TPTP Problem File: SLH0143^1.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_00617_026790__15652930_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1558 ( 567 unt; 287 typ; 0 def)
% Number of atoms : 3764 (1592 equ; 0 cnn)
% Maximal formula atoms : 10 ( 2 avg)
% Number of connectives : 13860 ( 530 ~; 62 |; 475 &;11277 @)
% ( 0 <=>;1516 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 42 ( 41 usr)
% Number of type conns : 657 ( 657 >; 0 *; 0 +; 0 <<)
% Number of symbols : 247 ( 246 usr; 21 con; 0-4 aty)
% Number of variables : 3804 ( 153 ^;3312 !; 339 ?;3804 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:07:38.433
%------------------------------------------------------------------------------
% Could-be-implicit typings (41)
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thf(sy_c_Digraph_Oreachable_001tf__a_001tf__b,type,
reachable_a_b: pre_pr7278220950009878019t_unit > a > a > $o ).
thf(sy_c_Digraph_Owf__digraph_Oarc_001tf__a_001tf__b,type,
wf_arc_a_b: pre_pr7278220950009878019t_unit > b > product_prod_a_a > $o ).
thf(sy_c_Digraph__Component_Oconnected_001tf__a_001tf__b,type,
digrap8783888973171253482ed_a_b: pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_Digraph__Component_Oinduce__subgraph_001tf__a_001tf__b,type,
digrap7873285959652527175ph_a_b: pre_pr7278220950009878019t_unit > set_a > pre_pr7278220950009878019t_unit ).
thf(sy_c_Digraph__Component_Opre__digraph_Oscc__of_001tf__a_001tf__b,type,
digrap2937667069914300949of_a_b: pre_pr7278220950009878019t_unit > a > set_a ).
thf(sy_c_Digraph__Component_Opre__digraph_Osccs__verts_001tf__a_001tf__b,type,
digrap2871191568752656621ts_a_b: pre_pr7278220950009878019t_unit > set_set_a ).
thf(sy_c_Digraph__Component_Ospanning_001tf__a_001tf__b,type,
digraph_spanning_a_b: pre_pr7278220950009878019t_unit > pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_Digraph__Component_Ospanning__tree_001tf__a_001tf__b,type,
digrap5718416180170401981ee_a_b: pre_pr7278220950009878019t_unit > pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_Directed__Tree__Additions_Odirected__tree_Oto__list__tree_001tf__a_001tf__b,type,
direct3773525127397338803ee_a_b: pre_pr7278220950009878019t_unit > pre_pr2882871181989701257t_unit ).
thf(sy_c_Euler_Opre__digraph_Oarc__set__balanced_001tf__a_001tf__b,type,
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thf(sy_c_Euler_Opre__digraph_Oeuler__trail_001t__List__Olist_Itf__a_J_001tf__b,type,
pre_eu4033079881512885387st_a_b: pre_pr2882871181989701257t_unit > list_a > list_b > list_a > $o ).
thf(sy_c_Euler_Opre__digraph_Oeuler__trail_001tf__a_001tf__a,type,
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thf(sy_c_Euler_Opre__digraph_Oeuler__trail_001tf__a_001tf__b,type,
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thf(sy_c_Euler_Opre__digraph_Oeuler__trail_001tf__b_001tf__a,type,
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thf(sy_c_Euler_Opre__digraph_Oeuler__trail_001tf__b_001tf__b,type,
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thf(sy_c_Graph__Additions_Owf__digraph_Ois__chain_H_001tf__a_001tf__b,type,
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graph_2957805489637798020ts_a_b: pre_pr7278220950009878019t_unit > set_a ).
thf(sy_c_Graph__Definitions_Owf__digraph_Ok__neighborhood_001tf__a_001tf__b,type,
graph_3921080825633621230od_a_b: pre_pr7278220950009878019t_unit > ( b > real ) > a > real > set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
minus_646659088055828811list_a: set_list_a > set_list_a > set_list_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
minus_5736297505244876581_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__b_J,type,
minus_minus_set_b: set_b > set_b > set_b ).
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_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_IKKBZ__Optimality_Odirected__tree_Obefore_001tf__a_001tf__b,type,
iKKBZ_7682935289300565975re_a_b: pre_pr7278220950009878019t_unit > list_a > list_a > $o ).
thf(sy_c_IKKBZ__Optimality_Odirected__tree_Oforward_001tf__a_001tf__b,type,
iKKBZ_4778857019735642799rd_a_b: pre_pr7278220950009878019t_unit > list_a > $o ).
thf(sy_c_IKKBZ__Optimality_Odirected__tree_Oforward__arcs_001tf__a_001tf__b,type,
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thf(sy_c_IKKBZ__Optimality_Odirected__tree_Ono__back_001tf__a_001tf__b,type,
iKKBZ_3684931046464919648ck_a_b: pre_pr7278220950009878019t_unit > list_a > $o ).
thf(sy_c_IKKBZ__Optimality_Odirected__tree_Ono__back__arcs_001tf__a_001tf__b,type,
iKKBZ_7773321254043928001cs_a_b: pre_pr7278220950009878019t_unit > list_a > $o ).
thf(sy_c_IKKBZ__Optimality_Odirected__tree_Oseq__conform_001tf__a_001tf__b,type,
iKKBZ_4622586873178280511rm_a_b: pre_pr7278220950009878019t_unit > list_a > $o ).
thf(sy_c_Kuratowski_Opre__digraph_Oinner__verts_001tf__a_001tf__b,type,
pre_inner_verts_a_b: pre_pr7278220950009878019t_unit > list_b > list_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_List_Oappend_001t__List__Olist_Itf__a_J,type,
append_list_a: list_list_a > list_list_a > list_list_a ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oappend_001t__Set__Oset_Itf__a_J,type,
append_set_a: list_set_a > list_set_a > list_set_a ).
thf(sy_c_List_Oappend_001tf__a,type,
append_a: list_a > list_a > list_a ).
thf(sy_c_List_Oappend_001tf__b,type,
append_b: list_b > list_b > list_b ).
thf(sy_c_List_Obutlast_001tf__a,type,
butlast_a: list_a > list_a ).
thf(sy_c_List_Odistinct_001t__List__Olist_Itf__a_J,type,
distinct_list_a: list_list_a > $o ).
thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_Odistinct_001t__Set__Oset_Itf__a_J,type,
distinct_set_a: list_set_a > $o ).
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_Olast_001tf__a,type,
last_a: list_a > a ).
thf(sy_c_List_Olast_001tf__b,type,
last_b: list_b > b ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__a_J,type,
cons_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__b_J,type,
cons_list_b: list_b > list_list_b > list_list_b ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Set__Oset_Itf__a_J,type,
cons_set_a: set_a > list_set_a > list_set_a ).
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_001t__List__Olist_Itf__a_J,type,
nil_list_a: list_list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__b_J,type,
nil_list_b: list_list_b ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__Set__Oset_Itf__a_J,type,
nil_set_a: list_set_a ).
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_001t__List__Olist_Itf__a_J,type,
hd_list_a: list_list_a > list_a ).
thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
hd_nat: list_nat > nat ).
thf(sy_c_List_Olist_Ohd_001t__Set__Oset_Itf__a_J,type,
hd_set_a: list_set_a > set_a ).
thf(sy_c_List_Olist_Ohd_001tf__a,type,
hd_a: list_a > a ).
thf(sy_c_List_Olist_Ohd_001tf__b,type,
hd_b: list_b > b ).
thf(sy_c_List_Olist_Omap_001tf__b_001tf__a,type,
map_b_a: ( b > a ) > list_b > list_a ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_Itf__a_J,type,
set_list_a2: list_list_a > set_list_a ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Set__Oset_Itf__a_J,type,
set_set_a2: list_set_a > set_set_a ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Olist_Oset_001tf__b,type,
set_b2: list_b > set_b ).
thf(sy_c_List_Olist_Otl_001t__List__Olist_Itf__a_J,type,
tl_list_a: list_list_a > list_list_a ).
thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
tl_nat: list_nat > list_nat ).
thf(sy_c_List_Olist_Otl_001t__Set__Oset_Itf__a_J,type,
tl_set_a: list_set_a > list_set_a ).
thf(sy_c_List_Olist_Otl_001tf__a,type,
tl_a: list_a > list_a ).
thf(sy_c_List_Olist_Otl_001tf__b,type,
tl_b: list_b > list_b ).
thf(sy_c_List_Onth_001t__List__Olist_Itf__a_J,type,
nth_list_a: list_list_a > nat > list_a ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Set__Oset_Itf__a_J,type,
nth_set_a: list_set_a > nat > set_a ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_List_Onth_001tf__b,type,
nth_b: list_b > nat > b ).
thf(sy_c_List_Orev_001tf__a,type,
rev_a: list_a > list_a ).
thf(sy_c_List_Orev_001tf__b,type,
rev_b: list_b > list_b ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
size_s349497388124573686list_a: list_list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
size_size_list_set_a: list_set_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__b_J,type,
size_size_list_b: list_b > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__b_J,type,
bot_bot_set_b: set_b ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_less_set_list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_less_set_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
ord_less_set_set_b: set_set_b > set_set_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__b_J,type,
ord_less_set_b: set_b > set_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_le8861187494160871172list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
ord_le3795704787696855135_set_b: set_set_b > set_set_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__b_J,type,
ord_less_eq_set_b: set_b > set_b > $o ).
thf(sy_c_Product__Type_OPair_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J_001t__List__Olist_Itf__a_J,type,
produc8111569692950616493list_a: ( a > a > $o ) > list_a > produc5032551385658279741list_a ).
thf(sy_c_Product__Type_OPair_001_062_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
produc8643929849434629545list_a: ( a > a ) > produc9164743771328383783list_a > produc1473018763691903991list_a ).
thf(sy_c_Product__Type_OPair_001_062_Itf__b_M_062_Itf__b_M_Eo_J_J_001t__List__Olist_Itf__b_J,type,
produc8193136575784045678list_b: ( b > b > $o ) > list_b > produc5185152304234826110list_b ).
thf(sy_c_Product__Type_OPair_001_062_Itf__b_Mtf__b_J_001t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__List__Olist_Itf__b_J_J,type,
produc748123367317244457list_b: ( b > b ) > produc3963297410138542439list_b > produc2395089919340105847list_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__List__Olist_Itf__a_J_J,type,
produc1910438886824740410list_a: list_a > produc9164743771328383783list_a > produc1553995403754578250list_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__b_J_001t__List__Olist_Itf__b_J,type,
produc1564554178308465111list_b: list_b > list_b > produc3963297410138542439list_b ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__b_J_001t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__List__Olist_Itf__b_J_J,type,
produc305491333965050169list_b: list_b > produc3963297410138542439list_b > produc8766925488660474953list_b ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__b_J_001t__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__b_J_Mt__List__Olist_Itf__b_J_J_J,type,
produc7106373121284446491list_b: list_b > produc8766925488660474953list_b > produc272433356463431595list_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_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_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_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_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Oinsert_001tf__b,type,
insert_b: b > set_b > set_b ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
set_or1222579329274155063t_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_Itf__a_J,type,
set_or6288561110385358355_set_a: set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_Itf__b_J,type,
set_or6288561114688587156_set_b: set_b > set_b > set_set_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__Tree_Opre__digraph_Oleaf_001tf__a_001tf__b,type,
shorte1213025427933718126af_a_b: pre_pr7278220950009878019t_unit > a > $o ).
thf(sy_c_Shortest__Path__Tree_Osubgraph_001tf__a_001tf__b,type,
shorte3657265928840388360ph_a_b: pre_pr7278220950009878019t_unit > pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_Stuff_ONOMATCH_001tf__a,type,
nOMATCH_a: a > a > $o ).
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_001t__List__Olist_Itf__a_J_001tf__b,type,
vertex6060786982766068989st_a_b: list_list_a > pre_pr2882871181989701257t_unit > $o ).
thf(sy_c_Vertex__Walk_Ovpath_001tf__a_001tf__b,type,
vertex_vpath_a_b: list_a > pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_Vertex__Walk_Ovwalk_001t__List__Olist_Itf__a_J_001tf__b,type,
vertex2966258834163962945st_a_b: list_list_a > pre_pr2882871181989701257t_unit > $o ).
thf(sy_c_Vertex__Walk_Ovwalk_001tf__a_001tf__b,type,
vertex_vwalk_a_b: list_a > pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_Weighted__Graph_Owf__digraph_Oawalk__cost_001tf__b,type,
weight7472181610322534790cost_b: ( b > real ) > list_b > real ).
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__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
member8191768239178080336list_a: produc9164743771328383783list_a > set_Pr4048851178543822343list_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_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__b_J,type,
member_set_b: set_b > set_set_b > $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_U,type,
u: list_a ).
thf(sy_v_V,type,
v: list_a ).
thf(sy_v_as,type,
as: list_a ).
thf(sy_v_bs,type,
bs: list_a ).
thf(sy_v_cs,type,
cs: list_a ).
thf(sy_v_i,type,
i: nat ).
thf(sy_v_j____,type,
j: nat ).
thf(sy_v_x____,type,
x: a ).
% Relevant facts (1270)
thf(fact_0_j__def_I2_J,axiom,
ord_less_nat @ j @ i ).
% j_def(2)
thf(fact_1_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_2_j__def_I1_J,axiom,
( ( nth_a @ ( append_a @ as @ u ) @ j )
= x ) ).
% j_def(1)
thf(fact_3__092_060open_062_Ias_A_064_AU_A_064_AV_A_064_Abs_A_064_Acs_J_A_B_Aj_A_061_Ax_092_060close_062,axiom,
( ( nth_a @ ( append_a @ as @ ( append_a @ u @ ( append_a @ v @ ( append_a @ bs @ cs ) ) ) ) @ j )
= x ) ).
% \<open>(as @ U @ V @ bs @ cs) ! j = x\<close>
thf(fact_4_x__def_I2_J,axiom,
member1426531477525435216od_a_a @ ( product_Pair_a_a @ x @ ( nth_a @ ( append_a @ as @ ( append_a @ u @ ( append_a @ v @ ( append_a @ bs @ cs ) ) ) ) @ i ) ) @ ( arcs_ends_a_b @ t ) ).
% x_def(2)
thf(fact_5__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062j_O_A_092_060lbrakk_062_Ias_A_064_AU_J_A_B_Aj_A_061_Ax_059_Aj_A_060_Ai_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [J: nat] :
( ( ( nth_a @ ( append_a @ as @ u ) @ J )
= x )
=> ~ ( ord_less_nat @ J @ i ) ) ).
% \<open>\<And>thesis. (\<And>j. \<lbrakk>(as @ U) ! j = x; j < i\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_6__C2_C_I1_J,axiom,
( i
= ( size_size_list_a @ ( append_a @ as @ u ) ) ) ).
% "2"(1)
thf(fact_7_assms_I2_J,axiom,
iKKBZ_4778857019735642799rd_a_b @ t @ v ).
% assms(2)
thf(fact_8_x__def_I1_J,axiom,
member_a @ x @ ( set_a2 @ u ) ).
% x_def(1)
thf(fact_9_assms_I3_J,axiom,
iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ as @ ( append_a @ u @ ( append_a @ bs @ ( append_a @ v @ cs ) ) ) ) ).
% assms(3)
thf(fact_10_loopfree_Oloopfree__digraph__axioms,axiom,
loopfree_digraph_a_b @ t ).
% loopfree.loopfree_digraph_axioms
thf(fact_11_nomulti_Onomulti__digraph__axioms,axiom,
nomulti_digraph_a_b @ t ).
% nomulti.nomulti_digraph_axioms
thf(fact_12__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_O_A_092_060lbrakk_062x_A_092_060in_062_Aset_AU_059_Ax_A_092_060rightarrow_062_092_060_094bsub_062T_092_060_094esub_062_A_Ias_A_064_AU_A_064_AV_A_064_Abs_A_064_Acs_J_A_B_Ai_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [X: a] :
( ( member_a @ X @ ( set_a2 @ u ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ ( nth_a @ ( append_a @ as @ ( append_a @ u @ ( append_a @ v @ ( append_a @ bs @ cs ) ) ) ) @ i ) ) @ ( arcs_ends_a_b @ t ) ) ) ).
% \<open>\<And>thesis. (\<And>x. \<lbrakk>x \<in> set U; x \<rightarrow>\<^bsub>T\<^esub> (as @ U @ V @ bs @ cs) ! i\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_13__092_060open_062_Ias_A_064_AU_A_064_AV_A_064_Abs_A_064_Acs_J_A_B_Ai_A_061_A_IV_A_064_Abs_A_064_Acs_J_A_B_A0_092_060close_062,axiom,
( ( nth_a @ ( append_a @ as @ ( append_a @ u @ ( append_a @ v @ ( append_a @ bs @ cs ) ) ) ) @ i )
= ( nth_a @ ( append_a @ v @ ( append_a @ bs @ cs ) ) @ zero_zero_nat ) ) ).
% \<open>(as @ U @ V @ bs @ cs) ! i = (V @ bs @ cs) ! 0\<close>
thf(fact_14_source__nmem__k__nh,axiom,
! [V: a,W: b > real,K: real] :
~ ( member_a @ V @ ( graph_3921080825633621230od_a_b @ t @ W @ V @ K ) ) ).
% source_nmem_k_nh
thf(fact_15__092_060open_062_Ias_A_064_AU_A_064_AV_A_064_Abs_A_064_Acs_J_A_B_Ai_A_061_Ahd_AV_092_060close_062,axiom,
( ( nth_a @ ( append_a @ as @ ( append_a @ u @ ( append_a @ v @ ( append_a @ bs @ cs ) ) ) ) @ i )
= ( hd_a @ v ) ) ).
% \<open>(as @ U @ V @ bs @ cs) ! i = hd V\<close>
thf(fact_16_cycle__free,axiom,
~ ? [X_1: list_b] : ( arc_pre_cycle_a_b @ t @ X_1 ) ).
% cycle_free
thf(fact_17_scc__of__eq,axiom,
! [U: a,V: a] :
( ( member_a @ U @ ( digrap2937667069914300949of_a_b @ t @ V ) )
=> ( ( digrap2937667069914300949of_a_b @ t @ U )
= ( digrap2937667069914300949of_a_b @ t @ V ) ) ) ).
% scc_of_eq
thf(fact_18_adj__in__verts_I2_J,axiom,
! [U: a,V: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( arcs_ends_a_b @ t ) )
=> ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% adj_in_verts(2)
thf(fact_19_adj__in__verts_I1_J,axiom,
! [U: a,V: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( arcs_ends_a_b @ t ) )
=> ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% adj_in_verts(1)
thf(fact_20_awalk__dom__if__uneq,axiom,
! [U: a,V: a,P: list_b] :
( ( U != V )
=> ( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ? [X: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ V ) @ ( arcs_ends_a_b @ t ) ) ) ) ).
% awalk_dom_if_uneq
thf(fact_21_adj__reachable__trans,axiom,
! [A: a,B: a,C: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( arcs_ends_a_b @ t ) )
=> ( ( reachable_a_b @ t @ B @ C )
=> ( reachable_a_b @ t @ A @ C ) ) ) ).
% adj_reachable_trans
thf(fact_22_unique__awalk__All,axiom,
! [U: a,V: a] :
( ? [P2: list_b] : ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ? [X: list_b] :
( ( arc_pre_awalk_a_b @ t @ U @ X @ V )
& ! [Y: list_b] :
( ( arc_pre_awalk_a_b @ t @ U @ Y @ V )
=> ( Y = X ) ) ) ) ).
% unique_awalk_All
thf(fact_23_awalk__ends__eqD,axiom,
! [U: a,P: list_b,V: a,W: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ U )
=> ( ( arc_pre_awalk_a_b @ t @ V @ P @ W )
=> ( V = W ) ) ) ).
% awalk_ends_eqD
thf(fact_24_reachable__trans,axiom,
! [U: a,V: a,W: a] :
( ( reachable_a_b @ t @ U @ V )
=> ( ( reachable_a_b @ t @ V @ W )
=> ( reachable_a_b @ t @ U @ W ) ) ) ).
% reachable_trans
thf(fact_25_awalk__last__in__verts,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% awalk_last_in_verts
thf(fact_26_awalk__hd__in__verts,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% awalk_hd_in_verts
thf(fact_27_reachable__in__verts_I1_J,axiom,
! [U: a,V: a] :
( ( reachable_a_b @ t @ U @ V )
=> ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% reachable_in_verts(1)
thf(fact_28_reachable__in__verts_I2_J,axiom,
! [U: a,V: a] :
( ( reachable_a_b @ t @ U @ V )
=> ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% reachable_in_verts(2)
thf(fact_29_reachable__awalkI,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( reachable_a_b @ t @ U @ V ) ) ).
% reachable_awalkI
thf(fact_30_reachable__awalk,axiom,
! [U: a,V: a] :
( ( reachable_a_b @ t @ U @ V )
= ( ? [P3: list_b] : ( arc_pre_awalk_a_b @ t @ U @ P3 @ V ) ) ) ).
% reachable_awalk
thf(fact_31_forward__split,axiom,
! [Xs: list_a,Ys: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ Xs @ Ys ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs ) ) ).
% forward_split
thf(fact_32_in__scc__of__self,axiom,
! [U: a] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( member_a @ U @ ( digrap2937667069914300949of_a_b @ t @ U ) ) ) ).
% in_scc_of_self
thf(fact_33_k__nh__reachable,axiom,
! [U: a,W: b > real,V: a,K: real] :
( ( member_a @ U @ ( graph_3921080825633621230od_a_b @ t @ W @ V @ K ) )
=> ( reachable_a_b @ t @ V @ U ) ) ).
% k_nh_reachable
thf(fact_34_reachable__via__child__impl__same,axiom,
! [X2: a,V: a,Y2: a,U: a] :
( ( reachable_a_b @ t @ X2 @ V )
=> ( ( reachable_a_b @ t @ Y2 @ V )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ X2 ) @ ( arcs_ends_a_b @ t ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ Y2 ) @ ( arcs_ends_a_b @ t ) )
=> ( X2 = Y2 ) ) ) ) ) ).
% reachable_via_child_impl_same
thf(fact_35_reachable__adj__trans,axiom,
! [A: a,B: a,C: a] :
( ( reachable_a_b @ t @ A @ B )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C ) @ ( arcs_ends_a_b @ t ) )
=> ( reachable_a_b @ t @ A @ C ) ) ) ).
% reachable_adj_trans
thf(fact_36_assms_I1_J,axiom,
? [X: a] :
( ( member_a @ X @ ( set_a2 @ u ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ ( hd_a @ v ) ) @ ( arcs_ends_a_b @ t ) ) ) ).
% assms(1)
thf(fact_37_reachable__induct,axiom,
! [U: a,V: a,P4: a > $o] :
( ( reachable_a_b @ t @ U @ V )
=> ( ( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( P4 @ U ) )
=> ( ! [X: a,Y3: a] :
( ( reachable_a_b @ t @ U @ X )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ ( arcs_ends_a_b @ t ) )
=> ( ( P4 @ X )
=> ( P4 @ Y3 ) ) ) )
=> ( P4 @ V ) ) ) ) ).
% reachable_induct
thf(fact_38_converse__reachable__induct,axiom,
! [U: a,V: a,P4: a > $o] :
( ( reachable_a_b @ t @ U @ V )
=> ( ( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( P4 @ V ) )
=> ( ! [X: a,Y3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ ( arcs_ends_a_b @ t ) )
=> ( ( reachable_a_b @ t @ Y3 @ V )
=> ( ( P4 @ Y3 )
=> ( P4 @ X ) ) ) )
=> ( P4 @ U ) ) ) ) ).
% converse_reachable_induct
thf(fact_39_converse__reachable__cases,axiom,
! [U: a,V: a] :
( ( reachable_a_b @ t @ U @ V )
=> ( ( ( U = V )
=> ~ ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) ) )
=> ~ ! [W2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ W2 ) @ ( arcs_ends_a_b @ t ) )
=> ~ ( reachable_a_b @ t @ W2 @ V ) ) ) ) ).
% converse_reachable_cases
thf(fact_40_hd__reach__all__forward,axiom,
! [Xs: list_a,X2: a] :
( ( member_a @ ( hd_a @ Xs ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs )
=> ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( reachable_a_b @ t @ ( hd_a @ Xs ) @ X2 ) ) ) ) ).
% hd_reach_all_forward
thf(fact_41_forward__arc__to__head_H,axiom,
! [Ys: list_a,X2: a,Y2: a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ Ys )
=> ( ~ ( member_a @ X2 @ ( set_a2 @ Ys ) )
=> ( ( member_a @ Y2 @ ( set_a2 @ Ys ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ ( arcs_ends_a_b @ t ) )
=> ( Y2
= ( hd_a @ Ys ) ) ) ) ) ) ).
% forward_arc_to_head'
thf(fact_42_before__alt1,axiom,
! [S1: list_a,S2: list_a] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ S1 ) )
& ? [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_a @ S2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ S1 @ I ) @ ( nth_a @ S2 @ J2 ) ) @ ( arcs_ends_a_b @ t ) ) ) ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ S1 ) )
& ? [Y4: a] :
( ( member_a @ Y4 @ ( set_a2 @ S2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ).
% before_alt1
thf(fact_43_mem__Collect__eq,axiom,
! [A: list_a,P4: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P4 ) )
= ( P4 @ A ) ) ).
% mem_Collect_eq
thf(fact_44_mem__Collect__eq,axiom,
! [A: set_a,P4: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P4 ) )
= ( P4 @ A ) ) ).
% mem_Collect_eq
thf(fact_45_mem__Collect__eq,axiom,
! [A: nat,P4: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P4 ) )
= ( P4 @ A ) ) ).
% mem_Collect_eq
thf(fact_46_mem__Collect__eq,axiom,
! [A: b,P4: b > $o] :
( ( member_b @ A @ ( collect_b @ P4 ) )
= ( P4 @ A ) ) ).
% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
! [A: a,P4: a > $o] :
( ( member_a @ A @ ( collect_a @ P4 ) )
= ( P4 @ A ) ) ).
% mem_Collect_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A2: set_list_a] :
( ( collect_list_a
@ ^ [X3: list_a] : ( member_list_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_49_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_50_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_51_Collect__mem__eq,axiom,
! [A2: set_b] :
( ( collect_b
@ ^ [X3: b] : ( member_b @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_52_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_53_forward__app,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ S1 )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ S2 )
=> ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ S1 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ ( hd_a @ S2 ) ) @ ( arcs_ends_a_b @ t ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ S1 @ S2 ) ) ) ) ) ).
% forward_app
thf(fact_54_forward__alt__aux2,axiom,
! [As: list_a,Bs: list_a,Xs: list_a,I2: nat] :
( ( ( append_a @ As @ Bs )
= Xs )
=> ( ( ( size_size_list_a @ As )
= I2 )
=> ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ As ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ ( nth_a @ Xs @ I2 ) ) @ ( arcs_ends_a_b @ t ) ) )
=> ? [J: nat] :
( ( ord_less_nat @ J @ I2 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ J ) @ ( nth_a @ Xs @ I2 ) ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ).
% forward_alt_aux2
thf(fact_55_reachable__refl,axiom,
! [V: a] :
( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( reachable_a_b @ t @ V @ V ) ) ).
% reachable_refl
thf(fact_56_reachable__adjI,axiom,
! [U: a,V: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( arcs_ends_a_b @ t ) )
=> ( reachable_a_b @ t @ U @ V ) ) ).
% reachable_adjI
thf(fact_57_reachable__arc__trans,axiom,
! [U: a,V: a,E: b,W: a] :
( ( reachable_a_b @ t @ U @ V )
=> ( ( wf_arc_a_b @ t @ E @ ( product_Pair_a_a @ V @ W ) )
=> ( reachable_a_b @ t @ U @ W ) ) ) ).
% reachable_arc_trans
thf(fact_58_directed__tree_Oforward_Ocong,axiom,
iKKBZ_4778857019735642799rd_a_b = iKKBZ_4778857019735642799rd_a_b ).
% directed_tree.forward.cong
thf(fact_59_closed__w__imp__cycle,axiom,
! [P: list_b] :
( ( arc_wf_closed_w_a_b @ t @ P )
=> ? [X_12: list_b] : ( arc_pre_cycle_a_b @ t @ X_12 ) ) ).
% closed_w_imp_cycle
thf(fact_60_before__arc__to__hd,axiom,
! [Xs: list_a,Ys: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ Xs @ Ys )
=> ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ ( hd_a @ Ys ) ) @ ( arcs_ends_a_b @ t ) ) ) ) ).
% before_arc_to_hd
thf(fact_61_before__ArcI,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S2 )
=> ? [X: a] :
( ( member_a @ X @ ( set_a2 @ S1 ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ S2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Xa ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% before_ArcI
thf(fact_62_mk__cycles__path__awalk,axiom,
! [U: a,C: list_b,N: nat] :
( ( arc_pre_awalk_a_b @ t @ U @ C @ U )
=> ( arc_pre_awalk_a_b @ t @ U @ ( shorte6374615165232202367path_b @ N @ C ) @ U ) ) ).
% mk_cycles_path_awalk
thf(fact_63_no__back__backarc__i__in__xs,axiom,
! [Ys: list_a,J3: nat,Xs: list_a,I2: nat] :
( ( iKKBZ_3684931046464919648ck_a_b @ t @ Ys )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_a @ ( append_a @ Xs @ Ys ) ) )
=> ( ( ord_less_nat @ I2 @ J3 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ ( append_a @ Xs @ Ys ) @ J3 ) @ ( nth_a @ ( append_a @ Xs @ Ys ) @ I2 ) ) @ ( arcs_ends_a_b @ t ) )
=> ( ( member_a @ ( nth_a @ Xs @ I2 ) @ ( set_a2 @ Xs ) )
& ( ( nth_a @ ( append_a @ Xs @ Ys ) @ I2 )
= ( nth_a @ Xs @ I2 ) ) ) ) ) ) ) ).
% no_back_backarc_i_in_xs
thf(fact_64_no__back__backarc__difsets_H,axiom,
! [Xs: list_a,Ys: list_a] :
( ( iKKBZ_3684931046464919648ck_a_b @ t @ Xs )
=> ( ( iKKBZ_3684931046464919648ck_a_b @ t @ Ys )
=> ( ? [I3: nat,J4: nat] :
( ( ord_less_nat @ I3 @ J4 )
& ( ord_less_nat @ J4 @ ( size_size_list_a @ ( append_a @ Xs @ Ys ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ ( append_a @ Xs @ Ys ) @ J4 ) @ ( nth_a @ ( append_a @ Xs @ Ys ) @ I3 ) ) @ ( arcs_ends_a_b @ t ) ) )
=> ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ).
