TPTP Problem File: SLH0142^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Query_Optimization/0013_IKKBZ_Optimality/prob_05231_239362__15964166_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1509 ( 435 unt; 252 typ; 0 def)
% Number of atoms : 3699 ( 996 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 12325 ( 521 ~; 61 |; 190 &;9672 @)
% ( 0 <=>;1881 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 7 avg)
% Number of types : 57 ( 56 usr)
% Number of type conns : 519 ( 519 >; 0 *; 0 +; 0 <<)
% Number of symbols : 197 ( 196 usr; 29 con; 0-6 aty)
% Number of variables : 3709 ( 227 ^;3308 !; 174 ?;3709 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:08:27.366
%------------------------------------------------------------------------------
% Could-be-implicit typings (56)
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J_J_J_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(ty_n_t__FSet__Ofset_It__List__Olist_Itf__a_J_J,type,
fset_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
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thf(ty_n_t__List__Olist_Itf__b_J,type,
list_b: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__FSet__Ofset_Itf__b_J,type,
fset_b: $tType ).
thf(ty_n_t__FSet__Ofset_Itf__a_J,type,
fset_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (196)
thf(sy_c_Digraph_Oarcs__ends_001tf__a_001tf__b,type,
arcs_ends_a_b: pre_pr7278220950009878019t_unit > set_Product_prod_a_a ).
thf(sy_c_Digraph_Oloopfree__digraph_001tf__a_001tf__b,type,
loopfree_digraph_a_b: pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_Digraph_Onomulti__digraph_001tf__a_001tf__b,type,
nomulti_digraph_a_b: pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_Dtree_Odtree_ONode_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
node_list_a_list_a: list_a > fset_P6656299774166858224list_a > dtree_list_a_list_a ).
thf(sy_c_Dtree_Odtree_ONode_001t__List__Olist_Itf__a_J_001tf__a,type,
node_list_a_a: list_a > fset_P8654898996138850416_a_a_a > dtree_list_a_a ).
thf(sy_c_Dtree_Odtree_ONode_001t__List__Olist_Itf__a_J_001tf__b,type,
node_list_a_b: list_a > fset_P2153231429829016240_a_b_b > dtree_list_a_b ).
thf(sy_c_Dtree_Odtree_ONode_001t__List__Olist_Itf__b_J_001t__List__Olist_Itf__a_J,type,
node_list_b_list_a: list_b > fset_P4021558052792023921list_a > dtree_list_b_list_a ).
thf(sy_c_Dtree_Odtree_ONode_001t__List__Olist_Itf__b_J_001tf__a,type,
node_list_b_a: list_b > fset_P724321373960676081_b_a_a > dtree_list_b_a ).
thf(sy_c_Dtree_Odtree_ONode_001t__List__Olist_Itf__b_J_001tf__b,type,
node_list_b_b: list_b > fset_P3446025844505617713_b_b_b > dtree_list_b_b ).
thf(sy_c_Dtree_Odtree_Odarcs_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
darcs_list_a_list_a: dtree_list_a_list_a > set_list_a ).
thf(sy_c_Dtree_Odtree_Odarcs_001t__List__Olist_Itf__a_J_001tf__a,type,
darcs_list_a_a: dtree_list_a_a > set_a ).
thf(sy_c_Dtree_Odtree_Odarcs_001t__List__Olist_Itf__a_J_001tf__b,type,
darcs_list_a_b: dtree_list_a_b > set_b ).
thf(sy_c_Dtree_Odtree_Odarcs_001t__List__Olist_Itf__b_J_001t__List__Olist_Itf__a_J,type,
darcs_list_b_list_a: dtree_list_b_list_a > set_list_a ).
thf(sy_c_Dtree_Odtree_Odarcs_001t__List__Olist_Itf__b_J_001tf__a,type,
darcs_list_b_a: dtree_list_b_a > set_a ).
thf(sy_c_Dtree_Odtree_Odarcs_001t__List__Olist_Itf__b_J_001tf__b,type,
darcs_list_b_b: dtree_list_b_b > set_b ).
thf(sy_c_Dtree_Odtree_Odverts_001t__List__Olist_Itf__a_J_001tf__b,type,
dverts_list_a_b: dtree_list_a_b > set_list_a ).
thf(sy_c_Dtree_Odtree_Oroot_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
root_list_a_list_a: dtree_list_a_list_a > list_a ).
thf(sy_c_Dtree_Odtree_Oroot_001t__List__Olist_Itf__a_J_001tf__a,type,
root_list_a_a: dtree_list_a_a > list_a ).
thf(sy_c_Dtree_Odtree_Oroot_001t__List__Olist_Itf__a_J_001tf__b,type,
root_list_a_b: dtree_list_a_b > list_a ).
thf(sy_c_Dtree_Odtree_Oroot_001t__List__Olist_Itf__b_J_001t__List__Olist_Itf__a_J,type,
root_list_b_list_a: dtree_list_b_list_a > list_b ).
thf(sy_c_Dtree_Odtree_Oroot_001t__List__Olist_Itf__b_J_001tf__a,type,
root_list_b_a: dtree_list_b_a > list_b ).
thf(sy_c_Dtree_Odtree_Oroot_001t__List__Olist_Itf__b_J_001tf__b,type,
root_list_b_b: dtree_list_b_b > list_b ).
thf(sy_c_Dtree_Odtree_Osucs_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_Dtree_Odtree_Osucs_001t__List__Olist_Itf__a_J_001tf__a,type,
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thf(sy_c_Dtree_Odtree_Osucs_001t__List__Olist_Itf__a_J_001tf__b,type,
sucs_list_a_b: dtree_list_a_b > fset_P2153231429829016240_a_b_b ).
thf(sy_c_Dtree_Odtree_Osucs_001t__List__Olist_Itf__b_J_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_Dtree_Odtree_Osucs_001t__List__Olist_Itf__b_J_001tf__a,type,
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thf(sy_c_Dtree_Odtree_Osucs_001t__List__Olist_Itf__b_J_001tf__b,type,
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thf(sy_c_Dtree_Ois__subtree_001t__List__Olist_Itf__a_J_001tf__b,type,
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thf(sy_c_Dtree_Omax__deg_001t__List__Olist_Itf__a_J_001tf__b,type,
max_deg_list_a_b: dtree_list_a_b > nat ).
thf(sy_c_Dtree_Onum__leaves_001t__List__Olist_Itf__a_J_001tf__b,type,
num_leaves_list_a_b: dtree_list_a_b > nat ).
thf(sy_c_Dtree_Ostrict__subtree_001t__List__Olist_Itf__a_J_001tf__b,type,
strict8995144569104247066st_a_b: dtree_list_a_b > dtree_list_a_b > $o ).
thf(sy_c_Dtree_Owf__darcs_001t__List__Olist_Itf__a_J_001tf__b,type,
wf_darcs_list_a_b: dtree_list_a_b > $o ).
thf(sy_c_Dtree_Owf__dverts_001t__List__Olist_Itf__a_J_001tf__b,type,
wf_dverts_list_a_b: dtree_list_a_b > $o ).
thf(sy_c_FSet_Ofcard_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
fcard_dtree_list_a_b: fset_dtree_list_a_b > nat ).
thf(sy_c_FSet_Ofcard_001t__List__Olist_Itf__a_J,type,
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fcard_8555586194327498616od_a_a: fset_P9143335661316304269od_a_a > nat ).
thf(sy_c_FSet_Ofcard_001tf__a,type,
fcard_a: fset_a > nat ).
thf(sy_c_FSet_Ofcard_001tf__b,type,
fcard_b: fset_b > nat ).
thf(sy_c_FSet_Ofinsert_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
finser8636396436308191404st_a_b: dtree_list_a_b > fset_dtree_list_a_b > fset_dtree_list_a_b ).
thf(sy_c_FSet_Ofinsert_001t__List__Olist_Itf__a_J,type,
finsert_list_a: list_a > fset_list_a > fset_list_a ).
thf(sy_c_FSet_Ofinsert_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J,type,
finser5913853959220252280list_a: produc7575571382841205696list_a > fset_P6656299774166858224list_a > fset_P6656299774166858224list_a ).
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finser1670425070513979513list_a: produc3332142494134932929list_a > fset_P4021558052792023921list_a > fset_P4021558052792023921list_a ).
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thf(sy_c_FSet_Ofinsert_001tf__b,type,
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thf(sy_c_FSet_Ofset_Ofset_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_FSet_Ofset_Ofset_001tf__a,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_IKKBZ_Odenormalize_001tf__a_001tf__b,type,
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thf(sy_c_IKKBZ_Oold_001tf__a_001tf__b,type,
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find_pos_a_b: ( list_a > real ) > list_a > dtree_list_a_b > produc9164743771328383783list_a ).
thf(sy_c_IKKBZ_Oold_Ofind__pos__aux_001tf__a_001tf__b,type,
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thf(sy_c_IKKBZ_Oranked__dtree_001tf__a_001tf__b,type,
ranked_dtree_a_b: dtree_list_a_b > compar7542523840845723048st_a_b > $o ).
thf(sy_c_IKKBZ_Oranked__dtree_Omerge1_001tf__a_001tf__b,type,
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thf(sy_c_IKKBZ_Oranked__dtree_Omerge_001tf__a_001tf__b,type,
ranked_merge_a_b: ( list_a > real ) > compar7542523840845723048st_a_b > dtree_list_a_b > dtree_list_a_b ).
thf(sy_c_IKKBZ_Oranked__dtree_Onormalize1_001tf__a_001tf__b,type,
ranked8905849569120154423e1_a_b: ( list_a > real ) > dtree_list_a_b > dtree_list_a_b ).
thf(sy_c_IKKBZ__Optimality_Odirected__tree_Obefore_001tf__a_001tf__b,type,
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thf(sy_c_IKKBZ__Optimality_Odirected__tree_Oforward_001tf__a_001tf__b,type,
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thf(sy_c_IKKBZ__Optimality_Odom__children_001tf__a_001tf__b,type,
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thf(sy_c_List_Oappend_001tf__a,type,
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thf(sy_c_List_Oappend_001tf__b,type,
append_b: list_b > list_b > list_b ).
thf(sy_c_List_Odistinct_001tf__a,type,
distinct_a: list_a > $o ).
thf(sy_c_List_Odistinct_001tf__b,type,
distinct_b: list_b > $o ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_OCons_001tf__b,type,
cons_b: b > list_b > list_b ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_ONil_001tf__b,type,
nil_b: list_b ).
thf(sy_c_List_Olist_Ohd_001tf__a,type,
hd_a: list_a > a ).
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thf(sy_c_List_Olist_Oset_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_List_Olist_Oset_001tf__b,type,
set_b2: list_b > set_b ).
thf(sy_c_List_Orev_001tf__a,type,
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thf(sy_c_List__Dtree_Ocombine_001tf__a_001tf__b,type,
list_combine_a_b: list_a > list_a > dtree_list_a_b > dtree_list_a_b ).
thf(sy_c_List__Dtree_Odlverts_001tf__a_001tf__b,type,
list_dlverts_a_b: dtree_list_a_b > set_a ).
thf(sy_c_List__Dtree_Olist__dtree_001tf__a_001tf__b,type,
list_list_dtree_a_b: dtree_list_a_b > $o ).
thf(sy_c_List__Dtree_Owf__dlverts_001tf__a_001tf__b,type,
list_wf_dlverts_a_b: dtree_list_a_b > $o ).
thf(sy_c_Nat_OSuc,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__b_M_Eo_J,type,
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produc2621617146629198007list_a: dtree_list_a_b > ( b > list_a ) > produc1920479565126685823list_a ).
thf(sy_c_Product__Type_OPair_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
produc783528831147138817st_a_b: dtree_list_a_b > dtree_list_a_b > produc1510363273921914569st_a_b ).
thf(sy_c_Product__Type_OPair_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_001tf__a,type,
produc7704165765595008945_a_b_a: dtree_list_a_b > a > produc6499617306661234687_a_b_a ).
thf(sy_c_Product__Type_OPair_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_001tf__b,type,
produc7704165765595008946_a_b_b: dtree_list_a_b > b > produc6499617310964463488_a_b_b ).
thf(sy_c_Product__Type_OPair_001t__Dtree__Odtree_It__List__Olist_Itf__b_J_Mt__List__Olist_Itf__a_J_J_001t__List__Olist_Itf__a_J,type,
produc2033296462658091379list_a: dtree_list_b_list_a > list_a > produc3332142494134932929list_a ).
thf(sy_c_Product__Type_OPair_001t__Dtree__Odtree_It__List__Olist_Itf__b_J_Mtf__a_J_001tf__a,type,
produc5561110458339973107_b_a_a: dtree_list_b_a > a > produc4356561999406198849_b_a_a ).
thf(sy_c_Product__Type_OPair_001t__Dtree__Odtree_It__List__Olist_Itf__b_J_Mtf__b_J_001tf__b,type,
produc4925461457734986419_b_b_b: dtree_list_b_b > b > produc3720913003104440961_b_b_b ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
produc148520996349637281st_a_b: list_a > dtree_list_a_b > produc111314985273491367st_a_b ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
produc6837034575241423639list_a: list_a > list_a > produc9164743771328383783list_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J,type,
produc673257793671328980st_a_b: list_a > produc111314985273491367st_a_b > produc7147531718898801626st_a_b ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J_J_J,type,
produc1848684973559390389st_a_b: list_a > produc3397603951383089160st_a_b > produc9008341577332299707st_a_b ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__b_J_001tf__a,type,
produc4145578316043568848st_b_a: list_b > a > produc1943741644644106336st_b_a ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__List__Olist_Itf__b_J,type,
produc7903367361620597084list_b: nat > list_b > produc7811952446676219690list_b ).
thf(sy_c_Product__Type_OPair_001tf__a_001t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mtf__a_J,type,
produc7119031474978700025st_b_a: a > produc1943741644644106336st_b_a > produc7945266988514096265st_b_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(sy_c_Product__Type_OPair_001tf__b_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_J_J,type,
produc5064203622704112514st_a_b: b > produc7147531718898801626st_a_b > produc3397603951383089160st_a_b ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_001tf__b,type,
produc5948858871325780166_a_b_b: produc6499617310964463488_a_b_b > dtree_list_a_b ).
thf(sy_c_Set_OCollect_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
collec2944820760411501129st_a_b: ( dtree_list_a_b > $o ) > set_dtree_list_a_b ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
collec3336397797384452498od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OCollect_001tf__b,type,
collect_b: ( b > $o ) > set_b ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
image_5965465251548763643st_a_b: ( produc6499617310964463488_a_b_b > dtree_list_a_b ) > set_Pr3443975907877334966_a_b_b > set_dtree_list_a_b ).
thf(sy_c_Shortest__Path_Owf__digraph_Omk__cycles__path_001tf__b,type,
shorte6374615165232202367path_b: nat > list_b > list_b ).
thf(sy_c_Shortest__Path_Owf__digraph_Omk__cycles__path__rel_001tf__b,type,
shorte5702012728047871812_rel_b: produc7811952446676219690list_b > produc7811952446676219690list_b > $o ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_Transitive__Closure_Otrancl_001tf__a,type,
transitive_trancl_a: set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Vertex__Walk_Ovpath_001tf__a_001tf__b,type,
vertex_vpath_a_b: list_a > pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J,type,
accp_P7720916649673260129list_b: ( produc7811952446676219690list_b > produc7811952446676219690list_b > $o ) > produc7811952446676219690list_b > $o ).
thf(sy_c_member_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
member551035911493665803st_a_b: dtree_list_a_b > set_dtree_list_a_b > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J,type,
member4695696432722591383_a_b_b: produc6499617310964463488_a_b_b > set_Pr3443975907877334966_a_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mtf__a_J,type,
member7370802231240916489st_b_a: produc1943741644644106336st_b_a > set_Pr2389355623220313408st_b_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J,type,
member8261005420521984321list_b: produc7811952446676219690list_b > set_Pr1349601357184307552list_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mtf__a_J_J,type,
member4827874839601930546st_b_a: produc7945266988514096265st_b_a > set_Pr6500140389540524009st_b_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_T,type,
t: pre_pr7278220950009878019t_unit ).
thf(sy_v_cmp,type,
cmp: compar7542523840845723048st_a_b ).
thf(sy_v_r____,type,
r: list_a ).
thf(sy_v_rank,type,
rank: list_a > real ).
thf(sy_v_t,type,
t2: dtree_list_a_b ).
thf(sy_v_xs____,type,
xs: fset_P2153231429829016240_a_b_b ).
% Relevant facts (1256)
thf(fact_0_merge1_Ocases,axiom,
! [X: dtree_list_a_b] :
~ ! [R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
( X
!= ( node_list_a_b @ R @ Xs ) ) ).
% merge1.cases
thf(fact_1__C2_Ohyps_C,axiom,
! [X2: produc6499617310964463488_a_b_b] :
( xs
!= ( finser2303212525150181944_a_b_b @ X2 @ bot_bo2248824169281960260_a_b_b ) ) ).
% "2.hyps"
thf(fact_2__C2_Oprems_C,axiom,
ord_less_eq_nat @ ( max_deg_list_a_b @ ( node_list_a_b @ r @ xs ) ) @ one_one_nat ).
% "2.prems"
thf(fact_3_normalize__full_Osimps_I2_J,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
( ! [X3: produc6499617310964463488_a_b_b] :
( Xs2
!= ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
=> ( ( iKKBZ_6959927528703686640ll_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
= ( node_list_a_b @ R2 @ Xs2 ) ) ) ).
% normalize_full.simps(2)
thf(fact_4_dtree_Oinject,axiom,
! [X1: list_a,X22: fset_P2153231429829016240_a_b_b,Y1: list_a,Y2: fset_P2153231429829016240_a_b_b] :
( ( ( node_list_a_b @ X1 @ X22 )
= ( node_list_a_b @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% dtree.inject
thf(fact_5_mdeg__root,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,V: list_a] :
( ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
= ( max_deg_list_a_b @ ( node_list_a_b @ V @ Xs2 ) ) ) ).
% mdeg_root
thf(fact_6_empty__if__mdeg__0,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
( ( ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
= zero_zero_nat )
=> ( Xs2 = bot_bo2248824169281960260_a_b_b ) ) ).
% empty_if_mdeg_0
thf(fact_7_empty__iff__mdeg__0,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
( ( ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
= zero_zero_nat )
= ( Xs2 = bot_bo2248824169281960260_a_b_b ) ) ).
% empty_iff_mdeg_0
thf(fact_8_nempty__if__mdeg__n0,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
( ( ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
!= zero_zero_nat )
=> ( Xs2 != bot_bo2248824169281960260_a_b_b ) ) ).
% nempty_if_mdeg_n0
thf(fact_9_normalize__full__wfdlverts,axiom,
! [T1: dtree_list_a_b] :
( ( list_wf_dlverts_a_b @ T1 )
=> ( list_wf_dlverts_a_b @ ( iKKBZ_6959927528703686640ll_a_b @ T1 ) ) ) ).
% normalize_full_wfdlverts
thf(fact_10_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_11_normalize__full__wfdarcs,axiom,
! [T1: dtree_list_a_b] :
( ( wf_darcs_list_a_b @ T1 )
=> ( wf_darcs_list_a_b @ ( iKKBZ_6959927528703686640ll_a_b @ T1 ) ) ) ).
% normalize_full_wfdarcs
thf(fact_12_wf__dlverts_Ocases,axiom,
! [X: dtree_list_a_b] :
~ ! [R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
( X
!= ( node_list_a_b @ R @ Xs ) ) ).
% wf_dlverts.cases
thf(fact_13_nat_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( suc @ X22 )
= ( suc @ Y2 ) )
= ( X22 = Y2 ) ) ).
% nat.inject
thf(fact_14_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_15_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_16_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_17_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_18_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_19_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_20_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_21_lift__Suc__mono__le,axiom,
! [F: nat > set_b,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_set_b @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_set_b @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_22_lift__Suc__mono__le,axiom,
! [F: nat > real,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_real @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_23_lift__Suc__mono__le,axiom,
! [F: nat > set_list_a,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_le8861187494160871172list_a @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_le8861187494160871172list_a @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_24_lift__Suc__mono__le,axiom,
! [F: nat > set_a,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_set_a @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_set_a @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_25_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_26_lift__Suc__antimono__le,axiom,
! [F: nat > set_b,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_set_b @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_set_b @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_27_lift__Suc__antimono__le,axiom,
! [F: nat > real,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_28_lift__Suc__antimono__le,axiom,
! [F: nat > set_list_a,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_le8861187494160871172list_a @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_le8861187494160871172list_a @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_29_lift__Suc__antimono__le,axiom,
! [F: nat > set_a,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_set_a @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_set_a @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_30_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_31_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_32_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_33_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_34_Suc__le__D,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M2 )
=> ? [M3: nat] :
( M2
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_35_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_36_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_37_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_38_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_39_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_40_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_41_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_42_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_43_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_44_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_45_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N3 )
=> ( P @ M4 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_46_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_47_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_48_mem__Collect__eq,axiom,
! [A: product_prod_a_a,P: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ A @ ( collec3336397797384452498od_a_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_49_mem__Collect__eq,axiom,
! [A: dtree_list_a_b,P: dtree_list_a_b > $o] :
( ( member551035911493665803st_a_b @ A @ ( collec2944820760411501129st_a_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_50_mem__Collect__eq,axiom,
! [A: list_a,P: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_51_mem__Collect__eq,axiom,
! [A: b,P: b > $o] :
( ( member_b @ A @ ( collect_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_52_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_53_Collect__mem__eq,axiom,
! [A2: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_54_Collect__mem__eq,axiom,
! [A2: set_dtree_list_a_b] :
( ( collec2944820760411501129st_a_b
@ ^ [X4: dtree_list_a_b] : ( member551035911493665803st_a_b @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_55_Collect__mem__eq,axiom,
! [A2: set_list_a] :
( ( collect_list_a
@ ^ [X4: list_a] : ( member_list_a @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_56_Collect__mem__eq,axiom,
! [A2: set_b] :
( ( collect_b
@ ^ [X4: b] : ( member_b @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_57_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_58_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R3: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R3 @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z: nat] :
( ( R3 @ X3 @ Y3 )
=> ( ( R3 @ Y3 @ Z )
=> ( R3 @ X3 @ Z ) ) )
=> ( ! [N3: nat] : ( R3 @ N3 @ ( suc @ N3 ) )
=> ( R3 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_59_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_60_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_61_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_62_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_63_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_64_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_65_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_66_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_67_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_68_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_69_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_70_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_71_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_72_empty__fset__if__mdeg__le1__not__single,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
( ! [X3: produc6499617310964463488_a_b_b] :
( Xs2
!= ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
=> ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) @ one_one_nat )
=> ( Xs2 = bot_bo2248824169281960260_a_b_b ) ) ) ).
% empty_fset_if_mdeg_le1_not_single
thf(fact_73_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_74_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_75_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_76_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_77_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_78_mdeg__1__singleton,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
( ( ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
= one_one_nat )
=> ? [X3: produc6499617310964463488_a_b_b] :
( Xs2
= ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) ) ) ).
% mdeg_1_singleton
thf(fact_79_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_80_dverts__mset_Ocases,axiom,
! [X: dtree_list_a_b] :
~ ! [R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
( X
!= ( node_list_a_b @ R @ Xs ) ) ).
% dverts_mset.cases
thf(fact_81_dtree_Oexhaust,axiom,
! [Y: dtree_list_a_b] :
~ ! [X12: list_a,X23: fset_P2153231429829016240_a_b_b] :
( Y
!= ( node_list_a_b @ X12 @ X23 ) ) ).
% dtree.exhaust
thf(fact_82_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_83_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_84_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_85_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_86_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_87_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_88_normalize__full_Ocases,axiom,
! [X: dtree_list_a_b] :
( ! [R: list_a,T12: dtree_list_a_b,E1: b] :
( X
!= ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
( ! [X2: produc6499617310964463488_a_b_b] :
( Xs
!= ( finser2303212525150181944_a_b_b @ X2 @ bot_bo2248824169281960260_a_b_b ) )
=> ! [R: list_a] :
( X
!= ( node_list_a_b @ R @ Xs ) ) ) ) ).
% normalize_full.cases
thf(fact_89_finsert__absorb2,axiom,
! [X: produc6499617310964463488_a_b_b,A2: fset_P2153231429829016240_a_b_b] :
( ( finser2303212525150181944_a_b_b @ X @ ( finser2303212525150181944_a_b_b @ X @ A2 ) )
= ( finser2303212525150181944_a_b_b @ X @ A2 ) ) ).
% finsert_absorb2
thf(fact_90_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_91_order__refl,axiom,
! [X: set_b] : ( ord_less_eq_set_b @ X @ X ) ).
% order_refl
thf(fact_92_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_93_order__refl,axiom,
! [X: set_list_a] : ( ord_le8861187494160871172list_a @ X @ X ) ).
% order_refl
thf(fact_94_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_95_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_96_dual__order_Orefl,axiom,
! [A: set_b] : ( ord_less_eq_set_b @ A @ A ) ).
% dual_order.refl
thf(fact_97_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_98_dual__order_Orefl,axiom,
! [A: set_list_a] : ( ord_le8861187494160871172list_a @ A @ A ) ).
% dual_order.refl
thf(fact_99_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_100_normalize__full__wfdverts,axiom,
! [T1: dtree_list_a_b] :
( ( list_wf_dlverts_a_b @ T1 )
=> ( wf_dverts_list_a_b @ ( iKKBZ_6959927528703686640ll_a_b @ T1 ) ) ) ).
% normalize_full_wfdverts
thf(fact_101_FSet_Ofset__induct,axiom,
! [P: fset_P2153231429829016240_a_b_b > $o,S: fset_P2153231429829016240_a_b_b] :
( ( P @ bot_bo2248824169281960260_a_b_b )
=> ( ! [X3: produc6499617310964463488_a_b_b,S2: fset_P2153231429829016240_a_b_b] :
( ( P @ S2 )
=> ( P @ ( finser2303212525150181944_a_b_b @ X3 @ S2 ) ) )
=> ( P @ S ) ) ) ).
% FSet.fset_induct
thf(fact_102_fset__exhaust,axiom,
! [S: fset_P2153231429829016240_a_b_b] :
( ( S != bot_bo2248824169281960260_a_b_b )
=> ~ ! [X3: produc6499617310964463488_a_b_b,S3: fset_P2153231429829016240_a_b_b] :
( S
!= ( finser2303212525150181944_a_b_b @ X3 @ S3 ) ) ) ).
% fset_exhaust
thf(fact_103_fdoubleton__eq__iff,axiom,
! [A: produc6499617310964463488_a_b_b,B: produc6499617310964463488_a_b_b,C: produc6499617310964463488_a_b_b,D: produc6499617310964463488_a_b_b] :
( ( ( finser2303212525150181944_a_b_b @ A @ ( finser2303212525150181944_a_b_b @ B @ bot_bo2248824169281960260_a_b_b ) )
= ( finser2303212525150181944_a_b_b @ C @ ( finser2303212525150181944_a_b_b @ D @ bot_bo2248824169281960260_a_b_b ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% fdoubleton_eq_iff
thf(fact_104_fempty__fsubsetI,axiom,
! [X: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ bot_bo2248824169281960260_a_b_b @ X ) ).
% fempty_fsubsetI
thf(fact_105_fsubset__fempty,axiom,
! [A2: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ bot_bo2248824169281960260_a_b_b )
= ( A2 = bot_bo2248824169281960260_a_b_b ) ) ).
% fsubset_fempty
thf(fact_106_wf__dverts__sub,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,Ys: fset_P2153231429829016240_a_b_b,R2: list_a] :
( ( ord_le8870638447146015504_a_b_b @ Xs2 @ Ys )
=> ( ( wf_dverts_list_a_b @ ( node_list_a_b @ R2 @ Ys ) )
=> ( wf_dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ).
% wf_dverts_sub
thf(fact_107_fsubset__finsertI2,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B2: fset_P2153231429829016240_a_b_b,B: produc6499617310964463488_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ B2 )
=> ( ord_le8870638447146015504_a_b_b @ A2 @ ( finser2303212525150181944_a_b_b @ B @ B2 ) ) ) ).
% fsubset_finsertI2
thf(fact_108_fsubset__finsertI,axiom,
! [B2: fset_P2153231429829016240_a_b_b,A: produc6499617310964463488_a_b_b] : ( ord_le8870638447146015504_a_b_b @ B2 @ ( finser2303212525150181944_a_b_b @ A @ B2 ) ) ).
% fsubset_finsertI
thf(fact_109_finsert__mono,axiom,
! [C2: fset_P2153231429829016240_a_b_b,D2: fset_P2153231429829016240_a_b_b,A: produc6499617310964463488_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ C2 @ D2 )
=> ( ord_le8870638447146015504_a_b_b @ ( finser2303212525150181944_a_b_b @ A @ C2 ) @ ( finser2303212525150181944_a_b_b @ A @ D2 ) ) ) ).
% finsert_mono
thf(fact_110_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_111_wf__dverts__if__wf__dlverts,axiom,
! [T: dtree_list_a_b] :
( ( list_wf_dlverts_a_b @ T )
=> ( wf_dverts_list_a_b @ T ) ) ).
% wf_dverts_if_wf_dlverts
thf(fact_112_fsubset__fsingletonD,axiom,
! [A2: fset_P2153231429829016240_a_b_b,X: produc6499617310964463488_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b ) )
=> ( ( A2 = bot_bo2248824169281960260_a_b_b )
| ( A2
= ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).
% fsubset_fsingletonD
thf(fact_113_wf__darcs__sub,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,Ys: fset_P2153231429829016240_a_b_b,R4: list_a,R2: list_a] :
( ( ord_le8870638447146015504_a_b_b @ Xs2 @ Ys )
=> ( ( wf_darcs_list_a_b @ ( node_list_a_b @ R4 @ Ys ) )
=> ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ).
% wf_darcs_sub
thf(fact_114_wf__dlverts__sub,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,Ys: fset_P2153231429829016240_a_b_b,R2: list_a] :
( ( ord_le8870638447146015504_a_b_b @ Xs2 @ Ys )
=> ( ( list_wf_dlverts_a_b @ ( node_list_a_b @ R2 @ Ys ) )
=> ( list_wf_dlverts_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ).
% wf_dlverts_sub
thf(fact_115_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_116_order__antisym__conv,axiom,
! [Y: set_b,X: set_b] :
( ( ord_less_eq_set_b @ Y @ X )
=> ( ( ord_less_eq_set_b @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_117_order__antisym__conv,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_118_order__antisym__conv,axiom,
! [Y: set_list_a,X: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y @ X )
=> ( ( ord_le8861187494160871172list_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_119_order__antisym__conv,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_120_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_121_linorder__le__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_122_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_123_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_124_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_125_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_126_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_b,C: set_b] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_127_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_128_ord__le__eq__subst,axiom,
! [A: set_b,B: set_b,F: set_b > nat,C: nat] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_129_ord__le__eq__subst,axiom,
! [A: set_b,B: set_b,F: set_b > real,C: real] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_130_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > set_b,C: set_b] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_131_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > set_a,C: set_a] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_132_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_133_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_134_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_135_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_136_ord__eq__le__subst,axiom,
! [A: set_b,F: nat > set_b,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_137_ord__eq__le__subst,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_138_ord__eq__le__subst,axiom,
! [A: nat,F: set_b > nat,B: set_b,C: set_b] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_139_ord__eq__le__subst,axiom,
! [A: real,F: set_b > real,B: set_b,C: set_b] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_140_ord__eq__le__subst,axiom,
! [A: set_b,F: real > set_b,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_141_ord__eq__le__subst,axiom,
! [A: set_a,F: real > set_a,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_142_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_143_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_144_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_145_order__eq__refl,axiom,
! [X: set_b,Y: set_b] :
( ( X = Y )
=> ( ord_less_eq_set_b @ X @ Y ) ) ).
