_o_a @ A @ C ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_72_ord__eq__le__trans,axiom,
% 2.63/1.50 ! [A: a,B: a,C: a] :
% 2.63/1.50 ( ( A = B )
% 2.63/1.50 => ( ( ord_less_eq_a @ B @ C )
% 2.63/1.50 => ( ord_less_eq_a @ A @ C ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_73_order__class_Oorder_Oeq__iff,axiom,
% 2.63/1.50 ( ( ^ [Y3: $o > a,Z: $o > a] : ( Y3 = Z ) )
% 2.63/1.50 = ( ^ [A3: $o > a,B2: $o > a] :
% 2.63/1.50 ( ( ord_less_eq_o_a @ A3 @ B2 )
% 2.63/1.50 & ( ord_less_eq_o_a @ B2 @ A3 ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_74_order__class_Oorder_Oeq__iff,axiom,
% 2.63/1.50 ( ( ^ [Y3: a,Z: a] : ( Y3 = Z ) )
% 2.63/1.50 = ( ^ [A3: a,B2: a] :
% 2.63/1.50 ( ( ord_less_eq_a @ A3 @ B2 )
% 2.63/1.50 & ( ord_less_eq_a @ B2 @ A3 ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_75_antisym__conv,axiom,
% 2.63/1.50 ! [Y: $o > a,X: $o > a] :
% 2.63/1.50 ( ( ord_less_eq_o_a @ Y @ X )
% 2.63/1.50 => ( ( ord_less_eq_o_a @ X @ Y )
% 2.63/1.50 = ( X = Y ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_76_antisym__conv,axiom,
% 2.63/1.50 ! [Y: a,X: a] :
% 2.63/1.50 ( ( ord_less_eq_a @ Y @ X )
% 2.63/1.50 => ( ( ord_less_eq_a @ X @ Y )
% 2.63/1.50 = ( X = Y ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_77_le__cases3,axiom,
% 2.63/1.50 ! [X: a,Y: a,Z2: a] :
% 2.63/1.50 ( ( ( ord_less_eq_a @ X @ Y )
% 2.63/1.50 => ~ ( ord_less_eq_a @ Y @ Z2 ) )
% 2.63/1.50 => ( ( ( ord_less_eq_a @ Y @ X )
% 2.63/1.50 => ~ ( ord_less_eq_a @ X @ Z2 ) )
% 2.63/1.50 => ( ( ( ord_less_eq_a @ X @ Z2 )
% 2.63/1.50 => ~ ( ord_less_eq_a @ Z2 @ Y ) )
% 2.63/1.50 => ( ( ( ord_less_eq_a @ Z2 @ Y )
% 2.63/1.50 => ~ ( ord_less_eq_a @ Y @ X ) )
% 2.63/1.50 => ( ( ( ord_less_eq_a @ Y @ Z2 )
% 2.63/1.50 => ~ ( ord_less_eq_a @ Z2 @ X ) )
% 2.63/1.50 => ~ ( ( ord_less_eq_a @ Z2 @ X )
% 2.63/1.50 => ~ ( ord_less_eq_a @ X @ Y ) ) ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_78_order_Otrans,axiom,
% 2.63/1.50 ! [A: $o > a,B: $o > a,C: $o > a] :
% 2.63/1.50 ( ( ord_less_eq_o_a @ A @ B )
% 2.63/1.50 => ( ( ord_less_eq_o_a @ B @ C )
% 2.63/1.50 => ( ord_less_eq_o_a @ A @ C ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_79_order_Otrans,axiom,
% 2.63/1.50 ! [A: a,B: a,C: a] :
% 2.63/1.50 ( ( ord_less_eq_a @ A @ B )
% 2.63/1.50 => ( ( ord_less_eq_a @ B @ C )
% 2.63/1.50 => ( ord_less_eq_a @ A @ C ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_80_le__cases,axiom,
% 2.63/1.50 ! [X: a,Y: a] :
% 2.63/1.50 ( ~ ( ord_less_eq_a @ X @ Y )
% 2.63/1.50 => ( ord_less_eq_a @ Y @ X ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_81_eq__refl,axiom,
% 2.63/1.50 ! [X: $o > a,Y: $o > a] :
% 2.63/1.50 ( ( X = Y )
% 2.63/1.50 => ( ord_less_eq_o_a @ X @ Y ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_82_eq__refl,axiom,
% 2.63/1.50 ! [X: a,Y: a] :
% 2.63/1.50 ( ( X = Y )
% 2.63/1.50 => ( ord_less_eq_a @ X @ Y ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_83_linear,axiom,
% 2.63/1.50 ! [X: a,Y: a] :
% 2.63/1.50 ( ( ord_less_eq_a @ X @ Y )
% 2.63/1.50 | ( ord_less_eq_a @ Y @ X ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_84_antisym,axiom,
% 2.63/1.50 ! [X: $o > a,Y: $o > a] :
% 2.63/1.50 ( ( ord_less_eq_o_a @ X @ Y )
% 2.63/1.50 => ( ( ord_less_eq_o_a @ Y @ X )
% 2.63/1.50 => ( X = Y ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_85_antisym,axiom,
% 2.63/1.50 ! [X: a,Y: a] :
% 2.63/1.50 ( ( ord_less_eq_a @ X @ Y )
% 2.63/1.50 => ( ( ord_less_eq_a @ Y @ X )
% 2.63/1.50 => ( X = Y ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_86_eq__iff,axiom,
% 2.63/1.50 ( ( ^ [Y3: $o > a,Z: $o > a] : ( Y3 = Z ) )
% 2.63/1.50 = ( ^ [X4: $o > a,Y4: $o > a] :
% 2.63/1.50 ( ( ord_less_eq_o_a @ X4 @ Y4 )
% 2.63/1.50 & ( ord_less_eq_o_a @ Y4 @ X4 ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_87_eq__iff,axiom,
% 2.63/1.50 ( ( ^ [Y3: a,Z: a] : ( Y3 = Z ) )
% 2.63/1.50 = ( ^ [X4: a,Y4: a] :
% 2.63/1.50 ( ( ord_less_eq_a @ X4 @ Y4 )
% 2.63/1.50 & ( ord_less_eq_a @ Y4 @ X4 ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_88_ord__le__eq__subst,axiom,
% 2.63/1.50 ! [A: a,B: a,F: a > $o > a,C: $o > a] :
% 2.63/1.50 ( ( ord_less_eq_a @ A @ B )
% 2.63/1.50 => ( ( ( F @ B )
% 2.63/1.50 = C )
% 2.63/1.50 => ( ! [X5: a,Y2: a] :
% 2.63/1.50 ( ( ord_less_eq_a @ X5 @ Y2 )
% 2.63/1.50 => ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 2.63/1.50 => ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_89_ord__le__eq__subst,axiom,
% 2.63/1.50 ! [A: $o > a,B: $o > a,F: ( $o > a ) > a,C: a] :
% 2.63/1.50 ( ( ord_less_eq_o_a @ A @ B )
% 2.63/1.50 => ( ( ( F @ B )
% 2.63/1.50 = C )
% 2.63/1.50 => ( ! [X5: $o > a,Y2: $o > a] :
% 2.63/1.50 ( ( ord_less_eq_o_a @ X5 @ Y2 )
% 2.63/1.50 => ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 2.63/1.50 => ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_90_ord__le__eq__subst,axiom,
% 2.63/1.50 ! [A: $o > a,B: $o > a,F: ( $o > a ) > $o > a,C: $o > a] :
% 2.63/1.50 ( ( ord_less_eq_o_a @ A @ B )
% 2.63/1.50 => ( ( ( F @ B )
% 2.63/1.50 = C )
% 2.63/1.50 => ( ! [X5: $o > a,Y2: $o > a] :
% 2.63/1.50 ( ( ord_less_eq_o_a @ X5 @ Y2 )
% 2.63/1.50 => ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 2.63/1.50 => ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_91_ord__le__eq__subst,axiom,
% 2.63/1.50 ! [A: a,B: a,F: a > a,C: a] :
% 2.63/1.50 ( ( ord_less_eq_a @ A @ B )
% 2.63/1.50 => ( ( ( F @ B )
% 2.63/1.50 = C )
% 2.63/1.50 => ( ! [X5: a,Y2: a] :
% 2.63/1.50 ( ( ord_less_eq_a @ X5 @ Y2 )
% 2.63/1.50 => ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 2.63/1.50 => ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_92_ord__eq__le__subst,axiom,
% 2.63/1.50 ! [A: $o > a,F: a > $o > a,B: a,C: a] :
% 2.63/1.50 ( ( A
% 2.63/1.50 = ( F @ B ) )
% 2.63/1.50 => ( ( ord_less_eq_a @ B @ C )
% 2.63/1.50 => ( ! [X5: a,Y2: a] :
% 2.63/1.50 ( ( ord_less_eq_a @ X5 @ Y2 )
% 2.63/1.50 => ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 2.63/1.50 => ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_93_ord__eq__le__subst,axiom,
% 2.63/1.50 ! [A: a,F: ( $o > a ) > a,B: $o > a,C: $o > a] :
% 2.63/1.50 ( ( A
% 2.63/1.50 = ( F @ B ) )
% 2.63/1.50 => ( ( ord_less_eq_o_a @ B @ C )
% 2.63/1.50 => ( ! [X5: $o > a,Y2: $o > a] :
% 2.63/1.50 ( ( ord_less_eq_o_a @ X5 @ Y2 )
% 2.63/1.50 => ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 2.63/1.50 => ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_94_ord__eq__le__subst,axiom,
% 2.63/1.50 ! [A: $o > a,F: ( $o > a ) > $o > a,B: $o > a,C: $o > a] :
% 2.63/1.50 ( ( A
% 2.63/1.50 = ( F @ B ) )
% 2.63/1.50 => ( ( ord_less_eq_o_a @ B @ C )
% 2.63/1.50 => ( ! [X5: $o > a,Y2: $o > a] :
% 2.63/1.50 ( ( ord_less_eq_o_a @ X5 @ Y2 )
% 2.63/1.50 => ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 2.63/1.50 => ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_95_ord__eq__le__subst,axiom,
% 2.63/1.50 ! [A: a,F: a > a,B: a,C: a] :
% 2.63/1.50 ( ( A
% 2.63/1.50 = ( F @ B ) )
% 2.63/1.50 => ( ( ord_less_eq_a @ B @ C )
% 2.63/1.50 => ( ! [X5: a,Y2: a] :
% 2.63/1.50 ( ( ord_less_eq_a @ X5 @ Y2 )
% 2.63/1.50 => ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
% 2.63/1.50 => ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_96_heap__top__max,axiom,
% 2.63/1.50 ! [T: tree_a] :
% 2.63/1.50 ( ( T != e_a )
% 2.63/1.50 => ( ( is_heap_a @ T )
% 2.63/1.50 => ( ( val_a @ T )
% 2.63/1.50 = ( lattic146396397_Max_a @ ( set_mset_a @ ( multiset_a2 @ T ) ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_97_of__list__tree_Osimps_I2_J,axiom,
% 2.63/1.50 ! [V: a,Tail: list_a] :
% 2.63/1.50 ( ( heapIm1637418125tree_a @ ( cons_a @ V @ Tail ) )
% 2.63/1.50 = ( t_a @ V @ ( heapIm1637418125tree_a @ Tail ) @ e_a ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_98_Greatest__equality,axiom,
% 2.63/1.50 ! [P: ( $o > a ) > $o,X: $o > a] :
% 2.63/1.50 ( ( P @ X )
% 2.63/1.50 => ( ! [Y2: $o > a] :
% 2.63/1.50 ( ( P @ Y2 )
% 2.63/1.50 => ( ord_less_eq_o_a @ Y2 @ X ) )
% 2.63/1.50 => ( ( order_Greatest_o_a @ P )
% 2.63/1.50 = X ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_99_Greatest__equality,axiom,
% 2.63/1.50 ! [P: a > $o,X: a] :
% 2.63/1.50 ( ( P @ X )
% 2.63/1.50 => ( ! [Y2: a] :
% 2.63/1.50 ( ( P @ Y2 )
% 2.63/1.50 => ( ord_less_eq_a @ Y2 @ X ) )
% 2.63/1.50 => ( ( order_Greatest_a @ P )
% 2.63/1.50 = X ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_100_GreatestI2__order,axiom,
% 2.63/1.50 ! [P: ( $o > a ) > $o,X: $o > a,Q: ( $o > a ) > $o] :
% 2.63/1.50 ( ( P @ X )
% 2.63/1.50 => ( ! [Y2: $o > a] :
% 2.63/1.50 ( ( P @ Y2 )
% 2.63/1.50 => ( ord_less_eq_o_a @ Y2 @ X ) )
% 2.63/1.50 => ( ! [X5: $o > a] :
% 2.63/1.50 ( ( P @ X5 )
% 2.63/1.50 => ( ! [Y5: $o > a] :
% 2.63/1.50 ( ( P @ Y5 )
% 2.63/1.50 => ( ord_less_eq_o_a @ Y5 @ X5 ) )
% 2.63/1.50 => ( Q @ X5 ) ) )
% 2.63/1.50 => ( Q @ ( order_Greatest_o_a @ P ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_101_GreatestI2__order,axiom,
% 2.63/1.50 ! [P: a > $o,X: a,Q: a > $o] :
% 2.63/1.50 ( ( P @ X )
% 2.63/1.50 => ( ! [Y2: a] :
% 2.63/1.50 ( ( P @ Y2 )
% 2.63/1.50 => ( ord_less_eq_a @ Y2 @ X ) )
% 2.63/1.50 => ( ! [X5: a] :
% 2.63/1.50 ( ( P @ X5 )
% 2.63/1.50 => ( ! [Y5: a] :
% 2.63/1.50 ( ( P @ Y5 )
% 2.63/1.50 => ( ord_less_eq_a @ Y5 @ X5 ) )
% 2.63/1.50 => ( Q @ X5 ) ) )
% 2.63/1.50 => ( Q @ ( order_Greatest_a @ P ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_102_le__rel__bool__arg__iff,axiom,
% 2.63/1.50 ( ord_less_eq_o_o_a
% 2.63/1.50 = ( ^ [X6: $o > $o > a,Y6: $o > $o > a] :
% 2.63/1.50 ( ( ord_less_eq_o_a @ ( X6 @ $false ) @ ( Y6 @ $false ) )
% 2.63/1.50 & ( ord_less_eq_o_a @ ( X6 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_103_le__rel__bool__arg__iff,axiom,
% 2.63/1.50 ( ord_less_eq_o_a
% 2.63/1.50 = ( ^ [X6: $o > a,Y6: $o > a] :
% 2.63/1.50 ( ( ord_less_eq_a @ ( X6 @ $false ) @ ( Y6 @ $false ) )
% 2.63/1.50 & ( ord_less_eq_a @ ( X6 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_104_verit__la__disequality,axiom,
% 2.63/1.50 ! [A: a,B: a] :
% 2.63/1.50 ( ( A = B )
% 2.63/1.50 | ~ ( ord_less_eq_a @ A @ B )
% 2.63/1.50 | ~ ( ord_less_eq_a @ B @ A ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_105_list_Oinject,axiom,
% 2.63/1.50 ! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
% 2.63/1.50 ( ( ( cons_a @ X21 @ X22 )
% 2.63/1.50 = ( cons_a @ Y21 @ Y22 ) )
% 2.63/1.50 = ( ( X21 = Y21 )
% 2.63/1.50 & ( X22 = Y22 ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_106_not__Cons__self2,axiom,
% 2.63/1.50 ! [X: a,Xs: list_a] :
% 2.63/1.50 ( ( cons_a @ X @ Xs )
% 2.63/1.50 != Xs ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_107_Heap__axioms__def,axiom,
% 2.63/1.50 ( heap_axioms_Tree_a_a
% 2.63/1.50 = ( ^ [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] :
% 2.63/1.50 ( ! [L3: tree_a] :
% 2.63/1.50 ( ( Multiset @ L3 )
% 2.63/1.50 = ( multiset_a2 @ ( As_tree @ L3 ) ) )
% 2.63/1.50 & ! [I: list_a] : ( is_heap_a @ ( As_tree @ ( Of_list @ I ) ) )
% 2.63/1.50 & ! [T2: tree_a] :
% 2.63/1.50 ( ( ( As_tree @ T2 )
% 2.63/1.50 = e_a )
% 2.63/1.50 = ( Is_empty @ T2 ) )
% 2.63/1.50 & ! [L3: tree_a,M: a,L4: tree_a] :
% 2.63/1.50 ( ~ ( Is_empty @ L3 )
% 2.63/1.50 => ( ( ( produc686083979Tree_a @ M @ L4 )
% 2.63/1.50 = ( Remove_max @ L3 ) )
% 2.63/1.50 => ( ( add_mset_a @ M @ ( Multiset @ L4 ) )
% 2.63/1.50 = ( Multiset @ L3 ) ) ) )
% 2.63/1.50 & ! [L3: tree_a,M: a,L4: tree_a] :
% 2.63/1.50 ( ~ ( Is_empty @ L3 )