% no_back_backarc_difsets'
thf(fact_65_no__back__backarc__difsets,axiom,
! [Xs: list_a,Ys: list_a,I2: nat,J3: nat] :
( ( iKKBZ_3684931046464919648ck_a_b @ t @ Xs )
=> ( ( iKKBZ_3684931046464919648ck_a_b @ t @ Ys )
=> ( ( ord_less_nat @ I2 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_a @ ( append_a @ Xs @ Ys ) ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ ( append_a @ Xs @ Ys ) @ J3 ) @ ( nth_a @ ( append_a @ Xs @ Ys ) @ I2 ) ) @ ( arcs_ends_a_b @ t ) )
=> ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ) ).
% no_back_backarc_difsets
thf(fact_66_hd__reach__all__forward_H,axiom,
! [Xs: list_a,X2: a] :
( ( ord_less_nat @ one_one_nat @ ( size_size_list_a @ Xs ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs )
=> ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( reachable_a_b @ t @ ( hd_a @ Xs ) @ X2 ) ) ) ) ).
% hd_reach_all_forward'
thf(fact_67_no__back__arcs__alt__aux1,axiom,
! [Xs: list_a,I2: nat,J3: nat] :
( ( iKKBZ_7773321254043928001cs_a_b @ t @ Xs )
=> ( ( ord_less_nat @ I2 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_a @ Xs ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ J3 ) @ ( nth_a @ Xs @ I2 ) ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% no_back_arcs_alt_aux1
thf(fact_68_before__alt2,axiom,
! [S1: list_a,S2: list_a] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ S1 ) )
=> ! [X3: a] :
( ( member_a @ X3 @ ( minus_minus_set_a @ ( minus_minus_set_a @ ( pre_ve642382030648772252t_unit @ t ) @ ( set_a2 @ S1 ) ) @ ( set_a2 @ S2 ) ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ S1 @ I ) @ X3 ) @ ( arcs_ends_a_b @ t ) ) ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ S1 ) )
=> ! [Y4: a] :
( ( member_a @ Y4 @ ( minus_minus_set_a @ ( minus_minus_set_a @ ( pre_ve642382030648772252t_unit @ t ) @ ( set_a2 @ S1 ) ) @ ( set_a2 @ S2 ) ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ).
% before_alt2
thf(fact_69_hd__in__verts__if__forward_H,axiom,
! [Xs: list_a] :
( ( ord_less_nat @ one_one_nat @ ( size_size_list_a @ Xs ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs )
=> ( member_a @ ( hd_a @ Xs ) @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% hd_in_verts_if_forward'
thf(fact_70_no__back__arcs__alt,axiom,
! [Xs: list_a] :
( ( iKKBZ_3684931046464919648ck_a_b @ t @ Xs )
= ( iKKBZ_7773321254043928001cs_a_b @ t @ Xs ) ) ).
% no_back_arcs_alt
thf(fact_71_no__back__arcs__alt__aux2,axiom,
! [Xs: list_a] :
( ( iKKBZ_3684931046464919648ck_a_b @ t @ Xs )
=> ( iKKBZ_7773321254043928001cs_a_b @ t @ Xs ) ) ).
% no_back_arcs_alt_aux2
thf(fact_72_before__forward1I,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S2 )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ S1 ) ) ).
% before_forward1I
thf(fact_73_before__forward2I,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S2 )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ S2 ) ) ).
% before_forward2I
thf(fact_74_before__no__back1I,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S2 )
=> ( iKKBZ_3684931046464919648ck_a_b @ t @ S1 ) ) ).
% before_no_back1I
thf(fact_75_before__no__back2I,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S2 )
=> ( iKKBZ_3684931046464919648ck_a_b @ t @ S2 ) ) ).
% before_no_back2I
thf(fact_76_no__back__before,axiom,
! [Xs: list_a,Ys: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ Xs @ Ys )
=> ( iKKBZ_3684931046464919648ck_a_b @ t @ ( append_a @ Xs @ Ys ) ) ) ).
% no_back_before
thf(fact_77_no__back__def,axiom,
! [Xs: list_a] :
( ( iKKBZ_3684931046464919648ck_a_b @ t @ Xs )
= ( ~ ? [I: nat,J2: nat] :
( ( ord_less_nat @ I @ J2 )
& ( ord_less_nat @ J2 @ ( size_size_list_a @ Xs ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ J2 ) @ ( nth_a @ Xs @ I ) ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% no_back_def
thf(fact_78_directed__tree_Obefore_Ocong,axiom,
iKKBZ_7682935289300565975re_a_b = iKKBZ_7682935289300565975re_a_b ).
% directed_tree.before.cong
thf(fact_79_directed__tree_Ono__back__arcs_Ocong,axiom,
iKKBZ_7773321254043928001cs_a_b = iKKBZ_7773321254043928001cs_a_b ).
% directed_tree.no_back_arcs.cong
thf(fact_80_directed__tree_Ono__back_Ocong,axiom,
iKKBZ_3684931046464919648ck_a_b = iKKBZ_3684931046464919648ck_a_b ).
% directed_tree.no_back.cong
thf(fact_81_no__back__backarc__j__in__ys,axiom,
! [Xs: list_a,J3: nat,Ys: list_a,I2: nat] :
( ( iKKBZ_3684931046464919648ck_a_b @ t @ Xs )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_a @ ( append_a @ Xs @ Ys ) ) )
=> ( ( ord_less_nat @ I2 @ J3 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ ( append_a @ Xs @ Ys ) @ J3 ) @ ( nth_a @ ( append_a @ Xs @ Ys ) @ I2 ) ) @ ( arcs_ends_a_b @ t ) )
=> ( ( member_a @ ( nth_a @ Ys @ ( minus_minus_nat @ J3 @ ( size_size_list_a @ Xs ) ) ) @ ( set_a2 @ Ys ) )
& ( ( nth_a @ ( append_a @ Xs @ Ys ) @ J3 )
= ( nth_a @ Ys @ ( minus_minus_nat @ J3 @ ( size_size_list_a @ Xs ) ) ) ) ) ) ) ) ) ).
% no_back_backarc_j_in_ys
thf(fact_82_no__back__backarc__app1,axiom,
! [J3: nat,Xs: list_a,Ys: list_a,I2: nat] :
( ( ord_less_nat @ J3 @ ( size_size_list_a @ ( append_a @ Xs @ Ys ) ) )
=> ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ J3 )
=> ( ( ord_less_nat @ I2 @ J3 )
=> ( ( iKKBZ_3684931046464919648ck_a_b @ t @ Ys )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ ( append_a @ Xs @ Ys ) @ J3 ) @ ( nth_a @ ( append_a @ Xs @ Ys ) @ I2 ) ) @ ( arcs_ends_a_b @ t ) )
=> ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) ) ) ) ) ) ) ).
% no_back_backarc_app1
thf(fact_83_no__back__backarc__app2,axiom,
! [Xs: list_a,I2: nat,J3: nat,Ys: list_a] :
( ( iKKBZ_3684931046464919648ck_a_b @ t @ Xs )
=> ( ( ord_less_nat @ I2 @ J3 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ ( append_a @ Xs @ Ys ) @ J3 ) @ ( nth_a @ ( append_a @ Xs @ Ys ) @ I2 ) ) @ ( arcs_ends_a_b @ t ) )
=> ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ J3 ) ) ) ) ).
% no_back_backarc_app2
thf(fact_84_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_85_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_86_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_87_move__mid__forward__if__noarc,axiom,
! [As: list_a,U2: list_a,Bs: list_a,Cs: list_a] :
( ( As != nil_a )
=> ( ~ ? [X: a] :
( ( member_a @ X @ ( set_a2 @ U2 ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Bs ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Xa ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ Cs ) ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ Bs @ ( append_a @ U2 @ Cs ) ) ) ) ) ) ) ).
% move_mid_forward_if_noarc
thf(fact_88_seq__conform__if__before,axiom,
! [Xs: list_a,Ys: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ Xs @ Ys )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ ( append_a @ Xs @ Ys ) ) ) ).
% seq_conform_if_before
thf(fact_89_no__back__alt,axiom,
! [Xs: list_a] :
( ( ! [I: nat,J2: nat] :
( ( ord_less_eq_nat @ J2 @ I )
| ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ J2 )
| ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ J2 ) @ ( nth_a @ Xs @ I ) ) @ ( arcs_ends_a_b @ t ) ) ) )
= ( iKKBZ_3684931046464919648ck_a_b @ t @ Xs ) ) ).
% no_back_alt
thf(fact_90_no__back__alt__aux,axiom,
! [Xs: list_a] :
( ! [I4: nat,J: nat] :
( ( ord_less_eq_nat @ J @ I4 )
| ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ J )
| ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ J ) @ ( nth_a @ Xs @ I4 ) ) @ ( arcs_ends_a_b @ t ) ) )
=> ( iKKBZ_3684931046464919648ck_a_b @ t @ Xs ) ) ).
% no_back_alt_aux
thf(fact_91_split__length__i,axiom,
! [I2: nat,Bs: list_a] :
( ( ord_less_eq_nat @ I2 @ ( size_size_list_a @ Bs ) )
=> ? [Xs2: list_a,Ys2: list_a] :
( ( ( append_a @ Xs2 @ Ys2 )
= Bs )
& ( ( size_size_list_a @ Xs2 )
= I2 ) ) ) ).
% split_length_i
thf(fact_92_split__length__i,axiom,
! [I2: nat,Bs: list_b] :
( ( ord_less_eq_nat @ I2 @ ( size_size_list_b @ Bs ) )
=> ? [Xs2: list_b,Ys2: list_b] :
( ( ( append_b @ Xs2 @ Ys2 )
= Bs )
& ( ( size_size_list_b @ Xs2 )
= I2 ) ) ) ).
% split_length_i
thf(fact_93_no__back__arcs_Osimps_I1_J,axiom,
iKKBZ_7773321254043928001cs_a_b @ t @ nil_a ).
% no_back_arcs.simps(1)
thf(fact_94_split__length__i__prefix,axiom,
! [As: list_a,I2: nat,Bs: list_a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ As ) @ I2 )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_a @ ( append_a @ As @ Bs ) ) )
=> ? [Xs2: list_a,Ys2: list_a] :
( ( ( append_a @ Xs2 @ Ys2 )
= Bs )
& ( ( size_size_list_a @ ( append_a @ As @ Xs2 ) )
= I2 ) ) ) ) ).
% split_length_i_prefix
thf(fact_95_split__length__i__prefix,axiom,
! [As: list_b,I2: nat,Bs: list_b] :
( ( ord_less_eq_nat @ ( size_size_list_b @ As ) @ I2 )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_b @ ( append_b @ As @ Bs ) ) )
=> ? [Xs2: list_b,Ys2: list_b] :
( ( ( append_b @ Xs2 @ Ys2 )
= Bs )
& ( ( size_size_list_b @ ( append_b @ As @ Xs2 ) )
= I2 ) ) ) ) ).
% split_length_i_prefix
thf(fact_96_before__conform2I,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S2 )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ S2 ) ) ).
% before_conform2I
thf(fact_97_before__conform1I,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S2 )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ S1 ) ) ).
% before_conform1I
thf(fact_98_seq__conform__alt,axiom,
! [Xs: list_a] :
( ( iKKBZ_4622586873178280511rm_a_b @ t @ Xs )
= ( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs )
& ( iKKBZ_3684931046464919648ck_a_b @ t @ Xs ) ) ) ).
% seq_conform_alt
thf(fact_99__C2_C_I2_J,axiom,
ord_less_eq_nat @ i @ ( minus_minus_nat @ ( size_size_list_a @ ( append_a @ as @ ( append_a @ u @ v ) ) ) @ one_one_nat ) ).
% "2"(2)
thf(fact_100_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_101_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_102_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_103_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_104_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_105_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_106_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_107_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_108_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_109_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_110_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_111_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_112_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_113_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_114_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_115_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_116_diff__diff__cancel,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_117_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_118_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_119_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_120_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_121_assms_I4_J,axiom,
member_nat @ i @ ( set_or1269000886237332187st_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( append_a @ as @ ( append_a @ u @ ( append_a @ v @ ( append_a @ bs @ cs ) ) ) ) ) @ one_one_nat ) ) ).
% assms(4)
thf(fact_122_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_123_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_124_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_125_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_126_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_127_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_128_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_129_diff__commute,axiom,
! [I2: nat,J3: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J3 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J3 ) ) ).
% diff_commute
thf(fact_130_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_131_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_132_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_133_le__trans,axiom,
! [I2: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ( ord_less_eq_nat @ J3 @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_134_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_135_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_136_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_137_Nat_Oex__has__greatest__nat,axiom,
! [P4: nat > $o,K: nat,B: nat] :
( ( P4 @ K )
=> ( ! [Y3: nat] :
( ( P4 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X: nat] :
( ( P4 @ X )
& ! [Y: nat] :
( ( P4 @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_138_directed__tree_Oseq__conform_Ocong,axiom,
iKKBZ_4622586873178280511rm_a_b = iKKBZ_4622586873178280511rm_a_b ).
% directed_tree.seq_conform.cong
thf(fact_139_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_140_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_141_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_142_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_143_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_144_diff__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ D @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_145_diff__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_146_diff__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_147_diff__eq__diff__less__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_eq_real @ A @ B )
= ( ord_less_eq_real @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_148_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_149_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_150_less__imp__diff__less,axiom,
! [J3: nat,K: nat,N: nat] :
( ( ord_less_nat @ J3 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J3 @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_151_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_152_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_153_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_154_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_155_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_156_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J3: nat] :
( ! [I4: nat,J: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ord_less_nat @ ( F @ I4 ) @ ( F @ J ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J3 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_157_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_158_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_159_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_160_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_161_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_162_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_163_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_164_ex__least__nat__le,axiom,
! [P4: nat > $o,N: nat] :
( ( P4 @ N )
=> ( ~ ( P4 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P4 @ I3 ) )
& ( P4 @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_165_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_166_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_167_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_168_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_169_linorder__neqE__nat,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_170_infinite__descent,axiom,
! [P4: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P4 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P4 @ M3 ) ) )
=> ( P4 @ N ) ) ).
% infinite_descent
thf(fact_171_nat__less__induct,axiom,
! [P4: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P4 @ M3 ) )
=> ( P4 @ N3 ) )
=> ( P4 @ N ) ) ).
% nat_less_induct
thf(fact_172_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_173_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_174_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_175_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_176_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_177_size__neq__size__imp__neq,axiom,
! [X2: list_a,Y2: list_a] :
( ( ( size_size_list_a @ X2 )
!= ( size_size_list_a @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_178_size__neq__size__imp__neq,axiom,
! [X2: list_b,Y2: list_b] :
( ( ( size_size_list_b @ X2 )
!= ( size_size_list_b @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_179_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_180_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_181_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_182_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_183_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_184_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_185_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: real,Z: real] : ( Y5 = Z ) )
= ( ^ [A3: real,B2: real] :
( ( minus_minus_real @ A3 @ B2 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_186_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_187_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_188_diff__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_189_diff__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_190_diff__eq__diff__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_191_diff__strict__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D @ C )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_192_infinite__descent0,axiom,
! [P4: nat > $o,N: nat] :
( ( P4 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P4 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P4 @ M3 ) ) ) )
=> ( P4 @ N ) ) ) ).
% infinite_descent0
thf(fact_193_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_194_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_195_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_196_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_197_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_198_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_199_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_200_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_201_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_202__C0_C,axiom,
member_nat @ i @ ( set_or1269000886237332187st_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( append_a @ as @ ( append_a @ u @ ( append_a @ bs @ ( append_a @ v @ cs ) ) ) ) ) @ one_one_nat ) ) ).
% "0"
thf(fact_203_length__greater__0__conv,axiom,
! [Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) )
= ( Xs != nil_a ) ) ).
% length_greater_0_conv
thf(fact_204_length__greater__0__conv,axiom,
! [Xs: list_b] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_b @ Xs ) )
= ( Xs != nil_b ) ) ).
% length_greater_0_conv
thf(fact_205_forward__app__aux,axiom,
! [S1: list_a,S2: list_a,I2: nat] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ S1 )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ S2 )
=> ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ S1 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ ( hd_a @ S2 ) ) @ ( arcs_ends_a_b @ t ) ) )
=> ( ( member_nat @ I2 @ ( set_or1269000886237332187st_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( append_a @ S1 @ S2 ) ) @ one_one_nat ) ) )
=> ? [J: nat] :
( ( ord_less_nat @ J @ I2 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ ( append_a @ S1 @ S2 ) @ J ) @ ( nth_a @ ( append_a @ S1 @ S2 ) @ I2 ) ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ).
% forward_app_aux
thf(fact_206_move__mid__forward__if__noarc__aux,axiom,
! [As: list_a,U2: list_a,Bs: list_a,Cs: list_a,I2: nat] :
( ( As != nil_a )
=> ( ~ ? [X: a] :
( ( member_a @ X @ ( set_a2 @ U2 ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Bs ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Xa ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ Cs ) ) ) )
=> ( ( member_nat @ I2 @ ( set_or1269000886237332187st_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( append_a @ As @ ( append_a @ Bs @ ( append_a @ U2 @ Cs ) ) ) ) @ one_one_nat ) ) )
=> ? [J: nat] :
( ( ord_less_nat @ J @ I2 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ ( append_a @ As @ ( append_a @ Bs @ ( append_a @ U2 @ Cs ) ) ) @ J ) @ ( nth_a @ ( append_a @ As @ ( append_a @ Bs @ ( append_a @ U2 @ Cs ) ) ) @ I2 ) ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ).
% move_mid_forward_if_noarc_aux
thf(fact_207_forward__alt__aux2_H,axiom,
! [Xs: list_a] :
( ! [X: nat] :
( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) )
=> ? [As2: list_a] :
( ? [Bs2: list_a] :
( ( append_a @ As2 @ Bs2 )
= Xs )
& ( ( size_size_list_a @ As2 )
= X )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ As2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ ( nth_a @ Xs @ X ) ) @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs ) ) ).
% forward_alt_aux2'
thf(fact_208_forward__alt__aux1_H,axiom,
! [Xs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs )
=> ! [X4: nat] :
( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) )
=> ? [As3: list_a] :
( ? [Bs3: list_a] :
( ( append_a @ As3 @ Bs3 )
= Xs )
& ( ( size_size_list_a @ As3 )
= X4 )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ As3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ ( nth_a @ Xs @ X4 ) ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ).
% forward_alt_aux1'
thf(fact_209_forward__alt,axiom,
! [Xs: list_a,X4: nat] :
( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) )
=> ? [As3: list_a,Bs3: list_a] :
( ( ( ( append_a @ As3 @ Bs3 )
= Xs )
& ( ( size_size_list_a @ As3 )
= X4 )
& ? [Y4: a] :
( ( member_a @ Y4 @ ( set_a2 @ As3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ ( nth_a @ Xs @ X4 ) ) @ ( arcs_ends_a_b @ t ) ) ) )
= ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs ) ) ) ).
% forward_alt
thf(fact_210_hd__append2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ).
% hd_append2
thf(fact_211_hd__append2,axiom,
! [Xs: list_b,Ys: list_b] :
( ( Xs != nil_b )
=> ( ( hd_b @ ( append_b @ Xs @ Ys ) )
= ( hd_b @ Xs ) ) ) ).
% hd_append2
thf(fact_212_forward__split__aux,axiom,
! [Xs: list_a,Ys: list_a,I2: nat] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ Xs @ Ys ) )
=> ( ( member_nat @ I2 @ ( set_or1269000886237332187st_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) )
=> ? [J: nat] :
( ( ord_less_nat @ J @ I2 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ J ) @ ( nth_a @ Xs @ I2 ) ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% forward_split_aux
thf(fact_213_forward__alt__aux1,axiom,
! [I2: nat,Xs: list_a,J3: nat] :
( ( member_nat @ I2 @ ( set_or1269000886237332187st_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) )
=> ( ( ord_less_nat @ J3 @ I2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ J3 ) @ ( nth_a @ Xs @ I2 ) ) @ ( arcs_ends_a_b @ t ) )
=> ? [As3: list_a] :
( ? [Bs3: list_a] :
( ( append_a @ As3 @ Bs3 )
= Xs )
& ( ( size_size_list_a @ As3 )
= I2 )
& ? [X: a] :
( ( member_a @ X @ ( set_a2 @ As3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ ( nth_a @ Xs @ I2 ) ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ).
% forward_alt_aux1
thf(fact_214_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_215_awalk__appendI,axiom,
! [U: a,P: list_b,V: a,Q: list_b,W: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( ( arc_pre_awalk_a_b @ t @ V @ Q @ W )
=> ( arc_pre_awalk_a_b @ t @ U @ ( append_b @ P @ Q ) @ W ) ) ) ).
% awalk_appendI
thf(fact_216_awalk__empty__ends,axiom,
! [U: a,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ nil_b @ V )
=> ( U = V ) ) ).
% awalk_empty_ends
thf(fact_217_awalk__ends,axiom,
! [U: a,P: list_b,V: a,U3: a,V2: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( ( arc_pre_awalk_a_b @ t @ U3 @ P @ V2 )
=> ( ( ( P != nil_b )
& ( U = U3 )
& ( V = V2 ) )
| ( ( P = nil_b )
& ( U = V )
& ( U3 = V2 ) ) ) ) ) ).
% awalk_ends
thf(fact_218_awalk__Nil__iff,axiom,
! [U: a,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ nil_b @ V )
= ( ( U = V )
& ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% awalk_Nil_iff
thf(fact_219_append_Oassoc,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( append_a @ ( append_a @ A @ B ) @ C )
= ( append_a @ A @ ( append_a @ B @ C ) ) ) ).
% append.assoc
thf(fact_220_append_Oassoc,axiom,
! [A: list_b,B: list_b,C: list_b] :
( ( append_b @ ( append_b @ A @ B ) @ C )
= ( append_b @ A @ ( append_b @ B @ C ) ) ) ).
% append.assoc
thf(fact_221_append__assoc,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( append_a @ ( append_a @ Xs @ Ys ) @ Zs )
= ( append_a @ Xs @ ( append_a @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_222_append__assoc,axiom,
! [Xs: list_b,Ys: list_b,Zs: list_b] :
( ( append_b @ ( append_b @ Xs @ Ys ) @ Zs )
= ( append_b @ Xs @ ( append_b @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_223_append__same__eq,axiom,
! [Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( append_a @ Ys @ Xs )
= ( append_a @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_224_append__same__eq,axiom,
! [Ys: list_b,Xs: list_b,Zs: list_b] :
( ( ( append_b @ Ys @ Xs )
= ( append_b @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_225_same__append__eq,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_226_same__append__eq,axiom,
! [Xs: list_b,Ys: list_b,Zs: list_b] :
( ( ( append_b @ Xs @ Ys )
= ( append_b @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_227_closed__w__def,axiom,
! [P: list_b] :
( ( arc_wf_closed_w_a_b @ t @ P )
= ( ? [U4: a] :
( ( arc_pre_awalk_a_b @ t @ U4 @ P @ U4 )
& ( ord_less_nat @ zero_zero_nat @ ( size_size_list_b @ P ) ) ) ) ) ).
% closed_w_def
thf(fact_228_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_229_append_Oright__neutral,axiom,
! [A: list_b] :
( ( append_b @ A @ nil_b )
= A ) ).
% append.right_neutral
thf(fact_230_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_231_append__Nil2,axiom,
! [Xs: list_b] :
( ( append_b @ Xs @ nil_b )
= Xs ) ).
% append_Nil2
thf(fact_232_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_233_append__self__conv,axiom,
! [Xs: list_b,Ys: list_b] :
( ( ( append_b @ Xs @ Ys )
= Xs )
= ( Ys = nil_b ) ) ).
% append_self_conv
thf(fact_234_self__append__conv,axiom,
! [Y2: list_a,Ys: list_a] :
( ( Y2
= ( append_a @ Y2 @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_235_self__append__conv,axiom,
! [Y2: list_b,Ys: list_b] :
( ( Y2
= ( append_b @ Y2 @ Ys ) )
= ( Ys = nil_b ) ) ).
% self_append_conv
thf(fact_236_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_237_append__self__conv2,axiom,
! [Xs: list_b,Ys: list_b] :
( ( ( append_b @ Xs @ Ys )
= Ys )
= ( Xs = nil_b ) ) ).
% append_self_conv2
thf(fact_238_self__append__conv2,axiom,
! [Y2: list_a,Xs: list_a] :
( ( Y2
= ( append_a @ Xs @ Y2 ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_239_self__append__conv2,axiom,
! [Y2: list_b,Xs: list_b] :
( ( Y2
= ( append_b @ Xs @ Y2 ) )
= ( Xs = nil_b ) ) ).
% self_append_conv2
thf(fact_240_Nil__is__append__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( nil_a
= ( append_a @ Xs @ Ys ) )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_241_Nil__is__append__conv,axiom,
! [Xs: list_b,Ys: list_b] :
( ( nil_b
= ( append_b @ Xs @ Ys ) )
= ( ( Xs = nil_b )
& ( Ys = nil_b ) ) ) ).
% Nil_is_append_conv
thf(fact_242_append__is__Nil__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= nil_a )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_243_append__is__Nil__conv,axiom,
! [Xs: list_b,Ys: list_b] :
( ( ( append_b @ Xs @ Ys )
= nil_b )
= ( ( Xs = nil_b )
& ( Ys = nil_b ) ) ) ).
% append_is_Nil_conv
thf(fact_244_append__eq__append__conv,axiom,
! [Xs: list_a,Ys: list_a,Us: list_a,Vs: list_a] :
( ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
| ( ( size_size_list_a @ Us )
= ( size_size_list_a @ Vs ) ) )
=> ( ( ( append_a @ Xs @ Us )
= ( append_a @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_245_append__eq__append__conv,axiom,
! [Xs: list_b,Ys: list_b,Us: list_b,Vs: list_b] :
( ( ( ( size_size_list_b @ Xs )
= ( size_size_list_b @ Ys ) )
| ( ( size_size_list_b @ Us )
= ( size_size_list_b @ Vs ) ) )
=> ( ( ( append_b @ Xs @ Us )
= ( append_b @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_246_forward__def,axiom,
! [Xs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) )
=> ? [J2: nat] :
( ( ord_less_nat @ J2 @ X3 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ J2 ) @ ( nth_a @ Xs @ X3 ) ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ).
% forward_def
thf(fact_247_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_248_length__0__conv,axiom,
! [Xs: list_b] :
( ( ( size_size_list_b @ Xs )
= zero_zero_nat )
= ( Xs = nil_b ) ) ).
% length_0_conv
thf(fact_249_subset__code_I1_J,axiom,
! [Xs: list_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ B3 )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
=> ( member_list_a @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_250_subset__code_I1_J,axiom,
! [Xs: list_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ B3 )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ ( set_set_a2 @ Xs ) )
=> ( member_set_a @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_251_subset__code_I1_J,axiom,
! [Xs: list_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B3 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_252_subset__code_I1_J,axiom,
! [Xs: list_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B3 )
= ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( member_a @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_253_subset__code_I1_J,axiom,
! [Xs: list_b,B3: set_b] :
( ( ord_less_eq_set_b @ ( set_b2 @ Xs ) @ B3 )
= ( ! [X3: b] :
( ( member_b @ X3 @ ( set_b2 @ Xs ) )
=> ( member_b @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_254_ex__nat__less,axiom,
! [N: nat,P4: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
& ( P4 @ M2 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
& ( P4 @ X3 ) ) ) ) ).
% ex_nat_less
thf(fact_255_all__nat__less,axiom,
! [N: nat,P4: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( P4 @ M2 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( P4 @ X3 ) ) ) ) ).
% all_nat_less
thf(fact_256_append__eq__appendI,axiom,
! [Xs: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us: list_a] :
( ( ( append_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_a @ Xs1 @ Us ) )
=> ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_257_append__eq__appendI,axiom,
! [Xs: list_b,Xs1: list_b,Zs: list_b,Ys: list_b,Us: list_b] :
( ( ( append_b @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_b @ Xs1 @ Us ) )
=> ( ( append_b @ Xs @ Ys )
= ( append_b @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_258_append__eq__append__conv2,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ts: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Ts ) )
= ( ? [Us2: list_a] :
( ( ( Xs
= ( append_a @ Zs @ Us2 ) )
& ( ( append_a @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_a @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_a @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_259_append__eq__append__conv2,axiom,
! [Xs: list_b,Ys: list_b,Zs: list_b,Ts: list_b] :
( ( ( append_b @ Xs @ Ys )
= ( append_b @ Zs @ Ts ) )
= ( ? [Us2: list_b] :
( ( ( Xs
= ( append_b @ Zs @ Us2 ) )
& ( ( append_b @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_b @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_b @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_260_neq__if__length__neq,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
!= ( size_size_list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_261_neq__if__length__neq,axiom,
! [Xs: list_b,Ys: list_b] :
( ( ( size_size_list_b @ Xs )
!= ( size_size_list_b @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_262_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_263_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_b] :
( ( size_size_list_b @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_264_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_265_append__Nil,axiom,
! [Ys: list_b] :
( ( append_b @ nil_b @ Ys )
= Ys ) ).
% append_Nil
thf(fact_266_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_267_append_Oleft__neutral,axiom,
! [A: list_b] :
( ( append_b @ nil_b @ A )
= A ) ).
% append.left_neutral
thf(fact_268_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_269_eq__Nil__appendI,axiom,
! [Xs: list_b,Ys: list_b] :
( ( Xs = Ys )
=> ( Xs
= ( append_b @ nil_b @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_270_length__induct,axiom,
! [P4: list_a > $o,Xs: list_a] :
( ! [Xs2: list_a] :
( ! [Ys3: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys3 ) @ ( size_size_list_a @ Xs2 ) )
=> ( P4 @ Ys3 ) )
=> ( P4 @ Xs2 ) )
=> ( P4 @ Xs ) ) ).
% length_induct
thf(fact_271_length__induct,axiom,
! [P4: list_b > $o,Xs: list_b] :
( ! [Xs2: list_b] :
( ! [Ys3: list_b] :
( ( ord_less_nat @ ( size_size_list_b @ Ys3 ) @ ( size_size_list_b @ Xs2 ) )
=> ( P4 @ Ys3 ) )
=> ( P4 @ Xs2 ) )
=> ( P4 @ Xs ) ) ).