% order_eq_refl
thf(fact_146_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_147_order__eq__refl,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( X = Y )
=> ( ord_le8861187494160871172list_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_148_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_149_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_150_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_151_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_152_order__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_153_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_b,C: set_b] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_b @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_154_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_155_order__subst2,axiom,
! [A: set_b,B: set_b,F: set_b > nat,C: nat] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_156_order__subst2,axiom,
! [A: set_b,B: set_b,F: set_b > real,C: real] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_157_order__subst2,axiom,
! [A: real,B: real,F: real > set_b,C: set_b] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_set_b @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_158_order__subst2,axiom,
! [A: real,B: real,F: real > set_a,C: set_a] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_159_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_160_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_161_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_162_order__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_163_order__subst1,axiom,
! [A: nat,F: set_b > nat,B: set_b,C: set_b] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_164_order__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_165_order__subst1,axiom,
! [A: set_b,F: nat > set_b,B: nat,C: nat] :
( ( ord_less_eq_set_b @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_166_order__subst1,axiom,
! [A: set_b,F: real > set_b,B: real,C: real] :
( ( ord_less_eq_set_b @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_167_order__subst1,axiom,
! [A: real,F: set_b > real,B: set_b,C: set_b] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_168_order__subst1,axiom,
! [A: real,F: set_a > real,B: set_a,C: set_a] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_169_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_170_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_b,Z2: set_b] : ( Y5 = Z2 ) )
= ( ^ [A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
& ( ord_less_eq_set_b @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_171_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z2: real] : ( Y5 = Z2 ) )
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
& ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_172_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_list_a,Z2: set_list_a] : ( Y5 = Z2 ) )
= ( ^ [A3: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ B3 )
& ( ord_le8861187494160871172list_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_173_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_174_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_175_antisym,axiom,
! [A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ord_less_eq_set_b @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_176_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_177_antisym,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_178_antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_179_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_180_dual__order_Otrans,axiom,
! [B: set_b,A: set_b,C: set_b] :
( ( ord_less_eq_set_b @ B @ A )
=> ( ( ord_less_eq_set_b @ C @ B )
=> ( ord_less_eq_set_b @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_181_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_182_dual__order_Otrans,axiom,
! [B: set_list_a,A: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ A )
=> ( ( ord_le8861187494160871172list_a @ C @ B )
=> ( ord_le8861187494160871172list_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_183_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_184_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_185_dual__order_Oantisym,axiom,
! [B: set_b,A: set_b] :
( ( ord_less_eq_set_b @ B @ A )
=> ( ( ord_less_eq_set_b @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_186_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_187_dual__order_Oantisym,axiom,
! [B: set_list_a,A: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ A )
=> ( ( ord_le8861187494160871172list_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_188_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_189_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_190_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_b,Z2: set_b] : ( Y5 = Z2 ) )
= ( ^ [A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ B3 @ A3 )
& ( ord_less_eq_set_b @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_191_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: real,Z2: real] : ( Y5 = Z2 ) )
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
& ( ord_less_eq_real @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_192_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_list_a,Z2: set_list_a] : ( Y5 = Z2 ) )
= ( ^ [A3: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ A3 )
& ( ord_le8861187494160871172list_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_193_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_194_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_195_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: real,B4: real] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_196_order__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_197_order__trans,axiom,
! [X: set_b,Y: set_b,Z3: set_b] :
( ( ord_less_eq_set_b @ X @ Y )
=> ( ( ord_less_eq_set_b @ Y @ Z3 )
=> ( ord_less_eq_set_b @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_198_order__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z3 )
=> ( ord_less_eq_real @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_199_order__trans,axiom,
! [X: set_list_a,Y: set_list_a,Z3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y )
=> ( ( ord_le8861187494160871172list_a @ Y @ Z3 )
=> ( ord_le8861187494160871172list_a @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_200_order__trans,axiom,
! [X: set_a,Y: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z3 )
=> ( ord_less_eq_set_a @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_201_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_202_order_Otrans,axiom,
! [A: set_b,B: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ord_less_eq_set_b @ A @ C ) ) ) ).
% order.trans
thf(fact_203_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_204_order_Otrans,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ord_le8861187494160871172list_a @ A @ C ) ) ) ).
% order.trans
thf(fact_205_order_Otrans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_206_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_207_order__antisym,axiom,
! [X: set_b,Y: set_b] :
( ( ord_less_eq_set_b @ X @ Y )
=> ( ( ord_less_eq_set_b @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_208_order__antisym,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_209_order__antisym,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y )
=> ( ( ord_le8861187494160871172list_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_210_order__antisym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_211_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_212_ord__le__eq__trans,axiom,
! [A: set_b,B: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_b @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_213_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_214_ord__le__eq__trans,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( B = C )
=> ( ord_le8861187494160871172list_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_215_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_216_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_217_ord__eq__le__trans,axiom,
! [A: set_b,B: set_b,C: set_b] :
( ( A = B )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ord_less_eq_set_b @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_218_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_219_ord__eq__le__trans,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( A = B )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ord_le8861187494160871172list_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_220_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_221_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
& ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_222_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_b,Z2: set_b] : ( Y5 = Z2 ) )
= ( ^ [X4: set_b,Y6: set_b] :
( ( ord_less_eq_set_b @ X4 @ Y6 )
& ( ord_less_eq_set_b @ Y6 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_223_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: real,Z2: real] : ( Y5 = Z2 ) )
= ( ^ [X4: real,Y6: real] :
( ( ord_less_eq_real @ X4 @ Y6 )
& ( ord_less_eq_real @ Y6 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_224_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_list_a,Z2: set_list_a] : ( Y5 = Z2 ) )
= ( ^ [X4: set_list_a,Y6: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y6 )
& ( ord_le8861187494160871172list_a @ Y6 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_225_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [X4: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y6 )
& ( ord_less_eq_set_a @ Y6 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_226_le__cases3,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_227_le__cases3,axiom,
! [X: real,Y: real,Z3: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z3 ) )
=> ( ( ( ord_less_eq_real @ X @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z3 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z3 @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_228_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_229_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_230_singleton__uneq,axiom,
! [R2: list_a,T: dtree_list_a_b,E: b] :
( ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T @ E ) @ bot_bo2248824169281960260_a_b_b ) )
!= T ) ).
% singleton_uneq
thf(fact_231_dtree__to__list_Ocases,axiom,
! [X: dtree_list_a_b] :
( ! [R: list_a,T2: dtree_list_a_b,E2: b] :
( X
!= ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T2 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
( ! [X2: produc6499617310964463488_a_b_b] :
( Xs
!= ( finser2303212525150181944_a_b_b @ X2 @ bot_bo2248824169281960260_a_b_b ) )
=> ! [R: list_a] :
( X
!= ( node_list_a_b @ R @ Xs ) ) ) ) ).
% dtree_to_list.cases
thf(fact_232_path__lverts_Ocases,axiom,
! [X: produc6499617306661234687_a_b_a] :
( ! [R: list_a,T2: dtree_list_a_b,E2: b,X3: a] :
( X
!= ( produc7704165765595008945_a_b_a @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T2 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) @ X3 ) )
=> ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
( ! [X2: produc6499617310964463488_a_b_b] :
( Xs
!= ( finser2303212525150181944_a_b_b @ X2 @ bot_bo2248824169281960260_a_b_b ) )
=> ! [R: list_a,X3: a] :
( X
!= ( produc7704165765595008945_a_b_a @ ( node_list_a_b @ R @ Xs ) @ X3 ) ) ) ) ).
% path_lverts.cases
thf(fact_233_finsert__commute,axiom,
! [X: produc6499617310964463488_a_b_b,Y: produc6499617310964463488_a_b_b,A2: fset_P2153231429829016240_a_b_b] :
( ( finser2303212525150181944_a_b_b @ X @ ( finser2303212525150181944_a_b_b @ Y @ A2 ) )
= ( finser2303212525150181944_a_b_b @ Y @ ( finser2303212525150181944_a_b_b @ X @ A2 ) ) ) ).
% finsert_commute
thf(fact_234_bot_Oextremum__uniqueI,axiom,
! [A: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A @ bot_bo2248824169281960260_a_b_b )
=> ( A = bot_bo2248824169281960260_a_b_b ) ) ).
% bot.extremum_uniqueI
thf(fact_235_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_236_bot_Oextremum__uniqueI,axiom,
! [A: set_b] :
( ( ord_less_eq_set_b @ A @ bot_bot_set_b )
=> ( A = bot_bot_set_b ) ) ).
% bot.extremum_uniqueI
thf(fact_237_bot_Oextremum__uniqueI,axiom,
! [A: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ bot_bot_set_list_a )
=> ( A = bot_bot_set_list_a ) ) ).
% bot.extremum_uniqueI
thf(fact_238_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_239_bot_Oextremum__unique,axiom,
! [A: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A @ bot_bo2248824169281960260_a_b_b )
= ( A = bot_bo2248824169281960260_a_b_b ) ) ).
% bot.extremum_unique
thf(fact_240_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_241_bot_Oextremum__unique,axiom,
! [A: set_b] :
( ( ord_less_eq_set_b @ A @ bot_bot_set_b )
= ( A = bot_bot_set_b ) ) ).
% bot.extremum_unique
thf(fact_242_bot_Oextremum__unique,axiom,
! [A: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ bot_bot_set_list_a )
= ( A = bot_bot_set_list_a ) ) ).
% bot.extremum_unique
thf(fact_243_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_244_bot_Oextremum,axiom,
! [A: fset_P2153231429829016240_a_b_b] : ( ord_le8870638447146015504_a_b_b @ bot_bo2248824169281960260_a_b_b @ A ) ).
% bot.extremum
thf(fact_245_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_246_bot_Oextremum,axiom,
! [A: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A ) ).
% bot.extremum
thf(fact_247_bot_Oextremum,axiom,
! [A: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A ) ).
% bot.extremum
thf(fact_248_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_249_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_250_finsert__not__fempty,axiom,
! [A: produc6499617310964463488_a_b_b,A2: fset_P2153231429829016240_a_b_b] :
( ( finser2303212525150181944_a_b_b @ A @ A2 )
!= bot_bo2248824169281960260_a_b_b ) ).
% finsert_not_fempty
thf(fact_251_fsingleton__inject,axiom,
! [A: produc6499617310964463488_a_b_b,B: produc6499617310964463488_a_b_b] :
( ( ( finser2303212525150181944_a_b_b @ A @ bot_bo2248824169281960260_a_b_b )
= ( finser2303212525150181944_a_b_b @ B @ bot_bo2248824169281960260_a_b_b ) )
=> ( A = B ) ) ).
% fsingleton_inject
thf(fact_252_denormalize_Ocases,axiom,
! [X: dtree_list_a_b] :
( ! [R: list_a,T2: dtree_list_a_b,E2: b] :
( X
!= ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T2 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
( ! [X2: produc6499617310964463488_a_b_b] :
( Xs
!= ( finser2303212525150181944_a_b_b @ X2 @ bot_bo2248824169281960260_a_b_b ) )
=> ! [R: list_a] :
( X
!= ( node_list_a_b @ R @ Xs ) ) ) ) ).
% denormalize.cases
thf(fact_253_fthe__felem__eq,axiom,
! [X: produc6499617310964463488_a_b_b] :
( ( fthe_e8731401527248658499_a_b_b @ ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b ) )
= X ) ).
% fthe_felem_eq
thf(fact_254_old_Oprod_Oinject,axiom,
! [A: dtree_list_a_b,B: b,A5: dtree_list_a_b,B5: b] :
( ( ( produc7704165765595008946_a_b_b @ A @ B )
= ( produc7704165765595008946_a_b_b @ A5 @ B5 ) )
= ( ( A = A5 )
& ( B = B5 ) ) ) ).
% old.prod.inject
thf(fact_255_old_Oprod_Oinject,axiom,
! [A: nat,B: list_b,A5: nat,B5: list_b] :
( ( ( produc7903367361620597084list_b @ A @ B )
= ( produc7903367361620597084list_b @ A5 @ B5 ) )
= ( ( A = A5 )
& ( B = B5 ) ) ) ).
% old.prod.inject
thf(fact_256_old_Oprod_Oinject,axiom,
! [A: a,B: produc1943741644644106336st_b_a,A5: a,B5: produc1943741644644106336st_b_a] :
( ( ( produc7119031474978700025st_b_a @ A @ B )
= ( produc7119031474978700025st_b_a @ A5 @ B5 ) )
= ( ( A = A5 )
& ( B = B5 ) ) ) ).
% old.prod.inject
thf(fact_257_old_Oprod_Oinject,axiom,
! [A: list_b,B: a,A5: list_b,B5: a] :
( ( ( produc4145578316043568848st_b_a @ A @ B )
= ( produc4145578316043568848st_b_a @ A5 @ B5 ) )
= ( ( A = A5 )
& ( B = B5 ) ) ) ).
% old.prod.inject
thf(fact_258_old_Oprod_Oinject,axiom,
! [A: a,B: a,A5: a,B5: a] :
( ( ( product_Pair_a_a @ A @ B )
= ( product_Pair_a_a @ A5 @ B5 ) )
= ( ( A = A5 )
& ( B = B5 ) ) ) ).
% old.prod.inject
thf(fact_259_prod_Oinject,axiom,
! [X1: dtree_list_a_b,X22: b,Y1: dtree_list_a_b,Y2: b] :
( ( ( produc7704165765595008946_a_b_b @ X1 @ X22 )
= ( produc7704165765595008946_a_b_b @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_260_prod_Oinject,axiom,
! [X1: nat,X22: list_b,Y1: nat,Y2: list_b] :
( ( ( produc7903367361620597084list_b @ X1 @ X22 )
= ( produc7903367361620597084list_b @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_261_prod_Oinject,axiom,
! [X1: a,X22: produc1943741644644106336st_b_a,Y1: a,Y2: produc1943741644644106336st_b_a] :
( ( ( produc7119031474978700025st_b_a @ X1 @ X22 )
= ( produc7119031474978700025st_b_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_262_prod_Oinject,axiom,
! [X1: list_b,X22: a,Y1: list_b,Y2: a] :
( ( ( produc4145578316043568848st_b_a @ X1 @ X22 )
= ( produc4145578316043568848st_b_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_263_prod_Oinject,axiom,
! [X1: a,X22: a,Y1: a,Y2: a] :
( ( ( product_Pair_a_a @ X1 @ X22 )
= ( product_Pair_a_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_264_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N3: nat] :
( X
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_265_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_266_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_267_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_268_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_269_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_270_mk__cycles__path_Ocases,axiom,
! [X: produc7811952446676219690list_b] :
( ! [C3: list_b] :
( X
!= ( produc7903367361620597084list_b @ zero_zero_nat @ C3 ) )
=> ~ ! [N3: nat,C3: list_b] :
( X
!= ( produc7903367361620597084list_b @ ( suc @ N3 ) @ C3 ) ) ) ).
% mk_cycles_path.cases
thf(fact_271_insert__between_Ocases,axiom,
! [X: produc9008341577332299707st_a_b] :
~ ! [V2: list_a,E2: b,X3: list_a,Y3: list_a,R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
( X
!= ( produc1848684973559390389st_a_b @ V2 @ ( produc5064203622704112514st_a_b @ E2 @ ( produc673257793671328980st_a_b @ X3 @ ( produc148520996349637281st_a_b @ Y3 @ ( node_list_a_b @ R @ Xs ) ) ) ) ) ) ).
% insert_between.cases
thf(fact_272_combine_Ocases,axiom,
! [X: produc7147531718898801626st_a_b] :
~ ! [X3: list_a,Y3: list_a,R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
( X
!= ( produc673257793671328980st_a_b @ X3 @ ( produc148520996349637281st_a_b @ Y3 @ ( node_list_a_b @ R @ Xs ) ) ) ) ).
% combine.cases
thf(fact_273_dtail_Ocases,axiom,
! [X: produc1920479565126685823list_a] :
~ ! [R: list_a,Xs: fset_P2153231429829016240_a_b_b,Def: b > list_a] :
( X
!= ( produc2621617146629198007list_a @ ( node_list_a_b @ R @ Xs ) @ Def ) ) ).
% dtail.cases
thf(fact_274_is__subtree_Ocases,axiom,
! [X: produc1510363273921914569st_a_b] :
~ ! [X3: dtree_list_a_b,R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
( X
!= ( produc783528831147138817st_a_b @ X3 @ ( node_list_a_b @ R @ Xs ) ) ) ).
% is_subtree.cases
thf(fact_275_old_Oprod_Oexhaust,axiom,
! [Y: produc6499617310964463488_a_b_b] :
~ ! [A4: dtree_list_a_b,B4: b] :
( Y
!= ( produc7704165765595008946_a_b_b @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_276_old_Oprod_Oexhaust,axiom,
! [Y: produc7811952446676219690list_b] :
~ ! [A4: nat,B4: list_b] :
( Y
!= ( produc7903367361620597084list_b @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_277_old_Oprod_Oexhaust,axiom,
! [Y: produc7945266988514096265st_b_a] :
~ ! [A4: a,B4: produc1943741644644106336st_b_a] :
( Y
!= ( produc7119031474978700025st_b_a @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_278_old_Oprod_Oexhaust,axiom,
! [Y: produc1943741644644106336st_b_a] :
~ ! [A4: list_b,B4: a] :
( Y
!= ( produc4145578316043568848st_b_a @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_279_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_a_a] :
~ ! [A4: a,B4: a] :
( Y
!= ( product_Pair_a_a @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_280_surj__pair,axiom,
! [P2: produc6499617310964463488_a_b_b] :
? [X3: dtree_list_a_b,Y3: b] :
( P2
= ( produc7704165765595008946_a_b_b @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_281_surj__pair,axiom,
! [P2: produc7811952446676219690list_b] :
? [X3: nat,Y3: list_b] :
( P2
= ( produc7903367361620597084list_b @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_282_surj__pair,axiom,
! [P2: produc7945266988514096265st_b_a] :
? [X3: a,Y3: produc1943741644644106336st_b_a] :
( P2
= ( produc7119031474978700025st_b_a @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_283_surj__pair,axiom,
! [P2: produc1943741644644106336st_b_a] :
? [X3: list_b,Y3: a] :
( P2
= ( produc4145578316043568848st_b_a @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_284_surj__pair,axiom,
! [P2: product_prod_a_a] :
? [X3: a,Y3: a] :
( P2
= ( product_Pair_a_a @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_285_prod__cases,axiom,
! [P: produc6499617310964463488_a_b_b > $o,P2: produc6499617310964463488_a_b_b] :
( ! [A4: dtree_list_a_b,B4: b] : ( P @ ( produc7704165765595008946_a_b_b @ A4 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_286_prod__cases,axiom,
! [P: produc7811952446676219690list_b > $o,P2: produc7811952446676219690list_b] :
( ! [A4: nat,B4: list_b] : ( P @ ( produc7903367361620597084list_b @ A4 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_287_prod__cases,axiom,
! [P: produc7945266988514096265st_b_a > $o,P2: produc7945266988514096265st_b_a] :
( ! [A4: a,B4: produc1943741644644106336st_b_a] : ( P @ ( produc7119031474978700025st_b_a @ A4 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_288_prod__cases,axiom,
! [P: produc1943741644644106336st_b_a > $o,P2: produc1943741644644106336st_b_a] :
( ! [A4: list_b,B4: a] : ( P @ ( produc4145578316043568848st_b_a @ A4 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_289_prod__cases,axiom,
! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
( ! [A4: a,B4: a] : ( P @ ( product_Pair_a_a @ A4 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_290_Pair__inject,axiom,
! [A: dtree_list_a_b,B: b,A5: dtree_list_a_b,B5: b] :
( ( ( produc7704165765595008946_a_b_b @ A @ B )
= ( produc7704165765595008946_a_b_b @ A5 @ B5 ) )
=> ~ ( ( A = A5 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_291_Pair__inject,axiom,
! [A: nat,B: list_b,A5: nat,B5: list_b] :
( ( ( produc7903367361620597084list_b @ A @ B )
= ( produc7903367361620597084list_b @ A5 @ B5 ) )
=> ~ ( ( A = A5 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_292_Pair__inject,axiom,
! [A: a,B: produc1943741644644106336st_b_a,A5: a,B5: produc1943741644644106336st_b_a] :
( ( ( produc7119031474978700025st_b_a @ A @ B )
= ( produc7119031474978700025st_b_a @ A5 @ B5 ) )
=> ~ ( ( A = A5 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_293_Pair__inject,axiom,
! [A: list_b,B: a,A5: list_b,B5: a] :
( ( ( produc4145578316043568848st_b_a @ A @ B )
= ( produc4145578316043568848st_b_a @ A5 @ B5 ) )
=> ~ ( ( A = A5 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_294_Pair__inject,axiom,
! [A: a,B: a,A5: a,B5: a] :
( ( ( product_Pair_a_a @ A @ B )
= ( product_Pair_a_a @ A5 @ B5 ) )
=> ~ ( ( A = A5 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_295_prod__cases3,axiom,
! [Y: produc7945266988514096265st_b_a] :
~ ! [A4: a,B4: list_b,C3: a] :
( Y
!= ( produc7119031474978700025st_b_a @ A4 @ ( produc4145578316043568848st_b_a @ B4 @ C3 ) ) ) ).
% prod_cases3
thf(fact_296_prod__induct3,axiom,
! [P: produc7945266988514096265st_b_a > $o,X: produc7945266988514096265st_b_a] :
( ! [A4: a,B4: list_b,C3: a] : ( P @ ( produc7119031474978700025st_b_a @ A4 @ ( produc4145578316043568848st_b_a @ B4 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_297_fcard0__if__mdeg__le1__not__single,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
( ! [X3: produc6499617310964463488_a_b_b] :
( Xs2
!= ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
=> ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) @ one_one_nat )
=> ( ( fcard_4742106318756258927_a_b_b @ Xs2 )
= zero_zero_nat ) ) ) ).
% fcard0_if_mdeg_le1_not_single
thf(fact_298_num__leaves__singleton,axiom,
! [R2: list_a,T: dtree_list_a_b,E: b] :
( ( num_leaves_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T @ E ) @ bot_bo2248824169281960260_a_b_b ) ) )
= ( num_leaves_list_a_b @ T ) ) ).
% num_leaves_singleton
thf(fact_299_mdeg__eq__child__if__singleton__gt1,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b] :
( ( ord_less_nat @ one_one_nat @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) )
= ( max_deg_list_a_b @ T1 ) ) ) ).
% mdeg_eq_child_if_singleton_gt1
thf(fact_300_singleton__if__mdeg__le1__elem,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: produc6499617310964463488_a_b_b] :
( ( ord_less_eq_nat @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) @ one_one_nat )
=> ( ( member4695696432722591383_a_b_b @ X @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( Xs2
= ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).
% singleton_if_mdeg_le1_elem
thf(fact_301_mdeg__child__sucs__le,axiom,
! [V: list_a,T: dtree_list_a_b,R2: list_a,E: b] : ( ord_less_eq_nat @ ( max_deg_list_a_b @ ( node_list_a_b @ V @ ( sucs_list_a_b @ T ) ) ) @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T @ E ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).
% mdeg_child_sucs_le
thf(fact_302_darcs__sub__if__children__sub,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,Ys: fset_P2153231429829016240_a_b_b,R2: list_a,V: list_a] :
( ( ord_le8870638447146015504_a_b_b @ Xs2 @ Ys )
=> ( ord_less_eq_set_b @ ( darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) @ ( darcs_list_a_b @ ( node_list_a_b @ V @ Ys ) ) ) ) ).
% darcs_sub_if_children_sub
thf(fact_303_old_Ofind__pos__aux_Ocases,axiom,
! [T: dtree_list_a_b,X: produc7147531718898801626st_a_b] :
( ( old_a_b @ T )
=> ( ! [V2: list_a,P3: list_a,R: list_a,T12: dtree_list_a_b,Uu: b] :
( X
!= ( produc673257793671328980st_a_b @ V2 @ ( produc148520996349637281st_a_b @ P3 @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ Uu ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) )
=> ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
( ! [X2: produc6499617310964463488_a_b_b] :
( Xs
!= ( finser2303212525150181944_a_b_b @ X2 @ bot_bo2248824169281960260_a_b_b ) )
=> ! [V2: list_a,P3: list_a,R: list_a] :
( X
!= ( produc673257793671328980st_a_b @ V2 @ ( produc148520996349637281st_a_b @ P3 @ ( node_list_a_b @ R @ Xs ) ) ) ) ) ) ) ).
% old.find_pos_aux.cases
thf(fact_304_normalize__full__darcs__sub,axiom,
! [T1: dtree_list_a_b] : ( ord_less_eq_set_b @ ( darcs_list_a_b @ ( iKKBZ_6959927528703686640ll_a_b @ T1 ) ) @ ( darcs_list_a_b @ T1 ) ) ).
% normalize_full_darcs_sub
thf(fact_305_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_306_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_307_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_308_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_309_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_310_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_311_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_312_bot__fset_Orep__eq,axiom,
( ( fset_P9138963618725001425_a_b_b @ bot_bo2248824169281960260_a_b_b )
= bot_bo4897374000430069834_a_b_b ) ).
% bot_fset.rep_eq
thf(fact_313_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_314_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_315_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_316_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_317_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_318_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_319_less__not__refl3,axiom,
! [S4: nat,T: nat] :
( ( ord_less_nat @ S4 @ T )
=> ( S4 != T ) ) ).
% less_not_refl3
thf(fact_320_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_321_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( P @ M4 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_322_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P @ M4 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_323_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_324_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_325_lift__Suc__mono__less,axiom,
! [F: nat > real,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_real @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_326_lift__Suc__mono__less,axiom,
! [F: nat > set_b,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_set_b @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_set_b @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_327_lift__Suc__mono__less,axiom,
! [F: nat > set_list_a,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_set_list_a @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_set_list_a @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_328_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_329_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_330_lift__Suc__mono__less__iff,axiom,
! [F: nat > set_b,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_set_b @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_set_b @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_331_lift__Suc__mono__less__iff,axiom,
! [F: nat > set_list_a,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_set_list_a @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_set_list_a @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_332_fcard__not0__if__elem,axiom,
! [Xs2: fset_P9143335661316304269od_a_a] :
( ? [X2: product_prod_a_a] : ( member1426531477525435216od_a_a @ X2 @ ( fset_P2369346144816688278od_a_a @ Xs2 ) )
=> ( ( fcard_8555586194327498616od_a_a @ Xs2 )
!= zero_zero_nat ) ) ).
% fcard_not0_if_elem
thf(fact_333_fcard__not0__if__elem,axiom,
! [Xs2: fset_dtree_list_a_b] :
( ? [X2: dtree_list_a_b] : ( member551035911493665803st_a_b @ X2 @ ( fset_dtree_list_a_b2 @ Xs2 ) )
=> ( ( fcard_dtree_list_a_b @ Xs2 )
!= zero_zero_nat ) ) ).
% fcard_not0_if_elem
thf(fact_334_fcard__not0__if__elem,axiom,
! [Xs2: fset_list_a] :
( ? [X2: list_a] : ( member_list_a @ X2 @ ( fset_list_a2 @ Xs2 ) )
=> ( ( fcard_list_a @ Xs2 )
!= zero_zero_nat ) ) ).
% fcard_not0_if_elem
thf(fact_335_fcard__not0__if__elem,axiom,
! [Xs2: fset_b] :
( ? [X2: b] : ( member_b @ X2 @ ( fset_b2 @ Xs2 ) )
=> ( ( fcard_b @ Xs2 )
!= zero_zero_nat ) ) ).
% fcard_not0_if_elem
thf(fact_336_fcard__not0__if__elem,axiom,
! [Xs2: fset_a] :
( ? [X2: a] : ( member_a @ X2 @ ( fset_a2 @ Xs2 ) )
=> ( ( fcard_a @ Xs2 )
!= zero_zero_nat ) ) ).
% fcard_not0_if_elem
thf(fact_337_fcard__not0__if__elem,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b] :
( ? [X2: produc6499617310964463488_a_b_b] : ( member4695696432722591383_a_b_b @ X2 @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( fcard_4742106318756258927_a_b_b @ Xs2 )
!= zero_zero_nat ) ) ).
% fcard_not0_if_elem
thf(fact_338_lt__ex,axiom,
! [X: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X ) ).
% lt_ex
thf(fact_339_gt__ex,axiom,
! [X: nat] :
? [X_12: nat] : ( ord_less_nat @ X @ X_12 ) ).
% gt_ex
thf(fact_340_gt__ex,axiom,
! [X: real] :
? [X_12: real] : ( ord_less_real @ X @ X_12 ) ).
% gt_ex
thf(fact_341_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z: real] :
( ( ord_less_real @ X @ Z )
& ( ord_less_real @ Z @ Y ) ) ) ).
% dense
thf(fact_342_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_343_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_344_less__imp__neq,axiom,
! [X: set_b,Y: set_b] :
( ( ord_less_set_b @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_345_less__imp__neq,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( ord_less_set_list_a @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_346_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_347_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_348_order_Oasym,axiom,
! [A: set_b,B: set_b] :
( ( ord_less_set_b @ A @ B )
=> ~ ( ord_less_set_b @ B @ A ) ) ).
% order.asym
thf(fact_349_order_Oasym,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_less_set_list_a @ A @ B )
=> ~ ( ord_less_set_list_a @ B @ A ) ) ).
% order.asym
thf(fact_350_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_351_ord__eq__less__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_352_ord__eq__less__trans,axiom,
! [A: set_b,B: set_b,C: set_b] :
( ( A = B )
=> ( ( ord_less_set_b @ B @ C )
=> ( ord_less_set_b @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_353_ord__eq__less__trans,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( A = B )
=> ( ( ord_less_set_list_a @ B @ C )
=> ( ord_less_set_list_a @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_354_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_355_ord__less__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_356_ord__less__eq__trans,axiom,
! [A: set_b,B: set_b,C: set_b] :
( ( ord_less_set_b @ A @ B )
=> ( ( B = C )
=> ( ord_less_set_b @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_357_ord__less__eq__trans,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_less_set_list_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_set_list_a @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_358_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X3 )
=> ( P @ Y4 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_359_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_360_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_361_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_362_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_363_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_364_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_365_dual__order_Oasym,axiom,
! [B: set_b,A: set_b] :
( ( ord_less_set_b @ B @ A )
=> ~ ( ord_less_set_b @ A @ B ) ) ).
% dual_order.asym
thf(fact_366_dual__order_Oasym,axiom,
! [B: set_list_a,A: set_list_a] :
( ( ord_less_set_list_a @ B @ A )
=> ~ ( ord_less_set_list_a @ A @ B ) ) ).
% dual_order.asym
thf(fact_367_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_368_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_369_dual__order_Oirrefl,axiom,
! [A: set_b] :
~ ( ord_less_set_b @ A @ A ) ).
% dual_order.irrefl
thf(fact_370_dual__order_Oirrefl,axiom,
! [A: set_list_a] :
~ ( ord_less_set_list_a @ A @ A ) ).
% dual_order.irrefl
thf(fact_371_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X5: nat] : ( P4 @ X5 ) )
= ( ^ [P5: nat > $o] :
? [N4: nat] :
( ( P5 @ N4 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ~ ( P5 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_372_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_373_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B4: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: real] : ( P @ A4 @ A4 )
=> ( ! [A4: real,B4: real] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_374_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_375_order_Ostrict__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_376_order_Ostrict__trans,axiom,
! [A: set_b,B: set_b,C: set_b] :
( ( ord_less_set_b @ A @ B )
=> ( ( ord_less_set_b @ B @ C )
=> ( ord_less_set_b @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_377_order_Ostrict__trans,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_less_set_list_a @ A @ B )
=> ( ( ord_less_set_list_a @ B @ C )
=> ( ord_less_set_list_a @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_378_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_379_not__less__iff__gr__or__eq,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ( ord_less_real @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_380_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_381_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_382_dual__order_Ostrict__trans,axiom,
! [B: set_b,A: set_b,C: set_b] :
( ( ord_less_set_b @ B @ A )
=> ( ( ord_less_set_b @ C @ B )
=> ( ord_less_set_b @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_383_dual__order_Ostrict__trans,axiom,
! [B: set_list_a,A: set_list_a,C: set_list_a] :
( ( ord_less_set_list_a @ B @ A )
=> ( ( ord_less_set_list_a @ C @ B )
=> ( ord_less_set_list_a @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_384_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_385_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_386_order_Ostrict__implies__not__eq,axiom,
! [A: set_b,B: set_b] :
( ( ord_less_set_b @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_387_order_Ostrict__implies__not__eq,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_less_set_list_a @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_388_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_389_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_390_dual__order_Ostrict__implies__not__eq,axiom,
! [B: set_b,A: set_b] :
( ( ord_less_set_b @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_391_dual__order_Ostrict__implies__not__eq,axiom,
! [B: set_list_a,A: set_list_a] :
( ( ord_less_set_list_a @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_392_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_393_fset__cong,axiom,
! [X: fset_P2153231429829016240_a_b_b,Y: fset_P2153231429829016240_a_b_b] :
( ( ( fset_P9138963618725001425_a_b_b @ X )
= ( fset_P9138963618725001425_a_b_b @ Y ) )
= ( X = Y ) ) ).
% fset_cong
thf(fact_394_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_395_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_396_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_397_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_398_order__less__asym,axiom,
! [X: set_b,Y: set_b] :
( ( ord_less_set_b @ X @ Y )
=> ~ ( ord_less_set_b @ Y @ X ) ) ).
% order_less_asym
thf(fact_399_order__less__asym,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( ord_less_set_list_a @ X @ Y )
=> ~ ( ord_less_set_list_a @ Y @ X ) ) ).
% order_less_asym
thf(fact_400_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_401_linorder__neq__iff,axiom,
! [X: real,Y: real] :
( ( X != Y )
= ( ( ord_less_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_402_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_403_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_404_order__less__asym_H,axiom,
! [A: set_b,B: set_b] :
( ( ord_less_set_b @ A @ B )
=> ~ ( ord_less_set_b @ B @ A ) ) ).
% order_less_asym'
thf(fact_405_order__less__asym_H,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_less_set_list_a @ A @ B )
=> ~ ( ord_less_set_list_a @ B @ A ) ) ).