% 2.63/1.50 => ( ( is_heap_a @ ( As_tree @ L3 ) )
% 2.63/1.50 => ( ( ( produc686083979Tree_a @ M @ L4 )
% 2.63/1.50 = ( Remove_max @ L3 ) )
% 2.63/1.50 => ( is_heap_a @ ( As_tree @ L4 ) ) ) ) )
% 2.63/1.50 & ! [T2: tree_a,M: a,T3: tree_a] :
% 2.63/1.50 ( ~ ( Is_empty @ T2 )
% 2.63/1.50 => ( ( ( produc686083979Tree_a @ M @ T3 )
% 2.63/1.50 = ( Remove_max @ T2 ) )
% 2.63/1.50 => ( M
% 2.63/1.50 = ( val_a @ ( As_tree @ T2 ) ) ) ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_108_Heap__axioms_Ointro,axiom,
% 2.63/1.50 ! [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] :
% 2.63/1.50 ( ! [L5: tree_a] :
% 2.63/1.50 ( ( Multiset2 @ L5 )
% 2.63/1.50 = ( multiset_a2 @ ( As_tree2 @ L5 ) ) )
% 2.63/1.50 => ( ! [I2: list_a] : ( is_heap_a @ ( As_tree2 @ ( Of_list2 @ I2 ) ) )
% 2.63/1.50 => ( ! [T4: tree_a] :
% 2.63/1.50 ( ( ( As_tree2 @ T4 )
% 2.63/1.50 = e_a )
% 2.63/1.50 = ( Is_empty2 @ T4 ) )
% 2.63/1.50 => ( ! [L5: tree_a,M2: a,L2: tree_a] :
% 2.63/1.50 ( ~ ( Is_empty2 @ L5 )
% 2.63/1.50 => ( ( ( produc686083979Tree_a @ M2 @ L2 )
% 2.63/1.50 = ( Remove_max2 @ L5 ) )
% 2.63/1.50 => ( ( add_mset_a @ M2 @ ( Multiset2 @ L2 ) )
% 2.63/1.50 = ( Multiset2 @ L5 ) ) ) )
% 2.63/1.50 => ( ! [L5: tree_a,M2: a,L2: tree_a] :
% 2.63/1.50 ( ~ ( Is_empty2 @ L5 )
% 2.63/1.50 => ( ( is_heap_a @ ( As_tree2 @ L5 ) )
% 2.63/1.50 => ( ( ( produc686083979Tree_a @ M2 @ L2 )
% 2.63/1.50 = ( Remove_max2 @ L5 ) )
% 2.63/1.50 => ( is_heap_a @ ( As_tree2 @ L2 ) ) ) ) )
% 2.63/1.50 => ( ! [T4: tree_a,M2: a,T5: tree_a] :
% 2.63/1.50 ( ~ ( Is_empty2 @ T4 )
% 2.63/1.50 => ( ( ( produc686083979Tree_a @ M2 @ T5 )
% 2.63/1.50 = ( Remove_max2 @ T4 ) )
% 2.63/1.50 => ( M2
% 2.63/1.50 = ( val_a @ ( As_tree2 @ T4 ) ) ) ) )
% 2.63/1.50 => ( heap_axioms_Tree_a_a @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 ) ) ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_109_Heap_Oremove__max__multiset_H,axiom,
% 2.63/1.50 ! [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,M3: a,L6: tree_a] :
% 2.63/1.50 ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 2.63/1.50 => ( ~ ( Is_empty2 @ L )
% 2.63/1.50 => ( ( ( produc686083979Tree_a @ M3 @ L6 )
% 2.63/1.50 = ( Remove_max2 @ L ) )
% 2.63/1.50 => ( ( add_mset_a @ M3 @ ( Multiset2 @ L6 ) )
% 2.63/1.50 = ( Multiset2 @ L ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_110_Heap_Oremove__max__val,axiom,
% 2.63/1.50 ! [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,T: tree_a,M3: a,T6: tree_a] :
% 2.63/1.50 ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 2.63/1.50 => ( ~ ( Is_empty2 @ T )
% 2.63/1.50 => ( ( ( produc686083979Tree_a @ M3 @ T6 )
% 2.63/1.50 = ( Remove_max2 @ T ) )
% 2.63/1.50 => ( M3
% 2.63/1.50 = ( val_a @ ( As_tree2 @ T ) ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_111_Heap_Oremove__max__is__heap,axiom,
% 2.63/1.50 ! [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,M3: a,L6: tree_a] :
% 2.63/1.50 ( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
% 2.63/1.50 => ( ~ ( Is_empty2 @ L )
% 2.63/1.50 => ( ( is_heap_a @ ( As_tree2 @ L ) )
% 2.63/1.50 => ( ( ( produc686083979Tree_a @ M3 @ L6 )
% 2.63/1.50 = ( Remove_max2 @ L ) )
% 2.63/1.50 => ( is_heap_a @ ( As_tree2 @ L6 ) ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_112_multi__self__add__other__not__self,axiom,
% 2.63/1.50 ! [M4: multiset_a,X: a] :
% 2.63/1.50 ( M4
% 2.63/1.50 != ( add_mset_a @ X @ M4 ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_113_add__mset__add__mset__same__iff,axiom,
% 2.63/1.50 ! [A: a,A2: multiset_a,B4: multiset_a] :
% 2.63/1.50 ( ( ( add_mset_a @ A @ A2 )
% 2.63/1.50 = ( add_mset_a @ A @ B4 ) )
% 2.63/1.50 = ( A2 = B4 ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_114_old_Oprod_Oinject,axiom,
% 2.63/1.50 ! [A: a,B: tree_a,A5: a,B5: tree_a] :
% 2.63/1.50 ( ( ( produc686083979Tree_a @ A @ B )
% 2.63/1.50 = ( produc686083979Tree_a @ A5 @ B5 ) )
% 2.63/1.50 = ( ( A = A5 )
% 2.63/1.50 & ( B = B5 ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_115_prod_Oinject,axiom,
% 2.63/1.50 ! [X12: a,X24: tree_a,Y1: a,Y24: tree_a] :
% 2.63/1.50 ( ( ( produc686083979Tree_a @ X12 @ X24 )
% 2.63/1.50 = ( produc686083979Tree_a @ Y1 @ Y24 ) )
% 2.63/1.50 = ( ( X12 = Y1 )
% 2.63/1.50 & ( X24 = Y24 ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_116_mset__add,axiom,
% 2.63/1.50 ! [A: a,A2: multiset_a] :
% 2.63/1.50 ( ( member_a @ A @ ( set_mset_a @ A2 ) )
% 2.63/1.50 => ~ ! [B6: multiset_a] :
% 2.63/1.50 ( A2
% 2.63/1.50 != ( add_mset_a @ A @ B6 ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_117_multi__member__split,axiom,
% 2.63/1.50 ! [X: a,M4: multiset_a] :
% 2.63/1.50 ( ( member_a @ X @ ( set_mset_a @ M4 ) )
% 2.63/1.50 => ? [A6: multiset_a] :
% 2.63/1.50 ( M4
% 2.63/1.50 = ( add_mset_a @ X @ A6 ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_118_surj__pair,axiom,
% 2.63/1.50 ! [P2: produc143150363Tree_a] :
% 2.63/1.50 ? [X5: a,Y2: tree_a] :
% 2.63/1.50 ( P2
% 2.63/1.50 = ( produc686083979Tree_a @ X5 @ Y2 ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_119_prod__cases,axiom,
% 2.63/1.50 ! [P: produc143150363Tree_a > $o,P2: produc143150363Tree_a] :
% 2.63/1.50 ( ! [A4: a,B3: tree_a] : ( P @ ( produc686083979Tree_a @ A4 @ B3 ) )
% 2.63/1.50 => ( P @ P2 ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_120_Pair__inject,axiom,
% 2.63/1.50 ! [A: a,B: tree_a,A5: a,B5: tree_a] :
% 2.63/1.50 ( ( ( produc686083979Tree_a @ A @ B )
% 2.63/1.50 = ( produc686083979Tree_a @ A5 @ B5 ) )
% 2.63/1.50 => ~ ( ( A = A5 )
% 2.63/1.50 => ( B != B5 ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_121_old_Oprod_Oexhaust,axiom,
% 2.63/1.50 ! [Y: produc143150363Tree_a] :
% 2.63/1.50 ~ ! [A4: a,B3: tree_a] :
% 2.63/1.50 ( Y
% 2.63/1.50 != ( produc686083979Tree_a @ A4 @ B3 ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_122_old_Oprod_Oinducts,axiom,
% 2.63/1.50 ! [P: produc143150363Tree_a > $o,Prod: produc143150363Tree_a] :
% 2.63/1.50 ( ! [A4: a,B3: tree_a] : ( P @ ( produc686083979Tree_a @ A4 @ B3 ) )
% 2.63/1.50 => ( P @ Prod ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_123_add__eq__conv__ex,axiom,
% 2.63/1.50 ! [A: a,M4: multiset_a,B: a,N: multiset_a] :
% 2.63/1.50 ( ( ( add_mset_a @ A @ M4 )
% 2.63/1.50 = ( add_mset_a @ B @ N ) )
% 2.63/1.50 = ( ( ( M4 = N )
% 2.63/1.50 & ( A = B ) )
% 2.63/1.50 | ? [K: multiset_a] :
% 2.63/1.50 ( ( M4
% 2.63/1.50 = ( add_mset_a @ B @ K ) )
% 2.63/1.50 & ( N
% 2.63/1.50 = ( add_mset_a @ A @ K ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_124_add__mset__commute,axiom,
% 2.63/1.50 ! [X: a,Y: a,M4: multiset_a] :
% 2.63/1.50 ( ( add_mset_a @ X @ ( add_mset_a @ Y @ M4 ) )
% 2.63/1.50 = ( add_mset_a @ Y @ ( add_mset_a @ X @ M4 ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_125_union__single__eq__member,axiom,
% 2.63/1.50 ! [X: a,M4: multiset_a,N: multiset_a] :
% 2.63/1.50 ( ( ( add_mset_a @ X @ M4 )
% 2.63/1.50 = N )
% 2.63/1.50 => ( member_a @ X @ ( set_mset_a @ N ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_126_insert__noteq__member,axiom,
% 2.63/1.50 ! [B: a,B4: multiset_a,C: a,C2: multiset_a] :
% 2.63/1.50 ( ( ( add_mset_a @ B @ B4 )
% 2.63/1.50 = ( add_mset_a @ C @ C2 ) )
% 2.63/1.50 => ( ( B != C )
% 2.63/1.50 => ( member_a @ C @ ( set_mset_a @ B4 ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_127_removeLeaf_Osimps_I1_J,axiom,
% 2.63/1.50 ! [V: a] :
% 2.63/1.50 ( ( heapIm837449470Leaf_a @ ( t_a @ V @ e_a @ e_a ) )
% 2.63/1.50 = ( produc686083979Tree_a @ V @ e_a ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_128_RemoveMax__axioms__def,axiom,
% 2.63/1.50 ( remove301631099ee_a_a
% 2.63/1.50 = ( ^ [Is_empty: tree_a > $o,Of_list: list_a > tree_a,Multiset: tree_a > multiset_a,Remove_max: tree_a > produc143150363Tree_a,Inv: tree_a > $o] :
% 2.63/1.50 ( ! [X4: list_a] : ( Inv @ ( Of_list @ X4 ) )
% 2.63/1.50 & ! [L3: tree_a,M: a,L4: tree_a] :
% 2.63/1.50 ( ~ ( Is_empty @ L3 )
% 2.63/1.50 => ( ( Inv @ L3 )
% 2.63/1.50 => ( ( ( produc686083979Tree_a @ M @ L4 )
% 2.63/1.50 = ( Remove_max @ L3 ) )
% 2.63/1.50 => ( M
% 2.63/1.50 = ( lattic146396397_Max_a @ ( set_Tree_a_a @ Multiset @ L3 ) ) ) ) ) )
% 2.63/1.50 & ! [L3: tree_a,M: a,L4: tree_a] :
% 2.63/1.50 ( ~ ( Is_empty @ L3 )
% 2.63/1.50 => ( ( Inv @ L3 )
% 2.63/1.50 => ( ( ( produc686083979Tree_a @ M @ L4 )
% 2.63/1.50 = ( Remove_max @ L3 ) )
% 2.63/1.50 => ( ( add_mset_a @ M @ ( Multiset @ L4 ) )
% 2.63/1.50 = ( Multiset @ L3 ) ) ) ) )
% 2.63/1.50 & ! [L3: tree_a,M: a,L4: tree_a] :
% 2.63/1.50 ( ~ ( Is_empty @ L3 )
% 2.63/1.50 => ( ( Inv @ L3 )
% 2.63/1.50 => ( ( ( produc686083979Tree_a @ M @ L4 )
% 2.63/1.50 = ( Remove_max @ L3 ) )
% 2.63/1.50 => ( Inv @ L4 ) ) ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_129_RemoveMax__axioms_Ointro,axiom,
% 2.63/1.50 ! [Inv2: tree_a > $o,Of_list2: list_a > tree_a,Is_empty2: tree_a > $o,Remove_max2: tree_a > produc143150363Tree_a,Multiset2: tree_a > multiset_a] :
% 2.63/1.50 ( ! [X5: list_a] : ( Inv2 @ ( Of_list2 @ X5 ) )
% 2.63/1.50 => ( ! [L5: tree_a,M2: a,L2: tree_a] :
% 2.63/1.50 ( ~ ( Is_empty2 @ L5 )
% 2.63/1.50 => ( ( Inv2 @ L5 )
% 2.63/1.50 => ( ( ( produc686083979Tree_a @ M2 @ L2 )
% 2.63/1.50 = ( Remove_max2 @ L5 ) )
% 2.63/1.50 => ( M2
% 2.63/1.50 = ( lattic146396397_Max_a @ ( set_Tree_a_a @ Multiset2 @ L5 ) ) ) ) ) )
% 2.63/1.50 => ( ! [L5: tree_a,M2: a,L2: tree_a] :
% 2.63/1.50 ( ~ ( Is_empty2 @ L5 )
% 2.63/1.50 => ( ( Inv2 @ L5 )
% 2.63/1.50 => ( ( ( produc686083979Tree_a @ M2 @ L2 )
% 2.63/1.50 = ( Remove_max2 @ L5 ) )
% 2.63/1.50 => ( ( add_mset_a @ M2 @ ( Multiset2 @ L2 ) )
% 2.63/1.50 = ( Multiset2 @ L5 ) ) ) ) )
% 2.63/1.50 => ( ! [L5: tree_a,M2: a,L2: tree_a] :
% 2.63/1.50 ( ~ ( Is_empty2 @ L5 )
% 2.63/1.50 => ( ( Inv2 @ L5 )
% 2.63/1.50 => ( ( ( produc686083979Tree_a @ M2 @ L2 )
% 2.63/1.50 = ( Remove_max2 @ L5 ) )
% 2.63/1.50 => ( Inv2 @ L2 ) ) ) )
% 2.63/1.50 => ( remove301631099ee_a_a @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv2 ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_130_RemoveMax_Oremove__max__max,axiom,
% 2.63/1.50 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,Remove_max2: tree_a > produc143150363Tree_a,Inv2: tree_a > $o,L: tree_a,M3: a,L6: tree_a] :
% 2.63/1.50 ( ( removeMax_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv2 )
% 2.63/1.50 => ( ~ ( Is_empty2 @ L )
% 2.63/1.50 => ( ( Inv2 @ L )
% 2.63/1.50 => ( ( ( produc686083979Tree_a @ M3 @ L6 )
% 2.63/1.50 = ( Remove_max2 @ L ) )
% 2.63/1.50 => ( M3
% 2.63/1.50 = ( lattic146396397_Max_a @ ( set_Tree_a_a @ Multiset2 @ L ) ) ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_131_multiset__induct__max,axiom,
% 2.63/1.50 ! [P: multiset_a > $o,M4: multiset_a] :
% 2.63/1.50 ( ( P @ zero_zero_multiset_a )
% 2.63/1.50 => ( ! [X5: a,M5: multiset_a] :
% 2.63/1.50 ( ( P @ M5 )
% 2.63/1.50 => ( ! [Xa: a] :
% 2.63/1.50 ( ( member_a @ Xa @ ( set_mset_a @ M5 ) )
% 2.63/1.50 => ( ord_less_eq_a @ Xa @ X5 ) )
% 2.63/1.50 => ( P @ ( add_mset_a @ X5 @ M5 ) ) ) )
% 2.63/1.50 => ( P @ M4 ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_132_add__mset__eq__singleton__iff,axiom,
% 2.63/1.50 ! [X: a,M4: multiset_a,Y: a] :
% 2.63/1.50 ( ( ( add_mset_a @ X @ M4 )
% 2.63/1.50 = ( add_mset_a @ Y @ zero_zero_multiset_a ) )
% 2.63/1.50 = ( ( M4 = zero_zero_multiset_a )
% 2.63/1.50 & ( X = Y ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_133_single__eq__add__mset,axiom,
