%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SWW470^2 : TPTP v9.2.1. Released v5.3.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n010.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 09:48:05 AM UTC 2026
% Result : Theorem 5.99s 2.18s
% Output : Refutation 5.99s
% Verified :
% SZS Type : Refutation
% Derivation depth : 15
% Number of leaves : 6
% Syntax : Number of formulae : 42 ( 12 unt; 0 typ; 0 def)
% Number of atoms : 364 ( 73 equ; 0 cnn)
% Maximal formula atoms : 7 ( 8 avg)
% Number of connectives : 585 ( 49 ~; 27 |; 26 &; 461 @)
% ( 0 <=>; 22 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 125 ( 125 >; 0 *; 0 +; 0 <<)
% Number of symbols : 286 ( 283 usr; 9 con; 0-6 aty)
% Number of variables : 195 ( 81 ^; 108 !; 6 ?; 195 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
nat: $tType ).
thf(type_def_7,type,
vname: $tType ).
thf(type_def_8,type,
state: $tType ).
thf(type_def_9,type,
com: $tType ).
thf(type_def_10,type,
loc: $tType ).
thf(type_def_11,type,
glb: $tType ).
thf(type_def_12,type,
hoare_1775062406iple_a: $tType ).
thf(type_def_13,type,
hoare_1167836817_state: $tType ).
thf(type_def_14,type,
x_a: $tType ).
thf(func_def_0,type,
x_a: $tType ).
thf(func_def_1,type,
com: $tType ).
thf(func_def_2,type,
glb: $tType ).
thf(func_def_3,type,
loc: $tType ).
thf(func_def_4,type,
state: $tType ).
thf(func_def_5,type,
vname: $tType ).
thf(func_def_6,type,
hoare_1775062406iple_a: $tType ).
thf(func_def_7,type,
hoare_1167836817_state: $tType ).
thf(func_def_8,type,
nat: $tType ).
thf(func_def_9,type,
big_la43341705in_nat: ( nat > $o ) > nat ).
thf(func_def_10,type,
big_se275732192ig_nat: ( nat > nat > nat ) > ( ( nat > $o ) > nat ) > $o ).
thf(func_def_11,type,
ass: vname > ( state > nat ) > com ).
thf(func_def_12,type,
local: loc > ( state > nat ) > com > com ).
thf(func_def_13,type,
skip: com ).
thf(func_def_14,type,
semi: com > com > com ).
thf(func_def_15,type,
glb_1: glb > vname ).
thf(func_def_16,type,
loc_1: loc > vname ).
thf(func_def_17,type,
finite2064891473iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > $o ).
thf(func_def_18,type,
finite2120172977le_a_o: ( hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ) > $o ).
thf(func_def_19,type,
finite856902323tate_o: ( hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ) > $o ).
thf(func_def_20,type,
finite389864113_nat_o: ( nat > ( nat > $o ) > nat > $o ) > $o ).
thf(func_def_21,type,
finite_finite_nat_o: ( ( nat > $o ) > $o ) > $o ).
thf(func_def_22,type,
finite2063573081iple_a: ( hoare_1775062406iple_a > $o ) > $o ).
thf(func_def_23,type,
finite1084549118_state: ( hoare_1167836817_state > $o ) > $o ).
thf(func_def_24,type,
finite_finite_nat: ( nat > $o ) > $o ).
thf(func_def_25,type,
finite1946188886iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
thf(func_def_26,type,
finite309220289_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(func_def_27,type,
finite_fold1Set_nat: ( nat > nat > nat ) > ( nat > $o ) > nat > $o ).
thf(func_def_28,type,
finite1790765286iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
thf(func_def_29,type,
finite1646097201_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
thf(func_def_30,type,
finite_fold1_nat: ( nat > nat > nat ) > ( nat > $o ) > nat ).
thf(func_def_31,type,
finite1544171829le_a_o: ( hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
thf(func_def_32,type,
finite1842721992iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
thf(func_def_33,type,
finite291020855tate_o: ( hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(func_def_34,type,
finite1731015960_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
thf(func_def_35,type,
finite326637109_nat_o: ( nat > ( nat > $o ) > nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(func_def_36,type,
finite_fold_nat_nat: ( nat > nat > nat ) > nat > ( nat > $o ) > nat ).
thf(func_def_37,type,
finite727644230iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
thf(func_def_38,type,
finite1316643734_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(func_def_39,type,
finite929467206at_nat: ( nat > nat > nat ) > nat > ( nat > $o ) > nat > $o ).
thf(func_def_40,type,
finite2078349315iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > ( ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ) > $o ).
thf(func_def_41,type,
finite1074406356_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > ( ( hoare_1167836817_state > $o ) > hoare_1167836817_state ) > $o ).
thf(func_def_42,type,
finite988810631ne_nat: ( nat > nat > nat ) > ( ( nat > $o ) > nat ) > $o ).
thf(func_def_43,type,
finite1358382848iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > ( ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ) > $o ).
thf(func_def_44,type,
finite806517911_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > ( ( hoare_1167836817_state > $o ) > hoare_1167836817_state ) > $o ).
thf(func_def_45,type,
finite795500164em_nat: ( nat > nat > nat ) > ( ( nat > $o ) > nat ) > $o ).
thf(func_def_46,type,
minus_1944206118le_a_o: ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
thf(func_def_47,type,
minus_2107060239tate_o: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(func_def_48,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(func_def_49,type,
times_times_nat: nat > nat > nat ).
thf(func_def_50,type,
the_Ho1155011127iple_a: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
thf(func_def_51,type,
the_Ho310147232_state: ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
thf(func_def_52,type,
the_nat: ( nat > $o ) > nat ).
thf(func_def_53,type,
hoare_Mirabelle_MGT: com > hoare_1167836817_state ).
thf(func_def_54,type,
hoare_1508237396rivs_a: ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > $o ).
thf(func_def_55,type,
hoare_123228589_state: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > $o ).
thf(func_def_56,type,
hoare_1766022166iple_a: ( x_a > state > $o ) > com > ( x_a > state > $o ) > hoare_1775062406iple_a ).
thf(func_def_57,type,
hoare_908217195_state: ( state > state > $o ) > com > ( state > state > $o ) > hoare_1167836817_state ).
thf(func_def_58,type,
hoare_1462269968alid_a: nat > hoare_1775062406iple_a > $o ).
thf(func_def_59,type,
hoare_56934129_state: nat > hoare_1167836817_state > $o ).
thf(func_def_60,type,
semila966743401le_a_o: ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
thf(func_def_61,type,
semila179895820tate_o: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(func_def_62,type,
semila1947288293_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(func_def_63,type,
semila854092349_inf_o: $o > $o > $o ).
thf(func_def_64,type,
semila80283416nf_nat: nat > nat > nat ).
thf(func_def_65,type,
semila13410563le_a_o: ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
thf(func_def_66,type,
semila1172322802tate_o: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(func_def_67,type,
semila848761471_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(func_def_68,type,
semila10642723_sup_o: $o > $o > $o ).
thf(func_def_69,type,
semila972727038up_nat: nat > nat > nat ).
thf(func_def_70,type,
evalc: com > state > state > $o ).
thf(func_def_71,type,
evaln: com > state > nat > state > $o ).
thf(func_def_72,type,
getlocs: state > loc > nat ).
thf(func_def_73,type,
update: state > vname > nat > state ).
thf(func_def_74,type,
bot_bot_nat_o_o: ( nat > $o ) > $o ).
thf(func_def_75,type,
bot_bo751897185le_a_o: hoare_1775062406iple_a > $o ).
thf(func_def_76,type,
bot_bo70021908tate_o: hoare_1167836817_state > $o ).
thf(func_def_77,type,
bot_bot_nat_o: nat > $o ).
thf(func_def_79,type,
bot_bot_nat: nat ).
thf(func_def_80,type,
ord_le1143225901le_a_o: ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > $o ).
thf(func_def_81,type,
ord_le827224136tate_o: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > $o ).
thf(func_def_82,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_83,type,
ord_less_eq_o: $o > $o > $o ).
thf(func_def_84,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_85,type,
partia126998524iple_a: hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
thf(func_def_86,type,
partia715677851_state: hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
thf(func_def_87,type,
partial_flat_lub_nat: nat > ( nat > $o ) > nat ).
thf(func_def_88,type,
collect_nat_o: ( ( nat > $o ) > $o ) > ( nat > $o ) > $o ).
thf(func_def_89,type,
collec676402587iple_a: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
thf(func_def_90,type,
collec1027672124_state: ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(func_def_91,type,
collect_nat: ( nat > $o ) > nat > $o ).
thf(func_def_92,type,
image_1170193413iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
thf(func_def_93,type,
image_1021683026_state: ( hoare_1775062406iple_a > hoare_1167836817_state ) > ( hoare_1775062406iple_a > $o ) > hoare_1167836817_state > $o ).
thf(func_def_94,type,
image_1806517641_a_nat: ( hoare_1775062406iple_a > nat ) > ( hoare_1775062406iple_a > $o ) > nat > $o ).
thf(func_def_95,type,
image_1802845250iple_a: ( hoare_1167836817_state > hoare_1775062406iple_a ) > ( hoare_1167836817_state > $o ) > hoare_1775062406iple_a > $o ).
thf(func_def_96,type,
image_31595733_state: ( hoare_1167836817_state > hoare_1167836817_state ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(func_def_97,type,
image_1476618182te_nat: ( hoare_1167836817_state > nat ) > ( hoare_1167836817_state > $o ) > nat > $o ).
thf(func_def_98,type,
image_43014529iple_a: ( nat > hoare_1775062406iple_a ) > ( nat > $o ) > hoare_1775062406iple_a > $o ).
thf(func_def_99,type,
image_2121260246_state: ( nat > hoare_1167836817_state ) > ( nat > $o ) > hoare_1167836817_state > $o ).
thf(func_def_100,type,
image_nat_nat: ( nat > nat ) > ( nat > $o ) > nat > $o ).
thf(func_def_101,type,
insert_nat_o: ( nat > $o ) > ( ( nat > $o ) > $o ) > ( nat > $o ) > $o ).
thf(func_def_102,type,
insert1281456128iple_a: hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
thf(func_def_103,type,
insert2134838167_state: hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(func_def_104,type,
insert_nat: nat > ( nat > $o ) > nat > $o ).
thf(func_def_105,type,
the_el1844711461iple_a: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
thf(func_def_106,type,
the_el323660082_state: ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
thf(func_def_107,type,
the_elem_nat: ( nat > $o ) > nat ).
thf(func_def_108,type,
fequal_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_109,type,
fequal_state: state > state > $o ).
thf(func_def_110,type,
fequal1288209029iple_a: hoare_1775062406iple_a > hoare_1775062406iple_a > $o ).
thf(func_def_111,type,
fequal1831255762_state: hoare_1167836817_state > hoare_1167836817_state > $o ).
thf(func_def_112,type,
fequal_nat: nat > nat > $o ).
thf(func_def_113,type,
member_nat_o: ( nat > $o ) > ( ( nat > $o ) > $o ) > $o ).
thf(func_def_114,type,
member2122167641iple_a: hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > $o ).
thf(func_def_115,type,
member2058392318_state: hoare_1167836817_state > ( hoare_1167836817_state > $o ) > $o ).
thf(func_def_116,type,
member_nat: nat > ( nat > $o ) > $o ).
thf(func_def_117,type,
g: hoare_1775062406iple_a > $o ).
thf(func_def_118,type,
p: x_a > state > $o ).
thf(func_def_119,type,
b: state > $o ).
thf(func_def_120,type,
c: com ).
thf(func_def_133,type,
sK0:
!>[X0: $tType] : ( ( X0 > $o ) > X0 ) ).
thf(func_def_134,type,
sK1:
!>[X0: $tType] : ( ( X0 > $o ) > X0 ) ).
thf(func_def_136,type,
sP2: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > hoare_1775062406iple_a > $o ).
thf(func_def_137,type,
sK3: state > state > com > state > state > com > nat ).
thf(func_def_138,type,
sK4: ( nat > $o ) > nat ).
thf(func_def_139,type,
sK5: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > hoare_1167836817_state ).
thf(func_def_140,type,
sK6: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > hoare_1167836817_state ).
thf(func_def_141,type,
sK7: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
thf(func_def_142,type,
sK8: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
thf(func_def_143,type,
sK9: ( nat > $o ) > nat ).
thf(func_def_144,type,
sK10: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
thf(func_def_145,type,
sK11: ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > nat ) > nat > hoare_1775062406iple_a ).
thf(func_def_146,type,
sK12: hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
thf(func_def_147,type,
sK13: nat > ( nat > $o ) > ( nat > nat > nat ) > nat ).
thf(func_def_148,type,
sK14: nat > ( nat > $o ) > ( nat > nat > nat ) > nat > $o ).
thf(func_def_149,type,
sK15: nat > ( nat > $o ) > ( nat > nat > nat ) > nat ).
thf(func_def_150,type,
sK16: nat > ( nat > $o ) > nat > ( nat > nat > nat ) > nat ).
thf(func_def_151,type,
sK17: nat > ( nat > $o ) > nat > ( nat > nat > nat ) > nat > $o ).
thf(func_def_152,type,
sK18: ( nat > $o ) > nat ).
thf(func_def_153,type,
sK19: ( nat > $o ) > nat > $o ).
thf(func_def_154,type,
sK20: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > hoare_1775062406iple_a > $o ).
thf(func_def_155,type,
sK21: ( hoare_1775062406iple_a > $o ) > ( ( hoare_1775062406iple_a > $o ) > $o ) > hoare_1775062406iple_a ).
thf(func_def_156,type,
sK22: ( hoare_1775062406iple_a > $o ) > ( ( hoare_1775062406iple_a > $o ) > $o ) > hoare_1775062406iple_a > $o ).
thf(func_def_157,type,
sK23: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > hoare_1167836817_state ) > hoare_1167836817_state > hoare_1167836817_state ).
thf(func_def_158,type,
sK24: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
thf(func_def_159,type,
sK25: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
thf(func_def_160,type,
sK26: ( nat > $o ) > nat ).
thf(func_def_161,type,
sK27: ( x_a > state > $o ) > ( x_a > state > $o ) > x_a ).
thf(func_def_162,type,
sK28: ( x_a > state > $o ) > ( x_a > state > $o ) > state ).
thf(func_def_163,type,
sK29: nat ).
thf(func_def_164,type,
sK30: nat ).
thf(func_def_165,type,
sK31: ( state > state > $o ) > ( state > state > $o ) > com > ( hoare_1167836817_state > $o ) > state ).
thf(func_def_166,type,
sK32: ( state > state > $o ) > ( state > state > $o ) > com > ( hoare_1167836817_state > $o ) > state ).
thf(func_def_167,type,
sK33: hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
thf(func_def_168,type,
sK34: ( ( nat > $o ) > $o ) > nat ).
thf(func_def_169,type,
sK35: ( ( nat > $o ) > $o ) > nat > $o ).
thf(func_def_170,type,
sK36: ( state > state > $o ) > com > ( state > state > $o ) > nat > state ).
thf(func_def_171,type,
sK37: ( state > state > $o ) > com > ( state > state > $o ) > nat > state ).
thf(func_def_172,type,
sK38: ( state > state > $o ) > com > ( state > state > $o ) > nat > state ).
thf(func_def_173,type,
sK39: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > hoare_1775062406iple_a > hoare_1775062406iple_a ).
thf(func_def_174,type,
sK40: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > hoare_1775062406iple_a > hoare_1775062406iple_a > $o ).
thf(func_def_175,type,
sK41: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > hoare_1775062406iple_a > hoare_1775062406iple_a ).
thf(func_def_176,type,
sK42: ( ( hoare_1775062406iple_a > $o ) > $o ) > hoare_1775062406iple_a ).
thf(func_def_177,type,
sK43: ( ( hoare_1775062406iple_a > $o ) > $o ) > hoare_1775062406iple_a > $o ).
thf(func_def_178,type,
sK44: ( nat > $o ) > nat > $o ).
thf(func_def_179,type,
sK45: ( nat > $o ) > nat ).
thf(func_def_180,type,
sK46: ( hoare_1167836817_state > $o ) > hoare_1167836817_state > hoare_1167836817_state > $o ).
thf(func_def_181,type,
sK47: ( nat > nat > nat ) > nat ).
thf(func_def_182,type,
sK48: ( nat > nat > nat ) > nat ).
thf(func_def_183,type,
sK49: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
thf(func_def_184,type,
sK50: hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > hoare_1167836817_state ).
thf(func_def_185,type,
sK51: hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > hoare_1167836817_state > $o ).
thf(func_def_186,type,
sK52: ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
thf(func_def_187,type,
sK53: ( nat > $o ) > nat ).
thf(func_def_188,type,
sK54: ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
thf(func_def_189,type,
sK55: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
thf(func_def_190,type,
sK56: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
thf(func_def_191,type,
sK57: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > hoare_1775062406iple_a > $o ).
thf(func_def_192,type,
sK58: ( nat > $o ) > nat ).
thf(func_def_193,type,
sK59: nat > ( nat > $o ) > nat ).
thf(func_def_194,type,
sK60: ( ( nat > $o ) > $o ) > nat > $o ).
thf(func_def_195,type,
sK61: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
thf(func_def_196,type,
sK62: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ).
thf(func_def_197,type,
sK63: ( ( hoare_1167836817_state > $o ) > $o ) > hoare_1167836817_state ).
thf(func_def_198,type,
sK64: ( ( hoare_1167836817_state > $o ) > $o ) > hoare_1167836817_state > $o ).
thf(func_def_199,type,
sK65: ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_200,type,
sK66: ( x_a > state > $o ) > ( x_a > state > $o ) > x_a ).
thf(func_def_201,type,
sK67: ( x_a > state > $o ) > ( x_a > state > $o ) > state ).
thf(func_def_202,type,
sK68: hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > hoare_1775062406iple_a ).
thf(func_def_203,type,
sK69: hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > hoare_1775062406iple_a > $o ).
thf(func_def_204,type,
sK70: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > ( hoare_1775062406iple_a > hoare_1775062406iple_a ) > hoare_1775062406iple_a ).
thf(func_def_205,type,
sK71: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > ( hoare_1775062406iple_a > hoare_1775062406iple_a ) > hoare_1775062406iple_a ).
thf(func_def_206,type,
sK72: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ).
thf(func_def_207,type,
sK73: ( nat > nat ) > nat ).
thf(func_def_208,type,
sK74: state > state > com > ( state > nat ) > loc > state ).
thf(func_def_209,type,
sK75: ( hoare_1167836817_state > $o ) > hoare_1167836817_state > hoare_1167836817_state > $o ).
thf(func_def_210,type,
sK76: hoare_1167836817_state > state > state > $o ).
thf(func_def_211,type,
sK77: hoare_1167836817_state > com ).
thf(func_def_212,type,
sK78: hoare_1167836817_state > state > state > $o ).
thf(func_def_213,type,
sK79: ( ( nat > $o ) > $o ) > nat ).
thf(func_def_214,type,
sK80: ( ( nat > $o ) > $o ) > nat > $o ).
thf(func_def_215,type,
sK81: ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > hoare_1167836817_state ) > hoare_1167836817_state > hoare_1775062406iple_a ).
thf(func_def_216,type,
sK82: state > state > com > nat ).
thf(func_def_217,type,
sK83: ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
thf(func_def_218,type,
sK84: ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
thf(func_def_219,type,
sK85: ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
thf(func_def_220,type,
sK86: state > nat > state > com > ( state > nat ) > loc > state ).
thf(func_def_221,type,
sK87: ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
thf(func_def_222,type,
sK88: ( ( hoare_1167836817_state > $o ) > $o ) > hoare_1167836817_state ).
thf(func_def_223,type,
sK89: ( ( hoare_1167836817_state > $o ) > $o ) > hoare_1167836817_state > $o ).
thf(func_def_224,type,
sK90: ( nat > $o ) > ( nat > hoare_1775062406iple_a ) > hoare_1775062406iple_a > nat ).
thf(func_def_225,type,
sK91: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
thf(func_def_226,type,
sK92: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
thf(func_def_227,type,
sK93: ( ( nat > $o ) > $o ) > nat ).
thf(func_def_228,type,
sK94: ( ( nat > $o ) > $o ) > nat ).
thf(func_def_229,type,
sK95: ( ( nat > $o ) > $o ) > nat > $o ).
thf(func_def_230,type,
sK96: ( nat > $o ) > nat > nat ).
thf(func_def_231,type,
sK97: ( nat > $o ) > ( nat > nat ) > nat > nat ).
thf(func_def_232,type,
sK98: ( state > state > $o ) > ( state > state > $o ) > state ).
thf(func_def_233,type,
sK99: ( state > state > $o ) > ( state > state > $o ) > state ).
thf(func_def_234,type,
sK100: ( x_a > state > $o ) > com > ( hoare_1775062406iple_a > $o ) > ( x_a > state > $o ) > x_a ).
thf(func_def_235,type,
sK101: ( x_a > state > $o ) > com > ( hoare_1775062406iple_a > $o ) > ( x_a > state > $o ) > state ).
thf(func_def_236,type,
sK102: ( x_a > state > $o ) > ( x_a > state > $o ) > ( x_a > state > $o ) > com > ( hoare_1775062406iple_a > $o ) > ( x_a > state > $o ) > state ).
thf(func_def_237,type,
sK103: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > ( hoare_1167836817_state > hoare_1167836817_state ) > hoare_1167836817_state ).
thf(func_def_238,type,
sK104: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > ( hoare_1167836817_state > hoare_1167836817_state ) > hoare_1167836817_state ).
thf(func_def_239,type,
sK105: ( state > state > $o ) > ( state > state > $o ) > ( state > state > $o ) > ( state > state > $o ) > state ).
thf(func_def_240,type,
sK106: ( state > state > $o ) > ( state > state > $o ) > ( state > state > $o ) > ( state > state > $o ) > state ).
thf(func_def_241,type,
sK107: ( state > state > $o ) > ( state > state > $o ) > ( state > state > $o ) > ( state > state > $o ) > state ).
thf(func_def_242,type,
sK108: hoare_1167836817_state > ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > hoare_1167836817_state ).
thf(func_def_243,type,
sK109: hoare_1167836817_state > ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > hoare_1167836817_state > $o ).
thf(func_def_244,type,
sK110: hoare_1167836817_state > ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > hoare_1167836817_state ).
thf(func_def_245,type,
sK111: ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(func_def_246,type,
sK112: ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
thf(func_def_247,type,
sK113: state > state > com > nat ).
thf(func_def_248,type,
sK114: ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
thf(func_def_249,type,
sK115: state > state > com > com > state ).
thf(func_def_250,type,
sK116: state > nat > state > com > com > state ).
thf(func_def_251,type,
sK117: ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_252,type,
sK118: ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_253,type,
sK119: ( x_a > state > $o ) > com > ( x_a > state > $o ) > nat > x_a ).
thf(func_def_254,type,
sK120: ( x_a > state > $o ) > com > ( x_a > state > $o ) > nat > state ).
thf(func_def_255,type,
sK121: ( x_a > state > $o ) > com > ( x_a > state > $o ) > nat > state ).
thf(func_def_256,type,
sK122: ( x_a > state > $o ) > ( x_a > state > $o ) > ( x_a > state > $o ) > ( x_a > state > $o ) > x_a ).
thf(func_def_257,type,
sK123: ( x_a > state > $o ) > ( x_a > state > $o ) > ( x_a > state > $o ) > ( x_a > state > $o ) > state ).
thf(func_def_258,type,
sK124: ( x_a > state > $o ) > ( x_a > state > $o ) > ( x_a > state > $o ) > ( x_a > state > $o ) > state ).
thf(func_def_259,type,
sK125: ( ( hoare_1167836817_state > $o ) > $o ) > hoare_1167836817_state ).
thf(func_def_260,type,
sK126: ( ( hoare_1167836817_state > $o ) > $o ) > hoare_1167836817_state ).
thf(func_def_261,type,
sK127: ( ( hoare_1167836817_state > $o ) > $o ) > hoare_1167836817_state > $o ).
thf(func_def_262,type,
sK128: ( ( hoare_1775062406iple_a > $o ) > $o ) > hoare_1775062406iple_a ).
thf(func_def_263,type,
sK129: ( ( hoare_1775062406iple_a > $o ) > $o ) > hoare_1775062406iple_a > $o ).
thf(func_def_264,type,
sK130: ( ( hoare_1775062406iple_a > $o ) > $o ) > hoare_1775062406iple_a ).
thf(func_def_265,type,
sK131: hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
thf(func_def_266,type,
sK132: ( x_a > state > $o ) > ( x_a > state > $o ) > com > ( hoare_1775062406iple_a > $o ) > x_a ).
thf(func_def_267,type,
sK133: ( x_a > state > $o ) > ( x_a > state > $o ) > com > ( hoare_1775062406iple_a > $o ) > state ).
thf(func_def_268,type,
sK134: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > hoare_1775062406iple_a ).
thf(func_def_269,type,
sK135: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > hoare_1775062406iple_a ).
thf(func_def_270,type,
sK136: vname > loc ).
thf(func_def_271,type,
sK137: vname > glb ).
thf(func_def_272,type,
sK138: hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > hoare_1775062406iple_a ).
thf(func_def_273,type,
sK139: hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > hoare_1775062406iple_a > $o ).
thf(func_def_274,type,
sK140: hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > hoare_1775062406iple_a ).
thf(func_def_275,type,
sK141: ( ( hoare_1775062406iple_a > $o ) > $o ) > hoare_1775062406iple_a ).
thf(func_def_276,type,
sK142: ( ( hoare_1775062406iple_a > $o ) > $o ) > hoare_1775062406iple_a > $o ).
thf(func_def_277,type,
sK143: ( nat > nat > nat ) > ( nat > nat ) > nat ).
thf(func_def_278,type,
sK144: ( nat > nat > nat ) > ( nat > nat ) > nat ).
thf(func_def_279,type,
sK145: ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
thf(func_def_280,type,
sK146: ( nat > $o ) > nat > nat > $o ).
thf(func_def_281,type,
sK147: ( state > state > $o ) > ( state > state > $o ) > state ).
thf(func_def_282,type,
sK148: ( state > state > $o ) > ( state > state > $o ) > state ).
thf(func_def_283,type,
sK149: hoare_1775062406iple_a > x_a > state > $o ).
thf(func_def_284,type,
sK150: hoare_1775062406iple_a > com ).
thf(func_def_285,type,
sK151: hoare_1775062406iple_a > x_a > state > $o ).
thf(func_def_286,type,
sK152: ( nat > $o ) > nat > nat > $o ).
thf(func_def_287,type,
sK153: hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
thf(func_def_288,type,
sK154: ( nat > $o ) > nat ).
thf(func_def_289,type,
sK155: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > hoare_1775062406iple_a ) > hoare_1775062406iple_a > hoare_1167836817_state ).
thf(func_def_290,type,
sK156: ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > hoare_1775062406iple_a ) > hoare_1775062406iple_a > hoare_1775062406iple_a ).
thf(func_def_291,type,
sK157: ( nat > $o ) > ( ( nat > $o ) > $o ) > nat ).
thf(func_def_292,type,
sK158: ( nat > $o ) > ( ( nat > $o ) > $o ) > nat > $o ).
thf(func_def_293,type,
sK159: nat ).
thf(func_def_294,type,
sK160: nat ).
thf(func_def_295,type,
sK161: ( ( nat > $o ) > $o ) > nat > $o ).
thf(func_def_296,type,
sK162: ( state > state > $o ) > com > ( hoare_1167836817_state > $o ) > ( state > state > $o ) > state ).
thf(func_def_297,type,
sK163: ( state > state > $o ) > com > ( hoare_1167836817_state > $o ) > ( state > state > $o ) > state ).
thf(func_def_298,type,
sK164: ( state > state > $o ) > ( state > state > $o ) > ( state > state > $o ) > com > ( hoare_1167836817_state > $o ) > ( state > state > $o ) > state ).
thf(func_def_299,type,
sK165: ( nat > nat ) > nat ).
thf(func_def_300,type,
sK166: ( nat > nat ) > nat ).
thf(func_def_301,type,
sK167: ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
thf(func_def_302,type,
sK168: ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
thf(func_def_303,type,
sK169: ( hoare_1167836817_state > $o ) > ( ( hoare_1167836817_state > $o ) > $o ) > hoare_1167836817_state ).
thf(func_def_304,type,
sK170: ( hoare_1167836817_state > $o ) > ( ( hoare_1167836817_state > $o ) > $o ) > hoare_1167836817_state > $o ).
thf(func_def_306,type,
ph172:
!>[X0: $tType] : X0 ).
thf(f5062,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f5061,f5057]) ).
thf(f5057,plain,
! [X1: $o] : ( $true = X1 ),
inference(trivial_inequality_removal,[status(thm)],[f5045]) ).
thf(f5045,plain,
! [X1: $o] :
( ( $true != $true )
| ( $true = X1 ) ),
inference(superposition,[status(thm)],[f4971,f4502]) ).
thf(f4502,plain,
! [X2: com,X3: x_a > state > $o,X0: hoare_1775062406iple_a > $o,X1: x_a > state > $o,X4: $o] :
( ( $true
= ( hoare_1508237396rivs_a @ X0
@ ( insert1281456128iple_a
@ ( hoare_1766022166iple_a
@ ^ [Y0: x_a,Y1: state] :
( ( X1 @ Y0 @ Y1 )
& X4 )
@ X2
@ X3 )
@ ( collec676402587iple_a
@ ^ [Y0: hoare_1775062406iple_a] : $false ) ) ) )
| ( $true = X4 ) ),
inference(definition_unfolding,[status(thm)],[f3708,f4102]) ).
thf(f4102,plain,
( bot_bo751897185le_a_o
= ( collec676402587iple_a
@ ^ [Y0: hoare_1775062406iple_a] : $false ) ),
inference(cnf_transformation,[status(thm)],[f1489]) ).
thf(f1489,plain,
( bot_bo751897185le_a_o
= ( collec676402587iple_a
@ ^ [Y0: hoare_1775062406iple_a] : $false ) ),
inference(fool_elimination,[status(thm)],[f1488]) ).
thf(f1488,plain,
( ( collec676402587iple_a
@ ^ [X0: hoare_1775062406iple_a] : $false )
= bot_bo751897185le_a_o ),
inference(rectify,[status(thm)],[f65]) ).
thf(f65,axiom,
( ( collec676402587iple_a
@ ^ [X52: hoare_1775062406iple_a] : $false )
= bot_bo751897185le_a_o ),
file('/export/starexec/sandbox/tmp/tmp.78tw4nyiuM/DTF2THF_30168.p',fact_64_empty__def) ).
thf(f3708,plain,
! [X2: com,X3: x_a > state > $o,X0: hoare_1775062406iple_a > $o,X1: x_a > state > $o,X4: $o] :
( ( $true
= ( hoare_1508237396rivs_a @ X0
@ ( insert1281456128iple_a
@ ( hoare_1766022166iple_a
@ ^ [Y0: x_a,Y1: state] :
( ( X1 @ Y0 @ Y1 )
& X4 )
@ X2
@ X3 )
@ bot_bo751897185le_a_o ) ) )
| ( $true = X4 ) ),
inference(cnf_transformation,[status(thm)],[f2287]) ).
thf(f2287,plain,
! [X0: hoare_1775062406iple_a > $o,X1: x_a > state > $o,X2: com,X3: x_a > state > $o,X4: $o] :
( ( $true
= ( hoare_1508237396rivs_a @ X0
@ ( insert1281456128iple_a
@ ( hoare_1766022166iple_a
@ ^ [Y0: x_a,Y1: state] :
( ( X1 @ Y0 @ Y1 )
& X4 )
@ X2
@ X3 )
@ bot_bo751897185le_a_o ) ) )
| ( ( $true
!= ( hoare_1508237396rivs_a @ X0 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ X1 @ X2 @ X3 ) @ bot_bo751897185le_a_o ) ) )
& ( $true = X4 ) ) ),
inference(ennf_transformation,[status(thm)],[f999]) ).
thf(f999,plain,
! [X0: hoare_1775062406iple_a > $o,X1: x_a > state > $o,X2: com,X3: x_a > state > $o,X4: $o] :
( ( ( $true = X4 )
=> ( $true
= ( hoare_1508237396rivs_a @ X0 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ X1 @ X2 @ X3 ) @ bot_bo751897185le_a_o ) ) ) )
=> ( $true
= ( hoare_1508237396rivs_a @ X0
@ ( insert1281456128iple_a
@ ( hoare_1766022166iple_a
@ ^ [Y0: x_a,Y1: state] :
( ( X1 @ Y0 @ Y1 )
& X4 )
@ X2
@ X3 )
@ bot_bo751897185le_a_o ) ) ) ),
inference(fool_elimination,[status(thm)],[f998]) ).
thf(f998,plain,
! [X0: hoare_1775062406iple_a > $o,X1: x_a > state > $o,X2: com,X3: x_a > state > $o,X4: $o] :
( ( X4
=> ( hoare_1508237396rivs_a @ X0 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ X1 @ X2 @ X3 ) @ bot_bo751897185le_a_o ) ) )
=> ( hoare_1508237396rivs_a @ X0
@ ( insert1281456128iple_a
@ ( hoare_1766022166iple_a
@ ^ [X5: x_a,X6: state] :
( ( X1 @ X5 @ X6 )
& X4 )
@ X2
@ X3 )
@ bot_bo751897185le_a_o ) ) ),
inference(rectify,[status(thm)],[f9]) ).
thf(f9,axiom,
! [X13: hoare_1775062406iple_a > $o,X14: x_a > state > $o,X15: com,X16: x_a > state > $o,X17: $o] :
( ( X17
=> ( hoare_1508237396rivs_a @ X13 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ X14 @ X15 @ X16 ) @ bot_bo751897185le_a_o ) ) )
=> ( hoare_1508237396rivs_a @ X13
@ ( insert1281456128iple_a
@ ( hoare_1766022166iple_a
@ ^ [X18: x_a,X19: state] :
( ( X14 @ X18 @ X19 )
& X17 )
@ X15
@ X16 )
@ bot_bo751897185le_a_o ) ) ),
file('/export/starexec/sandbox/tmp/tmp.78tw4nyiuM/DTF2THF_30168.p',fact_8_constant) ).
thf(f4971,plain,
! [X0: x_a > state > $o] :
( $true
!= ( hoare_1508237396rivs_a @ g
@ ( insert1281456128iple_a
@ ( hoare_1766022166iple_a @ X0 @ c
@ ^ [Y0: x_a,Y1: state] :
( ( p @ Y0 @ Y1 )
& ~ ( b @ Y1 ) ) )
@ ( collec676402587iple_a
@ ^ [Y0: hoare_1775062406iple_a] : $false ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f4970]) ).
thf(f4970,plain,
! [X0: x_a > state > $o] :
( ( $true = $false )
| ( $true
!= ( hoare_1508237396rivs_a @ g
@ ( insert1281456128iple_a
@ ( hoare_1766022166iple_a @ X0 @ c
@ ^ [Y0: x_a,Y1: state] :
( ( p @ Y0 @ Y1 )
& ~ ( b @ Y1 ) ) )
@ ( collec676402587iple_a
@ ^ [Y0: hoare_1775062406iple_a] : $false ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f4969]) ).
thf(f4969,plain,
! [X0: x_a > state > $o] :
( ( $true
!= ( hoare_1508237396rivs_a @ g
@ ( insert1281456128iple_a
@ ( hoare_1766022166iple_a @ X0 @ c
@ ^ [Y0: x_a,Y1: state] :
( ( p @ Y0 @ Y1 )
& ~ ( b @ Y1 ) ) )
@ ( collec676402587iple_a
@ ^ [Y0: hoare_1775062406iple_a] : $false ) ) ) )
| ( $true
= ( ^ [Y0: x_a,Y1: state] : $false
@ ( sK27 @ X0
@ ^ [Y0: x_a,Y1: state] : $false )
@ ( sK28 @ X0
@ ^ [Y0: x_a,Y1: state] : $false ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f4960]) ).
thf(f4960,plain,
! [X0: x_a > state > $o] :
( ( $true
!= ( hoare_1508237396rivs_a @ g
@ ( insert1281456128iple_a
@ ( hoare_1766022166iple_a @ X0 @ c
@ ^ [Y0: x_a,Y1: state] :
( ( p @ Y0 @ Y1 )
& ~ ( b @ Y1 ) ) )
@ ( collec676402587iple_a
@ ^ [Y0: hoare_1775062406iple_a] : $false ) ) ) )
| ( $true
= ( ^ [Y0: x_a,Y1: state] : $false
@ ( sK27 @ X0
@ ^ [Y0: x_a,Y1: state] : $false )
@ ( sK28 @ X0
@ ^ [Y0: x_a,Y1: state] : $false ) ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f4462,f4392]) ).
thf(f4392,plain,
! [X2: x_a > state > $o,X3: com,X0: x_a > state > $o,X1: hoare_1775062406iple_a > $o,X4: x_a > state > $o] :
( ( $true
= ( hoare_1508237396rivs_a @ X1
@ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ X0 @ X3 @ X4 )
@ ( collec676402587iple_a
@ ^ [Y0: hoare_1775062406iple_a] : $false ) ) ) )
| ( $true
!= ( hoare_1508237396rivs_a @ X1
@ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ X2 @ X3 @ X4 )
@ ( collec676402587iple_a
@ ^ [Y0: hoare_1775062406iple_a] : $false ) ) ) )
| ( ( X0 @ ( sK27 @ X2 @ X0 ) @ ( sK28 @ X2 @ X0 ) )
= $true ) ),
inference(definition_unfolding,[status(thm)],[f3466,f4102,f4102]) ).
thf(f3466,plain,
! [X2: x_a > state > $o,X3: com,X0: x_a > state > $o,X1: hoare_1775062406iple_a > $o,X4: x_a > state > $o] :
( ( $true
!= ( hoare_1508237396rivs_a @ X1 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ X2 @ X3 @ X4 ) @ bot_bo751897185le_a_o ) ) )
| ( $true
= ( hoare_1508237396rivs_a @ X1 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ X0 @ X3 @ X4 ) @ bot_bo751897185le_a_o ) ) )
| ( ( X0 @ ( sK27 @ X2 @ X0 ) @ ( sK28 @ X2 @ X0 ) )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f2903]) ).
thf(f2903,plain,
! [X0: x_a > state > $o,X1: hoare_1775062406iple_a > $o,X2: x_a > state > $o,X3: com,X4: x_a > state > $o] :
( ( $true
!= ( hoare_1508237396rivs_a @ X1 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ X2 @ X3 @ X4 ) @ bot_bo751897185le_a_o ) ) )
| ( $true
= ( hoare_1508237396rivs_a @ X1 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ X0 @ X3 @ X4 ) @ bot_bo751897185le_a_o ) ) )
| ( ( ( X2 @ ( sK27 @ X2 @ X0 ) @ ( sK28 @ X2 @ X0 ) )
!= $true )
& ( ( X0 @ ( sK27 @ X2 @ X0 ) @ ( sK28 @ X2 @ X0 ) )
= $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK27,sK28])],[f2657,f2902]) ).
thf(f2902,plain,
! [X0: x_a > state > $o,X2: x_a > state > $o] :
( ? [X5: x_a,X6: state] :
( ( $true
!= ( X2 @ X5 @ X6 ) )
& ( $true
= ( X0 @ X5 @ X6 ) ) )
=> ( ( ( X2 @ ( sK27 @ X2 @ X0 ) @ ( sK28 @ X2 @ X0 ) )
!= $true )
& ( ( X0 @ ( sK27 @ X2 @ X0 ) @ ( sK28 @ X2 @ X0 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2657,plain,
! [X0: x_a > state > $o,X1: hoare_1775062406iple_a > $o,X2: x_a > state > $o,X3: com,X4: x_a > state > $o] :
( ( $true
!= ( hoare_1508237396rivs_a @ X1 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ X2 @ X3 @ X4 ) @ bot_bo751897185le_a_o ) ) )
| ( $true
= ( hoare_1508237396rivs_a @ X1 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ X0 @ X3 @ X4 ) @ bot_bo751897185le_a_o ) ) )
| ? [X5: x_a,X6: state] :
( ( $true
!= ( X2 @ X5 @ X6 ) )
& ( $true
= ( X0 @ X5 @ X6 ) ) ) ),
inference(flattening,[status(thm)],[f2656]) ).
thf(f2656,plain,
! [X0: x_a > state > $o,X1: hoare_1775062406iple_a > $o,X2: x_a > state > $o,X3: com,X4: x_a > state > $o] :
( ( $true
= ( hoare_1508237396rivs_a @ X1 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ X0 @ X3 @ X4 ) @ bot_bo751897185le_a_o ) ) )
| ? [X5: x_a,X6: state] :
( ( $true
!= ( X2 @ X5 @ X6 ) )
& ( $true
= ( X0 @ X5 @ X6 ) ) )
| ( $true
!= ( hoare_1508237396rivs_a @ X1 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ X2 @ X3 @ X4 ) @ bot_bo751897185le_a_o ) ) ) ),
inference(ennf_transformation,[status(thm)],[f1527]) ).
thf(f1527,plain,
! [X0: x_a > state > $o,X1: hoare_1775062406iple_a > $o,X2: x_a > state > $o,X3: com,X4: x_a > state > $o] :
( ( $true
= ( hoare_1508237396rivs_a @ X1 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ X2 @ X3 @ X4 ) @ bot_bo751897185le_a_o ) ) )
=> ( ! [X5: x_a,X6: state] :
( ( $true
= ( X0 @ X5 @ X6 ) )
=> ( $true
= ( X2 @ X5 @ X6 ) ) )
=> ( $true
= ( hoare_1508237396rivs_a @ X1 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ X0 @ X3 @ X4 ) @ bot_bo751897185le_a_o ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f1526]) ).
thf(f1526,plain,
! [X0: x_a > state > $o,X1: hoare_1775062406iple_a > $o,X2: x_a > state > $o,X3: com,X4: x_a > state > $o] :
( ( hoare_1508237396rivs_a @ X1 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ X2 @ X3 @ X4 ) @ bot_bo751897185le_a_o ) )
=> ( ! [X5: x_a,X6: state] :
( ( X0 @ X5 @ X6 )
=> ( X2 @ X5 @ X6 ) )
=> ( hoare_1508237396rivs_a @ X1 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ X0 @ X3 @ X4 ) @ bot_bo751897185le_a_o ) ) ) ),
inference(rectify,[status(thm)],[f15]) ).
thf(f15,axiom,
! [X32: x_a > state > $o,X33: hoare_1775062406iple_a > $o,X34: x_a > state > $o,X35: com,X36: x_a > state > $o] :
( ( hoare_1508237396rivs_a @ X33 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ X34 @ X35 @ X36 ) @ bot_bo751897185le_a_o ) )
=> ( ! [X18: x_a,X19: state] :
( ( X32 @ X18 @ X19 )
=> ( X34 @ X18 @ X19 ) )
=> ( hoare_1508237396rivs_a @ X33 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ X32 @ X35 @ X36 ) @ bot_bo751897185le_a_o ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.78tw4nyiuM/DTF2THF_30168.p',fact_14_conseq1) ).
thf(f4462,plain,
( ( hoare_1508237396rivs_a @ g
@ ( insert1281456128iple_a
@ ( hoare_1766022166iple_a
@ ^ [Y0: x_a,Y1: state] : $false
@ c
@ ^ [Y0: x_a,Y1: state] :
( ( p @ Y0 @ Y1 )
& ~ ( b @ Y1 ) ) )
@ ( collec676402587iple_a
@ ^ [Y0: hoare_1775062406iple_a] : $false ) ) )
!= $true ),
inference(definition_unfolding,[status(thm)],[f3618,f4102]) ).
thf(f3618,plain,
( $true
!= ( hoare_1508237396rivs_a @ g
@ ( insert1281456128iple_a
@ ( hoare_1766022166iple_a
@ ^ [Y0: x_a,Y1: state] : $false
@ c
@ ^ [Y0: x_a,Y1: state] :
( ( p @ Y0 @ Y1 )
& ~ ( b @ Y1 ) ) )
@ bot_bo751897185le_a_o ) ) ),
inference(cnf_transformation,[status(thm)],[f2276]) ).
thf(f2276,plain,
( $true
!= ( hoare_1508237396rivs_a @ g
@ ( insert1281456128iple_a
@ ( hoare_1766022166iple_a
@ ^ [Y0: x_a,Y1: state] : $false
@ c
@ ^ [Y0: x_a,Y1: state] :
( ( p @ Y0 @ Y1 )
& ~ ( b @ Y1 ) ) )
@ bot_bo751897185le_a_o ) ) ),
inference(flattening,[status(thm)],[f2053]) ).
thf(f2053,plain,
( $true
!= ( hoare_1508237396rivs_a @ g
@ ( insert1281456128iple_a
@ ( hoare_1766022166iple_a
@ ^ [Y0: x_a,Y1: state] : $false
@ c
@ ^ [Y0: x_a,Y1: state] :
( ( p @ Y0 @ Y1 )
& ~ ( b @ Y1 ) ) )
@ bot_bo751897185le_a_o ) ) ),
inference(fool_elimination,[status(thm)],[f2052]) ).
thf(f2052,plain,
~ ( hoare_1508237396rivs_a @ g
@ ( insert1281456128iple_a
@ ( hoare_1766022166iple_a
@ ^ [X0: x_a,X1: state] : $false
@ c
@ ^ [X2: x_a,X3: state] :
( ( p @ X2 @ X3 )
& ~ ( b @ X3 ) ) )
@ bot_bo751897185le_a_o ) ),
inference(rectify,[status(thm)],[f712]) ).
thf(f712,negated_conjecture,
~ ( hoare_1508237396rivs_a @ g
@ ( insert1281456128iple_a
@ ( hoare_1766022166iple_a
@ ^ [X18: x_a,X19: state] : $false
@ c
@ ^ [X18: x_a,X19: state] :
( ( p @ X18 @ X19 )
& ~ ( b @ X19 ) ) )
@ bot_bo751897185le_a_o ) ),
inference(negated_conjecture,[status(cth)],[f711]) ).
thf(f711,conjecture,
( hoare_1508237396rivs_a @ g
@ ( insert1281456128iple_a
@ ( hoare_1766022166iple_a
@ ^ [X18: x_a,X19: state] : $false
@ c
@ ^ [X18: x_a,X19: state] :
( ( p @ X18 @ X19 )
& ~ ( b @ X19 ) ) )
@ bot_bo751897185le_a_o ) ),
file('/export/starexec/sandbox/tmp/tmp.78tw4nyiuM/DTF2THF_30168.p',conj_0) ).
thf(f5061,plain,
! [X0: x_a > state > $o,X1: hoare_1775062406iple_a > $o] :
( $true
!= ( hoare_1508237396rivs_a @ X1
@ ( insert1281456128iple_a
@ ( hoare_1766022166iple_a @ X0 @ c
@ ^ [Y0: x_a,Y1: state] :
( ( p @ Y0 @ Y1 )
& ~ ( b @ Y1 ) ) )
@ ( collec676402587iple_a
@ ^ [Y0: hoare_1775062406iple_a] : $false ) ) ) ),
inference(subsumption_resolution,[status(thm)],[f5056,f5057]) ).
thf(f5056,plain,
! [X0: x_a > state > $o,X1: hoare_1775062406iple_a > $o] :
( ( $true
!= ( hoare_1508237396rivs_a @ g @ X1 ) )
| ( $true
!= ( hoare_1508237396rivs_a @ X1
@ ( insert1281456128iple_a
@ ( hoare_1766022166iple_a @ X0 @ c
@ ^ [Y0: x_a,Y1: state] :
( ( p @ Y0 @ Y1 )
& ~ ( b @ Y1 ) ) )
@ ( collec676402587iple_a
@ ^ [Y0: hoare_1775062406iple_a] : $false ) ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f5048]) ).
thf(f5048,plain,
! [X0: x_a > state > $o,X1: hoare_1775062406iple_a > $o] :
( ( $true
!= ( hoare_1508237396rivs_a @ X1
@ ( insert1281456128iple_a
@ ( hoare_1766022166iple_a @ X0 @ c
@ ^ [Y0: x_a,Y1: state] :
( ( p @ Y0 @ Y1 )
& ~ ( b @ Y1 ) ) )
@ ( collec676402587iple_a
@ ^ [Y0: hoare_1775062406iple_a] : $false ) ) ) )
| ( $true
!= ( hoare_1508237396rivs_a @ g @ X1 ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f4971,f3556]) ).
thf(f3556,plain,
! [X2: hoare_1775062406iple_a > $o,X0: hoare_1775062406iple_a > $o,X1: hoare_1775062406iple_a > $o] :
( ( $true
= ( hoare_1508237396rivs_a @ X0 @ X2 ) )
| ( ( hoare_1508237396rivs_a @ X1 @ X2 )
!= $true )
| ( ( hoare_1508237396rivs_a @ X0 @ X1 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f2345]) ).
thf(f2345,plain,
! [X0: hoare_1775062406iple_a > $o,X1: hoare_1775062406iple_a > $o,X2: hoare_1775062406iple_a > $o] :
( ( $true
= ( hoare_1508237396rivs_a @ X0 @ X2 ) )
| ( ( hoare_1508237396rivs_a @ X0 @ X1 )
!= $true )
| ( ( hoare_1508237396rivs_a @ X1 @ X2 )
!= $true ) ),
inference(flattening,[status(thm)],[f2344]) ).
thf(f2344,plain,
! [X0: hoare_1775062406iple_a > $o,X1: hoare_1775062406iple_a > $o,X2: hoare_1775062406iple_a > $o] :
( ( $true
= ( hoare_1508237396rivs_a @ X0 @ X2 ) )
| ( ( hoare_1508237396rivs_a @ X0 @ X1 )
!= $true )
| ( ( hoare_1508237396rivs_a @ X1 @ X2 )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f927]) ).
thf(f927,plain,
! [X0: hoare_1775062406iple_a > $o,X1: hoare_1775062406iple_a > $o,X2: hoare_1775062406iple_a > $o] :
( ( ( hoare_1508237396rivs_a @ X1 @ X2 )
= $true )
=> ( ( ( hoare_1508237396rivs_a @ X0 @ X1 )
= $true )
=> ( $true
= ( hoare_1508237396rivs_a @ X0 @ X2 ) ) ) ),
inference(fool_elimination,[status(thm)],[f926]) ).
thf(f926,plain,
! [X0: hoare_1775062406iple_a > $o,X1: hoare_1775062406iple_a > $o,X2: hoare_1775062406iple_a > $o] :
( ( hoare_1508237396rivs_a @ X1 @ X2 )
=> ( ( hoare_1508237396rivs_a @ X0 @ X1 )
=> ( hoare_1508237396rivs_a @ X0 @ X2 ) ) ),
inference(rectify,[status(thm)],[f5]) ).
thf(f5,axiom,
! [X7: hoare_1775062406iple_a > $o,X8: hoare_1775062406iple_a > $o,X9: hoare_1775062406iple_a > $o] :
( ( hoare_1508237396rivs_a @ X8 @ X9 )
=> ( ( hoare_1508237396rivs_a @ X7 @ X8 )
=> ( hoare_1508237396rivs_a @ X7 @ X9 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.78tw4nyiuM/DTF2THF_30168.p',fact_4_cut) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SWW470^2 : TPTP v9.2.1. Released v5.3.0.
% 0.13/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n010.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 13:39:48 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.26/0.46 ---- Original DTF file ---
% 0.26/0.46 thf(spec,logic,$$dhol).
% 0.26/0.46 %------------------------------------------------------------------------------
% 0.26/0.46 % File : SWW470^2 : TPTP v9.2.1. Released v5.3.0.
% 0.26/0.46 % Domain : Software Verification
% 0.26/0.46 % Problem : Hoare's Logic with Procedures line 200, 500 axioms selected
% 0.26/0.46 % Version : Especial.
% 0.26/0.46 % English :
% 0.26/0.46
% 0.26/0.46 % Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% 0.26/0.46 % : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% 0.26/0.46 % Source : [Bla11]
% 0.26/0.46 % Names : hoare_500_thf_l200 [Bla11]
% 0.26/0.46
% 0.26/0.46 % Status : Theorem
% 0.26/0.46 % Rating : 0.33 v8.1.0, 0.50 v7.5.0, 1.00 v7.2.0, 0.75 v7.1.0, 0.00 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.80 v5.3.0
% 0.26/0.46 % Syntax : Number of formulae : 832 ( 178 unt; 121 typ; 0 def)
% 0.26/0.46 % Number of atoms : 3211 ( 690 equ; 62 cnn)
% 0.26/0.46 % Maximal formula atoms : 9 ( 4 avg)
% 0.26/0.46 % Number of connectives : 6257 ( 261 ~; 46 |; 141 &;4902 @)
% 0.26/0.46 % ( 145 <=>; 758 =>; 0 <=; 0 <~>)
% 0.26/0.46 % Maximal formula depth : 22 ( 7 avg)
% 0.26/0.46 % Number of types : 10 ( 9 usr)
% 0.26/0.46 % Number of type conns : 1667 (1667 >; 0 *; 0 +; 0 <<)
% 0.26/0.46 % Number of symbols : 119 ( 112 usr; 10 con; 0-4 aty)
% 0.26/0.46 % ( 0 !!; 4 ??; 0 @@+; 0 @@-)
% 0.26/0.46 % Number of variables : 2068 ( 140 ^;1880 !; 48 ?;2068 :)
% 0.26/0.46 % SPC : TH1_THM_EQU_NAR
% 0.26/0.46
% 0.26/0.46 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.26/0.46 % 2011-08-09 19:17:23
% 0.26/0.46 %------------------------------------------------------------------------------
% 0.26/0.46 %----Should-be-implicit typings (9)
% 0.26/0.46 thf(ty_ty_t__a,type,
% 0.26/0.46 x_a: $tType ).
% 0.26/0.46
% 0.26/0.46 thf(ty_ty_tc__Com__Ocom,type,
% 0.26/0.46 com: $tType ).
% 0.26/0.46
% 0.26/0.46 thf(ty_ty_tc__Com__Oglb,type,
% 0.26/0.46 glb: $tType ).
% 0.26/0.46
% 0.26/0.46 thf(ty_ty_tc__Com__Oloc,type,
% 0.26/0.46 loc: $tType ).
% 0.26/0.46
% 0.26/0.46 thf(ty_ty_tc__Com__Ostate,type,
% 0.26/0.46 state: $tType ).
% 0.26/0.46
% 0.26/0.46 thf(ty_ty_tc__Com__Ovname,type,
% 0.26/0.46 vname: $tType ).
% 0.26/0.46
% 0.26/0.46 thf(ty_ty_tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J,type,
% 0.26/0.46 hoare_1775062406iple_a: $tType ).
% 0.26/0.46
% 0.26/0.46 thf(ty_ty_tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostate_J,type,
% 0.26/0.46 hoare_1167836817_state: $tType ).
% 0.26/0.46
% 0.26/0.46 thf(ty_ty_tc__Nat__Onat,type,
% 0.26/0.46 nat: $tType ).
% 0.26/0.46
% 0.26/0.46 %----Explicit typings (115)
% 0.26/0.46 thf(sy_c_Big__Operators_Olattice__class_OSup__fin_000tc__Nat__Onat,type,
% 0.26/0.46 big_la43341705in_nat: ( nat > $o ) > nat ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Big__Operators_Osemilattice__big_000tc__Nat__Onat,type,
% 0.26/0.46 big_se275732192ig_nat: ( nat > nat > nat ) > ( ( nat > $o ) > nat ) > $o ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Com_Ocom_OAss,type,
% 0.26/0.46 ass: vname > ( state > nat ) > com ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Com_Ocom_OLocal,type,
% 0.26/0.46 local: loc > ( state > nat ) > com > com ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Com_Ocom_OSKIP,type,
% 0.26/0.46 skip: com ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Com_Ocom_OSemi,type,
% 0.26/0.46 semi: com > com > com ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Com_Ovname_OGlb,type,
% 0.26/0.46 glb_1: glb > vname ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Com_Ovname_OLoc,type,
% 0.26/0.46 loc_1: loc > vname ).
% 0.26/0.46
% 0.26/0.46 thf(sy_c_Finite__Set_Ocomp__fun__commute_000tc__Hoare____Mirabelle____srushsumbx__Ot,type,
% 0.26/0.46 finite2064891473iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Finite__Set_Ocomp__fun__idem_000tc__Hoare____Mirabelle____srushsumbx__Otrip,type,
% 0.26/0.47 finite2120172977le_a_o: ( hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Finite__Set_Ocomp__fun__idem_000tc__Hoare____Mirabelle____srushsumbx__Otrip_001,type,
% 0.26/0.47 finite856902323tate_o: ( hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Finite__Set_Ocomp__fun__idem_000tc__Nat__Onat_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 0.26/0.47 finite389864113_nat_o: ( nat > ( nat > $o ) > nat > $o ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Finite__Set_Ofinite_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 0.26/0.47 finite_finite_nat_o: ( ( nat > $o ) > $o ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Finite__Set_Ofinite_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_,type,
% 0.26/0.47 finite2063573081iple_a: ( hoare_1775062406iple_a > $o ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Finite__Set_Ofinite_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__C,type,
% 0.26/0.47 finite1084549118_state: ( hoare_1167836817_state > $o ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Finite__Set_Ofinite_000tc__Nat__Onat,type,
% 0.26/0.47 finite_finite_nat: ( nat > $o ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Finite__Set_Ofold1Set_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__,type,
% 0.26/0.47 finite1946188886iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Finite__Set_Ofold1Set_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc_,type,
% 0.26/0.47 finite309220289_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Finite__Set_Ofold1Set_000tc__Nat__Onat,type,
% 0.26/0.47 finite_fold1Set_nat: ( nat > nat > nat ) > ( nat > $o ) > nat > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Finite__Set_Ofold1_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J,type,
% 0.26/0.47 finite1790765286iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Finite__Set_Ofold1_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Co,type,
% 0.26/0.47 finite1646097201_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Finite__Set_Ofold1_000tc__Nat__Onat,type,
% 0.26/0.47 finite_fold1_nat: ( nat > nat > nat ) > ( nat > $o ) > nat ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Finite__Set_Ofold_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J_,type,
% 0.26/0.47 finite1544171829le_a_o: ( hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Finite__Set_Ofold_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J__002,type,
% 0.26/0.47 finite1842721992iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Finite__Set_Ofold_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com,type,
% 0.26/0.47 finite291020855tate_o: ( hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Finite__Set_Ofold_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com_003,type,
% 0.26/0.47 finite1731015960_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Finite__Set_Ofold_000tc__Nat__Onat_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 0.26/0.47 finite326637109_nat_o: ( nat > ( nat > $o ) > nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Finite__Set_Ofold_000tc__Nat__Onat_000tc__Nat__Onat,type,
% 0.26/0.47 finite_fold_nat_nat: ( nat > nat > nat ) > nat > ( nat > $o ) > nat ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Finite__Set_Ofold__graph_000tc__Hoare____Mirabelle____srushsumbx__Otriple_I,type,
% 0.26/0.47 finite727644230iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Finite__Set_Ofold__graph_000tc__Hoare____Mirabelle____srushsumbx__Otriple_I_004,type,
% 0.26/0.47 finite1316643734_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Finite__Set_Ofold__graph_000tc__Nat__Onat_000tc__Nat__Onat,type,
% 0.26/0.47 finite929467206at_nat: ( nat > nat > nat ) > nat > ( nat > $o ) > nat > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Finite__Set_Ofolding__one_000tc__Hoare____Mirabelle____srushsumbx__Otriple_,type,
% 0.26/0.47 finite2078349315iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > ( ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Finite__Set_Ofolding__one_000tc__Hoare____Mirabelle____srushsumbx__Otriple__005,type,
% 0.26/0.47 finite1074406356_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > ( ( hoare_1167836817_state > $o ) > hoare_1167836817_state ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Finite__Set_Ofolding__one_000tc__Nat__Onat,type,
% 0.26/0.47 finite988810631ne_nat: ( nat > nat > nat ) > ( ( nat > $o ) > nat ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Finite__Set_Ofolding__one__idem_000tc__Hoare____Mirabelle____srushsumbx__Ot,type,
% 0.26/0.47 finite1358382848iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > ( ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Finite__Set_Ofolding__one__idem_000tc__Hoare____Mirabelle____srushsumbx__Ot_006,type,
% 0.26/0.47 finite806517911_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > ( ( hoare_1167836817_state > $o ) > hoare_1167836817_state ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Finite__Set_Ofolding__one__idem_000tc__Nat__Onat,type,
% 0.26/0.47 finite795500164em_nat: ( nat > nat > nat ) > ( ( nat > $o ) > nat ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Hoare____Mirabelle____srushsumbx__,type,
% 0.26/0.47 minus_1944206118le_a_o: ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Hoare____Mirabelle____srushsumbx___007,type,
% 0.26/0.47 minus_2107060239tate_o: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 0.26/0.47 minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Groups_Otimes__class_Otimes_000tc__Nat__Onat,type,
% 0.26/0.47 times_times_nat: nat > nat > nat ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HOL_OThe_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J,type,
% 0.26/0.47 the_Ho1155011127iple_a: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HOL_OThe_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostate_,type,
% 0.26/0.47 the_Ho310147232_state: ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_HOL_OThe_000tc__Nat__Onat,type,
% 0.26/0.47 the_nat: ( nat > $o ) > nat ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Hoare__Mirabelle__srushsumbx_OMGT,type,
% 0.26/0.47 hoare_Mirabelle_MGT: com > hoare_1167836817_state ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Hoare__Mirabelle__srushsumbx_Ohoare__derivs_000t__a,type,
% 0.26/0.47 hoare_1508237396rivs_a: ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Hoare__Mirabelle__srushsumbx_Ohoare__derivs_000tc__Com__Ostate,type,
% 0.26/0.47 hoare_123228589_state: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Hoare__Mirabelle__srushsumbx_Otriple_Otriple_000t__a,type,
% 0.26/0.47 hoare_1766022166iple_a: ( x_a > state > $o ) > com > ( x_a > state > $o ) > hoare_1775062406iple_a ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Hoare__Mirabelle__srushsumbx_Otriple_Otriple_000tc__Com__Ostate,type,
% 0.26/0.47 hoare_908217195_state: ( state > state > $o ) > com > ( state > state > $o ) > hoare_1167836817_state ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Hoare__Mirabelle__srushsumbx_Otriple__valid_000t__a,type,
% 0.26/0.47 hoare_1462269968alid_a: nat > hoare_1775062406iple_a > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Hoare__Mirabelle__srushsumbx_Otriple__valid_000tc__Com__Ostate,type,
% 0.26/0.47 hoare_56934129_state: nat > hoare_1167836817_state > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Lattices_Osemilattice__inf__class_Oinf_000_062_Itc__Hoare____Mirabelle____s,type,
% 0.26/0.47 semila966743401le_a_o: ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Lattices_Osemilattice__inf__class_Oinf_000_062_Itc__Hoare____Mirabelle____s_008,type,
% 0.26/0.47 semila179895820tate_o: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Lattices_Osemilattice__inf__class_Oinf_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 0.26/0.47 semila1947288293_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Lattices_Osemilattice__inf__class_Oinf_000_Eo,type,
% 0.26/0.47 semila854092349_inf_o: $o > $o > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Lattices_Osemilattice__inf__class_Oinf_000tc__Nat__Onat,type,
% 0.26/0.47 semila80283416nf_nat: nat > nat > nat ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_062_Itc__Hoare____Mirabelle____s,type,
% 0.26/0.47 semila13410563le_a_o: ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_062_Itc__Hoare____Mirabelle____s_009,type,
% 0.26/0.47 semila1172322802tate_o: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 0.26/0.47 semila848761471_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_Eo,type,
% 0.26/0.47 semila10642723_sup_o: $o > $o > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000tc__Nat__Onat,type,
% 0.26/0.47 semila972727038up_nat: nat > nat > nat ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Natural_Oevalc,type,
% 0.26/0.47 evalc: com > state > state > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Natural_Oevaln,type,
% 0.26/0.47 evaln: com > state > nat > state > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Natural_Ogetlocs,type,
% 0.26/0.47 getlocs: state > loc > nat ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Natural_Oupdate,type,
% 0.26/0.47 update: state > vname > nat > state ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Orderings_Obot__class_Obot_000_062_I_062_Itc__Nat__Onat_M_Eo_J_M_Eo_J,type,
% 0.26/0.47 bot_bot_nat_o_o: ( nat > $o ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Hoare____Mirabelle____srushsumbx__O,type,
% 0.26/0.47 bot_bo751897185le_a_o: hoare_1775062406iple_a > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Hoare____Mirabelle____srushsumbx__O_010,type,
% 0.26/0.47 bot_bo70021908tate_o: hoare_1167836817_state > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 0.26/0.47 bot_bot_nat_o: nat > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Orderings_Obot__class_Obot_000_Eo,type,
% 0.26/0.47 bot_bot_o: $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Orderings_Obot__class_Obot_000tc__Nat__Onat,type,
% 0.26/0.47 bot_bot_nat: nat ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Hoare____Mirabelle____srushsum,type,
% 0.26/0.47 ord_le1143225901le_a_o: ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Hoare____Mirabelle____srushsum_011,type,
% 0.26/0.47 ord_le827224136tate_o: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 0.26/0.47 ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_000_Eo,type,
% 0.26/0.47 ord_less_eq_o: $o > $o > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__Nat__Onat,type,
% 0.26/0.47 ord_less_eq_nat: nat > nat > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Partial__Function_Oflat__lub_000tc__Hoare____Mirabelle____srushsumbx__Otrip,type,
% 0.26/0.47 partia126998524iple_a: hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Partial__Function_Oflat__lub_000tc__Hoare____Mirabelle____srushsumbx__Otrip_012,type,
% 0.26/0.47 partia715677851_state: hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Partial__Function_Oflat__lub_000tc__Nat__Onat,type,
% 0.26/0.47 partial_flat_lub_nat: nat > ( nat > $o ) > nat ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Set_OCollect_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 0.26/0.47 collect_nat_o: ( ( nat > $o ) > $o ) > ( nat > $o ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Set_OCollect_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J,type,
% 0.26/0.47 collec676402587iple_a: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Set_OCollect_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ost,type,
% 0.26/0.47 collec1027672124_state: ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Set_OCollect_000tc__Nat__Onat,type,
% 0.26/0.47 collect_nat: ( nat > $o ) > nat > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J_000tc__,type,
% 0.26/0.47 image_1170193413iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J_000tc___013,type,
% 0.26/0.47 image_1021683026_state: ( hoare_1775062406iple_a > hoare_1167836817_state ) > ( hoare_1775062406iple_a > $o ) > hoare_1167836817_state > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J_000tc___014,type,
% 0.26/0.47 image_1806517641_a_nat: ( hoare_1775062406iple_a > nat ) > ( hoare_1775062406iple_a > $o ) > nat > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostat,type,
% 0.26/0.47 image_1802845250iple_a: ( hoare_1167836817_state > hoare_1775062406iple_a ) > ( hoare_1167836817_state > $o ) > hoare_1775062406iple_a > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostat_015,type,
% 0.26/0.47 image_31595733_state: ( hoare_1167836817_state > hoare_1167836817_state ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostat_016,type,
% 0.26/0.47 image_1476618182te_nat: ( hoare_1167836817_state > nat ) > ( hoare_1167836817_state > $o ) > nat > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Hoare____Mirabelle____srushsumbx__Otripl,type,
% 0.26/0.47 image_43014529iple_a: ( nat > hoare_1775062406iple_a ) > ( nat > $o ) > hoare_1775062406iple_a > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Hoare____Mirabelle____srushsumbx__Otripl_017,type,
% 0.26/0.47 image_2121260246_state: ( nat > hoare_1167836817_state ) > ( nat > $o ) > hoare_1167836817_state > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Nat__Onat,type,
% 0.26/0.47 image_nat_nat: ( nat > nat ) > ( nat > $o ) > nat > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Set_Oinsert_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 0.26/0.47 insert_nat_o: ( nat > $o ) > ( ( nat > $o ) > $o ) > ( nat > $o ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Set_Oinsert_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J,type,
% 0.26/0.47 insert1281456128iple_a: hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Set_Oinsert_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Osta,type,
% 0.26/0.47 insert2134838167_state: hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Set_Oinsert_000tc__Nat__Onat,type,
% 0.26/0.47 insert_nat: nat > ( nat > $o ) > nat > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Set_Othe__elem_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J,type,
% 0.26/0.47 the_el1844711461iple_a: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Set_Othe__elem_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__O,type,
% 0.26/0.47 the_el323660082_state: ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_Set_Othe__elem_000tc__Nat__Onat,type,
% 0.26/0.47 the_elem_nat: ( nat > $o ) > nat ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_fequal_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 0.26/0.47 fequal_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_fequal_000tc__Com__Ostate,type,
% 0.26/0.47 fequal_state: state > state > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_fequal_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J,type,
% 0.26/0.47 fequal1288209029iple_a: hoare_1775062406iple_a > hoare_1775062406iple_a > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_fequal_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostate_J,type,
% 0.26/0.47 fequal1831255762_state: hoare_1167836817_state > hoare_1167836817_state > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_fequal_000tc__Nat__Onat,type,
% 0.26/0.47 fequal_nat: nat > nat > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_member_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 0.26/0.47 member_nat_o: ( nat > $o ) > ( ( nat > $o ) > $o ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_member_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J,type,
% 0.26/0.47 member2122167641iple_a: hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_member_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostate_J,type,
% 0.26/0.47 member2058392318_state: hoare_1167836817_state > ( hoare_1167836817_state > $o ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_c_member_000tc__Nat__Onat,type,
% 0.26/0.47 member_nat: nat > ( nat > $o ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_v_G,type,
% 0.26/0.47 g: hoare_1775062406iple_a > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_v_P,type,
% 0.26/0.47 p: x_a > state > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_v_b,type,
% 0.26/0.47 b: state > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_v_c,type,
% 0.26/0.47 c: com ).
% 0.26/0.47
% 0.26/0.47 %----Relevant facts (700)
% 0.26/0.47 thf(fact_0_empty,axiom,
% 0.26/0.47 ! [G_19: hoare_1775062406iple_a > $o] : ( hoare_1508237396rivs_a @ G_19 @ bot_bo751897185le_a_o ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_1_empty,axiom,
% 0.26/0.47 ! [G_19: hoare_1167836817_state > $o] : ( hoare_123228589_state @ G_19 @ bot_bo70021908tate_o ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_2_triple_Oinject,axiom,
% 0.26/0.47 ! [Fun1_2: state > state > $o,Com_4: com,Fun2_2: state > state > $o,Fun1_1: state > state > $o,Com_3: com,Fun2_1: state > state > $o] :
% 0.26/0.47 ( ( ( hoare_908217195_state @ Fun1_2 @ Com_4 @ Fun2_2 )
% 0.26/0.47 = ( hoare_908217195_state @ Fun1_1 @ Com_3 @ Fun2_1 ) )
% 0.26/0.47 <=> ( ( Fun1_2 = Fun1_1 )
% 0.26/0.47 & ( Com_4 = Com_3 )
% 0.26/0.47 & ( Fun2_2 = Fun2_1 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_3_triple_Oinject,axiom,
% 0.26/0.47 ! [Fun1_2: x_a > state > $o,Com_4: com,Fun2_2: x_a > state > $o,Fun1_1: x_a > state > $o,Com_3: com,Fun2_1: x_a > state > $o] :
% 0.26/0.47 ( ( ( hoare_1766022166iple_a @ Fun1_2 @ Com_4 @ Fun2_2 )
% 0.26/0.47 = ( hoare_1766022166iple_a @ Fun1_1 @ Com_3 @ Fun2_1 ) )
% 0.26/0.47 <=> ( ( Fun1_2 = Fun1_1 )
% 0.26/0.47 & ( Com_4 = Com_3 )
% 0.26/0.47 & ( Fun2_2 = Fun2_1 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_4_cut,axiom,
% 0.26/0.47 ! [G_18: hoare_1775062406iple_a > $o,G_17: hoare_1775062406iple_a > $o,Ts_4: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( hoare_1508237396rivs_a @ G_17 @ Ts_4 )
% 0.26/0.47 => ( ( hoare_1508237396rivs_a @ G_18 @ G_17 )
% 0.26/0.47 => ( hoare_1508237396rivs_a @ G_18 @ Ts_4 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_5_cut,axiom,
% 0.26/0.47 ! [G_18: hoare_1167836817_state > $o,G_17: hoare_1167836817_state > $o,Ts_4: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( hoare_123228589_state @ G_17 @ Ts_4 )
% 0.26/0.47 => ( ( hoare_123228589_state @ G_18 @ G_17 )
% 0.26/0.47 => ( hoare_123228589_state @ G_18 @ Ts_4 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_6_hoare__derivs_Oinsert,axiom,
% 0.26/0.47 ! [Ts_3: hoare_1775062406iple_a > $o,G_16: hoare_1775062406iple_a > $o,T_1: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( hoare_1508237396rivs_a @ G_16 @ ( insert1281456128iple_a @ T_1 @ bot_bo751897185le_a_o ) )
% 0.26/0.47 => ( ( hoare_1508237396rivs_a @ G_16 @ Ts_3 )
% 0.26/0.47 => ( hoare_1508237396rivs_a @ G_16 @ ( insert1281456128iple_a @ T_1 @ Ts_3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_7_hoare__derivs_Oinsert,axiom,
% 0.26/0.47 ! [Ts_3: hoare_1167836817_state > $o,G_16: hoare_1167836817_state > $o,T_1: hoare_1167836817_state] :
% 0.26/0.47 ( ( hoare_123228589_state @ G_16 @ ( insert2134838167_state @ T_1 @ bot_bo70021908tate_o ) )
% 0.26/0.47 => ( ( hoare_123228589_state @ G_16 @ Ts_3 )
% 0.26/0.47 => ( hoare_123228589_state @ G_16 @ ( insert2134838167_state @ T_1 @ Ts_3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_8_constant,axiom,
% 0.26/0.47 ! [G_15: hoare_1775062406iple_a > $o,P_33: x_a > state > $o,C_42: com,Q_15: x_a > state > $o,C_41: $o] :
% 0.26/0.47 ( ( C_41
% 0.26/0.47 => ( hoare_1508237396rivs_a @ G_15 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_33 @ C_42 @ Q_15 ) @ bot_bo751897185le_a_o ) ) )
% 0.26/0.47 => ( hoare_1508237396rivs_a @ G_15
% 0.26/0.47 @ ( insert1281456128iple_a
% 0.26/0.47 @ ( hoare_1766022166iple_a
% 0.26/0.47 @ ^ [Z: x_a,S: state] : ( (&) @ ( P_33 @ Z @ S ) @ C_41 )
% 0.26/0.47 @ C_42
% 0.26/0.47 @ Q_15 )
% 0.26/0.47 @ bot_bo751897185le_a_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_9_constant,axiom,
% 0.26/0.47 ! [G_15: hoare_1167836817_state > $o,P_33: state > state > $o,C_42: com,Q_15: state > state > $o,C_41: $o] :
% 0.26/0.47 ( ( C_41
% 0.26/0.47 => ( hoare_123228589_state @ G_15 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_33 @ C_42 @ Q_15 ) @ bot_bo70021908tate_o ) ) )
% 0.26/0.47 => ( hoare_123228589_state @ G_15
% 0.26/0.47 @ ( insert2134838167_state
% 0.26/0.47 @ ( hoare_908217195_state
% 0.26/0.47 @ ^ [Z: state,S: state] : ( (&) @ ( P_33 @ Z @ S ) @ C_41 )
% 0.26/0.47 @ C_42
% 0.26/0.47 @ Q_15 )
% 0.26/0.47 @ bot_bo70021908tate_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_10_escape,axiom,
% 0.26/0.47 ! [G_14: hoare_1775062406iple_a > $o,C_40: com,Q_14: x_a > state > $o,P_32: x_a > state > $o] :
% 0.26/0.47 ( ! [Z: x_a,S: state] :
% 0.26/0.47 ( ( P_32 @ Z @ S )
% 0.26/0.47 => ( hoare_1508237396rivs_a @ G_14
% 0.26/0.47 @ ( insert1281456128iple_a
% 0.26/0.47 @ ( hoare_1766022166iple_a
% 0.26/0.47 @ ^ [Za: x_a,S_5: state] : ( S_5 = S )
% 0.26/0.47 @ C_40
% 0.26/0.47 @ ^ [Z_23: x_a] : ( Q_14 @ Z ) )
% 0.26/0.47 @ bot_bo751897185le_a_o ) ) )
% 0.26/0.47 => ( hoare_1508237396rivs_a @ G_14 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_32 @ C_40 @ Q_14 ) @ bot_bo751897185le_a_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_11_escape,axiom,
% 0.26/0.47 ! [G_14: hoare_1167836817_state > $o,C_40: com,Q_14: state > state > $o,P_32: state > state > $o] :
% 0.26/0.47 ( ! [Z: state,S: state] :
% 0.26/0.47 ( ( P_32 @ Z @ S )
% 0.26/0.47 => ( hoare_123228589_state @ G_14
% 0.26/0.47 @ ( insert2134838167_state
% 0.26/0.47 @ ( hoare_908217195_state
% 0.26/0.47 @ ^ [Za: state,S_5: state] : ( S_5 = S )
% 0.26/0.47 @ C_40
% 0.26/0.47 @ ^ [Z_23: state] : ( Q_14 @ Z ) )
% 0.26/0.47 @ bot_bo70021908tate_o ) ) )
% 0.26/0.47 => ( hoare_123228589_state @ G_14 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_32 @ C_40 @ Q_14 ) @ bot_bo70021908tate_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_12_conseq2,axiom,
% 0.26/0.47 ! [Q_13: x_a > state > $o,G_13: hoare_1775062406iple_a > $o,P_31: x_a > state > $o,C_39: com,Q_12: x_a > state > $o] :
% 0.26/0.47 ( ( hoare_1508237396rivs_a @ G_13 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_31 @ C_39 @ Q_12 ) @ bot_bo751897185le_a_o ) )
% 0.26/0.47 => ( ! [Z: x_a,S: state] :
% 0.26/0.47 ( ( Q_12 @ Z @ S )
% 0.26/0.47 => ( Q_13 @ Z @ S ) )
% 0.26/0.47 => ( hoare_1508237396rivs_a @ G_13 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_31 @ C_39 @ Q_13 ) @ bot_bo751897185le_a_o ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_13_conseq2,axiom,
% 0.26/0.47 ! [Q_13: state > state > $o,G_13: hoare_1167836817_state > $o,P_31: state > state > $o,C_39: com,Q_12: state > state > $o] :
% 0.26/0.47 ( ( hoare_123228589_state @ G_13 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_31 @ C_39 @ Q_12 ) @ bot_bo70021908tate_o ) )
% 0.26/0.47 => ( ! [Z: state,S: state] :
% 0.26/0.47 ( ( Q_12 @ Z @ S )
% 0.26/0.47 => ( Q_13 @ Z @ S ) )
% 0.26/0.47 => ( hoare_123228589_state @ G_13 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_31 @ C_39 @ Q_13 ) @ bot_bo70021908tate_o ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_14_conseq1,axiom,
% 0.26/0.47 ! [P_30: x_a > state > $o,G_12: hoare_1775062406iple_a > $o,P_29: x_a > state > $o,C_38: com,Q_11: x_a > state > $o] :
% 0.26/0.47 ( ( hoare_1508237396rivs_a @ G_12 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_29 @ C_38 @ Q_11 ) @ bot_bo751897185le_a_o ) )
% 0.26/0.47 => ( ! [Z: x_a,S: state] :
% 0.26/0.47 ( ( P_30 @ Z @ S )
% 0.26/0.47 => ( P_29 @ Z @ S ) )
% 0.26/0.47 => ( hoare_1508237396rivs_a @ G_12 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_30 @ C_38 @ Q_11 ) @ bot_bo751897185le_a_o ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_15_conseq1,axiom,
% 0.26/0.47 ! [P_30: state > state > $o,G_12: hoare_1167836817_state > $o,P_29: state > state > $o,C_38: com,Q_11: state > state > $o] :
% 0.26/0.47 ( ( hoare_123228589_state @ G_12 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_29 @ C_38 @ Q_11 ) @ bot_bo70021908tate_o ) )
% 0.26/0.47 => ( ! [Z: state,S: state] :
% 0.26/0.47 ( ( P_30 @ Z @ S )
% 0.26/0.47 => ( P_29 @ Z @ S ) )
% 0.26/0.47 => ( hoare_123228589_state @ G_12 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_30 @ C_38 @ Q_11 ) @ bot_bo70021908tate_o ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_16_conseq12,axiom,
% 0.26/0.47 ! [Q_10: x_a > state > $o,P_28: x_a > state > $o,G_11: hoare_1775062406iple_a > $o,P_27: x_a > state > $o,C_37: com,Q_9: x_a > state > $o] :
% 0.26/0.47 ( ( hoare_1508237396rivs_a @ G_11 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_27 @ C_37 @ Q_9 ) @ bot_bo751897185le_a_o ) )
% 0.26/0.47 => ( ! [Z: x_a,S: state] :
% 0.26/0.47 ( ( P_28 @ Z @ S )
% 0.26/0.47 => ! [S_5: state] :
% 0.26/0.47 ( ! [Z_23: x_a] :
% 0.26/0.47 ( ( P_27 @ Z_23 @ S )
% 0.26/0.47 => ( Q_9 @ Z_23 @ S_5 ) )
% 0.26/0.47 => ( Q_10 @ Z @ S_5 ) ) )
% 0.26/0.47 => ( hoare_1508237396rivs_a @ G_11 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_28 @ C_37 @ Q_10 ) @ bot_bo751897185le_a_o ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_17_conseq12,axiom,
% 0.26/0.47 ! [Q_10: state > state > $o,P_28: state > state > $o,G_11: hoare_1167836817_state > $o,P_27: state > state > $o,C_37: com,Q_9: state > state > $o] :
% 0.26/0.47 ( ( hoare_123228589_state @ G_11 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_27 @ C_37 @ Q_9 ) @ bot_bo70021908tate_o ) )
% 0.26/0.47 => ( ! [Z: state,S: state] :
% 0.26/0.47 ( ( P_28 @ Z @ S )
% 0.26/0.47 => ! [S_5: state] :
% 0.26/0.47 ( ! [Z_23: state] :
% 0.26/0.47 ( ( P_27 @ Z_23 @ S )
% 0.26/0.47 => ( Q_9 @ Z_23 @ S_5 ) )
% 0.26/0.47 => ( Q_10 @ Z @ S_5 ) ) )
% 0.26/0.47 => ( hoare_123228589_state @ G_11 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_28 @ C_37 @ Q_10 ) @ bot_bo70021908tate_o ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_18_insertE,axiom,
% 0.26/0.47 ! [A_217: hoare_1775062406iple_a,B_99: hoare_1775062406iple_a,A_216: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( member2122167641iple_a @ A_217 @ ( insert1281456128iple_a @ B_99 @ A_216 ) )
% 0.26/0.47 => ( ( A_217 != B_99 )
% 0.26/0.47 => ( member2122167641iple_a @ A_217 @ A_216 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_19_insertE,axiom,
% 0.26/0.47 ! [A_217: nat,B_99: nat,A_216: nat > $o] :
% 0.26/0.47 ( ( member_nat @ A_217 @ ( insert_nat @ B_99 @ A_216 ) )
% 0.26/0.47 => ( ( A_217 != B_99 )
% 0.26/0.47 => ( member_nat @ A_217 @ A_216 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_20_insertE,axiom,
% 0.26/0.47 ! [A_217: hoare_1167836817_state,B_99: hoare_1167836817_state,A_216: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( member2058392318_state @ A_217 @ ( insert2134838167_state @ B_99 @ A_216 ) )
% 0.26/0.47 => ( ( A_217 != B_99 )
% 0.26/0.47 => ( member2058392318_state @ A_217 @ A_216 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_21_insertCI,axiom,
% 0.26/0.47 ! [B_98: hoare_1775062406iple_a,A_215: hoare_1775062406iple_a,B_97: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( ~ ( member2122167641iple_a @ A_215 @ B_97 )
% 0.26/0.47 => ( A_215 = B_98 ) )
% 0.26/0.47 => ( member2122167641iple_a @ A_215 @ ( insert1281456128iple_a @ B_98 @ B_97 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_22_insertCI,axiom,
% 0.26/0.47 ! [B_98: nat,A_215: nat,B_97: nat > $o] :
% 0.26/0.47 ( ( ~ ( member_nat @ A_215 @ B_97 )
% 0.26/0.47 => ( A_215 = B_98 ) )
% 0.26/0.47 => ( member_nat @ A_215 @ ( insert_nat @ B_98 @ B_97 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_23_insertCI,axiom,
% 0.26/0.47 ! [B_98: hoare_1167836817_state,A_215: hoare_1167836817_state,B_97: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( ~ ( member2058392318_state @ A_215 @ B_97 )
% 0.26/0.47 => ( A_215 = B_98 ) )
% 0.26/0.47 => ( member2058392318_state @ A_215 @ ( insert2134838167_state @ B_98 @ B_97 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_24_emptyE,axiom,
% 0.26/0.47 ! [A_214: hoare_1775062406iple_a] :
% 0.26/0.47 ~ ( member2122167641iple_a @ A_214 @ bot_bo751897185le_a_o ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_25_emptyE,axiom,
% 0.26/0.47 ! [A_214: hoare_1167836817_state] :
% 0.26/0.47 ~ ( member2058392318_state @ A_214 @ bot_bo70021908tate_o ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_26_emptyE,axiom,
% 0.26/0.47 ! [A_214: nat] :
% 0.26/0.47 ~ ( member_nat @ A_214 @ bot_bot_nat_o ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_27_singleton__conv2,axiom,
% 0.26/0.47 ! [A_213: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( collec676402587iple_a @ ( fequal1288209029iple_a @ A_213 ) )
% 0.26/0.47 = ( insert1281456128iple_a @ A_213 @ bot_bo751897185le_a_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_28_singleton__conv2,axiom,
% 0.26/0.47 ! [A_213: nat] :
% 0.26/0.47 ( ( collect_nat @ ( fequal_nat @ A_213 ) )
% 0.26/0.47 = ( insert_nat @ A_213 @ bot_bot_nat_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_29_singleton__conv2,axiom,
% 0.26/0.47 ! [A_213: nat > $o] :
% 0.26/0.47 ( ( collect_nat_o @ ( fequal_nat_o @ A_213 ) )
% 0.26/0.47 = ( insert_nat_o @ A_213 @ bot_bot_nat_o_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_30_singleton__conv2,axiom,
% 0.26/0.47 ! [A_213: hoare_1167836817_state] :
% 0.26/0.47 ( ( collec1027672124_state @ ( fequal1831255762_state @ A_213 ) )
% 0.26/0.47 = ( insert2134838167_state @ A_213 @ bot_bo70021908tate_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_31_singleton__conv,axiom,
% 0.26/0.47 ! [A_212: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( collec676402587iple_a
% 0.26/0.47 @ ^ [X_3: hoare_1775062406iple_a] : ( X_3 = A_212 ) )
% 0.26/0.47 = ( insert1281456128iple_a @ A_212 @ bot_bo751897185le_a_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_32_singleton__conv,axiom,
% 0.26/0.47 ! [A_212: nat] :
% 0.26/0.47 ( ( collect_nat
% 0.26/0.47 @ ^ [X_3: nat] : ( X_3 = A_212 ) )
% 0.26/0.47 = ( insert_nat @ A_212 @ bot_bot_nat_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_33_singleton__conv,axiom,
% 0.26/0.47 ! [A_212: nat > $o] :
% 0.26/0.47 ( ( collect_nat_o
% 0.26/0.47 @ ^ [X_3: nat > $o] : ( X_3 = A_212 ) )
% 0.26/0.47 = ( insert_nat_o @ A_212 @ bot_bot_nat_o_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_34_singleton__conv,axiom,
% 0.26/0.47 ! [A_212: hoare_1167836817_state] :
% 0.26/0.47 ( ( collec1027672124_state
% 0.26/0.47 @ ^ [X_3: hoare_1167836817_state] : ( X_3 = A_212 ) )
% 0.26/0.47 = ( insert2134838167_state @ A_212 @ bot_bo70021908tate_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_35_Collect__conv__if2,axiom,
% 0.26/0.47 ! [P_26: hoare_1775062406iple_a > $o,A_211: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( ( P_26 @ A_211 )
% 0.26/0.47 => ( ( collec676402587iple_a
% 0.26/0.47 @ ^ [X_3: hoare_1775062406iple_a] : ( (&) @ ( A_211 = X_3 ) @ ( P_26 @ X_3 ) ) )
% 0.26/0.47 = ( insert1281456128iple_a @ A_211 @ bot_bo751897185le_a_o ) ) )
% 0.26/0.47 & ( ~ ( P_26 @ A_211 )
% 0.26/0.47 => ( ( collec676402587iple_a
% 0.26/0.47 @ ^ [X_3: hoare_1775062406iple_a] : ( (&) @ ( A_211 = X_3 ) @ ( P_26 @ X_3 ) ) )
% 0.26/0.47 = bot_bo751897185le_a_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_36_Collect__conv__if2,axiom,
% 0.26/0.47 ! [P_26: nat > $o,A_211: nat] :
% 0.26/0.47 ( ( ( P_26 @ A_211 )
% 0.26/0.47 => ( ( collect_nat
% 0.26/0.47 @ ^ [X_3: nat] : ( (&) @ ( A_211 = X_3 ) @ ( P_26 @ X_3 ) ) )
% 0.26/0.47 = ( insert_nat @ A_211 @ bot_bot_nat_o ) ) )
% 0.26/0.47 & ( ~ ( P_26 @ A_211 )
% 0.26/0.47 => ( ( collect_nat
% 0.26/0.47 @ ^ [X_3: nat] : ( (&) @ ( A_211 = X_3 ) @ ( P_26 @ X_3 ) ) )
% 0.26/0.47 = bot_bot_nat_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_37_Collect__conv__if2,axiom,
% 0.26/0.47 ! [P_26: ( nat > $o ) > $o,A_211: nat > $o] :
% 0.26/0.47 ( ( ( P_26 @ A_211 )
% 0.26/0.47 => ( ( collect_nat_o
% 0.26/0.47 @ ^ [X_3: nat > $o] : ( (&) @ ( A_211 = X_3 ) @ ( P_26 @ X_3 ) ) )
% 0.26/0.47 = ( insert_nat_o @ A_211 @ bot_bot_nat_o_o ) ) )
% 0.26/0.47 & ( ~ ( P_26 @ A_211 )
% 0.26/0.47 => ( ( collect_nat_o
% 0.26/0.47 @ ^ [X_3: nat > $o] : ( (&) @ ( A_211 = X_3 ) @ ( P_26 @ X_3 ) ) )
% 0.26/0.47 = bot_bot_nat_o_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_38_Collect__conv__if2,axiom,
% 0.26/0.47 ! [P_26: hoare_1167836817_state > $o,A_211: hoare_1167836817_state] :
% 0.26/0.47 ( ( ( P_26 @ A_211 )
% 0.26/0.47 => ( ( collec1027672124_state
% 0.26/0.47 @ ^ [X_3: hoare_1167836817_state] : ( (&) @ ( A_211 = X_3 ) @ ( P_26 @ X_3 ) ) )
% 0.26/0.47 = ( insert2134838167_state @ A_211 @ bot_bo70021908tate_o ) ) )
% 0.26/0.47 & ( ~ ( P_26 @ A_211 )
% 0.26/0.47 => ( ( collec1027672124_state
% 0.26/0.47 @ ^ [X_3: hoare_1167836817_state] : ( (&) @ ( A_211 = X_3 ) @ ( P_26 @ X_3 ) ) )
% 0.26/0.47 = bot_bo70021908tate_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_39_Collect__conv__if,axiom,
% 0.26/0.47 ! [P_25: hoare_1775062406iple_a > $o,A_210: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( ( P_25 @ A_210 )
% 0.26/0.47 => ( ( collec676402587iple_a
% 0.26/0.47 @ ^ [X_3: hoare_1775062406iple_a] : ( (&) @ ( X_3 = A_210 ) @ ( P_25 @ X_3 ) ) )
% 0.26/0.47 = ( insert1281456128iple_a @ A_210 @ bot_bo751897185le_a_o ) ) )
% 0.26/0.47 & ( ~ ( P_25 @ A_210 )
% 0.26/0.47 => ( ( collec676402587iple_a
% 0.26/0.47 @ ^ [X_3: hoare_1775062406iple_a] : ( (&) @ ( X_3 = A_210 ) @ ( P_25 @ X_3 ) ) )
% 0.26/0.47 = bot_bo751897185le_a_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_40_Collect__conv__if,axiom,
% 0.26/0.47 ! [P_25: nat > $o,A_210: nat] :
% 0.26/0.47 ( ( ( P_25 @ A_210 )
% 0.26/0.47 => ( ( collect_nat
% 0.26/0.47 @ ^ [X_3: nat] : ( (&) @ ( X_3 = A_210 ) @ ( P_25 @ X_3 ) ) )
% 0.26/0.47 = ( insert_nat @ A_210 @ bot_bot_nat_o ) ) )
% 0.26/0.47 & ( ~ ( P_25 @ A_210 )
% 0.26/0.47 => ( ( collect_nat
% 0.26/0.47 @ ^ [X_3: nat] : ( (&) @ ( X_3 = A_210 ) @ ( P_25 @ X_3 ) ) )
% 0.26/0.47 = bot_bot_nat_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_41_Collect__conv__if,axiom,
% 0.26/0.47 ! [P_25: ( nat > $o ) > $o,A_210: nat > $o] :
% 0.26/0.47 ( ( ( P_25 @ A_210 )
% 0.26/0.47 => ( ( collect_nat_o
% 0.26/0.47 @ ^ [X_3: nat > $o] : ( (&) @ ( X_3 = A_210 ) @ ( P_25 @ X_3 ) ) )
% 0.26/0.47 = ( insert_nat_o @ A_210 @ bot_bot_nat_o_o ) ) )
% 0.26/0.47 & ( ~ ( P_25 @ A_210 )
% 0.26/0.47 => ( ( collect_nat_o
% 0.26/0.47 @ ^ [X_3: nat > $o] : ( (&) @ ( X_3 = A_210 ) @ ( P_25 @ X_3 ) ) )
% 0.26/0.47 = bot_bot_nat_o_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_42_Collect__conv__if,axiom,
% 0.26/0.47 ! [P_25: hoare_1167836817_state > $o,A_210: hoare_1167836817_state] :
% 0.26/0.47 ( ( ( P_25 @ A_210 )
% 0.26/0.47 => ( ( collec1027672124_state
% 0.26/0.47 @ ^ [X_3: hoare_1167836817_state] : ( (&) @ ( X_3 = A_210 ) @ ( P_25 @ X_3 ) ) )
% 0.26/0.47 = ( insert2134838167_state @ A_210 @ bot_bo70021908tate_o ) ) )
% 0.26/0.47 & ( ~ ( P_25 @ A_210 )
% 0.26/0.47 => ( ( collec1027672124_state
% 0.26/0.47 @ ^ [X_3: hoare_1167836817_state] : ( (&) @ ( X_3 = A_210 ) @ ( P_25 @ X_3 ) ) )
% 0.26/0.47 = bot_bo70021908tate_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_43_equals0D,axiom,
% 0.26/0.47 ! [A_209: hoare_1775062406iple_a,A_208: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( A_208 = bot_bo751897185le_a_o )
% 0.26/0.47 => ~ ( member2122167641iple_a @ A_209 @ A_208 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_44_equals0D,axiom,
% 0.26/0.47 ! [A_209: hoare_1167836817_state,A_208: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( A_208 = bot_bo70021908tate_o )
% 0.26/0.47 => ~ ( member2058392318_state @ A_209 @ A_208 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_45_equals0D,axiom,
% 0.26/0.47 ! [A_209: nat,A_208: nat > $o] :
% 0.26/0.47 ( ( A_208 = bot_bot_nat_o )
% 0.26/0.47 => ~ ( member_nat @ A_209 @ A_208 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_46_Collect__empty__eq,axiom,
% 0.26/0.47 ! [P_24: nat > $o] :
% 0.26/0.47 ( ( ( collect_nat @ P_24 )
% 0.26/0.47 = bot_bot_nat_o )
% 0.26/0.47 <=> ! [X_3: nat] :
% 0.26/0.47 ~ ( P_24 @ X_3 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_47_Collect__empty__eq,axiom,
% 0.26/0.47 ! [P_24: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( ( collec676402587iple_a @ P_24 )
% 0.26/0.47 = bot_bo751897185le_a_o )
% 0.26/0.47 <=> ! [X_3: hoare_1775062406iple_a] :
% 0.26/0.47 ~ ( P_24 @ X_3 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_48_Collect__empty__eq,axiom,
% 0.26/0.47 ! [P_24: ( nat > $o ) > $o] :
% 0.26/0.47 ( ( ( collect_nat_o @ P_24 )
% 0.26/0.47 = bot_bot_nat_o_o )
% 0.26/0.47 <=> ! [X_3: nat > $o] :
% 0.26/0.47 ~ ( P_24 @ X_3 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_49_Collect__empty__eq,axiom,
% 0.26/0.47 ! [P_24: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( ( collec1027672124_state @ P_24 )
% 0.26/0.47 = bot_bo70021908tate_o )
% 0.26/0.47 <=> ! [X_3: hoare_1167836817_state] :
% 0.26/0.47 ~ ( P_24 @ X_3 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_50_empty__iff,axiom,
% 0.26/0.47 ! [C_36: hoare_1775062406iple_a] :
% 0.26/0.47 ~ ( member2122167641iple_a @ C_36 @ bot_bo751897185le_a_o ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_51_empty__iff,axiom,
% 0.26/0.47 ! [C_36: hoare_1167836817_state] :
% 0.26/0.47 ~ ( member2058392318_state @ C_36 @ bot_bo70021908tate_o ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_52_empty__iff,axiom,
% 0.26/0.47 ! [C_36: nat] :
% 0.26/0.47 ~ ( member_nat @ C_36 @ bot_bot_nat_o ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_53_empty__Collect__eq,axiom,
% 0.26/0.47 ! [P_23: nat > $o] :
% 0.26/0.47 ( ( bot_bot_nat_o
% 0.26/0.47 = ( collect_nat @ P_23 ) )
% 0.26/0.47 <=> ! [X_3: nat] :
% 0.26/0.47 ~ ( P_23 @ X_3 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_54_empty__Collect__eq,axiom,
% 0.26/0.47 ! [P_23: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( bot_bo751897185le_a_o
% 0.26/0.47 = ( collec676402587iple_a @ P_23 ) )
% 0.26/0.47 <=> ! [X_3: hoare_1775062406iple_a] :
% 0.26/0.47 ~ ( P_23 @ X_3 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_55_empty__Collect__eq,axiom,
% 0.26/0.47 ! [P_23: ( nat > $o ) > $o] :
% 0.26/0.47 ( ( bot_bot_nat_o_o
% 0.26/0.47 = ( collect_nat_o @ P_23 ) )
% 0.26/0.47 <=> ! [X_3: nat > $o] :
% 0.26/0.47 ~ ( P_23 @ X_3 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_56_empty__Collect__eq,axiom,
% 0.26/0.47 ! [P_23: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( bot_bo70021908tate_o
% 0.26/0.47 = ( collec1027672124_state @ P_23 ) )
% 0.26/0.47 <=> ! [X_3: hoare_1167836817_state] :
% 0.26/0.47 ~ ( P_23 @ X_3 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_57_ex__in__conv,axiom,
% 0.26/0.47 ! [A_207: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ? [X_3: hoare_1775062406iple_a] : ( member2122167641iple_a @ X_3 @ A_207 )
% 0.26/0.47 <=> ( A_207 != bot_bo751897185le_a_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_58_ex__in__conv,axiom,
% 0.26/0.47 ! [A_207: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ? [X_3: hoare_1167836817_state] : ( member2058392318_state @ X_3 @ A_207 )
% 0.26/0.47 <=> ( A_207 != bot_bo70021908tate_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_59_ex__in__conv,axiom,
% 0.26/0.47 ! [A_207: nat > $o] :
% 0.26/0.47 ( ? [X_3: nat] : ( member_nat @ X_3 @ A_207 )
% 0.26/0.47 <=> ( A_207 != bot_bot_nat_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_60_all__not__in__conv,axiom,
% 0.26/0.47 ! [A_206: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ! [X_3: hoare_1775062406iple_a] :
% 0.26/0.47 ~ ( member2122167641iple_a @ X_3 @ A_206 )
% 0.26/0.47 <=> ( A_206 = bot_bo751897185le_a_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_61_all__not__in__conv,axiom,
% 0.26/0.47 ! [A_206: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ! [X_3: hoare_1167836817_state] :
% 0.26/0.47 ~ ( member2058392318_state @ X_3 @ A_206 )
% 0.26/0.47 <=> ( A_206 = bot_bo70021908tate_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_62_all__not__in__conv,axiom,
% 0.26/0.47 ! [A_206: nat > $o] :
% 0.26/0.47 ( ! [X_3: nat] :
% 0.26/0.47 ~ ( member_nat @ X_3 @ A_206 )
% 0.26/0.47 <=> ( A_206 = bot_bot_nat_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_63_empty__def,axiom,
% 0.26/0.47 ( bot_bot_nat_o
% 0.26/0.47 = ( collect_nat