% order_less_asym'
thf(fact_406_order__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_407_order__less__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_408_order__less__trans,axiom,
! [X: set_b,Y: set_b,Z3: set_b] :
( ( ord_less_set_b @ X @ Y )
=> ( ( ord_less_set_b @ Y @ Z3 )
=> ( ord_less_set_b @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_409_order__less__trans,axiom,
! [X: set_list_a,Y: set_list_a,Z3: set_list_a] :
( ( ord_less_set_list_a @ X @ Y )
=> ( ( ord_less_set_list_a @ Y @ Z3 )
=> ( ord_less_set_list_a @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_410_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_411_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_412_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_413_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_414_ord__eq__less__subst,axiom,
! [A: set_b,F: nat > set_b,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_b @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_415_ord__eq__less__subst,axiom,
! [A: set_b,F: real > set_b,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_b @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_416_ord__eq__less__subst,axiom,
! [A: nat,F: set_b > nat,B: set_b,C: set_b] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_b @ B @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_set_b @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_417_ord__eq__less__subst,axiom,
! [A: real,F: set_b > real,B: set_b,C: set_b] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_b @ B @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_set_b @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_418_ord__eq__less__subst,axiom,
! [A: set_list_a,F: nat > set_list_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_set_list_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_list_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_419_ord__eq__less__subst,axiom,
! [A: set_list_a,F: real > set_list_a,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_set_list_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_list_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_420_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_421_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_422_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_423_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_424_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_b,C: set_b] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_b @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_425_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > set_b,C: set_b] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_b @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_426_ord__less__eq__subst,axiom,
! [A: set_b,B: set_b,F: set_b > nat,C: nat] :
( ( ord_less_set_b @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_set_b @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_427_ord__less__eq__subst,axiom,
! [A: set_b,B: set_b,F: set_b > real,C: real] :
( ( ord_less_set_b @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_set_b @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_428_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_list_a,C: set_list_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_set_list_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_list_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_429_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > set_list_a,C: set_list_a] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_set_list_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_list_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_430_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_431_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_432_order__less__irrefl,axiom,
! [X: set_b] :
~ ( ord_less_set_b @ X @ X ) ).
% order_less_irrefl
thf(fact_433_order__less__irrefl,axiom,
! [X: set_list_a] :
~ ( ord_less_set_list_a @ X @ X ) ).
% order_less_irrefl
thf(fact_434_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_435_order__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_436_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_437_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_438_order__less__subst1,axiom,
! [A: nat,F: set_b > nat,B: set_b,C: set_b] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_b @ B @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_set_b @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_439_order__less__subst1,axiom,
! [A: real,F: set_b > real,B: set_b,C: set_b] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_set_b @ B @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_set_b @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_440_order__less__subst1,axiom,
! [A: set_b,F: nat > set_b,B: nat,C: nat] :
( ( ord_less_set_b @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_b @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_441_order__less__subst1,axiom,
! [A: set_b,F: real > set_b,B: real,C: real] :
( ( ord_less_set_b @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_b @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_442_order__less__subst1,axiom,
! [A: nat,F: set_list_a > nat,B: set_list_a,C: set_list_a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_list_a @ B @ C )
=> ( ! [X3: set_list_a,Y3: set_list_a] :
( ( ord_less_set_list_a @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_443_order__less__subst1,axiom,
! [A: real,F: set_list_a > real,B: set_list_a,C: set_list_a] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_set_list_a @ B @ C )
=> ( ! [X3: set_list_a,Y3: set_list_a] :
( ( ord_less_set_list_a @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_444_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_445_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_446_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_447_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_448_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_b,C: set_b] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_set_b @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_b @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_449_order__less__subst2,axiom,
! [A: real,B: real,F: real > set_b,C: set_b] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_set_b @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_b @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_450_order__less__subst2,axiom,
! [A: set_b,B: set_b,F: set_b > nat,C: nat] :
( ( ord_less_set_b @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_set_b @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_451_order__less__subst2,axiom,
! [A: set_b,B: set_b,F: set_b > real,C: real] :
( ( ord_less_set_b @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_set_b @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_452_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_list_a,C: set_list_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_set_list_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_set_list_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_list_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_453_order__less__subst2,axiom,
! [A: real,B: real,F: real > set_list_a,C: set_list_a] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_set_list_a @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_set_list_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_list_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_454_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_455_order__less__not__sym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_456_order__less__not__sym,axiom,
! [X: set_b,Y: set_b] :
( ( ord_less_set_b @ X @ Y )
=> ~ ( ord_less_set_b @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_457_order__less__not__sym,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( ord_less_set_list_a @ X @ Y )
=> ~ ( ord_less_set_list_a @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_458_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_459_order__less__imp__triv,axiom,
! [X: real,Y: real,P: $o] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_460_order__less__imp__triv,axiom,
! [X: set_b,Y: set_b,P: $o] :
( ( ord_less_set_b @ X @ Y )
=> ( ( ord_less_set_b @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_461_order__less__imp__triv,axiom,
! [X: set_list_a,Y: set_list_a,P: $o] :
( ( ord_less_set_list_a @ X @ Y )
=> ( ( ord_less_set_list_a @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_462_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_463_linorder__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_464_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_465_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_466_order__less__imp__not__eq,axiom,
! [X: set_b,Y: set_b] :
( ( ord_less_set_b @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_467_order__less__imp__not__eq,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( ord_less_set_list_a @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_468_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_469_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_470_order__less__imp__not__eq2,axiom,
! [X: set_b,Y: set_b] :
( ( ord_less_set_b @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_471_order__less__imp__not__eq2,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( ord_less_set_list_a @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_472_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_473_order__less__imp__not__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_474_order__less__imp__not__less,axiom,
! [X: set_b,Y: set_b] :
( ( ord_less_set_b @ X @ Y )
=> ~ ( ord_less_set_b @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_475_order__less__imp__not__less,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( ord_less_set_list_a @ X @ Y )
=> ~ ( ord_less_set_list_a @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_476_fcard1__if__le1__elem,axiom,
! [Xs2: fset_P9143335661316304269od_a_a,X: product_prod_a_a] :
( ( ord_less_eq_nat @ ( fcard_8555586194327498616od_a_a @ Xs2 ) @ one_one_nat )
=> ( ( member1426531477525435216od_a_a @ X @ ( fset_P2369346144816688278od_a_a @ Xs2 ) )
=> ( ( fcard_8555586194327498616od_a_a @ Xs2 )
= one_one_nat ) ) ) ).
% fcard1_if_le1_elem
thf(fact_477_fcard1__if__le1__elem,axiom,
! [Xs2: fset_dtree_list_a_b,X: dtree_list_a_b] :
( ( ord_less_eq_nat @ ( fcard_dtree_list_a_b @ Xs2 ) @ one_one_nat )
=> ( ( member551035911493665803st_a_b @ X @ ( fset_dtree_list_a_b2 @ Xs2 ) )
=> ( ( fcard_dtree_list_a_b @ Xs2 )
= one_one_nat ) ) ) ).
% fcard1_if_le1_elem
thf(fact_478_fcard1__if__le1__elem,axiom,
! [Xs2: fset_list_a,X: list_a] :
( ( ord_less_eq_nat @ ( fcard_list_a @ Xs2 ) @ one_one_nat )
=> ( ( member_list_a @ X @ ( fset_list_a2 @ Xs2 ) )
=> ( ( fcard_list_a @ Xs2 )
= one_one_nat ) ) ) ).
% fcard1_if_le1_elem
thf(fact_479_fcard1__if__le1__elem,axiom,
! [Xs2: fset_b,X: b] :
( ( ord_less_eq_nat @ ( fcard_b @ Xs2 ) @ one_one_nat )
=> ( ( member_b @ X @ ( fset_b2 @ Xs2 ) )
=> ( ( fcard_b @ Xs2 )
= one_one_nat ) ) ) ).
% fcard1_if_le1_elem
thf(fact_480_fcard1__if__le1__elem,axiom,
! [Xs2: fset_a,X: a] :
( ( ord_less_eq_nat @ ( fcard_a @ Xs2 ) @ one_one_nat )
=> ( ( member_a @ X @ ( fset_a2 @ Xs2 ) )
=> ( ( fcard_a @ Xs2 )
= one_one_nat ) ) ) ).
% fcard1_if_le1_elem
thf(fact_481_fcard1__if__le1__elem,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,X: produc6499617310964463488_a_b_b] :
( ( ord_less_eq_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) @ one_one_nat )
=> ( ( member4695696432722591383_a_b_b @ X @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( fcard_4742106318756258927_a_b_b @ Xs2 )
= one_one_nat ) ) ) ).
% fcard1_if_le1_elem
thf(fact_482_dtree_Osel_I2_J,axiom,
! [X1: list_a,X22: fset_P2153231429829016240_a_b_b] :
( ( sucs_list_a_b @ ( node_list_a_b @ X1 @ X22 ) )
= X22 ) ).
% dtree.sel(2)
thf(fact_483_num__leaves__ge__card,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] : ( ord_less_eq_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) @ ( num_leaves_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ).
% num_leaves_ge_card
thf(fact_484_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_485_order__le__imp__less__or__eq,axiom,
! [X: set_b,Y: set_b] :
( ( ord_less_eq_set_b @ X @ Y )
=> ( ( ord_less_set_b @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_486_order__le__imp__less__or__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_487_order__le__imp__less__or__eq,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y )
=> ( ( ord_less_set_list_a @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_488_order__le__imp__less__or__eq,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_set_a @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_489_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_490_linorder__le__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_491_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_492_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_493_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_494_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_495_order__less__le__subst2,axiom,
! [A: set_b,B: set_b,F: set_b > nat,C: nat] :
( ( ord_less_set_b @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_set_b @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_496_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_b,C: set_b] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_b @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_b @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_497_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > set_b,C: set_b] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_set_b @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_b @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_498_order__less__le__subst2,axiom,
! [A: set_b,B: set_b,F: set_b > real,C: real] :
( ( ord_less_set_b @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_set_b @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_499_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_500_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > set_a,C: set_a] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_501_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_502_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_503_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_504_order__less__le__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_505_order__less__le__subst1,axiom,
! [A: set_b,F: nat > set_b,B: nat,C: nat] :
( ( ord_less_set_b @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_b @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_506_order__less__le__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_507_order__less__le__subst1,axiom,
! [A: nat,F: set_b > nat,B: set_b,C: set_b] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_508_order__less__le__subst1,axiom,
! [A: real,F: set_b > real,B: set_b,C: set_b] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_509_order__less__le__subst1,axiom,
! [A: set_b,F: real > set_b,B: real,C: real] :
( ( ord_less_set_b @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_b @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_510_order__less__le__subst1,axiom,
! [A: set_a,F: real > set_a,B: real,C: real] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_511_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_512_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_513_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_514_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_515_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_b,C: set_b] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_b @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_b @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_516_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_517_order__le__less__subst2,axiom,
! [A: set_b,B: set_b,F: set_b > nat,C: nat] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_518_order__le__less__subst2,axiom,
! [A: set_b,B: set_b,F: set_b > real,C: real] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_519_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > set_b,C: set_b] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_set_b @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_b @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_520_order__le__less__subst2,axiom,
! [A: real,B: real,F: real > set_a,C: set_a] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_521_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_522_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_523_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_524_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_525_order__le__less__subst1,axiom,
! [A: nat,F: set_b > nat,B: set_b,C: set_b] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_b @ B @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_set_b @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_526_order__le__less__subst1,axiom,
! [A: set_b,F: nat > set_b,B: nat,C: nat] :
( ( ord_less_eq_set_b @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_b @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_527_order__le__less__subst1,axiom,
! [A: set_b,F: real > set_b,B: real,C: real] :
( ( ord_less_eq_set_b @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_b @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_528_order__le__less__subst1,axiom,
! [A: real,F: set_b > real,B: set_b,C: set_b] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_set_b @ B @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_set_b @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_529_order__le__less__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_530_order__le__less__subst1,axiom,
! [A: set_a,F: real > set_a,B: real,C: real] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_531_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_532_order__less__le__trans,axiom,
! [X: set_b,Y: set_b,Z3: set_b] :
( ( ord_less_set_b @ X @ Y )
=> ( ( ord_less_eq_set_b @ Y @ Z3 )
=> ( ord_less_set_b @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_533_order__less__le__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_534_order__less__le__trans,axiom,
! [X: set_list_a,Y: set_list_a,Z3: set_list_a] :
( ( ord_less_set_list_a @ X @ Y )
=> ( ( ord_le8861187494160871172list_a @ Y @ Z3 )
=> ( ord_less_set_list_a @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_535_order__less__le__trans,axiom,
! [X: set_a,Y: set_a,Z3: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z3 )
=> ( ord_less_set_a @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_536_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_537_order__le__less__trans,axiom,
! [X: set_b,Y: set_b,Z3: set_b] :
( ( ord_less_eq_set_b @ X @ Y )
=> ( ( ord_less_set_b @ Y @ Z3 )
=> ( ord_less_set_b @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_538_order__le__less__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_539_order__le__less__trans,axiom,
! [X: set_list_a,Y: set_list_a,Z3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y )
=> ( ( ord_less_set_list_a @ Y @ Z3 )
=> ( ord_less_set_list_a @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_540_order__le__less__trans,axiom,
! [X: set_a,Y: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ Z3 )
=> ( ord_less_set_a @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_541_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_542_order__neq__le__trans,axiom,
! [A: set_b,B: set_b] :
( ( A != B )
=> ( ( ord_less_eq_set_b @ A @ B )
=> ( ord_less_set_b @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_543_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_544_order__neq__le__trans,axiom,
! [A: set_list_a,B: set_list_a] :
( ( A != B )
=> ( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ord_less_set_list_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_545_order__neq__le__trans,axiom,
! [A: set_a,B: set_a] :
( ( A != B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_546_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_547_order__le__neq__trans,axiom,
! [A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_b @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_548_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_549_order__le__neq__trans,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_list_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_550_order__le__neq__trans,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_551_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_552_order__less__imp__le,axiom,
! [X: set_b,Y: set_b] :
( ( ord_less_set_b @ X @ Y )
=> ( ord_less_eq_set_b @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_553_order__less__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_554_order__less__imp__le,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( ord_less_set_list_a @ X @ Y )
=> ( ord_le8861187494160871172list_a @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_555_order__less__imp__le,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_556_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_557_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_558_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_559_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_560_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
& ( X4 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_561_order__less__le,axiom,
( ord_less_set_b
= ( ^ [X4: set_b,Y6: set_b] :
( ( ord_less_eq_set_b @ X4 @ Y6 )
& ( X4 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_562_order__less__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y6: real] :
( ( ord_less_eq_real @ X4 @ Y6 )
& ( X4 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_563_order__less__le,axiom,
( ord_less_set_list_a
= ( ^ [X4: set_list_a,Y6: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y6 )
& ( X4 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_564_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X4: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y6 )
& ( X4 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_565_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y6: nat] :
( ( ord_less_nat @ X4 @ Y6 )
| ( X4 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_566_order__le__less,axiom,
( ord_less_eq_set_b
= ( ^ [X4: set_b,Y6: set_b] :
( ( ord_less_set_b @ X4 @ Y6 )
| ( X4 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_567_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y6: real] :
( ( ord_less_real @ X4 @ Y6 )
| ( X4 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_568_order__le__less,axiom,
( ord_le8861187494160871172list_a
= ( ^ [X4: set_list_a,Y6: set_list_a] :
( ( ord_less_set_list_a @ X4 @ Y6 )
| ( X4 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_569_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X4: set_a,Y6: set_a] :
( ( ord_less_set_a @ X4 @ Y6 )
| ( X4 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_570_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_571_dual__order_Ostrict__implies__order,axiom,
! [B: set_b,A: set_b] :
( ( ord_less_set_b @ B @ A )
=> ( ord_less_eq_set_b @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_572_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_573_dual__order_Ostrict__implies__order,axiom,
! [B: set_list_a,A: set_list_a] :
( ( ord_less_set_list_a @ B @ A )
=> ( ord_le8861187494160871172list_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_574_dual__order_Ostrict__implies__order,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_575_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_576_order_Ostrict__implies__order,axiom,
! [A: set_b,B: set_b] :
( ( ord_less_set_b @ A @ B )
=> ( ord_less_eq_set_b @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_577_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_578_order_Ostrict__implies__order,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_less_set_list_a @ A @ B )
=> ( ord_le8861187494160871172list_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_579_order_Ostrict__implies__order,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_580_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_581_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_b
= ( ^ [B3: set_b,A3: set_b] :
( ( ord_less_eq_set_b @ B3 @ A3 )
& ~ ( ord_less_eq_set_b @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_582_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B3: real,A3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
& ~ ( ord_less_eq_real @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_583_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_list_a
= ( ^ [B3: set_list_a,A3: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ A3 )
& ~ ( ord_le8861187494160871172list_a @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_584_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ~ ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_585_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_586_dual__order_Ostrict__trans2,axiom,
! [B: set_b,A: set_b,C: set_b] :
( ( ord_less_set_b @ B @ A )
=> ( ( ord_less_eq_set_b @ C @ B )
=> ( ord_less_set_b @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_587_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_588_dual__order_Ostrict__trans2,axiom,
! [B: set_list_a,A: set_list_a,C: set_list_a] :
( ( ord_less_set_list_a @ B @ A )
=> ( ( ord_le8861187494160871172list_a @ C @ B )
=> ( ord_less_set_list_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_589_dual__order_Ostrict__trans2,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_590_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_591_dual__order_Ostrict__trans1,axiom,
! [B: set_b,A: set_b,C: set_b] :
( ( ord_less_eq_set_b @ B @ A )
=> ( ( ord_less_set_b @ C @ B )
=> ( ord_less_set_b @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_592_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_593_dual__order_Ostrict__trans1,axiom,
! [B: set_list_a,A: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ A )
=> ( ( ord_less_set_list_a @ C @ B )
=> ( ord_less_set_list_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_594_dual__order_Ostrict__trans1,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_595_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_596_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_b
= ( ^ [B3: set_b,A3: set_b] :
( ( ord_less_eq_set_b @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_597_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B3: real,A3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_598_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_list_a
= ( ^ [B3: set_list_a,A3: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_599_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_600_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_601_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_b
= ( ^ [B3: set_b,A3: set_b] :
( ( ord_less_set_b @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_602_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A3: real] :
( ( ord_less_real @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_603_dual__order_Oorder__iff__strict,axiom,
( ord_le8861187494160871172list_a
= ( ^ [B3: set_list_a,A3: set_list_a] :
( ( ord_less_set_list_a @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_604_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A3: set_a] :
( ( ord_less_set_a @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_605_dense__le__bounded,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z3 ) ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ) ).
% dense_le_bounded
thf(fact_606_dense__ge__bounded,axiom,
! [Z3: real,X: real,Y: real] :
( ( ord_less_real @ Z3 @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z3 @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ) ).
% dense_ge_bounded
thf(fact_607_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_608_order_Ostrict__iff__not,axiom,
( ord_less_set_b
= ( ^ [A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
& ~ ( ord_less_eq_set_b @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_609_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
& ~ ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_610_order_Ostrict__iff__not,axiom,
( ord_less_set_list_a
= ( ^ [A3: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ B3 )
& ~ ( ord_le8861187494160871172list_a @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_611_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ~ ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_612_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_613_order_Ostrict__trans2,axiom,
! [A: set_b,B: set_b,C: set_b] :
( ( ord_less_set_b @ A @ B )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ord_less_set_b @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_614_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_615_order_Ostrict__trans2,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_less_set_list_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ord_less_set_list_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_616_order_Ostrict__trans2,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_617_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_618_order_Ostrict__trans1,axiom,
! [A: set_b,B: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ord_less_set_b @ B @ C )
=> ( ord_less_set_b @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_619_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_620_order_Ostrict__trans1,axiom,
! [A: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_less_set_list_a @ B @ C )
=> ( ord_less_set_list_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_621_order_Ostrict__trans1,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_622_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_623_order_Ostrict__iff__order,axiom,
( ord_less_set_b
= ( ^ [A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_624_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_625_order_Ostrict__iff__order,axiom,
( ord_less_set_list_a
= ( ^ [A3: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_626_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_627_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_628_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_b
= ( ^ [A3: set_b,B3: set_b] :
( ( ord_less_set_b @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_629_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_630_order_Oorder__iff__strict,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A3: set_list_a,B3: set_list_a] :
( ( ord_less_set_list_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_631_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_632_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_633_not__le__imp__less,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_eq_real @ Y @ X )
=> ( ord_less_real @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_634_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
& ~ ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_635_less__le__not__le,axiom,
( ord_less_set_b
= ( ^ [X4: set_b,Y6: set_b] :
( ( ord_less_eq_set_b @ X4 @ Y6 )
& ~ ( ord_less_eq_set_b @ Y6 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_636_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y6: real] :
( ( ord_less_eq_real @ X4 @ Y6 )
& ~ ( ord_less_eq_real @ Y6 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_637_less__le__not__le,axiom,
( ord_less_set_list_a
= ( ^ [X4: set_list_a,Y6: set_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ Y6 )
& ~ ( ord_le8861187494160871172list_a @ Y6 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_638_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X4: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y6 )
& ~ ( ord_less_eq_set_a @ Y6 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_639_dense__le,axiom,
! [Y: real,Z3: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_eq_real @ X3 @ Z3 ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ).
% dense_le
thf(fact_640_dense__ge,axiom,
! [Z3: real,Y: real] :
( ! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ord_less_eq_real @ Y @ X3 ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ).
% dense_ge
thf(fact_641_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_642_antisym__conv2,axiom,
! [X: set_b,Y: set_b] :
( ( ord_less_eq_set_b @ X @ Y )
=> ( ( ~ ( ord_less_set_b @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_643_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_644_antisym__conv2,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y )
=> ( ( ~ ( ord_less_set_list_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_645_antisym__conv2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ~ ( ord_less_set_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_646_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_647_antisym__conv1,axiom,
! [X: set_b,Y: set_b] :
( ~ ( ord_less_set_b @ X @ Y )
=> ( ( ord_less_eq_set_b @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_648_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_649_antisym__conv1,axiom,
! [X: set_list_a,Y: set_list_a] :
( ~ ( ord_less_set_list_a @ X @ Y )
=> ( ( ord_le8861187494160871172list_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_650_antisym__conv1,axiom,
! [X: set_a,Y: set_a] :
( ~ ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_651_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_652_nless__le,axiom,
! [A: set_b,B: set_b] :
( ( ~ ( ord_less_set_b @ A @ B ) )
= ( ~ ( ord_less_eq_set_b @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_653_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_654_nless__le,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ~ ( ord_less_set_list_a @ A @ B ) )
= ( ~ ( ord_le8861187494160871172list_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_655_nless__le,axiom,
! [A: set_a,B: set_a] :
( ( ~ ( ord_less_set_a @ A @ B ) )
= ( ~ ( ord_less_eq_set_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_656_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_657_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_658_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_659_leD,axiom,
! [Y: set_b,X: set_b] :
( ( ord_less_eq_set_b @ Y @ X )
=> ~ ( ord_less_set_b @ X @ Y ) ) ).
% leD
thf(fact_660_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_661_leD,axiom,
! [Y: set_list_a,X: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y @ X )
=> ~ ( ord_less_set_list_a @ X @ Y ) ) ).
% leD
thf(fact_662_leD,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ~ ( ord_less_set_a @ X @ Y ) ) ).
% leD
thf(fact_663_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_664_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_665_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_666_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_667_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_668_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_669_less__eq__fset_Orep__eq,axiom,
( ord_le8870638447146015504_a_b_b
= ( ^ [X4: fset_P2153231429829016240_a_b_b,Xa: fset_P2153231429829016240_a_b_b] : ( ord_le1619362961161175062_a_b_b @ ( fset_P9138963618725001425_a_b_b @ X4 ) @ ( fset_P9138963618725001425_a_b_b @ Xa ) ) ) ) ).
% less_eq_fset.rep_eq
thf(fact_670_less__eq__fset_Orep__eq,axiom,
( ord_less_eq_fset_b
= ( ^ [X4: fset_b,Xa: fset_b] : ( ord_less_eq_set_b @ ( fset_b2 @ X4 ) @ ( fset_b2 @ Xa ) ) ) ) ).
% less_eq_fset.rep_eq
thf(fact_671_less__eq__fset_Orep__eq,axiom,
( ord_le510749213327159946list_a
= ( ^ [X4: fset_list_a,Xa: fset_list_a] : ( ord_le8861187494160871172list_a @ ( fset_list_a2 @ X4 ) @ ( fset_list_a2 @ Xa ) ) ) ) ).
% less_eq_fset.rep_eq
thf(fact_672_less__eq__fset_Orep__eq,axiom,
( ord_less_eq_fset_a
= ( ^ [X4: fset_a,Xa: fset_a] : ( ord_less_eq_set_a @ ( fset_a2 @ X4 ) @ ( fset_a2 @ Xa ) ) ) ) ).
% less_eq_fset.rep_eq
thf(fact_673_bot_Onot__eq__extremum,axiom,
! [A: fset_P2153231429829016240_a_b_b] :
( ( A != bot_bo2248824169281960260_a_b_b )
= ( ord_le6631730213922513156_a_b_b @ bot_bo2248824169281960260_a_b_b @ A ) ) ).
% bot.not_eq_extremum
thf(fact_674_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_675_bot_Onot__eq__extremum,axiom,
! [A: set_b] :
( ( A != bot_bot_set_b )
= ( ord_less_set_b @ bot_bot_set_b @ A ) ) ).
% bot.not_eq_extremum
thf(fact_676_bot_Onot__eq__extremum,axiom,
! [A: set_list_a] :
( ( A != bot_bot_set_list_a )
= ( ord_less_set_list_a @ bot_bot_set_list_a @ A ) ) ).
% bot.not_eq_extremum
thf(fact_677_bot_Oextremum__strict,axiom,
! [A: fset_P2153231429829016240_a_b_b] :
~ ( ord_le6631730213922513156_a_b_b @ A @ bot_bo2248824169281960260_a_b_b ) ).
% bot.extremum_strict
thf(fact_678_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_679_bot_Oextremum__strict,axiom,
! [A: set_b] :
~ ( ord_less_set_b @ A @ bot_bot_set_b ) ).
% bot.extremum_strict
thf(fact_680_bot_Oextremum__strict,axiom,
! [A: set_list_a] :
~ ( ord_less_set_list_a @ A @ bot_bot_set_list_a ) ).
% bot.extremum_strict
thf(fact_681_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_682_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_683_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_684_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_685_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_686_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_687_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_688_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_689_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_690_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_691_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_692_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_693_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_694_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_695_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_696_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M6: nat] :
( ( M
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_697_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_698_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_699_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_700_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_701_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_702_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_703_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_704_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_705_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_706_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_707_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_708_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_709_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_710_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_nat @ M5 @ N4 )
| ( M5 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_711_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_712_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
& ( M5 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_713_nempty__if__mdeg__gt__fcard,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
( ( ord_less_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
=> ( Xs2 != bot_bo2248824169281960260_a_b_b ) ) ).
% nempty_if_mdeg_gt_fcard
thf(fact_714_old_Omerge_Ocases,axiom,
! [T: dtree_list_a_b,X: dtree_list_a_b] :
( ( old_a_b @ T )
=> ~ ! [R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
( X
!= ( node_list_a_b @ R @ Xs ) ) ) ).
% old.merge.cases
thf(fact_715_disjoint__darcs__if__wf__aux3,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,T1: dtree_list_a_b,E12: b,T22: dtree_list_a_b,E22: b] :
( ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ~ ( member_b @ E22 @ ( darcs_list_a_b @ T1 ) ) ) ) ) ).
% disjoint_darcs_if_wf_aux3
thf(fact_716_disjoint__darcs__if__wf__aux1,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,T1: dtree_list_a_b,E12: b] :
( ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ~ ( member_b @ E12 @ ( darcs_list_a_b @ T1 ) ) ) ) ).
% disjoint_darcs_if_wf_aux1
thf(fact_717_singleton__if__fcard__le1__elem,axiom,
! [Xs2: fset_P9143335661316304269od_a_a,X: product_prod_a_a] :
( ( ord_less_eq_nat @ ( fcard_8555586194327498616od_a_a @ Xs2 ) @ one_one_nat )
=> ( ( member1426531477525435216od_a_a @ X @ ( fset_P2369346144816688278od_a_a @ Xs2 ) )
=> ( Xs2
= ( finser5399165410198402159od_a_a @ X @ bot_bo686543689707597305od_a_a ) ) ) ) ).
% singleton_if_fcard_le1_elem
thf(fact_718_singleton__if__fcard__le1__elem,axiom,
! [Xs2: fset_dtree_list_a_b,X: dtree_list_a_b] :
( ( ord_less_eq_nat @ ( fcard_dtree_list_a_b @ Xs2 ) @ one_one_nat )
=> ( ( member551035911493665803st_a_b @ X @ ( fset_dtree_list_a_b2 @ Xs2 ) )
=> ( Xs2
= ( finser8636396436308191404st_a_b @ X @ bot_bo4748119319284029112st_a_b ) ) ) ) ).
% singleton_if_fcard_le1_elem
thf(fact_719_singleton__if__fcard__le1__elem,axiom,
! [Xs2: fset_list_a,X: list_a] :
( ( ord_less_eq_nat @ ( fcard_list_a @ Xs2 ) @ one_one_nat )
=> ( ( member_list_a @ X @ ( fset_list_a2 @ Xs2 ) )
=> ( Xs2
= ( finsert_list_a @ X @ bot_bot_fset_list_a ) ) ) ) ).
% singleton_if_fcard_le1_elem
thf(fact_720_singleton__if__fcard__le1__elem,axiom,
! [Xs2: fset_b,X: b] :
( ( ord_less_eq_nat @ ( fcard_b @ Xs2 ) @ one_one_nat )
=> ( ( member_b @ X @ ( fset_b2 @ Xs2 ) )
=> ( Xs2
= ( finsert_b @ X @ bot_bot_fset_b ) ) ) ) ).
% singleton_if_fcard_le1_elem
thf(fact_721_singleton__if__fcard__le1__elem,axiom,
! [Xs2: fset_a,X: a] :
( ( ord_less_eq_nat @ ( fcard_a @ Xs2 ) @ one_one_nat )
=> ( ( member_a @ X @ ( fset_a2 @ Xs2 ) )
=> ( Xs2
= ( finsert_a @ X @ bot_bot_fset_a ) ) ) ) ).
% singleton_if_fcard_le1_elem
thf(fact_722_singleton__if__fcard__le1__elem,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,X: produc6499617310964463488_a_b_b] :
( ( ord_less_eq_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) @ one_one_nat )
=> ( ( member4695696432722591383_a_b_b @ X @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( Xs2
= ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).
% singleton_if_fcard_le1_elem
thf(fact_723_num__leaves__root,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,R4: list_a] :
( ( num_leaves_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
= ( num_leaves_list_a_b @ ( node_list_a_b @ R4 @ Xs2 ) ) ) ).
% num_leaves_root
thf(fact_724_wf__darcs__sucs,axiom,
! [T: dtree_list_a_b,X: produc6499617310964463488_a_b_b,R2: list_a] :
( ( wf_darcs_list_a_b @ T )
=> ( ( member4695696432722591383_a_b_b @ X @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T ) ) )
=> ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ).
% wf_darcs_sucs
thf(fact_725_fcard__fempty,axiom,
( ( fcard_4742106318756258927_a_b_b @ bot_bo2248824169281960260_a_b_b )
= zero_zero_nat ) ).
% fcard_fempty
thf(fact_726_fset__card__induct,axiom,
! [P: fset_P2153231429829016240_a_b_b > $o,S: fset_P2153231429829016240_a_b_b] :
( ( P @ bot_bo2248824169281960260_a_b_b )
=> ( ! [S2: fset_P2153231429829016240_a_b_b,T3: fset_P2153231429829016240_a_b_b] :
( ( ( suc @ ( fcard_4742106318756258927_a_b_b @ S2 ) )
= ( fcard_4742106318756258927_a_b_b @ T3 ) )
=> ( ( P @ S2 )
=> ( P @ T3 ) ) )
=> ( P @ S ) ) ) ).
% fset_card_induct
thf(fact_727_fcard__finsert__le,axiom,
! [A2: fset_P2153231429829016240_a_b_b,X: produc6499617310964463488_a_b_b] : ( ord_less_eq_nat @ ( fcard_4742106318756258927_a_b_b @ A2 ) @ ( fcard_4742106318756258927_a_b_b @ ( finser2303212525150181944_a_b_b @ X @ A2 ) ) ) ).
% fcard_finsert_le
thf(fact_728_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_729_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_730_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_731_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_732_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_733_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_734_fcard__mono,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B2: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( fcard_4742106318756258927_a_b_b @ A2 ) @ ( fcard_4742106318756258927_a_b_b @ B2 ) ) ) ).
% fcard_mono
thf(fact_735_fcard__seteq,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B2: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( fcard_4742106318756258927_a_b_b @ B2 ) @ ( fcard_4742106318756258927_a_b_b @ A2 ) )
=> ( A2 = B2 ) ) ) ).