% 2.63/1.50 ! [A: a,B: a,M4: multiset_a] :
% 2.63/1.50 ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
% 2.63/1.50 = ( add_mset_a @ B @ M4 ) )
% 2.63/1.50 = ( ( B = A )
% 2.63/1.50 & ( M4 = zero_zero_multiset_a ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_134_add__mset__eq__single,axiom,
% 2.63/1.50 ! [B: a,M4: multiset_a,A: a] :
% 2.63/1.50 ( ( ( add_mset_a @ B @ M4 )
% 2.63/1.50 = ( add_mset_a @ A @ zero_zero_multiset_a ) )
% 2.63/1.50 = ( ( B = A )
% 2.63/1.50 & ( M4 = zero_zero_multiset_a ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_135_single__eq__single,axiom,
% 2.63/1.50 ! [A: a,B: a] :
% 2.63/1.50 ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
% 2.63/1.50 = ( add_mset_a @ B @ zero_zero_multiset_a ) )
% 2.63/1.50 = ( A = B ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_136_multiset__cases,axiom,
% 2.63/1.50 ! [M4: multiset_a] :
% 2.63/1.50 ( ( M4 != zero_zero_multiset_a )
% 2.63/1.50 => ~ ! [X5: a,N2: multiset_a] :
% 2.63/1.50 ( M4
% 2.63/1.50 != ( add_mset_a @ X5 @ N2 ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_137_multiset__induct,axiom,
% 2.63/1.50 ! [P: multiset_a > $o,M4: multiset_a] :
% 2.63/1.50 ( ( P @ zero_zero_multiset_a )
% 2.63/1.50 => ( ! [X5: a,M5: multiset_a] :
% 2.63/1.50 ( ( P @ M5 )
% 2.63/1.50 => ( P @ ( add_mset_a @ X5 @ M5 ) ) )
% 2.63/1.50 => ( P @ M4 ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_138_multiset__induct2,axiom,
% 2.63/1.50 ! [P: multiset_a > multiset_a > $o,M4: multiset_a,N: multiset_a] :
% 2.63/1.50 ( ( P @ zero_zero_multiset_a @ zero_zero_multiset_a )
% 2.63/1.50 => ( ! [A4: a,M5: multiset_a,N2: multiset_a] :
% 2.63/1.50 ( ( P @ M5 @ N2 )
% 2.63/1.50 => ( P @ ( add_mset_a @ A4 @ M5 ) @ N2 ) )
% 2.63/1.50 => ( ! [A4: a,M5: multiset_a,N2: multiset_a] :
% 2.63/1.50 ( ( P @ M5 @ N2 )
% 2.63/1.50 => ( P @ M5 @ ( add_mset_a @ A4 @ N2 ) ) )
% 2.63/1.50 => ( P @ M4 @ N ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_139_empty__not__add__mset,axiom,
% 2.63/1.50 ! [A: a,A2: multiset_a] :
% 2.63/1.50 ( zero_zero_multiset_a
% 2.63/1.50 != ( add_mset_a @ A @ A2 ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_140_multi__nonempty__split,axiom,
% 2.63/1.50 ! [M4: multiset_a] :
% 2.63/1.50 ( ( M4 != zero_zero_multiset_a )
% 2.63/1.50 => ? [A6: multiset_a,A4: a] :
% 2.63/1.50 ( M4
% 2.63/1.50 = ( add_mset_a @ A4 @ A6 ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_141_multiset__nonemptyE,axiom,
% 2.63/1.50 ! [A2: multiset_a] :
% 2.63/1.50 ( ( A2 != zero_zero_multiset_a )
% 2.63/1.50 => ~ ! [X5: a] :
% 2.63/1.50 ~ ( member_a @ X5 @ ( set_mset_a @ A2 ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_142_RemoveMax_Oremove__max__inv,axiom,
% 2.63/1.50 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,Remove_max2: tree_a > produc143150363Tree_a,Inv2: tree_a > $o,L: tree_a,M3: a,L6: tree_a] :
% 2.63/1.50 ( ( removeMax_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv2 )
% 2.63/1.50 => ( ~ ( Is_empty2 @ L )
% 2.63/1.50 => ( ( Inv2 @ L )
% 2.63/1.50 => ( ( ( produc686083979Tree_a @ M3 @ L6 )
% 2.63/1.50 = ( Remove_max2 @ L ) )
% 2.63/1.50 => ( Inv2 @ L6 ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_143_multi__member__last,axiom,
% 2.63/1.50 ! [X: a] : ( member_a @ X @ ( set_mset_a @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_144_RemoveMax_Oremove__max__multiset,axiom,
% 2.63/1.50 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,Remove_max2: tree_a > produc143150363Tree_a,Inv2: tree_a > $o,L: tree_a,M3: a,L6: tree_a] :
% 2.63/1.50 ( ( removeMax_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv2 )
% 2.63/1.50 => ( ~ ( Is_empty2 @ L )
% 2.63/1.50 => ( ( Inv2 @ L )
% 2.63/1.50 => ( ( ( produc686083979Tree_a @ M3 @ L6 )
% 2.63/1.50 = ( Remove_max2 @ L ) )
% 2.63/1.50 => ( ( add_mset_a @ M3 @ ( Multiset2 @ L6 ) )
% 2.63/1.50 = ( Multiset2 @ L ) ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_145_multiset_Osimps_I1_J,axiom,
% 2.63/1.50 ( ( multiset_a2 @ e_a )
% 2.63/1.50 = zero_zero_multiset_a ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_146_multiset__induct__min,axiom,
% 2.63/1.50 ! [P: multiset_a > $o,M4: multiset_a] :
% 2.63/1.50 ( ( P @ zero_zero_multiset_a )
% 2.63/1.50 => ( ! [X5: a,M5: multiset_a] :
% 2.63/1.50 ( ( P @ M5 )
% 2.63/1.50 => ( ! [Xa: a] :
% 2.63/1.50 ( ( member_a @ Xa @ ( set_mset_a @ M5 ) )
% 2.63/1.50 => ( ord_less_eq_a @ X5 @ Xa ) )
% 2.63/1.50 => ( P @ ( add_mset_a @ X5 @ M5 ) ) ) )
% 2.63/1.50 => ( P @ M4 ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_147_Multiset_Ois__empty__def,axiom,
% 2.63/1.50 ( is_empty_a
% 2.63/1.50 = ( ^ [A7: multiset_a] : ( A7 = zero_zero_multiset_a ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_148_RemoveMax_Ossort_HInduct,axiom,
% 2.63/1.50 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,Remove_max2: tree_a > produc143150363Tree_a,Inv2: tree_a > $o,L: tree_a,P: tree_a > list_a > $o,Sl: list_a] :
% 2.63/1.50 ( ( removeMax_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv2 )
% 2.63/1.50 => ( ( Inv2 @ L )
% 2.63/1.50 => ( ( P @ L @ Sl )
% 2.63/1.50 => ( ! [L5: tree_a,Sl2: list_a,M2: a,L2: tree_a] :
% 2.63/1.50 ( ~ ( Is_empty2 @ L5 )
% 2.63/1.50 => ( ( Inv2 @ L5 )
% 2.63/1.50 => ( ( ( produc686083979Tree_a @ M2 @ L2 )
% 2.63/1.50 = ( Remove_max2 @ L5 ) )
% 2.63/1.50 => ( ( P @ L5 @ Sl2 )
% 2.63/1.50 => ( P @ L2 @ ( cons_a @ M2 @ Sl2 ) ) ) ) ) )
% 2.63/1.50 => ( P @ Empty @ ( ssort_Tree_a_a @ Is_empty2 @ Remove_max2 @ L @ Sl ) ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_149_RemoveMax_Ossort_H__dom_Ocases,axiom,
% 2.63/1.50 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,Remove_max2: tree_a > produc143150363Tree_a,Inv2: tree_a > $o,A: produc768687417list_a] :
% 2.63/1.50 ( ( removeMax_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv2 )
% 2.63/1.50 => ( ( ssort_dom_Tree_a_a @ Is_empty2 @ Remove_max2 @ A )
% 2.63/1.50 => ~ ! [L5: tree_a,Sl2: list_a] :
% 2.63/1.50 ( ( A
% 2.63/1.50 = ( produc1352981801list_a @ L5 @ Sl2 ) )
% 2.63/1.50 => ~ ( ~ ( Is_empty2 @ L5 )
% 2.63/1.50 => ! [M6: a,L7: tree_a] :
% 2.63/1.50 ( ( ( produc686083979Tree_a @ M6 @ L7 )
% 2.63/1.50 = ( Remove_max2 @ L5 ) )
% 2.63/1.50 => ( ssort_dom_Tree_a_a @ Is_empty2 @ Remove_max2 @ ( produc1352981801list_a @ L7 @ ( cons_a @ M6 @ Sl2 ) ) ) ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_150_RemoveMax_Ossort_H__dom_Osimps,axiom,
% 2.63/1.50 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,Remove_max2: tree_a > produc143150363Tree_a,Inv2: tree_a > $o,A: produc768687417list_a] :
% 2.63/1.50 ( ( removeMax_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv2 )
% 2.63/1.50 => ( ( ssort_dom_Tree_a_a @ Is_empty2 @ Remove_max2 @ A )
% 2.63/1.50 = ( ? [L3: tree_a,Sl3: list_a] :
% 2.63/1.50 ( ( A
% 2.63/1.50 = ( produc1352981801list_a @ L3 @ Sl3 ) )
% 2.63/1.50 & ! [X4: a,Y4: tree_a] :
% 2.63/1.50 ( ~ ( Is_empty2 @ L3 )
% 2.63/1.50 => ( ( ( produc686083979Tree_a @ X4 @ Y4 )
% 2.63/1.50 = ( Remove_max2 @ L3 ) )
% 2.63/1.50 => ( ssort_dom_Tree_a_a @ Is_empty2 @ Remove_max2 @ ( produc1352981801list_a @ Y4 @ ( cons_a @ X4 @ Sl3 ) ) ) ) ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_151_RemoveMax_Ossort_H__dom_Oinducts,axiom,
% 2.63/1.50 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,Remove_max2: tree_a > produc143150363Tree_a,Inv2: tree_a > $o,X: produc768687417list_a,P: produc768687417list_a > $o] :
% 2.63/1.50 ( ( removeMax_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv2 )
% 2.63/1.50 => ( ( ssort_dom_Tree_a_a @ Is_empty2 @ Remove_max2 @ X )
% 2.63/1.50 => ( ! [L5: tree_a,Sl2: list_a] :
% 2.63/1.50 ( ! [M6: a,L7: tree_a] :
% 2.63/1.50 ( ~ ( Is_empty2 @ L5 )
% 2.63/1.50 => ( ( ( produc686083979Tree_a @ M6 @ L7 )
% 2.63/1.50 = ( Remove_max2 @ L5 ) )
% 2.63/1.50 => ( ssort_dom_Tree_a_a @ Is_empty2 @ Remove_max2 @ ( produc1352981801list_a @ L7 @ ( cons_a @ M6 @ Sl2 ) ) ) ) )
% 2.63/1.50 => ( ! [M6: a,L7: tree_a] :
% 2.63/1.50 ( ~ ( Is_empty2 @ L5 )
% 2.63/1.50 => ( ( ( produc686083979Tree_a @ M6 @ L7 )
% 2.63/1.50 = ( Remove_max2 @ L5 ) )
% 2.63/1.50 => ( P @ ( produc1352981801list_a @ L7 @ ( cons_a @ M6 @ Sl2 ) ) ) ) )
% 2.63/1.50 => ( P @ ( produc1352981801list_a @ L5 @ Sl2 ) ) ) )
% 2.63/1.50 => ( P @ X ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_152_RemoveMax_Ossort_H__dom_Ointros,axiom,
% 2.63/1.50 ! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,Remove_max2: tree_a > produc143150363Tree_a,Inv2: tree_a > $o,L: tree_a,Sl: list_a] :
% 2.63/1.50 ( ( removeMax_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv2 )
% 2.63/1.50 => ( ! [M2: a,L2: tree_a] :
% 2.63/1.50 ( ~ ( Is_empty2 @ L )
% 2.63/1.50 => ( ( ( produc686083979Tree_a @ M2 @ L2 )
% 2.63/1.50 = ( Remove_max2 @ L ) )
% 2.63/1.50 => ( ssort_dom_Tree_a_a @ Is_empty2 @ Remove_max2 @ ( produc1352981801list_a @ L2 @ ( cons_a @ M2 @ Sl ) ) ) ) )
% 2.63/1.50 => ( ssort_dom_Tree_a_a @ Is_empty2 @ Remove_max2 @ ( produc1352981801list_a @ L @ Sl ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_153_multiset_Osimps_I2_J,axiom,
% 2.63/1.50 ! [V: a,L: tree_a,R: tree_a] :
% 2.63/1.50 ( ( multiset_a2 @ ( t_a @ V @ L @ R ) )
% 2.63/1.50 = ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ ( multiset_a2 @ L ) @ ( add_mset_a @ V @ zero_zero_multiset_a ) ) @ ( multiset_a2 @ R ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_154_single__subset__iff,axiom,
% 2.63/1.50 ! [A: a,M4: multiset_a] :
% 2.63/1.50 ( ( subseteq_mset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) @ M4 )
% 2.63/1.50 = ( member_a @ A @ ( set_mset_a @ M4 ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_155_subset__mset_Obot_Oextremum__unique,axiom,
% 2.63/1.50 ! [A: multiset_a] :
% 2.63/1.50 ( ( subseteq_mset_a @ A @ zero_zero_multiset_a )
% 2.63/1.50 = ( A = zero_zero_multiset_a ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_156_subset__mset_Ole__zero__eq,axiom,
% 2.63/1.50 ! [N3: multiset_a] :
% 2.63/1.50 ( ( subseteq_mset_a @ N3 @ zero_zero_multiset_a )
% 2.63/1.50 = ( N3 = zero_zero_multiset_a ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_157_subset__mset_Oadd__le__cancel__left,axiom,
% 2.63/1.50 ! [C: multiset_a,A: multiset_a,B: multiset_a] :
% 2.63/1.50 ( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C @ A ) @ ( plus_plus_multiset_a @ C @ B ) )
% 2.63/1.50 = ( subseteq_mset_a @ A @ B ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_158_subset__mset_Oadd__le__cancel__right,axiom,
% 2.63/1.50 ! [A: multiset_a,C: multiset_a,B: multiset_a] :
% 2.63/1.50 ( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ C ) )
% 2.63/1.50 = ( subseteq_mset_a @ A @ B ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_159_mset__subset__eq__mono__add__left__cancel,axiom,
% 2.63/1.50 ! [C2: multiset_a,A2: multiset_a,B4: multiset_a] :
% 2.63/1.50 ( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C2 @ A2 ) @ ( plus_plus_multiset_a @ C2 @ B4 ) )
% 2.63/1.50 = ( subseteq_mset_a @ A2 @ B4 ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_160_mset__subset__eq__mono__add__right__cancel,axiom,
% 2.63/1.50 ! [A2: multiset_a,C2: multiset_a,B4: multiset_a] :
% 2.63/1.50 ( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C2 ) @ ( plus_plus_multiset_a @ B4 @ C2 ) )
% 2.63/1.50 = ( subseteq_mset_a @ A2 @ B4 ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_161_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