% 0.26/0.47 @ ^ [X_3: nat] : $false ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_64_empty__def,axiom,
% 0.26/0.47 ( bot_bo751897185le_a_o
% 0.26/0.47 = ( collec676402587iple_a
% 0.26/0.47 @ ^ [X_3: hoare_1775062406iple_a] : $false ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_65_empty__def,axiom,
% 0.26/0.47 ( bot_bot_nat_o_o
% 0.26/0.47 = ( collect_nat_o
% 0.26/0.47 @ ^ [X_3: nat > $o] : $false ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_66_empty__def,axiom,
% 0.26/0.47 ( bot_bo70021908tate_o
% 0.26/0.47 = ( collec1027672124_state
% 0.26/0.47 @ ^ [X_3: hoare_1167836817_state] : $false ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_67_insert__absorb,axiom,
% 0.26/0.47 ! [A_205: hoare_1775062406iple_a,A_204: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( member2122167641iple_a @ A_205 @ A_204 )
% 0.26/0.47 => ( ( insert1281456128iple_a @ A_205 @ A_204 )
% 0.26/0.47 = A_204 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_68_insert__absorb,axiom,
% 0.26/0.47 ! [A_205: nat,A_204: nat > $o] :
% 0.26/0.47 ( ( member_nat @ A_205 @ A_204 )
% 0.26/0.47 => ( ( insert_nat @ A_205 @ A_204 )
% 0.26/0.47 = A_204 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_69_insert__absorb,axiom,
% 0.26/0.47 ! [A_205: hoare_1167836817_state,A_204: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( member2058392318_state @ A_205 @ A_204 )
% 0.26/0.47 => ( ( insert2134838167_state @ A_205 @ A_204 )
% 0.26/0.47 = A_204 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_70_insertI2,axiom,
% 0.26/0.47 ! [B_96: hoare_1775062406iple_a,A_203: hoare_1775062406iple_a,B_95: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( member2122167641iple_a @ A_203 @ B_95 )
% 0.26/0.47 => ( member2122167641iple_a @ A_203 @ ( insert1281456128iple_a @ B_96 @ B_95 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_71_insertI2,axiom,
% 0.26/0.47 ! [B_96: nat,A_203: nat,B_95: nat > $o] :
% 0.26/0.47 ( ( member_nat @ A_203 @ B_95 )
% 0.26/0.47 => ( member_nat @ A_203 @ ( insert_nat @ B_96 @ B_95 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_72_insertI2,axiom,
% 0.26/0.47 ! [B_96: hoare_1167836817_state,A_203: hoare_1167836817_state,B_95: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( member2058392318_state @ A_203 @ B_95 )
% 0.26/0.47 => ( member2058392318_state @ A_203 @ ( insert2134838167_state @ B_96 @ B_95 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_73_insert__ident,axiom,
% 0.26/0.47 ! [B_94: hoare_1775062406iple_a > $o,X_100: hoare_1775062406iple_a,A_202: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ~ ( member2122167641iple_a @ X_100 @ A_202 )
% 0.26/0.47 => ( ~ ( member2122167641iple_a @ X_100 @ B_94 )
% 0.26/0.47 => ( ( ( insert1281456128iple_a @ X_100 @ A_202 )
% 0.26/0.47 = ( insert1281456128iple_a @ X_100 @ B_94 ) )
% 0.26/0.47 <=> ( A_202 = B_94 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_74_insert__ident,axiom,
% 0.26/0.47 ! [B_94: nat > $o,X_100: nat,A_202: nat > $o] :
% 0.26/0.47 ( ~ ( member_nat @ X_100 @ A_202 )
% 0.26/0.47 => ( ~ ( member_nat @ X_100 @ B_94 )
% 0.26/0.47 => ( ( ( insert_nat @ X_100 @ A_202 )
% 0.26/0.47 = ( insert_nat @ X_100 @ B_94 ) )
% 0.26/0.47 <=> ( A_202 = B_94 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_75_insert__ident,axiom,
% 0.26/0.47 ! [B_94: hoare_1167836817_state > $o,X_100: hoare_1167836817_state,A_202: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ~ ( member2058392318_state @ X_100 @ A_202 )
% 0.26/0.47 => ( ~ ( member2058392318_state @ X_100 @ B_94 )
% 0.26/0.47 => ( ( ( insert2134838167_state @ X_100 @ A_202 )
% 0.26/0.47 = ( insert2134838167_state @ X_100 @ B_94 ) )
% 0.26/0.47 <=> ( A_202 = B_94 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_76_insert__code,axiom,
% 0.26/0.47 ! [Y_43: hoare_1775062406iple_a,A_201: hoare_1775062406iple_a > $o,X_99: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( insert1281456128iple_a @ Y_43 @ A_201 @ X_99 )
% 0.26/0.47 <=> ( ( Y_43 = X_99 )
% 0.26/0.47 | ( A_201 @ X_99 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_77_insert__code,axiom,
% 0.26/0.47 ! [Y_43: nat,A_201: nat > $o,X_99: nat] :
% 0.26/0.47 ( ( insert_nat @ Y_43 @ A_201 @ X_99 )
% 0.26/0.47 <=> ( ( Y_43 = X_99 )
% 0.26/0.47 | ( A_201 @ X_99 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_78_insert__code,axiom,
% 0.26/0.47 ! [Y_43: hoare_1167836817_state,A_201: hoare_1167836817_state > $o,X_99: hoare_1167836817_state] :
% 0.26/0.47 ( ( insert2134838167_state @ Y_43 @ A_201 @ X_99 )
% 0.26/0.47 <=> ( ( Y_43 = X_99 )
% 0.26/0.47 | ( A_201 @ X_99 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_79_insert__iff,axiom,
% 0.26/0.47 ! [A_200: hoare_1775062406iple_a,B_93: hoare_1775062406iple_a,A_199: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( member2122167641iple_a @ A_200 @ ( insert1281456128iple_a @ B_93 @ A_199 ) )
% 0.26/0.47 <=> ( ( A_200 = B_93 )
% 0.26/0.47 | ( member2122167641iple_a @ A_200 @ A_199 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_80_insert__iff,axiom,
% 0.26/0.47 ! [A_200: nat,B_93: nat,A_199: nat > $o] :
% 0.26/0.47 ( ( member_nat @ A_200 @ ( insert_nat @ B_93 @ A_199 ) )
% 0.26/0.47 <=> ( ( A_200 = B_93 )
% 0.26/0.47 | ( member_nat @ A_200 @ A_199 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_81_insert__iff,axiom,
% 0.26/0.47 ! [A_200: hoare_1167836817_state,B_93: hoare_1167836817_state,A_199: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( member2058392318_state @ A_200 @ ( insert2134838167_state @ B_93 @ A_199 ) )
% 0.26/0.47 <=> ( ( A_200 = B_93 )
% 0.26/0.47 | ( member2058392318_state @ A_200 @ A_199 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_82_insert__commute,axiom,
% 0.26/0.47 ! [X_98: hoare_1775062406iple_a,Y_42: hoare_1775062406iple_a,A_198: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( insert1281456128iple_a @ X_98 @ ( insert1281456128iple_a @ Y_42 @ A_198 ) )
% 0.26/0.47 = ( insert1281456128iple_a @ Y_42 @ ( insert1281456128iple_a @ X_98 @ A_198 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_83_insert__commute,axiom,
% 0.26/0.47 ! [X_98: nat,Y_42: nat,A_198: nat > $o] :
% 0.26/0.47 ( ( insert_nat @ X_98 @ ( insert_nat @ Y_42 @ A_198 ) )
% 0.26/0.47 = ( insert_nat @ Y_42 @ ( insert_nat @ X_98 @ A_198 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_84_insert__commute,axiom,
% 0.26/0.47 ! [X_98: hoare_1167836817_state,Y_42: hoare_1167836817_state,A_198: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( insert2134838167_state @ X_98 @ ( insert2134838167_state @ Y_42 @ A_198 ) )
% 0.26/0.47 = ( insert2134838167_state @ Y_42 @ ( insert2134838167_state @ X_98 @ A_198 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_85_insert__absorb2,axiom,
% 0.26/0.47 ! [X_97: hoare_1775062406iple_a,A_197: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( insert1281456128iple_a @ X_97 @ ( insert1281456128iple_a @ X_97 @ A_197 ) )
% 0.26/0.47 = ( insert1281456128iple_a @ X_97 @ A_197 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_86_insert__absorb2,axiom,
% 0.26/0.47 ! [X_97: nat,A_197: nat > $o] :
% 0.26/0.47 ( ( insert_nat @ X_97 @ ( insert_nat @ X_97 @ A_197 ) )
% 0.26/0.47 = ( insert_nat @ X_97 @ A_197 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_87_insert__absorb2,axiom,
% 0.26/0.47 ! [X_97: hoare_1167836817_state,A_197: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( insert2134838167_state @ X_97 @ ( insert2134838167_state @ X_97 @ A_197 ) )
% 0.26/0.47 = ( insert2134838167_state @ X_97 @ A_197 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_88_insert__Collect,axiom,
% 0.26/0.47 ! [A_196: hoare_1775062406iple_a,P_22: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( insert1281456128iple_a @ A_196 @ ( collec676402587iple_a @ P_22 ) )
% 0.26/0.47 = ( collec676402587iple_a
% 0.26/0.47 @ ^ [U_1: hoare_1775062406iple_a] : ( (=>) @ ( (~) @ ( U_1 = A_196 ) ) @ ( P_22 @ U_1 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_89_insert__Collect,axiom,
% 0.26/0.47 ! [A_196: nat,P_22: nat > $o] :
% 0.26/0.47 ( ( insert_nat @ A_196 @ ( collect_nat @ P_22 ) )
% 0.26/0.47 = ( collect_nat
% 0.26/0.47 @ ^ [U_1: nat] : ( (=>) @ ( (~) @ ( U_1 = A_196 ) ) @ ( P_22 @ U_1 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_90_insert__Collect,axiom,
% 0.26/0.47 ! [A_196: nat > $o,P_22: ( nat > $o ) > $o] :
% 0.26/0.47 ( ( insert_nat_o @ A_196 @ ( collect_nat_o @ P_22 ) )
% 0.26/0.47 = ( collect_nat_o
% 0.26/0.47 @ ^ [U_1: nat > $o] : ( (=>) @ ( (~) @ ( U_1 = A_196 ) ) @ ( P_22 @ U_1 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_91_insert__Collect,axiom,
% 0.26/0.47 ! [A_196: hoare_1167836817_state,P_22: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( insert2134838167_state @ A_196 @ ( collec1027672124_state @ P_22 ) )
% 0.26/0.47 = ( collec1027672124_state
% 0.26/0.47 @ ^ [U_1: hoare_1167836817_state] : ( (=>) @ ( (~) @ ( U_1 = A_196 ) ) @ ( P_22 @ U_1 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_92_insert__compr,axiom,
% 0.26/0.47 ! [A_195: hoare_1775062406iple_a,B_92: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( insert1281456128iple_a @ A_195 @ B_92 )
% 0.26/0.47 = ( collec676402587iple_a
% 0.26/0.47 @ ^ [X_3: hoare_1775062406iple_a] : ( (|) @ ( X_3 = A_195 ) @ ( member2122167641iple_a @ X_3 @ B_92 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_93_insert__compr,axiom,
% 0.26/0.47 ! [A_195: nat,B_92: nat > $o] :
% 0.26/0.47 ( ( insert_nat @ A_195 @ B_92 )
% 0.26/0.47 = ( collect_nat
% 0.26/0.47 @ ^ [X_3: nat] : ( (|) @ ( X_3 = A_195 ) @ ( member_nat @ X_3 @ B_92 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_94_insert__compr,axiom,
% 0.26/0.47 ! [A_195: nat > $o,B_92: ( nat > $o ) > $o] :
% 0.26/0.47 ( ( insert_nat_o @ A_195 @ B_92 )
% 0.26/0.47 = ( collect_nat_o
% 0.26/0.47 @ ^ [X_3: nat > $o] : ( (|) @ ( X_3 = A_195 ) @ ( member_nat_o @ X_3 @ B_92 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_95_insert__compr,axiom,
% 0.26/0.47 ! [A_195: hoare_1167836817_state,B_92: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( insert2134838167_state @ A_195 @ B_92 )
% 0.26/0.47 = ( collec1027672124_state
% 0.26/0.47 @ ^ [X_3: hoare_1167836817_state] : ( (|) @ ( X_3 = A_195 ) @ ( member2058392318_state @ X_3 @ B_92 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_96_insertI1,axiom,
% 0.26/0.47 ! [A_194: hoare_1775062406iple_a,B_91: hoare_1775062406iple_a > $o] : ( member2122167641iple_a @ A_194 @ ( insert1281456128iple_a @ A_194 @ B_91 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_97_insertI1,axiom,
% 0.26/0.47 ! [A_194: nat,B_91: nat > $o] : ( member_nat @ A_194 @ ( insert_nat @ A_194 @ B_91 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_98_insertI1,axiom,
% 0.26/0.47 ! [A_194: hoare_1167836817_state,B_91: hoare_1167836817_state > $o] : ( member2058392318_state @ A_194 @ ( insert2134838167_state @ A_194 @ B_91 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_99_insert__compr__raw,axiom,
% 0.26/0.47 ! [X_3: hoare_1775062406iple_a,Xa: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( insert1281456128iple_a @ X_3 @ Xa )
% 0.26/0.47 = ( collec676402587iple_a
% 0.26/0.47 @ ^ [Y_1: hoare_1775062406iple_a] : ( (|) @ ( Y_1 = X_3 ) @ ( member2122167641iple_a @ Y_1 @ Xa ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_100_insert__compr__raw,axiom,
% 0.26/0.47 ! [X_3: nat,Xa: nat > $o] :
% 0.26/0.47 ( ( insert_nat @ X_3 @ Xa )
% 0.26/0.47 = ( collect_nat
% 0.26/0.47 @ ^ [Y_1: nat] : ( (|) @ ( Y_1 = X_3 ) @ ( member_nat @ Y_1 @ Xa ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_101_insert__compr__raw,axiom,
% 0.26/0.47 ! [X_3: nat > $o,Xa: ( nat > $o ) > $o] :
% 0.26/0.47 ( ( insert_nat_o @ X_3 @ Xa )
% 0.26/0.47 = ( collect_nat_o
% 0.26/0.47 @ ^ [Y_1: nat > $o] : ( (|) @ ( Y_1 = X_3 ) @ ( member_nat_o @ Y_1 @ Xa ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_102_insert__compr__raw,axiom,
% 0.26/0.47 ! [X_3: hoare_1167836817_state,Xa: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( insert2134838167_state @ X_3 @ Xa )
% 0.26/0.47 = ( collec1027672124_state
% 0.26/0.47 @ ^ [Y_1: hoare_1167836817_state] : ( (|) @ ( Y_1 = X_3 ) @ ( member2058392318_state @ Y_1 @ Xa ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_103_singleton__inject,axiom,
% 0.26/0.47 ! [A_193: hoare_1775062406iple_a,B_90: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( ( insert1281456128iple_a @ A_193 @ bot_bo751897185le_a_o )
% 0.26/0.47 = ( insert1281456128iple_a @ B_90 @ bot_bo751897185le_a_o ) )
% 0.26/0.47 => ( A_193 = B_90 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_104_singleton__inject,axiom,
% 0.26/0.47 ! [A_193: nat,B_90: nat] :
% 0.26/0.47 ( ( ( insert_nat @ A_193 @ bot_bot_nat_o )
% 0.26/0.47 = ( insert_nat @ B_90 @ bot_bot_nat_o ) )
% 0.26/0.47 => ( A_193 = B_90 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_105_singleton__inject,axiom,
% 0.26/0.47 ! [A_193: hoare_1167836817_state,B_90: hoare_1167836817_state] :
% 0.26/0.47 ( ( ( insert2134838167_state @ A_193 @ bot_bo70021908tate_o )
% 0.26/0.47 = ( insert2134838167_state @ B_90 @ bot_bo70021908tate_o ) )
% 0.26/0.47 => ( A_193 = B_90 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_106_singletonE,axiom,
% 0.26/0.47 ! [B_89: hoare_1775062406iple_a,A_192: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( member2122167641iple_a @ B_89 @ ( insert1281456128iple_a @ A_192 @ bot_bo751897185le_a_o ) )
% 0.26/0.47 => ( B_89 = A_192 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_107_singletonE,axiom,
% 0.26/0.47 ! [B_89: nat,A_192: nat] :
% 0.26/0.47 ( ( member_nat @ B_89 @ ( insert_nat @ A_192 @ bot_bot_nat_o ) )
% 0.26/0.47 => ( B_89 = A_192 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_108_singletonE,axiom,
% 0.26/0.47 ! [B_89: hoare_1167836817_state,A_192: hoare_1167836817_state] :
% 0.26/0.47 ( ( member2058392318_state @ B_89 @ ( insert2134838167_state @ A_192 @ bot_bo70021908tate_o ) )
% 0.26/0.47 => ( B_89 = A_192 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_109_doubleton__eq__iff,axiom,
% 0.26/0.47 ! [A_191: hoare_1775062406iple_a,B_88: hoare_1775062406iple_a,C_35: hoare_1775062406iple_a,D_4: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( ( insert1281456128iple_a @ A_191 @ ( insert1281456128iple_a @ B_88 @ bot_bo751897185le_a_o ) )
% 0.26/0.47 = ( insert1281456128iple_a @ C_35 @ ( insert1281456128iple_a @ D_4 @ bot_bo751897185le_a_o ) ) )
% 0.26/0.47 <=> ( ( ( A_191 = C_35 )
% 0.26/0.47 & ( B_88 = D_4 ) )
% 0.26/0.47 | ( ( A_191 = D_4 )
% 0.26/0.47 & ( B_88 = C_35 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_110_doubleton__eq__iff,axiom,
% 0.26/0.47 ! [A_191: nat,B_88: nat,C_35: nat,D_4: nat] :
% 0.26/0.47 ( ( ( insert_nat @ A_191 @ ( insert_nat @ B_88 @ bot_bot_nat_o ) )
% 0.26/0.47 = ( insert_nat @ C_35 @ ( insert_nat @ D_4 @ bot_bot_nat_o ) ) )
% 0.26/0.47 <=> ( ( ( A_191 = C_35 )
% 0.26/0.47 & ( B_88 = D_4 ) )
% 0.26/0.47 | ( ( A_191 = D_4 )
% 0.26/0.47 & ( B_88 = C_35 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_111_doubleton__eq__iff,axiom,
% 0.26/0.47 ! [A_191: hoare_1167836817_state,B_88: hoare_1167836817_state,C_35: hoare_1167836817_state,D_4: hoare_1167836817_state] :
% 0.26/0.47 ( ( ( insert2134838167_state @ A_191 @ ( insert2134838167_state @ B_88 @ bot_bo70021908tate_o ) )
% 0.26/0.47 = ( insert2134838167_state @ C_35 @ ( insert2134838167_state @ D_4 @ bot_bo70021908tate_o ) ) )
% 0.26/0.47 <=> ( ( ( A_191 = C_35 )
% 0.26/0.47 & ( B_88 = D_4 ) )
% 0.26/0.47 | ( ( A_191 = D_4 )
% 0.26/0.47 & ( B_88 = C_35 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_112_singleton__iff,axiom,
% 0.26/0.47 ! [B_87: hoare_1775062406iple_a,A_190: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( member2122167641iple_a @ B_87 @ ( insert1281456128iple_a @ A_190 @ bot_bo751897185le_a_o ) )
% 0.26/0.47 <=> ( B_87 = A_190 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_113_singleton__iff,axiom,
% 0.26/0.47 ! [B_87: nat,A_190: nat] :
% 0.26/0.47 ( ( member_nat @ B_87 @ ( insert_nat @ A_190 @ bot_bot_nat_o ) )
% 0.26/0.47 <=> ( B_87 = A_190 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_114_singleton__iff,axiom,
% 0.26/0.47 ! [B_87: hoare_1167836817_state,A_190: hoare_1167836817_state] :
% 0.26/0.47 ( ( member2058392318_state @ B_87 @ ( insert2134838167_state @ A_190 @ bot_bo70021908tate_o ) )
% 0.26/0.47 <=> ( B_87 = A_190 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_115_insert__not__empty,axiom,
% 0.26/0.47 ! [A_189: hoare_1775062406iple_a,A_188: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( insert1281456128iple_a @ A_189 @ A_188 )
% 0.26/0.47 != bot_bo751897185le_a_o ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_116_insert__not__empty,axiom,
% 0.26/0.47 ! [A_189: nat,A_188: nat > $o] :
% 0.26/0.47 ( ( insert_nat @ A_189 @ A_188 )
% 0.26/0.47 != bot_bot_nat_o ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_117_insert__not__empty,axiom,
% 0.26/0.47 ! [A_189: hoare_1167836817_state,A_188: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( insert2134838167_state @ A_189 @ A_188 )
% 0.26/0.47 != bot_bo70021908tate_o ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_118_empty__not__insert,axiom,
% 0.26/0.47 ! [A_187: hoare_1775062406iple_a,A_186: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( bot_bo751897185le_a_o
% 0.26/0.47 != ( insert1281456128iple_a @ A_187 @ A_186 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_119_empty__not__insert,axiom,
% 0.26/0.47 ! [A_187: nat,A_186: nat > $o] :
% 0.26/0.47 ( bot_bot_nat_o
% 0.26/0.47 != ( insert_nat @ A_187 @ A_186 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_120_empty__not__insert,axiom,
% 0.26/0.47 ! [A_187: hoare_1167836817_state,A_186: hoare_1167836817_state > $o] :
% 0.26/0.47 ( bot_bo70021908tate_o
% 0.26/0.47 != ( insert2134838167_state @ A_187 @ A_186 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_121_the__elem__eq,axiom,
% 0.26/0.47 ! [X_96: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( the_el1844711461iple_a @ ( insert1281456128iple_a @ X_96 @ bot_bo751897185le_a_o ) )
% 0.26/0.47 = X_96 ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_122_the__elem__eq,axiom,
% 0.26/0.47 ! [X_96: nat] :
% 0.26/0.47 ( ( the_elem_nat @ ( insert_nat @ X_96 @ bot_bot_nat_o ) )
% 0.26/0.47 = X_96 ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_123_the__elem__eq,axiom,
% 0.26/0.47 ! [X_96: hoare_1167836817_state] :
% 0.26/0.47 ( ( the_el323660082_state @ ( insert2134838167_state @ X_96 @ bot_bo70021908tate_o ) )
% 0.26/0.47 = X_96 ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_124_bot__apply,axiom,
% 0.26/0.47 ! [X_95: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( bot_bo751897185le_a_o @ X_95 )
% 0.26/0.47 <=> bot_bot_o ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_125_bot__apply,axiom,
% 0.26/0.47 ! [X_95: nat] :
% 0.26/0.47 ( ( bot_bot_nat_o @ X_95 )
% 0.26/0.47 <=> bot_bot_o ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_126_bot__apply,axiom,
% 0.26/0.47 ! [X_95: hoare_1167836817_state] :
% 0.26/0.47 ( ( bot_bo70021908tate_o @ X_95 )
% 0.26/0.47 <=> bot_bot_o ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_127_bot__fun__def,axiom,
% 0.26/0.47 ! [X_3: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( bot_bo751897185le_a_o @ X_3 )
% 0.26/0.47 <=> bot_bot_o ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_128_bot__fun__def,axiom,
% 0.26/0.47 ! [X_3: nat] :
% 0.26/0.47 ( ( bot_bot_nat_o @ X_3 )
% 0.26/0.47 <=> bot_bot_o ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_129_bot__fun__def,axiom,
% 0.26/0.47 ! [X_3: hoare_1167836817_state] :
% 0.26/0.47 ( ( bot_bo70021908tate_o @ X_3 )
% 0.26/0.47 <=> bot_bot_o ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_130_hoare__derivs_OSkip,axiom,
% 0.26/0.47 ! [G_10: hoare_1775062406iple_a > $o,P_21: x_a > state > $o] : ( hoare_1508237396rivs_a @ G_10 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_21 @ skip @ P_21 ) @ bot_bo751897185le_a_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_131_hoare__derivs_OSkip,axiom,
% 0.26/0.47 ! [G_10: hoare_1167836817_state > $o,P_21: state > state > $o] : ( hoare_123228589_state @ G_10 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_21 @ skip @ P_21 ) @ bot_bo70021908tate_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_132_Comp,axiom,
% 0.26/0.47 ! [D_3: com,R_3: x_a > state > $o,G_9: hoare_1775062406iple_a > $o,P_20: x_a > state > $o,C_34: com,Q_8: x_a > state > $o] :
% 0.26/0.47 ( ( hoare_1508237396rivs_a @ G_9 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_20 @ C_34 @ Q_8 ) @ bot_bo751897185le_a_o ) )
% 0.26/0.47 => ( ( hoare_1508237396rivs_a @ G_9 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ Q_8 @ D_3 @ R_3 ) @ bot_bo751897185le_a_o ) )
% 0.26/0.47 => ( hoare_1508237396rivs_a @ G_9 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_20 @ ( semi @ C_34 @ D_3 ) @ R_3 ) @ bot_bo751897185le_a_o ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_133_Comp,axiom,
% 0.26/0.47 ! [D_3: com,R_3: state > state > $o,G_9: hoare_1167836817_state > $o,P_20: state > state > $o,C_34: com,Q_8: state > state > $o] :
% 0.26/0.47 ( ( hoare_123228589_state @ G_9 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_20 @ C_34 @ Q_8 ) @ bot_bo70021908tate_o ) )
% 0.26/0.47 => ( ( hoare_123228589_state @ G_9 @ ( insert2134838167_state @ ( hoare_908217195_state @ Q_8 @ D_3 @ R_3 ) @ bot_bo70021908tate_o ) )
% 0.26/0.47 => ( hoare_123228589_state @ G_9 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_20 @ ( semi @ C_34 @ D_3 ) @ R_3 ) @ bot_bo70021908tate_o ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_134_triple_Oexhaust,axiom,
% 0.26/0.47 ! [Y_41: hoare_1167836817_state] :
% 0.26/0.47 ~ ! [Fun1: state > state > $o,Com_2: com,Fun2: state > state > $o] :
% 0.26/0.47 ( Y_41
% 0.26/0.47 != ( hoare_908217195_state @ Fun1 @ Com_2 @ Fun2 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_135_triple_Oexhaust,axiom,
% 0.26/0.47 ! [Y_41: hoare_1775062406iple_a] :
% 0.26/0.47 ~ ! [Fun1: x_a > state > $o,Com_2: com,Fun2: x_a > state > $o] :
% 0.26/0.47 ( Y_41
% 0.26/0.47 != ( hoare_1766022166iple_a @ Fun1 @ Com_2 @ Fun2 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_136_Set_Oset__insert,axiom,
% 0.26/0.47 ! [X_94: hoare_1775062406iple_a,A_185: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( member2122167641iple_a @ X_94 @ A_185 )
% 0.26/0.47 => ~ ! [B_62: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( A_185
% 0.26/0.47 = ( insert1281456128iple_a @ X_94 @ B_62 ) )
% 0.26/0.47 => ( member2122167641iple_a @ X_94 @ B_62 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_137_Set_Oset__insert,axiom,
% 0.26/0.47 ! [X_94: nat,A_185: nat > $o] :
% 0.26/0.47 ( ( member_nat @ X_94 @ A_185 )
% 0.26/0.47 => ~ ! [B_62: nat > $o] :
% 0.26/0.47 ( ( A_185
% 0.26/0.47 = ( insert_nat @ X_94 @ B_62 ) )
% 0.26/0.47 => ( member_nat @ X_94 @ B_62 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_138_Set_Oset__insert,axiom,
% 0.26/0.47 ! [X_94: hoare_1167836817_state,A_185: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( member2058392318_state @ X_94 @ A_185 )
% 0.26/0.47 => ~ ! [B_62: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( A_185
% 0.26/0.47 = ( insert2134838167_state @ X_94 @ B_62 ) )
% 0.26/0.47 => ( member2058392318_state @ X_94 @ B_62 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_139_mk__disjoint__insert,axiom,
% 0.26/0.47 ! [A_184: hoare_1775062406iple_a,A_183: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( member2122167641iple_a @ A_184 @ A_183 )
% 0.26/0.47 => ? [B_62: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( A_183
% 0.26/0.47 = ( insert1281456128iple_a @ A_184 @ B_62 ) )
% 0.26/0.47 & ~ ( member2122167641iple_a @ A_184 @ B_62 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_140_mk__disjoint__insert,axiom,
% 0.26/0.47 ! [A_184: nat,A_183: nat > $o] :
% 0.26/0.47 ( ( member_nat @ A_184 @ A_183 )
% 0.26/0.47 => ? [B_62: nat > $o] :
% 0.26/0.47 ( ( A_183
% 0.26/0.47 = ( insert_nat @ A_184 @ B_62 ) )
% 0.26/0.47 & ~ ( member_nat @ A_184 @ B_62 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_141_mk__disjoint__insert,axiom,
% 0.26/0.47 ! [A_184: hoare_1167836817_state,A_183: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( member2058392318_state @ A_184 @ A_183 )
% 0.26/0.47 => ? [B_62: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( A_183
% 0.26/0.47 = ( insert2134838167_state @ A_184 @ B_62 ) )
% 0.26/0.47 & ~ ( member2058392318_state @ A_184 @ B_62 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_142_equals0I,axiom,
% 0.26/0.47 ! [A_182: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ! [Y_1: hoare_1775062406iple_a] :
% 0.26/0.47 ~ ( member2122167641iple_a @ Y_1 @ A_182 )
% 0.26/0.47 => ( A_182 = bot_bo751897185le_a_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_143_equals0I,axiom,
% 0.26/0.47 ! [A_182: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ! [Y_1: hoare_1167836817_state] :
% 0.26/0.47 ~ ( member2058392318_state @ Y_1 @ A_182 )
% 0.26/0.47 => ( A_182 = bot_bo70021908tate_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_144_equals0I,axiom,
% 0.26/0.47 ! [A_182: nat > $o] :
% 0.26/0.47 ( ! [Y_1: nat] :
% 0.26/0.47 ~ ( member_nat @ Y_1 @ A_182 )
% 0.26/0.47 => ( A_182 = bot_bot_nat_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_145_conseq,axiom,
% 0.26/0.47 ! [Q_6: x_a > state > $o,G_8: hoare_1775062406iple_a > $o,C_33: com,P_18: x_a > state > $o] :
% 0.26/0.47 ( ! [Z: x_a,S: state] :
% 0.26/0.47 ( ( P_18 @ Z @ S )
% 0.26/0.47 => ? [P_19: x_a > state > $o,Q_7: x_a > state > $o] :
% 0.26/0.47 ( ( hoare_1508237396rivs_a @ G_8 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_19 @ C_33 @ Q_7 ) @ bot_bo751897185le_a_o ) )
% 0.26/0.47 & ! [S_5: state] :
% 0.26/0.47 ( ! [Z_23: x_a] :
% 0.26/0.47 ( ( P_19 @ Z_23 @ S )
% 0.26/0.47 => ( Q_7 @ Z_23 @ S_5 ) )
% 0.26/0.47 => ( Q_6 @ Z @ S_5 ) ) ) )
% 0.26/0.47 => ( hoare_1508237396rivs_a @ G_8 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_18 @ C_33 @ Q_6 ) @ bot_bo751897185le_a_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_146_conseq,axiom,
% 0.26/0.47 ! [Q_6: state > state > $o,G_8: hoare_1167836817_state > $o,C_33: com,P_18: state > state > $o] :
% 0.26/0.47 ( ! [Z: state,S: state] :
% 0.26/0.47 ( ( P_18 @ Z @ S )
% 0.26/0.47 => ? [P_19: state > state > $o,Q_7: state > state > $o] :
% 0.26/0.47 ( ( hoare_123228589_state @ G_8 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_19 @ C_33 @ Q_7 ) @ bot_bo70021908tate_o ) )
% 0.26/0.47 & ! [S_5: state] :
% 0.26/0.47 ( ! [Z_23: state] :
% 0.26/0.47 ( ( P_19 @ Z_23 @ S )
% 0.26/0.47 => ( Q_7 @ Z_23 @ S_5 ) )
% 0.26/0.47 => ( Q_6 @ Z @ S_5 ) ) ) )
% 0.26/0.47 => ( hoare_123228589_state @ G_8 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_18 @ C_33 @ Q_6 ) @ bot_bo70021908tate_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_147_com_Osimps_I13_J,axiom,
% 0.26/0.47 ! [Com1: com,Com2: com] :
% 0.26/0.47 ( ( semi @ Com1 @ Com2 )
% 0.26/0.47 != skip ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_148_com_Osimps_I12_J,axiom,
% 0.26/0.47 ! [Com1: com,Com2: com] :
% 0.26/0.47 ( skip
% 0.26/0.47 != ( semi @ Com1 @ Com2 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_149_the__elem__def,axiom,
% 0.26/0.47 ! [X_93: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( the_el1844711461iple_a @ X_93 )
% 0.26/0.47 = ( the_Ho1155011127iple_a
% 0.26/0.47 @ ^ [X_3: hoare_1775062406iple_a] :
% 0.26/0.47 ( X_93
% 0.26/0.47 = ( insert1281456128iple_a @ X_3 @ bot_bo751897185le_a_o ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_150_the__elem__def,axiom,
% 0.26/0.47 ! [X_93: nat > $o] :
% 0.26/0.47 ( ( the_elem_nat @ X_93 )
% 0.26/0.47 = ( the_nat
% 0.26/0.47 @ ^ [X_3: nat] :
% 0.26/0.47 ( X_93
% 0.26/0.47 = ( insert_nat @ X_3 @ bot_bot_nat_o ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_151_the__elem__def,axiom,
% 0.26/0.47 ! [X_93: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( the_el323660082_state @ X_93 )
% 0.26/0.47 = ( the_Ho310147232_state
% 0.26/0.47 @ ^ [X_3: hoare_1167836817_state] :
% 0.26/0.47 ( X_93
% 0.26/0.47 = ( insert2134838167_state @ X_3 @ bot_bo70021908tate_o ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_152_com_Osimps_I3_J,axiom,
% 0.26/0.47 ! [Com1_1: com,Com2_1: com,Com1: com,Com2: com] :
% 0.26/0.47 ( ( ( semi @ Com1_1 @ Com2_1 )
% 0.26/0.47 = ( semi @ Com1 @ Com2 ) )
% 0.26/0.47 <=> ( ( Com1_1 = Com1 )
% 0.26/0.47 & ( Com2_1 = Com2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_153_nonempty__iff,axiom,
% 0.26/0.47 ! [A_181: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( A_181 != bot_bo751897185le_a_o )
% 0.26/0.47 <=> ? [X_3: hoare_1775062406iple_a,B_62: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( A_181
% 0.26/0.47 = ( insert1281456128iple_a @ X_3 @ B_62 ) )
% 0.26/0.47 & ~ ( member2122167641iple_a @ X_3 @ B_62 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_154_nonempty__iff,axiom,
% 0.26/0.47 ! [A_181: nat > $o] :
% 0.26/0.47 ( ( A_181 != bot_bot_nat_o )
% 0.26/0.47 <=> ? [X_3: nat,B_62: nat > $o] :
% 0.26/0.47 ( ( A_181
% 0.26/0.47 = ( insert_nat @ X_3 @ B_62 ) )
% 0.26/0.47 & ~ ( member_nat @ X_3 @ B_62 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_155_nonempty__iff,axiom,
% 0.26/0.47 ! [A_181: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( A_181 != bot_bo70021908tate_o )
% 0.26/0.47 <=> ? [X_3: hoare_1167836817_state,B_62: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( A_181
% 0.26/0.47 = ( insert2134838167_state @ X_3 @ B_62 ) )
% 0.26/0.47 & ~ ( member2058392318_state @ X_3 @ B_62 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_156_bot__empty__eq,axiom,
% 0.26/0.47 ! [X_3: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( bot_bo751897185le_a_o @ X_3 )
% 0.26/0.47 <=> ( member2122167641iple_a @ X_3 @ bot_bo751897185le_a_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_157_bot__empty__eq,axiom,
% 0.26/0.47 ! [X_3: hoare_1167836817_state] :
% 0.26/0.47 ( ( bot_bo70021908tate_o @ X_3 )
% 0.26/0.47 <=> ( member2058392318_state @ X_3 @ bot_bo70021908tate_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_158_bot__empty__eq,axiom,
% 0.26/0.47 ! [X_3: nat] :
% 0.26/0.47 ( ( bot_bot_nat_o @ X_3 )
% 0.26/0.47 <=> ( member_nat @ X_3 @ bot_bot_nat_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_159_Ass,axiom,
% 0.26/0.47 ! [G_7: hoare_1775062406iple_a > $o,P_17: x_a > state > $o,X_92: vname,A_180: state > nat] :
% 0.26/0.47 ( hoare_1508237396rivs_a @ G_7
% 0.26/0.47 @ ( insert1281456128iple_a
% 0.26/0.47 @ ( hoare_1766022166iple_a
% 0.26/0.47 @ ^ [Z: x_a,S: state] : ( P_17 @ Z @ ( update @ S @ X_92 @ ( A_180 @ S ) ) )
% 0.26/0.47 @ ( ass @ X_92 @ A_180 )
% 0.26/0.47 @ P_17 )
% 0.26/0.47 @ bot_bo751897185le_a_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_160_Ass,axiom,
% 0.26/0.47 ! [G_7: hoare_1167836817_state > $o,P_17: state > state > $o,X_92: vname,A_180: state > nat] :
% 0.26/0.47 ( hoare_123228589_state @ G_7
% 0.26/0.47 @ ( insert2134838167_state
% 0.26/0.47 @ ( hoare_908217195_state
% 0.26/0.47 @ ^ [Z: state,S: state] : ( P_17 @ Z @ ( update @ S @ X_92 @ ( A_180 @ S ) ) )
% 0.26/0.47 @ ( ass @ X_92 @ A_180 )
% 0.26/0.47 @ P_17 )
% 0.26/0.47 @ bot_bo70021908tate_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_161_image__constant__conv,axiom,
% 0.26/0.47 ! [C_32: hoare_1775062406iple_a,A_179: nat > $o] :
% 0.26/0.47 ( ( ( A_179 = bot_bot_nat_o )
% 0.26/0.47 => ( ( image_43014529iple_a
% 0.26/0.47 @ ^ [X_3: nat] : C_32
% 0.26/0.47 @ A_179 )
% 0.26/0.47 = bot_bo751897185le_a_o ) )
% 0.26/0.47 & ( ( A_179 != bot_bot_nat_o )
% 0.26/0.47 => ( ( image_43014529iple_a
% 0.26/0.47 @ ^ [X_3: nat] : C_32
% 0.26/0.47 @ A_179 )
% 0.26/0.47 = ( insert1281456128iple_a @ C_32 @ bot_bo751897185le_a_o ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_162_image__constant__conv,axiom,
% 0.26/0.47 ! [C_32: hoare_1775062406iple_a,A_179: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( ( A_179 = bot_bo70021908tate_o )
% 0.26/0.47 => ( ( image_1802845250iple_a
% 0.26/0.47 @ ^ [X_3: hoare_1167836817_state] : C_32
% 0.26/0.47 @ A_179 )
% 0.26/0.47 = bot_bo751897185le_a_o ) )
% 0.26/0.47 & ( ( A_179 != bot_bo70021908tate_o )
% 0.26/0.47 => ( ( image_1802845250iple_a
% 0.26/0.47 @ ^ [X_3: hoare_1167836817_state] : C_32
% 0.26/0.47 @ A_179 )
% 0.26/0.47 = ( insert1281456128iple_a @ C_32 @ bot_bo751897185le_a_o ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_163_image__constant__conv,axiom,
% 0.26/0.47 ! [C_32: nat,A_179: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( ( A_179 = bot_bo751897185le_a_o )
% 0.26/0.47 => ( ( image_1806517641_a_nat
% 0.26/0.47 @ ^ [X_3: hoare_1775062406iple_a] : C_32
% 0.26/0.47 @ A_179 )
% 0.26/0.47 = bot_bot_nat_o ) )
% 0.26/0.47 & ( ( A_179 != bot_bo751897185le_a_o )
% 0.26/0.47 => ( ( image_1806517641_a_nat
% 0.26/0.47 @ ^ [X_3: hoare_1775062406iple_a] : C_32
% 0.26/0.47 @ A_179 )
% 0.26/0.47 = ( insert_nat @ C_32 @ bot_bot_nat_o ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_164_image__constant__conv,axiom,
% 0.26/0.47 ! [C_32: hoare_1167836817_state,A_179: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( ( A_179 = bot_bo751897185le_a_o )
% 0.26/0.47 => ( ( image_1021683026_state
% 0.26/0.47 @ ^ [X_3: hoare_1775062406iple_a] : C_32
% 0.26/0.47 @ A_179 )
% 0.26/0.47 = bot_bo70021908tate_o ) )
% 0.26/0.47 & ( ( A_179 != bot_bo751897185le_a_o )
% 0.26/0.47 => ( ( image_1021683026_state
% 0.26/0.47 @ ^ [X_3: hoare_1775062406iple_a] : C_32
% 0.26/0.47 @ A_179 )
% 0.26/0.47 = ( insert2134838167_state @ C_32 @ bot_bo70021908tate_o ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_165_image__constant__conv,axiom,
% 0.26/0.47 ! [C_32: nat,A_179: nat > $o] :
% 0.26/0.47 ( ( ( A_179 = bot_bot_nat_o )
% 0.26/0.47 => ( ( image_nat_nat
% 0.26/0.47 @ ^ [X_3: nat] : C_32
% 0.26/0.47 @ A_179 )
% 0.26/0.47 = bot_bot_nat_o ) )
% 0.26/0.47 & ( ( A_179 != bot_bot_nat_o )
% 0.26/0.47 => ( ( image_nat_nat
% 0.26/0.47 @ ^ [X_3: nat] : C_32
% 0.26/0.47 @ A_179 )
% 0.26/0.47 = ( insert_nat @ C_32 @ bot_bot_nat_o ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_166_image__constant__conv,axiom,
% 0.26/0.47 ! [C_32: hoare_1167836817_state,A_179: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( ( A_179 = bot_bo70021908tate_o )
% 0.26/0.47 => ( ( image_31595733_state
% 0.26/0.47 @ ^ [X_3: hoare_1167836817_state] : C_32
% 0.26/0.47 @ A_179 )
% 0.26/0.47 = bot_bo70021908tate_o ) )
% 0.26/0.47 & ( ( A_179 != bot_bo70021908tate_o )
% 0.26/0.47 => ( ( image_31595733_state
% 0.26/0.47 @ ^ [X_3: hoare_1167836817_state] : C_32
% 0.26/0.47 @ A_179 )
% 0.26/0.47 = ( insert2134838167_state @ C_32 @ bot_bo70021908tate_o ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_167_image__constant__conv,axiom,
% 0.26/0.47 ! [C_32: hoare_1775062406iple_a,A_179: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( ( A_179 = bot_bo751897185le_a_o )
% 0.26/0.47 => ( ( image_1170193413iple_a
% 0.26/0.47 @ ^ [X_3: hoare_1775062406iple_a] : C_32
% 0.26/0.47 @ A_179 )
% 0.26/0.47 = bot_bo751897185le_a_o ) )
% 0.26/0.47 & ( ( A_179 != bot_bo751897185le_a_o )
% 0.26/0.47 => ( ( image_1170193413iple_a
% 0.26/0.47 @ ^ [X_3: hoare_1775062406iple_a] : C_32
% 0.26/0.47 @ A_179 )
% 0.26/0.47 = ( insert1281456128iple_a @ C_32 @ bot_bo751897185le_a_o ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_168_image__constant,axiom,
% 0.26/0.47 ! [C_31: hoare_1775062406iple_a,X_91: nat,A_178: nat > $o] :
% 0.26/0.47 ( ( member_nat @ X_91 @ A_178 )
% 0.26/0.47 => ( ( image_43014529iple_a
% 0.26/0.47 @ ^ [X_3: nat] : C_31
% 0.26/0.47 @ A_178 )
% 0.26/0.47 = ( insert1281456128iple_a @ C_31 @ bot_bo751897185le_a_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_169_image__constant,axiom,
% 0.26/0.47 ! [C_31: hoare_1775062406iple_a,X_91: hoare_1775062406iple_a,A_178: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( member2122167641iple_a @ X_91 @ A_178 )
% 0.26/0.47 => ( ( image_1170193413iple_a
% 0.26/0.47 @ ^ [X_3: hoare_1775062406iple_a] : C_31
% 0.26/0.47 @ A_178 )
% 0.26/0.47 = ( insert1281456128iple_a @ C_31 @ bot_bo751897185le_a_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_170_image__constant,axiom,
% 0.26/0.47 ! [C_31: hoare_1775062406iple_a,X_91: hoare_1167836817_state,A_178: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( member2058392318_state @ X_91 @ A_178 )
% 0.26/0.47 => ( ( image_1802845250iple_a
% 0.26/0.47 @ ^ [X_3: hoare_1167836817_state] : C_31
% 0.26/0.47 @ A_178 )
% 0.26/0.47 = ( insert1281456128iple_a @ C_31 @ bot_bo751897185le_a_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_171_image__constant,axiom,
% 0.26/0.47 ! [C_31: nat,X_91: nat,A_178: nat > $o] :
% 0.26/0.47 ( ( member_nat @ X_91 @ A_178 )
% 0.26/0.47 => ( ( image_nat_nat
% 0.26/0.47 @ ^ [X_3: nat] : C_31
% 0.26/0.47 @ A_178 )
% 0.26/0.47 = ( insert_nat @ C_31 @ bot_bot_nat_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_172_image__constant,axiom,
% 0.26/0.47 ! [C_31: nat,X_91: hoare_1775062406iple_a,A_178: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( member2122167641iple_a @ X_91 @ A_178 )
% 0.26/0.47 => ( ( image_1806517641_a_nat
% 0.26/0.47 @ ^ [X_3: hoare_1775062406iple_a] : C_31
% 0.26/0.47 @ A_178 )
% 0.26/0.47 = ( insert_nat @ C_31 @ bot_bot_nat_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_173_image__constant,axiom,
% 0.26/0.47 ! [C_31: nat,X_91: hoare_1167836817_state,A_178: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( member2058392318_state @ X_91 @ A_178 )
% 0.26/0.47 => ( ( image_1476618182te_nat
% 0.26/0.47 @ ^ [X_3: hoare_1167836817_state] : C_31
% 0.26/0.47 @ A_178 )
% 0.26/0.47 = ( insert_nat @ C_31 @ bot_bot_nat_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_174_image__constant,axiom,
% 0.26/0.47 ! [C_31: hoare_1167836817_state,X_91: hoare_1167836817_state,A_178: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( member2058392318_state @ X_91 @ A_178 )
% 0.26/0.47 => ( ( image_31595733_state
% 0.26/0.47 @ ^ [X_3: hoare_1167836817_state] : C_31
% 0.26/0.47 @ A_178 )
% 0.26/0.47 = ( insert2134838167_state @ C_31 @ bot_bo70021908tate_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_175_image__constant,axiom,
% 0.26/0.47 ! [C_31: hoare_1167836817_state,X_91: hoare_1775062406iple_a,A_178: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( member2122167641iple_a @ X_91 @ A_178 )
% 0.26/0.47 => ( ( image_1021683026_state
% 0.26/0.47 @ ^ [X_3: hoare_1775062406iple_a] : C_31
% 0.26/0.47 @ A_178 )
% 0.26/0.47 = ( insert2134838167_state @ C_31 @ bot_bo70021908tate_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_176_image__eqI,axiom,
% 0.26/0.47 ! [A_177: hoare_1167836817_state > $o,B_86: hoare_1167836817_state,F_65: hoare_1167836817_state > hoare_1167836817_state,X_90: hoare_1167836817_state] :
% 0.26/0.47 ( ( B_86
% 0.26/0.47 = ( F_65 @ X_90 ) )
% 0.26/0.47 => ( ( member2058392318_state @ X_90 @ A_177 )
% 0.26/0.47 => ( member2058392318_state @ B_86 @ ( image_31595733_state @ F_65 @ A_177 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_177_image__eqI,axiom,
% 0.26/0.47 ! [A_177: nat > $o,B_86: hoare_1167836817_state,F_65: nat > hoare_1167836817_state,X_90: nat] :
% 0.26/0.47 ( ( B_86
% 0.26/0.47 = ( F_65 @ X_90 ) )
% 0.26/0.47 => ( ( member_nat @ X_90 @ A_177 )
% 0.26/0.47 => ( member2058392318_state @ B_86 @ ( image_2121260246_state @ F_65 @ A_177 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_178_image__eqI,axiom,
% 0.26/0.47 ! [A_177: hoare_1775062406iple_a > $o,B_86: hoare_1167836817_state,F_65: hoare_1775062406iple_a > hoare_1167836817_state,X_90: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( B_86
% 0.26/0.47 = ( F_65 @ X_90 ) )
% 0.26/0.47 => ( ( member2122167641iple_a @ X_90 @ A_177 )
% 0.26/0.47 => ( member2058392318_state @ B_86 @ ( image_1021683026_state @ F_65 @ A_177 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_179_image__eqI,axiom,
% 0.26/0.47 ! [A_177: nat > $o,B_86: nat,F_65: nat > nat,X_90: nat] :
% 0.26/0.47 ( ( B_86
% 0.26/0.47 = ( F_65 @ X_90 ) )
% 0.26/0.47 => ( ( member_nat @ X_90 @ A_177 )
% 0.26/0.47 => ( member_nat @ B_86 @ ( image_nat_nat @ F_65 @ A_177 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_180_image__eqI,axiom,
% 0.26/0.47 ! [A_177: hoare_1775062406iple_a > $o,B_86: nat,F_65: hoare_1775062406iple_a > nat,X_90: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( B_86
% 0.26/0.47 = ( F_65 @ X_90 ) )
% 0.26/0.47 => ( ( member2122167641iple_a @ X_90 @ A_177 )
% 0.26/0.47 => ( member_nat @ B_86 @ ( image_1806517641_a_nat @ F_65 @ A_177 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_181_image__eqI,axiom,
% 0.26/0.47 ! [A_177: hoare_1167836817_state > $o,B_86: nat,F_65: hoare_1167836817_state > nat,X_90: hoare_1167836817_state] :
% 0.26/0.47 ( ( B_86
% 0.26/0.47 = ( F_65 @ X_90 ) )
% 0.26/0.47 => ( ( member2058392318_state @ X_90 @ A_177 )
% 0.26/0.47 => ( member_nat @ B_86 @ ( image_1476618182te_nat @ F_65 @ A_177 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_182_image__eqI,axiom,
% 0.26/0.47 ! [A_177: hoare_1775062406iple_a > $o,B_86: hoare_1775062406iple_a,F_65: hoare_1775062406iple_a > hoare_1775062406iple_a,X_90: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( B_86
% 0.26/0.47 = ( F_65 @ X_90 ) )
% 0.26/0.47 => ( ( member2122167641iple_a @ X_90 @ A_177 )
% 0.26/0.47 => ( member2122167641iple_a @ B_86 @ ( image_1170193413iple_a @ F_65 @ A_177 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_183_image__eqI,axiom,
% 0.26/0.47 ! [A_177: hoare_1167836817_state > $o,B_86: hoare_1775062406iple_a,F_65: hoare_1167836817_state > hoare_1775062406iple_a,X_90: hoare_1167836817_state] :
% 0.26/0.47 ( ( B_86
% 0.26/0.47 = ( F_65 @ X_90 ) )
% 0.26/0.47 => ( ( member2058392318_state @ X_90 @ A_177 )
% 0.26/0.47 => ( member2122167641iple_a @ B_86 @ ( image_1802845250iple_a @ F_65 @ A_177 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_184_image__eqI,axiom,
% 0.26/0.47 ! [A_177: nat > $o,B_86: hoare_1775062406iple_a,F_65: nat > hoare_1775062406iple_a,X_90: nat] :
% 0.26/0.47 ( ( B_86
% 0.26/0.47 = ( F_65 @ X_90 ) )
% 0.26/0.47 => ( ( member_nat @ X_90 @ A_177 )
% 0.26/0.47 => ( member2122167641iple_a @ B_86 @ ( image_43014529iple_a @ F_65 @ A_177 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_185_image__ident,axiom,
% 0.26/0.47 ! [Y_40: nat > $o] :
% 0.26/0.47 ( ( image_nat_nat
% 0.26/0.47 @ ^ [X_3: nat] : X_3
% 0.26/0.47 @ Y_40 )
% 0.26/0.47 = Y_40 ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_186_image__ident,axiom,
% 0.26/0.47 ! [Y_40: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( image_31595733_state
% 0.26/0.47 @ ^ [X_3: hoare_1167836817_state] : X_3
% 0.26/0.47 @ Y_40 )
% 0.26/0.47 = Y_40 ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_187_image__ident,axiom,
% 0.26/0.47 ! [Y_40: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( image_1170193413iple_a
% 0.26/0.47 @ ^ [X_3: hoare_1775062406iple_a] : X_3
% 0.26/0.47 @ Y_40 )
% 0.26/0.47 = Y_40 ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_188_com_Osimps_I1_J,axiom,
% 0.26/0.47 ! [Vname: vname,Fun_1: state > nat,Vname_1: vname,Fun: state > nat] :
% 0.26/0.47 ( ( ( ass @ Vname @ Fun_1 )
% 0.26/0.47 = ( ass @ Vname_1 @ Fun ) )
% 0.26/0.47 <=> ( ( Vname = Vname_1 )
% 0.26/0.47 & ( Fun_1 = Fun ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_189_rev__image__eqI,axiom,
% 0.26/0.47 ! [B_85: hoare_1167836817_state,F_64: hoare_1167836817_state > hoare_1167836817_state,X_89: hoare_1167836817_state,A_176: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( member2058392318_state @ X_89 @ A_176 )
% 0.26/0.47 => ( ( B_85
% 0.26/0.47 = ( F_64 @ X_89 ) )
% 0.26/0.47 => ( member2058392318_state @ B_85 @ ( image_31595733_state @ F_64 @ A_176 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_190_rev__image__eqI,axiom,
% 0.26/0.47 ! [B_85: hoare_1167836817_state,F_64: nat > hoare_1167836817_state,X_89: nat,A_176: nat > $o] :
% 0.26/0.47 ( ( member_nat @ X_89 @ A_176 )
% 0.26/0.47 => ( ( B_85
% 0.26/0.47 = ( F_64 @ X_89 ) )
% 0.26/0.47 => ( member2058392318_state @ B_85 @ ( image_2121260246_state @ F_64 @ A_176 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_191_rev__image__eqI,axiom,
% 0.26/0.47 ! [B_85: hoare_1167836817_state,F_64: hoare_1775062406iple_a > hoare_1167836817_state,X_89: hoare_1775062406iple_a,A_176: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( member2122167641iple_a @ X_89 @ A_176 )
% 0.26/0.47 => ( ( B_85
% 0.26/0.47 = ( F_64 @ X_89 ) )
% 0.26/0.47 => ( member2058392318_state @ B_85 @ ( image_1021683026_state @ F_64 @ A_176 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_192_rev__image__eqI,axiom,
% 0.26/0.47 ! [B_85: nat,F_64: nat > nat,X_89: nat,A_176: nat > $o] :
% 0.26/0.47 ( ( member_nat @ X_89 @ A_176 )
% 0.26/0.47 => ( ( B_85
% 0.26/0.47 = ( F_64 @ X_89 ) )
% 0.26/0.47 => ( member_nat @ B_85 @ ( image_nat_nat @ F_64 @ A_176 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_193_rev__image__eqI,axiom,
% 0.26/0.47 ! [B_85: nat,F_64: hoare_1775062406iple_a > nat,X_89: hoare_1775062406iple_a,A_176: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( member2122167641iple_a @ X_89 @ A_176 )
% 0.26/0.47 => ( ( B_85
% 0.26/0.47 = ( F_64 @ X_89 ) )
% 0.26/0.47 => ( member_nat @ B_85 @ ( image_1806517641_a_nat @ F_64 @ A_176 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_194_rev__image__eqI,axiom,
% 0.26/0.47 ! [B_85: nat,F_64: hoare_1167836817_state > nat,X_89: hoare_1167836817_state,A_176: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( member2058392318_state @ X_89 @ A_176 )
% 0.26/0.47 => ( ( B_85
% 0.26/0.47 = ( F_64 @ X_89 ) )
% 0.26/0.47 => ( member_nat @ B_85 @ ( image_1476618182te_nat @ F_64 @ A_176 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_195_rev__image__eqI,axiom,
% 0.26/0.47 ! [B_85: hoare_1775062406iple_a,F_64: hoare_1775062406iple_a > hoare_1775062406iple_a,X_89: hoare_1775062406iple_a,A_176: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( member2122167641iple_a @ X_89 @ A_176 )
% 0.26/0.47 => ( ( B_85
% 0.26/0.47 = ( F_64 @ X_89 ) )
% 0.26/0.47 => ( member2122167641iple_a @ B_85 @ ( image_1170193413iple_a @ F_64 @ A_176 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_196_rev__image__eqI,axiom,
% 0.26/0.47 ! [B_85: hoare_1775062406iple_a,F_64: hoare_1167836817_state > hoare_1775062406iple_a,X_89: hoare_1167836817_state,A_176: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( member2058392318_state @ X_89 @ A_176 )
% 0.26/0.47 => ( ( B_85
% 0.26/0.47 = ( F_64 @ X_89 ) )
% 0.26/0.47 => ( member2122167641iple_a @ B_85 @ ( image_1802845250iple_a @ F_64 @ A_176 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_197_rev__image__eqI,axiom,
% 0.26/0.47 ! [B_85: hoare_1775062406iple_a,F_64: nat > hoare_1775062406iple_a,X_89: nat,A_176: nat > $o] :
% 0.26/0.47 ( ( member_nat @ X_89 @ A_176 )
% 0.26/0.47 => ( ( B_85
% 0.26/0.47 = ( F_64 @ X_89 ) )
% 0.26/0.47 => ( member2122167641iple_a @ B_85 @ ( image_43014529iple_a @ F_64 @ A_176 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_198_imageI,axiom,
% 0.26/0.47 ! [F_63: hoare_1167836817_state > hoare_1167836817_state,X_88: hoare_1167836817_state,A_175: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( member2058392318_state @ X_88 @ A_175 )
% 0.26/0.47 => ( member2058392318_state @ ( F_63 @ X_88 ) @ ( image_31595733_state @ F_63 @ A_175 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_199_imageI,axiom,
% 0.26/0.47 ! [F_63: nat > hoare_1167836817_state,X_88: nat,A_175: nat > $o] :
% 0.26/0.47 ( ( member_nat @ X_88 @ A_175 )
% 0.26/0.47 => ( member2058392318_state @ ( F_63 @ X_88 ) @ ( image_2121260246_state @ F_63 @ A_175 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_200_imageI,axiom,
% 0.26/0.47 ! [F_63: hoare_1775062406iple_a > hoare_1167836817_state,X_88: hoare_1775062406iple_a,A_175: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( member2122167641iple_a @ X_88 @ A_175 )
% 0.26/0.47 => ( member2058392318_state @ ( F_63 @ X_88 ) @ ( image_1021683026_state @ F_63 @ A_175 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_201_imageI,axiom,
% 0.26/0.47 ! [F_63: nat > nat,X_88: nat,A_175: nat > $o] :
% 0.26/0.47 ( ( member_nat @ X_88 @ A_175 )
% 0.26/0.47 => ( member_nat @ ( F_63 @ X_88 ) @ ( image_nat_nat @ F_63 @ A_175 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_202_imageI,axiom,
% 0.26/0.47 ! [F_63: hoare_1775062406iple_a > nat,X_88: hoare_1775062406iple_a,A_175: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( member2122167641iple_a @ X_88 @ A_175 )
% 0.26/0.47 => ( member_nat @ ( F_63 @ X_88 ) @ ( image_1806517641_a_nat @ F_63 @ A_175 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_203_imageI,axiom,
% 0.26/0.47 ! [F_63: hoare_1167836817_state > nat,X_88: hoare_1167836817_state,A_175: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( member2058392318_state @ X_88 @ A_175 )
% 0.26/0.47 => ( member_nat @ ( F_63 @ X_88 ) @ ( image_1476618182te_nat @ F_63 @ A_175 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_204_imageI,axiom,
% 0.26/0.47 ! [F_63: hoare_1775062406iple_a > hoare_1775062406iple_a,X_88: hoare_1775062406iple_a,A_175: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( member2122167641iple_a @ X_88 @ A_175 )
% 0.26/0.47 => ( member2122167641iple_a @ ( F_63 @ X_88 ) @ ( image_1170193413iple_a @ F_63 @ A_175 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_205_imageI,axiom,
% 0.26/0.47 ! [F_63: hoare_1167836817_state > hoare_1775062406iple_a,X_88: hoare_1167836817_state,A_175: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( member2058392318_state @ X_88 @ A_175 )
% 0.26/0.47 => ( member2122167641iple_a @ ( F_63 @ X_88 ) @ ( image_1802845250iple_a @ F_63 @ A_175 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_206_imageI,axiom,
% 0.26/0.47 ! [F_63: nat > hoare_1775062406iple_a,X_88: nat,A_175: nat > $o] :
% 0.26/0.47 ( ( member_nat @ X_88 @ A_175 )
% 0.26/0.47 => ( member2122167641iple_a @ ( F_63 @ X_88 ) @ ( image_43014529iple_a @ F_63 @ A_175 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_207_image__iff,axiom,
% 0.26/0.47 ! [Z_22: hoare_1167836817_state,F_62: hoare_1167836817_state > hoare_1167836817_state,A_174: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( member2058392318_state @ Z_22 @ ( image_31595733_state @ F_62 @ A_174 ) )
% 0.26/0.47 <=> ? [X_3: hoare_1167836817_state] :
% 0.26/0.47 ( ( member2058392318_state @ X_3 @ A_174 )
% 0.26/0.47 & ( Z_22
% 0.26/0.47 = ( F_62 @ X_3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_208_image__iff,axiom,
% 0.26/0.47 ! [Z_22: hoare_1167836817_state,F_62: hoare_1775062406iple_a > hoare_1167836817_state,A_174: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( member2058392318_state @ Z_22 @ ( image_1021683026_state @ F_62 @ A_174 ) )
% 0.26/0.47 <=> ? [X_3: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( member2122167641iple_a @ X_3 @ A_174 )
% 0.26/0.47 & ( Z_22
% 0.26/0.47 = ( F_62 @ X_3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_209_image__iff,axiom,
% 0.26/0.47 ! [Z_22: nat,F_62: nat > nat,A_174: nat > $o] :
% 0.26/0.47 ( ( member_nat @ Z_22 @ ( image_nat_nat @ F_62 @ A_174 ) )
% 0.26/0.47 <=> ? [X_3: nat] :
% 0.26/0.47 ( ( member_nat @ X_3 @ A_174 )
% 0.26/0.47 & ( Z_22
% 0.26/0.47 = ( F_62 @ X_3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_210_image__iff,axiom,
% 0.26/0.47 ! [Z_22: nat,F_62: hoare_1775062406iple_a > nat,A_174: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( member_nat @ Z_22 @ ( image_1806517641_a_nat @ F_62 @ A_174 ) )
% 0.26/0.47 <=> ? [X_3: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( member2122167641iple_a @ X_3 @ A_174 )
% 0.26/0.47 & ( Z_22
% 0.26/0.47 = ( F_62 @ X_3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_211_image__iff,axiom,
% 0.26/0.47 ! [Z_22: hoare_1775062406iple_a,F_62: hoare_1775062406iple_a > hoare_1775062406iple_a,A_174: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( member2122167641iple_a @ Z_22 @ ( image_1170193413iple_a @ F_62 @ A_174 ) )
% 0.26/0.47 <=> ? [X_3: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( member2122167641iple_a @ X_3 @ A_174 )
% 0.26/0.47 & ( Z_22
% 0.26/0.47 = ( F_62 @ X_3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_212_image__iff,axiom,
% 0.26/0.47 ! [Z_22: hoare_1775062406iple_a,F_62: hoare_1167836817_state > hoare_1775062406iple_a,A_174: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( member2122167641iple_a @ Z_22 @ ( image_1802845250iple_a @ F_62 @ A_174 ) )
% 0.26/0.47 <=> ? [X_3: hoare_1167836817_state] :
% 0.26/0.47 ( ( member2058392318_state @ X_3 @ A_174 )
% 0.26/0.47 & ( Z_22
% 0.26/0.47 = ( F_62 @ X_3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_213_image__iff,axiom,
% 0.26/0.47 ! [Z_22: hoare_1775062406iple_a,F_62: nat > hoare_1775062406iple_a,A_174: nat > $o] :
% 0.26/0.47 ( ( member2122167641iple_a @ Z_22 @ ( image_43014529iple_a @ F_62 @ A_174 ) )
% 0.26/0.47 <=> ? [X_3: nat] :
% 0.26/0.47 ( ( member_nat @ X_3 @ A_174 )
% 0.26/0.47 & ( Z_22
% 0.26/0.47 = ( F_62 @ X_3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_214_com_Osimps_I24_J,axiom,
% 0.26/0.47 ! [Vname: vname,Fun_1: state > nat,Com1: com,Com2: com] :
% 0.26/0.47 ( ( ass @ Vname @ Fun_1 )
% 0.26/0.47 != ( semi @ Com1 @ Com2 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_215_com_Osimps_I25_J,axiom,
% 0.26/0.47 ! [Com1: com,Com2: com,Vname: vname,Fun_1: state > nat] :
% 0.26/0.47 ( ( semi @ Com1 @ Com2 )
% 0.26/0.47 != ( ass @ Vname @ Fun_1 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_216_com_Osimps_I8_J,axiom,
% 0.26/0.47 ! [Vname_1: vname,Fun: state > nat] :
% 0.26/0.47 ( skip
% 0.26/0.47 != ( ass @ Vname_1 @ Fun ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_217_com_Osimps_I9_J,axiom,
% 0.26/0.47 ! [Vname_1: vname,Fun: state > nat] :
% 0.26/0.47 ( ( ass @ Vname_1 @ Fun )
% 0.26/0.47 != skip ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_218_image__is__empty,axiom,
% 0.26/0.47 ! [F_61: hoare_1167836817_state > hoare_1775062406iple_a,A_173: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( ( image_1802845250iple_a @ F_61 @ A_173 )
% 0.26/0.47 = bot_bo751897185le_a_o )
% 0.26/0.47 <=> ( A_173 = bot_bo70021908tate_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_219_image__is__empty,axiom,
% 0.26/0.47 ! [F_61: nat > hoare_1775062406iple_a,A_173: nat > $o] :
% 0.26/0.47 ( ( ( image_43014529iple_a @ F_61 @ A_173 )
% 0.26/0.47 = bot_bo751897185le_a_o )
% 0.26/0.47 <=> ( A_173 = bot_bot_nat_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_220_image__is__empty,axiom,
% 0.26/0.47 ! [F_61: hoare_1775062406iple_a > hoare_1167836817_state,A_173: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( ( image_1021683026_state @ F_61 @ A_173 )
% 0.26/0.47 = bot_bo70021908tate_o )
% 0.26/0.47 <=> ( A_173 = bot_bo751897185le_a_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_221_image__is__empty,axiom,
% 0.26/0.47 ! [F_61: hoare_1775062406iple_a > nat,A_173: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( ( image_1806517641_a_nat @ F_61 @ A_173 )
% 0.26/0.47 = bot_bot_nat_o )
% 0.26/0.47 <=> ( A_173 = bot_bo751897185le_a_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_222_image__empty,axiom,
% 0.26/0.47 ! [F_60: hoare_1775062406iple_a > hoare_1167836817_state] :
% 0.26/0.47 ( ( image_1021683026_state @ F_60 @ bot_bo751897185le_a_o )
% 0.26/0.47 = bot_bo70021908tate_o ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_223_image__empty,axiom,
% 0.26/0.47 ! [F_60: hoare_1775062406iple_a > nat] :
% 0.26/0.47 ( ( image_1806517641_a_nat @ F_60 @ bot_bo751897185le_a_o )
% 0.26/0.47 = bot_bot_nat_o ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_224_image__empty,axiom,
% 0.26/0.47 ! [F_60: hoare_1167836817_state > hoare_1775062406iple_a] :
% 0.26/0.47 ( ( image_1802845250iple_a @ F_60 @ bot_bo70021908tate_o )
% 0.26/0.47 = bot_bo751897185le_a_o ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_225_image__empty,axiom,
% 0.26/0.47 ! [F_60: nat > hoare_1775062406iple_a] :
% 0.26/0.47 ( ( image_43014529iple_a @ F_60 @ bot_bot_nat_o )
% 0.26/0.47 = bot_bo751897185le_a_o ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_226_empty__is__image,axiom,
% 0.26/0.47 ! [F_59: hoare_1167836817_state > hoare_1775062406iple_a,A_172: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( bot_bo751897185le_a_o
% 0.26/0.47 = ( image_1802845250iple_a @ F_59 @ A_172 ) )
% 0.26/0.47 <=> ( A_172 = bot_bo70021908tate_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_227_empty__is__image,axiom,
% 0.26/0.47 ! [F_59: nat > hoare_1775062406iple_a,A_172: nat > $o] :
% 0.26/0.47 ( ( bot_bo751897185le_a_o
% 0.26/0.47 = ( image_43014529iple_a @ F_59 @ A_172 ) )
% 0.26/0.47 <=> ( A_172 = bot_bot_nat_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_228_empty__is__image,axiom,
% 0.26/0.47 ! [F_59: hoare_1775062406iple_a > hoare_1167836817_state,A_172: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( bot_bo70021908tate_o
% 0.26/0.47 = ( image_1021683026_state @ F_59 @ A_172 ) )
% 0.26/0.47 <=> ( A_172 = bot_bo751897185le_a_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_229_empty__is__image,axiom,
% 0.26/0.47 ! [F_59: hoare_1775062406iple_a > nat,A_172: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( bot_bot_nat_o
% 0.26/0.47 = ( image_1806517641_a_nat @ F_59 @ A_172 ) )
% 0.26/0.47 <=> ( A_172 = bot_bo751897185le_a_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_230_mem__def,axiom,
% 0.26/0.47 ! [X_87: hoare_1775062406iple_a,A_171: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( member2122167641iple_a @ X_87 @ A_171 )
% 0.26/0.47 <=> ( A_171 @ X_87 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_231_mem__def,axiom,
% 0.26/0.47 ! [X_87: nat,A_171: nat > $o] :
% 0.26/0.47 ( ( member_nat @ X_87 @ A_171 )
% 0.26/0.47 <=> ( A_171 @ X_87 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_232_Collect__def,axiom,
% 0.26/0.47 ! [P_16: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( collec676402587iple_a @ P_16 )
% 0.26/0.47 = P_16 ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_233_Collect__def,axiom,
% 0.26/0.47 ! [P_16: nat > $o] :
% 0.26/0.47 ( ( collect_nat @ P_16 )
% 0.26/0.47 = P_16 ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_234_insert__image,axiom,
% 0.26/0.47 ! [F_58: hoare_1775062406iple_a > hoare_1775062406iple_a,X_86: hoare_1775062406iple_a,A_170: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( member2122167641iple_a @ X_86 @ A_170 )
% 0.26/0.47 => ( ( insert1281456128iple_a @ ( F_58 @ X_86 ) @ ( image_1170193413iple_a @ F_58 @ A_170 ) )
% 0.26/0.47 = ( image_1170193413iple_a @ F_58 @ A_170 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_235_insert__image,axiom,
% 0.26/0.47 ! [F_58: nat > hoare_1775062406iple_a,X_86: nat,A_170: nat > $o] :
% 0.26/0.47 ( ( member_nat @ X_86 @ A_170 )
% 0.26/0.47 => ( ( insert1281456128iple_a @ ( F_58 @ X_86 ) @ ( image_43014529iple_a @ F_58 @ A_170 ) )
% 0.26/0.47 = ( image_43014529iple_a @ F_58 @ A_170 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_236_image__insert,axiom,
% 0.26/0.47 ! [F_57: hoare_1775062406iple_a > hoare_1167836817_state,A_169: hoare_1775062406iple_a,B_84: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( image_1021683026_state @ F_57 @ ( insert1281456128iple_a @ A_169 @ B_84 ) )
% 0.26/0.47 = ( insert2134838167_state @ ( F_57 @ A_169 ) @ ( image_1021683026_state @ F_57 @ B_84 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_237_image__insert,axiom,
% 0.26/0.47 ! [F_57: hoare_1775062406iple_a > nat,A_169: hoare_1775062406iple_a,B_84: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( image_1806517641_a_nat @ F_57 @ ( insert1281456128iple_a @ A_169 @ B_84 ) )
% 0.26/0.47 = ( insert_nat @ ( F_57 @ A_169 ) @ ( image_1806517641_a_nat @ F_57 @ B_84 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_238_image__insert,axiom,
% 0.26/0.47 ! [F_57: hoare_1167836817_state > hoare_1775062406iple_a,A_169: hoare_1167836817_state,B_84: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( image_1802845250iple_a @ F_57 @ ( insert2134838167_state @ A_169 @ B_84 ) )
% 0.26/0.47 = ( insert1281456128iple_a @ ( F_57 @ A_169 ) @ ( image_1802845250iple_a @ F_57 @ B_84 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_239_image__insert,axiom,
% 0.26/0.47 ! [F_57: nat > hoare_1775062406iple_a,A_169: nat,B_84: nat > $o] :
% 0.26/0.47 ( ( image_43014529iple_a @ F_57 @ ( insert_nat @ A_169 @ B_84 ) )
% 0.26/0.47 = ( insert1281456128iple_a @ ( F_57 @ A_169 ) @ ( image_43014529iple_a @ F_57 @ B_84 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_240_the__sym__eq__trivial,axiom,
% 0.26/0.47 ! [X_85: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( the_Ho1155011127iple_a @ ( fequal1288209029iple_a @ X_85 ) )
% 0.26/0.47 = X_85 ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_241_the__eq__trivial,axiom,
% 0.26/0.47 ! [A_168: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( the_Ho1155011127iple_a
% 0.26/0.47 @ ^ [X_3: hoare_1775062406iple_a] : ( X_3 = A_168 ) )
% 0.26/0.47 = A_168 ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_242_If__def,axiom,
% 0.26/0.47 ! [X_84: hoare_1775062406iple_a,Y_39: hoare_1775062406iple_a,P_15: $o] :
% 0.26/0.47 ( ( P_15
% 0.26/0.47 => ( X_84
% 0.26/0.47 = ( the_Ho1155011127iple_a
% 0.26/0.47 @ ^ [Z_21: hoare_1775062406iple_a] : ( (&) @ ( (=>) @ P_15 @ ( Z_21 = X_84 ) ) @ ( (=>) @ ( (~) @ P_15 ) @ ( Z_21 = Y_39 ) ) ) ) ) )
% 0.26/0.47 & ( ~ P_15
% 0.26/0.47 => ( Y_39
% 0.26/0.47 = ( the_Ho1155011127iple_a
% 0.26/0.47 @ ^ [Z_21: hoare_1775062406iple_a] : ( (&) @ ( (=>) @ P_15 @ ( Z_21 = X_84 ) ) @ ( (=>) @ ( (~) @ P_15 ) @ ( Z_21 = Y_39 ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_243_fold1Set__sing,axiom,
% 0.26/0.47 ! [F_56: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,A_167: hoare_1167836817_state,B_83: hoare_1167836817_state] :
% 0.26/0.47 ( ( finite309220289_state @ F_56 @ ( insert2134838167_state @ A_167 @ bot_bo70021908tate_o ) @ B_83 )
% 0.26/0.47 <=> ( A_167 = B_83 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_244_fold1Set__sing,axiom,
% 0.26/0.47 ! [F_56: nat > nat > nat,A_167: nat,B_83: nat] :
% 0.26/0.47 ( ( finite_fold1Set_nat @ F_56 @ ( insert_nat @ A_167 @ bot_bot_nat_o ) @ B_83 )
% 0.26/0.47 <=> ( A_167 = B_83 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_245_fold1Set__sing,axiom,
% 0.26/0.47 ! [F_56: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,A_167: hoare_1775062406iple_a,B_83: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( finite1946188886iple_a @ F_56 @ ( insert1281456128iple_a @ A_167 @ bot_bo751897185le_a_o ) @ B_83 )
% 0.26/0.47 <=> ( A_167 = B_83 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_246_the__equality,axiom,
% 0.26/0.47 ! [P_14: hoare_1775062406iple_a > $o,A_166: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( P_14 @ A_166 )
% 0.26/0.47 => ( ! [X_3: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( P_14 @ X_3 )
% 0.26/0.47 => ( X_3 = A_166 ) )
% 0.26/0.47 => ( ( the_Ho1155011127iple_a @ P_14 )
% 0.26/0.47 = A_166 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_247_folding__one_Osingleton,axiom,
% 0.26/0.47 ! [X_83: hoare_1167836817_state,F_55: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_54: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
% 0.26/0.47 ( ( finite1074406356_state @ F_55 @ F_54 )
% 0.26/0.47 => ( ( F_54 @ ( insert2134838167_state @ X_83 @ bot_bo70021908tate_o ) )
% 0.26/0.47 = X_83 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_248_folding__one_Osingleton,axiom,
% 0.26/0.47 ! [X_83: nat,F_55: nat > nat > nat,F_54: ( nat > $o ) > nat] :
% 0.26/0.47 ( ( finite988810631ne_nat @ F_55 @ F_54 )
% 0.26/0.47 => ( ( F_54 @ ( insert_nat @ X_83 @ bot_bot_nat_o ) )
% 0.26/0.47 = X_83 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_249_folding__one_Osingleton,axiom,
% 0.26/0.47 ! [X_83: hoare_1775062406iple_a,F_55: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_54: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
% 0.26/0.47 ( ( finite2078349315iple_a @ F_55 @ F_54 )
% 0.26/0.47 => ( ( F_54 @ ( insert1281456128iple_a @ X_83 @ bot_bo751897185le_a_o ) )
% 0.26/0.47 = X_83 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_250_hoare__derivs_OLocal,axiom,
% 0.26/0.47 ! [A_165: state > nat,G_6: hoare_1167836817_state > $o,P_13: state > state > $o,C_30: com,Q_5: state > state > $o,X_82: loc,S_6: state] :
% 0.26/0.47 ( ( hoare_123228589_state @ G_6
% 0.26/0.47 @ ( insert2134838167_state
% 0.26/0.47 @ ( hoare_908217195_state @ P_13 @ C_30
% 0.26/0.47 @ ^ [Z: state,S: state] : ( Q_5 @ Z @ ( update @ S @ ( loc_1 @ X_82 ) @ ( getlocs @ S_6 @ X_82 ) ) ) )
% 0.26/0.47 @ bot_bo70021908tate_o ) )
% 0.26/0.47 => ( hoare_123228589_state @ G_6
% 0.26/0.47 @ ( insert2134838167_state
% 0.26/0.47 @ ( hoare_908217195_state
% 0.26/0.47 @ ^ [Z: state,S: state] : ( (&) @ ( S_6 = S ) @ ( P_13 @ Z @ ( update @ S @ ( loc_1 @ X_82 ) @ ( A_165 @ S ) ) ) )
% 0.26/0.47 @ ( local @ X_82 @ A_165 @ C_30 )
% 0.26/0.47 @ Q_5 )
% 0.26/0.47 @ bot_bo70021908tate_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_251_hoare__derivs_OLocal,axiom,
% 0.26/0.47 ! [A_165: state > nat,G_6: hoare_1775062406iple_a > $o,P_13: x_a > state > $o,C_30: com,Q_5: x_a > state > $o,X_82: loc,S_6: state] :
% 0.26/0.47 ( ( hoare_1508237396rivs_a @ G_6
% 0.26/0.47 @ ( insert1281456128iple_a
% 0.26/0.47 @ ( hoare_1766022166iple_a @ P_13 @ C_30
% 0.26/0.47 @ ^ [Z: x_a,S: state] : ( Q_5 @ Z @ ( update @ S @ ( loc_1 @ X_82 ) @ ( getlocs @ S_6 @ X_82 ) ) ) )
% 0.26/0.47 @ bot_bo751897185le_a_o ) )
% 0.26/0.47 => ( hoare_1508237396rivs_a @ G_6
% 0.26/0.47 @ ( insert1281456128iple_a
% 0.26/0.47 @ ( hoare_1766022166iple_a
% 0.26/0.47 @ ^ [Z: x_a,S: state] : ( (&) @ ( S_6 = S ) @ ( P_13 @ Z @ ( update @ S @ ( loc_1 @ X_82 ) @ ( A_165 @ S ) ) ) )
% 0.26/0.47 @ ( local @ X_82 @ A_165 @ C_30 )
% 0.26/0.47 @ Q_5 )
% 0.26/0.47 @ bot_bo751897185le_a_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_252_vname_Osimps_I2_J,axiom,
% 0.26/0.47 ! [Loc_2: loc,Loc_1: loc] :
% 0.26/0.47 ( ( ( loc_1 @ Loc_2 )
% 0.26/0.47 = ( loc_1 @ Loc_1 ) )
% 0.26/0.47 <=> ( Loc_2 = Loc_1 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_253_com_Osimps_I2_J,axiom,
% 0.26/0.47 ! [Loc_2: loc,Fun_1: state > nat,Com_1: com,Loc_1: loc,Fun: state > nat,Com: com] :
% 0.26/0.47 ( ( ( local @ Loc_2 @ Fun_1 @ Com_1 )
% 0.26/0.47 = ( local @ Loc_1 @ Fun @ Com ) )
% 0.26/0.47 <=> ( ( Loc_2 = Loc_1 )
% 0.26/0.47 & ( Fun_1 = Fun )
% 0.26/0.47 & ( Com_1 = Com ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_254_com_Osimps_I34_J,axiom,
% 0.26/0.47 ! [Loc_2: loc,Fun_1: state > nat,Com_1: com,Com1: com,Com2: com] :
% 0.26/0.47 ( ( local @ Loc_2 @ Fun_1 @ Com_1 )
% 0.26/0.47 != ( semi @ Com1 @ Com2 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_255_com_Osimps_I35_J,axiom,
% 0.26/0.47 ! [Com1: com,Com2: com,Loc_2: loc,Fun_1: state > nat,Com_1: com] :
% 0.26/0.47 ( ( semi @ Com1 @ Com2 )
% 0.26/0.47 != ( local @ Loc_2 @ Fun_1 @ Com_1 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_256_com_Osimps_I23_J,axiom,
% 0.26/0.47 ! [Loc_1: loc,Fun: state > nat,Com: com,Vname: vname,Fun_1: state > nat] :
% 0.26/0.47 ( ( local @ Loc_1 @ Fun @ Com )
% 0.26/0.47 != ( ass @ Vname @ Fun_1 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_257_com_Osimps_I22_J,axiom,
% 0.26/0.47 ! [Vname: vname,Fun_1: state > nat,Loc_1: loc,Fun: state > nat,Com: com] :
% 0.26/0.47 ( ( ass @ Vname @ Fun_1 )
% 0.26/0.47 != ( local @ Loc_1 @ Fun @ Com ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_258_com_Osimps_I11_J,axiom,
% 0.26/0.47 ! [Loc_1: loc,Fun: state > nat,Com: com] :
% 0.26/0.47 ( ( local @ Loc_1 @ Fun @ Com )
% 0.26/0.47 != skip ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_259_com_Osimps_I10_J,axiom,
% 0.26/0.47 ! [Loc_1: loc,Fun: state > nat,Com: com] :
% 0.26/0.47 ( skip
% 0.26/0.47 != ( local @ Loc_1 @ Fun @ Com ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_260_empty__fold1SetE,axiom,
% 0.26/0.47 ! [F_53: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,X_81: hoare_1167836817_state] :
% 0.26/0.47 ~ ( finite309220289_state @ F_53 @ bot_bo70021908tate_o @ X_81 ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_261_empty__fold1SetE,axiom,
% 0.26/0.47 ! [F_53: nat > nat > nat,X_81: nat] :
% 0.26/0.47 ~ ( finite_fold1Set_nat @ F_53 @ bot_bot_nat_o @ X_81 ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_262_empty__fold1SetE,axiom,
% 0.26/0.47 ! [F_53: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,X_81: hoare_1775062406iple_a] :
% 0.26/0.47 ~ ( finite1946188886iple_a @ F_53 @ bot_bo751897185le_a_o @ X_81 ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_263_fold1Set__nonempty,axiom,
% 0.26/0.47 ! [F_52: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,A_164: hoare_1167836817_state > $o,X_80: hoare_1167836817_state] :
% 0.26/0.47 ( ( finite309220289_state @ F_52 @ A_164 @ X_80 )
% 0.26/0.47 => ( A_164 != bot_bo70021908tate_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_264_fold1Set__nonempty,axiom,
% 0.26/0.47 ! [F_52: nat > nat > nat,A_164: nat > $o,X_80: nat] :
% 0.26/0.47 ( ( finite_fold1Set_nat @ F_52 @ A_164 @ X_80 )
% 0.26/0.47 => ( A_164 != bot_bot_nat_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_265_fold1Set__nonempty,axiom,
% 0.26/0.47 ! [F_52: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,A_164: hoare_1775062406iple_a > $o,X_80: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( finite1946188886iple_a @ F_52 @ A_164 @ X_80 )
% 0.26/0.47 => ( A_164 != bot_bo751897185le_a_o ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_266_theI,axiom,
% 0.26/0.47 ! [P_12: hoare_1775062406iple_a > $o,A_163: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( P_12 @ A_163 )
% 0.26/0.47 => ( ! [X_3: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( P_12 @ X_3 )
% 0.26/0.47 => ( X_3 = A_163 ) )
% 0.26/0.47 => ( P_12 @ ( the_Ho1155011127iple_a @ P_12 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_267_the1__equality,axiom,
% 0.26/0.47 ! [A_162: hoare_1775062406iple_a,P_11: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ? [X_3: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( P_11 @ X_3 )
% 0.26/0.47 & ! [Y_1: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( P_11 @ Y_1 )
% 0.26/0.47 => ( Y_1 = X_3 ) ) )
% 0.26/0.47 => ( ( P_11 @ A_162 )
% 0.26/0.47 => ( ( the_Ho1155011127iple_a @ P_11 )
% 0.26/0.47 = A_162 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_268_theI_H,axiom,
% 0.26/0.47 ! [P_10: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ? [X_3: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( P_10 @ X_3 )
% 0.26/0.47 & ! [Y_1: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( P_10 @ Y_1 )
% 0.26/0.47 => ( Y_1 = X_3 ) ) )
% 0.26/0.47 => ( P_10 @ ( the_Ho1155011127iple_a @ P_10 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_269_evalc_OLocal,axiom,
% 0.26/0.47 ! [C_9: com,S0: state,Y_37: loc,A_158: state > nat,S1: state] :
% 0.26/0.47 ( ( evalc @ C_9 @ ( update @ S0 @ ( loc_1 @ Y_37 ) @ ( A_158 @ S0 ) ) @ S1 )
% 0.26/0.47 => ( evalc @ ( local @ Y_37 @ A_158 @ C_9 ) @ S0 @ ( update @ S1 @ ( loc_1 @ Y_37 ) @ ( getlocs @ S0 @ Y_37 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_270_evaln_OLocal,axiom,
% 0.26/0.47 ! [C_9: com,S0: state,Y_37: loc,A_158: state > nat,N_3: nat,S1: state] :
% 0.26/0.47 ( ( evaln @ C_9 @ ( update @ S0 @ ( loc_1 @ Y_37 ) @ ( A_158 @ S0 ) ) @ N_3 @ S1 )
% 0.26/0.47 => ( evaln @ ( local @ Y_37 @ A_158 @ C_9 ) @ S0 @ N_3 @ ( update @ S1 @ ( loc_1 @ Y_37 ) @ ( getlocs @ S0 @ Y_37 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_271_fold1Set_Ointros,axiom,
% 0.26/0.47 ! [F_51: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,A_161: hoare_1167836817_state,A_160: hoare_1167836817_state > $o,X_79: hoare_1167836817_state] :
% 0.26/0.47 ( ( finite1316643734_state @ F_51 @ A_161 @ A_160 @ X_79 )
% 0.26/0.47 => ( ~ ( member2058392318_state @ A_161 @ A_160 )
% 0.26/0.47 => ( finite309220289_state @ F_51 @ ( insert2134838167_state @ A_161 @ A_160 ) @ X_79 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_272_fold1Set_Ointros,axiom,
% 0.26/0.47 ! [F_51: nat > nat > nat,A_161: nat,A_160: nat > $o,X_79: nat] :
% 0.26/0.47 ( ( finite929467206at_nat @ F_51 @ A_161 @ A_160 @ X_79 )
% 0.26/0.47 => ( ~ ( member_nat @ A_161 @ A_160 )
% 0.26/0.47 => ( finite_fold1Set_nat @ F_51 @ ( insert_nat @ A_161 @ A_160 ) @ X_79 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_273_fold1Set_Ointros,axiom,
% 0.26/0.47 ! [F_51: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,A_161: hoare_1775062406iple_a,A_160: hoare_1775062406iple_a > $o,X_79: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( finite727644230iple_a @ F_51 @ A_161 @ A_160 @ X_79 )
% 0.26/0.47 => ( ~ ( member2122167641iple_a @ A_161 @ A_160 )
% 0.26/0.47 => ( finite1946188886iple_a @ F_51 @ ( insert1281456128iple_a @ A_161 @ A_160 ) @ X_79 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_274_evaln_OSemi,axiom,
% 0.26/0.47 ! [C1: com,S2: state,C0: com,S0: state,N_3: nat,S1: state] :
% 0.26/0.47 ( ( evaln @ C0 @ S0 @ N_3 @ S1 )
% 0.26/0.47 => ( ( evaln @ C1 @ S1 @ N_3 @ S2 )
% 0.26/0.47 => ( evaln @ ( semi @ C0 @ C1 ) @ S0 @ N_3 @ S2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_275_evaln_OSkip,axiom,
% 0.26/0.47 ! [S_2: state,N_3: nat] : ( evaln @ skip @ S_2 @ N_3 @ S_2 ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_276_evaln__elim__cases_I1_J,axiom,
% 0.26/0.47 ! [S_2: state,N_3: nat,T: state] :
% 0.26/0.47 ( ( evaln @ skip @ S_2 @ N_3 @ T )
% 0.26/0.47 => ( T = S_2 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_277_evalc_OSemi,axiom,
% 0.26/0.47 ! [C1: com,S2: state,C0: com,S0: state,S1: state] :
% 0.26/0.47 ( ( evalc @ C0 @ S0 @ S1 )
% 0.26/0.47 => ( ( evalc @ C1 @ S1 @ S2 )
% 0.26/0.47 => ( evalc @ ( semi @ C0 @ C1 ) @ S0 @ S2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_278_evalc_OSkip,axiom,
% 0.26/0.47 ! [S_2: state] : ( evalc @ skip @ S_2 @ S_2 ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_279_evalc__elim__cases_I1_J,axiom,
% 0.26/0.47 ! [S_2: state,T: state] :
% 0.26/0.47 ( ( evalc @ skip @ S_2 @ T )
% 0.26/0.47 => ( T = S_2 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_280_evaln_OAssign,axiom,
% 0.26/0.47 ! [X_78: vname,A_158: state > nat,S_2: state,N_3: nat] : ( evaln @ ( ass @ X_78 @ A_158 ) @ S_2 @ N_3 @ ( update @ S_2 @ X_78 @ ( A_158 @ S_2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_281_evaln__elim__cases_I2_J,axiom,
% 0.26/0.47 ! [X_78: vname,A_158: state > nat,S_2: state,N_3: nat,T: state] :
% 0.26/0.47 ( ( evaln @ ( ass @ X_78 @ A_158 ) @ S_2 @ N_3 @ T )
% 0.26/0.47 => ( T
% 0.26/0.47 = ( update @ S_2 @ X_78 @ ( A_158 @ S_2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_282_evalc_OAssign,axiom,
% 0.26/0.47 ! [X_78: vname,A_158: state > nat,S_2: state] : ( evalc @ ( ass @ X_78 @ A_158 ) @ S_2 @ ( update @ S_2 @ X_78 @ ( A_158 @ S_2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_283_evalc__elim__cases_I2_J,axiom,
% 0.26/0.47 ! [X_78: vname,A_158: state > nat,S_2: state,T: state] :
% 0.26/0.47 ( ( evalc @ ( ass @ X_78 @ A_158 ) @ S_2 @ T )
% 0.26/0.47 => ( T
% 0.26/0.47 = ( update @ S_2 @ X_78 @ ( A_158 @ S_2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_284_eval__eq,axiom,
% 0.26/0.47 ! [C_9: com,S_2: state,T: state] :
% 0.26/0.47 ( ( evalc @ C_9 @ S_2 @ T )
% 0.26/0.47 <=> ? [N_1: nat] : ( evaln @ C_9 @ S_2 @ N_1 @ T ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_285_com__det,axiom,
% 0.26/0.47 ! [U: state,C_9: com,S_2: state,T: state] :
% 0.26/0.47 ( ( evalc @ C_9 @ S_2 @ T )
% 0.26/0.47 => ( ( evalc @ C_9 @ S_2 @ U )
% 0.26/0.47 => ( U = T ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_286_evaln__evalc,axiom,
% 0.26/0.47 ! [C_9: com,S_2: state,N_3: nat,T: state] :
% 0.26/0.47 ( ( evaln @ C_9 @ S_2 @ N_3 @ T )
% 0.26/0.47 => ( evalc @ C_9 @ S_2 @ T ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_287_empty__fold__graphE,axiom,
% 0.26/0.47 ! [F_50: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,Z_20: hoare_1775062406iple_a,X_77: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( finite727644230iple_a @ F_50 @ Z_20 @ bot_bo751897185le_a_o @ X_77 )
% 0.26/0.47 => ( X_77 = Z_20 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_288_fold__graph_OemptyI,axiom,
% 0.26/0.47 ! [F_49: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,Z_19: hoare_1775062406iple_a] : ( finite727644230iple_a @ F_49 @ Z_19 @ bot_bo751897185le_a_o @ Z_19 ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_289_fold__graph_OinsertI,axiom,
% 0.26/0.47 ! [F_48: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,Z_18: hoare_1775062406iple_a,Y_38: hoare_1775062406iple_a,X_76: hoare_1775062406iple_a,A_159: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ~ ( member2122167641iple_a @ X_76 @ A_159 )
% 0.26/0.47 => ( ( finite727644230iple_a @ F_48 @ Z_18 @ A_159 @ Y_38 )
% 0.26/0.47 => ( finite727644230iple_a @ F_48 @ Z_18 @ ( insert1281456128iple_a @ X_76 @ A_159 ) @ ( F_48 @ X_76 @ Y_38 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_290_evalc__elim__cases_I3_J,axiom,
% 0.26/0.47 ! [Y_37: loc,A_158: state > nat,C_9: com,S_2: state,T: state] :
% 0.26/0.47 ( ( evalc @ ( local @ Y_37 @ A_158 @ C_9 ) @ S_2 @ T )
% 0.26/0.47 => ~ ! [S1_1: state] :
% 0.26/0.47 ( ( T
% 0.26/0.47 = ( update @ S1_1 @ ( loc_1 @ Y_37 ) @ ( getlocs @ S_2 @ Y_37 ) ) )
% 0.26/0.47 => ~ ( evalc @ C_9 @ ( update @ S_2 @ ( loc_1 @ Y_37 ) @ ( A_158 @ S_2 ) ) @ S1_1 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_291_evaln__elim__cases_I3_J,axiom,
% 0.26/0.47 ! [Y_37: loc,A_158: state > nat,C_9: com,S_2: state,N_3: nat,T: state] :
% 0.26/0.47 ( ( evaln @ ( local @ Y_37 @ A_158 @ C_9 ) @ S_2 @ N_3 @ T )
% 0.26/0.47 => ~ ! [S1_1: state] :
% 0.26/0.47 ( ( T
% 0.26/0.47 = ( update @ S1_1 @ ( loc_1 @ Y_37 ) @ ( getlocs @ S_2 @ Y_37 ) ) )
% 0.26/0.47 => ~ ( evaln @ C_9 @ ( update @ S_2 @ ( loc_1 @ Y_37 ) @ ( A_158 @ S_2 ) ) @ N_3 @ S1_1 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_292_evalc__elim__cases_I4_J,axiom,
% 0.26/0.47 ! [C1: com,C2: com,S_2: state,T: state] :
% 0.26/0.47 ( ( evalc @ ( semi @ C1 @ C2 ) @ S_2 @ T )
% 0.26/0.47 => ~ ! [S1_1: state] :
% 0.26/0.47 ( ( evalc @ C1 @ S_2 @ S1_1 )
% 0.26/0.47 => ~ ( evalc @ C2 @ S1_1 @ T ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_293_evaln__elim__cases_I4_J,axiom,
% 0.26/0.47 ! [C1: com,C2: com,S_2: state,N_3: nat,T: state] :
% 0.26/0.47 ( ( evaln @ ( semi @ C1 @ C2 ) @ S_2 @ N_3 @ T )
% 0.26/0.47 => ~ ! [S1_1: state] :
% 0.26/0.47 ( ( evaln @ C1 @ S_2 @ N_3 @ S1_1 )
% 0.26/0.47 => ~ ( evaln @ C2 @ S1_1 @ N_3 @ T ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_294_insert__fold1SetE,axiom,
% 0.26/0.47 ! [F_47: nat > nat > nat,A_157: nat,X_75: nat > $o,X_74: nat] :
% 0.26/0.47 ( ( finite_fold1Set_nat @ F_47 @ ( insert_nat @ A_157 @ X_75 ) @ X_74 )
% 0.26/0.47 => ~ ! [A_43: nat,A_98: nat > $o] :
% 0.26/0.47 ( ( ( insert_nat @ A_157 @ X_75 )
% 0.26/0.47 = ( insert_nat @ A_43 @ A_98 ) )
% 0.26/0.47 => ( ( finite929467206at_nat @ F_47 @ A_43 @ A_98 @ X_74 )
% 0.26/0.47 => ( member_nat @ A_43 @ A_98 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_295_insert__fold1SetE,axiom,
% 0.26/0.47 ! [F_47: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,A_157: hoare_1167836817_state,X_75: hoare_1167836817_state > $o,X_74: hoare_1167836817_state] :
% 0.26/0.47 ( ( finite309220289_state @ F_47 @ ( insert2134838167_state @ A_157 @ X_75 ) @ X_74 )
% 0.26/0.47 => ~ ! [A_43: hoare_1167836817_state,A_98: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( ( insert2134838167_state @ A_157 @ X_75 )
% 0.26/0.47 = ( insert2134838167_state @ A_43 @ A_98 ) )
% 0.26/0.47 => ( ( finite1316643734_state @ F_47 @ A_43 @ A_98 @ X_74 )
% 0.26/0.47 => ( member2058392318_state @ A_43 @ A_98 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_296_insert__fold1SetE,axiom,
% 0.26/0.47 ! [F_47: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,A_157: hoare_1775062406iple_a,X_75: hoare_1775062406iple_a > $o,X_74: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( finite1946188886iple_a @ F_47 @ ( insert1281456128iple_a @ A_157 @ X_75 ) @ X_74 )
% 0.26/0.47 => ~ ! [A_43: hoare_1775062406iple_a,A_98: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( ( insert1281456128iple_a @ A_157 @ X_75 )
% 0.26/0.47 = ( insert1281456128iple_a @ A_43 @ A_98 ) )
% 0.26/0.47 => ( ( finite727644230iple_a @ F_47 @ A_43 @ A_98 @ X_74 )
% 0.26/0.47 => ( member2122167641iple_a @ A_43 @ A_98 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_297_MGT__def,axiom,
% 0.26/0.47 ! [C_9: com] :
% 0.26/0.47 ( ( hoare_Mirabelle_MGT @ C_9 )
% 0.26/0.47 = ( hoare_908217195_state @ fequal_state @ C_9 @ ( evalc @ C_9 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_298_evalc__evaln,axiom,
% 0.26/0.47 ! [C_9: com,S_2: state,T: state] :
% 0.26/0.47 ( ( evalc @ C_9 @ S_2 @ T )
% 0.26/0.47 => ? [N_1: nat] : ( evaln @ C_9 @ S_2 @ N_1 @ T ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_299_fold1Set_Osimps,axiom,
% 0.26/0.47 ! [F_46: nat > nat > nat,A1_1: nat > $o,A2_1: nat] :
% 0.26/0.47 ( ( finite_fold1Set_nat @ F_46 @ A1_1 @ A2_1 )
% 0.26/0.47 <=> ? [A_43: nat,A_98: nat > $o,X_3: nat] :
% 0.26/0.47 ( ( A1_1
% 0.26/0.47 = ( insert_nat @ A_43 @ A_98 ) )
% 0.26/0.47 & ( A2_1 = X_3 )
% 0.26/0.47 & ( finite929467206at_nat @ F_46 @ A_43 @ A_98 @ X_3 )
% 0.26/0.47 & ~ ( member_nat @ A_43 @ A_98 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_300_fold1Set_Osimps,axiom,
% 0.26/0.47 ! [F_46: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,A1_1: hoare_1167836817_state > $o,A2_1: hoare_1167836817_state] :
% 0.26/0.47 ( ( finite309220289_state @ F_46 @ A1_1 @ A2_1 )
% 0.26/0.47 <=> ? [A_43: hoare_1167836817_state,A_98: hoare_1167836817_state > $o,X_3: hoare_1167836817_state] :
% 0.26/0.47 ( ( A1_1
% 0.26/0.47 = ( insert2134838167_state @ A_43 @ A_98 ) )
% 0.26/0.47 & ( A2_1 = X_3 )
% 0.26/0.47 & ( finite1316643734_state @ F_46 @ A_43 @ A_98 @ X_3 )
% 0.26/0.47 & ~ ( member2058392318_state @ A_43 @ A_98 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_301_fold1Set_Osimps,axiom,
% 0.26/0.47 ! [F_46: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,A1_1: hoare_1775062406iple_a > $o,A2_1: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( finite1946188886iple_a @ F_46 @ A1_1 @ A2_1 )
% 0.26/0.47 <=> ? [A_43: hoare_1775062406iple_a,A_98: hoare_1775062406iple_a > $o,X_3: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( A1_1
% 0.26/0.47 = ( insert1281456128iple_a @ A_43 @ A_98 ) )
% 0.26/0.47 & ( A2_1 = X_3 )
% 0.26/0.47 & ( finite727644230iple_a @ F_46 @ A_43 @ A_98 @ X_3 )
% 0.26/0.47 & ~ ( member2122167641iple_a @ A_43 @ A_98 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_302_fold__graph_Osimps,axiom,
% 0.26/0.47 ! [F_45: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,Z_17: hoare_1775062406iple_a,A1: hoare_1775062406iple_a > $o,A2: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( finite727644230iple_a @ F_45 @ Z_17 @ A1 @ A2 )
% 0.26/0.47 <=> ( ( ( A1 = bot_bo751897185le_a_o )
% 0.26/0.47 & ( A2 = Z_17 ) )
% 0.26/0.47 | ? [X_3: hoare_1775062406iple_a,A_98: hoare_1775062406iple_a > $o,Y_1: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( A1
% 0.26/0.47 = ( insert1281456128iple_a @ X_3 @ A_98 ) )
% 0.26/0.47 & ( A2
% 0.26/0.47 = ( F_45 @ X_3 @ Y_1 ) )
% 0.26/0.47 & ~ ( member2122167641iple_a @ X_3 @ A_98 )
% 0.26/0.47 & ( finite727644230iple_a @ F_45 @ Z_17 @ A_98 @ Y_1 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_303_evaln__max2,axiom,
% 0.26/0.47 ! [C2: com,S2: state,N2: nat,T2: state,C1: com,S1: state,N1: nat,T1: state] :
% 0.26/0.47 ( ( evaln @ C1 @ S1 @ N1 @ T1 )
% 0.26/0.47 => ( ( evaln @ C2 @ S2 @ N2 @ T2 )
% 0.26/0.47 => ? [N_1: nat] :
% 0.26/0.47 ( ( evaln @ C1 @ S1 @ N_1 @ T1 )
% 0.26/0.47 & ( evaln @ C2 @ S2 @ N_1 @ T2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_304_triple__valid__def2,axiom,
% 0.26/0.47 ! [N_5: nat,P_9: state > state > $o,C_29: com,Q_4: state > state > $o] :
% 0.26/0.47 ( ( hoare_56934129_state @ N_5 @ ( hoare_908217195_state @ P_9 @ C_29 @ Q_4 ) )
% 0.26/0.47 <=> ! [Z: state,S: state] :
% 0.26/0.47 ( ( P_9 @ Z @ S )
% 0.26/0.47 => ! [S_5: state] :
% 0.26/0.47 ( ( evaln @ C_29 @ S @ N_5 @ S_5 )
% 0.26/0.47 => ( Q_4 @ Z @ S_5 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_305_triple__valid__def2,axiom,
% 0.26/0.47 ! [N_5: nat,P_9: x_a > state > $o,C_29: com,Q_4: x_a > state > $o] :
% 0.26/0.47 ( ( hoare_1462269968alid_a @ N_5 @ ( hoare_1766022166iple_a @ P_9 @ C_29 @ Q_4 ) )
% 0.26/0.47 <=> ! [Z: x_a,S: state] :
% 0.26/0.47 ( ( P_9 @ Z @ S )
% 0.26/0.47 => ! [S_5: state] :
% 0.26/0.47 ( ( evaln @ C_29 @ S @ N_5 @ S_5 )
% 0.26/0.47 => ( Q_4 @ Z @ S_5 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_306_folding__one_Oinsert,axiom,
% 0.26/0.47 ! [X_73: nat,A_156: nat > $o,F_44: nat > nat > nat,F_43: ( nat > $o ) > nat] :
% 0.26/0.47 ( ( finite988810631ne_nat @ F_44 @ F_43 )
% 0.26/0.47 => ( ( finite_finite_nat @ A_156 )
% 0.26/0.47 => ( ~ ( member_nat @ X_73 @ A_156 )
% 0.26/0.47 => ( ( A_156 != bot_bot_nat_o )
% 0.26/0.47 => ( ( F_43 @ ( insert_nat @ X_73 @ A_156 ) )
% 0.26/0.47 = ( F_44 @ X_73 @ ( F_43 @ A_156 ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_307_folding__one_Oinsert,axiom,
% 0.26/0.47 ! [X_73: hoare_1167836817_state,A_156: hoare_1167836817_state > $o,F_44: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_43: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
% 0.26/0.47 ( ( finite1074406356_state @ F_44 @ F_43 )
% 0.26/0.47 => ( ( finite1084549118_state @ A_156 )
% 0.26/0.47 => ( ~ ( member2058392318_state @ X_73 @ A_156 )
% 0.26/0.47 => ( ( A_156 != bot_bo70021908tate_o )
% 0.26/0.47 => ( ( F_43 @ ( insert2134838167_state @ X_73 @ A_156 ) )
% 0.26/0.47 = ( F_44 @ X_73 @ ( F_43 @ A_156 ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_308_folding__one_Oinsert,axiom,
% 0.26/0.47 ! [X_73: hoare_1775062406iple_a,A_156: hoare_1775062406iple_a > $o,F_44: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_43: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
% 0.26/0.47 ( ( finite2078349315iple_a @ F_44 @ F_43 )
% 0.26/0.47 => ( ( finite2063573081iple_a @ A_156 )
% 0.26/0.47 => ( ~ ( member2122167641iple_a @ X_73 @ A_156 )
% 0.26/0.47 => ( ( A_156 != bot_bo751897185le_a_o )
% 0.26/0.47 => ( ( F_43 @ ( insert1281456128iple_a @ X_73 @ A_156 ) )
% 0.26/0.47 = ( F_44 @ X_73 @ ( F_43 @ A_156 ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_309_finite__Collect__conjI,axiom,
% 0.26/0.47 ! [Q_3: hoare_1775062406iple_a > $o,P_8: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( ( finite2063573081iple_a @ ( collec676402587iple_a @ P_8 ) )
% 0.26/0.47 | ( finite2063573081iple_a @ ( collec676402587iple_a @ Q_3 ) ) )
% 0.26/0.47 => ( finite2063573081iple_a
% 0.26/0.47 @ ( collec676402587iple_a
% 0.26/0.47 @ ^ [X_3: hoare_1775062406iple_a] : ( (&) @ ( P_8 @ X_3 ) @ ( Q_3 @ X_3 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_310_finite__Collect__conjI,axiom,
% 0.26/0.47 ! [Q_3: nat > $o,P_8: nat > $o] :
% 0.26/0.47 ( ( ( finite_finite_nat @ ( collect_nat @ P_8 ) )
% 0.26/0.47 | ( finite_finite_nat @ ( collect_nat @ Q_3 ) ) )
% 0.26/0.47 => ( finite_finite_nat
% 0.26/0.47 @ ( collect_nat
% 0.26/0.47 @ ^ [X_3: nat] : ( (&) @ ( P_8 @ X_3 ) @ ( Q_3 @ X_3 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_311_finite_OemptyI,axiom,
% 0.26/0.47 finite_finite_nat @ bot_bot_nat_o ).