% fcard_seteq
thf(fact_736_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_737_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_738_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_739_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_740_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_741_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_742_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_743_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_744_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_745_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_746_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_747_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_748_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_749_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_750_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_751_singleton__if__mdeg__le1__elem__suc,axiom,
! [T: dtree_list_a_b,X: produc6499617310964463488_a_b_b] :
( ( ord_less_eq_nat @ ( max_deg_list_a_b @ T ) @ one_one_nat )
=> ( ( member4695696432722591383_a_b_b @ X @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T ) ) )
=> ( ( sucs_list_a_b @ T )
= ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).
% singleton_if_mdeg_le1_elem_suc
thf(fact_752_fcard0__if__mdeg__le1__not__single__suc,axiom,
! [T: dtree_list_a_b] :
( ! [X3: produc6499617310964463488_a_b_b] :
( ( sucs_list_a_b @ T )
!= ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
=> ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ T ) @ one_one_nat )
=> ( ( fcard_4742106318756258927_a_b_b @ ( sucs_list_a_b @ T ) )
= zero_zero_nat ) ) ) ).
% fcard0_if_mdeg_le1_not_single_suc
thf(fact_753_mdeg0__if__fcard0,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
( ( ( fcard_4742106318756258927_a_b_b @ Xs2 )
= zero_zero_nat )
=> ( ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
= zero_zero_nat ) ) ).
% mdeg0_if_fcard0
thf(fact_754_mdeg0__iff__fcard0,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
( ( ( fcard_4742106318756258927_a_b_b @ Xs2 )
= zero_zero_nat )
= ( ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
= zero_zero_nat ) ) ).
% mdeg0_iff_fcard0
thf(fact_755_mdeg__ge__fcard,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] : ( ord_less_eq_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ).
% mdeg_ge_fcard
thf(fact_756_mdeg__eq__fcard__if__empty,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
( ( Xs2 = bot_bo2248824169281960260_a_b_b )
=> ( ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
= ( fcard_4742106318756258927_a_b_b @ Xs2 ) ) ) ).
% mdeg_eq_fcard_if_empty
thf(fact_757_mdeg__child__sucs__eq__if__gt1,axiom,
! [R2: list_a,T: dtree_list_a_b,E: b,V: list_a] :
( ( ord_less_nat @ one_one_nat @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T @ E ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T @ E ) @ bot_bo2248824169281960260_a_b_b ) ) )
= ( max_deg_list_a_b @ ( node_list_a_b @ V @ ( sucs_list_a_b @ T ) ) ) ) ) ).
% mdeg_child_sucs_eq_if_gt1
thf(fact_758_fcard__single__1__iff,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b] :
( ( ( fcard_4742106318756258927_a_b_b @ Xs2 )
= one_one_nat )
= ( ? [X4: produc6499617310964463488_a_b_b] :
( Xs2
= ( finser2303212525150181944_a_b_b @ X4 @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).
% fcard_single_1_iff
thf(fact_759_fcard__single__1,axiom,
! [X: produc6499617310964463488_a_b_b] :
( ( fcard_4742106318756258927_a_b_b @ ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b ) )
= one_one_nat ) ).
% fcard_single_1
thf(fact_760_disjoint__darcs__if__wf__aux4,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,T1: dtree_list_a_b,E12: b,T22: dtree_list_a_b,E22: b] :
( ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( ( produc7704165765595008946_a_b_b @ T1 @ E12 )
!= ( produc7704165765595008946_a_b_b @ T22 @ E22 ) )
=> ( E12 != E22 ) ) ) ) ) ).
% disjoint_darcs_if_wf_aux4
thf(fact_761_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_762_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_763_num__leaves__ge1,axiom,
! [T: dtree_list_a_b] : ( ord_less_eq_nat @ one_one_nat @ ( num_leaves_list_a_b @ T ) ) ).
% num_leaves_ge1
thf(fact_764_singleton__uneq_H,axiom,
! [R2: list_a,T: dtree_list_a_b,E: b,V: list_a] :
( ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T @ E ) @ bot_bo2248824169281960260_a_b_b ) )
!= ( node_list_a_b @ V @ ( sucs_list_a_b @ T ) ) ) ).
% singleton_uneq'
thf(fact_765_empty__fset__if__fcard__le1__not__singleton,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b] :
( ! [X3: produc6499617310964463488_a_b_b] :
( Xs2
!= ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
=> ( ( ord_less_eq_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) @ one_one_nat )
=> ( Xs2 = bot_bo2248824169281960260_a_b_b ) ) ) ).
% empty_fset_if_fcard_le1_not_singleton
thf(fact_766_mdeg__ge__child,axiom,
! [T1: dtree_list_a_b,E12: b,Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ord_less_eq_nat @ ( max_deg_list_a_b @ T1 ) @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ).
% mdeg_ge_child
thf(fact_767_mdeg__gt__0__if__nempty,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
( ( Xs2 != bot_bo2248824169281960260_a_b_b )
=> ( ord_less_nat @ zero_zero_nat @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ).
% mdeg_gt_0_if_nempty
thf(fact_768_fcard0__if__le1__not__singleton,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b] :
( ! [X3: produc6499617310964463488_a_b_b] :
( Xs2
!= ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
=> ( ( ord_less_eq_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) @ one_one_nat )
=> ( ( fcard_4742106318756258927_a_b_b @ Xs2 )
= zero_zero_nat ) ) ) ).
% fcard0_if_le1_not_singleton
thf(fact_769_num__leaves__1__if__mdeg__1,axiom,
! [T: dtree_list_a_b] :
( ( ord_less_eq_nat @ ( max_deg_list_a_b @ T ) @ one_one_nat )
=> ( ( num_leaves_list_a_b @ T )
= one_one_nat ) ) ).
% num_leaves_1_if_mdeg_1
thf(fact_770_empty__fset__if__mdeg__le1__not__single__suc,axiom,
! [T: dtree_list_a_b] :
( ! [X3: produc6499617310964463488_a_b_b] :
( ( sucs_list_a_b @ T )
!= ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
=> ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ T ) @ one_one_nat )
=> ( ( sucs_list_a_b @ T )
= bot_bo2248824169281960260_a_b_b ) ) ) ).
% empty_fset_if_mdeg_le1_not_single_suc
thf(fact_771_old_Ofind__pos_Ocases,axiom,
! [T: dtree_list_a_b,X: produc111314985273491367st_a_b] :
( ( old_a_b @ T )
=> ( ! [V2: list_a,R: list_a,T12: dtree_list_a_b,Uu: b] :
( X
!= ( produc148520996349637281st_a_b @ V2 @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ Uu ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
( ! [X2: produc6499617310964463488_a_b_b] :
( Xs
!= ( finser2303212525150181944_a_b_b @ X2 @ bot_bo2248824169281960260_a_b_b ) )
=> ! [V2: list_a,R: list_a] :
( X
!= ( produc148520996349637281st_a_b @ V2 @ ( node_list_a_b @ R @ Xs ) ) ) ) ) ) ).
% old.find_pos.cases
thf(fact_772_dtree__size__skip__decr,axiom,
! [R2: list_a,T1: dtree_list_a_b,V: list_a,E12: b] : ( ord_less_nat @ ( size_s415192292648992904st_a_b @ ( node_list_a_b @ R2 @ ( sucs_list_a_b @ T1 ) ) ) @ ( size_s415192292648992904st_a_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).
% dtree_size_skip_decr
thf(fact_773_child__mdeg__gt1__if__sub__fcard__gt1,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,V: list_a,Ys: fset_P2153231429829016240_a_b_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ ( node_list_a_b @ V @ Ys ) )
=> ( ( ( node_list_a_b @ R2 @ Xs2 )
!= ( node_list_a_b @ V @ Ys ) )
=> ( ( ord_less_nat @ one_one_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) )
=> ? [T12: dtree_list_a_b,E23: b] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E23 ) @ ( fset_P9138963618725001425_a_b_b @ Ys ) )
& ( ord_less_nat @ one_one_nat @ ( max_deg_list_a_b @ T12 ) ) ) ) ) ) ).
% child_mdeg_gt1_if_sub_fcard_gt1
thf(fact_774_darcs__combine__sub__orig,axiom,
! [X: list_a,Y: list_a,T1: dtree_list_a_b] : ( ord_less_eq_set_b @ ( darcs_list_a_b @ ( list_combine_a_b @ X @ Y @ T1 ) ) @ ( darcs_list_a_b @ T1 ) ) ).
% darcs_combine_sub_orig
thf(fact_775_subsetI,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A2 )
=> ( member1426531477525435216od_a_a @ X3 @ B2 ) )
=> ( ord_le746702958409616551od_a_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_776_subsetI,axiom,
! [A2: set_dtree_list_a_b,B2: set_dtree_list_a_b] :
( ! [X3: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X3 @ A2 )
=> ( member551035911493665803st_a_b @ X3 @ B2 ) )
=> ( ord_le7599451563663638410st_a_b @ A2 @ B2 ) ) ).
% subsetI
thf(fact_777_subsetI,axiom,
! [A2: set_b,B2: set_b] :
( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( member_b @ X3 @ B2 ) )
=> ( ord_less_eq_set_b @ A2 @ B2 ) ) ).
% subsetI
thf(fact_778_subsetI,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ( member_list_a @ X3 @ B2 ) )
=> ( ord_le8861187494160871172list_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_779_subsetI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_780_subset__empty,axiom,
! [A2: set_b] :
( ( ord_less_eq_set_b @ A2 @ bot_bot_set_b )
= ( A2 = bot_bot_set_b ) ) ).
% subset_empty
thf(fact_781_subset__empty,axiom,
! [A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ bot_bot_set_list_a )
= ( A2 = bot_bot_set_list_a ) ) ).
% subset_empty
thf(fact_782_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_783_empty__subsetI,axiom,
! [A2: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A2 ) ).
% empty_subsetI
thf(fact_784_empty__subsetI,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A2 ) ).
% empty_subsetI
thf(fact_785_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_786_subset__antisym,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_787_subset__antisym,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_788_subset__antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_789_old_Ofind__pos_Osimps_I2_J,axiom,
! [T: dtree_list_a_b,Xs2: fset_P2153231429829016240_a_b_b,Rank: list_a > real,V: list_a,R2: list_a] :
( ( old_a_b @ T )
=> ( ! [X3: produc6499617310964463488_a_b_b] :
( Xs2
!= ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
=> ( ( find_pos_a_b @ Rank @ V @ ( node_list_a_b @ R2 @ Xs2 ) )
= ( produc6837034575241423639list_a @ R2 @ R2 ) ) ) ) ).
% old.find_pos.simps(2)
thf(fact_790_all__not__in__conv,axiom,
! [A2: set_Product_prod_a_a] :
( ( ! [X4: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ X4 @ A2 ) )
= ( A2 = bot_bo3357376287454694259od_a_a ) ) ).
% all_not_in_conv
thf(fact_791_all__not__in__conv,axiom,
! [A2: set_dtree_list_a_b] :
( ( ! [X4: dtree_list_a_b] :
~ ( member551035911493665803st_a_b @ X4 @ A2 ) )
= ( A2 = bot_bo798015271861357502st_a_b ) ) ).
% all_not_in_conv
thf(fact_792_all__not__in__conv,axiom,
! [A2: set_list_a] :
( ( ! [X4: list_a] :
~ ( member_list_a @ X4 @ A2 ) )
= ( A2 = bot_bot_set_list_a ) ) ).
% all_not_in_conv
thf(fact_793_all__not__in__conv,axiom,
! [A2: set_b] :
( ( ! [X4: b] :
~ ( member_b @ X4 @ A2 ) )
= ( A2 = bot_bot_set_b ) ) ).
% all_not_in_conv
thf(fact_794_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X4: a] :
~ ( member_a @ X4 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_795_empty__iff,axiom,
! [C: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ C @ bot_bo3357376287454694259od_a_a ) ).
% empty_iff
thf(fact_796_empty__iff,axiom,
! [C: dtree_list_a_b] :
~ ( member551035911493665803st_a_b @ C @ bot_bo798015271861357502st_a_b ) ).
% empty_iff
thf(fact_797_empty__iff,axiom,
! [C: list_a] :
~ ( member_list_a @ C @ bot_bot_set_list_a ) ).
% empty_iff
thf(fact_798_empty__iff,axiom,
! [C: b] :
~ ( member_b @ C @ bot_bot_set_b ) ).
% empty_iff
thf(fact_799_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_800_psubsetI,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_b @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_801_psubsetI,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_list_a @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_802_psubsetI,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_803_psubsetE,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_set_b @ A2 @ B2 )
=> ~ ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ord_less_eq_set_b @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_804_psubsetE,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( ord_less_set_list_a @ A2 @ B2 )
=> ~ ( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ord_le8861187494160871172list_a @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_805_psubsetE,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_806_psubset__eq,axiom,
( ord_less_set_b
= ( ^ [A6: set_b,B6: set_b] :
( ( ord_less_eq_set_b @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_807_psubset__eq,axiom,
( ord_less_set_list_a
= ( ^ [A6: set_list_a,B6: set_list_a] :
( ( ord_le8861187494160871172list_a @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_808_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_809_not__psubset__empty,axiom,
! [A2: set_b] :
~ ( ord_less_set_b @ A2 @ bot_bot_set_b ) ).
% not_psubset_empty
thf(fact_810_not__psubset__empty,axiom,
! [A2: set_list_a] :
~ ( ord_less_set_list_a @ A2 @ bot_bot_set_list_a ) ).
% not_psubset_empty
thf(fact_811_psubset__imp__subset,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_set_b @ A2 @ B2 )
=> ( ord_less_eq_set_b @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_812_psubset__imp__subset,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( ord_less_set_list_a @ A2 @ B2 )
=> ( ord_le8861187494160871172list_a @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_813_psubset__imp__subset,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_814_psubset__subset__trans,axiom,
! [A2: set_b,B2: set_b,C2: set_b] :
( ( ord_less_set_b @ A2 @ B2 )
=> ( ( ord_less_eq_set_b @ B2 @ C2 )
=> ( ord_less_set_b @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_815_psubset__subset__trans,axiom,
! [A2: set_list_a,B2: set_list_a,C2: set_list_a] :
( ( ord_less_set_list_a @ A2 @ B2 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ C2 )
=> ( ord_less_set_list_a @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_816_psubset__subset__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_set_a @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_817_subset__not__subset__eq,axiom,
( ord_less_set_b
= ( ^ [A6: set_b,B6: set_b] :
( ( ord_less_eq_set_b @ A6 @ B6 )
& ~ ( ord_less_eq_set_b @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_818_subset__not__subset__eq,axiom,
( ord_less_set_list_a
= ( ^ [A6: set_list_a,B6: set_list_a] :
( ( ord_le8861187494160871172list_a @ A6 @ B6 )
& ~ ( ord_le8861187494160871172list_a @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_819_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ~ ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_820_subset__psubset__trans,axiom,
! [A2: set_b,B2: set_b,C2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( ord_less_set_b @ B2 @ C2 )
=> ( ord_less_set_b @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_821_subset__psubset__trans,axiom,
! [A2: set_list_a,B2: set_list_a,C2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ( ord_less_set_list_a @ B2 @ C2 )
=> ( ord_less_set_list_a @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_822_subset__psubset__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C2 )
=> ( ord_less_set_a @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_823_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A6: set_b,B6: set_b] :
( ( ord_less_set_b @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_824_subset__iff__psubset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A6: set_list_a,B6: set_list_a] :
( ( ord_less_set_list_a @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_825_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_set_a @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_826_subtree__size__le,axiom,
! [T1: dtree_list_a_b,T22: dtree_list_a_b] :
( ( is_subtree_list_a_b @ T1 @ T22 )
=> ( ord_less_eq_nat @ ( size_s415192292648992904st_a_b @ T1 ) @ ( size_s415192292648992904st_a_b @ T22 ) ) ) ).
% subtree_size_le
thf(fact_827_subtree__size__decr,axiom,
! [T1: dtree_list_a_b,T22: dtree_list_a_b] :
( ( is_subtree_list_a_b @ T1 @ T22 )
=> ( ( T1 != T22 )
=> ( ord_less_nat @ ( size_s415192292648992904st_a_b @ T1 ) @ ( size_s415192292648992904st_a_b @ T22 ) ) ) ) ).
% subtree_size_decr
thf(fact_828_subtree__eq__if__trans__eq2,axiom,
! [T1: dtree_list_a_b,T22: dtree_list_a_b,T32: dtree_list_a_b] :
( ( is_subtree_list_a_b @ T1 @ T22 )
=> ( ( is_subtree_list_a_b @ T22 @ T32 )
=> ( ( T1 = T32 )
=> ( T22 = T32 ) ) ) ) ).
% subtree_eq_if_trans_eq2
thf(fact_829_subtree__eq__if__trans__eq1,axiom,
! [T1: dtree_list_a_b,T22: dtree_list_a_b,T32: dtree_list_a_b] :
( ( is_subtree_list_a_b @ T1 @ T22 )
=> ( ( is_subtree_list_a_b @ T22 @ T32 )
=> ( ( T1 = T32 )
=> ( T1 = T22 ) ) ) ) ).
% subtree_eq_if_trans_eq1
thf(fact_830_subtree__antisym,axiom,
! [T1: dtree_list_a_b,T22: dtree_list_a_b] :
( ( is_subtree_list_a_b @ T1 @ T22 )
=> ( ( is_subtree_list_a_b @ T22 @ T1 )
=> ( T1 = T22 ) ) ) ).
% subtree_antisym
thf(fact_831_subtree__trans,axiom,
! [X: dtree_list_a_b,Y: dtree_list_a_b,Z3: dtree_list_a_b] :
( ( is_subtree_list_a_b @ X @ Y )
=> ( ( is_subtree_list_a_b @ Y @ Z3 )
=> ( is_subtree_list_a_b @ X @ Z3 ) ) ) ).
% subtree_trans
thf(fact_832_self__subtree,axiom,
! [T: dtree_list_a_b] : ( is_subtree_list_a_b @ T @ T ) ).
% self_subtree
thf(fact_833_ex__in__conv,axiom,
! [A2: set_Product_prod_a_a] :
( ( ? [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ A2 ) )
= ( A2 != bot_bo3357376287454694259od_a_a ) ) ).
% ex_in_conv
thf(fact_834_ex__in__conv,axiom,
! [A2: set_dtree_list_a_b] :
( ( ? [X4: dtree_list_a_b] : ( member551035911493665803st_a_b @ X4 @ A2 ) )
= ( A2 != bot_bo798015271861357502st_a_b ) ) ).
% ex_in_conv
thf(fact_835_ex__in__conv,axiom,
! [A2: set_list_a] :
( ( ? [X4: list_a] : ( member_list_a @ X4 @ A2 ) )
= ( A2 != bot_bot_set_list_a ) ) ).
% ex_in_conv
thf(fact_836_ex__in__conv,axiom,
! [A2: set_b] :
( ( ? [X4: b] : ( member_b @ X4 @ A2 ) )
= ( A2 != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_837_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X4: a] : ( member_a @ X4 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_838_equals0I,axiom,
! [A2: set_Product_prod_a_a] :
( ! [Y3: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ Y3 @ A2 )
=> ( A2 = bot_bo3357376287454694259od_a_a ) ) ).
% equals0I
thf(fact_839_equals0I,axiom,
! [A2: set_dtree_list_a_b] :
( ! [Y3: dtree_list_a_b] :
~ ( member551035911493665803st_a_b @ Y3 @ A2 )
=> ( A2 = bot_bo798015271861357502st_a_b ) ) ).
% equals0I
thf(fact_840_equals0I,axiom,
! [A2: set_list_a] :
( ! [Y3: list_a] :
~ ( member_list_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_list_a ) ) ).
% equals0I
thf(fact_841_equals0I,axiom,
! [A2: set_b] :
( ! [Y3: b] :
~ ( member_b @ Y3 @ A2 )
=> ( A2 = bot_bot_set_b ) ) ).
% equals0I
thf(fact_842_equals0I,axiom,
! [A2: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_843_equals0D,axiom,
! [A2: set_Product_prod_a_a,A: product_prod_a_a] :
( ( A2 = bot_bo3357376287454694259od_a_a )
=> ~ ( member1426531477525435216od_a_a @ A @ A2 ) ) ).
% equals0D
thf(fact_844_equals0D,axiom,
! [A2: set_dtree_list_a_b,A: dtree_list_a_b] :
( ( A2 = bot_bo798015271861357502st_a_b )
=> ~ ( member551035911493665803st_a_b @ A @ A2 ) ) ).
% equals0D
thf(fact_845_equals0D,axiom,
! [A2: set_list_a,A: list_a] :
( ( A2 = bot_bot_set_list_a )
=> ~ ( member_list_a @ A @ A2 ) ) ).
% equals0D
thf(fact_846_equals0D,axiom,
! [A2: set_b,A: b] :
( ( A2 = bot_bot_set_b )
=> ~ ( member_b @ A @ A2 ) ) ).
% equals0D
thf(fact_847_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_848_emptyE,axiom,
! [A: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ).
% emptyE
thf(fact_849_emptyE,axiom,
! [A: dtree_list_a_b] :
~ ( member551035911493665803st_a_b @ A @ bot_bo798015271861357502st_a_b ) ).
% emptyE
thf(fact_850_emptyE,axiom,
! [A: list_a] :
~ ( member_list_a @ A @ bot_bot_set_list_a ) ).
% emptyE
thf(fact_851_emptyE,axiom,
! [A: b] :
~ ( member_b @ A @ bot_bot_set_b ) ).
% emptyE
thf(fact_852_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_853_size__neq__size__imp__neq,axiom,
! [X: char,Y: char] :
( ( ( size_size_char @ X )
!= ( size_size_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_854_size__neq__size__imp__neq,axiom,
! [X: dtree_list_a_b,Y: dtree_list_a_b] :
( ( ( size_s415192292648992904st_a_b @ X )
!= ( size_s415192292648992904st_a_b @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_855_wf__dlverts__subtree,axiom,
! [T: dtree_list_a_b,T1: dtree_list_a_b] :
( ( list_wf_dlverts_a_b @ T )
=> ( ( is_subtree_list_a_b @ T1 @ T )
=> ( list_wf_dlverts_a_b @ T1 ) ) ) ).
% wf_dlverts_subtree
thf(fact_856_not__pfsubset__fempty,axiom,
! [A2: fset_P2153231429829016240_a_b_b] :
~ ( ord_le6631730213922513156_a_b_b @ A2 @ bot_bo2248824169281960260_a_b_b ) ).
% not_pfsubset_fempty
thf(fact_857_dtree__size__eq__root,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,R4: list_a] :
( ( size_s415192292648992904st_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
= ( size_s415192292648992904st_a_b @ ( node_list_a_b @ R4 @ Xs2 ) ) ) ).
% dtree_size_eq_root
thf(fact_858_dtree_Osize__neq,axiom,
! [X: dtree_list_a_b] :
( ( size_s415192292648992904st_a_b @ X )
!= zero_zero_nat ) ).
% dtree.size_neq
thf(fact_859_darcs__subtree__subset,axiom,
! [X: dtree_list_a_b,Y: dtree_list_a_b] :
( ( is_subtree_list_a_b @ X @ Y )
=> ( ord_less_eq_set_b @ ( darcs_list_a_b @ X ) @ ( darcs_list_a_b @ Y ) ) ) ).
% darcs_subtree_subset
thf(fact_860_mdeg__ge__sub,axiom,
! [T1: dtree_list_a_b,T22: dtree_list_a_b] :
( ( is_subtree_list_a_b @ T1 @ T22 )
=> ( ord_less_eq_nat @ ( max_deg_list_a_b @ T1 ) @ ( max_deg_list_a_b @ T22 ) ) ) ).
% mdeg_ge_sub
thf(fact_861_pfsubset__fcard__mono,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B2: fset_P2153231429829016240_a_b_b] :
( ( ord_le6631730213922513156_a_b_b @ A2 @ B2 )
=> ( ord_less_nat @ ( fcard_4742106318756258927_a_b_b @ A2 ) @ ( fcard_4742106318756258927_a_b_b @ B2 ) ) ) ).
% pfsubset_fcard_mono
thf(fact_862_Collect__mono__iff,axiom,
! [P: b > $o,Q: b > $o] :
( ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) )
= ( ! [X4: b] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_863_Collect__mono__iff,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) )
= ( ! [X4: list_a] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_864_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_865_set__eq__subset,axiom,
( ( ^ [Y5: set_b,Z2: set_b] : ( Y5 = Z2 ) )
= ( ^ [A6: set_b,B6: set_b] :
( ( ord_less_eq_set_b @ A6 @ B6 )
& ( ord_less_eq_set_b @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_866_set__eq__subset,axiom,
( ( ^ [Y5: set_list_a,Z2: set_list_a] : ( Y5 = Z2 ) )
= ( ^ [A6: set_list_a,B6: set_list_a] :
( ( ord_le8861187494160871172list_a @ A6 @ B6 )
& ( ord_le8861187494160871172list_a @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_867_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_868_subset__trans,axiom,
! [A2: set_b,B2: set_b,C2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( ord_less_eq_set_b @ B2 @ C2 )
=> ( ord_less_eq_set_b @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_869_subset__trans,axiom,
! [A2: set_list_a,B2: set_list_a,C2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ C2 )
=> ( ord_le8861187494160871172list_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_870_subset__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_871_Collect__mono,axiom,
! [P: b > $o,Q: b > $o] :
( ! [X3: b] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).
% Collect_mono
thf(fact_872_Collect__mono,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ! [X3: list_a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) ) ) ).
% Collect_mono
thf(fact_873_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_874_subset__refl,axiom,
! [A2: set_b] : ( ord_less_eq_set_b @ A2 @ A2 ) ).
% subset_refl
thf(fact_875_subset__refl,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_876_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_877_subset__iff,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
! [T4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T4 @ A6 )
=> ( member1426531477525435216od_a_a @ T4 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_878_subset__iff,axiom,
( ord_le7599451563663638410st_a_b
= ( ^ [A6: set_dtree_list_a_b,B6: set_dtree_list_a_b] :
! [T4: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ T4 @ A6 )
=> ( member551035911493665803st_a_b @ T4 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_879_subset__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A6: set_b,B6: set_b] :
! [T4: b] :
( ( member_b @ T4 @ A6 )
=> ( member_b @ T4 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_880_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A6: set_list_a,B6: set_list_a] :
! [T4: list_a] :
( ( member_list_a @ T4 @ A6 )
=> ( member_list_a @ T4 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_881_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [T4: a] :
( ( member_a @ T4 @ A6 )
=> ( member_a @ T4 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_882_equalityD2,axiom,
! [A2: set_b,B2: set_b] :
( ( A2 = B2 )
=> ( ord_less_eq_set_b @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_883_equalityD2,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( A2 = B2 )
=> ( ord_le8861187494160871172list_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_884_equalityD2,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_885_equalityD1,axiom,
! [A2: set_b,B2: set_b] :
( ( A2 = B2 )
=> ( ord_less_eq_set_b @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_886_equalityD1,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( A2 = B2 )
=> ( ord_le8861187494160871172list_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_887_equalityD1,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_888_subset__eq,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A6 )
=> ( member1426531477525435216od_a_a @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_889_subset__eq,axiom,
( ord_le7599451563663638410st_a_b
= ( ^ [A6: set_dtree_list_a_b,B6: set_dtree_list_a_b] :
! [X4: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X4 @ A6 )
=> ( member551035911493665803st_a_b @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_890_subset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A6: set_b,B6: set_b] :
! [X4: b] :
( ( member_b @ X4 @ A6 )
=> ( member_b @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_891_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A6: set_list_a,B6: set_list_a] :
! [X4: list_a] :
( ( member_list_a @ X4 @ A6 )
=> ( member_list_a @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_892_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [X4: a] :
( ( member_a @ X4 @ A6 )
=> ( member_a @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_893_equalityE,axiom,
! [A2: set_b,B2: set_b] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ~ ( ord_less_eq_set_b @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_894_equalityE,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( A2 = B2 )
=> ~ ( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ~ ( ord_le8861187494160871172list_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_895_equalityE,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_896_subsetD,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ( ( member1426531477525435216od_a_a @ C @ A2 )
=> ( member1426531477525435216od_a_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_897_subsetD,axiom,
! [A2: set_dtree_list_a_b,B2: set_dtree_list_a_b,C: dtree_list_a_b] :
( ( ord_le7599451563663638410st_a_b @ A2 @ B2 )
=> ( ( member551035911493665803st_a_b @ C @ A2 )
=> ( member551035911493665803st_a_b @ C @ B2 ) ) ) ).
% subsetD
thf(fact_898_subsetD,axiom,
! [A2: set_b,B2: set_b,C: b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( member_b @ C @ A2 )
=> ( member_b @ C @ B2 ) ) ) ).
% subsetD
thf(fact_899_subsetD,axiom,
! [A2: set_list_a,B2: set_list_a,C: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ( member_list_a @ C @ A2 )
=> ( member_list_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_900_subsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_901_in__mono,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,X: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B2 )
=> ( ( member1426531477525435216od_a_a @ X @ A2 )
=> ( member1426531477525435216od_a_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_902_in__mono,axiom,
! [A2: set_dtree_list_a_b,B2: set_dtree_list_a_b,X: dtree_list_a_b] :
( ( ord_le7599451563663638410st_a_b @ A2 @ B2 )
=> ( ( member551035911493665803st_a_b @ X @ A2 )
=> ( member551035911493665803st_a_b @ X @ B2 ) ) ) ).
% in_mono
thf(fact_903_in__mono,axiom,
! [A2: set_b,B2: set_b,X: b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( member_b @ X @ A2 )
=> ( member_b @ X @ B2 ) ) ) ).
% in_mono
thf(fact_904_in__mono,axiom,
! [A2: set_list_a,B2: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ( member_list_a @ X @ A2 )
=> ( member_list_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_905_in__mono,axiom,
! [A2: set_a,B2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_906_fcard__pfsubset,axiom,
! [A2: fset_P2153231429829016240_a_b_b,B2: fset_P2153231429829016240_a_b_b] :
( ( ord_le8870638447146015504_a_b_b @ A2 @ B2 )
=> ( ( ord_less_nat @ ( fcard_4742106318756258927_a_b_b @ A2 ) @ ( fcard_4742106318756258927_a_b_b @ B2 ) )
=> ( ord_le6631730213922513156_a_b_b @ A2 @ B2 ) ) ) ).
% fcard_pfsubset
thf(fact_907_size__le__if__child__subset,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,Ys: fset_P2153231429829016240_a_b_b,R2: list_a,V: list_a] :
( ( ord_le8870638447146015504_a_b_b @ Xs2 @ Ys )
=> ( ord_less_eq_nat @ ( size_s415192292648992904st_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) @ ( size_s415192292648992904st_a_b @ ( node_list_a_b @ V @ Ys ) ) ) ) ).
% size_le_if_child_subset
thf(fact_908_size__le__if__sucs__subset,axiom,
! [T1: dtree_list_a_b,T22: dtree_list_a_b] :
( ( ord_le8870638447146015504_a_b_b @ ( sucs_list_a_b @ T1 ) @ ( sucs_list_a_b @ T22 ) )
=> ( ord_less_eq_nat @ ( size_s415192292648992904st_a_b @ T1 ) @ ( size_s415192292648992904st_a_b @ T22 ) ) ) ).
% size_le_if_sucs_subset
thf(fact_909_dtree__size__decr__aux,axiom,
! [X: dtree_list_a_b,Y: b,Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ Y ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ord_less_nat @ ( size_s415192292648992904st_a_b @ X ) @ ( size_s415192292648992904st_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ).
% dtree_size_decr_aux
thf(fact_910_old_Ofind__pos_Oelims,axiom,
! [T: dtree_list_a_b,Rank: list_a > real,X: list_a,Xa2: dtree_list_a_b,Y: produc9164743771328383783list_a] :
( ( old_a_b @ T )
=> ( ( ( find_pos_a_b @ Rank @ X @ Xa2 )
= Y )
=> ( ! [R: list_a,T12: dtree_list_a_b] :
( ? [Uu: b] :
( Xa2
= ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ Uu ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( Y
!= ( find_pos_aux_a_b @ Rank @ X @ R @ T12 ) ) )
=> ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
( ! [X2: produc6499617310964463488_a_b_b] :
( Xs
!= ( finser2303212525150181944_a_b_b @ X2 @ bot_bo2248824169281960260_a_b_b ) )
=> ! [R: list_a] :
( ( Xa2
= ( node_list_a_b @ R @ Xs ) )
=> ( Y
!= ( produc6837034575241423639list_a @ R @ R ) ) ) ) ) ) ) ).
% old.find_pos.elims
thf(fact_911_old_Ofind__pos_Osimps_I1_J,axiom,
! [T: dtree_list_a_b,Rank: list_a > real,V: list_a,R2: list_a,T1: dtree_list_a_b,Uu2: b] :
( ( old_a_b @ T )
=> ( ( find_pos_a_b @ Rank @ V @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ Uu2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
= ( find_pos_aux_a_b @ Rank @ V @ R2 @ T1 ) ) ) ).