% 2.63/1.50 ! [X: multiset_a,Y: multiset_a] :
% 2.63/1.50 ( ( zero_zero_multiset_a
% 2.63/1.50 = ( plus_plus_multiset_a @ X @ Y ) )
% 2.63/1.50 = ( ( X = zero_zero_multiset_a )
% 2.63/1.50 & ( Y = zero_zero_multiset_a ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_162_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
% 2.63/1.50 ! [X: multiset_a,Y: multiset_a] :
% 2.63/1.50 ( ( ( plus_plus_multiset_a @ X @ Y )
% 2.63/1.50 = zero_zero_multiset_a )
% 2.63/1.50 = ( ( X = zero_zero_multiset_a )
% 2.63/1.50 & ( Y = zero_zero_multiset_a ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_163_union__eq__empty,axiom,
% 2.63/1.50 ! [M4: multiset_a,N: multiset_a] :
% 2.63/1.50 ( ( ( plus_plus_multiset_a @ M4 @ N )
% 2.63/1.50 = zero_zero_multiset_a )
% 2.63/1.50 = ( ( M4 = zero_zero_multiset_a )
% 2.63/1.50 & ( N = zero_zero_multiset_a ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_164_empty__eq__union,axiom,
% 2.63/1.50 ! [M4: multiset_a,N: multiset_a] :
% 2.63/1.50 ( ( zero_zero_multiset_a
% 2.63/1.50 = ( plus_plus_multiset_a @ M4 @ N ) )
% 2.63/1.50 = ( ( M4 = zero_zero_multiset_a )
% 2.63/1.50 & ( N = zero_zero_multiset_a ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_165_union__mset__add__mset__right,axiom,
% 2.63/1.50 ! [A2: multiset_a,A: a,B4: multiset_a] :
% 2.63/1.50 ( ( plus_plus_multiset_a @ A2 @ ( add_mset_a @ A @ B4 ) )
% 2.63/1.50 = ( add_mset_a @ A @ ( plus_plus_multiset_a @ A2 @ B4 ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_166_union__mset__add__mset__left,axiom,
% 2.63/1.50 ! [A: a,A2: multiset_a,B4: multiset_a] :
% 2.63/1.50 ( ( plus_plus_multiset_a @ ( add_mset_a @ A @ A2 ) @ B4 )
% 2.63/1.50 = ( add_mset_a @ A @ ( plus_plus_multiset_a @ A2 @ B4 ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_167_subset__mset_Oadd__le__same__cancel1,axiom,
% 2.63/1.50 ! [B: multiset_a,A: multiset_a] :
% 2.63/1.50 ( ( subseteq_mset_a @ ( plus_plus_multiset_a @ B @ A ) @ B )
% 2.63/1.50 = ( subseteq_mset_a @ A @ zero_zero_multiset_a ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_168_subset__mset_Oadd__le__same__cancel2,axiom,
% 2.63/1.50 ! [A: multiset_a,B: multiset_a] :
% 2.63/1.50 ( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ B ) @ B )
% 2.63/1.50 = ( subseteq_mset_a @ A @ zero_zero_multiset_a ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_169_subset__mset_Ole__add__same__cancel1,axiom,
% 2.63/1.50 ! [A: multiset_a,B: multiset_a] :
% 2.63/1.50 ( ( subseteq_mset_a @ A @ ( plus_plus_multiset_a @ A @ B ) )
% 2.63/1.50 = ( subseteq_mset_a @ zero_zero_multiset_a @ B ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_170_subset__mset_Ole__add__same__cancel2,axiom,
% 2.63/1.50 ! [A: multiset_a,B: multiset_a] :
% 2.63/1.50 ( ( subseteq_mset_a @ A @ ( plus_plus_multiset_a @ B @ A ) )
% 2.63/1.50 = ( subseteq_mset_a @ zero_zero_multiset_a @ B ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_171_add__mset__subseteq__single__iff,axiom,
% 2.63/1.50 ! [A: a,M4: multiset_a,B: a] :
% 2.63/1.50 ( ( subseteq_mset_a @ ( add_mset_a @ A @ M4 ) @ ( add_mset_a @ B @ zero_zero_multiset_a ) )
% 2.63/1.50 = ( ( M4 = zero_zero_multiset_a )
% 2.63/1.50 & ( A = B ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_172_union__assoc,axiom,
% 2.63/1.50 ! [M4: multiset_a,N: multiset_a,K2: multiset_a] :
% 2.63/1.50 ( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ M4 @ N ) @ K2 )
% 2.63/1.50 = ( plus_plus_multiset_a @ M4 @ ( plus_plus_multiset_a @ N @ K2 ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_173_union__lcomm,axiom,
% 2.63/1.50 ! [M4: multiset_a,N: multiset_a,K2: multiset_a] :
% 2.63/1.50 ( ( plus_plus_multiset_a @ M4 @ ( plus_plus_multiset_a @ N @ K2 ) )
% 2.63/1.50 = ( plus_plus_multiset_a @ N @ ( plus_plus_multiset_a @ M4 @ K2 ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_174_union__commute,axiom,
% 2.63/1.50 ( plus_plus_multiset_a
% 2.63/1.50 = ( ^ [M7: multiset_a,N4: multiset_a] : ( plus_plus_multiset_a @ N4 @ M7 ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_175_union__left__cancel,axiom,
% 2.63/1.50 ! [K2: multiset_a,M4: multiset_a,N: multiset_a] :
% 2.63/1.50 ( ( ( plus_plus_multiset_a @ K2 @ M4 )
% 2.63/1.50 = ( plus_plus_multiset_a @ K2 @ N ) )
% 2.63/1.50 = ( M4 = N ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_176_union__right__cancel,axiom,
% 2.63/1.50 ! [M4: multiset_a,K2: multiset_a,N: multiset_a] :
% 2.63/1.50 ( ( ( plus_plus_multiset_a @ M4 @ K2 )
% 2.63/1.50 = ( plus_plus_multiset_a @ N @ K2 ) )
% 2.63/1.50 = ( M4 = N ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_177_subset__mset_Oadd__mono,axiom,
% 2.63/1.50 ! [A: multiset_a,B: multiset_a,C: multiset_a,D: multiset_a] :
% 2.63/1.50 ( ( subseteq_mset_a @ A @ B )
% 2.63/1.50 => ( ( subseteq_mset_a @ C @ D )
% 2.63/1.50 => ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ D ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_178_subset__mset_Oless__eqE,axiom,
% 2.63/1.50 ! [A: multiset_a,B: multiset_a] :
% 2.63/1.50 ( ( subseteq_mset_a @ A @ B )
% 2.63/1.50 => ~ ! [C3: multiset_a] :
% 2.63/1.50 ( B
% 2.63/1.50 != ( plus_plus_multiset_a @ A @ C3 ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_179_subset__mset_Ole__iff__add,axiom,
% 2.63/1.50 ( subseteq_mset_a
% 2.63/1.50 = ( ^ [A3: multiset_a,B2: multiset_a] :
% 2.63/1.50 ? [C4: multiset_a] :
% 2.63/1.50 ( B2
% 2.63/1.50 = ( plus_plus_multiset_a @ A3 @ C4 ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_180_mset__subset__eq__add__left,axiom,
% 2.63/1.50 ! [A2: multiset_a,B4: multiset_a] : ( subseteq_mset_a @ A2 @ ( plus_plus_multiset_a @ A2 @ B4 ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_181_mset__subset__eq__mono__add,axiom,
% 2.63/1.50 ! [A2: multiset_a,B4: multiset_a,C2: multiset_a,D2: multiset_a] :
% 2.63/1.50 ( ( subseteq_mset_a @ A2 @ B4 )
% 2.63/1.50 => ( ( subseteq_mset_a @ C2 @ D2 )
% 2.63/1.50 => ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C2 ) @ ( plus_plus_multiset_a @ B4 @ D2 ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_182_mset__subset__eq__add__right,axiom,
% 2.63/1.50 ! [B4: multiset_a,A2: multiset_a] : ( subseteq_mset_a @ B4 @ ( plus_plus_multiset_a @ A2 @ B4 ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_183_multi__union__self__other__eq,axiom,
% 2.63/1.50 ! [A2: multiset_a,X7: multiset_a,Y7: multiset_a] :
% 2.63/1.50 ( ( ( plus_plus_multiset_a @ A2 @ X7 )
% 2.63/1.50 = ( plus_plus_multiset_a @ A2 @ Y7 ) )
% 2.63/1.50 => ( X7 = Y7 ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_184_subset__mset_Oadd__left__mono,axiom,
% 2.63/1.50 ! [A: multiset_a,B: multiset_a,C: multiset_a] :
% 2.63/1.50 ( ( subseteq_mset_a @ A @ B )
% 2.63/1.50 => ( subseteq_mset_a @ ( plus_plus_multiset_a @ C @ A ) @ ( plus_plus_multiset_a @ C @ B ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_185_mset__subset__eq__exists__conv,axiom,
% 2.63/1.50 ( subseteq_mset_a
% 2.63/1.50 = ( ^ [A7: multiset_a,B7: multiset_a] :
% 2.63/1.50 ? [C5: multiset_a] :
% 2.63/1.50 ( B7
% 2.63/1.50 = ( plus_plus_multiset_a @ A7 @ C5 ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_186_subset__mset_Oadd__right__mono,axiom,
% 2.63/1.50 ! [A: multiset_a,B: multiset_a,C: multiset_a] :
% 2.63/1.50 ( ( subseteq_mset_a @ A @ B )
% 2.63/1.50 => ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_187_subset__mset_Oadd__le__imp__le__left,axiom,
% 2.63/1.50 ! [C: multiset_a,A: multiset_a,B: multiset_a] :
% 2.63/1.50 ( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C @ A ) @ ( plus_plus_multiset_a @ C @ B ) )
% 2.63/1.50 => ( subseteq_mset_a @ A @ B ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_188_subset__mset_Oadd__le__imp__le__right,axiom,
% 2.63/1.50 ! [A: multiset_a,C: multiset_a,B: multiset_a] :
% 2.63/1.50 ( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ C ) )
% 2.63/1.50 => ( subseteq_mset_a @ A @ B ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_189_union__iff,axiom,
% 2.63/1.50 ! [A: a,A2: multiset_a,B4: multiset_a] :
% 2.63/1.50 ( ( member_a @ A @ ( set_mset_a @ ( plus_plus_multiset_a @ A2 @ B4 ) ) )
% 2.63/1.50 = ( ( member_a @ A @ ( set_mset_a @ A2 ) )
% 2.63/1.50 | ( member_a @ A @ ( set_mset_a @ B4 ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_190_mset__subset__eqD,axiom,
% 2.63/1.50 ! [A2: multiset_a,B4: multiset_a,X: a] :
% 2.63/1.50 ( ( subseteq_mset_a @ A2 @ B4 )
% 2.63/1.50 => ( ( member_a @ X @ ( set_mset_a @ A2 ) )
% 2.63/1.50 => ( member_a @ X @ ( set_mset_a @ B4 ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_191_set__mset__mono,axiom,
% 2.63/1.50 ! [A2: multiset_a,B4: multiset_a] :
% 2.63/1.50 ( ( subseteq_mset_a @ A2 @ B4 )
% 2.63/1.50 => ( ord_less_eq_set_a @ ( set_mset_a @ A2 ) @ ( set_mset_a @ B4 ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_192_mset__subset__eq__add__mset__cancel,axiom,
% 2.63/1.50 ! [A: a,A2: multiset_a,B4: multiset_a] :
% 2.63/1.50 ( ( subseteq_mset_a @ ( add_mset_a @ A @ A2 ) @ ( add_mset_a @ A @ B4 ) )
% 2.63/1.50 = ( subseteq_mset_a @ A2 @ B4 ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_193_add__mono__thms__linordered__semiring_I3_J,axiom,
% 2.63/1.50 ! [I3: multiset_a,J: multiset_a,K3: multiset_a,L: multiset_a] :
% 2.63/1.50 ( ( ( ord_le1199012836iset_a @ I3 @ J )
% 2.63/1.50 & ( K3 = L ) )
% 2.63/1.50 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I3 @ K3 ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_194_add__mono__thms__linordered__semiring_I2_J,axiom,
% 2.63/1.50 ! [I3: multiset_a,J: multiset_a,K3: multiset_a,L: multiset_a] :
% 2.63/1.50 ( ( ( I3 = J )
% 2.63/1.50 & ( ord_le1199012836iset_a @ K3 @ L ) )
% 2.63/1.50 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I3 @ K3 ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_195_add__mono__thms__linordered__semiring_I1_J,axiom,
% 2.63/1.50 ! [I3: multiset_a,J: multiset_a,K3: multiset_a,L: multiset_a] :
% 2.63/1.50 ( ( ( ord_le1199012836iset_a @ I3 @ J )
% 2.63/1.50 & ( ord_le1199012836iset_a @ K3 @ L ) )
% 2.63/1.50 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I3 @ K3 ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_196_add__mono,axiom,
% 2.63/1.50 ! [A: multiset_a,B: multiset_a,C: multiset_a,D: multiset_a] :
% 2.63/1.50 ( ( ord_le1199012836iset_a @ A @ B )
% 2.63/1.50 => ( ( ord_le1199012836iset_a @ C @ D )
% 2.63/1.50 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ D ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_197_add__left__mono,axiom,
% 2.63/1.50 ! [A: multiset_a,B: multiset_a,C: multiset_a] :
% 2.63/1.50 ( ( ord_le1199012836iset_a @ A @ B )
% 2.63/1.50 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ C @ A ) @ ( plus_plus_multiset_a @ C @ B ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_198_add__right__mono,axiom,
% 2.63/1.50 ! [A: multiset_a,B: multiset_a,C: multiset_a] :
% 2.63/1.50 ( ( ord_le1199012836iset_a @ A @ B )
% 2.63/1.50 => ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_199_empty__neutral_I2_J,axiom,
% 2.63/1.50 ! [X: multiset_a] :
% 2.63/1.50 ( ( plus_plus_multiset_a @ X @ zero_zero_multiset_a )
% 2.63/1.50 = X ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_200_empty__neutral_I1_J,axiom,
% 2.63/1.50 ! [X: multiset_a] :
% 2.63/1.50 ( ( plus_plus_multiset_a @ zero_zero_multiset_a @ X )
% 2.63/1.50 = X ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_201_empty__le,axiom,