% 0.26/0.47
% 0.26/0.47 thf(fact_312_finite_OemptyI,axiom,
% 0.26/0.47 finite1084549118_state @ bot_bo70021908tate_o ).
% 0.26/0.47
% 0.26/0.47 thf(fact_313_finite_OemptyI,axiom,
% 0.26/0.47 finite2063573081iple_a @ bot_bo751897185le_a_o ).
% 0.26/0.47
% 0.26/0.47 thf(fact_314_finite_OinsertI,axiom,
% 0.26/0.47 ! [A_155: nat,A_154: nat > $o] :
% 0.26/0.47 ( ( finite_finite_nat @ A_154 )
% 0.26/0.47 => ( finite_finite_nat @ ( insert_nat @ A_155 @ A_154 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_315_finite_OinsertI,axiom,
% 0.26/0.47 ! [A_155: hoare_1167836817_state,A_154: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( finite1084549118_state @ A_154 )
% 0.26/0.47 => ( finite1084549118_state @ ( insert2134838167_state @ A_155 @ A_154 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_316_finite_OinsertI,axiom,
% 0.26/0.47 ! [A_155: hoare_1775062406iple_a,A_154: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( finite2063573081iple_a @ A_154 )
% 0.26/0.47 => ( finite2063573081iple_a @ ( insert1281456128iple_a @ A_155 @ A_154 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_317_finite__Collect__disjI,axiom,
% 0.26/0.47 ! [P_7: hoare_1775062406iple_a > $o,Q_2: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( finite2063573081iple_a
% 0.26/0.47 @ ( collec676402587iple_a
% 0.26/0.47 @ ^ [X_3: hoare_1775062406iple_a] : ( (|) @ ( P_7 @ X_3 ) @ ( Q_2 @ X_3 ) ) ) )
% 0.26/0.47 <=> ( ( finite2063573081iple_a @ ( collec676402587iple_a @ P_7 ) )
% 0.26/0.47 & ( finite2063573081iple_a @ ( collec676402587iple_a @ Q_2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_318_finite__Collect__disjI,axiom,
% 0.26/0.47 ! [P_7: nat > $o,Q_2: nat > $o] :
% 0.26/0.47 ( ( finite_finite_nat
% 0.26/0.47 @ ( collect_nat
% 0.26/0.47 @ ^ [X_3: nat] : ( (|) @ ( P_7 @ X_3 ) @ ( Q_2 @ X_3 ) ) ) )
% 0.26/0.47 <=> ( ( finite_finite_nat @ ( collect_nat @ P_7 ) )
% 0.26/0.47 & ( finite_finite_nat @ ( collect_nat @ Q_2 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_319_vname_Osimps_I1_J,axiom,
% 0.26/0.47 ! [Glb_1: glb,Glb_2: glb] :
% 0.26/0.47 ( ( ( glb_1 @ Glb_1 )
% 0.26/0.47 = ( glb_1 @ Glb_2 ) )
% 0.26/0.47 <=> ( Glb_1 = Glb_2 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_320_finite__insert,axiom,
% 0.26/0.47 ! [A_153: nat,A_152: nat > $o] :
% 0.26/0.47 ( ( finite_finite_nat @ ( insert_nat @ A_153 @ A_152 ) )
% 0.26/0.47 <=> ( finite_finite_nat @ A_152 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_321_finite__insert,axiom,
% 0.26/0.47 ! [A_153: hoare_1167836817_state,A_152: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( finite1084549118_state @ ( insert2134838167_state @ A_153 @ A_152 ) )
% 0.26/0.47 <=> ( finite1084549118_state @ A_152 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_322_finite__insert,axiom,
% 0.26/0.47 ! [A_153: hoare_1775062406iple_a,A_152: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( finite2063573081iple_a @ ( insert1281456128iple_a @ A_153 @ A_152 ) )
% 0.26/0.47 <=> ( finite2063573081iple_a @ A_152 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_323_vname_Osimps_I4_J,axiom,
% 0.26/0.47 ! [Loc_1: loc,Glb_1: glb] :
% 0.26/0.47 ( ( loc_1 @ Loc_1 )
% 0.26/0.47 != ( glb_1 @ Glb_1 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_324_vname_Osimps_I3_J,axiom,
% 0.26/0.47 ! [Glb_1: glb,Loc_1: loc] :
% 0.26/0.47 ( ( glb_1 @ Glb_1 )
% 0.26/0.47 != ( loc_1 @ Loc_1 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_325_folding__one_Oclosed,axiom,
% 0.26/0.47 ! [A_151: nat > $o,F_42: nat > nat > nat,F_41: ( nat > $o ) > nat] :
% 0.26/0.47 ( ( finite988810631ne_nat @ F_42 @ F_41 )
% 0.26/0.47 => ( ( finite_finite_nat @ A_151 )
% 0.26/0.47 => ( ( A_151 != bot_bot_nat_o )
% 0.26/0.47 => ( ! [X_3: nat,Y_1: nat] : ( member_nat @ ( F_42 @ X_3 @ Y_1 ) @ ( insert_nat @ X_3 @ ( insert_nat @ Y_1 @ bot_bot_nat_o ) ) )
% 0.26/0.47 => ( member_nat @ ( F_41 @ A_151 ) @ A_151 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_326_folding__one_Oclosed,axiom,
% 0.26/0.47 ! [A_151: hoare_1167836817_state > $o,F_42: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_41: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
% 0.26/0.47 ( ( finite1074406356_state @ F_42 @ F_41 )
% 0.26/0.47 => ( ( finite1084549118_state @ A_151 )
% 0.26/0.47 => ( ( A_151 != bot_bo70021908tate_o )
% 0.26/0.47 => ( ! [X_3: hoare_1167836817_state,Y_1: hoare_1167836817_state] : ( member2058392318_state @ ( F_42 @ X_3 @ Y_1 ) @ ( insert2134838167_state @ X_3 @ ( insert2134838167_state @ Y_1 @ bot_bo70021908tate_o ) ) )
% 0.26/0.47 => ( member2058392318_state @ ( F_41 @ A_151 ) @ A_151 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_327_folding__one_Oclosed,axiom,
% 0.26/0.47 ! [A_151: hoare_1775062406iple_a > $o,F_42: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_41: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
% 0.26/0.47 ( ( finite2078349315iple_a @ F_42 @ F_41 )
% 0.26/0.47 => ( ( finite2063573081iple_a @ A_151 )
% 0.26/0.47 => ( ( A_151 != bot_bo751897185le_a_o )
% 0.26/0.47 => ( ! [X_3: hoare_1775062406iple_a,Y_1: hoare_1775062406iple_a] : ( member2122167641iple_a @ ( F_42 @ X_3 @ Y_1 ) @ ( insert1281456128iple_a @ X_3 @ ( insert1281456128iple_a @ Y_1 @ bot_bo751897185le_a_o ) ) )
% 0.26/0.47 => ( member2122167641iple_a @ ( F_41 @ A_151 ) @ A_151 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_328_finite__nonempty__imp__fold1Set,axiom,
% 0.26/0.47 ! [F_40: nat > nat > nat,A_150: nat > $o] :
% 0.26/0.47 ( ( finite_finite_nat @ A_150 )
% 0.26/0.47 => ( ( A_150 != bot_bot_nat_o )
% 0.26/0.47 => ( ?? @ nat @ ( finite_fold1Set_nat @ F_40 @ A_150 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_329_finite__nonempty__imp__fold1Set,axiom,
% 0.26/0.47 ! [F_40: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,A_150: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( finite2063573081iple_a @ A_150 )
% 0.26/0.47 => ( ( A_150 != bot_bo751897185le_a_o )
% 0.26/0.47 => ( ?? @ hoare_1775062406iple_a @ ( finite1946188886iple_a @ F_40 @ A_150 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_330_finite__nonempty__imp__fold1Set,axiom,
% 0.26/0.47 ! [F_40: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,A_150: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( finite1084549118_state @ A_150 )
% 0.26/0.47 => ( ( A_150 != bot_bo70021908tate_o )
% 0.26/0.47 => ( ?? @ hoare_1167836817_state @ ( finite309220289_state @ F_40 @ A_150 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_331_finite__induct,axiom,
% 0.26/0.47 ! [P_6: ( nat > $o ) > $o,F_39: nat > $o] :
% 0.26/0.47 ( ( finite_finite_nat @ F_39 )
% 0.26/0.47 => ( ( P_6 @ bot_bot_nat_o )
% 0.26/0.47 => ( ! [X_3: nat,F_3: nat > $o] :
% 0.26/0.47 ( ( finite_finite_nat @ F_3 )
% 0.26/0.47 => ( ~ ( member_nat @ X_3 @ F_3 )
% 0.26/0.47 => ( ( P_6 @ F_3 )
% 0.26/0.47 => ( P_6 @ ( insert_nat @ X_3 @ F_3 ) ) ) ) )
% 0.26/0.47 => ( P_6 @ F_39 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_332_finite__induct,axiom,
% 0.26/0.47 ! [P_6: ( hoare_1167836817_state > $o ) > $o,F_39: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( finite1084549118_state @ F_39 )
% 0.26/0.47 => ( ( P_6 @ bot_bo70021908tate_o )
% 0.26/0.47 => ( ! [X_3: hoare_1167836817_state,F_3: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( finite1084549118_state @ F_3 )
% 0.26/0.47 => ( ~ ( member2058392318_state @ X_3 @ F_3 )
% 0.26/0.47 => ( ( P_6 @ F_3 )
% 0.26/0.47 => ( P_6 @ ( insert2134838167_state @ X_3 @ F_3 ) ) ) ) )
% 0.26/0.47 => ( P_6 @ F_39 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_333_finite__induct,axiom,
% 0.26/0.47 ! [P_6: ( hoare_1775062406iple_a > $o ) > $o,F_39: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( finite2063573081iple_a @ F_39 )
% 0.26/0.47 => ( ( P_6 @ bot_bo751897185le_a_o )
% 0.26/0.47 => ( ! [X_3: hoare_1775062406iple_a,F_3: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( finite2063573081iple_a @ F_3 )
% 0.26/0.47 => ( ~ ( member2122167641iple_a @ X_3 @ F_3 )
% 0.26/0.47 => ( ( P_6 @ F_3 )
% 0.26/0.47 => ( P_6 @ ( insert1281456128iple_a @ X_3 @ F_3 ) ) ) ) )
% 0.26/0.47 => ( P_6 @ F_39 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_334_finite_Osimps,axiom,
% 0.26/0.47 ! [A_149: nat > $o] :
% 0.26/0.47 ( ( finite_finite_nat @ A_149 )
% 0.26/0.47 <=> ( ( A_149 = bot_bot_nat_o )
% 0.26/0.47 | ? [A_98: nat > $o,A_43: nat] :
% 0.26/0.47 ( ( A_149
% 0.26/0.47 = ( insert_nat @ A_43 @ A_98 ) )
% 0.26/0.47 & ( finite_finite_nat @ A_98 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_335_finite_Osimps,axiom,
% 0.26/0.47 ! [A_149: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( finite1084549118_state @ A_149 )
% 0.26/0.47 <=> ( ( A_149 = bot_bo70021908tate_o )
% 0.26/0.47 | ? [A_98: hoare_1167836817_state > $o,A_43: hoare_1167836817_state] :
% 0.26/0.47 ( ( A_149
% 0.26/0.47 = ( insert2134838167_state @ A_43 @ A_98 ) )
% 0.26/0.47 & ( finite1084549118_state @ A_98 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_336_finite_Osimps,axiom,
% 0.26/0.47 ! [A_149: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( finite2063573081iple_a @ A_149 )
% 0.26/0.47 <=> ( ( A_149 = bot_bo751897185le_a_o )
% 0.26/0.47 | ? [A_98: hoare_1775062406iple_a > $o,A_43: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( A_149
% 0.26/0.47 = ( insert1281456128iple_a @ A_43 @ A_98 ) )
% 0.26/0.47 & ( finite2063573081iple_a @ A_98 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_337_finite__imp__fold__graph,axiom,
% 0.26/0.47 ! [F_38: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,Z_16: hoare_1775062406iple_a,A_148: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( finite2063573081iple_a @ A_148 )
% 0.26/0.47 => ( ?? @ hoare_1775062406iple_a @ ( finite727644230iple_a @ F_38 @ Z_16 @ A_148 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_338_folding__one__idem_Oinsert__idem,axiom,
% 0.26/0.47 ! [X_72: nat,A_147: nat > $o,F_37: nat > nat > nat,F_36: ( nat > $o ) > nat] :
% 0.26/0.47 ( ( finite795500164em_nat @ F_37 @ F_36 )
% 0.26/0.47 => ( ( finite_finite_nat @ A_147 )
% 0.26/0.47 => ( ( A_147 != bot_bot_nat_o )
% 0.26/0.47 => ( ( F_36 @ ( insert_nat @ X_72 @ A_147 ) )
% 0.26/0.47 = ( F_37 @ X_72 @ ( F_36 @ A_147 ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_339_folding__one__idem_Oinsert__idem,axiom,
% 0.26/0.47 ! [X_72: hoare_1167836817_state,A_147: hoare_1167836817_state > $o,F_37: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_36: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
% 0.26/0.47 ( ( finite806517911_state @ F_37 @ F_36 )
% 0.26/0.47 => ( ( finite1084549118_state @ A_147 )
% 0.26/0.47 => ( ( A_147 != bot_bo70021908tate_o )
% 0.26/0.47 => ( ( F_36 @ ( insert2134838167_state @ X_72 @ A_147 ) )
% 0.26/0.47 = ( F_37 @ X_72 @ ( F_36 @ A_147 ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_340_folding__one__idem_Oinsert__idem,axiom,
% 0.26/0.47 ! [X_72: hoare_1775062406iple_a,A_147: hoare_1775062406iple_a > $o,F_37: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_36: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
% 0.26/0.47 ( ( finite1358382848iple_a @ F_37 @ F_36 )
% 0.26/0.47 => ( ( finite2063573081iple_a @ A_147 )
% 0.26/0.47 => ( ( A_147 != bot_bo751897185le_a_o )
% 0.26/0.47 => ( ( F_36 @ ( insert1281456128iple_a @ X_72 @ A_147 ) )
% 0.26/0.47 = ( F_37 @ X_72 @ ( F_36 @ A_147 ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_341_finite__ne__induct,axiom,
% 0.26/0.47 ! [P_5: ( nat > $o ) > $o,F_35: nat > $o] :
% 0.26/0.47 ( ( finite_finite_nat @ F_35 )
% 0.26/0.47 => ( ( F_35 != bot_bot_nat_o )
% 0.26/0.47 => ( ! [X_3: nat] : ( P_5 @ ( insert_nat @ X_3 @ bot_bot_nat_o ) )
% 0.26/0.47 => ( ! [X_3: nat,F_3: nat > $o] :
% 0.26/0.47 ( ( finite_finite_nat @ F_3 )
% 0.26/0.47 => ( ( F_3 != bot_bot_nat_o )
% 0.26/0.47 => ( ~ ( member_nat @ X_3 @ F_3 )
% 0.26/0.47 => ( ( P_5 @ F_3 )
% 0.26/0.47 => ( P_5 @ ( insert_nat @ X_3 @ F_3 ) ) ) ) ) )
% 0.26/0.47 => ( P_5 @ F_35 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_342_finite__ne__induct,axiom,
% 0.26/0.47 ! [P_5: ( hoare_1167836817_state > $o ) > $o,F_35: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( finite1084549118_state @ F_35 )
% 0.26/0.47 => ( ( F_35 != bot_bo70021908tate_o )
% 0.26/0.47 => ( ! [X_3: hoare_1167836817_state] : ( P_5 @ ( insert2134838167_state @ X_3 @ bot_bo70021908tate_o ) )
% 0.26/0.47 => ( ! [X_3: hoare_1167836817_state,F_3: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( finite1084549118_state @ F_3 )
% 0.26/0.47 => ( ( F_3 != bot_bo70021908tate_o )
% 0.26/0.47 => ( ~ ( member2058392318_state @ X_3 @ F_3 )
% 0.26/0.47 => ( ( P_5 @ F_3 )
% 0.26/0.47 => ( P_5 @ ( insert2134838167_state @ X_3 @ F_3 ) ) ) ) ) )
% 0.26/0.47 => ( P_5 @ F_35 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_343_finite__ne__induct,axiom,
% 0.26/0.47 ! [P_5: ( hoare_1775062406iple_a > $o ) > $o,F_35: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( finite2063573081iple_a @ F_35 )
% 0.26/0.47 => ( ( F_35 != bot_bo751897185le_a_o )
% 0.26/0.47 => ( ! [X_3: hoare_1775062406iple_a] : ( P_5 @ ( insert1281456128iple_a @ X_3 @ bot_bo751897185le_a_o ) )
% 0.26/0.47 => ( ! [X_3: hoare_1775062406iple_a,F_3: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( finite2063573081iple_a @ F_3 )
% 0.26/0.47 => ( ( F_3 != bot_bo751897185le_a_o )
% 0.26/0.47 => ( ~ ( member2122167641iple_a @ X_3 @ F_3 )
% 0.26/0.47 => ( ( P_5 @ F_3 )
% 0.26/0.47 => ( P_5 @ ( insert1281456128iple_a @ X_3 @ F_3 ) ) ) ) ) )
% 0.26/0.47 => ( P_5 @ F_35 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_344_vname_Oexhaust,axiom,
% 0.26/0.47 ! [Y: vname] :
% 0.26/0.47 ( ! [Glb: glb] :
% 0.26/0.47 ( Y
% 0.26/0.47 != ( glb_1 @ Glb ) )
% 0.26/0.47 => ~ ! [Loc: loc] :
% 0.26/0.47 ( Y
% 0.26/0.47 != ( loc_1 @ Loc ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_345_folding__one_Oremove,axiom,
% 0.26/0.47 ! [X_71: nat,A_146: nat > $o,F_34: nat > nat > nat,F_33: ( nat > $o ) > nat] :
% 0.26/0.47 ( ( finite988810631ne_nat @ F_34 @ F_33 )
% 0.26/0.47 => ( ( finite_finite_nat @ A_146 )
% 0.26/0.47 => ( ( member_nat @ X_71 @ A_146 )
% 0.26/0.47 => ( ( ( ( minus_minus_nat_o @ A_146 @ ( insert_nat @ X_71 @ bot_bot_nat_o ) )
% 0.26/0.47 = bot_bot_nat_o )
% 0.26/0.47 => ( ( F_33 @ A_146 )
% 0.26/0.47 = X_71 ) )
% 0.26/0.47 & ( ( ( minus_minus_nat_o @ A_146 @ ( insert_nat @ X_71 @ bot_bot_nat_o ) )
% 0.26/0.47 != bot_bot_nat_o )
% 0.26/0.47 => ( ( F_33 @ A_146 )
% 0.26/0.47 = ( F_34 @ X_71 @ ( F_33 @ ( minus_minus_nat_o @ A_146 @ ( insert_nat @ X_71 @ bot_bot_nat_o ) ) ) ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_346_folding__one_Oremove,axiom,
% 0.26/0.47 ! [X_71: hoare_1167836817_state,A_146: hoare_1167836817_state > $o,F_34: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_33: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
% 0.26/0.47 ( ( finite1074406356_state @ F_34 @ F_33 )
% 0.26/0.47 => ( ( finite1084549118_state @ A_146 )
% 0.26/0.47 => ( ( member2058392318_state @ X_71 @ A_146 )
% 0.26/0.47 => ( ( ( ( minus_2107060239tate_o @ A_146 @ ( insert2134838167_state @ X_71 @ bot_bo70021908tate_o ) )
% 0.26/0.47 = bot_bo70021908tate_o )
% 0.26/0.47 => ( ( F_33 @ A_146 )
% 0.26/0.47 = X_71 ) )
% 0.26/0.47 & ( ( ( minus_2107060239tate_o @ A_146 @ ( insert2134838167_state @ X_71 @ bot_bo70021908tate_o ) )
% 0.26/0.47 != bot_bo70021908tate_o )
% 0.26/0.47 => ( ( F_33 @ A_146 )
% 0.26/0.47 = ( F_34 @ X_71 @ ( F_33 @ ( minus_2107060239tate_o @ A_146 @ ( insert2134838167_state @ X_71 @ bot_bo70021908tate_o ) ) ) ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_347_folding__one_Oremove,axiom,
% 0.26/0.47 ! [X_71: hoare_1775062406iple_a,A_146: hoare_1775062406iple_a > $o,F_34: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_33: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
% 0.26/0.47 ( ( finite2078349315iple_a @ F_34 @ F_33 )
% 0.26/0.47 => ( ( finite2063573081iple_a @ A_146 )
% 0.26/0.47 => ( ( member2122167641iple_a @ X_71 @ A_146 )
% 0.26/0.47 => ( ( ( ( minus_1944206118le_a_o @ A_146 @ ( insert1281456128iple_a @ X_71 @ bot_bo751897185le_a_o ) )
% 0.26/0.47 = bot_bo751897185le_a_o )
% 0.26/0.47 => ( ( F_33 @ A_146 )
% 0.26/0.47 = X_71 ) )
% 0.26/0.47 & ( ( ( minus_1944206118le_a_o @ A_146 @ ( insert1281456128iple_a @ X_71 @ bot_bo751897185le_a_o ) )
% 0.26/0.47 != bot_bo751897185le_a_o )
% 0.26/0.47 => ( ( F_33 @ A_146 )
% 0.26/0.47 = ( F_34 @ X_71 @ ( F_33 @ ( minus_1944206118le_a_o @ A_146 @ ( insert1281456128iple_a @ X_71 @ bot_bo751897185le_a_o ) ) ) ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_348_folding__one_Oinsert__remove,axiom,
% 0.26/0.47 ! [X_70: nat,A_145: nat > $o,F_32: nat > nat > nat,F_31: ( nat > $o ) > nat] :
% 0.26/0.47 ( ( finite988810631ne_nat @ F_32 @ F_31 )
% 0.26/0.47 => ( ( finite_finite_nat @ A_145 )
% 0.26/0.47 => ( ( ( ( minus_minus_nat_o @ A_145 @ ( insert_nat @ X_70 @ bot_bot_nat_o ) )
% 0.26/0.47 = bot_bot_nat_o )
% 0.26/0.47 => ( ( F_31 @ ( insert_nat @ X_70 @ A_145 ) )
% 0.26/0.47 = X_70 ) )
% 0.26/0.47 & ( ( ( minus_minus_nat_o @ A_145 @ ( insert_nat @ X_70 @ bot_bot_nat_o ) )
% 0.26/0.47 != bot_bot_nat_o )
% 0.26/0.47 => ( ( F_31 @ ( insert_nat @ X_70 @ A_145 ) )
% 0.26/0.47 = ( F_32 @ X_70 @ ( F_31 @ ( minus_minus_nat_o @ A_145 @ ( insert_nat @ X_70 @ bot_bot_nat_o ) ) ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_349_folding__one_Oinsert__remove,axiom,
% 0.26/0.47 ! [X_70: hoare_1167836817_state,A_145: hoare_1167836817_state > $o,F_32: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_31: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
% 0.26/0.47 ( ( finite1074406356_state @ F_32 @ F_31 )
% 0.26/0.47 => ( ( finite1084549118_state @ A_145 )
% 0.26/0.47 => ( ( ( ( minus_2107060239tate_o @ A_145 @ ( insert2134838167_state @ X_70 @ bot_bo70021908tate_o ) )
% 0.26/0.47 = bot_bo70021908tate_o )
% 0.26/0.47 => ( ( F_31 @ ( insert2134838167_state @ X_70 @ A_145 ) )
% 0.26/0.47 = X_70 ) )
% 0.26/0.47 & ( ( ( minus_2107060239tate_o @ A_145 @ ( insert2134838167_state @ X_70 @ bot_bo70021908tate_o ) )
% 0.26/0.47 != bot_bo70021908tate_o )
% 0.26/0.47 => ( ( F_31 @ ( insert2134838167_state @ X_70 @ A_145 ) )
% 0.26/0.47 = ( F_32 @ X_70 @ ( F_31 @ ( minus_2107060239tate_o @ A_145 @ ( insert2134838167_state @ X_70 @ bot_bo70021908tate_o ) ) ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_350_folding__one_Oinsert__remove,axiom,
% 0.26/0.47 ! [X_70: hoare_1775062406iple_a,A_145: hoare_1775062406iple_a > $o,F_32: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_31: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
% 0.26/0.47 ( ( finite2078349315iple_a @ F_32 @ F_31 )
% 0.26/0.47 => ( ( finite2063573081iple_a @ A_145 )
% 0.26/0.47 => ( ( ( ( minus_1944206118le_a_o @ A_145 @ ( insert1281456128iple_a @ X_70 @ bot_bo751897185le_a_o ) )
% 0.26/0.47 = bot_bo751897185le_a_o )
% 0.26/0.47 => ( ( F_31 @ ( insert1281456128iple_a @ X_70 @ A_145 ) )
% 0.26/0.47 = X_70 ) )
% 0.26/0.47 & ( ( ( minus_1944206118le_a_o @ A_145 @ ( insert1281456128iple_a @ X_70 @ bot_bo751897185le_a_o ) )
% 0.26/0.47 != bot_bo751897185le_a_o )
% 0.26/0.47 => ( ( F_31 @ ( insert1281456128iple_a @ X_70 @ A_145 ) )
% 0.26/0.47 = ( F_32 @ X_70 @ ( F_31 @ ( minus_1944206118le_a_o @ A_145 @ ( insert1281456128iple_a @ X_70 @ bot_bo751897185le_a_o ) ) ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_351_DiffE,axiom,
% 0.26/0.47 ! [C_28: hoare_1775062406iple_a,A_144: hoare_1775062406iple_a > $o,B_82: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( member2122167641iple_a @ C_28 @ ( minus_1944206118le_a_o @ A_144 @ B_82 ) )
% 0.26/0.47 => ~ ( ( member2122167641iple_a @ C_28 @ A_144 )
% 0.26/0.47 => ( member2122167641iple_a @ C_28 @ B_82 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_352_DiffE,axiom,
% 0.26/0.47 ! [C_28: nat,A_144: nat > $o,B_82: nat > $o] :
% 0.26/0.47 ( ( member_nat @ C_28 @ ( minus_minus_nat_o @ A_144 @ B_82 ) )
% 0.26/0.47 => ~ ( ( member_nat @ C_28 @ A_144 )
% 0.26/0.47 => ( member_nat @ C_28 @ B_82 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_353_DiffI,axiom,
% 0.26/0.47 ! [B_81: hoare_1775062406iple_a > $o,C_27: hoare_1775062406iple_a,A_143: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( member2122167641iple_a @ C_27 @ A_143 )
% 0.26/0.47 => ( ~ ( member2122167641iple_a @ C_27 @ B_81 )
% 0.26/0.47 => ( member2122167641iple_a @ C_27 @ ( minus_1944206118le_a_o @ A_143 @ B_81 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_354_DiffI,axiom,
% 0.26/0.47 ! [B_81: nat > $o,C_27: nat,A_143: nat > $o] :
% 0.26/0.47 ( ( member_nat @ C_27 @ A_143 )
% 0.26/0.47 => ( ~ ( member_nat @ C_27 @ B_81 )
% 0.26/0.47 => ( member_nat @ C_27 @ ( minus_minus_nat_o @ A_143 @ B_81 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_355_finite__Diff,axiom,
% 0.26/0.47 ! [B_80: nat > $o,A_142: nat > $o] :
% 0.26/0.47 ( ( finite_finite_nat @ A_142 )
% 0.26/0.47 => ( finite_finite_nat @ ( minus_minus_nat_o @ A_142 @ B_80 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_356_DiffD2,axiom,
% 0.26/0.47 ! [C_26: hoare_1775062406iple_a,A_141: hoare_1775062406iple_a > $o,B_79: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( member2122167641iple_a @ C_26 @ ( minus_1944206118le_a_o @ A_141 @ B_79 ) )
% 0.26/0.47 => ~ ( member2122167641iple_a @ C_26 @ B_79 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_357_DiffD2,axiom,
% 0.26/0.47 ! [C_26: nat,A_141: nat > $o,B_79: nat > $o] :
% 0.26/0.47 ( ( member_nat @ C_26 @ ( minus_minus_nat_o @ A_141 @ B_79 ) )
% 0.26/0.47 => ~ ( member_nat @ C_26 @ B_79 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_358_DiffD1,axiom,
% 0.26/0.47 ! [C_25: hoare_1775062406iple_a,A_140: hoare_1775062406iple_a > $o,B_78: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( member2122167641iple_a @ C_25 @ ( minus_1944206118le_a_o @ A_140 @ B_78 ) )
% 0.26/0.47 => ( member2122167641iple_a @ C_25 @ A_140 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_359_DiffD1,axiom,
% 0.26/0.47 ! [C_25: nat,A_140: nat > $o,B_78: nat > $o] :
% 0.26/0.47 ( ( member_nat @ C_25 @ ( minus_minus_nat_o @ A_140 @ B_78 ) )
% 0.26/0.47 => ( member_nat @ C_25 @ A_140 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_360_Diff__iff,axiom,
% 0.26/0.47 ! [C_24: hoare_1775062406iple_a,A_139: hoare_1775062406iple_a > $o,B_77: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( member2122167641iple_a @ C_24 @ ( minus_1944206118le_a_o @ A_139 @ B_77 ) )
% 0.26/0.47 <=> ( ( member2122167641iple_a @ C_24 @ A_139 )
% 0.26/0.47 & ~ ( member2122167641iple_a @ C_24 @ B_77 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_361_Diff__iff,axiom,
% 0.26/0.47 ! [C_24: nat,A_139: nat > $o,B_77: nat > $o] :
% 0.26/0.47 ( ( member_nat @ C_24 @ ( minus_minus_nat_o @ A_139 @ B_77 ) )
% 0.26/0.47 <=> ( ( member_nat @ C_24 @ A_139 )
% 0.26/0.47 & ~ ( member_nat @ C_24 @ B_77 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_362_set__diff__eq,axiom,
% 0.26/0.47 ! [A_138: hoare_1775062406iple_a > $o,B_76: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( minus_1944206118le_a_o @ A_138 @ B_76 )
% 0.26/0.47 = ( collec676402587iple_a
% 0.26/0.47 @ ^ [X_3: hoare_1775062406iple_a] : ( (&) @ ( member2122167641iple_a @ X_3 @ A_138 ) @ ( (~) @ ( member2122167641iple_a @ X_3 @ B_76 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_363_set__diff__eq,axiom,
% 0.26/0.47 ! [A_138: nat > $o,B_76: nat > $o] :
% 0.26/0.47 ( ( minus_minus_nat_o @ A_138 @ B_76 )
% 0.26/0.47 = ( collect_nat
% 0.26/0.47 @ ^ [X_3: nat] : ( (&) @ ( member_nat @ X_3 @ A_138 ) @ ( (~) @ ( member_nat @ X_3 @ B_76 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_364_Diff__cancel,axiom,
% 0.26/0.47 ! [A_137: nat > $o] :
% 0.26/0.47 ( ( minus_minus_nat_o @ A_137 @ A_137 )
% 0.26/0.47 = bot_bot_nat_o ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_365_Diff__cancel,axiom,
% 0.26/0.47 ! [A_137: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( minus_2107060239tate_o @ A_137 @ A_137 )
% 0.26/0.47 = bot_bo70021908tate_o ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_366_Diff__cancel,axiom,
% 0.26/0.47 ! [A_137: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( minus_1944206118le_a_o @ A_137 @ A_137 )
% 0.26/0.47 = bot_bo751897185le_a_o ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_367_Diff__empty,axiom,
% 0.26/0.47 ! [A_136: nat > $o] :
% 0.26/0.47 ( ( minus_minus_nat_o @ A_136 @ bot_bot_nat_o )
% 0.26/0.47 = A_136 ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_368_Diff__empty,axiom,
% 0.26/0.47 ! [A_136: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( minus_2107060239tate_o @ A_136 @ bot_bo70021908tate_o )
% 0.26/0.47 = A_136 ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_369_Diff__empty,axiom,
% 0.26/0.47 ! [A_136: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( minus_1944206118le_a_o @ A_136 @ bot_bo751897185le_a_o )
% 0.26/0.47 = A_136 ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_370_empty__Diff,axiom,
% 0.26/0.47 ! [A_135: nat > $o] :
% 0.26/0.47 ( ( minus_minus_nat_o @ bot_bot_nat_o @ A_135 )
% 0.26/0.47 = bot_bot_nat_o ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_371_empty__Diff,axiom,
% 0.26/0.47 ! [A_135: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( minus_2107060239tate_o @ bot_bo70021908tate_o @ A_135 )
% 0.26/0.47 = bot_bo70021908tate_o ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_372_empty__Diff,axiom,
% 0.26/0.47 ! [A_135: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( minus_1944206118le_a_o @ bot_bo751897185le_a_o @ A_135 )
% 0.26/0.47 = bot_bo751897185le_a_o ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_373_finite__Diff2,axiom,
% 0.26/0.47 ! [A_134: nat > $o,B_75: nat > $o] :
% 0.26/0.47 ( ( finite_finite_nat @ B_75 )
% 0.26/0.47 => ( ( finite_finite_nat @ ( minus_minus_nat_o @ A_134 @ B_75 ) )
% 0.26/0.47 <=> ( finite_finite_nat @ A_134 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_374_insert__Diff1,axiom,
% 0.26/0.47 ! [A_133: nat > $o,X_69: nat,B_74: nat > $o] :
% 0.26/0.47 ( ( member_nat @ X_69 @ B_74 )
% 0.26/0.47 => ( ( minus_minus_nat_o @ ( insert_nat @ X_69 @ A_133 ) @ B_74 )
% 0.26/0.47 = ( minus_minus_nat_o @ A_133 @ B_74 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_375_insert__Diff1,axiom,
% 0.26/0.47 ! [A_133: hoare_1167836817_state > $o,X_69: hoare_1167836817_state,B_74: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( member2058392318_state @ X_69 @ B_74 )
% 0.26/0.47 => ( ( minus_2107060239tate_o @ ( insert2134838167_state @ X_69 @ A_133 ) @ B_74 )
% 0.26/0.47 = ( minus_2107060239tate_o @ A_133 @ B_74 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_376_insert__Diff1,axiom,
% 0.26/0.47 ! [A_133: hoare_1775062406iple_a > $o,X_69: hoare_1775062406iple_a,B_74: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( member2122167641iple_a @ X_69 @ B_74 )
% 0.26/0.47 => ( ( minus_1944206118le_a_o @ ( insert1281456128iple_a @ X_69 @ A_133 ) @ B_74 )
% 0.26/0.47 = ( minus_1944206118le_a_o @ A_133 @ B_74 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_377_insert__Diff__if,axiom,
% 0.26/0.47 ! [A_132: nat > $o,X_68: nat,B_73: nat > $o] :
% 0.26/0.47 ( ( ( member_nat @ X_68 @ B_73 )
% 0.26/0.47 => ( ( minus_minus_nat_o @ ( insert_nat @ X_68 @ A_132 ) @ B_73 )
% 0.26/0.47 = ( minus_minus_nat_o @ A_132 @ B_73 ) ) )
% 0.26/0.47 & ( ~ ( member_nat @ X_68 @ B_73 )
% 0.26/0.47 => ( ( minus_minus_nat_o @ ( insert_nat @ X_68 @ A_132 ) @ B_73 )
% 0.26/0.47 = ( insert_nat @ X_68 @ ( minus_minus_nat_o @ A_132 @ B_73 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_378_insert__Diff__if,axiom,
% 0.26/0.47 ! [A_132: hoare_1167836817_state > $o,X_68: hoare_1167836817_state,B_73: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( ( member2058392318_state @ X_68 @ B_73 )
% 0.26/0.47 => ( ( minus_2107060239tate_o @ ( insert2134838167_state @ X_68 @ A_132 ) @ B_73 )
% 0.26/0.47 = ( minus_2107060239tate_o @ A_132 @ B_73 ) ) )
% 0.26/0.47 & ( ~ ( member2058392318_state @ X_68 @ B_73 )
% 0.26/0.47 => ( ( minus_2107060239tate_o @ ( insert2134838167_state @ X_68 @ A_132 ) @ B_73 )
% 0.26/0.47 = ( insert2134838167_state @ X_68 @ ( minus_2107060239tate_o @ A_132 @ B_73 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_379_insert__Diff__if,axiom,
% 0.26/0.47 ! [A_132: hoare_1775062406iple_a > $o,X_68: hoare_1775062406iple_a,B_73: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( ( member2122167641iple_a @ X_68 @ B_73 )
% 0.26/0.47 => ( ( minus_1944206118le_a_o @ ( insert1281456128iple_a @ X_68 @ A_132 ) @ B_73 )
% 0.26/0.47 = ( minus_1944206118le_a_o @ A_132 @ B_73 ) ) )
% 0.26/0.47 & ( ~ ( member2122167641iple_a @ X_68 @ B_73 )
% 0.26/0.47 => ( ( minus_1944206118le_a_o @ ( insert1281456128iple_a @ X_68 @ A_132 ) @ B_73 )
% 0.26/0.47 = ( insert1281456128iple_a @ X_68 @ ( minus_1944206118le_a_o @ A_132 @ B_73 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_380_insert__Diff,axiom,
% 0.26/0.47 ! [A_131: nat,A_130: nat > $o] :
% 0.26/0.47 ( ( member_nat @ A_131 @ A_130 )
% 0.26/0.47 => ( ( insert_nat @ A_131 @ ( minus_minus_nat_o @ A_130 @ ( insert_nat @ A_131 @ bot_bot_nat_o ) ) )
% 0.26/0.47 = A_130 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_381_insert__Diff,axiom,
% 0.26/0.47 ! [A_131: hoare_1167836817_state,A_130: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( member2058392318_state @ A_131 @ A_130 )
% 0.26/0.47 => ( ( insert2134838167_state @ A_131 @ ( minus_2107060239tate_o @ A_130 @ ( insert2134838167_state @ A_131 @ bot_bo70021908tate_o ) ) )
% 0.26/0.47 = A_130 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_382_insert__Diff,axiom,
% 0.26/0.47 ! [A_131: hoare_1775062406iple_a,A_130: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( member2122167641iple_a @ A_131 @ A_130 )
% 0.26/0.47 => ( ( insert1281456128iple_a @ A_131 @ ( minus_1944206118le_a_o @ A_130 @ ( insert1281456128iple_a @ A_131 @ bot_bo751897185le_a_o ) ) )
% 0.26/0.47 = A_130 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_383_Diff__insert__absorb,axiom,
% 0.26/0.47 ! [X_67: nat,A_129: nat > $o] :
% 0.26/0.47 ( ~ ( member_nat @ X_67 @ A_129 )
% 0.26/0.47 => ( ( minus_minus_nat_o @ ( insert_nat @ X_67 @ A_129 ) @ ( insert_nat @ X_67 @ bot_bot_nat_o ) )
% 0.26/0.47 = A_129 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_384_Diff__insert__absorb,axiom,
% 0.26/0.47 ! [X_67: hoare_1167836817_state,A_129: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ~ ( member2058392318_state @ X_67 @ A_129 )
% 0.26/0.47 => ( ( minus_2107060239tate_o @ ( insert2134838167_state @ X_67 @ A_129 ) @ ( insert2134838167_state @ X_67 @ bot_bo70021908tate_o ) )
% 0.26/0.47 = A_129 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_385_Diff__insert__absorb,axiom,
% 0.26/0.47 ! [X_67: hoare_1775062406iple_a,A_129: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ~ ( member2122167641iple_a @ X_67 @ A_129 )
% 0.26/0.47 => ( ( minus_1944206118le_a_o @ ( insert1281456128iple_a @ X_67 @ A_129 ) @ ( insert1281456128iple_a @ X_67 @ bot_bo751897185le_a_o ) )
% 0.26/0.47 = A_129 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_386_insert__Diff__single,axiom,
% 0.26/0.47 ! [A_128: nat,A_127: nat > $o] :
% 0.26/0.47 ( ( insert_nat @ A_128 @ ( minus_minus_nat_o @ A_127 @ ( insert_nat @ A_128 @ bot_bot_nat_o ) ) )
% 0.26/0.47 = ( insert_nat @ A_128 @ A_127 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_387_insert__Diff__single,axiom,
% 0.26/0.47 ! [A_128: hoare_1167836817_state,A_127: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( insert2134838167_state @ A_128 @ ( minus_2107060239tate_o @ A_127 @ ( insert2134838167_state @ A_128 @ bot_bo70021908tate_o ) ) )
% 0.26/0.47 = ( insert2134838167_state @ A_128 @ A_127 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_388_insert__Diff__single,axiom,
% 0.26/0.47 ! [A_128: hoare_1775062406iple_a,A_127: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( insert1281456128iple_a @ A_128 @ ( minus_1944206118le_a_o @ A_127 @ ( insert1281456128iple_a @ A_128 @ bot_bo751897185le_a_o ) ) )
% 0.26/0.47 = ( insert1281456128iple_a @ A_128 @ A_127 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_389_Diff__insert2,axiom,
% 0.26/0.47 ! [A_126: nat > $o,A_125: nat,B_72: nat > $o] :
% 0.26/0.47 ( ( minus_minus_nat_o @ A_126 @ ( insert_nat @ A_125 @ B_72 ) )
% 0.26/0.47 = ( minus_minus_nat_o @ ( minus_minus_nat_o @ A_126 @ ( insert_nat @ A_125 @ bot_bot_nat_o ) ) @ B_72 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_390_Diff__insert2,axiom,
% 0.26/0.47 ! [A_126: hoare_1167836817_state > $o,A_125: hoare_1167836817_state,B_72: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( minus_2107060239tate_o @ A_126 @ ( insert2134838167_state @ A_125 @ B_72 ) )
% 0.26/0.47 = ( minus_2107060239tate_o @ ( minus_2107060239tate_o @ A_126 @ ( insert2134838167_state @ A_125 @ bot_bo70021908tate_o ) ) @ B_72 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_391_Diff__insert2,axiom,
% 0.26/0.47 ! [A_126: hoare_1775062406iple_a > $o,A_125: hoare_1775062406iple_a,B_72: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( minus_1944206118le_a_o @ A_126 @ ( insert1281456128iple_a @ A_125 @ B_72 ) )
% 0.26/0.47 = ( minus_1944206118le_a_o @ ( minus_1944206118le_a_o @ A_126 @ ( insert1281456128iple_a @ A_125 @ bot_bo751897185le_a_o ) ) @ B_72 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_392_Diff__insert,axiom,
% 0.26/0.47 ! [A_124: nat > $o,A_123: nat,B_71: nat > $o] :
% 0.26/0.47 ( ( minus_minus_nat_o @ A_124 @ ( insert_nat @ A_123 @ B_71 ) )
% 0.26/0.47 = ( minus_minus_nat_o @ ( minus_minus_nat_o @ A_124 @ B_71 ) @ ( insert_nat @ A_123 @ bot_bot_nat_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_393_Diff__insert,axiom,
% 0.26/0.47 ! [A_124: hoare_1167836817_state > $o,A_123: hoare_1167836817_state,B_71: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( minus_2107060239tate_o @ A_124 @ ( insert2134838167_state @ A_123 @ B_71 ) )
% 0.26/0.47 = ( minus_2107060239tate_o @ ( minus_2107060239tate_o @ A_124 @ B_71 ) @ ( insert2134838167_state @ A_123 @ bot_bo70021908tate_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_394_Diff__insert,axiom,
% 0.26/0.47 ! [A_124: hoare_1775062406iple_a > $o,A_123: hoare_1775062406iple_a,B_71: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( minus_1944206118le_a_o @ A_124 @ ( insert1281456128iple_a @ A_123 @ B_71 ) )
% 0.26/0.47 = ( minus_1944206118le_a_o @ ( minus_1944206118le_a_o @ A_124 @ B_71 ) @ ( insert1281456128iple_a @ A_123 @ bot_bo751897185le_a_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_395_finite__Diff__insert,axiom,
% 0.26/0.47 ! [A_122: nat > $o,A_121: nat,B_70: nat > $o] :
% 0.26/0.47 ( ( finite_finite_nat @ ( minus_minus_nat_o @ A_122 @ ( insert_nat @ A_121 @ B_70 ) ) )
% 0.26/0.47 <=> ( finite_finite_nat @ ( minus_minus_nat_o @ A_122 @ B_70 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_396_finite__Diff__insert,axiom,
% 0.26/0.47 ! [A_122: hoare_1167836817_state > $o,A_121: hoare_1167836817_state,B_70: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( finite1084549118_state @ ( minus_2107060239tate_o @ A_122 @ ( insert2134838167_state @ A_121 @ B_70 ) ) )
% 0.26/0.47 <=> ( finite1084549118_state @ ( minus_2107060239tate_o @ A_122 @ B_70 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_397_finite__Diff__insert,axiom,
% 0.26/0.47 ! [A_122: hoare_1775062406iple_a > $o,A_121: hoare_1775062406iple_a,B_70: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( finite2063573081iple_a @ ( minus_1944206118le_a_o @ A_122 @ ( insert1281456128iple_a @ A_121 @ B_70 ) ) )
% 0.26/0.47 <=> ( finite2063573081iple_a @ ( minus_1944206118le_a_o @ A_122 @ B_70 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_398_folding__one__idem_Oin__idem,axiom,
% 0.26/0.47 ! [X_66: hoare_1775062406iple_a,A_120: hoare_1775062406iple_a > $o,F_30: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_29: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
% 0.26/0.47 ( ( finite1358382848iple_a @ F_30 @ F_29 )
% 0.26/0.47 => ( ( finite2063573081iple_a @ A_120 )
% 0.26/0.47 => ( ( member2122167641iple_a @ X_66 @ A_120 )
% 0.26/0.47 => ( ( F_30 @ X_66 @ ( F_29 @ A_120 ) )
% 0.26/0.47 = ( F_29 @ A_120 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_399_folding__one__idem_Oin__idem,axiom,
% 0.26/0.47 ! [X_66: nat,A_120: nat > $o,F_30: nat > nat > nat,F_29: ( nat > $o ) > nat] :
% 0.26/0.47 ( ( finite795500164em_nat @ F_30 @ F_29 )
% 0.26/0.47 => ( ( finite_finite_nat @ A_120 )
% 0.26/0.47 => ( ( member_nat @ X_66 @ A_120 )
% 0.26/0.47 => ( ( F_30 @ X_66 @ ( F_29 @ A_120 ) )
% 0.26/0.47 = ( F_29 @ A_120 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_400_folding__one__idem_Ohom__commute,axiom,
% 0.26/0.47 ! [N_4: nat > $o,H_1: nat > nat,F_28: nat > nat > nat,F_27: ( nat > $o ) > nat] :
% 0.26/0.47 ( ( finite795500164em_nat @ F_28 @ F_27 )
% 0.26/0.47 => ( ! [X_3: nat,Y_1: nat] :
% 0.26/0.47 ( ( H_1 @ ( F_28 @ X_3 @ Y_1 ) )
% 0.26/0.47 = ( F_28 @ ( H_1 @ X_3 ) @ ( H_1 @ Y_1 ) ) )
% 0.26/0.47 => ( ( finite_finite_nat @ N_4 )
% 0.26/0.47 => ( ( N_4 != bot_bot_nat_o )
% 0.26/0.47 => ( ( H_1 @ ( F_27 @ N_4 ) )
% 0.26/0.47 = ( F_27 @ ( image_nat_nat @ H_1 @ N_4 ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_401_folding__one__idem_Ohom__commute,axiom,
% 0.26/0.47 ! [N_4: hoare_1167836817_state > $o,H_1: hoare_1167836817_state > hoare_1167836817_state,F_28: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_27: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
% 0.26/0.47 ( ( finite806517911_state @ F_28 @ F_27 )
% 0.26/0.47 => ( ! [X_3: hoare_1167836817_state,Y_1: hoare_1167836817_state] :
% 0.26/0.47 ( ( H_1 @ ( F_28 @ X_3 @ Y_1 ) )
% 0.26/0.47 = ( F_28 @ ( H_1 @ X_3 ) @ ( H_1 @ Y_1 ) ) )
% 0.26/0.47 => ( ( finite1084549118_state @ N_4 )
% 0.26/0.47 => ( ( N_4 != bot_bo70021908tate_o )
% 0.26/0.47 => ( ( H_1 @ ( F_27 @ N_4 ) )
% 0.26/0.47 = ( F_27 @ ( image_31595733_state @ H_1 @ N_4 ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_402_folding__one__idem_Ohom__commute,axiom,
% 0.26/0.47 ! [N_4: hoare_1775062406iple_a > $o,H_1: hoare_1775062406iple_a > hoare_1775062406iple_a,F_28: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_27: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
% 0.26/0.47 ( ( finite1358382848iple_a @ F_28 @ F_27 )
% 0.26/0.47 => ( ! [X_3: hoare_1775062406iple_a,Y_1: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( H_1 @ ( F_28 @ X_3 @ Y_1 ) )
% 0.26/0.47 = ( F_28 @ ( H_1 @ X_3 ) @ ( H_1 @ Y_1 ) ) )
% 0.26/0.47 => ( ( finite2063573081iple_a @ N_4 )
% 0.26/0.47 => ( ( N_4 != bot_bo751897185le_a_o )
% 0.26/0.47 => ( ( H_1 @ ( F_27 @ N_4 ) )
% 0.26/0.47 = ( F_27 @ ( image_1170193413iple_a @ H_1 @ N_4 ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_403_finite__empty__induct,axiom,
% 0.26/0.47 ! [P_4: ( nat > $o ) > $o,A_119: nat > $o] :
% 0.26/0.47 ( ( finite_finite_nat @ A_119 )
% 0.26/0.47 => ( ( P_4 @ A_119 )
% 0.26/0.47 => ( ! [A_43: nat,A_98: nat > $o] :
% 0.26/0.47 ( ( finite_finite_nat @ A_98 )
% 0.26/0.47 => ( ( member_nat @ A_43 @ A_98 )
% 0.26/0.47 => ( ( P_4 @ A_98 )
% 0.26/0.47 => ( P_4 @ ( minus_minus_nat_o @ A_98 @ ( insert_nat @ A_43 @ bot_bot_nat_o ) ) ) ) ) )
% 0.26/0.47 => ( P_4 @ bot_bot_nat_o ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_404_finite__empty__induct,axiom,
% 0.26/0.47 ! [P_4: ( hoare_1167836817_state > $o ) > $o,A_119: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( finite1084549118_state @ A_119 )
% 0.26/0.47 => ( ( P_4 @ A_119 )
% 0.26/0.47 => ( ! [A_43: hoare_1167836817_state,A_98: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( finite1084549118_state @ A_98 )
% 0.26/0.47 => ( ( member2058392318_state @ A_43 @ A_98 )
% 0.26/0.47 => ( ( P_4 @ A_98 )
% 0.26/0.47 => ( P_4 @ ( minus_2107060239tate_o @ A_98 @ ( insert2134838167_state @ A_43 @ bot_bo70021908tate_o ) ) ) ) ) )
% 0.26/0.47 => ( P_4 @ bot_bo70021908tate_o ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_405_finite__empty__induct,axiom,
% 0.26/0.47 ! [P_4: ( hoare_1775062406iple_a > $o ) > $o,A_119: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( finite2063573081iple_a @ A_119 )
% 0.26/0.47 => ( ( P_4 @ A_119 )
% 0.26/0.47 => ( ! [A_43: hoare_1775062406iple_a,A_98: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( finite2063573081iple_a @ A_98 )
% 0.26/0.47 => ( ( member2122167641iple_a @ A_43 @ A_98 )
% 0.26/0.47 => ( ( P_4 @ A_98 )
% 0.26/0.47 => ( P_4 @ ( minus_1944206118le_a_o @ A_98 @ ( insert1281456128iple_a @ A_43 @ bot_bo751897185le_a_o ) ) ) ) ) )
% 0.26/0.47 => ( P_4 @ bot_bo751897185le_a_o ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_406_comp__fun__idem__remove,axiom,
% 0.26/0.47 ( finite389864113_nat_o
% 0.26/0.47 @ ^ [X_3: nat,A_98: nat > $o] : ( minus_minus_nat_o @ A_98 @ ( insert_nat @ X_3 @ bot_bot_nat_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_407_comp__fun__idem__remove,axiom,
% 0.26/0.47 ( finite856902323tate_o
% 0.26/0.47 @ ^ [X_3: hoare_1167836817_state,A_98: hoare_1167836817_state > $o] : ( minus_2107060239tate_o @ A_98 @ ( insert2134838167_state @ X_3 @ bot_bo70021908tate_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_408_comp__fun__idem__remove,axiom,
% 0.26/0.47 ( finite2120172977le_a_o
% 0.26/0.47 @ ^ [X_3: hoare_1775062406iple_a,A_98: hoare_1775062406iple_a > $o] : ( minus_1944206118le_a_o @ A_98 @ ( insert1281456128iple_a @ X_3 @ bot_bo751897185le_a_o ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_409_comp__fun__commute_Ofold__graph__insertE__aux,axiom,
% 0.26/0.47 ! [A_118: hoare_1775062406iple_a,Z_15: hoare_1775062406iple_a,A_117: hoare_1775062406iple_a > $o,Y_35: hoare_1775062406iple_a,F_26: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a] :
% 0.26/0.47 ( ( finite2064891473iple_a @ F_26 )
% 0.26/0.47 => ( ( finite727644230iple_a @ F_26 @ Z_15 @ A_117 @ Y_35 )
% 0.26/0.47 => ( ( member2122167641iple_a @ A_118 @ A_117 )
% 0.26/0.47 => ? [Y_36: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( Y_35
% 0.26/0.47 = ( F_26 @ A_118 @ Y_36 ) )
% 0.26/0.47 & ( finite727644230iple_a @ F_26 @ Z_15 @ ( minus_1944206118le_a_o @ A_117 @ ( insert1281456128iple_a @ A_118 @ bot_bo751897185le_a_o ) ) @ Y_36 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_410_fold__graph__permute__diff,axiom,
% 0.26/0.47 ! [A_116: nat,B_69: nat,A_115: nat > $o,X_65: nat] :
% 0.26/0.47 ( ( finite929467206at_nat @ times_times_nat @ B_69 @ A_115 @ X_65 )
% 0.26/0.47 => ( ( member_nat @ A_116 @ A_115 )
% 0.26/0.47 => ( ~ ( member_nat @ B_69 @ A_115 )
% 0.26/0.47 => ( finite929467206at_nat @ times_times_nat @ A_116 @ ( insert_nat @ B_69 @ ( minus_minus_nat_o @ A_115 @ ( insert_nat @ A_116 @ bot_bot_nat_o ) ) ) @ X_65 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_411_comp__fun__idem__insert,axiom,
% 0.26/0.47 finite389864113_nat_o @ insert_nat ).
% 0.26/0.47
% 0.26/0.47 thf(fact_412_comp__fun__idem__insert,axiom,
% 0.26/0.47 finite856902323tate_o @ insert2134838167_state ).
% 0.26/0.47
% 0.26/0.47 thf(fact_413_comp__fun__idem__insert,axiom,
% 0.26/0.47 finite2120172977le_a_o @ insert1281456128iple_a ).
% 0.26/0.47
% 0.26/0.47 thf(fact_414_comp__fun__commute_Ofold__graph__determ,axiom,
% 0.26/0.47 ! [Y_34: hoare_1775062406iple_a,Z_14: hoare_1775062406iple_a,A_114: hoare_1775062406iple_a > $o,X_64: hoare_1775062406iple_a,F_25: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a] :
% 0.26/0.47 ( ( finite2064891473iple_a @ F_25 )
% 0.26/0.47 => ( ( finite727644230iple_a @ F_25 @ Z_14 @ A_114 @ X_64 )
% 0.26/0.47 => ( ( finite727644230iple_a @ F_25 @ Z_14 @ A_114 @ Y_34 )
% 0.26/0.47 => ( Y_34 = X_64 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_415_fold__graph__insert__swap,axiom,
% 0.26/0.47 ! [Z_13: nat,B_68: nat,A_113: nat > $o,Y_33: nat] :
% 0.26/0.47 ( ( finite929467206at_nat @ times_times_nat @ B_68 @ A_113 @ Y_33 )
% 0.26/0.47 => ( ~ ( member_nat @ B_68 @ A_113 )
% 0.26/0.47 => ( finite929467206at_nat @ times_times_nat @ Z_13 @ ( insert_nat @ B_68 @ A_113 ) @ ( times_times_nat @ Z_13 @ Y_33 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_416_comp__fun__commute_Ofold__graph__insertE,axiom,
% 0.26/0.47 ! [Z_12: hoare_1775062406iple_a,X_63: hoare_1775062406iple_a,A_112: hoare_1775062406iple_a > $o,V: hoare_1775062406iple_a,F_24: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a] :
% 0.26/0.47 ( ( finite2064891473iple_a @ F_24 )
% 0.26/0.47 => ( ( finite727644230iple_a @ F_24 @ Z_12 @ ( insert1281456128iple_a @ X_63 @ A_112 ) @ V )
% 0.26/0.47 => ( ~ ( member2122167641iple_a @ X_63 @ A_112 )
% 0.26/0.47 => ~ ! [Y_1: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( V
% 0.26/0.47 = ( F_24 @ X_63 @ Y_1 ) )
% 0.26/0.47 => ~ ( finite727644230iple_a @ F_24 @ Z_12 @ A_112 @ Y_1 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_417_fold1__insert,axiom,
% 0.26/0.47 ! [X_62: nat,A_111: nat > $o] :
% 0.26/0.47 ( ( A_111 != bot_bot_nat_o )
% 0.26/0.47 => ( ( finite_finite_nat @ A_111 )
% 0.26/0.47 => ( ~ ( member_nat @ X_62 @ A_111 )
% 0.26/0.47 => ( ( finite_fold1_nat @ times_times_nat @ ( insert_nat @ X_62 @ A_111 ) )
% 0.26/0.47 = ( times_times_nat @ X_62 @ ( finite_fold1_nat @ times_times_nat @ A_111 ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_418_fold1__eq__fold,axiom,
% 0.26/0.47 ! [A_110: nat,A_109: nat > $o] :
% 0.26/0.47 ( ( finite_finite_nat @ A_109 )
% 0.26/0.47 => ( ~ ( member_nat @ A_110 @ A_109 )
% 0.26/0.47 => ( ( finite_fold1_nat @ times_times_nat @ ( insert_nat @ A_110 @ A_109 ) )
% 0.26/0.47 = ( finite_fold_nat_nat @ times_times_nat @ A_110 @ A_109 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_419_fold1__singleton,axiom,
% 0.26/0.47 ! [F_23: nat > nat > nat,A_108: nat] :
% 0.26/0.47 ( ( finite_fold1_nat @ F_23 @ ( insert_nat @ A_108 @ bot_bot_nat_o ) )
% 0.26/0.47 = A_108 ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_420_fold1__singleton,axiom,
% 0.26/0.47 ! [F_23: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,A_108: hoare_1167836817_state] :
% 0.26/0.47 ( ( finite1646097201_state @ F_23 @ ( insert2134838167_state @ A_108 @ bot_bo70021908tate_o ) )
% 0.26/0.47 = A_108 ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_421_fold1__singleton,axiom,
% 0.26/0.47 ! [F_23: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,A_108: hoare_1775062406iple_a] :
% 0.26/0.47 ( ( finite1790765286iple_a @ F_23 @ ( insert1281456128iple_a @ A_108 @ bot_bo751897185le_a_o ) )
% 0.26/0.47 = A_108 ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_422_fold1__singleton__def,axiom,
% 0.26/0.47 ! [A_107: nat,G_5: ( nat > $o ) > nat,F_22: nat > nat > nat] :
% 0.26/0.47 ( ( G_5
% 0.26/0.47 = ( finite_fold1_nat @ F_22 ) )
% 0.26/0.47 => ( ( G_5 @ ( insert_nat @ A_107 @ bot_bot_nat_o ) )
% 0.26/0.47 = A_107 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_423_fold1__singleton__def,axiom,
% 0.26/0.47 ! [A_107: hoare_1167836817_state,G_5: ( hoare_1167836817_state > $o ) > hoare_1167836817_state,F_22: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state] :
% 0.26/0.47 ( ( G_5
% 0.26/0.47 = ( finite1646097201_state @ F_22 ) )
% 0.26/0.47 => ( ( G_5 @ ( insert2134838167_state @ A_107 @ bot_bo70021908tate_o ) )
% 0.26/0.47 = A_107 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_424_fold1__singleton__def,axiom,
% 0.26/0.47 ! [A_107: hoare_1775062406iple_a,G_5: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a,F_22: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a] :
% 0.26/0.47 ( ( G_5
% 0.26/0.47 = ( finite1790765286iple_a @ F_22 ) )
% 0.26/0.47 => ( ( G_5 @ ( insert1281456128iple_a @ A_107 @ bot_bo751897185le_a_o ) )
% 0.26/0.47 = A_107 ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_425_comp__fun__commute_Ofold__equality,axiom,
% 0.26/0.47 ! [Z_11: hoare_1775062406iple_a,A_106: hoare_1775062406iple_a > $o,Y_32: hoare_1775062406iple_a,F_21: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a] :
% 0.26/0.47 ( ( finite2064891473iple_a @ F_21 )
% 0.26/0.47 => ( ( finite727644230iple_a @ F_21 @ Z_11 @ A_106 @ Y_32 )
% 0.26/0.47 => ( ( finite1842721992iple_a @ F_21 @ Z_11 @ A_106 )
% 0.26/0.47 = Y_32 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_426_fold__def,axiom,
% 0.26/0.47 ! [F_20: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,Z_10: hoare_1775062406iple_a,A_105: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( finite1842721992iple_a @ F_20 @ Z_10 @ A_105 )
% 0.26/0.47 = ( the_Ho1155011127iple_a @ ( finite727644230iple_a @ F_20 @ Z_10 @ A_105 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_427_folding__one_Oeq__fold,axiom,
% 0.26/0.47 ! [A_104: nat > $o,F_19: nat > nat > nat,F_18: ( nat > $o ) > nat] :
% 0.26/0.47 ( ( finite988810631ne_nat @ F_19 @ F_18 )
% 0.26/0.47 => ( ( finite_finite_nat @ A_104 )
% 0.26/0.47 => ( ( F_18 @ A_104 )
% 0.26/0.47 = ( finite_fold1_nat @ F_19 @ A_104 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_428_folding__one_Oeq__fold,axiom,
% 0.26/0.47 ! [A_104: hoare_1775062406iple_a > $o,F_19: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_18: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
% 0.26/0.47 ( ( finite2078349315iple_a @ F_19 @ F_18 )
% 0.26/0.47 => ( ( finite2063573081iple_a @ A_104 )
% 0.26/0.47 => ( ( F_18 @ A_104 )
% 0.26/0.47 = ( finite1790765286iple_a @ F_19 @ A_104 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_429_folding__one_Oeq__fold_H,axiom,
% 0.26/0.47 ! [X_61: nat,A_103: nat > $o,F_17: nat > nat > nat,F_16: ( nat > $o ) > nat] :
% 0.26/0.47 ( ( finite988810631ne_nat @ F_17 @ F_16 )
% 0.26/0.47 => ( ( finite_finite_nat @ A_103 )
% 0.26/0.47 => ( ~ ( member_nat @ X_61 @ A_103 )
% 0.26/0.47 => ( ( F_16 @ ( insert_nat @ X_61 @ A_103 ) )
% 0.26/0.47 = ( finite_fold_nat_nat @ F_17 @ X_61 @ A_103 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_430_folding__one_Oeq__fold_H,axiom,
% 0.26/0.47 ! [X_61: hoare_1167836817_state,A_103: hoare_1167836817_state > $o,F_17: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_16: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
% 0.26/0.47 ( ( finite1074406356_state @ F_17 @ F_16 )
% 0.26/0.47 => ( ( finite1084549118_state @ A_103 )
% 0.26/0.47 => ( ~ ( member2058392318_state @ X_61 @ A_103 )
% 0.26/0.47 => ( ( F_16 @ ( insert2134838167_state @ X_61 @ A_103 ) )
% 0.26/0.47 = ( finite1731015960_state @ F_17 @ X_61 @ A_103 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_431_folding__one_Oeq__fold_H,axiom,
% 0.26/0.47 ! [X_61: hoare_1775062406iple_a,A_103: hoare_1775062406iple_a > $o,F_17: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_16: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
% 0.26/0.47 ( ( finite2078349315iple_a @ F_17 @ F_16 )
% 0.26/0.47 => ( ( finite2063573081iple_a @ A_103 )
% 0.26/0.47 => ( ~ ( member2122167641iple_a @ X_61 @ A_103 )
% 0.26/0.47 => ( ( F_16 @ ( insert1281456128iple_a @ X_61 @ A_103 ) )
% 0.26/0.47 = ( finite1842721992iple_a @ F_17 @ X_61 @ A_103 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_432_folding__one__idem_Oeq__fold__idem_H,axiom,
% 0.26/0.47 ! [A_102: nat,A_101: nat > $o,F_15: nat > nat > nat,F_14: ( nat > $o ) > nat] :
% 0.26/0.47 ( ( finite795500164em_nat @ F_15 @ F_14 )
% 0.26/0.47 => ( ( finite_finite_nat @ A_101 )
% 0.26/0.47 => ( ( F_14 @ ( insert_nat @ A_102 @ A_101 ) )
% 0.26/0.47 = ( finite_fold_nat_nat @ F_15 @ A_102 @ A_101 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_433_folding__one__idem_Oeq__fold__idem_H,axiom,
% 0.26/0.47 ! [A_102: hoare_1167836817_state,A_101: hoare_1167836817_state > $o,F_15: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_14: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
% 0.26/0.47 ( ( finite806517911_state @ F_15 @ F_14 )
% 0.26/0.47 => ( ( finite1084549118_state @ A_101 )
% 0.26/0.47 => ( ( F_14 @ ( insert2134838167_state @ A_102 @ A_101 ) )
% 0.26/0.47 = ( finite1731015960_state @ F_15 @ A_102 @ A_101 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_434_folding__one__idem_Oeq__fold__idem_H,axiom,
% 0.26/0.47 ! [A_102: hoare_1775062406iple_a,A_101: hoare_1775062406iple_a > $o,F_15: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_14: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
% 0.26/0.47 ( ( finite1358382848iple_a @ F_15 @ F_14 )
% 0.26/0.47 => ( ( finite2063573081iple_a @ A_101 )
% 0.26/0.47 => ( ( F_14 @ ( insert1281456128iple_a @ A_102 @ A_101 ) )
% 0.26/0.47 = ( finite1842721992iple_a @ F_15 @ A_102 @ A_101 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_435_comp__fun__commute_Ofold__graph__fold,axiom,
% 0.26/0.47 ! [Z_9: hoare_1775062406iple_a,A_100: hoare_1775062406iple_a > $o,F_13: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a] :
% 0.26/0.47 ( ( finite2064891473iple_a @ F_13 )
% 0.26/0.47 => ( ( finite2063573081iple_a @ A_100 )
% 0.26/0.47 => ( finite727644230iple_a @ F_13 @ Z_9 @ A_100 @ ( finite1842721992iple_a @ F_13 @ Z_9 @ A_100 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_436_fold1__def,axiom,
% 0.26/0.47 ! [F_12: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,A_99: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( finite1790765286iple_a @ F_12 @ A_99 )
% 0.26/0.47 = ( the_Ho1155011127iple_a @ ( finite1946188886iple_a @ F_12 @ A_99 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_437_minus__fold__remove,axiom,
% 0.26/0.47 ! [B_67: nat > $o,A_97: nat > $o] :
% 0.26/0.47 ( ( finite_finite_nat @ A_97 )
% 0.26/0.47 => ( ( minus_minus_nat_o @ B_67 @ A_97 )
% 0.26/0.47 = ( finite326637109_nat_o
% 0.26/0.47 @ ^ [X_3: nat,A_98: nat > $o] : ( minus_minus_nat_o @ A_98 @ ( insert_nat @ X_3 @ bot_bot_nat_o ) )
% 0.26/0.47 @ B_67
% 0.26/0.47 @ A_97 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_438_minus__fold__remove,axiom,
% 0.26/0.47 ! [B_67: hoare_1167836817_state > $o,A_97: hoare_1167836817_state > $o] :
% 0.26/0.47 ( ( finite1084549118_state @ A_97 )
% 0.26/0.47 => ( ( minus_2107060239tate_o @ B_67 @ A_97 )
% 0.26/0.47 = ( finite291020855tate_o
% 0.26/0.47 @ ^ [X_3: hoare_1167836817_state,A_98: hoare_1167836817_state > $o] : ( minus_2107060239tate_o @ A_98 @ ( insert2134838167_state @ X_3 @ bot_bo70021908tate_o ) )
% 0.26/0.47 @ B_67
% 0.26/0.47 @ A_97 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_439_minus__fold__remove,axiom,
% 0.26/0.47 ! [B_67: hoare_1775062406iple_a > $o,A_97: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( finite2063573081iple_a @ A_97 )
% 0.26/0.47 => ( ( minus_1944206118le_a_o @ B_67 @ A_97 )
% 0.26/0.47 = ( finite1544171829le_a_o
% 0.26/0.47 @ ^ [X_3: hoare_1775062406iple_a,A_98: hoare_1775062406iple_a > $o] : ( minus_1944206118le_a_o @ A_98 @ ( insert1281456128iple_a @ X_3 @ bot_bo751897185le_a_o ) )
% 0.26/0.47 @ B_67
% 0.26/0.47 @ A_97 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_440_fold1__in,axiom,
% 0.26/0.47 ! [A_96: nat > $o] :
% 0.26/0.47 ( ( finite_finite_nat @ A_96 )
% 0.26/0.47 => ( ( A_96 != bot_bot_nat_o )
% 0.26/0.47 => ( ! [X_3: nat,Y_1: nat] : ( member_nat @ ( times_times_nat @ X_3 @ Y_1 ) @ ( insert_nat @ X_3 @ ( insert_nat @ Y_1 @ bot_bot_nat_o ) ) )
% 0.26/0.47 => ( member_nat @ ( finite_fold1_nat @ times_times_nat @ A_96 ) @ A_96 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_441_semilattice__big_OF__eq,axiom,
% 0.26/0.47 ! [A_95: nat > $o,F_11: nat > nat > nat,F_10: ( nat > $o ) > nat] :
% 0.26/0.47 ( ( big_se275732192ig_nat @ F_11 @ F_10 )
% 0.26/0.47 => ( ( finite_finite_nat @ A_95 )
% 0.26/0.47 => ( ( F_10 @ A_95 )
% 0.26/0.47 = ( finite_fold1_nat @ F_11 @ A_95 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_442_folding__one__idem_Osubset__idem,axiom,
% 0.26/0.47 ! [B_66: nat > $o,A_94: nat > $o,F_9: nat > nat > nat,F_8: ( nat > $o ) > nat] :
% 0.26/0.47 ( ( finite795500164em_nat @ F_9 @ F_8 )
% 0.26/0.47 => ( ( finite_finite_nat @ A_94 )
% 0.26/0.47 => ( ( B_66 != bot_bot_nat_o )
% 0.26/0.47 => ( ( ord_less_eq_nat_o @ B_66 @ A_94 )
% 0.26/0.47 => ( ( F_9 @ ( F_8 @ B_66 ) @ ( F_8 @ A_94 ) )
% 0.26/0.47 = ( F_8 @ A_94 ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_443_folding__one__idem_Osubset__idem,axiom,
% 0.26/0.47 ! [B_66: hoare_1167836817_state > $o,A_94: hoare_1167836817_state > $o,F_9: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_8: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
% 0.26/0.47 ( ( finite806517911_state @ F_9 @ F_8 )
% 0.26/0.47 => ( ( finite1084549118_state @ A_94 )
% 0.26/0.47 => ( ( B_66 != bot_bo70021908tate_o )
% 0.26/0.47 => ( ( ord_le827224136tate_o @ B_66 @ A_94 )
% 0.26/0.47 => ( ( F_9 @ ( F_8 @ B_66 ) @ ( F_8 @ A_94 ) )
% 0.26/0.47 = ( F_8 @ A_94 ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_444_folding__one__idem_Osubset__idem,axiom,
% 0.26/0.47 ! [B_66: hoare_1775062406iple_a > $o,A_94: hoare_1775062406iple_a > $o,F_9: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_8: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
% 0.26/0.47 ( ( finite1358382848iple_a @ F_9 @ F_8 )
% 0.26/0.47 => ( ( finite2063573081iple_a @ A_94 )
% 0.26/0.47 => ( ( B_66 != bot_bo751897185le_a_o )
% 0.26/0.47 => ( ( ord_le1143225901le_a_o @ B_66 @ A_94 )
% 0.26/0.47 => ( ( F_9 @ ( F_8 @ B_66 ) @ ( F_8 @ A_94 ) )
% 0.26/0.47 = ( F_8 @ A_94 ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_445_order__refl,axiom,
% 0.26/0.47 ! [X_60: nat] : ( ord_less_eq_nat @ X_60 @ X_60 ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_446_subsetD,axiom,
% 0.26/0.47 ! [C_23: hoare_1775062406iple_a,A_93: hoare_1775062406iple_a > $o,B_65: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( ord_le1143225901le_a_o @ A_93 @ B_65 )
% 0.26/0.47 => ( ( member2122167641iple_a @ C_23 @ A_93 )
% 0.26/0.47 => ( member2122167641iple_a @ C_23 @ B_65 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_447_subsetD,axiom,
% 0.26/0.47 ! [C_23: nat,A_93: nat > $o,B_65: nat > $o] :
% 0.26/0.47 ( ( ord_less_eq_nat_o @ A_93 @ B_65 )
% 0.26/0.47 => ( ( member_nat @ C_23 @ A_93 )
% 0.26/0.47 => ( member_nat @ C_23 @ B_65 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_448_UnCI,axiom,
% 0.26/0.47 ! [A_92: hoare_1775062406iple_a > $o,C_22: hoare_1775062406iple_a,B_64: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( ~ ( member2122167641iple_a @ C_22 @ B_64 )
% 0.26/0.47 => ( member2122167641iple_a @ C_22 @ A_92 ) )
% 0.26/0.47 => ( member2122167641iple_a @ C_22 @ ( semila13410563le_a_o @ A_92 @ B_64 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_449_UnCI,axiom,
% 0.26/0.47 ! [A_92: nat > $o,C_22: nat,B_64: nat > $o] :
% 0.26/0.47 ( ( ~ ( member_nat @ C_22 @ B_64 )
% 0.26/0.47 => ( member_nat @ C_22 @ A_92 ) )
% 0.26/0.47 => ( member_nat @ C_22 @ ( semila848761471_nat_o @ A_92 @ B_64 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_450_UnE,axiom,
% 0.26/0.47 ! [C_21: hoare_1775062406iple_a,A_91: hoare_1775062406iple_a > $o,B_63: hoare_1775062406iple_a > $o] :
% 0.26/0.47 ( ( member2122167641iple_a @ C_21 @ ( semila13410563le_a_o @ A_91 @ B_63 ) )
% 0.26/0.47 => ( ~ ( member2122167641iple_a @ C_21 @ A_91 )
% 0.26/0.47 => ( member2122167641iple_a @ C_21 @ B_63 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_451_UnE,axiom,
% 0.26/0.47 ! [C_21: nat,A_91: nat > $o,B_63: nat > $o] :
% 0.26/0.47 ( ( member_nat @ C_21 @ ( semila848761471_nat_o @ A_91 @ B_63 ) )
% 0.26/0.47 => ( ~ ( member_nat @ C_21 @ A_91 )
% 0.26/0.47 => ( member_nat @ C_21 @ B_63 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_452_empty__subsetI,axiom,
% 0.26/0.47 ! [A_90: nat > $o] : ( ord_less_eq_nat_o @ bot_bot_nat_o @ A_90 ) ).