% old.find_pos.simps(1)
thf(fact_912_ranked__dtree_Ochild__mdeg__gt1__if__sub__fcard__gt1,axiom,
! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,V: list_a,Ys: fset_P2153231429829016240_a_b_b] :
( ( ranked_dtree_a_b @ T @ Cmp )
=> ( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ ( node_list_a_b @ V @ Ys ) )
=> ( ( ( node_list_a_b @ R2 @ Xs2 )
!= ( node_list_a_b @ V @ Ys ) )
=> ( ( ord_less_nat @ one_one_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) )
=> ? [T12: dtree_list_a_b,E23: b] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E23 ) @ ( fset_P9138963618725001425_a_b_b @ Ys ) )
& ( ord_less_nat @ one_one_nat @ ( max_deg_list_a_b @ T12 ) ) ) ) ) ) ) ).
% ranked_dtree.child_mdeg_gt1_if_sub_fcard_gt1
thf(fact_913_subtree__child__if__dvert__notr__mdeg__le1,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,V: list_a] :
( ( ord_less_eq_nat @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) @ one_one_nat )
=> ( ( V != R2 )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
=> ? [R5: list_a,E2: b,Zs: fset_P2153231429829016240_a_b_b] : ( is_subtree_list_a_b @ ( node_list_a_b @ R5 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ( node_list_a_b @ V @ Zs ) @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ) ).
% subtree_child_if_dvert_notr_mdeg_le1
thf(fact_914_psubset__trans,axiom,
! [A2: set_b,B2: set_b,C2: set_b] :
( ( ord_less_set_b @ A2 @ B2 )
=> ( ( ord_less_set_b @ B2 @ C2 )
=> ( ord_less_set_b @ A2 @ C2 ) ) ) ).
% psubset_trans
thf(fact_915_psubset__trans,axiom,
! [A2: set_list_a,B2: set_list_a,C2: set_list_a] :
( ( ord_less_set_list_a @ A2 @ B2 )
=> ( ( ord_less_set_list_a @ B2 @ C2 )
=> ( ord_less_set_list_a @ A2 @ C2 ) ) ) ).
% psubset_trans
thf(fact_916_psubsetD,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ A2 @ B2 )
=> ( ( member1426531477525435216od_a_a @ C @ A2 )
=> ( member1426531477525435216od_a_a @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_917_psubsetD,axiom,
! [A2: set_dtree_list_a_b,B2: set_dtree_list_a_b,C: dtree_list_a_b] :
( ( ord_le4535551246020252542st_a_b @ A2 @ B2 )
=> ( ( member551035911493665803st_a_b @ C @ A2 )
=> ( member551035911493665803st_a_b @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_918_psubsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_919_psubsetD,axiom,
! [A2: set_b,B2: set_b,C: b] :
( ( ord_less_set_b @ A2 @ B2 )
=> ( ( member_b @ C @ A2 )
=> ( member_b @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_920_psubsetD,axiom,
! [A2: set_list_a,B2: set_list_a,C: list_a] :
( ( ord_less_set_list_a @ A2 @ B2 )
=> ( ( member_list_a @ C @ A2 )
=> ( member_list_a @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_921_ranked__dtree_Oranked__dtree__subtree,axiom,
! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,X: dtree_list_a_b] :
( ( ranked_dtree_a_b @ T @ Cmp )
=> ( ( is_subtree_list_a_b @ X @ T )
=> ( ranked_dtree_a_b @ X @ Cmp ) ) ) ).
% ranked_dtree.ranked_dtree_subtree
thf(fact_922_dverts__nempty,axiom,
! [T: dtree_list_a_b] :
( ( dverts_list_a_b @ T )
!= bot_bot_set_list_a ) ).
% dverts_nempty
thf(fact_923_dtree_Oset__intros_I1_J,axiom,
! [X1: list_a,X22: fset_P2153231429829016240_a_b_b] : ( member_list_a @ X1 @ ( dverts_list_a_b @ ( node_list_a_b @ X1 @ X22 ) ) ) ).
% dtree.set_intros(1)
thf(fact_924_ranked__dtree_Omerge1_Ocases,axiom,
! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,X: dtree_list_a_b] :
( ( ranked_dtree_a_b @ T @ Cmp )
=> ~ ! [R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
( X
!= ( node_list_a_b @ R @ Xs ) ) ) ).
% ranked_dtree.merge1.cases
thf(fact_925_old_Ofind__pos__y__in__dverts,axiom,
! [T: dtree_list_a_b,X: list_a,Y: list_a,Rank: list_a > real,V: list_a,P2: list_a,T1: dtree_list_a_b] :
( ( old_a_b @ T )
=> ( ( ( produc6837034575241423639list_a @ X @ Y )
= ( find_pos_aux_a_b @ Rank @ V @ P2 @ T1 ) )
=> ( member_list_a @ Y @ ( dverts_list_a_b @ T1 ) ) ) ) ).
% old.find_pos_y_in_dverts
thf(fact_926_old_Ofind__pos__x__in__dverts,axiom,
! [T: dtree_list_a_b,X: list_a,Y: list_a,Rank: list_a > real,V: list_a,P2: list_a,T1: dtree_list_a_b] :
( ( old_a_b @ T )
=> ( ( ( produc6837034575241423639list_a @ X @ Y )
= ( find_pos_aux_a_b @ Rank @ V @ P2 @ T1 ) )
=> ( ( member_list_a @ X @ ( dverts_list_a_b @ T1 ) )
| ( P2 = X ) ) ) ) ).
% old.find_pos_x_in_dverts
thf(fact_927_subtree__root__if__dverts,axiom,
! [X: list_a,T: dtree_list_a_b] :
( ( member_list_a @ X @ ( dverts_list_a_b @ T ) )
=> ? [Xs: fset_P2153231429829016240_a_b_b] : ( is_subtree_list_a_b @ ( node_list_a_b @ X @ Xs ) @ T ) ) ).
% subtree_root_if_dverts
thf(fact_928_dverts__subtree__subset,axiom,
! [X: dtree_list_a_b,Y: dtree_list_a_b] :
( ( is_subtree_list_a_b @ X @ Y )
=> ( ord_le8861187494160871172list_a @ ( dverts_list_a_b @ X ) @ ( dverts_list_a_b @ Y ) ) ) ).
% dverts_subtree_subset
thf(fact_929_ranked__dtree_Oranked__dtree__rec__suc,axiom,
! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,X: dtree_list_a_b,E: b] :
( ( ranked_dtree_a_b @ T @ Cmp )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T ) ) )
=> ( ranked_dtree_a_b @ X @ Cmp ) ) ) ).
% ranked_dtree.ranked_dtree_rec_suc
thf(fact_930_ranked__dtree_Oranked__dtree__rec,axiom,
! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b,E: b] :
( ( ranked_dtree_a_b @ T @ Cmp )
=> ( ( ( node_list_a_b @ R2 @ Xs2 )
= T )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ranked_dtree_a_b @ X @ Cmp ) ) ) ) ).
% ranked_dtree.ranked_dtree_rec
thf(fact_931_ranked__dtree_Onormalize1_Ocases,axiom,
! [T: dtree_list_a_b,Cmp: compar7542523840845723048st_a_b,X: dtree_list_a_b] :
( ( ranked_dtree_a_b @ T @ Cmp )
=> ( ! [R: list_a,T12: dtree_list_a_b,E2: b] :
( X
!= ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
( ! [X2: produc6499617310964463488_a_b_b] :
( Xs
!= ( finser2303212525150181944_a_b_b @ X2 @ bot_bo2248824169281960260_a_b_b ) )
=> ! [R: list_a] :
( X
!= ( node_list_a_b @ R @ Xs ) ) ) ) ) ).
% ranked_dtree.normalize1.cases
thf(fact_932_root__not__child__if__wf__dverts,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,T1: dtree_list_a_b,E12: b] :
( ( wf_dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ~ ( member_list_a @ R2 @ ( dverts_list_a_b @ T1 ) ) ) ) ).
% root_not_child_if_wf_dverts
thf(fact_933_single__subtree__root__dverts,axiom,
! [V22: list_a,T22: dtree_list_a_b,E22: b,T1: dtree_list_a_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ V22 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T1 )
=> ( member_list_a @ V22 @ ( dverts_list_a_b @ T1 ) ) ) ).
% single_subtree_root_dverts
thf(fact_934_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C4: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_935_subtree__child__if__dvert__notroot__mdeg__le1,axiom,
! [T: dtree_list_a_b,V: list_a] :
( ( ord_less_eq_nat @ ( max_deg_list_a_b @ T ) @ one_one_nat )
=> ( ( V
!= ( root_list_a_b @ T ) )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ T ) )
=> ? [R5: list_a,E2: b,Zs: fset_P2153231429829016240_a_b_b] : ( is_subtree_list_a_b @ ( node_list_a_b @ R5 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ( node_list_a_b @ V @ Zs ) @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T ) ) ) ) ).
% subtree_child_if_dvert_notroot_mdeg_le1
thf(fact_936_bot__empty__eq,axiom,
( bot_bo4160289986317612842_a_a_o
= ( ^ [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ bot_bo3357376287454694259od_a_a ) ) ) ).
% bot_empty_eq
thf(fact_937_bot__empty__eq,axiom,
( bot_bo84193016852448327_a_b_o
= ( ^ [X4: dtree_list_a_b] : ( member551035911493665803st_a_b @ X4 @ bot_bo798015271861357502st_a_b ) ) ) ).
% bot_empty_eq
thf(fact_938_bot__empty__eq,axiom,
( bot_bot_list_a_o
= ( ^ [X4: list_a] : ( member_list_a @ X4 @ bot_bot_set_list_a ) ) ) ).
% bot_empty_eq
thf(fact_939_bot__empty__eq,axiom,
( bot_bot_b_o
= ( ^ [X4: b] : ( member_b @ X4 @ bot_bot_set_b ) ) ) ).
% bot_empty_eq
thf(fact_940_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X4: a] : ( member_a @ X4 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_941_dtree_Ocollapse,axiom,
! [Dtree: dtree_list_a_b] :
( ( node_list_a_b @ ( root_list_a_b @ Dtree ) @ ( sucs_list_a_b @ Dtree ) )
= Dtree ) ).
% dtree.collapse
thf(fact_942_dtree_Oset__sel_I1_J,axiom,
! [A: dtree_list_a_b] : ( member_list_a @ ( root_list_a_b @ A ) @ ( dverts_list_a_b @ A ) ) ).
% dtree.set_sel(1)
thf(fact_943_dtree_Oexpand,axiom,
! [Dtree: dtree_list_a_b,Dtree2: dtree_list_a_b] :
( ( ( ( root_list_a_b @ Dtree )
= ( root_list_a_b @ Dtree2 ) )
& ( ( sucs_list_a_b @ Dtree )
= ( sucs_list_a_b @ Dtree2 ) ) )
=> ( Dtree = Dtree2 ) ) ).
% dtree.expand
thf(fact_944_dtree_Osel_I1_J,axiom,
! [X1: list_a,X22: fset_P2153231429829016240_a_b_b] :
( ( root_list_a_b @ ( node_list_a_b @ X1 @ X22 ) )
= X1 ) ).
% dtree.sel(1)
thf(fact_945_dtree_Oexhaust__sel,axiom,
! [Dtree: dtree_list_a_b] :
( Dtree
= ( node_list_a_b @ ( root_list_a_b @ Dtree ) @ ( sucs_list_a_b @ Dtree ) ) ) ).
% dtree.exhaust_sel
thf(fact_946_subrelI,axiom,
! [R2: set_Pr3443975907877334966_a_b_b,S4: set_Pr3443975907877334966_a_b_b] :
( ! [X3: dtree_list_a_b,Y3: b] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X3 @ Y3 ) @ R2 )
=> ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X3 @ Y3 ) @ S4 ) )
=> ( ord_le1619362961161175062_a_b_b @ R2 @ S4 ) ) ).
% subrelI
thf(fact_947_subrelI,axiom,
! [R2: set_Pr1349601357184307552list_b,S4: set_Pr1349601357184307552list_b] :
( ! [X3: nat,Y3: list_b] :
( ( member8261005420521984321list_b @ ( produc7903367361620597084list_b @ X3 @ Y3 ) @ R2 )
=> ( member8261005420521984321list_b @ ( produc7903367361620597084list_b @ X3 @ Y3 ) @ S4 ) )
=> ( ord_le972014486225453504list_b @ R2 @ S4 ) ) ).
% subrelI
thf(fact_948_subrelI,axiom,
! [R2: set_Pr6500140389540524009st_b_a,S4: set_Pr6500140389540524009st_b_a] :
( ! [X3: a,Y3: produc1943741644644106336st_b_a] :
( ( member4827874839601930546st_b_a @ ( produc7119031474978700025st_b_a @ X3 @ Y3 ) @ R2 )
=> ( member4827874839601930546st_b_a @ ( produc7119031474978700025st_b_a @ X3 @ Y3 ) @ S4 ) )
=> ( ord_le1497144413764237193st_b_a @ R2 @ S4 ) ) ).
% subrelI
thf(fact_949_subrelI,axiom,
! [R2: set_Pr2389355623220313408st_b_a,S4: set_Pr2389355623220313408st_b_a] :
( ! [X3: list_b,Y3: a] :
( ( member7370802231240916489st_b_a @ ( produc4145578316043568848st_b_a @ X3 @ Y3 ) @ R2 )
=> ( member7370802231240916489st_b_a @ ( produc4145578316043568848st_b_a @ X3 @ Y3 ) @ S4 ) )
=> ( ord_le5459107721870501088st_b_a @ R2 @ S4 ) ) ).
% subrelI
thf(fact_950_subrelI,axiom,
! [R2: set_Product_prod_a_a,S4: set_Product_prod_a_a] :
( ! [X3: a,Y3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ S4 ) )
=> ( ord_le746702958409616551od_a_a @ R2 @ S4 ) ) ).
% subrelI
thf(fact_951_wf__dlverts__sucs,axiom,
! [T: dtree_list_a_b,X: produc6499617310964463488_a_b_b] :
( ( list_wf_dlverts_a_b @ T )
=> ( ( member4695696432722591383_a_b_b @ X @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T ) ) )
=> ( list_wf_dlverts_a_b @ ( node_list_a_b @ ( root_list_a_b @ T ) @ ( finser2303212525150181944_a_b_b @ X @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ).
% wf_dlverts_sucs
thf(fact_952_single__subtree__child__root__dverts,axiom,
! [V22: list_a,T22: dtree_list_a_b,E22: b,T1: dtree_list_a_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ V22 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T1 )
=> ( member_list_a @ ( root_list_a_b @ T22 ) @ ( dverts_list_a_b @ T1 ) ) ) ).
% single_subtree_child_root_dverts
thf(fact_953_combine__darcs__sub,axiom,
! [R2: list_b,T1: dtree_list_b_b,E12: b] : ( ord_less_eq_set_b @ ( darcs_list_b_b @ ( node_list_b_b @ ( append_b @ R2 @ ( root_list_b_b @ T1 ) ) @ ( sucs_list_b_b @ T1 ) ) ) @ ( darcs_list_b_b @ ( node_list_b_b @ R2 @ ( finser8747880254144935225_b_b_b @ ( produc4925461457734986419_b_b_b @ T1 @ E12 ) @ bot_bo3541618583958561733_b_b_b ) ) ) ) ).
% combine_darcs_sub
thf(fact_954_combine__darcs__sub,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b] : ( ord_less_eq_set_b @ ( darcs_list_a_b @ ( node_list_a_b @ ( append_a @ R2 @ ( root_list_a_b @ T1 ) ) @ ( sucs_list_a_b @ T1 ) ) ) @ ( darcs_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).
% combine_darcs_sub
thf(fact_955_combine__darcs__sub,axiom,
! [R2: list_a,T1: dtree_list_a_list_a,E12: list_a] : ( ord_le8861187494160871172list_a @ ( darcs_list_a_list_a @ ( node_list_a_list_a @ ( append_a @ R2 @ ( root_list_a_list_a @ T1 ) ) @ ( sucs_list_a_list_a @ T1 ) ) ) @ ( darcs_list_a_list_a @ ( node_list_a_list_a @ R2 @ ( finser5913853959220252280list_a @ ( produc6276725351364364146list_a @ T1 @ E12 ) @ bot_bo7860097226086663300list_a ) ) ) ) ).
% combine_darcs_sub
thf(fact_956_combine__darcs__sub,axiom,
! [R2: list_b,T1: dtree_list_b_list_a,E12: list_a] : ( ord_le8861187494160871172list_a @ ( darcs_list_b_list_a @ ( node_list_b_list_a @ ( append_b @ R2 @ ( root_list_b_list_a @ T1 ) ) @ ( sucs_list_b_list_a @ T1 ) ) ) @ ( darcs_list_b_list_a @ ( node_list_b_list_a @ R2 @ ( finser1670425070513979513list_a @ ( produc2033296462658091379list_a @ T1 @ E12 ) @ bot_bo5225355504711828997list_a ) ) ) ) ).
% combine_darcs_sub
thf(fact_957_combine__darcs__sub,axiom,
! [R2: list_a,T1: dtree_list_a_a,E12: a] : ( ord_less_eq_set_a @ ( darcs_list_a_a @ ( node_list_a_a @ ( append_a @ R2 @ ( root_list_a_a @ T1 ) ) @ ( sucs_list_a_a @ T1 ) ) ) @ ( darcs_list_a_a @ ( node_list_a_a @ R2 @ ( finser2938861521451939832_a_a_a @ ( produc8339814766199995634_a_a_a @ T1 @ E12 ) @ bot_bo8750491735591794436_a_a_a ) ) ) ) ).
% combine_darcs_sub
thf(fact_958_combine__darcs__sub,axiom,
! [R2: list_b,T1: dtree_list_b_a,E12: a] : ( ord_less_eq_set_a @ ( darcs_list_b_a @ ( node_list_b_a @ ( append_b @ R2 @ ( root_list_b_a @ T1 ) ) @ ( sucs_list_b_a @ T1 ) ) ) @ ( darcs_list_b_a @ ( node_list_b_a @ R2 @ ( finser160157213591917305_b_a_a @ ( produc5561110458339973107_b_a_a @ T1 @ E12 ) @ bot_bo819914113413620101_b_a_a ) ) ) ) ).
% combine_darcs_sub
thf(fact_959_combine__wf__arcs,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b] :
( ( wf_darcs_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( wf_darcs_list_a_b @ ( node_list_a_b @ ( append_a @ R2 @ ( root_list_a_b @ T1 ) ) @ ( sucs_list_a_b @ T1 ) ) ) ) ).
% combine_wf_arcs
thf(fact_960_combine__wf__dlverts,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b] :
( ( list_wf_dlverts_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( list_wf_dlverts_a_b @ ( node_list_a_b @ ( append_a @ R2 @ ( root_list_a_b @ T1 ) ) @ ( sucs_list_a_b @ T1 ) ) ) ) ).
% combine_wf_dlverts
thf(fact_961_dtree__size__skip__decr1,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b] : ( ord_less_nat @ ( size_s415192292648992904st_a_b @ ( node_list_a_b @ ( append_a @ R2 @ ( root_list_a_b @ T1 ) ) @ ( sucs_list_a_b @ T1 ) ) ) @ ( size_s415192292648992904st_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).
% dtree_size_skip_decr1
thf(fact_962_split__length__i,axiom,
! [I: nat,Bs: list_a] :
( ( ord_less_eq_nat @ I @ ( size_size_list_a @ Bs ) )
=> ? [Xs: list_a,Ys2: list_a] :
( ( ( append_a @ Xs @ Ys2 )
= Bs )
& ( ( size_size_list_a @ Xs )
= I ) ) ) ).
% split_length_i
thf(fact_963_split__length__i,axiom,
! [I: nat,Bs: list_b] :
( ( ord_less_eq_nat @ I @ ( size_size_list_b @ Bs ) )
=> ? [Xs: list_b,Ys2: list_b] :
( ( ( append_b @ Xs @ Ys2 )
= Bs )
& ( ( size_size_list_b @ Xs )
= I ) ) ) ).
% split_length_i
thf(fact_964_split__length__i__prefix,axiom,
! [As: list_a,I: nat,Bs: list_a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ As ) @ I )
=> ( ( ord_less_nat @ I @ ( size_size_list_a @ ( append_a @ As @ Bs ) ) )
=> ? [Xs: list_a,Ys2: list_a] :
( ( ( append_a @ Xs @ Ys2 )
= Bs )
& ( ( size_size_list_a @ ( append_a @ As @ Xs ) )
= I ) ) ) ) ).
% split_length_i_prefix
thf(fact_965_split__length__i__prefix,axiom,
! [As: list_b,I: nat,Bs: list_b] :
( ( ord_less_eq_nat @ ( size_size_list_b @ As ) @ I )
=> ( ( ord_less_nat @ I @ ( size_size_list_b @ ( append_b @ As @ Bs ) ) )
=> ? [Xs: list_b,Ys2: list_b] :
( ( ( append_b @ Xs @ Ys2 )
= Bs )
& ( ( size_size_list_b @ ( append_b @ As @ Xs ) )
= I ) ) ) ) ).
% split_length_i_prefix
thf(fact_966_normalize__full_Osimps_I1_J,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b] :
( ( iKKBZ_6959927528703686640ll_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) )
= ( iKKBZ_6959927528703686640ll_a_b @ ( node_list_a_b @ ( append_a @ R2 @ ( root_list_a_b @ T1 ) ) @ ( sucs_list_a_b @ T1 ) ) ) ) ).
% normalize_full.simps(1)
thf(fact_967_normalize__full_Oelims,axiom,
! [X: dtree_list_a_b,Y: dtree_list_a_b] :
( ( ( iKKBZ_6959927528703686640ll_a_b @ X )
= Y )
=> ( ! [R: list_a,T12: dtree_list_a_b] :
( ? [E1: b] :
( X
= ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( Y
!= ( iKKBZ_6959927528703686640ll_a_b @ ( node_list_a_b @ ( append_a @ R @ ( root_list_a_b @ T12 ) ) @ ( sucs_list_a_b @ T12 ) ) ) ) )
=> ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
( ! [X2: produc6499617310964463488_a_b_b] :
( Xs
!= ( finser2303212525150181944_a_b_b @ X2 @ bot_bo2248824169281960260_a_b_b ) )
=> ! [R: list_a] :
( ( X
= ( node_list_a_b @ R @ Xs ) )
=> ( Y
!= ( node_list_a_b @ R @ Xs ) ) ) ) ) ) ).
% normalize_full.elims
thf(fact_968_combine__uneq,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b] :
( ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) )
!= ( node_list_a_b @ ( append_a @ R2 @ ( root_list_a_b @ T1 ) ) @ ( sucs_list_a_b @ T1 ) ) ) ).
% combine_uneq
thf(fact_969_size__combine__decr,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b] : ( ord_less_nat @ ( size_s415192292648992904st_a_b @ ( node_list_a_b @ ( append_a @ R2 @ ( root_list_a_b @ T1 ) ) @ ( sucs_list_a_b @ T1 ) ) ) @ ( size_s415192292648992904st_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).
% size_combine_decr
thf(fact_970_combine__denormalize__eq,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b] :
( ( denormalize_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) )
= ( denormalize_a_b @ ( node_list_a_b @ ( append_a @ R2 @ ( root_list_a_b @ T1 ) ) @ ( sucs_list_a_b @ T1 ) ) ) ) ).
% combine_denormalize_eq
thf(fact_971_combine__distinct,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b,V: list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( dverts_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( distinct_a @ X3 ) )
=> ( ( list_wf_dlverts_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ ( append_a @ R2 @ ( root_list_a_b @ T1 ) ) @ ( sucs_list_a_b @ T1 ) ) ) )
=> ( distinct_a @ V ) ) ) ) ).
% combine_distinct
thf(fact_972_combine__nempty__if__wf__dlverts,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b] :
( ( list_wf_dlverts_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( append_a @ R2 @ ( root_list_a_b @ T1 ) )
!= nil_a ) ) ).
% combine_nempty_if_wf_dlverts
thf(fact_973_combine__dlverts__eq,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b] :
( ( list_dlverts_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) )
= ( list_dlverts_a_b @ ( node_list_a_b @ ( append_a @ R2 @ ( root_list_a_b @ T1 ) ) @ ( sucs_list_a_b @ T1 ) ) ) ) ).
% combine_dlverts_eq
thf(fact_974_distinct__mid__unique2,axiom,
! [Xs2: list_b,U: list_b,Ys: list_b,As: list_b,Bs: list_b] :
( ( distinct_b @ ( append_b @ Xs2 @ ( append_b @ U @ Ys ) ) )
=> ( ( U != nil_b )
=> ( ( ( append_b @ Xs2 @ ( append_b @ U @ Ys ) )
= ( append_b @ As @ ( append_b @ U @ Bs ) ) )
=> ( Ys = Bs ) ) ) ) ).
% distinct_mid_unique2
thf(fact_975_distinct__mid__unique2,axiom,
! [Xs2: list_a,U: list_a,Ys: list_a,As: list_a,Bs: list_a] :
( ( distinct_a @ ( append_a @ Xs2 @ ( append_a @ U @ Ys ) ) )
=> ( ( U != nil_a )
=> ( ( ( append_a @ Xs2 @ ( append_a @ U @ Ys ) )
= ( append_a @ As @ ( append_a @ U @ Bs ) ) )
=> ( Ys = Bs ) ) ) ) ).
% distinct_mid_unique2
thf(fact_976_distinct__mid__unique1,axiom,
! [Xs2: list_b,U: list_b,Ys: list_b,As: list_b,Bs: list_b] :
( ( distinct_b @ ( append_b @ Xs2 @ ( append_b @ U @ Ys ) ) )
=> ( ( U != nil_b )
=> ( ( ( append_b @ Xs2 @ ( append_b @ U @ Ys ) )
= ( append_b @ As @ ( append_b @ U @ Bs ) ) )
=> ( As = Xs2 ) ) ) ) ).
% distinct_mid_unique1
thf(fact_977_distinct__mid__unique1,axiom,
! [Xs2: list_a,U: list_a,Ys: list_a,As: list_a,Bs: list_a] :
( ( distinct_a @ ( append_a @ Xs2 @ ( append_a @ U @ Ys ) ) )
=> ( ( U != nil_a )
=> ( ( ( append_a @ Xs2 @ ( append_a @ U @ Ys ) )
= ( append_a @ As @ ( append_a @ U @ Bs ) ) )
=> ( As = Xs2 ) ) ) ) ).
% distinct_mid_unique1
thf(fact_978_normalize__full__dlverts__eq,axiom,
! [T1: dtree_list_a_b] :
( ( list_dlverts_a_b @ ( iKKBZ_6959927528703686640ll_a_b @ T1 ) )
= ( list_dlverts_a_b @ T1 ) ) ).
% normalize_full_dlverts_eq
thf(fact_979_normalize__full__denormalize__eq,axiom,
! [T1: dtree_list_a_b] :
( ( denormalize_a_b @ ( iKKBZ_6959927528703686640ll_a_b @ T1 ) )
= ( denormalize_a_b @ T1 ) ) ).
% normalize_full_denormalize_eq
thf(fact_980_denormalize__nempty__if__wf,axiom,
! [T: dtree_list_a_b] :
( ( list_wf_dlverts_a_b @ T )
=> ( ( denormalize_a_b @ T )
!= nil_a ) ) ).
% denormalize_nempty_if_wf
thf(fact_981_denormalize__distinct,axiom,
! [T1: dtree_list_a_b] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( dverts_list_a_b @ T1 ) )
=> ( distinct_a @ X3 ) )
=> ( ( list_wf_dlverts_a_b @ T1 )
=> ( distinct_a @ ( denormalize_a_b @ T1 ) ) ) ) ).
% denormalize_distinct
thf(fact_982_dlverts__nempty__aux,axiom,
! [T: dtree_list_a_b] :
( ~ ( member_list_a @ nil_a @ ( dverts_list_a_b @ T ) )
=> ( ( list_dlverts_a_b @ T )
!= bot_bot_set_a ) ) ).
% dlverts_nempty_aux
thf(fact_983_subtree__in__dlverts,axiom,
! [T1: dtree_list_a_b,T22: dtree_list_a_b] :
( ( is_subtree_list_a_b @ T1 @ T22 )
=> ( ord_less_eq_set_a @ ( list_dlverts_a_b @ T1 ) @ ( list_dlverts_a_b @ T22 ) ) ) ).
% subtree_in_dlverts
thf(fact_984_dlverts__nempty__if__wf,axiom,
! [T: dtree_list_a_b] :
( ( list_wf_dlverts_a_b @ T )
=> ( ( list_dlverts_a_b @ T )
!= bot_bot_set_a ) ) ).
% dlverts_nempty_if_wf
thf(fact_985_empty__notin__wf__dlverts,axiom,
! [T: dtree_list_a_b] :
( ( list_wf_dlverts_a_b @ T )
=> ~ ( member_list_a @ nil_a @ ( dverts_list_a_b @ T ) ) ) ).
% empty_notin_wf_dlverts
thf(fact_986_denormalize_Osimps_I2_J,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
( ! [X3: produc6499617310964463488_a_b_b] :
( Xs2
!= ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
=> ( ( denormalize_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
= R2 ) ) ).
% denormalize.simps(2)
thf(fact_987_child__in__dlverts,axiom,
! [T1: dtree_list_a_b,E: b,Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ord_less_eq_set_a @ ( list_dlverts_a_b @ T1 ) @ ( list_dlverts_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ).
% child_in_dlverts
thf(fact_988_suc__in__dlverts,axiom,
! [T1: dtree_list_a_b,E: b,T22: dtree_list_a_b] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E ) @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T22 ) ) )
=> ( ord_less_eq_set_a @ ( list_dlverts_a_b @ T1 ) @ ( list_dlverts_a_b @ T22 ) ) ) ).
% suc_in_dlverts
thf(fact_989_denormalize_Osimps_I1_J,axiom,
! [R2: list_a,T: dtree_list_a_b,E: b] :
( ( denormalize_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T @ E ) @ bot_bo2248824169281960260_a_b_b ) ) )
= ( append_a @ R2 @ ( denormalize_a_b @ T ) ) ) ).
% denormalize.simps(1)
thf(fact_990_denormalize_Oelims,axiom,
! [X: dtree_list_a_b,Y: list_a] :
( ( ( denormalize_a_b @ X )
= Y )
=> ( ! [R: list_a,T2: dtree_list_a_b] :
( ? [E2: b] :
( X
= ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T2 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( Y
!= ( append_a @ R @ ( denormalize_a_b @ T2 ) ) ) )
=> ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
( ! [X2: produc6499617310964463488_a_b_b] :
( Xs
!= ( finser2303212525150181944_a_b_b @ X2 @ bot_bo2248824169281960260_a_b_b ) )
=> ! [R: list_a] :
( ( X
= ( node_list_a_b @ R @ Xs ) )
=> ( Y != R ) ) ) ) ) ).
% denormalize.elims
thf(fact_991_length__greater__0__conv,axiom,
! [Xs2: list_b] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_b @ Xs2 ) )
= ( Xs2 != nil_b ) ) ).
% length_greater_0_conv
thf(fact_992_length__greater__0__conv,axiom,
! [Xs2: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs2 ) )
= ( Xs2 != nil_a ) ) ).
% length_greater_0_conv
thf(fact_993_length__0__conv,axiom,
! [Xs2: list_b] :
( ( ( size_size_list_b @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_b ) ) ).
% length_0_conv
thf(fact_994_length__0__conv,axiom,
! [Xs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_a ) ) ).
% length_0_conv
thf(fact_995_mk__cycles__path_Oelims,axiom,
! [X: nat,Xa2: list_b,Y: list_b] :
( ( ( shorte6374615165232202367path_b @ X @ Xa2 )
= Y )
=> ( ( ( X = zero_zero_nat )
=> ( Y != nil_b ) )
=> ~ ! [N3: nat] :
( ( X
= ( suc @ N3 ) )
=> ( Y
!= ( append_b @ Xa2 @ ( shorte6374615165232202367path_b @ N3 @ Xa2 ) ) ) ) ) ) ).
% mk_cycles_path.elims
thf(fact_996_append__eq__append__conv,axiom,
! [Xs2: list_a,Ys: list_a,Us: list_a,Vs: list_a] :
( ( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
| ( ( size_size_list_a @ Us )
= ( size_size_list_a @ Vs ) ) )
=> ( ( ( append_a @ Xs2 @ Us )
= ( append_a @ Ys @ Vs ) )
= ( ( Xs2 = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_997_append__eq__append__conv,axiom,
! [Xs2: list_b,Ys: list_b,Us: list_b,Vs: list_b] :
( ( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_b @ Ys ) )
| ( ( size_size_list_b @ Us )
= ( size_size_list_b @ Vs ) ) )
=> ( ( ( append_b @ Xs2 @ Us )
= ( append_b @ Ys @ Vs ) )
= ( ( Xs2 = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_998_mk__cycles__path_Osimps_I2_J,axiom,
! [N: nat,C: list_b] :
( ( shorte6374615165232202367path_b @ ( suc @ N ) @ C )
= ( append_b @ C @ ( shorte6374615165232202367path_b @ N @ C ) ) ) ).
% mk_cycles_path.simps(2)
thf(fact_999_mk__cycles__path_Osimps_I1_J,axiom,
! [C: list_b] :
( ( shorte6374615165232202367path_b @ zero_zero_nat @ C )
= nil_b ) ).