% 2.63/1.50 ! [A2: multiset_a] : ( subseteq_mset_a @ zero_zero_multiset_a @ A2 ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_202_subset__mset_Ozero__le,axiom,
% 2.63/1.50 ! [X: multiset_a] : ( subseteq_mset_a @ zero_zero_multiset_a @ X ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_203_subset__mset_Obot_Oextremum,axiom,
% 2.63/1.50 ! [A: multiset_a] : ( subseteq_mset_a @ zero_zero_multiset_a @ A ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_204_subset__mset_Oadd__decreasing,axiom,
% 2.63/1.50 ! [A: multiset_a,C: multiset_a,B: multiset_a] :
% 2.63/1.50 ( ( subseteq_mset_a @ A @ zero_zero_multiset_a )
% 2.63/1.50 => ( ( subseteq_mset_a @ C @ B )
% 2.63/1.50 => ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C ) @ B ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_205_subset__mset_Oadd__increasing,axiom,
% 2.63/1.50 ! [A: multiset_a,B: multiset_a,C: multiset_a] :
% 2.63/1.50 ( ( subseteq_mset_a @ zero_zero_multiset_a @ A )
% 2.63/1.50 => ( ( subseteq_mset_a @ B @ C )
% 2.63/1.50 => ( subseteq_mset_a @ B @ ( plus_plus_multiset_a @ A @ C ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_206_subset__mset_Oadd__decreasing2,axiom,
% 2.63/1.50 ! [C: multiset_a,A: multiset_a,B: multiset_a] :
% 2.63/1.50 ( ( subseteq_mset_a @ C @ zero_zero_multiset_a )
% 2.63/1.50 => ( ( subseteq_mset_a @ A @ B )
% 2.63/1.50 => ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C ) @ B ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_207_subset__mset_Oadd__increasing2,axiom,
% 2.63/1.50 ! [C: multiset_a,B: multiset_a,A: multiset_a] :
% 2.63/1.50 ( ( subseteq_mset_a @ zero_zero_multiset_a @ C )
% 2.63/1.50 => ( ( subseteq_mset_a @ B @ A )
% 2.63/1.50 => ( subseteq_mset_a @ B @ ( plus_plus_multiset_a @ A @ C ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_208_subset__mset_Oadd__nonneg__nonneg,axiom,
% 2.63/1.50 ! [A: multiset_a,B: multiset_a] :
% 2.63/1.50 ( ( subseteq_mset_a @ zero_zero_multiset_a @ A )
% 2.63/1.50 => ( ( subseteq_mset_a @ zero_zero_multiset_a @ B )
% 2.63/1.50 => ( subseteq_mset_a @ zero_zero_multiset_a @ ( plus_plus_multiset_a @ A @ B ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_209_subset__mset_Oadd__nonpos__nonpos,axiom,
% 2.63/1.50 ! [A: multiset_a,B: multiset_a] :
% 2.63/1.50 ( ( subseteq_mset_a @ A @ zero_zero_multiset_a )
% 2.63/1.50 => ( ( subseteq_mset_a @ B @ zero_zero_multiset_a )
% 2.63/1.50 => ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ B ) @ zero_zero_multiset_a ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_210_subset__mset_Oadd__nonneg__eq__0__iff,axiom,
% 2.63/1.50 ! [X: multiset_a,Y: multiset_a] :
% 2.63/1.50 ( ( subseteq_mset_a @ zero_zero_multiset_a @ X )
% 2.63/1.50 => ( ( subseteq_mset_a @ zero_zero_multiset_a @ Y )
% 2.63/1.50 => ( ( ( plus_plus_multiset_a @ X @ Y )
% 2.63/1.50 = zero_zero_multiset_a )
% 2.63/1.50 = ( ( X = zero_zero_multiset_a )
% 2.63/1.50 & ( Y = zero_zero_multiset_a ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_211_subset__mset_Oadd__nonpos__eq__0__iff,axiom,
% 2.63/1.50 ! [X: multiset_a,Y: multiset_a] :
% 2.63/1.50 ( ( subseteq_mset_a @ X @ zero_zero_multiset_a )
% 2.63/1.50 => ( ( subseteq_mset_a @ Y @ zero_zero_multiset_a )
% 2.63/1.50 => ( ( ( plus_plus_multiset_a @ X @ Y )
% 2.63/1.50 = zero_zero_multiset_a )
% 2.63/1.50 = ( ( X = zero_zero_multiset_a )
% 2.63/1.50 & ( Y = zero_zero_multiset_a ) ) ) ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_212_subset__mset_Obot_Oextremum__uniqueI,axiom,
% 2.63/1.50 ! [A: multiset_a] :
% 2.63/1.50 ( ( subseteq_mset_a @ A @ zero_zero_multiset_a )
% 2.63/1.50 => ( A = zero_zero_multiset_a ) ) ).
% 2.63/1.50
% 2.63/1.50 thf(fact_213_verit__sum__simplify,axiom,
% 2.63/1.50 ! [A: multiset_a] :
% 2.63/1.50 ( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
% 2.63/1.50 = A ) ).
% 2.63/1.50
% 2.63/1.50 thf(conj_0,conjecture,
% 2.63/1.50 ( ( multiset_a2 @ ( heapIm1091024090Down_a @ ( t_a @ v @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) ) ) )
% 2.63/1.50 = ( multiset_a2 @ ( t_a @ v @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) ) ) ) ).
% 2.63/1.50 ------------------------
% 2.63/1.53 % (6764)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_6481 for (2999ds/183Mi)
% 2.63/1.53 % (6765)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_6481 for (2999ds/4Mi)
% 2.63/1.53 % (6770)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_6481 for (2999ds/18Mi)
% 2.63/1.53 % (6767)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_6481 for (2999ds/2Mi)
% 2.63/1.53 % (6765)Instruction limit reached!
% 2.63/1.53 % (6765)------------------------------
% 2.63/1.53 % (6765)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.53 % (6765)Termination reason: Unknown
% 2.63/1.53 % (6765)Termination phase: shuffling
% 2.63/1.53
% 2.63/1.53 % (6765)Memory used [KB]: 1407
% 2.63/1.53 % (6765)Time elapsed: 0.003 s
% 2.63/1.53 % (6765)Instructions burned: 6 (million)
% 2.63/1.53 % (6765)------------------------------
% 2.63/1.53 % (6765)------------------------------
% 2.63/1.53 % (6767)Instruction limit reached!
% 2.63/1.53 % (6767)------------------------------
% 2.63/1.53 % (6767)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.53 % (6767)Termination reason: Unknown
% 2.63/1.53 % (6767)Termination phase: shuffling
% 2.63/1.53
% 2.63/1.53 % (6767)Memory used [KB]: 1407
% 2.63/1.53 % (6767)Time elapsed: 0.003 s
% 2.63/1.53 % (6767)Instructions burned: 3 (million)
% 2.63/1.53 % (6767)------------------------------
% 2.63/1.53 % (6767)------------------------------
% 2.63/1.54 % (6770)Instruction limit reached!
% 2.63/1.54 % (6770)------------------------------
% 2.63/1.54 % (6770)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.54 % (6770)Termination reason: Unknown
% 2.63/1.54 % (6770)Termination phase: shuffling
% 2.63/1.54
% 2.63/1.54 % (6770)Memory used [KB]: 1663
% 2.63/1.54 % (6770)Time elapsed: 0.008 s
% 2.63/1.54 % (6770)Instructions burned: 19 (million)
% 2.63/1.54 % (6770)------------------------------
% 2.63/1.54 % (6770)------------------------------
% 2.63/1.54 % (6766)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_6481 for (2999ds/27Mi)
% 2.63/1.54 % (6768)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_6481 for (2999ds/2Mi)
% 2.63/1.54 % (6769)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_6481 for (2999ds/275Mi)
% 2.63/1.54 % (6768)Instruction limit reached!
% 2.63/1.54 % (6768)------------------------------
% 2.63/1.54 % (6768)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.54 % (6768)Termination reason: Unknown
% 2.63/1.54 % (6768)Termination phase: shuffling
% 2.63/1.54
% 2.63/1.54 % (6768)Memory used [KB]: 1407
% 2.63/1.54 % (6768)Time elapsed: 0.004 s
% 2.63/1.54 % (6768)Instructions burned: 3 (million)
% 2.63/1.54 % (6768)------------------------------
% 2.63/1.54 % (6768)------------------------------
% 2.63/1.54 % (6771)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_6481 for (2999ds/3Mi)
% 2.63/1.54 % (6771)Instruction limit reached!
% 2.63/1.54 % (6771)------------------------------
% 2.63/1.54 % (6771)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.54 % (6771)Termination reason: Unknown
% 2.63/1.54 % (6771)Termination phase: shuffling
% 2.63/1.54
% 2.63/1.54 % (6771)Memory used [KB]: 1407
% 2.63/1.54 % (6771)Time elapsed: 0.003 s
% 2.63/1.54 % (6771)Instructions burned: 4 (million)
% 2.63/1.54 % (6771)------------------------------
% 2.63/1.54 % (6771)------------------------------
% 2.63/1.54 % (6772)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_6481 for (2999ds/37Mi)
% 2.63/1.55 % (6773)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_6481 for (2999ds/15Mi)
% 2.63/1.55 % (6773)Instruction limit reached!
% 2.63/1.55 % (6773)------------------------------
% 2.63/1.55 % (6773)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.55 % (6773)Termination reason: Unknown
% 2.63/1.55 % (6773)Termination phase: shuffling
% 2.63/1.55
% 2.63/1.55 % (6773)Memory used [KB]: 1535
% 2.63/1.55 % (6773)Time elapsed: 0.006 s
% 2.63/1.55 % (6773)Instructions burned: 15 (million)
% 2.63/1.55 % (6773)------------------------------
% 2.63/1.55 % (6773)------------------------------
% 2.63/1.55 % (6775)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_6481 for (2999ds/1041Mi)
% 2.63/1.55 % (6766)Instruction limit reached!
% 2.63/1.55 % (6766)------------------------------
% 2.63/1.55 % (6766)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.55 % (6766)Termination reason: Unknown
% 2.63/1.55 % (6766)Termination phase: Property scanning
% 2.63/1.55
% 2.63/1.55 % (6766)Memory used [KB]: 1791
% 2.63/1.55 % (6766)Time elapsed: 0.019 s
% 2.63/1.55 % (6766)Instructions burned: 28 (million)
% 2.63/1.55 % (6766)------------------------------
% 2.63/1.55 % (6766)------------------------------
% 2.63/1.56 % (6772)Instruction limit reached!
% 2.63/1.56 % (6772)------------------------------
% 2.63/1.56 % (6772)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.56 % (6772)Termination reason: Unknown
% 2.63/1.56 % (6772)Termination phase: shuffling
% 2.63/1.56
% 2.63/1.56 % (6772)Memory used [KB]: 2174
% 2.63/1.56 % (6772)Time elapsed: 0.013 s
% 2.63/1.56 % (6772)Instructions burned: 38 (million)
% 2.63/1.56 % (6772)------------------------------
% 2.63/1.56 % (6772)------------------------------
% 2.63/1.56 % (6774)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_6481 for (2999ds/3Mi)
% 2.63/1.56 % (6774)Instruction limit reached!
% 2.63/1.56 % (6774)------------------------------
% 2.63/1.56 % (6774)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.56 % (6774)Termination reason: Unknown
% 2.63/1.56 % (6774)Termination phase: shuffling
% 2.63/1.56
% 2.63/1.56 % (6774)Memory used [KB]: 1407
% 2.63/1.56 % (6774)Time elapsed: 0.004 s
% 2.63/1.56 % (6774)Instructions burned: 3 (million)
% 2.63/1.56 % (6774)------------------------------
% 2.63/1.56 % (6774)------------------------------
% 2.63/1.56 % (6776)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_6481 for (2999ds/7Mi)
% 2.63/1.56 % (6776)Instruction limit reached!
% 2.63/1.56 % (6776)------------------------------
% 2.63/1.56 % (6776)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.56 % (6776)Termination reason: Unknown
% 2.63/1.56 % (6776)Termination phase: shuffling
% 2.63/1.56
% 2.63/1.56 % (6776)Memory used [KB]: 1535
% 2.63/1.56 % (6776)Time elapsed: 0.004 s
% 2.63/1.56 % (6776)Instructions burned: 10 (million)
% 2.63/1.56 % (6776)------------------------------
% 2.63/1.56 % (6776)------------------------------
% 2.63/1.57 % (6778)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_6481 for (2999ds/3Mi)
% 2.63/1.57 % (6778)Instruction limit reached!
% 2.63/1.57 % (6778)------------------------------
% 2.63/1.57 % (6778)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.57 % (6778)Termination reason: Unknown
% 2.63/1.57 % (6778)Termination phase: shuffling
% 2.63/1.57
% 2.63/1.57 % (6778)Memory used [KB]: 1407
% 2.63/1.57 % (6778)Time elapsed: 0.003 s
% 2.63/1.57 % (6778)Instructions burned: 4 (million)
% 2.63/1.57 % (6778)------------------------------
% 2.63/1.57 % (6778)------------------------------
% 2.63/1.57 % (6777)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_6481 for (2999ds/16Mi)
% 2.63/1.57 % (6780)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_6481 for (2999ds/7Mi)
% 2.63/1.57 % (6780)Instruction limit reached!