% 0.26/0.47
% 0.26/0.47 thf(fact_453_empty__subsetI,axiom,
% 0.26/0.47 ! [A_90: hoare_1167836817_state > $o] : ( ord_le827224136tate_o @ bot_bo70021908tate_o @ A_90 ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_454_empty__subsetI,axiom,
% 0.26/0.48 ! [A_90: hoare_1775062406iple_a > $o] : ( ord_le1143225901le_a_o @ bot_bo751897185le_a_o @ A_90 ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_455_finite__Collect__subsets,axiom,
% 0.26/0.48 ! [A_89: nat > $o] :
% 0.26/0.48 ( ( finite_finite_nat @ A_89 )
% 0.26/0.48 => ( finite_finite_nat_o
% 0.26/0.48 @ ( collect_nat_o
% 0.26/0.48 @ ^ [B_62: nat > $o] : ( ord_less_eq_nat_o @ B_62 @ A_89 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_456_sup__le__fold__sup,axiom,
% 0.26/0.48 ! [B_61: nat,A_88: nat,A_87: nat > $o] :
% 0.26/0.48 ( ( finite_finite_nat @ A_87 )
% 0.26/0.48 => ( ( member_nat @ A_88 @ A_87 )
% 0.26/0.48 => ( ord_less_eq_nat @ ( semila972727038up_nat @ A_88 @ B_61 ) @ ( finite_fold_nat_nat @ semila972727038up_nat @ B_61 @ A_87 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_457_subset__empty,axiom,
% 0.26/0.48 ! [A_86: nat > $o] :
% 0.26/0.48 ( ( ord_less_eq_nat_o @ A_86 @ bot_bot_nat_o )
% 0.26/0.48 <=> ( A_86 = bot_bot_nat_o ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_458_subset__empty,axiom,
% 0.26/0.48 ! [A_86: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( ord_le827224136tate_o @ A_86 @ bot_bo70021908tate_o )
% 0.26/0.48 <=> ( A_86 = bot_bo70021908tate_o ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_459_subset__empty,axiom,
% 0.26/0.48 ! [A_86: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( ord_le1143225901le_a_o @ A_86 @ bot_bo751897185le_a_o )
% 0.26/0.48 <=> ( A_86 = bot_bo751897185le_a_o ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_460_rev__finite__subset,axiom,
% 0.26/0.48 ! [A_85: nat > $o,B_60: nat > $o] :
% 0.26/0.48 ( ( finite_finite_nat @ B_60 )
% 0.26/0.48 => ( ( ord_less_eq_nat_o @ A_85 @ B_60 )
% 0.26/0.48 => ( finite_finite_nat @ A_85 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_461_finite__subset,axiom,
% 0.26/0.48 ! [A_84: nat > $o,B_59: nat > $o] :
% 0.26/0.48 ( ( ord_less_eq_nat_o @ A_84 @ B_59 )
% 0.26/0.48 => ( ( finite_finite_nat @ B_59 )
% 0.26/0.48 => ( finite_finite_nat @ A_84 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_462_subset__insertI,axiom,
% 0.26/0.48 ! [B_58: nat > $o,A_83: nat] : ( ord_less_eq_nat_o @ B_58 @ ( insert_nat @ A_83 @ B_58 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_463_subset__insertI,axiom,
% 0.26/0.48 ! [B_58: hoare_1167836817_state > $o,A_83: hoare_1167836817_state] : ( ord_le827224136tate_o @ B_58 @ ( insert2134838167_state @ A_83 @ B_58 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_464_subset__insertI,axiom,
% 0.26/0.48 ! [B_58: hoare_1775062406iple_a > $o,A_83: hoare_1775062406iple_a] : ( ord_le1143225901le_a_o @ B_58 @ ( insert1281456128iple_a @ A_83 @ B_58 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_465_insert__subset,axiom,
% 0.26/0.48 ! [X_59: nat,A_82: nat > $o,B_57: nat > $o] :
% 0.26/0.48 ( ( ord_less_eq_nat_o @ ( insert_nat @ X_59 @ A_82 ) @ B_57 )
% 0.26/0.48 <=> ( ( member_nat @ X_59 @ B_57 )
% 0.26/0.48 & ( ord_less_eq_nat_o @ A_82 @ B_57 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_466_insert__subset,axiom,
% 0.26/0.48 ! [X_59: hoare_1167836817_state,A_82: hoare_1167836817_state > $o,B_57: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( ord_le827224136tate_o @ ( insert2134838167_state @ X_59 @ A_82 ) @ B_57 )
% 0.26/0.48 <=> ( ( member2058392318_state @ X_59 @ B_57 )
% 0.26/0.48 & ( ord_le827224136tate_o @ A_82 @ B_57 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_467_insert__subset,axiom,
% 0.26/0.48 ! [X_59: hoare_1775062406iple_a,A_82: hoare_1775062406iple_a > $o,B_57: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( ord_le1143225901le_a_o @ ( insert1281456128iple_a @ X_59 @ A_82 ) @ B_57 )
% 0.26/0.48 <=> ( ( member2122167641iple_a @ X_59 @ B_57 )
% 0.26/0.48 & ( ord_le1143225901le_a_o @ A_82 @ B_57 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_468_subset__insert,axiom,
% 0.26/0.48 ! [B_56: nat > $o,X_58: nat,A_81: nat > $o] :
% 0.26/0.48 ( ~ ( member_nat @ X_58 @ A_81 )
% 0.26/0.48 => ( ( ord_less_eq_nat_o @ A_81 @ ( insert_nat @ X_58 @ B_56 ) )
% 0.26/0.48 <=> ( ord_less_eq_nat_o @ A_81 @ B_56 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_469_subset__insert,axiom,
% 0.26/0.48 ! [B_56: hoare_1167836817_state > $o,X_58: hoare_1167836817_state,A_81: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ~ ( member2058392318_state @ X_58 @ A_81 )
% 0.26/0.48 => ( ( ord_le827224136tate_o @ A_81 @ ( insert2134838167_state @ X_58 @ B_56 ) )
% 0.26/0.48 <=> ( ord_le827224136tate_o @ A_81 @ B_56 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_470_subset__insert,axiom,
% 0.26/0.48 ! [B_56: hoare_1775062406iple_a > $o,X_58: hoare_1775062406iple_a,A_81: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ~ ( member2122167641iple_a @ X_58 @ A_81 )
% 0.26/0.48 => ( ( ord_le1143225901le_a_o @ A_81 @ ( insert1281456128iple_a @ X_58 @ B_56 ) )
% 0.26/0.48 <=> ( ord_le1143225901le_a_o @ A_81 @ B_56 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_471_subset__insertI2,axiom,
% 0.26/0.48 ! [B_55: nat,A_80: nat > $o,B_54: nat > $o] :
% 0.26/0.48 ( ( ord_less_eq_nat_o @ A_80 @ B_54 )
% 0.26/0.48 => ( ord_less_eq_nat_o @ A_80 @ ( insert_nat @ B_55 @ B_54 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_472_subset__insertI2,axiom,
% 0.26/0.48 ! [B_55: hoare_1167836817_state,A_80: hoare_1167836817_state > $o,B_54: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( ord_le827224136tate_o @ A_80 @ B_54 )
% 0.26/0.48 => ( ord_le827224136tate_o @ A_80 @ ( insert2134838167_state @ B_55 @ B_54 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_473_subset__insertI2,axiom,
% 0.26/0.48 ! [B_55: hoare_1775062406iple_a,A_80: hoare_1775062406iple_a > $o,B_54: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( ord_le1143225901le_a_o @ A_80 @ B_54 )
% 0.26/0.48 => ( ord_le1143225901le_a_o @ A_80 @ ( insert1281456128iple_a @ B_55 @ B_54 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_474_insert__mono,axiom,
% 0.26/0.48 ! [A_79: nat,C_20: nat > $o,D_2: nat > $o] :
% 0.26/0.48 ( ( ord_less_eq_nat_o @ C_20 @ D_2 )
% 0.26/0.48 => ( ord_less_eq_nat_o @ ( insert_nat @ A_79 @ C_20 ) @ ( insert_nat @ A_79 @ D_2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_475_insert__mono,axiom,
% 0.26/0.48 ! [A_79: hoare_1167836817_state,C_20: hoare_1167836817_state > $o,D_2: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( ord_le827224136tate_o @ C_20 @ D_2 )
% 0.26/0.48 => ( ord_le827224136tate_o @ ( insert2134838167_state @ A_79 @ C_20 ) @ ( insert2134838167_state @ A_79 @ D_2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_476_insert__mono,axiom,
% 0.26/0.48 ! [A_79: hoare_1775062406iple_a,C_20: hoare_1775062406iple_a > $o,D_2: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( ord_le1143225901le_a_o @ C_20 @ D_2 )
% 0.26/0.48 => ( ord_le1143225901le_a_o @ ( insert1281456128iple_a @ A_79 @ C_20 ) @ ( insert1281456128iple_a @ A_79 @ D_2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_477_le__supE,axiom,
% 0.26/0.48 ! [A_78: nat,B_53: nat,X_57: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ ( semila972727038up_nat @ A_78 @ B_53 ) @ X_57 )
% 0.26/0.48 => ~ ( ( ord_less_eq_nat @ A_78 @ X_57 )
% 0.26/0.48 => ~ ( ord_less_eq_nat @ B_53 @ X_57 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_478_sup__mono,axiom,
% 0.26/0.48 ! [B_52: nat,D_1: nat,A_77: nat,C_19: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ A_77 @ C_19 )
% 0.26/0.48 => ( ( ord_less_eq_nat @ B_52 @ D_1 )
% 0.26/0.48 => ( ord_less_eq_nat @ ( semila972727038up_nat @ A_77 @ B_52 ) @ ( semila972727038up_nat @ C_19 @ D_1 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_479_sup__least,axiom,
% 0.26/0.48 ! [Z_8: nat,Y_31: nat,X_56: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ Y_31 @ X_56 )
% 0.26/0.48 => ( ( ord_less_eq_nat @ Z_8 @ X_56 )
% 0.26/0.48 => ( ord_less_eq_nat @ ( semila972727038up_nat @ Y_31 @ Z_8 ) @ X_56 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_480_le__supI,axiom,
% 0.26/0.48 ! [B_51: nat,A_76: nat,X_55: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ A_76 @ X_55 )
% 0.26/0.48 => ( ( ord_less_eq_nat @ B_51 @ X_55 )
% 0.26/0.48 => ( ord_less_eq_nat @ ( semila972727038up_nat @ A_76 @ B_51 ) @ X_55 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_481_sup__absorb1,axiom,
% 0.26/0.48 ! [Y_30: nat,X_54: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ Y_30 @ X_54 )
% 0.26/0.48 => ( ( semila972727038up_nat @ X_54 @ Y_30 )
% 0.26/0.48 = X_54 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_482_sup__absorb2,axiom,
% 0.26/0.48 ! [X_53: nat,Y_29: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ X_53 @ Y_29 )
% 0.26/0.48 => ( ( semila972727038up_nat @ X_53 @ Y_29 )
% 0.26/0.48 = Y_29 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_483_le__supI2,axiom,
% 0.26/0.48 ! [A_75: nat,X_52: nat,B_50: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ X_52 @ B_50 )
% 0.26/0.48 => ( ord_less_eq_nat @ X_52 @ ( semila972727038up_nat @ A_75 @ B_50 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_484_le__supI1,axiom,
% 0.26/0.48 ! [B_49: nat,X_51: nat,A_74: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ X_51 @ A_74 )
% 0.26/0.48 => ( ord_less_eq_nat @ X_51 @ ( semila972727038up_nat @ A_74 @ B_49 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_485_le__sup__iff,axiom,
% 0.26/0.48 ! [X_50: nat,Y_28: nat,Z_7: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ ( semila972727038up_nat @ X_50 @ Y_28 ) @ Z_7 )
% 0.26/0.48 <=> ( ( ord_less_eq_nat @ X_50 @ Z_7 )
% 0.26/0.48 & ( ord_less_eq_nat @ Y_28 @ Z_7 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_486_le__iff__sup,axiom,
% 0.26/0.48 ! [X_49: nat,Y_27: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ X_49 @ Y_27 )
% 0.26/0.48 <=> ( ( semila972727038up_nat @ X_49 @ Y_27 )
% 0.26/0.48 = Y_27 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_487_sup__ge2,axiom,
% 0.26/0.48 ! [Y_26: nat,X_48: nat] : ( ord_less_eq_nat @ Y_26 @ ( semila972727038up_nat @ X_48 @ Y_26 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_488_inf__sup__ord_I4_J,axiom,
% 0.26/0.48 ! [Y_25: nat,X_47: nat] : ( ord_less_eq_nat @ Y_25 @ ( semila972727038up_nat @ X_47 @ Y_25 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_489_sup__ge1,axiom,
% 0.26/0.48 ! [X_46: nat,Y_24: nat] : ( ord_less_eq_nat @ X_46 @ ( semila972727038up_nat @ X_46 @ Y_24 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_490_inf__sup__ord_I3_J,axiom,
% 0.26/0.48 ! [X_45: nat,Y_23: nat] : ( ord_less_eq_nat @ X_45 @ ( semila972727038up_nat @ X_45 @ Y_23 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_491_sup__eq__bot__iff,axiom,
% 0.26/0.48 ! [X_44: $o,Y_22: $o] :
% 0.26/0.48 ( ( ( semila10642723_sup_o @ X_44 @ Y_22 )
% 0.26/0.48 <=> bot_bot_o )
% 0.26/0.48 <=> ( ( X_44
% 0.26/0.48 <=> bot_bot_o )
% 0.26/0.48 & ( Y_22
% 0.26/0.48 <=> bot_bot_o ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_492_sup__eq__bot__iff,axiom,
% 0.26/0.48 ! [X_44: nat > $o,Y_22: nat > $o] :
% 0.26/0.48 ( ( ( semila848761471_nat_o @ X_44 @ Y_22 )
% 0.26/0.48 = bot_bot_nat_o )
% 0.26/0.48 <=> ( ( X_44 = bot_bot_nat_o )
% 0.26/0.48 & ( Y_22 = bot_bot_nat_o ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_493_sup__eq__bot__iff,axiom,
% 0.26/0.48 ! [X_44: hoare_1167836817_state > $o,Y_22: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( ( semila1172322802tate_o @ X_44 @ Y_22 )
% 0.26/0.48 = bot_bo70021908tate_o )
% 0.26/0.48 <=> ( ( X_44 = bot_bo70021908tate_o )
% 0.26/0.48 & ( Y_22 = bot_bo70021908tate_o ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_494_sup__eq__bot__iff,axiom,
% 0.26/0.48 ! [X_44: hoare_1775062406iple_a > $o,Y_22: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( ( semila13410563le_a_o @ X_44 @ Y_22 )
% 0.26/0.48 = bot_bo751897185le_a_o )
% 0.26/0.48 <=> ( ( X_44 = bot_bo751897185le_a_o )
% 0.26/0.48 & ( Y_22 = bot_bo751897185le_a_o ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_495_sup__bot__right,axiom,
% 0.26/0.48 ! [X_43: $o] :
% 0.26/0.48 ( ( semila10642723_sup_o @ X_43 @ bot_bot_o )
% 0.26/0.48 <=> X_43 ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_496_sup__bot__right,axiom,
% 0.26/0.48 ! [X_43: nat > $o] :
% 0.26/0.48 ( ( semila848761471_nat_o @ X_43 @ bot_bot_nat_o )
% 0.26/0.48 = X_43 ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_497_sup__bot__right,axiom,
% 0.26/0.48 ! [X_43: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( semila1172322802tate_o @ X_43 @ bot_bo70021908tate_o )
% 0.26/0.48 = X_43 ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_498_sup__bot__right,axiom,
% 0.26/0.48 ! [X_43: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( semila13410563le_a_o @ X_43 @ bot_bo751897185le_a_o )
% 0.26/0.48 = X_43 ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_499_sup__bot__left,axiom,
% 0.26/0.48 ! [X_42: $o] :
% 0.26/0.48 ( ( semila10642723_sup_o @ bot_bot_o @ X_42 )
% 0.26/0.48 <=> X_42 ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_500_sup__bot__left,axiom,
% 0.26/0.48 ! [X_42: nat > $o] :
% 0.26/0.48 ( ( semila848761471_nat_o @ bot_bot_nat_o @ X_42 )
% 0.26/0.48 = X_42 ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_501_sup__bot__left,axiom,
% 0.26/0.48 ! [X_42: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( semila1172322802tate_o @ bot_bo70021908tate_o @ X_42 )
% 0.26/0.48 = X_42 ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_502_sup__bot__left,axiom,
% 0.26/0.48 ! [X_42: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( semila13410563le_a_o @ bot_bo751897185le_a_o @ X_42 )
% 0.26/0.48 = X_42 ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_503_Un__empty__left,axiom,
% 0.26/0.48 ! [B_48: nat > $o] :
% 0.26/0.48 ( ( semila848761471_nat_o @ bot_bot_nat_o @ B_48 )
% 0.26/0.48 = B_48 ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_504_Un__empty__left,axiom,
% 0.26/0.48 ! [B_48: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( semila1172322802tate_o @ bot_bo70021908tate_o @ B_48 )
% 0.26/0.48 = B_48 ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_505_Un__empty__left,axiom,
% 0.26/0.48 ! [B_48: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( semila13410563le_a_o @ bot_bo751897185le_a_o @ B_48 )
% 0.26/0.48 = B_48 ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_506_Un__empty__right,axiom,
% 0.26/0.48 ! [A_73: nat > $o] :
% 0.26/0.48 ( ( semila848761471_nat_o @ A_73 @ bot_bot_nat_o )
% 0.26/0.48 = A_73 ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_507_Un__empty__right,axiom,
% 0.26/0.48 ! [A_73: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( semila1172322802tate_o @ A_73 @ bot_bo70021908tate_o )
% 0.26/0.48 = A_73 ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_508_Un__empty__right,axiom,
% 0.26/0.48 ! [A_73: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( semila13410563le_a_o @ A_73 @ bot_bo751897185le_a_o )
% 0.26/0.48 = A_73 ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_509_Un__empty,axiom,
% 0.26/0.48 ! [A_72: nat > $o,B_47: nat > $o] :
% 0.26/0.48 ( ( ( semila848761471_nat_o @ A_72 @ B_47 )
% 0.26/0.48 = bot_bot_nat_o )
% 0.26/0.48 <=> ( ( A_72 = bot_bot_nat_o )
% 0.26/0.48 & ( B_47 = bot_bot_nat_o ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_510_Un__empty,axiom,
% 0.26/0.48 ! [A_72: hoare_1167836817_state > $o,B_47: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( ( semila1172322802tate_o @ A_72 @ B_47 )
% 0.26/0.48 = bot_bo70021908tate_o )
% 0.26/0.48 <=> ( ( A_72 = bot_bo70021908tate_o )
% 0.26/0.48 & ( B_47 = bot_bo70021908tate_o ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_511_Un__empty,axiom,
% 0.26/0.48 ! [A_72: hoare_1775062406iple_a > $o,B_47: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( ( semila13410563le_a_o @ A_72 @ B_47 )
% 0.26/0.48 = bot_bo751897185le_a_o )
% 0.26/0.48 <=> ( ( A_72 = bot_bo751897185le_a_o )
% 0.26/0.48 & ( B_47 = bot_bo751897185le_a_o ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_512_finite__Un,axiom,
% 0.26/0.48 ! [F_7: nat > $o,G_4: nat > $o] :
% 0.26/0.48 ( ( finite_finite_nat @ ( semila848761471_nat_o @ F_7 @ G_4 ) )
% 0.26/0.48 <=> ( ( finite_finite_nat @ F_7 )
% 0.26/0.48 & ( finite_finite_nat @ G_4 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_513_finite__UnI,axiom,
% 0.26/0.48 ! [G_3: nat > $o,F_6: nat > $o] :
% 0.26/0.48 ( ( finite_finite_nat @ F_6 )
% 0.26/0.48 => ( ( finite_finite_nat @ G_3 )
% 0.26/0.48 => ( finite_finite_nat @ ( semila848761471_nat_o @ F_6 @ G_3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_514_linorder__le__cases,axiom,
% 0.26/0.48 ! [X_41: nat,Y_21: nat] :
% 0.26/0.48 ( ~ ( ord_less_eq_nat @ X_41 @ Y_21 )
% 0.26/0.48 => ( ord_less_eq_nat @ Y_21 @ X_41 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_515_xt1_I6_J,axiom,
% 0.26/0.48 ! [Z_6: nat,Y_20: nat,X_40: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ Y_20 @ X_40 )
% 0.26/0.48 => ( ( ord_less_eq_nat @ Z_6 @ Y_20 )
% 0.26/0.48 => ( ord_less_eq_nat @ Z_6 @ X_40 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_516_xt1_I5_J,axiom,
% 0.26/0.48 ! [Y_19: nat,X_39: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ Y_19 @ X_39 )
% 0.26/0.48 => ( ( ord_less_eq_nat @ X_39 @ Y_19 )
% 0.26/0.48 => ( X_39 = Y_19 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_517_order__trans,axiom,
% 0.26/0.48 ! [Z_5: nat,X_38: nat,Y_18: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ X_38 @ Y_18 )
% 0.26/0.48 => ( ( ord_less_eq_nat @ Y_18 @ Z_5 )
% 0.26/0.48 => ( ord_less_eq_nat @ X_38 @ Z_5 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_518_order__antisym,axiom,
% 0.26/0.48 ! [X_37: nat,Y_17: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ X_37 @ Y_17 )
% 0.26/0.48 => ( ( ord_less_eq_nat @ Y_17 @ X_37 )
% 0.26/0.48 => ( X_37 = Y_17 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_519_xt1_I4_J,axiom,
% 0.26/0.48 ! [C_18: nat,B_46: nat,A_71: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ B_46 @ A_71 )
% 0.26/0.48 => ( ( B_46 = C_18 )
% 0.26/0.48 => ( ord_less_eq_nat @ C_18 @ A_71 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_520_ord__le__eq__trans,axiom,
% 0.26/0.48 ! [C_17: nat,A_70: nat,B_45: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ A_70 @ B_45 )
% 0.26/0.48 => ( ( B_45 = C_17 )
% 0.26/0.48 => ( ord_less_eq_nat @ A_70 @ C_17 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_521_xt1_I3_J,axiom,
% 0.26/0.48 ! [C_16: nat,A_69: nat,B_44: nat] :
% 0.26/0.48 ( ( A_69 = B_44 )
% 0.26/0.48 => ( ( ord_less_eq_nat @ C_16 @ B_44 )
% 0.26/0.48 => ( ord_less_eq_nat @ C_16 @ A_69 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_522_ord__eq__le__trans,axiom,
% 0.26/0.48 ! [C_15: nat,A_68: nat,B_43: nat] :
% 0.26/0.48 ( ( A_68 = B_43 )
% 0.26/0.48 => ( ( ord_less_eq_nat @ B_43 @ C_15 )
% 0.26/0.48 => ( ord_less_eq_nat @ A_68 @ C_15 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_523_order__antisym__conv,axiom,
% 0.26/0.48 ! [Y_16: nat,X_36: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ Y_16 @ X_36 )
% 0.26/0.48 => ( ( ord_less_eq_nat @ X_36 @ Y_16 )
% 0.26/0.48 <=> ( X_36 = Y_16 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_524_order__eq__refl,axiom,
% 0.26/0.48 ! [X_35: nat,Y_15: nat] :
% 0.26/0.48 ( ( X_35 = Y_15 )
% 0.26/0.48 => ( ord_less_eq_nat @ X_35 @ Y_15 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_525_order__eq__iff,axiom,
% 0.26/0.48 ! [X_34: nat,Y_14: nat] :
% 0.26/0.48 ( ( X_34 = Y_14 )
% 0.26/0.48 <=> ( ( ord_less_eq_nat @ X_34 @ Y_14 )
% 0.26/0.48 & ( ord_less_eq_nat @ Y_14 @ X_34 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_526_linorder__linear,axiom,
% 0.26/0.48 ! [X_33: nat,Y_13: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ X_33 @ Y_13 )
% 0.26/0.48 | ( ord_less_eq_nat @ Y_13 @ X_33 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_527_Collect__disj__eq,axiom,
% 0.26/0.48 ! [P_3: hoare_1775062406iple_a > $o,Q_1: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( collec676402587iple_a
% 0.26/0.48 @ ^ [X_3: hoare_1775062406iple_a] : ( (|) @ ( P_3 @ X_3 ) @ ( Q_1 @ X_3 ) ) )
% 0.26/0.48 = ( semila13410563le_a_o @ ( collec676402587iple_a @ P_3 ) @ ( collec676402587iple_a @ Q_1 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_528_Collect__disj__eq,axiom,
% 0.26/0.48 ! [P_3: nat > $o,Q_1: nat > $o] :
% 0.26/0.48 ( ( collect_nat
% 0.26/0.48 @ ^ [X_3: nat] : ( (|) @ ( P_3 @ X_3 ) @ ( Q_1 @ X_3 ) ) )
% 0.26/0.48 = ( semila848761471_nat_o @ ( collect_nat @ P_3 ) @ ( collect_nat @ Q_1 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_529_set__mp,axiom,
% 0.26/0.48 ! [X_32: hoare_1775062406iple_a,A_67: hoare_1775062406iple_a > $o,B_42: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( ord_le1143225901le_a_o @ A_67 @ B_42 )
% 0.26/0.48 => ( ( member2122167641iple_a @ X_32 @ A_67 )
% 0.26/0.48 => ( member2122167641iple_a @ X_32 @ B_42 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_530_set__mp,axiom,
% 0.26/0.48 ! [X_32: nat,A_67: nat > $o,B_42: nat > $o] :
% 0.26/0.48 ( ( ord_less_eq_nat_o @ A_67 @ B_42 )
% 0.26/0.48 => ( ( member_nat @ X_32 @ A_67 )
% 0.26/0.48 => ( member_nat @ X_32 @ B_42 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_531_set__rev__mp,axiom,
% 0.26/0.48 ! [B_41: hoare_1775062406iple_a > $o,X_31: hoare_1775062406iple_a,A_66: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( member2122167641iple_a @ X_31 @ A_66 )
% 0.26/0.48 => ( ( ord_le1143225901le_a_o @ A_66 @ B_41 )
% 0.26/0.48 => ( member2122167641iple_a @ X_31 @ B_41 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_532_set__rev__mp,axiom,
% 0.26/0.48 ! [B_41: nat > $o,X_31: nat,A_66: nat > $o] :
% 0.26/0.48 ( ( member_nat @ X_31 @ A_66 )
% 0.26/0.48 => ( ( ord_less_eq_nat_o @ A_66 @ B_41 )
% 0.26/0.48 => ( member_nat @ X_31 @ B_41 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_533_in__mono,axiom,
% 0.26/0.48 ! [X_30: hoare_1775062406iple_a,A_65: hoare_1775062406iple_a > $o,B_40: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( ord_le1143225901le_a_o @ A_65 @ B_40 )
% 0.26/0.48 => ( ( member2122167641iple_a @ X_30 @ A_65 )
% 0.26/0.48 => ( member2122167641iple_a @ X_30 @ B_40 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_534_in__mono,axiom,
% 0.26/0.48 ! [X_30: nat,A_65: nat > $o,B_40: nat > $o] :
% 0.26/0.48 ( ( ord_less_eq_nat_o @ A_65 @ B_40 )
% 0.26/0.48 => ( ( member_nat @ X_30 @ A_65 )
% 0.26/0.48 => ( member_nat @ X_30 @ B_40 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_535_UnI2,axiom,
% 0.26/0.48 ! [A_64: hoare_1775062406iple_a > $o,C_14: hoare_1775062406iple_a,B_39: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( member2122167641iple_a @ C_14 @ B_39 )
% 0.26/0.48 => ( member2122167641iple_a @ C_14 @ ( semila13410563le_a_o @ A_64 @ B_39 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_536_UnI2,axiom,
% 0.26/0.48 ! [A_64: nat > $o,C_14: nat,B_39: nat > $o] :
% 0.26/0.48 ( ( member_nat @ C_14 @ B_39 )
% 0.26/0.48 => ( member_nat @ C_14 @ ( semila848761471_nat_o @ A_64 @ B_39 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_537_UnI1,axiom,
% 0.26/0.48 ! [B_38: hoare_1775062406iple_a > $o,C_13: hoare_1775062406iple_a,A_63: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( member2122167641iple_a @ C_13 @ A_63 )
% 0.26/0.48 => ( member2122167641iple_a @ C_13 @ ( semila13410563le_a_o @ A_63 @ B_38 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_538_UnI1,axiom,
% 0.26/0.48 ! [B_38: nat > $o,C_13: nat,A_63: nat > $o] :
% 0.26/0.48 ( ( member_nat @ C_13 @ A_63 )
% 0.26/0.48 => ( member_nat @ C_13 @ ( semila848761471_nat_o @ A_63 @ B_38 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_539_Un__iff,axiom,
% 0.26/0.48 ! [C_12: hoare_1775062406iple_a,A_62: hoare_1775062406iple_a > $o,B_37: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( member2122167641iple_a @ C_12 @ ( semila13410563le_a_o @ A_62 @ B_37 ) )
% 0.26/0.48 <=> ( ( member2122167641iple_a @ C_12 @ A_62 )
% 0.26/0.48 | ( member2122167641iple_a @ C_12 @ B_37 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_540_Un__iff,axiom,
% 0.26/0.48 ! [C_12: nat,A_62: nat > $o,B_37: nat > $o] :
% 0.26/0.48 ( ( member_nat @ C_12 @ ( semila848761471_nat_o @ A_62 @ B_37 ) )
% 0.26/0.48 <=> ( ( member_nat @ C_12 @ A_62 )
% 0.26/0.48 | ( member_nat @ C_12 @ B_37 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_541_Un__def,axiom,
% 0.26/0.48 ! [A_61: hoare_1775062406iple_a > $o,B_36: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( semila13410563le_a_o @ A_61 @ B_36 )
% 0.26/0.48 = ( collec676402587iple_a
% 0.26/0.48 @ ^ [X_3: hoare_1775062406iple_a] : ( (|) @ ( member2122167641iple_a @ X_3 @ A_61 ) @ ( member2122167641iple_a @ X_3 @ B_36 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_542_Un__def,axiom,
% 0.26/0.48 ! [A_61: nat > $o,B_36: nat > $o] :
% 0.26/0.48 ( ( semila848761471_nat_o @ A_61 @ B_36 )
% 0.26/0.48 = ( collect_nat
% 0.26/0.48 @ ^ [X_3: nat] : ( (|) @ ( member_nat @ X_3 @ A_61 ) @ ( member_nat @ X_3 @ B_36 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_543_pred__subset__eq,axiom,
% 0.26/0.48 ! [R_2: hoare_1775062406iple_a > $o,S_4: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( ord_le1143225901le_a_o
% 0.26/0.48 @ ^ [X_3: hoare_1775062406iple_a] : ( member2122167641iple_a @ X_3 @ R_2 )
% 0.26/0.48 @ ^ [X_3: hoare_1775062406iple_a] : ( member2122167641iple_a @ X_3 @ S_4 ) )
% 0.26/0.48 <=> ( ord_le1143225901le_a_o @ R_2 @ S_4 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_544_pred__subset__eq,axiom,
% 0.26/0.48 ! [R_2: nat > $o,S_4: nat > $o] :
% 0.26/0.48 ( ( ord_less_eq_nat_o
% 0.26/0.48 @ ^ [X_3: nat] : ( member_nat @ X_3 @ R_2 )
% 0.26/0.48 @ ^ [X_3: nat] : ( member_nat @ X_3 @ S_4 ) )
% 0.26/0.48 <=> ( ord_less_eq_nat_o @ R_2 @ S_4 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_545_sup__Un__eq,axiom,
% 0.26/0.48 ! [R_1: hoare_1775062406iple_a > $o,S_3: hoare_1775062406iple_a > $o,X_3: hoare_1775062406iple_a] :
% 0.26/0.48 ( ( semila13410563le_a_o
% 0.26/0.48 @ ^ [Y_1: hoare_1775062406iple_a] : ( member2122167641iple_a @ Y_1 @ R_1 )
% 0.26/0.48 @ ^ [Y_1: hoare_1775062406iple_a] : ( member2122167641iple_a @ Y_1 @ S_3 )
% 0.26/0.48 @ X_3 )
% 0.26/0.48 <=> ( member2122167641iple_a @ X_3 @ ( semila13410563le_a_o @ R_1 @ S_3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_546_sup__Un__eq,axiom,
% 0.26/0.48 ! [R_1: nat > $o,S_3: nat > $o,X_3: nat] :
% 0.26/0.48 ( ( semila848761471_nat_o
% 0.26/0.48 @ ^ [Y_1: nat] : ( member_nat @ Y_1 @ R_1 )
% 0.26/0.48 @ ^ [Y_1: nat] : ( member_nat @ Y_1 @ S_3 )
% 0.26/0.48 @ X_3 )
% 0.26/0.48 <=> ( member_nat @ X_3 @ ( semila848761471_nat_o @ R_1 @ S_3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_547_bot__least,axiom,
% 0.26/0.48 ! [A_60: $o] : ( ord_less_eq_o @ bot_bot_o @ A_60 ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_548_bot__least,axiom,
% 0.26/0.48 ! [A_60: nat > $o] : ( ord_less_eq_nat_o @ bot_bot_nat_o @ A_60 ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_549_bot__least,axiom,
% 0.26/0.48 ! [A_60: hoare_1167836817_state > $o] : ( ord_le827224136tate_o @ bot_bo70021908tate_o @ A_60 ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_550_bot__least,axiom,
% 0.26/0.48 ! [A_60: hoare_1775062406iple_a > $o] : ( ord_le1143225901le_a_o @ bot_bo751897185le_a_o @ A_60 ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_551_bot__least,axiom,
% 0.26/0.48 ! [A_60: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A_60 ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_552_bot__unique,axiom,
% 0.26/0.48 ! [A_59: $o] :
% 0.26/0.48 ( ( ord_less_eq_o @ A_59 @ bot_bot_o )
% 0.26/0.48 <=> ( A_59
% 0.26/0.48 <=> bot_bot_o ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_553_bot__unique,axiom,
% 0.26/0.48 ! [A_59: nat > $o] :
% 0.26/0.48 ( ( ord_less_eq_nat_o @ A_59 @ bot_bot_nat_o )
% 0.26/0.48 <=> ( A_59 = bot_bot_nat_o ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_554_bot__unique,axiom,
% 0.26/0.48 ! [A_59: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( ord_le827224136tate_o @ A_59 @ bot_bo70021908tate_o )
% 0.26/0.48 <=> ( A_59 = bot_bo70021908tate_o ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_555_bot__unique,axiom,
% 0.26/0.48 ! [A_59: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( ord_le1143225901le_a_o @ A_59 @ bot_bo751897185le_a_o )
% 0.26/0.48 <=> ( A_59 = bot_bo751897185le_a_o ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_556_bot__unique,axiom,
% 0.26/0.48 ! [A_59: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ A_59 @ bot_bot_nat )
% 0.26/0.48 <=> ( A_59 = bot_bot_nat ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_557_le__bot,axiom,
% 0.26/0.48 ! [A_58: $o] :
% 0.26/0.48 ( ( ord_less_eq_o @ A_58 @ bot_bot_o )
% 0.26/0.48 => ( A_58
% 0.26/0.48 <=> bot_bot_o ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_558_le__bot,axiom,
% 0.26/0.48 ! [A_58: nat > $o] :
% 0.26/0.48 ( ( ord_less_eq_nat_o @ A_58 @ bot_bot_nat_o )
% 0.26/0.48 => ( A_58 = bot_bot_nat_o ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_559_le__bot,axiom,
% 0.26/0.48 ! [A_58: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( ord_le827224136tate_o @ A_58 @ bot_bo70021908tate_o )
% 0.26/0.48 => ( A_58 = bot_bo70021908tate_o ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_560_le__bot,axiom,
% 0.26/0.48 ! [A_58: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( ord_le1143225901le_a_o @ A_58 @ bot_bo751897185le_a_o )
% 0.26/0.48 => ( A_58 = bot_bo751897185le_a_o ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_561_le__bot,axiom,
% 0.26/0.48 ! [A_58: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ A_58 @ bot_bot_nat )
% 0.26/0.48 => ( A_58 = bot_bot_nat ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_562_Un__insert__right,axiom,
% 0.26/0.48 ! [A_57: nat > $o,A_56: nat,B_35: nat > $o] :
% 0.26/0.48 ( ( semila848761471_nat_o @ A_57 @ ( insert_nat @ A_56 @ B_35 ) )
% 0.26/0.48 = ( insert_nat @ A_56 @ ( semila848761471_nat_o @ A_57 @ B_35 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_563_Un__insert__right,axiom,
% 0.26/0.48 ! [A_57: hoare_1167836817_state > $o,A_56: hoare_1167836817_state,B_35: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( semila1172322802tate_o @ A_57 @ ( insert2134838167_state @ A_56 @ B_35 ) )
% 0.26/0.48 = ( insert2134838167_state @ A_56 @ ( semila1172322802tate_o @ A_57 @ B_35 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_564_Un__insert__right,axiom,
% 0.26/0.48 ! [A_57: hoare_1775062406iple_a > $o,A_56: hoare_1775062406iple_a,B_35: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( semila13410563le_a_o @ A_57 @ ( insert1281456128iple_a @ A_56 @ B_35 ) )
% 0.26/0.48 = ( insert1281456128iple_a @ A_56 @ ( semila13410563le_a_o @ A_57 @ B_35 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_565_Un__insert__left,axiom,
% 0.26/0.48 ! [A_55: nat,B_34: nat > $o,C_11: nat > $o] :
% 0.26/0.48 ( ( semila848761471_nat_o @ ( insert_nat @ A_55 @ B_34 ) @ C_11 )
% 0.26/0.48 = ( insert_nat @ A_55 @ ( semila848761471_nat_o @ B_34 @ C_11 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_566_Un__insert__left,axiom,
% 0.26/0.48 ! [A_55: hoare_1167836817_state,B_34: hoare_1167836817_state > $o,C_11: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( semila1172322802tate_o @ ( insert2134838167_state @ A_55 @ B_34 ) @ C_11 )
% 0.26/0.48 = ( insert2134838167_state @ A_55 @ ( semila1172322802tate_o @ B_34 @ C_11 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_567_Un__insert__left,axiom,
% 0.26/0.48 ! [A_55: hoare_1775062406iple_a,B_34: hoare_1775062406iple_a > $o,C_11: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( semila13410563le_a_o @ ( insert1281456128iple_a @ A_55 @ B_34 ) @ C_11 )
% 0.26/0.48 = ( insert1281456128iple_a @ A_55 @ ( semila13410563le_a_o @ B_34 @ C_11 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_568_weaken,axiom,
% 0.26/0.48 ! [Ts_2: hoare_1167836817_state > $o,G_2: hoare_1167836817_state > $o,Ts_1: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( hoare_123228589_state @ G_2 @ Ts_1 )
% 0.26/0.48 => ( ( ord_le827224136tate_o @ Ts_2 @ Ts_1 )
% 0.26/0.48 => ( hoare_123228589_state @ G_2 @ Ts_2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_569_weaken,axiom,
% 0.26/0.48 ! [Ts_2: hoare_1775062406iple_a > $o,G_2: hoare_1775062406iple_a > $o,Ts_1: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( hoare_1508237396rivs_a @ G_2 @ Ts_1 )
% 0.26/0.48 => ( ( ord_le1143225901le_a_o @ Ts_2 @ Ts_1 )
% 0.26/0.48 => ( hoare_1508237396rivs_a @ G_2 @ Ts_2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_570_asm,axiom,
% 0.26/0.48 ! [Ts: hoare_1167836817_state > $o,G_1: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( ord_le827224136tate_o @ Ts @ G_1 )
% 0.26/0.48 => ( hoare_123228589_state @ G_1 @ Ts ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_571_asm,axiom,
% 0.26/0.48 ! [Ts: hoare_1775062406iple_a > $o,G_1: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( ord_le1143225901le_a_o @ Ts @ G_1 )
% 0.26/0.48 => ( hoare_1508237396rivs_a @ G_1 @ Ts ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_572_insert__def,axiom,
% 0.26/0.48 ! [A_54: nat,B_33: nat > $o] :
% 0.26/0.48 ( ( insert_nat @ A_54 @ B_33 )
% 0.26/0.48 = ( semila848761471_nat_o
% 0.26/0.48 @ ( collect_nat
% 0.26/0.48 @ ^ [X_3: nat] : ( X_3 = A_54 ) )
% 0.26/0.48 @ B_33 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_573_insert__def,axiom,
% 0.26/0.48 ! [A_54: hoare_1167836817_state,B_33: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( insert2134838167_state @ A_54 @ B_33 )
% 0.26/0.48 = ( semila1172322802tate_o
% 0.26/0.48 @ ( collec1027672124_state
% 0.26/0.48 @ ^ [X_3: hoare_1167836817_state] : ( X_3 = A_54 ) )
% 0.26/0.48 @ B_33 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_574_insert__def,axiom,
% 0.26/0.48 ! [A_54: hoare_1775062406iple_a,B_33: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( insert1281456128iple_a @ A_54 @ B_33 )
% 0.26/0.48 = ( semila13410563le_a_o
% 0.26/0.48 @ ( collec676402587iple_a
% 0.26/0.48 @ ^ [X_3: hoare_1775062406iple_a] : ( X_3 = A_54 ) )
% 0.26/0.48 @ B_33 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_575_insert__is__Un,axiom,
% 0.26/0.48 ! [A_53: nat,A_52: nat > $o] :
% 0.26/0.48 ( ( insert_nat @ A_53 @ A_52 )
% 0.26/0.48 = ( semila848761471_nat_o @ ( insert_nat @ A_53 @ bot_bot_nat_o ) @ A_52 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_576_insert__is__Un,axiom,
% 0.26/0.48 ! [A_53: hoare_1167836817_state,A_52: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( insert2134838167_state @ A_53 @ A_52 )
% 0.26/0.48 = ( semila1172322802tate_o @ ( insert2134838167_state @ A_53 @ bot_bo70021908tate_o ) @ A_52 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_577_insert__is__Un,axiom,
% 0.26/0.48 ! [A_53: hoare_1775062406iple_a,A_52: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( insert1281456128iple_a @ A_53 @ A_52 )
% 0.26/0.48 = ( semila13410563le_a_o @ ( insert1281456128iple_a @ A_53 @ bot_bo751897185le_a_o ) @ A_52 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_578_fold__sup__insert,axiom,
% 0.26/0.48 ! [B_32: nat,A_51: nat,A_50: nat > $o] :
% 0.26/0.48 ( ( finite_finite_nat @ A_50 )
% 0.26/0.48 => ( ( finite_fold_nat_nat @ semila972727038up_nat @ B_32 @ ( insert_nat @ A_51 @ A_50 ) )
% 0.26/0.48 = ( semila972727038up_nat @ A_51 @ ( finite_fold_nat_nat @ semila972727038up_nat @ B_32 @ A_50 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_579_union__fold__insert,axiom,
% 0.26/0.48 ! [B_31: nat > $o,A_49: nat > $o] :
% 0.26/0.48 ( ( finite_finite_nat @ A_49 )
% 0.26/0.48 => ( ( semila848761471_nat_o @ A_49 @ B_31 )
% 0.26/0.48 = ( finite326637109_nat_o @ insert_nat @ B_31 @ A_49 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_580_union__fold__insert,axiom,
% 0.26/0.48 ! [B_31: hoare_1167836817_state > $o,A_49: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( finite1084549118_state @ A_49 )
% 0.26/0.48 => ( ( semila1172322802tate_o @ A_49 @ B_31 )
% 0.26/0.48 = ( finite291020855tate_o @ insert2134838167_state @ B_31 @ A_49 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_581_union__fold__insert,axiom,
% 0.26/0.48 ! [B_31: hoare_1775062406iple_a > $o,A_49: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( finite2063573081iple_a @ A_49 )
% 0.26/0.48 => ( ( semila13410563le_a_o @ A_49 @ B_31 )
% 0.26/0.48 = ( finite1544171829le_a_o @ insert1281456128iple_a @ B_31 @ A_49 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_582_subset__singletonD,axiom,
% 0.26/0.48 ! [A_48: nat > $o,X_29: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat_o @ A_48 @ ( insert_nat @ X_29 @ bot_bot_nat_o ) )
% 0.26/0.48 => ( ( A_48 = bot_bot_nat_o )
% 0.26/0.48 | ( A_48
% 0.26/0.48 = ( insert_nat @ X_29 @ bot_bot_nat_o ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_583_subset__singletonD,axiom,
% 0.26/0.48 ! [A_48: hoare_1167836817_state > $o,X_29: hoare_1167836817_state] :
% 0.26/0.48 ( ( ord_le827224136tate_o @ A_48 @ ( insert2134838167_state @ X_29 @ bot_bo70021908tate_o ) )
% 0.26/0.48 => ( ( A_48 = bot_bo70021908tate_o )
% 0.26/0.48 | ( A_48
% 0.26/0.48 = ( insert2134838167_state @ X_29 @ bot_bo70021908tate_o ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_584_subset__singletonD,axiom,
% 0.26/0.48 ! [A_48: hoare_1775062406iple_a > $o,X_29: hoare_1775062406iple_a] :
% 0.26/0.48 ( ( ord_le1143225901le_a_o @ A_48 @ ( insert1281456128iple_a @ X_29 @ bot_bo751897185le_a_o ) )
% 0.26/0.48 => ( ( A_48 = bot_bo751897185le_a_o )
% 0.26/0.48 | ( A_48
% 0.26/0.48 = ( insert1281456128iple_a @ X_29 @ bot_bo751897185le_a_o ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_585_diff__single__insert,axiom,
% 0.26/0.48 ! [A_47: nat > $o,X_28: nat,B_30: nat > $o] :
% 0.26/0.48 ( ( ord_less_eq_nat_o @ ( minus_minus_nat_o @ A_47 @ ( insert_nat @ X_28 @ bot_bot_nat_o ) ) @ B_30 )
% 0.26/0.48 => ( ( member_nat @ X_28 @ A_47 )
% 0.26/0.48 => ( ord_less_eq_nat_o @ A_47 @ ( insert_nat @ X_28 @ B_30 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_586_diff__single__insert,axiom,
% 0.26/0.48 ! [A_47: hoare_1167836817_state > $o,X_28: hoare_1167836817_state,B_30: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( ord_le827224136tate_o @ ( minus_2107060239tate_o @ A_47 @ ( insert2134838167_state @ X_28 @ bot_bo70021908tate_o ) ) @ B_30 )
% 0.26/0.48 => ( ( member2058392318_state @ X_28 @ A_47 )
% 0.26/0.48 => ( ord_le827224136tate_o @ A_47 @ ( insert2134838167_state @ X_28 @ B_30 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_587_diff__single__insert,axiom,
% 0.26/0.48 ! [A_47: hoare_1775062406iple_a > $o,X_28: hoare_1775062406iple_a,B_30: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( ord_le1143225901le_a_o @ ( minus_1944206118le_a_o @ A_47 @ ( insert1281456128iple_a @ X_28 @ bot_bo751897185le_a_o ) ) @ B_30 )
% 0.26/0.48 => ( ( member2122167641iple_a @ X_28 @ A_47 )
% 0.26/0.48 => ( ord_le1143225901le_a_o @ A_47 @ ( insert1281456128iple_a @ X_28 @ B_30 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_588_subset__insert__iff,axiom,
% 0.26/0.48 ! [A_46: nat > $o,X_27: nat,B_29: nat > $o] :
% 0.26/0.48 ( ( ord_less_eq_nat_o @ A_46 @ ( insert_nat @ X_27 @ B_29 ) )
% 0.26/0.48 <=> ( ( ( member_nat @ X_27 @ A_46 )
% 0.26/0.48 => ( ord_less_eq_nat_o @ ( minus_minus_nat_o @ A_46 @ ( insert_nat @ X_27 @ bot_bot_nat_o ) ) @ B_29 ) )
% 0.26/0.48 & ( ~ ( member_nat @ X_27 @ A_46 )
% 0.26/0.48 => ( ord_less_eq_nat_o @ A_46 @ B_29 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_589_subset__insert__iff,axiom,
% 0.26/0.48 ! [A_46: hoare_1167836817_state > $o,X_27: hoare_1167836817_state,B_29: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( ord_le827224136tate_o @ A_46 @ ( insert2134838167_state @ X_27 @ B_29 ) )
% 0.26/0.48 <=> ( ( ( member2058392318_state @ X_27 @ A_46 )
% 0.26/0.48 => ( ord_le827224136tate_o @ ( minus_2107060239tate_o @ A_46 @ ( insert2134838167_state @ X_27 @ bot_bo70021908tate_o ) ) @ B_29 ) )
% 0.26/0.48 & ( ~ ( member2058392318_state @ X_27 @ A_46 )
% 0.26/0.48 => ( ord_le827224136tate_o @ A_46 @ B_29 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_590_subset__insert__iff,axiom,
% 0.26/0.48 ! [A_46: hoare_1775062406iple_a > $o,X_27: hoare_1775062406iple_a,B_29: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( ord_le1143225901le_a_o @ A_46 @ ( insert1281456128iple_a @ X_27 @ B_29 ) )
% 0.26/0.48 <=> ( ( ( member2122167641iple_a @ X_27 @ A_46 )
% 0.26/0.48 => ( ord_le1143225901le_a_o @ ( minus_1944206118le_a_o @ A_46 @ ( insert1281456128iple_a @ X_27 @ bot_bo751897185le_a_o ) ) @ B_29 ) )
% 0.26/0.48 & ( ~ ( member2122167641iple_a @ X_27 @ A_46 )
% 0.26/0.48 => ( ord_le1143225901le_a_o @ A_46 @ B_29 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_591_folding__one__idem_Ounion__idem,axiom,
% 0.26/0.48 ! [B_28: nat > $o,A_45: nat > $o,F_5: nat > nat > nat,F_4: ( nat > $o ) > nat] :
% 0.26/0.48 ( ( finite795500164em_nat @ F_5 @ F_4 )
% 0.26/0.48 => ( ( finite_finite_nat @ A_45 )
% 0.26/0.48 => ( ( A_45 != bot_bot_nat_o )
% 0.26/0.48 => ( ( finite_finite_nat @ B_28 )
% 0.26/0.48 => ( ( B_28 != bot_bot_nat_o )
% 0.26/0.48 => ( ( F_4 @ ( semila848761471_nat_o @ A_45 @ B_28 ) )
% 0.26/0.48 = ( F_5 @ ( F_4 @ A_45 ) @ ( F_4 @ B_28 ) ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_592_folding__one__idem_Ounion__idem,axiom,
% 0.26/0.48 ! [B_28: hoare_1167836817_state > $o,A_45: hoare_1167836817_state > $o,F_5: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_4: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
% 0.26/0.48 ( ( finite806517911_state @ F_5 @ F_4 )
% 0.26/0.48 => ( ( finite1084549118_state @ A_45 )
% 0.26/0.48 => ( ( A_45 != bot_bo70021908tate_o )
% 0.26/0.48 => ( ( finite1084549118_state @ B_28 )
% 0.26/0.48 => ( ( B_28 != bot_bo70021908tate_o )
% 0.26/0.48 => ( ( F_4 @ ( semila1172322802tate_o @ A_45 @ B_28 ) )
% 0.26/0.48 = ( F_5 @ ( F_4 @ A_45 ) @ ( F_4 @ B_28 ) ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_593_folding__one__idem_Ounion__idem,axiom,
% 0.26/0.48 ! [B_28: hoare_1775062406iple_a > $o,A_45: hoare_1775062406iple_a > $o,F_5: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_4: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
% 0.26/0.48 ( ( finite1358382848iple_a @ F_5 @ F_4 )
% 0.26/0.48 => ( ( finite2063573081iple_a @ A_45 )
% 0.26/0.48 => ( ( A_45 != bot_bo751897185le_a_o )
% 0.26/0.48 => ( ( finite2063573081iple_a @ B_28 )
% 0.26/0.48 => ( ( B_28 != bot_bo751897185le_a_o )
% 0.26/0.48 => ( ( F_4 @ ( semila13410563le_a_o @ A_45 @ B_28 ) )
% 0.26/0.48 = ( F_5 @ ( F_4 @ A_45 ) @ ( F_4 @ B_28 ) ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_594_fold__sup__le__sup,axiom,
% 0.26/0.48 ! [C_10: nat,B_27: nat,A_44: nat > $o] :
% 0.26/0.48 ( ( finite_finite_nat @ A_44 )
% 0.26/0.48 => ( ! [X_3: nat] :
% 0.26/0.48 ( ( member_nat @ X_3 @ A_44 )
% 0.26/0.48 => ( ord_less_eq_nat @ X_3 @ B_27 ) )
% 0.26/0.48 => ( ord_less_eq_nat @ ( finite_fold_nat_nat @ semila972727038up_nat @ C_10 @ A_44 ) @ ( semila972727038up_nat @ B_27 @ C_10 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_595_finite__subset__induct,axiom,
% 0.26/0.48 ! [P_2: ( nat > $o ) > $o,A_42: nat > $o,F_2: nat > $o] :
% 0.26/0.48 ( ( finite_finite_nat @ F_2 )
% 0.26/0.48 => ( ( ord_less_eq_nat_o @ F_2 @ A_42 )
% 0.26/0.48 => ( ( P_2 @ bot_bot_nat_o )
% 0.26/0.48 => ( ! [A_43: nat,F_3: nat > $o] :
% 0.26/0.48 ( ( finite_finite_nat @ F_3 )
% 0.26/0.48 => ( ( member_nat @ A_43 @ A_42 )
% 0.26/0.48 => ( ~ ( member_nat @ A_43 @ F_3 )
% 0.26/0.48 => ( ( P_2 @ F_3 )
% 0.26/0.48 => ( P_2 @ ( insert_nat @ A_43 @ F_3 ) ) ) ) ) )
% 0.26/0.48 => ( P_2 @ F_2 ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_596_finite__subset__induct,axiom,
% 0.26/0.48 ! [P_2: ( hoare_1167836817_state > $o ) > $o,A_42: hoare_1167836817_state > $o,F_2: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( finite1084549118_state @ F_2 )
% 0.26/0.48 => ( ( ord_le827224136tate_o @ F_2 @ A_42 )
% 0.26/0.48 => ( ( P_2 @ bot_bo70021908tate_o )
% 0.26/0.48 => ( ! [A_43: hoare_1167836817_state,F_3: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( finite1084549118_state @ F_3 )
% 0.26/0.48 => ( ( member2058392318_state @ A_43 @ A_42 )
% 0.26/0.48 => ( ~ ( member2058392318_state @ A_43 @ F_3 )
% 0.26/0.48 => ( ( P_2 @ F_3 )
% 0.26/0.48 => ( P_2 @ ( insert2134838167_state @ A_43 @ F_3 ) ) ) ) ) )
% 0.26/0.48 => ( P_2 @ F_2 ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_597_finite__subset__induct,axiom,
% 0.26/0.48 ! [P_2: ( hoare_1775062406iple_a > $o ) > $o,A_42: hoare_1775062406iple_a > $o,F_2: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( finite2063573081iple_a @ F_2 )
% 0.26/0.48 => ( ( ord_le1143225901le_a_o @ F_2 @ A_42 )
% 0.26/0.48 => ( ( P_2 @ bot_bo751897185le_a_o )
% 0.26/0.48 => ( ! [A_43: hoare_1775062406iple_a,F_3: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( finite2063573081iple_a @ F_3 )
% 0.26/0.48 => ( ( member2122167641iple_a @ A_43 @ A_42 )
% 0.26/0.48 => ( ~ ( member2122167641iple_a @ A_43 @ F_3 )
% 0.26/0.48 => ( ( P_2 @ F_3 )
% 0.26/0.48 => ( P_2 @ ( insert1281456128iple_a @ A_43 @ F_3 ) ) ) ) ) )
% 0.26/0.48 => ( P_2 @ F_2 ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_598_subsetI,axiom,
% 0.26/0.48 ! [B_26: hoare_1775062406iple_a > $o,A_41: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ! [X_3: hoare_1775062406iple_a] :
% 0.26/0.48 ( ( member2122167641iple_a @ X_3 @ A_41 )
% 0.26/0.48 => ( member2122167641iple_a @ X_3 @ B_26 ) )
% 0.26/0.48 => ( ord_le1143225901le_a_o @ A_41 @ B_26 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_599_subsetI,axiom,
% 0.26/0.48 ! [B_26: nat > $o,A_41: nat > $o] :
% 0.26/0.48 ( ! [X_3: nat] :
% 0.26/0.48 ( ( member_nat @ X_3 @ A_41 )
% 0.26/0.48 => ( member_nat @ X_3 @ B_26 ) )
% 0.26/0.48 => ( ord_less_eq_nat_o @ A_41 @ B_26 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_600_evaln__nonstrict,axiom,
% 0.26/0.48 ! [M_1: nat,C_9: com,S_2: state,N_3: nat,T: state] :
% 0.26/0.48 ( ( evaln @ C_9 @ S_2 @ N_3 @ T )
% 0.26/0.48 => ( ( ord_less_eq_nat @ N_3 @ M_1 )
% 0.26/0.48 => ( evaln @ C_9 @ S_2 @ M_1 @ T ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_601_finite__Collect__le__nat,axiom,
% 0.26/0.48 ! [K: nat] :
% 0.26/0.48 ( finite_finite_nat
% 0.26/0.48 @ ( collect_nat
% 0.26/0.48 @ ^ [N_1: nat] : ( ord_less_eq_nat @ N_1 @ K ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_602_flat__lub__def,axiom,
% 0.26/0.48 ! [A_40: hoare_1775062406iple_a > $o,B_25: hoare_1775062406iple_a] :
% 0.26/0.48 ( ( ( ord_le1143225901le_a_o @ A_40 @ ( insert1281456128iple_a @ B_25 @ bot_bo751897185le_a_o ) )
% 0.26/0.48 => ( ( partia126998524iple_a @ B_25 @ A_40 )
% 0.26/0.48 = B_25 ) )
% 0.26/0.48 & ( ~ ( ord_le1143225901le_a_o @ A_40 @ ( insert1281456128iple_a @ B_25 @ bot_bo751897185le_a_o ) )
% 0.26/0.48 => ( ( partia126998524iple_a @ B_25 @ A_40 )
% 0.26/0.48 = ( the_Ho1155011127iple_a
% 0.26/0.48 @ ^ [X_3: hoare_1775062406iple_a] : ( member2122167641iple_a @ X_3 @ ( minus_1944206118le_a_o @ A_40 @ ( insert1281456128iple_a @ B_25 @ bot_bo751897185le_a_o ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_603_flat__lub__def,axiom,
% 0.26/0.48 ! [A_40: nat > $o,B_25: nat] :
% 0.26/0.48 ( ( ( ord_less_eq_nat_o @ A_40 @ ( insert_nat @ B_25 @ bot_bot_nat_o ) )
% 0.26/0.48 => ( ( partial_flat_lub_nat @ B_25 @ A_40 )
% 0.26/0.48 = B_25 ) )
% 0.26/0.48 & ( ~ ( ord_less_eq_nat_o @ A_40 @ ( insert_nat @ B_25 @ bot_bot_nat_o ) )
% 0.26/0.48 => ( ( partial_flat_lub_nat @ B_25 @ A_40 )
% 0.26/0.48 = ( the_nat
% 0.26/0.48 @ ^ [X_3: nat] : ( member_nat @ X_3 @ ( minus_minus_nat_o @ A_40 @ ( insert_nat @ B_25 @ bot_bot_nat_o ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_604_flat__lub__def,axiom,
% 0.26/0.48 ! [A_40: hoare_1167836817_state > $o,B_25: hoare_1167836817_state] :
% 0.26/0.48 ( ( ( ord_le827224136tate_o @ A_40 @ ( insert2134838167_state @ B_25 @ bot_bo70021908tate_o ) )
% 0.26/0.48 => ( ( partia715677851_state @ B_25 @ A_40 )
% 0.26/0.48 = B_25 ) )
% 0.26/0.48 & ( ~ ( ord_le827224136tate_o @ A_40 @ ( insert2134838167_state @ B_25 @ bot_bo70021908tate_o ) )
% 0.26/0.48 => ( ( partia715677851_state @ B_25 @ A_40 )
% 0.26/0.48 = ( the_Ho310147232_state
% 0.26/0.48 @ ^ [X_3: hoare_1167836817_state] : ( member2058392318_state @ X_3 @ ( minus_2107060239tate_o @ A_40 @ ( insert2134838167_state @ B_25 @ bot_bo70021908tate_o ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_605_finite__nat__set__iff__bounded__le,axiom,
% 0.26/0.48 ! [N_2: nat > $o] :
% 0.26/0.48 ( ( finite_finite_nat @ N_2 )
% 0.26/0.48 <=> ? [M: nat] :
% 0.26/0.48 ! [X_3: nat] :
% 0.26/0.48 ( ( member_nat @ X_3 @ N_2 )
% 0.26/0.48 => ( ord_less_eq_nat @ X_3 @ M ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_606_finite__less__ub,axiom,
% 0.26/0.48 ! [U: nat,F_1: nat > nat] :
% 0.26/0.48 ( ! [N_1: nat] : ( ord_less_eq_nat @ N_1 @ ( F_1 @ N_1 ) )
% 0.26/0.48 => ( finite_finite_nat
% 0.26/0.48 @ ( collect_nat
% 0.26/0.48 @ ^ [N_1: nat] : ( ord_less_eq_nat @ ( F_1 @ N_1 ) @ U ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_607_Sup__fin_Oremove,axiom,
% 0.26/0.48 ! [X_26: nat,A_39: nat > $o] :
% 0.26/0.48 ( ( finite_finite_nat @ A_39 )
% 0.26/0.48 => ( ( member_nat @ X_26 @ A_39 )
% 0.26/0.48 => ( ( ( ( minus_minus_nat_o @ A_39 @ ( insert_nat @ X_26 @ bot_bot_nat_o ) )
% 0.26/0.48 = bot_bot_nat_o )
% 0.26/0.48 => ( ( big_la43341705in_nat @ A_39 )
% 0.26/0.48 = X_26 ) )
% 0.26/0.48 & ( ( ( minus_minus_nat_o @ A_39 @ ( insert_nat @ X_26 @ bot_bot_nat_o ) )
% 0.26/0.48 != bot_bot_nat_o )
% 0.26/0.48 => ( ( big_la43341705in_nat @ A_39 )
% 0.26/0.48 = ( semila972727038up_nat @ X_26 @ ( big_la43341705in_nat @ ( minus_minus_nat_o @ A_39 @ ( insert_nat @ X_26 @ bot_bot_nat_o ) ) ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_608_Sup__fin_Osingleton,axiom,
% 0.26/0.48 ! [X_25: nat] :
% 0.26/0.48 ( ( big_la43341705in_nat @ ( insert_nat @ X_25 @ bot_bot_nat_o ) )
% 0.26/0.48 = X_25 ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_609_Sup__fin_Oin__idem,axiom,
% 0.26/0.48 ! [X_24: nat,A_38: nat > $o] :
% 0.26/0.48 ( ( finite_finite_nat @ A_38 )
% 0.26/0.48 => ( ( member_nat @ X_24 @ A_38 )
% 0.26/0.48 => ( ( semila972727038up_nat @ X_24 @ ( big_la43341705in_nat @ A_38 ) )
% 0.26/0.48 = ( big_la43341705in_nat @ A_38 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_610_Sup__fin_OF__eq,axiom,
% 0.26/0.48 ! [A_37: nat > $o] :
% 0.26/0.48 ( ( finite_finite_nat @ A_37 )
% 0.26/0.48 => ( ( big_la43341705in_nat @ A_37 )
% 0.26/0.48 = ( finite_fold1_nat @ semila972727038up_nat @ A_37 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_611_Sup__fin_Oinsert__idem,axiom,
% 0.26/0.48 ! [X_23: nat,A_36: nat > $o] :
% 0.26/0.48 ( ( finite_finite_nat @ A_36 )
% 0.26/0.48 => ( ( A_36 != bot_bot_nat_o )
% 0.26/0.48 => ( ( big_la43341705in_nat @ ( insert_nat @ X_23 @ A_36 ) )
% 0.26/0.48 = ( semila972727038up_nat @ X_23 @ ( big_la43341705in_nat @ A_36 ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_612_Sup__fin_Oinsert,axiom,
% 0.26/0.48 ! [X_22: nat,A_35: nat > $o] :
% 0.26/0.48 ( ( finite_finite_nat @ A_35 )
% 0.26/0.48 => ( ~ ( member_nat @ X_22 @ A_35 )
% 0.26/0.48 => ( ( A_35 != bot_bot_nat_o )
% 0.26/0.48 => ( ( big_la43341705in_nat @ ( insert_nat @ X_22 @ A_35 ) )
% 0.26/0.48 = ( semila972727038up_nat @ X_22 @ ( big_la43341705in_nat @ A_35 ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_613_Sup__fin_Osubset__idem,axiom,
% 0.26/0.48 ! [B_24: nat > $o,A_34: nat > $o] :
% 0.26/0.48 ( ( finite_finite_nat @ A_34 )
% 0.26/0.48 => ( ( B_24 != bot_bot_nat_o )
% 0.26/0.48 => ( ( ord_less_eq_nat_o @ B_24 @ A_34 )
% 0.26/0.48 => ( ( semila972727038up_nat @ ( big_la43341705in_nat @ B_24 ) @ ( big_la43341705in_nat @ A_34 ) )
% 0.26/0.48 = ( big_la43341705in_nat @ A_34 ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_614_Sup__fin_Ounion__idem,axiom,
% 0.26/0.48 ! [B_23: nat > $o,A_33: nat > $o] :
% 0.26/0.48 ( ( finite_finite_nat @ A_33 )
% 0.26/0.48 => ( ( A_33 != bot_bot_nat_o )
% 0.26/0.48 => ( ( finite_finite_nat @ B_23 )
% 0.26/0.48 => ( ( B_23 != bot_bot_nat_o )
% 0.26/0.48 => ( ( big_la43341705in_nat @ ( semila848761471_nat_o @ A_33 @ B_23 ) )
% 0.26/0.48 = ( semila972727038up_nat @ ( big_la43341705in_nat @ A_33 ) @ ( big_la43341705in_nat @ B_23 ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_615_Sup__fin_Oeq__fold__idem_H,axiom,
% 0.26/0.48 ! [A_32: nat,A_31: nat > $o] :
% 0.26/0.48 ( ( finite_finite_nat @ A_31 )
% 0.26/0.48 => ( ( big_la43341705in_nat @ ( insert_nat @ A_32 @ A_31 ) )
% 0.26/0.48 = ( finite_fold_nat_nat @ semila972727038up_nat @ A_32 @ A_31 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_616_Sup__fin_Oeq__fold_H,axiom,
% 0.26/0.48 ! [X_21: nat,A_30: nat > $o] :
% 0.26/0.48 ( ( finite_finite_nat @ A_30 )
% 0.26/0.48 => ( ~ ( member_nat @ X_21 @ A_30 )
% 0.26/0.48 => ( ( big_la43341705in_nat @ ( insert_nat @ X_21 @ A_30 ) )
% 0.26/0.48 = ( finite_fold_nat_nat @ semila972727038up_nat @ X_21 @ A_30 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_617_Sup__fin_Oinsert__remove,axiom,
% 0.26/0.48 ! [X_20: nat,A_29: nat > $o] :
% 0.26/0.48 ( ( finite_finite_nat @ A_29 )
% 0.26/0.48 => ( ( ( ( minus_minus_nat_o @ A_29 @ ( insert_nat @ X_20 @ bot_bot_nat_o ) )
% 0.26/0.48 = bot_bot_nat_o )
% 0.26/0.48 => ( ( big_la43341705in_nat @ ( insert_nat @ X_20 @ A_29 ) )
% 0.26/0.48 = X_20 ) )
% 0.26/0.48 & ( ( ( minus_minus_nat_o @ A_29 @ ( insert_nat @ X_20 @ bot_bot_nat_o ) )
% 0.26/0.48 != bot_bot_nat_o )
% 0.26/0.48 => ( ( big_la43341705in_nat @ ( insert_nat @ X_20 @ A_29 ) )
% 0.26/0.48 = ( semila972727038up_nat @ X_20 @ ( big_la43341705in_nat @ ( minus_minus_nat_o @ A_29 @ ( insert_nat @ X_20 @ bot_bot_nat_o ) ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_618_Sup__fin_Ohom__commute,axiom,
% 0.26/0.48 ! [N: nat > $o,H: nat > nat] :
% 0.26/0.48 ( ! [X_3: nat,Y_1: nat] :
% 0.26/0.48 ( ( H @ ( semila972727038up_nat @ X_3 @ Y_1 ) )
% 0.26/0.48 = ( semila972727038up_nat @ ( H @ X_3 ) @ ( H @ Y_1 ) ) )
% 0.26/0.48 => ( ( finite_finite_nat @ N )
% 0.26/0.48 => ( ( N != bot_bot_nat_o )
% 0.26/0.48 => ( ( H @ ( big_la43341705in_nat @ N ) )
% 0.26/0.48 = ( big_la43341705in_nat @ ( image_nat_nat @ H @ N ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_619_Sup__fin_Oclosed,axiom,
% 0.26/0.48 ! [A_28: nat > $o] :
% 0.26/0.48 ( ( finite_finite_nat @ A_28 )
% 0.26/0.48 => ( ( A_28 != bot_bot_nat_o )
% 0.26/0.48 => ( ! [X_3: nat,Y_1: nat] : ( member_nat @ ( semila972727038up_nat @ X_3 @ Y_1 ) @ ( insert_nat @ X_3 @ ( insert_nat @ Y_1 @ bot_bot_nat_o ) ) )
% 0.26/0.48 => ( member_nat @ ( big_la43341705in_nat @ A_28 ) @ A_28 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_620_Sup__fin_Ounion__disjoint,axiom,
% 0.26/0.48 ! [B_22: nat > $o,A_27: nat > $o] :
% 0.26/0.48 ( ( finite_finite_nat @ A_27 )
% 0.26/0.48 => ( ( A_27 != bot_bot_nat_o )
% 0.26/0.48 => ( ( finite_finite_nat @ B_22 )
% 0.26/0.48 => ( ( B_22 != bot_bot_nat_o )
% 0.26/0.48 => ( ( ( semila1947288293_nat_o @ A_27 @ B_22 )
% 0.26/0.48 = bot_bot_nat_o )
% 0.26/0.48 => ( ( big_la43341705in_nat @ ( semila848761471_nat_o @ A_27 @ B_22 ) )
% 0.26/0.48 = ( semila972727038up_nat @ ( big_la43341705in_nat @ A_27 ) @ ( big_la43341705in_nat @ B_22 ) ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_621_IntI,axiom,
% 0.26/0.48 ! [B_21: hoare_1775062406iple_a > $o,C_8: hoare_1775062406iple_a,A_26: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( member2122167641iple_a @ C_8 @ A_26 )
% 0.26/0.48 => ( ( member2122167641iple_a @ C_8 @ B_21 )
% 0.26/0.48 => ( member2122167641iple_a @ C_8 @ ( semila966743401le_a_o @ A_26 @ B_21 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_622_IntI,axiom,
% 0.26/0.48 ! [B_21: nat > $o,C_8: nat,A_26: nat > $o] :
% 0.26/0.48 ( ( member_nat @ C_8 @ A_26 )
% 0.26/0.48 => ( ( member_nat @ C_8 @ B_21 )
% 0.26/0.48 => ( member_nat @ C_8 @ ( semila1947288293_nat_o @ A_26 @ B_21 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_623_IntE,axiom,
% 0.26/0.48 ! [C_7: hoare_1775062406iple_a,A_25: hoare_1775062406iple_a > $o,B_20: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( member2122167641iple_a @ C_7 @ ( semila966743401le_a_o @ A_25 @ B_20 ) )
% 0.26/0.48 => ~ ( ( member2122167641iple_a @ C_7 @ A_25 )
% 0.26/0.48 => ~ ( member2122167641iple_a @ C_7 @ B_20 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_624_IntE,axiom,
% 0.26/0.48 ! [C_7: nat,A_25: nat > $o,B_20: nat > $o] :
% 0.26/0.48 ( ( member_nat @ C_7 @ ( semila1947288293_nat_o @ A_25 @ B_20 ) )
% 0.26/0.48 => ~ ( ( member_nat @ C_7 @ A_25 )
% 0.26/0.48 => ~ ( member_nat @ C_7 @ B_20 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_625_finite__Int,axiom,
% 0.26/0.48 ! [G: nat > $o,F: nat > $o] :
% 0.26/0.48 ( ( ( finite_finite_nat @ F )
% 0.26/0.48 | ( finite_finite_nat @ G ) )
% 0.26/0.48 => ( finite_finite_nat @ ( semila1947288293_nat_o @ F @ G ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_626_le__infE,axiom,
% 0.26/0.48 ! [X_19: nat,A_24: nat,B_19: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ X_19 @ ( semila80283416nf_nat @ A_24 @ B_19 ) )
% 0.26/0.48 => ~ ( ( ord_less_eq_nat @ X_19 @ A_24 )
% 0.26/0.48 => ~ ( ord_less_eq_nat @ X_19 @ B_19 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_627_inf__mono,axiom,
% 0.26/0.48 ! [B_18: nat,D: nat,A_23: nat,C_6: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ A_23 @ C_6 )
% 0.26/0.48 => ( ( ord_less_eq_nat @ B_18 @ D )
% 0.26/0.48 => ( ord_less_eq_nat @ ( semila80283416nf_nat @ A_23 @ B_18 ) @ ( semila80283416nf_nat @ C_6 @ D ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_628_inf__greatest,axiom,
% 0.26/0.48 ! [Z_4: nat,X_18: nat,Y_12: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ X_18 @ Y_12 )
% 0.26/0.48 => ( ( ord_less_eq_nat @ X_18 @ Z_4 )
% 0.26/0.48 => ( ord_less_eq_nat @ X_18 @ ( semila80283416nf_nat @ Y_12 @ Z_4 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_629_le__infI,axiom,
% 0.26/0.48 ! [B_17: nat,X_17: nat,A_22: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ X_17 @ A_22 )
% 0.26/0.48 => ( ( ord_less_eq_nat @ X_17 @ B_17 )
% 0.26/0.48 => ( ord_less_eq_nat @ X_17 @ ( semila80283416nf_nat @ A_22 @ B_17 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_630_inf__absorb2,axiom,
% 0.26/0.48 ! [Y_11: nat,X_16: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ Y_11 @ X_16 )
% 0.26/0.48 => ( ( semila80283416nf_nat @ X_16 @ Y_11 )
% 0.26/0.48 = Y_11 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_631_inf__absorb1,axiom,
% 0.26/0.48 ! [X_15: nat,Y_10: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ X_15 @ Y_10 )
% 0.26/0.48 => ( ( semila80283416nf_nat @ X_15 @ Y_10 )
% 0.26/0.48 = X_15 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_632_le__infI2,axiom,
% 0.26/0.48 ! [A_21: nat,B_16: nat,X_14: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ B_16 @ X_14 )
% 0.26/0.48 => ( ord_less_eq_nat @ ( semila80283416nf_nat @ A_21 @ B_16 ) @ X_14 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_633_le__infI1,axiom,
% 0.26/0.48 ! [B_15: nat,A_20: nat,X_13: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ A_20 @ X_13 )
% 0.26/0.48 => ( ord_less_eq_nat @ ( semila80283416nf_nat @ A_20 @ B_15 ) @ X_13 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_634_le__inf__iff,axiom,
% 0.26/0.48 ! [X_12: nat,Y_9: nat,Z_3: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ X_12 @ ( semila80283416nf_nat @ Y_9 @ Z_3 ) )
% 0.26/0.48 <=> ( ( ord_less_eq_nat @ X_12 @ Y_9 )
% 0.26/0.48 & ( ord_less_eq_nat @ X_12 @ Z_3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_635_le__iff__inf,axiom,
% 0.26/0.48 ! [X_11: nat,Y_8: nat] :
% 0.26/0.48 ( ( ord_less_eq_nat @ X_11 @ Y_8 )
% 0.26/0.48 <=> ( ( semila80283416nf_nat @ X_11 @ Y_8 )
% 0.26/0.48 = X_11 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_636_inf__le2,axiom,
% 0.26/0.48 ! [X_10: nat,Y_7: nat] : ( ord_less_eq_nat @ ( semila80283416nf_nat @ X_10 @ Y_7 ) @ Y_7 ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_637_inf__sup__ord_I2_J,axiom,
% 0.26/0.48 ! [X_9: nat,Y_6: nat] : ( ord_less_eq_nat @ ( semila80283416nf_nat @ X_9 @ Y_6 ) @ Y_6 ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_638_inf__le1,axiom,
% 0.26/0.48 ! [X_8: nat,Y_5: nat] : ( ord_less_eq_nat @ ( semila80283416nf_nat @ X_8 @ Y_5 ) @ X_8 ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_639_inf__sup__ord_I1_J,axiom,
% 0.26/0.48 ! [X_7: nat,Y_4: nat] : ( ord_less_eq_nat @ ( semila80283416nf_nat @ X_7 @ Y_4 ) @ X_7 ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_640_distrib__sup__le,axiom,
% 0.26/0.48 ! [X_6: nat,Y_3: nat,Z_2: nat] : ( ord_less_eq_nat @ ( semila972727038up_nat @ X_6 @ ( semila80283416nf_nat @ Y_3 @ Z_2 ) ) @ ( semila80283416nf_nat @ ( semila972727038up_nat @ X_6 @ Y_3 ) @ ( semila972727038up_nat @ X_6 @ Z_2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_641_distrib__inf__le,axiom,
% 0.26/0.48 ! [X_5: nat,Y_2: nat,Z_1: nat] : ( ord_less_eq_nat @ ( semila972727038up_nat @ ( semila80283416nf_nat @ X_5 @ Y_2 ) @ ( semila80283416nf_nat @ X_5 @ Z_1 ) ) @ ( semila80283416nf_nat @ X_5 @ ( semila972727038up_nat @ Y_2 @ Z_1 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_642_Int__insert__left__if1,axiom,
% 0.26/0.48 ! [B_14: nat > $o,A_19: nat,C_5: nat > $o] :
% 0.26/0.48 ( ( member_nat @ A_19 @ C_5 )
% 0.26/0.48 => ( ( semila1947288293_nat_o @ ( insert_nat @ A_19 @ B_14 ) @ C_5 )
% 0.26/0.48 = ( insert_nat @ A_19 @ ( semila1947288293_nat_o @ B_14 @ C_5 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_643_Int__insert__left__if1,axiom,
% 0.26/0.48 ! [B_14: hoare_1167836817_state > $o,A_19: hoare_1167836817_state,C_5: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( member2058392318_state @ A_19 @ C_5 )
% 0.26/0.48 => ( ( semila179895820tate_o @ ( insert2134838167_state @ A_19 @ B_14 ) @ C_5 )
% 0.26/0.48 = ( insert2134838167_state @ A_19 @ ( semila179895820tate_o @ B_14 @ C_5 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_644_Int__insert__left__if1,axiom,
% 0.26/0.48 ! [B_14: hoare_1775062406iple_a > $o,A_19: hoare_1775062406iple_a,C_5: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( member2122167641iple_a @ A_19 @ C_5 )
% 0.26/0.48 => ( ( semila966743401le_a_o @ ( insert1281456128iple_a @ A_19 @ B_14 ) @ C_5 )
% 0.26/0.48 = ( insert1281456128iple_a @ A_19 @ ( semila966743401le_a_o @ B_14 @ C_5 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_645_Int__insert__right__if1,axiom,
% 0.26/0.48 ! [B_13: nat > $o,A_18: nat,A_17: nat > $o] :
% 0.26/0.48 ( ( member_nat @ A_18 @ A_17 )
% 0.26/0.48 => ( ( semila1947288293_nat_o @ A_17 @ ( insert_nat @ A_18 @ B_13 ) )
% 0.26/0.48 = ( insert_nat @ A_18 @ ( semila1947288293_nat_o @ A_17 @ B_13 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_646_Int__insert__right__if1,axiom,
% 0.26/0.48 ! [B_13: hoare_1167836817_state > $o,A_18: hoare_1167836817_state,A_17: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( member2058392318_state @ A_18 @ A_17 )
% 0.26/0.48 => ( ( semila179895820tate_o @ A_17 @ ( insert2134838167_state @ A_18 @ B_13 ) )
% 0.26/0.48 = ( insert2134838167_state @ A_18 @ ( semila179895820tate_o @ A_17 @ B_13 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_647_Int__insert__right__if1,axiom,
% 0.26/0.48 ! [B_13: hoare_1775062406iple_a > $o,A_18: hoare_1775062406iple_a,A_17: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( member2122167641iple_a @ A_18 @ A_17 )
% 0.26/0.48 => ( ( semila966743401le_a_o @ A_17 @ ( insert1281456128iple_a @ A_18 @ B_13 ) )
% 0.26/0.48 = ( insert1281456128iple_a @ A_18 @ ( semila966743401le_a_o @ A_17 @ B_13 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_648_Int__insert__left__if0,axiom,
% 0.26/0.48 ! [B_12: nat > $o,A_16: nat,C_4: nat > $o] :
% 0.26/0.48 ( ~ ( member_nat @ A_16 @ C_4 )
% 0.26/0.48 => ( ( semila1947288293_nat_o @ ( insert_nat @ A_16 @ B_12 ) @ C_4 )
% 0.26/0.48 = ( semila1947288293_nat_o @ B_12 @ C_4 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_649_Int__insert__left__if0,axiom,
% 0.26/0.48 ! [B_12: hoare_1167836817_state > $o,A_16: hoare_1167836817_state,C_4: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ~ ( member2058392318_state @ A_16 @ C_4 )
% 0.26/0.48 => ( ( semila179895820tate_o @ ( insert2134838167_state @ A_16 @ B_12 ) @ C_4 )
% 0.26/0.48 = ( semila179895820tate_o @ B_12 @ C_4 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_650_Int__insert__left__if0,axiom,
% 0.26/0.48 ! [B_12: hoare_1775062406iple_a > $o,A_16: hoare_1775062406iple_a,C_4: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ~ ( member2122167641iple_a @ A_16 @ C_4 )
% 0.26/0.48 => ( ( semila966743401le_a_o @ ( insert1281456128iple_a @ A_16 @ B_12 ) @ C_4 )
% 0.26/0.48 = ( semila966743401le_a_o @ B_12 @ C_4 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_651_Int__insert__right__if0,axiom,
% 0.26/0.48 ! [B_11: nat > $o,A_15: nat,A_14: nat > $o] :
% 0.26/0.48 ( ~ ( member_nat @ A_15 @ A_14 )
% 0.26/0.48 => ( ( semila1947288293_nat_o @ A_14 @ ( insert_nat @ A_15 @ B_11 ) )
% 0.26/0.48 = ( semila1947288293_nat_o @ A_14 @ B_11 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_652_Int__insert__right__if0,axiom,
% 0.26/0.48 ! [B_11: hoare_1167836817_state > $o,A_15: hoare_1167836817_state,A_14: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ~ ( member2058392318_state @ A_15 @ A_14 )
% 0.26/0.48 => ( ( semila179895820tate_o @ A_14 @ ( insert2134838167_state @ A_15 @ B_11 ) )
% 0.26/0.48 = ( semila179895820tate_o @ A_14 @ B_11 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_653_Int__insert__right__if0,axiom,
% 0.26/0.48 ! [B_11: hoare_1775062406iple_a > $o,A_15: hoare_1775062406iple_a,A_14: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ~ ( member2122167641iple_a @ A_15 @ A_14 )
% 0.26/0.48 => ( ( semila966743401le_a_o @ A_14 @ ( insert1281456128iple_a @ A_15 @ B_11 ) )
% 0.26/0.48 = ( semila966743401le_a_o @ A_14 @ B_11 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_654_insert__inter__insert,axiom,
% 0.26/0.48 ! [A_13: nat,A_12: nat > $o,B_10: nat > $o] :
% 0.26/0.48 ( ( semila1947288293_nat_o @ ( insert_nat @ A_13 @ A_12 ) @ ( insert_nat @ A_13 @ B_10 ) )
% 0.26/0.48 = ( insert_nat @ A_13 @ ( semila1947288293_nat_o @ A_12 @ B_10 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_655_insert__inter__insert,axiom,
% 0.26/0.48 ! [A_13: hoare_1167836817_state,A_12: hoare_1167836817_state > $o,B_10: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( semila179895820tate_o @ ( insert2134838167_state @ A_13 @ A_12 ) @ ( insert2134838167_state @ A_13 @ B_10 ) )
% 0.26/0.48 = ( insert2134838167_state @ A_13 @ ( semila179895820tate_o @ A_12 @ B_10 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_656_insert__inter__insert,axiom,
% 0.26/0.48 ! [A_13: hoare_1775062406iple_a,A_12: hoare_1775062406iple_a > $o,B_10: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( semila966743401le_a_o @ ( insert1281456128iple_a @ A_13 @ A_12 ) @ ( insert1281456128iple_a @ A_13 @ B_10 ) )
% 0.26/0.48 = ( insert1281456128iple_a @ A_13 @ ( semila966743401le_a_o @ A_12 @ B_10 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_657_Int__insert__left,axiom,
% 0.26/0.48 ! [B_9: nat > $o,A_11: nat,C_3: nat > $o] :
% 0.26/0.48 ( ( ( member_nat @ A_11 @ C_3 )
% 0.26/0.48 => ( ( semila1947288293_nat_o @ ( insert_nat @ A_11 @ B_9 ) @ C_3 )
% 0.26/0.48 = ( insert_nat @ A_11 @ ( semila1947288293_nat_o @ B_9 @ C_3 ) ) ) )
% 0.26/0.48 & ( ~ ( member_nat @ A_11 @ C_3 )
% 0.26/0.48 => ( ( semila1947288293_nat_o @ ( insert_nat @ A_11 @ B_9 ) @ C_3 )
% 0.26/0.48 = ( semila1947288293_nat_o @ B_9 @ C_3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_658_Int__insert__left,axiom,
% 0.26/0.48 ! [B_9: hoare_1167836817_state > $o,A_11: hoare_1167836817_state,C_3: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( ( member2058392318_state @ A_11 @ C_3 )
% 0.26/0.48 => ( ( semila179895820tate_o @ ( insert2134838167_state @ A_11 @ B_9 ) @ C_3 )
% 0.26/0.48 = ( insert2134838167_state @ A_11 @ ( semila179895820tate_o @ B_9 @ C_3 ) ) ) )
% 0.26/0.48 & ( ~ ( member2058392318_state @ A_11 @ C_3 )
% 0.26/0.48 => ( ( semila179895820tate_o @ ( insert2134838167_state @ A_11 @ B_9 ) @ C_3 )
% 0.26/0.48 = ( semila179895820tate_o @ B_9 @ C_3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_659_Int__insert__left,axiom,
% 0.26/0.48 ! [B_9: hoare_1775062406iple_a > $o,A_11: hoare_1775062406iple_a,C_3: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( ( member2122167641iple_a @ A_11 @ C_3 )
% 0.26/0.48 => ( ( semila966743401le_a_o @ ( insert1281456128iple_a @ A_11 @ B_9 ) @ C_3 )
% 0.26/0.48 = ( insert1281456128iple_a @ A_11 @ ( semila966743401le_a_o @ B_9 @ C_3 ) ) ) )
% 0.26/0.48 & ( ~ ( member2122167641iple_a @ A_11 @ C_3 )
% 0.26/0.48 => ( ( semila966743401le_a_o @ ( insert1281456128iple_a @ A_11 @ B_9 ) @ C_3 )
% 0.26/0.48 = ( semila966743401le_a_o @ B_9 @ C_3 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_660_Int__insert__right,axiom,
% 0.26/0.48 ! [B_8: nat > $o,A_10: nat,A_9: nat > $o] :
% 0.26/0.48 ( ( ( member_nat @ A_10 @ A_9 )
% 0.26/0.48 => ( ( semila1947288293_nat_o @ A_9 @ ( insert_nat @ A_10 @ B_8 ) )
% 0.26/0.48 = ( insert_nat @ A_10 @ ( semila1947288293_nat_o @ A_9 @ B_8 ) ) ) )
% 0.26/0.48 & ( ~ ( member_nat @ A_10 @ A_9 )
% 0.26/0.48 => ( ( semila1947288293_nat_o @ A_9 @ ( insert_nat @ A_10 @ B_8 ) )
% 0.26/0.48 = ( semila1947288293_nat_o @ A_9 @ B_8 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_661_Int__insert__right,axiom,
% 0.26/0.48 ! [B_8: hoare_1167836817_state > $o,A_10: hoare_1167836817_state,A_9: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( ( member2058392318_state @ A_10 @ A_9 )
% 0.26/0.48 => ( ( semila179895820tate_o @ A_9 @ ( insert2134838167_state @ A_10 @ B_8 ) )
% 0.26/0.48 = ( insert2134838167_state @ A_10 @ ( semila179895820tate_o @ A_9 @ B_8 ) ) ) )
% 0.26/0.48 & ( ~ ( member2058392318_state @ A_10 @ A_9 )
% 0.26/0.48 => ( ( semila179895820tate_o @ A_9 @ ( insert2134838167_state @ A_10 @ B_8 ) )
% 0.26/0.48 = ( semila179895820tate_o @ A_9 @ B_8 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_662_Int__insert__right,axiom,
% 0.26/0.48 ! [B_8: hoare_1775062406iple_a > $o,A_10: hoare_1775062406iple_a,A_9: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( ( member2122167641iple_a @ A_10 @ A_9 )
% 0.26/0.48 => ( ( semila966743401le_a_o @ A_9 @ ( insert1281456128iple_a @ A_10 @ B_8 ) )
% 0.26/0.48 = ( insert1281456128iple_a @ A_10 @ ( semila966743401le_a_o @ A_9 @ B_8 ) ) ) )
% 0.26/0.48 & ( ~ ( member2122167641iple_a @ A_10 @ A_9 )
% 0.26/0.48 => ( ( semila966743401le_a_o @ A_9 @ ( insert1281456128iple_a @ A_10 @ B_8 ) )
% 0.26/0.48 = ( semila966743401le_a_o @ A_9 @ B_8 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_663_inf__Int__eq,axiom,
% 0.26/0.48 ! [R: hoare_1775062406iple_a > $o,S_1: hoare_1775062406iple_a > $o,X_3: hoare_1775062406iple_a] :
% 0.26/0.48 ( ( semila966743401le_a_o
% 0.26/0.48 @ ^ [Y_1: hoare_1775062406iple_a] : ( member2122167641iple_a @ Y_1 @ R )
% 0.26/0.48 @ ^ [Y_1: hoare_1775062406iple_a] : ( member2122167641iple_a @ Y_1 @ S_1 )
% 0.26/0.48 @ X_3 )
% 0.26/0.48 <=> ( member2122167641iple_a @ X_3 @ ( semila966743401le_a_o @ R @ S_1 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_664_inf__Int__eq,axiom,
% 0.26/0.48 ! [R: nat > $o,S_1: nat > $o,X_3: nat] :
% 0.26/0.48 ( ( semila1947288293_nat_o
% 0.26/0.48 @ ^ [Y_1: nat] : ( member_nat @ Y_1 @ R )
% 0.26/0.48 @ ^ [Y_1: nat] : ( member_nat @ Y_1 @ S_1 )
% 0.26/0.48 @ X_3 )
% 0.26/0.48 <=> ( member_nat @ X_3 @ ( semila1947288293_nat_o @ R @ S_1 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_665_Collect__conj__eq,axiom,
% 0.26/0.48 ! [P_1: hoare_1775062406iple_a > $o,Q: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( collec676402587iple_a
% 0.26/0.48 @ ^ [X_3: hoare_1775062406iple_a] : ( (&) @ ( P_1 @ X_3 ) @ ( Q @ X_3 ) ) )
% 0.26/0.48 = ( semila966743401le_a_o @ ( collec676402587iple_a @ P_1 ) @ ( collec676402587iple_a @ Q ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_666_Collect__conj__eq,axiom,
% 0.26/0.48 ! [P_1: nat > $o,Q: nat > $o] :
% 0.26/0.48 ( ( collect_nat
% 0.26/0.48 @ ^ [X_3: nat] : ( (&) @ ( P_1 @ X_3 ) @ ( Q @ X_3 ) ) )
% 0.26/0.48 = ( semila1947288293_nat_o @ ( collect_nat @ P_1 ) @ ( collect_nat @ Q ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_667_Int__Collect,axiom,
% 0.26/0.48 ! [X_4: hoare_1775062406iple_a,A_8: hoare_1775062406iple_a > $o,P: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( member2122167641iple_a @ X_4 @ ( semila966743401le_a_o @ A_8 @ ( collec676402587iple_a @ P ) ) )
% 0.26/0.48 <=> ( ( member2122167641iple_a @ X_4 @ A_8 )
% 0.26/0.48 & ( P @ X_4 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_668_Int__Collect,axiom,
% 0.26/0.48 ! [X_4: nat,A_8: nat > $o,P: nat > $o] :
% 0.26/0.48 ( ( member_nat @ X_4 @ ( semila1947288293_nat_o @ A_8 @ ( collect_nat @ P ) ) )
% 0.26/0.48 <=> ( ( member_nat @ X_4 @ A_8 )
% 0.26/0.48 & ( P @ X_4 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_669_Int__def,axiom,
% 0.26/0.48 ! [A_7: hoare_1775062406iple_a > $o,B_7: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( semila966743401le_a_o @ A_7 @ B_7 )
% 0.26/0.48 = ( collec676402587iple_a
% 0.26/0.48 @ ^ [X_3: hoare_1775062406iple_a] : ( (&) @ ( member2122167641iple_a @ X_3 @ A_7 ) @ ( member2122167641iple_a @ X_3 @ B_7 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_670_Int__def,axiom,
% 0.26/0.48 ! [A_7: nat > $o,B_7: nat > $o] :
% 0.26/0.48 ( ( semila1947288293_nat_o @ A_7 @ B_7 )
% 0.26/0.48 = ( collect_nat
% 0.26/0.48 @ ^ [X_3: nat] : ( (&) @ ( member_nat @ X_3 @ A_7 ) @ ( member_nat @ X_3 @ B_7 ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_671_Int__iff,axiom,
% 0.26/0.48 ! [C_2: hoare_1775062406iple_a,A_6: hoare_1775062406iple_a > $o,B_6: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( member2122167641iple_a @ C_2 @ ( semila966743401le_a_o @ A_6 @ B_6 ) )
% 0.26/0.48 <=> ( ( member2122167641iple_a @ C_2 @ A_6 )
% 0.26/0.48 & ( member2122167641iple_a @ C_2 @ B_6 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_672_Int__iff,axiom,
% 0.26/0.48 ! [C_2: nat,A_6: nat > $o,B_6: nat > $o] :
% 0.26/0.48 ( ( member_nat @ C_2 @ ( semila1947288293_nat_o @ A_6 @ B_6 ) )
% 0.26/0.48 <=> ( ( member_nat @ C_2 @ A_6 )
% 0.26/0.48 & ( member_nat @ C_2 @ B_6 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_673_IntD1,axiom,
% 0.26/0.48 ! [C_1: hoare_1775062406iple_a,A_5: hoare_1775062406iple_a > $o,B_5: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( member2122167641iple_a @ C_1 @ ( semila966743401le_a_o @ A_5 @ B_5 ) )
% 0.26/0.48 => ( member2122167641iple_a @ C_1 @ A_5 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_674_IntD1,axiom,
% 0.26/0.48 ! [C_1: nat,A_5: nat > $o,B_5: nat > $o] :
% 0.26/0.48 ( ( member_nat @ C_1 @ ( semila1947288293_nat_o @ A_5 @ B_5 ) )
% 0.26/0.48 => ( member_nat @ C_1 @ A_5 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_675_IntD2,axiom,
% 0.26/0.48 ! [C: hoare_1775062406iple_a,A_4: hoare_1775062406iple_a > $o,B_4: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( member2122167641iple_a @ C @ ( semila966743401le_a_o @ A_4 @ B_4 ) )
% 0.26/0.48 => ( member2122167641iple_a @ C @ B_4 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_676_IntD2,axiom,
% 0.26/0.48 ! [C: nat,A_4: nat > $o,B_4: nat > $o] :
% 0.26/0.48 ( ( member_nat @ C @ ( semila1947288293_nat_o @ A_4 @ B_4 ) )
% 0.26/0.48 => ( member_nat @ C @ B_4 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_677_disjoint__iff__not__equal,axiom,
% 0.26/0.48 ! [A_3: nat > $o,B_3: nat > $o] :
% 0.26/0.48 ( ( ( semila1947288293_nat_o @ A_3 @ B_3 )
% 0.26/0.48 = bot_bot_nat_o )
% 0.26/0.48 <=> ! [X_3: nat] :
% 0.26/0.48 ( ( member_nat @ X_3 @ A_3 )
% 0.26/0.48 => ! [Xa: nat] :
% 0.26/0.48 ( ( member_nat @ Xa @ B_3 )
% 0.26/0.48 => ( X_3 != Xa ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_678_disjoint__iff__not__equal,axiom,
% 0.26/0.48 ! [A_3: hoare_1167836817_state > $o,B_3: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( ( semila179895820tate_o @ A_3 @ B_3 )
% 0.26/0.48 = bot_bo70021908tate_o )
% 0.26/0.48 <=> ! [X_3: hoare_1167836817_state] :
% 0.26/0.48 ( ( member2058392318_state @ X_3 @ A_3 )
% 0.26/0.48 => ! [Xa: hoare_1167836817_state] :
% 0.26/0.48 ( ( member2058392318_state @ Xa @ B_3 )
% 0.26/0.48 => ( X_3 != Xa ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_679_disjoint__iff__not__equal,axiom,
% 0.26/0.48 ! [A_3: hoare_1775062406iple_a > $o,B_3: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( ( semila966743401le_a_o @ A_3 @ B_3 )
% 0.26/0.48 = bot_bo751897185le_a_o )
% 0.26/0.48 <=> ! [X_3: hoare_1775062406iple_a] :
% 0.26/0.48 ( ( member2122167641iple_a @ X_3 @ A_3 )
% 0.26/0.48 => ! [Xa: hoare_1775062406iple_a] :
% 0.26/0.48 ( ( member2122167641iple_a @ Xa @ B_3 )
% 0.26/0.48 => ( X_3 != Xa ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_680_Int__empty__right,axiom,
% 0.26/0.48 ! [A_2: nat > $o] :
% 0.26/0.48 ( ( semila1947288293_nat_o @ A_2 @ bot_bot_nat_o )
% 0.26/0.48 = bot_bot_nat_o ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_681_Int__empty__right,axiom,
% 0.26/0.48 ! [A_2: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( semila179895820tate_o @ A_2 @ bot_bo70021908tate_o )
% 0.26/0.48 = bot_bo70021908tate_o ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_682_Int__empty__right,axiom,
% 0.26/0.48 ! [A_2: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( semila966743401le_a_o @ A_2 @ bot_bo751897185le_a_o )
% 0.26/0.48 = bot_bo751897185le_a_o ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_683_Int__empty__left,axiom,
% 0.26/0.48 ! [B_2: nat > $o] :
% 0.26/0.48 ( ( semila1947288293_nat_o @ bot_bot_nat_o @ B_2 )
% 0.26/0.48 = bot_bot_nat_o ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_684_Int__empty__left,axiom,
% 0.26/0.48 ! [B_2: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( semila179895820tate_o @ bot_bo70021908tate_o @ B_2 )
% 0.26/0.48 = bot_bo70021908tate_o ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_685_Int__empty__left,axiom,
% 0.26/0.48 ! [B_2: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( semila966743401le_a_o @ bot_bo751897185le_a_o @ B_2 )
% 0.26/0.48 = bot_bo751897185le_a_o ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_686_inf__bot__left,axiom,
% 0.26/0.48 ! [X_2: $o] :
% 0.26/0.48 ( ( semila854092349_inf_o @ bot_bot_o @ X_2 )
% 0.26/0.48 <=> bot_bot_o ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_687_inf__bot__left,axiom,
% 0.26/0.48 ! [X_2: nat > $o] :
% 0.26/0.48 ( ( semila1947288293_nat_o @ bot_bot_nat_o @ X_2 )
% 0.26/0.48 = bot_bot_nat_o ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_688_inf__bot__left,axiom,
% 0.26/0.48 ! [X_2: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( semila179895820tate_o @ bot_bo70021908tate_o @ X_2 )
% 0.26/0.48 = bot_bo70021908tate_o ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_689_inf__bot__left,axiom,
% 0.26/0.48 ! [X_2: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( semila966743401le_a_o @ bot_bo751897185le_a_o @ X_2 )
% 0.26/0.48 = bot_bo751897185le_a_o ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_690_inf__bot__right,axiom,
% 0.26/0.48 ! [X_1: $o] :
% 0.26/0.48 ( ( semila854092349_inf_o @ X_1 @ bot_bot_o )
% 0.26/0.48 <=> bot_bot_o ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_691_inf__bot__right,axiom,
% 0.26/0.48 ! [X_1: nat > $o] :
% 0.26/0.48 ( ( semila1947288293_nat_o @ X_1 @ bot_bot_nat_o )
% 0.26/0.48 = bot_bot_nat_o ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_692_inf__bot__right,axiom,
% 0.26/0.48 ! [X_1: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( semila179895820tate_o @ X_1 @ bot_bo70021908tate_o )
% 0.26/0.48 = bot_bo70021908tate_o ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_693_inf__bot__right,axiom,
% 0.26/0.48 ! [X_1: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( semila966743401le_a_o @ X_1 @ bot_bo751897185le_a_o )
% 0.26/0.48 = bot_bo751897185le_a_o ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_694_Diff__triv,axiom,
% 0.26/0.48 ! [A_1: nat > $o,B_1: nat > $o] :
% 0.26/0.48 ( ( ( semila1947288293_nat_o @ A_1 @ B_1 )
% 0.26/0.48 = bot_bot_nat_o )
% 0.26/0.48 => ( ( minus_minus_nat_o @ A_1 @ B_1 )
% 0.26/0.48 = A_1 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_695_Diff__triv,axiom,
% 0.26/0.48 ! [A_1: hoare_1167836817_state > $o,B_1: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( ( semila179895820tate_o @ A_1 @ B_1 )
% 0.26/0.48 = bot_bo70021908tate_o )
% 0.26/0.48 => ( ( minus_2107060239tate_o @ A_1 @ B_1 )
% 0.26/0.48 = A_1 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_696_Diff__triv,axiom,
% 0.26/0.48 ! [A_1: hoare_1775062406iple_a > $o,B_1: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( ( semila966743401le_a_o @ A_1 @ B_1 )
% 0.26/0.48 = bot_bo751897185le_a_o )
% 0.26/0.48 => ( ( minus_1944206118le_a_o @ A_1 @ B_1 )
% 0.26/0.48 = A_1 ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_697_Diff__disjoint,axiom,
% 0.26/0.48 ! [A: nat > $o,B: nat > $o] :
% 0.26/0.48 ( ( semila1947288293_nat_o @ A @ ( minus_minus_nat_o @ B @ A ) )
% 0.26/0.48 = bot_bot_nat_o ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_698_Diff__disjoint,axiom,
% 0.26/0.48 ! [A: hoare_1167836817_state > $o,B: hoare_1167836817_state > $o] :
% 0.26/0.48 ( ( semila179895820tate_o @ A @ ( minus_2107060239tate_o @ B @ A ) )
% 0.26/0.48 = bot_bo70021908tate_o ) ).
% 0.26/0.48
% 0.26/0.48 thf(fact_699_Diff__disjoint,axiom,
% 0.26/0.48 ! [A: hoare_1775062406iple_a > $o,B: hoare_1775062406iple_a > $o] :
% 0.26/0.48 ( ( semila966743401le_a_o @ A @ ( minus_1944206118le_a_o @ B @ A ) )
% 0.26/0.48 = bot_bo751897185le_a_o ) ).
% 0.26/0.48
% 0.26/0.48 %----Helper facts (10)
% 0.26/0.48 thf(help_fequal_1_1_fequal_000tc__Nat__Onat_T,axiom,
% 0.26/0.48 ! [X: nat,Y: nat] :
% 0.26/0.48 ( ~ ( fequal_nat @ X @ Y )
% 0.26/0.48 | ( X = Y ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(help_fequal_2_1_fequal_000tc__Nat__Onat_T,axiom,
% 0.26/0.48 ! [X: nat,Y: nat] :
% 0.26/0.48 ( ( X != Y )
% 0.26/0.48 | ( fequal_nat @ X @ Y ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(help_fequal_1_1_fequal_000tc__Com__Ostate_T,axiom,
% 0.26/0.48 ! [X: state,Y: state] :
% 0.26/0.48 ( ~ ( fequal_state @ X @ Y )
% 0.26/0.48 | ( X = Y ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(help_fequal_2_1_fequal_000tc__Com__Ostate_T,axiom,
% 0.26/0.48 ! [X: state,Y: state] :
% 0.26/0.48 ( ( X != Y )
% 0.26/0.48 | ( fequal_state @ X @ Y ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(help_fequal_1_1_fequal_000_062_Itc__Nat__Onat_M_Eo_J_T,axiom,
% 0.26/0.48 ! [X: nat > $o,Y: nat > $o] :
% 0.26/0.48 ( ~ ( fequal_nat_o @ X @ Y )
% 0.26/0.48 | ( X = Y ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(help_fequal_2_1_fequal_000_062_Itc__Nat__Onat_M_Eo_J_T,axiom,
% 0.26/0.48 ! [X: nat > $o,Y: nat > $o] :
% 0.26/0.48 ( ( X != Y )
% 0.26/0.48 | ( fequal_nat_o @ X @ Y ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(help_fequal_1_1_fequal_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J_,axiom,
% 0.26/0.48 ! [X: hoare_1775062406iple_a,Y: hoare_1775062406iple_a] :
% 0.26/0.48 ( ~ ( fequal1288209029iple_a @ X @ Y )
% 0.26/0.48 | ( X = Y ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(help_fequal_2_1_fequal_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J_,axiom,
% 0.26/0.48 ! [X: hoare_1775062406iple_a,Y: hoare_1775062406iple_a] :
% 0.26/0.48 ( ( X != Y )
% 0.26/0.48 | ( fequal1288209029iple_a @ X @ Y ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(help_fequal_1_1_fequal_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com,axiom,
% 0.26/0.48 ! [X: hoare_1167836817_state,Y: hoare_1167836817_state] :
% 0.26/0.48 ( ~ ( fequal1831255762_state @ X @ Y )
% 0.26/0.48 | ( X = Y ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(help_fequal_2_1_fequal_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com,axiom,
% 0.26/0.48 ! [X: hoare_1167836817_state,Y: hoare_1167836817_state] :
% 0.26/0.48 ( ( X != Y )
% 0.26/0.48 | ( fequal1831255762_state @ X @ Y ) ) ).
% 0.26/0.48
% 0.26/0.48 %----Conjectures (1)
% 0.26/0.48 thf(conj_0,conjecture,
% 0.26/0.48 ( hoare_1508237396rivs_a @ g
% 0.26/0.48 @ ( insert1281456128iple_a
% 0.26/0.48 @ ( hoare_1766022166iple_a
% 0.26/0.48 @ ^ [Z: x_a,S: state] : $false
% 0.26/0.48 @ c
% 0.26/0.48 @ ^ [Z: x_a,S: state] : ( (&) @ ( p @ Z @ S ) @ ( (~) @ ( b @ S ) ) ) )
% 0.26/0.48 @ bot_bo751897185le_a_o ) ) ).
% 0.26/0.48
% 0.26/0.48 %------------------------------------------------------------------------------
% 0.26/0.48 ------------------------
% 3.31/1.61 ---- Embedded in THF ---
% 3.31/1.61 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.31/1.61 %%% Generated on Tue Feb 24 13:39:49 EST 2026
% 3.31/1.61 %%% using '$$dhol' embedding, version 1.3.0.
% 3.31/1.61 %%% Logic specification used:
% 3.31/1.61 %%% thf(spec, logic, $$dhol).
% 3.31/1.61
% 3.31/1.61 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.78tw4nyiuM/DTF2DTF30168.p
% See solution above
% 3.31/1.62 ------------------------
% 3.31/1.66 ---- Cleaned THF ---
% 3.31/1.67 thf(ty_ty_t__a,type,
% 3.31/1.67 x_a: $tType ).
% 3.31/1.67
% 3.31/1.67 thf(ty_ty_tc__Com__Ocom,type,
% 3.31/1.67 com: $tType ).
% 3.31/1.67
% 3.31/1.67 thf(ty_ty_tc__Com__Oglb,type,
% 3.31/1.67 glb: $tType ).
% 3.31/1.67
% 3.31/1.67 thf(ty_ty_tc__Com__Oloc,type,
% 3.31/1.67 loc: $tType ).
% 3.31/1.67
% 3.31/1.67 thf(ty_ty_tc__Com__Ostate,type,
% 3.31/1.67 state: $tType ).
% 3.31/1.67
% 3.31/1.67 thf(ty_ty_tc__Com__Ovname,type,
% 3.31/1.67 vname: $tType ).
% 3.31/1.67
% 3.31/1.67 thf(ty_ty_tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J,type,
% 3.31/1.67 hoare_1775062406iple_a: $tType ).
% 3.31/1.67
% 3.31/1.67 thf(ty_ty_tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostate_J,type,
% 3.31/1.67 hoare_1167836817_state: $tType ).
% 3.31/1.67
% 3.31/1.67 thf(ty_ty_tc__Nat__Onat,type,
% 3.31/1.67 nat: $tType ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Big__Operators_Olattice__class_OSup__fin_000tc__Nat__Onat,type,
% 3.31/1.67 big_la43341705in_nat: ( nat > $o ) > nat ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Big__Operators_Osemilattice__big_000tc__Nat__Onat,type,
% 3.31/1.67 big_se275732192ig_nat: ( nat > nat > nat ) > ( ( nat > $o ) > nat ) > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Com_Ocom_OAss,type,
% 3.31/1.67 ass: vname > ( state > nat ) > com ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Com_Ocom_OLocal,type,
% 3.31/1.67 local: loc > ( state > nat ) > com > com ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Com_Ocom_OSKIP,type,
% 3.31/1.67 skip: com ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Com_Ocom_OSemi,type,
% 3.31/1.67 semi: com > com > com ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Com_Ovname_OGlb,type,
% 3.31/1.67 glb_1: glb > vname ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Com_Ovname_OLoc,type,
% 3.31/1.67 loc_1: loc > vname ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Finite__Set_Ocomp__fun__commute_000tc__Hoare____Mirabelle____srushsumbx__Ot,type,
% 3.31/1.67 finite2064891473iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Finite__Set_Ocomp__fun__idem_000tc__Hoare____Mirabelle____srushsumbx__Otrip,type,
% 3.31/1.67 finite2120172977le_a_o: ( hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ) > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Finite__Set_Ocomp__fun__idem_000tc__Hoare____Mirabelle____srushsumbx__Otrip_001,type,
% 3.31/1.67 finite856902323tate_o: ( hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ) > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Finite__Set_Ocomp__fun__idem_000tc__Nat__Onat_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 3.31/1.67 finite389864113_nat_o: ( nat > ( nat > $o ) > nat > $o ) > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Finite__Set_Ofinite_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 3.31/1.67 finite_finite_nat_o: ( ( nat > $o ) > $o ) > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Finite__Set_Ofinite_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_,type,
% 3.31/1.67 finite2063573081iple_a: ( hoare_1775062406iple_a > $o ) > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Finite__Set_Ofinite_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__C,type,
% 3.31/1.67 finite1084549118_state: ( hoare_1167836817_state > $o ) > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Finite__Set_Ofinite_000tc__Nat__Onat,type,
% 3.31/1.67 finite_finite_nat: ( nat > $o ) > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Finite__Set_Ofold1Set_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__,type,
% 3.31/1.67 finite1946188886iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Finite__Set_Ofold1Set_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc_,type,
% 3.31/1.67 finite309220289_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Finite__Set_Ofold1Set_000tc__Nat__Onat,type,
% 3.31/1.67 finite_fold1Set_nat: ( nat > nat > nat ) > ( nat > $o ) > nat > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Finite__Set_Ofold1_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J,type,
% 3.31/1.67 finite1790765286iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Finite__Set_Ofold1_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Co,type,
% 3.31/1.67 finite1646097201_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Finite__Set_Ofold1_000tc__Nat__Onat,type,
% 3.31/1.67 finite_fold1_nat: ( nat > nat > nat ) > ( nat > $o ) > nat ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Finite__Set_Ofold_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J_,type,
% 3.31/1.67 finite1544171829le_a_o: ( hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Finite__Set_Ofold_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J__002,type,
% 3.31/1.67 finite1842721992iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Finite__Set_Ofold_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com,type,
% 3.31/1.67 finite291020855tate_o: ( hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Finite__Set_Ofold_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com_003,type,
% 3.31/1.67 finite1731015960_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Finite__Set_Ofold_000tc__Nat__Onat_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 3.31/1.67 finite326637109_nat_o: ( nat > ( nat > $o ) > nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Finite__Set_Ofold_000tc__Nat__Onat_000tc__Nat__Onat,type,
% 3.31/1.67 finite_fold_nat_nat: ( nat > nat > nat ) > nat > ( nat > $o ) > nat ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Finite__Set_Ofold__graph_000tc__Hoare____Mirabelle____srushsumbx__Otriple_I,type,
% 3.31/1.67 finite727644230iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Finite__Set_Ofold__graph_000tc__Hoare____Mirabelle____srushsumbx__Otriple_I_004,type,
% 3.31/1.67 finite1316643734_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Finite__Set_Ofold__graph_000tc__Nat__Onat_000tc__Nat__Onat,type,
% 3.31/1.67 finite929467206at_nat: ( nat > nat > nat ) > nat > ( nat > $o ) > nat > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Finite__Set_Ofolding__one_000tc__Hoare____Mirabelle____srushsumbx__Otriple_,type,
% 3.31/1.67 finite2078349315iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > ( ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ) > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Finite__Set_Ofolding__one_000tc__Hoare____Mirabelle____srushsumbx__Otriple__005,type,
% 3.31/1.67 finite1074406356_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > ( ( hoare_1167836817_state > $o ) > hoare_1167836817_state ) > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Finite__Set_Ofolding__one_000tc__Nat__Onat,type,
% 3.31/1.67 finite988810631ne_nat: ( nat > nat > nat ) > ( ( nat > $o ) > nat ) > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Finite__Set_Ofolding__one__idem_000tc__Hoare____Mirabelle____srushsumbx__Ot,type,
% 3.31/1.67 finite1358382848iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a ) > ( ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ) > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Finite__Set_Ofolding__one__idem_000tc__Hoare____Mirabelle____srushsumbx__Ot_006,type,
% 3.31/1.67 finite806517911_state: ( hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state ) > ( ( hoare_1167836817_state > $o ) > hoare_1167836817_state ) > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Finite__Set_Ofolding__one__idem_000tc__Nat__Onat,type,
% 3.31/1.67 finite795500164em_nat: ( nat > nat > nat ) > ( ( nat > $o ) > nat ) > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Hoare____Mirabelle____srushsumbx__,type,
% 3.31/1.67 minus_1944206118le_a_o: ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Hoare____Mirabelle____srushsumbx___007,type,
% 3.31/1.67 minus_2107060239tate_o: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 3.31/1.67 minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Groups_Otimes__class_Otimes_000tc__Nat__Onat,type,
% 3.31/1.67 times_times_nat: nat > nat > nat ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_HOL_OThe_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J,type,
% 3.31/1.67 the_Ho1155011127iple_a: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_HOL_OThe_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostate_,type,
% 3.31/1.67 the_Ho310147232_state: ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_HOL_OThe_000tc__Nat__Onat,type,
% 3.31/1.67 the_nat: ( nat > $o ) > nat ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Hoare__Mirabelle__srushsumbx_OMGT,type,
% 3.31/1.67 hoare_Mirabelle_MGT: com > hoare_1167836817_state ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Hoare__Mirabelle__srushsumbx_Ohoare__derivs_000t__a,type,
% 3.31/1.67 hoare_1508237396rivs_a: ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Hoare__Mirabelle__srushsumbx_Ohoare__derivs_000tc__Com__Ostate,type,
% 3.31/1.67 hoare_123228589_state: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Hoare__Mirabelle__srushsumbx_Otriple_Otriple_000t__a,type,
% 3.31/1.67 hoare_1766022166iple_a: ( x_a > state > $o ) > com > ( x_a > state > $o ) > hoare_1775062406iple_a ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Hoare__Mirabelle__srushsumbx_Otriple_Otriple_000tc__Com__Ostate,type,
% 3.31/1.67 hoare_908217195_state: ( state > state > $o ) > com > ( state > state > $o ) > hoare_1167836817_state ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Hoare__Mirabelle__srushsumbx_Otriple__valid_000t__a,type,
% 3.31/1.67 hoare_1462269968alid_a: nat > hoare_1775062406iple_a > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Hoare__Mirabelle__srushsumbx_Otriple__valid_000tc__Com__Ostate,type,
% 3.31/1.67 hoare_56934129_state: nat > hoare_1167836817_state > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Lattices_Osemilattice__inf__class_Oinf_000_062_Itc__Hoare____Mirabelle____s,type,
% 3.31/1.67 semila966743401le_a_o: ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Lattices_Osemilattice__inf__class_Oinf_000_062_Itc__Hoare____Mirabelle____s_008,type,
% 3.31/1.67 semila179895820tate_o: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Lattices_Osemilattice__inf__class_Oinf_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 3.31/1.67 semila1947288293_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Lattices_Osemilattice__inf__class_Oinf_000_Eo,type,
% 3.31/1.67 semila854092349_inf_o: $o > $o > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Lattices_Osemilattice__inf__class_Oinf_000tc__Nat__Onat,type,
% 3.31/1.67 semila80283416nf_nat: nat > nat > nat ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_062_Itc__Hoare____Mirabelle____s,type,
% 3.31/1.67 semila13410563le_a_o: ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_062_Itc__Hoare____Mirabelle____s_009,type,
% 3.31/1.67 semila1172322802tate_o: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 3.31/1.67 semila848761471_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_Eo,type,
% 3.31/1.67 semila10642723_sup_o: $o > $o > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000tc__Nat__Onat,type,
% 3.31/1.67 semila972727038up_nat: nat > nat > nat ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Natural_Oevalc,type,
% 3.31/1.67 evalc: com > state > state > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Natural_Oevaln,type,
% 3.31/1.67 evaln: com > state > nat > state > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Natural_Ogetlocs,type,
% 3.31/1.67 getlocs: state > loc > nat ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Natural_Oupdate,type,
% 3.31/1.67 update: state > vname > nat > state ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Orderings_Obot__class_Obot_000_062_I_062_Itc__Nat__Onat_M_Eo_J_M_Eo_J,type,
% 3.31/1.67 bot_bot_nat_o_o: ( nat > $o ) > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Hoare____Mirabelle____srushsumbx__O,type,
% 3.31/1.67 bot_bo751897185le_a_o: hoare_1775062406iple_a > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Hoare____Mirabelle____srushsumbx__O_010,type,
% 3.31/1.67 bot_bo70021908tate_o: hoare_1167836817_state > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 3.31/1.67 bot_bot_nat_o: nat > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Orderings_Obot__class_Obot_000_Eo,type,
% 3.31/1.67 bot_bot_o: $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Orderings_Obot__class_Obot_000tc__Nat__Onat,type,
% 3.31/1.67 bot_bot_nat: nat ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Hoare____Mirabelle____srushsum,type,
% 3.31/1.67 ord_le1143225901le_a_o: ( hoare_1775062406iple_a > $o ) > ( hoare_1775062406iple_a > $o ) > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Hoare____Mirabelle____srushsum_011,type,
% 3.31/1.67 ord_le827224136tate_o: ( hoare_1167836817_state > $o ) > ( hoare_1167836817_state > $o ) > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 3.31/1.67 ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Orderings_Oord__class_Oless__eq_000_Eo,type,
% 3.31/1.67 ord_less_eq_o: $o > $o > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__Nat__Onat,type,
% 3.31/1.67 ord_less_eq_nat: nat > nat > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Partial__Function_Oflat__lub_000tc__Hoare____Mirabelle____srushsumbx__Otrip,type,
% 3.31/1.67 partia126998524iple_a: hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Partial__Function_Oflat__lub_000tc__Hoare____Mirabelle____srushsumbx__Otrip_012,type,
% 3.31/1.67 partia715677851_state: hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Partial__Function_Oflat__lub_000tc__Nat__Onat,type,
% 3.31/1.67 partial_flat_lub_nat: nat > ( nat > $o ) > nat ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Set_OCollect_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 3.31/1.67 collect_nat_o: ( ( nat > $o ) > $o ) > ( nat > $o ) > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Set_OCollect_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J,type,
% 3.31/1.67 collec676402587iple_a: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Set_OCollect_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ost,type,
% 3.31/1.67 collec1027672124_state: ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Set_OCollect_000tc__Nat__Onat,type,
% 3.31/1.67 collect_nat: ( nat > $o ) > nat > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J_000tc__,type,
% 3.31/1.67 image_1170193413iple_a: ( hoare_1775062406iple_a > hoare_1775062406iple_a ) > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J_000tc___013,type,
% 3.31/1.67 image_1021683026_state: ( hoare_1775062406iple_a > hoare_1167836817_state ) > ( hoare_1775062406iple_a > $o ) > hoare_1167836817_state > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J_000tc___014,type,
% 3.31/1.67 image_1806517641_a_nat: ( hoare_1775062406iple_a > nat ) > ( hoare_1775062406iple_a > $o ) > nat > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostat,type,
% 3.31/1.67 image_1802845250iple_a: ( hoare_1167836817_state > hoare_1775062406iple_a ) > ( hoare_1167836817_state > $o ) > hoare_1775062406iple_a > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostat_015,type,
% 3.31/1.67 image_31595733_state: ( hoare_1167836817_state > hoare_1167836817_state ) > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostat_016,type,
% 3.31/1.67 image_1476618182te_nat: ( hoare_1167836817_state > nat ) > ( hoare_1167836817_state > $o ) > nat > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Hoare____Mirabelle____srushsumbx__Otripl,type,
% 3.31/1.67 image_43014529iple_a: ( nat > hoare_1775062406iple_a ) > ( nat > $o ) > hoare_1775062406iple_a > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Hoare____Mirabelle____srushsumbx__Otripl_017,type,
% 3.31/1.67 image_2121260246_state: ( nat > hoare_1167836817_state ) > ( nat > $o ) > hoare_1167836817_state > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Nat__Onat,type,
% 3.31/1.67 image_nat_nat: ( nat > nat ) > ( nat > $o ) > nat > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Set_Oinsert_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 3.31/1.67 insert_nat_o: ( nat > $o ) > ( ( nat > $o ) > $o ) > ( nat > $o ) > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Set_Oinsert_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J,type,
% 3.31/1.67 insert1281456128iple_a: hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Set_Oinsert_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Osta,type,
% 3.31/1.67 insert2134838167_state: hoare_1167836817_state > ( hoare_1167836817_state > $o ) > hoare_1167836817_state > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Set_Oinsert_000tc__Nat__Onat,type,
% 3.31/1.67 insert_nat: nat > ( nat > $o ) > nat > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Set_Othe__elem_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J,type,
% 3.31/1.67 the_el1844711461iple_a: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Set_Othe__elem_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__O,type,
% 3.31/1.67 the_el323660082_state: ( hoare_1167836817_state > $o ) > hoare_1167836817_state ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_Set_Othe__elem_000tc__Nat__Onat,type,
% 3.31/1.67 the_elem_nat: ( nat > $o ) > nat ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_fequal_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 3.31/1.67 fequal_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_fequal_000tc__Com__Ostate,type,
% 3.31/1.67 fequal_state: state > state > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_fequal_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J,type,
% 3.31/1.67 fequal1288209029iple_a: hoare_1775062406iple_a > hoare_1775062406iple_a > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_fequal_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostate_J,type,
% 3.31/1.67 fequal1831255762_state: hoare_1167836817_state > hoare_1167836817_state > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_fequal_000tc__Nat__Onat,type,
% 3.31/1.67 fequal_nat: nat > nat > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_member_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 3.31/1.67 member_nat_o: ( nat > $o ) > ( ( nat > $o ) > $o ) > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_member_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J,type,
% 3.31/1.67 member2122167641iple_a: hoare_1775062406iple_a > ( hoare_1775062406iple_a > $o ) > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_member_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com__Ostate_J,type,
% 3.31/1.67 member2058392318_state: hoare_1167836817_state > ( hoare_1167836817_state > $o ) > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_c_member_000tc__Nat__Onat,type,
% 3.31/1.67 member_nat: nat > ( nat > $o ) > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_v_G,type,
% 3.31/1.67 g: hoare_1775062406iple_a > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_v_P,type,
% 3.31/1.67 p: x_a > state > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_v_b,type,
% 3.31/1.67 b: state > $o ).
% 3.31/1.67
% 3.31/1.67 thf(sy_v_c,type,
% 3.31/1.67 c: com ).