% mk_cycles_path.simps(1)
thf(fact_1000_list_Osize_I3_J,axiom,
( ( size_size_list_b @ nil_b )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_1001_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_1002_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C4: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_1003_mk__cycles__path_Opelims,axiom,
! [X: nat,Xa2: list_b,Y: list_b] :
( ( ( shorte6374615165232202367path_b @ X @ Xa2 )
= Y )
=> ( ( accp_P7720916649673260129list_b @ shorte5702012728047871812_rel_b @ ( produc7903367361620597084list_b @ X @ Xa2 ) )
=> ( ( ( X = zero_zero_nat )
=> ( ( Y = nil_b )
=> ~ ( accp_P7720916649673260129list_b @ shorte5702012728047871812_rel_b @ ( produc7903367361620597084list_b @ zero_zero_nat @ Xa2 ) ) ) )
=> ~ ! [N3: nat] :
( ( X
= ( suc @ N3 ) )
=> ( ( Y
= ( append_b @ Xa2 @ ( shorte6374615165232202367path_b @ N3 @ Xa2 ) ) )
=> ~ ( accp_P7720916649673260129list_b @ shorte5702012728047871812_rel_b @ ( produc7903367361620597084list_b @ ( suc @ N3 ) @ Xa2 ) ) ) ) ) ) ) ).
% mk_cycles_path.pelims
thf(fact_1004_ex__subtree__if__in__lverts,axiom,
! [V: a,T1: dtree_list_a_b] :
( ( member_a @ V @ ( list_dlverts_a_b @ T1 ) )
=> ? [T23: dtree_list_a_b] :
( ( is_subtree_list_a_b @ T23 @ T1 )
& ( member_a @ V @ ( set_a2 @ ( root_list_a_b @ T23 ) ) ) ) ) ).
% ex_subtree_if_in_lverts
thf(fact_1005_set__empty2,axiom,
! [Xs2: list_b] :
( ( bot_bot_set_b
= ( set_b2 @ Xs2 ) )
= ( Xs2 = nil_b ) ) ).
% set_empty2
thf(fact_1006_set__empty2,axiom,
! [Xs2: list_a] :
( ( bot_bot_set_a
= ( set_a2 @ Xs2 ) )
= ( Xs2 = nil_a ) ) ).
% set_empty2
thf(fact_1007_set__empty,axiom,
! [Xs2: list_b] :
( ( ( set_b2 @ Xs2 )
= bot_bot_set_b )
= ( Xs2 = nil_b ) ) ).
% set_empty
thf(fact_1008_set__empty,axiom,
! [Xs2: list_a] :
( ( ( set_a2 @ Xs2 )
= bot_bot_set_a )
= ( Xs2 = nil_a ) ) ).
% set_empty
thf(fact_1009_subset__code_I1_J,axiom,
! [Xs2: list_P1396940483166286381od_a_a,B2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs2 ) @ B2 )
= ( ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ ( set_Product_prod_a_a2 @ Xs2 ) )
=> ( member1426531477525435216od_a_a @ X4 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_1010_subset__code_I1_J,axiom,
! [Xs2: list_dtree_list_a_b,B2: set_dtree_list_a_b] :
( ( ord_le7599451563663638410st_a_b @ ( set_dtree_list_a_b2 @ Xs2 ) @ B2 )
= ( ! [X4: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X4 @ ( set_dtree_list_a_b2 @ Xs2 ) )
=> ( member551035911493665803st_a_b @ X4 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_1011_subset__code_I1_J,axiom,
! [Xs2: list_b,B2: set_b] :
( ( ord_less_eq_set_b @ ( set_b2 @ Xs2 ) @ B2 )
= ( ! [X4: b] :
( ( member_b @ X4 @ ( set_b2 @ Xs2 ) )
=> ( member_b @ X4 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_1012_subset__code_I1_J,axiom,
! [Xs2: list_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs2 ) @ B2 )
= ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Xs2 ) )
=> ( member_list_a @ X4 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_1013_subset__code_I1_J,axiom,
! [Xs2: list_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ B2 )
= ( ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs2 ) )
=> ( member_a @ X4 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_1014_ranked__dtree__with__orig_Oranked__dtree__orig__subtree,axiom,
! [T: dtree_list_a_b,Rank: list_a > real,Cost: list_a > real,Cmp: compar7542523840845723048st_a_b,T5: pre_pr7278220950009878019t_unit,Root: a,X: dtree_list_a_b] :
( ( iKKBZ_7928045550548935486ig_a_b @ T @ Rank @ Cost @ Cmp @ T5 @ Root )
=> ( ( is_subtree_list_a_b @ X @ T )
=> ( iKKBZ_7928045550548935486ig_a_b @ X @ Rank @ Cost @ Cmp @ T5 @ Root ) ) ) ).
% ranked_dtree_with_orig.ranked_dtree_orig_subtree
thf(fact_1015_ranked__dtree__with__orig_Oaxioms_I1_J,axiom,
! [T: dtree_list_a_b,Rank: list_a > real,Cost: list_a > real,Cmp: compar7542523840845723048st_a_b,T5: pre_pr7278220950009878019t_unit,Root: a] :
( ( iKKBZ_7928045550548935486ig_a_b @ T @ Rank @ Cost @ Cmp @ T5 @ Root )
=> ( ranked_dtree_a_b @ T @ Cmp ) ) ).
% ranked_dtree_with_orig.axioms(1)
thf(fact_1016_ranked__dtree__with__orig_Odverts__same__if__set__subtree,axiom,
! [T: dtree_list_a_b,Rank: list_a > real,Cost: list_a > real,Cmp: compar7542523840845723048st_a_b,T5: pre_pr7278220950009878019t_unit,Root: a,T1: dtree_list_a_b,V1: list_a,X: a,V22: list_a] :
( ( iKKBZ_7928045550548935486ig_a_b @ T @ Rank @ Cost @ Cmp @ T5 @ Root )
=> ( ( is_subtree_list_a_b @ T1 @ T )
=> ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V22 ) )
=> ( ( member_list_a @ V22 @ ( dverts_list_a_b @ T ) )
=> ( V1 = V22 ) ) ) ) ) ) ) ).
% ranked_dtree_with_orig.dverts_same_if_set_subtree
thf(fact_1017_ranked__dtree__with__orig_Overts__distinct,axiom,
! [T: dtree_list_a_b,Rank: list_a > real,Cost: list_a > real,Cmp: compar7542523840845723048st_a_b,T5: pre_pr7278220950009878019t_unit,Root: a,V: list_a] :
( ( iKKBZ_7928045550548935486ig_a_b @ T @ Rank @ Cost @ Cmp @ T5 @ Root )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ T ) )
=> ( distinct_a @ V ) ) ) ).
% ranked_dtree_with_orig.verts_distinct
thf(fact_1018_ranked__dtree__with__orig_Onormalize__full__dlverts__eq,axiom,
! [T: dtree_list_a_b,Rank: list_a > real,Cost: list_a > real,Cmp: compar7542523840845723048st_a_b,T5: pre_pr7278220950009878019t_unit,Root: a,T1: dtree_list_a_b] :
( ( iKKBZ_7928045550548935486ig_a_b @ T @ Rank @ Cost @ Cmp @ T5 @ Root )
=> ( ( list_dlverts_a_b @ ( iKKBZ_6959927528703686640ll_a_b @ T1 ) )
= ( list_dlverts_a_b @ T1 ) ) ) ).
% ranked_dtree_with_orig.normalize_full_dlverts_eq
thf(fact_1019_empty__set,axiom,
( bot_bot_set_b
= ( set_b2 @ nil_b ) ) ).
% empty_set
thf(fact_1020_empty__set,axiom,
( bot_bot_set_a
= ( set_a2 @ nil_a ) ) ).
% empty_set
thf(fact_1021_ranked__dtree__with__orig_Onormalize__full__wfdlverts,axiom,
! [T: dtree_list_a_b,Rank: list_a > real,Cost: list_a > real,Cmp: compar7542523840845723048st_a_b,T5: pre_pr7278220950009878019t_unit,Root: a,T1: dtree_list_a_b] :
( ( iKKBZ_7928045550548935486ig_a_b @ T @ Rank @ Cost @ Cmp @ T5 @ Root )
=> ( ( list_wf_dlverts_a_b @ T1 )
=> ( list_wf_dlverts_a_b @ ( iKKBZ_6959927528703686640ll_a_b @ T1 ) ) ) ) ).
% ranked_dtree_with_orig.normalize_full_wfdlverts
thf(fact_1022_ranked__dtree__with__orig_Onormalize__full__wfdarcs,axiom,
! [T: dtree_list_a_b,Rank: list_a > real,Cost: list_a > real,Cmp: compar7542523840845723048st_a_b,T5: pre_pr7278220950009878019t_unit,Root: a,T1: dtree_list_a_b] :
( ( iKKBZ_7928045550548935486ig_a_b @ T @ Rank @ Cost @ Cmp @ T5 @ Root )
=> ( ( wf_darcs_list_a_b @ T1 )
=> ( wf_darcs_list_a_b @ ( iKKBZ_6959927528703686640ll_a_b @ T1 ) ) ) ) ).
% ranked_dtree_with_orig.normalize_full_wfdarcs
thf(fact_1023_ranked__dtree__with__orig_Onormalize__full__denormalize__eq,axiom,
! [T: dtree_list_a_b,Rank: list_a > real,Cost: list_a > real,Cmp: compar7542523840845723048st_a_b,T5: pre_pr7278220950009878019t_unit,Root: a,T1: dtree_list_a_b] :
( ( iKKBZ_7928045550548935486ig_a_b @ T @ Rank @ Cost @ Cmp @ T5 @ Root )
=> ( ( denormalize_a_b @ ( iKKBZ_6959927528703686640ll_a_b @ T1 ) )
= ( denormalize_a_b @ T1 ) ) ) ).
% ranked_dtree_with_orig.normalize_full_denormalize_eq
thf(fact_1024_list__in__verts__iff__lverts,axiom,
! [X: a,T: dtree_list_a_b] :
( ( member_a @ X @ ( list_dlverts_a_b @ T ) )
= ( ? [X4: list_a] :
( ( member_list_a @ X4 @ ( dverts_list_a_b @ T ) )
& ( member_a @ X @ ( set_a2 @ X4 ) ) ) ) ) ).
% list_in_verts_iff_lverts
thf(fact_1025_list__in__verts__if__lverts,axiom,
! [X: a,T: dtree_list_a_b] :
( ( member_a @ X @ ( list_dlverts_a_b @ T ) )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ ( dverts_list_a_b @ T ) )
& ( member_a @ X @ ( set_a2 @ X3 ) ) ) ) ).
% list_in_verts_if_lverts
thf(fact_1026_lverts__if__in__verts,axiom,
! [V: list_a,T: dtree_list_a_b,X: a] :
( ( member_list_a @ V @ ( dverts_list_a_b @ T ) )
=> ( ( member_a @ X @ ( set_a2 @ V ) )
=> ( member_a @ X @ ( list_dlverts_a_b @ T ) ) ) ) ).
% lverts_if_in_verts
thf(fact_1027_dverts__same__if__set__wf,axiom,
! [T: dtree_list_a_b,V1: list_a,V22: list_a,X: a] :
( ( list_wf_dlverts_a_b @ T )
=> ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T ) )
=> ( ( member_list_a @ V22 @ ( dverts_list_a_b @ T ) )
=> ( ( member_a @ X @ ( set_a2 @ V1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V22 ) )
=> ( V1 = V22 ) ) ) ) ) ) ).
% dverts_same_if_set_wf
thf(fact_1028_ranked__dtree__with__orig_Oranked__dtree__orig__rec,axiom,
! [T: dtree_list_a_b,Rank: list_a > real,Cost: list_a > real,Cmp: compar7542523840845723048st_a_b,T5: pre_pr7278220950009878019t_unit,Root: a,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b,E: b] :
( ( iKKBZ_7928045550548935486ig_a_b @ T @ Rank @ Cost @ Cmp @ T5 @ Root )
=> ( ( ( node_list_a_b @ R2 @ Xs2 )
= T )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( iKKBZ_7928045550548935486ig_a_b @ X @ Rank @ Cost @ Cmp @ T5 @ Root ) ) ) ) ).
% ranked_dtree_with_orig.ranked_dtree_orig_rec
thf(fact_1029_ranked__dtree__with__orig_Overts__distinct__subtree,axiom,
! [T: dtree_list_a_b,Rank: list_a > real,Cost: list_a > real,Cmp: compar7542523840845723048st_a_b,T5: pre_pr7278220950009878019t_unit,Root: a,Tn: dtree_list_a_b,V: list_a] :
( ( iKKBZ_7928045550548935486ig_a_b @ T @ Rank @ Cost @ Cmp @ T5 @ Root )
=> ( ( is_subtree_list_a_b @ Tn @ T )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ Tn ) )
=> ( distinct_a @ V ) ) ) ) ).
% ranked_dtree_with_orig.verts_distinct_subtree
thf(fact_1030_ranked__dtree__with__orig_Onormalize__full__darcs__sub,axiom,
! [T: dtree_list_a_b,Rank: list_a > real,Cost: list_a > real,Cmp: compar7542523840845723048st_a_b,T5: pre_pr7278220950009878019t_unit,Root: a,T1: dtree_list_a_b] :
( ( iKKBZ_7928045550548935486ig_a_b @ T @ Rank @ Cost @ Cmp @ T5 @ Root )
=> ( ord_less_eq_set_b @ ( darcs_list_a_b @ ( iKKBZ_6959927528703686640ll_a_b @ T1 ) ) @ ( darcs_list_a_b @ T1 ) ) ) ).
% ranked_dtree_with_orig.normalize_full_darcs_sub
thf(fact_1031_length__pos__if__in__set,axiom,
! [X: product_prod_a_a,Xs2: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3885678630836030617od_a_a @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_1032_length__pos__if__in__set,axiom,
! [X: dtree_list_a_b,Xs2: list_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X @ ( set_dtree_list_a_b2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s7183927777527390232st_a_b @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_1033_length__pos__if__in__set,axiom,
! [X: list_a,Xs2: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s349497388124573686list_a @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_1034_length__pos__if__in__set,axiom,
! [X: b,Xs2: list_b] :
( ( member_b @ X @ ( set_b2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_b @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_1035_length__pos__if__in__set,axiom,
! [X: a,Xs2: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_1036_subtree__root__if__dlverts,axiom,
! [X: a,T: dtree_list_a_b] :
( ( member_a @ X @ ( list_dlverts_a_b @ T ) )
=> ? [R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R @ Xs ) @ T )
& ( member_a @ X @ ( set_a2 @ R ) ) ) ) ).
% subtree_root_if_dlverts
thf(fact_1037_root__if__same__lvert__wf,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: a,V: list_a] :
( ( list_wf_dlverts_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ( member_a @ X @ ( set_a2 @ R2 ) )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
=> ( ( member_a @ X @ ( set_a2 @ V ) )
=> ( V = R2 ) ) ) ) ) ).
% root_if_same_lvert_wf
thf(fact_1038_ranked__dtree__with__orig_Onormalize__full__wfdverts,axiom,
! [T: dtree_list_a_b,Rank: list_a > real,Cost: list_a > real,Cmp: compar7542523840845723048st_a_b,T5: pre_pr7278220950009878019t_unit,Root: a,T1: dtree_list_a_b] :
( ( iKKBZ_7928045550548935486ig_a_b @ T @ Rank @ Cost @ Cmp @ T5 @ Root )
=> ( ( list_wf_dlverts_a_b @ T1 )
=> ( wf_dverts_list_a_b @ ( iKKBZ_6959927528703686640ll_a_b @ T1 ) ) ) ) ).
% ranked_dtree_with_orig.normalize_full_wfdverts
thf(fact_1039_cas_Ocases,axiom,
! [X: produc7945266988514096265st_b_a] :
( ! [U2: a,V2: a] :
( X
!= ( produc7119031474978700025st_b_a @ U2 @ ( produc4145578316043568848st_b_a @ nil_b @ V2 ) ) )
=> ~ ! [U2: a,E2: b,Es: list_b,V2: a] :
( X
!= ( produc7119031474978700025st_b_a @ U2 @ ( produc4145578316043568848st_b_a @ ( cons_b @ E2 @ Es ) @ V2 ) ) ) ) ).
% cas.cases
thf(fact_1040_normalize1__size__decr,axiom,
! [T1: dtree_list_a_b] :
( ( ( ranked8905849569120154423e1_a_b @ rank @ T1 )
!= T1 )
=> ( ord_less_nat @ ( size_s415192292648992904st_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) ) @ ( size_s415192292648992904st_a_b @ T1 ) ) ) ).
% normalize1_size_decr
thf(fact_1041_num__leaves__normalize1__le,axiom,
! [T1: dtree_list_a_b] : ( ord_less_eq_nat @ ( num_leaves_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) ) @ ( num_leaves_list_a_b @ T1 ) ) ).
% num_leaves_normalize1_le
thf(fact_1042_normalize1__size__le,axiom,
! [T1: dtree_list_a_b] : ( ord_less_eq_nat @ ( size_s415192292648992904st_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) ) @ ( size_s415192292648992904st_a_b @ T1 ) ) ).
% normalize1_size_le
thf(fact_1043_normalize1__root__nempty,axiom,
! [T1: dtree_list_a_b] :
( ( ( root_list_a_b @ T1 )
!= nil_a )
=> ( ( root_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) )
!= nil_a ) ) ).
% normalize1_root_nempty
thf(fact_1044_wf__dlverts__normalize1,axiom,
! [T1: dtree_list_a_b] :
( ( list_wf_dlverts_a_b @ T1 )
=> ( list_wf_dlverts_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) ) ) ).
% wf_dlverts_normalize1
thf(fact_1045_wf__darcs__normalize1,axiom,
! [T1: dtree_list_a_b] :
( ( wf_darcs_list_a_b @ T1 )
=> ( wf_darcs_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) ) ) ).
% wf_darcs_normalize1
thf(fact_1046_normalize1__mdeg__le,axiom,
! [T1: dtree_list_a_b] : ( ord_less_eq_nat @ ( max_deg_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) ) @ ( max_deg_list_a_b @ T1 ) ) ).
% normalize1_mdeg_le
thf(fact_1047_normalize1__darcs__sub,axiom,
! [T1: dtree_list_a_b] : ( ord_less_eq_set_b @ ( darcs_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) ) @ ( darcs_list_a_b @ T1 ) ) ).
% normalize1_darcs_sub
thf(fact_1048_normalize1__denormalize__eq,axiom,
! [T1: dtree_list_a_b] :
( ( list_wf_dlverts_a_b @ T1 )
=> ( ( denormalize_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) )
= ( denormalize_a_b @ T1 ) ) ) ).
% normalize1_denormalize_eq
thf(fact_1049_normalize1__denormalize__eq_H,axiom,
! [T1: dtree_list_a_b] :
( ( wf_darcs_list_a_b @ T1 )
=> ( ( denormalize_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) )
= ( denormalize_a_b @ T1 ) ) ) ).
% normalize1_denormalize_eq'
thf(fact_1050_normalize1__dverts__app__contr,axiom,
! [V: list_a,T1: dtree_list_a_b] :
( ( member_list_a @ V @ ( dverts_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) ) )
=> ( ~ ( member_list_a @ V @ ( dverts_list_a_b @ T1 ) )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ ( dverts_list_a_b @ T1 ) )
& ? [Xa3: list_a] :
( ( member_list_a @ Xa3 @ ( dverts_list_a_b @ T1 ) )
& ( ( append_a @ X3 @ Xa3 )
= V )
& ( ord_less_real @ ( rank @ ( rev_a @ Xa3 ) ) @ ( rank @ ( rev_a @ X3 ) ) ) ) ) ) ) ).
% normalize1_dverts_app_contr
thf(fact_1051_root__normalize1__eq2,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,R2: list_a] :
( ! [X3: produc6499617310964463488_a_b_b] :
( Xs2
!= ( finser2303212525150181944_a_b_b @ X3 @ bot_bo2248824169281960260_a_b_b ) )
=> ( ( root_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ ( node_list_a_b @ R2 @ Xs2 ) ) )
= R2 ) ) ).
% root_normalize1_eq2
thf(fact_1052_normalize1__mdeg__eq_H,axiom,
! [T1: dtree_list_a_b] :
( ( list_wf_dlverts_a_b @ T1 )
=> ( ( ( max_deg_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) )
= ( max_deg_list_a_b @ T1 ) )
| ( ( ( max_deg_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) )
= zero_zero_nat )
& ( ( max_deg_list_a_b @ T1 )
= one_one_nat ) ) ) ) ).
% normalize1_mdeg_eq'
thf(fact_1053_normalize1__mdeg__eq,axiom,
! [T1: dtree_list_a_b] :
( ( wf_darcs_list_a_b @ T1 )
=> ( ( ( max_deg_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) )
= ( max_deg_list_a_b @ T1 ) )
| ( ( ( max_deg_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) )
= zero_zero_nat )
& ( ( max_deg_list_a_b @ T1 )
= one_one_nat ) ) ) ) ).
% normalize1_mdeg_eq
thf(fact_1054_sub__contr__if__new__contr,axiom,
! [T1: dtree_list_a_b,R2: list_a] :
( ~ ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R2 ) ) )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) ) ) ) @ ( rank @ ( rev_a @ R2 ) ) )
=> ? [V2: list_a,T23: dtree_list_a_b,E23: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ V2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E23 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T1 )
& ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T23 ) ) ) @ ( rank @ ( rev_a @ V2 ) ) ) ) ) ) ).
% sub_contr_if_new_contr
thf(fact_1055_contr__if__normalize1__uneq,axiom,
! [T1: dtree_list_a_b] :
( ( ( ranked8905849569120154423e1_a_b @ rank @ T1 )
!= T1 )
=> ? [V2: list_a,T23: dtree_list_a_b,E23: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ V2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E23 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T1 )
& ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T23 ) ) ) @ ( rank @ ( rev_a @ V2 ) ) ) ) ) ).
% contr_if_normalize1_uneq
thf(fact_1056_contr__before__normalize1,axiom,
! [V: list_a,T1: dtree_list_a_b,E12: b,T32: dtree_list_a_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( ranked8905849569120154423e1_a_b @ rank @ T32 ) )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ V ) ) )
=> ? [V3: list_a,T23: dtree_list_a_b,E23: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ V3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E23 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T32 )
& ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T23 ) ) ) @ ( rank @ ( rev_a @ V3 ) ) ) ) ) ) ).
% contr_before_normalize1
thf(fact_1057_root__normalize1__eq1,axiom,
! [T1: dtree_list_a_b,R2: list_a,E12: b] :
( ~ ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R2 ) ) )
=> ( ( root_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
= R2 ) ) ).
% root_normalize1_eq1
thf(fact_1058_root__normalize1__eq1_H,axiom,
! [T1: dtree_list_a_b,R2: list_a,E12: b] :
( ~ ( ord_less_eq_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R2 ) ) )
=> ( ( root_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
= R2 ) ) ).
% root_normalize1_eq1'
thf(fact_1059_child__contr__if__new__contr,axiom,
! [T1: dtree_list_a_b,R2: list_a] :
( ~ ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R2 ) ) )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) ) ) ) @ ( rank @ ( rev_a @ R2 ) ) )
=> ? [T23: dtree_list_a_b,E23: b] :
( ( ( sucs_list_a_b @ T1 )
= ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E23 ) @ bot_bo2248824169281960260_a_b_b ) )
& ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T23 ) ) ) @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) ) ) ) ) ).
% child_contr_if_new_contr
thf(fact_1060_normalize1__dverts__contr__subtree,axiom,
! [V: list_a,T1: dtree_list_a_b] :
( ( member_list_a @ V @ ( dverts_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) ) )
=> ( ~ ( member_list_a @ V @ ( dverts_list_a_b @ T1 ) )
=> ? [V23: list_a,T23: dtree_list_a_b,E23: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ V23 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E23 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T1 )
& ( ( append_a @ V23 @ ( root_list_a_b @ T23 ) )
= V )
& ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T23 ) ) ) @ ( rank @ ( rev_a @ V23 ) ) ) ) ) ) ).
% normalize1_dverts_contr_subtree
thf(fact_1061_normalize1__uneq__if__contr,axiom,
! [R1: list_a,T1: dtree_list_a_b,E12: b,T22: dtree_list_a_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T22 )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R1 ) ) )
=> ( ( wf_darcs_list_a_b @ T22 )
=> ( T22
!= ( ranked8905849569120154423e1_a_b @ rank @ T22 ) ) ) ) ) ).
% normalize1_uneq_if_contr
thf(fact_1062_sorted__ranks__if__normalize1__eq,axiom,
! [T22: dtree_list_a_b,R1: list_a,T1: dtree_list_a_b,E12: b] :
( ( wf_darcs_list_a_b @ T22 )
=> ( ( is_subtree_list_a_b @ ( node_list_a_b @ R1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T22 )
=> ( ( T22
= ( ranked8905849569120154423e1_a_b @ rank @ T22 ) )
=> ( ord_less_eq_real @ ( rank @ ( rev_a @ R1 ) ) @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) ) ) ) ) ).
% sorted_ranks_if_normalize1_eq
thf(fact_1063_normalize1__dlverts__eq,axiom,
! [T1: dtree_list_a_b] :
( ( list_dlverts_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) )
= ( list_dlverts_a_b @ T1 ) ) ).
% normalize1_dlverts_eq
thf(fact_1064_normalize1_Osimps_I1_J,axiom,
! [T1: dtree_list_a_b,R2: list_a,E: b] :
( ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R2 ) ) )
=> ( ( ranked8905849569120154423e1_a_b @ rank @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E ) @ bot_bo2248824169281960260_a_b_b ) ) )
= ( node_list_a_b @ ( append_a @ R2 @ ( root_list_a_b @ T1 ) ) @ ( sucs_list_a_b @ T1 ) ) ) )
& ( ~ ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R2 ) ) )
=> ( ( ranked8905849569120154423e1_a_b @ rank @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E ) @ bot_bo2248824169281960260_a_b_b ) ) )
= ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) @ E ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ).
% normalize1.simps(1)
thf(fact_1065_num__leaves__normalize1__eq,axiom,
! [T1: dtree_list_a_b] :
( ( wf_darcs_list_a_b @ T1 )
=> ( ( num_leaves_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) )
= ( num_leaves_list_a_b @ T1 ) ) ) ).
% num_leaves_normalize1_eq
thf(fact_1066_dom__children__normalize1__1,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b,T22: dtree_list_a_b,E22: b] :
( ( iKKBZ_3908525916494739553en_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t )
=> ( ( ( sucs_list_a_b @ T1 )
= ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ bot_bo2248824169281960260_a_b_b ) )
=> ( ( list_wf_dlverts_a_b @ T1 )
=> ( ( ( ranked8905849569120154423e1_a_b @ rank @ T1 )
= ( node_list_a_b @ ( append_a @ ( root_list_a_b @ T1 ) @ ( root_list_a_b @ T22 ) ) @ ( sucs_list_a_b @ T22 ) ) )
=> ( ( ( max_deg_list_a_b @ T1 )
= one_one_nat )
=> ( iKKBZ_3908525916494739553en_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t ) ) ) ) ) ) ).
% dom_children_normalize1_1
thf(fact_1067_merge__new__contr__fcard__gt1,axiom,
! [T1: dtree_list_a_b,V: list_a,T22: dtree_list_a_b,E22: b] :
( ! [R12: list_a,T23: dtree_list_a_b,E23: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R12 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E23 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T1 )
=> ( ord_less_eq_real @ ( rank @ ( rev_a @ R12 ) ) @ ( rank @ ( rev_a @ ( root_list_a_b @ T23 ) ) ) ) )
=> ( ( ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ bot_bo2248824169281960260_a_b_b ) )
= ( ranked_merge_a_b @ rank @ cmp @ T1 ) )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T22 ) ) ) @ ( rank @ ( rev_a @ V ) ) )
=> ( ord_less_nat @ one_one_nat @ ( fcard_4742106318756258927_a_b_b @ ( sucs_list_a_b @ T1 ) ) ) ) ) ) ).
% merge_new_contr_fcard_gt1
thf(fact_1068_max__deg1__gt__1__if__new__contr,axiom,
! [T0: dtree_list_a_b,R2: list_a,T1: dtree_list_a_b,E12: b] :
( ! [R12: list_a,T23: dtree_list_a_b,E23: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R12 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E23 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T0 )
=> ( ord_less_eq_real @ ( rank @ ( rev_a @ R12 ) ) @ ( rank @ ( rev_a @ ( root_list_a_b @ T23 ) ) ) ) )
=> ( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( ranked_merge1_a_b @ rank @ cmp @ T0 ) )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R2 ) ) )
=> ( ord_less_nat @ one_one_nat @ ( max_deg_list_a_b @ T0 ) ) ) ) ) ).
% max_deg1_gt_1_if_new_contr
thf(fact_1069_no__back__arcs_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ~ ! [X3: a,Xs: list_a] :
( X
!= ( cons_a @ X3 @ Xs ) ) ) ).
% no_back_arcs.cases
thf(fact_1070_forward__arcs_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X3: a] :
( X
!= ( cons_a @ X3 @ nil_a ) )
=> ~ ! [X3: a,V2: a,Va: list_a] :
( X
!= ( cons_a @ X3 @ ( cons_a @ V2 @ Va ) ) ) ) ) ).
% forward_arcs.cases
thf(fact_1071_before__conform1I,axiom,
! [S1: list_a,S22: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S22 )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ S1 ) ) ).
% before_conform1I
thf(fact_1072_before__conform2I,axiom,
! [S1: list_a,S22: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S22 )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ S22 ) ) ).
% before_conform2I
thf(fact_1073_seq__conform__if__before,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ Xs2 @ Ys )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ ( append_a @ Xs2 @ Ys ) ) ) ).
% seq_conform_if_before
thf(fact_1074_seq__conform__single,axiom,
! [X: a] : ( iKKBZ_4622586873178280511rm_a_b @ t @ ( cons_a @ X @ nil_a ) ) ).
% seq_conform_single
thf(fact_1075_normalize__full__dom__preserv,axiom,
! [T1: dtree_list_a_b] :
( ( iKKBZ_3908525916494739553en_a_b @ T1 @ t )
=> ( iKKBZ_3908525916494739553en_a_b @ ( iKKBZ_6959927528703686640ll_a_b @ T1 ) @ t ) ) ).
% normalize_full_dom_preserv
thf(fact_1076_merge1__dverts__sub,axiom,
! [T1: dtree_list_a_b] : ( ord_le8861187494160871172list_a @ ( dverts_list_a_b @ ( ranked_merge1_a_b @ rank @ cmp @ T1 ) ) @ ( dverts_list_a_b @ T1 ) ) ).
% merge1_dverts_sub
thf(fact_1077_merge__empty,axiom,
! [R2: list_a] :
( ( ranked_merge_a_b @ rank @ cmp @ ( node_list_a_b @ R2 @ bot_bo2248824169281960260_a_b_b ) )
= ( node_list_a_b @ R2 @ bot_bo2248824169281960260_a_b_b ) ) ).
% merge_empty
thf(fact_1078_subtree__merge1__orig,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,T1: dtree_list_a_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ ( ranked_merge1_a_b @ rank @ cmp @ T1 ) )
=> ? [Ys2: fset_P2153231429829016240_a_b_b] : ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Ys2 ) @ T1 ) ) ).
% subtree_merge1_orig
thf(fact_1079_merge1__mdeg__le,axiom,
! [T1: dtree_list_a_b] : ( ord_less_eq_nat @ ( max_deg_list_a_b @ ( ranked_merge1_a_b @ rank @ cmp @ T1 ) ) @ ( max_deg_list_a_b @ T1 ) ) ).
% merge1_mdeg_le
thf(fact_1080_num__leaves__merge1__le,axiom,
! [T1: dtree_list_a_b] : ( ord_less_eq_nat @ ( num_leaves_list_a_b @ ( ranked_merge1_a_b @ rank @ cmp @ T1 ) ) @ ( num_leaves_list_a_b @ T1 ) ) ).
% num_leaves_merge1_le
thf(fact_1081_merge1__mdeg__gt1__if__uneq,axiom,
! [T1: dtree_list_a_b] :
( ( ( ranked_merge1_a_b @ rank @ cmp @ T1 )
!= T1 )
=> ( ord_less_nat @ one_one_nat @ ( max_deg_list_a_b @ T1 ) ) ) ).
% merge1_mdeg_gt1_if_uneq
thf(fact_1082_merge1__eq__if__mdeg__le1,axiom,
! [T1: dtree_list_a_b] :
( ( ord_less_eq_nat @ ( max_deg_list_a_b @ T1 ) @ one_one_nat )
=> ( ( ranked_merge1_a_b @ rank @ cmp @ T1 )
= T1 ) ) ).
% merge1_eq_if_mdeg_le1
thf(fact_1083_merge__mdeg__le__1,axiom,
! [T1: dtree_list_a_b] : ( ord_less_eq_nat @ ( max_deg_list_a_b @ ( ranked_merge_a_b @ rank @ cmp @ T1 ) ) @ one_one_nat ) ).