% length_induct
thf(fact_272_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_273_list_Osize_I3_J,axiom,
( ( size_size_list_b @ nil_b )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_274_hd__in__set,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( member_list_a @ ( hd_list_a @ Xs ) @ ( set_list_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_275_hd__in__set,axiom,
! [Xs: list_set_a] :
( ( Xs != nil_set_a )
=> ( member_set_a @ ( hd_set_a @ Xs ) @ ( set_set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_276_hd__in__set,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( member_nat @ ( hd_nat @ Xs ) @ ( set_nat2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_277_hd__in__set,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( member_a @ ( hd_a @ Xs ) @ ( set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_278_hd__in__set,axiom,
! [Xs: list_b] :
( ( Xs != nil_b )
=> ( member_b @ ( hd_b @ Xs ) @ ( set_b2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_279_list_Oset__sel_I1_J,axiom,
! [A: list_list_a] :
( ( A != nil_list_a )
=> ( member_list_a @ ( hd_list_a @ A ) @ ( set_list_a2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_280_list_Oset__sel_I1_J,axiom,
! [A: list_set_a] :
( ( A != nil_set_a )
=> ( member_set_a @ ( hd_set_a @ A ) @ ( set_set_a2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_281_list_Oset__sel_I1_J,axiom,
! [A: list_nat] :
( ( A != nil_nat )
=> ( member_nat @ ( hd_nat @ A ) @ ( set_nat2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_282_list_Oset__sel_I1_J,axiom,
! [A: list_a] :
( ( A != nil_a )
=> ( member_a @ ( hd_a @ A ) @ ( set_a2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_283_list_Oset__sel_I1_J,axiom,
! [A: list_b] :
( ( A != nil_b )
=> ( member_b @ ( hd_b @ A ) @ ( set_b2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_284_longest__common__prefix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ps: list_a,Xs3: list_a,Ys4: list_a] :
( ( Xs
= ( append_a @ Ps @ Xs3 ) )
& ( Ys
= ( append_a @ Ps @ Ys4 ) )
& ( ( Xs3 = nil_a )
| ( Ys4 = nil_a )
| ( ( hd_a @ Xs3 )
!= ( hd_a @ Ys4 ) ) ) ) ).
% longest_common_prefix
thf(fact_285_longest__common__prefix,axiom,
! [Xs: list_b,Ys: list_b] :
? [Ps: list_b,Xs3: list_b,Ys4: list_b] :
( ( Xs
= ( append_b @ Ps @ Xs3 ) )
& ( Ys
= ( append_b @ Ps @ Ys4 ) )
& ( ( Xs3 = nil_b )
| ( Ys4 = nil_b )
| ( ( hd_b @ Xs3 )
!= ( hd_b @ Ys4 ) ) ) ) ).
% longest_common_prefix
thf(fact_286_hd__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Ys ) ) )
& ( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ) ).
% hd_append
thf(fact_287_hd__append,axiom,
! [Xs: list_b,Ys: list_b] :
( ( ( Xs = nil_b )
=> ( ( hd_b @ ( append_b @ Xs @ Ys ) )
= ( hd_b @ Ys ) ) )
& ( ( Xs != nil_b )
=> ( ( hd_b @ ( append_b @ Xs @ Ys ) )
= ( hd_b @ Xs ) ) ) ) ).
% hd_append
thf(fact_288_nth__equalityI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ Xs @ I4 )
= ( nth_a @ Ys @ I4 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_289_nth__equalityI,axiom,
! [Xs: list_b,Ys: list_b] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_b @ Ys ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_b @ Xs ) )
=> ( ( nth_b @ Xs @ I4 )
= ( nth_b @ Ys @ I4 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_290_Skolem__list__nth,axiom,
! [K: nat,P4: nat > a > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ? [X5: a] : ( P4 @ I @ X5 ) ) )
= ( ? [Xs4: list_a] :
( ( ( size_size_list_a @ Xs4 )
= K )
& ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ( P4 @ I @ ( nth_a @ Xs4 @ I ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_291_Skolem__list__nth,axiom,
! [K: nat,P4: nat > b > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ? [X5: b] : ( P4 @ I @ X5 ) ) )
= ( ? [Xs4: list_b] :
( ( ( size_size_list_b @ Xs4 )
= K )
& ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ( P4 @ I @ ( nth_b @ Xs4 @ I ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_292_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_a,Z: list_a] : ( Y5 = Z ) )
= ( ^ [Xs4: list_a,Ys5: list_a] :
( ( ( size_size_list_a @ Xs4 )
= ( size_size_list_a @ Ys5 ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs4 ) )
=> ( ( nth_a @ Xs4 @ I )
= ( nth_a @ Ys5 @ I ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_293_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_b,Z: list_b] : ( Y5 = Z ) )
= ( ^ [Xs4: list_b,Ys5: list_b] :
( ( ( size_size_list_b @ Xs4 )
= ( size_size_list_b @ Ys5 ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_b @ Xs4 ) )
=> ( ( nth_b @ Xs4 @ I )
= ( nth_b @ Ys5 @ I ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_294_length__pos__if__in__set,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s349497388124573686list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_295_length__pos__if__in__set,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_set_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_296_length__pos__if__in__set,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_297_length__pos__if__in__set,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_298_length__pos__if__in__set,axiom,
! [X2: b,Xs: list_b] :
( ( member_b @ X2 @ ( set_b2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_b @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_299_nth__mem,axiom,
! [N: nat,Xs: list_list_a] :
( ( ord_less_nat @ N @ ( size_s349497388124573686list_a @ Xs ) )
=> ( member_list_a @ ( nth_list_a @ Xs @ N ) @ ( set_list_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_300_nth__mem,axiom,
! [N: nat,Xs: list_set_a] :
( ( ord_less_nat @ N @ ( size_size_list_set_a @ Xs ) )
=> ( member_set_a @ ( nth_set_a @ Xs @ N ) @ ( set_set_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_301_nth__mem,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_302_nth__mem,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( member_a @ ( nth_a @ Xs @ N ) @ ( set_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_303_nth__mem,axiom,
! [N: nat,Xs: list_b] :
( ( ord_less_nat @ N @ ( size_size_list_b @ Xs ) )
=> ( member_b @ ( nth_b @ Xs @ N ) @ ( set_b2 @ Xs ) ) ) ).
% nth_mem
thf(fact_304_list__ball__nth,axiom,
! [N: nat,Xs: list_a,P4: a > $o] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( P4 @ X ) )
=> ( P4 @ ( nth_a @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_305_list__ball__nth,axiom,
! [N: nat,Xs: list_b,P4: b > $o] :
( ( ord_less_nat @ N @ ( size_size_list_b @ Xs ) )
=> ( ! [X: b] :
( ( member_b @ X @ ( set_b2 @ Xs ) )
=> ( P4 @ X ) )
=> ( P4 @ ( nth_b @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_306_in__set__conv__nth,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ Xs ) )
& ( ( nth_list_a @ Xs @ I )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_307_in__set__conv__nth,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_set_a @ Xs ) )
& ( ( nth_set_a @ Xs @ I )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_308_in__set__conv__nth,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_309_in__set__conv__nth,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ I )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_310_in__set__conv__nth,axiom,
! [X2: b,Xs: list_b] :
( ( member_b @ X2 @ ( set_b2 @ Xs ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_b @ Xs ) )
& ( ( nth_b @ Xs @ I )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_311_all__nth__imp__all__set,axiom,
! [Xs: list_list_a,P4: list_a > $o,X2: list_a] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( P4 @ ( nth_list_a @ Xs @ I4 ) ) )
=> ( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ( P4 @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_312_all__nth__imp__all__set,axiom,
! [Xs: list_set_a,P4: set_a > $o,X2: set_a] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_set_a @ Xs ) )
=> ( P4 @ ( nth_set_a @ Xs @ I4 ) ) )
=> ( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
=> ( P4 @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_313_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P4: nat > $o,X2: nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( P4 @ ( nth_nat @ Xs @ I4 ) ) )
=> ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( P4 @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_314_all__nth__imp__all__set,axiom,
! [Xs: list_a,P4: a > $o,X2: a] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( P4 @ ( nth_a @ Xs @ I4 ) ) )
=> ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( P4 @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_315_all__nth__imp__all__set,axiom,
! [Xs: list_b,P4: b > $o,X2: b] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_b @ Xs ) )
=> ( P4 @ ( nth_b @ Xs @ I4 ) ) )
=> ( ( member_b @ X2 @ ( set_b2 @ Xs ) )
=> ( P4 @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_316_all__set__conv__all__nth,axiom,
! [Xs: list_a,P4: a > $o] :
( ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( P4 @ X3 ) ) )
= ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( P4 @ ( nth_a @ Xs @ I ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_317_all__set__conv__all__nth,axiom,
! [Xs: list_b,P4: b > $o] :
( ( ! [X3: b] :
( ( member_b @ X3 @ ( set_b2 @ Xs ) )
=> ( P4 @ X3 ) ) )
= ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_b @ Xs ) )
=> ( P4 @ ( nth_b @ Xs @ I ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_318_hd__conv__nth,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ Xs )
= ( nth_a @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_319_hd__conv__nth,axiom,
! [Xs: list_b] :
( ( Xs != nil_b )
=> ( ( hd_b @ Xs )
= ( nth_b @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_320_nth__append,axiom,
! [N: nat,Xs: list_a,Ys: list_a] :
( ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
= ( nth_a @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
= ( nth_a @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_321_nth__append,axiom,
! [N: nat,Xs: list_b,Ys: list_b] :
( ( ( ord_less_nat @ N @ ( size_size_list_b @ Xs ) )
=> ( ( nth_b @ ( append_b @ Xs @ Ys ) @ N )
= ( nth_b @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_b @ Xs ) )
=> ( ( nth_b @ ( append_b @ Xs @ Ys ) @ N )
= ( nth_b @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_b @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_322__092_060open_062_092_060And_062thesis_O_A_092_060lbrakk_062i_A_060_Alength_A_Ias_A_064_AU_J_A_092_060Longrightarrow_062_Athesis_059_A_092_060lbrakk_062i_A_061_Alength_A_Ias_A_064_AU_J_059_Ai_A_092_060le_062_Alength_A_Ias_A_064_AU_A_064_AV_J_A_N_A1_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_059_Ai_A_092_060in_062_A_123length_A_Ias_A_064_AU_J_A_L_A1_O_Olength_A_Ias_A_064_AU_A_064_AV_J_A_N_A1_125_A_092_060Longrightarrow_062_Athesis_059_Ai_A_092_060in_062_A_123length_A_Ias_A_064_AU_A_064_AV_J_O_Olength_A_Ias_A_064_AU_A_064_AV_A_064_Abs_J_A_N_A1_125_A_092_060Longrightarrow_062_Athesis_059_Alength_A_Ias_A_064_AU_A_064_AV_A_064_Abs_J_A_092_060le_062_Ai_A_092_060Longrightarrow_062_Athesis_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
( ~ ( ord_less_nat @ i @ ( size_size_list_a @ ( append_a @ as @ u ) ) )
=> ( ( ( i
= ( size_size_list_a @ ( append_a @ as @ u ) ) )
=> ~ ( ord_less_eq_nat @ i @ ( minus_minus_nat @ ( size_size_list_a @ ( append_a @ as @ ( append_a @ u @ v ) ) ) @ one_one_nat ) ) )
=> ( ~ ( member_nat @ i @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ ( size_size_list_a @ ( append_a @ as @ u ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ ( append_a @ as @ ( append_a @ u @ v ) ) ) @ one_one_nat ) ) )
=> ( ~ ( member_nat @ i @ ( set_or1269000886237332187st_nat @ ( size_size_list_a @ ( append_a @ as @ ( append_a @ u @ v ) ) ) @ ( minus_minus_nat @ ( size_size_list_a @ ( append_a @ as @ ( append_a @ u @ ( append_a @ v @ bs ) ) ) ) @ one_one_nat ) ) )
=> ( ord_less_eq_nat @ ( size_size_list_a @ ( append_a @ as @ ( append_a @ u @ ( append_a @ v @ bs ) ) ) ) @ i ) ) ) ) ) ).
% \<open>\<And>thesis. \<lbrakk>i < length (as @ U) \<Longrightarrow> thesis; \<lbrakk>i = length (as @ U); i \<le> length (as @ U @ V) - 1\<rbrakk> \<Longrightarrow> thesis; i \<in> {length (as @ U) + 1..length (as @ U @ V) - 1} \<Longrightarrow> thesis; i \<in> {length (as @ U @ V)..length (as @ U @ V @ bs) - 1} \<Longrightarrow> thesis; length (as @ U @ V @ bs) \<le> i \<Longrightarrow> thesis\<rbrakk> \<Longrightarrow> thesis\<close>
thf(fact_323_no__back__arcs_Oelims_I2_J,axiom,
! [X2: list_a] :
( ( iKKBZ_7773321254043928001cs_a_b @ t @ X2 )
=> ( ( X2 != nil_a )
=> ~ ! [X: a,Xs2: list_a] :
( ( X2
= ( cons_a @ X @ Xs2 ) )
=> ~ ( ~ ? [Y: a] :
( ( member_a @ Y @ ( set_a2 @ Xs2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_7773321254043928001cs_a_b @ t @ Xs2 ) ) ) ) ) ).
% no_back_arcs.elims(2)
thf(fact_324_no__back__arcs_Oelims_I1_J,axiom,
! [X2: list_a,Y2: $o] :
( ( ( iKKBZ_7773321254043928001cs_a_b @ t @ X2 )
= Y2 )
=> ( ( ( X2 = nil_a )
=> ~ Y2 )
=> ~ ! [X: a,Xs2: list_a] :
( ( X2
= ( cons_a @ X @ Xs2 ) )
=> ( Y2
= ( ~ ( ~ ? [Y4: a] :
( ( member_a @ Y4 @ ( set_a2 @ Xs2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_7773321254043928001cs_a_b @ t @ Xs2 ) ) ) ) ) ) ) ).
% no_back_arcs.elims(1)
thf(fact_325_no__back__insert__aux,axiom,
! [X2: a,Xs: list_a] :
( ! [I4: nat,J: nat] :
( ( ord_less_eq_nat @ J @ I4 )
| ( ord_less_eq_nat @ ( size_size_list_a @ ( cons_a @ X2 @ Xs ) ) @ J )
| ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ ( cons_a @ X2 @ Xs ) @ J ) @ ( nth_a @ ( cons_a @ X2 @ Xs ) @ I4 ) ) @ ( arcs_ends_a_b @ t ) ) )
=> ! [I3: nat,J4: nat] :
( ( ord_less_eq_nat @ J4 @ I3 )
| ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ J4 )
| ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ J4 ) @ ( nth_a @ Xs @ I3 ) ) @ ( arcs_ends_a_b @ t ) ) ) ) ).
% no_back_insert_aux
thf(fact_326_scc__of__in__sccs__verts,axiom,
! [U: a] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( member_set_a @ ( digrap2937667069914300949of_a_b @ t @ U ) @ ( digrap2871191568752656621ts_a_b @ t ) ) ) ).
% scc_of_in_sccs_verts
thf(fact_327_atLeastatMost__subset__iff,axiom,
! [A: set_a,B: set_a,C: set_a,D: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or6288561110385358355_set_a @ A @ B ) @ ( set_or6288561110385358355_set_a @ C @ D ) )
= ( ~ ( ord_less_eq_set_a @ A @ B )
| ( ( ord_less_eq_set_a @ C @ A )
& ( ord_less_eq_set_a @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_328_atLeastatMost__subset__iff,axiom,
! [A: set_b,B: set_b,C: set_b,D: set_b] :
( ( ord_le3795704787696855135_set_b @ ( set_or6288561114688587156_set_b @ A @ B ) @ ( set_or6288561114688587156_set_b @ C @ D ) )
= ( ~ ( ord_less_eq_set_b @ A @ B )
| ( ( ord_less_eq_set_b @ C @ A )
& ( ord_less_eq_set_b @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_329_atLeastatMost__subset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_330_atLeastatMost__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_331_atLeastAtMost__iff,axiom,
! [I2: set_a,L: set_a,U: set_a] :
( ( member_set_a @ I2 @ ( set_or6288561110385358355_set_a @ L @ U ) )
= ( ( ord_less_eq_set_a @ L @ I2 )
& ( ord_less_eq_set_a @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_332_atLeastAtMost__iff,axiom,
! [I2: set_b,L: set_b,U: set_b] :
( ( member_set_b @ I2 @ ( set_or6288561114688587156_set_b @ L @ U ) )
= ( ( ord_less_eq_set_b @ L @ I2 )
& ( ord_less_eq_set_b @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_333_atLeastAtMost__iff,axiom,
! [I2: real,L: real,U: real] :
( ( member_real @ I2 @ ( set_or1222579329274155063t_real @ L @ U ) )
= ( ( ord_less_eq_real @ L @ I2 )
& ( ord_less_eq_real @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_334_atLeastAtMost__iff,axiom,
! [I2: nat,L: nat,U: nat] :
( ( member_nat @ I2 @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I2 )
& ( ord_less_eq_nat @ I2 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_335_Icc__eq__Icc,axiom,
! [L: set_a,H: set_a,L2: set_a,H2: set_a] :
( ( ( set_or6288561110385358355_set_a @ L @ H )
= ( set_or6288561110385358355_set_a @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_set_a @ L @ H )
& ~ ( ord_less_eq_set_a @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_336_Icc__eq__Icc,axiom,
! [L: set_b,H: set_b,L2: set_b,H2: set_b] :
( ( ( set_or6288561114688587156_set_b @ L @ H )
= ( set_or6288561114688587156_set_b @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_set_b @ L @ H )
& ~ ( ord_less_eq_set_b @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_337_Icc__eq__Icc,axiom,
! [L: real,H: real,L2: real,H2: real] :
( ( ( set_or1222579329274155063t_real @ L @ H )
= ( set_or1222579329274155063t_real @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_real @ L @ H )
& ~ ( ord_less_eq_real @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_338_Icc__eq__Icc,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or1269000886237332187st_nat @ L @ H )
= ( set_or1269000886237332187st_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_nat @ L @ H )
& ~ ( ord_less_eq_nat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_339_forward__arcs_Ocases,axiom,
! [X2: list_a] :
( ( X2 != nil_a )
=> ( ! [X: a] :
( X2
!= ( cons_a @ X @ nil_a ) )
=> ~ ! [X: a,V3: a,Va: list_a] :
( X2
!= ( cons_a @ X @ ( cons_a @ V3 @ Va ) ) ) ) ) ).
% forward_arcs.cases
thf(fact_340_no__back__arcs_Ocases,axiom,
! [X2: list_a] :
( ( X2 != nil_a )
=> ~ ! [X: a,Xs2: list_a] :
( X2
!= ( cons_a @ X @ Xs2 ) ) ) ).
% no_back_arcs.cases
thf(fact_341_no__back__insert,axiom,
! [X2: a,Xs: list_a] :
( ( iKKBZ_3684931046464919648ck_a_b @ t @ ( cons_a @ X2 @ Xs ) )
=> ( iKKBZ_3684931046464919648ck_a_b @ t @ Xs ) ) ).
% no_back_insert
thf(fact_342_sccs__verts__subsets,axiom,
! [S3: set_a] :
( ( member_set_a @ S3 @ ( digrap2871191568752656621ts_a_b @ t ) )
=> ( ord_less_eq_set_a @ S3 @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% sccs_verts_subsets
thf(fact_343_two__elems__if__length__gt1,axiom,
! [Xs: list_a] :
( ( ord_less_nat @ one_one_nat @ ( size_size_list_a @ Xs ) )
=> ? [X: a,Y3: a,Ys2: list_a] :
( ( cons_a @ X @ ( cons_a @ Y3 @ Ys2 ) )
= Xs ) ) ).
% two_elems_if_length_gt1
thf(fact_344_two__elems__if__length__gt1,axiom,
! [Xs: list_b] :
( ( ord_less_nat @ one_one_nat @ ( size_size_list_b @ Xs ) )
=> ? [X: b,Y3: b,Ys2: list_b] :
( ( cons_b @ X @ ( cons_b @ Y3 @ Ys2 ) )
= Xs ) ) ).
% two_elems_if_length_gt1
thf(fact_345_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_346_add__right__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_347_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_348_add__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_349_list_Oinject,axiom,
! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
( ( ( cons_a @ X21 @ X22 )
= ( cons_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_350_list_Oinject,axiom,
! [X21: b,X22: list_b,Y21: b,Y22: list_b] :
( ( ( cons_b @ X21 @ X22 )
= ( cons_b @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_351_forward__single,axiom,
! [X2: a] : ( iKKBZ_4778857019735642799rd_a_b @ t @ ( cons_a @ X2 @ nil_a ) ) ).
% forward_single
thf(fact_352_no__back__single,axiom,
! [X2: a] : ( iKKBZ_3684931046464919648ck_a_b @ t @ ( cons_a @ X2 @ nil_a ) ) ).
% no_back_single
thf(fact_353_no__back__arcs__single,axiom,
! [X2: a] : ( iKKBZ_7773321254043928001cs_a_b @ t @ ( cons_a @ X2 @ nil_a ) ) ).
% no_back_arcs_single
thf(fact_354_seq__conform__single,axiom,
! [X2: a] : ( iKKBZ_4622586873178280511rm_a_b @ t @ ( cons_a @ X2 @ nil_a ) ) ).
% seq_conform_single
thf(fact_355_hd__in__verts__if__forward,axiom,
! [X2: a,Y2: a,Xs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs ) ) )
=> ( member_a @ ( hd_a @ ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs ) ) ) @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% hd_in_verts_if_forward
thf(fact_356_no__arc__fst__if__no__back,axiom,
! [X2: a,Xs: list_a,Y2: a] :
( ( iKKBZ_3684931046464919648ck_a_b @ t @ ( cons_a @ X2 @ Xs ) )
=> ( ( member_a @ Y2 @ ( set_a2 @ Xs ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y2 @ X2 ) @ ( arcs_ends_a_b @ t ) ) ) ) ).
% no_arc_fst_if_no_back
thf(fact_357_no__back__arcs_Osimps_I2_J,axiom,
! [X2: a,Xs: list_a] :
( ( iKKBZ_7773321254043928001cs_a_b @ t @ ( cons_a @ X2 @ Xs ) )
= ( ~ ? [Y4: a] :
( ( member_a @ Y4 @ ( set_a2 @ Xs ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X2 ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_7773321254043928001cs_a_b @ t @ Xs ) ) ) ).
% no_back_arcs.simps(2)
thf(fact_358_no__back__arcs_Oelims_I3_J,axiom,
! [X2: list_a] :
( ~ ( iKKBZ_7773321254043928001cs_a_b @ t @ X2 )
=> ~ ! [X: a,Xs2: list_a] :
( ( X2
= ( cons_a @ X @ Xs2 ) )
=> ( ~ ? [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Xs2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_7773321254043928001cs_a_b @ t @ Xs2 ) ) ) ) ).
% no_back_arcs.elims(3)
thf(fact_359_hd__reach__all__forward_H_H,axiom,
! [X2: a,Y2: a,Xs: list_a,Z2: a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs ) ) )
=> ( ( member_a @ Z2 @ ( set_a2 @ ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs ) ) ) )
=> ( reachable_a_b @ t @ ( hd_a @ ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs ) ) ) @ Z2 ) ) ) ).
% hd_reach_all_forward''
thf(fact_360_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_361_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_362_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_363_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_364_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_365_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_366_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_367_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_368_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_369_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_370_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_371_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_372_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_373_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_374_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_375_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y2: nat] :
( ( ( plus_plus_nat @ X2 @ Y2 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_376_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y2: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y2 ) )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_377_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_378_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_379_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_380_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_381_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_382_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_383_add__diff__cancel,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_384_diff__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_385_add__diff__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_386_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_387_add__diff__cancel__left_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_388_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_389_add__diff__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_390_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_391_add__diff__cancel__right_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_392_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_393_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_394_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_395_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_396_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_397_diff__diff__left,axiom,
! [I2: nat,J3: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J3 ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J3 @ K ) ) ) ).
% diff_diff_left
thf(fact_398_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_399_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_400_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_401_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_402_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_403_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_404_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_405_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_406_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_407_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_408_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_409_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_410_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_411_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_412_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_413_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_414_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_415_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_416_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_417_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_418_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_419_append1__eq__conv,axiom,
! [Xs: list_a,X2: a,Ys: list_a,Y2: a] :
( ( ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y2 @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_420_append1__eq__conv,axiom,
! [Xs: list_b,X2: b,Ys: list_b,Y2: b] :
( ( ( append_b @ Xs @ ( cons_b @ X2 @ nil_b ) )
= ( append_b @ Ys @ ( cons_b @ Y2 @ nil_b ) ) )
= ( ( Xs = Ys )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_421_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_422_nth__Cons__0,axiom,
! [X2: a,Xs: list_a] :
( ( nth_a @ ( cons_a @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_423_nth__Cons__0,axiom,
! [X2: b,Xs: list_b] :
( ( nth_b @ ( cons_b @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_424_length__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( size_size_list_a @ ( append_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) ) ) ).
% length_append
thf(fact_425_length__append,axiom,
! [Xs: list_b,Ys: list_b] :
( ( size_size_list_b @ ( append_b @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_b @ Xs ) @ ( size_size_list_b @ Ys ) ) ) ).
% length_append
thf(fact_426_Nat_Odiff__diff__right,axiom,
! [K: nat,J3: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J3 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J3 ) ) ) ).
% Nat.diff_diff_right
thf(fact_427_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J3: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J3 @ K ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J3 @ I2 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_428_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J3: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J3 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J3 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_429_nth__append__length,axiom,
! [Xs: list_a,X2: a,Ys: list_a] :
( ( nth_a @ ( append_a @ Xs @ ( cons_a @ X2 @ Ys ) ) @ ( size_size_list_a @ Xs ) )
= X2 ) ).
% nth_append_length
thf(fact_430_nth__append__length,axiom,
! [Xs: list_b,X2: b,Ys: list_b] :
( ( nth_b @ ( append_b @ Xs @ ( cons_b @ X2 @ Ys ) ) @ ( size_size_list_b @ Xs ) )
= X2 ) ).
% nth_append_length
thf(fact_431_nth__append__length__plus,axiom,
! [Xs: list_a,Ys: list_a,N: nat] :
( ( nth_a @ ( append_a @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ N ) )
= ( nth_a @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_432_nth__append__length__plus,axiom,
! [Xs: list_b,Ys: list_b,N: nat] :
( ( nth_b @ ( append_b @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_b @ Xs ) @ N ) )
= ( nth_b @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_433_nth__Cons__pos,axiom,
! [N: nat,X2: a,Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_a @ ( cons_a @ X2 @ Xs ) @ N )
= ( nth_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_434_nth__Cons__pos,axiom,
! [N: nat,X2: b,Xs: list_b] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_b @ ( cons_b @ X2 @ Xs ) @ N )
= ( nth_b @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_435_transpose_Ocases,axiom,
! [X2: list_list_a] :
( ( X2 != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X2
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X: a,Xs2: list_a,Xss: list_list_a] :
( X2
!= ( cons_list_a @ ( cons_a @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_436_transpose_Ocases,axiom,
! [X2: list_list_b] :
( ( X2 != nil_list_b )
=> ( ! [Xss: list_list_b] :
( X2
!= ( cons_list_b @ nil_b @ Xss ) )
=> ~ ! [X: b,Xs2: list_b,Xss: list_list_b] :
( X2
!= ( cons_list_b @ ( cons_b @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_437_not__Cons__self2,axiom,
! [X2: a,Xs: list_a] :
( ( cons_a @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_438_not__Cons__self2,axiom,
! [X2: b,Xs: list_b] :
( ( cons_b @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_439_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_440_add__right__imp__eq,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_441_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_442_add__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_443_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_444_add_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_445_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_446_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A3: real,B2: real] : ( plus_plus_real @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_447_add_Oright__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_448_add_Oleft__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_449_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_450_add_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_451_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_452_group__cancel_Oadd2,axiom,
! [B3: real,K: real,B: real,A: real] :
( ( B3
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B3 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_453_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_454_group__cancel_Oadd1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_455_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J3: nat,K: nat,L: nat] :
( ( ( I2 = J3 )
& ( K = L ) )
=> ( ( plus_plus_nat @ I2 @ K )
= ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_456_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: real,J3: real,K: real,L: real] :
( ( ( I2 = J3 )
& ( K = L ) )
=> ( ( plus_plus_real @ I2 @ K )
= ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_457_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_458_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_459_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_460_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: nat,J3: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J3 )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_461_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: real,J3: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I2 @ J3 )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_462_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: nat,J3: nat,K: nat,L: nat] :
( ( ( I2 = J3 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_463_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: real,J3: real,K: real,L: real] :
( ( ( I2 = J3 )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_464_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: nat,J3: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J3 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_465_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: real,J3: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I2 @ J3 )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_466_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_467_add__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_468_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_469_add__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_470_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_471_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_472_add__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_473_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
? [C3: nat] :
( B2
= ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_474_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_475_add__le__imp__le__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_476_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_477_add__le__imp__le__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_478_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_479_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_480_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_481_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_482_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_483_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_484_add__less__imp__less__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_485_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_486_add__less__imp__less__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_487_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_488_add__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_489_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_490_add__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_491_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_492_add__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_493_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: nat,J3: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J3 )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_494_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: real,J3: real,K: real,L: real] :
( ( ( ord_less_real @ I2 @ J3 )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_495_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: nat,J3: nat,K: nat,L: nat] :
( ( ( I2 = J3 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_496_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: real,J3: real,K: real,L: real] :
( ( ( I2 = J3 )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_497_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: nat,J3: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J3 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_498_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: real,J3: real,K: real,L: real] :
( ( ( ord_less_real @ I2 @ J3 )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_499_group__cancel_Osub1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( minus_minus_real @ A2 @ B )
= ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_500_diff__eq__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( minus_minus_real @ A @ B )
= C )
= ( A
= ( plus_plus_real @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_501_eq__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( A
= ( minus_minus_real @ C @ B ) )
= ( ( plus_plus_real @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_502_add__diff__eq,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_503_diff__diff__eq2,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_504_diff__add__eq,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_505_diff__add__eq__diff__diff__swap,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_506_add__implies__diff,axiom,
! [C: real,B: real,A: real] :
( ( ( plus_plus_real @ C @ B )
= A )
=> ( C
= ( minus_minus_real @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_507_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_508_diff__diff__eq,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_509_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_510_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_511_list_Odistinct_I1_J,axiom,
! [X21: b,X22: list_b] :
( nil_b
!= ( cons_b @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_512_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_513_list_OdiscI,axiom,
! [List: list_b,X21: b,X22: list_b] :
( ( List
= ( cons_b @ X21 @ X22 ) )
=> ( List != nil_b ) ) ).