% 3.31/1.67
% 3.31/1.67 thf(fact_0_empty,axiom,
% 3.31/1.67 ! [G_19: hoare_1775062406iple_a > $o] : ( hoare_1508237396rivs_a @ G_19 @ bot_bo751897185le_a_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_1_empty,axiom,
% 3.31/1.67 ! [G_19: hoare_1167836817_state > $o] : ( hoare_123228589_state @ G_19 @ bot_bo70021908tate_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_2_triple_Oinject,axiom,
% 3.31/1.67 ! [Fun1_2: state > state > $o,Com_4: com,Fun2_2: state > state > $o,Fun1_1: state > state > $o,Com_3: com,Fun2_1: state > state > $o] :
% 3.31/1.67 ( ( ( hoare_908217195_state @ Fun1_2 @ Com_4 @ Fun2_2 )
% 3.31/1.67 = ( hoare_908217195_state @ Fun1_1 @ Com_3 @ Fun2_1 ) )
% 3.31/1.67 <=> ( ( Fun1_2 = Fun1_1 )
% 3.31/1.67 & ( Com_4 = Com_3 )
% 3.31/1.67 & ( Fun2_2 = Fun2_1 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_3_triple_Oinject,axiom,
% 3.31/1.67 ! [Fun1_2: x_a > state > $o,Com_4: com,Fun2_2: x_a > state > $o,Fun1_1: x_a > state > $o,Com_3: com,Fun2_1: x_a > state > $o] :
% 3.31/1.67 ( ( ( hoare_1766022166iple_a @ Fun1_2 @ Com_4 @ Fun2_2 )
% 3.31/1.67 = ( hoare_1766022166iple_a @ Fun1_1 @ Com_3 @ Fun2_1 ) )
% 3.31/1.67 <=> ( ( Fun1_2 = Fun1_1 )
% 3.31/1.67 & ( Com_4 = Com_3 )
% 3.31/1.67 & ( Fun2_2 = Fun2_1 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_4_cut,axiom,
% 3.31/1.67 ! [G_18: hoare_1775062406iple_a > $o,G_17: hoare_1775062406iple_a > $o,Ts_4: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( hoare_1508237396rivs_a @ G_17 @ Ts_4 )
% 3.31/1.67 => ( ( hoare_1508237396rivs_a @ G_18 @ G_17 )
% 3.31/1.67 => ( hoare_1508237396rivs_a @ G_18 @ Ts_4 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_5_cut,axiom,
% 3.31/1.67 ! [G_18: hoare_1167836817_state > $o,G_17: hoare_1167836817_state > $o,Ts_4: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( hoare_123228589_state @ G_17 @ Ts_4 )
% 3.31/1.67 => ( ( hoare_123228589_state @ G_18 @ G_17 )
% 3.31/1.67 => ( hoare_123228589_state @ G_18 @ Ts_4 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_6_hoare__derivs_Oinsert,axiom,
% 3.31/1.67 ! [Ts_3: hoare_1775062406iple_a > $o,G_16: hoare_1775062406iple_a > $o,T_1: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( hoare_1508237396rivs_a @ G_16 @ ( insert1281456128iple_a @ T_1 @ bot_bo751897185le_a_o ) )
% 3.31/1.67 => ( ( hoare_1508237396rivs_a @ G_16 @ Ts_3 )
% 3.31/1.67 => ( hoare_1508237396rivs_a @ G_16 @ ( insert1281456128iple_a @ T_1 @ Ts_3 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_7_hoare__derivs_Oinsert,axiom,
% 3.31/1.67 ! [Ts_3: hoare_1167836817_state > $o,G_16: hoare_1167836817_state > $o,T_1: hoare_1167836817_state] :
% 3.31/1.67 ( ( hoare_123228589_state @ G_16 @ ( insert2134838167_state @ T_1 @ bot_bo70021908tate_o ) )
% 3.31/1.67 => ( ( hoare_123228589_state @ G_16 @ Ts_3 )
% 3.31/1.67 => ( hoare_123228589_state @ G_16 @ ( insert2134838167_state @ T_1 @ Ts_3 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_8_constant,axiom,
% 3.31/1.67 ! [G_15: hoare_1775062406iple_a > $o,P_33: x_a > state > $o,C_42: com,Q_15: x_a > state > $o,C_41: $o] :
% 3.31/1.67 ( ( C_41
% 3.31/1.67 => ( hoare_1508237396rivs_a @ G_15 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_33 @ C_42 @ Q_15 ) @ bot_bo751897185le_a_o ) ) )
% 3.31/1.67 => ( hoare_1508237396rivs_a @ G_15
% 3.31/1.67 @ ( insert1281456128iple_a
% 3.31/1.67 @ ( hoare_1766022166iple_a
% 3.31/1.67 @ ^ [Z: x_a,S: state] : ( (&) @ ( P_33 @ Z @ S ) @ C_41 )
% 3.31/1.67 @ C_42
% 3.31/1.67 @ Q_15 )
% 3.31/1.67 @ bot_bo751897185le_a_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_9_constant,axiom,
% 3.31/1.67 ! [G_15: hoare_1167836817_state > $o,P_33: state > state > $o,C_42: com,Q_15: state > state > $o,C_41: $o] :
% 3.31/1.67 ( ( C_41
% 3.31/1.67 => ( hoare_123228589_state @ G_15 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_33 @ C_42 @ Q_15 ) @ bot_bo70021908tate_o ) ) )
% 3.31/1.67 => ( hoare_123228589_state @ G_15
% 3.31/1.67 @ ( insert2134838167_state
% 3.31/1.67 @ ( hoare_908217195_state
% 3.31/1.67 @ ^ [Z: state,S: state] : ( (&) @ ( P_33 @ Z @ S ) @ C_41 )
% 3.31/1.67 @ C_42
% 3.31/1.67 @ Q_15 )
% 3.31/1.67 @ bot_bo70021908tate_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_10_escape,axiom,
% 3.31/1.67 ! [G_14: hoare_1775062406iple_a > $o,C_40: com,Q_14: x_a > state > $o,P_32: x_a > state > $o] :
% 3.31/1.67 ( ! [Z: x_a,S: state] :
% 3.31/1.67 ( ( P_32 @ Z @ S )
% 3.31/1.67 => ( hoare_1508237396rivs_a @ G_14
% 3.31/1.67 @ ( insert1281456128iple_a
% 3.31/1.67 @ ( hoare_1766022166iple_a
% 3.31/1.67 @ ^ [Za: x_a,S_5: state] : ( S_5 = S )
% 3.31/1.67 @ C_40
% 3.31/1.67 @ ^ [Z_23: x_a] : ( Q_14 @ Z ) )
% 3.31/1.67 @ bot_bo751897185le_a_o ) ) )
% 3.31/1.67 => ( hoare_1508237396rivs_a @ G_14 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_32 @ C_40 @ Q_14 ) @ bot_bo751897185le_a_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_11_escape,axiom,
% 3.31/1.67 ! [G_14: hoare_1167836817_state > $o,C_40: com,Q_14: state > state > $o,P_32: state > state > $o] :
% 3.31/1.67 ( ! [Z: state,S: state] :
% 3.31/1.67 ( ( P_32 @ Z @ S )
% 3.31/1.67 => ( hoare_123228589_state @ G_14
% 3.31/1.67 @ ( insert2134838167_state
% 3.31/1.67 @ ( hoare_908217195_state
% 3.31/1.67 @ ^ [Za: state,S_5: state] : ( S_5 = S )
% 3.31/1.67 @ C_40
% 3.31/1.67 @ ^ [Z_23: state] : ( Q_14 @ Z ) )
% 3.31/1.67 @ bot_bo70021908tate_o ) ) )
% 3.31/1.67 => ( hoare_123228589_state @ G_14 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_32 @ C_40 @ Q_14 ) @ bot_bo70021908tate_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_12_conseq2,axiom,
% 3.31/1.67 ! [Q_13: x_a > state > $o,G_13: hoare_1775062406iple_a > $o,P_31: x_a > state > $o,C_39: com,Q_12: x_a > state > $o] :
% 3.31/1.67 ( ( hoare_1508237396rivs_a @ G_13 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_31 @ C_39 @ Q_12 ) @ bot_bo751897185le_a_o ) )
% 3.31/1.67 => ( ! [Z: x_a,S: state] :
% 3.31/1.67 ( ( Q_12 @ Z @ S )
% 3.31/1.67 => ( Q_13 @ Z @ S ) )
% 3.31/1.67 => ( hoare_1508237396rivs_a @ G_13 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_31 @ C_39 @ Q_13 ) @ bot_bo751897185le_a_o ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_13_conseq2,axiom,
% 3.31/1.67 ! [Q_13: state > state > $o,G_13: hoare_1167836817_state > $o,P_31: state > state > $o,C_39: com,Q_12: state > state > $o] :
% 3.31/1.67 ( ( hoare_123228589_state @ G_13 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_31 @ C_39 @ Q_12 ) @ bot_bo70021908tate_o ) )
% 3.31/1.67 => ( ! [Z: state,S: state] :
% 3.31/1.67 ( ( Q_12 @ Z @ S )
% 3.31/1.67 => ( Q_13 @ Z @ S ) )
% 3.31/1.67 => ( hoare_123228589_state @ G_13 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_31 @ C_39 @ Q_13 ) @ bot_bo70021908tate_o ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_14_conseq1,axiom,
% 3.31/1.67 ! [P_30: x_a > state > $o,G_12: hoare_1775062406iple_a > $o,P_29: x_a > state > $o,C_38: com,Q_11: x_a > state > $o] :
% 3.31/1.67 ( ( hoare_1508237396rivs_a @ G_12 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_29 @ C_38 @ Q_11 ) @ bot_bo751897185le_a_o ) )
% 3.31/1.67 => ( ! [Z: x_a,S: state] :
% 3.31/1.67 ( ( P_30 @ Z @ S )
% 3.31/1.67 => ( P_29 @ Z @ S ) )
% 3.31/1.67 => ( hoare_1508237396rivs_a @ G_12 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_30 @ C_38 @ Q_11 ) @ bot_bo751897185le_a_o ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_15_conseq1,axiom,
% 3.31/1.67 ! [P_30: state > state > $o,G_12: hoare_1167836817_state > $o,P_29: state > state > $o,C_38: com,Q_11: state > state > $o] :
% 3.31/1.67 ( ( hoare_123228589_state @ G_12 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_29 @ C_38 @ Q_11 ) @ bot_bo70021908tate_o ) )
% 3.31/1.67 => ( ! [Z: state,S: state] :
% 3.31/1.67 ( ( P_30 @ Z @ S )
% 3.31/1.67 => ( P_29 @ Z @ S ) )
% 3.31/1.67 => ( hoare_123228589_state @ G_12 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_30 @ C_38 @ Q_11 ) @ bot_bo70021908tate_o ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_16_conseq12,axiom,
% 3.31/1.67 ! [Q_10: x_a > state > $o,P_28: x_a > state > $o,G_11: hoare_1775062406iple_a > $o,P_27: x_a > state > $o,C_37: com,Q_9: x_a > state > $o] :
% 3.31/1.67 ( ( hoare_1508237396rivs_a @ G_11 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_27 @ C_37 @ Q_9 ) @ bot_bo751897185le_a_o ) )
% 3.31/1.67 => ( ! [Z: x_a,S: state] :
% 3.31/1.67 ( ( P_28 @ Z @ S )
% 3.31/1.67 => ! [S_5: state] :
% 3.31/1.67 ( ! [Z_23: x_a] :
% 3.31/1.67 ( ( P_27 @ Z_23 @ S )
% 3.31/1.67 => ( Q_9 @ Z_23 @ S_5 ) )
% 3.31/1.67 => ( Q_10 @ Z @ S_5 ) ) )
% 3.31/1.67 => ( hoare_1508237396rivs_a @ G_11 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_28 @ C_37 @ Q_10 ) @ bot_bo751897185le_a_o ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_17_conseq12,axiom,
% 3.31/1.67 ! [Q_10: state > state > $o,P_28: state > state > $o,G_11: hoare_1167836817_state > $o,P_27: state > state > $o,C_37: com,Q_9: state > state > $o] :
% 3.31/1.67 ( ( hoare_123228589_state @ G_11 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_27 @ C_37 @ Q_9 ) @ bot_bo70021908tate_o ) )
% 3.31/1.67 => ( ! [Z: state,S: state] :
% 3.31/1.67 ( ( P_28 @ Z @ S )
% 3.31/1.67 => ! [S_5: state] :
% 3.31/1.67 ( ! [Z_23: state] :
% 3.31/1.67 ( ( P_27 @ Z_23 @ S )
% 3.31/1.67 => ( Q_9 @ Z_23 @ S_5 ) )
% 3.31/1.67 => ( Q_10 @ Z @ S_5 ) ) )
% 3.31/1.67 => ( hoare_123228589_state @ G_11 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_28 @ C_37 @ Q_10 ) @ bot_bo70021908tate_o ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_18_insertE,axiom,
% 3.31/1.67 ! [A_217: hoare_1775062406iple_a,B_99: hoare_1775062406iple_a,A_216: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ A_217 @ ( insert1281456128iple_a @ B_99 @ A_216 ) )
% 3.31/1.67 => ( ( A_217 != B_99 )
% 3.31/1.67 => ( member2122167641iple_a @ A_217 @ A_216 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_19_insertE,axiom,
% 3.31/1.67 ! [A_217: nat,B_99: nat,A_216: nat > $o] :
% 3.31/1.67 ( ( member_nat @ A_217 @ ( insert_nat @ B_99 @ A_216 ) )
% 3.31/1.67 => ( ( A_217 != B_99 )
% 3.31/1.67 => ( member_nat @ A_217 @ A_216 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_20_insertE,axiom,
% 3.31/1.67 ! [A_217: hoare_1167836817_state,B_99: hoare_1167836817_state,A_216: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( member2058392318_state @ A_217 @ ( insert2134838167_state @ B_99 @ A_216 ) )
% 3.31/1.67 => ( ( A_217 != B_99 )
% 3.31/1.67 => ( member2058392318_state @ A_217 @ A_216 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_21_insertCI,axiom,
% 3.31/1.67 ! [B_98: hoare_1775062406iple_a,A_215: hoare_1775062406iple_a,B_97: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ~ ( member2122167641iple_a @ A_215 @ B_97 )
% 3.31/1.67 => ( A_215 = B_98 ) )
% 3.31/1.67 => ( member2122167641iple_a @ A_215 @ ( insert1281456128iple_a @ B_98 @ B_97 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_22_insertCI,axiom,
% 3.31/1.67 ! [B_98: nat,A_215: nat,B_97: nat > $o] :
% 3.31/1.67 ( ( ~ ( member_nat @ A_215 @ B_97 )
% 3.31/1.67 => ( A_215 = B_98 ) )
% 3.31/1.67 => ( member_nat @ A_215 @ ( insert_nat @ B_98 @ B_97 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_23_insertCI,axiom,
% 3.31/1.67 ! [B_98: hoare_1167836817_state,A_215: hoare_1167836817_state,B_97: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( ~ ( member2058392318_state @ A_215 @ B_97 )
% 3.31/1.67 => ( A_215 = B_98 ) )
% 3.31/1.67 => ( member2058392318_state @ A_215 @ ( insert2134838167_state @ B_98 @ B_97 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_24_emptyE,axiom,
% 3.31/1.67 ! [A_214: hoare_1775062406iple_a] :
% 3.31/1.67 ~ ( member2122167641iple_a @ A_214 @ bot_bo751897185le_a_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_25_emptyE,axiom,
% 3.31/1.67 ! [A_214: hoare_1167836817_state] :
% 3.31/1.67 ~ ( member2058392318_state @ A_214 @ bot_bo70021908tate_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_26_emptyE,axiom,
% 3.31/1.67 ! [A_214: nat] :
% 3.31/1.67 ~ ( member_nat @ A_214 @ bot_bot_nat_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_27_singleton__conv2,axiom,
% 3.31/1.67 ! [A_213: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( collec676402587iple_a @ ( fequal1288209029iple_a @ A_213 ) )
% 3.31/1.67 = ( insert1281456128iple_a @ A_213 @ bot_bo751897185le_a_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_28_singleton__conv2,axiom,
% 3.31/1.67 ! [A_213: nat] :
% 3.31/1.67 ( ( collect_nat @ ( fequal_nat @ A_213 ) )
% 3.31/1.67 = ( insert_nat @ A_213 @ bot_bot_nat_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_29_singleton__conv2,axiom,
% 3.31/1.67 ! [A_213: nat > $o] :
% 3.31/1.67 ( ( collect_nat_o @ ( fequal_nat_o @ A_213 ) )
% 3.31/1.67 = ( insert_nat_o @ A_213 @ bot_bot_nat_o_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_30_singleton__conv2,axiom,
% 3.31/1.67 ! [A_213: hoare_1167836817_state] :
% 3.31/1.67 ( ( collec1027672124_state @ ( fequal1831255762_state @ A_213 ) )
% 3.31/1.67 = ( insert2134838167_state @ A_213 @ bot_bo70021908tate_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_31_singleton__conv,axiom,
% 3.31/1.67 ! [A_212: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( collec676402587iple_a
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] : ( X_3 = A_212 ) )
% 3.31/1.67 = ( insert1281456128iple_a @ A_212 @ bot_bo751897185le_a_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_32_singleton__conv,axiom,
% 3.31/1.67 ! [A_212: nat] :
% 3.31/1.67 ( ( collect_nat
% 3.31/1.67 @ ^ [X_3: nat] : ( X_3 = A_212 ) )
% 3.31/1.67 = ( insert_nat @ A_212 @ bot_bot_nat_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_33_singleton__conv,axiom,
% 3.31/1.67 ! [A_212: nat > $o] :
% 3.31/1.67 ( ( collect_nat_o
% 3.31/1.67 @ ^ [X_3: nat > $o] : ( X_3 = A_212 ) )
% 3.31/1.67 = ( insert_nat_o @ A_212 @ bot_bot_nat_o_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_34_singleton__conv,axiom,
% 3.31/1.67 ! [A_212: hoare_1167836817_state] :
% 3.31/1.67 ( ( collec1027672124_state
% 3.31/1.67 @ ^ [X_3: hoare_1167836817_state] : ( X_3 = A_212 ) )
% 3.31/1.67 = ( insert2134838167_state @ A_212 @ bot_bo70021908tate_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_35_Collect__conv__if2,axiom,
% 3.31/1.67 ! [P_26: hoare_1775062406iple_a > $o,A_211: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( ( P_26 @ A_211 )
% 3.31/1.67 => ( ( collec676402587iple_a
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] : ( (&) @ ( A_211 = X_3 ) @ ( P_26 @ X_3 ) ) )
% 3.31/1.67 = ( insert1281456128iple_a @ A_211 @ bot_bo751897185le_a_o ) ) )
% 3.31/1.67 & ( ~ ( P_26 @ A_211 )
% 3.31/1.67 => ( ( collec676402587iple_a
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] : ( (&) @ ( A_211 = X_3 ) @ ( P_26 @ X_3 ) ) )
% 3.31/1.67 = bot_bo751897185le_a_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_36_Collect__conv__if2,axiom,
% 3.31/1.67 ! [P_26: nat > $o,A_211: nat] :
% 3.31/1.67 ( ( ( P_26 @ A_211 )
% 3.31/1.67 => ( ( collect_nat
% 3.31/1.67 @ ^ [X_3: nat] : ( (&) @ ( A_211 = X_3 ) @ ( P_26 @ X_3 ) ) )
% 3.31/1.67 = ( insert_nat @ A_211 @ bot_bot_nat_o ) ) )
% 3.31/1.67 & ( ~ ( P_26 @ A_211 )
% 3.31/1.67 => ( ( collect_nat
% 3.31/1.67 @ ^ [X_3: nat] : ( (&) @ ( A_211 = X_3 ) @ ( P_26 @ X_3 ) ) )
% 3.31/1.67 = bot_bot_nat_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_37_Collect__conv__if2,axiom,
% 3.31/1.67 ! [P_26: ( nat > $o ) > $o,A_211: nat > $o] :
% 3.31/1.67 ( ( ( P_26 @ A_211 )
% 3.31/1.67 => ( ( collect_nat_o
% 3.31/1.67 @ ^ [X_3: nat > $o] : ( (&) @ ( A_211 = X_3 ) @ ( P_26 @ X_3 ) ) )
% 3.31/1.67 = ( insert_nat_o @ A_211 @ bot_bot_nat_o_o ) ) )
% 3.31/1.67 & ( ~ ( P_26 @ A_211 )
% 3.31/1.67 => ( ( collect_nat_o
% 3.31/1.67 @ ^ [X_3: nat > $o] : ( (&) @ ( A_211 = X_3 ) @ ( P_26 @ X_3 ) ) )
% 3.31/1.67 = bot_bot_nat_o_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_38_Collect__conv__if2,axiom,
% 3.31/1.67 ! [P_26: hoare_1167836817_state > $o,A_211: hoare_1167836817_state] :
% 3.31/1.67 ( ( ( P_26 @ A_211 )
% 3.31/1.67 => ( ( collec1027672124_state
% 3.31/1.67 @ ^ [X_3: hoare_1167836817_state] : ( (&) @ ( A_211 = X_3 ) @ ( P_26 @ X_3 ) ) )
% 3.31/1.67 = ( insert2134838167_state @ A_211 @ bot_bo70021908tate_o ) ) )
% 3.31/1.67 & ( ~ ( P_26 @ A_211 )
% 3.31/1.67 => ( ( collec1027672124_state
% 3.31/1.67 @ ^ [X_3: hoare_1167836817_state] : ( (&) @ ( A_211 = X_3 ) @ ( P_26 @ X_3 ) ) )
% 3.31/1.67 = bot_bo70021908tate_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_39_Collect__conv__if,axiom,
% 3.31/1.67 ! [P_25: hoare_1775062406iple_a > $o,A_210: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( ( P_25 @ A_210 )
% 3.31/1.67 => ( ( collec676402587iple_a
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] : ( (&) @ ( X_3 = A_210 ) @ ( P_25 @ X_3 ) ) )
% 3.31/1.67 = ( insert1281456128iple_a @ A_210 @ bot_bo751897185le_a_o ) ) )
% 3.31/1.67 & ( ~ ( P_25 @ A_210 )
% 3.31/1.67 => ( ( collec676402587iple_a
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] : ( (&) @ ( X_3 = A_210 ) @ ( P_25 @ X_3 ) ) )
% 3.31/1.67 = bot_bo751897185le_a_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_40_Collect__conv__if,axiom,
% 3.31/1.67 ! [P_25: nat > $o,A_210: nat] :
% 3.31/1.67 ( ( ( P_25 @ A_210 )
% 3.31/1.67 => ( ( collect_nat
% 3.31/1.67 @ ^ [X_3: nat] : ( (&) @ ( X_3 = A_210 ) @ ( P_25 @ X_3 ) ) )
% 3.31/1.67 = ( insert_nat @ A_210 @ bot_bot_nat_o ) ) )
% 3.31/1.67 & ( ~ ( P_25 @ A_210 )
% 3.31/1.67 => ( ( collect_nat
% 3.31/1.67 @ ^ [X_3: nat] : ( (&) @ ( X_3 = A_210 ) @ ( P_25 @ X_3 ) ) )
% 3.31/1.67 = bot_bot_nat_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_41_Collect__conv__if,axiom,
% 3.31/1.67 ! [P_25: ( nat > $o ) > $o,A_210: nat > $o] :
% 3.31/1.67 ( ( ( P_25 @ A_210 )
% 3.31/1.67 => ( ( collect_nat_o
% 3.31/1.67 @ ^ [X_3: nat > $o] : ( (&) @ ( X_3 = A_210 ) @ ( P_25 @ X_3 ) ) )
% 3.31/1.67 = ( insert_nat_o @ A_210 @ bot_bot_nat_o_o ) ) )
% 3.31/1.67 & ( ~ ( P_25 @ A_210 )
% 3.31/1.67 => ( ( collect_nat_o
% 3.31/1.67 @ ^ [X_3: nat > $o] : ( (&) @ ( X_3 = A_210 ) @ ( P_25 @ X_3 ) ) )
% 3.31/1.67 = bot_bot_nat_o_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_42_Collect__conv__if,axiom,
% 3.31/1.67 ! [P_25: hoare_1167836817_state > $o,A_210: hoare_1167836817_state] :
% 3.31/1.67 ( ( ( P_25 @ A_210 )
% 3.31/1.67 => ( ( collec1027672124_state
% 3.31/1.67 @ ^ [X_3: hoare_1167836817_state] : ( (&) @ ( X_3 = A_210 ) @ ( P_25 @ X_3 ) ) )
% 3.31/1.67 = ( insert2134838167_state @ A_210 @ bot_bo70021908tate_o ) ) )
% 3.31/1.67 & ( ~ ( P_25 @ A_210 )
% 3.31/1.67 => ( ( collec1027672124_state
% 3.31/1.67 @ ^ [X_3: hoare_1167836817_state] : ( (&) @ ( X_3 = A_210 ) @ ( P_25 @ X_3 ) ) )
% 3.31/1.67 = bot_bo70021908tate_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_43_equals0D,axiom,
% 3.31/1.67 ! [A_209: hoare_1775062406iple_a,A_208: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( A_208 = bot_bo751897185le_a_o )
% 3.31/1.67 => ~ ( member2122167641iple_a @ A_209 @ A_208 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_44_equals0D,axiom,
% 3.31/1.67 ! [A_209: hoare_1167836817_state,A_208: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( A_208 = bot_bo70021908tate_o )
% 3.31/1.67 => ~ ( member2058392318_state @ A_209 @ A_208 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_45_equals0D,axiom,
% 3.31/1.67 ! [A_209: nat,A_208: nat > $o] :
% 3.31/1.67 ( ( A_208 = bot_bot_nat_o )
% 3.31/1.67 => ~ ( member_nat @ A_209 @ A_208 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_46_Collect__empty__eq,axiom,
% 3.31/1.67 ! [P_24: nat > $o] :
% 3.31/1.67 ( ( ( collect_nat @ P_24 )
% 3.31/1.67 = bot_bot_nat_o )
% 3.31/1.67 <=> ! [X_3: nat] :
% 3.31/1.67 ~ ( P_24 @ X_3 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_47_Collect__empty__eq,axiom,
% 3.31/1.67 ! [P_24: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ( collec676402587iple_a @ P_24 )
% 3.31/1.67 = bot_bo751897185le_a_o )
% 3.31/1.67 <=> ! [X_3: hoare_1775062406iple_a] :
% 3.31/1.67 ~ ( P_24 @ X_3 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_48_Collect__empty__eq,axiom,
% 3.31/1.67 ! [P_24: ( nat > $o ) > $o] :
% 3.31/1.67 ( ( ( collect_nat_o @ P_24 )
% 3.31/1.67 = bot_bot_nat_o_o )
% 3.31/1.67 <=> ! [X_3: nat > $o] :
% 3.31/1.67 ~ ( P_24 @ X_3 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_49_Collect__empty__eq,axiom,
% 3.31/1.67 ! [P_24: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( ( collec1027672124_state @ P_24 )
% 3.31/1.67 = bot_bo70021908tate_o )
% 3.31/1.67 <=> ! [X_3: hoare_1167836817_state] :
% 3.31/1.67 ~ ( P_24 @ X_3 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_50_empty__iff,axiom,
% 3.31/1.67 ! [C_36: hoare_1775062406iple_a] :
% 3.31/1.67 ~ ( member2122167641iple_a @ C_36 @ bot_bo751897185le_a_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_51_empty__iff,axiom,
% 3.31/1.67 ! [C_36: hoare_1167836817_state] :
% 3.31/1.67 ~ ( member2058392318_state @ C_36 @ bot_bo70021908tate_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_52_empty__iff,axiom,
% 3.31/1.67 ! [C_36: nat] :
% 3.31/1.67 ~ ( member_nat @ C_36 @ bot_bot_nat_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_53_empty__Collect__eq,axiom,
% 3.31/1.67 ! [P_23: nat > $o] :
% 3.31/1.67 ( ( bot_bot_nat_o
% 3.31/1.67 = ( collect_nat @ P_23 ) )
% 3.31/1.67 <=> ! [X_3: nat] :
% 3.31/1.67 ~ ( P_23 @ X_3 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_54_empty__Collect__eq,axiom,
% 3.31/1.67 ! [P_23: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( bot_bo751897185le_a_o
% 3.31/1.67 = ( collec676402587iple_a @ P_23 ) )
% 3.31/1.67 <=> ! [X_3: hoare_1775062406iple_a] :
% 3.31/1.67 ~ ( P_23 @ X_3 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_55_empty__Collect__eq,axiom,
% 3.31/1.67 ! [P_23: ( nat > $o ) > $o] :
% 3.31/1.67 ( ( bot_bot_nat_o_o
% 3.31/1.67 = ( collect_nat_o @ P_23 ) )
% 3.31/1.67 <=> ! [X_3: nat > $o] :
% 3.31/1.67 ~ ( P_23 @ X_3 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_56_empty__Collect__eq,axiom,
% 3.31/1.67 ! [P_23: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( bot_bo70021908tate_o
% 3.31/1.67 = ( collec1027672124_state @ P_23 ) )
% 3.31/1.67 <=> ! [X_3: hoare_1167836817_state] :
% 3.31/1.67 ~ ( P_23 @ X_3 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_57_ex__in__conv,axiom,
% 3.31/1.67 ! [A_207: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ? [X_3: hoare_1775062406iple_a] : ( member2122167641iple_a @ X_3 @ A_207 )
% 3.31/1.67 <=> ( A_207 != bot_bo751897185le_a_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_58_ex__in__conv,axiom,
% 3.31/1.67 ! [A_207: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ? [X_3: hoare_1167836817_state] : ( member2058392318_state @ X_3 @ A_207 )
% 3.31/1.67 <=> ( A_207 != bot_bo70021908tate_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_59_ex__in__conv,axiom,
% 3.31/1.67 ! [A_207: nat > $o] :
% 3.31/1.67 ( ? [X_3: nat] : ( member_nat @ X_3 @ A_207 )
% 3.31/1.67 <=> ( A_207 != bot_bot_nat_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_60_all__not__in__conv,axiom,
% 3.31/1.67 ! [A_206: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ! [X_3: hoare_1775062406iple_a] :
% 3.31/1.67 ~ ( member2122167641iple_a @ X_3 @ A_206 )
% 3.31/1.67 <=> ( A_206 = bot_bo751897185le_a_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_61_all__not__in__conv,axiom,
% 3.31/1.67 ! [A_206: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ! [X_3: hoare_1167836817_state] :
% 3.31/1.67 ~ ( member2058392318_state @ X_3 @ A_206 )
% 3.31/1.67 <=> ( A_206 = bot_bo70021908tate_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_62_all__not__in__conv,axiom,
% 3.31/1.67 ! [A_206: nat > $o] :
% 3.31/1.67 ( ! [X_3: nat] :
% 3.31/1.67 ~ ( member_nat @ X_3 @ A_206 )
% 3.31/1.67 <=> ( A_206 = bot_bot_nat_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_63_empty__def,axiom,
% 3.31/1.67 ( bot_bot_nat_o
% 3.31/1.67 = ( collect_nat
% 3.31/1.67 @ ^ [X_3: nat] : $false ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_64_empty__def,axiom,
% 3.31/1.67 ( bot_bo751897185le_a_o
% 3.31/1.67 = ( collec676402587iple_a
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] : $false ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_65_empty__def,axiom,
% 3.31/1.67 ( bot_bot_nat_o_o
% 3.31/1.67 = ( collect_nat_o
% 3.31/1.67 @ ^ [X_3: nat > $o] : $false ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_66_empty__def,axiom,
% 3.31/1.67 ( bot_bo70021908tate_o
% 3.31/1.67 = ( collec1027672124_state
% 3.31/1.67 @ ^ [X_3: hoare_1167836817_state] : $false ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_67_insert__absorb,axiom,
% 3.31/1.67 ! [A_205: hoare_1775062406iple_a,A_204: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ A_205 @ A_204 )
% 3.31/1.67 => ( ( insert1281456128iple_a @ A_205 @ A_204 )
% 3.31/1.67 = A_204 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_68_insert__absorb,axiom,
% 3.31/1.67 ! [A_205: nat,A_204: nat > $o] :
% 3.31/1.67 ( ( member_nat @ A_205 @ A_204 )
% 3.31/1.67 => ( ( insert_nat @ A_205 @ A_204 )
% 3.31/1.67 = A_204 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_69_insert__absorb,axiom,
% 3.31/1.67 ! [A_205: hoare_1167836817_state,A_204: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( member2058392318_state @ A_205 @ A_204 )
% 3.31/1.67 => ( ( insert2134838167_state @ A_205 @ A_204 )
% 3.31/1.67 = A_204 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_70_insertI2,axiom,
% 3.31/1.67 ! [B_96: hoare_1775062406iple_a,A_203: hoare_1775062406iple_a,B_95: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ A_203 @ B_95 )
% 3.31/1.67 => ( member2122167641iple_a @ A_203 @ ( insert1281456128iple_a @ B_96 @ B_95 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_71_insertI2,axiom,
% 3.31/1.67 ! [B_96: nat,A_203: nat,B_95: nat > $o] :
% 3.31/1.67 ( ( member_nat @ A_203 @ B_95 )
% 3.31/1.67 => ( member_nat @ A_203 @ ( insert_nat @ B_96 @ B_95 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_72_insertI2,axiom,
% 3.31/1.67 ! [B_96: hoare_1167836817_state,A_203: hoare_1167836817_state,B_95: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( member2058392318_state @ A_203 @ B_95 )
% 3.31/1.67 => ( member2058392318_state @ A_203 @ ( insert2134838167_state @ B_96 @ B_95 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_73_insert__ident,axiom,
% 3.31/1.67 ! [B_94: hoare_1775062406iple_a > $o,X_100: hoare_1775062406iple_a,A_202: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ~ ( member2122167641iple_a @ X_100 @ A_202 )
% 3.31/1.67 => ( ~ ( member2122167641iple_a @ X_100 @ B_94 )
% 3.31/1.67 => ( ( ( insert1281456128iple_a @ X_100 @ A_202 )
% 3.31/1.67 = ( insert1281456128iple_a @ X_100 @ B_94 ) )
% 3.31/1.67 <=> ( A_202 = B_94 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_74_insert__ident,axiom,
% 3.31/1.67 ! [B_94: nat > $o,X_100: nat,A_202: nat > $o] :
% 3.31/1.67 ( ~ ( member_nat @ X_100 @ A_202 )
% 3.31/1.67 => ( ~ ( member_nat @ X_100 @ B_94 )
% 3.31/1.67 => ( ( ( insert_nat @ X_100 @ A_202 )
% 3.31/1.67 = ( insert_nat @ X_100 @ B_94 ) )
% 3.31/1.67 <=> ( A_202 = B_94 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_75_insert__ident,axiom,
% 3.31/1.67 ! [B_94: hoare_1167836817_state > $o,X_100: hoare_1167836817_state,A_202: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ~ ( member2058392318_state @ X_100 @ A_202 )
% 3.31/1.67 => ( ~ ( member2058392318_state @ X_100 @ B_94 )
% 3.31/1.67 => ( ( ( insert2134838167_state @ X_100 @ A_202 )
% 3.31/1.67 = ( insert2134838167_state @ X_100 @ B_94 ) )
% 3.31/1.67 <=> ( A_202 = B_94 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_76_insert__code,axiom,
% 3.31/1.67 ! [Y_43: hoare_1775062406iple_a,A_201: hoare_1775062406iple_a > $o,X_99: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( insert1281456128iple_a @ Y_43 @ A_201 @ X_99 )
% 3.31/1.67 <=> ( ( Y_43 = X_99 )
% 3.31/1.67 | ( A_201 @ X_99 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_77_insert__code,axiom,
% 3.31/1.67 ! [Y_43: nat,A_201: nat > $o,X_99: nat] :
% 3.31/1.67 ( ( insert_nat @ Y_43 @ A_201 @ X_99 )
% 3.31/1.67 <=> ( ( Y_43 = X_99 )
% 3.31/1.67 | ( A_201 @ X_99 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_78_insert__code,axiom,
% 3.31/1.67 ! [Y_43: hoare_1167836817_state,A_201: hoare_1167836817_state > $o,X_99: hoare_1167836817_state] :
% 3.31/1.67 ( ( insert2134838167_state @ Y_43 @ A_201 @ X_99 )
% 3.31/1.67 <=> ( ( Y_43 = X_99 )
% 3.31/1.67 | ( A_201 @ X_99 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_79_insert__iff,axiom,
% 3.31/1.67 ! [A_200: hoare_1775062406iple_a,B_93: hoare_1775062406iple_a,A_199: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ A_200 @ ( insert1281456128iple_a @ B_93 @ A_199 ) )
% 3.31/1.67 <=> ( ( A_200 = B_93 )
% 3.31/1.67 | ( member2122167641iple_a @ A_200 @ A_199 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_80_insert__iff,axiom,
% 3.31/1.67 ! [A_200: nat,B_93: nat,A_199: nat > $o] :
% 3.31/1.67 ( ( member_nat @ A_200 @ ( insert_nat @ B_93 @ A_199 ) )
% 3.31/1.67 <=> ( ( A_200 = B_93 )
% 3.31/1.67 | ( member_nat @ A_200 @ A_199 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_81_insert__iff,axiom,
% 3.31/1.67 ! [A_200: hoare_1167836817_state,B_93: hoare_1167836817_state,A_199: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( member2058392318_state @ A_200 @ ( insert2134838167_state @ B_93 @ A_199 ) )
% 3.31/1.67 <=> ( ( A_200 = B_93 )
% 3.31/1.67 | ( member2058392318_state @ A_200 @ A_199 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_82_insert__commute,axiom,
% 3.31/1.67 ! [X_98: hoare_1775062406iple_a,Y_42: hoare_1775062406iple_a,A_198: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( insert1281456128iple_a @ X_98 @ ( insert1281456128iple_a @ Y_42 @ A_198 ) )
% 3.31/1.67 = ( insert1281456128iple_a @ Y_42 @ ( insert1281456128iple_a @ X_98 @ A_198 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_83_insert__commute,axiom,
% 3.31/1.67 ! [X_98: nat,Y_42: nat,A_198: nat > $o] :
% 3.31/1.67 ( ( insert_nat @ X_98 @ ( insert_nat @ Y_42 @ A_198 ) )
% 3.31/1.67 = ( insert_nat @ Y_42 @ ( insert_nat @ X_98 @ A_198 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_84_insert__commute,axiom,
% 3.31/1.67 ! [X_98: hoare_1167836817_state,Y_42: hoare_1167836817_state,A_198: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( insert2134838167_state @ X_98 @ ( insert2134838167_state @ Y_42 @ A_198 ) )
% 3.31/1.67 = ( insert2134838167_state @ Y_42 @ ( insert2134838167_state @ X_98 @ A_198 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_85_insert__absorb2,axiom,
% 3.31/1.67 ! [X_97: hoare_1775062406iple_a,A_197: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( insert1281456128iple_a @ X_97 @ ( insert1281456128iple_a @ X_97 @ A_197 ) )
% 3.31/1.67 = ( insert1281456128iple_a @ X_97 @ A_197 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_86_insert__absorb2,axiom,
% 3.31/1.67 ! [X_97: nat,A_197: nat > $o] :
% 3.31/1.67 ( ( insert_nat @ X_97 @ ( insert_nat @ X_97 @ A_197 ) )
% 3.31/1.67 = ( insert_nat @ X_97 @ A_197 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_87_insert__absorb2,axiom,
% 3.31/1.67 ! [X_97: hoare_1167836817_state,A_197: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( insert2134838167_state @ X_97 @ ( insert2134838167_state @ X_97 @ A_197 ) )
% 3.31/1.67 = ( insert2134838167_state @ X_97 @ A_197 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_88_insert__Collect,axiom,
% 3.31/1.67 ! [A_196: hoare_1775062406iple_a,P_22: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( insert1281456128iple_a @ A_196 @ ( collec676402587iple_a @ P_22 ) )
% 3.31/1.67 = ( collec676402587iple_a
% 3.31/1.67 @ ^ [U_1: hoare_1775062406iple_a] : ( (=>) @ ( (~) @ ( U_1 = A_196 ) ) @ ( P_22 @ U_1 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_89_insert__Collect,axiom,
% 3.31/1.67 ! [A_196: nat,P_22: nat > $o] :
% 3.31/1.67 ( ( insert_nat @ A_196 @ ( collect_nat @ P_22 ) )
% 3.31/1.67 = ( collect_nat
% 3.31/1.67 @ ^ [U_1: nat] : ( (=>) @ ( (~) @ ( U_1 = A_196 ) ) @ ( P_22 @ U_1 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_90_insert__Collect,axiom,
% 3.31/1.67 ! [A_196: nat > $o,P_22: ( nat > $o ) > $o] :
% 3.31/1.67 ( ( insert_nat_o @ A_196 @ ( collect_nat_o @ P_22 ) )
% 3.31/1.67 = ( collect_nat_o
% 3.31/1.67 @ ^ [U_1: nat > $o] : ( (=>) @ ( (~) @ ( U_1 = A_196 ) ) @ ( P_22 @ U_1 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_91_insert__Collect,axiom,
% 3.31/1.67 ! [A_196: hoare_1167836817_state,P_22: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( insert2134838167_state @ A_196 @ ( collec1027672124_state @ P_22 ) )
% 3.31/1.67 = ( collec1027672124_state
% 3.31/1.67 @ ^ [U_1: hoare_1167836817_state] : ( (=>) @ ( (~) @ ( U_1 = A_196 ) ) @ ( P_22 @ U_1 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_92_insert__compr,axiom,
% 3.31/1.67 ! [A_195: hoare_1775062406iple_a,B_92: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( insert1281456128iple_a @ A_195 @ B_92 )
% 3.31/1.67 = ( collec676402587iple_a
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] : ( (|) @ ( X_3 = A_195 ) @ ( member2122167641iple_a @ X_3 @ B_92 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_93_insert__compr,axiom,
% 3.31/1.67 ! [A_195: nat,B_92: nat > $o] :
% 3.31/1.67 ( ( insert_nat @ A_195 @ B_92 )
% 3.31/1.67 = ( collect_nat
% 3.31/1.67 @ ^ [X_3: nat] : ( (|) @ ( X_3 = A_195 ) @ ( member_nat @ X_3 @ B_92 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_94_insert__compr,axiom,
% 3.31/1.67 ! [A_195: nat > $o,B_92: ( nat > $o ) > $o] :
% 3.31/1.67 ( ( insert_nat_o @ A_195 @ B_92 )
% 3.31/1.67 = ( collect_nat_o
% 3.31/1.67 @ ^ [X_3: nat > $o] : ( (|) @ ( X_3 = A_195 ) @ ( member_nat_o @ X_3 @ B_92 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_95_insert__compr,axiom,
% 3.31/1.67 ! [A_195: hoare_1167836817_state,B_92: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( insert2134838167_state @ A_195 @ B_92 )
% 3.31/1.67 = ( collec1027672124_state
% 3.31/1.67 @ ^ [X_3: hoare_1167836817_state] : ( (|) @ ( X_3 = A_195 ) @ ( member2058392318_state @ X_3 @ B_92 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_96_insertI1,axiom,
% 3.31/1.67 ! [A_194: hoare_1775062406iple_a,B_91: hoare_1775062406iple_a > $o] : ( member2122167641iple_a @ A_194 @ ( insert1281456128iple_a @ A_194 @ B_91 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_97_insertI1,axiom,
% 3.31/1.67 ! [A_194: nat,B_91: nat > $o] : ( member_nat @ A_194 @ ( insert_nat @ A_194 @ B_91 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_98_insertI1,axiom,
% 3.31/1.67 ! [A_194: hoare_1167836817_state,B_91: hoare_1167836817_state > $o] : ( member2058392318_state @ A_194 @ ( insert2134838167_state @ A_194 @ B_91 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_99_insert__compr__raw,axiom,
% 3.31/1.67 ! [X_3: hoare_1775062406iple_a,Xa: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( insert1281456128iple_a @ X_3 @ Xa )
% 3.31/1.67 = ( collec676402587iple_a
% 3.31/1.67 @ ^ [Y_1: hoare_1775062406iple_a] : ( (|) @ ( Y_1 = X_3 ) @ ( member2122167641iple_a @ Y_1 @ Xa ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_100_insert__compr__raw,axiom,
% 3.31/1.67 ! [X_3: nat,Xa: nat > $o] :
% 3.31/1.67 ( ( insert_nat @ X_3 @ Xa )
% 3.31/1.67 = ( collect_nat
% 3.31/1.67 @ ^ [Y_1: nat] : ( (|) @ ( Y_1 = X_3 ) @ ( member_nat @ Y_1 @ Xa ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_101_insert__compr__raw,axiom,
% 3.31/1.67 ! [X_3: nat > $o,Xa: ( nat > $o ) > $o] :
% 3.31/1.67 ( ( insert_nat_o @ X_3 @ Xa )
% 3.31/1.67 = ( collect_nat_o
% 3.31/1.67 @ ^ [Y_1: nat > $o] : ( (|) @ ( Y_1 = X_3 ) @ ( member_nat_o @ Y_1 @ Xa ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_102_insert__compr__raw,axiom,
% 3.31/1.67 ! [X_3: hoare_1167836817_state,Xa: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( insert2134838167_state @ X_3 @ Xa )
% 3.31/1.67 = ( collec1027672124_state
% 3.31/1.67 @ ^ [Y_1: hoare_1167836817_state] : ( (|) @ ( Y_1 = X_3 ) @ ( member2058392318_state @ Y_1 @ Xa ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_103_singleton__inject,axiom,
% 3.31/1.67 ! [A_193: hoare_1775062406iple_a,B_90: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( ( insert1281456128iple_a @ A_193 @ bot_bo751897185le_a_o )
% 3.31/1.67 = ( insert1281456128iple_a @ B_90 @ bot_bo751897185le_a_o ) )
% 3.31/1.67 => ( A_193 = B_90 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_104_singleton__inject,axiom,
% 3.31/1.67 ! [A_193: nat,B_90: nat] :
% 3.31/1.67 ( ( ( insert_nat @ A_193 @ bot_bot_nat_o )
% 3.31/1.67 = ( insert_nat @ B_90 @ bot_bot_nat_o ) )
% 3.31/1.67 => ( A_193 = B_90 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_105_singleton__inject,axiom,
% 3.31/1.67 ! [A_193: hoare_1167836817_state,B_90: hoare_1167836817_state] :
% 3.31/1.67 ( ( ( insert2134838167_state @ A_193 @ bot_bo70021908tate_o )
% 3.31/1.67 = ( insert2134838167_state @ B_90 @ bot_bo70021908tate_o ) )
% 3.31/1.67 => ( A_193 = B_90 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_106_singletonE,axiom,
% 3.31/1.67 ! [B_89: hoare_1775062406iple_a,A_192: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( member2122167641iple_a @ B_89 @ ( insert1281456128iple_a @ A_192 @ bot_bo751897185le_a_o ) )
% 3.31/1.67 => ( B_89 = A_192 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_107_singletonE,axiom,
% 3.31/1.67 ! [B_89: nat,A_192: nat] :
% 3.31/1.67 ( ( member_nat @ B_89 @ ( insert_nat @ A_192 @ bot_bot_nat_o ) )
% 3.31/1.67 => ( B_89 = A_192 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_108_singletonE,axiom,
% 3.31/1.67 ! [B_89: hoare_1167836817_state,A_192: hoare_1167836817_state] :
% 3.31/1.67 ( ( member2058392318_state @ B_89 @ ( insert2134838167_state @ A_192 @ bot_bo70021908tate_o ) )
% 3.31/1.67 => ( B_89 = A_192 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_109_doubleton__eq__iff,axiom,
% 3.31/1.67 ! [A_191: hoare_1775062406iple_a,B_88: hoare_1775062406iple_a,C_35: hoare_1775062406iple_a,D_4: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( ( insert1281456128iple_a @ A_191 @ ( insert1281456128iple_a @ B_88 @ bot_bo751897185le_a_o ) )
% 3.31/1.67 = ( insert1281456128iple_a @ C_35 @ ( insert1281456128iple_a @ D_4 @ bot_bo751897185le_a_o ) ) )
% 3.31/1.67 <=> ( ( ( A_191 = C_35 )
% 3.31/1.67 & ( B_88 = D_4 ) )
% 3.31/1.67 | ( ( A_191 = D_4 )
% 3.31/1.67 & ( B_88 = C_35 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_110_doubleton__eq__iff,axiom,
% 3.31/1.67 ! [A_191: nat,B_88: nat,C_35: nat,D_4: nat] :
% 3.31/1.67 ( ( ( insert_nat @ A_191 @ ( insert_nat @ B_88 @ bot_bot_nat_o ) )
% 3.31/1.67 = ( insert_nat @ C_35 @ ( insert_nat @ D_4 @ bot_bot_nat_o ) ) )
% 3.31/1.67 <=> ( ( ( A_191 = C_35 )
% 3.31/1.67 & ( B_88 = D_4 ) )
% 3.31/1.67 | ( ( A_191 = D_4 )
% 3.31/1.67 & ( B_88 = C_35 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_111_doubleton__eq__iff,axiom,
% 3.31/1.67 ! [A_191: hoare_1167836817_state,B_88: hoare_1167836817_state,C_35: hoare_1167836817_state,D_4: hoare_1167836817_state] :
% 3.31/1.67 ( ( ( insert2134838167_state @ A_191 @ ( insert2134838167_state @ B_88 @ bot_bo70021908tate_o ) )
% 3.31/1.67 = ( insert2134838167_state @ C_35 @ ( insert2134838167_state @ D_4 @ bot_bo70021908tate_o ) ) )
% 3.31/1.67 <=> ( ( ( A_191 = C_35 )
% 3.31/1.67 & ( B_88 = D_4 ) )
% 3.31/1.67 | ( ( A_191 = D_4 )
% 3.31/1.67 & ( B_88 = C_35 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_112_singleton__iff,axiom,
% 3.31/1.67 ! [B_87: hoare_1775062406iple_a,A_190: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( member2122167641iple_a @ B_87 @ ( insert1281456128iple_a @ A_190 @ bot_bo751897185le_a_o ) )
% 3.31/1.67 <=> ( B_87 = A_190 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_113_singleton__iff,axiom,
% 3.31/1.67 ! [B_87: nat,A_190: nat] :
% 3.31/1.67 ( ( member_nat @ B_87 @ ( insert_nat @ A_190 @ bot_bot_nat_o ) )
% 3.31/1.67 <=> ( B_87 = A_190 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_114_singleton__iff,axiom,
% 3.31/1.67 ! [B_87: hoare_1167836817_state,A_190: hoare_1167836817_state] :
% 3.31/1.67 ( ( member2058392318_state @ B_87 @ ( insert2134838167_state @ A_190 @ bot_bo70021908tate_o ) )
% 3.31/1.67 <=> ( B_87 = A_190 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_115_insert__not__empty,axiom,
% 3.31/1.67 ! [A_189: hoare_1775062406iple_a,A_188: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( insert1281456128iple_a @ A_189 @ A_188 )
% 3.31/1.67 != bot_bo751897185le_a_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_116_insert__not__empty,axiom,
% 3.31/1.67 ! [A_189: nat,A_188: nat > $o] :
% 3.31/1.67 ( ( insert_nat @ A_189 @ A_188 )
% 3.31/1.67 != bot_bot_nat_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_117_insert__not__empty,axiom,
% 3.31/1.67 ! [A_189: hoare_1167836817_state,A_188: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( insert2134838167_state @ A_189 @ A_188 )
% 3.31/1.67 != bot_bo70021908tate_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_118_empty__not__insert,axiom,
% 3.31/1.67 ! [A_187: hoare_1775062406iple_a,A_186: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( bot_bo751897185le_a_o
% 3.31/1.67 != ( insert1281456128iple_a @ A_187 @ A_186 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_119_empty__not__insert,axiom,
% 3.31/1.67 ! [A_187: nat,A_186: nat > $o] :
% 3.31/1.67 ( bot_bot_nat_o
% 3.31/1.67 != ( insert_nat @ A_187 @ A_186 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_120_empty__not__insert,axiom,
% 3.31/1.67 ! [A_187: hoare_1167836817_state,A_186: hoare_1167836817_state > $o] :
% 3.31/1.67 ( bot_bo70021908tate_o
% 3.31/1.67 != ( insert2134838167_state @ A_187 @ A_186 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_121_the__elem__eq,axiom,
% 3.31/1.67 ! [X_96: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( the_el1844711461iple_a @ ( insert1281456128iple_a @ X_96 @ bot_bo751897185le_a_o ) )
% 3.31/1.67 = X_96 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_122_the__elem__eq,axiom,
% 3.31/1.67 ! [X_96: nat] :
% 3.31/1.67 ( ( the_elem_nat @ ( insert_nat @ X_96 @ bot_bot_nat_o ) )
% 3.31/1.67 = X_96 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_123_the__elem__eq,axiom,
% 3.31/1.67 ! [X_96: hoare_1167836817_state] :
% 3.31/1.67 ( ( the_el323660082_state @ ( insert2134838167_state @ X_96 @ bot_bo70021908tate_o ) )
% 3.31/1.67 = X_96 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_124_bot__apply,axiom,
% 3.31/1.67 ! [X_95: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( bot_bo751897185le_a_o @ X_95 )
% 3.31/1.67 <=> bot_bot_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_125_bot__apply,axiom,
% 3.31/1.67 ! [X_95: nat] :
% 3.31/1.67 ( ( bot_bot_nat_o @ X_95 )
% 3.31/1.67 <=> bot_bot_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_126_bot__apply,axiom,
% 3.31/1.67 ! [X_95: hoare_1167836817_state] :
% 3.31/1.67 ( ( bot_bo70021908tate_o @ X_95 )
% 3.31/1.67 <=> bot_bot_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_127_bot__fun__def,axiom,
% 3.31/1.67 ! [X_3: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( bot_bo751897185le_a_o @ X_3 )
% 3.31/1.67 <=> bot_bot_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_128_bot__fun__def,axiom,
% 3.31/1.67 ! [X_3: nat] :
% 3.31/1.67 ( ( bot_bot_nat_o @ X_3 )
% 3.31/1.67 <=> bot_bot_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_129_bot__fun__def,axiom,
% 3.31/1.67 ! [X_3: hoare_1167836817_state] :
% 3.31/1.67 ( ( bot_bo70021908tate_o @ X_3 )
% 3.31/1.67 <=> bot_bot_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_130_hoare__derivs_OSkip,axiom,
% 3.31/1.67 ! [G_10: hoare_1775062406iple_a > $o,P_21: x_a > state > $o] : ( hoare_1508237396rivs_a @ G_10 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_21 @ skip @ P_21 ) @ bot_bo751897185le_a_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_131_hoare__derivs_OSkip,axiom,
% 3.31/1.67 ! [G_10: hoare_1167836817_state > $o,P_21: state > state > $o] : ( hoare_123228589_state @ G_10 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_21 @ skip @ P_21 ) @ bot_bo70021908tate_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_132_Comp,axiom,
% 3.31/1.67 ! [D_3: com,R_3: x_a > state > $o,G_9: hoare_1775062406iple_a > $o,P_20: x_a > state > $o,C_34: com,Q_8: x_a > state > $o] :
% 3.31/1.67 ( ( hoare_1508237396rivs_a @ G_9 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_20 @ C_34 @ Q_8 ) @ bot_bo751897185le_a_o ) )
% 3.31/1.67 => ( ( hoare_1508237396rivs_a @ G_9 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ Q_8 @ D_3 @ R_3 ) @ bot_bo751897185le_a_o ) )
% 3.31/1.67 => ( hoare_1508237396rivs_a @ G_9 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_20 @ ( semi @ C_34 @ D_3 ) @ R_3 ) @ bot_bo751897185le_a_o ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_133_Comp,axiom,
% 3.31/1.67 ! [D_3: com,R_3: state > state > $o,G_9: hoare_1167836817_state > $o,P_20: state > state > $o,C_34: com,Q_8: state > state > $o] :
% 3.31/1.67 ( ( hoare_123228589_state @ G_9 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_20 @ C_34 @ Q_8 ) @ bot_bo70021908tate_o ) )
% 3.31/1.67 => ( ( hoare_123228589_state @ G_9 @ ( insert2134838167_state @ ( hoare_908217195_state @ Q_8 @ D_3 @ R_3 ) @ bot_bo70021908tate_o ) )
% 3.31/1.67 => ( hoare_123228589_state @ G_9 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_20 @ ( semi @ C_34 @ D_3 ) @ R_3 ) @ bot_bo70021908tate_o ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_134_triple_Oexhaust,axiom,
% 3.31/1.67 ! [Y_41: hoare_1167836817_state] :
% 3.31/1.67 ~ ! [Fun1: state > state > $o,Com_2: com,Fun2: state > state > $o] :
% 3.31/1.67 ( Y_41
% 3.31/1.67 != ( hoare_908217195_state @ Fun1 @ Com_2 @ Fun2 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_135_triple_Oexhaust,axiom,
% 3.31/1.67 ! [Y_41: hoare_1775062406iple_a] :
% 3.31/1.67 ~ ! [Fun1: x_a > state > $o,Com_2: com,Fun2: x_a > state > $o] :
% 3.31/1.67 ( Y_41
% 3.31/1.67 != ( hoare_1766022166iple_a @ Fun1 @ Com_2 @ Fun2 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_136_Set_Oset__insert,axiom,
% 3.31/1.67 ! [X_94: hoare_1775062406iple_a,A_185: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ X_94 @ A_185 )
% 3.31/1.67 => ~ ! [B_62: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( A_185
% 3.31/1.67 = ( insert1281456128iple_a @ X_94 @ B_62 ) )
% 3.31/1.67 => ( member2122167641iple_a @ X_94 @ B_62 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_137_Set_Oset__insert,axiom,
% 3.31/1.67 ! [X_94: nat,A_185: nat > $o] :
% 3.31/1.67 ( ( member_nat @ X_94 @ A_185 )
% 3.31/1.67 => ~ ! [B_62: nat > $o] :
% 3.31/1.67 ( ( A_185
% 3.31/1.67 = ( insert_nat @ X_94 @ B_62 ) )
% 3.31/1.67 => ( member_nat @ X_94 @ B_62 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_138_Set_Oset__insert,axiom,
% 3.31/1.67 ! [X_94: hoare_1167836817_state,A_185: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( member2058392318_state @ X_94 @ A_185 )
% 3.31/1.67 => ~ ! [B_62: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( A_185
% 3.31/1.67 = ( insert2134838167_state @ X_94 @ B_62 ) )
% 3.31/1.67 => ( member2058392318_state @ X_94 @ B_62 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_139_mk__disjoint__insert,axiom,
% 3.31/1.67 ! [A_184: hoare_1775062406iple_a,A_183: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ A_184 @ A_183 )
% 3.31/1.67 => ? [B_62: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( A_183
% 3.31/1.67 = ( insert1281456128iple_a @ A_184 @ B_62 ) )
% 3.31/1.67 & ~ ( member2122167641iple_a @ A_184 @ B_62 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_140_mk__disjoint__insert,axiom,
% 3.31/1.67 ! [A_184: nat,A_183: nat > $o] :
% 3.31/1.67 ( ( member_nat @ A_184 @ A_183 )
% 3.31/1.67 => ? [B_62: nat > $o] :
% 3.31/1.67 ( ( A_183
% 3.31/1.67 = ( insert_nat @ A_184 @ B_62 ) )
% 3.31/1.67 & ~ ( member_nat @ A_184 @ B_62 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_141_mk__disjoint__insert,axiom,
% 3.31/1.67 ! [A_184: hoare_1167836817_state,A_183: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( member2058392318_state @ A_184 @ A_183 )
% 3.31/1.67 => ? [B_62: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( A_183
% 3.31/1.67 = ( insert2134838167_state @ A_184 @ B_62 ) )
% 3.31/1.67 & ~ ( member2058392318_state @ A_184 @ B_62 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_142_equals0I,axiom,
% 3.31/1.67 ! [A_182: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ! [Y_1: hoare_1775062406iple_a] :
% 3.31/1.67 ~ ( member2122167641iple_a @ Y_1 @ A_182 )
% 3.31/1.67 => ( A_182 = bot_bo751897185le_a_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_143_equals0I,axiom,
% 3.31/1.67 ! [A_182: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ! [Y_1: hoare_1167836817_state] :
% 3.31/1.67 ~ ( member2058392318_state @ Y_1 @ A_182 )
% 3.31/1.67 => ( A_182 = bot_bo70021908tate_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_144_equals0I,axiom,
% 3.31/1.67 ! [A_182: nat > $o] :
% 3.31/1.67 ( ! [Y_1: nat] :
% 3.31/1.67 ~ ( member_nat @ Y_1 @ A_182 )
% 3.31/1.67 => ( A_182 = bot_bot_nat_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_145_conseq,axiom,
% 3.31/1.67 ! [Q_6: x_a > state > $o,G_8: hoare_1775062406iple_a > $o,C_33: com,P_18: x_a > state > $o] :
% 3.31/1.67 ( ! [Z: x_a,S: state] :
% 3.31/1.67 ( ( P_18 @ Z @ S )
% 3.31/1.67 => ? [P_19: x_a > state > $o,Q_7: x_a > state > $o] :
% 3.31/1.67 ( ( hoare_1508237396rivs_a @ G_8 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_19 @ C_33 @ Q_7 ) @ bot_bo751897185le_a_o ) )
% 3.31/1.67 & ! [S_5: state] :
% 3.31/1.67 ( ! [Z_23: x_a] :
% 3.31/1.67 ( ( P_19 @ Z_23 @ S )
% 3.31/1.67 => ( Q_7 @ Z_23 @ S_5 ) )
% 3.31/1.67 => ( Q_6 @ Z @ S_5 ) ) ) )
% 3.31/1.67 => ( hoare_1508237396rivs_a @ G_8 @ ( insert1281456128iple_a @ ( hoare_1766022166iple_a @ P_18 @ C_33 @ Q_6 ) @ bot_bo751897185le_a_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_146_conseq,axiom,
% 3.31/1.67 ! [Q_6: state > state > $o,G_8: hoare_1167836817_state > $o,C_33: com,P_18: state > state > $o] :
% 3.31/1.67 ( ! [Z: state,S: state] :
% 3.31/1.67 ( ( P_18 @ Z @ S )
% 3.31/1.67 => ? [P_19: state > state > $o,Q_7: state > state > $o] :
% 3.31/1.67 ( ( hoare_123228589_state @ G_8 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_19 @ C_33 @ Q_7 ) @ bot_bo70021908tate_o ) )
% 3.31/1.67 & ! [S_5: state] :
% 3.31/1.67 ( ! [Z_23: state] :
% 3.31/1.67 ( ( P_19 @ Z_23 @ S )
% 3.31/1.67 => ( Q_7 @ Z_23 @ S_5 ) )
% 3.31/1.67 => ( Q_6 @ Z @ S_5 ) ) ) )
% 3.31/1.67 => ( hoare_123228589_state @ G_8 @ ( insert2134838167_state @ ( hoare_908217195_state @ P_18 @ C_33 @ Q_6 ) @ bot_bo70021908tate_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_147_com_Osimps_I13_J,axiom,
% 3.31/1.67 ! [Com1: com,Com2: com] :
% 3.31/1.67 ( ( semi @ Com1 @ Com2 )
% 3.31/1.67 != skip ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_148_com_Osimps_I12_J,axiom,
% 3.31/1.67 ! [Com1: com,Com2: com] :
% 3.31/1.67 ( skip
% 3.31/1.67 != ( semi @ Com1 @ Com2 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_149_the__elem__def,axiom,
% 3.31/1.67 ! [X_93: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( the_el1844711461iple_a @ X_93 )
% 3.31/1.67 = ( the_Ho1155011127iple_a
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] :
% 3.31/1.67 ( X_93
% 3.31/1.67 = ( insert1281456128iple_a @ X_3 @ bot_bo751897185le_a_o ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_150_the__elem__def,axiom,
% 3.31/1.67 ! [X_93: nat > $o] :
% 3.31/1.67 ( ( the_elem_nat @ X_93 )
% 3.31/1.67 = ( the_nat
% 3.31/1.67 @ ^ [X_3: nat] :
% 3.31/1.67 ( X_93
% 3.31/1.67 = ( insert_nat @ X_3 @ bot_bot_nat_o ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_151_the__elem__def,axiom,
% 3.31/1.67 ! [X_93: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( the_el323660082_state @ X_93 )
% 3.31/1.67 = ( the_Ho310147232_state
% 3.31/1.67 @ ^ [X_3: hoare_1167836817_state] :
% 3.31/1.67 ( X_93
% 3.31/1.67 = ( insert2134838167_state @ X_3 @ bot_bo70021908tate_o ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_152_com_Osimps_I3_J,axiom,
% 3.31/1.67 ! [Com1_1: com,Com2_1: com,Com1: com,Com2: com] :
% 3.31/1.67 ( ( ( semi @ Com1_1 @ Com2_1 )
% 3.31/1.67 = ( semi @ Com1 @ Com2 ) )
% 3.31/1.67 <=> ( ( Com1_1 = Com1 )
% 3.31/1.67 & ( Com2_1 = Com2 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_153_nonempty__iff,axiom,
% 3.31/1.67 ! [A_181: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( A_181 != bot_bo751897185le_a_o )
% 3.31/1.67 <=> ? [X_3: hoare_1775062406iple_a,B_62: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( A_181
% 3.31/1.67 = ( insert1281456128iple_a @ X_3 @ B_62 ) )
% 3.31/1.67 & ~ ( member2122167641iple_a @ X_3 @ B_62 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_154_nonempty__iff,axiom,
% 3.31/1.67 ! [A_181: nat > $o] :
% 3.31/1.67 ( ( A_181 != bot_bot_nat_o )
% 3.31/1.67 <=> ? [X_3: nat,B_62: nat > $o] :
% 3.31/1.67 ( ( A_181
% 3.31/1.67 = ( insert_nat @ X_3 @ B_62 ) )
% 3.31/1.67 & ~ ( member_nat @ X_3 @ B_62 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_155_nonempty__iff,axiom,
% 3.31/1.67 ! [A_181: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( A_181 != bot_bo70021908tate_o )
% 3.31/1.67 <=> ? [X_3: hoare_1167836817_state,B_62: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( A_181
% 3.31/1.67 = ( insert2134838167_state @ X_3 @ B_62 ) )
% 3.31/1.67 & ~ ( member2058392318_state @ X_3 @ B_62 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_156_bot__empty__eq,axiom,
% 3.31/1.67 ! [X_3: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( bot_bo751897185le_a_o @ X_3 )
% 3.31/1.67 <=> ( member2122167641iple_a @ X_3 @ bot_bo751897185le_a_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_157_bot__empty__eq,axiom,
% 3.31/1.67 ! [X_3: hoare_1167836817_state] :
% 3.31/1.67 ( ( bot_bo70021908tate_o @ X_3 )
% 3.31/1.67 <=> ( member2058392318_state @ X_3 @ bot_bo70021908tate_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_158_bot__empty__eq,axiom,
% 3.31/1.67 ! [X_3: nat] :
% 3.31/1.67 ( ( bot_bot_nat_o @ X_3 )
% 3.31/1.67 <=> ( member_nat @ X_3 @ bot_bot_nat_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_159_Ass,axiom,
% 3.31/1.67 ! [G_7: hoare_1775062406iple_a > $o,P_17: x_a > state > $o,X_92: vname,A_180: state > nat] :
% 3.31/1.67 ( hoare_1508237396rivs_a @ G_7
% 3.31/1.67 @ ( insert1281456128iple_a
% 3.31/1.67 @ ( hoare_1766022166iple_a
% 3.31/1.67 @ ^ [Z: x_a,S: state] : ( P_17 @ Z @ ( update @ S @ X_92 @ ( A_180 @ S ) ) )
% 3.31/1.67 @ ( ass @ X_92 @ A_180 )
% 3.31/1.67 @ P_17 )
% 3.31/1.67 @ bot_bo751897185le_a_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_160_Ass,axiom,
% 3.31/1.67 ! [G_7: hoare_1167836817_state > $o,P_17: state > state > $o,X_92: vname,A_180: state > nat] :
% 3.31/1.67 ( hoare_123228589_state @ G_7
% 3.31/1.67 @ ( insert2134838167_state
% 3.31/1.67 @ ( hoare_908217195_state
% 3.31/1.67 @ ^ [Z: state,S: state] : ( P_17 @ Z @ ( update @ S @ X_92 @ ( A_180 @ S ) ) )
% 3.31/1.67 @ ( ass @ X_92 @ A_180 )
% 3.31/1.67 @ P_17 )
% 3.31/1.67 @ bot_bo70021908tate_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_161_image__constant__conv,axiom,
% 3.31/1.67 ! [C_32: hoare_1775062406iple_a,A_179: nat > $o] :
% 3.31/1.67 ( ( ( A_179 = bot_bot_nat_o )
% 3.31/1.67 => ( ( image_43014529iple_a
% 3.31/1.67 @ ^ [X_3: nat] : C_32
% 3.31/1.67 @ A_179 )
% 3.31/1.67 = bot_bo751897185le_a_o ) )
% 3.31/1.67 & ( ( A_179 != bot_bot_nat_o )
% 3.31/1.67 => ( ( image_43014529iple_a
% 3.31/1.67 @ ^ [X_3: nat] : C_32
% 3.31/1.67 @ A_179 )
% 3.31/1.67 = ( insert1281456128iple_a @ C_32 @ bot_bo751897185le_a_o ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_162_image__constant__conv,axiom,
% 3.31/1.67 ! [C_32: hoare_1775062406iple_a,A_179: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( ( A_179 = bot_bo70021908tate_o )
% 3.31/1.67 => ( ( image_1802845250iple_a
% 3.31/1.67 @ ^ [X_3: hoare_1167836817_state] : C_32
% 3.31/1.67 @ A_179 )
% 3.31/1.67 = bot_bo751897185le_a_o ) )
% 3.31/1.67 & ( ( A_179 != bot_bo70021908tate_o )
% 3.31/1.67 => ( ( image_1802845250iple_a
% 3.31/1.67 @ ^ [X_3: hoare_1167836817_state] : C_32
% 3.31/1.67 @ A_179 )
% 3.31/1.67 = ( insert1281456128iple_a @ C_32 @ bot_bo751897185le_a_o ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_163_image__constant__conv,axiom,
% 3.31/1.67 ! [C_32: nat,A_179: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ( A_179 = bot_bo751897185le_a_o )
% 3.31/1.67 => ( ( image_1806517641_a_nat
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] : C_32
% 3.31/1.67 @ A_179 )
% 3.31/1.67 = bot_bot_nat_o ) )
% 3.31/1.67 & ( ( A_179 != bot_bo751897185le_a_o )
% 3.31/1.67 => ( ( image_1806517641_a_nat
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] : C_32
% 3.31/1.67 @ A_179 )
% 3.31/1.67 = ( insert_nat @ C_32 @ bot_bot_nat_o ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_164_image__constant__conv,axiom,
% 3.31/1.67 ! [C_32: hoare_1167836817_state,A_179: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ( A_179 = bot_bo751897185le_a_o )
% 3.31/1.67 => ( ( image_1021683026_state
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] : C_32
% 3.31/1.67 @ A_179 )
% 3.31/1.67 = bot_bo70021908tate_o ) )
% 3.31/1.67 & ( ( A_179 != bot_bo751897185le_a_o )
% 3.31/1.67 => ( ( image_1021683026_state
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] : C_32
% 3.31/1.67 @ A_179 )
% 3.31/1.67 = ( insert2134838167_state @ C_32 @ bot_bo70021908tate_o ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_165_image__constant__conv,axiom,
% 3.31/1.67 ! [C_32: nat,A_179: nat > $o] :
% 3.31/1.67 ( ( ( A_179 = bot_bot_nat_o )
% 3.31/1.67 => ( ( image_nat_nat
% 3.31/1.67 @ ^ [X_3: nat] : C_32
% 3.31/1.67 @ A_179 )
% 3.31/1.67 = bot_bot_nat_o ) )
% 3.31/1.67 & ( ( A_179 != bot_bot_nat_o )
% 3.31/1.67 => ( ( image_nat_nat
% 3.31/1.67 @ ^ [X_3: nat] : C_32
% 3.31/1.67 @ A_179 )
% 3.31/1.67 = ( insert_nat @ C_32 @ bot_bot_nat_o ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_166_image__constant__conv,axiom,
% 3.31/1.67 ! [C_32: hoare_1167836817_state,A_179: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( ( A_179 = bot_bo70021908tate_o )
% 3.31/1.67 => ( ( image_31595733_state
% 3.31/1.67 @ ^ [X_3: hoare_1167836817_state] : C_32
% 3.31/1.67 @ A_179 )
% 3.31/1.67 = bot_bo70021908tate_o ) )
% 3.31/1.67 & ( ( A_179 != bot_bo70021908tate_o )
% 3.31/1.67 => ( ( image_31595733_state
% 3.31/1.67 @ ^ [X_3: hoare_1167836817_state] : C_32
% 3.31/1.67 @ A_179 )
% 3.31/1.67 = ( insert2134838167_state @ C_32 @ bot_bo70021908tate_o ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_167_image__constant__conv,axiom,
% 3.31/1.67 ! [C_32: hoare_1775062406iple_a,A_179: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ( A_179 = bot_bo751897185le_a_o )
% 3.31/1.67 => ( ( image_1170193413iple_a
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] : C_32
% 3.31/1.67 @ A_179 )
% 3.31/1.67 = bot_bo751897185le_a_o ) )
% 3.31/1.67 & ( ( A_179 != bot_bo751897185le_a_o )
% 3.31/1.67 => ( ( image_1170193413iple_a
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] : C_32
% 3.31/1.67 @ A_179 )
% 3.31/1.67 = ( insert1281456128iple_a @ C_32 @ bot_bo751897185le_a_o ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_168_image__constant,axiom,
% 3.31/1.67 ! [C_31: hoare_1775062406iple_a,X_91: nat,A_178: nat > $o] :
% 3.31/1.67 ( ( member_nat @ X_91 @ A_178 )
% 3.31/1.67 => ( ( image_43014529iple_a
% 3.31/1.67 @ ^ [X_3: nat] : C_31
% 3.31/1.67 @ A_178 )
% 3.31/1.67 = ( insert1281456128iple_a @ C_31 @ bot_bo751897185le_a_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_169_image__constant,axiom,
% 3.31/1.67 ! [C_31: hoare_1775062406iple_a,X_91: hoare_1775062406iple_a,A_178: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ X_91 @ A_178 )
% 3.31/1.67 => ( ( image_1170193413iple_a
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] : C_31
% 3.31/1.67 @ A_178 )
% 3.31/1.67 = ( insert1281456128iple_a @ C_31 @ bot_bo751897185le_a_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_170_image__constant,axiom,
% 3.31/1.67 ! [C_31: hoare_1775062406iple_a,X_91: hoare_1167836817_state,A_178: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( member2058392318_state @ X_91 @ A_178 )
% 3.31/1.67 => ( ( image_1802845250iple_a
% 3.31/1.67 @ ^ [X_3: hoare_1167836817_state] : C_31
% 3.31/1.67 @ A_178 )
% 3.31/1.67 = ( insert1281456128iple_a @ C_31 @ bot_bo751897185le_a_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_171_image__constant,axiom,
% 3.31/1.67 ! [C_31: nat,X_91: nat,A_178: nat > $o] :
% 3.31/1.67 ( ( member_nat @ X_91 @ A_178 )
% 3.31/1.67 => ( ( image_nat_nat
% 3.31/1.67 @ ^ [X_3: nat] : C_31
% 3.31/1.67 @ A_178 )
% 3.31/1.67 = ( insert_nat @ C_31 @ bot_bot_nat_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_172_image__constant,axiom,
% 3.31/1.67 ! [C_31: nat,X_91: hoare_1775062406iple_a,A_178: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ X_91 @ A_178 )
% 3.31/1.67 => ( ( image_1806517641_a_nat
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] : C_31
% 3.31/1.67 @ A_178 )
% 3.31/1.67 = ( insert_nat @ C_31 @ bot_bot_nat_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_173_image__constant,axiom,
% 3.31/1.67 ! [C_31: nat,X_91: hoare_1167836817_state,A_178: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( member2058392318_state @ X_91 @ A_178 )
% 3.31/1.67 => ( ( image_1476618182te_nat
% 3.31/1.67 @ ^ [X_3: hoare_1167836817_state] : C_31
% 3.31/1.67 @ A_178 )
% 3.31/1.67 = ( insert_nat @ C_31 @ bot_bot_nat_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_174_image__constant,axiom,
% 3.31/1.67 ! [C_31: hoare_1167836817_state,X_91: hoare_1167836817_state,A_178: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( member2058392318_state @ X_91 @ A_178 )
% 3.31/1.67 => ( ( image_31595733_state
% 3.31/1.67 @ ^ [X_3: hoare_1167836817_state] : C_31
% 3.31/1.67 @ A_178 )
% 3.31/1.67 = ( insert2134838167_state @ C_31 @ bot_bo70021908tate_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_175_image__constant,axiom,
% 3.31/1.67 ! [C_31: hoare_1167836817_state,X_91: hoare_1775062406iple_a,A_178: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ X_91 @ A_178 )
% 3.31/1.67 => ( ( image_1021683026_state
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] : C_31
% 3.31/1.67 @ A_178 )
% 3.31/1.67 = ( insert2134838167_state @ C_31 @ bot_bo70021908tate_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_176_image__eqI,axiom,
% 3.31/1.67 ! [A_177: hoare_1167836817_state > $o,B_86: hoare_1167836817_state,F_65: hoare_1167836817_state > hoare_1167836817_state,X_90: hoare_1167836817_state] :
% 3.31/1.67 ( ( B_86
% 3.31/1.67 = ( F_65 @ X_90 ) )
% 3.31/1.67 => ( ( member2058392318_state @ X_90 @ A_177 )
% 3.31/1.67 => ( member2058392318_state @ B_86 @ ( image_31595733_state @ F_65 @ A_177 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_177_image__eqI,axiom,
% 3.31/1.67 ! [A_177: nat > $o,B_86: hoare_1167836817_state,F_65: nat > hoare_1167836817_state,X_90: nat] :
% 3.31/1.67 ( ( B_86
% 3.31/1.67 = ( F_65 @ X_90 ) )
% 3.31/1.67 => ( ( member_nat @ X_90 @ A_177 )
% 3.31/1.67 => ( member2058392318_state @ B_86 @ ( image_2121260246_state @ F_65 @ A_177 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_178_image__eqI,axiom,
% 3.31/1.67 ! [A_177: hoare_1775062406iple_a > $o,B_86: hoare_1167836817_state,F_65: hoare_1775062406iple_a > hoare_1167836817_state,X_90: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( B_86
% 3.31/1.67 = ( F_65 @ X_90 ) )
% 3.31/1.67 => ( ( member2122167641iple_a @ X_90 @ A_177 )
% 3.31/1.67 => ( member2058392318_state @ B_86 @ ( image_1021683026_state @ F_65 @ A_177 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_179_image__eqI,axiom,
% 3.31/1.67 ! [A_177: nat > $o,B_86: nat,F_65: nat > nat,X_90: nat] :
% 3.31/1.67 ( ( B_86
% 3.31/1.67 = ( F_65 @ X_90 ) )
% 3.31/1.67 => ( ( member_nat @ X_90 @ A_177 )
% 3.31/1.67 => ( member_nat @ B_86 @ ( image_nat_nat @ F_65 @ A_177 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_180_image__eqI,axiom,
% 3.31/1.67 ! [A_177: hoare_1775062406iple_a > $o,B_86: nat,F_65: hoare_1775062406iple_a > nat,X_90: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( B_86
% 3.31/1.67 = ( F_65 @ X_90 ) )
% 3.31/1.67 => ( ( member2122167641iple_a @ X_90 @ A_177 )
% 3.31/1.67 => ( member_nat @ B_86 @ ( image_1806517641_a_nat @ F_65 @ A_177 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_181_image__eqI,axiom,
% 3.31/1.67 ! [A_177: hoare_1167836817_state > $o,B_86: nat,F_65: hoare_1167836817_state > nat,X_90: hoare_1167836817_state] :
% 3.31/1.67 ( ( B_86
% 3.31/1.67 = ( F_65 @ X_90 ) )
% 3.31/1.67 => ( ( member2058392318_state @ X_90 @ A_177 )
% 3.31/1.67 => ( member_nat @ B_86 @ ( image_1476618182te_nat @ F_65 @ A_177 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_182_image__eqI,axiom,
% 3.31/1.67 ! [A_177: hoare_1775062406iple_a > $o,B_86: hoare_1775062406iple_a,F_65: hoare_1775062406iple_a > hoare_1775062406iple_a,X_90: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( B_86
% 3.31/1.67 = ( F_65 @ X_90 ) )
% 3.31/1.67 => ( ( member2122167641iple_a @ X_90 @ A_177 )
% 3.31/1.67 => ( member2122167641iple_a @ B_86 @ ( image_1170193413iple_a @ F_65 @ A_177 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_183_image__eqI,axiom,
% 3.31/1.67 ! [A_177: hoare_1167836817_state > $o,B_86: hoare_1775062406iple_a,F_65: hoare_1167836817_state > hoare_1775062406iple_a,X_90: hoare_1167836817_state] :
% 3.31/1.67 ( ( B_86
% 3.31/1.67 = ( F_65 @ X_90 ) )
% 3.31/1.67 => ( ( member2058392318_state @ X_90 @ A_177 )
% 3.31/1.67 => ( member2122167641iple_a @ B_86 @ ( image_1802845250iple_a @ F_65 @ A_177 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_184_image__eqI,axiom,
% 3.31/1.67 ! [A_177: nat > $o,B_86: hoare_1775062406iple_a,F_65: nat > hoare_1775062406iple_a,X_90: nat] :
% 3.31/1.67 ( ( B_86
% 3.31/1.67 = ( F_65 @ X_90 ) )
% 3.31/1.67 => ( ( member_nat @ X_90 @ A_177 )
% 3.31/1.67 => ( member2122167641iple_a @ B_86 @ ( image_43014529iple_a @ F_65 @ A_177 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_185_image__ident,axiom,
% 3.31/1.67 ! [Y_40: nat > $o] :
% 3.31/1.67 ( ( image_nat_nat
% 3.31/1.67 @ ^ [X_3: nat] : X_3
% 3.31/1.67 @ Y_40 )
% 3.31/1.67 = Y_40 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_186_image__ident,axiom,
% 3.31/1.67 ! [Y_40: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( image_31595733_state
% 3.31/1.67 @ ^ [X_3: hoare_1167836817_state] : X_3
% 3.31/1.67 @ Y_40 )
% 3.31/1.67 = Y_40 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_187_image__ident,axiom,
% 3.31/1.67 ! [Y_40: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( image_1170193413iple_a
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] : X_3
% 3.31/1.67 @ Y_40 )
% 3.31/1.67 = Y_40 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_188_com_Osimps_I1_J,axiom,
% 3.31/1.67 ! [Vname: vname,Fun_1: state > nat,Vname_1: vname,Fun: state > nat] :
% 3.31/1.67 ( ( ( ass @ Vname @ Fun_1 )
% 3.31/1.67 = ( ass @ Vname_1 @ Fun ) )
% 3.31/1.67 <=> ( ( Vname = Vname_1 )
% 3.31/1.67 & ( Fun_1 = Fun ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_189_rev__image__eqI,axiom,
% 3.31/1.67 ! [B_85: hoare_1167836817_state,F_64: hoare_1167836817_state > hoare_1167836817_state,X_89: hoare_1167836817_state,A_176: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( member2058392318_state @ X_89 @ A_176 )
% 3.31/1.67 => ( ( B_85
% 3.31/1.67 = ( F_64 @ X_89 ) )
% 3.31/1.67 => ( member2058392318_state @ B_85 @ ( image_31595733_state @ F_64 @ A_176 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_190_rev__image__eqI,axiom,
% 3.31/1.67 ! [B_85: hoare_1167836817_state,F_64: nat > hoare_1167836817_state,X_89: nat,A_176: nat > $o] :
% 3.31/1.67 ( ( member_nat @ X_89 @ A_176 )
% 3.31/1.67 => ( ( B_85
% 3.31/1.67 = ( F_64 @ X_89 ) )
% 3.31/1.67 => ( member2058392318_state @ B_85 @ ( image_2121260246_state @ F_64 @ A_176 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_191_rev__image__eqI,axiom,
% 3.31/1.67 ! [B_85: hoare_1167836817_state,F_64: hoare_1775062406iple_a > hoare_1167836817_state,X_89: hoare_1775062406iple_a,A_176: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ X_89 @ A_176 )
% 3.31/1.67 => ( ( B_85
% 3.31/1.67 = ( F_64 @ X_89 ) )
% 3.31/1.67 => ( member2058392318_state @ B_85 @ ( image_1021683026_state @ F_64 @ A_176 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_192_rev__image__eqI,axiom,
% 3.31/1.67 ! [B_85: nat,F_64: nat > nat,X_89: nat,A_176: nat > $o] :
% 3.31/1.67 ( ( member_nat @ X_89 @ A_176 )
% 3.31/1.67 => ( ( B_85
% 3.31/1.67 = ( F_64 @ X_89 ) )
% 3.31/1.67 => ( member_nat @ B_85 @ ( image_nat_nat @ F_64 @ A_176 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_193_rev__image__eqI,axiom,
% 3.31/1.67 ! [B_85: nat,F_64: hoare_1775062406iple_a > nat,X_89: hoare_1775062406iple_a,A_176: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ X_89 @ A_176 )
% 3.31/1.67 => ( ( B_85
% 3.31/1.67 = ( F_64 @ X_89 ) )
% 3.31/1.67 => ( member_nat @ B_85 @ ( image_1806517641_a_nat @ F_64 @ A_176 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_194_rev__image__eqI,axiom,
% 3.31/1.67 ! [B_85: nat,F_64: hoare_1167836817_state > nat,X_89: hoare_1167836817_state,A_176: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( member2058392318_state @ X_89 @ A_176 )
% 3.31/1.67 => ( ( B_85
% 3.31/1.67 = ( F_64 @ X_89 ) )
% 3.31/1.67 => ( member_nat @ B_85 @ ( image_1476618182te_nat @ F_64 @ A_176 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_195_rev__image__eqI,axiom,
% 3.31/1.67 ! [B_85: hoare_1775062406iple_a,F_64: hoare_1775062406iple_a > hoare_1775062406iple_a,X_89: hoare_1775062406iple_a,A_176: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ X_89 @ A_176 )
% 3.31/1.67 => ( ( B_85
% 3.31/1.67 = ( F_64 @ X_89 ) )
% 3.31/1.67 => ( member2122167641iple_a @ B_85 @ ( image_1170193413iple_a @ F_64 @ A_176 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_196_rev__image__eqI,axiom,
% 3.31/1.67 ! [B_85: hoare_1775062406iple_a,F_64: hoare_1167836817_state > hoare_1775062406iple_a,X_89: hoare_1167836817_state,A_176: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( member2058392318_state @ X_89 @ A_176 )
% 3.31/1.67 => ( ( B_85
% 3.31/1.67 = ( F_64 @ X_89 ) )
% 3.31/1.67 => ( member2122167641iple_a @ B_85 @ ( image_1802845250iple_a @ F_64 @ A_176 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_197_rev__image__eqI,axiom,
% 3.31/1.67 ! [B_85: hoare_1775062406iple_a,F_64: nat > hoare_1775062406iple_a,X_89: nat,A_176: nat > $o] :
% 3.31/1.67 ( ( member_nat @ X_89 @ A_176 )
% 3.31/1.67 => ( ( B_85
% 3.31/1.67 = ( F_64 @ X_89 ) )
% 3.31/1.67 => ( member2122167641iple_a @ B_85 @ ( image_43014529iple_a @ F_64 @ A_176 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_198_imageI,axiom,
% 3.31/1.67 ! [F_63: hoare_1167836817_state > hoare_1167836817_state,X_88: hoare_1167836817_state,A_175: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( member2058392318_state @ X_88 @ A_175 )
% 3.31/1.67 => ( member2058392318_state @ ( F_63 @ X_88 ) @ ( image_31595733_state @ F_63 @ A_175 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_199_imageI,axiom,
% 3.31/1.67 ! [F_63: nat > hoare_1167836817_state,X_88: nat,A_175: nat > $o] :
% 3.31/1.67 ( ( member_nat @ X_88 @ A_175 )
% 3.31/1.67 => ( member2058392318_state @ ( F_63 @ X_88 ) @ ( image_2121260246_state @ F_63 @ A_175 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_200_imageI,axiom,
% 3.31/1.67 ! [F_63: hoare_1775062406iple_a > hoare_1167836817_state,X_88: hoare_1775062406iple_a,A_175: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ X_88 @ A_175 )
% 3.31/1.67 => ( member2058392318_state @ ( F_63 @ X_88 ) @ ( image_1021683026_state @ F_63 @ A_175 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_201_imageI,axiom,
% 3.31/1.67 ! [F_63: nat > nat,X_88: nat,A_175: nat > $o] :
% 3.31/1.67 ( ( member_nat @ X_88 @ A_175 )
% 3.31/1.67 => ( member_nat @ ( F_63 @ X_88 ) @ ( image_nat_nat @ F_63 @ A_175 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_202_imageI,axiom,
% 3.31/1.67 ! [F_63: hoare_1775062406iple_a > nat,X_88: hoare_1775062406iple_a,A_175: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ X_88 @ A_175 )
% 3.31/1.67 => ( member_nat @ ( F_63 @ X_88 ) @ ( image_1806517641_a_nat @ F_63 @ A_175 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_203_imageI,axiom,
% 3.31/1.67 ! [F_63: hoare_1167836817_state > nat,X_88: hoare_1167836817_state,A_175: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( member2058392318_state @ X_88 @ A_175 )
% 3.31/1.67 => ( member_nat @ ( F_63 @ X_88 ) @ ( image_1476618182te_nat @ F_63 @ A_175 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_204_imageI,axiom,
% 3.31/1.67 ! [F_63: hoare_1775062406iple_a > hoare_1775062406iple_a,X_88: hoare_1775062406iple_a,A_175: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ X_88 @ A_175 )
% 3.31/1.67 => ( member2122167641iple_a @ ( F_63 @ X_88 ) @ ( image_1170193413iple_a @ F_63 @ A_175 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_205_imageI,axiom,
% 3.31/1.67 ! [F_63: hoare_1167836817_state > hoare_1775062406iple_a,X_88: hoare_1167836817_state,A_175: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( member2058392318_state @ X_88 @ A_175 )
% 3.31/1.67 => ( member2122167641iple_a @ ( F_63 @ X_88 ) @ ( image_1802845250iple_a @ F_63 @ A_175 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_206_imageI,axiom,
% 3.31/1.67 ! [F_63: nat > hoare_1775062406iple_a,X_88: nat,A_175: nat > $o] :
% 3.31/1.67 ( ( member_nat @ X_88 @ A_175 )
% 3.31/1.67 => ( member2122167641iple_a @ ( F_63 @ X_88 ) @ ( image_43014529iple_a @ F_63 @ A_175 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_207_image__iff,axiom,
% 3.31/1.67 ! [Z_22: hoare_1167836817_state,F_62: hoare_1167836817_state > hoare_1167836817_state,A_174: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( member2058392318_state @ Z_22 @ ( image_31595733_state @ F_62 @ A_174 ) )
% 3.31/1.67 <=> ? [X_3: hoare_1167836817_state] :
% 3.31/1.67 ( ( member2058392318_state @ X_3 @ A_174 )
% 3.31/1.67 & ( Z_22
% 3.31/1.67 = ( F_62 @ X_3 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_208_image__iff,axiom,
% 3.31/1.67 ! [Z_22: hoare_1167836817_state,F_62: hoare_1775062406iple_a > hoare_1167836817_state,A_174: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2058392318_state @ Z_22 @ ( image_1021683026_state @ F_62 @ A_174 ) )
% 3.31/1.67 <=> ? [X_3: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( member2122167641iple_a @ X_3 @ A_174 )
% 3.31/1.67 & ( Z_22
% 3.31/1.67 = ( F_62 @ X_3 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_209_image__iff,axiom,
% 3.31/1.67 ! [Z_22: nat,F_62: nat > nat,A_174: nat > $o] :
% 3.31/1.67 ( ( member_nat @ Z_22 @ ( image_nat_nat @ F_62 @ A_174 ) )
% 3.31/1.67 <=> ? [X_3: nat] :
% 3.31/1.67 ( ( member_nat @ X_3 @ A_174 )
% 3.31/1.67 & ( Z_22
% 3.31/1.67 = ( F_62 @ X_3 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_210_image__iff,axiom,
% 3.31/1.67 ! [Z_22: nat,F_62: hoare_1775062406iple_a > nat,A_174: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member_nat @ Z_22 @ ( image_1806517641_a_nat @ F_62 @ A_174 ) )
% 3.31/1.67 <=> ? [X_3: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( member2122167641iple_a @ X_3 @ A_174 )
% 3.31/1.67 & ( Z_22
% 3.31/1.67 = ( F_62 @ X_3 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_211_image__iff,axiom,
% 3.31/1.67 ! [Z_22: hoare_1775062406iple_a,F_62: hoare_1775062406iple_a > hoare_1775062406iple_a,A_174: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ Z_22 @ ( image_1170193413iple_a @ F_62 @ A_174 ) )
% 3.31/1.67 <=> ? [X_3: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( member2122167641iple_a @ X_3 @ A_174 )
% 3.31/1.67 & ( Z_22
% 3.31/1.67 = ( F_62 @ X_3 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_212_image__iff,axiom,
% 3.31/1.67 ! [Z_22: hoare_1775062406iple_a,F_62: hoare_1167836817_state > hoare_1775062406iple_a,A_174: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ Z_22 @ ( image_1802845250iple_a @ F_62 @ A_174 ) )
% 3.31/1.67 <=> ? [X_3: hoare_1167836817_state] :
% 3.31/1.67 ( ( member2058392318_state @ X_3 @ A_174 )
% 3.31/1.67 & ( Z_22
% 3.31/1.67 = ( F_62 @ X_3 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_213_image__iff,axiom,
% 3.31/1.67 ! [Z_22: hoare_1775062406iple_a,F_62: nat > hoare_1775062406iple_a,A_174: nat > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ Z_22 @ ( image_43014529iple_a @ F_62 @ A_174 ) )
% 3.31/1.67 <=> ? [X_3: nat] :
% 3.31/1.67 ( ( member_nat @ X_3 @ A_174 )
% 3.31/1.67 & ( Z_22
% 3.31/1.67 = ( F_62 @ X_3 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_214_com_Osimps_I24_J,axiom,
% 3.31/1.67 ! [Vname: vname,Fun_1: state > nat,Com1: com,Com2: com] :
% 3.31/1.67 ( ( ass @ Vname @ Fun_1 )
% 3.31/1.67 != ( semi @ Com1 @ Com2 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_215_com_Osimps_I25_J,axiom,
% 3.31/1.67 ! [Com1: com,Com2: com,Vname: vname,Fun_1: state > nat] :
% 3.31/1.67 ( ( semi @ Com1 @ Com2 )