% merge_mdeg_le_1
thf(fact_1084_dverts__if__subtree__merge1,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,T1: dtree_list_a_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ ( ranked_merge1_a_b @ rank @ cmp @ T1 ) )
=> ( member_list_a @ R2 @ ( dverts_list_a_b @ T1 ) ) ) ).
% dverts_if_subtree_merge1
thf(fact_1085_merge__cases,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
? [T2: dtree_list_a_b,E2: b] :
( ( ( ranked_merge_a_b @ rank @ cmp @ ( node_list_a_b @ R2 @ Xs2 ) )
= ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T2 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
| ( ( ranked_merge_a_b @ rank @ cmp @ ( node_list_a_b @ R2 @ Xs2 ) )
= ( node_list_a_b @ R2 @ bot_bo2248824169281960260_a_b_b ) ) ) ).
% merge_cases
thf(fact_1086_merge1__child__in__orig,axiom,
! [R2: list_a,Ys: fset_P2153231429829016240_a_b_b,Xs2: fset_P2153231429829016240_a_b_b,T1: dtree_list_a_b,E12: b] :
( ( ( ranked_merge1_a_b @ rank @ cmp @ ( node_list_a_b @ R2 @ Ys ) )
= ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ? [T23: dtree_list_a_b] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E12 ) @ ( fset_P9138963618725001425_a_b_b @ Ys ) )
& ( ( root_list_a_b @ T23 )
= ( root_list_a_b @ T1 ) ) ) ) ) ).
% merge1_child_in_orig
thf(fact_1087_merge__mdeg__le1__sub,axiom,
! [T1: dtree_list_a_b,T22: dtree_list_a_b] :
( ( is_subtree_list_a_b @ T1 @ ( ranked_merge_a_b @ rank @ cmp @ T22 ) )
=> ( ord_less_eq_nat @ ( max_deg_list_a_b @ T1 ) @ one_one_nat ) ) ).
% merge_mdeg_le1_sub
thf(fact_1088_merge__empty__sucs,axiom,
! [T1: dtree_list_a_b] :
( ( ( sucs_list_a_b @ T1 )
= bot_bo2248824169281960260_a_b_b )
=> ( ( ranked_merge_a_b @ rank @ cmp @ T1 )
= ( node_list_a_b @ ( root_list_a_b @ T1 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ).
% merge_empty_sucs
thf(fact_1089_merge1__fcard__le,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] : ( ord_less_eq_nat @ ( fcard_4742106318756258927_a_b_b @ ( sucs_list_a_b @ ( ranked_merge1_a_b @ rank @ cmp @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) ) ).
% merge1_fcard_le
thf(fact_1090_merge__fcard__le1,axiom,
! [T1: dtree_list_a_b] : ( ord_less_eq_nat @ ( fcard_4742106318756258927_a_b_b @ ( sucs_list_a_b @ ( ranked_merge_a_b @ rank @ cmp @ T1 ) ) ) @ one_one_nat ) ).
% merge_fcard_le1
thf(fact_1091_num__leaves__merge1__lt,axiom,
! [T1: dtree_list_a_b] :
( ( ord_less_nat @ one_one_nat @ ( max_deg_list_a_b @ T1 ) )
=> ( ord_less_nat @ ( num_leaves_list_a_b @ ( ranked_merge1_a_b @ rank @ cmp @ T1 ) ) @ ( num_leaves_list_a_b @ T1 ) ) ) ).
% num_leaves_merge1_lt
thf(fact_1092_merge__cases__sucs,axiom,
! [T1: dtree_list_a_b] :
? [T2: dtree_list_a_b,E2: b] :
( ( ( ranked_merge_a_b @ rank @ cmp @ T1 )
= ( node_list_a_b @ ( root_list_a_b @ T1 ) @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T2 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
| ( ( ranked_merge_a_b @ rank @ cmp @ T1 )
= ( node_list_a_b @ ( root_list_a_b @ T1 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ).
% merge_cases_sucs
thf(fact_1093_merge__root__child__eq,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b,T22: dtree_list_a_b,E22: b] :
( ( ( ranked_merge_a_b @ rank @ cmp @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) )
= ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( root_list_a_b @ T1 )
= ( root_list_a_b @ T22 ) ) ) ).
% merge_root_child_eq
thf(fact_1094_merge1__subtree__if__mdeg__gt1,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,T1: dtree_list_a_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ ( ranked_merge1_a_b @ rank @ cmp @ T1 ) )
=> ( ( ord_less_nat @ one_one_nat @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) )
=> ? [Ys2: fset_P2153231429829016240_a_b_b] :
( ( ( ranked_merge1_a_b @ rank @ cmp @ ( node_list_a_b @ R2 @ Ys2 ) )
= ( node_list_a_b @ R2 @ Xs2 ) )
& ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Ys2 ) @ T1 ) ) ) ) ).
% merge1_subtree_if_mdeg_gt1
thf(fact_1095_merge__subtree__fcard__le1,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,T1: dtree_list_a_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ ( ranked_merge_a_b @ rank @ cmp @ T1 ) )
=> ( ord_less_eq_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) @ one_one_nat ) ) ).
% merge_subtree_fcard_le1
thf(fact_1096_merge1__not__merge__if__fcard__gt1,axiom,
! [R2: list_a,Ys: fset_P2153231429829016240_a_b_b,Xs2: fset_P2153231429829016240_a_b_b] :
( ( ( ranked_merge1_a_b @ rank @ cmp @ ( node_list_a_b @ R2 @ Ys ) )
= ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ( ord_less_nat @ one_one_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) )
=> ( ( ranked_merge_a_b @ rank @ cmp @ ( node_list_a_b @ R2 @ Ys ) )
!= ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ).
% merge1_not_merge_if_fcard_gt1
thf(fact_1097_dom__children__normalize1__preserv,axiom,
! [T1: dtree_list_a_b] :
( ( ord_less_eq_nat @ ( max_deg_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) ) @ one_one_nat )
=> ( ( iKKBZ_3908525916494739553en_a_b @ T1 @ t )
=> ( ( list_wf_dlverts_a_b @ T1 )
=> ( iKKBZ_3908525916494739553en_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) @ t ) ) ) ) ).
% dom_children_normalize1_preserv
thf(fact_1098_merge__fcard__le1__sub,axiom,
! [T1: dtree_list_a_b,T22: dtree_list_a_b] :
( ( is_subtree_list_a_b @ T1 @ ( ranked_merge_a_b @ rank @ cmp @ T22 ) )
=> ( ord_less_eq_nat @ ( fcard_4742106318756258927_a_b_b @ ( sucs_list_a_b @ T1 ) ) @ one_one_nat ) ) ).
% merge_fcard_le1_sub
thf(fact_1099_dom__children__combine,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b] :
( ( iKKBZ_3908525916494739553en_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t )
=> ( iKKBZ_3908525916494739553en_a_b @ ( node_list_a_b @ ( append_a @ R2 @ ( root_list_a_b @ T1 ) ) @ ( sucs_list_a_b @ T1 ) ) @ t ) ) ).
% dom_children_combine
thf(fact_1100_merge1__elem__in__img__if__fcard__gt1,axiom,
! [R2: list_a,Ys: fset_P2153231429829016240_a_b_b,Xs2: fset_P2153231429829016240_a_b_b,T22: dtree_list_a_b,E22: b] :
( ( ( ranked_merge1_a_b @ rank @ cmp @ ( node_list_a_b @ R2 @ Ys ) )
= ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ( ord_less_nat @ one_one_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ? [T12: dtree_list_a_b] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Ys ) )
& ( ( ranked_merge1_a_b @ rank @ cmp @ T12 )
= T22 ) ) ) ) ) ).
% merge1_elem_in_img_if_fcard_gt1
thf(fact_1101_merge1__subtree__if__fcard__gt1,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,T1: dtree_list_a_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ ( ranked_merge1_a_b @ rank @ cmp @ T1 ) )
=> ( ( ord_less_nat @ one_one_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) )
=> ? [Ys2: fset_P2153231429829016240_a_b_b] :
( ( ( ranked_merge1_a_b @ rank @ cmp @ ( node_list_a_b @ R2 @ Ys2 ) )
= ( node_list_a_b @ R2 @ Xs2 ) )
& ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Ys2 ) @ T1 )
& ( ord_less_eq_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) @ ( fcard_4742106318756258927_a_b_b @ Ys2 ) ) ) ) ) ).
% merge1_subtree_if_fcard_gt1
thf(fact_1102_merge__single__root,axiom,
! [T22: dtree_list_a_b,E22: b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ ( ranked_merge_a_b @ rank @ cmp @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) )
=> ( ( ranked_merge_a_b @ rank @ cmp @ ( node_list_a_b @ R2 @ Xs2 ) )
= ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).
% merge_single_root
thf(fact_1103_merge__single__root__sucs,axiom,
! [T22: dtree_list_a_b,E22: b,T1: dtree_list_a_b] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ ( ranked_merge_a_b @ rank @ cmp @ T1 ) ) ) )
=> ( ( ranked_merge_a_b @ rank @ cmp @ T1 )
= ( node_list_a_b @ ( root_list_a_b @ T1 ) @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).
% merge_single_root_sucs
thf(fact_1104_merge__strict__subtree__nocontr2,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,R1: list_a,T1: dtree_list_a_b,E12: b] :
( ! [R12: list_a,T12: dtree_list_a_b,E1: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R12 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ord_less_eq_real @ ( rank @ ( rev_a @ R12 ) ) @ ( rank @ ( rev_a @ ( root_list_a_b @ T12 ) ) ) ) )
=> ( ( strict8995144569104247066st_a_b @ ( node_list_a_b @ R1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( ranked_merge_a_b @ rank @ cmp @ ( node_list_a_b @ R2 @ Xs2 ) ) )
=> ( ord_less_eq_real @ ( rank @ ( rev_a @ R1 ) ) @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) ) ) ) ).
% merge_strict_subtree_nocontr2
thf(fact_1105_merge__strict__subtree__nocontr__sucs2,axiom,
! [T22: dtree_list_a_b,R1: list_a,T1: dtree_list_a_b,E12: b] :
( ! [R12: list_a,T12: dtree_list_a_b,E1: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R12 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T22 )
=> ( ord_less_eq_real @ ( rank @ ( rev_a @ R12 ) ) @ ( rank @ ( rev_a @ ( root_list_a_b @ T12 ) ) ) ) )
=> ( ( strict8995144569104247066st_a_b @ ( node_list_a_b @ R1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( ranked_merge_a_b @ rank @ cmp @ T22 ) )
=> ( ord_less_eq_real @ ( rank @ ( rev_a @ R1 ) ) @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) ) ) ) ).
% merge_strict_subtree_nocontr_sucs2
thf(fact_1106_merge1__subtree__if__new__contr,axiom,
! [T0: dtree_list_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,V: list_a,T1: dtree_list_a_b,E12: b] :
( ! [R12: list_a,T23: dtree_list_a_b,E23: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R12 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E23 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T0 )
=> ( ord_less_eq_real @ ( rank @ ( rev_a @ R12 ) ) @ ( rank @ ( rev_a @ ( root_list_a_b @ T23 ) ) ) ) )
=> ( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ ( ranked_merge1_a_b @ rank @ cmp @ T0 ) )
=> ( ( is_subtree_list_a_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ V ) ) )
=> ? [Ys2: fset_P2153231429829016240_a_b_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Ys2 ) @ T0 )
& ( ( ranked_merge1_a_b @ rank @ cmp @ ( node_list_a_b @ R2 @ Ys2 ) )
= ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ) ) ) ).
% merge1_subtree_if_new_contr
thf(fact_1107_merge__root__if__contr,axiom,
! [T1: dtree_list_a_b,V: list_a,T22: dtree_list_a_b,E22: b] :
( ! [R12: list_a,T23: dtree_list_a_b,E23: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R12 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E23 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T1 )
=> ( ord_less_eq_real @ ( rank @ ( rev_a @ R12 ) ) @ ( rank @ ( rev_a @ ( root_list_a_b @ T23 ) ) ) ) )
=> ( ( is_subtree_list_a_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( ranked_merge_a_b @ rank @ cmp @ T1 ) )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T22 ) ) ) @ ( rank @ ( rev_a @ V ) ) )
=> ( ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ bot_bo2248824169281960260_a_b_b ) )
= ( ranked_merge_a_b @ rank @ cmp @ T1 ) ) ) ) ) ).
% merge_root_if_contr
thf(fact_1108_dom__children__normalize1,axiom,
! [R0: list_a,T1: dtree_list_a_b,E12: b] :
( ( iKKBZ_3908525916494739553en_a_b @ ( node_list_a_b @ R0 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t )
=> ( ( list_wf_dlverts_a_b @ T1 )
=> ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ T1 ) @ one_one_nat )
=> ( iKKBZ_3908525916494739553en_a_b @ ( node_list_a_b @ R0 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t ) ) ) ) ).
% dom_children_normalize1
thf(fact_1109_merge1__root__eq,axiom,
! [T1: dtree_list_a_b] :
( ( root_list_a_b @ ( ranked_merge1_a_b @ rank @ cmp @ T1 ) )
= ( root_list_a_b @ T1 ) ) ).
% merge1_root_eq
thf(fact_1110_merge__root__eq,axiom,
! [T1: dtree_list_a_b] :
( ( root_list_a_b @ ( ranked_merge_a_b @ rank @ cmp @ T1 ) )
= ( root_list_a_b @ T1 ) ) ).
% merge_root_eq
thf(fact_1111_loopfree_Oloopfree__digraph__axioms,axiom,
loopfree_digraph_a_b @ t ).
% loopfree.loopfree_digraph_axioms
thf(fact_1112_nomulti_Onomulti__digraph__axioms,axiom,
nomulti_digraph_a_b @ t ).
% nomulti.nomulti_digraph_axioms
thf(fact_1113_source__nmem__k__nh,axiom,
! [V: a,W2: b > real,K: real] :
~ ( member_a @ V @ ( graph_3921080825633621230od_a_b @ t @ W2 @ V @ K ) ) ).
% source_nmem_k_nh
thf(fact_1114_combine__forward,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b,V: list_a] :
( ( iKKBZ_3908525916494739553en_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( dverts_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ X3 ) )
=> ( ( list_wf_dlverts_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ ( append_a @ R2 @ ( root_list_a_b @ T1 ) ) @ ( sucs_list_a_b @ T1 ) ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ V ) ) ) ) ) ).
% combine_forward
thf(fact_1115_forward__split,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ Xs2 @ Ys ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs2 ) ) ).
% forward_split
thf(fact_1116_before__forward1I,axiom,
! [S1: list_a,S22: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S22 )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ S1 ) ) ).
% before_forward1I
thf(fact_1117_before__forward2I,axiom,
! [S1: list_a,S22: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S22 )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ S22 ) ) ).
% before_forward2I
thf(fact_1118_seq__conform__if__dstnct__fwd,axiom,
! [Xs2: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs2 )
=> ( ( distinct_a @ Xs2 )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ Xs2 ) ) ) ).
% seq_conform_if_dstnct_fwd
thf(fact_1119_forward__cons,axiom,
! [X: a,Xs2: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( rev_a @ ( cons_a @ X @ Xs2 ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( rev_a @ Xs2 ) ) ) ).
% forward_cons
thf(fact_1120_forward__single,axiom,
! [X: a] : ( iKKBZ_4778857019735642799rd_a_b @ t @ ( cons_a @ X @ nil_a ) ) ).
% forward_single
thf(fact_1121_move__mid__backward__if__noarc,axiom,
! [U: list_a,V4: list_a,As: list_a,Bs: list_a,Cs: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ U @ V4 )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ U @ ( append_a @ Bs @ ( append_a @ V4 @ Cs ) ) ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ U @ ( append_a @ V4 @ ( append_a @ Bs @ Cs ) ) ) ) ) ) ) ).
% move_mid_backward_if_noarc
thf(fact_1122_normalize__full__forward,axiom,
! [T1: dtree_list_a_b] :
( ( iKKBZ_3908525916494739553en_a_b @ T1 @ t )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( dverts_list_a_b @ T1 ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ X3 ) )
=> ( ( list_wf_dlverts_a_b @ T1 )
=> ! [X2: list_a] :
( ( member_list_a @ X2 @ ( dverts_list_a_b @ ( iKKBZ_6959927528703686640ll_a_b @ T1 ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ X2 ) ) ) ) ) ).
% normalize_full_forward
thf(fact_1123_merge__singleton__sucs,axiom,
! [T1: dtree_list_a_b] :
( ( list_list_dtree_a_b @ ( node_list_a_b @ ( root_list_a_b @ T1 ) @ ( sucs_list_a_b @ T1 ) ) )
=> ( ( ( sucs_list_a_b @ T1 )
!= bot_bo2248824169281960260_a_b_b )
=> ? [T2: dtree_list_a_b,E2: b] :
( ( ranked_merge_a_b @ rank @ cmp @ T1 )
= ( node_list_a_b @ ( root_list_a_b @ T1 ) @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T2 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ).
% merge_singleton_sucs
thf(fact_1124_not__single__subtree__if__nwf__sucs,axiom,
! [T22: dtree_list_a_b,R1: list_a,T1: dtree_list_a_b,E12: b] :
( ~ ( list_list_dtree_a_b @ T22 )
=> ~ ( is_subtree_list_a_b @ ( node_list_a_b @ R1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( ranked_merge_a_b @ rank @ cmp @ T22 ) ) ) ).
% not_single_subtree_if_nwf_sucs
thf(fact_1125_merge__empty__if__nwf,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
( ~ ( list_list_dtree_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ( ranked_merge_a_b @ rank @ cmp @ ( node_list_a_b @ R2 @ Xs2 ) )
= ( node_list_a_b @ R2 @ bot_bo2248824169281960260_a_b_b ) ) ) ).
% merge_empty_if_nwf
thf(fact_1126_merge__nempty__sucs,axiom,
! [T1: dtree_list_a_b] :
( ( list_list_dtree_a_b @ T1 )
=> ( ( ( sucs_list_a_b @ T1 )
!= bot_bo2248824169281960260_a_b_b )
=> ( ( sucs_list_a_b @ ( ranked_merge_a_b @ rank @ cmp @ T1 ) )
!= bot_bo2248824169281960260_a_b_b ) ) ) ).
% merge_nempty_sucs
thf(fact_1127_merge__empty__if__nwf__sucs,axiom,
! [T1: dtree_list_a_b] :
( ~ ( list_list_dtree_a_b @ T1 )
=> ( ( ranked_merge_a_b @ rank @ cmp @ T1 )
= ( node_list_a_b @ ( root_list_a_b @ T1 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ).
% merge_empty_if_nwf_sucs
thf(fact_1128_merge__nempty,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
( ( list_list_dtree_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ( Xs2 != bot_bo2248824169281960260_a_b_b )
=> ( ( sucs_list_a_b @ ( ranked_merge_a_b @ rank @ cmp @ ( node_list_a_b @ R2 @ Xs2 ) ) )
!= bot_bo2248824169281960260_a_b_b ) ) ) ).
% merge_nempty
thf(fact_1129_merge__dom__children__if__ndisjoint,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
( ~ ( list_list_dtree_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( iKKBZ_3908525916494739553en_a_b @ ( ranked_merge_a_b @ rank @ cmp @ ( node_list_a_b @ R2 @ Xs2 ) ) @ t ) ) ).
% merge_dom_children_if_ndisjoint
thf(fact_1130_merge__singleton,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
( ( list_list_dtree_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ( Xs2 != bot_bo2248824169281960260_a_b_b )
=> ? [T2: dtree_list_a_b,E2: b] :
( ( ranked_merge_a_b @ rank @ cmp @ ( node_list_a_b @ R2 @ Xs2 ) )
= ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T2 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ).
% merge_singleton
thf(fact_1131_merge__disjoint__if__child,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b,T22: dtree_list_a_b,E22: b] :
( ( ( ranked_merge_a_b @ rank @ cmp @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) )
= ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( list_list_dtree_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).
% merge_disjoint_if_child
thf(fact_1132_not__single__subtree__if__nwf,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,R1: list_a,T1: dtree_list_a_b,E12: b] :
( ~ ( list_list_dtree_a_b @ ( node_list_a_b @ R2 @ Xs2 ) )
=> ~ ( is_subtree_list_a_b @ ( node_list_a_b @ R1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( ranked_merge_a_b @ rank @ cmp @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ).
% not_single_subtree_if_nwf
thf(fact_1133_merge1__dom__contr,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b] :
( ! [R12: list_a,T23: dtree_list_a_b,E23: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R12 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E23 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ( ord_less_eq_real @ ( rank @ ( rev_a @ R12 ) ) @ ( rank @ ( rev_a @ ( root_list_a_b @ T23 ) ) ) ) )
=> ( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( ranked_merge1_a_b @ rank @ cmp @ t2 ) )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R2 ) ) )
=> ( ( ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) )
= one_one_nat )
=> ( iKKBZ_3908525916494739553en_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t ) ) ) ) ) ).
% merge1_dom_contr
thf(fact_1134_normalize1__dom__contr,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( ranked8905849569120154423e1_a_b @ rank @ t2 ) )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R2 ) ) )
=> ( ( ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) )
= one_one_nat )
=> ( iKKBZ_3908525916494739553en_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t ) ) ) ) ).
% normalize1_dom_contr
thf(fact_1135_v__in__comb__if__in__dlverts,axiom,
! [V: a,X: list_a,Y: list_a] :
( ( member_a @ V @ ( list_dlverts_a_b @ t2 ) )
=> ( member_a @ V @ ( list_dlverts_a_b @ ( list_combine_a_b @ X @ Y @ t2 ) ) ) ) ).
% v_in_comb_if_in_dlverts
thf(fact_1136_v__in__dlverts__if__in__comb,axiom,
! [V: a,X: list_a,Y: list_a] :
( ( member_a @ V @ ( list_dlverts_a_b @ ( list_combine_a_b @ X @ Y @ t2 ) ) )
=> ( member_a @ V @ ( list_dlverts_a_b @ t2 ) ) ) ).
% v_in_dlverts_if_in_comb
thf(fact_1137_list__dtree__axioms,axiom,
list_list_dtree_a_b @ t2 ).
% list_dtree_axioms
thf(fact_1138_list__dtree__comb,axiom,
! [X: list_a,Y: list_a] : ( list_list_dtree_a_b @ ( list_combine_a_b @ X @ Y @ t2 ) ) ).
% list_dtree_comb
thf(fact_1139_wf__dlverts__combine,axiom,
! [X: list_a,Y: list_a] : ( list_wf_dlverts_a_b @ ( list_combine_a_b @ X @ Y @ t2 ) ) ).
% wf_dlverts_combine
thf(fact_1140_wf__lverts,axiom,
list_wf_dlverts_a_b @ t2 ).
% wf_lverts
thf(fact_1141_wf__darcs__combine,axiom,
! [X: list_a,Y: list_a] : ( wf_darcs_list_a_b @ ( list_combine_a_b @ X @ Y @ t2 ) ) ).
% wf_darcs_combine
thf(fact_1142_wf__arcs,axiom,
wf_darcs_list_a_b @ t2 ).
% wf_arcs
thf(fact_1143_wf__verts,axiom,
wf_dverts_list_a_b @ t2 ).
% wf_verts
thf(fact_1144_ranked__dtree__axioms,axiom,
ranked_dtree_a_b @ t2 @ cmp ).
% ranked_dtree_axioms
thf(fact_1145_dverts__same__if__set__subtree,axiom,
! [T1: dtree_list_a_b,V1: list_a,X: a,V22: list_a] :
( ( is_subtree_list_a_b @ T1 @ t2 )
=> ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V22 ) )
=> ( ( member_list_a @ V22 @ ( dverts_list_a_b @ t2 ) )
=> ( V1 = V22 ) ) ) ) ) ) ).
% dverts_same_if_set_subtree
thf(fact_1146_list__dtree__sub,axiom,
! [X: dtree_list_a_b] :
( ( is_subtree_list_a_b @ X @ t2 )
=> ( list_list_dtree_a_b @ X ) ) ).
% list_dtree_sub
thf(fact_1147_verts__distinct,axiom,
! [V: list_a] :
( ( member_list_a @ V @ ( dverts_list_a_b @ t2 ) )
=> ( distinct_a @ V ) ) ).
% verts_distinct
thf(fact_1148_ranked__dtree__subtree,axiom,
! [X: dtree_list_a_b] :
( ( is_subtree_list_a_b @ X @ t2 )
=> ( ranked_dtree_a_b @ X @ cmp ) ) ).
% ranked_dtree_subtree
thf(fact_1149_arc__uneq__if__subtree__uneq,axiom,
! [X1: dtree_list_a_b,E12: b,Xs2: fset_P2153231429829016240_a_b_b,X22: dtree_list_a_b,E22: b,R2: list_a] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X1 @ E12 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X22 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( X1 != X22 )
=> ( ( ( node_list_a_b @ R2 @ Xs2 )
= t2 )
=> ( E12 != E22 ) ) ) ) ) ).
% arc_uneq_if_subtree_uneq
thf(fact_1150_subtree__uneq__if__arc__uneq,axiom,
! [X1: dtree_list_a_b,E12: b,Xs2: fset_P2153231429829016240_a_b_b,X22: dtree_list_a_b,E22: b,R2: list_a] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X1 @ E12 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X22 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( E12 != E22 )
=> ( ( ( node_list_a_b @ R2 @ Xs2 )
= t2 )
=> ( X1 != X22 ) ) ) ) ) ).
% subtree_uneq_if_arc_uneq
thf(fact_1151_verts__distinct__subtree,axiom,
! [Tn: dtree_list_a_b,V: list_a] :
( ( is_subtree_list_a_b @ Tn @ t2 )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ Tn ) )
=> ( distinct_a @ V ) ) ) ).
% verts_distinct_subtree
thf(fact_1152_verts__forward,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( dverts_list_a_b @ t2 ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ X2 ) ) ).
% verts_forward
thf(fact_1153_list__dtree__normalize1,axiom,
list_list_dtree_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ t2 ) ).
% list_dtree_normalize1
thf(fact_1154_verts__conform,axiom,
! [V: list_a] :
( ( member_list_a @ V @ ( dverts_list_a_b @ t2 ) )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ V ) ) ).
% verts_conform
thf(fact_1155_child__arc__not__subtree,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b,E12: b] :
( ( ( node_list_a_b @ R2 @ Xs2 )
= t2 )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E12 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ~ ( member_b @ E12 @ ( darcs_list_a_b @ X ) ) ) ) ).
% child_arc_not_subtree
thf(fact_1156_subtree__root__not__root,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b,E: b] :
( ( t2
= ( node_list_a_b @ R2 @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( root_list_a_b @ X )
!= R2 ) ) ) ).
% subtree_root_not_root
thf(fact_1157_list__dtree__rec,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b,E: b] :
( ( ( node_list_a_b @ R2 @ Xs2 )
= t2 )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( list_list_dtree_a_b @ X ) ) ) ).
% list_dtree_rec
thf(fact_1158_ranked__dtree__normalize1,axiom,
ranked_dtree_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ t2 ) @ cmp ).
% ranked_dtree_normalize1
thf(fact_1159_ranked__dtree__merge1,axiom,
ranked_dtree_a_b @ ( ranked_merge1_a_b @ rank @ cmp @ t2 ) @ cmp ).
% ranked_dtree_merge1
thf(fact_1160_merge__ranked__dtree,axiom,
ranked_dtree_a_b @ ( ranked_merge_a_b @ rank @ cmp @ t2 ) @ cmp ).
% merge_ranked_dtree
thf(fact_1161_list__dtree__rec__suc,axiom,
! [X: dtree_list_a_b,E: b] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ t2 ) ) )
=> ( list_list_dtree_a_b @ X ) ) ).
% list_dtree_rec_suc
thf(fact_1162_ranked__dtree__rec,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b,E: b] :
( ( ( node_list_a_b @ R2 @ Xs2 )
= t2 )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ranked_dtree_a_b @ X @ cmp ) ) ) ).
% ranked_dtree_rec
thf(fact_1163_ranked__dtree__rec__suc,axiom,
! [X: dtree_list_a_b,E: b] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ t2 ) ) )
=> ( ranked_dtree_a_b @ X @ cmp ) ) ).
% ranked_dtree_rec_suc
thf(fact_1164_normalize1__verts__distinct,axiom,
! [V: list_a] :
( ( member_list_a @ V @ ( dverts_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ t2 ) ) )
=> ( distinct_a @ V ) ) ).
% normalize1_verts_distinct
thf(fact_1165_distinct__normalize1,axiom,
! [V: list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( dverts_list_a_b @ t2 ) )
=> ( distinct_a @ X3 ) )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ t2 ) ) )
=> ( distinct_a @ V ) ) ) ).
% distinct_normalize1
thf(fact_1166_verts__conform__subtree,axiom,
! [Tn: dtree_list_a_b,V: list_a] :
( ( is_subtree_list_a_b @ Tn @ t2 )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ Tn ) )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ V ) ) ) ).
% verts_conform_subtree
thf(fact_1167_merge__list__dtree,axiom,
list_list_dtree_a_b @ ( ranked_merge_a_b @ rank @ cmp @ t2 ) ).
% merge_list_dtree
thf(fact_1168_wf__dlverts__merge1,axiom,
list_wf_dlverts_a_b @ ( ranked_merge1_a_b @ rank @ cmp @ t2 ) ).
% wf_dlverts_merge1
thf(fact_1169_merge__wf__dlverts,axiom,
list_wf_dlverts_a_b @ ( ranked_merge_a_b @ rank @ cmp @ t2 ) ).
% merge_wf_dlverts
thf(fact_1170_wf__darcs__merge1,axiom,
wf_darcs_list_a_b @ ( ranked_merge1_a_b @ rank @ cmp @ t2 ) ).
% wf_darcs_merge1
thf(fact_1171_merge__wf__darcs,axiom,
wf_darcs_list_a_b @ ( ranked_merge_a_b @ rank @ cmp @ t2 ) ).
% merge_wf_darcs
thf(fact_1172_distinct__merge1,axiom,
! [V: list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( dverts_list_a_b @ t2 ) )
=> ( distinct_a @ X3 ) )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ ( ranked_merge1_a_b @ rank @ cmp @ t2 ) ) )
=> ( distinct_a @ V ) ) ) ).
% distinct_merge1
thf(fact_1173_merge1__verts__distinct,axiom,
! [V: list_a] :
( ( member_list_a @ V @ ( dverts_list_a_b @ ( ranked_merge1_a_b @ rank @ cmp @ t2 ) ) )
=> ( distinct_a @ V ) ) ).
% merge1_verts_distinct
thf(fact_1174_distinct__merge,axiom,
! [V: list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( dverts_list_a_b @ t2 ) )
=> ( distinct_a @ X3 ) )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ ( ranked_merge_a_b @ rank @ cmp @ t2 ) ) )
=> ( distinct_a @ V ) ) ) ).
% distinct_merge
thf(fact_1175_normalize1__verts__conform,axiom,
! [V: list_a] :
( ( member_list_a @ V @ ( dverts_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ t2 ) ) )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ V ) ) ).
% normalize1_verts_conform
thf(fact_1176_merge1__subtree__dlverts__supset,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ ( ranked_merge1_a_b @ rank @ cmp @ t2 ) )
=> ? [Ys2: fset_P2153231429829016240_a_b_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Ys2 ) @ t2 )
& ( ord_less_eq_set_a @ ( list_dlverts_a_b @ ( node_list_a_b @ R2 @ Ys2 ) ) @ ( list_dlverts_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ) ).
% merge1_subtree_dlverts_supset
thf(fact_1177_merge1__verts__conform,axiom,
! [V: list_a] :
( ( member_list_a @ V @ ( dverts_list_a_b @ ( ranked_merge1_a_b @ rank @ cmp @ t2 ) ) )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ V ) ) ).
% merge1_verts_conform
thf(fact_1178_dom__mdeg__le1__normalize1,axiom,
( ( ord_less_eq_nat @ ( max_deg_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ t2 ) ) @ one_one_nat )
=> ( ( ( ranked8905849569120154423e1_a_b @ rank @ t2 )
!= t2 )
=> ( iKKBZ_3908525916494739553en_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ t2 ) @ t ) ) ) ).
% dom_mdeg_le1_normalize1
thf(fact_1179_distint__verts__singleton__subtree,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ( distinct_a @ ( append_a @ R2 @ ( root_list_a_b @ T1 ) ) ) ) ).
% distint_verts_singleton_subtree
thf(fact_1180_dlverts__comb__id,axiom,
! [X: list_a,Y: list_a] :
( ( list_dlverts_a_b @ ( list_combine_a_b @ X @ Y @ t2 ) )
= ( list_dlverts_a_b @ t2 ) ) ).
% dlverts_comb_id
thf(fact_1181_dom__mdeg__le1__merge1,axiom,
( ( ord_less_eq_nat @ ( max_deg_list_a_b @ ( ranked_merge1_a_b @ rank @ cmp @ t2 ) ) @ one_one_nat )
=> ( ( ( ranked_merge1_a_b @ rank @ cmp @ t2 )
!= t2 )
=> ( iKKBZ_3908525916494739553en_a_b @ ( ranked_merge1_a_b @ rank @ cmp @ t2 ) @ t ) ) ) ).