% list.discI
thf(fact_514_list_Oexhaust,axiom,
! [Y2: list_a] :
( ( Y2 != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y2
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_515_list_Oexhaust,axiom,
! [Y2: list_b] :
( ( Y2 != nil_b )
=> ~ ! [X212: b,X222: list_b] :
( Y2
!= ( cons_b @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_516_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y4: a,Ys5: list_a] :
( Xs
= ( cons_a @ Y4 @ Ys5 ) ) ) ) ).
% neq_Nil_conv
thf(fact_517_neq__Nil__conv,axiom,
! [Xs: list_b] :
( ( Xs != nil_b )
= ( ? [Y4: b,Ys5: list_b] :
( Xs
= ( cons_b @ Y4 @ Ys5 ) ) ) ) ).
% neq_Nil_conv
thf(fact_518_list__induct2_H,axiom,
! [P4: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P4 @ nil_a @ nil_a )
=> ( ! [X: a,Xs2: list_a] : ( P4 @ ( cons_a @ X @ Xs2 ) @ nil_a )
=> ( ! [Y3: a,Ys2: list_a] : ( P4 @ nil_a @ ( cons_a @ Y3 @ Ys2 ) )
=> ( ! [X: a,Xs2: list_a,Y3: a,Ys2: list_a] :
( ( P4 @ Xs2 @ Ys2 )
=> ( P4 @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) )
=> ( P4 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_519_list__induct2_H,axiom,
! [P4: list_a > list_b > $o,Xs: list_a,Ys: list_b] :
( ( P4 @ nil_a @ nil_b )
=> ( ! [X: a,Xs2: list_a] : ( P4 @ ( cons_a @ X @ Xs2 ) @ nil_b )
=> ( ! [Y3: b,Ys2: list_b] : ( P4 @ nil_a @ ( cons_b @ Y3 @ Ys2 ) )
=> ( ! [X: a,Xs2: list_a,Y3: b,Ys2: list_b] :
( ( P4 @ Xs2 @ Ys2 )
=> ( P4 @ ( cons_a @ X @ Xs2 ) @ ( cons_b @ Y3 @ Ys2 ) ) )
=> ( P4 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_520_list__induct2_H,axiom,
! [P4: list_b > list_a > $o,Xs: list_b,Ys: list_a] :
( ( P4 @ nil_b @ nil_a )
=> ( ! [X: b,Xs2: list_b] : ( P4 @ ( cons_b @ X @ Xs2 ) @ nil_a )
=> ( ! [Y3: a,Ys2: list_a] : ( P4 @ nil_b @ ( cons_a @ Y3 @ Ys2 ) )
=> ( ! [X: b,Xs2: list_b,Y3: a,Ys2: list_a] :
( ( P4 @ Xs2 @ Ys2 )
=> ( P4 @ ( cons_b @ X @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) )
=> ( P4 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_521_list__induct2_H,axiom,
! [P4: list_b > list_b > $o,Xs: list_b,Ys: list_b] :
( ( P4 @ nil_b @ nil_b )
=> ( ! [X: b,Xs2: list_b] : ( P4 @ ( cons_b @ X @ Xs2 ) @ nil_b )
=> ( ! [Y3: b,Ys2: list_b] : ( P4 @ nil_b @ ( cons_b @ Y3 @ Ys2 ) )
=> ( ! [X: b,Xs2: list_b,Y3: b,Ys2: list_b] :
( ( P4 @ Xs2 @ Ys2 )
=> ( P4 @ ( cons_b @ X @ Xs2 ) @ ( cons_b @ Y3 @ Ys2 ) ) )
=> ( P4 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_522_list__nonempty__induct,axiom,
! [Xs: list_a,P4: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X: a] : ( P4 @ ( cons_a @ X @ nil_a ) )
=> ( ! [X: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P4 @ Xs2 )
=> ( P4 @ ( cons_a @ X @ Xs2 ) ) ) )
=> ( P4 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_523_list__nonempty__induct,axiom,
! [Xs: list_b,P4: list_b > $o] :
( ( Xs != nil_b )
=> ( ! [X: b] : ( P4 @ ( cons_b @ X @ nil_b ) )
=> ( ! [X: b,Xs2: list_b] :
( ( Xs2 != nil_b )
=> ( ( P4 @ Xs2 )
=> ( P4 @ ( cons_b @ X @ Xs2 ) ) ) )
=> ( P4 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_524_list_Oset__intros_I2_J,axiom,
! [Y2: list_a,X22: list_list_a,X21: list_a] :
( ( member_list_a @ Y2 @ ( set_list_a2 @ X22 ) )
=> ( member_list_a @ Y2 @ ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_525_list_Oset__intros_I2_J,axiom,
! [Y2: set_a,X22: list_set_a,X21: set_a] :
( ( member_set_a @ Y2 @ ( set_set_a2 @ X22 ) )
=> ( member_set_a @ Y2 @ ( set_set_a2 @ ( cons_set_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_526_list_Oset__intros_I2_J,axiom,
! [Y2: nat,X22: list_nat,X21: nat] :
( ( member_nat @ Y2 @ ( set_nat2 @ X22 ) )
=> ( member_nat @ Y2 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_527_list_Oset__intros_I2_J,axiom,
! [Y2: a,X22: list_a,X21: a] :
( ( member_a @ Y2 @ ( set_a2 @ X22 ) )
=> ( member_a @ Y2 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_528_list_Oset__intros_I2_J,axiom,
! [Y2: b,X22: list_b,X21: b] :
( ( member_b @ Y2 @ ( set_b2 @ X22 ) )
=> ( member_b @ Y2 @ ( set_b2 @ ( cons_b @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_529_list_Oset__intros_I1_J,axiom,
! [X21: list_a,X22: list_list_a] : ( member_list_a @ X21 @ ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_530_list_Oset__intros_I1_J,axiom,
! [X21: set_a,X22: list_set_a] : ( member_set_a @ X21 @ ( set_set_a2 @ ( cons_set_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_531_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_532_list_Oset__intros_I1_J,axiom,
! [X21: a,X22: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_533_list_Oset__intros_I1_J,axiom,
! [X21: b,X22: list_b] : ( member_b @ X21 @ ( set_b2 @ ( cons_b @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_534_list_Oset__cases,axiom,
! [E: list_a,A: list_list_a] :
( ( member_list_a @ E @ ( set_list_a2 @ A ) )
=> ( ! [Z22: list_list_a] :
( A
!= ( cons_list_a @ E @ Z22 ) )
=> ~ ! [Z1: list_a,Z22: list_list_a] :
( ( A
= ( cons_list_a @ Z1 @ Z22 ) )
=> ~ ( member_list_a @ E @ ( set_list_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_535_list_Oset__cases,axiom,
! [E: set_a,A: list_set_a] :
( ( member_set_a @ E @ ( set_set_a2 @ A ) )
=> ( ! [Z22: list_set_a] :
( A
!= ( cons_set_a @ E @ Z22 ) )
=> ~ ! [Z1: set_a,Z22: list_set_a] :
( ( A
= ( cons_set_a @ Z1 @ Z22 ) )
=> ~ ( member_set_a @ E @ ( set_set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_536_list_Oset__cases,axiom,
! [E: nat,A: list_nat] :
( ( member_nat @ E @ ( set_nat2 @ A ) )
=> ( ! [Z22: list_nat] :
( A
!= ( cons_nat @ E @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_537_list_Oset__cases,axiom,
! [E: a,A: list_a] :
( ( member_a @ E @ ( set_a2 @ A ) )
=> ( ! [Z22: list_a] :
( A
!= ( cons_a @ E @ Z22 ) )
=> ~ ! [Z1: a,Z22: list_a] :
( ( A
= ( cons_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_538_list_Oset__cases,axiom,
! [E: b,A: list_b] :
( ( member_b @ E @ ( set_b2 @ A ) )
=> ( ! [Z22: list_b] :
( A
!= ( cons_b @ E @ Z22 ) )
=> ~ ! [Z1: b,Z22: list_b] :
( ( A
= ( cons_b @ Z1 @ Z22 ) )
=> ~ ( member_b @ E @ ( set_b2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_539_set__ConsD,axiom,
! [Y2: list_a,X2: list_a,Xs: list_list_a] :
( ( member_list_a @ Y2 @ ( set_list_a2 @ ( cons_list_a @ X2 @ Xs ) ) )
=> ( ( Y2 = X2 )
| ( member_list_a @ Y2 @ ( set_list_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_540_set__ConsD,axiom,
! [Y2: set_a,X2: set_a,Xs: list_set_a] :
( ( member_set_a @ Y2 @ ( set_set_a2 @ ( cons_set_a @ X2 @ Xs ) ) )
=> ( ( Y2 = X2 )
| ( member_set_a @ Y2 @ ( set_set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_541_set__ConsD,axiom,
! [Y2: nat,X2: nat,Xs: list_nat] :
( ( member_nat @ Y2 @ ( set_nat2 @ ( cons_nat @ X2 @ Xs ) ) )
=> ( ( Y2 = X2 )
| ( member_nat @ Y2 @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_542_set__ConsD,axiom,
! [Y2: a,X2: a,Xs: list_a] :
( ( member_a @ Y2 @ ( set_a2 @ ( cons_a @ X2 @ Xs ) ) )
=> ( ( Y2 = X2 )
| ( member_a @ Y2 @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_543_set__ConsD,axiom,
! [Y2: b,X2: b,Xs: list_b] :
( ( member_b @ Y2 @ ( set_b2 @ ( cons_b @ X2 @ Xs ) ) )
=> ( ( Y2 = X2 )
| ( member_b @ Y2 @ ( set_b2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_544_set__subset__Cons,axiom,
! [Xs: list_a,X2: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_545_set__subset__Cons,axiom,
! [Xs: list_b,X2: b] : ( ord_less_eq_set_b @ ( set_b2 @ Xs ) @ ( set_b2 @ ( cons_b @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_546_Cons__eq__appendI,axiom,
! [X2: a,Xs1: list_a,Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( cons_a @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X2 @ Xs )
= ( append_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_547_Cons__eq__appendI,axiom,
! [X2: b,Xs1: list_b,Ys: list_b,Xs: list_b,Zs: list_b] :
( ( ( cons_b @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_b @ Xs1 @ Zs ) )
=> ( ( cons_b @ X2 @ Xs )
= ( append_b @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_548_append__Cons,axiom,
! [X2: a,Xs: list_a,Ys: list_a] :
( ( append_a @ ( cons_a @ X2 @ Xs ) @ Ys )
= ( cons_a @ X2 @ ( append_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_549_append__Cons,axiom,
! [X2: b,Xs: list_b,Ys: list_b] :
( ( append_b @ ( cons_b @ X2 @ Xs ) @ Ys )
= ( cons_b @ X2 @ ( append_b @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_550_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_551_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_552_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_553_trans__less__add2,axiom,
! [I2: nat,J3: nat,M: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M @ J3 ) ) ) ).
% trans_less_add2
thf(fact_554_trans__less__add1,axiom,
! [I2: nat,J3: nat,M: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J3 @ M ) ) ) ).
% trans_less_add1
thf(fact_555_add__less__mono1,axiom,
! [I2: nat,J3: nat,K: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J3 @ K ) ) ) ).
% add_less_mono1
thf(fact_556_not__add__less2,axiom,
! [J3: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J3 @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_557_not__add__less1,axiom,
! [I2: nat,J3: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J3 ) @ I2 ) ).
% not_add_less1
thf(fact_558_add__less__mono,axiom,
! [I2: nat,J3: nat,K: nat,L: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ) ).
% add_less_mono
thf(fact_559_add__lessD1,axiom,
! [I2: nat,J3: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J3 ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_560_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_561_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_562_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_563_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_564_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_565_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_566_add__le__mono,axiom,
! [I2: nat,J3: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ) ).
% add_le_mono
thf(fact_567_add__le__mono1,axiom,
! [I2: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J3 @ K ) ) ) ).
% add_le_mono1
thf(fact_568_trans__le__add1,axiom,
! [I2: nat,J3: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J3 @ M ) ) ) ).
% trans_le_add1
thf(fact_569_trans__le__add2,axiom,
! [I2: nat,J3: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J3 ) ) ) ).
% trans_le_add2
thf(fact_570_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_571_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_572_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_573_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_574_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_575_list_Osel_I1_J,axiom,
! [X21: a,X22: list_a] :
( ( hd_a @ ( cons_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_576_list_Osel_I1_J,axiom,
! [X21: b,X22: list_b] :
( ( hd_b @ ( cons_b @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_577_add__nonpos__eq__0__iff,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y2 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X2 @ Y2 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_578_add__nonpos__eq__0__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
=> ( ( ( plus_plus_real @ X2 @ Y2 )
= zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_579_add__nonneg__eq__0__iff,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ( ( plus_plus_nat @ X2 @ Y2 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_580_add__nonneg__eq__0__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ( plus_plus_real @ X2 @ Y2 )
= zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_581_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_582_add__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_583_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_584_add__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_585_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_586_add__increasing2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_587_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_588_add__decreasing2,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_589_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_590_add__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_591_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_592_add__decreasing,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_593_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: nat,J3: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J3 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_594_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: real,J3: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I2 @ J3 )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_595_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: nat,J3: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J3 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_596_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: real,J3: real,K: real,L: real] :
( ( ( ord_less_real @ I2 @ J3 )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_597_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_598_add__le__less__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_599_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_600_add__less__le__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_601_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_602_add__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_603_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_604_add__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_605_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_606_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_607_pos__add__strict,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_608_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_609_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_610_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_611_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_612_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_613_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_614_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_615_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_616_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_617_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_618_le__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
= ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_619_diff__le__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
= ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_620_diff__less__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
= ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_621_less__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
= ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_622_rev__induct,axiom,
! [P4: list_a > $o,Xs: list_a] :
( ( P4 @ nil_a )
=> ( ! [X: a,Xs2: list_a] :
( ( P4 @ Xs2 )
=> ( P4 @ ( append_a @ Xs2 @ ( cons_a @ X @ nil_a ) ) ) )
=> ( P4 @ Xs ) ) ) ).
% rev_induct
thf(fact_623_rev__induct,axiom,
! [P4: list_b > $o,Xs: list_b] :
( ( P4 @ nil_b )
=> ( ! [X: b,Xs2: list_b] :
( ( P4 @ Xs2 )
=> ( P4 @ ( append_b @ Xs2 @ ( cons_b @ X @ nil_b ) ) ) )
=> ( P4 @ Xs ) ) ) ).
% rev_induct
thf(fact_624_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys2: list_a,Y3: a] :
( Xs
!= ( append_a @ Ys2 @ ( cons_a @ Y3 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_625_rev__exhaust,axiom,
! [Xs: list_b] :
( ( Xs != nil_b )
=> ~ ! [Ys2: list_b,Y3: b] :
( Xs
!= ( append_b @ Ys2 @ ( cons_b @ Y3 @ nil_b ) ) ) ) ).
% rev_exhaust
thf(fact_626_Cons__eq__append__conv,axiom,
! [X2: a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X2 @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X2 @ Xs )
= Zs ) )
| ? [Ys6: list_a] :
( ( ( cons_a @ X2 @ Ys6 )
= Ys )
& ( Xs
= ( append_a @ Ys6 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_627_Cons__eq__append__conv,axiom,
! [X2: b,Xs: list_b,Ys: list_b,Zs: list_b] :
( ( ( cons_b @ X2 @ Xs )
= ( append_b @ Ys @ Zs ) )
= ( ( ( Ys = nil_b )
& ( ( cons_b @ X2 @ Xs )
= Zs ) )
| ? [Ys6: list_b] :
( ( ( cons_b @ X2 @ Ys6 )
= Ys )
& ( Xs
= ( append_b @ Ys6 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_628_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X2: a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X2 @ Xs ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X2 @ Xs ) ) )
| ? [Ys6: list_a] :
( ( Ys
= ( cons_a @ X2 @ Ys6 ) )
& ( ( append_a @ Ys6 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_629_append__eq__Cons__conv,axiom,
! [Ys: list_b,Zs: list_b,X2: b,Xs: list_b] :
( ( ( append_b @ Ys @ Zs )
= ( cons_b @ X2 @ Xs ) )
= ( ( ( Ys = nil_b )
& ( Zs
= ( cons_b @ X2 @ Xs ) ) )
| ? [Ys6: list_b] :
( ( Ys
= ( cons_b @ X2 @ Ys6 ) )
& ( ( append_b @ Ys6 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_630_rev__nonempty__induct,axiom,
! [Xs: list_a,P4: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X: a] : ( P4 @ ( cons_a @ X @ nil_a ) )
=> ( ! [X: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P4 @ Xs2 )
=> ( P4 @ ( append_a @ Xs2 @ ( cons_a @ X @ nil_a ) ) ) ) )
=> ( P4 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_631_rev__nonempty__induct,axiom,
! [Xs: list_b,P4: list_b > $o] :
( ( Xs != nil_b )
=> ( ! [X: b] : ( P4 @ ( cons_b @ X @ nil_b ) )
=> ( ! [X: b,Xs2: list_b] :
( ( Xs2 != nil_b )
=> ( ( P4 @ Xs2 )
=> ( P4 @ ( append_b @ Xs2 @ ( cons_b @ X @ nil_b ) ) ) ) )
=> ( P4 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_632_split__list,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ? [Ys2: list_list_a,Zs2: list_list_a] :
( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_633_split__list,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
=> ? [Ys2: list_set_a,Zs2: list_set_a] :
( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_634_split__list,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs2: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_635_split__list,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_636_split__list,axiom,
! [X2: b,Xs: list_b] :
( ( member_b @ X2 @ ( set_b2 @ Xs ) )
=> ? [Ys2: list_b,Zs2: list_b] :
( Xs
= ( append_b @ Ys2 @ ( cons_b @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_637_split__list__last,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ? [Ys2: list_list_a,Zs2: list_list_a] :
( ( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X2 @ Zs2 ) ) )
& ~ ( member_list_a @ X2 @ ( set_list_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_638_split__list__last,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
=> ? [Ys2: list_set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X2 @ Zs2 ) ) )
& ~ ( member_set_a @ X2 @ ( set_set_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_639_split__list__last,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs2 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_640_split__list__last,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
& ~ ( member_a @ X2 @ ( set_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_641_split__list__last,axiom,
! [X2: b,Xs: list_b] :
( ( member_b @ X2 @ ( set_b2 @ Xs ) )
=> ? [Ys2: list_b,Zs2: list_b] :
( ( Xs
= ( append_b @ Ys2 @ ( cons_b @ X2 @ Zs2 ) ) )
& ~ ( member_b @ X2 @ ( set_b2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_642_split__list__prop,axiom,
! [Xs: list_a,P4: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P4 @ X4 ) )
=> ? [Ys2: list_a,X: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
& ( P4 @ X ) ) ) ).
% split_list_prop
thf(fact_643_split__list__prop,axiom,
! [Xs: list_b,P4: b > $o] :
( ? [X4: b] :
( ( member_b @ X4 @ ( set_b2 @ Xs ) )
& ( P4 @ X4 ) )
=> ? [Ys2: list_b,X: b] :
( ? [Zs2: list_b] :
( Xs
= ( append_b @ Ys2 @ ( cons_b @ X @ Zs2 ) ) )
& ( P4 @ X ) ) ) ).
% split_list_prop
thf(fact_644_split__list__first,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ? [Ys2: list_list_a,Zs2: list_list_a] :
( ( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X2 @ Zs2 ) ) )
& ~ ( member_list_a @ X2 @ ( set_list_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_645_split__list__first,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
=> ? [Ys2: list_set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X2 @ Zs2 ) ) )
& ~ ( member_set_a @ X2 @ ( set_set_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_646_split__list__first,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs2 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_647_split__list__first,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
& ~ ( member_a @ X2 @ ( set_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_648_split__list__first,axiom,
! [X2: b,Xs: list_b] :
( ( member_b @ X2 @ ( set_b2 @ Xs ) )
=> ? [Ys2: list_b,Zs2: list_b] :
( ( Xs
= ( append_b @ Ys2 @ ( cons_b @ X2 @ Zs2 ) ) )
& ~ ( member_b @ X2 @ ( set_b2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_649_split__list__propE,axiom,
! [Xs: list_a,P4: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P4 @ X4 ) )
=> ~ ! [Ys2: list_a,X: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
=> ~ ( P4 @ X ) ) ) ).
% split_list_propE
thf(fact_650_split__list__propE,axiom,
! [Xs: list_b,P4: b > $o] :
( ? [X4: b] :
( ( member_b @ X4 @ ( set_b2 @ Xs ) )
& ( P4 @ X4 ) )
=> ~ ! [Ys2: list_b,X: b] :
( ? [Zs2: list_b] :
( Xs
= ( append_b @ Ys2 @ ( cons_b @ X @ Zs2 ) ) )
=> ~ ( P4 @ X ) ) ) ).
% split_list_propE
thf(fact_651_append__Cons__eq__iff,axiom,
! [X2: list_a,Xs: list_list_a,Ys: list_list_a,Xs5: list_list_a,Ys7: list_list_a] :
( ~ ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ( ~ ( member_list_a @ X2 @ ( set_list_a2 @ Ys ) )
=> ( ( ( append_list_a @ Xs @ ( cons_list_a @ X2 @ Ys ) )
= ( append_list_a @ Xs5 @ ( cons_list_a @ X2 @ Ys7 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys7 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_652_append__Cons__eq__iff,axiom,
! [X2: set_a,Xs: list_set_a,Ys: list_set_a,Xs5: list_set_a,Ys7: list_set_a] :
( ~ ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
=> ( ~ ( member_set_a @ X2 @ ( set_set_a2 @ Ys ) )
=> ( ( ( append_set_a @ Xs @ ( cons_set_a @ X2 @ Ys ) )
= ( append_set_a @ Xs5 @ ( cons_set_a @ X2 @ Ys7 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys7 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_653_append__Cons__eq__iff,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat,Xs5: list_nat,Ys7: list_nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ~ ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
=> ( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys ) )
= ( append_nat @ Xs5 @ ( cons_nat @ X2 @ Ys7 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys7 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_654_append__Cons__eq__iff,axiom,
! [X2: a,Xs: list_a,Ys: list_a,Xs5: list_a,Ys7: list_a] :
( ~ ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ~ ( member_a @ X2 @ ( set_a2 @ Ys ) )
=> ( ( ( append_a @ Xs @ ( cons_a @ X2 @ Ys ) )
= ( append_a @ Xs5 @ ( cons_a @ X2 @ Ys7 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys7 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_655_append__Cons__eq__iff,axiom,
! [X2: b,Xs: list_b,Ys: list_b,Xs5: list_b,Ys7: list_b] :
( ~ ( member_b @ X2 @ ( set_b2 @ Xs ) )
=> ( ~ ( member_b @ X2 @ ( set_b2 @ Ys ) )
=> ( ( ( append_b @ Xs @ ( cons_b @ X2 @ Ys ) )
= ( append_b @ Xs5 @ ( cons_b @ X2 @ Ys7 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys7 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_656_in__set__conv__decomp,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
= ( ? [Ys5: list_list_a,Zs3: list_list_a] :
( Xs
= ( append_list_a @ Ys5 @ ( cons_list_a @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_657_in__set__conv__decomp,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
= ( ? [Ys5: list_set_a,Zs3: list_set_a] :
( Xs
= ( append_set_a @ Ys5 @ ( cons_set_a @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_658_in__set__conv__decomp,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys5: list_nat,Zs3: list_nat] :
( Xs
= ( append_nat @ Ys5 @ ( cons_nat @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_659_in__set__conv__decomp,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
= ( ? [Ys5: list_a,Zs3: list_a] :
( Xs
= ( append_a @ Ys5 @ ( cons_a @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_660_in__set__conv__decomp,axiom,
! [X2: b,Xs: list_b] :
( ( member_b @ X2 @ ( set_b2 @ Xs ) )
= ( ? [Ys5: list_b,Zs3: list_b] :
( Xs
= ( append_b @ Ys5 @ ( cons_b @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_661_split__list__last__prop,axiom,
! [Xs: list_a,P4: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P4 @ X4 ) )
=> ? [Ys2: list_a,X: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
& ( P4 @ X )
& ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Zs2 ) )
=> ~ ( P4 @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_662_split__list__last__prop,axiom,
! [Xs: list_b,P4: b > $o] :
( ? [X4: b] :
( ( member_b @ X4 @ ( set_b2 @ Xs ) )
& ( P4 @ X4 ) )
=> ? [Ys2: list_b,X: b,Zs2: list_b] :
( ( Xs
= ( append_b @ Ys2 @ ( cons_b @ X @ Zs2 ) ) )
& ( P4 @ X )
& ! [Xa2: b] :
( ( member_b @ Xa2 @ ( set_b2 @ Zs2 ) )
=> ~ ( P4 @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_663_split__list__first__prop,axiom,
! [Xs: list_a,P4: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P4 @ X4 ) )
=> ? [Ys2: list_a,X: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
& ( P4 @ X )
& ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Ys2 ) )
=> ~ ( P4 @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_664_split__list__first__prop,axiom,
! [Xs: list_b,P4: b > $o] :
( ? [X4: b] :
( ( member_b @ X4 @ ( set_b2 @ Xs ) )
& ( P4 @ X4 ) )
=> ? [Ys2: list_b,X: b] :
( ? [Zs2: list_b] :
( Xs
= ( append_b @ Ys2 @ ( cons_b @ X @ Zs2 ) ) )
& ( P4 @ X )
& ! [Xa2: b] :
( ( member_b @ Xa2 @ ( set_b2 @ Ys2 ) )
=> ~ ( P4 @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_665_split__list__last__propE,axiom,
! [Xs: list_a,P4: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P4 @ X4 ) )
=> ~ ! [Ys2: list_a,X: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
=> ( ( P4 @ X )
=> ~ ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Zs2 ) )
=> ~ ( P4 @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_666_split__list__last__propE,axiom,
! [Xs: list_b,P4: b > $o] :
( ? [X4: b] :
( ( member_b @ X4 @ ( set_b2 @ Xs ) )
& ( P4 @ X4 ) )
=> ~ ! [Ys2: list_b,X: b,Zs2: list_b] :
( ( Xs
= ( append_b @ Ys2 @ ( cons_b @ X @ Zs2 ) ) )
=> ( ( P4 @ X )
=> ~ ! [Xa2: b] :
( ( member_b @ Xa2 @ ( set_b2 @ Zs2 ) )
=> ~ ( P4 @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_667_split__list__first__propE,axiom,
! [Xs: list_a,P4: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P4 @ X4 ) )
=> ~ ! [Ys2: list_a,X: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
=> ( ( P4 @ X )
=> ~ ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Ys2 ) )
=> ~ ( P4 @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_668_split__list__first__propE,axiom,
! [Xs: list_b,P4: b > $o] :
( ? [X4: b] :
( ( member_b @ X4 @ ( set_b2 @ Xs ) )
& ( P4 @ X4 ) )
=> ~ ! [Ys2: list_b,X: b] :
( ? [Zs2: list_b] :
( Xs
= ( append_b @ Ys2 @ ( cons_b @ X @ Zs2 ) ) )
=> ( ( P4 @ X )
=> ~ ! [Xa2: b] :
( ( member_b @ Xa2 @ ( set_b2 @ Ys2 ) )
=> ~ ( P4 @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_669_in__set__conv__decomp__last,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
= ( ? [Ys5: list_list_a,Zs3: list_list_a] :
( ( Xs
= ( append_list_a @ Ys5 @ ( cons_list_a @ X2 @ Zs3 ) ) )
& ~ ( member_list_a @ X2 @ ( set_list_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_670_in__set__conv__decomp__last,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
= ( ? [Ys5: list_set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys5 @ ( cons_set_a @ X2 @ Zs3 ) ) )
& ~ ( member_set_a @ X2 @ ( set_set_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_671_in__set__conv__decomp__last,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys5: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys5 @ ( cons_nat @ X2 @ Zs3 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_672_in__set__conv__decomp__last,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
= ( ? [Ys5: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys5 @ ( cons_a @ X2 @ Zs3 ) ) )
& ~ ( member_a @ X2 @ ( set_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_673_in__set__conv__decomp__last,axiom,
! [X2: b,Xs: list_b] :
( ( member_b @ X2 @ ( set_b2 @ Xs ) )
= ( ? [Ys5: list_b,Zs3: list_b] :
( ( Xs
= ( append_b @ Ys5 @ ( cons_b @ X2 @ Zs3 ) ) )
& ~ ( member_b @ X2 @ ( set_b2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_674_in__set__conv__decomp__first,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
= ( ? [Ys5: list_list_a,Zs3: list_list_a] :
( ( Xs
= ( append_list_a @ Ys5 @ ( cons_list_a @ X2 @ Zs3 ) ) )
& ~ ( member_list_a @ X2 @ ( set_list_a2 @ Ys5 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_675_in__set__conv__decomp__first,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
= ( ? [Ys5: list_set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys5 @ ( cons_set_a @ X2 @ Zs3 ) ) )
& ~ ( member_set_a @ X2 @ ( set_set_a2 @ Ys5 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_676_in__set__conv__decomp__first,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys5: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys5 @ ( cons_nat @ X2 @ Zs3 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Ys5 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_677_in__set__conv__decomp__first,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
= ( ? [Ys5: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys5 @ ( cons_a @ X2 @ Zs3 ) ) )
& ~ ( member_a @ X2 @ ( set_a2 @ Ys5 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_678_in__set__conv__decomp__first,axiom,
! [X2: b,Xs: list_b] :
( ( member_b @ X2 @ ( set_b2 @ Xs ) )
= ( ? [Ys5: list_b,Zs3: list_b] :
( ( Xs
= ( append_b @ Ys5 @ ( cons_b @ X2 @ Zs3 ) ) )
& ~ ( member_b @ X2 @ ( set_b2 @ Ys5 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_679_split__list__last__prop__iff,axiom,