% 3.31/1.67 != ( ass @ Vname @ Fun_1 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_216_com_Osimps_I8_J,axiom,
% 3.31/1.67 ! [Vname_1: vname,Fun: state > nat] :
% 3.31/1.67 ( skip
% 3.31/1.67 != ( ass @ Vname_1 @ Fun ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_217_com_Osimps_I9_J,axiom,
% 3.31/1.67 ! [Vname_1: vname,Fun: state > nat] :
% 3.31/1.67 ( ( ass @ Vname_1 @ Fun )
% 3.31/1.67 != skip ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_218_image__is__empty,axiom,
% 3.31/1.67 ! [F_61: hoare_1167836817_state > hoare_1775062406iple_a,A_173: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( ( image_1802845250iple_a @ F_61 @ A_173 )
% 3.31/1.67 = bot_bo751897185le_a_o )
% 3.31/1.67 <=> ( A_173 = bot_bo70021908tate_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_219_image__is__empty,axiom,
% 3.31/1.67 ! [F_61: nat > hoare_1775062406iple_a,A_173: nat > $o] :
% 3.31/1.67 ( ( ( image_43014529iple_a @ F_61 @ A_173 )
% 3.31/1.67 = bot_bo751897185le_a_o )
% 3.31/1.67 <=> ( A_173 = bot_bot_nat_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_220_image__is__empty,axiom,
% 3.31/1.67 ! [F_61: hoare_1775062406iple_a > hoare_1167836817_state,A_173: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ( image_1021683026_state @ F_61 @ A_173 )
% 3.31/1.67 = bot_bo70021908tate_o )
% 3.31/1.67 <=> ( A_173 = bot_bo751897185le_a_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_221_image__is__empty,axiom,
% 3.31/1.67 ! [F_61: hoare_1775062406iple_a > nat,A_173: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ( image_1806517641_a_nat @ F_61 @ A_173 )
% 3.31/1.67 = bot_bot_nat_o )
% 3.31/1.67 <=> ( A_173 = bot_bo751897185le_a_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_222_image__empty,axiom,
% 3.31/1.67 ! [F_60: hoare_1775062406iple_a > hoare_1167836817_state] :
% 3.31/1.67 ( ( image_1021683026_state @ F_60 @ bot_bo751897185le_a_o )
% 3.31/1.67 = bot_bo70021908tate_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_223_image__empty,axiom,
% 3.31/1.67 ! [F_60: hoare_1775062406iple_a > nat] :
% 3.31/1.67 ( ( image_1806517641_a_nat @ F_60 @ bot_bo751897185le_a_o )
% 3.31/1.67 = bot_bot_nat_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_224_image__empty,axiom,
% 3.31/1.67 ! [F_60: hoare_1167836817_state > hoare_1775062406iple_a] :
% 3.31/1.67 ( ( image_1802845250iple_a @ F_60 @ bot_bo70021908tate_o )
% 3.31/1.67 = bot_bo751897185le_a_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_225_image__empty,axiom,
% 3.31/1.67 ! [F_60: nat > hoare_1775062406iple_a] :
% 3.31/1.67 ( ( image_43014529iple_a @ F_60 @ bot_bot_nat_o )
% 3.31/1.67 = bot_bo751897185le_a_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_226_empty__is__image,axiom,
% 3.31/1.67 ! [F_59: hoare_1167836817_state > hoare_1775062406iple_a,A_172: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( bot_bo751897185le_a_o
% 3.31/1.67 = ( image_1802845250iple_a @ F_59 @ A_172 ) )
% 3.31/1.67 <=> ( A_172 = bot_bo70021908tate_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_227_empty__is__image,axiom,
% 3.31/1.67 ! [F_59: nat > hoare_1775062406iple_a,A_172: nat > $o] :
% 3.31/1.67 ( ( bot_bo751897185le_a_o
% 3.31/1.67 = ( image_43014529iple_a @ F_59 @ A_172 ) )
% 3.31/1.67 <=> ( A_172 = bot_bot_nat_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_228_empty__is__image,axiom,
% 3.31/1.67 ! [F_59: hoare_1775062406iple_a > hoare_1167836817_state,A_172: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( bot_bo70021908tate_o
% 3.31/1.67 = ( image_1021683026_state @ F_59 @ A_172 ) )
% 3.31/1.67 <=> ( A_172 = bot_bo751897185le_a_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_229_empty__is__image,axiom,
% 3.31/1.67 ! [F_59: hoare_1775062406iple_a > nat,A_172: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( bot_bot_nat_o
% 3.31/1.67 = ( image_1806517641_a_nat @ F_59 @ A_172 ) )
% 3.31/1.67 <=> ( A_172 = bot_bo751897185le_a_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_230_mem__def,axiom,
% 3.31/1.67 ! [X_87: hoare_1775062406iple_a,A_171: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ X_87 @ A_171 )
% 3.31/1.67 <=> ( A_171 @ X_87 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_231_mem__def,axiom,
% 3.31/1.67 ! [X_87: nat,A_171: nat > $o] :
% 3.31/1.67 ( ( member_nat @ X_87 @ A_171 )
% 3.31/1.67 <=> ( A_171 @ X_87 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_232_Collect__def,axiom,
% 3.31/1.67 ! [P_16: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( collec676402587iple_a @ P_16 )
% 3.31/1.67 = P_16 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_233_Collect__def,axiom,
% 3.31/1.67 ! [P_16: nat > $o] :
% 3.31/1.67 ( ( collect_nat @ P_16 )
% 3.31/1.67 = P_16 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_234_insert__image,axiom,
% 3.31/1.67 ! [F_58: hoare_1775062406iple_a > hoare_1775062406iple_a,X_86: hoare_1775062406iple_a,A_170: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ X_86 @ A_170 )
% 3.31/1.67 => ( ( insert1281456128iple_a @ ( F_58 @ X_86 ) @ ( image_1170193413iple_a @ F_58 @ A_170 ) )
% 3.31/1.67 = ( image_1170193413iple_a @ F_58 @ A_170 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_235_insert__image,axiom,
% 3.31/1.67 ! [F_58: nat > hoare_1775062406iple_a,X_86: nat,A_170: nat > $o] :
% 3.31/1.67 ( ( member_nat @ X_86 @ A_170 )
% 3.31/1.67 => ( ( insert1281456128iple_a @ ( F_58 @ X_86 ) @ ( image_43014529iple_a @ F_58 @ A_170 ) )
% 3.31/1.67 = ( image_43014529iple_a @ F_58 @ A_170 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_236_image__insert,axiom,
% 3.31/1.67 ! [F_57: hoare_1775062406iple_a > hoare_1167836817_state,A_169: hoare_1775062406iple_a,B_84: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( image_1021683026_state @ F_57 @ ( insert1281456128iple_a @ A_169 @ B_84 ) )
% 3.31/1.67 = ( insert2134838167_state @ ( F_57 @ A_169 ) @ ( image_1021683026_state @ F_57 @ B_84 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_237_image__insert,axiom,
% 3.31/1.67 ! [F_57: hoare_1775062406iple_a > nat,A_169: hoare_1775062406iple_a,B_84: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( image_1806517641_a_nat @ F_57 @ ( insert1281456128iple_a @ A_169 @ B_84 ) )
% 3.31/1.67 = ( insert_nat @ ( F_57 @ A_169 ) @ ( image_1806517641_a_nat @ F_57 @ B_84 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_238_image__insert,axiom,
% 3.31/1.67 ! [F_57: hoare_1167836817_state > hoare_1775062406iple_a,A_169: hoare_1167836817_state,B_84: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( image_1802845250iple_a @ F_57 @ ( insert2134838167_state @ A_169 @ B_84 ) )
% 3.31/1.67 = ( insert1281456128iple_a @ ( F_57 @ A_169 ) @ ( image_1802845250iple_a @ F_57 @ B_84 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_239_image__insert,axiom,
% 3.31/1.67 ! [F_57: nat > hoare_1775062406iple_a,A_169: nat,B_84: nat > $o] :
% 3.31/1.67 ( ( image_43014529iple_a @ F_57 @ ( insert_nat @ A_169 @ B_84 ) )
% 3.31/1.67 = ( insert1281456128iple_a @ ( F_57 @ A_169 ) @ ( image_43014529iple_a @ F_57 @ B_84 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_240_the__sym__eq__trivial,axiom,
% 3.31/1.67 ! [X_85: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( the_Ho1155011127iple_a @ ( fequal1288209029iple_a @ X_85 ) )
% 3.31/1.67 = X_85 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_241_the__eq__trivial,axiom,
% 3.31/1.67 ! [A_168: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( the_Ho1155011127iple_a
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] : ( X_3 = A_168 ) )
% 3.31/1.67 = A_168 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_242_If__def,axiom,
% 3.31/1.67 ! [X_84: hoare_1775062406iple_a,Y_39: hoare_1775062406iple_a,P_15: $o] :
% 3.31/1.67 ( ( P_15
% 3.31/1.67 => ( X_84
% 3.31/1.67 = ( the_Ho1155011127iple_a
% 3.31/1.67 @ ^ [Z_21: hoare_1775062406iple_a] : ( (&) @ ( (=>) @ P_15 @ ( Z_21 = X_84 ) ) @ ( (=>) @ ( (~) @ P_15 ) @ ( Z_21 = Y_39 ) ) ) ) ) )
% 3.31/1.67 & ( ~ P_15
% 3.31/1.67 => ( Y_39
% 3.31/1.67 = ( the_Ho1155011127iple_a
% 3.31/1.67 @ ^ [Z_21: hoare_1775062406iple_a] : ( (&) @ ( (=>) @ P_15 @ ( Z_21 = X_84 ) ) @ ( (=>) @ ( (~) @ P_15 ) @ ( Z_21 = Y_39 ) ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_243_fold1Set__sing,axiom,
% 3.31/1.67 ! [F_56: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,A_167: hoare_1167836817_state,B_83: hoare_1167836817_state] :
% 3.31/1.67 ( ( finite309220289_state @ F_56 @ ( insert2134838167_state @ A_167 @ bot_bo70021908tate_o ) @ B_83 )
% 3.31/1.67 <=> ( A_167 = B_83 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_244_fold1Set__sing,axiom,
% 3.31/1.67 ! [F_56: nat > nat > nat,A_167: nat,B_83: nat] :
% 3.31/1.67 ( ( finite_fold1Set_nat @ F_56 @ ( insert_nat @ A_167 @ bot_bot_nat_o ) @ B_83 )
% 3.31/1.67 <=> ( A_167 = B_83 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_245_fold1Set__sing,axiom,
% 3.31/1.67 ! [F_56: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,A_167: hoare_1775062406iple_a,B_83: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( finite1946188886iple_a @ F_56 @ ( insert1281456128iple_a @ A_167 @ bot_bo751897185le_a_o ) @ B_83 )
% 3.31/1.67 <=> ( A_167 = B_83 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_246_the__equality,axiom,
% 3.31/1.67 ! [P_14: hoare_1775062406iple_a > $o,A_166: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( P_14 @ A_166 )
% 3.31/1.67 => ( ! [X_3: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( P_14 @ X_3 )
% 3.31/1.67 => ( X_3 = A_166 ) )
% 3.31/1.67 => ( ( the_Ho1155011127iple_a @ P_14 )
% 3.31/1.67 = A_166 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_247_folding__one_Osingleton,axiom,
% 3.31/1.67 ! [X_83: hoare_1167836817_state,F_55: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_54: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
% 3.31/1.67 ( ( finite1074406356_state @ F_55 @ F_54 )
% 3.31/1.67 => ( ( F_54 @ ( insert2134838167_state @ X_83 @ bot_bo70021908tate_o ) )
% 3.31/1.67 = X_83 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_248_folding__one_Osingleton,axiom,
% 3.31/1.67 ! [X_83: nat,F_55: nat > nat > nat,F_54: ( nat > $o ) > nat] :
% 3.31/1.67 ( ( finite988810631ne_nat @ F_55 @ F_54 )
% 3.31/1.67 => ( ( F_54 @ ( insert_nat @ X_83 @ bot_bot_nat_o ) )
% 3.31/1.67 = X_83 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_249_folding__one_Osingleton,axiom,
% 3.31/1.67 ! [X_83: hoare_1775062406iple_a,F_55: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_54: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
% 3.31/1.67 ( ( finite2078349315iple_a @ F_55 @ F_54 )
% 3.31/1.67 => ( ( F_54 @ ( insert1281456128iple_a @ X_83 @ bot_bo751897185le_a_o ) )
% 3.31/1.67 = X_83 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_250_hoare__derivs_OLocal,axiom,
% 3.31/1.67 ! [A_165: state > nat,G_6: hoare_1167836817_state > $o,P_13: state > state > $o,C_30: com,Q_5: state > state > $o,X_82: loc,S_6: state] :
% 3.31/1.67 ( ( hoare_123228589_state @ G_6
% 3.31/1.67 @ ( insert2134838167_state
% 3.31/1.67 @ ( hoare_908217195_state @ P_13 @ C_30
% 3.31/1.67 @ ^ [Z: state,S: state] : ( Q_5 @ Z @ ( update @ S @ ( loc_1 @ X_82 ) @ ( getlocs @ S_6 @ X_82 ) ) ) )
% 3.31/1.67 @ bot_bo70021908tate_o ) )
% 3.31/1.67 => ( hoare_123228589_state @ G_6
% 3.31/1.67 @ ( insert2134838167_state
% 3.31/1.67 @ ( hoare_908217195_state
% 3.31/1.67 @ ^ [Z: state,S: state] : ( (&) @ ( S_6 = S ) @ ( P_13 @ Z @ ( update @ S @ ( loc_1 @ X_82 ) @ ( A_165 @ S ) ) ) )
% 3.31/1.67 @ ( local @ X_82 @ A_165 @ C_30 )
% 3.31/1.67 @ Q_5 )
% 3.31/1.67 @ bot_bo70021908tate_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_251_hoare__derivs_OLocal,axiom,
% 3.31/1.67 ! [A_165: state > nat,G_6: hoare_1775062406iple_a > $o,P_13: x_a > state > $o,C_30: com,Q_5: x_a > state > $o,X_82: loc,S_6: state] :
% 3.31/1.67 ( ( hoare_1508237396rivs_a @ G_6
% 3.31/1.67 @ ( insert1281456128iple_a
% 3.31/1.67 @ ( hoare_1766022166iple_a @ P_13 @ C_30
% 3.31/1.67 @ ^ [Z: x_a,S: state] : ( Q_5 @ Z @ ( update @ S @ ( loc_1 @ X_82 ) @ ( getlocs @ S_6 @ X_82 ) ) ) )
% 3.31/1.67 @ bot_bo751897185le_a_o ) )
% 3.31/1.67 => ( hoare_1508237396rivs_a @ G_6
% 3.31/1.67 @ ( insert1281456128iple_a
% 3.31/1.67 @ ( hoare_1766022166iple_a
% 3.31/1.67 @ ^ [Z: x_a,S: state] : ( (&) @ ( S_6 = S ) @ ( P_13 @ Z @ ( update @ S @ ( loc_1 @ X_82 ) @ ( A_165 @ S ) ) ) )
% 3.31/1.67 @ ( local @ X_82 @ A_165 @ C_30 )
% 3.31/1.67 @ Q_5 )
% 3.31/1.67 @ bot_bo751897185le_a_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_252_vname_Osimps_I2_J,axiom,
% 3.31/1.67 ! [Loc_2: loc,Loc_1: loc] :
% 3.31/1.67 ( ( ( loc_1 @ Loc_2 )
% 3.31/1.67 = ( loc_1 @ Loc_1 ) )
% 3.31/1.67 <=> ( Loc_2 = Loc_1 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_253_com_Osimps_I2_J,axiom,
% 3.31/1.67 ! [Loc_2: loc,Fun_1: state > nat,Com_1: com,Loc_1: loc,Fun: state > nat,Com: com] :
% 3.31/1.67 ( ( ( local @ Loc_2 @ Fun_1 @ Com_1 )
% 3.31/1.67 = ( local @ Loc_1 @ Fun @ Com ) )
% 3.31/1.67 <=> ( ( Loc_2 = Loc_1 )
% 3.31/1.67 & ( Fun_1 = Fun )
% 3.31/1.67 & ( Com_1 = Com ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_254_com_Osimps_I34_J,axiom,
% 3.31/1.67 ! [Loc_2: loc,Fun_1: state > nat,Com_1: com,Com1: com,Com2: com] :
% 3.31/1.67 ( ( local @ Loc_2 @ Fun_1 @ Com_1 )
% 3.31/1.67 != ( semi @ Com1 @ Com2 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_255_com_Osimps_I35_J,axiom,
% 3.31/1.67 ! [Com1: com,Com2: com,Loc_2: loc,Fun_1: state > nat,Com_1: com] :
% 3.31/1.67 ( ( semi @ Com1 @ Com2 )
% 3.31/1.67 != ( local @ Loc_2 @ Fun_1 @ Com_1 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_256_com_Osimps_I23_J,axiom,
% 3.31/1.67 ! [Loc_1: loc,Fun: state > nat,Com: com,Vname: vname,Fun_1: state > nat] :
% 3.31/1.67 ( ( local @ Loc_1 @ Fun @ Com )
% 3.31/1.67 != ( ass @ Vname @ Fun_1 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_257_com_Osimps_I22_J,axiom,
% 3.31/1.67 ! [Vname: vname,Fun_1: state > nat,Loc_1: loc,Fun: state > nat,Com: com] :
% 3.31/1.67 ( ( ass @ Vname @ Fun_1 )
% 3.31/1.67 != ( local @ Loc_1 @ Fun @ Com ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_258_com_Osimps_I11_J,axiom,
% 3.31/1.67 ! [Loc_1: loc,Fun: state > nat,Com: com] :
% 3.31/1.67 ( ( local @ Loc_1 @ Fun @ Com )
% 3.31/1.67 != skip ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_259_com_Osimps_I10_J,axiom,
% 3.31/1.67 ! [Loc_1: loc,Fun: state > nat,Com: com] :
% 3.31/1.67 ( skip
% 3.31/1.67 != ( local @ Loc_1 @ Fun @ Com ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_260_empty__fold1SetE,axiom,
% 3.31/1.67 ! [F_53: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,X_81: hoare_1167836817_state] :
% 3.31/1.67 ~ ( finite309220289_state @ F_53 @ bot_bo70021908tate_o @ X_81 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_261_empty__fold1SetE,axiom,
% 3.31/1.67 ! [F_53: nat > nat > nat,X_81: nat] :
% 3.31/1.67 ~ ( finite_fold1Set_nat @ F_53 @ bot_bot_nat_o @ X_81 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_262_empty__fold1SetE,axiom,
% 3.31/1.67 ! [F_53: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,X_81: hoare_1775062406iple_a] :
% 3.31/1.67 ~ ( finite1946188886iple_a @ F_53 @ bot_bo751897185le_a_o @ X_81 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_263_fold1Set__nonempty,axiom,
% 3.31/1.67 ! [F_52: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,A_164: hoare_1167836817_state > $o,X_80: hoare_1167836817_state] :
% 3.31/1.67 ( ( finite309220289_state @ F_52 @ A_164 @ X_80 )
% 3.31/1.67 => ( A_164 != bot_bo70021908tate_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_264_fold1Set__nonempty,axiom,
% 3.31/1.67 ! [F_52: nat > nat > nat,A_164: nat > $o,X_80: nat] :
% 3.31/1.67 ( ( finite_fold1Set_nat @ F_52 @ A_164 @ X_80 )
% 3.31/1.67 => ( A_164 != bot_bot_nat_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_265_fold1Set__nonempty,axiom,
% 3.31/1.67 ! [F_52: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,A_164: hoare_1775062406iple_a > $o,X_80: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( finite1946188886iple_a @ F_52 @ A_164 @ X_80 )
% 3.31/1.67 => ( A_164 != bot_bo751897185le_a_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_266_theI,axiom,
% 3.31/1.67 ! [P_12: hoare_1775062406iple_a > $o,A_163: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( P_12 @ A_163 )
% 3.31/1.67 => ( ! [X_3: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( P_12 @ X_3 )
% 3.31/1.67 => ( X_3 = A_163 ) )
% 3.31/1.67 => ( P_12 @ ( the_Ho1155011127iple_a @ P_12 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_267_the1__equality,axiom,
% 3.31/1.67 ! [A_162: hoare_1775062406iple_a,P_11: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ? [X_3: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( P_11 @ X_3 )
% 3.31/1.67 & ! [Y_1: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( P_11 @ Y_1 )
% 3.31/1.67 => ( Y_1 = X_3 ) ) )
% 3.31/1.67 => ( ( P_11 @ A_162 )
% 3.31/1.67 => ( ( the_Ho1155011127iple_a @ P_11 )
% 3.31/1.67 = A_162 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_268_theI_H,axiom,
% 3.31/1.67 ! [P_10: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ? [X_3: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( P_10 @ X_3 )
% 3.31/1.67 & ! [Y_1: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( P_10 @ Y_1 )
% 3.31/1.67 => ( Y_1 = X_3 ) ) )
% 3.31/1.67 => ( P_10 @ ( the_Ho1155011127iple_a @ P_10 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_269_evalc_OLocal,axiom,
% 3.31/1.67 ! [C_9: com,S0: state,Y_37: loc,A_158: state > nat,S1: state] :
% 3.31/1.67 ( ( evalc @ C_9 @ ( update @ S0 @ ( loc_1 @ Y_37 ) @ ( A_158 @ S0 ) ) @ S1 )
% 3.31/1.67 => ( evalc @ ( local @ Y_37 @ A_158 @ C_9 ) @ S0 @ ( update @ S1 @ ( loc_1 @ Y_37 ) @ ( getlocs @ S0 @ Y_37 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_270_evaln_OLocal,axiom,
% 3.31/1.67 ! [C_9: com,S0: state,Y_37: loc,A_158: state > nat,N_3: nat,S1: state] :
% 3.31/1.67 ( ( evaln @ C_9 @ ( update @ S0 @ ( loc_1 @ Y_37 ) @ ( A_158 @ S0 ) ) @ N_3 @ S1 )
% 3.31/1.67 => ( evaln @ ( local @ Y_37 @ A_158 @ C_9 ) @ S0 @ N_3 @ ( update @ S1 @ ( loc_1 @ Y_37 ) @ ( getlocs @ S0 @ Y_37 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_271_fold1Set_Ointros,axiom,
% 3.31/1.67 ! [F_51: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,A_161: hoare_1167836817_state,A_160: hoare_1167836817_state > $o,X_79: hoare_1167836817_state] :
% 3.31/1.67 ( ( finite1316643734_state @ F_51 @ A_161 @ A_160 @ X_79 )
% 3.31/1.67 => ( ~ ( member2058392318_state @ A_161 @ A_160 )
% 3.31/1.67 => ( finite309220289_state @ F_51 @ ( insert2134838167_state @ A_161 @ A_160 ) @ X_79 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_272_fold1Set_Ointros,axiom,
% 3.31/1.67 ! [F_51: nat > nat > nat,A_161: nat,A_160: nat > $o,X_79: nat] :
% 3.31/1.67 ( ( finite929467206at_nat @ F_51 @ A_161 @ A_160 @ X_79 )
% 3.31/1.67 => ( ~ ( member_nat @ A_161 @ A_160 )
% 3.31/1.67 => ( finite_fold1Set_nat @ F_51 @ ( insert_nat @ A_161 @ A_160 ) @ X_79 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_273_fold1Set_Ointros,axiom,
% 3.31/1.67 ! [F_51: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,A_161: hoare_1775062406iple_a,A_160: hoare_1775062406iple_a > $o,X_79: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( finite727644230iple_a @ F_51 @ A_161 @ A_160 @ X_79 )
% 3.31/1.67 => ( ~ ( member2122167641iple_a @ A_161 @ A_160 )
% 3.31/1.67 => ( finite1946188886iple_a @ F_51 @ ( insert1281456128iple_a @ A_161 @ A_160 ) @ X_79 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_274_evaln_OSemi,axiom,
% 3.31/1.67 ! [C1: com,S2: state,C0: com,S0: state,N_3: nat,S1: state] :
% 3.31/1.67 ( ( evaln @ C0 @ S0 @ N_3 @ S1 )
% 3.31/1.67 => ( ( evaln @ C1 @ S1 @ N_3 @ S2 )
% 3.31/1.67 => ( evaln @ ( semi @ C0 @ C1 ) @ S0 @ N_3 @ S2 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_275_evaln_OSkip,axiom,
% 3.31/1.67 ! [S_2: state,N_3: nat] : ( evaln @ skip @ S_2 @ N_3 @ S_2 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_276_evaln__elim__cases_I1_J,axiom,
% 3.31/1.67 ! [S_2: state,N_3: nat,T: state] :
% 3.31/1.67 ( ( evaln @ skip @ S_2 @ N_3 @ T )
% 3.31/1.67 => ( T = S_2 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_277_evalc_OSemi,axiom,
% 3.31/1.67 ! [C1: com,S2: state,C0: com,S0: state,S1: state] :
% 3.31/1.67 ( ( evalc @ C0 @ S0 @ S1 )
% 3.31/1.67 => ( ( evalc @ C1 @ S1 @ S2 )
% 3.31/1.67 => ( evalc @ ( semi @ C0 @ C1 ) @ S0 @ S2 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_278_evalc_OSkip,axiom,
% 3.31/1.67 ! [S_2: state] : ( evalc @ skip @ S_2 @ S_2 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_279_evalc__elim__cases_I1_J,axiom,
% 3.31/1.67 ! [S_2: state,T: state] :
% 3.31/1.67 ( ( evalc @ skip @ S_2 @ T )
% 3.31/1.67 => ( T = S_2 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_280_evaln_OAssign,axiom,
% 3.31/1.67 ! [X_78: vname,A_158: state > nat,S_2: state,N_3: nat] : ( evaln @ ( ass @ X_78 @ A_158 ) @ S_2 @ N_3 @ ( update @ S_2 @ X_78 @ ( A_158 @ S_2 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_281_evaln__elim__cases_I2_J,axiom,
% 3.31/1.67 ! [X_78: vname,A_158: state > nat,S_2: state,N_3: nat,T: state] :
% 3.31/1.67 ( ( evaln @ ( ass @ X_78 @ A_158 ) @ S_2 @ N_3 @ T )
% 3.31/1.67 => ( T
% 3.31/1.67 = ( update @ S_2 @ X_78 @ ( A_158 @ S_2 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_282_evalc_OAssign,axiom,
% 3.31/1.67 ! [X_78: vname,A_158: state > nat,S_2: state] : ( evalc @ ( ass @ X_78 @ A_158 ) @ S_2 @ ( update @ S_2 @ X_78 @ ( A_158 @ S_2 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_283_evalc__elim__cases_I2_J,axiom,
% 3.31/1.67 ! [X_78: vname,A_158: state > nat,S_2: state,T: state] :
% 3.31/1.67 ( ( evalc @ ( ass @ X_78 @ A_158 ) @ S_2 @ T )
% 3.31/1.67 => ( T
% 3.31/1.67 = ( update @ S_2 @ X_78 @ ( A_158 @ S_2 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_284_eval__eq,axiom,
% 3.31/1.67 ! [C_9: com,S_2: state,T: state] :
% 3.31/1.67 ( ( evalc @ C_9 @ S_2 @ T )
% 3.31/1.67 <=> ? [N_1: nat] : ( evaln @ C_9 @ S_2 @ N_1 @ T ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_285_com__det,axiom,
% 3.31/1.67 ! [U: state,C_9: com,S_2: state,T: state] :
% 3.31/1.67 ( ( evalc @ C_9 @ S_2 @ T )
% 3.31/1.67 => ( ( evalc @ C_9 @ S_2 @ U )
% 3.31/1.67 => ( U = T ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_286_evaln__evalc,axiom,
% 3.31/1.67 ! [C_9: com,S_2: state,N_3: nat,T: state] :
% 3.31/1.67 ( ( evaln @ C_9 @ S_2 @ N_3 @ T )
% 3.31/1.67 => ( evalc @ C_9 @ S_2 @ T ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_287_empty__fold__graphE,axiom,
% 3.31/1.67 ! [F_50: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,Z_20: hoare_1775062406iple_a,X_77: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( finite727644230iple_a @ F_50 @ Z_20 @ bot_bo751897185le_a_o @ X_77 )
% 3.31/1.67 => ( X_77 = Z_20 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_288_fold__graph_OemptyI,axiom,
% 3.31/1.67 ! [F_49: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,Z_19: hoare_1775062406iple_a] : ( finite727644230iple_a @ F_49 @ Z_19 @ bot_bo751897185le_a_o @ Z_19 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_289_fold__graph_OinsertI,axiom,
% 3.31/1.67 ! [F_48: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,Z_18: hoare_1775062406iple_a,Y_38: hoare_1775062406iple_a,X_76: hoare_1775062406iple_a,A_159: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ~ ( member2122167641iple_a @ X_76 @ A_159 )
% 3.31/1.67 => ( ( finite727644230iple_a @ F_48 @ Z_18 @ A_159 @ Y_38 )
% 3.31/1.67 => ( finite727644230iple_a @ F_48 @ Z_18 @ ( insert1281456128iple_a @ X_76 @ A_159 ) @ ( F_48 @ X_76 @ Y_38 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_290_evalc__elim__cases_I3_J,axiom,
% 3.31/1.67 ! [Y_37: loc,A_158: state > nat,C_9: com,S_2: state,T: state] :
% 3.31/1.67 ( ( evalc @ ( local @ Y_37 @ A_158 @ C_9 ) @ S_2 @ T )
% 3.31/1.67 => ~ ! [S1_1: state] :
% 3.31/1.67 ( ( T
% 3.31/1.67 = ( update @ S1_1 @ ( loc_1 @ Y_37 ) @ ( getlocs @ S_2 @ Y_37 ) ) )
% 3.31/1.67 => ~ ( evalc @ C_9 @ ( update @ S_2 @ ( loc_1 @ Y_37 ) @ ( A_158 @ S_2 ) ) @ S1_1 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_291_evaln__elim__cases_I3_J,axiom,
% 3.31/1.67 ! [Y_37: loc,A_158: state > nat,C_9: com,S_2: state,N_3: nat,T: state] :
% 3.31/1.67 ( ( evaln @ ( local @ Y_37 @ A_158 @ C_9 ) @ S_2 @ N_3 @ T )
% 3.31/1.67 => ~ ! [S1_1: state] :
% 3.31/1.67 ( ( T
% 3.31/1.67 = ( update @ S1_1 @ ( loc_1 @ Y_37 ) @ ( getlocs @ S_2 @ Y_37 ) ) )
% 3.31/1.67 => ~ ( evaln @ C_9 @ ( update @ S_2 @ ( loc_1 @ Y_37 ) @ ( A_158 @ S_2 ) ) @ N_3 @ S1_1 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_292_evalc__elim__cases_I4_J,axiom,
% 3.31/1.67 ! [C1: com,C2: com,S_2: state,T: state] :
% 3.31/1.67 ( ( evalc @ ( semi @ C1 @ C2 ) @ S_2 @ T )
% 3.31/1.67 => ~ ! [S1_1: state] :
% 3.31/1.67 ( ( evalc @ C1 @ S_2 @ S1_1 )
% 3.31/1.67 => ~ ( evalc @ C2 @ S1_1 @ T ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_293_evaln__elim__cases_I4_J,axiom,
% 3.31/1.67 ! [C1: com,C2: com,S_2: state,N_3: nat,T: state] :
% 3.31/1.67 ( ( evaln @ ( semi @ C1 @ C2 ) @ S_2 @ N_3 @ T )
% 3.31/1.67 => ~ ! [S1_1: state] :
% 3.31/1.67 ( ( evaln @ C1 @ S_2 @ N_3 @ S1_1 )
% 3.31/1.67 => ~ ( evaln @ C2 @ S1_1 @ N_3 @ T ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_294_insert__fold1SetE,axiom,
% 3.31/1.67 ! [F_47: nat > nat > nat,A_157: nat,X_75: nat > $o,X_74: nat] :
% 3.31/1.67 ( ( finite_fold1Set_nat @ F_47 @ ( insert_nat @ A_157 @ X_75 ) @ X_74 )
% 3.31/1.67 => ~ ! [A_43: nat,A_98: nat > $o] :
% 3.31/1.67 ( ( ( insert_nat @ A_157 @ X_75 )
% 3.31/1.67 = ( insert_nat @ A_43 @ A_98 ) )
% 3.31/1.67 => ( ( finite929467206at_nat @ F_47 @ A_43 @ A_98 @ X_74 )
% 3.31/1.67 => ( member_nat @ A_43 @ A_98 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_295_insert__fold1SetE,axiom,
% 3.31/1.67 ! [F_47: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,A_157: hoare_1167836817_state,X_75: hoare_1167836817_state > $o,X_74: hoare_1167836817_state] :
% 3.31/1.67 ( ( finite309220289_state @ F_47 @ ( insert2134838167_state @ A_157 @ X_75 ) @ X_74 )
% 3.31/1.67 => ~ ! [A_43: hoare_1167836817_state,A_98: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( ( insert2134838167_state @ A_157 @ X_75 )
% 3.31/1.67 = ( insert2134838167_state @ A_43 @ A_98 ) )
% 3.31/1.67 => ( ( finite1316643734_state @ F_47 @ A_43 @ A_98 @ X_74 )
% 3.31/1.67 => ( member2058392318_state @ A_43 @ A_98 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_296_insert__fold1SetE,axiom,
% 3.31/1.67 ! [F_47: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,A_157: hoare_1775062406iple_a,X_75: hoare_1775062406iple_a > $o,X_74: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( finite1946188886iple_a @ F_47 @ ( insert1281456128iple_a @ A_157 @ X_75 ) @ X_74 )
% 3.31/1.67 => ~ ! [A_43: hoare_1775062406iple_a,A_98: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ( insert1281456128iple_a @ A_157 @ X_75 )
% 3.31/1.67 = ( insert1281456128iple_a @ A_43 @ A_98 ) )
% 3.31/1.67 => ( ( finite727644230iple_a @ F_47 @ A_43 @ A_98 @ X_74 )
% 3.31/1.67 => ( member2122167641iple_a @ A_43 @ A_98 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_297_MGT__def,axiom,
% 3.31/1.67 ! [C_9: com] :
% 3.31/1.67 ( ( hoare_Mirabelle_MGT @ C_9 )
% 3.31/1.67 = ( hoare_908217195_state @ fequal_state @ C_9 @ ( evalc @ C_9 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_298_evalc__evaln,axiom,
% 3.31/1.67 ! [C_9: com,S_2: state,T: state] :
% 3.31/1.67 ( ( evalc @ C_9 @ S_2 @ T )
% 3.31/1.67 => ? [N_1: nat] : ( evaln @ C_9 @ S_2 @ N_1 @ T ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_299_fold1Set_Osimps,axiom,
% 3.31/1.67 ! [F_46: nat > nat > nat,A1_1: nat > $o,A2_1: nat] :
% 3.31/1.67 ( ( finite_fold1Set_nat @ F_46 @ A1_1 @ A2_1 )
% 3.31/1.67 <=> ? [A_43: nat,A_98: nat > $o,X_3: nat] :
% 3.31/1.67 ( ( A1_1
% 3.31/1.67 = ( insert_nat @ A_43 @ A_98 ) )
% 3.31/1.67 & ( A2_1 = X_3 )
% 3.31/1.67 & ( finite929467206at_nat @ F_46 @ A_43 @ A_98 @ X_3 )
% 3.31/1.67 & ~ ( member_nat @ A_43 @ A_98 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_300_fold1Set_Osimps,axiom,
% 3.31/1.67 ! [F_46: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,A1_1: hoare_1167836817_state > $o,A2_1: hoare_1167836817_state] :
% 3.31/1.67 ( ( finite309220289_state @ F_46 @ A1_1 @ A2_1 )
% 3.31/1.67 <=> ? [A_43: hoare_1167836817_state,A_98: hoare_1167836817_state > $o,X_3: hoare_1167836817_state] :
% 3.31/1.67 ( ( A1_1
% 3.31/1.67 = ( insert2134838167_state @ A_43 @ A_98 ) )
% 3.31/1.67 & ( A2_1 = X_3 )
% 3.31/1.67 & ( finite1316643734_state @ F_46 @ A_43 @ A_98 @ X_3 )
% 3.31/1.67 & ~ ( member2058392318_state @ A_43 @ A_98 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_301_fold1Set_Osimps,axiom,
% 3.31/1.67 ! [F_46: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,A1_1: hoare_1775062406iple_a > $o,A2_1: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( finite1946188886iple_a @ F_46 @ A1_1 @ A2_1 )
% 3.31/1.67 <=> ? [A_43: hoare_1775062406iple_a,A_98: hoare_1775062406iple_a > $o,X_3: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( A1_1
% 3.31/1.67 = ( insert1281456128iple_a @ A_43 @ A_98 ) )
% 3.31/1.67 & ( A2_1 = X_3 )
% 3.31/1.67 & ( finite727644230iple_a @ F_46 @ A_43 @ A_98 @ X_3 )
% 3.31/1.67 & ~ ( member2122167641iple_a @ A_43 @ A_98 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_302_fold__graph_Osimps,axiom,
% 3.31/1.67 ! [F_45: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,Z_17: hoare_1775062406iple_a,A1: hoare_1775062406iple_a > $o,A2: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( finite727644230iple_a @ F_45 @ Z_17 @ A1 @ A2 )
% 3.31/1.67 <=> ( ( ( A1 = bot_bo751897185le_a_o )
% 3.31/1.67 & ( A2 = Z_17 ) )
% 3.31/1.67 | ? [X_3: hoare_1775062406iple_a,A_98: hoare_1775062406iple_a > $o,Y_1: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( A1
% 3.31/1.67 = ( insert1281456128iple_a @ X_3 @ A_98 ) )
% 3.31/1.67 & ( A2
% 3.31/1.67 = ( F_45 @ X_3 @ Y_1 ) )
% 3.31/1.67 & ~ ( member2122167641iple_a @ X_3 @ A_98 )
% 3.31/1.67 & ( finite727644230iple_a @ F_45 @ Z_17 @ A_98 @ Y_1 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_303_evaln__max2,axiom,
% 3.31/1.67 ! [C2: com,S2: state,N2: nat,T2: state,C1: com,S1: state,N1: nat,T1: state] :
% 3.31/1.67 ( ( evaln @ C1 @ S1 @ N1 @ T1 )
% 3.31/1.67 => ( ( evaln @ C2 @ S2 @ N2 @ T2 )
% 3.31/1.67 => ? [N_1: nat] :
% 3.31/1.67 ( ( evaln @ C1 @ S1 @ N_1 @ T1 )
% 3.31/1.67 & ( evaln @ C2 @ S2 @ N_1 @ T2 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_304_triple__valid__def2,axiom,
% 3.31/1.67 ! [N_5: nat,P_9: state > state > $o,C_29: com,Q_4: state > state > $o] :
% 3.31/1.67 ( ( hoare_56934129_state @ N_5 @ ( hoare_908217195_state @ P_9 @ C_29 @ Q_4 ) )
% 3.31/1.67 <=> ! [Z: state,S: state] :
% 3.31/1.67 ( ( P_9 @ Z @ S )
% 3.31/1.67 => ! [S_5: state] :
% 3.31/1.67 ( ( evaln @ C_29 @ S @ N_5 @ S_5 )
% 3.31/1.67 => ( Q_4 @ Z @ S_5 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_305_triple__valid__def2,axiom,
% 3.31/1.67 ! [N_5: nat,P_9: x_a > state > $o,C_29: com,Q_4: x_a > state > $o] :
% 3.31/1.67 ( ( hoare_1462269968alid_a @ N_5 @ ( hoare_1766022166iple_a @ P_9 @ C_29 @ Q_4 ) )
% 3.31/1.67 <=> ! [Z: x_a,S: state] :
% 3.31/1.67 ( ( P_9 @ Z @ S )
% 3.31/1.67 => ! [S_5: state] :
% 3.31/1.67 ( ( evaln @ C_29 @ S @ N_5 @ S_5 )
% 3.31/1.67 => ( Q_4 @ Z @ S_5 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_306_folding__one_Oinsert,axiom,
% 3.31/1.67 ! [X_73: nat,A_156: nat > $o,F_44: nat > nat > nat,F_43: ( nat > $o ) > nat] :
% 3.31/1.67 ( ( finite988810631ne_nat @ F_44 @ F_43 )
% 3.31/1.67 => ( ( finite_finite_nat @ A_156 )
% 3.31/1.67 => ( ~ ( member_nat @ X_73 @ A_156 )
% 3.31/1.67 => ( ( A_156 != bot_bot_nat_o )
% 3.31/1.67 => ( ( F_43 @ ( insert_nat @ X_73 @ A_156 ) )
% 3.31/1.67 = ( F_44 @ X_73 @ ( F_43 @ A_156 ) ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_307_folding__one_Oinsert,axiom,
% 3.31/1.67 ! [X_73: hoare_1167836817_state,A_156: hoare_1167836817_state > $o,F_44: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_43: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
% 3.31/1.67 ( ( finite1074406356_state @ F_44 @ F_43 )
% 3.31/1.67 => ( ( finite1084549118_state @ A_156 )
% 3.31/1.67 => ( ~ ( member2058392318_state @ X_73 @ A_156 )
% 3.31/1.67 => ( ( A_156 != bot_bo70021908tate_o )
% 3.31/1.67 => ( ( F_43 @ ( insert2134838167_state @ X_73 @ A_156 ) )
% 3.31/1.67 = ( F_44 @ X_73 @ ( F_43 @ A_156 ) ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_308_folding__one_Oinsert,axiom,
% 3.31/1.67 ! [X_73: hoare_1775062406iple_a,A_156: hoare_1775062406iple_a > $o,F_44: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_43: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
% 3.31/1.67 ( ( finite2078349315iple_a @ F_44 @ F_43 )
% 3.31/1.67 => ( ( finite2063573081iple_a @ A_156 )
% 3.31/1.67 => ( ~ ( member2122167641iple_a @ X_73 @ A_156 )
% 3.31/1.67 => ( ( A_156 != bot_bo751897185le_a_o )
% 3.31/1.67 => ( ( F_43 @ ( insert1281456128iple_a @ X_73 @ A_156 ) )
% 3.31/1.67 = ( F_44 @ X_73 @ ( F_43 @ A_156 ) ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_309_finite__Collect__conjI,axiom,
% 3.31/1.67 ! [Q_3: hoare_1775062406iple_a > $o,P_8: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ( finite2063573081iple_a @ ( collec676402587iple_a @ P_8 ) )
% 3.31/1.67 | ( finite2063573081iple_a @ ( collec676402587iple_a @ Q_3 ) ) )
% 3.31/1.67 => ( finite2063573081iple_a
% 3.31/1.67 @ ( collec676402587iple_a
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] : ( (&) @ ( P_8 @ X_3 ) @ ( Q_3 @ X_3 ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_310_finite__Collect__conjI,axiom,
% 3.31/1.67 ! [Q_3: nat > $o,P_8: nat > $o] :
% 3.31/1.67 ( ( ( finite_finite_nat @ ( collect_nat @ P_8 ) )
% 3.31/1.67 | ( finite_finite_nat @ ( collect_nat @ Q_3 ) ) )
% 3.31/1.67 => ( finite_finite_nat
% 3.31/1.67 @ ( collect_nat
% 3.31/1.67 @ ^ [X_3: nat] : ( (&) @ ( P_8 @ X_3 ) @ ( Q_3 @ X_3 ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_311_finite_OemptyI,axiom,
% 3.31/1.67 finite_finite_nat @ bot_bot_nat_o ).
% 3.31/1.67
% 3.31/1.67 thf(fact_312_finite_OemptyI,axiom,
% 3.31/1.67 finite1084549118_state @ bot_bo70021908tate_o ).
% 3.31/1.67
% 3.31/1.67 thf(fact_313_finite_OemptyI,axiom,
% 3.31/1.67 finite2063573081iple_a @ bot_bo751897185le_a_o ).
% 3.31/1.67
% 3.31/1.67 thf(fact_314_finite_OinsertI,axiom,
% 3.31/1.67 ! [A_155: nat,A_154: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ A_154 )
% 3.31/1.67 => ( finite_finite_nat @ ( insert_nat @ A_155 @ A_154 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_315_finite_OinsertI,axiom,
% 3.31/1.67 ! [A_155: hoare_1167836817_state,A_154: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( finite1084549118_state @ A_154 )
% 3.31/1.67 => ( finite1084549118_state @ ( insert2134838167_state @ A_155 @ A_154 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_316_finite_OinsertI,axiom,
% 3.31/1.67 ! [A_155: hoare_1775062406iple_a,A_154: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( finite2063573081iple_a @ A_154 )
% 3.31/1.67 => ( finite2063573081iple_a @ ( insert1281456128iple_a @ A_155 @ A_154 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_317_finite__Collect__disjI,axiom,
% 3.31/1.67 ! [P_7: hoare_1775062406iple_a > $o,Q_2: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( finite2063573081iple_a
% 3.31/1.67 @ ( collec676402587iple_a
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] : ( (|) @ ( P_7 @ X_3 ) @ ( Q_2 @ X_3 ) ) ) )
% 3.31/1.67 <=> ( ( finite2063573081iple_a @ ( collec676402587iple_a @ P_7 ) )
% 3.31/1.67 & ( finite2063573081iple_a @ ( collec676402587iple_a @ Q_2 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_318_finite__Collect__disjI,axiom,
% 3.31/1.67 ! [P_7: nat > $o,Q_2: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat
% 3.31/1.67 @ ( collect_nat
% 3.31/1.67 @ ^ [X_3: nat] : ( (|) @ ( P_7 @ X_3 ) @ ( Q_2 @ X_3 ) ) ) )
% 3.31/1.67 <=> ( ( finite_finite_nat @ ( collect_nat @ P_7 ) )
% 3.31/1.67 & ( finite_finite_nat @ ( collect_nat @ Q_2 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_319_vname_Osimps_I1_J,axiom,
% 3.31/1.67 ! [Glb_1: glb,Glb_2: glb] :
% 3.31/1.67 ( ( ( glb_1 @ Glb_1 )
% 3.31/1.67 = ( glb_1 @ Glb_2 ) )
% 3.31/1.67 <=> ( Glb_1 = Glb_2 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_320_finite__insert,axiom,
% 3.31/1.67 ! [A_153: nat,A_152: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ ( insert_nat @ A_153 @ A_152 ) )
% 3.31/1.67 <=> ( finite_finite_nat @ A_152 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_321_finite__insert,axiom,
% 3.31/1.67 ! [A_153: hoare_1167836817_state,A_152: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( finite1084549118_state @ ( insert2134838167_state @ A_153 @ A_152 ) )
% 3.31/1.67 <=> ( finite1084549118_state @ A_152 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_322_finite__insert,axiom,
% 3.31/1.67 ! [A_153: hoare_1775062406iple_a,A_152: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( finite2063573081iple_a @ ( insert1281456128iple_a @ A_153 @ A_152 ) )
% 3.31/1.67 <=> ( finite2063573081iple_a @ A_152 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_323_vname_Osimps_I4_J,axiom,
% 3.31/1.67 ! [Loc_1: loc,Glb_1: glb] :
% 3.31/1.67 ( ( loc_1 @ Loc_1 )
% 3.31/1.67 != ( glb_1 @ Glb_1 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_324_vname_Osimps_I3_J,axiom,
% 3.31/1.67 ! [Glb_1: glb,Loc_1: loc] :
% 3.31/1.67 ( ( glb_1 @ Glb_1 )
% 3.31/1.67 != ( loc_1 @ Loc_1 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_325_folding__one_Oclosed,axiom,
% 3.31/1.67 ! [A_151: nat > $o,F_42: nat > nat > nat,F_41: ( nat > $o ) > nat] :
% 3.31/1.67 ( ( finite988810631ne_nat @ F_42 @ F_41 )
% 3.31/1.67 => ( ( finite_finite_nat @ A_151 )
% 3.31/1.67 => ( ( A_151 != bot_bot_nat_o )
% 3.31/1.67 => ( ! [X_3: nat,Y_1: nat] : ( member_nat @ ( F_42 @ X_3 @ Y_1 ) @ ( insert_nat @ X_3 @ ( insert_nat @ Y_1 @ bot_bot_nat_o ) ) )
% 3.31/1.67 => ( member_nat @ ( F_41 @ A_151 ) @ A_151 ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_326_folding__one_Oclosed,axiom,
% 3.31/1.67 ! [A_151: hoare_1167836817_state > $o,F_42: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_41: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
% 3.31/1.67 ( ( finite1074406356_state @ F_42 @ F_41 )
% 3.31/1.67 => ( ( finite1084549118_state @ A_151 )
% 3.31/1.67 => ( ( A_151 != bot_bo70021908tate_o )
% 3.31/1.67 => ( ! [X_3: hoare_1167836817_state,Y_1: hoare_1167836817_state] : ( member2058392318_state @ ( F_42 @ X_3 @ Y_1 ) @ ( insert2134838167_state @ X_3 @ ( insert2134838167_state @ Y_1 @ bot_bo70021908tate_o ) ) )
% 3.31/1.67 => ( member2058392318_state @ ( F_41 @ A_151 ) @ A_151 ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_327_folding__one_Oclosed,axiom,
% 3.31/1.67 ! [A_151: hoare_1775062406iple_a > $o,F_42: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_41: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
% 3.31/1.67 ( ( finite2078349315iple_a @ F_42 @ F_41 )
% 3.31/1.67 => ( ( finite2063573081iple_a @ A_151 )
% 3.31/1.67 => ( ( A_151 != bot_bo751897185le_a_o )
% 3.31/1.67 => ( ! [X_3: hoare_1775062406iple_a,Y_1: hoare_1775062406iple_a] : ( member2122167641iple_a @ ( F_42 @ X_3 @ Y_1 ) @ ( insert1281456128iple_a @ X_3 @ ( insert1281456128iple_a @ Y_1 @ bot_bo751897185le_a_o ) ) )
% 3.31/1.67 => ( member2122167641iple_a @ ( F_41 @ A_151 ) @ A_151 ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_328_finite__nonempty__imp__fold1Set,axiom,
% 3.31/1.67 ! [F_40: nat > nat > nat,A_150: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ A_150 )
% 3.31/1.67 => ( ( A_150 != bot_bot_nat_o )
% 3.31/1.67 => ( ?? @ nat @ ( finite_fold1Set_nat @ F_40 @ A_150 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_329_finite__nonempty__imp__fold1Set,axiom,
% 3.31/1.67 ! [F_40: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,A_150: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( finite2063573081iple_a @ A_150 )
% 3.31/1.67 => ( ( A_150 != bot_bo751897185le_a_o )
% 3.31/1.67 => ( ?? @ hoare_1775062406iple_a @ ( finite1946188886iple_a @ F_40 @ A_150 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_330_finite__nonempty__imp__fold1Set,axiom,
% 3.31/1.67 ! [F_40: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,A_150: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( finite1084549118_state @ A_150 )
% 3.31/1.67 => ( ( A_150 != bot_bo70021908tate_o )
% 3.31/1.67 => ( ?? @ hoare_1167836817_state @ ( finite309220289_state @ F_40 @ A_150 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_331_finite__induct,axiom,
% 3.31/1.67 ! [P_6: ( nat > $o ) > $o,F_39: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ F_39 )
% 3.31/1.67 => ( ( P_6 @ bot_bot_nat_o )
% 3.31/1.67 => ( ! [X_3: nat,F_3: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ F_3 )
% 3.31/1.67 => ( ~ ( member_nat @ X_3 @ F_3 )
% 3.31/1.67 => ( ( P_6 @ F_3 )
% 3.31/1.67 => ( P_6 @ ( insert_nat @ X_3 @ F_3 ) ) ) ) )
% 3.31/1.67 => ( P_6 @ F_39 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_332_finite__induct,axiom,
% 3.31/1.67 ! [P_6: ( hoare_1167836817_state > $o ) > $o,F_39: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( finite1084549118_state @ F_39 )
% 3.31/1.67 => ( ( P_6 @ bot_bo70021908tate_o )
% 3.31/1.67 => ( ! [X_3: hoare_1167836817_state,F_3: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( finite1084549118_state @ F_3 )
% 3.31/1.67 => ( ~ ( member2058392318_state @ X_3 @ F_3 )
% 3.31/1.67 => ( ( P_6 @ F_3 )
% 3.31/1.67 => ( P_6 @ ( insert2134838167_state @ X_3 @ F_3 ) ) ) ) )
% 3.31/1.67 => ( P_6 @ F_39 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_333_finite__induct,axiom,
% 3.31/1.67 ! [P_6: ( hoare_1775062406iple_a > $o ) > $o,F_39: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( finite2063573081iple_a @ F_39 )
% 3.31/1.67 => ( ( P_6 @ bot_bo751897185le_a_o )
% 3.31/1.67 => ( ! [X_3: hoare_1775062406iple_a,F_3: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( finite2063573081iple_a @ F_3 )
% 3.31/1.67 => ( ~ ( member2122167641iple_a @ X_3 @ F_3 )
% 3.31/1.67 => ( ( P_6 @ F_3 )
% 3.31/1.67 => ( P_6 @ ( insert1281456128iple_a @ X_3 @ F_3 ) ) ) ) )
% 3.31/1.67 => ( P_6 @ F_39 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_334_finite_Osimps,axiom,
% 3.31/1.67 ! [A_149: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ A_149 )
% 3.31/1.67 <=> ( ( A_149 = bot_bot_nat_o )
% 3.31/1.67 | ? [A_98: nat > $o,A_43: nat] :
% 3.31/1.67 ( ( A_149
% 3.31/1.67 = ( insert_nat @ A_43 @ A_98 ) )
% 3.31/1.67 & ( finite_finite_nat @ A_98 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_335_finite_Osimps,axiom,
% 3.31/1.67 ! [A_149: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( finite1084549118_state @ A_149 )
% 3.31/1.67 <=> ( ( A_149 = bot_bo70021908tate_o )
% 3.31/1.67 | ? [A_98: hoare_1167836817_state > $o,A_43: hoare_1167836817_state] :
% 3.31/1.67 ( ( A_149
% 3.31/1.67 = ( insert2134838167_state @ A_43 @ A_98 ) )
% 3.31/1.67 & ( finite1084549118_state @ A_98 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_336_finite_Osimps,axiom,
% 3.31/1.67 ! [A_149: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( finite2063573081iple_a @ A_149 )
% 3.31/1.67 <=> ( ( A_149 = bot_bo751897185le_a_o )
% 3.31/1.67 | ? [A_98: hoare_1775062406iple_a > $o,A_43: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( A_149
% 3.31/1.67 = ( insert1281456128iple_a @ A_43 @ A_98 ) )
% 3.31/1.67 & ( finite2063573081iple_a @ A_98 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_337_finite__imp__fold__graph,axiom,
% 3.31/1.67 ! [F_38: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,Z_16: hoare_1775062406iple_a,A_148: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( finite2063573081iple_a @ A_148 )
% 3.31/1.67 => ( ?? @ hoare_1775062406iple_a @ ( finite727644230iple_a @ F_38 @ Z_16 @ A_148 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_338_folding__one__idem_Oinsert__idem,axiom,
% 3.31/1.67 ! [X_72: nat,A_147: nat > $o,F_37: nat > nat > nat,F_36: ( nat > $o ) > nat] :
% 3.31/1.67 ( ( finite795500164em_nat @ F_37 @ F_36 )
% 3.31/1.67 => ( ( finite_finite_nat @ A_147 )
% 3.31/1.67 => ( ( A_147 != bot_bot_nat_o )
% 3.31/1.67 => ( ( F_36 @ ( insert_nat @ X_72 @ A_147 ) )
% 3.31/1.67 = ( F_37 @ X_72 @ ( F_36 @ A_147 ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_339_folding__one__idem_Oinsert__idem,axiom,
% 3.31/1.67 ! [X_72: hoare_1167836817_state,A_147: hoare_1167836817_state > $o,F_37: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_36: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
% 3.31/1.67 ( ( finite806517911_state @ F_37 @ F_36 )
% 3.31/1.67 => ( ( finite1084549118_state @ A_147 )
% 3.31/1.67 => ( ( A_147 != bot_bo70021908tate_o )
% 3.31/1.67 => ( ( F_36 @ ( insert2134838167_state @ X_72 @ A_147 ) )
% 3.31/1.67 = ( F_37 @ X_72 @ ( F_36 @ A_147 ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_340_folding__one__idem_Oinsert__idem,axiom,
% 3.31/1.67 ! [X_72: hoare_1775062406iple_a,A_147: hoare_1775062406iple_a > $o,F_37: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_36: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
% 3.31/1.67 ( ( finite1358382848iple_a @ F_37 @ F_36 )
% 3.31/1.67 => ( ( finite2063573081iple_a @ A_147 )
% 3.31/1.67 => ( ( A_147 != bot_bo751897185le_a_o )
% 3.31/1.67 => ( ( F_36 @ ( insert1281456128iple_a @ X_72 @ A_147 ) )
% 3.31/1.67 = ( F_37 @ X_72 @ ( F_36 @ A_147 ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_341_finite__ne__induct,axiom,
% 3.31/1.67 ! [P_5: ( nat > $o ) > $o,F_35: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ F_35 )
% 3.31/1.67 => ( ( F_35 != bot_bot_nat_o )
% 3.31/1.67 => ( ! [X_3: nat] : ( P_5 @ ( insert_nat @ X_3 @ bot_bot_nat_o ) )
% 3.31/1.67 => ( ! [X_3: nat,F_3: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ F_3 )
% 3.31/1.67 => ( ( F_3 != bot_bot_nat_o )
% 3.31/1.67 => ( ~ ( member_nat @ X_3 @ F_3 )
% 3.31/1.67 => ( ( P_5 @ F_3 )
% 3.31/1.67 => ( P_5 @ ( insert_nat @ X_3 @ F_3 ) ) ) ) ) )
% 3.31/1.67 => ( P_5 @ F_35 ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_342_finite__ne__induct,axiom,
% 3.31/1.67 ! [P_5: ( hoare_1167836817_state > $o ) > $o,F_35: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( finite1084549118_state @ F_35 )
% 3.31/1.67 => ( ( F_35 != bot_bo70021908tate_o )
% 3.31/1.67 => ( ! [X_3: hoare_1167836817_state] : ( P_5 @ ( insert2134838167_state @ X_3 @ bot_bo70021908tate_o ) )
% 3.31/1.67 => ( ! [X_3: hoare_1167836817_state,F_3: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( finite1084549118_state @ F_3 )
% 3.31/1.67 => ( ( F_3 != bot_bo70021908tate_o )
% 3.31/1.67 => ( ~ ( member2058392318_state @ X_3 @ F_3 )
% 3.31/1.67 => ( ( P_5 @ F_3 )
% 3.31/1.67 => ( P_5 @ ( insert2134838167_state @ X_3 @ F_3 ) ) ) ) ) )
% 3.31/1.67 => ( P_5 @ F_35 ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_343_finite__ne__induct,axiom,
% 3.31/1.67 ! [P_5: ( hoare_1775062406iple_a > $o ) > $o,F_35: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( finite2063573081iple_a @ F_35 )
% 3.31/1.67 => ( ( F_35 != bot_bo751897185le_a_o )
% 3.31/1.67 => ( ! [X_3: hoare_1775062406iple_a] : ( P_5 @ ( insert1281456128iple_a @ X_3 @ bot_bo751897185le_a_o ) )
% 3.31/1.67 => ( ! [X_3: hoare_1775062406iple_a,F_3: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( finite2063573081iple_a @ F_3 )
% 3.31/1.67 => ( ( F_3 != bot_bo751897185le_a_o )
% 3.31/1.67 => ( ~ ( member2122167641iple_a @ X_3 @ F_3 )
% 3.31/1.67 => ( ( P_5 @ F_3 )
% 3.31/1.67 => ( P_5 @ ( insert1281456128iple_a @ X_3 @ F_3 ) ) ) ) ) )
% 3.31/1.67 => ( P_5 @ F_35 ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_344_vname_Oexhaust,axiom,
% 3.31/1.67 ! [Y: vname] :
% 3.31/1.67 ( ! [Glb: glb] :
% 3.31/1.67 ( Y
% 3.31/1.67 != ( glb_1 @ Glb ) )
% 3.31/1.67 => ~ ! [Loc: loc] :
% 3.31/1.67 ( Y
% 3.31/1.67 != ( loc_1 @ Loc ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_345_folding__one_Oremove,axiom,
% 3.31/1.67 ! [X_71: nat,A_146: nat > $o,F_34: nat > nat > nat,F_33: ( nat > $o ) > nat] :
% 3.31/1.67 ( ( finite988810631ne_nat @ F_34 @ F_33 )
% 3.31/1.67 => ( ( finite_finite_nat @ A_146 )
% 3.31/1.67 => ( ( member_nat @ X_71 @ A_146 )
% 3.31/1.67 => ( ( ( ( minus_minus_nat_o @ A_146 @ ( insert_nat @ X_71 @ bot_bot_nat_o ) )
% 3.31/1.67 = bot_bot_nat_o )
% 3.31/1.67 => ( ( F_33 @ A_146 )
% 3.31/1.67 = X_71 ) )
% 3.31/1.67 & ( ( ( minus_minus_nat_o @ A_146 @ ( insert_nat @ X_71 @ bot_bot_nat_o ) )
% 3.31/1.67 != bot_bot_nat_o )
% 3.31/1.67 => ( ( F_33 @ A_146 )
% 3.31/1.67 = ( F_34 @ X_71 @ ( F_33 @ ( minus_minus_nat_o @ A_146 @ ( insert_nat @ X_71 @ bot_bot_nat_o ) ) ) ) ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_346_folding__one_Oremove,axiom,
% 3.31/1.67 ! [X_71: hoare_1167836817_state,A_146: hoare_1167836817_state > $o,F_34: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_33: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
% 3.31/1.67 ( ( finite1074406356_state @ F_34 @ F_33 )
% 3.31/1.67 => ( ( finite1084549118_state @ A_146 )
% 3.31/1.67 => ( ( member2058392318_state @ X_71 @ A_146 )
% 3.31/1.67 => ( ( ( ( minus_2107060239tate_o @ A_146 @ ( insert2134838167_state @ X_71 @ bot_bo70021908tate_o ) )
% 3.31/1.67 = bot_bo70021908tate_o )
% 3.31/1.67 => ( ( F_33 @ A_146 )
% 3.31/1.67 = X_71 ) )
% 3.31/1.67 & ( ( ( minus_2107060239tate_o @ A_146 @ ( insert2134838167_state @ X_71 @ bot_bo70021908tate_o ) )
% 3.31/1.67 != bot_bo70021908tate_o )
% 3.31/1.67 => ( ( F_33 @ A_146 )
% 3.31/1.67 = ( F_34 @ X_71 @ ( F_33 @ ( minus_2107060239tate_o @ A_146 @ ( insert2134838167_state @ X_71 @ bot_bo70021908tate_o ) ) ) ) ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_347_folding__one_Oremove,axiom,
% 3.31/1.67 ! [X_71: hoare_1775062406iple_a,A_146: hoare_1775062406iple_a > $o,F_34: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_33: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
% 3.31/1.67 ( ( finite2078349315iple_a @ F_34 @ F_33 )
% 3.31/1.67 => ( ( finite2063573081iple_a @ A_146 )
% 3.31/1.67 => ( ( member2122167641iple_a @ X_71 @ A_146 )
% 3.31/1.67 => ( ( ( ( minus_1944206118le_a_o @ A_146 @ ( insert1281456128iple_a @ X_71 @ bot_bo751897185le_a_o ) )
% 3.31/1.67 = bot_bo751897185le_a_o )
% 3.31/1.67 => ( ( F_33 @ A_146 )
% 3.31/1.67 = X_71 ) )
% 3.31/1.67 & ( ( ( minus_1944206118le_a_o @ A_146 @ ( insert1281456128iple_a @ X_71 @ bot_bo751897185le_a_o ) )
% 3.31/1.67 != bot_bo751897185le_a_o )
% 3.31/1.67 => ( ( F_33 @ A_146 )
% 3.31/1.67 = ( F_34 @ X_71 @ ( F_33 @ ( minus_1944206118le_a_o @ A_146 @ ( insert1281456128iple_a @ X_71 @ bot_bo751897185le_a_o ) ) ) ) ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_348_folding__one_Oinsert__remove,axiom,
% 3.31/1.67 ! [X_70: nat,A_145: nat > $o,F_32: nat > nat > nat,F_31: ( nat > $o ) > nat] :
% 3.31/1.67 ( ( finite988810631ne_nat @ F_32 @ F_31 )
% 3.31/1.67 => ( ( finite_finite_nat @ A_145 )
% 3.31/1.67 => ( ( ( ( minus_minus_nat_o @ A_145 @ ( insert_nat @ X_70 @ bot_bot_nat_o ) )
% 3.31/1.67 = bot_bot_nat_o )
% 3.31/1.67 => ( ( F_31 @ ( insert_nat @ X_70 @ A_145 ) )
% 3.31/1.67 = X_70 ) )
% 3.31/1.67 & ( ( ( minus_minus_nat_o @ A_145 @ ( insert_nat @ X_70 @ bot_bot_nat_o ) )
% 3.31/1.67 != bot_bot_nat_o )
% 3.31/1.67 => ( ( F_31 @ ( insert_nat @ X_70 @ A_145 ) )
% 3.31/1.67 = ( F_32 @ X_70 @ ( F_31 @ ( minus_minus_nat_o @ A_145 @ ( insert_nat @ X_70 @ bot_bot_nat_o ) ) ) ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_349_folding__one_Oinsert__remove,axiom,
% 3.31/1.67 ! [X_70: hoare_1167836817_state,A_145: hoare_1167836817_state > $o,F_32: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_31: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
% 3.31/1.67 ( ( finite1074406356_state @ F_32 @ F_31 )
% 3.31/1.67 => ( ( finite1084549118_state @ A_145 )
% 3.31/1.67 => ( ( ( ( minus_2107060239tate_o @ A_145 @ ( insert2134838167_state @ X_70 @ bot_bo70021908tate_o ) )
% 3.31/1.67 = bot_bo70021908tate_o )
% 3.31/1.67 => ( ( F_31 @ ( insert2134838167_state @ X_70 @ A_145 ) )
% 3.31/1.67 = X_70 ) )
% 3.31/1.67 & ( ( ( minus_2107060239tate_o @ A_145 @ ( insert2134838167_state @ X_70 @ bot_bo70021908tate_o ) )
% 3.31/1.67 != bot_bo70021908tate_o )
% 3.31/1.67 => ( ( F_31 @ ( insert2134838167_state @ X_70 @ A_145 ) )
% 3.31/1.67 = ( F_32 @ X_70 @ ( F_31 @ ( minus_2107060239tate_o @ A_145 @ ( insert2134838167_state @ X_70 @ bot_bo70021908tate_o ) ) ) ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_350_folding__one_Oinsert__remove,axiom,
% 3.31/1.67 ! [X_70: hoare_1775062406iple_a,A_145: hoare_1775062406iple_a > $o,F_32: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_31: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
% 3.31/1.67 ( ( finite2078349315iple_a @ F_32 @ F_31 )
% 3.31/1.67 => ( ( finite2063573081iple_a @ A_145 )
% 3.31/1.67 => ( ( ( ( minus_1944206118le_a_o @ A_145 @ ( insert1281456128iple_a @ X_70 @ bot_bo751897185le_a_o ) )
% 3.31/1.67 = bot_bo751897185le_a_o )
% 3.31/1.67 => ( ( F_31 @ ( insert1281456128iple_a @ X_70 @ A_145 ) )
% 3.31/1.67 = X_70 ) )
% 3.31/1.67 & ( ( ( minus_1944206118le_a_o @ A_145 @ ( insert1281456128iple_a @ X_70 @ bot_bo751897185le_a_o ) )
% 3.31/1.67 != bot_bo751897185le_a_o )
% 3.31/1.67 => ( ( F_31 @ ( insert1281456128iple_a @ X_70 @ A_145 ) )
% 3.31/1.67 = ( F_32 @ X_70 @ ( F_31 @ ( minus_1944206118le_a_o @ A_145 @ ( insert1281456128iple_a @ X_70 @ bot_bo751897185le_a_o ) ) ) ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_351_DiffE,axiom,
% 3.31/1.67 ! [C_28: hoare_1775062406iple_a,A_144: hoare_1775062406iple_a > $o,B_82: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ C_28 @ ( minus_1944206118le_a_o @ A_144 @ B_82 ) )
% 3.31/1.67 => ~ ( ( member2122167641iple_a @ C_28 @ A_144 )
% 3.31/1.67 => ( member2122167641iple_a @ C_28 @ B_82 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_352_DiffE,axiom,
% 3.31/1.67 ! [C_28: nat,A_144: nat > $o,B_82: nat > $o] :
% 3.31/1.67 ( ( member_nat @ C_28 @ ( minus_minus_nat_o @ A_144 @ B_82 ) )
% 3.31/1.67 => ~ ( ( member_nat @ C_28 @ A_144 )
% 3.31/1.67 => ( member_nat @ C_28 @ B_82 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_353_DiffI,axiom,
% 3.31/1.67 ! [B_81: hoare_1775062406iple_a > $o,C_27: hoare_1775062406iple_a,A_143: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ C_27 @ A_143 )
% 3.31/1.67 => ( ~ ( member2122167641iple_a @ C_27 @ B_81 )
% 3.31/1.67 => ( member2122167641iple_a @ C_27 @ ( minus_1944206118le_a_o @ A_143 @ B_81 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_354_DiffI,axiom,
% 3.31/1.67 ! [B_81: nat > $o,C_27: nat,A_143: nat > $o] :
% 3.31/1.67 ( ( member_nat @ C_27 @ A_143 )
% 3.31/1.67 => ( ~ ( member_nat @ C_27 @ B_81 )
% 3.31/1.67 => ( member_nat @ C_27 @ ( minus_minus_nat_o @ A_143 @ B_81 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_355_finite__Diff,axiom,
% 3.31/1.67 ! [B_80: nat > $o,A_142: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ A_142 )
% 3.31/1.67 => ( finite_finite_nat @ ( minus_minus_nat_o @ A_142 @ B_80 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_356_DiffD2,axiom,
% 3.31/1.67 ! [C_26: hoare_1775062406iple_a,A_141: hoare_1775062406iple_a > $o,B_79: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ C_26 @ ( minus_1944206118le_a_o @ A_141 @ B_79 ) )
% 3.31/1.67 => ~ ( member2122167641iple_a @ C_26 @ B_79 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_357_DiffD2,axiom,
% 3.31/1.67 ! [C_26: nat,A_141: nat > $o,B_79: nat > $o] :
% 3.31/1.67 ( ( member_nat @ C_26 @ ( minus_minus_nat_o @ A_141 @ B_79 ) )
% 3.31/1.67 => ~ ( member_nat @ C_26 @ B_79 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_358_DiffD1,axiom,
% 3.31/1.67 ! [C_25: hoare_1775062406iple_a,A_140: hoare_1775062406iple_a > $o,B_78: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ C_25 @ ( minus_1944206118le_a_o @ A_140 @ B_78 ) )
% 3.31/1.67 => ( member2122167641iple_a @ C_25 @ A_140 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_359_DiffD1,axiom,
% 3.31/1.67 ! [C_25: nat,A_140: nat > $o,B_78: nat > $o] :
% 3.31/1.67 ( ( member_nat @ C_25 @ ( minus_minus_nat_o @ A_140 @ B_78 ) )
% 3.31/1.67 => ( member_nat @ C_25 @ A_140 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_360_Diff__iff,axiom,
% 3.31/1.67 ! [C_24: hoare_1775062406iple_a,A_139: hoare_1775062406iple_a > $o,B_77: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ C_24 @ ( minus_1944206118le_a_o @ A_139 @ B_77 ) )
% 3.31/1.67 <=> ( ( member2122167641iple_a @ C_24 @ A_139 )
% 3.31/1.67 & ~ ( member2122167641iple_a @ C_24 @ B_77 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_361_Diff__iff,axiom,
% 3.31/1.67 ! [C_24: nat,A_139: nat > $o,B_77: nat > $o] :
% 3.31/1.67 ( ( member_nat @ C_24 @ ( minus_minus_nat_o @ A_139 @ B_77 ) )
% 3.31/1.67 <=> ( ( member_nat @ C_24 @ A_139 )
% 3.31/1.67 & ~ ( member_nat @ C_24 @ B_77 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_362_set__diff__eq,axiom,
% 3.31/1.67 ! [A_138: hoare_1775062406iple_a > $o,B_76: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( minus_1944206118le_a_o @ A_138 @ B_76 )
% 3.31/1.67 = ( collec676402587iple_a
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] : ( (&) @ ( member2122167641iple_a @ X_3 @ A_138 ) @ ( (~) @ ( member2122167641iple_a @ X_3 @ B_76 ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_363_set__diff__eq,axiom,
% 3.31/1.67 ! [A_138: nat > $o,B_76: nat > $o] :
% 3.31/1.67 ( ( minus_minus_nat_o @ A_138 @ B_76 )
% 3.31/1.67 = ( collect_nat
% 3.31/1.67 @ ^ [X_3: nat] : ( (&) @ ( member_nat @ X_3 @ A_138 ) @ ( (~) @ ( member_nat @ X_3 @ B_76 ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_364_Diff__cancel,axiom,
% 3.31/1.67 ! [A_137: nat > $o] :
% 3.31/1.67 ( ( minus_minus_nat_o @ A_137 @ A_137 )
% 3.31/1.67 = bot_bot_nat_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_365_Diff__cancel,axiom,
% 3.31/1.67 ! [A_137: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( minus_2107060239tate_o @ A_137 @ A_137 )
% 3.31/1.67 = bot_bo70021908tate_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_366_Diff__cancel,axiom,
% 3.31/1.67 ! [A_137: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( minus_1944206118le_a_o @ A_137 @ A_137 )
% 3.31/1.67 = bot_bo751897185le_a_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_367_Diff__empty,axiom,
% 3.31/1.67 ! [A_136: nat > $o] :
% 3.31/1.67 ( ( minus_minus_nat_o @ A_136 @ bot_bot_nat_o )
% 3.31/1.67 = A_136 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_368_Diff__empty,axiom,
% 3.31/1.67 ! [A_136: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( minus_2107060239tate_o @ A_136 @ bot_bo70021908tate_o )
% 3.31/1.67 = A_136 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_369_Diff__empty,axiom,
% 3.31/1.67 ! [A_136: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( minus_1944206118le_a_o @ A_136 @ bot_bo751897185le_a_o )
% 3.31/1.67 = A_136 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_370_empty__Diff,axiom,
% 3.31/1.67 ! [A_135: nat > $o] :
% 3.31/1.67 ( ( minus_minus_nat_o @ bot_bot_nat_o @ A_135 )
% 3.31/1.67 = bot_bot_nat_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_371_empty__Diff,axiom,
% 3.31/1.67 ! [A_135: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( minus_2107060239tate_o @ bot_bo70021908tate_o @ A_135 )
% 3.31/1.67 = bot_bo70021908tate_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_372_empty__Diff,axiom,
% 3.31/1.67 ! [A_135: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( minus_1944206118le_a_o @ bot_bo751897185le_a_o @ A_135 )
% 3.31/1.67 = bot_bo751897185le_a_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_373_finite__Diff2,axiom,
% 3.31/1.67 ! [A_134: nat > $o,B_75: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ B_75 )
% 3.31/1.67 => ( ( finite_finite_nat @ ( minus_minus_nat_o @ A_134 @ B_75 ) )
% 3.31/1.67 <=> ( finite_finite_nat @ A_134 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_374_insert__Diff1,axiom,
% 3.31/1.67 ! [A_133: nat > $o,X_69: nat,B_74: nat > $o] :
% 3.31/1.67 ( ( member_nat @ X_69 @ B_74 )
% 3.31/1.67 => ( ( minus_minus_nat_o @ ( insert_nat @ X_69 @ A_133 ) @ B_74 )
% 3.31/1.67 = ( minus_minus_nat_o @ A_133 @ B_74 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_375_insert__Diff1,axiom,
% 3.31/1.67 ! [A_133: hoare_1167836817_state > $o,X_69: hoare_1167836817_state,B_74: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( member2058392318_state @ X_69 @ B_74 )
% 3.31/1.67 => ( ( minus_2107060239tate_o @ ( insert2134838167_state @ X_69 @ A_133 ) @ B_74 )
% 3.31/1.67 = ( minus_2107060239tate_o @ A_133 @ B_74 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_376_insert__Diff1,axiom,
% 3.31/1.67 ! [A_133: hoare_1775062406iple_a > $o,X_69: hoare_1775062406iple_a,B_74: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ X_69 @ B_74 )
% 3.31/1.67 => ( ( minus_1944206118le_a_o @ ( insert1281456128iple_a @ X_69 @ A_133 ) @ B_74 )
% 3.31/1.67 = ( minus_1944206118le_a_o @ A_133 @ B_74 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_377_insert__Diff__if,axiom,
% 3.31/1.67 ! [A_132: nat > $o,X_68: nat,B_73: nat > $o] :
% 3.31/1.67 ( ( ( member_nat @ X_68 @ B_73 )
% 3.31/1.67 => ( ( minus_minus_nat_o @ ( insert_nat @ X_68 @ A_132 ) @ B_73 )
% 3.31/1.67 = ( minus_minus_nat_o @ A_132 @ B_73 ) ) )
% 3.31/1.67 & ( ~ ( member_nat @ X_68 @ B_73 )
% 3.31/1.67 => ( ( minus_minus_nat_o @ ( insert_nat @ X_68 @ A_132 ) @ B_73 )
% 3.31/1.67 = ( insert_nat @ X_68 @ ( minus_minus_nat_o @ A_132 @ B_73 ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_378_insert__Diff__if,axiom,
% 3.31/1.67 ! [A_132: hoare_1167836817_state > $o,X_68: hoare_1167836817_state,B_73: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( ( member2058392318_state @ X_68 @ B_73 )
% 3.31/1.67 => ( ( minus_2107060239tate_o @ ( insert2134838167_state @ X_68 @ A_132 ) @ B_73 )
% 3.31/1.67 = ( minus_2107060239tate_o @ A_132 @ B_73 ) ) )
% 3.31/1.67 & ( ~ ( member2058392318_state @ X_68 @ B_73 )
% 3.31/1.67 => ( ( minus_2107060239tate_o @ ( insert2134838167_state @ X_68 @ A_132 ) @ B_73 )
% 3.31/1.67 = ( insert2134838167_state @ X_68 @ ( minus_2107060239tate_o @ A_132 @ B_73 ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_379_insert__Diff__if,axiom,
% 3.31/1.67 ! [A_132: hoare_1775062406iple_a > $o,X_68: hoare_1775062406iple_a,B_73: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ( member2122167641iple_a @ X_68 @ B_73 )
% 3.31/1.67 => ( ( minus_1944206118le_a_o @ ( insert1281456128iple_a @ X_68 @ A_132 ) @ B_73 )
% 3.31/1.67 = ( minus_1944206118le_a_o @ A_132 @ B_73 ) ) )
% 3.31/1.67 & ( ~ ( member2122167641iple_a @ X_68 @ B_73 )
% 3.31/1.67 => ( ( minus_1944206118le_a_o @ ( insert1281456128iple_a @ X_68 @ A_132 ) @ B_73 )
% 3.31/1.67 = ( insert1281456128iple_a @ X_68 @ ( minus_1944206118le_a_o @ A_132 @ B_73 ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_380_insert__Diff,axiom,
% 3.31/1.67 ! [A_131: nat,A_130: nat > $o] :
% 3.31/1.67 ( ( member_nat @ A_131 @ A_130 )
% 3.31/1.67 => ( ( insert_nat @ A_131 @ ( minus_minus_nat_o @ A_130 @ ( insert_nat @ A_131 @ bot_bot_nat_o ) ) )
% 3.31/1.67 = A_130 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_381_insert__Diff,axiom,
% 3.31/1.67 ! [A_131: hoare_1167836817_state,A_130: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( member2058392318_state @ A_131 @ A_130 )
% 3.31/1.67 => ( ( insert2134838167_state @ A_131 @ ( minus_2107060239tate_o @ A_130 @ ( insert2134838167_state @ A_131 @ bot_bo70021908tate_o ) ) )
% 3.31/1.67 = A_130 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_382_insert__Diff,axiom,
% 3.31/1.67 ! [A_131: hoare_1775062406iple_a,A_130: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ A_131 @ A_130 )
% 3.31/1.67 => ( ( insert1281456128iple_a @ A_131 @ ( minus_1944206118le_a_o @ A_130 @ ( insert1281456128iple_a @ A_131 @ bot_bo751897185le_a_o ) ) )
% 3.31/1.67 = A_130 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_383_Diff__insert__absorb,axiom,
% 3.31/1.67 ! [X_67: nat,A_129: nat > $o] :
% 3.31/1.67 ( ~ ( member_nat @ X_67 @ A_129 )
% 3.31/1.67 => ( ( minus_minus_nat_o @ ( insert_nat @ X_67 @ A_129 ) @ ( insert_nat @ X_67 @ bot_bot_nat_o ) )
% 3.31/1.67 = A_129 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_384_Diff__insert__absorb,axiom,
% 3.31/1.67 ! [X_67: hoare_1167836817_state,A_129: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ~ ( member2058392318_state @ X_67 @ A_129 )
% 3.31/1.67 => ( ( minus_2107060239tate_o @ ( insert2134838167_state @ X_67 @ A_129 ) @ ( insert2134838167_state @ X_67 @ bot_bo70021908tate_o ) )
% 3.31/1.67 = A_129 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_385_Diff__insert__absorb,axiom,
% 3.31/1.67 ! [X_67: hoare_1775062406iple_a,A_129: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ~ ( member2122167641iple_a @ X_67 @ A_129 )
% 3.31/1.67 => ( ( minus_1944206118le_a_o @ ( insert1281456128iple_a @ X_67 @ A_129 ) @ ( insert1281456128iple_a @ X_67 @ bot_bo751897185le_a_o ) )
% 3.31/1.67 = A_129 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_386_insert__Diff__single,axiom,
% 3.31/1.67 ! [A_128: nat,A_127: nat > $o] :
% 3.31/1.67 ( ( insert_nat @ A_128 @ ( minus_minus_nat_o @ A_127 @ ( insert_nat @ A_128 @ bot_bot_nat_o ) ) )
% 3.31/1.67 = ( insert_nat @ A_128 @ A_127 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_387_insert__Diff__single,axiom,
% 3.31/1.67 ! [A_128: hoare_1167836817_state,A_127: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( insert2134838167_state @ A_128 @ ( minus_2107060239tate_o @ A_127 @ ( insert2134838167_state @ A_128 @ bot_bo70021908tate_o ) ) )
% 3.31/1.67 = ( insert2134838167_state @ A_128 @ A_127 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_388_insert__Diff__single,axiom,
% 3.31/1.67 ! [A_128: hoare_1775062406iple_a,A_127: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( insert1281456128iple_a @ A_128 @ ( minus_1944206118le_a_o @ A_127 @ ( insert1281456128iple_a @ A_128 @ bot_bo751897185le_a_o ) ) )
% 3.31/1.67 = ( insert1281456128iple_a @ A_128 @ A_127 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_389_Diff__insert2,axiom,
% 3.31/1.67 ! [A_126: nat > $o,A_125: nat,B_72: nat > $o] :
% 3.31/1.67 ( ( minus_minus_nat_o @ A_126 @ ( insert_nat @ A_125 @ B_72 ) )
% 3.31/1.67 = ( minus_minus_nat_o @ ( minus_minus_nat_o @ A_126 @ ( insert_nat @ A_125 @ bot_bot_nat_o ) ) @ B_72 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_390_Diff__insert2,axiom,
% 3.31/1.67 ! [A_126: hoare_1167836817_state > $o,A_125: hoare_1167836817_state,B_72: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( minus_2107060239tate_o @ A_126 @ ( insert2134838167_state @ A_125 @ B_72 ) )
% 3.31/1.67 = ( minus_2107060239tate_o @ ( minus_2107060239tate_o @ A_126 @ ( insert2134838167_state @ A_125 @ bot_bo70021908tate_o ) ) @ B_72 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_391_Diff__insert2,axiom,
% 3.31/1.67 ! [A_126: hoare_1775062406iple_a > $o,A_125: hoare_1775062406iple_a,B_72: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( minus_1944206118le_a_o @ A_126 @ ( insert1281456128iple_a @ A_125 @ B_72 ) )
% 3.31/1.67 = ( minus_1944206118le_a_o @ ( minus_1944206118le_a_o @ A_126 @ ( insert1281456128iple_a @ A_125 @ bot_bo751897185le_a_o ) ) @ B_72 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_392_Diff__insert,axiom,
% 3.31/1.67 ! [A_124: nat > $o,A_123: nat,B_71: nat > $o] :
% 3.31/1.67 ( ( minus_minus_nat_o @ A_124 @ ( insert_nat @ A_123 @ B_71 ) )
% 3.31/1.67 = ( minus_minus_nat_o @ ( minus_minus_nat_o @ A_124 @ B_71 ) @ ( insert_nat @ A_123 @ bot_bot_nat_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_393_Diff__insert,axiom,
% 3.31/1.67 ! [A_124: hoare_1167836817_state > $o,A_123: hoare_1167836817_state,B_71: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( minus_2107060239tate_o @ A_124 @ ( insert2134838167_state @ A_123 @ B_71 ) )
% 3.31/1.67 = ( minus_2107060239tate_o @ ( minus_2107060239tate_o @ A_124 @ B_71 ) @ ( insert2134838167_state @ A_123 @ bot_bo70021908tate_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_394_Diff__insert,axiom,
% 3.31/1.67 ! [A_124: hoare_1775062406iple_a > $o,A_123: hoare_1775062406iple_a,B_71: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( minus_1944206118le_a_o @ A_124 @ ( insert1281456128iple_a @ A_123 @ B_71 ) )
% 3.31/1.67 = ( minus_1944206118le_a_o @ ( minus_1944206118le_a_o @ A_124 @ B_71 ) @ ( insert1281456128iple_a @ A_123 @ bot_bo751897185le_a_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_395_finite__Diff__insert,axiom,
% 3.31/1.67 ! [A_122: nat > $o,A_121: nat,B_70: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ ( minus_minus_nat_o @ A_122 @ ( insert_nat @ A_121 @ B_70 ) ) )
% 3.31/1.67 <=> ( finite_finite_nat @ ( minus_minus_nat_o @ A_122 @ B_70 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_396_finite__Diff__insert,axiom,
% 3.31/1.67 ! [A_122: hoare_1167836817_state > $o,A_121: hoare_1167836817_state,B_70: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( finite1084549118_state @ ( minus_2107060239tate_o @ A_122 @ ( insert2134838167_state @ A_121 @ B_70 ) ) )
% 3.31/1.67 <=> ( finite1084549118_state @ ( minus_2107060239tate_o @ A_122 @ B_70 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_397_finite__Diff__insert,axiom,
% 3.31/1.67 ! [A_122: hoare_1775062406iple_a > $o,A_121: hoare_1775062406iple_a,B_70: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( finite2063573081iple_a @ ( minus_1944206118le_a_o @ A_122 @ ( insert1281456128iple_a @ A_121 @ B_70 ) ) )
% 3.31/1.67 <=> ( finite2063573081iple_a @ ( minus_1944206118le_a_o @ A_122 @ B_70 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_398_folding__one__idem_Oin__idem,axiom,
% 3.31/1.67 ! [X_66: hoare_1775062406iple_a,A_120: hoare_1775062406iple_a > $o,F_30: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_29: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
% 3.31/1.67 ( ( finite1358382848iple_a @ F_30 @ F_29 )
% 3.31/1.67 => ( ( finite2063573081iple_a @ A_120 )
% 3.31/1.67 => ( ( member2122167641iple_a @ X_66 @ A_120 )
% 3.31/1.67 => ( ( F_30 @ X_66 @ ( F_29 @ A_120 ) )
% 3.31/1.67 = ( F_29 @ A_120 ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_399_folding__one__idem_Oin__idem,axiom,
% 3.31/1.67 ! [X_66: nat,A_120: nat > $o,F_30: nat > nat > nat,F_29: ( nat > $o ) > nat] :
% 3.31/1.67 ( ( finite795500164em_nat @ F_30 @ F_29 )
% 3.31/1.67 => ( ( finite_finite_nat @ A_120 )
% 3.31/1.67 => ( ( member_nat @ X_66 @ A_120 )
% 3.31/1.67 => ( ( F_30 @ X_66 @ ( F_29 @ A_120 ) )
% 3.31/1.67 = ( F_29 @ A_120 ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_400_folding__one__idem_Ohom__commute,axiom,
% 3.31/1.67 ! [N_4: nat > $o,H_1: nat > nat,F_28: nat > nat > nat,F_27: ( nat > $o ) > nat] :
% 3.31/1.67 ( ( finite795500164em_nat @ F_28 @ F_27 )
% 3.31/1.67 => ( ! [X_3: nat,Y_1: nat] :
% 3.31/1.67 ( ( H_1 @ ( F_28 @ X_3 @ Y_1 ) )
% 3.31/1.67 = ( F_28 @ ( H_1 @ X_3 ) @ ( H_1 @ Y_1 ) ) )
% 3.31/1.67 => ( ( finite_finite_nat @ N_4 )
% 3.31/1.67 => ( ( N_4 != bot_bot_nat_o )
% 3.31/1.67 => ( ( H_1 @ ( F_27 @ N_4 ) )
% 3.31/1.67 = ( F_27 @ ( image_nat_nat @ H_1 @ N_4 ) ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_401_folding__one__idem_Ohom__commute,axiom,
% 3.31/1.67 ! [N_4: hoare_1167836817_state > $o,H_1: hoare_1167836817_state > hoare_1167836817_state,F_28: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_27: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
% 3.31/1.67 ( ( finite806517911_state @ F_28 @ F_27 )
% 3.31/1.67 => ( ! [X_3: hoare_1167836817_state,Y_1: hoare_1167836817_state] :
% 3.31/1.67 ( ( H_1 @ ( F_28 @ X_3 @ Y_1 ) )
% 3.31/1.67 = ( F_28 @ ( H_1 @ X_3 ) @ ( H_1 @ Y_1 ) ) )
% 3.31/1.67 => ( ( finite1084549118_state @ N_4 )
% 3.31/1.67 => ( ( N_4 != bot_bo70021908tate_o )
% 3.31/1.67 => ( ( H_1 @ ( F_27 @ N_4 ) )
% 3.31/1.67 = ( F_27 @ ( image_31595733_state @ H_1 @ N_4 ) ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_402_folding__one__idem_Ohom__commute,axiom,
% 3.31/1.67 ! [N_4: hoare_1775062406iple_a > $o,H_1: hoare_1775062406iple_a > hoare_1775062406iple_a,F_28: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_27: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
% 3.31/1.67 ( ( finite1358382848iple_a @ F_28 @ F_27 )
% 3.31/1.67 => ( ! [X_3: hoare_1775062406iple_a,Y_1: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( H_1 @ ( F_28 @ X_3 @ Y_1 ) )
% 3.31/1.67 = ( F_28 @ ( H_1 @ X_3 ) @ ( H_1 @ Y_1 ) ) )
% 3.31/1.67 => ( ( finite2063573081iple_a @ N_4 )
% 3.31/1.67 => ( ( N_4 != bot_bo751897185le_a_o )
% 3.31/1.67 => ( ( H_1 @ ( F_27 @ N_4 ) )
% 3.31/1.67 = ( F_27 @ ( image_1170193413iple_a @ H_1 @ N_4 ) ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_403_finite__empty__induct,axiom,
% 3.31/1.67 ! [P_4: ( nat > $o ) > $o,A_119: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ A_119 )
% 3.31/1.67 => ( ( P_4 @ A_119 )
% 3.31/1.67 => ( ! [A_43: nat,A_98: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ A_98 )
% 3.31/1.67 => ( ( member_nat @ A_43 @ A_98 )
% 3.31/1.67 => ( ( P_4 @ A_98 )
% 3.31/1.67 => ( P_4 @ ( minus_minus_nat_o @ A_98 @ ( insert_nat @ A_43 @ bot_bot_nat_o ) ) ) ) ) )
% 3.31/1.67 => ( P_4 @ bot_bot_nat_o ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_404_finite__empty__induct,axiom,
% 3.31/1.67 ! [P_4: ( hoare_1167836817_state > $o ) > $o,A_119: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( finite1084549118_state @ A_119 )
% 3.31/1.67 => ( ( P_4 @ A_119 )
% 3.31/1.67 => ( ! [A_43: hoare_1167836817_state,A_98: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( finite1084549118_state @ A_98 )
% 3.31/1.67 => ( ( member2058392318_state @ A_43 @ A_98 )
% 3.31/1.67 => ( ( P_4 @ A_98 )
% 3.31/1.67 => ( P_4 @ ( minus_2107060239tate_o @ A_98 @ ( insert2134838167_state @ A_43 @ bot_bo70021908tate_o ) ) ) ) ) )
% 3.31/1.67 => ( P_4 @ bot_bo70021908tate_o ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_405_finite__empty__induct,axiom,
% 3.31/1.67 ! [P_4: ( hoare_1775062406iple_a > $o ) > $o,A_119: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( finite2063573081iple_a @ A_119 )
% 3.31/1.67 => ( ( P_4 @ A_119 )
% 3.31/1.67 => ( ! [A_43: hoare_1775062406iple_a,A_98: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( finite2063573081iple_a @ A_98 )
% 3.31/1.67 => ( ( member2122167641iple_a @ A_43 @ A_98 )
% 3.31/1.67 => ( ( P_4 @ A_98 )
% 3.31/1.67 => ( P_4 @ ( minus_1944206118le_a_o @ A_98 @ ( insert1281456128iple_a @ A_43 @ bot_bo751897185le_a_o ) ) ) ) ) )
% 3.31/1.67 => ( P_4 @ bot_bo751897185le_a_o ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_406_comp__fun__idem__remove,axiom,
% 3.31/1.67 ( finite389864113_nat_o
% 3.31/1.67 @ ^ [X_3: nat,A_98: nat > $o] : ( minus_minus_nat_o @ A_98 @ ( insert_nat @ X_3 @ bot_bot_nat_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_407_comp__fun__idem__remove,axiom,
% 3.31/1.67 ( finite856902323tate_o
% 3.31/1.67 @ ^ [X_3: hoare_1167836817_state,A_98: hoare_1167836817_state > $o] : ( minus_2107060239tate_o @ A_98 @ ( insert2134838167_state @ X_3 @ bot_bo70021908tate_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_408_comp__fun__idem__remove,axiom,
% 3.31/1.67 ( finite2120172977le_a_o
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a,A_98: hoare_1775062406iple_a > $o] : ( minus_1944206118le_a_o @ A_98 @ ( insert1281456128iple_a @ X_3 @ bot_bo751897185le_a_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_409_comp__fun__commute_Ofold__graph__insertE__aux,axiom,
% 3.31/1.67 ! [A_118: hoare_1775062406iple_a,Z_15: hoare_1775062406iple_a,A_117: hoare_1775062406iple_a > $o,Y_35: hoare_1775062406iple_a,F_26: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a] :
% 3.31/1.67 ( ( finite2064891473iple_a @ F_26 )
% 3.31/1.67 => ( ( finite727644230iple_a @ F_26 @ Z_15 @ A_117 @ Y_35 )
% 3.31/1.67 => ( ( member2122167641iple_a @ A_118 @ A_117 )
% 3.31/1.67 => ? [Y_36: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( Y_35
% 3.31/1.67 = ( F_26 @ A_118 @ Y_36 ) )
% 3.31/1.67 & ( finite727644230iple_a @ F_26 @ Z_15 @ ( minus_1944206118le_a_o @ A_117 @ ( insert1281456128iple_a @ A_118 @ bot_bo751897185le_a_o ) ) @ Y_36 ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_410_fold__graph__permute__diff,axiom,
% 3.31/1.67 ! [A_116: nat,B_69: nat,A_115: nat > $o,X_65: nat] :
% 3.31/1.67 ( ( finite929467206at_nat @ times_times_nat @ B_69 @ A_115 @ X_65 )
% 3.31/1.67 => ( ( member_nat @ A_116 @ A_115 )
% 3.31/1.67 => ( ~ ( member_nat @ B_69 @ A_115 )
% 3.31/1.67 => ( finite929467206at_nat @ times_times_nat @ A_116 @ ( insert_nat @ B_69 @ ( minus_minus_nat_o @ A_115 @ ( insert_nat @ A_116 @ bot_bot_nat_o ) ) ) @ X_65 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_411_comp__fun__idem__insert,axiom,
% 3.31/1.67 finite389864113_nat_o @ insert_nat ).
% 3.31/1.67
% 3.31/1.67 thf(fact_412_comp__fun__idem__insert,axiom,
% 3.31/1.67 finite856902323tate_o @ insert2134838167_state ).
% 3.31/1.67
% 3.31/1.67 thf(fact_413_comp__fun__idem__insert,axiom,
% 3.31/1.67 finite2120172977le_a_o @ insert1281456128iple_a ).