% dom_mdeg_le1_merge1
thf(fact_1182_dom__children__child__self,axiom,
! [T1: dtree_list_a_b,T22: dtree_list_a_b,E22: b,R2: list_a,E12: b] :
( ( iKKBZ_3908525916494739553en_a_b @ T1 @ t )
=> ( ( ( sucs_list_a_b @ T1 )
= ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ bot_bo2248824169281960260_a_b_b ) )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T22 ) ) ) @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) )
=> ( ( t2
= ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( iKKBZ_3908525916494739553en_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t ) ) ) ) ) ).
% dom_children_child_self
thf(fact_1183_contr__before,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R2 ) ) )
=> ( iKKBZ_7682935289300565975re_a_b @ t @ R2 @ ( root_list_a_b @ T1 ) ) ) ) ).
% contr_before
thf(fact_1184_contr__forward,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R2 ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ R2 @ ( root_list_a_b @ T1 ) ) ) ) ) ).
% contr_forward
thf(fact_1185_contr__seq__conform,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R2 ) ) )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ ( append_a @ R2 @ ( root_list_a_b @ T1 ) ) ) ) ) ).
% contr_seq_conform
thf(fact_1186_dom__contr__subtree,axiom,
! [Tn: dtree_list_a_b,R2: list_a,T1: dtree_list_a_b,E12: b] :
( ( is_subtree_list_a_b @ Tn @ t2 )
=> ( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ Tn )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R2 ) ) )
=> ( ( ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) )
= one_one_nat )
=> ( iKKBZ_3908525916494739553en_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t ) ) ) ) ) ).
% dom_contr_subtree
thf(fact_1187_dom__contr,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R2 ) ) )
=> ( ( ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) )
= one_one_nat )
=> ( iKKBZ_3908525916494739553en_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t ) ) ) ) ).
% dom_contr
thf(fact_1188_dom__mdeg__le1,axiom,
! [V: list_a,T22: dtree_list_a_b,E22: b] :
( ( ord_less_eq_nat @ ( max_deg_list_a_b @ t2 ) @ one_one_nat )
=> ( ( is_subtree_list_a_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T22 ) ) ) @ ( rank @ ( rev_a @ V ) ) )
=> ( iKKBZ_3908525916494739553en_a_b @ t2 @ t ) ) ) ) ).
% dom_mdeg_le1
thf(fact_1189_dom__contr_H,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R2 ) ) )
=> ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) ) @ one_one_nat )
=> ( iKKBZ_3908525916494739553en_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t ) ) ) ) ).
% dom_contr'
thf(fact_1190_merge1__darcs__eq,axiom,
( ( darcs_list_a_b @ ( ranked_merge1_a_b @ rank @ cmp @ t2 ) )
= ( darcs_list_a_b @ t2 ) ) ).
% merge1_darcs_eq
thf(fact_1191_merge1__dlverts__eq,axiom,
( ( list_dlverts_a_b @ ( ranked_merge1_a_b @ rank @ cmp @ t2 ) )
= ( list_dlverts_a_b @ t2 ) ) ).
% merge1_dlverts_eq
thf(fact_1192_merge1__dverts__eq,axiom,
( ( dverts_list_a_b @ ( ranked_merge1_a_b @ rank @ cmp @ t2 ) )
= ( dverts_list_a_b @ t2 ) ) ).
% merge1_dverts_eq
thf(fact_1193_normalize1__subtree__same__hd,axiom,
! [V: list_a,T1: dtree_list_a_b,E12: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( ranked8905849569120154423e1_a_b @ rank @ t2 ) )
=> ? [T33: dtree_list_a_b,E3: b] :
( ( ( is_subtree_list_a_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T33 @ E3 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
& ( ( hd_a @ ( root_list_a_b @ T1 ) )
= ( hd_a @ ( root_list_a_b @ T33 ) ) ) )
| ? [V23: list_a] :
( ( V
= ( append_a @ V23 @ ( root_list_a_b @ T33 ) ) )
& ( ( sucs_list_a_b @ T33 )
= ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) )
& ( is_subtree_list_a_b @ ( node_list_a_b @ V23 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T33 @ E3 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
& ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T33 ) ) ) @ ( rank @ ( rev_a @ V23 ) ) ) ) ) ) ).
% normalize1_subtree_same_hd
thf(fact_1194_dom__between__child__roots,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R2 ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ R2 ) )
& ? [Xa3: a] :
( ( member_a @ Xa3 @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Xa3 ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ).
% dom_between_child_roots
thf(fact_1195_loopfree_Oadj__not__same,axiom,
! [A: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A ) @ ( arcs_ends_a_b @ t ) ) ).
% loopfree.adj_not_same
thf(fact_1196_before__ArcI,axiom,
! [S1: list_a,S22: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S22 )
=> ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ S1 ) )
& ? [Xa3: a] :
( ( member_a @ Xa3 @ ( set_a2 @ S22 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Xa3 ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% before_ArcI
thf(fact_1197_normalize1__hd__root__eq_H,axiom,
! [T1: dtree_list_a_b] :
( ( list_wf_dlverts_a_b @ T1 )
=> ( ( hd_a @ ( root_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) ) )
= ( hd_a @ ( root_list_a_b @ T1 ) ) ) ) ).
% normalize1_hd_root_eq'
thf(fact_1198_forward__arc__to__head_H,axiom,
! [Ys: list_a,X: a,Y: a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ Ys )
=> ( ~ ( member_a @ X @ ( set_a2 @ Ys ) )
=> ( ( member_a @ Y @ ( set_a2 @ Ys ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( Y
= ( hd_a @ Ys ) ) ) ) ) ) ).
% forward_arc_to_head'
thf(fact_1199_before__arc__to__hd,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ Xs2 @ Ys )
=> ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ ( hd_a @ Ys ) ) @ ( arcs_ends_a_b @ t ) ) ) ) ).
% before_arc_to_hd
thf(fact_1200_dlverts__arc__in__dlverts,axiom,
! [T1: dtree_list_a_b,X: a,Y: a] :
( ( is_subtree_list_a_b @ T1 @ t2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( ( member_a @ X @ ( list_dlverts_a_b @ T1 ) )
=> ( member_a @ Y @ ( list_dlverts_a_b @ T1 ) ) ) ) ) ).
% dlverts_arc_in_dlverts
thf(fact_1201_move__mid__forward__if__noarc,axiom,
! [As: list_a,U: list_a,Bs: list_a,Cs: list_a] :
( ( As != nil_a )
=> ( ~ ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ U ) )
& ? [Xa3: a] :
( ( member_a @ Xa3 @ ( set_a2 @ Bs ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Xa3 ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ U @ ( append_a @ Bs @ Cs ) ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ Bs @ ( append_a @ U @ Cs ) ) ) ) ) ) ) ).
% move_mid_forward_if_noarc
thf(fact_1202_arc__to__lst__if__forward,axiom,
! [X: a,Xs2: list_a,Y: a,Ys: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( rev_a @ ( cons_a @ X @ Xs2 ) ) )
=> ( ( Xs2
= ( cons_a @ Y @ Ys ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ X ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% arc_to_lst_if_forward
thf(fact_1203_forward__app,axiom,
! [S1: list_a,S22: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ S1 )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ S22 )
=> ( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ S1 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ ( hd_a @ S22 ) ) @ ( arcs_ends_a_b @ t ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ S1 @ S22 ) ) ) ) ) ).
% forward_app
thf(fact_1204_no__back__arc__if__fwd__dstct,axiom,
! [As: list_a,Bs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ Bs ) )
=> ( ( distinct_a @ ( append_a @ As @ Bs ) )
=> ~ ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Bs ) )
& ? [Xa4: a] :
( ( member_a @ Xa4 @ ( set_a2 @ As ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Xa4 ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ).
% no_back_arc_if_fwd_dstct
thf(fact_1205_dverts__arc__in__dverts,axiom,
! [T1: dtree_list_a_b,V1: list_a,X: a,Y: a,V22: list_a] :
( ( is_subtree_list_a_b @ T1 @ t2 )
=> ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V1 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( ( member_a @ Y @ ( set_a2 @ V22 ) )
=> ( ( member_list_a @ V22 @ ( dverts_list_a_b @ t2 ) )
=> ( member_list_a @ V22 @ ( dverts_list_a_b @ T1 ) ) ) ) ) ) ) ) ).
% dverts_arc_in_dverts
thf(fact_1206_arc__in__dlverts,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: a,Y: a] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ t2 )
=> ( ( member_a @ X @ ( set_a2 @ R2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( member_a @ Y @ ( list_dlverts_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ) ) ).
% arc_in_dlverts
thf(fact_1207_arc__in__dlverts__subtree,axiom,
! [Tn: dtree_list_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: a,Y: a] :
( ( is_subtree_list_a_b @ Tn @ t2 )
=> ( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ Tn )
=> ( ( member_a @ X @ ( set_a2 @ R2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( member_a @ Y @ ( list_dlverts_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ) ) ) ).
% arc_in_dlverts_subtree
thf(fact_1208_arc__to__dverts__in__subtree,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: a,Y: a,V: list_a] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ t2 )
=> ( ( member_a @ X @ ( set_a2 @ R2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( ( member_a @ Y @ ( set_a2 @ V ) )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ t2 ) )
=> ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ) ) ) ) ).
% arc_to_dverts_in_subtree
thf(fact_1209_dverts__arc__in__dlverts,axiom,
! [T1: dtree_list_a_b,V1: list_a,X: a,Y: a] :
( ( is_subtree_list_a_b @ T1 @ t2 )
=> ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V1 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( member_a @ Y @ ( list_dlverts_a_b @ T1 ) ) ) ) ) ) ).
% dverts_arc_in_dlverts
thf(fact_1210_normalize1__arc__in__dlverts,axiom,
! [V: list_a,Ys: fset_P2153231429829016240_a_b_b,X: a,Y: a] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ V @ Ys ) @ ( ranked8905849569120154423e1_a_b @ rank @ t2 ) )
=> ( ( member_a @ X @ ( set_a2 @ V ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( member_a @ Y @ ( list_dlverts_a_b @ ( node_list_a_b @ V @ Ys ) ) ) ) ) ) ).
% normalize1_arc_in_dlverts
thf(fact_1211_merge1__arc__in__dlverts,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: a,Y: a] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ ( ranked_merge1_a_b @ rank @ cmp @ t2 ) )
=> ( ( member_a @ X @ ( set_a2 @ R2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( member_a @ Y @ ( list_dlverts_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ) ) ).
% merge1_arc_in_dlverts
thf(fact_1212_dom__self__contr,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R2 ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ R2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ ( hd_a @ ( root_list_a_b @ T1 ) ) ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% dom_self_contr
thf(fact_1213_normalize1__hd__root__eq,axiom,
! [T1: dtree_list_a_b] :
( ( ( root_list_a_b @ T1 )
!= nil_a )
=> ( ( hd_a @ ( root_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) ) )
= ( hd_a @ ( root_list_a_b @ T1 ) ) ) ) ).
% normalize1_hd_root_eq
thf(fact_1214_merge1__hd__root__eq,axiom,
! [T1: dtree_list_a_b] :
( ( hd_a @ ( root_list_a_b @ ( ranked_merge1_a_b @ rank @ cmp @ T1 ) ) )
= ( hd_a @ ( root_list_a_b @ T1 ) ) ) ).
% merge1_hd_root_eq
thf(fact_1215_merge__hd__root__eq,axiom,
! [T1: dtree_list_a_b] :
( ( hd_a @ ( root_list_a_b @ ( ranked_merge_a_b @ rank @ cmp @ T1 ) ) )
= ( hd_a @ ( root_list_a_b @ T1 ) ) ) ).
% merge_hd_root_eq
thf(fact_1216_loopfree_OvpathI__arc,axiom,
! [A: a,B: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( arcs_ends_a_b @ t ) )
=> ( vertex_vpath_a_b @ ( cons_a @ A @ ( cons_a @ B @ nil_a ) ) @ t ) ) ).
% loopfree.vpathI_arc
thf(fact_1217_subtree__rank__ge__if__mdeg__le1__nocontr,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b,V: list_a,Y: a] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) ) @ one_one_nat )
=> ( ~ ? [V23: list_a,T23: dtree_list_a_b] :
( ? [E23: b] : ( is_subtree_list_a_b @ ( node_list_a_b @ V23 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E23 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T1 )
& ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T23 ) ) ) @ ( rank @ ( rev_a @ V23 ) ) ) )
=> ( ( V != R2 )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ t2 ) )
=> ( ( member_a @ Y @ ( set_a2 @ V ) )
=> ( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ R2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
=> ( ord_less_eq_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ V ) ) ) ) ) ) ) ) ) ) ).
% subtree_rank_ge_if_mdeg_le1_nocontr
thf(fact_1218_reachable1__not__reverse,axiom,
! [X: a,Y: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ).
% reachable1_not_reverse
thf(fact_1219_reachable1__from__outside__dom,axiom,
! [X: a,Y: a,Ys: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( ~ ( member_a @ X @ ( set_a2 @ Ys ) )
=> ( ( member_a @ Y @ ( set_a2 @ Ys ) )
=> ? [X6: a,X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Ys ) )
& ~ ( member_a @ X6 @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X6 @ X3 ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ).
% reachable1_from_outside_dom
thf(fact_1220_reachable1__append__old__if__arc,axiom,
! [Xs2: list_a,Ys: list_a,Z3: a,Y: a] :
( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs2 ) )
& ? [Xa4: a] :
( ( member_a @ Xa4 @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Xa4 ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ( ~ ( member_a @ Z3 @ ( set_a2 @ Xs2 ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs2 )
=> ( ( member_a @ Y @ ( set_a2 @ ( append_a @ Xs2 @ Ys ) ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z3 @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z3 @ X3 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ) ).
% reachable1_append_old_if_arc
thf(fact_1221_hd__reachable1__from__outside_H,axiom,
! [X: a,Y: a,Ys: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ Ys )
=> ( ~ ( member_a @ X @ ( set_a2 @ Ys ) )
=> ( ( member_a @ Y @ ( set_a2 @ Ys ) )
=> ( ? [X3: a] : ( member_a @ X3 @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ ( hd_a @ Ys ) ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ).
% hd_reachable1_from_outside'
thf(fact_1222_dlverts__reach1__in__dlverts,axiom,
! [X: a,Y: a,T1: dtree_list_a_b] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( ( is_subtree_list_a_b @ T1 @ t2 )
=> ( ( member_a @ X @ ( list_dlverts_a_b @ T1 ) )
=> ( member_a @ Y @ ( list_dlverts_a_b @ T1 ) ) ) ) ) ).
% dlverts_reach1_in_dlverts
thf(fact_1223_no__back__reach1__if__fwd__dstct,axiom,
! [As: list_a,Bs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ Bs ) )
=> ( ( distinct_a @ ( append_a @ As @ Bs ) )
=> ~ ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Bs ) )
& ? [Xa4: a] :
( ( member_a @ Xa4 @ ( set_a2 @ As ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Xa4 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ).
% no_back_reach1_if_fwd_dstct
thf(fact_1224_dverts__reach1__in__dverts,axiom,
! [T1: dtree_list_a_b,V1: list_a,X: a,Y: a,V22: list_a] :
( ( is_subtree_list_a_b @ T1 @ t2 )
=> ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V1 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( ( member_a @ Y @ ( set_a2 @ V22 ) )
=> ( ( member_list_a @ V22 @ ( dverts_list_a_b @ t2 ) )
=> ( member_list_a @ V22 @ ( dverts_list_a_b @ T1 ) ) ) ) ) ) ) ) ).
% dverts_reach1_in_dverts
thf(fact_1225_dverts__reachable1__if__dom__children,axiom,
! [T1: dtree_list_a_b,V: list_a] :
( ( iKKBZ_3908525916494739553en_a_b @ T1 @ t )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ T1 ) )
=> ( ( V
!= ( root_list_a_b @ T1 ) )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( dverts_list_a_b @ T1 ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ X3 ) )
=> ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ V ) )
=> ? [Xa3: a] :
( ( member_a @ Xa3 @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa3 @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ) ).
% dverts_reachable1_if_dom_children
thf(fact_1226_dverts__reach1__in__dverts__r,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,V: list_a] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ t2 )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ t2 ) )
=> ( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ R2 ) )
& ? [Xa4: a] :
( ( member_a @ Xa4 @ ( set_a2 @ V ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Xa4 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ) ) ).
% dverts_reach1_in_dverts_r
thf(fact_1227_dverts__reach1__in__dverts__root,axiom,
! [T1: dtree_list_a_b,V: list_a] :
( ( is_subtree_list_a_b @ T1 @ t2 )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ t2 ) )
=> ( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
& ? [Xa4: a] :
( ( member_a @ Xa4 @ ( set_a2 @ V ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Xa4 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( member_list_a @ V @ ( dverts_list_a_b @ T1 ) ) ) ) ) ).
% dverts_reach1_in_dverts_root
thf(fact_1228_dverts__reach1__in__dlverts,axiom,
! [T1: dtree_list_a_b,V1: list_a,X: a,Y: a] :
( ( is_subtree_list_a_b @ T1 @ t2 )
=> ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V1 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( member_a @ Y @ ( list_dlverts_a_b @ T1 ) ) ) ) ) ) ).
% dverts_reach1_in_dlverts
thf(fact_1229_subtree__dverts__reachable1__if__mdeg__gt1,axiom,
! [T1: dtree_list_a_b,V: list_a] :
( ( is_subtree_list_a_b @ T1 @ t2 )
=> ( ( ord_less_nat @ one_one_nat @ ( max_deg_list_a_b @ T1 ) )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ T1 ) )
=> ( ( V
!= ( root_list_a_b @ T1 ) )
=> ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ V ) )
=> ? [Xa3: a] :
( ( member_a @ Xa3 @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa3 @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ) ).
% subtree_dverts_reachable1_if_mdeg_gt1
thf(fact_1230_subtree__y__reach__if__mdeg__gt1__notroot__reach,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b,V: list_a,Y: a] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ( ( ord_less_nat @ one_one_nat @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( ( V != R2 )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ t2 ) )
=> ( ( V
!= ( root_list_a_b @ T1 ) )
=> ( ( member_a @ Y @ ( set_a2 @ V ) )
=> ( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ R2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ) ) ) ).
% subtree_y_reach_if_mdeg_gt1_notroot_reach
thf(fact_1231_subtree__eqroot__if__mdeg__gt1__reach,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b,V: list_a] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ( ( ord_less_nat @ one_one_nat @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ t2 ) )
=> ( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ V ) )
& ~ ? [Xa3: a] :
( ( member_a @ Xa3 @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa3 @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
& ? [Xa4: a] :
( ( member_a @ Xa4 @ ( set_a2 @ R2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa4 @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( ( V != R2 )
=> ( ( root_list_a_b @ T1 )
= V ) ) ) ) ) ) ).
% subtree_eqroot_if_mdeg_gt1_reach
thf(fact_1232_subtree__dverts__reachable1__if__mdeg__gt1__singleton,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b,V: list_a] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ( ( ord_less_nat @ one_one_nat @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ T1 ) )
=> ( ( V
!= ( root_list_a_b @ T1 ) )
=> ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ V ) )
=> ? [Xa3: a] :
( ( member_a @ Xa3 @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa3 @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ) ).
% subtree_dverts_reachable1_if_mdeg_gt1_singleton
thf(fact_1233_subtree__rank__ge__if__reach_H,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ! [X2: list_a] :
( ( member_list_a @ X2 @ ( dverts_list_a_b @ t2 ) )
=> ( ? [Xa4: a] :
( ( member_a @ Xa4 @ ( set_a2 @ X2 ) )
& ~ ? [Xb: a] :
( ( member_a @ Xb @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xb @ Xa4 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
& ? [Xb2: a] :
( ( member_a @ Xb2 @ ( set_a2 @ R2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xb2 @ Xa4 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
& ( X2 != R2 ) )
=> ( ord_less_eq_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ X2 ) ) ) ) ) ) ).
% subtree_rank_ge_if_reach'
thf(fact_1234_subtree__rank__ge__if__reach,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b,V: list_a] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ( ( V != R2 )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ t2 ) )
=> ( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ V ) )
& ~ ? [Xa3: a] :
( ( member_a @ Xa3 @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa3 @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
& ? [Xa4: a] :
( ( member_a @ Xa4 @ ( set_a2 @ R2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa4 @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( ord_less_eq_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ V ) ) ) ) ) ) ) ).
% subtree_rank_ge_if_reach
thf(fact_1235_subtree__dverts__reachable1__if__mdeg__le1__subcontr,axiom,
! [T1: dtree_list_a_b,V22: list_a,T22: dtree_list_a_b,E22: b,V: list_a] :
( ( is_subtree_list_a_b @ T1 @ t2 )
=> ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ T1 ) @ one_one_nat )
=> ( ( is_subtree_list_a_b @ ( node_list_a_b @ V22 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T1 )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T22 ) ) ) @ ( rank @ ( rev_a @ V22 ) ) )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ T1 ) )
=> ( ( V
!= ( root_list_a_b @ T1 ) )
=> ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ V ) )
=> ? [Xa3: a] :
( ( member_a @ Xa3 @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa3 @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ) ) ) ).
% subtree_dverts_reachable1_if_mdeg_le1_subcontr
thf(fact_1236_subtree__y__reach__if__mdeg__le1__notroot__subcontr,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b,V22: list_a,T22: dtree_list_a_b,E22: b,V: list_a,Y: a] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) ) @ one_one_nat )
=> ( ( is_subtree_list_a_b @ ( node_list_a_b @ V22 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E22 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T1 )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T22 ) ) ) @ ( rank @ ( rev_a @ V22 ) ) )
=> ( ( V != R2 )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ t2 ) )
=> ( ( V
!= ( root_list_a_b @ T1 ) )
=> ( ( member_a @ Y @ ( set_a2 @ V ) )
=> ( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ R2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% subtree_y_reach_if_mdeg_le1_notroot_subcontr
thf(fact_1237_subtree__rank__ge__if__mdeg__gt1__reach,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b,V: list_a] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ( ( ord_less_nat @ one_one_nat @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ t2 ) )
=> ( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ V ) )
& ~ ? [Xa3: a] :
( ( member_a @ Xa3 @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa3 @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
& ? [Xa4: a] :
( ( member_a @ Xa4 @ ( set_a2 @ R2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa4 @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( ( V != R2 )
=> ( ord_less_eq_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ V ) ) ) ) ) ) ) ) ).
% subtree_rank_ge_if_mdeg_gt1_reach
thf(fact_1238_subtree__rank__ge__if__mdeg__le1,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b,V: list_a] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) ) @ one_one_nat )
=> ( ( V != R2 )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ t2 ) )
=> ( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ V ) )
& ~ ? [Xa3: a] :
( ( member_a @ Xa3 @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa3 @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
& ? [Xa4: a] :
( ( member_a @ Xa4 @ ( set_a2 @ R2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa4 @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( ord_less_eq_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ V ) ) ) ) ) ) ) ) ).
% subtree_rank_ge_if_mdeg_le1
thf(fact_1239_subtree__rank__ge__if__mdeg__le1_H,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b,V: list_a,Y: a] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ( ( ord_less_eq_nat @ ( max_deg_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) ) @ one_one_nat )
=> ( ( V != R2 )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ t2 ) )
=> ( ( member_a @ Y @ ( set_a2 @ V ) )
=> ( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ R2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
=> ( ~ ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
=> ( ord_less_eq_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ V ) ) ) ) ) ) ) ) ) ) ).
% subtree_rank_ge_if_mdeg_le1'
thf(fact_1240_normalize1__arc__in__dlverts_H,axiom,
! [R6: list_a,Xs3: fset_P2153231429829016240_a_b_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R6 @ Xs3 ) @ ( ranked8905849569120154423e1_a_b @ rank @ t2 ) )
=> ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ R6 ) )
=> ! [Y4: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y4 ) @ ( arcs_ends_a_b @ t ) )
=> ( ( member_a @ Y4 @ ( set_a2 @ R6 ) )
| ? [Xa3: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Xa3 @ ( fset_P9138963618725001425_a_b_b @ Xs3 ) )
& ( member_a @ Y4 @ ( list_dlverts_a_b @ ( produc5948858871325780166_a_b_b @ Xa3 ) ) ) ) ) ) ) ) ).
% normalize1_arc_in_dlverts'
thf(fact_1241_arc__in__dlverts__subtree_H,axiom,
! [Tn: dtree_list_a_b] :
( ( is_subtree_list_a_b @ Tn @ t2 )
=> ! [R6: list_a,Xs3: fset_P2153231429829016240_a_b_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R6 @ Xs3 ) @ Tn )
=> ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ R6 ) )
=> ! [Y4: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y4 ) @ ( arcs_ends_a_b @ t ) )
=> ( ( member_a @ Y4 @ ( set_a2 @ R6 ) )
| ? [Xa3: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Xa3 @ ( fset_P9138963618725001425_a_b_b @ Xs3 ) )
& ( member_a @ Y4 @ ( list_dlverts_a_b @ ( produc5948858871325780166_a_b_b @ Xa3 ) ) ) ) ) ) ) ) ) ).
% arc_in_dlverts_subtree'
thf(fact_1242_merge__dom__contr__if__nocontr__mdeg__le1,axiom,
! [R2: list_a,T1: dtree_list_a_b,E12: b] :
( ! [R12: list_a,T23: dtree_list_a_b] :
( ? [E23: b] : ( is_subtree_list_a_b @ ( node_list_a_b @ R12 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E23 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ( ord_less_eq_real @ ( rank @ ( rev_a @ R12 ) ) @ ( rank @ ( rev_a @ ( root_list_a_b @ T23 ) ) ) ) )
=> ( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( ranked_merge_a_b @ rank @ cmp @ t2 ) )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R2 ) ) )
=> ( ! [X3: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X3 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ t2 ) ) ) )
=> ( ord_less_eq_nat @ ( max_deg_list_a_b @ X3 ) @ one_one_nat ) )
=> ( iKKBZ_3908525916494739553en_a_b @ ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E12 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t ) ) ) ) ) ).
% merge_dom_contr_if_nocontr_mdeg_le1
thf(fact_1243_merge__dom__sub__contr__if__nocontr,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,T1: dtree_list_a_b] :
( ! [R12: list_a,T23: dtree_list_a_b] :
( ? [E23: b] : ( is_subtree_list_a_b @ ( node_list_a_b @ R12 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E23 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ( ord_less_eq_real @ ( rank @ ( rev_a @ R12 ) ) @ ( rank @ ( rev_a @ ( root_list_a_b @ T23 ) ) ) ) )
=> ( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ ( ranked_merge_a_b @ rank @ cmp @ t2 ) )
=> ( ( member551035911493665803st_a_b @ T1 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( ? [V5: list_a,T24: dtree_list_a_b] :
( ? [E24: b] : ( is_subtree_list_a_b @ ( node_list_a_b @ V5 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T24 @ E24 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( node_list_a_b @ R2 @ Xs2 ) )
& ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T24 ) ) ) @ ( rank @ ( rev_a @ V5 ) ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ R2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ ( hd_a @ ( root_list_a_b @ T1 ) ) ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ).
% merge_dom_sub_contr_if_nocontr
thf(fact_1244_darcs__child__subset,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b] :
( ( ( node_list_a_b @ R2 @ Xs2 )
= t2 )
=> ( ( member551035911493665803st_a_b @ X @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( ord_less_set_b @ ( darcs_list_a_b @ X ) @ ( darcs_list_a_b @ t2 ) ) ) ) ).
% darcs_child_subset
thf(fact_1245_dverts__child__subset,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b] :
( ( ( node_list_a_b @ R2 @ Xs2 )
= t2 )
=> ( ( member551035911493665803st_a_b @ X @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( ord_less_set_list_a @ ( dverts_list_a_b @ X ) @ ( dverts_list_a_b @ t2 ) ) ) ) ).
% dverts_child_subset
thf(fact_1246_root__not__subtree,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b] :
( ( ( node_list_a_b @ R2 @ Xs2 )
= t2 )
=> ( ( member551035911493665803st_a_b @ X @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ~ ( member_list_a @ R2 @ ( dverts_list_a_b @ X ) ) ) ) ).
% root_not_subtree
thf(fact_1247_verts__child__if__merge__child,axiom,
! [T1: dtree_list_a_b,T0: dtree_list_a_b,X: list_a] :
( ( member551035911493665803st_a_b @ T1 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ ( ranked_merge_a_b @ rank @ cmp @ T0 ) ) ) ) )
=> ( ( member_list_a @ X @ ( dverts_list_a_b @ T1 ) )
=> ? [X3: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X3 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T0 ) ) ) )
& ( member_list_a @ X @ ( dverts_list_a_b @ X3 ) ) ) ) ) ).
% verts_child_if_merge_child
thf(fact_1248_distint__verts__subtree,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,T1: dtree_list_a_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ t2 )
=> ( ( member551035911493665803st_a_b @ T1 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( distinct_a @ ( append_a @ R2 @ ( root_list_a_b @ T1 ) ) ) ) ) ).
% distint_verts_subtree
thf(fact_1249_merge__dom__children,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
( ( iKKBZ_3908525916494739553en_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ t )
=> ( ! [X3: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X3 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( list_wf_dlverts_a_b @ X3 ) )
=> ( iKKBZ_3908525916494739553en_a_b @ ( ranked_merge_a_b @ rank @ cmp @ ( node_list_a_b @ R2 @ Xs2 ) ) @ t ) ) ) ).
% merge_dom_children
thf(fact_1250_merge__dom__children__sucs,axiom,
! [T0: dtree_list_a_b] :
( ( iKKBZ_3908525916494739553en_a_b @ T0 @ t )
=> ( ! [X3: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X3 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T0 ) ) ) )
=> ( list_wf_dlverts_a_b @ X3 ) )
=> ( iKKBZ_3908525916494739553en_a_b @ ( ranked_merge_a_b @ rank @ cmp @ T0 ) @ t ) ) ) ).
% merge_dom_children_sucs
thf(fact_1251_merge__single__root1,axiom,
! [T22: dtree_list_a_b,R2: list_a,Xs2: fset_P2153231429829016240_a_b_b] :
( ( member551035911493665803st_a_b @ T22 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ ( ranked_merge_a_b @ rank @ cmp @ ( node_list_a_b @ R2 @ Xs2 ) ) ) ) ) )
=> ? [E23: b] :
( ( ranked_merge_a_b @ rank @ cmp @ ( node_list_a_b @ R2 @ Xs2 ) )
= ( node_list_a_b @ R2 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E23 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).
% merge_single_root1
thf(fact_1252_merge1__childdeg__gt1__if__fcard__gt1,axiom,
! [T1: dtree_list_a_b] :
( ( ord_less_nat @ one_one_nat @ ( fcard_4742106318756258927_a_b_b @ ( sucs_list_a_b @ ( ranked_merge1_a_b @ rank @ cmp @ T1 ) ) ) )
=> ? [X3: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X3 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T1 ) ) ) )
& ( ord_less_nat @ one_one_nat @ ( max_deg_list_a_b @ X3 ) ) ) ) ).
% merge1_childdeg_gt1_if_fcard_gt1
thf(fact_1253_merge__single__root1__sucs,axiom,
! [T22: dtree_list_a_b,T1: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ T22 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ ( ranked_merge_a_b @ rank @ cmp @ T1 ) ) ) ) )
=> ? [E23: b] :
( ( ranked_merge_a_b @ rank @ cmp @ T1 )
= ( node_list_a_b @ ( root_list_a_b @ T1 ) @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E23 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).
% merge_single_root1_sucs
thf(fact_1254_merge1__childdeg__gt1__if__fcard__gt1__sub,axiom,
! [R2: list_a,Xs2: fset_P2153231429829016240_a_b_b,T1: dtree_list_a_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Xs2 ) @ ( ranked_merge1_a_b @ rank @ cmp @ T1 ) )
=> ( ( ord_less_nat @ one_one_nat @ ( fcard_4742106318756258927_a_b_b @ Xs2 ) )
=> ? [Ys2: fset_P2153231429829016240_a_b_b] :
( ( ( ranked_merge1_a_b @ rank @ cmp @ ( node_list_a_b @ R2 @ Ys2 ) )
= ( node_list_a_b @ R2 @ Xs2 ) )
& ( is_subtree_list_a_b @ ( node_list_a_b @ R2 @ Ys2 ) @ T1 )
& ? [X3: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X3 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Ys2 ) ) )
& ( ord_less_nat @ one_one_nat @ ( max_deg_list_a_b @ X3 ) ) ) ) ) ) ).
% merge1_childdeg_gt1_if_fcard_gt1_sub
thf(fact_1255_dverts__merge__sub,axiom,
! [T0: dtree_list_a_b] :
( ! [X3: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X3 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ T0 ) ) ) )
=> ( ord_less_eq_nat @ ( max_deg_list_a_b @ X3 ) @ one_one_nat ) )
=> ( ord_le8861187494160871172list_a @ ( dverts_list_a_b @ ( ranked_merge_a_b @ rank @ cmp @ T0 ) ) @ ( dverts_list_a_b @ T0 ) ) ) ).
% dverts_merge_sub
% Conjectures (1)
thf(conj_0,conjecture,
( ( max_deg_list_a_b @ ( iKKBZ_6959927528703686640ll_a_b @ ( node_list_a_b @ r @ xs ) ) )
= zero_zero_nat ) ).
%------------------------------------------------------------------------------