! [Xs: list_a,P4: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P4 @ X3 ) ) )
= ( ? [Ys5: list_a,X3: a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys5 @ ( cons_a @ X3 @ Zs3 ) ) )
& ( P4 @ X3 )
& ! [Y4: a] :
( ( member_a @ Y4 @ ( set_a2 @ Zs3 ) )
=> ~ ( P4 @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_680_split__list__last__prop__iff,axiom,
! [Xs: list_b,P4: b > $o] :
( ( ? [X3: b] :
( ( member_b @ X3 @ ( set_b2 @ Xs ) )
& ( P4 @ X3 ) ) )
= ( ? [Ys5: list_b,X3: b,Zs3: list_b] :
( ( Xs
= ( append_b @ Ys5 @ ( cons_b @ X3 @ Zs3 ) ) )
& ( P4 @ X3 )
& ! [Y4: b] :
( ( member_b @ Y4 @ ( set_b2 @ Zs3 ) )
=> ~ ( P4 @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_681_split__list__first__prop__iff,axiom,
! [Xs: list_a,P4: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P4 @ X3 ) ) )
= ( ? [Ys5: list_a,X3: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys5 @ ( cons_a @ X3 @ Zs3 ) ) )
& ( P4 @ X3 )
& ! [Y4: a] :
( ( member_a @ Y4 @ ( set_a2 @ Ys5 ) )
=> ~ ( P4 @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_682_split__list__first__prop__iff,axiom,
! [Xs: list_b,P4: b > $o] :
( ( ? [X3: b] :
( ( member_b @ X3 @ ( set_b2 @ Xs ) )
& ( P4 @ X3 ) ) )
= ( ? [Ys5: list_b,X3: b] :
( ? [Zs3: list_b] :
( Xs
= ( append_b @ Ys5 @ ( cons_b @ X3 @ Zs3 ) ) )
& ( P4 @ X3 )
& ! [Y4: b] :
( ( member_b @ Y4 @ ( set_b2 @ Ys5 ) )
=> ~ ( P4 @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_683_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_a,P4: list_a > list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P4 @ nil_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X: a,Xs2: list_a,Y3: a,Ys2: list_a,Z3: a,Zs2: list_a,W2: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P4 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P4 @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_a @ W2 @ Ws2 ) ) ) ) ) )
=> ( P4 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_684_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_b,P4: list_a > list_a > list_a > list_b > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_b @ Ws ) )
=> ( ( P4 @ nil_a @ nil_a @ nil_a @ nil_b )
=> ( ! [X: a,Xs2: list_a,Y3: a,Ys2: list_a,Z3: a,Zs2: list_a,W2: b,Ws2: list_b] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_b @ Ws2 ) )
=> ( ( P4 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P4 @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_b @ W2 @ Ws2 ) ) ) ) ) )
=> ( P4 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_685_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_b,Ws: list_a,P4: list_a > list_a > list_b > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_b @ Zs ) )
=> ( ( ( size_size_list_b @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P4 @ nil_a @ nil_a @ nil_b @ nil_a )
=> ( ! [X: a,Xs2: list_a,Y3: a,Ys2: list_a,Z3: b,Zs2: list_b,W2: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_b @ Zs2 ) )
=> ( ( ( size_size_list_b @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P4 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P4 @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_b @ Z3 @ Zs2 ) @ ( cons_a @ W2 @ Ws2 ) ) ) ) ) )
=> ( P4 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_686_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_b,Ws: list_b,P4: list_a > list_a > list_b > list_b > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_b @ Zs ) )
=> ( ( ( size_size_list_b @ Zs )
= ( size_size_list_b @ Ws ) )
=> ( ( P4 @ nil_a @ nil_a @ nil_b @ nil_b )
=> ( ! [X: a,Xs2: list_a,Y3: a,Ys2: list_a,Z3: b,Zs2: list_b,W2: b,Ws2: list_b] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_b @ Zs2 ) )
=> ( ( ( size_size_list_b @ Zs2 )
= ( size_size_list_b @ Ws2 ) )
=> ( ( P4 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P4 @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_b @ Z3 @ Zs2 ) @ ( cons_b @ W2 @ Ws2 ) ) ) ) ) )
=> ( P4 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_687_list__induct4,axiom,
! [Xs: list_a,Ys: list_b,Zs: list_a,Ws: list_a,P4: list_a > list_b > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P4 @ nil_a @ nil_b @ nil_a @ nil_a )
=> ( ! [X: a,Xs2: list_a,Y3: b,Ys2: list_b,Z3: a,Zs2: list_a,W2: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_b @ Ys2 ) )
=> ( ( ( size_size_list_b @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P4 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P4 @ ( cons_a @ X @ Xs2 ) @ ( cons_b @ Y3 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_a @ W2 @ Ws2 ) ) ) ) ) )
=> ( P4 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_688_list__induct4,axiom,
! [Xs: list_a,Ys: list_b,Zs: list_a,Ws: list_b,P4: list_a > list_b > list_a > list_b > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_b @ Ws ) )
=> ( ( P4 @ nil_a @ nil_b @ nil_a @ nil_b )
=> ( ! [X: a,Xs2: list_a,Y3: b,Ys2: list_b,Z3: a,Zs2: list_a,W2: b,Ws2: list_b] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_b @ Ys2 ) )
=> ( ( ( size_size_list_b @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_b @ Ws2 ) )
=> ( ( P4 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P4 @ ( cons_a @ X @ Xs2 ) @ ( cons_b @ Y3 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_b @ W2 @ Ws2 ) ) ) ) ) )
=> ( P4 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_689_list__induct4,axiom,
! [Xs: list_a,Ys: list_b,Zs: list_b,Ws: list_a,P4: list_a > list_b > list_b > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_b @ Zs ) )
=> ( ( ( size_size_list_b @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P4 @ nil_a @ nil_b @ nil_b @ nil_a )
=> ( ! [X: a,Xs2: list_a,Y3: b,Ys2: list_b,Z3: b,Zs2: list_b,W2: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_b @ Ys2 ) )
=> ( ( ( size_size_list_b @ Ys2 )
= ( size_size_list_b @ Zs2 ) )
=> ( ( ( size_size_list_b @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P4 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P4 @ ( cons_a @ X @ Xs2 ) @ ( cons_b @ Y3 @ Ys2 ) @ ( cons_b @ Z3 @ Zs2 ) @ ( cons_a @ W2 @ Ws2 ) ) ) ) ) )
=> ( P4 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_690_list__induct4,axiom,
! [Xs: list_a,Ys: list_b,Zs: list_b,Ws: list_b,P4: list_a > list_b > list_b > list_b > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_b @ Zs ) )
=> ( ( ( size_size_list_b @ Zs )
= ( size_size_list_b @ Ws ) )
=> ( ( P4 @ nil_a @ nil_b @ nil_b @ nil_b )
=> ( ! [X: a,Xs2: list_a,Y3: b,Ys2: list_b,Z3: b,Zs2: list_b,W2: b,Ws2: list_b] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_b @ Ys2 ) )
=> ( ( ( size_size_list_b @ Ys2 )
= ( size_size_list_b @ Zs2 ) )
=> ( ( ( size_size_list_b @ Zs2 )
= ( size_size_list_b @ Ws2 ) )
=> ( ( P4 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P4 @ ( cons_a @ X @ Xs2 ) @ ( cons_b @ Y3 @ Ys2 ) @ ( cons_b @ Z3 @ Zs2 ) @ ( cons_b @ W2 @ Ws2 ) ) ) ) ) )
=> ( P4 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_691_list__induct4,axiom,
! [Xs: list_b,Ys: list_a,Zs: list_a,Ws: list_a,P4: list_b > list_a > list_a > list_a > $o] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P4 @ nil_b @ nil_a @ nil_a @ nil_a )
=> ( ! [X: b,Xs2: list_b,Y3: a,Ys2: list_a,Z3: a,Zs2: list_a,W2: a,Ws2: list_a] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P4 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P4 @ ( cons_b @ X @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_a @ W2 @ Ws2 ) ) ) ) ) )
=> ( P4 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_692_list__induct4,axiom,
! [Xs: list_b,Ys: list_a,Zs: list_a,Ws: list_b,P4: list_b > list_a > list_a > list_b > $o] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_b @ Ws ) )
=> ( ( P4 @ nil_b @ nil_a @ nil_a @ nil_b )
=> ( ! [X: b,Xs2: list_b,Y3: a,Ys2: list_a,Z3: a,Zs2: list_a,W2: b,Ws2: list_b] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_b @ Ws2 ) )
=> ( ( P4 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P4 @ ( cons_b @ X @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_b @ W2 @ Ws2 ) ) ) ) ) )
=> ( P4 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_693_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,P4: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P4 @ nil_a @ nil_a @ nil_a )
=> ( ! [X: a,Xs2: list_a,Y3: a,Ys2: list_a,Z3: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P4 @ Xs2 @ Ys2 @ Zs2 )
=> ( P4 @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P4 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_694_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_b,P4: list_a > list_a > list_b > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_b @ Zs ) )
=> ( ( P4 @ nil_a @ nil_a @ nil_b )
=> ( ! [X: a,Xs2: list_a,Y3: a,Ys2: list_a,Z3: b,Zs2: list_b] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_b @ Zs2 ) )
=> ( ( P4 @ Xs2 @ Ys2 @ Zs2 )
=> ( P4 @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_b @ Z3 @ Zs2 ) ) ) ) )
=> ( P4 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_695_list__induct3,axiom,
! [Xs: list_a,Ys: list_b,Zs: list_a,P4: list_a > list_b > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P4 @ nil_a @ nil_b @ nil_a )
=> ( ! [X: a,Xs2: list_a,Y3: b,Ys2: list_b,Z3: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_b @ Ys2 ) )
=> ( ( ( size_size_list_b @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P4 @ Xs2 @ Ys2 @ Zs2 )
=> ( P4 @ ( cons_a @ X @ Xs2 ) @ ( cons_b @ Y3 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P4 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_696_list__induct3,axiom,
! [Xs: list_a,Ys: list_b,Zs: list_b,P4: list_a > list_b > list_b > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_b @ Zs ) )
=> ( ( P4 @ nil_a @ nil_b @ nil_b )
=> ( ! [X: a,Xs2: list_a,Y3: b,Ys2: list_b,Z3: b,Zs2: list_b] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_b @ Ys2 ) )
=> ( ( ( size_size_list_b @ Ys2 )
= ( size_size_list_b @ Zs2 ) )
=> ( ( P4 @ Xs2 @ Ys2 @ Zs2 )
=> ( P4 @ ( cons_a @ X @ Xs2 ) @ ( cons_b @ Y3 @ Ys2 ) @ ( cons_b @ Z3 @ Zs2 ) ) ) ) )
=> ( P4 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_697_list__induct3,axiom,
! [Xs: list_b,Ys: list_a,Zs: list_a,P4: list_b > list_a > list_a > $o] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P4 @ nil_b @ nil_a @ nil_a )
=> ( ! [X: b,Xs2: list_b,Y3: a,Ys2: list_a,Z3: a,Zs2: list_a] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P4 @ Xs2 @ Ys2 @ Zs2 )
=> ( P4 @ ( cons_b @ X @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P4 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_698_list__induct3,axiom,
! [Xs: list_b,Ys: list_a,Zs: list_b,P4: list_b > list_a > list_b > $o] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_b @ Zs ) )
=> ( ( P4 @ nil_b @ nil_a @ nil_b )
=> ( ! [X: b,Xs2: list_b,Y3: a,Ys2: list_a,Z3: b,Zs2: list_b] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_b @ Zs2 ) )
=> ( ( P4 @ Xs2 @ Ys2 @ Zs2 )
=> ( P4 @ ( cons_b @ X @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_b @ Z3 @ Zs2 ) ) ) ) )
=> ( P4 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_699_list__induct3,axiom,
! [Xs: list_b,Ys: list_b,Zs: list_a,P4: list_b > list_b > list_a > $o] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P4 @ nil_b @ nil_b @ nil_a )
=> ( ! [X: b,Xs2: list_b,Y3: b,Ys2: list_b,Z3: a,Zs2: list_a] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_b @ Ys2 ) )
=> ( ( ( size_size_list_b @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P4 @ Xs2 @ Ys2 @ Zs2 )
=> ( P4 @ ( cons_b @ X @ Xs2 ) @ ( cons_b @ Y3 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P4 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_700_list__induct3,axiom,
! [Xs: list_b,Ys: list_b,Zs: list_b,P4: list_b > list_b > list_b > $o] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_b @ Ys ) )
=> ( ( ( size_size_list_b @ Ys )
= ( size_size_list_b @ Zs ) )
=> ( ( P4 @ nil_b @ nil_b @ nil_b )
=> ( ! [X: b,Xs2: list_b,Y3: b,Ys2: list_b,Z3: b,Zs2: list_b] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_b @ Ys2 ) )
=> ( ( ( size_size_list_b @ Ys2 )
= ( size_size_list_b @ Zs2 ) )
=> ( ( P4 @ Xs2 @ Ys2 @ Zs2 )
=> ( P4 @ ( cons_b @ X @ Xs2 ) @ ( cons_b @ Y3 @ Ys2 ) @ ( cons_b @ Z3 @ Zs2 ) ) ) ) )
=> ( P4 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_701_list__induct2,axiom,
! [Xs: list_a,Ys: list_a,P4: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P4 @ nil_a @ nil_a )
=> ( ! [X: a,Xs2: list_a,Y3: a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P4 @ Xs2 @ Ys2 )
=> ( P4 @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) ) )
=> ( P4 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_702_list__induct2,axiom,
! [Xs: list_a,Ys: list_b,P4: list_a > list_b > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_b @ Ys ) )
=> ( ( P4 @ nil_a @ nil_b )
=> ( ! [X: a,Xs2: list_a,Y3: b,Ys2: list_b] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_b @ Ys2 ) )
=> ( ( P4 @ Xs2 @ Ys2 )
=> ( P4 @ ( cons_a @ X @ Xs2 ) @ ( cons_b @ Y3 @ Ys2 ) ) ) )
=> ( P4 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_703_list__induct2,axiom,
! [Xs: list_b,Ys: list_a,P4: list_b > list_a > $o] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P4 @ nil_b @ nil_a )
=> ( ! [X: b,Xs2: list_b,Y3: a,Ys2: list_a] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P4 @ Xs2 @ Ys2 )
=> ( P4 @ ( cons_b @ X @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) ) )
=> ( P4 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_704_list__induct2,axiom,
! [Xs: list_b,Ys: list_b,P4: list_b > list_b > $o] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_b @ Ys ) )
=> ( ( P4 @ nil_b @ nil_b )
=> ( ! [X: b,Xs2: list_b,Y3: b,Ys2: list_b] :
( ( ( size_size_list_b @ Xs2 )
= ( size_size_list_b @ Ys2 ) )
=> ( ( P4 @ Xs2 @ Ys2 )
=> ( P4 @ ( cons_b @ X @ Xs2 ) @ ( cons_b @ Y3 @ Ys2 ) ) ) )
=> ( P4 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_705_less__imp__add__positive,axiom,
! [I2: nat,J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I2 @ K2 )
= J3 ) ) ) ).
% less_imp_add_positive
thf(fact_706_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_707_impossible__Cons,axiom,
! [Xs: list_a,Ys: list_a,X2: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
=> ( Xs
!= ( cons_a @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_708_impossible__Cons,axiom,
! [Xs: list_b,Ys: list_b,X2: b] :
( ( ord_less_eq_nat @ ( size_size_list_b @ Xs ) @ ( size_size_list_b @ Ys ) )
=> ( Xs
!= ( cons_b @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_709_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_710_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_711_less__diff__conv,axiom,
! [I2: nat,J3: nat,K: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J3 @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J3 ) ) ).
% less_diff_conv
thf(fact_712_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ( ( minus_minus_nat @ J3 @ I2 )
= K )
= ( J3
= ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_713_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J3: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J3 @ I2 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J3 @ K ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_714_Nat_Odiff__add__assoc,axiom,
! [K: nat,J3: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J3 ) @ K )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J3 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_715_Nat_Ole__diff__conv2,axiom,
! [K: nat,J3: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J3 @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J3 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_716_le__diff__conv,axiom,
! [J3: nat,K: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J3 @ K ) @ I2 )
= ( ord_less_eq_nat @ J3 @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% le_diff_conv
thf(fact_717_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_718_add__neg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_nonpos
thf(fact_719_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_720_add__nonneg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_721_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_722_add__nonpos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_neg
thf(fact_723_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_724_add__pos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_725_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_726_add__strict__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_727_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_728_add__strict__increasing2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_729_same__length__different,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != Ys )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ? [Pre: list_a,X: a,Xs3: list_a,Y3: a,Ys4: list_a] :
( ( X != Y3 )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X @ nil_a ) @ Xs3 ) ) )
& ( Ys
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y3 @ nil_a ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_730_same__length__different,axiom,
! [Xs: list_b,Ys: list_b] :
( ( Xs != Ys )
=> ( ( ( size_size_list_b @ Xs )
= ( size_size_list_b @ Ys ) )
=> ? [Pre: list_b,X: b,Xs3: list_b,Y3: b,Ys4: list_b] :
( ( X != Y3 )
& ( Xs
= ( append_b @ Pre @ ( append_b @ ( cons_b @ X @ nil_b ) @ Xs3 ) ) )
& ( Ys
= ( append_b @ Pre @ ( append_b @ ( cons_b @ Y3 @ nil_b ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_731_nat__diff__split,axiom,
! [P4: nat > $o,A: nat,B: nat] :
( ( P4 @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P4 @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P4 @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_732_nat__diff__split__asm,axiom,
! [P4: nat > $o,A: nat,B: nat] :
( ( P4 @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P4 @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P4 @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_733_less__diff__conv2,axiom,
! [K: nat,J3: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J3 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J3 @ K ) @ I2 )
= ( ord_less_nat @ J3 @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_734_bounded__Max__nat,axiom,
! [P4: nat > $o,X2: nat,M5: nat] :
( ( P4 @ X2 )
=> ( ! [X: nat] :
( ( P4 @ X )
=> ( ord_less_eq_nat @ X @ M5 ) )
=> ~ ! [M4: nat] :
( ( P4 @ M4 )
=> ~ ! [X4: nat] :
( ( P4 @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_735_nth__Cons_H,axiom,
! [N: nat,X2: a,Xs: list_a] :
( ( ( N = zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X2 @ Xs ) @ N )
= X2 ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X2 @ Xs ) @ N )
= ( nth_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_736_nth__Cons_H,axiom,
! [N: nat,X2: b,Xs: list_b] :
( ( ( N = zero_zero_nat )
=> ( ( nth_b @ ( cons_b @ X2 @ Xs ) @ N )
= X2 ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_b @ ( cons_b @ X2 @ Xs ) @ N )
= ( nth_b @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_737_nth__equal__first__eq,axiom,
! [X2: list_a,Xs: list_list_a,N: nat] :
( ~ ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( ( nth_list_a @ ( cons_list_a @ X2 @ Xs ) @ N )
= X2 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_738_nth__equal__first__eq,axiom,
! [X2: set_a,Xs: list_set_a,N: nat] :
( ~ ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_set_a @ Xs ) )
=> ( ( ( nth_set_a @ ( cons_set_a @ X2 @ Xs ) @ N )
= X2 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_739_nth__equal__first__eq,axiom,
! [X2: nat,Xs: list_nat,N: nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
= X2 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_740_nth__equal__first__eq,axiom,
! [X2: a,Xs: list_a,N: nat] :
( ~ ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( ( nth_a @ ( cons_a @ X2 @ Xs ) @ N )
= X2 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_741_nth__equal__first__eq,axiom,
! [X2: b,Xs: list_b,N: nat] :
( ~ ( member_b @ X2 @ ( set_b2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_b @ Xs ) )
=> ( ( ( nth_b @ ( cons_b @ X2 @ Xs ) @ N )
= X2 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_742_nth__non__equal__first__eq,axiom,
! [X2: a,Y2: a,Xs: list_a,N: nat] :
( ( X2 != Y2 )
=> ( ( ( nth_a @ ( cons_a @ X2 @ Xs ) @ N )
= Y2 )
= ( ( ( nth_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_743_nth__non__equal__first__eq,axiom,
! [X2: b,Y2: b,Xs: list_b,N: nat] :
( ( X2 != Y2 )
=> ( ( ( nth_b @ ( cons_b @ X2 @ Xs ) @ N )
= Y2 )
= ( ( ( nth_b @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_744_atLeastatMost__psubset__iff,axiom,
! [A: set_a,B: set_a,C: set_a,D: set_a] :
( ( ord_less_set_set_a @ ( set_or6288561110385358355_set_a @ A @ B ) @ ( set_or6288561110385358355_set_a @ C @ D ) )
= ( ( ~ ( ord_less_eq_set_a @ A @ B )
| ( ( ord_less_eq_set_a @ C @ A )
& ( ord_less_eq_set_a @ B @ D )
& ( ( ord_less_set_a @ C @ A )
| ( ord_less_set_a @ B @ D ) ) ) )
& ( ord_less_eq_set_a @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_745_atLeastatMost__psubset__iff,axiom,
! [A: set_b,B: set_b,C: set_b,D: set_b] :
( ( ord_less_set_set_b @ ( set_or6288561114688587156_set_b @ A @ B ) @ ( set_or6288561114688587156_set_b @ C @ D ) )
= ( ( ~ ( ord_less_eq_set_b @ A @ B )
| ( ( ord_less_eq_set_b @ C @ A )
& ( ord_less_eq_set_b @ B @ D )
& ( ( ord_less_set_b @ C @ A )
| ( ord_less_set_b @ B @ D ) ) ) )
& ( ord_less_eq_set_b @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_746_atLeastatMost__psubset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ( ~ ( ord_less_eq_real @ A @ B )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D )
& ( ( ord_less_real @ C @ A )
| ( ord_less_real @ B @ D ) ) ) )
& ( ord_less_eq_real @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_747_atLeastatMost__psubset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D )
& ( ( ord_less_nat @ C @ A )
| ( ord_less_nat @ B @ D ) ) ) )
& ( ord_less_eq_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_748_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_749_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_750_le__add__diff__inverse2,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_751_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_752_le__add__diff__inverse,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_753_forward__arcs_Oelims_I1_J,axiom,
! [X2: list_a,Y2: $o] :
( ( ( iKKBZ_4180558001818622352cs_a_b @ t @ X2 )
= Y2 )
=> ( ( ( X2 = nil_a )
=> ~ Y2 )
=> ( ( ? [X: a] :
( X2
= ( cons_a @ X @ nil_a ) )
=> ~ Y2 )
=> ~ ! [X: a,V3: a,Va: list_a] :
( ( X2
= ( cons_a @ X @ ( cons_a @ V3 @ Va ) ) )
=> ( Y2
= ( ~ ( ? [Y4: a] :
( ( member_a @ Y4 @ ( set_a2 @ ( cons_a @ V3 @ Va ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ V3 @ Va ) ) ) ) ) ) ) ) ) ).
% forward_arcs.elims(1)
thf(fact_754_forward__arcs_Oelims_I2_J,axiom,
! [X2: list_a] :
( ( iKKBZ_4180558001818622352cs_a_b @ t @ X2 )
=> ( ( X2 != nil_a )
=> ( ! [X: a] :
( X2
!= ( cons_a @ X @ nil_a ) )
=> ~ ! [X: a,V3: a,Va: list_a] :
( ( X2
= ( cons_a @ X @ ( cons_a @ V3 @ Va ) ) )
=> ~ ( ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ ( cons_a @ V3 @ Va ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ V3 @ Va ) ) ) ) ) ) ) ).
% forward_arcs.elims(2)
thf(fact_755_in__set__inner__verts__appendI__r,axiom,
! [U: a,Q: list_b,P: list_b] :
( ( member_a @ U @ ( set_a2 @ ( pre_inner_verts_a_b @ t @ Q ) ) )
=> ( member_a @ U @ ( set_a2 @ ( pre_inner_verts_a_b @ t @ ( append_b @ P @ Q ) ) ) ) ) ).
% in_set_inner_verts_appendI_r
thf(fact_756_in__set__inner__verts__appendI__l,axiom,
! [U: a,P: list_b,Q: list_b] :
( ( member_a @ U @ ( set_a2 @ ( pre_inner_verts_a_b @ t @ P ) ) )
=> ( member_a @ U @ ( set_a2 @ ( pre_inner_verts_a_b @ t @ ( append_b @ P @ Q ) ) ) ) ) ).
% in_set_inner_verts_appendI_l
thf(fact_757_forward__arcs__split,axiom,
! [Ys: list_a,Xs: list_a] :
( ( iKKBZ_4180558001818622352cs_a_b @ t @ ( append_a @ Ys @ Xs ) )
=> ( iKKBZ_4180558001818622352cs_a_b @ t @ Xs ) ) ).
% forward_arcs_split
thf(fact_758_forward__arcs_Osimps_I1_J,axiom,
iKKBZ_4180558001818622352cs_a_b @ t @ nil_a ).
% forward_arcs.simps(1)
thf(fact_759_forward__arcs_Osimps_I2_J,axiom,
! [X2: a] : ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ X2 @ nil_a ) ) ).
% forward_arcs.simps(2)
thf(fact_760_forward__arcs__single,axiom,
! [X2: a] : ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ X2 @ nil_a ) ) ).
% forward_arcs_single
thf(fact_761_forward__arcs_Oelims_I3_J,axiom,
! [X2: list_a] :
( ~ ( iKKBZ_4180558001818622352cs_a_b @ t @ X2 )
=> ~ ! [X: a,V3: a,Va: list_a] :
( ( X2
= ( cons_a @ X @ ( cons_a @ V3 @ Va ) ) )
=> ( ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ ( cons_a @ V3 @ Va ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ V3 @ Va ) ) ) ) ) ).
% forward_arcs.elims(3)
thf(fact_762_forward__arcs_Osimps_I3_J,axiom,
! [X2: a,V: a,Va2: list_a] :
( ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ X2 @ ( cons_a @ V @ Va2 ) ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ ( cons_a @ V @ Va2 ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ X2 ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ V @ Va2 ) ) ) ) ).
% forward_arcs.simps(3)
thf(fact_763_inner__verts__singleton,axiom,
! [X2: b] :
( ( pre_inner_verts_a_b @ t @ ( cons_b @ X2 @ nil_b ) )
= nil_a ) ).
% inner_verts_singleton
thf(fact_764_inner__verts__Nil,axiom,
( ( pre_inner_verts_a_b @ t @ nil_b )
= nil_a ) ).
% inner_verts_Nil
thf(fact_765_directed__tree_Oforward__arcs_Ocong,axiom,
iKKBZ_4180558001818622352cs_a_b = iKKBZ_4180558001818622352cs_a_b ).
% directed_tree.forward_arcs.cong
thf(fact_766_linorder__neqE__linordered__idom,axiom,
! [X2: real,Y2: real] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ Y2 @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_767_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_768_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_769_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_770_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_771_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_772_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_773_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_774_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_775_add__less__zeroD,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ ( plus_plus_real @ X2 @ Y2 ) @ zero_zero_real )
=> ( ( ord_less_real @ X2 @ zero_zero_real )
| ( ord_less_real @ Y2 @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_776_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_777_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_778_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_779_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_780_add__le__imp__le__diff,axiom,
! [I2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_781_add__le__imp__le__diff,axiom,
! [I2: real,K: real,N: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
=> ( ord_less_eq_real @ I2 @ ( minus_minus_real @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_782_add__le__add__imp__diff__le,axiom,
! [I2: nat,K: nat,N: nat,J3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J3 @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J3 @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J3 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_783_add__le__add__imp__diff__le,axiom,
! [I2: real,K: real,N: real,J3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J3 @ K ) )
=> ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J3 @ K ) )
=> ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J3 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_784_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_785_add__mono1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% add_mono1
thf(fact_786_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_787_less__add__one,axiom,
! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% less_add_one
thf(fact_788_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_789_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: real,B: real] :
( ~ ( ord_less_real @ A @ B )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_790_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_791_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_792_forward__arcs__alt__aux1,axiom,
! [Xs: list_a,I2: nat] :
( ( iKKBZ_4180558001818622352cs_a_b @ t @ Xs )
=> ( ( member_nat @ I2 @ ( set_or1269000886237332187st_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( rev_a @ Xs ) ) @ one_one_nat ) ) )
=> ? [J: nat] :
( ( ord_less_nat @ J @ I2 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ ( rev_a @ Xs ) @ J ) @ ( nth_a @ ( rev_a @ Xs ) @ I2 ) ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% forward_arcs_alt_aux1
thf(fact_793_loopfree__digraph_OvpathI__arc,axiom,
! [G: pre_pr2882871181989701257t_unit,A: list_a,B: list_a] :
( ( loopfr7852502256416881111st_a_b @ G )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A @ B ) @ ( arcs_ends_list_a_b @ G ) )
=> ( vertex6060786982766068989st_a_b @ ( cons_list_a @ A @ ( cons_list_a @ B @ nil_list_a ) ) @ G ) ) ) ).
% loopfree_digraph.vpathI_arc
thf(fact_794_loopfree__digraph_OvpathI__arc,axiom,
! [G: pre_pr7278220950009878019t_unit,A: a,B: a] :
( ( loopfree_digraph_a_b @ G )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( arcs_ends_a_b @ G ) )
=> ( vertex_vpath_a_b @ ( cons_a @ A @ ( cons_a @ B @ nil_a ) ) @ G ) ) ) ).
% loopfree_digraph.vpathI_arc
thf(fact_795_vwalk__wf__digraph__consI,axiom,
! [P: list_a,A: a] :
( ( vertex_vwalk_a_b @ P @ t )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ ( hd_a @ P ) ) @ ( arcs_ends_a_b @ t ) )
=> ( vertex_vwalk_a_b @ ( cons_a @ A @ P ) @ t ) ) ) ).
% vwalk_wf_digraph_consI
thf(fact_796_psubsetI,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_set_a @ A2 @ B3 ) ) ) ).
% psubsetI
thf(fact_797_psubsetI,axiom,
! [A2: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_set_b @ A2 @ B3 ) ) ) ).
% psubsetI
thf(fact_798_cas_Ocases,axiom,
! [X2: produc7945266988514096265st_b_a] :
( ! [U5: a,V3: a] :
( X2
!= ( produc7119031474978700025st_b_a @ U5 @ ( produc4145578316043568848st_b_a @ nil_b @ V3 ) ) )
=> ~ ! [U5: a,E2: b,Es: list_b,V3: a] :
( X2
!= ( produc7119031474978700025st_b_a @ U5 @ ( produc4145578316043568848st_b_a @ ( cons_b @ E2 @ Es ) @ V3 ) ) ) ) ).