% 3.31/1.67
% 3.31/1.67 thf(fact_414_comp__fun__commute_Ofold__graph__determ,axiom,
% 3.31/1.67 ! [Y_34: hoare_1775062406iple_a,Z_14: hoare_1775062406iple_a,A_114: hoare_1775062406iple_a > $o,X_64: hoare_1775062406iple_a,F_25: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a] :
% 3.31/1.67 ( ( finite2064891473iple_a @ F_25 )
% 3.31/1.67 => ( ( finite727644230iple_a @ F_25 @ Z_14 @ A_114 @ X_64 )
% 3.31/1.67 => ( ( finite727644230iple_a @ F_25 @ Z_14 @ A_114 @ Y_34 )
% 3.31/1.67 => ( Y_34 = X_64 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_415_fold__graph__insert__swap,axiom,
% 3.31/1.67 ! [Z_13: nat,B_68: nat,A_113: nat > $o,Y_33: nat] :
% 3.31/1.67 ( ( finite929467206at_nat @ times_times_nat @ B_68 @ A_113 @ Y_33 )
% 3.31/1.67 => ( ~ ( member_nat @ B_68 @ A_113 )
% 3.31/1.67 => ( finite929467206at_nat @ times_times_nat @ Z_13 @ ( insert_nat @ B_68 @ A_113 ) @ ( times_times_nat @ Z_13 @ Y_33 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_416_comp__fun__commute_Ofold__graph__insertE,axiom,
% 3.31/1.67 ! [Z_12: hoare_1775062406iple_a,X_63: hoare_1775062406iple_a,A_112: hoare_1775062406iple_a > $o,V: hoare_1775062406iple_a,F_24: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a] :
% 3.31/1.67 ( ( finite2064891473iple_a @ F_24 )
% 3.31/1.67 => ( ( finite727644230iple_a @ F_24 @ Z_12 @ ( insert1281456128iple_a @ X_63 @ A_112 ) @ V )
% 3.31/1.67 => ( ~ ( member2122167641iple_a @ X_63 @ A_112 )
% 3.31/1.67 => ~ ! [Y_1: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( V
% 3.31/1.67 = ( F_24 @ X_63 @ Y_1 ) )
% 3.31/1.67 => ~ ( finite727644230iple_a @ F_24 @ Z_12 @ A_112 @ Y_1 ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_417_fold1__insert,axiom,
% 3.31/1.67 ! [X_62: nat,A_111: nat > $o] :
% 3.31/1.67 ( ( A_111 != bot_bot_nat_o )
% 3.31/1.67 => ( ( finite_finite_nat @ A_111 )
% 3.31/1.67 => ( ~ ( member_nat @ X_62 @ A_111 )
% 3.31/1.67 => ( ( finite_fold1_nat @ times_times_nat @ ( insert_nat @ X_62 @ A_111 ) )
% 3.31/1.67 = ( times_times_nat @ X_62 @ ( finite_fold1_nat @ times_times_nat @ A_111 ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_418_fold1__eq__fold,axiom,
% 3.31/1.67 ! [A_110: nat,A_109: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ A_109 )
% 3.31/1.67 => ( ~ ( member_nat @ A_110 @ A_109 )
% 3.31/1.67 => ( ( finite_fold1_nat @ times_times_nat @ ( insert_nat @ A_110 @ A_109 ) )
% 3.31/1.67 = ( finite_fold_nat_nat @ times_times_nat @ A_110 @ A_109 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_419_fold1__singleton,axiom,
% 3.31/1.67 ! [F_23: nat > nat > nat,A_108: nat] :
% 3.31/1.67 ( ( finite_fold1_nat @ F_23 @ ( insert_nat @ A_108 @ bot_bot_nat_o ) )
% 3.31/1.67 = A_108 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_420_fold1__singleton,axiom,
% 3.31/1.67 ! [F_23: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,A_108: hoare_1167836817_state] :
% 3.31/1.67 ( ( finite1646097201_state @ F_23 @ ( insert2134838167_state @ A_108 @ bot_bo70021908tate_o ) )
% 3.31/1.67 = A_108 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_421_fold1__singleton,axiom,
% 3.31/1.67 ! [F_23: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,A_108: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( finite1790765286iple_a @ F_23 @ ( insert1281456128iple_a @ A_108 @ bot_bo751897185le_a_o ) )
% 3.31/1.67 = A_108 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_422_fold1__singleton__def,axiom,
% 3.31/1.67 ! [A_107: nat,G_5: ( nat > $o ) > nat,F_22: nat > nat > nat] :
% 3.31/1.67 ( ( G_5
% 3.31/1.67 = ( finite_fold1_nat @ F_22 ) )
% 3.31/1.67 => ( ( G_5 @ ( insert_nat @ A_107 @ bot_bot_nat_o ) )
% 3.31/1.67 = A_107 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_423_fold1__singleton__def,axiom,
% 3.31/1.67 ! [A_107: hoare_1167836817_state,G_5: ( hoare_1167836817_state > $o ) > hoare_1167836817_state,F_22: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state] :
% 3.31/1.67 ( ( G_5
% 3.31/1.67 = ( finite1646097201_state @ F_22 ) )
% 3.31/1.67 => ( ( G_5 @ ( insert2134838167_state @ A_107 @ bot_bo70021908tate_o ) )
% 3.31/1.67 = A_107 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_424_fold1__singleton__def,axiom,
% 3.31/1.67 ! [A_107: hoare_1775062406iple_a,G_5: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a,F_22: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a] :
% 3.31/1.67 ( ( G_5
% 3.31/1.67 = ( finite1790765286iple_a @ F_22 ) )
% 3.31/1.67 => ( ( G_5 @ ( insert1281456128iple_a @ A_107 @ bot_bo751897185le_a_o ) )
% 3.31/1.67 = A_107 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_425_comp__fun__commute_Ofold__equality,axiom,
% 3.31/1.67 ! [Z_11: hoare_1775062406iple_a,A_106: hoare_1775062406iple_a > $o,Y_32: hoare_1775062406iple_a,F_21: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a] :
% 3.31/1.67 ( ( finite2064891473iple_a @ F_21 )
% 3.31/1.67 => ( ( finite727644230iple_a @ F_21 @ Z_11 @ A_106 @ Y_32 )
% 3.31/1.67 => ( ( finite1842721992iple_a @ F_21 @ Z_11 @ A_106 )
% 3.31/1.67 = Y_32 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_426_fold__def,axiom,
% 3.31/1.67 ! [F_20: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,Z_10: hoare_1775062406iple_a,A_105: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( finite1842721992iple_a @ F_20 @ Z_10 @ A_105 )
% 3.31/1.67 = ( the_Ho1155011127iple_a @ ( finite727644230iple_a @ F_20 @ Z_10 @ A_105 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_427_folding__one_Oeq__fold,axiom,
% 3.31/1.67 ! [A_104: nat > $o,F_19: nat > nat > nat,F_18: ( nat > $o ) > nat] :
% 3.31/1.67 ( ( finite988810631ne_nat @ F_19 @ F_18 )
% 3.31/1.67 => ( ( finite_finite_nat @ A_104 )
% 3.31/1.67 => ( ( F_18 @ A_104 )
% 3.31/1.67 = ( finite_fold1_nat @ F_19 @ A_104 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_428_folding__one_Oeq__fold,axiom,
% 3.31/1.67 ! [A_104: hoare_1775062406iple_a > $o,F_19: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_18: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
% 3.31/1.67 ( ( finite2078349315iple_a @ F_19 @ F_18 )
% 3.31/1.67 => ( ( finite2063573081iple_a @ A_104 )
% 3.31/1.67 => ( ( F_18 @ A_104 )
% 3.31/1.67 = ( finite1790765286iple_a @ F_19 @ A_104 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_429_folding__one_Oeq__fold_H,axiom,
% 3.31/1.67 ! [X_61: nat,A_103: nat > $o,F_17: nat > nat > nat,F_16: ( nat > $o ) > nat] :
% 3.31/1.67 ( ( finite988810631ne_nat @ F_17 @ F_16 )
% 3.31/1.67 => ( ( finite_finite_nat @ A_103 )
% 3.31/1.67 => ( ~ ( member_nat @ X_61 @ A_103 )
% 3.31/1.67 => ( ( F_16 @ ( insert_nat @ X_61 @ A_103 ) )
% 3.31/1.67 = ( finite_fold_nat_nat @ F_17 @ X_61 @ A_103 ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_430_folding__one_Oeq__fold_H,axiom,
% 3.31/1.67 ! [X_61: hoare_1167836817_state,A_103: hoare_1167836817_state > $o,F_17: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_16: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
% 3.31/1.67 ( ( finite1074406356_state @ F_17 @ F_16 )
% 3.31/1.67 => ( ( finite1084549118_state @ A_103 )
% 3.31/1.67 => ( ~ ( member2058392318_state @ X_61 @ A_103 )
% 3.31/1.67 => ( ( F_16 @ ( insert2134838167_state @ X_61 @ A_103 ) )
% 3.31/1.67 = ( finite1731015960_state @ F_17 @ X_61 @ A_103 ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_431_folding__one_Oeq__fold_H,axiom,
% 3.31/1.67 ! [X_61: hoare_1775062406iple_a,A_103: hoare_1775062406iple_a > $o,F_17: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_16: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
% 3.31/1.67 ( ( finite2078349315iple_a @ F_17 @ F_16 )
% 3.31/1.67 => ( ( finite2063573081iple_a @ A_103 )
% 3.31/1.67 => ( ~ ( member2122167641iple_a @ X_61 @ A_103 )
% 3.31/1.67 => ( ( F_16 @ ( insert1281456128iple_a @ X_61 @ A_103 ) )
% 3.31/1.67 = ( finite1842721992iple_a @ F_17 @ X_61 @ A_103 ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_432_folding__one__idem_Oeq__fold__idem_H,axiom,
% 3.31/1.67 ! [A_102: nat,A_101: nat > $o,F_15: nat > nat > nat,F_14: ( nat > $o ) > nat] :
% 3.31/1.67 ( ( finite795500164em_nat @ F_15 @ F_14 )
% 3.31/1.67 => ( ( finite_finite_nat @ A_101 )
% 3.31/1.67 => ( ( F_14 @ ( insert_nat @ A_102 @ A_101 ) )
% 3.31/1.67 = ( finite_fold_nat_nat @ F_15 @ A_102 @ A_101 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_433_folding__one__idem_Oeq__fold__idem_H,axiom,
% 3.31/1.67 ! [A_102: hoare_1167836817_state,A_101: hoare_1167836817_state > $o,F_15: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_14: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
% 3.31/1.67 ( ( finite806517911_state @ F_15 @ F_14 )
% 3.31/1.67 => ( ( finite1084549118_state @ A_101 )
% 3.31/1.67 => ( ( F_14 @ ( insert2134838167_state @ A_102 @ A_101 ) )
% 3.31/1.67 = ( finite1731015960_state @ F_15 @ A_102 @ A_101 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_434_folding__one__idem_Oeq__fold__idem_H,axiom,
% 3.31/1.67 ! [A_102: hoare_1775062406iple_a,A_101: hoare_1775062406iple_a > $o,F_15: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_14: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
% 3.31/1.67 ( ( finite1358382848iple_a @ F_15 @ F_14 )
% 3.31/1.67 => ( ( finite2063573081iple_a @ A_101 )
% 3.31/1.67 => ( ( F_14 @ ( insert1281456128iple_a @ A_102 @ A_101 ) )
% 3.31/1.67 = ( finite1842721992iple_a @ F_15 @ A_102 @ A_101 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_435_comp__fun__commute_Ofold__graph__fold,axiom,
% 3.31/1.67 ! [Z_9: hoare_1775062406iple_a,A_100: hoare_1775062406iple_a > $o,F_13: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a] :
% 3.31/1.67 ( ( finite2064891473iple_a @ F_13 )
% 3.31/1.67 => ( ( finite2063573081iple_a @ A_100 )
% 3.31/1.67 => ( finite727644230iple_a @ F_13 @ Z_9 @ A_100 @ ( finite1842721992iple_a @ F_13 @ Z_9 @ A_100 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_436_fold1__def,axiom,
% 3.31/1.67 ! [F_12: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,A_99: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( finite1790765286iple_a @ F_12 @ A_99 )
% 3.31/1.67 = ( the_Ho1155011127iple_a @ ( finite1946188886iple_a @ F_12 @ A_99 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_437_minus__fold__remove,axiom,
% 3.31/1.67 ! [B_67: nat > $o,A_97: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ A_97 )
% 3.31/1.67 => ( ( minus_minus_nat_o @ B_67 @ A_97 )
% 3.31/1.67 = ( finite326637109_nat_o
% 3.31/1.67 @ ^ [X_3: nat,A_98: nat > $o] : ( minus_minus_nat_o @ A_98 @ ( insert_nat @ X_3 @ bot_bot_nat_o ) )
% 3.31/1.67 @ B_67
% 3.31/1.67 @ A_97 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_438_minus__fold__remove,axiom,
% 3.31/1.67 ! [B_67: hoare_1167836817_state > $o,A_97: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( finite1084549118_state @ A_97 )
% 3.31/1.67 => ( ( minus_2107060239tate_o @ B_67 @ A_97 )
% 3.31/1.67 = ( finite291020855tate_o
% 3.31/1.67 @ ^ [X_3: hoare_1167836817_state,A_98: hoare_1167836817_state > $o] : ( minus_2107060239tate_o @ A_98 @ ( insert2134838167_state @ X_3 @ bot_bo70021908tate_o ) )
% 3.31/1.67 @ B_67
% 3.31/1.67 @ A_97 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_439_minus__fold__remove,axiom,
% 3.31/1.67 ! [B_67: hoare_1775062406iple_a > $o,A_97: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( finite2063573081iple_a @ A_97 )
% 3.31/1.67 => ( ( minus_1944206118le_a_o @ B_67 @ A_97 )
% 3.31/1.67 = ( finite1544171829le_a_o
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a,A_98: hoare_1775062406iple_a > $o] : ( minus_1944206118le_a_o @ A_98 @ ( insert1281456128iple_a @ X_3 @ bot_bo751897185le_a_o ) )
% 3.31/1.67 @ B_67
% 3.31/1.67 @ A_97 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_440_fold1__in,axiom,
% 3.31/1.67 ! [A_96: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ A_96 )
% 3.31/1.67 => ( ( A_96 != bot_bot_nat_o )
% 3.31/1.67 => ( ! [X_3: nat,Y_1: nat] : ( member_nat @ ( times_times_nat @ X_3 @ Y_1 ) @ ( insert_nat @ X_3 @ ( insert_nat @ Y_1 @ bot_bot_nat_o ) ) )
% 3.31/1.67 => ( member_nat @ ( finite_fold1_nat @ times_times_nat @ A_96 ) @ A_96 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_441_semilattice__big_OF__eq,axiom,
% 3.31/1.67 ! [A_95: nat > $o,F_11: nat > nat > nat,F_10: ( nat > $o ) > nat] :
% 3.31/1.67 ( ( big_se275732192ig_nat @ F_11 @ F_10 )
% 3.31/1.67 => ( ( finite_finite_nat @ A_95 )
% 3.31/1.67 => ( ( F_10 @ A_95 )
% 3.31/1.67 = ( finite_fold1_nat @ F_11 @ A_95 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_442_folding__one__idem_Osubset__idem,axiom,
% 3.31/1.67 ! [B_66: nat > $o,A_94: nat > $o,F_9: nat > nat > nat,F_8: ( nat > $o ) > nat] :
% 3.31/1.67 ( ( finite795500164em_nat @ F_9 @ F_8 )
% 3.31/1.67 => ( ( finite_finite_nat @ A_94 )
% 3.31/1.67 => ( ( B_66 != bot_bot_nat_o )
% 3.31/1.67 => ( ( ord_less_eq_nat_o @ B_66 @ A_94 )
% 3.31/1.67 => ( ( F_9 @ ( F_8 @ B_66 ) @ ( F_8 @ A_94 ) )
% 3.31/1.67 = ( F_8 @ A_94 ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_443_folding__one__idem_Osubset__idem,axiom,
% 3.31/1.67 ! [B_66: hoare_1167836817_state > $o,A_94: hoare_1167836817_state > $o,F_9: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_8: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
% 3.31/1.67 ( ( finite806517911_state @ F_9 @ F_8 )
% 3.31/1.67 => ( ( finite1084549118_state @ A_94 )
% 3.31/1.67 => ( ( B_66 != bot_bo70021908tate_o )
% 3.31/1.67 => ( ( ord_le827224136tate_o @ B_66 @ A_94 )
% 3.31/1.67 => ( ( F_9 @ ( F_8 @ B_66 ) @ ( F_8 @ A_94 ) )
% 3.31/1.67 = ( F_8 @ A_94 ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_444_folding__one__idem_Osubset__idem,axiom,
% 3.31/1.67 ! [B_66: hoare_1775062406iple_a > $o,A_94: hoare_1775062406iple_a > $o,F_9: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_8: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
% 3.31/1.67 ( ( finite1358382848iple_a @ F_9 @ F_8 )
% 3.31/1.67 => ( ( finite2063573081iple_a @ A_94 )
% 3.31/1.67 => ( ( B_66 != bot_bo751897185le_a_o )
% 3.31/1.67 => ( ( ord_le1143225901le_a_o @ B_66 @ A_94 )
% 3.31/1.67 => ( ( F_9 @ ( F_8 @ B_66 ) @ ( F_8 @ A_94 ) )
% 3.31/1.67 = ( F_8 @ A_94 ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_445_order__refl,axiom,
% 3.31/1.67 ! [X_60: nat] : ( ord_less_eq_nat @ X_60 @ X_60 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_446_subsetD,axiom,
% 3.31/1.67 ! [C_23: hoare_1775062406iple_a,A_93: hoare_1775062406iple_a > $o,B_65: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ord_le1143225901le_a_o @ A_93 @ B_65 )
% 3.31/1.67 => ( ( member2122167641iple_a @ C_23 @ A_93 )
% 3.31/1.67 => ( member2122167641iple_a @ C_23 @ B_65 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_447_subsetD,axiom,
% 3.31/1.67 ! [C_23: nat,A_93: nat > $o,B_65: nat > $o] :
% 3.31/1.67 ( ( ord_less_eq_nat_o @ A_93 @ B_65 )
% 3.31/1.67 => ( ( member_nat @ C_23 @ A_93 )
% 3.31/1.67 => ( member_nat @ C_23 @ B_65 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_448_UnCI,axiom,
% 3.31/1.67 ! [A_92: hoare_1775062406iple_a > $o,C_22: hoare_1775062406iple_a,B_64: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ~ ( member2122167641iple_a @ C_22 @ B_64 )
% 3.31/1.67 => ( member2122167641iple_a @ C_22 @ A_92 ) )
% 3.31/1.67 => ( member2122167641iple_a @ C_22 @ ( semila13410563le_a_o @ A_92 @ B_64 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_449_UnCI,axiom,
% 3.31/1.67 ! [A_92: nat > $o,C_22: nat,B_64: nat > $o] :
% 3.31/1.67 ( ( ~ ( member_nat @ C_22 @ B_64 )
% 3.31/1.67 => ( member_nat @ C_22 @ A_92 ) )
% 3.31/1.67 => ( member_nat @ C_22 @ ( semila848761471_nat_o @ A_92 @ B_64 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_450_UnE,axiom,
% 3.31/1.67 ! [C_21: hoare_1775062406iple_a,A_91: hoare_1775062406iple_a > $o,B_63: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ C_21 @ ( semila13410563le_a_o @ A_91 @ B_63 ) )
% 3.31/1.67 => ( ~ ( member2122167641iple_a @ C_21 @ A_91 )
% 3.31/1.67 => ( member2122167641iple_a @ C_21 @ B_63 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_451_UnE,axiom,
% 3.31/1.67 ! [C_21: nat,A_91: nat > $o,B_63: nat > $o] :
% 3.31/1.67 ( ( member_nat @ C_21 @ ( semila848761471_nat_o @ A_91 @ B_63 ) )
% 3.31/1.67 => ( ~ ( member_nat @ C_21 @ A_91 )
% 3.31/1.67 => ( member_nat @ C_21 @ B_63 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_452_empty__subsetI,axiom,
% 3.31/1.67 ! [A_90: nat > $o] : ( ord_less_eq_nat_o @ bot_bot_nat_o @ A_90 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_453_empty__subsetI,axiom,
% 3.31/1.67 ! [A_90: hoare_1167836817_state > $o] : ( ord_le827224136tate_o @ bot_bo70021908tate_o @ A_90 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_454_empty__subsetI,axiom,
% 3.31/1.67 ! [A_90: hoare_1775062406iple_a > $o] : ( ord_le1143225901le_a_o @ bot_bo751897185le_a_o @ A_90 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_455_finite__Collect__subsets,axiom,
% 3.31/1.67 ! [A_89: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ A_89 )
% 3.31/1.67 => ( finite_finite_nat_o
% 3.31/1.67 @ ( collect_nat_o
% 3.31/1.67 @ ^ [B_62: nat > $o] : ( ord_less_eq_nat_o @ B_62 @ A_89 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_456_sup__le__fold__sup,axiom,
% 3.31/1.67 ! [B_61: nat,A_88: nat,A_87: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ A_87 )
% 3.31/1.67 => ( ( member_nat @ A_88 @ A_87 )
% 3.31/1.67 => ( ord_less_eq_nat @ ( semila972727038up_nat @ A_88 @ B_61 ) @ ( finite_fold_nat_nat @ semila972727038up_nat @ B_61 @ A_87 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_457_subset__empty,axiom,
% 3.31/1.67 ! [A_86: nat > $o] :
% 3.31/1.67 ( ( ord_less_eq_nat_o @ A_86 @ bot_bot_nat_o )
% 3.31/1.67 <=> ( A_86 = bot_bot_nat_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_458_subset__empty,axiom,
% 3.31/1.67 ! [A_86: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( ord_le827224136tate_o @ A_86 @ bot_bo70021908tate_o )
% 3.31/1.67 <=> ( A_86 = bot_bo70021908tate_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_459_subset__empty,axiom,
% 3.31/1.67 ! [A_86: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ord_le1143225901le_a_o @ A_86 @ bot_bo751897185le_a_o )
% 3.31/1.67 <=> ( A_86 = bot_bo751897185le_a_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_460_rev__finite__subset,axiom,
% 3.31/1.67 ! [A_85: nat > $o,B_60: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ B_60 )
% 3.31/1.67 => ( ( ord_less_eq_nat_o @ A_85 @ B_60 )
% 3.31/1.67 => ( finite_finite_nat @ A_85 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_461_finite__subset,axiom,
% 3.31/1.67 ! [A_84: nat > $o,B_59: nat > $o] :
% 3.31/1.67 ( ( ord_less_eq_nat_o @ A_84 @ B_59 )
% 3.31/1.67 => ( ( finite_finite_nat @ B_59 )
% 3.31/1.67 => ( finite_finite_nat @ A_84 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_462_subset__insertI,axiom,
% 3.31/1.67 ! [B_58: nat > $o,A_83: nat] : ( ord_less_eq_nat_o @ B_58 @ ( insert_nat @ A_83 @ B_58 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_463_subset__insertI,axiom,
% 3.31/1.67 ! [B_58: hoare_1167836817_state > $o,A_83: hoare_1167836817_state] : ( ord_le827224136tate_o @ B_58 @ ( insert2134838167_state @ A_83 @ B_58 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_464_subset__insertI,axiom,
% 3.31/1.67 ! [B_58: hoare_1775062406iple_a > $o,A_83: hoare_1775062406iple_a] : ( ord_le1143225901le_a_o @ B_58 @ ( insert1281456128iple_a @ A_83 @ B_58 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_465_insert__subset,axiom,
% 3.31/1.67 ! [X_59: nat,A_82: nat > $o,B_57: nat > $o] :
% 3.31/1.67 ( ( ord_less_eq_nat_o @ ( insert_nat @ X_59 @ A_82 ) @ B_57 )
% 3.31/1.67 <=> ( ( member_nat @ X_59 @ B_57 )
% 3.31/1.67 & ( ord_less_eq_nat_o @ A_82 @ B_57 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_466_insert__subset,axiom,
% 3.31/1.67 ! [X_59: hoare_1167836817_state,A_82: hoare_1167836817_state > $o,B_57: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( ord_le827224136tate_o @ ( insert2134838167_state @ X_59 @ A_82 ) @ B_57 )
% 3.31/1.67 <=> ( ( member2058392318_state @ X_59 @ B_57 )
% 3.31/1.67 & ( ord_le827224136tate_o @ A_82 @ B_57 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_467_insert__subset,axiom,
% 3.31/1.67 ! [X_59: hoare_1775062406iple_a,A_82: hoare_1775062406iple_a > $o,B_57: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ord_le1143225901le_a_o @ ( insert1281456128iple_a @ X_59 @ A_82 ) @ B_57 )
% 3.31/1.67 <=> ( ( member2122167641iple_a @ X_59 @ B_57 )
% 3.31/1.67 & ( ord_le1143225901le_a_o @ A_82 @ B_57 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_468_subset__insert,axiom,
% 3.31/1.67 ! [B_56: nat > $o,X_58: nat,A_81: nat > $o] :
% 3.31/1.67 ( ~ ( member_nat @ X_58 @ A_81 )
% 3.31/1.67 => ( ( ord_less_eq_nat_o @ A_81 @ ( insert_nat @ X_58 @ B_56 ) )
% 3.31/1.67 <=> ( ord_less_eq_nat_o @ A_81 @ B_56 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_469_subset__insert,axiom,
% 3.31/1.67 ! [B_56: hoare_1167836817_state > $o,X_58: hoare_1167836817_state,A_81: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ~ ( member2058392318_state @ X_58 @ A_81 )
% 3.31/1.67 => ( ( ord_le827224136tate_o @ A_81 @ ( insert2134838167_state @ X_58 @ B_56 ) )
% 3.31/1.67 <=> ( ord_le827224136tate_o @ A_81 @ B_56 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_470_subset__insert,axiom,
% 3.31/1.67 ! [B_56: hoare_1775062406iple_a > $o,X_58: hoare_1775062406iple_a,A_81: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ~ ( member2122167641iple_a @ X_58 @ A_81 )
% 3.31/1.67 => ( ( ord_le1143225901le_a_o @ A_81 @ ( insert1281456128iple_a @ X_58 @ B_56 ) )
% 3.31/1.67 <=> ( ord_le1143225901le_a_o @ A_81 @ B_56 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_471_subset__insertI2,axiom,
% 3.31/1.67 ! [B_55: nat,A_80: nat > $o,B_54: nat > $o] :
% 3.31/1.67 ( ( ord_less_eq_nat_o @ A_80 @ B_54 )
% 3.31/1.67 => ( ord_less_eq_nat_o @ A_80 @ ( insert_nat @ B_55 @ B_54 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_472_subset__insertI2,axiom,
% 3.31/1.67 ! [B_55: hoare_1167836817_state,A_80: hoare_1167836817_state > $o,B_54: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( ord_le827224136tate_o @ A_80 @ B_54 )
% 3.31/1.67 => ( ord_le827224136tate_o @ A_80 @ ( insert2134838167_state @ B_55 @ B_54 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_473_subset__insertI2,axiom,
% 3.31/1.67 ! [B_55: hoare_1775062406iple_a,A_80: hoare_1775062406iple_a > $o,B_54: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ord_le1143225901le_a_o @ A_80 @ B_54 )
% 3.31/1.67 => ( ord_le1143225901le_a_o @ A_80 @ ( insert1281456128iple_a @ B_55 @ B_54 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_474_insert__mono,axiom,
% 3.31/1.67 ! [A_79: nat,C_20: nat > $o,D_2: nat > $o] :
% 3.31/1.67 ( ( ord_less_eq_nat_o @ C_20 @ D_2 )
% 3.31/1.67 => ( ord_less_eq_nat_o @ ( insert_nat @ A_79 @ C_20 ) @ ( insert_nat @ A_79 @ D_2 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_475_insert__mono,axiom,
% 3.31/1.67 ! [A_79: hoare_1167836817_state,C_20: hoare_1167836817_state > $o,D_2: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( ord_le827224136tate_o @ C_20 @ D_2 )
% 3.31/1.67 => ( ord_le827224136tate_o @ ( insert2134838167_state @ A_79 @ C_20 ) @ ( insert2134838167_state @ A_79 @ D_2 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_476_insert__mono,axiom,
% 3.31/1.67 ! [A_79: hoare_1775062406iple_a,C_20: hoare_1775062406iple_a > $o,D_2: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ord_le1143225901le_a_o @ C_20 @ D_2 )
% 3.31/1.67 => ( ord_le1143225901le_a_o @ ( insert1281456128iple_a @ A_79 @ C_20 ) @ ( insert1281456128iple_a @ A_79 @ D_2 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_477_le__supE,axiom,
% 3.31/1.67 ! [A_78: nat,B_53: nat,X_57: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ ( semila972727038up_nat @ A_78 @ B_53 ) @ X_57 )
% 3.31/1.67 => ~ ( ( ord_less_eq_nat @ A_78 @ X_57 )
% 3.31/1.67 => ~ ( ord_less_eq_nat @ B_53 @ X_57 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_478_sup__mono,axiom,
% 3.31/1.67 ! [B_52: nat,D_1: nat,A_77: nat,C_19: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ A_77 @ C_19 )
% 3.31/1.67 => ( ( ord_less_eq_nat @ B_52 @ D_1 )
% 3.31/1.67 => ( ord_less_eq_nat @ ( semila972727038up_nat @ A_77 @ B_52 ) @ ( semila972727038up_nat @ C_19 @ D_1 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_479_sup__least,axiom,
% 3.31/1.67 ! [Z_8: nat,Y_31: nat,X_56: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ Y_31 @ X_56 )
% 3.31/1.67 => ( ( ord_less_eq_nat @ Z_8 @ X_56 )
% 3.31/1.67 => ( ord_less_eq_nat @ ( semila972727038up_nat @ Y_31 @ Z_8 ) @ X_56 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_480_le__supI,axiom,
% 3.31/1.67 ! [B_51: nat,A_76: nat,X_55: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ A_76 @ X_55 )
% 3.31/1.67 => ( ( ord_less_eq_nat @ B_51 @ X_55 )
% 3.31/1.67 => ( ord_less_eq_nat @ ( semila972727038up_nat @ A_76 @ B_51 ) @ X_55 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_481_sup__absorb1,axiom,
% 3.31/1.67 ! [Y_30: nat,X_54: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ Y_30 @ X_54 )
% 3.31/1.67 => ( ( semila972727038up_nat @ X_54 @ Y_30 )
% 3.31/1.67 = X_54 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_482_sup__absorb2,axiom,
% 3.31/1.67 ! [X_53: nat,Y_29: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ X_53 @ Y_29 )
% 3.31/1.67 => ( ( semila972727038up_nat @ X_53 @ Y_29 )
% 3.31/1.67 = Y_29 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_483_le__supI2,axiom,
% 3.31/1.67 ! [A_75: nat,X_52: nat,B_50: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ X_52 @ B_50 )
% 3.31/1.67 => ( ord_less_eq_nat @ X_52 @ ( semila972727038up_nat @ A_75 @ B_50 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_484_le__supI1,axiom,
% 3.31/1.67 ! [B_49: nat,X_51: nat,A_74: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ X_51 @ A_74 )
% 3.31/1.67 => ( ord_less_eq_nat @ X_51 @ ( semila972727038up_nat @ A_74 @ B_49 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_485_le__sup__iff,axiom,
% 3.31/1.67 ! [X_50: nat,Y_28: nat,Z_7: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ ( semila972727038up_nat @ X_50 @ Y_28 ) @ Z_7 )
% 3.31/1.67 <=> ( ( ord_less_eq_nat @ X_50 @ Z_7 )
% 3.31/1.67 & ( ord_less_eq_nat @ Y_28 @ Z_7 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_486_le__iff__sup,axiom,
% 3.31/1.67 ! [X_49: nat,Y_27: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ X_49 @ Y_27 )
% 3.31/1.67 <=> ( ( semila972727038up_nat @ X_49 @ Y_27 )
% 3.31/1.67 = Y_27 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_487_sup__ge2,axiom,
% 3.31/1.67 ! [Y_26: nat,X_48: nat] : ( ord_less_eq_nat @ Y_26 @ ( semila972727038up_nat @ X_48 @ Y_26 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_488_inf__sup__ord_I4_J,axiom,
% 3.31/1.67 ! [Y_25: nat,X_47: nat] : ( ord_less_eq_nat @ Y_25 @ ( semila972727038up_nat @ X_47 @ Y_25 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_489_sup__ge1,axiom,
% 3.31/1.67 ! [X_46: nat,Y_24: nat] : ( ord_less_eq_nat @ X_46 @ ( semila972727038up_nat @ X_46 @ Y_24 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_490_inf__sup__ord_I3_J,axiom,
% 3.31/1.67 ! [X_45: nat,Y_23: nat] : ( ord_less_eq_nat @ X_45 @ ( semila972727038up_nat @ X_45 @ Y_23 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_491_sup__eq__bot__iff,axiom,
% 3.31/1.67 ! [X_44: $o,Y_22: $o] :
% 3.31/1.67 ( ( ( semila10642723_sup_o @ X_44 @ Y_22 )
% 3.31/1.67 <=> bot_bot_o )
% 3.31/1.67 <=> ( ( X_44
% 3.31/1.67 <=> bot_bot_o )
% 3.31/1.67 & ( Y_22
% 3.31/1.67 <=> bot_bot_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_492_sup__eq__bot__iff,axiom,
% 3.31/1.67 ! [X_44: nat > $o,Y_22: nat > $o] :
% 3.31/1.67 ( ( ( semila848761471_nat_o @ X_44 @ Y_22 )
% 3.31/1.67 = bot_bot_nat_o )
% 3.31/1.67 <=> ( ( X_44 = bot_bot_nat_o )
% 3.31/1.67 & ( Y_22 = bot_bot_nat_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_493_sup__eq__bot__iff,axiom,
% 3.31/1.67 ! [X_44: hoare_1167836817_state > $o,Y_22: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( ( semila1172322802tate_o @ X_44 @ Y_22 )
% 3.31/1.67 = bot_bo70021908tate_o )
% 3.31/1.67 <=> ( ( X_44 = bot_bo70021908tate_o )
% 3.31/1.67 & ( Y_22 = bot_bo70021908tate_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_494_sup__eq__bot__iff,axiom,
% 3.31/1.67 ! [X_44: hoare_1775062406iple_a > $o,Y_22: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ( semila13410563le_a_o @ X_44 @ Y_22 )
% 3.31/1.67 = bot_bo751897185le_a_o )
% 3.31/1.67 <=> ( ( X_44 = bot_bo751897185le_a_o )
% 3.31/1.67 & ( Y_22 = bot_bo751897185le_a_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_495_sup__bot__right,axiom,
% 3.31/1.67 ! [X_43: $o] :
% 3.31/1.67 ( ( semila10642723_sup_o @ X_43 @ bot_bot_o )
% 3.31/1.67 <=> X_43 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_496_sup__bot__right,axiom,
% 3.31/1.67 ! [X_43: nat > $o] :
% 3.31/1.67 ( ( semila848761471_nat_o @ X_43 @ bot_bot_nat_o )
% 3.31/1.67 = X_43 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_497_sup__bot__right,axiom,
% 3.31/1.67 ! [X_43: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( semila1172322802tate_o @ X_43 @ bot_bo70021908tate_o )
% 3.31/1.67 = X_43 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_498_sup__bot__right,axiom,
% 3.31/1.67 ! [X_43: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( semila13410563le_a_o @ X_43 @ bot_bo751897185le_a_o )
% 3.31/1.67 = X_43 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_499_sup__bot__left,axiom,
% 3.31/1.67 ! [X_42: $o] :
% 3.31/1.67 ( ( semila10642723_sup_o @ bot_bot_o @ X_42 )
% 3.31/1.67 <=> X_42 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_500_sup__bot__left,axiom,
% 3.31/1.67 ! [X_42: nat > $o] :
% 3.31/1.67 ( ( semila848761471_nat_o @ bot_bot_nat_o @ X_42 )
% 3.31/1.67 = X_42 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_501_sup__bot__left,axiom,
% 3.31/1.67 ! [X_42: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( semila1172322802tate_o @ bot_bo70021908tate_o @ X_42 )
% 3.31/1.67 = X_42 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_502_sup__bot__left,axiom,
% 3.31/1.67 ! [X_42: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( semila13410563le_a_o @ bot_bo751897185le_a_o @ X_42 )
% 3.31/1.67 = X_42 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_503_Un__empty__left,axiom,
% 3.31/1.67 ! [B_48: nat > $o] :
% 3.31/1.67 ( ( semila848761471_nat_o @ bot_bot_nat_o @ B_48 )
% 3.31/1.67 = B_48 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_504_Un__empty__left,axiom,
% 3.31/1.67 ! [B_48: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( semila1172322802tate_o @ bot_bo70021908tate_o @ B_48 )
% 3.31/1.67 = B_48 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_505_Un__empty__left,axiom,
% 3.31/1.67 ! [B_48: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( semila13410563le_a_o @ bot_bo751897185le_a_o @ B_48 )
% 3.31/1.67 = B_48 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_506_Un__empty__right,axiom,
% 3.31/1.67 ! [A_73: nat > $o] :
% 3.31/1.67 ( ( semila848761471_nat_o @ A_73 @ bot_bot_nat_o )
% 3.31/1.67 = A_73 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_507_Un__empty__right,axiom,
% 3.31/1.67 ! [A_73: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( semila1172322802tate_o @ A_73 @ bot_bo70021908tate_o )
% 3.31/1.67 = A_73 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_508_Un__empty__right,axiom,
% 3.31/1.67 ! [A_73: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( semila13410563le_a_o @ A_73 @ bot_bo751897185le_a_o )
% 3.31/1.67 = A_73 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_509_Un__empty,axiom,
% 3.31/1.67 ! [A_72: nat > $o,B_47: nat > $o] :
% 3.31/1.67 ( ( ( semila848761471_nat_o @ A_72 @ B_47 )
% 3.31/1.67 = bot_bot_nat_o )
% 3.31/1.67 <=> ( ( A_72 = bot_bot_nat_o )
% 3.31/1.67 & ( B_47 = bot_bot_nat_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_510_Un__empty,axiom,
% 3.31/1.67 ! [A_72: hoare_1167836817_state > $o,B_47: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( ( semila1172322802tate_o @ A_72 @ B_47 )
% 3.31/1.67 = bot_bo70021908tate_o )
% 3.31/1.67 <=> ( ( A_72 = bot_bo70021908tate_o )
% 3.31/1.67 & ( B_47 = bot_bo70021908tate_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_511_Un__empty,axiom,
% 3.31/1.67 ! [A_72: hoare_1775062406iple_a > $o,B_47: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ( semila13410563le_a_o @ A_72 @ B_47 )
% 3.31/1.67 = bot_bo751897185le_a_o )
% 3.31/1.67 <=> ( ( A_72 = bot_bo751897185le_a_o )
% 3.31/1.67 & ( B_47 = bot_bo751897185le_a_o ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_512_finite__Un,axiom,
% 3.31/1.67 ! [F_7: nat > $o,G_4: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ ( semila848761471_nat_o @ F_7 @ G_4 ) )
% 3.31/1.67 <=> ( ( finite_finite_nat @ F_7 )
% 3.31/1.67 & ( finite_finite_nat @ G_4 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_513_finite__UnI,axiom,
% 3.31/1.67 ! [G_3: nat > $o,F_6: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ F_6 )
% 3.31/1.67 => ( ( finite_finite_nat @ G_3 )
% 3.31/1.67 => ( finite_finite_nat @ ( semila848761471_nat_o @ F_6 @ G_3 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_514_linorder__le__cases,axiom,
% 3.31/1.67 ! [X_41: nat,Y_21: nat] :
% 3.31/1.67 ( ~ ( ord_less_eq_nat @ X_41 @ Y_21 )
% 3.31/1.67 => ( ord_less_eq_nat @ Y_21 @ X_41 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_515_xt1_I6_J,axiom,
% 3.31/1.67 ! [Z_6: nat,Y_20: nat,X_40: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ Y_20 @ X_40 )
% 3.31/1.67 => ( ( ord_less_eq_nat @ Z_6 @ Y_20 )
% 3.31/1.67 => ( ord_less_eq_nat @ Z_6 @ X_40 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_516_xt1_I5_J,axiom,
% 3.31/1.67 ! [Y_19: nat,X_39: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ Y_19 @ X_39 )
% 3.31/1.67 => ( ( ord_less_eq_nat @ X_39 @ Y_19 )
% 3.31/1.67 => ( X_39 = Y_19 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_517_order__trans,axiom,
% 3.31/1.67 ! [Z_5: nat,X_38: nat,Y_18: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ X_38 @ Y_18 )
% 3.31/1.67 => ( ( ord_less_eq_nat @ Y_18 @ Z_5 )
% 3.31/1.67 => ( ord_less_eq_nat @ X_38 @ Z_5 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_518_order__antisym,axiom,
% 3.31/1.67 ! [X_37: nat,Y_17: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ X_37 @ Y_17 )
% 3.31/1.67 => ( ( ord_less_eq_nat @ Y_17 @ X_37 )
% 3.31/1.67 => ( X_37 = Y_17 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_519_xt1_I4_J,axiom,
% 3.31/1.67 ! [C_18: nat,B_46: nat,A_71: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ B_46 @ A_71 )
% 3.31/1.67 => ( ( B_46 = C_18 )
% 3.31/1.67 => ( ord_less_eq_nat @ C_18 @ A_71 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_520_ord__le__eq__trans,axiom,
% 3.31/1.67 ! [C_17: nat,A_70: nat,B_45: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ A_70 @ B_45 )
% 3.31/1.67 => ( ( B_45 = C_17 )
% 3.31/1.67 => ( ord_less_eq_nat @ A_70 @ C_17 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_521_xt1_I3_J,axiom,
% 3.31/1.67 ! [C_16: nat,A_69: nat,B_44: nat] :
% 3.31/1.67 ( ( A_69 = B_44 )
% 3.31/1.67 => ( ( ord_less_eq_nat @ C_16 @ B_44 )
% 3.31/1.67 => ( ord_less_eq_nat @ C_16 @ A_69 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_522_ord__eq__le__trans,axiom,
% 3.31/1.67 ! [C_15: nat,A_68: nat,B_43: nat] :
% 3.31/1.67 ( ( A_68 = B_43 )
% 3.31/1.67 => ( ( ord_less_eq_nat @ B_43 @ C_15 )
% 3.31/1.67 => ( ord_less_eq_nat @ A_68 @ C_15 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_523_order__antisym__conv,axiom,
% 3.31/1.67 ! [Y_16: nat,X_36: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ Y_16 @ X_36 )
% 3.31/1.67 => ( ( ord_less_eq_nat @ X_36 @ Y_16 )
% 3.31/1.67 <=> ( X_36 = Y_16 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_524_order__eq__refl,axiom,
% 3.31/1.67 ! [X_35: nat,Y_15: nat] :
% 3.31/1.67 ( ( X_35 = Y_15 )
% 3.31/1.67 => ( ord_less_eq_nat @ X_35 @ Y_15 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_525_order__eq__iff,axiom,
% 3.31/1.67 ! [X_34: nat,Y_14: nat] :
% 3.31/1.67 ( ( X_34 = Y_14 )
% 3.31/1.67 <=> ( ( ord_less_eq_nat @ X_34 @ Y_14 )
% 3.31/1.67 & ( ord_less_eq_nat @ Y_14 @ X_34 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_526_linorder__linear,axiom,
% 3.31/1.67 ! [X_33: nat,Y_13: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ X_33 @ Y_13 )
% 3.31/1.67 | ( ord_less_eq_nat @ Y_13 @ X_33 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_527_Collect__disj__eq,axiom,
% 3.31/1.67 ! [P_3: hoare_1775062406iple_a > $o,Q_1: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( collec676402587iple_a
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] : ( (|) @ ( P_3 @ X_3 ) @ ( Q_1 @ X_3 ) ) )
% 3.31/1.67 = ( semila13410563le_a_o @ ( collec676402587iple_a @ P_3 ) @ ( collec676402587iple_a @ Q_1 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_528_Collect__disj__eq,axiom,
% 3.31/1.67 ! [P_3: nat > $o,Q_1: nat > $o] :
% 3.31/1.67 ( ( collect_nat
% 3.31/1.67 @ ^ [X_3: nat] : ( (|) @ ( P_3 @ X_3 ) @ ( Q_1 @ X_3 ) ) )
% 3.31/1.67 = ( semila848761471_nat_o @ ( collect_nat @ P_3 ) @ ( collect_nat @ Q_1 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_529_set__mp,axiom,
% 3.31/1.67 ! [X_32: hoare_1775062406iple_a,A_67: hoare_1775062406iple_a > $o,B_42: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ord_le1143225901le_a_o @ A_67 @ B_42 )
% 3.31/1.67 => ( ( member2122167641iple_a @ X_32 @ A_67 )
% 3.31/1.67 => ( member2122167641iple_a @ X_32 @ B_42 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_530_set__mp,axiom,
% 3.31/1.67 ! [X_32: nat,A_67: nat > $o,B_42: nat > $o] :
% 3.31/1.67 ( ( ord_less_eq_nat_o @ A_67 @ B_42 )
% 3.31/1.67 => ( ( member_nat @ X_32 @ A_67 )
% 3.31/1.67 => ( member_nat @ X_32 @ B_42 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_531_set__rev__mp,axiom,
% 3.31/1.67 ! [B_41: hoare_1775062406iple_a > $o,X_31: hoare_1775062406iple_a,A_66: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ X_31 @ A_66 )
% 3.31/1.67 => ( ( ord_le1143225901le_a_o @ A_66 @ B_41 )
% 3.31/1.67 => ( member2122167641iple_a @ X_31 @ B_41 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_532_set__rev__mp,axiom,
% 3.31/1.67 ! [B_41: nat > $o,X_31: nat,A_66: nat > $o] :
% 3.31/1.67 ( ( member_nat @ X_31 @ A_66 )
% 3.31/1.67 => ( ( ord_less_eq_nat_o @ A_66 @ B_41 )
% 3.31/1.67 => ( member_nat @ X_31 @ B_41 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_533_in__mono,axiom,
% 3.31/1.67 ! [X_30: hoare_1775062406iple_a,A_65: hoare_1775062406iple_a > $o,B_40: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ord_le1143225901le_a_o @ A_65 @ B_40 )
% 3.31/1.67 => ( ( member2122167641iple_a @ X_30 @ A_65 )
% 3.31/1.67 => ( member2122167641iple_a @ X_30 @ B_40 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_534_in__mono,axiom,
% 3.31/1.67 ! [X_30: nat,A_65: nat > $o,B_40: nat > $o] :
% 3.31/1.67 ( ( ord_less_eq_nat_o @ A_65 @ B_40 )
% 3.31/1.67 => ( ( member_nat @ X_30 @ A_65 )
% 3.31/1.67 => ( member_nat @ X_30 @ B_40 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_535_UnI2,axiom,
% 3.31/1.67 ! [A_64: hoare_1775062406iple_a > $o,C_14: hoare_1775062406iple_a,B_39: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ C_14 @ B_39 )
% 3.31/1.67 => ( member2122167641iple_a @ C_14 @ ( semila13410563le_a_o @ A_64 @ B_39 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_536_UnI2,axiom,
% 3.31/1.67 ! [A_64: nat > $o,C_14: nat,B_39: nat > $o] :
% 3.31/1.67 ( ( member_nat @ C_14 @ B_39 )
% 3.31/1.67 => ( member_nat @ C_14 @ ( semila848761471_nat_o @ A_64 @ B_39 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_537_UnI1,axiom,
% 3.31/1.67 ! [B_38: hoare_1775062406iple_a > $o,C_13: hoare_1775062406iple_a,A_63: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ C_13 @ A_63 )
% 3.31/1.67 => ( member2122167641iple_a @ C_13 @ ( semila13410563le_a_o @ A_63 @ B_38 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_538_UnI1,axiom,
% 3.31/1.67 ! [B_38: nat > $o,C_13: nat,A_63: nat > $o] :
% 3.31/1.67 ( ( member_nat @ C_13 @ A_63 )
% 3.31/1.67 => ( member_nat @ C_13 @ ( semila848761471_nat_o @ A_63 @ B_38 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_539_Un__iff,axiom,
% 3.31/1.67 ! [C_12: hoare_1775062406iple_a,A_62: hoare_1775062406iple_a > $o,B_37: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ C_12 @ ( semila13410563le_a_o @ A_62 @ B_37 ) )
% 3.31/1.67 <=> ( ( member2122167641iple_a @ C_12 @ A_62 )
% 3.31/1.67 | ( member2122167641iple_a @ C_12 @ B_37 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_540_Un__iff,axiom,
% 3.31/1.67 ! [C_12: nat,A_62: nat > $o,B_37: nat > $o] :
% 3.31/1.67 ( ( member_nat @ C_12 @ ( semila848761471_nat_o @ A_62 @ B_37 ) )
% 3.31/1.67 <=> ( ( member_nat @ C_12 @ A_62 )
% 3.31/1.67 | ( member_nat @ C_12 @ B_37 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_541_Un__def,axiom,
% 3.31/1.67 ! [A_61: hoare_1775062406iple_a > $o,B_36: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( semila13410563le_a_o @ A_61 @ B_36 )
% 3.31/1.67 = ( collec676402587iple_a
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] : ( (|) @ ( member2122167641iple_a @ X_3 @ A_61 ) @ ( member2122167641iple_a @ X_3 @ B_36 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_542_Un__def,axiom,
% 3.31/1.67 ! [A_61: nat > $o,B_36: nat > $o] :
% 3.31/1.67 ( ( semila848761471_nat_o @ A_61 @ B_36 )
% 3.31/1.67 = ( collect_nat
% 3.31/1.67 @ ^ [X_3: nat] : ( (|) @ ( member_nat @ X_3 @ A_61 ) @ ( member_nat @ X_3 @ B_36 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_543_pred__subset__eq,axiom,
% 3.31/1.67 ! [R_2: hoare_1775062406iple_a > $o,S_4: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ord_le1143225901le_a_o
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] : ( member2122167641iple_a @ X_3 @ R_2 )
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] : ( member2122167641iple_a @ X_3 @ S_4 ) )
% 3.31/1.67 <=> ( ord_le1143225901le_a_o @ R_2 @ S_4 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_544_pred__subset__eq,axiom,
% 3.31/1.67 ! [R_2: nat > $o,S_4: nat > $o] :
% 3.31/1.67 ( ( ord_less_eq_nat_o
% 3.31/1.67 @ ^ [X_3: nat] : ( member_nat @ X_3 @ R_2 )
% 3.31/1.67 @ ^ [X_3: nat] : ( member_nat @ X_3 @ S_4 ) )
% 3.31/1.67 <=> ( ord_less_eq_nat_o @ R_2 @ S_4 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_545_sup__Un__eq,axiom,
% 3.31/1.67 ! [R_1: hoare_1775062406iple_a > $o,S_3: hoare_1775062406iple_a > $o,X_3: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( semila13410563le_a_o
% 3.31/1.67 @ ^ [Y_1: hoare_1775062406iple_a] : ( member2122167641iple_a @ Y_1 @ R_1 )
% 3.31/1.67 @ ^ [Y_1: hoare_1775062406iple_a] : ( member2122167641iple_a @ Y_1 @ S_3 )
% 3.31/1.67 @ X_3 )
% 3.31/1.67 <=> ( member2122167641iple_a @ X_3 @ ( semila13410563le_a_o @ R_1 @ S_3 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_546_sup__Un__eq,axiom,
% 3.31/1.67 ! [R_1: nat > $o,S_3: nat > $o,X_3: nat] :
% 3.31/1.67 ( ( semila848761471_nat_o
% 3.31/1.67 @ ^ [Y_1: nat] : ( member_nat @ Y_1 @ R_1 )
% 3.31/1.67 @ ^ [Y_1: nat] : ( member_nat @ Y_1 @ S_3 )
% 3.31/1.67 @ X_3 )
% 3.31/1.67 <=> ( member_nat @ X_3 @ ( semila848761471_nat_o @ R_1 @ S_3 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_547_bot__least,axiom,
% 3.31/1.67 ! [A_60: $o] : ( ord_less_eq_o @ bot_bot_o @ A_60 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_548_bot__least,axiom,
% 3.31/1.67 ! [A_60: nat > $o] : ( ord_less_eq_nat_o @ bot_bot_nat_o @ A_60 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_549_bot__least,axiom,
% 3.31/1.67 ! [A_60: hoare_1167836817_state > $o] : ( ord_le827224136tate_o @ bot_bo70021908tate_o @ A_60 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_550_bot__least,axiom,
% 3.31/1.67 ! [A_60: hoare_1775062406iple_a > $o] : ( ord_le1143225901le_a_o @ bot_bo751897185le_a_o @ A_60 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_551_bot__least,axiom,
% 3.31/1.67 ! [A_60: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A_60 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_552_bot__unique,axiom,
% 3.31/1.67 ! [A_59: $o] :
% 3.31/1.67 ( ( ord_less_eq_o @ A_59 @ bot_bot_o )
% 3.31/1.67 <=> ( A_59
% 3.31/1.67 <=> bot_bot_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_553_bot__unique,axiom,
% 3.31/1.67 ! [A_59: nat > $o] :
% 3.31/1.67 ( ( ord_less_eq_nat_o @ A_59 @ bot_bot_nat_o )
% 3.31/1.67 <=> ( A_59 = bot_bot_nat_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_554_bot__unique,axiom,
% 3.31/1.67 ! [A_59: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( ord_le827224136tate_o @ A_59 @ bot_bo70021908tate_o )
% 3.31/1.67 <=> ( A_59 = bot_bo70021908tate_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_555_bot__unique,axiom,
% 3.31/1.67 ! [A_59: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ord_le1143225901le_a_o @ A_59 @ bot_bo751897185le_a_o )
% 3.31/1.67 <=> ( A_59 = bot_bo751897185le_a_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_556_bot__unique,axiom,
% 3.31/1.67 ! [A_59: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ A_59 @ bot_bot_nat )
% 3.31/1.67 <=> ( A_59 = bot_bot_nat ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_557_le__bot,axiom,
% 3.31/1.67 ! [A_58: $o] :
% 3.31/1.67 ( ( ord_less_eq_o @ A_58 @ bot_bot_o )
% 3.31/1.67 => ( A_58
% 3.31/1.67 <=> bot_bot_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_558_le__bot,axiom,
% 3.31/1.67 ! [A_58: nat > $o] :
% 3.31/1.67 ( ( ord_less_eq_nat_o @ A_58 @ bot_bot_nat_o )
% 3.31/1.67 => ( A_58 = bot_bot_nat_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_559_le__bot,axiom,
% 3.31/1.67 ! [A_58: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( ord_le827224136tate_o @ A_58 @ bot_bo70021908tate_o )
% 3.31/1.67 => ( A_58 = bot_bo70021908tate_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_560_le__bot,axiom,
% 3.31/1.67 ! [A_58: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ord_le1143225901le_a_o @ A_58 @ bot_bo751897185le_a_o )
% 3.31/1.67 => ( A_58 = bot_bo751897185le_a_o ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_561_le__bot,axiom,
% 3.31/1.67 ! [A_58: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ A_58 @ bot_bot_nat )
% 3.31/1.67 => ( A_58 = bot_bot_nat ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_562_Un__insert__right,axiom,
% 3.31/1.67 ! [A_57: nat > $o,A_56: nat,B_35: nat > $o] :
% 3.31/1.67 ( ( semila848761471_nat_o @ A_57 @ ( insert_nat @ A_56 @ B_35 ) )
% 3.31/1.67 = ( insert_nat @ A_56 @ ( semila848761471_nat_o @ A_57 @ B_35 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_563_Un__insert__right,axiom,
% 3.31/1.67 ! [A_57: hoare_1167836817_state > $o,A_56: hoare_1167836817_state,B_35: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( semila1172322802tate_o @ A_57 @ ( insert2134838167_state @ A_56 @ B_35 ) )
% 3.31/1.67 = ( insert2134838167_state @ A_56 @ ( semila1172322802tate_o @ A_57 @ B_35 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_564_Un__insert__right,axiom,
% 3.31/1.67 ! [A_57: hoare_1775062406iple_a > $o,A_56: hoare_1775062406iple_a,B_35: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( semila13410563le_a_o @ A_57 @ ( insert1281456128iple_a @ A_56 @ B_35 ) )
% 3.31/1.67 = ( insert1281456128iple_a @ A_56 @ ( semila13410563le_a_o @ A_57 @ B_35 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_565_Un__insert__left,axiom,
% 3.31/1.67 ! [A_55: nat,B_34: nat > $o,C_11: nat > $o] :
% 3.31/1.67 ( ( semila848761471_nat_o @ ( insert_nat @ A_55 @ B_34 ) @ C_11 )
% 3.31/1.67 = ( insert_nat @ A_55 @ ( semila848761471_nat_o @ B_34 @ C_11 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_566_Un__insert__left,axiom,
% 3.31/1.67 ! [A_55: hoare_1167836817_state,B_34: hoare_1167836817_state > $o,C_11: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( semila1172322802tate_o @ ( insert2134838167_state @ A_55 @ B_34 ) @ C_11 )
% 3.31/1.67 = ( insert2134838167_state @ A_55 @ ( semila1172322802tate_o @ B_34 @ C_11 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_567_Un__insert__left,axiom,
% 3.31/1.67 ! [A_55: hoare_1775062406iple_a,B_34: hoare_1775062406iple_a > $o,C_11: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( semila13410563le_a_o @ ( insert1281456128iple_a @ A_55 @ B_34 ) @ C_11 )
% 3.31/1.67 = ( insert1281456128iple_a @ A_55 @ ( semila13410563le_a_o @ B_34 @ C_11 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_568_weaken,axiom,
% 3.31/1.67 ! [Ts_2: hoare_1167836817_state > $o,G_2: hoare_1167836817_state > $o,Ts_1: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( hoare_123228589_state @ G_2 @ Ts_1 )
% 3.31/1.67 => ( ( ord_le827224136tate_o @ Ts_2 @ Ts_1 )
% 3.31/1.67 => ( hoare_123228589_state @ G_2 @ Ts_2 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_569_weaken,axiom,
% 3.31/1.67 ! [Ts_2: hoare_1775062406iple_a > $o,G_2: hoare_1775062406iple_a > $o,Ts_1: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( hoare_1508237396rivs_a @ G_2 @ Ts_1 )
% 3.31/1.67 => ( ( ord_le1143225901le_a_o @ Ts_2 @ Ts_1 )
% 3.31/1.67 => ( hoare_1508237396rivs_a @ G_2 @ Ts_2 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_570_asm,axiom,
% 3.31/1.67 ! [Ts: hoare_1167836817_state > $o,G_1: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( ord_le827224136tate_o @ Ts @ G_1 )
% 3.31/1.67 => ( hoare_123228589_state @ G_1 @ Ts ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_571_asm,axiom,
% 3.31/1.67 ! [Ts: hoare_1775062406iple_a > $o,G_1: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ord_le1143225901le_a_o @ Ts @ G_1 )
% 3.31/1.67 => ( hoare_1508237396rivs_a @ G_1 @ Ts ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_572_insert__def,axiom,
% 3.31/1.67 ! [A_54: nat,B_33: nat > $o] :
% 3.31/1.67 ( ( insert_nat @ A_54 @ B_33 )
% 3.31/1.67 = ( semila848761471_nat_o
% 3.31/1.67 @ ( collect_nat
% 3.31/1.67 @ ^ [X_3: nat] : ( X_3 = A_54 ) )
% 3.31/1.67 @ B_33 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_573_insert__def,axiom,
% 3.31/1.67 ! [A_54: hoare_1167836817_state,B_33: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( insert2134838167_state @ A_54 @ B_33 )
% 3.31/1.67 = ( semila1172322802tate_o
% 3.31/1.67 @ ( collec1027672124_state
% 3.31/1.67 @ ^ [X_3: hoare_1167836817_state] : ( X_3 = A_54 ) )
% 3.31/1.67 @ B_33 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_574_insert__def,axiom,
% 3.31/1.67 ! [A_54: hoare_1775062406iple_a,B_33: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( insert1281456128iple_a @ A_54 @ B_33 )
% 3.31/1.67 = ( semila13410563le_a_o
% 3.31/1.67 @ ( collec676402587iple_a
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] : ( X_3 = A_54 ) )
% 3.31/1.67 @ B_33 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_575_insert__is__Un,axiom,
% 3.31/1.67 ! [A_53: nat,A_52: nat > $o] :
% 3.31/1.67 ( ( insert_nat @ A_53 @ A_52 )
% 3.31/1.67 = ( semila848761471_nat_o @ ( insert_nat @ A_53 @ bot_bot_nat_o ) @ A_52 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_576_insert__is__Un,axiom,
% 3.31/1.67 ! [A_53: hoare_1167836817_state,A_52: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( insert2134838167_state @ A_53 @ A_52 )
% 3.31/1.67 = ( semila1172322802tate_o @ ( insert2134838167_state @ A_53 @ bot_bo70021908tate_o ) @ A_52 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_577_insert__is__Un,axiom,
% 3.31/1.67 ! [A_53: hoare_1775062406iple_a,A_52: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( insert1281456128iple_a @ A_53 @ A_52 )
% 3.31/1.67 = ( semila13410563le_a_o @ ( insert1281456128iple_a @ A_53 @ bot_bo751897185le_a_o ) @ A_52 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_578_fold__sup__insert,axiom,
% 3.31/1.67 ! [B_32: nat,A_51: nat,A_50: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ A_50 )
% 3.31/1.67 => ( ( finite_fold_nat_nat @ semila972727038up_nat @ B_32 @ ( insert_nat @ A_51 @ A_50 ) )
% 3.31/1.67 = ( semila972727038up_nat @ A_51 @ ( finite_fold_nat_nat @ semila972727038up_nat @ B_32 @ A_50 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_579_union__fold__insert,axiom,
% 3.31/1.67 ! [B_31: nat > $o,A_49: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ A_49 )
% 3.31/1.67 => ( ( semila848761471_nat_o @ A_49 @ B_31 )
% 3.31/1.67 = ( finite326637109_nat_o @ insert_nat @ B_31 @ A_49 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_580_union__fold__insert,axiom,
% 3.31/1.67 ! [B_31: hoare_1167836817_state > $o,A_49: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( finite1084549118_state @ A_49 )
% 3.31/1.67 => ( ( semila1172322802tate_o @ A_49 @ B_31 )
% 3.31/1.67 = ( finite291020855tate_o @ insert2134838167_state @ B_31 @ A_49 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_581_union__fold__insert,axiom,
% 3.31/1.67 ! [B_31: hoare_1775062406iple_a > $o,A_49: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( finite2063573081iple_a @ A_49 )
% 3.31/1.67 => ( ( semila13410563le_a_o @ A_49 @ B_31 )
% 3.31/1.67 = ( finite1544171829le_a_o @ insert1281456128iple_a @ B_31 @ A_49 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_582_subset__singletonD,axiom,
% 3.31/1.67 ! [A_48: nat > $o,X_29: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat_o @ A_48 @ ( insert_nat @ X_29 @ bot_bot_nat_o ) )
% 3.31/1.67 => ( ( A_48 = bot_bot_nat_o )
% 3.31/1.67 | ( A_48
% 3.31/1.67 = ( insert_nat @ X_29 @ bot_bot_nat_o ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_583_subset__singletonD,axiom,
% 3.31/1.67 ! [A_48: hoare_1167836817_state > $o,X_29: hoare_1167836817_state] :
% 3.31/1.67 ( ( ord_le827224136tate_o @ A_48 @ ( insert2134838167_state @ X_29 @ bot_bo70021908tate_o ) )
% 3.31/1.67 => ( ( A_48 = bot_bo70021908tate_o )
% 3.31/1.67 | ( A_48
% 3.31/1.67 = ( insert2134838167_state @ X_29 @ bot_bo70021908tate_o ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_584_subset__singletonD,axiom,
% 3.31/1.67 ! [A_48: hoare_1775062406iple_a > $o,X_29: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( ord_le1143225901le_a_o @ A_48 @ ( insert1281456128iple_a @ X_29 @ bot_bo751897185le_a_o ) )
% 3.31/1.67 => ( ( A_48 = bot_bo751897185le_a_o )
% 3.31/1.67 | ( A_48
% 3.31/1.67 = ( insert1281456128iple_a @ X_29 @ bot_bo751897185le_a_o ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_585_diff__single__insert,axiom,
% 3.31/1.67 ! [A_47: nat > $o,X_28: nat,B_30: nat > $o] :
% 3.31/1.67 ( ( ord_less_eq_nat_o @ ( minus_minus_nat_o @ A_47 @ ( insert_nat @ X_28 @ bot_bot_nat_o ) ) @ B_30 )
% 3.31/1.67 => ( ( member_nat @ X_28 @ A_47 )
% 3.31/1.67 => ( ord_less_eq_nat_o @ A_47 @ ( insert_nat @ X_28 @ B_30 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_586_diff__single__insert,axiom,
% 3.31/1.67 ! [A_47: hoare_1167836817_state > $o,X_28: hoare_1167836817_state,B_30: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( ord_le827224136tate_o @ ( minus_2107060239tate_o @ A_47 @ ( insert2134838167_state @ X_28 @ bot_bo70021908tate_o ) ) @ B_30 )
% 3.31/1.67 => ( ( member2058392318_state @ X_28 @ A_47 )
% 3.31/1.67 => ( ord_le827224136tate_o @ A_47 @ ( insert2134838167_state @ X_28 @ B_30 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_587_diff__single__insert,axiom,
% 3.31/1.67 ! [A_47: hoare_1775062406iple_a > $o,X_28: hoare_1775062406iple_a,B_30: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ord_le1143225901le_a_o @ ( minus_1944206118le_a_o @ A_47 @ ( insert1281456128iple_a @ X_28 @ bot_bo751897185le_a_o ) ) @ B_30 )
% 3.31/1.67 => ( ( member2122167641iple_a @ X_28 @ A_47 )
% 3.31/1.67 => ( ord_le1143225901le_a_o @ A_47 @ ( insert1281456128iple_a @ X_28 @ B_30 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_588_subset__insert__iff,axiom,
% 3.31/1.67 ! [A_46: nat > $o,X_27: nat,B_29: nat > $o] :
% 3.31/1.67 ( ( ord_less_eq_nat_o @ A_46 @ ( insert_nat @ X_27 @ B_29 ) )
% 3.31/1.67 <=> ( ( ( member_nat @ X_27 @ A_46 )
% 3.31/1.67 => ( ord_less_eq_nat_o @ ( minus_minus_nat_o @ A_46 @ ( insert_nat @ X_27 @ bot_bot_nat_o ) ) @ B_29 ) )
% 3.31/1.67 & ( ~ ( member_nat @ X_27 @ A_46 )
% 3.31/1.67 => ( ord_less_eq_nat_o @ A_46 @ B_29 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_589_subset__insert__iff,axiom,
% 3.31/1.67 ! [A_46: hoare_1167836817_state > $o,X_27: hoare_1167836817_state,B_29: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( ord_le827224136tate_o @ A_46 @ ( insert2134838167_state @ X_27 @ B_29 ) )
% 3.31/1.67 <=> ( ( ( member2058392318_state @ X_27 @ A_46 )
% 3.31/1.67 => ( ord_le827224136tate_o @ ( minus_2107060239tate_o @ A_46 @ ( insert2134838167_state @ X_27 @ bot_bo70021908tate_o ) ) @ B_29 ) )
% 3.31/1.67 & ( ~ ( member2058392318_state @ X_27 @ A_46 )
% 3.31/1.67 => ( ord_le827224136tate_o @ A_46 @ B_29 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_590_subset__insert__iff,axiom,
% 3.31/1.67 ! [A_46: hoare_1775062406iple_a > $o,X_27: hoare_1775062406iple_a,B_29: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ord_le1143225901le_a_o @ A_46 @ ( insert1281456128iple_a @ X_27 @ B_29 ) )
% 3.31/1.67 <=> ( ( ( member2122167641iple_a @ X_27 @ A_46 )
% 3.31/1.67 => ( ord_le1143225901le_a_o @ ( minus_1944206118le_a_o @ A_46 @ ( insert1281456128iple_a @ X_27 @ bot_bo751897185le_a_o ) ) @ B_29 ) )
% 3.31/1.67 & ( ~ ( member2122167641iple_a @ X_27 @ A_46 )
% 3.31/1.67 => ( ord_le1143225901le_a_o @ A_46 @ B_29 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_591_folding__one__idem_Ounion__idem,axiom,
% 3.31/1.67 ! [B_28: nat > $o,A_45: nat > $o,F_5: nat > nat > nat,F_4: ( nat > $o ) > nat] :
% 3.31/1.67 ( ( finite795500164em_nat @ F_5 @ F_4 )
% 3.31/1.67 => ( ( finite_finite_nat @ A_45 )
% 3.31/1.67 => ( ( A_45 != bot_bot_nat_o )
% 3.31/1.67 => ( ( finite_finite_nat @ B_28 )
% 3.31/1.67 => ( ( B_28 != bot_bot_nat_o )
% 3.31/1.67 => ( ( F_4 @ ( semila848761471_nat_o @ A_45 @ B_28 ) )
% 3.31/1.67 = ( F_5 @ ( F_4 @ A_45 ) @ ( F_4 @ B_28 ) ) ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_592_folding__one__idem_Ounion__idem,axiom,
% 3.31/1.67 ! [B_28: hoare_1167836817_state > $o,A_45: hoare_1167836817_state > $o,F_5: hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state,F_4: ( hoare_1167836817_state > $o ) > hoare_1167836817_state] :
% 3.31/1.67 ( ( finite806517911_state @ F_5 @ F_4 )
% 3.31/1.67 => ( ( finite1084549118_state @ A_45 )
% 3.31/1.67 => ( ( A_45 != bot_bo70021908tate_o )
% 3.31/1.67 => ( ( finite1084549118_state @ B_28 )
% 3.31/1.67 => ( ( B_28 != bot_bo70021908tate_o )
% 3.31/1.67 => ( ( F_4 @ ( semila1172322802tate_o @ A_45 @ B_28 ) )
% 3.31/1.67 = ( F_5 @ ( F_4 @ A_45 ) @ ( F_4 @ B_28 ) ) ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_593_folding__one__idem_Ounion__idem,axiom,
% 3.31/1.67 ! [B_28: hoare_1775062406iple_a > $o,A_45: hoare_1775062406iple_a > $o,F_5: hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a,F_4: ( hoare_1775062406iple_a > $o ) > hoare_1775062406iple_a] :
% 3.31/1.67 ( ( finite1358382848iple_a @ F_5 @ F_4 )
% 3.31/1.67 => ( ( finite2063573081iple_a @ A_45 )
% 3.31/1.67 => ( ( A_45 != bot_bo751897185le_a_o )
% 3.31/1.67 => ( ( finite2063573081iple_a @ B_28 )
% 3.31/1.67 => ( ( B_28 != bot_bo751897185le_a_o )
% 3.31/1.67 => ( ( F_4 @ ( semila13410563le_a_o @ A_45 @ B_28 ) )
% 3.31/1.67 = ( F_5 @ ( F_4 @ A_45 ) @ ( F_4 @ B_28 ) ) ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_594_fold__sup__le__sup,axiom,
% 3.31/1.67 ! [C_10: nat,B_27: nat,A_44: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ A_44 )
% 3.31/1.67 => ( ! [X_3: nat] :
% 3.31/1.67 ( ( member_nat @ X_3 @ A_44 )
% 3.31/1.67 => ( ord_less_eq_nat @ X_3 @ B_27 ) )
% 3.31/1.67 => ( ord_less_eq_nat @ ( finite_fold_nat_nat @ semila972727038up_nat @ C_10 @ A_44 ) @ ( semila972727038up_nat @ B_27 @ C_10 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_595_finite__subset__induct,axiom,
% 3.31/1.67 ! [P_2: ( nat > $o ) > $o,A_42: nat > $o,F_2: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ F_2 )
% 3.31/1.67 => ( ( ord_less_eq_nat_o @ F_2 @ A_42 )
% 3.31/1.67 => ( ( P_2 @ bot_bot_nat_o )
% 3.31/1.67 => ( ! [A_43: nat,F_3: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ F_3 )
% 3.31/1.67 => ( ( member_nat @ A_43 @ A_42 )
% 3.31/1.67 => ( ~ ( member_nat @ A_43 @ F_3 )
% 3.31/1.67 => ( ( P_2 @ F_3 )
% 3.31/1.67 => ( P_2 @ ( insert_nat @ A_43 @ F_3 ) ) ) ) ) )
% 3.31/1.67 => ( P_2 @ F_2 ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_596_finite__subset__induct,axiom,
% 3.31/1.67 ! [P_2: ( hoare_1167836817_state > $o ) > $o,A_42: hoare_1167836817_state > $o,F_2: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( finite1084549118_state @ F_2 )
% 3.31/1.67 => ( ( ord_le827224136tate_o @ F_2 @ A_42 )
% 3.31/1.67 => ( ( P_2 @ bot_bo70021908tate_o )
% 3.31/1.67 => ( ! [A_43: hoare_1167836817_state,F_3: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( finite1084549118_state @ F_3 )
% 3.31/1.67 => ( ( member2058392318_state @ A_43 @ A_42 )
% 3.31/1.67 => ( ~ ( member2058392318_state @ A_43 @ F_3 )
% 3.31/1.67 => ( ( P_2 @ F_3 )
% 3.31/1.67 => ( P_2 @ ( insert2134838167_state @ A_43 @ F_3 ) ) ) ) ) )
% 3.31/1.67 => ( P_2 @ F_2 ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_597_finite__subset__induct,axiom,
% 3.31/1.67 ! [P_2: ( hoare_1775062406iple_a > $o ) > $o,A_42: hoare_1775062406iple_a > $o,F_2: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( finite2063573081iple_a @ F_2 )
% 3.31/1.67 => ( ( ord_le1143225901le_a_o @ F_2 @ A_42 )
% 3.31/1.67 => ( ( P_2 @ bot_bo751897185le_a_o )
% 3.31/1.67 => ( ! [A_43: hoare_1775062406iple_a,F_3: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( finite2063573081iple_a @ F_3 )
% 3.31/1.67 => ( ( member2122167641iple_a @ A_43 @ A_42 )
% 3.31/1.67 => ( ~ ( member2122167641iple_a @ A_43 @ F_3 )
% 3.31/1.67 => ( ( P_2 @ F_3 )
% 3.31/1.67 => ( P_2 @ ( insert1281456128iple_a @ A_43 @ F_3 ) ) ) ) ) )
% 3.31/1.67 => ( P_2 @ F_2 ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_598_subsetI,axiom,
% 3.31/1.67 ! [B_26: hoare_1775062406iple_a > $o,A_41: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ! [X_3: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( member2122167641iple_a @ X_3 @ A_41 )
% 3.31/1.67 => ( member2122167641iple_a @ X_3 @ B_26 ) )
% 3.31/1.67 => ( ord_le1143225901le_a_o @ A_41 @ B_26 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_599_subsetI,axiom,
% 3.31/1.67 ! [B_26: nat > $o,A_41: nat > $o] :
% 3.31/1.67 ( ! [X_3: nat] :
% 3.31/1.67 ( ( member_nat @ X_3 @ A_41 )
% 3.31/1.67 => ( member_nat @ X_3 @ B_26 ) )
% 3.31/1.67 => ( ord_less_eq_nat_o @ A_41 @ B_26 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_600_evaln__nonstrict,axiom,
% 3.31/1.67 ! [M_1: nat,C_9: com,S_2: state,N_3: nat,T: state] :
% 3.31/1.67 ( ( evaln @ C_9 @ S_2 @ N_3 @ T )
% 3.31/1.67 => ( ( ord_less_eq_nat @ N_3 @ M_1 )
% 3.31/1.67 => ( evaln @ C_9 @ S_2 @ M_1 @ T ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_601_finite__Collect__le__nat,axiom,
% 3.31/1.67 ! [K: nat] :
% 3.31/1.67 ( finite_finite_nat
% 3.31/1.67 @ ( collect_nat
% 3.31/1.67 @ ^ [N_1: nat] : ( ord_less_eq_nat @ N_1 @ K ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_602_flat__lub__def,axiom,
% 3.31/1.67 ! [A_40: hoare_1775062406iple_a > $o,B_25: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( ( ord_le1143225901le_a_o @ A_40 @ ( insert1281456128iple_a @ B_25 @ bot_bo751897185le_a_o ) )
% 3.31/1.67 => ( ( partia126998524iple_a @ B_25 @ A_40 )
% 3.31/1.67 = B_25 ) )
% 3.31/1.67 & ( ~ ( ord_le1143225901le_a_o @ A_40 @ ( insert1281456128iple_a @ B_25 @ bot_bo751897185le_a_o ) )
% 3.31/1.67 => ( ( partia126998524iple_a @ B_25 @ A_40 )
% 3.31/1.67 = ( the_Ho1155011127iple_a
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] : ( member2122167641iple_a @ X_3 @ ( minus_1944206118le_a_o @ A_40 @ ( insert1281456128iple_a @ B_25 @ bot_bo751897185le_a_o ) ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_603_flat__lub__def,axiom,
% 3.31/1.67 ! [A_40: nat > $o,B_25: nat] :
% 3.31/1.67 ( ( ( ord_less_eq_nat_o @ A_40 @ ( insert_nat @ B_25 @ bot_bot_nat_o ) )
% 3.31/1.67 => ( ( partial_flat_lub_nat @ B_25 @ A_40 )
% 3.31/1.67 = B_25 ) )
% 3.31/1.67 & ( ~ ( ord_less_eq_nat_o @ A_40 @ ( insert_nat @ B_25 @ bot_bot_nat_o ) )
% 3.31/1.67 => ( ( partial_flat_lub_nat @ B_25 @ A_40 )
% 3.31/1.67 = ( the_nat
% 3.31/1.67 @ ^ [X_3: nat] : ( member_nat @ X_3 @ ( minus_minus_nat_o @ A_40 @ ( insert_nat @ B_25 @ bot_bot_nat_o ) ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_604_flat__lub__def,axiom,
% 3.31/1.67 ! [A_40: hoare_1167836817_state > $o,B_25: hoare_1167836817_state] :
% 3.31/1.67 ( ( ( ord_le827224136tate_o @ A_40 @ ( insert2134838167_state @ B_25 @ bot_bo70021908tate_o ) )
% 3.31/1.67 => ( ( partia715677851_state @ B_25 @ A_40 )
% 3.31/1.67 = B_25 ) )
% 3.31/1.67 & ( ~ ( ord_le827224136tate_o @ A_40 @ ( insert2134838167_state @ B_25 @ bot_bo70021908tate_o ) )
% 3.31/1.67 => ( ( partia715677851_state @ B_25 @ A_40 )
% 3.31/1.67 = ( the_Ho310147232_state
% 3.31/1.67 @ ^ [X_3: hoare_1167836817_state] : ( member2058392318_state @ X_3 @ ( minus_2107060239tate_o @ A_40 @ ( insert2134838167_state @ B_25 @ bot_bo70021908tate_o ) ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_605_finite__nat__set__iff__bounded__le,axiom,
% 3.31/1.67 ! [N_2: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ N_2 )
% 3.31/1.67 <=> ? [M: nat] :
% 3.31/1.67 ! [X_3: nat] :
% 3.31/1.67 ( ( member_nat @ X_3 @ N_2 )
% 3.31/1.67 => ( ord_less_eq_nat @ X_3 @ M ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_606_finite__less__ub,axiom,
% 3.31/1.67 ! [U: nat,F_1: nat > nat] :
% 3.31/1.67 ( ! [N_1: nat] : ( ord_less_eq_nat @ N_1 @ ( F_1 @ N_1 ) )
% 3.31/1.67 => ( finite_finite_nat
% 3.31/1.67 @ ( collect_nat
% 3.31/1.67 @ ^ [N_1: nat] : ( ord_less_eq_nat @ ( F_1 @ N_1 ) @ U ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_607_Sup__fin_Oremove,axiom,
% 3.31/1.67 ! [X_26: nat,A_39: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ A_39 )
% 3.31/1.67 => ( ( member_nat @ X_26 @ A_39 )
% 3.31/1.67 => ( ( ( ( minus_minus_nat_o @ A_39 @ ( insert_nat @ X_26 @ bot_bot_nat_o ) )
% 3.31/1.67 = bot_bot_nat_o )
% 3.31/1.67 => ( ( big_la43341705in_nat @ A_39 )
% 3.31/1.67 = X_26 ) )
% 3.31/1.67 & ( ( ( minus_minus_nat_o @ A_39 @ ( insert_nat @ X_26 @ bot_bot_nat_o ) )
% 3.31/1.67 != bot_bot_nat_o )
% 3.31/1.67 => ( ( big_la43341705in_nat @ A_39 )
% 3.31/1.67 = ( semila972727038up_nat @ X_26 @ ( big_la43341705in_nat @ ( minus_minus_nat_o @ A_39 @ ( insert_nat @ X_26 @ bot_bot_nat_o ) ) ) ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_608_Sup__fin_Osingleton,axiom,
% 3.31/1.67 ! [X_25: nat] :
% 3.31/1.67 ( ( big_la43341705in_nat @ ( insert_nat @ X_25 @ bot_bot_nat_o ) )
% 3.31/1.67 = X_25 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_609_Sup__fin_Oin__idem,axiom,
% 3.31/1.67 ! [X_24: nat,A_38: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ A_38 )
% 3.31/1.67 => ( ( member_nat @ X_24 @ A_38 )
% 3.31/1.67 => ( ( semila972727038up_nat @ X_24 @ ( big_la43341705in_nat @ A_38 ) )
% 3.31/1.67 = ( big_la43341705in_nat @ A_38 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_610_Sup__fin_OF__eq,axiom,
% 3.31/1.67 ! [A_37: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ A_37 )
% 3.31/1.67 => ( ( big_la43341705in_nat @ A_37 )
% 3.31/1.67 = ( finite_fold1_nat @ semila972727038up_nat @ A_37 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_611_Sup__fin_Oinsert__idem,axiom,
% 3.31/1.67 ! [X_23: nat,A_36: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ A_36 )
% 3.31/1.67 => ( ( A_36 != bot_bot_nat_o )
% 3.31/1.67 => ( ( big_la43341705in_nat @ ( insert_nat @ X_23 @ A_36 ) )
% 3.31/1.67 = ( semila972727038up_nat @ X_23 @ ( big_la43341705in_nat @ A_36 ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_612_Sup__fin_Oinsert,axiom,
% 3.31/1.67 ! [X_22: nat,A_35: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ A_35 )
% 3.31/1.67 => ( ~ ( member_nat @ X_22 @ A_35 )
% 3.31/1.67 => ( ( A_35 != bot_bot_nat_o )
% 3.31/1.67 => ( ( big_la43341705in_nat @ ( insert_nat @ X_22 @ A_35 ) )
% 3.31/1.67 = ( semila972727038up_nat @ X_22 @ ( big_la43341705in_nat @ A_35 ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_613_Sup__fin_Osubset__idem,axiom,
% 3.31/1.67 ! [B_24: nat > $o,A_34: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ A_34 )
% 3.31/1.67 => ( ( B_24 != bot_bot_nat_o )
% 3.31/1.67 => ( ( ord_less_eq_nat_o @ B_24 @ A_34 )
% 3.31/1.67 => ( ( semila972727038up_nat @ ( big_la43341705in_nat @ B_24 ) @ ( big_la43341705in_nat @ A_34 ) )
% 3.31/1.67 = ( big_la43341705in_nat @ A_34 ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_614_Sup__fin_Ounion__idem,axiom,
% 3.31/1.67 ! [B_23: nat > $o,A_33: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ A_33 )
% 3.31/1.67 => ( ( A_33 != bot_bot_nat_o )
% 3.31/1.67 => ( ( finite_finite_nat @ B_23 )
% 3.31/1.67 => ( ( B_23 != bot_bot_nat_o )
% 3.31/1.67 => ( ( big_la43341705in_nat @ ( semila848761471_nat_o @ A_33 @ B_23 ) )
% 3.31/1.67 = ( semila972727038up_nat @ ( big_la43341705in_nat @ A_33 ) @ ( big_la43341705in_nat @ B_23 ) ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_615_Sup__fin_Oeq__fold__idem_H,axiom,
% 3.31/1.67 ! [A_32: nat,A_31: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ A_31 )
% 3.31/1.67 => ( ( big_la43341705in_nat @ ( insert_nat @ A_32 @ A_31 ) )
% 3.31/1.67 = ( finite_fold_nat_nat @ semila972727038up_nat @ A_32 @ A_31 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_616_Sup__fin_Oeq__fold_H,axiom,
% 3.31/1.67 ! [X_21: nat,A_30: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ A_30 )
% 3.31/1.67 => ( ~ ( member_nat @ X_21 @ A_30 )
% 3.31/1.67 => ( ( big_la43341705in_nat @ ( insert_nat @ X_21 @ A_30 ) )
% 3.31/1.67 = ( finite_fold_nat_nat @ semila972727038up_nat @ X_21 @ A_30 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_617_Sup__fin_Oinsert__remove,axiom,
% 3.31/1.67 ! [X_20: nat,A_29: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ A_29 )
% 3.31/1.67 => ( ( ( ( minus_minus_nat_o @ A_29 @ ( insert_nat @ X_20 @ bot_bot_nat_o ) )
% 3.31/1.67 = bot_bot_nat_o )
% 3.31/1.67 => ( ( big_la43341705in_nat @ ( insert_nat @ X_20 @ A_29 ) )
% 3.31/1.67 = X_20 ) )
% 3.31/1.67 & ( ( ( minus_minus_nat_o @ A_29 @ ( insert_nat @ X_20 @ bot_bot_nat_o ) )
% 3.31/1.67 != bot_bot_nat_o )
% 3.31/1.67 => ( ( big_la43341705in_nat @ ( insert_nat @ X_20 @ A_29 ) )
% 3.31/1.67 = ( semila972727038up_nat @ X_20 @ ( big_la43341705in_nat @ ( minus_minus_nat_o @ A_29 @ ( insert_nat @ X_20 @ bot_bot_nat_o ) ) ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_618_Sup__fin_Ohom__commute,axiom,
% 3.31/1.67 ! [N: nat > $o,H: nat > nat] :
% 3.31/1.67 ( ! [X_3: nat,Y_1: nat] :
% 3.31/1.67 ( ( H @ ( semila972727038up_nat @ X_3 @ Y_1 ) )
% 3.31/1.67 = ( semila972727038up_nat @ ( H @ X_3 ) @ ( H @ Y_1 ) ) )
% 3.31/1.67 => ( ( finite_finite_nat @ N )
% 3.31/1.67 => ( ( N != bot_bot_nat_o )
% 3.31/1.67 => ( ( H @ ( big_la43341705in_nat @ N ) )
% 3.31/1.67 = ( big_la43341705in_nat @ ( image_nat_nat @ H @ N ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_619_Sup__fin_Oclosed,axiom,
% 3.31/1.67 ! [A_28: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ A_28 )
% 3.31/1.67 => ( ( A_28 != bot_bot_nat_o )
% 3.31/1.67 => ( ! [X_3: nat,Y_1: nat] : ( member_nat @ ( semila972727038up_nat @ X_3 @ Y_1 ) @ ( insert_nat @ X_3 @ ( insert_nat @ Y_1 @ bot_bot_nat_o ) ) )
% 3.31/1.67 => ( member_nat @ ( big_la43341705in_nat @ A_28 ) @ A_28 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_620_Sup__fin_Ounion__disjoint,axiom,
% 3.31/1.67 ! [B_22: nat > $o,A_27: nat > $o] :
% 3.31/1.67 ( ( finite_finite_nat @ A_27 )
% 3.31/1.67 => ( ( A_27 != bot_bot_nat_o )
% 3.31/1.67 => ( ( finite_finite_nat @ B_22 )
% 3.31/1.67 => ( ( B_22 != bot_bot_nat_o )
% 3.31/1.67 => ( ( ( semila1947288293_nat_o @ A_27 @ B_22 )
% 3.31/1.67 = bot_bot_nat_o )
% 3.31/1.67 => ( ( big_la43341705in_nat @ ( semila848761471_nat_o @ A_27 @ B_22 ) )
% 3.31/1.67 = ( semila972727038up_nat @ ( big_la43341705in_nat @ A_27 ) @ ( big_la43341705in_nat @ B_22 ) ) ) ) ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_621_IntI,axiom,
% 3.31/1.67 ! [B_21: hoare_1775062406iple_a > $o,C_8: hoare_1775062406iple_a,A_26: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ C_8 @ A_26 )
% 3.31/1.67 => ( ( member2122167641iple_a @ C_8 @ B_21 )
% 3.31/1.67 => ( member2122167641iple_a @ C_8 @ ( semila966743401le_a_o @ A_26 @ B_21 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_622_IntI,axiom,
% 3.31/1.67 ! [B_21: nat > $o,C_8: nat,A_26: nat > $o] :
% 3.31/1.67 ( ( member_nat @ C_8 @ A_26 )
% 3.31/1.67 => ( ( member_nat @ C_8 @ B_21 )
% 3.31/1.67 => ( member_nat @ C_8 @ ( semila1947288293_nat_o @ A_26 @ B_21 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_623_IntE,axiom,
% 3.31/1.67 ! [C_7: hoare_1775062406iple_a,A_25: hoare_1775062406iple_a > $o,B_20: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ C_7 @ ( semila966743401le_a_o @ A_25 @ B_20 ) )
% 3.31/1.67 => ~ ( ( member2122167641iple_a @ C_7 @ A_25 )
% 3.31/1.67 => ~ ( member2122167641iple_a @ C_7 @ B_20 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_624_IntE,axiom,
% 3.31/1.67 ! [C_7: nat,A_25: nat > $o,B_20: nat > $o] :
% 3.31/1.67 ( ( member_nat @ C_7 @ ( semila1947288293_nat_o @ A_25 @ B_20 ) )
% 3.31/1.67 => ~ ( ( member_nat @ C_7 @ A_25 )
% 3.31/1.67 => ~ ( member_nat @ C_7 @ B_20 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_625_finite__Int,axiom,
% 3.31/1.67 ! [G: nat > $o,F: nat > $o] :
% 3.31/1.67 ( ( ( finite_finite_nat @ F )
% 3.31/1.67 | ( finite_finite_nat @ G ) )
% 3.31/1.67 => ( finite_finite_nat @ ( semila1947288293_nat_o @ F @ G ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_626_le__infE,axiom,
% 3.31/1.67 ! [X_19: nat,A_24: nat,B_19: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ X_19 @ ( semila80283416nf_nat @ A_24 @ B_19 ) )
% 3.31/1.67 => ~ ( ( ord_less_eq_nat @ X_19 @ A_24 )
% 3.31/1.67 => ~ ( ord_less_eq_nat @ X_19 @ B_19 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_627_inf__mono,axiom,
% 3.31/1.67 ! [B_18: nat,D: nat,A_23: nat,C_6: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ A_23 @ C_6 )
% 3.31/1.67 => ( ( ord_less_eq_nat @ B_18 @ D )
% 3.31/1.67 => ( ord_less_eq_nat @ ( semila80283416nf_nat @ A_23 @ B_18 ) @ ( semila80283416nf_nat @ C_6 @ D ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_628_inf__greatest,axiom,
% 3.31/1.67 ! [Z_4: nat,X_18: nat,Y_12: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ X_18 @ Y_12 )
% 3.31/1.67 => ( ( ord_less_eq_nat @ X_18 @ Z_4 )
% 3.31/1.67 => ( ord_less_eq_nat @ X_18 @ ( semila80283416nf_nat @ Y_12 @ Z_4 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_629_le__infI,axiom,
% 3.31/1.67 ! [B_17: nat,X_17: nat,A_22: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ X_17 @ A_22 )
% 3.31/1.67 => ( ( ord_less_eq_nat @ X_17 @ B_17 )
% 3.31/1.67 => ( ord_less_eq_nat @ X_17 @ ( semila80283416nf_nat @ A_22 @ B_17 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_630_inf__absorb2,axiom,
% 3.31/1.67 ! [Y_11: nat,X_16: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ Y_11 @ X_16 )
% 3.31/1.67 => ( ( semila80283416nf_nat @ X_16 @ Y_11 )
% 3.31/1.67 = Y_11 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_631_inf__absorb1,axiom,
% 3.31/1.67 ! [X_15: nat,Y_10: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ X_15 @ Y_10 )
% 3.31/1.67 => ( ( semila80283416nf_nat @ X_15 @ Y_10 )
% 3.31/1.67 = X_15 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_632_le__infI2,axiom,
% 3.31/1.67 ! [A_21: nat,B_16: nat,X_14: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ B_16 @ X_14 )
% 3.31/1.67 => ( ord_less_eq_nat @ ( semila80283416nf_nat @ A_21 @ B_16 ) @ X_14 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_633_le__infI1,axiom,
% 3.31/1.67 ! [B_15: nat,A_20: nat,X_13: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ A_20 @ X_13 )
% 3.31/1.67 => ( ord_less_eq_nat @ ( semila80283416nf_nat @ A_20 @ B_15 ) @ X_13 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_634_le__inf__iff,axiom,
% 3.31/1.67 ! [X_12: nat,Y_9: nat,Z_3: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ X_12 @ ( semila80283416nf_nat @ Y_9 @ Z_3 ) )
% 3.31/1.67 <=> ( ( ord_less_eq_nat @ X_12 @ Y_9 )
% 3.31/1.67 & ( ord_less_eq_nat @ X_12 @ Z_3 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_635_le__iff__inf,axiom,
% 3.31/1.67 ! [X_11: nat,Y_8: nat] :
% 3.31/1.67 ( ( ord_less_eq_nat @ X_11 @ Y_8 )
% 3.31/1.67 <=> ( ( semila80283416nf_nat @ X_11 @ Y_8 )
% 3.31/1.67 = X_11 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_636_inf__le2,axiom,
% 3.31/1.67 ! [X_10: nat,Y_7: nat] : ( ord_less_eq_nat @ ( semila80283416nf_nat @ X_10 @ Y_7 ) @ Y_7 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_637_inf__sup__ord_I2_J,axiom,
% 3.31/1.67 ! [X_9: nat,Y_6: nat] : ( ord_less_eq_nat @ ( semila80283416nf_nat @ X_9 @ Y_6 ) @ Y_6 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_638_inf__le1,axiom,
% 3.31/1.67 ! [X_8: nat,Y_5: nat] : ( ord_less_eq_nat @ ( semila80283416nf_nat @ X_8 @ Y_5 ) @ X_8 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_639_inf__sup__ord_I1_J,axiom,
% 3.31/1.67 ! [X_7: nat,Y_4: nat] : ( ord_less_eq_nat @ ( semila80283416nf_nat @ X_7 @ Y_4 ) @ X_7 ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_640_distrib__sup__le,axiom,
% 3.31/1.67 ! [X_6: nat,Y_3: nat,Z_2: nat] : ( ord_less_eq_nat @ ( semila972727038up_nat @ X_6 @ ( semila80283416nf_nat @ Y_3 @ Z_2 ) ) @ ( semila80283416nf_nat @ ( semila972727038up_nat @ X_6 @ Y_3 ) @ ( semila972727038up_nat @ X_6 @ Z_2 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_641_distrib__inf__le,axiom,
% 3.31/1.67 ! [X_5: nat,Y_2: nat,Z_1: nat] : ( ord_less_eq_nat @ ( semila972727038up_nat @ ( semila80283416nf_nat @ X_5 @ Y_2 ) @ ( semila80283416nf_nat @ X_5 @ Z_1 ) ) @ ( semila80283416nf_nat @ X_5 @ ( semila972727038up_nat @ Y_2 @ Z_1 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_642_Int__insert__left__if1,axiom,
% 3.31/1.67 ! [B_14: nat > $o,A_19: nat,C_5: nat > $o] :
% 3.31/1.67 ( ( member_nat @ A_19 @ C_5 )
% 3.31/1.67 => ( ( semila1947288293_nat_o @ ( insert_nat @ A_19 @ B_14 ) @ C_5 )
% 3.31/1.67 = ( insert_nat @ A_19 @ ( semila1947288293_nat_o @ B_14 @ C_5 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_643_Int__insert__left__if1,axiom,
% 3.31/1.67 ! [B_14: hoare_1167836817_state > $o,A_19: hoare_1167836817_state,C_5: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( member2058392318_state @ A_19 @ C_5 )
% 3.31/1.67 => ( ( semila179895820tate_o @ ( insert2134838167_state @ A_19 @ B_14 ) @ C_5 )
% 3.31/1.67 = ( insert2134838167_state @ A_19 @ ( semila179895820tate_o @ B_14 @ C_5 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_644_Int__insert__left__if1,axiom,
% 3.31/1.67 ! [B_14: hoare_1775062406iple_a > $o,A_19: hoare_1775062406iple_a,C_5: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ A_19 @ C_5 )
% 3.31/1.67 => ( ( semila966743401le_a_o @ ( insert1281456128iple_a @ A_19 @ B_14 ) @ C_5 )
% 3.31/1.67 = ( insert1281456128iple_a @ A_19 @ ( semila966743401le_a_o @ B_14 @ C_5 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_645_Int__insert__right__if1,axiom,
% 3.31/1.67 ! [B_13: nat > $o,A_18: nat,A_17: nat > $o] :
% 3.31/1.67 ( ( member_nat @ A_18 @ A_17 )
% 3.31/1.67 => ( ( semila1947288293_nat_o @ A_17 @ ( insert_nat @ A_18 @ B_13 ) )
% 3.31/1.67 = ( insert_nat @ A_18 @ ( semila1947288293_nat_o @ A_17 @ B_13 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_646_Int__insert__right__if1,axiom,
% 3.31/1.67 ! [B_13: hoare_1167836817_state > $o,A_18: hoare_1167836817_state,A_17: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( member2058392318_state @ A_18 @ A_17 )
% 3.31/1.67 => ( ( semila179895820tate_o @ A_17 @ ( insert2134838167_state @ A_18 @ B_13 ) )
% 3.31/1.67 = ( insert2134838167_state @ A_18 @ ( semila179895820tate_o @ A_17 @ B_13 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_647_Int__insert__right__if1,axiom,
% 3.31/1.67 ! [B_13: hoare_1775062406iple_a > $o,A_18: hoare_1775062406iple_a,A_17: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ A_18 @ A_17 )
% 3.31/1.67 => ( ( semila966743401le_a_o @ A_17 @ ( insert1281456128iple_a @ A_18 @ B_13 ) )
% 3.31/1.67 = ( insert1281456128iple_a @ A_18 @ ( semila966743401le_a_o @ A_17 @ B_13 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_648_Int__insert__left__if0,axiom,
% 3.31/1.67 ! [B_12: nat > $o,A_16: nat,C_4: nat > $o] :
% 3.31/1.67 ( ~ ( member_nat @ A_16 @ C_4 )
% 3.31/1.67 => ( ( semila1947288293_nat_o @ ( insert_nat @ A_16 @ B_12 ) @ C_4 )
% 3.31/1.67 = ( semila1947288293_nat_o @ B_12 @ C_4 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_649_Int__insert__left__if0,axiom,
% 3.31/1.67 ! [B_12: hoare_1167836817_state > $o,A_16: hoare_1167836817_state,C_4: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ~ ( member2058392318_state @ A_16 @ C_4 )
% 3.31/1.67 => ( ( semila179895820tate_o @ ( insert2134838167_state @ A_16 @ B_12 ) @ C_4 )
% 3.31/1.67 = ( semila179895820tate_o @ B_12 @ C_4 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_650_Int__insert__left__if0,axiom,
% 3.31/1.67 ! [B_12: hoare_1775062406iple_a > $o,A_16: hoare_1775062406iple_a,C_4: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ~ ( member2122167641iple_a @ A_16 @ C_4 )
% 3.31/1.67 => ( ( semila966743401le_a_o @ ( insert1281456128iple_a @ A_16 @ B_12 ) @ C_4 )
% 3.31/1.67 = ( semila966743401le_a_o @ B_12 @ C_4 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_651_Int__insert__right__if0,axiom,
% 3.31/1.67 ! [B_11: nat > $o,A_15: nat,A_14: nat > $o] :
% 3.31/1.67 ( ~ ( member_nat @ A_15 @ A_14 )
% 3.31/1.67 => ( ( semila1947288293_nat_o @ A_14 @ ( insert_nat @ A_15 @ B_11 ) )
% 3.31/1.67 = ( semila1947288293_nat_o @ A_14 @ B_11 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_652_Int__insert__right__if0,axiom,
% 3.31/1.67 ! [B_11: hoare_1167836817_state > $o,A_15: hoare_1167836817_state,A_14: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ~ ( member2058392318_state @ A_15 @ A_14 )
% 3.31/1.67 => ( ( semila179895820tate_o @ A_14 @ ( insert2134838167_state @ A_15 @ B_11 ) )
% 3.31/1.67 = ( semila179895820tate_o @ A_14 @ B_11 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_653_Int__insert__right__if0,axiom,
% 3.31/1.67 ! [B_11: hoare_1775062406iple_a > $o,A_15: hoare_1775062406iple_a,A_14: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ~ ( member2122167641iple_a @ A_15 @ A_14 )
% 3.31/1.67 => ( ( semila966743401le_a_o @ A_14 @ ( insert1281456128iple_a @ A_15 @ B_11 ) )
% 3.31/1.67 = ( semila966743401le_a_o @ A_14 @ B_11 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_654_insert__inter__insert,axiom,
% 3.31/1.67 ! [A_13: nat,A_12: nat > $o,B_10: nat > $o] :
% 3.31/1.67 ( ( semila1947288293_nat_o @ ( insert_nat @ A_13 @ A_12 ) @ ( insert_nat @ A_13 @ B_10 ) )
% 3.31/1.67 = ( insert_nat @ A_13 @ ( semila1947288293_nat_o @ A_12 @ B_10 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_655_insert__inter__insert,axiom,
% 3.31/1.67 ! [A_13: hoare_1167836817_state,A_12: hoare_1167836817_state > $o,B_10: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( semila179895820tate_o @ ( insert2134838167_state @ A_13 @ A_12 ) @ ( insert2134838167_state @ A_13 @ B_10 ) )
% 3.31/1.67 = ( insert2134838167_state @ A_13 @ ( semila179895820tate_o @ A_12 @ B_10 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_656_insert__inter__insert,axiom,
% 3.31/1.67 ! [A_13: hoare_1775062406iple_a,A_12: hoare_1775062406iple_a > $o,B_10: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( semila966743401le_a_o @ ( insert1281456128iple_a @ A_13 @ A_12 ) @ ( insert1281456128iple_a @ A_13 @ B_10 ) )
% 3.31/1.67 = ( insert1281456128iple_a @ A_13 @ ( semila966743401le_a_o @ A_12 @ B_10 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_657_Int__insert__left,axiom,
% 3.31/1.67 ! [B_9: nat > $o,A_11: nat,C_3: nat > $o] :
% 3.31/1.67 ( ( ( member_nat @ A_11 @ C_3 )
% 3.31/1.67 => ( ( semila1947288293_nat_o @ ( insert_nat @ A_11 @ B_9 ) @ C_3 )
% 3.31/1.67 = ( insert_nat @ A_11 @ ( semila1947288293_nat_o @ B_9 @ C_3 ) ) ) )
% 3.31/1.67 & ( ~ ( member_nat @ A_11 @ C_3 )
% 3.31/1.67 => ( ( semila1947288293_nat_o @ ( insert_nat @ A_11 @ B_9 ) @ C_3 )
% 3.31/1.67 = ( semila1947288293_nat_o @ B_9 @ C_3 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_658_Int__insert__left,axiom,
% 3.31/1.67 ! [B_9: hoare_1167836817_state > $o,A_11: hoare_1167836817_state,C_3: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( ( member2058392318_state @ A_11 @ C_3 )
% 3.31/1.67 => ( ( semila179895820tate_o @ ( insert2134838167_state @ A_11 @ B_9 ) @ C_3 )
% 3.31/1.67 = ( insert2134838167_state @ A_11 @ ( semila179895820tate_o @ B_9 @ C_3 ) ) ) )
% 3.31/1.67 & ( ~ ( member2058392318_state @ A_11 @ C_3 )
% 3.31/1.67 => ( ( semila179895820tate_o @ ( insert2134838167_state @ A_11 @ B_9 ) @ C_3 )
% 3.31/1.67 = ( semila179895820tate_o @ B_9 @ C_3 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_659_Int__insert__left,axiom,
% 3.31/1.67 ! [B_9: hoare_1775062406iple_a > $o,A_11: hoare_1775062406iple_a,C_3: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ( member2122167641iple_a @ A_11 @ C_3 )
% 3.31/1.67 => ( ( semila966743401le_a_o @ ( insert1281456128iple_a @ A_11 @ B_9 ) @ C_3 )
% 3.31/1.67 = ( insert1281456128iple_a @ A_11 @ ( semila966743401le_a_o @ B_9 @ C_3 ) ) ) )
% 3.31/1.67 & ( ~ ( member2122167641iple_a @ A_11 @ C_3 )
% 3.31/1.67 => ( ( semila966743401le_a_o @ ( insert1281456128iple_a @ A_11 @ B_9 ) @ C_3 )
% 3.31/1.67 = ( semila966743401le_a_o @ B_9 @ C_3 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_660_Int__insert__right,axiom,
% 3.31/1.67 ! [B_8: nat > $o,A_10: nat,A_9: nat > $o] :
% 3.31/1.67 ( ( ( member_nat @ A_10 @ A_9 )
% 3.31/1.67 => ( ( semila1947288293_nat_o @ A_9 @ ( insert_nat @ A_10 @ B_8 ) )
% 3.31/1.67 = ( insert_nat @ A_10 @ ( semila1947288293_nat_o @ A_9 @ B_8 ) ) ) )
% 3.31/1.67 & ( ~ ( member_nat @ A_10 @ A_9 )
% 3.31/1.67 => ( ( semila1947288293_nat_o @ A_9 @ ( insert_nat @ A_10 @ B_8 ) )
% 3.31/1.67 = ( semila1947288293_nat_o @ A_9 @ B_8 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_661_Int__insert__right,axiom,
% 3.31/1.67 ! [B_8: hoare_1167836817_state > $o,A_10: hoare_1167836817_state,A_9: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( ( member2058392318_state @ A_10 @ A_9 )
% 3.31/1.67 => ( ( semila179895820tate_o @ A_9 @ ( insert2134838167_state @ A_10 @ B_8 ) )
% 3.31/1.67 = ( insert2134838167_state @ A_10 @ ( semila179895820tate_o @ A_9 @ B_8 ) ) ) )
% 3.31/1.67 & ( ~ ( member2058392318_state @ A_10 @ A_9 )
% 3.31/1.67 => ( ( semila179895820tate_o @ A_9 @ ( insert2134838167_state @ A_10 @ B_8 ) )
% 3.31/1.67 = ( semila179895820tate_o @ A_9 @ B_8 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_662_Int__insert__right,axiom,
% 3.31/1.67 ! [B_8: hoare_1775062406iple_a > $o,A_10: hoare_1775062406iple_a,A_9: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ( member2122167641iple_a @ A_10 @ A_9 )
% 3.31/1.67 => ( ( semila966743401le_a_o @ A_9 @ ( insert1281456128iple_a @ A_10 @ B_8 ) )
% 3.31/1.67 = ( insert1281456128iple_a @ A_10 @ ( semila966743401le_a_o @ A_9 @ B_8 ) ) ) )
% 3.31/1.67 & ( ~ ( member2122167641iple_a @ A_10 @ A_9 )
% 3.31/1.67 => ( ( semila966743401le_a_o @ A_9 @ ( insert1281456128iple_a @ A_10 @ B_8 ) )
% 3.31/1.67 = ( semila966743401le_a_o @ A_9 @ B_8 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_663_inf__Int__eq,axiom,
% 3.31/1.67 ! [R: hoare_1775062406iple_a > $o,S_1: hoare_1775062406iple_a > $o,X_3: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( semila966743401le_a_o
% 3.31/1.67 @ ^ [Y_1: hoare_1775062406iple_a] : ( member2122167641iple_a @ Y_1 @ R )
% 3.31/1.67 @ ^ [Y_1: hoare_1775062406iple_a] : ( member2122167641iple_a @ Y_1 @ S_1 )
% 3.31/1.67 @ X_3 )
% 3.31/1.67 <=> ( member2122167641iple_a @ X_3 @ ( semila966743401le_a_o @ R @ S_1 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_664_inf__Int__eq,axiom,
% 3.31/1.67 ! [R: nat > $o,S_1: nat > $o,X_3: nat] :
% 3.31/1.67 ( ( semila1947288293_nat_o
% 3.31/1.67 @ ^ [Y_1: nat] : ( member_nat @ Y_1 @ R )
% 3.31/1.67 @ ^ [Y_1: nat] : ( member_nat @ Y_1 @ S_1 )
% 3.31/1.67 @ X_3 )
% 3.31/1.67 <=> ( member_nat @ X_3 @ ( semila1947288293_nat_o @ R @ S_1 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_665_Collect__conj__eq,axiom,
% 3.31/1.67 ! [P_1: hoare_1775062406iple_a > $o,Q: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( collec676402587iple_a
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] : ( (&) @ ( P_1 @ X_3 ) @ ( Q @ X_3 ) ) )
% 3.31/1.67 = ( semila966743401le_a_o @ ( collec676402587iple_a @ P_1 ) @ ( collec676402587iple_a @ Q ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_666_Collect__conj__eq,axiom,
% 3.31/1.67 ! [P_1: nat > $o,Q: nat > $o] :
% 3.31/1.67 ( ( collect_nat
% 3.31/1.67 @ ^ [X_3: nat] : ( (&) @ ( P_1 @ X_3 ) @ ( Q @ X_3 ) ) )
% 3.31/1.67 = ( semila1947288293_nat_o @ ( collect_nat @ P_1 ) @ ( collect_nat @ Q ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_667_Int__Collect,axiom,
% 3.31/1.67 ! [X_4: hoare_1775062406iple_a,A_8: hoare_1775062406iple_a > $o,P: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ X_4 @ ( semila966743401le_a_o @ A_8 @ ( collec676402587iple_a @ P ) ) )
% 3.31/1.67 <=> ( ( member2122167641iple_a @ X_4 @ A_8 )
% 3.31/1.67 & ( P @ X_4 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_668_Int__Collect,axiom,
% 3.31/1.67 ! [X_4: nat,A_8: nat > $o,P: nat > $o] :
% 3.31/1.67 ( ( member_nat @ X_4 @ ( semila1947288293_nat_o @ A_8 @ ( collect_nat @ P ) ) )
% 3.31/1.67 <=> ( ( member_nat @ X_4 @ A_8 )
% 3.31/1.67 & ( P @ X_4 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_669_Int__def,axiom,
% 3.31/1.67 ! [A_7: hoare_1775062406iple_a > $o,B_7: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( semila966743401le_a_o @ A_7 @ B_7 )
% 3.31/1.67 = ( collec676402587iple_a
% 3.31/1.67 @ ^ [X_3: hoare_1775062406iple_a] : ( (&) @ ( member2122167641iple_a @ X_3 @ A_7 ) @ ( member2122167641iple_a @ X_3 @ B_7 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_670_Int__def,axiom,
% 3.31/1.67 ! [A_7: nat > $o,B_7: nat > $o] :
% 3.31/1.67 ( ( semila1947288293_nat_o @ A_7 @ B_7 )
% 3.31/1.67 = ( collect_nat
% 3.31/1.67 @ ^ [X_3: nat] : ( (&) @ ( member_nat @ X_3 @ A_7 ) @ ( member_nat @ X_3 @ B_7 ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_671_Int__iff,axiom,
% 3.31/1.67 ! [C_2: hoare_1775062406iple_a,A_6: hoare_1775062406iple_a > $o,B_6: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ C_2 @ ( semila966743401le_a_o @ A_6 @ B_6 ) )
% 3.31/1.67 <=> ( ( member2122167641iple_a @ C_2 @ A_6 )
% 3.31/1.67 & ( member2122167641iple_a @ C_2 @ B_6 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_672_Int__iff,axiom,
% 3.31/1.67 ! [C_2: nat,A_6: nat > $o,B_6: nat > $o] :
% 3.31/1.67 ( ( member_nat @ C_2 @ ( semila1947288293_nat_o @ A_6 @ B_6 ) )
% 3.31/1.67 <=> ( ( member_nat @ C_2 @ A_6 )
% 3.31/1.67 & ( member_nat @ C_2 @ B_6 ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_673_IntD1,axiom,
% 3.31/1.67 ! [C_1: hoare_1775062406iple_a,A_5: hoare_1775062406iple_a > $o,B_5: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ C_1 @ ( semila966743401le_a_o @ A_5 @ B_5 ) )
% 3.31/1.67 => ( member2122167641iple_a @ C_1 @ A_5 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_674_IntD1,axiom,
% 3.31/1.67 ! [C_1: nat,A_5: nat > $o,B_5: nat > $o] :
% 3.31/1.67 ( ( member_nat @ C_1 @ ( semila1947288293_nat_o @ A_5 @ B_5 ) )
% 3.31/1.67 => ( member_nat @ C_1 @ A_5 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_675_IntD2,axiom,
% 3.31/1.67 ! [C: hoare_1775062406iple_a,A_4: hoare_1775062406iple_a > $o,B_4: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( member2122167641iple_a @ C @ ( semila966743401le_a_o @ A_4 @ B_4 ) )
% 3.31/1.67 => ( member2122167641iple_a @ C @ B_4 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_676_IntD2,axiom,
% 3.31/1.67 ! [C: nat,A_4: nat > $o,B_4: nat > $o] :
% 3.31/1.67 ( ( member_nat @ C @ ( semila1947288293_nat_o @ A_4 @ B_4 ) )
% 3.31/1.67 => ( member_nat @ C @ B_4 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_677_disjoint__iff__not__equal,axiom,
% 3.31/1.67 ! [A_3: nat > $o,B_3: nat > $o] :
% 3.31/1.67 ( ( ( semila1947288293_nat_o @ A_3 @ B_3 )
% 3.31/1.67 = bot_bot_nat_o )
% 3.31/1.67 <=> ! [X_3: nat] :
% 3.31/1.67 ( ( member_nat @ X_3 @ A_3 )
% 3.31/1.67 => ! [Xa: nat] :
% 3.31/1.67 ( ( member_nat @ Xa @ B_3 )
% 3.31/1.67 => ( X_3 != Xa ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_678_disjoint__iff__not__equal,axiom,
% 3.31/1.67 ! [A_3: hoare_1167836817_state > $o,B_3: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( ( semila179895820tate_o @ A_3 @ B_3 )
% 3.31/1.67 = bot_bo70021908tate_o )
% 3.31/1.67 <=> ! [X_3: hoare_1167836817_state] :
% 3.31/1.67 ( ( member2058392318_state @ X_3 @ A_3 )
% 3.31/1.67 => ! [Xa: hoare_1167836817_state] :
% 3.31/1.67 ( ( member2058392318_state @ Xa @ B_3 )
% 3.31/1.67 => ( X_3 != Xa ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_679_disjoint__iff__not__equal,axiom,
% 3.31/1.67 ! [A_3: hoare_1775062406iple_a > $o,B_3: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ( semila966743401le_a_o @ A_3 @ B_3 )
% 3.31/1.67 = bot_bo751897185le_a_o )
% 3.31/1.67 <=> ! [X_3: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( member2122167641iple_a @ X_3 @ A_3 )
% 3.31/1.67 => ! [Xa: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( member2122167641iple_a @ Xa @ B_3 )
% 3.31/1.67 => ( X_3 != Xa ) ) ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_680_Int__empty__right,axiom,
% 3.31/1.67 ! [A_2: nat > $o] :
% 3.31/1.67 ( ( semila1947288293_nat_o @ A_2 @ bot_bot_nat_o )
% 3.31/1.67 = bot_bot_nat_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_681_Int__empty__right,axiom,
% 3.31/1.67 ! [A_2: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( semila179895820tate_o @ A_2 @ bot_bo70021908tate_o )
% 3.31/1.67 = bot_bo70021908tate_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_682_Int__empty__right,axiom,
% 3.31/1.67 ! [A_2: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( semila966743401le_a_o @ A_2 @ bot_bo751897185le_a_o )
% 3.31/1.67 = bot_bo751897185le_a_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_683_Int__empty__left,axiom,
% 3.31/1.67 ! [B_2: nat > $o] :
% 3.31/1.67 ( ( semila1947288293_nat_o @ bot_bot_nat_o @ B_2 )
% 3.31/1.67 = bot_bot_nat_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_684_Int__empty__left,axiom,
% 3.31/1.67 ! [B_2: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( semila179895820tate_o @ bot_bo70021908tate_o @ B_2 )
% 3.31/1.67 = bot_bo70021908tate_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_685_Int__empty__left,axiom,
% 3.31/1.67 ! [B_2: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( semila966743401le_a_o @ bot_bo751897185le_a_o @ B_2 )
% 3.31/1.67 = bot_bo751897185le_a_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_686_inf__bot__left,axiom,
% 3.31/1.67 ! [X_2: $o] :
% 3.31/1.67 ( ( semila854092349_inf_o @ bot_bot_o @ X_2 )
% 3.31/1.67 <=> bot_bot_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_687_inf__bot__left,axiom,
% 3.31/1.67 ! [X_2: nat > $o] :
% 3.31/1.67 ( ( semila1947288293_nat_o @ bot_bot_nat_o @ X_2 )
% 3.31/1.67 = bot_bot_nat_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_688_inf__bot__left,axiom,
% 3.31/1.67 ! [X_2: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( semila179895820tate_o @ bot_bo70021908tate_o @ X_2 )
% 3.31/1.67 = bot_bo70021908tate_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_689_inf__bot__left,axiom,
% 3.31/1.67 ! [X_2: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( semila966743401le_a_o @ bot_bo751897185le_a_o @ X_2 )
% 3.31/1.67 = bot_bo751897185le_a_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_690_inf__bot__right,axiom,
% 3.31/1.67 ! [X_1: $o] :
% 3.31/1.67 ( ( semila854092349_inf_o @ X_1 @ bot_bot_o )
% 3.31/1.67 <=> bot_bot_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_691_inf__bot__right,axiom,
% 3.31/1.67 ! [X_1: nat > $o] :
% 3.31/1.67 ( ( semila1947288293_nat_o @ X_1 @ bot_bot_nat_o )
% 3.31/1.67 = bot_bot_nat_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_692_inf__bot__right,axiom,
% 3.31/1.67 ! [X_1: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( semila179895820tate_o @ X_1 @ bot_bo70021908tate_o )
% 3.31/1.67 = bot_bo70021908tate_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_693_inf__bot__right,axiom,
% 3.31/1.67 ! [X_1: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( semila966743401le_a_o @ X_1 @ bot_bo751897185le_a_o )
% 3.31/1.67 = bot_bo751897185le_a_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_694_Diff__triv,axiom,
% 3.31/1.67 ! [A_1: nat > $o,B_1: nat > $o] :
% 3.31/1.67 ( ( ( semila1947288293_nat_o @ A_1 @ B_1 )
% 3.31/1.67 = bot_bot_nat_o )
% 3.31/1.67 => ( ( minus_minus_nat_o @ A_1 @ B_1 )
% 3.31/1.67 = A_1 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_695_Diff__triv,axiom,
% 3.31/1.67 ! [A_1: hoare_1167836817_state > $o,B_1: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( ( semila179895820tate_o @ A_1 @ B_1 )
% 3.31/1.67 = bot_bo70021908tate_o )
% 3.31/1.67 => ( ( minus_2107060239tate_o @ A_1 @ B_1 )
% 3.31/1.67 = A_1 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_696_Diff__triv,axiom,
% 3.31/1.67 ! [A_1: hoare_1775062406iple_a > $o,B_1: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( ( semila966743401le_a_o @ A_1 @ B_1 )
% 3.31/1.67 = bot_bo751897185le_a_o )
% 3.31/1.67 => ( ( minus_1944206118le_a_o @ A_1 @ B_1 )
% 3.31/1.67 = A_1 ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_697_Diff__disjoint,axiom,
% 3.31/1.67 ! [A: nat > $o,B: nat > $o] :
% 3.31/1.67 ( ( semila1947288293_nat_o @ A @ ( minus_minus_nat_o @ B @ A ) )
% 3.31/1.67 = bot_bot_nat_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_698_Diff__disjoint,axiom,
% 3.31/1.67 ! [A: hoare_1167836817_state > $o,B: hoare_1167836817_state > $o] :
% 3.31/1.67 ( ( semila179895820tate_o @ A @ ( minus_2107060239tate_o @ B @ A ) )
% 3.31/1.67 = bot_bo70021908tate_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(fact_699_Diff__disjoint,axiom,
% 3.31/1.67 ! [A: hoare_1775062406iple_a > $o,B: hoare_1775062406iple_a > $o] :
% 3.31/1.67 ( ( semila966743401le_a_o @ A @ ( minus_1944206118le_a_o @ B @ A ) )
% 3.31/1.67 = bot_bo751897185le_a_o ) ).