% 2.63/1.57 % (6780)------------------------------
% 2.63/1.57 % (6780)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.57 % (6780)Termination reason: Unknown
% 2.63/1.57 % (6780)Termination phase: shuffling
% 2.63/1.57
% 2.63/1.57 % (6780)Memory used [KB]: 1407
% 2.63/1.57 % (6780)Time elapsed: 0.004 s
% 2.63/1.57 % (6780)Instructions burned: 9 (million)
% 2.63/1.57 % (6780)------------------------------
% 2.63/1.57 % (6780)------------------------------
% 2.63/1.58 % (6779)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_6481 for (2999ds/3Mi)
% 2.63/1.58 % (6781)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_6481 for (2999ds/3Mi)
% 2.63/1.58 % (6779)Instruction limit reached!
% 2.63/1.58 % (6779)------------------------------
% 2.63/1.58 % (6779)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.58 % (6779)Termination reason: Unknown
% 2.63/1.58 % (6779)Termination phase: shuffling
% 2.63/1.58
% 2.63/1.58 % (6779)Memory used [KB]: 1407
% 2.63/1.58 % (6779)Time elapsed: 0.004 s
% 2.63/1.58 % (6779)Instructions burned: 3 (million)
% 2.63/1.58 % (6779)------------------------------
% 2.63/1.58 % (6779)------------------------------
% 2.63/1.58 % (6781)Instruction limit reached!
% 2.63/1.58 % (6781)------------------------------
% 2.63/1.58 % (6781)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.58 % (6781)Termination reason: Unknown
% 2.63/1.58 % (6781)Termination phase: shuffling
% 2.63/1.58
% 2.63/1.58 % (6781)Memory used [KB]: 1407
% 2.63/1.58 % (6781)Time elapsed: 0.003 s
% 2.63/1.58 % (6781)Instructions burned: 4 (million)
% 2.63/1.58 % (6781)------------------------------
% 2.63/1.58 % (6781)------------------------------
% 2.63/1.58 % (6764)Instruction limit reached!
% 2.63/1.58 % (6764)------------------------------
% 2.63/1.58 % (6764)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.58 % (6764)Termination reason: Unknown
% 2.63/1.58 % (6764)Termination phase: Saturation
% 2.63/1.58
% 2.63/1.58 % (6764)Memory used [KB]: 7164
% 2.63/1.58 % (6764)Time elapsed: 0.053 s
% 2.63/1.58 % (6764)Instructions burned: 183 (million)
% 2.63/1.58 % (6764)------------------------------
% 2.63/1.58 % (6764)------------------------------
% 2.63/1.58 % (6777)Instruction limit reached!
% 2.63/1.58 % (6777)------------------------------
% 2.63/1.58 % (6777)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.58 % (6777)Termination reason: Unknown
% 2.63/1.58 % (6777)Termination phase: shuffling
% 2.63/1.58
% 2.63/1.58 % (6777)Memory used [KB]: 1663
% 2.63/1.58 % (6777)Time elapsed: 0.012 s
% 2.63/1.58 % (6777)Instructions burned: 17 (million)
% 2.63/1.58 % (6777)------------------------------
% 2.63/1.58 % (6777)------------------------------
% 2.63/1.58 % (6782)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_6481 for (2999ds/4Mi)
% 2.63/1.59 % (6782)Instruction limit reached!
% 2.63/1.59 % (6782)------------------------------
% 2.63/1.59 % (6782)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.59 % (6782)Termination reason: Unknown
% 2.63/1.59 % (6782)Termination phase: shuffling
% 2.63/1.59
% 2.63/1.59 % (6782)Memory used [KB]: 1407
% 2.63/1.59 % (6782)Time elapsed: 0.003 s
% 2.63/1.59 % (6782)Instructions burned: 6 (million)
% 2.63/1.59 % (6782)------------------------------
% 2.63/1.59 % (6782)------------------------------
% 2.63/1.59 % (6784)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_6481 for (2999ds/710Mi)
% 2.63/1.59 % (6785)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_6481 for (2999ds/6Mi)
% 2.63/1.59 % (6769)First to succeed.
% 2.63/1.59 % (6785)Instruction limit reached!
% 2.63/1.59 % (6785)------------------------------
% 2.63/1.59 % (6785)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.59 % (6785)Termination reason: Unknown
% 2.63/1.59 % (6785)Termination phase: shuffling
% 2.63/1.59
% 2.63/1.59 % (6785)Memory used [KB]: 1407
% 2.63/1.59 % (6785)Time elapsed: 0.003 s
% 2.63/1.59 % (6785)Instructions burned: 6 (million)
% 2.63/1.59 % (6785)------------------------------
% 2.63/1.59 % (6785)------------------------------
% 2.63/1.59 % (6769)Refutation found. Thanks to Tanya!
% 2.63/1.59 % SZS status Theorem for DTF2THF_6481
% 2.63/1.59 % SZS output start Proof for DTF2THF_6481
% 2.63/1.59 thf(type_def_5, type, multiset_a: $tType).
% 2.63/1.59 thf(type_def_7, type, tree_a: $tType).
% 2.63/1.59 thf(type_def_8, type, list_a: $tType).
% 2.63/1.59 thf(type_def_9, type, produc143150363Tree_a: $tType).
% 2.63/1.59 thf(type_def_10, type, a: $tType).
% 2.63/1.59 thf(type_def_11, type, set_a: $tType).
% 2.63/1.59 thf(type_def_12, type, produc768687417list_a: $tType).
% 2.63/1.59 thf(func_def_0, type, produc768687417list_a: $tType).
% 2.63/1.59 thf(func_def_1, type, produc143150363Tree_a: $tType).
% 2.63/1.59 thf(func_def_2, type, multiset_a: $tType).
% 2.63/1.59 thf(func_def_3, type, list_a: $tType).
% 2.63/1.59 thf(func_def_4, type, tree_a: $tType).
% 2.63/1.59 thf(func_def_5, type, set_a: $tType).
% 2.63/1.59 thf(func_def_6, type, a: $tType).
% 2.63/1.59 thf(func_def_7, type, plus_plus_multiset_a: multiset_a > multiset_a > multiset_a).
% 2.63/1.59 thf(func_def_8, type, zero_zero_multiset_a: multiset_a).
% 2.63/1.59 thf(func_def_9, type, heapIm970322378pify_a: tree_a > tree_a).
% 2.63/1.59 thf(func_def_10, type, heapIm229596386mpty_a: tree_a > $o).
% 2.63/1.59 thf(func_def_11, type, heapIm1057938560list_a: list_a > tree_a).
% 2.63/1.59 thf(func_def_12, type, heapIm1140443833left_a: tree_a > tree_a).
% 2.63/1.59 thf(func_def_13, type, heapIm1637418125tree_a: list_a > tree_a).
% 2.63/1.59 thf(func_def_14, type, heapIm837449470Leaf_a: tree_a > produc143150363Tree_a).
% 2.63/1.59 thf(func_def_15, type, heapIm1257206334ight_a: tree_a > tree_a).
% 2.63/1.59 thf(func_def_16, type, heapIm1091024090Down_a: tree_a > tree_a).
% 2.63/1.59 thf(func_def_17, 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).
% 2.63/1.59 thf(func_def_18, 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).
% 2.63/1.59 thf(func_def_19, type, e_a: tree_a).
% 2.63/1.59 thf(func_def_20, type, t_a: a > tree_a > tree_a > tree_a).
% 2.63/1.59 thf(func_def_21, type, in_tree_a: a > tree_a > $o).
% 2.63/1.59 thf(func_def_22, type, is_heap_a: tree_a > $o).
% 2.63/1.59 thf(func_def_23, type, multiset_a2: tree_a > multiset_a).
% 2.63/1.59 thf(func_def_24, type, val_a: tree_a > a).
% 2.63/1.59 thf(func_def_25, type, lattic146396397_Max_a: set_a > a).
% 2.63/1.59 thf(func_def_26, type, cons_a: a > list_a > list_a).
% 2.63/1.59 thf(func_def_27, type, add_mset_a: a > multiset_a > multiset_a).
% 2.63/1.59 thf(func_def_28, type, is_empty_a: multiset_a > $o).
% 2.63/1.59 thf(func_def_29, type, set_mset_a: multiset_a > set_a).
% 2.63/1.59 thf(func_def_30, type, subseteq_mset_a: multiset_a > multiset_a > $o).
% 2.63/1.59 thf(func_def_31, type, ord_less_eq_o_o_a: ($o > $o > a) > ($o > $o > a) > $o).
% 2.63/1.59 thf(func_def_32, type, ord_less_eq_o_a: ($o > a) > ($o > a) > $o).
% 2.63/1.59 thf(func_def_33, type, ord_le1199012836iset_a: multiset_a > multiset_a > $o).
% 2.63/1.59 thf(func_def_34, type, ord_less_eq_set_a: set_a > set_a > $o).
% 2.63/1.59 thf(func_def_35, type, ord_less_eq_a: a > a > $o).
% 2.63/1.59 thf(func_def_36, type, order_Greatest_o_a: (($o > a) > $o) > $o > a).
% 2.63/1.59 thf(func_def_37, type, order_Greatest_a: (a > $o) > a).
% 2.63/1.59 thf(func_def_38, type, produc1352981801list_a: tree_a > list_a > produc768687417list_a).
% 2.63/1.59 thf(func_def_39, type, produc686083979Tree_a: a > tree_a > produc143150363Tree_a).
% 2.63/1.59 thf(func_def_40, type, set_Tree_a_a: (tree_a > multiset_a) > tree_a > set_a).
% 2.63/1.59 thf(func_def_41, type, removeMax_Tree_a_a: tree_a > (tree_a > $o) > (list_a > tree_a) > (tree_a > multiset_a) > (tree_a > produc143150363Tree_a) > (tree_a > $o) > $o).
% 2.63/1.59 thf(func_def_42, type, ssort_Tree_a_a: (tree_a > $o) > (tree_a > produc143150363Tree_a) > tree_a > list_a > list_a).
% 2.63/1.59 thf(func_def_43, type, ssort_dom_Tree_a_a: (tree_a > $o) > (tree_a > produc143150363Tree_a) > produc768687417list_a > $o).
% 2.63/1.59 thf(func_def_44, type, remove301631099ee_a_a: (tree_a > $o) > (list_a > tree_a) > (tree_a > multiset_a) > (tree_a > produc143150363Tree_a) > (tree_a > $o) > $o).
% 2.63/1.59 thf(func_def_45, type, collect_a: (a > $o) > set_a).
% 2.63/1.59 thf(func_def_46, type, member_a: a > set_a > $o).
% 2.63/1.59 thf(func_def_47, type, l1: tree_a).
% 2.63/1.59 thf(func_def_48, type, l2: tree_a).
% 2.63/1.59 thf(func_def_49, type, r1: tree_a).
% 2.63/1.59 thf(func_def_50, type, r2: tree_a).
% 2.63/1.59 thf(func_def_51, type, v1: a).
% 2.63/1.59 thf(func_def_52, type, v2: a).
% 2.63/1.59 thf(func_def_53, type, v: a).
% 2.63/1.59 thf(func_def_71, type, sP0: a > a > a > $o).
% 2.63/1.59 thf(func_def_72, type, sP1: a > a > a > $o).
% 2.63/1.59 thf(func_def_73, type, sP2: a > a > a > $o).
% 2.63/1.59 thf(func_def_74, type, sP3: (tree_a > $o) > $o).
% 2.63/1.59 thf(func_def_75, type, sP4: (tree_a > $o) > $o).
% 2.63/1.59 thf(func_def_76, type, sP5: (tree_a > $o) > $o).
% 2.63/1.59 thf(func_def_77, type, sK6: multiset_a > a > multiset_a).
% 2.63/1.59 thf(func_def_78, type, sK7: (a > a) > a).
% 2.63/1.59 thf(func_def_79, type, sK8: (a > a) > a).
% 2.63/1.59 thf(func_def_80, type, sK9: (a > a) > a).
% 2.63/1.59 thf(func_def_81, type, sK10: (a > a) > a).
% 2.63/1.59 thf(func_def_82, type, sK11: multiset_a > a).
% 2.63/1.59 thf(func_def_83, type, sK12: multiset_a > multiset_a).
% 2.63/1.59 thf(func_def_84, type, sK13: (multiset_a > $o) > multiset_a).
% 2.63/1.59 thf(func_def_85, type, sK14: (multiset_a > $o) > a).
% 2.63/1.59 thf(func_def_86, type, sK15: tree_a > a).
% 2.63/1.59 thf(func_def_87, type, sK16: tree_a > tree_a).
% 2.63/1.59 thf(func_def_88, type, sK17: tree_a > tree_a).
% 2.63/1.59 thf(func_def_89, type, sK18: tree_a > a).
% 2.63/1.59 thf(func_def_90, type, sK19: tree_a > a).
% 2.63/1.59 thf(func_def_91, type, sK20: tree_a > tree_a).
% 2.63/1.59 thf(func_def_92, type, sK21: tree_a > a).
% 2.63/1.59 thf(func_def_93, type, sK22: tree_a > a).
% 2.63/1.59 thf(func_def_94, type, sK23: tree_a > a).
% 2.63/1.59 thf(func_def_95, type, sK24: tree_a > tree_a).
% 2.63/1.59 thf(func_def_96, type, sK25: tree_a > tree_a).
% 2.63/1.59 thf(func_def_97, type, sK26: tree_a > tree_a).
% 2.63/1.59 thf(func_def_98, type, sK27: tree_a > tree_a).
% 2.63/1.59 thf(func_def_99, type, sK28: tree_a > tree_a).
% 2.63/1.59 thf(func_def_100, type, sK29: tree_a > a).
% 2.63/1.59 thf(func_def_101, type, sK30: tree_a > a).
% 2.63/1.59 thf(func_def_102, type, sK31: tree_a > a).
% 2.63/1.59 thf(func_def_103, type, sK32: tree_a > a).
% 2.63/1.59 thf(func_def_104, type, sK33: tree_a > tree_a).
% 2.63/1.59 thf(func_def_105, type, sK34: tree_a > tree_a).
% 2.63/1.59 thf(func_def_106, type, sK35: tree_a > a).
% 2.63/1.59 thf(func_def_107, type, sK36: tree_a > tree_a).
% 2.63/1.59 thf(func_def_108, type, sK37: tree_a > tree_a).
% 2.63/1.59 thf(func_def_109, type, sK38: tree_a > tree_a).
% 2.63/1.59 thf(func_def_110, type, sK39: tree_a > a).
% 2.63/1.59 thf(func_def_111, type, sK40: tree_a > tree_a).
% 2.63/1.59 thf(func_def_112, type, sK41: tree_a > a).
% 2.63/1.59 thf(func_def_113, type, sK42: tree_a > a).
% 2.63/1.59 thf(func_def_114, type, sK43: tree_a > a).
% 2.63/1.59 thf(func_def_115, type, sK44: tree_a > tree_a).
% 2.63/1.59 thf(func_def_116, type, sK45: tree_a > tree_a).
% 2.63/1.59 thf(func_def_117, type, sK46: tree_a > tree_a).
% 2.63/1.59 thf(func_def_118, type, sK47: tree_a > a).
% 2.63/1.59 thf(func_def_119, type, sK48: tree_a > tree_a).
% 2.63/1.59 thf(func_def_120, type, sK49: tree_a > a).