% cas.cases
thf(fact_799_subsetI,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ! [X: list_a] :
( ( member_list_a @ X @ A2 )
=> ( member_list_a @ X @ B3 ) )
=> ( ord_le8861187494160871172list_a @ A2 @ B3 ) ) ).
% subsetI
thf(fact_800_subsetI,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A2 )
=> ( member_set_a @ X @ B3 ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B3 ) ) ).
% subsetI
thf(fact_801_subsetI,axiom,
! [A2: set_nat,B3: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B3 ) )
=> ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% subsetI
thf(fact_802_subsetI,axiom,
! [A2: set_a,B3: set_a] :
( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( member_a @ X @ B3 ) )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% subsetI
thf(fact_803_subsetI,axiom,
! [A2: set_b,B3: set_b] :
( ! [X: b] :
( ( member_b @ X @ A2 )
=> ( member_b @ X @ B3 ) )
=> ( ord_less_eq_set_b @ A2 @ B3 ) ) ).
% subsetI
thf(fact_804_subset__antisym,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_805_subset__antisym,axiom,
! [A2: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ( ord_less_eq_set_b @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_806_DiffI,axiom,
! [C: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ A2 )
=> ( ~ ( member_list_a @ C @ B3 )
=> ( member_list_a @ C @ ( minus_646659088055828811list_a @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_807_DiffI,axiom,
! [C: set_a,A2: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ A2 )
=> ( ~ ( member_set_a @ C @ B3 )
=> ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_808_DiffI,axiom,
! [C: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ~ ( member_nat @ C @ B3 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_809_DiffI,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ A2 )
=> ( ~ ( member_a @ C @ B3 )
=> ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_810_DiffI,axiom,
! [C: b,A2: set_b,B3: set_b] :
( ( member_b @ C @ A2 )
=> ( ~ ( member_b @ C @ B3 )
=> ( member_b @ C @ ( minus_minus_set_b @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_811_Diff__iff,axiom,
! [C: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A2 @ B3 ) )
= ( ( member_list_a @ C @ A2 )
& ~ ( member_list_a @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_812_Diff__iff,axiom,
! [C: set_a,A2: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B3 ) )
= ( ( member_set_a @ C @ A2 )
& ~ ( member_set_a @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_813_Diff__iff,axiom,
! [C: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
= ( ( member_nat @ C @ A2 )
& ~ ( member_nat @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_814_Diff__iff,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) )
= ( ( member_a @ C @ A2 )
& ~ ( member_a @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_815_Diff__iff,axiom,
! [C: b,A2: set_b,B3: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B3 ) )
= ( ( member_b @ C @ A2 )
& ~ ( member_b @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_816_Diff__idemp,axiom,
! [A2: set_a,B3: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ B3 )
= ( minus_minus_set_a @ A2 @ B3 ) ) ).
% Diff_idemp
thf(fact_817_Diff__idemp,axiom,
! [A2: set_b,B3: set_b] :
( ( minus_minus_set_b @ ( minus_minus_set_b @ A2 @ B3 ) @ B3 )
= ( minus_minus_set_b @ A2 @ B3 ) ) ).
% Diff_idemp
thf(fact_818_rev__is__rev__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( rev_a @ Xs )
= ( rev_a @ Ys ) )
= ( Xs = Ys ) ) ).
% rev_is_rev_conv
thf(fact_819_rev__rev__ident,axiom,
! [Xs: list_a] :
( ( rev_a @ ( rev_a @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_820_forward__cons,axiom,
! [X2: a,Xs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( rev_a @ ( cons_a @ X2 @ Xs ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( rev_a @ Xs ) ) ) ).
% forward_cons
thf(fact_821_forward__arcs__alt__aux2,axiom,
! [Xs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( rev_a @ Xs ) )
=> ( iKKBZ_4180558001818622352cs_a_b @ t @ Xs ) ) ).
% forward_arcs_alt_aux2
thf(fact_822_forward__arcs__alt_H,axiom,
! [Xs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( rev_a @ Xs ) )
= ( iKKBZ_4180558001818622352cs_a_b @ t @ Xs ) ) ).
% forward_arcs_alt'
thf(fact_823_forward__arcs__alt,axiom,
! [Xs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs )
= ( iKKBZ_4180558001818622352cs_a_b @ t @ ( rev_a @ Xs ) ) ) ).
% forward_arcs_alt
thf(fact_824_seq__conform__def,axiom,
! [Xs: list_a] :
( ( iKKBZ_4622586873178280511rm_a_b @ t @ Xs )
= ( ( iKKBZ_4180558001818622352cs_a_b @ t @ ( rev_a @ Xs ) )
& ( iKKBZ_7773321254043928001cs_a_b @ t @ Xs ) ) ) ).
% seq_conform_def
thf(fact_825_arc__to__lst__if__forward,axiom,
! [X2: a,Xs: list_a,Y2: a,Ys: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( rev_a @ ( cons_a @ X2 @ Xs ) ) )
=> ( ( Xs
= ( cons_a @ Y2 @ Ys ) )
=> ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ X2 ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% arc_to_lst_if_forward
thf(fact_826_rev__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rev_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rev_is_Nil_conv
thf(fact_827_rev__is__Nil__conv,axiom,
! [Xs: list_b] :
( ( ( rev_b @ Xs )
= nil_b )
= ( Xs = nil_b ) ) ).
% rev_is_Nil_conv
thf(fact_828_Nil__is__rev__conv,axiom,
! [Xs: list_a] :
( ( nil_a
= ( rev_a @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_rev_conv
thf(fact_829_Nil__is__rev__conv,axiom,
! [Xs: list_b] :
( ( nil_b
= ( rev_b @ Xs ) )
= ( Xs = nil_b ) ) ).
% Nil_is_rev_conv
thf(fact_830_set__rev,axiom,
! [Xs: list_a] :
( ( set_a2 @ ( rev_a @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% set_rev
thf(fact_831_set__rev,axiom,
! [Xs: list_b] :
( ( set_b2 @ ( rev_b @ Xs ) )
= ( set_b2 @ Xs ) ) ).
% set_rev
thf(fact_832_rev__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( rev_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( rev_a @ Ys ) @ ( rev_a @ Xs ) ) ) ).
% rev_append
thf(fact_833_rev__append,axiom,
! [Xs: list_b,Ys: list_b] :
( ( rev_b @ ( append_b @ Xs @ Ys ) )
= ( append_b @ ( rev_b @ Ys ) @ ( rev_b @ Xs ) ) ) ).
% rev_append
thf(fact_834_length__rev,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( rev_a @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_rev
thf(fact_835_length__rev,axiom,
! [Xs: list_b] :
( ( size_size_list_b @ ( rev_b @ Xs ) )
= ( size_size_list_b @ Xs ) ) ).
% length_rev
thf(fact_836_hd__reach__all__forward__arcs,axiom,
! [Xs: list_a,X2: a] :
( ( member_a @ ( hd_a @ ( rev_a @ Xs ) ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( iKKBZ_4180558001818622352cs_a_b @ t @ Xs )
=> ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( reachable_a_b @ t @ ( hd_a @ ( rev_a @ Xs ) ) @ X2 ) ) ) ) ).
% hd_reach_all_forward_arcs
thf(fact_837_rev__singleton__conv,axiom,
! [Xs: list_a,X2: a] :
( ( ( rev_a @ Xs )
= ( cons_a @ X2 @ nil_a ) )
= ( Xs
= ( cons_a @ X2 @ nil_a ) ) ) ).
% rev_singleton_conv
thf(fact_838_rev__singleton__conv,axiom,
! [Xs: list_b,X2: b] :
( ( ( rev_b @ Xs )
= ( cons_b @ X2 @ nil_b ) )
= ( Xs
= ( cons_b @ X2 @ nil_b ) ) ) ).
% rev_singleton_conv
thf(fact_839_singleton__rev__conv,axiom,
! [X2: a,Xs: list_a] :
( ( ( cons_a @ X2 @ nil_a )
= ( rev_a @ Xs ) )
= ( ( cons_a @ X2 @ nil_a )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_840_singleton__rev__conv,axiom,
! [X2: b,Xs: list_b] :
( ( ( cons_b @ X2 @ nil_b )
= ( rev_b @ Xs ) )
= ( ( cons_b @ X2 @ nil_b )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_841_rev__eq__Cons__iff,axiom,
! [Xs: list_a,Y2: a,Ys: list_a] :
( ( ( rev_a @ Xs )
= ( cons_a @ Y2 @ Ys ) )
= ( Xs
= ( append_a @ ( rev_a @ Ys ) @ ( cons_a @ Y2 @ nil_a ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_842_rev__eq__Cons__iff,axiom,
! [Xs: list_b,Y2: b,Ys: list_b] :
( ( ( rev_b @ Xs )
= ( cons_b @ Y2 @ Ys ) )
= ( Xs
= ( append_b @ ( rev_b @ Ys ) @ ( cons_b @ Y2 @ nil_b ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_843_vwalk__Cons__Cons,axiom,
! [U: a,V: a,Ws: list_a] :
( ( vertex_vwalk_a_b @ ( cons_a @ U @ ( cons_a @ V @ Ws ) ) @ t )
= ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( arcs_ends_a_b @ t ) )
& ( vertex_vwalk_a_b @ ( cons_a @ V @ Ws ) @ t ) ) ) ).
% vwalk_Cons_Cons
thf(fact_844_map__tailrec__rev_Ocases,axiom,
! [X2: produc2395089919340105847list_b] :
( ! [F2: b > b,Bs3: list_b] :
( X2
!= ( produc748123367317244457list_b @ F2 @ ( produc1564554178308465111list_b @ nil_b @ Bs3 ) ) )
=> ~ ! [F2: b > b,A4: b,As3: list_b,Bs3: list_b] :
( X2
!= ( produc748123367317244457list_b @ F2 @ ( produc1564554178308465111list_b @ ( cons_b @ A4 @ As3 ) @ Bs3 ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_845_map__tailrec__rev_Ocases,axiom,
! [X2: produc1473018763691903991list_a] :
( ! [F2: a > a,Bs3: list_a] :
( X2
!= ( produc8643929849434629545list_a @ F2 @ ( produc6837034575241423639list_a @ nil_a @ Bs3 ) ) )
=> ~ ! [F2: a > a,A4: a,As3: list_a,Bs3: list_a] :
( X2
!= ( produc8643929849434629545list_a @ F2 @ ( produc6837034575241423639list_a @ ( cons_a @ A4 @ As3 ) @ Bs3 ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_846_rev_Osimps_I1_J,axiom,
( ( rev_a @ nil_a )
= nil_a ) ).
% rev.simps(1)
thf(fact_847_rev_Osimps_I1_J,axiom,
( ( rev_b @ nil_b )
= nil_b ) ).
% rev.simps(1)
thf(fact_848_rev__swap,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( rev_a @ Xs )
= Ys )
= ( Xs
= ( rev_a @ Ys ) ) ) ).
% rev_swap
thf(fact_849_sorted__wrt_Ocases,axiom,
! [X2: produc5032551385658279741list_a] :
( ! [P5: a > a > $o] :
( X2
!= ( produc8111569692950616493list_a @ P5 @ nil_a ) )
=> ~ ! [P5: a > a > $o,X: a,Ys2: list_a] :
( X2
!= ( produc8111569692950616493list_a @ P5 @ ( cons_a @ X @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_850_sorted__wrt_Ocases,axiom,
! [X2: produc5185152304234826110list_b] :
( ! [P5: b > b > $o] :
( X2
!= ( produc8193136575784045678list_b @ P5 @ nil_b ) )
=> ~ ! [P5: b > b > $o,X: b,Ys2: list_b] :
( X2
!= ( produc8193136575784045678list_b @ P5 @ ( cons_b @ X @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_851_successively_Ocases,axiom,
! [X2: produc5032551385658279741list_a] :
( ! [P5: a > a > $o] :
( X2
!= ( produc8111569692950616493list_a @ P5 @ nil_a ) )
=> ( ! [P5: a > a > $o,X: a] :
( X2
!= ( produc8111569692950616493list_a @ P5 @ ( cons_a @ X @ nil_a ) ) )
=> ~ ! [P5: a > a > $o,X: a,Y3: a,Xs2: list_a] :
( X2
!= ( produc8111569692950616493list_a @ P5 @ ( cons_a @ X @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_852_successively_Ocases,axiom,
! [X2: produc5185152304234826110list_b] :
( ! [P5: b > b > $o] :
( X2
!= ( produc8193136575784045678list_b @ P5 @ nil_b ) )
=> ( ! [P5: b > b > $o,X: b] :
( X2
!= ( produc8193136575784045678list_b @ P5 @ ( cons_b @ X @ nil_b ) ) )
=> ~ ! [P5: b > b > $o,X: b,Y3: b,Xs2: list_b] :
( X2
!= ( produc8193136575784045678list_b @ P5 @ ( cons_b @ X @ ( cons_b @ Y3 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_853_rev_Osimps_I2_J,axiom,
! [X2: a,Xs: list_a] :
( ( rev_a @ ( cons_a @ X2 @ Xs ) )
= ( append_a @ ( rev_a @ Xs ) @ ( cons_a @ X2 @ nil_a ) ) ) ).
% rev.simps(2)
thf(fact_854_rev_Osimps_I2_J,axiom,
! [X2: b,Xs: list_b] :
( ( rev_b @ ( cons_b @ X2 @ Xs ) )
= ( append_b @ ( rev_b @ Xs ) @ ( cons_b @ X2 @ nil_b ) ) ) ).
% rev.simps(2)
thf(fact_855_in__mono,axiom,
! [A2: set_list_a,B3: set_list_a,X2: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( member_list_a @ X2 @ A2 )
=> ( member_list_a @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_856_in__mono,axiom,
! [A2: set_set_a,B3: set_set_a,X2: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( member_set_a @ X2 @ A2 )
=> ( member_set_a @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_857_in__mono,axiom,
! [A2: set_nat,B3: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_858_in__mono,axiom,
! [A2: set_a,B3: set_a,X2: a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_859_in__mono,axiom,
! [A2: set_b,B3: set_b,X2: b] :
( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ( member_b @ X2 @ A2 )
=> ( member_b @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_860_subsetD,axiom,
! [A2: set_list_a,B3: set_list_a,C: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( member_list_a @ C @ A2 )
=> ( member_list_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_861_subsetD,axiom,
! [A2: set_set_a,B3: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_862_subsetD,axiom,
! [A2: set_nat,B3: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_863_subsetD,axiom,
! [A2: set_a,B3: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_864_subsetD,axiom,
! [A2: set_b,B3: set_b,C: b] :
( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ( member_b @ C @ A2 )
=> ( member_b @ C @ B3 ) ) ) ).
% subsetD
thf(fact_865_equalityE,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ~ ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_866_equalityE,axiom,
! [A2: set_b,B3: set_b] :
( ( A2 = B3 )
=> ~ ( ( ord_less_eq_set_b @ A2 @ B3 )
=> ~ ( ord_less_eq_set_b @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_867_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A5: set_list_a,B4: set_list_a] :
! [X3: list_a] :
( ( member_list_a @ X3 @ A5 )
=> ( member_list_a @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_868_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B4: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A5 )
=> ( member_set_a @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_869_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ( member_nat @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_870_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B4: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ( member_a @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_871_subset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A5: set_b,B4: set_b] :
! [X3: b] :
( ( member_b @ X3 @ A5 )
=> ( member_b @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_872_equalityD1,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_873_equalityD1,axiom,
! [A2: set_b,B3: set_b] :
( ( A2 = B3 )
=> ( ord_less_eq_set_b @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_874_equalityD2,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ( ord_less_eq_set_a @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_875_equalityD2,axiom,
! [A2: set_b,B3: set_b] :
( ( A2 = B3 )
=> ( ord_less_eq_set_b @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_876_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A5: set_list_a,B4: set_list_a] :
! [T2: list_a] :
( ( member_list_a @ T2 @ A5 )
=> ( member_list_a @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_877_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B4: set_set_a] :
! [T2: set_a] :
( ( member_set_a @ T2 @ A5 )
=> ( member_set_a @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_878_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A5 )
=> ( member_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_879_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B4: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A5 )
=> ( member_a @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_880_subset__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A5: set_b,B4: set_b] :
! [T2: b] :
( ( member_b @ T2 @ A5 )
=> ( member_b @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_881_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_882_subset__refl,axiom,
! [A2: set_b] : ( ord_less_eq_set_b @ A2 @ A2 ) ).
% subset_refl
thf(fact_883_Collect__mono,axiom,
! [P4: a > $o,Q2: a > $o] :
( ! [X: a] :
( ( P4 @ X )
=> ( Q2 @ X ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P4 ) @ ( collect_a @ Q2 ) ) ) ).
% Collect_mono
thf(fact_884_Collect__mono,axiom,
! [P4: b > $o,Q2: b > $o] :
( ! [X: b] :
( ( P4 @ X )
=> ( Q2 @ X ) )
=> ( ord_less_eq_set_b @ ( collect_b @ P4 ) @ ( collect_b @ Q2 ) ) ) ).
% Collect_mono
thf(fact_885_subset__trans,axiom,
! [A2: set_a,B3: set_a,C4: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C4 )
=> ( ord_less_eq_set_a @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_886_subset__trans,axiom,
! [A2: set_b,B3: set_b,C4: set_b] :
( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ( ord_less_eq_set_b @ B3 @ C4 )
=> ( ord_less_eq_set_b @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_887_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z: set_a] : ( Y5 = Z ) )
= ( ^ [A5: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A5 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_888_set__eq__subset,axiom,
( ( ^ [Y5: set_b,Z: set_b] : ( Y5 = Z ) )
= ( ^ [A5: set_b,B4: set_b] :
( ( ord_less_eq_set_b @ A5 @ B4 )
& ( ord_less_eq_set_b @ B4 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_889_Collect__mono__iff,axiom,
! [P4: a > $o,Q2: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P4 ) @ ( collect_a @ Q2 ) )
= ( ! [X3: a] :
( ( P4 @ X3 )
=> ( Q2 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_890_Collect__mono__iff,axiom,
! [P4: b > $o,Q2: b > $o] :
( ( ord_less_eq_set_b @ ( collect_b @ P4 ) @ ( collect_b @ Q2 ) )
= ( ! [X3: b] :
( ( P4 @ X3 )
=> ( Q2 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_891_DiffE,axiom,
! [C: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A2 @ B3 ) )
=> ~ ( ( member_list_a @ C @ A2 )
=> ( member_list_a @ C @ B3 ) ) ) ).
% DiffE
thf(fact_892_DiffE,axiom,
! [C: set_a,A2: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B3 ) )
=> ~ ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B3 ) ) ) ).
% DiffE
thf(fact_893_DiffE,axiom,
! [C: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B3 ) ) ) ).
% DiffE
thf(fact_894_DiffE,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B3 ) ) ) ).
% DiffE
thf(fact_895_DiffE,axiom,
! [C: b,A2: set_b,B3: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B3 ) )
=> ~ ( ( member_b @ C @ A2 )
=> ( member_b @ C @ B3 ) ) ) ).
% DiffE
thf(fact_896_DiffD1,axiom,
! [C: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A2 @ B3 ) )
=> ( member_list_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_897_DiffD1,axiom,
! [C: set_a,A2: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B3 ) )
=> ( member_set_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_898_DiffD1,axiom,
! [C: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
=> ( member_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_899_DiffD1,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) )
=> ( member_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_900_DiffD1,axiom,
! [C: b,A2: set_b,B3: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B3 ) )
=> ( member_b @ C @ A2 ) ) ).
% DiffD1
thf(fact_901_DiffD2,axiom,
! [C: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A2 @ B3 ) )
=> ~ ( member_list_a @ C @ B3 ) ) ).
% DiffD2
thf(fact_902_DiffD2,axiom,
! [C: set_a,A2: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B3 ) )
=> ~ ( member_set_a @ C @ B3 ) ) ).
% DiffD2
thf(fact_903_DiffD2,axiom,
! [C: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
=> ~ ( member_nat @ C @ B3 ) ) ).
% DiffD2
thf(fact_904_DiffD2,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) )
=> ~ ( member_a @ C @ B3 ) ) ).
% DiffD2
thf(fact_905_DiffD2,axiom,
! [C: b,A2: set_b,B3: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B3 ) )
=> ~ ( member_b @ C @ B3 ) ) ).
% DiffD2
thf(fact_906_Diff__mono,axiom,
! [A2: set_a,C4: set_a,D3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ C4 )
=> ( ( ord_less_eq_set_a @ D3 @ B3 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ ( minus_minus_set_a @ C4 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_907_Diff__mono,axiom,
! [A2: set_b,C4: set_b,D3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A2 @ C4 )
=> ( ( ord_less_eq_set_b @ D3 @ B3 )
=> ( ord_less_eq_set_b @ ( minus_minus_set_b @ A2 @ B3 ) @ ( minus_minus_set_b @ C4 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_908_Diff__subset,axiom,
! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ A2 ) ).
% Diff_subset
thf(fact_909_Diff__subset,axiom,
! [A2: set_b,B3: set_b] : ( ord_less_eq_set_b @ ( minus_minus_set_b @ A2 @ B3 ) @ A2 ) ).
% Diff_subset
thf(fact_910_double__diff,axiom,
! [A2: set_a,B3: set_a,C4: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C4 )
=> ( ( minus_minus_set_a @ B3 @ ( minus_minus_set_a @ C4 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_911_double__diff,axiom,
! [A2: set_b,B3: set_b,C4: set_b] :
( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ( ord_less_eq_set_b @ B3 @ C4 )
=> ( ( minus_minus_set_b @ B3 @ ( minus_minus_set_b @ C4 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_912_psubsetE,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_set_a @ A2 @ B3 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ).
% psubsetE
thf(fact_913_psubsetE,axiom,
! [A2: set_b,B3: set_b] :
( ( ord_less_set_b @ A2 @ B3 )
=> ~ ( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ord_less_eq_set_b @ B3 @ A2 ) ) ) ).
% psubsetE
thf(fact_914_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_915_psubset__eq,axiom,
( ord_less_set_b
= ( ^ [A5: set_b,B4: set_b] :
( ( ord_less_eq_set_b @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_916_psubset__imp__subset,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_set_a @ A2 @ B3 )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_917_psubset__imp__subset,axiom,
! [A2: set_b,B3: set_b] :
( ( ord_less_set_b @ A2 @ B3 )
=> ( ord_less_eq_set_b @ A2 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_918_psubset__subset__trans,axiom,
! [A2: set_a,B3: set_a,C4: set_a] :
( ( ord_less_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C4 )
=> ( ord_less_set_a @ A2 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_919_psubset__subset__trans,axiom,
! [A2: set_b,B3: set_b,C4: set_b] :
( ( ord_less_set_b @ A2 @ B3 )
=> ( ( ord_less_eq_set_b @ B3 @ C4 )
=> ( ord_less_set_b @ A2 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_920_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A5 @ B4 )
& ~ ( ord_less_eq_set_a @ B4 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_921_subset__not__subset__eq,axiom,
( ord_less_set_b
= ( ^ [A5: set_b,B4: set_b] :
( ( ord_less_eq_set_b @ A5 @ B4 )
& ~ ( ord_less_eq_set_b @ B4 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_922_subset__psubset__trans,axiom,
! [A2: set_a,B3: set_a,C4: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_set_a @ B3 @ C4 )
=> ( ord_less_set_a @ A2 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_923_subset__psubset__trans,axiom,
! [A2: set_b,B3: set_b,C4: set_b] :
( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ( ord_less_set_b @ B3 @ C4 )
=> ( ord_less_set_b @ A2 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_924_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B4: set_a] :
( ( ord_less_set_a @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_925_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A5: set_b,B4: set_b] :
( ( ord_less_set_b @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_926_psubset__imp__ex__mem,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( ord_less_set_list_a @ A2 @ B3 )
=> ? [B5: list_a] : ( member_list_a @ B5 @ ( minus_646659088055828811list_a @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_927_psubset__imp__ex__mem,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B3 )
=> ? [B5: set_a] : ( member_set_a @ B5 @ ( minus_5736297505244876581_set_a @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_928_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A2 @ B3 )
=> ? [B5: nat] : ( member_nat @ B5 @ ( minus_minus_set_nat @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_929_psubset__imp__ex__mem,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_set_a @ A2 @ B3 )
=> ? [B5: a] : ( member_a @ B5 @ ( minus_minus_set_a @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_930_psubset__imp__ex__mem,axiom,
! [A2: set_b,B3: set_b] :
( ( ord_less_set_b @ A2 @ B3 )
=> ? [B5: b] : ( member_b @ B5 @ ( minus_minus_set_b @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_931_vwalk__singleton,axiom,
! [U: a,G: pre_pr7278220950009878019t_unit] :
( ( vertex_vwalk_a_b @ ( cons_a @ U @ nil_a ) @ G )
= ( member_a @ U @ ( pre_ve642382030648772252t_unit @ G ) ) ) ).
% vwalk_singleton
thf(fact_932_vwalk__singleton,axiom,
! [U: list_a,G: pre_pr2882871181989701257t_unit] :
( ( vertex2966258834163962945st_a_b @ ( cons_list_a @ U @ nil_list_a ) @ G )
= ( member_list_a @ U @ ( pre_ve1830060048215441954t_unit @ G ) ) ) ).
% vwalk_singleton
thf(fact_933_to__list__tree__dom__iff,axiom,
! [X2: a,Y2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ ( arcs_ends_a_b @ t ) )
= ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X2 @ nil_a ) @ ( cons_a @ Y2 @ nil_a ) ) @ ( arcs_ends_list_a_b @ ( direct3773525127397338803ee_a_b @ t ) ) ) ) ).
% to_list_tree_dom_iff
thf(fact_934_vwalk__consI,axiom,
! [P: list_a,G: pre_pr7278220950009878019t_unit,A: a] :
( ( vertex_vwalk_a_b @ P @ G )
=> ( ( member_a @ A @ ( pre_ve642382030648772252t_unit @ G ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ ( hd_a @ P ) ) @ ( arcs_ends_a_b @ G ) )
=> ( vertex_vwalk_a_b @ ( cons_a @ A @ P ) @ G ) ) ) ) ).
% vwalk_consI
thf(fact_935_vwalk__consI,axiom,
! [P: list_list_a,G: pre_pr2882871181989701257t_unit,A: list_a] :
( ( vertex2966258834163962945st_a_b @ P @ G )
=> ( ( member_list_a @ A @ ( pre_ve1830060048215441954t_unit @ G ) )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A @ ( hd_list_a @ P ) ) @ ( arcs_ends_list_a_b @ G ) )
=> ( vertex2966258834163962945st_a_b @ ( cons_list_a @ A @ P ) @ G ) ) ) ) ).
% vwalk_consI
thf(fact_936_vwalk__consE,axiom,
! [A: list_a,P: list_list_a,G: pre_pr2882871181989701257t_unit] :
( ( vertex2966258834163962945st_a_b @ ( cons_list_a @ A @ P ) @ G )
=> ( ( P != nil_list_a )
=> ~ ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A @ ( hd_list_a @ P ) ) @ ( arcs_ends_list_a_b @ G ) )
=> ~ ( vertex2966258834163962945st_a_b @ P @ G ) ) ) ) ).
% vwalk_consE
thf(fact_937_vwalk__consE,axiom,
! [A: a,P: list_a,G: pre_pr7278220950009878019t_unit] :
( ( vertex_vwalk_a_b @ ( cons_a @ A @ P ) @ G )
=> ( ( P != nil_a )
=> ~ ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ ( hd_a @ P ) ) @ ( arcs_ends_a_b @ G ) )
=> ~ ( vertex_vwalk_a_b @ P @ G ) ) ) ) ).
% vwalk_consE
thf(fact_938_to__list__tree__nempty,axiom,
! [V: list_a] :
( ( member_list_a @ V @ ( pre_ve1830060048215441954t_unit @ ( direct3773525127397338803ee_a_b @ t ) ) )
=> ( V != nil_a ) ) ).
% to_list_tree_nempty
thf(fact_939_to__list__tree__single,axiom,
! [V: list_a] :
( ( member_list_a @ V @ ( pre_ve1830060048215441954t_unit @ ( direct3773525127397338803ee_a_b @ t ) ) )
=> ? [X: a] :
( ( V
= ( cons_a @ X @ nil_a ) )
& ( member_a @ X @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% to_list_tree_single
thf(fact_940_vwalk__to__vpath_Ocases,axiom,
! [X2: list_a] :
( ( X2 != nil_a )
=> ~ ! [X: a,Xs2: list_a] :
( X2
!= ( cons_a @ X @ Xs2 ) ) ) ).
% vwalk_to_vpath.cases
thf(fact_941_vwalk__to__vpath_Ocases,axiom,
! [X2: list_b] :
( ( X2 != nil_b )
=> ~ ! [X: b,Xs2: list_b] :
( X2
!= ( cons_b @ X @ Xs2 ) ) ) ).
% vwalk_to_vpath.cases
thf(fact_942_vwalkI__append__l,axiom,
! [P: list_a,Q: list_a,G: pre_pr7278220950009878019t_unit] :
( ( P != nil_a )
=> ( ( vertex_vwalk_a_b @ ( append_a @ P @ Q ) @ G )
=> ( vertex_vwalk_a_b @ P @ G ) ) ) ).
% vwalkI_append_l
thf(fact_943_vwalkI__append__r,axiom,
! [Q: list_a,P: list_a,G: pre_pr7278220950009878019t_unit] :
( ( Q != nil_a )
=> ( ( vertex_vwalk_a_b @ ( append_a @ P @ Q ) @ G )
=> ( vertex_vwalk_a_b @ Q @ G ) ) ) ).
% vwalkI_append_r
thf(fact_944_vwalk__verts__in__verts,axiom,
! [P: list_a,G: pre_pr7278220950009878019t_unit,U: a] :
( ( vertex_vwalk_a_b @ P @ G )
=> ( ( member_a @ U @ ( set_a2 @ P ) )
=> ( member_a @ U @ ( pre_ve642382030648772252t_unit @ G ) ) ) ) ).
% vwalk_verts_in_verts
thf(fact_945_vwalk__verts__in__verts,axiom,
! [P: list_list_a,G: pre_pr2882871181989701257t_unit,U: list_a] :
( ( vertex2966258834163962945st_a_b @ P @ G )
=> ( ( member_list_a @ U @ ( set_list_a2 @ P ) )
=> ( member_list_a @ U @ ( pre_ve1830060048215441954t_unit @ G ) ) ) ) ).