% 3.31/1.67
% 3.31/1.67 thf(help_fequal_1_1_fequal_000tc__Nat__Onat_T,axiom,
% 3.31/1.67 ! [X: nat,Y: nat] :
% 3.31/1.67 ( ~ ( fequal_nat @ X @ Y )
% 3.31/1.67 | ( X = Y ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(help_fequal_2_1_fequal_000tc__Nat__Onat_T,axiom,
% 3.31/1.67 ! [X: nat,Y: nat] :
% 3.31/1.67 ( ( X != Y )
% 3.31/1.67 | ( fequal_nat @ X @ Y ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(help_fequal_1_1_fequal_000tc__Com__Ostate_T,axiom,
% 3.31/1.67 ! [X: state,Y: state] :
% 3.31/1.67 ( ~ ( fequal_state @ X @ Y )
% 3.31/1.67 | ( X = Y ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(help_fequal_2_1_fequal_000tc__Com__Ostate_T,axiom,
% 3.31/1.67 ! [X: state,Y: state] :
% 3.31/1.67 ( ( X != Y )
% 3.31/1.67 | ( fequal_state @ X @ Y ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(help_fequal_1_1_fequal_000_062_Itc__Nat__Onat_M_Eo_J_T,axiom,
% 3.31/1.67 ! [X: nat > $o,Y: nat > $o] :
% 3.31/1.67 ( ~ ( fequal_nat_o @ X @ Y )
% 3.31/1.67 | ( X = Y ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(help_fequal_2_1_fequal_000_062_Itc__Nat__Onat_M_Eo_J_T,axiom,
% 3.31/1.67 ! [X: nat > $o,Y: nat > $o] :
% 3.31/1.67 ( ( X != Y )
% 3.31/1.67 | ( fequal_nat_o @ X @ Y ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(help_fequal_1_1_fequal_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J_,axiom,
% 3.31/1.67 ! [X: hoare_1775062406iple_a,Y: hoare_1775062406iple_a] :
% 3.31/1.67 ( ~ ( fequal1288209029iple_a @ X @ Y )
% 3.31/1.67 | ( X = Y ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(help_fequal_2_1_fequal_000tc__Hoare____Mirabelle____srushsumbx__Otriple_It__a_J_,axiom,
% 3.31/1.67 ! [X: hoare_1775062406iple_a,Y: hoare_1775062406iple_a] :
% 3.31/1.67 ( ( X != Y )
% 3.31/1.67 | ( fequal1288209029iple_a @ X @ Y ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(help_fequal_1_1_fequal_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com,axiom,
% 3.31/1.67 ! [X: hoare_1167836817_state,Y: hoare_1167836817_state] :
% 3.31/1.67 ( ~ ( fequal1831255762_state @ X @ Y )
% 3.31/1.67 | ( X = Y ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(help_fequal_2_1_fequal_000tc__Hoare____Mirabelle____srushsumbx__Otriple_Itc__Com,axiom,
% 3.31/1.67 ! [X: hoare_1167836817_state,Y: hoare_1167836817_state] :
% 3.31/1.67 ( ( X != Y )
% 3.31/1.67 | ( fequal1831255762_state @ X @ Y ) ) ).
% 3.31/1.67
% 3.31/1.67 thf(conj_0,conjecture,
% 3.31/1.67 ( hoare_1508237396rivs_a @ g
% 3.31/1.67 @ ( insert1281456128iple_a
% 3.31/1.67 @ ( hoare_1766022166iple_a
% 3.31/1.67 @ ^ [Z: x_a,S: state] : $false
% 3.31/1.67 @ c
% 3.31/1.67 @ ^ [Z: x_a,S: state] : ( (&) @ ( p @ Z @ S ) @ ( (~) @ ( b @ S ) ) ) )
% 3.31/1.67 @ bot_bo751897185le_a_o ) ) ).
% 3.31/1.67 ------------------------
% 3.39/1.73 % (30476)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_30168 for (2999ds/183Mi)
% 3.39/1.73 % (30477)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_30168 for (2999ds/4Mi)
% 3.39/1.73 % (30478)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_30168 for (2999ds/27Mi)
% 3.39/1.73 % (30477)Instruction limit reached!
% 3.39/1.73 % (30477)------------------------------
% 3.39/1.73 % (30477)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.73 % (30477)Termination reason: Unknown
% 3.39/1.73 % (30477)Termination phase: shuffling
% 3.39/1.73
% 3.39/1.73 % (30477)Memory used [KB]: 2302
% 3.39/1.73 % (30477)Time elapsed: 0.003 s
% 3.39/1.73 % (30477)Instructions burned: 4 (million)
% 3.39/1.73 % (30477)------------------------------
% 3.39/1.73 % (30477)------------------------------
% 3.39/1.74 % (30479)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_30168 for (2999ds/2Mi)
% 3.39/1.74 % (30480)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_30168 for (2999ds/2Mi)
% 3.39/1.74 % (30481)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_30168 for (2999ds/275Mi)
% 3.39/1.74 % (30482)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_30168 for (2999ds/18Mi)
% 3.39/1.74 % (30478)Instruction limit reached!
% 3.39/1.74 % (30478)------------------------------
% 3.39/1.74 % (30478)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.74 % (30478)Termination reason: Unknown
% 3.39/1.74 % (30478)Termination phase: shuffling
% 3.39/1.74
% 3.39/1.74 % (30478)Memory used [KB]: 3582
% 3.39/1.74 % (30478)Time elapsed: 0.011 s
% 3.39/1.74 % (30478)Instructions burned: 28 (million)
% 3.39/1.74 % (30478)------------------------------
% 3.39/1.74 % (30478)------------------------------
% 3.39/1.74 % (30479)Instruction limit reached!
% 3.39/1.74 % (30479)------------------------------
% 3.39/1.74 % (30479)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.74 % (30479)Termination reason: Unknown
% 3.39/1.74 % (30479)Termination phase: shuffling
% 3.39/1.74
% 3.39/1.74 % (30479)Memory used [KB]: 2302
% 3.39/1.74 % (30479)Time elapsed: 0.004 s
% 3.39/1.74 % (30479)Instructions burned: 2 (million)
% 3.39/1.74 % (30479)------------------------------
% 3.39/1.74 % (30479)------------------------------
% 3.39/1.74 % (30480)Instruction limit reached!
% 3.39/1.74 % (30480)------------------------------
% 3.39/1.74 % (30480)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.74 % (30480)Termination reason: Unknown
% 3.39/1.74 % (30480)Termination phase: shuffling
% 3.39/1.74
% 3.39/1.74 % (30480)Memory used [KB]: 2174
% 3.39/1.74 % (30480)Time elapsed: 0.004 s
% 3.39/1.74 % (30480)Instructions burned: 2 (million)
% 3.39/1.74 % (30480)------------------------------
% 3.39/1.74 % (30480)------------------------------
% 3.39/1.74 % (30483)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_30168 for (2999ds/3Mi)
% 3.39/1.74 % (30483)Instruction limit reached!
% 3.39/1.74 % (30483)------------------------------
% 3.39/1.74 % (30483)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.74 % (30483)Termination reason: Unknown
% 3.39/1.74 % (30483)Termination phase: shuffling
% 3.39/1.74
% 3.39/1.74 % (30483)Memory used [KB]: 2302
% 3.39/1.74 % (30483)Time elapsed: 0.002 s
% 3.39/1.74 % (30483)Instructions burned: 3 (million)
% 3.39/1.74 % (30483)------------------------------
% 3.39/1.74 % (30483)------------------------------
% 3.39/1.75 % (30484)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_30168 for (2999ds/37Mi)
% 3.39/1.75 % (30482)Instruction limit reached!
% 3.39/1.75 % (30482)------------------------------
% 3.39/1.75 % (30482)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.75 % (30482)Termination reason: Unknown
% 3.39/1.75 % (30482)Termination phase: shuffling
% 3.39/1.75
% 3.39/1.75 % (30482)Memory used [KB]: 3198
% 3.39/1.75 % (30482)Time elapsed: 0.014 s
% 3.39/1.75 % (30482)Instructions burned: 18 (million)
% 3.39/1.75 % (30482)------------------------------
% 3.39/1.75 % (30482)------------------------------
% 3.39/1.75 % (30487)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_30168 for (2999ds/1041Mi)
% 3.39/1.76 % (30485)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_30168 for (2999ds/15Mi)
% 3.39/1.76 % (30486)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_30168 for (2999ds/3Mi)
% 3.39/1.76 % (30484)Instruction limit reached!
% 3.39/1.76 % (30484)------------------------------
% 3.39/1.76 % (30484)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.76 % (30484)Termination reason: Unknown
% 3.39/1.76 % (30484)Termination phase: shuffling
% 3.39/1.76
% 3.39/1.76 % (30484)Memory used [KB]: 3837
% 3.39/1.76 % (30484)Time elapsed: 0.012 s
% 3.39/1.76 % (30484)Instructions burned: 37 (million)
% 3.39/1.76 % (30484)------------------------------
% 3.39/1.76 % (30484)------------------------------
% 3.39/1.76 % (30486)Instruction limit reached!
% 3.39/1.76 % (30486)------------------------------
% 3.39/1.76 % (30486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.76 % (30486)Termination reason: Unknown
% 3.39/1.76 % (30486)Termination phase: shuffling
% 3.39/1.76
% 3.39/1.76 % (30486)Memory used [KB]: 2302
% 3.39/1.76 % (30486)Time elapsed: 0.005 s
% 3.39/1.76 % (30486)Instructions burned: 4 (million)
% 3.39/1.76 % (30486)------------------------------
% 3.39/1.76 % (30486)------------------------------
% 3.39/1.77 % (30485)Instruction limit reached!
% 3.39/1.77 % (30485)------------------------------
% 3.39/1.77 % (30485)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.77 % (30485)Termination reason: Unknown
% 3.39/1.77 % (30485)Termination phase: shuffling
% 3.39/1.77
% 3.39/1.77 % (30485)Memory used [KB]: 2942
% 3.39/1.77 % (30485)Time elapsed: 0.011 s
% 3.39/1.77 % (30485)Instructions burned: 16 (million)
% 3.39/1.77 % (30485)------------------------------
% 3.39/1.77 % (30485)------------------------------
% 3.39/1.77 % (30488)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_30168 for (2999ds/7Mi)
% 3.39/1.77 % (30489)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_30168 for (2999ds/16Mi)
% 3.39/1.77 % (30488)Instruction limit reached!
% 3.39/1.77 % (30488)------------------------------
% 3.39/1.77 % (30488)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.77 % (30488)Termination reason: Unknown
% 3.39/1.77 % (30488)Termination phase: shuffling
% 3.39/1.77
% 3.39/1.77 % (30488)Memory used [KB]: 2558
% 3.39/1.77 % (30488)Time elapsed: 0.007 s
% 3.39/1.77 % (30488)Instructions burned: 7 (million)
% 3.39/1.77 % (30488)------------------------------
% 3.39/1.77 % (30488)------------------------------
% 3.39/1.77 % (30489)Instruction limit reached!
% 3.39/1.77 % (30489)------------------------------
% 3.39/1.77 % (30489)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.77 % (30489)Termination reason: Unknown
% 3.39/1.77 % (30489)Termination phase: shuffling
% 3.39/1.77
% 3.39/1.77 % (30489)Memory used [KB]: 3070
% 3.39/1.77 % (30489)Time elapsed: 0.007 s
% 3.39/1.77 % (30489)Instructions burned: 17 (million)
% 3.39/1.77 % (30489)------------------------------
% 3.39/1.77 % (30489)------------------------------
% 3.39/1.78 % (30490)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_30168 for (2999ds/3Mi)
% 3.39/1.78 % (30491)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_30168 for (2999ds/3Mi)
% 3.39/1.78 % (30491)Instruction limit reached!
% 3.39/1.78 % (30491)------------------------------
% 3.39/1.78 % (30491)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.78 % (30491)Termination reason: Unknown
% 3.39/1.78 % (30491)Termination phase: shuffling
% 3.39/1.78
% 3.39/1.78 % (30491)Memory used [KB]: 2302
% 3.39/1.78 % (30491)Time elapsed: 0.003 s
% 3.39/1.78 % (30491)Instructions burned: 3 (million)
% 3.39/1.78 % (30491)------------------------------
% 3.39/1.78 % (30491)------------------------------
% 3.39/1.78 % (30490)Instruction limit reached!
% 3.39/1.78 % (30490)------------------------------
% 3.39/1.78 % (30490)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.78 % (30490)Termination reason: Unknown
% 3.39/1.78 % (30490)Termination phase: shuffling
% 3.39/1.78
% 3.39/1.78 % (30490)Memory used [KB]: 2302
% 3.39/1.78 % (30490)Time elapsed: 0.004 s
% 3.39/1.78 % (30490)Instructions burned: 3 (million)
% 3.39/1.78 % (30490)------------------------------
% 3.39/1.78 % (30490)------------------------------
% 3.39/1.78 % (30476)Instruction limit reached!
% 3.39/1.78 % (30476)------------------------------
% 3.39/1.78 % (30476)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.78 % (30476)Termination reason: Unknown
% 3.39/1.78 % (30476)Termination phase: Function definition elimination
% 3.39/1.78
% 3.39/1.78 % (30476)Memory used [KB]: 6268
% 3.39/1.78 % (30476)Time elapsed: 0.056 s
% 3.39/1.78 % (30476)Instructions burned: 184 (million)
% 3.39/1.78 % (30476)------------------------------
% 3.39/1.78 % (30476)------------------------------
% 3.39/1.78 % (30493)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_30168 for (2999ds/3Mi)
% 3.39/1.79 % (30493)Instruction limit reached!
% 3.39/1.79 % (30493)------------------------------
% 3.39/1.79 % (30493)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.79 % (30493)Termination reason: Unknown
% 3.39/1.79 % (30493)Termination phase: shuffling
% 3.39/1.79
% 3.39/1.79 % (30493)Memory used [KB]: 2430
% 3.39/1.79 % (30493)Time elapsed: 0.003 s
% 3.39/1.79 % (30493)Instructions burned: 6 (million)
% 3.39/1.79 % (30493)------------------------------
% 3.39/1.79 % (30493)------------------------------
% 3.39/1.79 % (30494)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_30168 for (2999ds/4Mi)
% 3.39/1.79 % (30492)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_30168 for (2999ds/7Mi)
% 3.39/1.79 % (30494)Instruction limit reached!
% 3.39/1.79 % (30494)------------------------------
% 3.39/1.79 % (30494)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.79 % (30494)Termination reason: Unknown
% 3.39/1.79 % (30494)Termination phase: shuffling
% 3.39/1.79
% 3.39/1.79 % (30494)Memory used [KB]: 2430
% 3.39/1.79 % (30494)Time elapsed: 0.004 s
% 3.39/1.79 % (30494)Instructions burned: 6 (million)
% 3.39/1.79 % (30494)------------------------------
% 3.39/1.79 % (30494)------------------------------
% 3.39/1.79 % (30496)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_30168 for (2998ds/710Mi)
% 3.39/1.79 % (30492)Instruction limit reached!
% 3.39/1.79 % (30492)------------------------------
% 3.39/1.79 % (30492)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.79 % (30492)Termination reason: Unknown
% 3.39/1.79 % (30492)Termination phase: shuffling
% 3.39/1.79
% 3.39/1.79 % (30492)Memory used [KB]: 2558
% 3.39/1.79 % (30492)Time elapsed: 0.007 s
% 3.39/1.79 % (30492)Instructions burned: 7 (million)
% 3.39/1.79 % (30492)------------------------------
% 3.39/1.79 % (30492)------------------------------
% 3.39/1.80 % (30497)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_30168 for (2998ds/6Mi)
% 3.39/1.80 % (30495)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_30168 for (2999ds/18Mi)
% 3.39/1.80 % (30497)Instruction limit reached!
% 3.39/1.80 % (30497)------------------------------
% 3.39/1.80 % (30497)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.80 % (30497)Termination reason: Unknown
% 3.39/1.80 % (30497)Termination phase: shuffling
% 3.39/1.80
% 3.39/1.80 % (30497)Memory used [KB]: 2686
% 3.39/1.80 % (30497)Time elapsed: 0.004 s
% 3.39/1.80 % (30497)Instructions burned: 8 (million)
% 3.39/1.80 % (30497)------------------------------
% 3.39/1.80 % (30497)------------------------------
% 3.39/1.80 % (30498)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_30168 for (2998ds/902Mi)
% 3.39/1.81 % (30500)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_30168 for (2998ds/5Mi)
% 3.39/1.81 % (30495)Instruction limit reached!
% 3.39/1.81 % (30495)------------------------------
% 3.39/1.81 % (30495)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.81 % (30495)Termination reason: Unknown
% 3.39/1.81 % (30495)Termination phase: shuffling
% 3.39/1.81
% 3.39/1.81 % (30495)Memory used [KB]: 3198
% 3.39/1.81 % (30495)Time elapsed: 0.014 s
% 3.39/1.81 % (30495)Instructions burned: 18 (million)
% 3.39/1.81 % (30495)------------------------------
% 3.39/1.81 % (30495)------------------------------
% 3.39/1.81 % (30500)Instruction limit reached!
% 3.39/1.81 % (30500)------------------------------
% 3.39/1.81 % (30500)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.81 % (30500)Termination reason: Unknown
% 3.39/1.81 % (30500)Termination phase: shuffling
% 3.39/1.81
% 3.39/1.81 % (30500)Memory used [KB]: 2430
% 3.39/1.81 % (30500)Time elapsed: 0.003 s
% 3.39/1.81 % (30500)Instructions burned: 6 (million)
% 3.39/1.81 % (30500)------------------------------
% 3.39/1.81 % (30500)------------------------------
% 3.39/1.81 % (30499)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_30168 for (2998ds/21Mi)
% 3.39/1.82 % (30502)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_30168 for (2998ds/377Mi)
% 3.39/1.82 % (30499)Instruction limit reached!
% 3.39/1.82 % (30499)------------------------------
% 3.39/1.82 % (30499)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.82 % (30499)Termination reason: Unknown
% 3.39/1.82 % (30499)Termination phase: shuffling
% 3.39/1.82
% 3.39/1.82 % (30499)Memory used [KB]: 3326
% 3.39/1.82 % (30499)Time elapsed: 0.011 s
% 3.39/1.82 % (30499)Instructions burned: 23 (million)
% 3.39/1.82 % (30499)------------------------------
% 3.39/1.82 % (30499)------------------------------
% 3.39/1.83 % (30501)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_30168 for (2998ds/6Mi)
% 3.39/1.83 % (30503)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_30168 for (2998ds/779Mi)
% 3.39/1.83 % (30501)Instruction limit reached!
% 3.39/1.83 % (30501)------------------------------
% 3.39/1.83 % (30501)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.83 % (30501)Termination reason: Unknown
% 3.39/1.83 % (30501)Termination phase: shuffling
% 3.39/1.83
% 3.39/1.83 % (30501)Memory used [KB]: 2430
% 3.39/1.83 % (30501)Time elapsed: 0.006 s
% 3.39/1.83 % (30501)Instructions burned: 6 (million)
% 3.39/1.83 % (30501)------------------------------
% 3.39/1.83 % (30501)------------------------------
% 3.39/1.85 % (30504)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_30168 for (2998ds/19Mi)
% 3.39/1.86 % (30504)Instruction limit reached!
% 3.39/1.86 % (30504)------------------------------
% 3.39/1.86 % (30504)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.86 % (30504)Termination reason: Unknown
% 3.39/1.86 % (30504)Termination phase: shuffling
% 3.39/1.86
% 3.39/1.86 % (30504)Memory used [KB]: 3070
% 3.39/1.86 % (30504)Time elapsed: 0.010 s
% 3.39/1.86 % (30504)Instructions burned: 19 (million)
% 3.39/1.86 % (30504)------------------------------
% 3.39/1.86 % (30504)------------------------------
% 3.39/1.87 % (30505)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_30168 for (2998ds/879Mi)
% 3.39/1.88 % (30481)Instruction limit reached!
% 3.39/1.88 % (30481)------------------------------
% 3.39/1.88 % (30481)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.88 % (30481)Termination reason: Unknown
% 3.39/1.88 % (30481)Termination phase: Saturation
% 3.39/1.88
% 3.39/1.88 % (30481)Memory used [KB]: 10490
% 3.39/1.88 % (30481)Time elapsed: 0.141 s
% 3.39/1.88 % (30481)Instructions burned: 276 (million)
% 3.39/1.88 % (30481)------------------------------
% 3.39/1.88 % (30481)------------------------------
% 3.39/1.89 % (30506)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_30168 for (2998ds/17Mi)
% 3.39/1.89 % (30506)Instruction limit reached!
% 3.39/1.89 % (30506)------------------------------
% 3.39/1.89 % (30506)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.89 % (30506)Termination reason: Unknown
% 3.39/1.89 % (30506)Termination phase: shuffling
% 3.39/1.89
% 3.39/1.89 % (30506)Memory used [KB]: 3198
% 3.39/1.89 % (30506)Time elapsed: 0.008 s
% 3.39/1.89 % (30506)Instructions burned: 19 (million)
% 3.39/1.89 % (30506)------------------------------
% 3.39/1.89 % (30506)------------------------------
% 3.39/1.90 % (30507)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_30168 for (2997ds/3Mi)
% 3.39/1.91 % (30507)Instruction limit reached!
% 3.39/1.91 % (30507)------------------------------
% 3.39/1.91 % (30507)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.91 % (30507)Termination reason: Unknown
% 3.39/1.91 % (30507)Termination phase: shuffling
% 3.39/1.91
% 3.39/1.91 % (30507)Memory used [KB]: 2302
% 3.39/1.91 % (30507)Time elapsed: 0.003 s
% 3.39/1.91 % (30507)Instructions burned: 3 (million)
% 3.39/1.91 % (30507)------------------------------
% 3.39/1.91 % (30507)------------------------------
% 3.39/1.92 % (30508)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_30168 for (2997ds/30Mi)
% 3.39/1.93 % (30508)Instruction limit reached!
% 3.39/1.93 % (30508)------------------------------
% 3.39/1.93 % (30508)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.93 % (30508)Termination reason: Unknown
% 3.39/1.93 % (30508)Termination phase: shuffling
% 3.39/1.93
% 3.39/1.93 % (30508)Memory used [KB]: 3709
% 3.39/1.93 % (30508)Time elapsed: 0.013 s
% 3.39/1.93 % (30508)Instructions burned: 30 (million)
% 3.39/1.93 % (30508)------------------------------
% 3.39/1.93 % (30508)------------------------------
% 3.39/1.93 % (30502)Instruction limit reached!
% 3.39/1.93 % (30502)------------------------------
% 3.39/1.93 % (30502)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.93 % (30502)Termination reason: Unknown
% 3.39/1.93 % (30502)Termination phase: Saturation
% 3.39/1.93
% 3.39/1.93 % (30502)Memory used [KB]: 11769
% 3.39/1.93 % (30502)Time elapsed: 0.112 s
% 3.39/1.93 % (30502)Instructions burned: 377 (million)
% 3.39/1.93 % (30502)------------------------------
% 3.39/1.93 % (30502)------------------------------
% 3.39/1.94 % (30509)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_30168 for (2997ds/127Mi)
% 3.39/1.94 % (30510)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_30168 for (2997ds/100Mi)
% 3.39/1.97 % (30510)Instruction limit reached!
% 3.39/1.97 % (30510)------------------------------
% 3.39/1.97 % (30510)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.97 % (30510)Termination reason: Unknown
% 3.39/1.97 % (30510)Termination phase: shuffling
% 3.39/1.97
% 3.39/1.97 % (30510)Memory used [KB]: 5756
% 3.39/1.97 % (30510)Time elapsed: 0.029 s
% 3.39/1.97 % (30510)Instructions burned: 100 (million)
% 3.39/1.97 % (30510)------------------------------
% 3.39/1.97 % (30510)------------------------------
% 3.39/1.98 % (30511)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_30168 for (2997ds/3Mi)
% 3.39/1.98 % (30509)Instruction limit reached!
% 3.39/1.98 % (30509)------------------------------
% 3.39/1.98 % (30509)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.98 % (30509)Termination reason: Unknown
% 3.39/1.98 % (30509)Termination phase: Naming
% 3.39/1.98
% 3.39/1.98 % (30509)Memory used [KB]: 5628
% 3.39/1.98 % (30509)Time elapsed: 0.044 s
% 3.39/1.98 % (30509)Instructions burned: 127 (million)
% 3.39/1.98 % (30509)------------------------------
% 3.39/1.98 % (30509)------------------------------
% 3.39/1.98 % (30511)Instruction limit reached!
% 3.39/1.98 % (30511)------------------------------
% 3.39/1.98 % (30511)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/1.98 % (30511)Termination reason: Unknown
% 3.39/1.98 % (30511)Termination phase: shuffling
% 3.39/1.98
% 3.39/1.98 % (30511)Memory used [KB]: 2430
% 3.39/1.98 % (30511)Time elapsed: 0.003 s
% 3.39/1.98 % (30511)Instructions burned: 6 (million)
% 3.39/1.98 % (30511)------------------------------
% 3.39/1.98 % (30511)------------------------------
% 3.39/1.99 % (30513)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_30168 for (2997ds/86Mi)
% 3.39/1.99 % (30512)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_30168 for (2997ds/20Mi)
% 3.39/2.00 % (30512)Instruction limit reached!
% 3.39/2.00 % (30512)------------------------------
% 3.39/2.00 % (30512)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/2.00 % (30512)Termination reason: Unknown
% 3.39/2.00 % (30512)Termination phase: shuffling
% 3.39/2.00
% 3.39/2.00 % (30512)Memory used [KB]: 3326
% 3.39/2.00 % (30512)Time elapsed: 0.009 s
% 3.39/2.00 % (30512)Instructions burned: 20 (million)
% 3.39/2.00 % (30512)------------------------------
% 3.39/2.00 % (30512)------------------------------
% 3.39/2.01 % (30514)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_30168 for (2996ds/107Mi)
% 3.39/2.01 % (30496)Instruction limit reached!
% 3.39/2.01 % (30496)------------------------------
% 3.39/2.01 % (30496)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/2.01 % (30496)Termination reason: Unknown
% 3.39/2.01 % (30496)Termination phase: Saturation
% 3.39/2.01
% 3.39/2.01 % (30496)Memory used [KB]: 10874
% 3.39/2.01 % (30496)Time elapsed: 0.222 s
% 3.39/2.01 % (30496)Instructions burned: 711 (million)
% 3.39/2.01 % (30496)------------------------------
% 3.39/2.01 % (30496)------------------------------
% 3.39/2.02 % (30513)Instruction limit reached!
% 3.39/2.02 % (30513)------------------------------
% 3.39/2.02 % (30513)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.39/2.02 % (30513)Termination reason: Unknown
% 3.39/2.02 % (30513)Termination phase: Property scanning
% 3.39/2.02
% 3.39/2.02 % (30513)Memory used [KB]: 4989
% 3.39/2.02 % (30513)Time elapsed: 0.029 s
% 3.39/2.02 % (30513)Instructions burned: 89 (million)
% 3.39/2.02 % (30513)------------------------------
% 3.39/2.02 % (30513)------------------------------
% 5.47/2.02 % (30515)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_30168 for (2996ds/39Mi)
% 5.47/2.03 % (30516)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_30168 for (2996ds/448Mi)
% 5.47/2.04 % (30515)Instruction limit reached!
% 5.47/2.04 % (30515)------------------------------
% 5.47/2.04 % (30515)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.47/2.04 % (30515)Termination reason: Unknown
% 5.47/2.04 % (30515)Termination phase: shuffling
% 5.47/2.04
% 5.47/2.04 % (30515)Memory used [KB]: 4093
% 5.47/2.04 % (30515)Time elapsed: 0.013 s
% 5.47/2.04 % (30515)Instructions burned: 42 (million)
% 5.47/2.04 % (30515)------------------------------
% 5.47/2.04 % (30515)------------------------------
% 5.47/2.04 % (30514)Instruction limit reached!
% 5.47/2.04 % (30514)------------------------------
% 5.47/2.04 % (30514)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.47/2.04 % (30514)Termination reason: Unknown
% 5.47/2.04 % (30514)Termination phase: SInE selection
% 5.47/2.04
% 5.47/2.04 % (30514)Memory used [KB]: 4989
% 5.47/2.04 % (30514)Time elapsed: 0.038 s
% 5.47/2.04 % (30514)Instructions burned: 109 (million)
% 5.47/2.04 % (30514)------------------------------
% 5.47/2.04 % (30514)------------------------------
% 5.47/2.05 % (30517)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_30168 for (2996ds/46Mi)
% 5.47/2.05 % (30503)Instruction limit reached!
% 5.47/2.05 % (30503)------------------------------
% 5.47/2.05 % (30503)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.47/2.05 % (30503)Termination reason: Unknown
% 5.47/2.05 % (30503)Termination phase: Saturation
% 5.47/2.05
% 5.47/2.05 % (30503)Memory used [KB]: 14200
% 5.47/2.05 % (30503)Time elapsed: 0.219 s
% 5.47/2.05 % (30503)Instructions burned: 779 (million)
% 5.47/2.05 % (30503)------------------------------
% 5.47/2.05 % (30503)------------------------------
% 5.47/2.05 % (30518)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_30168 for (2996ds/98Mi)
% 5.47/2.06 % (30519)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_30168 for (2996ds/507Mi)
% 5.47/2.06 % (30487)Instruction limit reached!
% 5.47/2.06 % (30487)------------------------------
% 5.47/2.06 % (30487)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.47/2.06 % (30487)Termination reason: Unknown
% 5.47/2.06 % (30487)Termination phase: Saturation
% 5.47/2.06
% 5.47/2.06 % (30487)Memory used [KB]: 13432
% 5.47/2.06 % (30487)Time elapsed: 0.311 s
% 5.47/2.06 % (30487)Instructions burned: 1041 (million)
% 5.47/2.06 % (30487)------------------------------
% 5.47/2.06 % (30487)------------------------------
% 5.47/2.06 % (30517)Instruction limit reached!
% 5.47/2.06 % (30517)------------------------------
% 5.47/2.06 % (30517)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.47/2.06 % (30517)Termination reason: Unknown
% 5.47/2.06 % (30517)Termination phase: shuffling
% 5.47/2.06
% 5.47/2.06 % (30517)Memory used [KB]: 4349
% 5.47/2.06 % (30517)Time elapsed: 0.017 s
% 5.47/2.06 % (30517)Instructions burned: 46 (million)
% 5.47/2.06 % (30517)------------------------------
% 5.47/2.06 % (30517)------------------------------
% 5.47/2.07 % (30520)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_30168 for (2996ds/149Mi)
% 5.47/2.07 % (30521)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_30168 for (2996ds/39Mi)
% 5.47/2.07 % (30498)Instruction limit reached!
% 5.47/2.07 % (30498)------------------------------
% 5.47/2.07 % (30498)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.47/2.07 % (30498)Termination reason: Unknown
% 5.47/2.07 % (30498)Termination phase: Saturation
% 5.47/2.07
% 5.47/2.07 % (30498)Memory used [KB]: 11897
% 5.47/2.07 % (30498)Time elapsed: 0.275 s
% 5.47/2.07 % (30498)Instructions burned: 902 (million)
% 5.47/2.07 % (30498)------------------------------
% 5.47/2.07 % (30498)------------------------------
% 5.47/2.08 % (30518)Instruction limit reached!
% 5.47/2.08 % (30518)------------------------------
% 5.47/2.08 % (30518)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.47/2.08 % (30518)Termination reason: Unknown
% 5.47/2.08 % (30518)Termination phase: shuffling
% 5.47/2.08
% 5.47/2.08 % (30518)Memory used [KB]: 5756
% 5.47/2.08 % (30518)Time elapsed: 0.032 s
% 5.47/2.08 % (30518)Instructions burned: 100 (million)
% 5.47/2.08 % (30518)------------------------------
% 5.47/2.08 % (30518)------------------------------
% 5.47/2.08 % (30522)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_30168 for (2996ds/6Mi)
% 5.47/2.09 % (30521)Instruction limit reached!
% 5.47/2.09 % (30521)------------------------------
% 5.47/2.09 % (30521)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.47/2.09 % (30521)Termination reason: Unknown
% 5.47/2.09 % (30521)Termination phase: shuffling
% 5.47/2.09
% 5.47/2.09 % (30521)Memory used [KB]: 4221
% 5.47/2.09 % (30521)Time elapsed: 0.015 s
% 5.47/2.09 % (30521)Instructions burned: 41 (million)
% 5.47/2.09 % (30521)------------------------------
% 5.47/2.09 % (30521)------------------------------
% 5.47/2.09 % (30522)Instruction limit reached!
% 5.47/2.09 % (30522)------------------------------
% 5.47/2.09 % (30522)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.47/2.09 % (30522)Termination reason: Unknown
% 5.47/2.09 % (30522)Termination phase: shuffling
% 5.47/2.09
% 5.47/2.09 % (30522)Memory used [KB]: 2302
% 5.47/2.09 % (30522)Time elapsed: 0.003 s
% 5.47/2.09 % (30522)Instructions burned: 6 (million)
% 5.47/2.09 % (30522)------------------------------
% 5.47/2.09 % (30522)------------------------------
% 5.47/2.10 % (30524)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_30168 for (2995ds/47Mi)
% 5.47/2.10 % (30525)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_30168 for (2995ds/44Mi)
% 5.47/2.10 % (30523)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_30168 for (2995ds/1087Mi)
% 5.47/2.11 % (30525)Instruction limit reached!
% 5.47/2.11 % (30525)------------------------------
% 5.47/2.11 % (30525)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.47/2.11 % (30525)Termination reason: Unknown
% 5.47/2.11 % (30525)Termination phase: shuffling
% 5.47/2.11
% 5.47/2.11 % (30525)Memory used [KB]: 4093
% 5.47/2.11 % (30525)Time elapsed: 0.014 s
% 5.47/2.11 % (30525)Instructions burned: 46 (million)
% 5.47/2.11 % (30525)------------------------------
% 5.47/2.11 % (30525)------------------------------
% 5.47/2.11 % (30524)Instruction limit reached!
% 5.47/2.11 % (30524)------------------------------
% 5.47/2.11 % (30524)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.47/2.11 % (30524)Termination reason: Unknown
% 5.47/2.11 % (30524)Termination phase: shuffling
% 5.47/2.11
% 5.47/2.11 % (30524)Memory used [KB]: 4349
% 5.47/2.11 % (30524)Time elapsed: 0.017 s
% 5.47/2.11 % (30524)Instructions burned: 47 (million)
% 5.47/2.11 % (30524)------------------------------
% 5.47/2.11 % (30524)------------------------------
% 5.47/2.12 % (30520)Instruction limit reached!
% 5.47/2.12 % (30520)------------------------------
% 5.47/2.12 % (30520)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.47/2.12 % (30520)Termination reason: Unknown
% 5.47/2.12 % (30520)Termination phase: Property scanning
% 5.47/2.12
% 5.47/2.12 % (30520)Memory used [KB]: 6140
% 5.47/2.12 % (30520)Time elapsed: 0.047 s
% 5.47/2.12 % (30520)Instructions burned: 152 (million)
% 5.47/2.12 % (30520)------------------------------
% 5.47/2.12 % (30520)------------------------------
% 5.47/2.12 % (30505)Instruction limit reached!
% 5.47/2.12 % (30505)------------------------------
% 5.47/2.12 % (30505)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.47/2.12 % (30505)Termination reason: Unknown
% 5.47/2.12 % (30505)Termination phase: Saturation
% 5.47/2.12
% 5.47/2.12 % (30505)Memory used [KB]: 10362
% 5.47/2.12 % (30505)Time elapsed: 0.252 s
% 5.47/2.12 % (30505)Instructions burned: 879 (million)
% 5.47/2.12 % (30505)------------------------------
% 5.47/2.12 % (30505)------------------------------
% 5.47/2.12 % (30526)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_30168 for (2995ds/432Mi)
% 5.47/2.12 % (30527)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_30168 for (2995ds/62Mi)
% 5.99/2.13 % (30528)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_30168 for (2995ds/31Mi)
% 5.99/2.13 % (30529)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_30168 for (2995ds/96Mi)
% 5.99/2.14 % (30528)Instruction limit reached!
% 5.99/2.14 % (30528)------------------------------
% 5.99/2.14 % (30528)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.99/2.14 % (30528)Termination reason: Unknown
% 5.99/2.14 % (30528)Termination phase: shuffling
% 5.99/2.14
% 5.99/2.14 % (30528)Memory used [KB]: 3837
% 5.99/2.14 % (30528)Time elapsed: 0.013 s
% 5.99/2.14 % (30528)Instructions burned: 33 (million)
% 5.99/2.14 % (30528)------------------------------
% 5.99/2.14 % (30528)------------------------------
% 5.99/2.14 % (30527)Instruction limit reached!
% 5.99/2.14 % (30527)------------------------------
% 5.99/2.14 % (30527)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.99/2.14 % (30527)Termination reason: Unknown
% 5.99/2.14 % (30527)Termination phase: shuffling
% 5.99/2.14
% 5.99/2.14 % (30527)Memory used [KB]: 4733
% 5.99/2.14 % (30527)Time elapsed: 0.020 s
% 5.99/2.14 % (30527)Instructions burned: 64 (million)
% 5.99/2.14 % (30527)------------------------------
% 5.99/2.14 % (30527)------------------------------
% 5.99/2.15 % (30516)Instruction limit reached!
% 5.99/2.15 % (30516)------------------------------
% 5.99/2.15 % (30516)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.99/2.15 % (30516)Termination reason: Unknown
% 5.99/2.15 % (30516)Termination phase: Saturation
% 5.99/2.15
% 5.99/2.15 % (30516)Memory used [KB]: 11769
% 5.99/2.15 % (30516)Time elapsed: 0.121 s
% 5.99/2.15 % (30516)Instructions burned: 449 (million)
% 5.99/2.15 % (30516)------------------------------
% 5.99/2.15 % (30516)------------------------------
% 5.99/2.15 % (30530)lrs+1002_1:4_au=on:cbe=off:cnfonf=lazy_pi_sigma_gen:hud=15:inj=on:nwc=4.0:s2a=on:sac=on:sp=reverse_arity:i=453:si=on:rtra=on_0 on DTF2THF_30168 for (2995ds/453Mi)
% 5.99/2.15 % (30531)lrs-1004_1:1_au=on:bsd=on:cbe=off:cnfonf=off:fe=axiom:fsd=on:i=71:si=on:rtra=on_0 on DTF2THF_30168 for (2995ds/71Mi)
% 5.99/2.16 % (30532)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_30168 for (2995ds/83Mi)
% 5.99/2.16 % (30529)Instruction limit reached!
% 5.99/2.16 % (30529)------------------------------
% 5.99/2.16 % (30529)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.99/2.16 % (30529)Termination reason: Unknown
% 5.99/2.16 % (30529)Termination phase: Property scanning
% 5.99/2.16
% 5.99/2.16 % (30529)Memory used [KB]: 4989
% 5.99/2.16 % (30529)Time elapsed: 0.034 s
% 5.99/2.16 % (30529)Instructions burned: 97 (million)
% 5.99/2.16 % (30529)------------------------------
% 5.99/2.16 % (30529)------------------------------
% 5.99/2.17 % (30533)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_30168 for (2995ds/11Mi)
% 5.99/2.17 % (30531)Instruction limit reached!
% 5.99/2.17 % (30531)------------------------------
% 5.99/2.17 % (30531)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.99/2.17 % (30531)Termination reason: Unknown
% 5.99/2.17 % (30531)Termination phase: shuffling
% 5.99/2.17
% 5.99/2.17 % (30531)Memory used [KB]: 5117
% 5.99/2.17 % (30531)Time elapsed: 0.025 s
% 5.99/2.17 % (30531)Instructions burned: 71 (million)
% 5.99/2.17 % (30531)------------------------------
% 5.99/2.17 % (30531)------------------------------
% 5.99/2.18 % (30533)Instruction limit reached!
% 5.99/2.18 % (30533)------------------------------
% 5.99/2.18 % (30533)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.99/2.18 % (30533)Termination reason: Unknown
% 5.99/2.18 % (30533)Termination phase: shuffling
% 5.99/2.18
% 5.99/2.18 % (30533)Memory used [KB]: 2686
% 5.99/2.18 % (30533)Time elapsed: 0.005 s
% 5.99/2.18 % (30533)Instructions burned: 12 (million)
% 5.99/2.18 % (30533)------------------------------
% 5.99/2.18 % (30533)------------------------------
% 5.99/2.18 % (30519)First to succeed.
% 5.99/2.18 % (30519)Refutation found. Thanks to Tanya!
% 5.99/2.18 % SZS status Theorem for DTF2THF_30168
% 5.99/2.18 % SZS output start Proof for DTF2THF_30168
% 5.99/2.18 thf(type_def_5, type, nat: $tType).
% 5.99/2.18 thf(type_def_7, type, vname: $tType).
% 5.99/2.18 thf(type_def_8, type, state: $tType).
% 5.99/2.18 thf(type_def_9, type, com: $tType).
% 5.99/2.18 thf(type_def_10, type, loc: $tType).
% 5.99/2.18 thf(type_def_11, type, glb: $tType).
% 5.99/2.18 thf(type_def_12, type, hoare_1775062406iple_a: $tType).
% 5.99/2.18 thf(type_def_13, type, hoare_1167836817_state: $tType).
% 5.99/2.18 thf(type_def_14, type, x_a: $tType).
% 5.99/2.18 thf(func_def_0, type, x_a: $tType).
% 5.99/2.18 thf(func_def_1, type, com: $tType).
% 5.99/2.18 thf(func_def_2, type, glb: $tType).
% 5.99/2.18 thf(func_def_3, type, loc: $tType).
% 5.99/2.18 thf(func_def_4, type, state: $tType).
% 5.99/2.18 thf(func_def_5, type, vname: $tType).
% 5.99/2.18 thf(func_def_6, type, hoare_1775062406iple_a: $tType).
% 5.99/2.18 thf(func_def_7, type, hoare_1167836817_state: $tType).
% 5.99/2.18 thf(func_def_8, type, nat: $tType).
% 5.99/2.18 thf(func_def_9, type, big_la43341705in_nat: (nat > $o) > nat).
% 5.99/2.18 thf(func_def_10, type, big_se275732192ig_nat: (nat > nat > nat) > ((nat > $o) > nat) > $o).
% 5.99/2.18 thf(func_def_11, type, ass: vname > (state > nat) > com).
% 5.99/2.18 thf(func_def_12, type, local: loc > (state > nat) > com > com).
% 5.99/2.18 thf(func_def_13, type, skip: com).
% 5.99/2.18 thf(func_def_14, type, semi: com > com > com).
% 5.99/2.18 thf(func_def_15, type, glb_1: glb > vname).
% 5.99/2.18 thf(func_def_16, type, loc_1: loc > vname).
% 5.99/2.18 thf(func_def_17, type, finite2064891473iple_a: (hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a) > $o).
% 5.99/2.18 thf(func_def_18, type, finite2120172977le_a_o: (hoare_1775062406iple_a > (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a > $o) > $o).
% 5.99/2.18 thf(func_def_19, type, finite856902323tate_o: (hoare_1167836817_state > (hoare_1167836817_state > $o) > hoare_1167836817_state > $o) > $o).
% 5.99/2.18 thf(func_def_20, type, finite389864113_nat_o: (nat > (nat > $o) > nat > $o) > $o).
% 5.99/2.18 thf(func_def_21, type, finite_finite_nat_o: ((nat > $o) > $o) > $o).
% 5.99/2.18 thf(func_def_22, type, finite2063573081iple_a: (hoare_1775062406iple_a > $o) > $o).
% 5.99/2.18 thf(func_def_23, type, finite1084549118_state: (hoare_1167836817_state > $o) > $o).
% 5.99/2.18 thf(func_def_24, type, finite_finite_nat: (nat > $o) > $o).
% 5.99/2.18 thf(func_def_25, type, finite1946188886iple_a: (hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a) > (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a > $o).
% 5.99/2.18 thf(func_def_26, type, finite309220289_state: (hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state) > (hoare_1167836817_state > $o) > hoare_1167836817_state > $o).
% 5.99/2.18 thf(func_def_27, type, finite_fold1Set_nat: (nat > nat > nat) > (nat > $o) > nat > $o).
% 5.99/2.18 thf(func_def_28, type, finite1790765286iple_a: (hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a) > (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_29, type, finite1646097201_state: (hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state) > (hoare_1167836817_state > $o) > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_30, type, finite_fold1_nat: (nat > nat > nat) > (nat > $o) > nat).
% 5.99/2.18 thf(func_def_31, type, finite1544171829le_a_o: (hoare_1775062406iple_a > (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a > $o) > (hoare_1775062406iple_a > $o) > (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a > $o).
% 5.99/2.18 thf(func_def_32, type, finite1842721992iple_a: (hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a) > hoare_1775062406iple_a > (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_33, type, finite291020855tate_o: (hoare_1167836817_state > (hoare_1167836817_state > $o) > hoare_1167836817_state > $o) > (hoare_1167836817_state > $o) > (hoare_1167836817_state > $o) > hoare_1167836817_state > $o).
% 5.99/2.18 thf(func_def_34, type, finite1731015960_state: (hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state) > hoare_1167836817_state > (hoare_1167836817_state > $o) > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_35, type, finite326637109_nat_o: (nat > (nat > $o) > nat > $o) > (nat > $o) > (nat > $o) > nat > $o).
% 5.99/2.18 thf(func_def_36, type, finite_fold_nat_nat: (nat > nat > nat) > nat > (nat > $o) > nat).
% 5.99/2.18 thf(func_def_37, type, finite727644230iple_a: (hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a) > hoare_1775062406iple_a > (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a > $o).
% 5.99/2.18 thf(func_def_38, type, finite1316643734_state: (hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state) > hoare_1167836817_state > (hoare_1167836817_state > $o) > hoare_1167836817_state > $o).
% 5.99/2.18 thf(func_def_39, type, finite929467206at_nat: (nat > nat > nat) > nat > (nat > $o) > nat > $o).
% 5.99/2.18 thf(func_def_40, type, finite2078349315iple_a: (hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a) > ((hoare_1775062406iple_a > $o) > hoare_1775062406iple_a) > $o).
% 5.99/2.18 thf(func_def_41, type, finite1074406356_state: (hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state) > ((hoare_1167836817_state > $o) > hoare_1167836817_state) > $o).
% 5.99/2.18 thf(func_def_42, type, finite988810631ne_nat: (nat > nat > nat) > ((nat > $o) > nat) > $o).
% 5.99/2.18 thf(func_def_43, type, finite1358382848iple_a: (hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a) > ((hoare_1775062406iple_a > $o) > hoare_1775062406iple_a) > $o).
% 5.99/2.18 thf(func_def_44, type, finite806517911_state: (hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state) > ((hoare_1167836817_state > $o) > hoare_1167836817_state) > $o).
% 5.99/2.18 thf(func_def_45, type, finite795500164em_nat: (nat > nat > nat) > ((nat > $o) > nat) > $o).
% 5.99/2.18 thf(func_def_46, type, minus_1944206118le_a_o: (hoare_1775062406iple_a > $o) > (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a > $o).
% 5.99/2.18 thf(func_def_47, type, minus_2107060239tate_o: (hoare_1167836817_state > $o) > (hoare_1167836817_state > $o) > hoare_1167836817_state > $o).
% 5.99/2.18 thf(func_def_48, type, minus_minus_nat_o: (nat > $o) > (nat > $o) > nat > $o).
% 5.99/2.18 thf(func_def_49, type, times_times_nat: nat > nat > nat).
% 5.99/2.18 thf(func_def_50, type, the_Ho1155011127iple_a: (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_51, type, the_Ho310147232_state: (hoare_1167836817_state > $o) > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_52, type, the_nat: (nat > $o) > nat).
% 5.99/2.18 thf(func_def_53, type, hoare_Mirabelle_MGT: com > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_54, type, hoare_1508237396rivs_a: (hoare_1775062406iple_a > $o) > (hoare_1775062406iple_a > $o) > $o).
% 5.99/2.18 thf(func_def_55, type, hoare_123228589_state: (hoare_1167836817_state > $o) > (hoare_1167836817_state > $o) > $o).
% 5.99/2.18 thf(func_def_56, type, hoare_1766022166iple_a: (x_a > state > $o) > com > (x_a > state > $o) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_57, type, hoare_908217195_state: (state > state > $o) > com > (state > state > $o) > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_58, type, hoare_1462269968alid_a: nat > hoare_1775062406iple_a > $o).
% 5.99/2.18 thf(func_def_59, type, hoare_56934129_state: nat > hoare_1167836817_state > $o).
% 5.99/2.18 thf(func_def_60, type, semila966743401le_a_o: (hoare_1775062406iple_a > $o) > (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a > $o).
% 5.99/2.18 thf(func_def_61, type, semila179895820tate_o: (hoare_1167836817_state > $o) > (hoare_1167836817_state > $o) > hoare_1167836817_state > $o).
% 5.99/2.18 thf(func_def_62, type, semila1947288293_nat_o: (nat > $o) > (nat > $o) > nat > $o).
% 5.99/2.18 thf(func_def_63, type, semila854092349_inf_o: $o > $o > $o).
% 5.99/2.18 thf(func_def_64, type, semila80283416nf_nat: nat > nat > nat).
% 5.99/2.18 thf(func_def_65, type, semila13410563le_a_o: (hoare_1775062406iple_a > $o) > (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a > $o).
% 5.99/2.18 thf(func_def_66, type, semila1172322802tate_o: (hoare_1167836817_state > $o) > (hoare_1167836817_state > $o) > hoare_1167836817_state > $o).
% 5.99/2.18 thf(func_def_67, type, semila848761471_nat_o: (nat > $o) > (nat > $o) > nat > $o).
% 5.99/2.18 thf(func_def_68, type, semila10642723_sup_o: $o > $o > $o).
% 5.99/2.18 thf(func_def_69, type, semila972727038up_nat: nat > nat > nat).
% 5.99/2.18 thf(func_def_70, type, evalc: com > state > state > $o).
% 5.99/2.18 thf(func_def_71, type, evaln: com > state > nat > state > $o).
% 5.99/2.18 thf(func_def_72, type, getlocs: state > loc > nat).
% 5.99/2.18 thf(func_def_73, type, update: state > vname > nat > state).
% 5.99/2.18 thf(func_def_74, type, bot_bot_nat_o_o: (nat > $o) > $o).
% 5.99/2.18 thf(func_def_75, type, bot_bo751897185le_a_o: hoare_1775062406iple_a > $o).
% 5.99/2.18 thf(func_def_76, type, bot_bo70021908tate_o: hoare_1167836817_state > $o).
% 5.99/2.18 thf(func_def_77, type, bot_bot_nat_o: nat > $o).
% 5.99/2.18 thf(func_def_79, type, bot_bot_nat: nat).
% 5.99/2.18 thf(func_def_80, type, ord_le1143225901le_a_o: (hoare_1775062406iple_a > $o) > (hoare_1775062406iple_a > $o) > $o).
% 5.99/2.18 thf(func_def_81, type, ord_le827224136tate_o: (hoare_1167836817_state > $o) > (hoare_1167836817_state > $o) > $o).
% 5.99/2.18 thf(func_def_82, type, ord_less_eq_nat_o: (nat > $o) > (nat > $o) > $o).
% 5.99/2.18 thf(func_def_83, type, ord_less_eq_o: $o > $o > $o).
% 5.99/2.18 thf(func_def_84, type, ord_less_eq_nat: nat > nat > $o).
% 5.99/2.18 thf(func_def_85, type, partia126998524iple_a: hoare_1775062406iple_a > (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_86, type, partia715677851_state: hoare_1167836817_state > (hoare_1167836817_state > $o) > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_87, type, partial_flat_lub_nat: nat > (nat > $o) > nat).
% 5.99/2.18 thf(func_def_88, type, collect_nat_o: ((nat > $o) > $o) > (nat > $o) > $o).
% 5.99/2.18 thf(func_def_89, type, collec676402587iple_a: (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a > $o).
% 5.99/2.18 thf(func_def_90, type, collec1027672124_state: (hoare_1167836817_state > $o) > hoare_1167836817_state > $o).
% 5.99/2.18 thf(func_def_91, type, collect_nat: (nat > $o) > nat > $o).
% 5.99/2.18 thf(func_def_92, type, image_1170193413iple_a: (hoare_1775062406iple_a > hoare_1775062406iple_a) > (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a > $o).
% 5.99/2.18 thf(func_def_93, type, image_1021683026_state: (hoare_1775062406iple_a > hoare_1167836817_state) > (hoare_1775062406iple_a > $o) > hoare_1167836817_state > $o).
% 5.99/2.18 thf(func_def_94, type, image_1806517641_a_nat: (hoare_1775062406iple_a > nat) > (hoare_1775062406iple_a > $o) > nat > $o).
% 5.99/2.18 thf(func_def_95, type, image_1802845250iple_a: (hoare_1167836817_state > hoare_1775062406iple_a) > (hoare_1167836817_state > $o) > hoare_1775062406iple_a > $o).
% 5.99/2.18 thf(func_def_96, type, image_31595733_state: (hoare_1167836817_state > hoare_1167836817_state) > (hoare_1167836817_state > $o) > hoare_1167836817_state > $o).
% 5.99/2.18 thf(func_def_97, type, image_1476618182te_nat: (hoare_1167836817_state > nat) > (hoare_1167836817_state > $o) > nat > $o).
% 5.99/2.18 thf(func_def_98, type, image_43014529iple_a: (nat > hoare_1775062406iple_a) > (nat > $o) > hoare_1775062406iple_a > $o).
% 5.99/2.18 thf(func_def_99, type, image_2121260246_state: (nat > hoare_1167836817_state) > (nat > $o) > hoare_1167836817_state > $o).
% 5.99/2.18 thf(func_def_100, type, image_nat_nat: (nat > nat) > (nat > $o) > nat > $o).
% 5.99/2.18 thf(func_def_101, type, insert_nat_o: (nat > $o) > ((nat > $o) > $o) > (nat > $o) > $o).
% 5.99/2.18 thf(func_def_102, type, insert1281456128iple_a: hoare_1775062406iple_a > (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a > $o).
% 5.99/2.18 thf(func_def_103, type, insert2134838167_state: hoare_1167836817_state > (hoare_1167836817_state > $o) > hoare_1167836817_state > $o).
% 5.99/2.18 thf(func_def_104, type, insert_nat: nat > (nat > $o) > nat > $o).
% 5.99/2.18 thf(func_def_105, type, the_el1844711461iple_a: (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_106, type, the_el323660082_state: (hoare_1167836817_state > $o) > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_107, type, the_elem_nat: (nat > $o) > nat).
% 5.99/2.18 thf(func_def_108, type, fequal_nat_o: (nat > $o) > (nat > $o) > $o).
% 5.99/2.18 thf(func_def_109, type, fequal_state: state > state > $o).
% 5.99/2.18 thf(func_def_110, type, fequal1288209029iple_a: hoare_1775062406iple_a > hoare_1775062406iple_a > $o).
% 5.99/2.18 thf(func_def_111, type, fequal1831255762_state: hoare_1167836817_state > hoare_1167836817_state > $o).
% 5.99/2.18 thf(func_def_112, type, fequal_nat: nat > nat > $o).
% 5.99/2.18 thf(func_def_113, type, member_nat_o: (nat > $o) > ((nat > $o) > $o) > $o).
% 5.99/2.18 thf(func_def_114, type, member2122167641iple_a: hoare_1775062406iple_a > (hoare_1775062406iple_a > $o) > $o).
% 5.99/2.18 thf(func_def_115, type, member2058392318_state: hoare_1167836817_state > (hoare_1167836817_state > $o) > $o).
% 5.99/2.18 thf(func_def_116, type, member_nat: nat > (nat > $o) > $o).
% 5.99/2.18 thf(func_def_117, type, g: hoare_1775062406iple_a > $o).
% 5.99/2.18 thf(func_def_118, type, p: x_a > state > $o).
% 5.99/2.18 thf(func_def_119, type, b: state > $o).
% 5.99/2.18 thf(func_def_120, type, c: com).
% 5.99/2.18 thf(func_def_133, type, sK0: !>[X0: $tType]:((X0 > $o) > X0)).
% 5.99/2.18 thf(func_def_134, type, sK1: !>[X0: $tType]:((X0 > $o) > X0)).
% 5.99/2.18 thf(func_def_136, type, sP2: (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a > (hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a) > hoare_1775062406iple_a > $o).
% 5.99/2.18 thf(func_def_137, type, sK3: state > state > com > state > state > com > nat).
% 5.99/2.18 thf(func_def_138, type, sK4: (nat > $o) > nat).
% 5.99/2.18 thf(func_def_139, type, sK5: (hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state) > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_140, type, sK6: (hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state) > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_141, type, sK7: (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_142, type, sK8: (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_143, type, sK9: (nat > $o) > nat).
% 5.99/2.18 thf(func_def_144, type, sK10: (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_145, type, sK11: (hoare_1775062406iple_a > $o) > (hoare_1775062406iple_a > nat) > nat > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_146, type, sK12: hoare_1775062406iple_a > (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_147, type, sK13: nat > (nat > $o) > (nat > nat > nat) > nat).
% 5.99/2.18 thf(func_def_148, type, sK14: nat > (nat > $o) > (nat > nat > nat) > nat > $o).
% 5.99/2.18 thf(func_def_149, type, sK15: nat > (nat > $o) > (nat > nat > nat) > nat).
% 5.99/2.18 thf(func_def_150, type, sK16: nat > (nat > $o) > nat > (nat > nat > nat) > nat).
% 5.99/2.18 thf(func_def_151, type, sK17: nat > (nat > $o) > nat > (nat > nat > nat) > nat > $o).
% 5.99/2.18 thf(func_def_152, type, sK18: (nat > $o) > nat).
% 5.99/2.18 thf(func_def_153, type, sK19: (nat > $o) > nat > $o).
% 5.99/2.18 thf(func_def_154, type, sK20: (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a > hoare_1775062406iple_a > $o).
% 5.99/2.18 thf(func_def_155, type, sK21: (hoare_1775062406iple_a > $o) > ((hoare_1775062406iple_a > $o) > $o) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_156, type, sK22: (hoare_1775062406iple_a > $o) > ((hoare_1775062406iple_a > $o) > $o) > hoare_1775062406iple_a > $o).
% 5.99/2.18 thf(func_def_157, type, sK23: (hoare_1167836817_state > $o) > (hoare_1167836817_state > hoare_1167836817_state) > hoare_1167836817_state > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_158, type, sK24: (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_159, type, sK25: (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a > $o).
% 5.99/2.18 thf(func_def_160, type, sK26: (nat > $o) > nat).
% 5.99/2.18 thf(func_def_161, type, sK27: (x_a > state > $o) > (x_a > state > $o) > x_a).
% 5.99/2.18 thf(func_def_162, type, sK28: (x_a > state > $o) > (x_a > state > $o) > state).
% 5.99/2.18 thf(func_def_163, type, sK29: nat).
% 5.99/2.18 thf(func_def_164, type, sK30: nat).
% 5.99/2.18 thf(func_def_165, type, sK31: (state > state > $o) > (state > state > $o) > com > (hoare_1167836817_state > $o) > state).
% 5.99/2.18 thf(func_def_166, type, sK32: (state > state > $o) > (state > state > $o) > com > (hoare_1167836817_state > $o) > state).
% 5.99/2.18 thf(func_def_167, type, sK33: hoare_1775062406iple_a > (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_168, type, sK34: ((nat > $o) > $o) > nat).
% 5.99/2.18 thf(func_def_169, type, sK35: ((nat > $o) > $o) > nat > $o).
% 5.99/2.18 thf(func_def_170, type, sK36: (state > state > $o) > com > (state > state > $o) > nat > state).
% 5.99/2.18 thf(func_def_171, type, sK37: (state > state > $o) > com > (state > state > $o) > nat > state).
% 5.99/2.18 thf(func_def_172, type, sK38: (state > state > $o) > com > (state > state > $o) > nat > state).
% 5.99/2.18 thf(func_def_173, type, sK39: (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a > (hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a) > hoare_1775062406iple_a > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_174, type, sK40: (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a > (hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a) > hoare_1775062406iple_a > hoare_1775062406iple_a > $o).
% 5.99/2.18 thf(func_def_175, type, sK41: (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a > (hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a) > hoare_1775062406iple_a > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_176, type, sK42: ((hoare_1775062406iple_a > $o) > $o) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_177, type, sK43: ((hoare_1775062406iple_a > $o) > $o) > hoare_1775062406iple_a > $o).
% 5.99/2.18 thf(func_def_178, type, sK44: (nat > $o) > nat > $o).
% 5.99/2.18 thf(func_def_179, type, sK45: (nat > $o) > nat).
% 5.99/2.18 thf(func_def_180, type, sK46: (hoare_1167836817_state > $o) > hoare_1167836817_state > hoare_1167836817_state > $o).
% 5.99/2.18 thf(func_def_181, type, sK47: (nat > nat > nat) > nat).
% 5.99/2.18 thf(func_def_182, type, sK48: (nat > nat > nat) > nat).
% 5.99/2.18 thf(func_def_183, type, sK49: (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_184, type, sK50: hoare_1167836817_state > (hoare_1167836817_state > $o) > hoare_1167836817_state > (hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state) > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_185, type, sK51: hoare_1167836817_state > (hoare_1167836817_state > $o) > hoare_1167836817_state > (hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state) > hoare_1167836817_state > $o).
% 5.99/2.18 thf(func_def_186, type, sK52: (hoare_1167836817_state > $o) > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_187, type, sK53: (nat > $o) > nat).
% 5.99/2.18 thf(func_def_188, type, sK54: (hoare_1775062406iple_a > $o) > (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_189, type, sK55: (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a > $o).
% 5.99/2.18 thf(func_def_190, type, sK56: (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_191, type, sK57: (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a > hoare_1775062406iple_a > $o).
% 5.99/2.18 thf(func_def_192, type, sK58: (nat > $o) > nat).
% 5.99/2.18 thf(func_def_193, type, sK59: nat > (nat > $o) > nat).
% 5.99/2.18 thf(func_def_194, type, sK60: ((nat > $o) > $o) > nat > $o).
% 5.99/2.18 thf(func_def_195, type, sK61: (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_196, type, sK62: (hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a) > hoare_1775062406iple_a > (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_197, type, sK63: ((hoare_1167836817_state > $o) > $o) > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_198, type, sK64: ((hoare_1167836817_state > $o) > $o) > hoare_1167836817_state > $o).
% 5.99/2.18 thf(func_def_199, type, sK65: (nat > $o) > (nat > $o) > nat).
% 5.99/2.18 thf(func_def_200, type, sK66: (x_a > state > $o) > (x_a > state > $o) > x_a).
% 5.99/2.18 thf(func_def_201, type, sK67: (x_a > state > $o) > (x_a > state > $o) > state).
% 5.99/2.18 thf(func_def_202, type, sK68: hoare_1775062406iple_a > (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a > (hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_203, type, sK69: hoare_1775062406iple_a > (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a > (hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a) > hoare_1775062406iple_a > $o).
% 5.99/2.18 thf(func_def_204, type, sK70: (hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a) > (hoare_1775062406iple_a > hoare_1775062406iple_a) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_205, type, sK71: (hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a) > (hoare_1775062406iple_a > hoare_1775062406iple_a) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_206, type, sK72: (hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a) > hoare_1775062406iple_a > (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_207, type, sK73: (nat > nat) > nat).
% 5.99/2.18 thf(func_def_208, type, sK74: state > state > com > (state > nat) > loc > state).
% 5.99/2.18 thf(func_def_209, type, sK75: (hoare_1167836817_state > $o) > hoare_1167836817_state > hoare_1167836817_state > $o).
% 5.99/2.18 thf(func_def_210, type, sK76: hoare_1167836817_state > state > state > $o).
% 5.99/2.18 thf(func_def_211, type, sK77: hoare_1167836817_state > com).
% 5.99/2.18 thf(func_def_212, type, sK78: hoare_1167836817_state > state > state > $o).
% 5.99/2.18 thf(func_def_213, type, sK79: ((nat > $o) > $o) > nat).
% 5.99/2.18 thf(func_def_214, type, sK80: ((nat > $o) > $o) > nat > $o).
% 5.99/2.18 thf(func_def_215, type, sK81: (hoare_1775062406iple_a > $o) > (hoare_1775062406iple_a > hoare_1167836817_state) > hoare_1167836817_state > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_216, type, sK82: state > state > com > nat).
% 5.99/2.18 thf(func_def_217, type, sK83: (hoare_1167836817_state > $o) > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_218, type, sK84: (hoare_1775062406iple_a > $o) > (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_219, type, sK85: (hoare_1775062406iple_a > $o) > (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_220, type, sK86: state > nat > state > com > (state > nat) > loc > state).
% 5.99/2.18 thf(func_def_221, type, sK87: (hoare_1167836817_state > $o) > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_222, type, sK88: ((hoare_1167836817_state > $o) > $o) > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_223, type, sK89: ((hoare_1167836817_state > $o) > $o) > hoare_1167836817_state > $o).
% 5.99/2.18 thf(func_def_224, type, sK90: (nat > $o) > (nat > hoare_1775062406iple_a) > hoare_1775062406iple_a > nat).
% 5.99/2.18 thf(func_def_225, type, sK91: (hoare_1167836817_state > $o) > (hoare_1167836817_state > $o) > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_226, type, sK92: (hoare_1167836817_state > $o) > (hoare_1167836817_state > $o) > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_227, type, sK93: ((nat > $o) > $o) > nat).
% 5.99/2.18 thf(func_def_228, type, sK94: ((nat > $o) > $o) > nat).
% 5.99/2.18 thf(func_def_229, type, sK95: ((nat > $o) > $o) > nat > $o).
% 5.99/2.18 thf(func_def_230, type, sK96: (nat > $o) > nat > nat).
% 5.99/2.18 thf(func_def_231, type, sK97: (nat > $o) > (nat > nat) > nat > nat).
% 5.99/2.18 thf(func_def_232, type, sK98: (state > state > $o) > (state > state > $o) > state).
% 5.99/2.18 thf(func_def_233, type, sK99: (state > state > $o) > (state > state > $o) > state).
% 5.99/2.18 thf(func_def_234, type, sK100: (x_a > state > $o) > com > (hoare_1775062406iple_a > $o) > (x_a > state > $o) > x_a).
% 5.99/2.18 thf(func_def_235, type, sK101: (x_a > state > $o) > com > (hoare_1775062406iple_a > $o) > (x_a > state > $o) > state).
% 5.99/2.18 thf(func_def_236, type, sK102: (x_a > state > $o) > (x_a > state > $o) > (x_a > state > $o) > com > (hoare_1775062406iple_a > $o) > (x_a > state > $o) > state).
% 5.99/2.18 thf(func_def_237, type, sK103: (hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state) > (hoare_1167836817_state > hoare_1167836817_state) > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_238, type, sK104: (hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state) > (hoare_1167836817_state > hoare_1167836817_state) > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_239, type, sK105: (state > state > $o) > (state > state > $o) > (state > state > $o) > (state > state > $o) > state).
% 5.99/2.18 thf(func_def_240, type, sK106: (state > state > $o) > (state > state > $o) > (state > state > $o) > (state > state > $o) > state).
% 5.99/2.18 thf(func_def_241, type, sK107: (state > state > $o) > (state > state > $o) > (state > state > $o) > (state > state > $o) > state).
% 5.99/2.18 thf(func_def_242, type, sK108: hoare_1167836817_state > (hoare_1167836817_state > $o) > (hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state) > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_243, type, sK109: hoare_1167836817_state > (hoare_1167836817_state > $o) > (hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state) > hoare_1167836817_state > $o).
% 5.99/2.18 thf(func_def_244, type, sK110: hoare_1167836817_state > (hoare_1167836817_state > $o) > (hoare_1167836817_state > hoare_1167836817_state > hoare_1167836817_state) > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_245, type, sK111: (hoare_1167836817_state > $o) > hoare_1167836817_state > $o).
% 5.99/2.18 thf(func_def_246, type, sK112: (hoare_1167836817_state > $o) > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_247, type, sK113: state > state > com > nat).
% 5.99/2.18 thf(func_def_248, type, sK114: (hoare_1167836817_state > $o) > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_249, type, sK115: state > state > com > com > state).
% 5.99/2.18 thf(func_def_250, type, sK116: state > nat > state > com > com > state).
% 5.99/2.18 thf(func_def_251, type, sK117: (nat > $o) > (nat > $o) > nat).
% 5.99/2.18 thf(func_def_252, type, sK118: (nat > $o) > (nat > $o) > nat).
% 5.99/2.18 thf(func_def_253, type, sK119: (x_a > state > $o) > com > (x_a > state > $o) > nat > x_a).
% 5.99/2.18 thf(func_def_254, type, sK120: (x_a > state > $o) > com > (x_a > state > $o) > nat > state).
% 5.99/2.18 thf(func_def_255, type, sK121: (x_a > state > $o) > com > (x_a > state > $o) > nat > state).
% 5.99/2.18 thf(func_def_256, type, sK122: (x_a > state > $o) > (x_a > state > $o) > (x_a > state > $o) > (x_a > state > $o) > x_a).
% 5.99/2.18 thf(func_def_257, type, sK123: (x_a > state > $o) > (x_a > state > $o) > (x_a > state > $o) > (x_a > state > $o) > state).
% 5.99/2.18 thf(func_def_258, type, sK124: (x_a > state > $o) > (x_a > state > $o) > (x_a > state > $o) > (x_a > state > $o) > state).
% 5.99/2.18 thf(func_def_259, type, sK125: ((hoare_1167836817_state > $o) > $o) > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_260, type, sK126: ((hoare_1167836817_state > $o) > $o) > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_261, type, sK127: ((hoare_1167836817_state > $o) > $o) > hoare_1167836817_state > $o).
% 5.99/2.18 thf(func_def_262, type, sK128: ((hoare_1775062406iple_a > $o) > $o) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_263, type, sK129: ((hoare_1775062406iple_a > $o) > $o) > hoare_1775062406iple_a > $o).
% 5.99/2.18 thf(func_def_264, type, sK130: ((hoare_1775062406iple_a > $o) > $o) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_265, type, sK131: hoare_1775062406iple_a > (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_266, type, sK132: (x_a > state > $o) > (x_a > state > $o) > com > (hoare_1775062406iple_a > $o) > x_a).
% 5.99/2.18 thf(func_def_267, type, sK133: (x_a > state > $o) > (x_a > state > $o) > com > (hoare_1775062406iple_a > $o) > state).
% 5.99/2.18 thf(func_def_268, type, sK134: (hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_269, type, sK135: (hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_270, type, sK136: vname > loc).
% 5.99/2.18 thf(func_def_271, type, sK137: vname > glb).
% 5.99/2.18 thf(func_def_272, type, sK138: hoare_1775062406iple_a > (hoare_1775062406iple_a > $o) > (hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_273, type, sK139: hoare_1775062406iple_a > (hoare_1775062406iple_a > $o) > (hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a) > hoare_1775062406iple_a > $o).
% 5.99/2.18 thf(func_def_274, type, sK140: hoare_1775062406iple_a > (hoare_1775062406iple_a > $o) > (hoare_1775062406iple_a > hoare_1775062406iple_a > hoare_1775062406iple_a) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_275, type, sK141: ((hoare_1775062406iple_a > $o) > $o) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_276, type, sK142: ((hoare_1775062406iple_a > $o) > $o) > hoare_1775062406iple_a > $o).
% 5.99/2.18 thf(func_def_277, type, sK143: (nat > nat > nat) > (nat > nat) > nat).
% 5.99/2.18 thf(func_def_278, type, sK144: (nat > nat > nat) > (nat > nat) > nat).
% 5.99/2.18 thf(func_def_279, type, sK145: (hoare_1167836817_state > $o) > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_280, type, sK146: (nat > $o) > nat > nat > $o).
% 5.99/2.18 thf(func_def_281, type, sK147: (state > state > $o) > (state > state > $o) > state).
% 5.99/2.18 thf(func_def_282, type, sK148: (state > state > $o) > (state > state > $o) > state).
% 5.99/2.18 thf(func_def_283, type, sK149: hoare_1775062406iple_a > x_a > state > $o).
% 5.99/2.18 thf(func_def_284, type, sK150: hoare_1775062406iple_a > com).
% 5.99/2.18 thf(func_def_285, type, sK151: hoare_1775062406iple_a > x_a > state > $o).
% 5.99/2.18 thf(func_def_286, type, sK152: (nat > $o) > nat > nat > $o).
% 5.99/2.18 thf(func_def_287, type, sK153: hoare_1775062406iple_a > (hoare_1775062406iple_a > $o) > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_288, type, sK154: (nat > $o) > nat).
% 5.99/2.18 thf(func_def_289, type, sK155: (hoare_1167836817_state > $o) > (hoare_1167836817_state > hoare_1775062406iple_a) > hoare_1775062406iple_a > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_290, type, sK156: (hoare_1775062406iple_a > $o) > (hoare_1775062406iple_a > hoare_1775062406iple_a) > hoare_1775062406iple_a > hoare_1775062406iple_a).
% 5.99/2.18 thf(func_def_291, type, sK157: (nat > $o) > ((nat > $o) > $o) > nat).
% 5.99/2.18 thf(func_def_292, type, sK158: (nat > $o) > ((nat > $o) > $o) > nat > $o).
% 5.99/2.18 thf(func_def_293, type, sK159: nat).
% 5.99/2.18 thf(func_def_294, type, sK160: nat).
% 5.99/2.18 thf(func_def_295, type, sK161: ((nat > $o) > $o) > nat > $o).
% 5.99/2.18 thf(func_def_296, type, sK162: (state > state > $o) > com > (hoare_1167836817_state > $o) > (state > state > $o) > state).
% 5.99/2.18 thf(func_def_297, type, sK163: (state > state > $o) > com > (hoare_1167836817_state > $o) > (state > state > $o) > state).
% 5.99/2.18 thf(func_def_298, type, sK164: (state > state > $o) > (state > state > $o) > (state > state > $o) > com > (hoare_1167836817_state > $o) > (state > state > $o) > state).
% 5.99/2.18 thf(func_def_299, type, sK165: (nat > nat) > nat).
% 5.99/2.18 thf(func_def_300, type, sK166: (nat > nat) > nat).
% 5.99/2.18 thf(func_def_301, type, sK167: (hoare_1167836817_state > $o) > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_302, type, sK168: (hoare_1167836817_state > $o) > hoare_1167836817_state > $o).
% 5.99/2.18 thf(func_def_303, type, sK169: (hoare_1167836817_state > $o) > ((hoare_1167836817_state > $o) > $o) > hoare_1167836817_state).
% 5.99/2.18 thf(func_def_304, type, sK170: (hoare_1167836817_state > $o) > ((hoare_1167836817_state > $o) > $o) > hoare_1167836817_state > $o).
% 5.99/2.18 thf(func_def_306, type, ph172: !>[X0: $tType]:(X0)).
% 5.99/2.18 thf(f5062,plain,(
% 5.99/2.18 $false),
% 5.99/2.18 inference(subsumption_resolution,[status(thm)],[f5061,f5057])).
% 5.99/2.18 thf(f5057,plain,(
% 5.99/2.18 ( ! [X1 : $o] : (($true = X1)) )),
% 5.99/2.18 inference(trivial_inequality_removal,[status(thm)],[f5045])).
% 5.99/2.18 thf(f5045,plain,(
% 5.99/2.18 ( ! [X1 : $o] : (($true != $true) | ($true = X1)) )),
% 5.99/2.18 inference(superposition,[status(thm)],[f4971,f4502])).
% 5.99/2.18 thf(f4502,plain,(
% 5.99/2.18 ( ! [X2 : com,X3 : x_a > state > $o,X0 : hoare_1775062406iple_a > $o,X1 : x_a > state > $o,X4 : $o] : (($true = (hoare_1508237396rivs_a @ X0 @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ (^[Y0 : x_a]: ((^[Y1 : state]: ((X1 @ Y0 @ Y1) & X4)))) @ X2 @ X3) @ (collec676402587iple_a @ (^[Y0 : hoare_1775062406iple_a]: ($false)))))) | ($true = X4)) )),
% 5.99/2.18 inference(definition_unfolding,[status(thm)],[f3708,f4102])).
% 5.99/2.18 thf(f4102,plain,(
% 5.99/2.18 (bot_bo751897185le_a_o = (collec676402587iple_a @ (^[Y0 : hoare_1775062406iple_a]: ($false))))),
% 5.99/2.18 inference(cnf_transformation,[status(thm)],[f1489])).
% 5.99/2.18 thf(f1489,plain,(
% 5.99/2.18 (bot_bo751897185le_a_o = (collec676402587iple_a @ (^[Y0 : hoare_1775062406iple_a]: ($false))))),
% 5.99/2.18 inference(fool_elimination,[status(thm)],[f1488])).
% 5.99/2.18 thf(f1488,plain,(
% 5.99/2.18 ((collec676402587iple_a @ (^[X0 : hoare_1775062406iple_a] : ($false))) = bot_bo751897185le_a_o)),
% 5.99/2.18 inference(rectify,[status(thm)],[f65])).
% 5.99/2.18 thf(f65,axiom,(
% 5.99/2.18 ((collec676402587iple_a @ (^[X52 : hoare_1775062406iple_a] : ($false))) = bot_bo751897185le_a_o)),
% 5.99/2.18 file('/export/starexec/sandbox/tmp/tmp.78tw4nyiuM/DTF2THF_30168.p',fact_64_empty__def)).
% 5.99/2.18 thf(f3708,plain,(
% 5.99/2.18 ( ! [X2 : com,X3 : x_a > state > $o,X0 : hoare_1775062406iple_a > $o,X1 : x_a > state > $o,X4 : $o] : (($true = (hoare_1508237396rivs_a @ X0 @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ (^[Y0 : x_a]: ((^[Y1 : state]: ((X1 @ Y0 @ Y1) & X4)))) @ X2 @ X3) @ bot_bo751897185le_a_o))) | ($true = X4)) )),
% 5.99/2.18 inference(cnf_transformation,[status(thm)],[f2287])).
% 5.99/2.18 thf(f2287,plain,(
% 5.99/2.18 ! [X0 : hoare_1775062406iple_a > $o,X1 : x_a > state > $o,X2 : com,X3 : x_a > state > $o,X4 : $o] : (($true = (hoare_1508237396rivs_a @ X0 @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ (^[Y0 : x_a]: ((^[Y1 : state]: ((X1 @ Y0 @ Y1) & X4)))) @ X2 @ X3) @ bot_bo751897185le_a_o))) | (($true != (hoare_1508237396rivs_a @ X0 @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ X1 @ X2 @ X3) @ bot_bo751897185le_a_o))) & ($true = X4)))),
% 5.99/2.18 inference(ennf_transformation,[status(thm)],[f999])).
% 5.99/2.18 thf(f999,plain,(
% 5.99/2.18 ! [X0 : hoare_1775062406iple_a > $o,X1 : x_a > state > $o,X2 : com,X3 : x_a > state > $o,X4 : $o] : ((($true = X4) => ($true = (hoare_1508237396rivs_a @ X0 @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ X1 @ X2 @ X3) @ bot_bo751897185le_a_o)))) => ($true = (hoare_1508237396rivs_a @ X0 @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ (^[Y0 : x_a]: ((^[Y1 : state]: ((X1 @ Y0 @ Y1) & X4)))) @ X2 @ X3) @ bot_bo751897185le_a_o))))),
% 5.99/2.18 inference(fool_elimination,[status(thm)],[f998])).
% 5.99/2.18 thf(f998,plain,(
% 5.99/2.18 ! [X0 : hoare_1775062406iple_a > $o,X1 : x_a > state > $o,X2 : com,X3 : x_a > state > $o,X4 : $o] : ((X4 => (hoare_1508237396rivs_a @ X0 @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ X1 @ X2 @ X3) @ bot_bo751897185le_a_o))) => (hoare_1508237396rivs_a @ X0 @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ (^[X5 : x_a, X6 : state] : ((X1 @ X5 @ X6) & X4)) @ X2 @ X3) @ bot_bo751897185le_a_o)))),
% 5.99/2.18 inference(rectify,[status(thm)],[f9])).
% 5.99/2.18 thf(f9,axiom,(
% 5.99/2.18 ! [X13 : hoare_1775062406iple_a > $o,X14 : x_a > state > $o,X15 : com,X16 : x_a > state > $o,X17 : $o] : ((X17 => (hoare_1508237396rivs_a @ X13 @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ X14 @ X15 @ X16) @ bot_bo751897185le_a_o))) => (hoare_1508237396rivs_a @ X13 @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ (^[X18 : x_a, X19 : state] : ((X14 @ X18 @ X19) & X17)) @ X15 @ X16) @ bot_bo751897185le_a_o)))),
% 5.99/2.18 file('/export/starexec/sandbox/tmp/tmp.78tw4nyiuM/DTF2THF_30168.p',fact_8_constant)).
% 5.99/2.18 thf(f4971,plain,(
% 5.99/2.18 ( ! [X0 : x_a > state > $o] : (($true != (hoare_1508237396rivs_a @ g @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ X0 @ c @ (^[Y0 : x_a]: ((^[Y1 : state]: ((p @ Y0 @ Y1) & (~ (b @ Y1))))))) @ (collec676402587iple_a @ (^[Y0 : hoare_1775062406iple_a]: ($false))))))) )),
% 5.99/2.18 inference(trivial_inequality_removal,[status(thm)],[f4970])).
% 5.99/2.18 thf(f4970,plain,(
% 5.99/2.18 ( ! [X0 : x_a > state > $o] : (($true = $false) | ($true != (hoare_1508237396rivs_a @ g @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ X0 @ c @ (^[Y0 : x_a]: ((^[Y1 : state]: ((p @ Y0 @ Y1) & (~ (b @ Y1))))))) @ (collec676402587iple_a @ (^[Y0 : hoare_1775062406iple_a]: ($false))))))) )),
% 5.99/2.18 inference(beta_eta_normalization,[status(thm)],[f4969])).
% 5.99/2.18 thf(f4969,plain,(
% 5.99/2.18 ( ! [X0 : x_a > state > $o] : (($true != (hoare_1508237396rivs_a @ g @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ X0 @ c @ (^[Y0 : x_a]: ((^[Y1 : state]: ((p @ Y0 @ Y1) & (~ (b @ Y1))))))) @ (collec676402587iple_a @ (^[Y0 : hoare_1775062406iple_a]: ($false)))))) | ($true = ((^[Y0 : x_a]: ((^[Y1 : state]: ($false)))) @ (sK27 @ X0 @ (^[Y0 : x_a]: ((^[Y1 : state]: ($false))))) @ (sK28 @ X0 @ (^[Y0 : x_a]: ((^[Y1 : state]: ($false)))))))) )),
% 5.99/2.18 inference(trivial_inequality_removal,[status(thm)],[f4960])).
% 5.99/2.18 thf(f4960,plain,(
% 5.99/2.18 ( ! [X0 : x_a > state > $o] : (($true != (hoare_1508237396rivs_a @ g @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ X0 @ c @ (^[Y0 : x_a]: ((^[Y1 : state]: ((p @ Y0 @ Y1) & (~ (b @ Y1))))))) @ (collec676402587iple_a @ (^[Y0 : hoare_1775062406iple_a]: ($false)))))) | ($true = ((^[Y0 : x_a]: ((^[Y1 : state]: ($false)))) @ (sK27 @ X0 @ (^[Y0 : x_a]: ((^[Y1 : state]: ($false))))) @ (sK28 @ X0 @ (^[Y0 : x_a]: ((^[Y1 : state]: ($false))))))) | ($true != $true)) )),
% 5.99/2.18 inference(superposition,[status(thm)],[f4462,f4392])).
% 5.99/2.18 thf(f4392,plain,(
% 5.99/2.18 ( ! [X2 : x_a > state > $o,X3 : com,X0 : x_a > state > $o,X1 : hoare_1775062406iple_a > $o,X4 : x_a > state > $o] : (($true = (hoare_1508237396rivs_a @ X1 @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ X0 @ X3 @ X4) @ (collec676402587iple_a @ (^[Y0 : hoare_1775062406iple_a]: ($false)))))) | ($true != (hoare_1508237396rivs_a @ X1 @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ X2 @ X3 @ X4) @ (collec676402587iple_a @ (^[Y0 : hoare_1775062406iple_a]: ($false)))))) | ((X0 @ (sK27 @ X2 @ X0) @ (sK28 @ X2 @ X0)) = $true)) )),
% 5.99/2.18 inference(definition_unfolding,[status(thm)],[f3466,f4102,f4102])).
% 5.99/2.18 thf(f3466,plain,(
% 5.99/2.18 ( ! [X2 : x_a > state > $o,X3 : com,X0 : x_a > state > $o,X1 : hoare_1775062406iple_a > $o,X4 : x_a > state > $o] : (($true != (hoare_1508237396rivs_a @ X1 @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ X2 @ X3 @ X4) @ bot_bo751897185le_a_o))) | ($true = (hoare_1508237396rivs_a @ X1 @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ X0 @ X3 @ X4) @ bot_bo751897185le_a_o))) | ((X0 @ (sK27 @ X2 @ X0) @ (sK28 @ X2 @ X0)) = $true)) )),
% 5.99/2.18 inference(cnf_transformation,[status(thm)],[f2903])).
% 5.99/2.18 thf(f2903,plain,(
% 5.99/2.18 ! [X0 : x_a > state > $o,X1 : hoare_1775062406iple_a > $o,X2 : x_a > state > $o,X3 : com,X4 : x_a > state > $o] : (($true != (hoare_1508237396rivs_a @ X1 @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ X2 @ X3 @ X4) @ bot_bo751897185le_a_o))) | ($true = (hoare_1508237396rivs_a @ X1 @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ X0 @ X3 @ X4) @ bot_bo751897185le_a_o))) | (((X2 @ (sK27 @ X2 @ X0) @ (sK28 @ X2 @ X0)) != $true) & ((X0 @ (sK27 @ X2 @ X0) @ (sK28 @ X2 @ X0)) = $true)))),
% 5.99/2.18 inference(skolemisation,[status(esa),new_symbols(skolem,[sK27,sK28])],[f2657,f2902])).
% 5.99/2.18 thf(f2902,plain,(
% 5.99/2.18 ! [X0 : x_a > state > $o,X2 : x_a > state > $o] : (? [X5 : x_a,X6 : state] : (($true != (X2 @ X5 @ X6)) & ($true = (X0 @ X5 @ X6))) => (((X2 @ (sK27 @ X2 @ X0) @ (sK28 @ X2 @ X0)) != $true) & ((X0 @ (sK27 @ X2 @ X0) @ (sK28 @ X2 @ X0)) = $true)))),
% 5.99/2.18 introduced(definition,[],[choice_axiom])).
% 5.99/2.18 thf(f2657,plain,(
% 5.99/2.18 ! [X0 : x_a > state > $o,X1 : hoare_1775062406iple_a > $o,X2 : x_a > state > $o,X3 : com,X4 : x_a > state > $o] : (($true != (hoare_1508237396rivs_a @ X1 @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ X2 @ X3 @ X4) @ bot_bo751897185le_a_o))) | ($true = (hoare_1508237396rivs_a @ X1 @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ X0 @ X3 @ X4) @ bot_bo751897185le_a_o))) | ? [X5 : x_a,X6 : state] : (($true != (X2 @ X5 @ X6)) & ($true = (X0 @ X5 @ X6))))),
% 5.99/2.18 inference(flattening,[status(thm)],[f2656])).
% 5.99/2.18 thf(f2656,plain,(
% 5.99/2.18 ! [X0 : x_a > state > $o,X1 : hoare_1775062406iple_a > $o,X2 : x_a > state > $o,X3 : com,X4 : x_a > state > $o] : ((($true = (hoare_1508237396rivs_a @ X1 @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ X0 @ X3 @ X4) @ bot_bo751897185le_a_o))) | ? [X5 : x_a,X6 : state] : (($true != (X2 @ X5 @ X6)) & ($true = (X0 @ X5 @ X6)))) | ($true != (hoare_1508237396rivs_a @ X1 @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ X2 @ X3 @ X4) @ bot_bo751897185le_a_o))))),
% 5.99/2.18 inference(ennf_transformation,[status(thm)],[f1527])).
% 5.99/2.18 thf(f1527,plain,(
% 5.99/2.18 ! [X0 : x_a > state > $o,X1 : hoare_1775062406iple_a > $o,X2 : x_a > state > $o,X3 : com,X4 : x_a > state > $o] : (($true = (hoare_1508237396rivs_a @ X1 @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ X2 @ X3 @ X4) @ bot_bo751897185le_a_o))) => (! [X5 : x_a,X6 : state] : (($true = (X0 @ X5 @ X6)) => ($true = (X2 @ X5 @ X6))) => ($true = (hoare_1508237396rivs_a @ X1 @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ X0 @ X3 @ X4) @ bot_bo751897185le_a_o)))))),
% 5.99/2.18 inference(fool_elimination,[status(thm)],[f1526])).
% 5.99/2.18 thf(f1526,plain,(
% 5.99/2.18 ! [X0 : x_a > state > $o,X1 : hoare_1775062406iple_a > $o,X2 : x_a > state > $o,X3 : com,X4 : x_a > state > $o] : ((hoare_1508237396rivs_a @ X1 @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ X2 @ X3 @ X4) @ bot_bo751897185le_a_o)) => (! [X5 : x_a,X6 : state] : ((X0 @ X5 @ X6) => (X2 @ X5 @ X6)) => (hoare_1508237396rivs_a @ X1 @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ X0 @ X3 @ X4) @ bot_bo751897185le_a_o))))),
% 5.99/2.18 inference(rectify,[status(thm)],[f15])).
% 5.99/2.18 thf(f15,axiom,(
% 5.99/2.18 ! [X32 : x_a > state > $o,X33 : hoare_1775062406iple_a > $o,X34 : x_a > state > $o,X35 : com,X36 : x_a > state > $o] : ((hoare_1508237396rivs_a @ X33 @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ X34 @ X35 @ X36) @ bot_bo751897185le_a_o)) => (! [X18 : x_a,X19 : state] : ((X32 @ X18 @ X19) => (X34 @ X18 @ X19)) => (hoare_1508237396rivs_a @ X33 @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ X32 @ X35 @ X36) @ bot_bo751897185le_a_o))))),
% 5.99/2.18 file('/export/starexec/sandbox/tmp/tmp.78tw4nyiuM/DTF2THF_30168.p',fact_14_conseq1)).
% 5.99/2.18 thf(f4462,plain,(
% 5.99/2.18 ((hoare_1508237396rivs_a @ g @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ (^[Y0 : x_a]: ((^[Y1 : state]: ($false)))) @ c @ (^[Y0 : x_a]: ((^[Y1 : state]: ((p @ Y0 @ Y1) & (~ (b @ Y1))))))) @ (collec676402587iple_a @ (^[Y0 : hoare_1775062406iple_a]: ($false))))) != $true)),
% 5.99/2.18 inference(definition_unfolding,[status(thm)],[f3618,f4102])).
% 5.99/2.18 thf(f3618,plain,(
% 5.99/2.18 ($true != (hoare_1508237396rivs_a @ g @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ (^[Y0 : x_a]: ((^[Y1 : state]: ($false)))) @ c @ (^[Y0 : x_a]: ((^[Y1 : state]: ((p @ Y0 @ Y1) & (~ (b @ Y1))))))) @ bot_bo751897185le_a_o)))),
% 5.99/2.18 inference(cnf_transformation,[status(thm)],[f2276])).
% 5.99/2.18 thf(f2276,plain,(
% 5.99/2.18 ($true != (hoare_1508237396rivs_a @ g @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ (^[Y0 : x_a]: ((^[Y1 : state]: ($false)))) @ c @ (^[Y0 : x_a]: ((^[Y1 : state]: ((p @ Y0 @ Y1) & (~ (b @ Y1))))))) @ bot_bo751897185le_a_o)))),
% 5.99/2.18 inference(flattening,[status(thm)],[f2053])).
% 5.99/2.18 thf(f2053,plain,(
% 5.99/2.18 ~($true = (hoare_1508237396rivs_a @ g @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ (^[Y0 : x_a]: ((^[Y1 : state]: ($false)))) @ c @ (^[Y0 : x_a]: ((^[Y1 : state]: ((p @ Y0 @ Y1) & (~ (b @ Y1))))))) @ bot_bo751897185le_a_o)))),
% 5.99/2.18 inference(fool_elimination,[status(thm)],[f2052])).
% 5.99/2.18 thf(f2052,plain,(
% 5.99/2.18 ~(hoare_1508237396rivs_a @ g @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ (^[X0 : x_a, X1 : state] : ($false)) @ c @ (^[X2 : x_a, X3 : state] : ((p @ X2 @ X3) & (~ (b @ X3))))) @ bot_bo751897185le_a_o))),
% 5.99/2.18 inference(rectify,[status(thm)],[f712])).
% 5.99/2.18 thf(f712,negated_conjecture,(
% 5.99/2.18 ~(hoare_1508237396rivs_a @ g @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ (^[X18 : x_a, X19 : state] : ($false)) @ c @ (^[X18 : x_a, X19 : state] : ((p @ X18 @ X19) & (~ (b @ X19))))) @ bot_bo751897185le_a_o))),
% 5.99/2.18 inference(negated_conjecture,[status(cth)],[f711])).
% 5.99/2.18 thf(f711,conjecture,(
% 5.99/2.18 (hoare_1508237396rivs_a @ g @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ (^[X18 : x_a, X19 : state] : ($false)) @ c @ (^[X18 : x_a, X19 : state] : ((p @ X18 @ X19) & (~ (b @ X19))))) @ bot_bo751897185le_a_o))),
% 5.99/2.18 file('/export/starexec/sandbox/tmp/tmp.78tw4nyiuM/DTF2THF_30168.p',conj_0)).
% 5.99/2.18 thf(f5061,plain,(
% 5.99/2.18 ( ! [X0 : x_a > state > $o,X1 : hoare_1775062406iple_a > $o] : (($true != (hoare_1508237396rivs_a @ X1 @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ X0 @ c @ (^[Y0 : x_a]: ((^[Y1 : state]: ((p @ Y0 @ Y1) & (~ (b @ Y1))))))) @ (collec676402587iple_a @ (^[Y0 : hoare_1775062406iple_a]: ($false))))))) )),
% 5.99/2.18 inference(subsumption_resolution,[status(thm)],[f5056,f5057])).
% 5.99/2.18 thf(f5056,plain,(
% 5.99/2.18 ( ! [X0 : x_a > state > $o,X1 : hoare_1775062406iple_a > $o] : (($true != (hoare_1508237396rivs_a @ g @ X1)) | ($true != (hoare_1508237396rivs_a @ X1 @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ X0 @ c @ (^[Y0 : x_a]: ((^[Y1 : state]: ((p @ Y0 @ Y1) & (~ (b @ Y1))))))) @ (collec676402587iple_a @ (^[Y0 : hoare_1775062406iple_a]: ($false))))))) )),
% 5.99/2.18 inference(trivial_inequality_removal,[status(thm)],[f5048])).
% 5.99/2.18 thf(f5048,plain,(
% 5.99/2.18 ( ! [X0 : x_a > state > $o,X1 : hoare_1775062406iple_a > $o] : (($true != (hoare_1508237396rivs_a @ X1 @ (insert1281456128iple_a @ (hoare_1766022166iple_a @ X0 @ c @ (^[Y0 : x_a]: ((^[Y1 : state]: ((p @ Y0 @ Y1) & (~ (b @ Y1))))))) @ (collec676402587iple_a @ (^[Y0 : hoare_1775062406iple_a]: ($false)))))) | ($true != (hoare_1508237396rivs_a @ g @ X1)) | ($true != $true)) )),
% 5.99/2.18 inference(superposition,[status(thm)],[f4971,f3556])).
% 5.99/2.18 thf(f3556,plain,(
% 5.99/2.18 ( ! [X2 : hoare_1775062406iple_a > $o,X0 : hoare_1775062406iple_a > $o,X1 : hoare_1775062406iple_a > $o] : (($true = (hoare_1508237396rivs_a @ X0 @ X2)) | ((hoare_1508237396rivs_a @ X1 @ X2) != $true) | ((hoare_1508237396rivs_a @ X0 @ X1) != $true)) )),
% 5.99/2.18 inference(cnf_transformation,[status(thm)],[f2345])).
% 5.99/2.18 thf(f2345,plain,(
% 5.99/2.18 ! [X0 : hoare_1775062406iple_a > $o,X1 : hoare_1775062406iple_a > $o,X2 : hoare_1775062406iple_a > $o] : (($true = (hoare_1508237396rivs_a @ X0 @ X2)) | ((hoare_1508237396rivs_a @ X0 @ X1) != $true) | ((hoare_1508237396rivs_a @ X1 @ X2) != $true))),
% 5.99/2.18 inference(flattening,[status(thm)],[f2344])).
% 5.99/2.18 thf(f2344,plain,(
% 5.99/2.18 ! [X0 : hoare_1775062406iple_a > $o,X1 : hoare_1775062406iple_a > $o,X2 : hoare_1775062406iple_a > $o] : ((($true = (hoare_1508237396rivs_a @ X0 @ X2)) | ((hoare_1508237396rivs_a @ X0 @ X1) != $true)) | ((hoare_1508237396rivs_a @ X1 @ X2) != $true))),
% 5.99/2.18 inference(ennf_transformation,[status(thm)],[f927])).
% 5.99/2.18 thf(f927,plain,(
% 5.99/2.18 ! [X0 : hoare_1775062406iple_a > $o,X1 : hoare_1775062406iple_a > $o,X2 : hoare_1775062406iple_a > $o] : (((hoare_1508237396rivs_a @ X1 @ X2) = $true) => (((hoare_1508237396rivs_a @ X0 @ X1) = $true) => ($true = (hoare_1508237396rivs_a @ X0 @ X2))))),
% 5.99/2.18 inference(fool_elimination,[status(thm)],[f926])).
% 5.99/2.18 thf(f926,plain,(
% 5.99/2.18 ! [X0 : hoare_1775062406iple_a > $o,X1 : hoare_1775062406iple_a > $o,X2 : hoare_1775062406iple_a > $o] : ((hoare_1508237396rivs_a @ X1 @ X2) => ((hoare_1508237396rivs_a @ X0 @ X1) => (hoare_1508237396rivs_a @ X0 @ X2)))),
% 5.99/2.18 inference(rectify,[status(thm)],[f5])).
% 5.99/2.18 thf(f5,axiom,(
% 5.99/2.18 ! [X7 : hoare_1775062406iple_a > $o,X8 : hoare_1775062406iple_a > $o,X9 : hoare_1775062406iple_a > $o] : ((hoare_1508237396rivs_a @ X8 @ X9) => ((hoare_1508237396rivs_a @ X7 @ X8) => (hoare_1508237396rivs_a @ X7 @ X9)))),
% 5.99/2.18 file('/export/starexec/sandbox/tmp/tmp.78tw4nyiuM/DTF2THF_30168.p',fact_4_cut)).
% 5.99/2.18 % SZS output end Proof for DTF2THF_30168
% 5.99/2.18 % (30519)------------------------------
% 5.99/2.18 % (30519)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.99/2.18 % (30519)Termination reason: Refutation
% 5.99/2.18
% 5.99/2.18 % (30519)Memory used [KB]: 12537
% 5.99/2.18 % (30519)Time elapsed: 0.122 s
% 5.99/2.18 % (30519)Instructions burned: 393 (million)
% 5.99/2.18 % (30519)------------------------------
% 5.99/2.18 % (30519)------------------------------
% 5.99/2.18 % (30475)Success in time 0.497 s
%------------------------------------------------------------------------------