% 2.63/1.59 thf(func_def_121, type, sK50: tree_a > tree_a).
% 2.63/1.59 thf(func_def_122, type, sK51: tree_a > a).
% 2.63/1.59 thf(func_def_123, type, sK52: tree_a > tree_a).
% 2.63/1.59 thf(func_def_124, type, sK53: tree_a > a).
% 2.63/1.59 thf(func_def_125, type, sK54: (multiset_a > multiset_a > $o) > multiset_a).
% 2.63/1.59 thf(func_def_126, type, sK55: (multiset_a > multiset_a > $o) > a).
% 2.63/1.59 thf(func_def_127, type, sK56: (multiset_a > multiset_a > $o) > multiset_a).
% 2.63/1.59 thf(func_def_128, type, sK57: (multiset_a > multiset_a > $o) > a).
% 2.63/1.59 thf(func_def_129, type, sK58: (multiset_a > multiset_a > $o) > multiset_a).
% 2.63/1.59 thf(func_def_130, type, sK59: (multiset_a > multiset_a > $o) > multiset_a).
% 2.63/1.59 thf(func_def_131, type, sK60: (a > a) > a).
% 2.63/1.59 thf(func_def_132, type, sK61: (a > a) > a).
% 2.63/1.59 thf(func_def_133, type, sK62: tree_a > tree_a).
% 2.63/1.59 thf(func_def_134, type, sK63: tree_a > tree_a).
% 2.63/1.59 thf(func_def_135, type, sK64: tree_a > a).
% 2.63/1.59 thf(func_def_136, type, sK65: tree_a > a).
% 2.63/1.59 thf(func_def_137, type, sK66: tree_a > tree_a).
% 2.63/1.59 thf(func_def_138, type, sK67: tree_a > tree_a).
% 2.63/1.59 thf(func_def_139, type, sK68: tree_a > a).
% 2.63/1.59 thf(func_def_140, type, sK69: tree_a > tree_a).
% 2.63/1.59 thf(func_def_141, type, sK70: tree_a > a).
% 2.63/1.59 thf(func_def_142, type, sK71: tree_a > tree_a).
% 2.63/1.59 thf(func_def_143, type, sK72: tree_a > a).
% 2.63/1.59 thf(func_def_144, type, sK73: tree_a > tree_a).
% 2.63/1.59 thf(func_def_145, type, sK74: tree_a > a).
% 2.63/1.59 thf(func_def_146, type, sK75: tree_a > tree_a).
% 2.63/1.59 thf(func_def_147, type, sK76: tree_a > a).
% 2.63/1.59 thf(func_def_148, type, sK77: tree_a > a).
% 2.63/1.59 thf(func_def_149, type, sK78: tree_a > tree_a).
% 2.63/1.59 thf(func_def_150, type, sK79: tree_a > tree_a).
% 2.63/1.59 thf(func_def_151, type, sK80: tree_a > a).
% 2.63/1.59 thf(func_def_152, type, sK81: (tree_a > $o) > tree_a).
% 2.63/1.59 thf(func_def_153, type, sK82: (tree_a > $o) > a).
% 2.63/1.59 thf(func_def_154, type, sK83: (tree_a > $o) > tree_a).
% 2.63/1.59 thf(func_def_155, type, sK84: (multiset_a > $o) > a).
% 2.63/1.59 thf(func_def_156, type, sK85: (multiset_a > $o) > multiset_a).
% 2.63/1.59 thf(func_def_157, type, sK86: multiset_a > a).
% 2.63/1.60 thf(func_def_158, type, sK87: multiset_a > multiset_a).
% 2.63/1.60 thf(func_def_159, type, sK88: multiset_a > a).
% 2.63/1.60 thf(func_def_160, type, sK89: (a > a > $o) > a).
% 2.63/1.60 thf(func_def_161, type, sK90: (a > a > $o) > a).
% 2.63/1.60 thf(func_def_162, type, sK91: (a > a > $o) > a).
% 2.63/1.60 thf(func_def_163, type, sK92: (a > a > $o) > a).
% 2.63/1.60 thf(func_def_164, type, sK93: (multiset_a > $o) > a).
% 2.63/1.60 thf(func_def_165, type, sK94: (multiset_a > $o) > multiset_a).
% 2.63/1.60 thf(func_def_166, type, sK95: (a > a) > a).
% 2.63/1.60 thf(func_def_167, type, sK96: (a > a) > a).
% 2.63/1.60 thf(func_def_168, type, sK97: multiset_a > a > multiset_a).
% 2.63/1.60 thf(func_def_169, type, sK98: tree_a > a > tree_a).
% 2.63/1.60 thf(func_def_170, type, sK99: tree_a > a > tree_a).
% 2.63/1.60 thf(func_def_171, type, sK100: tree_a > a > a).
% 2.63/1.60 thf(func_def_172, type, sK101: multiset_a > a > a > multiset_a > multiset_a).
% 2.63/1.60 thf(func_def_173, type, sK102: (tree_a > $o) > tree_a).
% 2.63/1.60 thf(func_def_174, type, sK103: (tree_a > $o) > tree_a).
% 2.63/1.60 thf(func_def_175, type, sK104: (tree_a > $o) > tree_a).
% 2.63/1.60 thf(func_def_176, type, sK105: (tree_a > $o) > a).
% 2.63/1.60 thf(func_def_177, type, sK106: (tree_a > $o) > a).
% 2.63/1.60 thf(func_def_178, type, sK107: (tree_a > $o) > tree_a).
% 2.63/1.60 thf(func_def_179, type, sK108: (tree_a > $o) > a).
% 2.63/1.60 thf(func_def_180, type, sK109: (tree_a > $o) > tree_a).
% 2.63/1.60 thf(func_def_181, type, sK110: (tree_a > $o) > a).
% 2.63/1.60 thf(func_def_182, type, sK111: (tree_a > $o) > tree_a).
% 2.63/1.60 thf(func_def_183, type, sK112: (tree_a > $o) > a).
% 2.63/1.60 thf(func_def_184, type, sK113: (tree_a > $o) > tree_a).
% 2.63/1.60 thf(func_def_185, type, sK114: (tree_a > $o) > tree_a).
% 2.63/1.60 thf(func_def_186, type, sK115: (tree_a > $o) > a).
% 2.63/1.60 thf(func_def_187, type, sK116: (tree_a > $o) > a).
% 2.63/1.60 thf(func_def_188, type, sK117: (tree_a > $o) > tree_a).
% 2.63/1.60 thf(func_def_189, type, sK118: (tree_a > $o) > tree_a).
% 2.63/1.60 thf(func_def_190, type, sK119: (tree_a > $o) > a).
% 2.63/1.60 thf(func_def_191, type, sK120: (tree_a > $o) > a).
% 2.63/1.60 thf(func_def_192, type, sK121: (tree_a > $o) > tree_a).
% 2.63/1.60 thf(func_def_193, type, sK122: (tree_a > $o) > a).
% 2.63/1.60 thf(func_def_194, type, sK123: (tree_a > $o) > tree_a).
% 2.63/1.60 thf(func_def_195, type, sK124: (tree_a > $o) > a).
% 2.63/1.60 thf(func_def_197, type, ph126: !>[X0: $tType]:(X0)).
% 2.63/1.60 thf(f1080,plain,(
% 2.63/1.60 $false),
% 2.63/1.60 inference(subsumption_resolution,[status(thm)],[f1079,f902])).
% 2.63/1.60 thf(f902,plain,(
% 2.63/1.60 ((ord_less_eq_a @ v2 @ v1) = $true)),
% 2.63/1.60 inference(cnf_transformation,[status(thm)],[f323])).
% 2.63/1.60 thf(f323,plain,(
% 2.63/1.60 ((ord_less_eq_a @ v2 @ v1) = $true)),
% 2.63/1.60 inference(fool_elimination,[status(thm)],[f322])).
% 2.63/1.60 thf(f322,plain,(
% 2.63/1.60 (ord_less_eq_a @ v2 @ v1)),
% 2.63/1.60 inference(rectify,[status(thm)],[f1])).
% 2.63/1.60 thf(f1,axiom,(
% 2.63/1.60 (ord_less_eq_a @ v2 @ v1)),
% 2.63/1.60 file('/export/starexec/sandbox/tmp/tmp.geO6X9CZKJ/DTF2THF_6481.p',fact_0__092_060open_062v2_A_092_060le_062_Av1_092_060close_062)).
% 2.63/1.60 thf(f1079,plain,(
% 2.63/1.60 ((ord_less_eq_a @ v2 @ v1) != $true)),
% 2.63/1.60 inference(forward_demodulation,[status(thm)],[f1078,f794])).
% 2.63/1.60 thf(f794,plain,(
% 2.63/1.60 ( ! [X2 : tree_a,X0 : a,X1 : tree_a] : (((val_a @ (t_a @ X0 @ X1 @ X2)) = X0)) )),
% 2.63/1.60 inference(cnf_transformation,[status(thm)],[f574])).
% 2.63/1.60 thf(f574,plain,(
% 2.63/1.60 ! [X0 : a,X1 : tree_a,X2 : tree_a] : ((val_a @ (t_a @ X0 @ X1 @ X2)) = X0)),
% 2.63/1.60 inference(rectify,[status(thm)],[f31])).
% 2.63/1.60 thf(f31,axiom,(
% 2.63/1.60 ! [X1 : a,X38 : tree_a,X39 : tree_a] : ((val_a @ (t_a @ X1 @ X38 @ X39)) = X1)),
% 2.63/1.60 file('/export/starexec/sandbox/tmp/tmp.geO6X9CZKJ/DTF2THF_6481.p',fact_30_val_Osimps)).
% 2.63/1.60 thf(f1078,plain,(
% 2.63/1.60 ((ord_less_eq_a @ v2 @ (val_a @ (t_a @ v1 @ l1 @ r1))) != $true)),
% 2.63/1.60 inference(forward_demodulation,[status(thm)],[f1077,f794])).
% 2.63/1.60 thf(f1077,plain,(
% 2.63/1.60 ((ord_less_eq_a @ (val_a @ (t_a @ v2 @ l2 @ r2)) @ (val_a @ (t_a @ v1 @ l1 @ r1))) != $true)),
% 2.63/1.60 inference(subsumption_resolution,[status(thm)],[f1076,f815])).
% 2.63/1.60 thf(f815,plain,(
% 2.63/1.60 ((ord_less_eq_a @ v1 @ v) = $true)),
% 2.63/1.60 inference(cnf_transformation,[status(thm)],[f415])).
% 2.63/1.60 thf(f415,plain,(
% 2.63/1.60 ((ord_less_eq_a @ v1 @ v) = $true)),
% 2.63/1.60 inference(fool_elimination,[status(thm)],[f414])).
% 2.63/1.60 thf(f414,plain,(
% 2.63/1.60 (ord_less_eq_a @ v1 @ v)),
% 2.63/1.60 inference(rectify,[status(thm)],[f2])).
% 2.63/1.60 thf(f2,axiom,(
% 2.63/1.60 (ord_less_eq_a @ v1 @ v)),
% 2.63/1.60 file('/export/starexec/sandbox/tmp/tmp.geO6X9CZKJ/DTF2THF_6481.p',fact_1_True)).
% 2.63/1.60 thf(f1076,plain,(
% 2.63/1.60 ((ord_less_eq_a @ v1 @ v) != $true) | ((ord_less_eq_a @ (val_a @ (t_a @ v2 @ l2 @ r2)) @ (val_a @ (t_a @ v1 @ l1 @ r1))) != $true)),
% 2.63/1.60 inference(forward_demodulation,[status(thm)],[f1075,f794])).
% 2.63/1.60 thf(f1075,plain,(
% 2.63/1.60 ((ord_less_eq_a @ (val_a @ (t_a @ v1 @ l1 @ r1)) @ v) != $true) | ((ord_less_eq_a @ (val_a @ (t_a @ v2 @ l2 @ r2)) @ (val_a @ (t_a @ v1 @ l1 @ r1))) != $true)),
% 2.63/1.60 inference(trivial_inequality_removal,[status(thm)],[f1069])).
% 2.63/1.60 thf(f1069,plain,(
% 2.63/1.60 ((ord_less_eq_a @ (val_a @ (t_a @ v2 @ l2 @ r2)) @ (val_a @ (t_a @ v1 @ l1 @ r1))) != $true) | ((multiset_a2 @ (t_a @ v @ (t_a @ v1 @ l1 @ r1) @ (t_a @ v2 @ l2 @ r2))) != (multiset_a2 @ (t_a @ v @ (t_a @ v1 @ l1 @ r1) @ (t_a @ v2 @ l2 @ r2)))) | ((ord_less_eq_a @ (val_a @ (t_a @ v1 @ l1 @ r1)) @ v) != $true)),
% 2.63/1.60 inference(superposition,[status(thm)],[f875,f977])).
% 2.63/1.60 thf(f977,plain,(
% 2.63/1.60 ( ! [X2 : a,X3 : tree_a,X0 : a,X1 : tree_a,X6 : tree_a,X4 : a,X5 : tree_a] : (((t_a @ X4 @ (t_a @ X2 @ X5 @ X6) @ (t_a @ X0 @ X3 @ X1)) = (heapIm1091024090Down_a @ (t_a @ X4 @ (t_a @ X2 @ X5 @ X6) @ (t_a @ X0 @ X3 @ X1)))) | ((ord_less_eq_a @ (val_a @ (t_a @ X0 @ X3 @ X1)) @ (val_a @ (t_a @ X2 @ X5 @ X6))) != $true) | ((ord_less_eq_a @ (val_a @ (t_a @ X2 @ X5 @ X6)) @ X4) != $true)) )),
% 2.63/1.60 inference(cnf_transformation,[status(thm)],[f644])).
% 2.63/1.60 thf(f644,plain,(
% 2.63/1.60 ! [X0 : a,X1 : tree_a,X2 : a,X3 : tree_a,X4 : a,X5 : tree_a,X6 : tree_a] : ((((ord_less_eq_a @ (val_a @ (t_a @ X0 @ X3 @ X1)) @ (val_a @ (t_a @ X2 @ X5 @ X6))) != $true) | ((((ord_less_eq_a @ (val_a @ (t_a @ X2 @ X5 @ X6)) @ X4) = $true) | ((t_a @ (val_a @ (t_a @ X2 @ X5 @ X6)) @ (heapIm1091024090Down_a @ (t_a @ X4 @ (heapIm1140443833left_a @ (t_a @ X2 @ X5 @ X6)) @ (heapIm1257206334ight_a @ (t_a @ X2 @ X5 @ X6)))) @ (t_a @ X0 @ X3 @ X1)) = (heapIm1091024090Down_a @ (t_a @ X4 @ (t_a @ X2 @ X5 @ X6) @ (t_a @ X0 @ X3 @ X1))))) & (((ord_less_eq_a @ (val_a @ (t_a @ X2 @ X5 @ X6)) @ X4) != $true) | ((t_a @ X4 @ (t_a @ X2 @ X5 @ X6) @ (t_a @ X0 @ X3 @ X1)) = (heapIm1091024090Down_a @ (t_a @ X4 @ (t_a @ X2 @ X5 @ X6) @ (t_a @ X0 @ X3 @ X1))))))) & (((ord_less_eq_a @ (val_a @ (t_a @ X0 @ X3 @ X1)) @ (val_a @ (t_a @ X2 @ X5 @ X6))) = $true) | ((((ord_less_eq_a @ (val_a @ (t_a @ X0 @ X3 @ X1)) @ X4) != $true) | ((t_a @ X4 @ (t_a @ X2 @ X5 @ X6) @ (t_a @ X0 @ X3 @ X1)) = (heapIm1091024090Down_a @ (t_a @ X4 @ (t_a @ X2 @ X5 @ X6) @ (t_a @ X0 @ X3 @ X1))))) & (((ord_less_eq_a @ (val_a @ (t_a @ X0 @ X3 @ X1)) @ X4) = $true) | ((heapIm1091024090Down_a @ (t_a @ X4 @ (t_a @ X2 @ X5 @ X6) @ (t_a @ X0 @ X3 @ X1))) = (t_a @ (val_a @ (t_a @ X0 @ X3 @ X1)) @ (t_a @ X2 @ X5 @ X6) @ (heapIm1091024090Down_a @ (t_a @ X4 @ (heapIm1140443833left_a @ (t_a @ X0 @ X3 @ X1)) @ (heapIm1257206334ight_a @ (t_a @ X0 @ X3 @ X1))))))))))),
% 2.63/1.60 inference(ennf_transformation,[status(thm)],[f589])).