% vwalk_verts_in_verts
thf(fact_946_vpath__self,axiom,
! [U: a,G: pre_pr7278220950009878019t_unit] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ G ) )
=> ( vertex_vpath_a_b @ ( cons_a @ U @ nil_a ) @ G ) ) ).
% vpath_self
thf(fact_947_vpath__self,axiom,
! [U: list_a,G: pre_pr2882871181989701257t_unit] :
( ( member_list_a @ U @ ( pre_ve1830060048215441954t_unit @ G ) )
=> ( vertex6060786982766068989st_a_b @ ( cons_list_a @ U @ nil_list_a ) @ G ) ) ).
% vpath_self
thf(fact_948_vwalk__induct,axiom,
! [P: list_a,G: pre_pr7278220950009878019t_unit,P4: list_a > $o] :
( ( vertex_vwalk_a_b @ P @ G )
=> ( ! [U5: a] :
( ( member_a @ U5 @ ( pre_ve642382030648772252t_unit @ G ) )
=> ( P4 @ ( cons_a @ U5 @ nil_a ) ) )
=> ( ! [U5: a,V3: a,Es: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U5 @ V3 ) @ ( arcs_ends_a_b @ G ) )
=> ( ( P4 @ ( cons_a @ V3 @ Es ) )
=> ( P4 @ ( cons_a @ U5 @ ( cons_a @ V3 @ Es ) ) ) ) )
=> ( P4 @ P ) ) ) ) ).
% vwalk_induct
thf(fact_949_vwalk__induct,axiom,
! [P: list_list_a,G: pre_pr2882871181989701257t_unit,P4: list_list_a > $o] :
( ( vertex2966258834163962945st_a_b @ P @ G )
=> ( ! [U5: list_a] :
( ( member_list_a @ U5 @ ( pre_ve1830060048215441954t_unit @ G ) )
=> ( P4 @ ( cons_list_a @ U5 @ nil_list_a ) ) )
=> ( ! [U5: list_a,V3: list_a,Es: list_list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ U5 @ V3 ) @ ( arcs_ends_list_a_b @ G ) )
=> ( ( P4 @ ( cons_list_a @ V3 @ Es ) )
=> ( P4 @ ( cons_list_a @ U5 @ ( cons_list_a @ V3 @ Es ) ) ) ) )
=> ( P4 @ P ) ) ) ) ).
% vwalk_induct
thf(fact_950_inner__verts__Cons,axiom,
! [U: a,E: b,Es2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ ( cons_b @ E @ Es2 ) @ V )
=> ( ( ( Es2 != nil_b )
=> ( ( pre_inner_verts_a_b @ t @ ( cons_b @ E @ Es2 ) )
= ( cons_a @ ( pre_he5236287464308401016t_unit @ t @ E ) @ ( pre_inner_verts_a_b @ t @ Es2 ) ) ) )
& ( ( Es2 = nil_b )
=> ( ( pre_inner_verts_a_b @ t @ ( cons_b @ E @ Es2 ) )
= nil_a ) ) ) ) ).
% inner_verts_Cons
thf(fact_951_merge__in__verts,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( graph_2957805489637798020ts_a_b @ t ) )
=> ( member_a @ X2 @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% merge_in_verts
thf(fact_952_double__eq__0__iff,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_953_last__merge__is__merge,axiom,
! [Y2: a] :
( ( member_a @ Y2 @ ( graph_2659413520663303054ts_a_b @ t ) )
=> ( member_a @ Y2 @ ( graph_2957805489637798020ts_a_b @ t ) ) ) ).
% last_merge_is_merge
thf(fact_954_last__merge__alt,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( graph_2659413520663303054ts_a_b @ t ) )
=> ! [Z4: a] :
( ( ( reachable_a_b @ t @ X2 @ Z4 )
& ( Z4 != X2 ) )
=> ~ ( member_a @ Z4 @ ( graph_2957805489637798020ts_a_b @ t ) ) ) ) ).
% last_merge_alt
thf(fact_955_head__add__vert,axiom,
! [U: a] :
( ( pre_he5236287464308401016t_unit @ ( pre_add_vert_a_b @ t @ U ) )
= ( pre_he5236287464308401016t_unit @ t ) ) ).
% head_add_vert
thf(fact_956_merge__in__supergraph,axiom,
! [C4: pre_pr7278220950009878019t_unit,X2: a] :
( ( shorte3657265928840388360ph_a_b @ C4 @ t )
=> ( ( member_a @ X2 @ ( graph_2957805489637798020ts_a_b @ C4 ) )
=> ( member_a @ X2 @ ( graph_2957805489637798020ts_a_b @ t ) ) ) ) ).
% merge_in_supergraph
thf(fact_957_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 ) )
=> ~ ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Bs ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ As ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Xa2 ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ).
% no_back_arc_if_fwd_dstct
thf(fact_958_no__back__if__distinct__forward,axiom,
! [Xs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs )
=> ( ( distinct_a @ Xs )
=> ( iKKBZ_3684931046464919648ck_a_b @ t @ Xs ) ) ) ).
% no_back_if_distinct_forward
thf(fact_959_seq__conform__if__dstnct__fwd,axiom,
! [Xs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs )
=> ( ( distinct_a @ Xs )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ Xs ) ) ) ).
% seq_conform_if_dstnct_fwd
thf(fact_960_distinct__rev,axiom,
! [Xs: list_a] :
( ( distinct_a @ ( rev_a @ Xs ) )
= ( distinct_a @ Xs ) ) ).
% distinct_rev
thf(fact_961_distinct__rev,axiom,
! [Xs: list_b] :
( ( distinct_b @ ( rev_b @ Xs ) )
= ( distinct_b @ Xs ) ) ).
% distinct_rev
thf(fact_962_subgraph__no__last__merge__chain,axiom,
! [C4: pre_pr7278220950009878019t_unit] :
( ( shorte3657265928840388360ph_a_b @ C4 @ t )
=> ( graph_8150681439568091980in_a_b @ C4 ) ) ).
% subgraph_no_last_merge_chain
thf(fact_963_distinct__length__2__or__more,axiom,
! [A: a,B: a,Xs: list_a] :
( ( distinct_a @ ( cons_a @ A @ ( cons_a @ B @ Xs ) ) )
= ( ( A != B )
& ( distinct_a @ ( cons_a @ A @ Xs ) )
& ( distinct_a @ ( cons_a @ B @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_964_distinct__length__2__or__more,axiom,
! [A: b,B: b,Xs: list_b] :
( ( distinct_b @ ( cons_b @ A @ ( cons_b @ B @ Xs ) ) )
= ( ( A != B )
& ( distinct_b @ ( cons_b @ A @ Xs ) )
& ( distinct_b @ ( cons_b @ B @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_965_distinct_Osimps_I1_J,axiom,
distinct_a @ nil_a ).
% distinct.simps(1)
thf(fact_966_distinct_Osimps_I1_J,axiom,
distinct_b @ nil_b ).
% distinct.simps(1)
thf(fact_967_distinct__singleton,axiom,
! [X2: a] : ( distinct_a @ ( cons_a @ X2 @ nil_a ) ) ).
% distinct_singleton
thf(fact_968_distinct__singleton,axiom,
! [X2: b] : ( distinct_b @ ( cons_b @ X2 @ nil_b ) ) ).
% distinct_singleton
thf(fact_969_distinct_Osimps_I2_J,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( distinct_list_a @ ( cons_list_a @ X2 @ Xs ) )
= ( ~ ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
& ( distinct_list_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_970_distinct_Osimps_I2_J,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( distinct_set_a @ ( cons_set_a @ X2 @ Xs ) )
= ( ~ ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
& ( distinct_set_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_971_distinct_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( distinct_nat @ ( cons_nat @ X2 @ Xs ) )
= ( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
& ( distinct_nat @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_972_distinct_Osimps_I2_J,axiom,
! [X2: a,Xs: list_a] :
( ( distinct_a @ ( cons_a @ X2 @ Xs ) )
= ( ~ ( member_a @ X2 @ ( set_a2 @ Xs ) )
& ( distinct_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_973_distinct_Osimps_I2_J,axiom,
! [X2: b,Xs: list_b] :
( ( distinct_b @ ( cons_b @ X2 @ Xs ) )
= ( ~ ( member_b @ X2 @ ( set_b2 @ Xs ) )
& ( distinct_b @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_974_not__distinct__decomp,axiom,
! [Ws: list_a] :
( ~ ( distinct_a @ Ws )
=> ? [Xs2: list_a,Ys2: list_a,Zs2: list_a,Y3: a] :
( Ws
= ( append_a @ Xs2 @ ( append_a @ ( cons_a @ Y3 @ nil_a ) @ ( append_a @ Ys2 @ ( append_a @ ( cons_a @ Y3 @ nil_a ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_975_not__distinct__decomp,axiom,
! [Ws: list_b] :
( ~ ( distinct_b @ Ws )
=> ? [Xs2: list_b,Ys2: list_b,Zs2: list_b,Y3: b] :
( Ws
= ( append_b @ Xs2 @ ( append_b @ ( cons_b @ Y3 @ nil_b ) @ ( append_b @ Ys2 @ ( append_b @ ( cons_b @ Y3 @ nil_b ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_976_not__distinct__conv__prefix,axiom,
! [As: list_list_a] :
( ( ~ ( distinct_list_a @ As ) )
= ( ? [Xs4: list_list_a,Y4: list_a,Ys5: list_list_a] :
( ( member_list_a @ Y4 @ ( set_list_a2 @ Xs4 ) )
& ( distinct_list_a @ Xs4 )
& ( As
= ( append_list_a @ Xs4 @ ( cons_list_a @ Y4 @ Ys5 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_977_not__distinct__conv__prefix,axiom,
! [As: list_set_a] :
( ( ~ ( distinct_set_a @ As ) )
= ( ? [Xs4: list_set_a,Y4: set_a,Ys5: list_set_a] :
( ( member_set_a @ Y4 @ ( set_set_a2 @ Xs4 ) )
& ( distinct_set_a @ Xs4 )
& ( As
= ( append_set_a @ Xs4 @ ( cons_set_a @ Y4 @ Ys5 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_978_not__distinct__conv__prefix,axiom,
! [As: list_nat] :
( ( ~ ( distinct_nat @ As ) )
= ( ? [Xs4: list_nat,Y4: nat,Ys5: list_nat] :
( ( member_nat @ Y4 @ ( set_nat2 @ Xs4 ) )
& ( distinct_nat @ Xs4 )
& ( As
= ( append_nat @ Xs4 @ ( cons_nat @ Y4 @ Ys5 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_979_not__distinct__conv__prefix,axiom,
! [As: list_a] :
( ( ~ ( distinct_a @ As ) )
= ( ? [Xs4: list_a,Y4: a,Ys5: list_a] :
( ( member_a @ Y4 @ ( set_a2 @ Xs4 ) )
& ( distinct_a @ Xs4 )
& ( As
= ( append_a @ Xs4 @ ( cons_a @ Y4 @ Ys5 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_980_not__distinct__conv__prefix,axiom,
! [As: list_b] :
( ( ~ ( distinct_b @ As ) )
= ( ? [Xs4: list_b,Y4: b,Ys5: list_b] :
( ( member_b @ Y4 @ ( set_b2 @ Xs4 ) )
& ( distinct_b @ Xs4 )
& ( As
= ( append_b @ Xs4 @ ( cons_b @ Y4 @ Ys5 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_981_distinct__conv__nth,axiom,
( distinct_a
= ( ^ [Xs4: list_a] :
! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs4 ) )
=> ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_a @ Xs4 ) )
=> ( ( I != J2 )
=> ( ( nth_a @ Xs4 @ I )
!= ( nth_a @ Xs4 @ J2 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_982_distinct__conv__nth,axiom,
( distinct_b
= ( ^ [Xs4: list_b] :
! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_b @ Xs4 ) )
=> ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_b @ Xs4 ) )
=> ( ( I != J2 )
=> ( ( nth_b @ Xs4 @ I )
!= ( nth_b @ Xs4 @ J2 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_983_nth__eq__iff__index__eq,axiom,
! [Xs: list_a,I2: nat,J3: nat] :
( ( distinct_a @ Xs )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_a @ Xs ) )
=> ( ( ( nth_a @ Xs @ I2 )
= ( nth_a @ Xs @ J3 ) )
= ( I2 = J3 ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_984_nth__eq__iff__index__eq,axiom,
! [Xs: list_b,I2: nat,J3: nat] :
( ( distinct_b @ Xs )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_b @ Xs ) )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_b @ Xs ) )
=> ( ( ( nth_b @ Xs @ I2 )
= ( nth_b @ Xs @ J3 ) )
= ( I2 = J3 ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_985_distinct__Ex1,axiom,
! [Xs: list_list_a,X2: list_a] :
( ( distinct_list_a @ Xs )
=> ( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ? [X: nat] :
( ( ord_less_nat @ X @ ( size_s349497388124573686list_a @ Xs ) )
& ( ( nth_list_a @ Xs @ X )
= X2 )
& ! [Y: nat] :
( ( ( ord_less_nat @ Y @ ( size_s349497388124573686list_a @ Xs ) )
& ( ( nth_list_a @ Xs @ Y )
= X2 ) )
=> ( Y = X ) ) ) ) ) ).
% distinct_Ex1
thf(fact_986_distinct__Ex1,axiom,
! [Xs: list_set_a,X2: set_a] :
( ( distinct_set_a @ Xs )
=> ( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
=> ? [X: nat] :
( ( ord_less_nat @ X @ ( size_size_list_set_a @ Xs ) )
& ( ( nth_set_a @ Xs @ X )
= X2 )
& ! [Y: nat] :
( ( ( ord_less_nat @ Y @ ( size_size_list_set_a @ Xs ) )
& ( ( nth_set_a @ Xs @ Y )
= X2 ) )
=> ( Y = X ) ) ) ) ) ).
% distinct_Ex1
thf(fact_987_distinct__Ex1,axiom,
! [Xs: list_nat,X2: nat] :
( ( distinct_nat @ Xs )
=> ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ? [X: nat] :
( ( ord_less_nat @ X @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ X )
= X2 )
& ! [Y: nat] :
( ( ( ord_less_nat @ Y @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ Y )
= X2 ) )
=> ( Y = X ) ) ) ) ) ).
% distinct_Ex1
thf(fact_988_distinct__Ex1,axiom,
! [Xs: list_a,X2: a] :
( ( distinct_a @ Xs )
=> ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ? [X: nat] :
( ( ord_less_nat @ X @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ X )
= X2 )
& ! [Y: nat] :
( ( ( ord_less_nat @ Y @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ Y )
= X2 ) )
=> ( Y = X ) ) ) ) ) ).
% distinct_Ex1
thf(fact_989_distinct__Ex1,axiom,
! [Xs: list_b,X2: b] :
( ( distinct_b @ Xs )
=> ( ( member_b @ X2 @ ( set_b2 @ Xs ) )
=> ? [X: nat] :
( ( ord_less_nat @ X @ ( size_size_list_b @ Xs ) )
& ( ( nth_b @ Xs @ X )
= X2 )
& ! [Y: nat] :
( ( ( ord_less_nat @ Y @ ( size_size_list_b @ Xs ) )
& ( ( nth_b @ Xs @ Y )
= X2 ) )
=> ( Y = X ) ) ) ) ) ).
% distinct_Ex1
thf(fact_990_awalk__not__distinct__decomp,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( ~ ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) )
=> ? [Q3: list_b,R: list_b,S4: list_b] :
( ( P
= ( append_b @ Q3 @ ( append_b @ R @ S4 ) ) )
& ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ Q3 ) )
& ( ord_less_nat @ zero_zero_nat @ ( size_size_list_b @ R ) )
& ? [W2: a] :
( ( arc_pre_awalk_a_b @ t @ U @ Q3 @ W2 )
& ( arc_pre_awalk_a_b @ t @ W2 @ R @ W2 )
& ( arc_pre_awalk_a_b @ t @ W2 @ S4 @ V ) ) ) ) ) ).
% awalk_not_distinct_decomp
thf(fact_991_del__vert__add__vert,axiom,
! [U: a] :
( ~ ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( pre_del_vert_a_b @ ( pre_add_vert_a_b @ t @ U ) @ U )
= t ) ) ).
% del_vert_add_vert
thf(fact_992_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 ) )
=> ~ ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Bs ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ As ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Xa2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ).
% no_back_reach1_if_fwd_dstct
thf(fact_993_awalk__verts__non__Nil,axiom,
! [U: a,P: list_b] :
( ( arc_pr7493981781705774526ts_a_b @ t @ U @ P )
!= nil_a ) ).
% awalk_verts_non_Nil
thf(fact_994_awalk__verts__ne__eq,axiom,
! [P: list_b,U: a,V: a] :
( ( P != nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ P )
= ( arc_pr7493981781705774526ts_a_b @ t @ V @ P ) ) ) ).
% awalk_verts_ne_eq
thf(fact_995_head__del__vert,axiom,
! [U: a] :
( ( pre_he5236287464308401016t_unit @ ( pre_del_vert_a_b @ t @ U ) )
= ( pre_he5236287464308401016t_unit @ t ) ) ).
% head_del_vert
thf(fact_996_hd__in__awalk__verts_I1_J,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( member_a @ U @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) ) ) ).
% hd_in_awalk_verts(1)
thf(fact_997_distinct__verts__imp__distinct,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) )
=> ( distinct_b @ P ) ) ) ).
% distinct_verts_imp_distinct
thf(fact_998_reachable1__not__reverse,axiom,
! [X2: a,Y2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y2 @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ).
% reachable1_not_reverse
thf(fact_999_awhd__append,axiom,
! [U: a,P: list_b,Q: list_b] :
( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ ( append_b @ P @ Q ) ) )
= ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ Q ) ) @ P ) ) ) ).
% awhd_append
thf(fact_1000_awalk__imp__vwalk,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( vertex_vwalk_a_b @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) @ t ) ) ).
% awalk_imp_vwalk
thf(fact_1001_awalk__verts__arc2,axiom,
! [U: a,P: list_b,V: a,E: b] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( ( member_b @ E @ ( set_b2 @ P ) )
=> ( member_a @ ( pre_he5236287464308401016t_unit @ t @ E ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) ) ) ) ).
% awalk_verts_arc2
thf(fact_1002_awalk__verts__reachable__from,axiom,
! [U: a,P: list_b,V: a,W: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) )
=> ( reachable_a_b @ t @ U @ W ) ) ) ).
% awalk_verts_reachable_from
thf(fact_1003_awalk__verts__reachable__to,axiom,
! [U: a,P: list_b,V: a,W: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) )
=> ( reachable_a_b @ t @ W @ V ) ) ) ).
% awalk_verts_reachable_to
thf(fact_1004_awalk__decomp,axiom,
! [U: a,P: list_b,V: a,W: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) )
=> ? [Q3: list_b,R: list_b] :
( ( P
= ( append_b @ Q3 @ R ) )
& ( arc_pre_awalk_a_b @ t @ U @ Q3 @ W )
& ( arc_pre_awalk_a_b @ t @ W @ R @ V ) ) ) ) ).
% awalk_decomp
thf(fact_1005_rotate__awalkE,axiom,
! [U: a,P: list_b,W: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ U )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) )
=> ~ ! [Q3: list_b,R: list_b] :
( ( P
= ( append_b @ Q3 @ R ) )
=> ( ( arc_pre_awalk_a_b @ t @ W @ ( append_b @ R @ Q3 ) @ W )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ W @ ( append_b @ R @ Q3 ) ) )
!= ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) ) ) ) ) ) ).
% rotate_awalkE
thf(fact_1006_awalk__verts__append__distinct,axiom,
! [R2: a,P1: list_b,P22: list_b] :
( ? [X_1: a] : ( arc_pre_awalk_a_b @ t @ R2 @ ( append_b @ P1 @ P22 ) @ X_1 )
=> ( ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ R2 @ ( append_b @ P1 @ P22 ) ) )
=> ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ R2 @ P1 ) ) ) ) ).
% awalk_verts_append_distinct
thf(fact_1007_awalk__verts_Osimps_I1_J,axiom,
! [U: a] :
( ( arc_pr7493981781705774526ts_a_b @ t @ U @ nil_b )
= ( cons_a @ U @ nil_a ) ) ).
% awalk_verts.simps(1)
thf(fact_1008_reachable1__in__verts_I1_J,axiom,
! [U: a,V: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% reachable1_in_verts(1)
thf(fact_1009_reachable1__in__verts_I2_J,axiom,
! [U: a,V: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% reachable1_in_verts(2)
thf(fact_1010_awalk__verts__subset__if__p__sub,axiom,
! [U: a,P1: list_b,V: a,P22: list_b] :
( ( arc_pre_awalk_a_b @ t @ U @ P1 @ V )
=> ( ( arc_pre_awalk_a_b @ t @ U @ P22 @ V )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P1 ) @ ( set_b2 @ P22 ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P1 ) ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P22 ) ) ) ) ) ) ).
% awalk_verts_subset_if_p_sub
thf(fact_1011_reachable1__from__outside__dom,axiom,
! [X2: a,Y2: a,Ys: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( ~ ( member_a @ X2 @ ( set_a2 @ Ys ) )
=> ( ( member_a @ Y2 @ ( set_a2 @ Ys ) )
=> ? [X6: a,X: a] :
( ( member_a @ X @ ( set_a2 @ Ys ) )
& ~ ( member_a @ X6 @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X6 @ X ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ).
% reachable1_from_outside_dom
thf(fact_1012_reachable__reachable1__trans,axiom,
! [U: a,V: a,W: a] :
( ( reachable_a_b @ t @ U @ V )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ V @ W ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ W ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% reachable_reachable1_trans
thf(fact_1013_reachable1__reachable__trans,axiom,
! [U: a,V: a,W: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( ( reachable_a_b @ t @ V @ W )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ W ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% reachable1_reachable_trans
thf(fact_1014_awalk__del__vert,axiom,
! [U: a,P: list_b,V: a,X2: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( ~ ( member_a @ X2 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) )
=> ( arc_pre_awalk_a_b @ ( pre_del_vert_a_b @ t @ X2 ) @ U @ P @ V ) ) ) ).
% awalk_del_vert
thf(fact_1015_dominated__notin__awalk,axiom,
! [U: a,V: a,R2: a,P: list_b] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( arcs_ends_a_b @ t ) )
=> ( ( arc_pre_awalk_a_b @ t @ R2 @ P @ U )
=> ~ ( member_a @ V @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ R2 @ P ) ) ) ) ) ).
% dominated_notin_awalk
thf(fact_1016_awalk__verts__dom__if__uneq,axiom,
! [U: a,V: a,P: list_b] :
( ( U != V )
=> ( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ? [X: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ V ) @ ( arcs_ends_a_b @ t ) )
& ( member_a @ X @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) ) ) ) ) ).
% awalk_verts_dom_if_uneq
thf(fact_1017_awalk__verts__append3,axiom,
! [U: a,P: list_b,E: b,Q: list_b,R2: a,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ ( append_b @ P @ ( cons_b @ E @ Q ) ) @ R2 )
=> ( ( arc_pre_awalk_a_b @ t @ V @ Q @ R2 )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ ( append_b @ P @ ( cons_b @ E @ Q ) ) )
= ( append_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) @ ( arc_pr7493981781705774526ts_a_b @ t @ V @ Q ) ) ) ) ) ).
% awalk_verts_append3
thf(fact_1018_reachable1__awalkI,axiom,
! [V: a,P: list_b,W: a] :
( ( arc_pre_awalk_a_b @ t @ V @ P @ W )
=> ( ( P != nil_b )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ V @ W ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% reachable1_awalkI
thf(fact_1019_reachable1__awalk,axiom,
! [U: a,V: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
= ( ? [P3: list_b] :
( ( arc_pre_awalk_a_b @ t @ U @ P3 @ V )
& ( P3 != nil_b ) ) ) ) ).
% reachable1_awalk
thf(fact_1020_awalk__cyc__decompE_H,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( ~ ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) )
=> ~ ! [Q3: list_b,R: list_b,S4: list_b] :
( ( P
= ( append_b @ Q3 @ ( append_b @ R @ S4 ) ) )
=> ( ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ Q3 ) )
=> ( ? [W2: a] :
( ( arc_pre_awalk_a_b @ t @ U @ Q3 @ W2 )
& ( arc_pre_awalk_a_b @ t @ W2 @ R @ W2 )
& ( arc_pre_awalk_a_b @ t @ W2 @ S4 @ V ) )
=> ~ ( arc_wf_closed_w_a_b @ t @ R ) ) ) ) ) ) ).
% awalk_cyc_decompE'
thf(fact_1021_reachable1__append__old__if__arc,axiom,
! [Xs: list_a,Ys: list_a,Z2: a,Y2: a] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Xa2 ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ( ~ ( member_a @ Z2 @ ( set_a2 @ Xs ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs )
=> ( ( member_a @ Y2 @ ( set_a2 @ ( append_a @ Xs @ Ys ) ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z2 @ Y2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z2 @ X ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ) ).
% reachable1_append_old_if_arc
thf(fact_1022_hd__reachable1__from__outside_H,axiom,
! [X2: a,Y2: a,Ys: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ Ys )
=> ( ~ ( member_a @ X2 @ ( set_a2 @ Ys ) )
=> ( ( member_a @ Y2 @ ( set_a2 @ Ys ) )
=> ( ? [X: a] : ( member_a @ X @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ ( hd_a @ Ys ) ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ).
% hd_reachable1_from_outside'
thf(fact_1023_awhd__of__awalk,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) )
= U ) ) ).
% awhd_of_awalk
thf(fact_1024_reachable__neq__reachable1,axiom,
! [V: a,W: a] :
( ( reachable_a_b @ t @ V @ W )
=> ( ( V != W )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ V @ W ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% reachable_neq_reachable1
thf(fact_1025_reachable1__reachable,axiom,
! [V: a,W: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ V @ W ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( reachable_a_b @ t @ V @ W ) ) ).
% reachable1_reachable
thf(fact_1026_is__awalk__cyc__decomp_Oelims_I3_J,axiom,
! [X2: list_b,Xa3: produc8766925488660474953list_b] :
( ~ ( arc_wf7293661141070756729mp_a_b @ t @ X2 @ Xa3 )
=> ~ ! [Q3: list_b,R: list_b,S4: list_b] :
( ( Xa3
= ( produc305491333965050169list_b @ Q3 @ ( produc1564554178308465111list_b @ R @ S4 ) ) )
=> ( ( X2
= ( append_b @ Q3 @ ( append_b @ R @ S4 ) ) )
& ? [U6: a,V4: a] :
( ( arc_pre_awalk_a_b @ t @ U6 @ Q3 @ V4 )
& ( arc_pre_awalk_a_b @ t @ V4 @ R @ V4 )
& ? [X_1: a] : ( arc_pre_awalk_a_b @ t @ V4 @ S4 @ X_1 ) )
& ( ord_less_nat @ zero_zero_nat @ ( size_size_list_b @ R ) )
& ? [U6: a] : ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U6 @ Q3 ) ) ) ) ) ).
% is_awalk_cyc_decomp.elims(3)
thf(fact_1027_is__awalk__cyc__decomp_Oelims_I2_J,axiom,
! [X2: list_b,Xa3: produc8766925488660474953list_b] :
( ( arc_wf7293661141070756729mp_a_b @ t @ X2 @ Xa3 )
=> ~ ! [Q3: list_b,R: list_b,S4: list_b] :
( ( Xa3
= ( produc305491333965050169list_b @ Q3 @ ( produc1564554178308465111list_b @ R @ S4 ) ) )
=> ~ ( ( X2
= ( append_b @ Q3 @ ( append_b @ R @ S4 ) ) )
& ? [U5: a,V3: a] :
( ( arc_pre_awalk_a_b @ t @ U5 @ Q3 @ V3 )
& ( arc_pre_awalk_a_b @ t @ V3 @ R @ V3 )
& ? [X_12: a] : ( arc_pre_awalk_a_b @ t @ V3 @ S4 @ X_12 ) )
& ( ord_less_nat @ zero_zero_nat @ ( size_size_list_b @ R ) )
& ? [U5: a] : ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U5 @ Q3 ) ) ) ) ) ).
% is_awalk_cyc_decomp.elims(2)
thf(fact_1028_is__awalk__cyc__decomp_Oelims_I1_J,axiom,
! [X2: list_b,Xa3: produc8766925488660474953list_b,Y2: $o] :
( ( ( arc_wf7293661141070756729mp_a_b @ t @ X2 @ Xa3 )
= Y2 )
=> ~ ! [Q3: list_b,R: list_b,S4: list_b] :
( ( Xa3
= ( produc305491333965050169list_b @ Q3 @ ( produc1564554178308465111list_b @ R @ S4 ) ) )
=> ( Y2
= ( ~ ( ( X2
= ( append_b @ Q3 @ ( append_b @ R @ S4 ) ) )
& ? [U4: a,V5: a] :
( ( arc_pre_awalk_a_b @ t @ U4 @ Q3 @ V5 )
& ( arc_pre_awalk_a_b @ t @ V5 @ R @ V5 )
& ? [X5: a] : ( arc_pre_awalk_a_b @ t @ V5 @ S4 @ X5 ) )
& ( ord_less_nat @ zero_zero_nat @ ( size_size_list_b @ R ) )
& ? [U4: a] : ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U4 @ Q3 ) ) ) ) ) ) ) ).
% is_awalk_cyc_decomp.elims(1)
thf(fact_1029_is__awalk__cyc__decomp_Ocases,axiom,
! [X2: produc272433356463431595list_b] :
~ ! [P6: list_b,Q3: list_b,R: list_b,S4: list_b] :
( X2
!= ( produc7106373121284446491list_b @ P6 @ ( produc305491333965050169list_b @ Q3 @ ( produc1564554178308465111list_b @ R @ S4 ) ) ) ) ).
% is_awalk_cyc_decomp.cases
thf(fact_1030_is__awalk__cyc__decomp_Osimps,axiom,
! [P: list_b,Q: list_b,R2: list_b,S: list_b] :
( ( arc_wf7293661141070756729mp_a_b @ t @ P @ ( produc305491333965050169list_b @ Q @ ( produc1564554178308465111list_b @ R2 @ S ) ) )
= ( ( P
= ( append_b @ Q @ ( append_b @ R2 @ S ) ) )
& ? [U4: a,V5: a,W3: a] :
( ( arc_pre_awalk_a_b @ t @ U4 @ Q @ V5 )
& ( arc_pre_awalk_a_b @ t @ V5 @ R2 @ V5 )
& ( arc_pre_awalk_a_b @ t @ V5 @ S @ W3 ) )
& ( ord_less_nat @ zero_zero_nat @ ( size_size_list_b @ R2 ) )
& ? [U4: a] : ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U4 @ Q ) ) ) ) ).