% 2.63/1.60 thf(f589,plain,(
% 2.63/1.60 ! [X0 : a,X1 : tree_a,X2 : a,X3 : tree_a,X4 : a,X5 : tree_a,X6 : tree_a] : ((((ord_less_eq_a @ (val_a @ (t_a @ X0 @ X3 @ X1)) @ (val_a @ (t_a @ X2 @ X5 @ X6))) != $true) => ((((ord_less_eq_a @ (val_a @ (t_a @ X0 @ X3 @ X1)) @ X4) = $true) => ((t_a @ X4 @ (t_a @ X2 @ X5 @ X6) @ (t_a @ X0 @ X3 @ X1)) = (heapIm1091024090Down_a @ (t_a @ X4 @ (t_a @ X2 @ X5 @ X6) @ (t_a @ X0 @ X3 @ X1))))) & (((ord_less_eq_a @ (val_a @ (t_a @ X0 @ X3 @ X1)) @ X4) != $true) => ((heapIm1091024090Down_a @ (t_a @ X4 @ (t_a @ X2 @ X5 @ X6) @ (t_a @ X0 @ X3 @ X1))) = (t_a @ (val_a @ (t_a @ X0 @ X3 @ X1)) @ (t_a @ X2 @ X5 @ X6) @ (heapIm1091024090Down_a @ (t_a @ X4 @ (heapIm1140443833left_a @ (t_a @ X0 @ X3 @ X1)) @ (heapIm1257206334ight_a @ (t_a @ X0 @ X3 @ X1))))))))) & (((ord_less_eq_a @ (val_a @ (t_a @ X0 @ X3 @ X1)) @ (val_a @ (t_a @ X2 @ X5 @ X6))) = $true) => ((((ord_less_eq_a @ (val_a @ (t_a @ X2 @ X5 @ X6)) @ X4) != $true) => ((t_a @ (val_a @ (t_a @ X2 @ X5 @ X6)) @ (heapIm1091024090Down_a @ (t_a @ X4 @ (heapIm1140443833left_a @ (t_a @ X2 @ X5 @ X6)) @ (heapIm1257206334ight_a @ (t_a @ X2 @ X5 @ X6)))) @ (t_a @ X0 @ X3 @ X1)) = (heapIm1091024090Down_a @ (t_a @ X4 @ (t_a @ X2 @ X5 @ X6) @ (t_a @ X0 @ X3 @ X1))))) & (((ord_less_eq_a @ (val_a @ (t_a @ X2 @ X5 @ X6)) @ X4) = $true) => ((t_a @ X4 @ (t_a @ X2 @ X5 @ X6) @ (t_a @ X0 @ X3 @ X1)) = (heapIm1091024090Down_a @ (t_a @ X4 @ (t_a @ X2 @ X5 @ X6) @ (t_a @ X0 @ X3 @ X1))))))))),
% 2.63/1.60 inference(flattening,[status(thm)],[f391])).
% 2.63/1.60 thf(f391,plain,(
% 2.63/1.60 ! [X0 : a,X1 : tree_a,X2 : a,X3 : tree_a,X4 : a,X5 : tree_a,X6 : tree_a] : ((~((ord_less_eq_a @ (val_a @ (t_a @ X0 @ X3 @ X1)) @ (val_a @ (t_a @ X2 @ X5 @ X6))) = $true) => ((~((ord_less_eq_a @ (val_a @ (t_a @ X0 @ X3 @ X1)) @ X4) = $true) => ((heapIm1091024090Down_a @ (t_a @ X4 @ (t_a @ X2 @ X5 @ X6) @ (t_a @ X0 @ X3 @ X1))) = (t_a @ (val_a @ (t_a @ X0 @ X3 @ X1)) @ (t_a @ X2 @ X5 @ X6) @ (heapIm1091024090Down_a @ (t_a @ X4 @ (heapIm1140443833left_a @ (t_a @ X0 @ X3 @ X1)) @ (heapIm1257206334ight_a @ (t_a @ X0 @ X3 @ X1))))))) & (((ord_less_eq_a @ (val_a @ (t_a @ X0 @ X3 @ X1)) @ X4) = $true) => ((t_a @ X4 @ (t_a @ X2 @ X5 @ X6) @ (t_a @ X0 @ X3 @ X1)) = (heapIm1091024090Down_a @ (t_a @ X4 @ (t_a @ X2 @ X5 @ X6) @ (t_a @ X0 @ X3 @ X1))))))) & (((ord_less_eq_a @ (val_a @ (t_a @ X0 @ X3 @ X1)) @ (val_a @ (t_a @ X2 @ X5 @ X6))) = $true) => ((((ord_less_eq_a @ (val_a @ (t_a @ X2 @ X5 @ X6)) @ X4) = $true) => ((t_a @ X4 @ (t_a @ X2 @ X5 @ X6) @ (t_a @ X0 @ X3 @ X1)) = (heapIm1091024090Down_a @ (t_a @ X4 @ (t_a @ X2 @ X5 @ X6) @ (t_a @ X0 @ X3 @ X1))))) & (~((ord_less_eq_a @ (val_a @ (t_a @ X2 @ X5 @ X6)) @ X4) = $true) => ((t_a @ (val_a @ (t_a @ X2 @ X5 @ X6)) @ (heapIm1091024090Down_a @ (t_a @ X4 @ (heapIm1140443833left_a @ (t_a @ X2 @ X5 @ X6)) @ (heapIm1257206334ight_a @ (t_a @ X2 @ X5 @ X6)))) @ (t_a @ X0 @ X3 @ X1)) = (heapIm1091024090Down_a @ (t_a @ X4 @ (t_a @ X2 @ X5 @ X6) @ (t_a @ X0 @ X3 @ X1))))))))),
% 2.63/1.60 inference(fool_elimination,[status(thm)],[f390])).
% 2.63/1.60 thf(f390,plain,(
% 2.63/1.60 ! [X0 : a,X1 : tree_a,X2 : a,X3 : tree_a,X4 : a,X5 : tree_a,X6 : tree_a] : ((~(ord_less_eq_a @ (val_a @ (t_a @ X0 @ X3 @ X1)) @ (val_a @ (t_a @ X2 @ X5 @ X6))) => ((~(ord_less_eq_a @ (val_a @ (t_a @ X0 @ X3 @ X1)) @ X4) => ((heapIm1091024090Down_a @ (t_a @ X4 @ (t_a @ X2 @ X5 @ X6) @ (t_a @ X0 @ X3 @ X1))) = (t_a @ (val_a @ (t_a @ X0 @ X3 @ X1)) @ (t_a @ X2 @ X5 @ X6) @ (heapIm1091024090Down_a @ (t_a @ X4 @ (heapIm1140443833left_a @ (t_a @ X0 @ X3 @ X1)) @ (heapIm1257206334ight_a @ (t_a @ X0 @ X3 @ X1))))))) & ((ord_less_eq_a @ (val_a @ (t_a @ X0 @ X3 @ X1)) @ X4) => ((t_a @ X4 @ (t_a @ X2 @ X5 @ X6) @ (t_a @ X0 @ X3 @ X1)) = (heapIm1091024090Down_a @ (t_a @ X4 @ (t_a @ X2 @ X5 @ X6) @ (t_a @ X0 @ X3 @ X1))))))) & ((ord_less_eq_a @ (val_a @ (t_a @ X0 @ X3 @ X1)) @ (val_a @ (t_a @ X2 @ X5 @ X6))) => (((ord_less_eq_a @ (val_a @ (t_a @ X2 @ X5 @ X6)) @ X4) => ((t_a @ X4 @ (t_a @ X2 @ X5 @ X6) @ (t_a @ X0 @ X3 @ X1)) = (heapIm1091024090Down_a @ (t_a @ X4 @ (t_a @ X2 @ X5 @ X6) @ (t_a @ X0 @ X3 @ X1))))) & (~(ord_less_eq_a @ (val_a @ (t_a @ X2 @ X5 @ X6)) @ X4) => ((t_a @ (val_a @ (t_a @ X2 @ X5 @ X6)) @ (heapIm1091024090Down_a @ (t_a @ X4 @ (heapIm1140443833left_a @ (t_a @ X2 @ X5 @ X6)) @ (heapIm1257206334ight_a @ (t_a @ X2 @ X5 @ X6)))) @ (t_a @ X0 @ X3 @ X1)) = (heapIm1091024090Down_a @ (t_a @ X4 @ (t_a @ X2 @ X5 @ X6) @ (t_a @ X0 @ X3 @ X1))))))))),
% 2.63/1.60 inference(rectify,[status(thm)],[f33])).
% 2.63/1.60 thf(f33,axiom,(
% 2.63/1.60 ! [X18 : a,X20 : tree_a,X15 : a,X19 : tree_a,X1 : a,X16 : tree_a,X17 : tree_a] : ((~(ord_less_eq_a @ (val_a @ (t_a @ X18 @ X19 @ X20)) @ (val_a @ (t_a @ X15 @ X16 @ X17))) => ((~(ord_less_eq_a @ (val_a @ (t_a @ X18 @ X19 @ X20)) @ X1) => ((heapIm1091024090Down_a @ (t_a @ X1 @ (t_a @ X15 @ X16 @ X17) @ (t_a @ X18 @ X19 @ X20))) = (t_a @ (val_a @ (t_a @ X18 @ X19 @ X20)) @ (t_a @ X15 @ X16 @ X17) @ (heapIm1091024090Down_a @ (t_a @ X1 @ (heapIm1140443833left_a @ (t_a @ X18 @ X19 @ X20)) @ (heapIm1257206334ight_a @ (t_a @ X18 @ X19 @ X20))))))) & ((ord_less_eq_a @ (val_a @ (t_a @ X18 @ X19 @ X20)) @ X1) => ((t_a @ X1 @ (t_a @ X15 @ X16 @ X17) @ (t_a @ X18 @ X19 @ X20)) = (heapIm1091024090Down_a @ (t_a @ X1 @ (t_a @ X15 @ X16 @ X17) @ (t_a @ X18 @ X19 @ X20))))))) & ((ord_less_eq_a @ (val_a @ (t_a @ X18 @ X19 @ X20)) @ (val_a @ (t_a @ X15 @ X16 @ X17))) => (((ord_less_eq_a @ (val_a @ (t_a @ X15 @ X16 @ X17)) @ X1) => ((t_a @ X1 @ (t_a @ X15 @ X16 @ X17) @ (t_a @ X18 @ X19 @ X20)) = (heapIm1091024090Down_a @ (t_a @ X1 @ (t_a @ X15 @ X16 @ X17) @ (t_a @ X18 @ X19 @ X20))))) & (~(ord_less_eq_a @ (val_a @ (t_a @ X15 @ X16 @ X17)) @ X1) => ((heapIm1091024090Down_a @ (t_a @ X1 @ (t_a @ X15 @ X16 @ X17) @ (t_a @ X18 @ X19 @ X20))) = (t_a @ (val_a @ (t_a @ X15 @ X16 @ X17)) @ (heapIm1091024090Down_a @ (t_a @ X1 @ (heapIm1140443833left_a @ (t_a @ X15 @ X16 @ X17)) @ (heapIm1257206334ight_a @ (t_a @ X15 @ X16 @ X17)))) @ (t_a @ X18 @ X19 @ X20)))))))),
% 2.63/1.60 file('/export/starexec/sandbox/tmp/tmp.geO6X9CZKJ/DTF2THF_6481.p',fact_32_siftDown_Osimps_I6_J)).
% 2.63/1.60 thf(f875,plain,(
% 2.63/1.60 ((multiset_a2 @ (heapIm1091024090Down_a @ (t_a @ v @ (t_a @ v1 @ l1 @ r1) @ (t_a @ v2 @ l2 @ r2)))) != (multiset_a2 @ (t_a @ v @ (t_a @ v1 @ l1 @ r1) @ (t_a @ v2 @ l2 @ r2))))),
% 2.63/1.60 inference(cnf_transformation,[status(thm)],[f558])).
% 2.63/1.60 thf(f558,plain,(
% 2.63/1.60 ((multiset_a2 @ (heapIm1091024090Down_a @ (t_a @ v @ (t_a @ v1 @ l1 @ r1) @ (t_a @ v2 @ l2 @ r2)))) != (multiset_a2 @ (t_a @ v @ (t_a @ v1 @ l1 @ r1) @ (t_a @ v2 @ l2 @ r2))))),
% 2.63/1.60 inference(flattening,[status(thm)],[f216])).
% 2.63/1.60 thf(f216,negated_conjecture,(
% 2.63/1.60 ~((multiset_a2 @ (heapIm1091024090Down_a @ (t_a @ v @ (t_a @ v1 @ l1 @ r1) @ (t_a @ v2 @ l2 @ r2)))) = (multiset_a2 @ (t_a @ v @ (t_a @ v1 @ l1 @ r1) @ (t_a @ v2 @ l2 @ r2))))),
% 2.63/1.60 inference(negated_conjecture,[status(cth)],[f215])).
% 2.63/1.60 thf(f215,conjecture,(
% 2.63/1.60 ((multiset_a2 @ (heapIm1091024090Down_a @ (t_a @ v @ (t_a @ v1 @ l1 @ r1) @ (t_a @ v2 @ l2 @ r2)))) = (multiset_a2 @ (t_a @ v @ (t_a @ v1 @ l1 @ r1) @ (t_a @ v2 @ l2 @ r2))))),
% 2.63/1.60 file('/export/starexec/sandbox/tmp/tmp.geO6X9CZKJ/DTF2THF_6481.p',conj_0)).
% 2.63/1.60 % SZS output end Proof for DTF2THF_6481
% 2.63/1.60 % (6769)------------------------------
% 2.63/1.60 % (6769)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.60 % (6769)Termination reason: Refutation
% 2.63/1.60
% 2.63/1.60 % (6769)Memory used [KB]: 6908
% 2.63/1.60 % (6769)Time elapsed: 0.058 s
% 2.63/1.60 % (6769)Instructions burned: 96 (million)
% 2.63/1.60 % (6769)------------------------------
% 2.63/1.60 % (6769)------------------------------
% 2.63/1.60 % (6763)Success in time 0.082 s
%------------------------------------------------------------------------------