% is_awalk_cyc_decomp.simps
thf(fact_1031_awalk__cyc__decompE,axiom,
! [P: list_b,Q: list_b,R2: list_b,S: list_b,U: a,V: a] :
( ( ( arc_wf4740610840468824943mp_a_b @ t @ P )
= ( produc305491333965050169list_b @ Q @ ( produc1564554178308465111list_b @ R2 @ S ) ) )
=> ( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( ~ ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) )
=> ~ ( ( P
= ( append_b @ Q @ ( append_b @ R2 @ S ) ) )
=> ( ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ Q ) )
=> ( ? [W2: a] :
( ( arc_pre_awalk_a_b @ t @ U @ Q @ W2 )
& ( arc_pre_awalk_a_b @ t @ W2 @ R2 @ W2 )
& ( arc_pre_awalk_a_b @ t @ W2 @ S @ V ) )
=> ~ ( arc_wf_closed_w_a_b @ t @ R2 ) ) ) ) ) ) ) ).
% awalk_cyc_decompE
thf(fact_1032_awalk__to__apath__induct,axiom,
! [U: a,P: list_b,V: a,P4: list_b > $o] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( ! [P6: list_b] :
( ( arc_pre_awalk_a_b @ t @ U @ P6 @ V )
=> ( ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P6 ) )
=> ( P4 @ P6 ) ) )
=> ( ! [P6: list_b,Q3: list_b,R: list_b,S4: list_b] :
( ( arc_pre_awalk_a_b @ t @ U @ P6 @ V )
=> ( ( ( arc_wf4740610840468824943mp_a_b @ t @ P6 )
= ( produc305491333965050169list_b @ Q3 @ ( produc1564554178308465111list_b @ R @ S4 ) ) )
=> ( ~ ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P6 ) )
=> ( ( P4 @ ( append_b @ Q3 @ S4 ) )
=> ( P4 @ P6 ) ) ) ) )
=> ( P4 @ P ) ) ) ) ).
% awalk_to_apath_induct
thf(fact_1033_leaf__not__mem__awalk,axiom,
! [X2: a,U: a,P: list_b,V: a] :
( ( shorte1213025427933718126af_a_b @ t @ X2 )
=> ( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( ( V != X2 )
=> ~ ( member_a @ X2 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) ) ) ) ) ).
% leaf_not_mem_awalk
thf(fact_1034_awalk__cyc__decomp__has__prop,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( ~ ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) )
=> ( arc_wf7293661141070756729mp_a_b @ t @ P @ ( arc_wf4740610840468824943mp_a_b @ t @ P ) ) ) ) ).
% awalk_cyc_decomp_has_prop
thf(fact_1035_step__awalk__to__apath,axiom,
! [U: a,P: list_b,V: a,Q: list_b,R2: list_b,S: list_b] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( ( ( arc_wf4740610840468824943mp_a_b @ t @ P )
= ( produc305491333965050169list_b @ Q @ ( produc1564554178308465111list_b @ R2 @ S ) ) )
=> ( ~ ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) )
=> ( ( arc_wf446166946845163101th_a_b @ t @ P )
= ( arc_wf446166946845163101th_a_b @ t @ ( append_b @ Q @ S ) ) ) ) ) ) ).
% step_awalk_to_apath
thf(fact_1036_awalk__to__apath__verts__subset,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ ( arc_wf446166946845163101th_a_b @ t @ P ) ) ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) ) ) ).
% awalk_to_apath_verts_subset
thf(fact_1037_awalk__to__apath__subset,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( ord_less_eq_set_b @ ( set_b2 @ ( arc_wf446166946845163101th_a_b @ t @ P ) ) @ ( set_b2 @ P ) ) ) ).
% awalk_to_apath_subset
thf(fact_1038_not__distinct__if__head__eq__tail,axiom,
! [P: b,U: a,E: b,R2: a,Ps2: list_b,P22: list_b,V: a] :
( ( ( pre_ta4931606617599662728t_unit @ t @ P )
= U )
=> ( ( ( pre_he5236287464308401016t_unit @ t @ E )
= U )
=> ( ( arc_pre_awalk_a_b @ t @ R2 @ ( append_b @ Ps2 @ ( append_b @ ( cons_b @ P @ nil_b ) @ ( cons_b @ E @ P22 ) ) ) @ V )
=> ~ ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ R2 @ ( append_b @ Ps2 @ ( append_b @ ( cons_b @ P @ nil_b ) @ ( cons_b @ E @ P22 ) ) ) ) ) ) ) ) ).
% not_distinct_if_head_eq_tail
thf(fact_1039_rotate__trailE_H,axiom,
! [U: a,P: list_b,W: a] :
( ( arc_pre_trail_a_b @ t @ U @ P @ U )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) )
=> ~ ! [Q3: list_b] :
( ( arc_pre_trail_a_b @ t @ W @ Q3 @ W )
=> ( ( ( set_b2 @ Q3 )
= ( set_b2 @ P ) )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ W @ Q3 ) )
!= ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) ) ) ) ) ) ).
% rotate_trailE'
thf(fact_1040_tail__del__vert,axiom,
! [U: a] :
( ( pre_ta4931606617599662728t_unit @ ( pre_del_vert_a_b @ t @ U ) )
= ( pre_ta4931606617599662728t_unit @ t ) ) ).
% tail_del_vert
thf(fact_1041_tail__add__vert,axiom,
! [U: a] :
( ( pre_ta4931606617599662728t_unit @ ( pre_add_vert_a_b @ t @ U ) )
= ( pre_ta4931606617599662728t_unit @ t ) ) ).
% tail_add_vert
thf(fact_1042_trail__Nil__iff,axiom,
! [U: a,V: a] :
( ( arc_pre_trail_a_b @ t @ U @ nil_b @ V )
= ( ( U = V )
& ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% trail_Nil_iff
thf(fact_1043_trail__def,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_trail_a_b @ t @ U @ P @ V )
= ( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
& ( distinct_b @ P ) ) ) ).
% trail_def
thf(fact_1044_awalk__verts__arc1,axiom,
! [E: b,P: list_b,U: a] :
( ( member_b @ E @ ( set_b2 @ P ) )
=> ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ E ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) ) ) ).
% awalk_verts_arc1
thf(fact_1045_rotate__trailE,axiom,
! [U: a,P: list_b,W: a] :
( ( arc_pre_trail_a_b @ t @ U @ P @ U )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) )
=> ~ ! [Q3: list_b,R: list_b] :
( ( P
= ( append_b @ Q3 @ R ) )
=> ( ( arc_pre_trail_a_b @ t @ W @ ( append_b @ R @ Q3 ) @ W )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ W @ ( append_b @ R @ Q3 ) ) )
!= ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) ) ) ) ) ) ).
% rotate_trailE
thf(fact_1046_awalk__verts__arc1__app,axiom,
! [E: b,R2: a,P1: list_b,P22: list_b] : ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ E ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ R2 @ ( append_b @ P1 @ ( cons_b @ E @ P22 ) ) ) ) ) ).
% awalk_verts_arc1_app
thf(fact_1047_awalk__verts_Osimps_I2_J,axiom,
! [U: a,E: b,Es2: list_b] :
( ( arc_pr7493981781705774526ts_a_b @ t @ U @ ( cons_b @ E @ Es2 ) )
= ( cons_a @ ( pre_ta4931606617599662728t_unit @ t @ E ) @ ( arc_pr7493981781705774526ts_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E ) @ Es2 ) ) ) ).
% awalk_verts.simps(2)
thf(fact_1048_awalk__vertex__props,axiom,
! [U: a,P: list_b,V: a,P4: a > $o,Q2: a > $o] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( ( P != nil_b )
=> ( ! [W2: a] :
( ( member_a @ W2 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) )
=> ( ( P4 @ W2 )
| ( Q2 @ W2 ) ) )
=> ( ( P4 @ U )
=> ( ( Q2 @ V )
=> ? [X: b] :
( ( member_b @ X @ ( set_b2 @ P ) )
& ( P4 @ ( pre_ta4931606617599662728t_unit @ t @ X ) )
& ( Q2 @ ( pre_he5236287464308401016t_unit @ t @ X ) ) ) ) ) ) ) ) ).
% awalk_vertex_props
thf(fact_1049_arc__balancedI__trail,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_trail_a_b @ t @ U @ P @ V )
=> ( pre_ar5931435604406180204ed_a_b @ t @ U @ ( set_b2 @ P ) @ V ) ) ).
% arc_balancedI_trail
thf(fact_1050_awalk__induce,axiom,
! [U: a,P: list_b,V: a,S3: set_a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) @ S3 )
=> ( arc_pre_awalk_a_b @ ( digrap7873285959652527175ph_a_b @ t @ S3 ) @ U @ P @ V ) ) ) ).
% awalk_induce
thf(fact_1051_cycle__conv,axiom,
! [P: list_b] :
( ( arc_pre_cycle_a_b @ t @ P )
= ( ? [U4: a] :
( ( arc_pre_awalk_a_b @ t @ U4 @ P @ U4 )
& ( distinct_a @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U4 @ P ) ) )
& ( distinct_b @ P )
& ( P != nil_b ) ) ) ) ).
% cycle_conv
thf(fact_1052_trail__Cons__iff,axiom,
! [U: a,E: b,Es2: list_b,W: a] :
( ( arc_pre_trail_a_b @ t @ U @ ( cons_b @ E @ Es2 ) @ W )
= ( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( U
= ( pre_ta4931606617599662728t_unit @ t @ E ) )
& ~ ( member_b @ E @ ( set_b2 @ Es2 ) )
& ( arc_pre_trail_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E ) @ Es2 @ W ) ) ) ).
% trail_Cons_iff
thf(fact_1053_two__in__arcs__contr,axiom,
! [E1: b,E22: b] :
( ( member_b @ E1 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( member_b @ E22 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( E1 != E22 )
=> ( ( pre_he5236287464308401016t_unit @ t @ E1 )
!= ( pre_he5236287464308401016t_unit @ t @ E22 ) ) ) ) ) ).
% two_in_arcs_contr
thf(fact_1054_awalk__verts__induce,axiom,
! [S3: set_a] :
( ( arc_pr7493981781705774526ts_a_b @ ( digrap7873285959652527175ph_a_b @ t @ S3 ) )
= ( arc_pr7493981781705774526ts_a_b @ t ) ) ).
% awalk_verts_induce
thf(fact_1055_arcs__add__vert,axiom,
! [U: a] :
( ( pre_ar1395965042833527383t_unit @ ( pre_add_vert_a_b @ t @ U ) )
= ( pre_ar1395965042833527383t_unit @ t ) ) ).
% arcs_add_vert
thf(fact_1056_head__in__verts,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( member_a @ ( pre_he5236287464308401016t_unit @ t @ E ) @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% head_in_verts
thf(fact_1057_tail__in__verts,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ E ) @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% tail_in_verts
thf(fact_1058_loopfree_Ono__loops,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( pre_ta4931606617599662728t_unit @ t @ E )
!= ( pre_he5236287464308401016t_unit @ t @ E ) ) ) ).
% loopfree.no_loops
thf(fact_1059_nomulti_Ono__multi__alt,axiom,
! [E1: b,E22: b] :
( ( member_b @ E1 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( member_b @ E22 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( E1 != E22 )
=> ( ( ( pre_he5236287464308401016t_unit @ t @ E1 )
!= ( pre_he5236287464308401016t_unit @ t @ E22 ) )
| ( ( pre_ta4931606617599662728t_unit @ t @ E1 )
!= ( pre_ta4931606617599662728t_unit @ t @ E22 ) ) ) ) ) ) ).
% nomulti.no_multi_alt
thf(fact_1060_All__arcs__in__path,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ? [P6: list_b,U5: a,V3: a] :
( ( arc_pre_awalk_a_b @ t @ U5 @ P6 @ V3 )
& ( member_b @ E @ ( set_b2 @ P6 ) ) ) ) ).
% All_arcs_in_path
thf(fact_1061_reachable__induce__ss,axiom,
! [S3: set_a,U: a,V: a,T3: set_a] :
( ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ t @ S3 ) @ U @ V )
=> ( ( ord_less_eq_set_a @ S3 @ T3 )
=> ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ t @ T3 ) @ U @ V ) ) ) ).
% reachable_induce_ss
thf(fact_1062_reachable__induce__subgraphD,axiom,
! [S3: set_a,U: a,V: a] :
( ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ t @ S3 ) @ U @ V )
=> ( ( ord_less_eq_set_a @ S3 @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( reachable_a_b @ t @ U @ V ) ) ) ).
% reachable_induce_subgraphD
thf(fact_1063_dominates__induce__ss,axiom,
! [U: a,V: a,S3: set_a,T3: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( arcs_ends_a_b @ ( digrap7873285959652527175ph_a_b @ t @ S3 ) ) )
=> ( ( ord_less_eq_set_a @ S3 @ T3 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( arcs_ends_a_b @ ( digrap7873285959652527175ph_a_b @ t @ T3 ) ) ) ) ) ).
% dominates_induce_ss
thf(fact_1064_in__arcs__imp__in__arcs__ends,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( pre_ta4931606617599662728t_unit @ t @ E ) @ ( pre_he5236287464308401016t_unit @ t @ E ) ) @ ( arcs_ends_a_b @ t ) ) ) ).
% in_arcs_imp_in_arcs_ends
thf(fact_1065_unique__arc_I1_J,axiom,
! [U: a,V: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( arcs_ends_a_b @ t ) )
=> ? [X: b] :
( ( member_b @ X @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( ( pre_ta4931606617599662728t_unit @ t @ X )
= U )
& ( ( pre_he5236287464308401016t_unit @ t @ X )
= V )
& ! [Y: b] :
( ( ( member_b @ Y @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( ( pre_ta4931606617599662728t_unit @ t @ Y )
= U )
& ( ( pre_he5236287464308401016t_unit @ t @ Y )
= V ) )
=> ( Y = X ) ) ) ) ).
% unique_arc(1)
thf(fact_1066_unique__arc_I2_J,axiom,
! [U: a,V: a] :
( ~ ? [E2: b] :
( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( ( pre_ta4931606617599662728t_unit @ t @ E2 )
= U )
& ( ( pre_he5236287464308401016t_unit @ t @ E2 )
= V ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( arcs_ends_a_b @ t ) ) ) ).
% unique_arc(2)
thf(fact_1067_awalk__Cons__iff,axiom,
! [U: a,E: b,Es2: list_b,W: a] :
( ( arc_pre_awalk_a_b @ t @ U @ ( cons_b @ E @ Es2 ) @ W )
= ( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( U
= ( pre_ta4931606617599662728t_unit @ t @ E ) )
& ( arc_pre_awalk_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E ) @ Es2 @ W ) ) ) ).
% awalk_Cons_iff
thf(fact_1068_arcE,axiom,
! [E: b,U: a,V: a] :
( ( wf_arc_a_b @ t @ E @ ( product_Pair_a_a @ U @ V ) )
=> ~ ( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( ( pre_ta4931606617599662728t_unit @ t @ E )
= U )
=> ( ( pre_he5236287464308401016t_unit @ t @ E )
!= V ) ) ) ) ).
% arcE
thf(fact_1069_distinct__tl__verts__imp__distinct,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( ( distinct_a @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) )
=> ( distinct_b @ P ) ) ) ).
% distinct_tl_verts_imp_distinct
thf(fact_1070_arc__implies__awalk,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( arc_pre_awalk_a_b @ t @ ( pre_ta4931606617599662728t_unit @ t @ E ) @ ( cons_b @ E @ nil_b ) @ ( pre_he5236287464308401016t_unit @ t @ E ) ) ) ).
% arc_implies_awalk
thf(fact_1071_awalk__verts__in__verts,axiom,
! [U: a,P: list_b,V: a] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P ) @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( member_a @ V @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) )
=> ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ) ).
% awalk_verts_in_verts
thf(fact_1072_cycle__altdef,axiom,
! [P: list_b] :
( ( arc_pre_cycle_a_b @ t @ P )
= ( ( arc_wf_closed_w_a_b @ t @ P )
& ? [U4: a] : ( distinct_a @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U4 @ P ) ) ) ) ) ).
% cycle_altdef
thf(fact_1073_awhd__in__verts,axiom,
! [U: a,P: list_b] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P ) @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( member_a @ ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% awhd_in_verts
thf(fact_1074_cycle__def,axiom,
! [P: list_b] :
( ( arc_pre_cycle_a_b @ t @ P )
= ( ? [U4: a] :
( ( arc_pre_awalk_a_b @ t @ U4 @ P @ U4 )
& ( distinct_a @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U4 @ P ) ) )
& ( P != nil_b ) ) ) ) ).
% cycle_def
thf(fact_1075_tl__append2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_1076_tl__append2,axiom,
! [Xs: list_b,Ys: list_b] :
( ( Xs != nil_b )
=> ( ( tl_b @ ( append_b @ Xs @ Ys ) )
= ( append_b @ ( tl_b @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_1077_list_Ocollapse,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_1078_list_Ocollapse,axiom,
! [List: list_b] :
( ( List != nil_b )
=> ( ( cons_b @ ( hd_b @ List ) @ ( tl_b @ List ) )
= List ) ) ).
% list.collapse
thf(fact_1079_hd__Cons__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs ) @ ( tl_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_1080_hd__Cons__tl,axiom,
! [Xs: list_b] :
( ( Xs != nil_b )
=> ( ( cons_b @ ( hd_b @ Xs ) @ ( tl_b @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_1081_length__tl,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( tl_a @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) ).
% length_tl
thf(fact_1082_length__tl,axiom,
! [Xs: list_b] :
( ( size_size_list_b @ ( tl_b @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_b @ Xs ) @ one_one_nat ) ) ).
% length_tl
thf(fact_1083_tl__append__if,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( tl_a @ Ys ) ) )
& ( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_1084_tl__append__if,axiom,
! [Xs: list_b,Ys: list_b] :
( ( ( Xs = nil_b )
=> ( ( tl_b @ ( append_b @ Xs @ Ys ) )
= ( tl_b @ Ys ) ) )
& ( ( Xs != nil_b )
=> ( ( tl_b @ ( append_b @ Xs @ Ys ) )
= ( append_b @ ( tl_b @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_1085_list_Osel_I3_J,axiom,
! [X21: a,X22: list_a] :
( ( tl_a @ ( cons_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_1086_list_Osel_I3_J,axiom,
! [X21: b,X22: list_b] :
( ( tl_b @ ( cons_b @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_1087_list_Osel_I2_J,axiom,
( ( tl_a @ nil_a )
= nil_a ) ).
% list.sel(2)
thf(fact_1088_list_Osel_I2_J,axiom,
( ( tl_b @ nil_b )
= nil_b ) ).
% list.sel(2)
thf(fact_1089_list_Oset__sel_I2_J,axiom,
! [A: list_list_a,X2: list_a] :
( ( A != nil_list_a )
=> ( ( member_list_a @ X2 @ ( set_list_a2 @ ( tl_list_a @ A ) ) )
=> ( member_list_a @ X2 @ ( set_list_a2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_1090_list_Oset__sel_I2_J,axiom,
! [A: list_set_a,X2: set_a] :
( ( A != nil_set_a )
=> ( ( member_set_a @ X2 @ ( set_set_a2 @ ( tl_set_a @ A ) ) )
=> ( member_set_a @ X2 @ ( set_set_a2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_1091_list_Oset__sel_I2_J,axiom,
! [A: list_nat,X2: nat] :
( ( A != nil_nat )
=> ( ( member_nat @ X2 @ ( set_nat2 @ ( tl_nat @ A ) ) )
=> ( member_nat @ X2 @ ( set_nat2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_1092_list_Oset__sel_I2_J,axiom,
! [A: list_a,X2: a] :
( ( A != nil_a )
=> ( ( member_a @ X2 @ ( set_a2 @ ( tl_a @ A ) ) )
=> ( member_a @ X2 @ ( set_a2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_1093_list_Oset__sel_I2_J,axiom,
! [A: list_b,X2: b] :
( ( A != nil_b )
=> ( ( member_b @ X2 @ ( set_b2 @ ( tl_b @ A ) ) )
=> ( member_b @ X2 @ ( set_b2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_1094_Nil__tl,axiom,
! [Xs: list_a] :
( ( nil_a
= ( tl_a @ Xs ) )
= ( ( Xs = nil_a )
| ? [X3: a] :
( Xs
= ( cons_a @ X3 @ nil_a ) ) ) ) ).
% Nil_tl
thf(fact_1095_Nil__tl,axiom,
! [Xs: list_b] :
( ( nil_b
= ( tl_b @ Xs ) )
= ( ( Xs = nil_b )
| ? [X3: b] :
( Xs
= ( cons_b @ X3 @ nil_b ) ) ) ) ).
% Nil_tl
thf(fact_1096_tl__Nil,axiom,
! [Xs: list_a] :
( ( ( tl_a @ Xs )
= nil_a )
= ( ( Xs = nil_a )
| ? [X3: a] :
( Xs
= ( cons_a @ X3 @ nil_a ) ) ) ) ).
% tl_Nil
thf(fact_1097_tl__Nil,axiom,
! [Xs: list_b] :
( ( ( tl_b @ Xs )
= nil_b )
= ( ( Xs = nil_b )
| ? [X3: b] :
( Xs
= ( cons_b @ X3 @ nil_b ) ) ) ) ).
% tl_Nil
thf(fact_1098_distinct__tl,axiom,
! [Xs: list_a] :
( ( distinct_a @ Xs )
=> ( distinct_a @ ( tl_a @ Xs ) ) ) ).
% distinct_tl
thf(fact_1099_distinct__tl,axiom,
! [Xs: list_b] :
( ( distinct_b @ Xs )
=> ( distinct_b @ ( tl_b @ Xs ) ) ) ).
% distinct_tl
thf(fact_1100_list_Oexpand,axiom,
! [List: list_a,List2: list_a] :
( ( ( List = nil_a )
= ( List2 = nil_a ) )
=> ( ( ( List != nil_a )
=> ( ( List2 != nil_a )
=> ( ( ( hd_a @ List )
= ( hd_a @ List2 ) )
& ( ( tl_a @ List )
= ( tl_a @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_1101_list_Oexpand,axiom,
! [List: list_b,List2: list_b] :
( ( ( List = nil_b )
= ( List2 = nil_b ) )
=> ( ( ( List != nil_b )
=> ( ( List2 != nil_b )
=> ( ( ( hd_b @ List )
= ( hd_b @ List2 ) )
& ( ( tl_b @ List )
= ( tl_b @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_1102_list_Oexhaust__sel,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( List
= ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_1103_list_Oexhaust__sel,axiom,
! [List: list_b] :
( ( List != nil_b )
=> ( List
= ( cons_b @ ( hd_b @ List ) @ ( tl_b @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_1104_euler__trail__def,axiom,
! [U: a,P: list_b,V: a] :
( ( pre_euler_trail_a_b @ t @ U @ P @ V )
= ( ( arc_pre_trail_a_b @ t @ U @ P @ V )
& ( ( set_b2 @ P )
= ( pre_ar1395965042833527383t_unit @ t ) )
& ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) )
= ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% euler_trail_def
thf(fact_1105_awalk__induct__raw,axiom,
! [U: a,P: list_b,V: a,P4: a > list_b > a > $o] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( ! [W1: a] :
( ( member_a @ W1 @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( P4 @ W1 @ nil_b @ W1 ) )
=> ( ! [W1: a,W22: a,E2: b,Es: list_b] :
( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( ( arc_to_ends_a_b @ t @ E2 )
= ( product_Pair_a_a @ W1 @ W22 ) )
=> ( ( P4 @ W22 @ Es @ V )
=> ( P4 @ W1 @ ( cons_b @ E2 @ Es ) @ V ) ) ) )
=> ( P4 @ U @ P @ V ) ) ) ) ).
% awalk_induct_raw
thf(fact_1106_trail__connected,axiom,
! [U: a,P: list_b,V: a] :
( ( digrap8783888973171253482ed_a_b @ t )
=> ( ( arc_pre_trail_a_b @ t @ U @ P @ V )
=> ( ( ( set_b2 @ P )
!= ( pre_ar1395965042833527383t_unit @ t ) )
=> ? [E2: b] :
( ( member_b @ E2 @ ( minus_minus_set_b @ ( pre_ar1395965042833527383t_unit @ t ) @ ( set_b2 @ P ) ) )
& ( ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ E2 ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) )
| ( member_a @ ( pre_he5236287464308401016t_unit @ t @ E2 ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) ) ) ) ) ) ) ).
% trail_connected
thf(fact_1107_connected,axiom,
digrap8783888973171253482ed_a_b @ t ).
% connected
thf(fact_1108_nomulti_Ono__multi__arcs,axiom,
! [E1: b,E22: b] :
( ( member_b @ E1 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( member_b @ E22 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( ( arc_to_ends_a_b @ t @ E1 )
= ( arc_to_ends_a_b @ t @ E22 ) )
=> ( E1 = E22 ) ) ) ) ).
% nomulti.no_multi_arcs
thf(fact_1109_ends__del__vert,axiom,
! [U: a] :
( ( arc_to_ends_a_b @ ( pre_del_vert_a_b @ t @ U ) )
= ( arc_to_ends_a_b @ t ) ) ).
% ends_del_vert
thf(fact_1110_euler__imp__connected,axiom,
! [U: a,P: list_b,V: a] :
( ( pre_euler_trail_a_b @ t @ U @ P @ V )
=> ( digrap8783888973171253482ed_a_b @ t ) ) ).
% euler_imp_connected
thf(fact_1111_dominatesI,axiom,
! [A: b,U: a,V: a] :
( ( ( arc_to_ends_a_b @ t @ A )
= ( product_Pair_a_a @ U @ V ) )
=> ( ( member_b @ A @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( arcs_ends_a_b @ t ) ) ) ) ).
% dominatesI
thf(fact_1112_awalk__ConsI,axiom,
! [V: a,Es2: list_b,W: a,E: b,U: a] :
( ( arc_pre_awalk_a_b @ t @ V @ Es2 @ W )
=> ( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( ( arc_to_ends_a_b @ t @ E )
= ( product_Pair_a_a @ U @ V ) )
=> ( arc_pre_awalk_a_b @ t @ U @ ( cons_b @ E @ Es2 ) @ W ) ) ) ) ).
% awalk_ConsI
thf(fact_1113_euler__trail__conv__connected,axiom,
! [U: a,P: list_b,V: a] :
( ( digrap8783888973171253482ed_a_b @ t )
=> ( ( pre_euler_trail_a_b @ t @ U @ P @ V )
= ( ( arc_pre_trail_a_b @ t @ U @ P @ V )
& ( ( set_b2 @ P )
= ( pre_ar1395965042833527383t_unit @ t ) ) ) ) ) ).
% euler_trail_conv_connected
thf(fact_1114_awalk__connected,axiom,
! [U: a,P: list_b,V: a] :
( ( digrap8783888973171253482ed_a_b @ t )
=> ( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( ( ( set_b2 @ P )
!= ( pre_ar1395965042833527383t_unit @ t ) )
=> ? [E2: b] :
( ( member_b @ E2 @ ( minus_minus_set_b @ ( pre_ar1395965042833527383t_unit @ t ) @ ( set_b2 @ P ) ) )
& ( ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ E2 ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) )
| ( member_a @ ( pre_he5236287464308401016t_unit @ t @ E2 ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) ) ) ) ) ) ) ).
% awalk_connected
thf(fact_1115_spanning__tree__imp__connected,axiom,
! [H3: pre_pr7278220950009878019t_unit] :
( ( digrap5718416180170401981ee_a_b @ H3 @ t )
=> ( digrap8783888973171253482ed_a_b @ t ) ) ).
% spanning_tree_imp_connected
thf(fact_1116_connected__spanning__imp__connected,axiom,
! [H3: pre_pr7278220950009878019t_unit] :
( ( digraph_spanning_a_b @ H3 @ t )
=> ( ( digrap8783888973171253482ed_a_b @ H3 )
=> ( digrap8783888973171253482ed_a_b @ t ) ) ) ).
% connected_spanning_imp_connected
thf(fact_1117_pre__digraph_Oeuler__trail__def,axiom,
( pre_euler_trail_a_a
= ( ^ [G2: pre_pr3327329314391289540t_unit,U4: a,P3: list_a,V5: a] :
( ( arc_pre_trail_a_a @ G2 @ U4 @ P3 @ V5 )
& ( ( set_a2 @ P3 )
= ( pre_ar6668445444069714712t_unit @ G2 ) )
& ( ( set_a2 @ ( arc_pr7493981781705774525ts_a_a @ G2 @ U4 @ P3 ) )
= ( pre_ve5914862431884959581t_unit @ G2 ) ) ) ) ) ).
% pre_digraph.euler_trail_def
thf(fact_1118_pre__digraph_Oeuler__trail__def,axiom,
( pre_euler_trail_b_a
= ( ^ [G2: pre_pr3994228789931197893t_unit,U4: b,P3: list_a,V5: b] :
( ( arc_pre_trail_b_a @ G2 @ U4 @ P3 @ V5 )
& ( ( set_a2 @ P3 )
= ( pre_ar2913695170082820505t_unit @ G2 ) )
& ( ( set_b2 @ ( arc_pr4706526199733098492ts_b_a @ G2 @ U4 @ P3 ) )
= ( pre_ve2160112157898065374t_unit @ G2 ) ) ) ) ) ).
% pre_digraph.euler_trail_def
thf(fact_1119_pre__digraph_Oeuler__trail__def,axiom,
( pre_euler_trail_b_b
= ( ^ [G2: pre_pr7945120425549786372t_unit,U4: b,P3: list_b,V5: b] :
( ( arc_pre_trail_b_b @ G2 @ U4 @ P3 @ V5 )
& ( ( set_b2 @ P3 )
= ( pre_ar6864586805701408984t_unit @ G2 ) )
& ( ( set_b2 @ ( arc_pr4706526199733098493ts_b_b @ G2 @ U4 @ P3 ) )
= ( pre_ve6111003793516653853t_unit @ G2 ) ) ) ) ) ).
% pre_digraph.euler_trail_def
thf(fact_1120_pre__digraph_Oeuler__trail__def,axiom,
( pre_eu4033079881512885387st_a_b
= ( ^ [G2: pre_pr2882871181989701257t_unit,U4: list_a,P3: list_b,V5: list_a] :
( ( arc_pr7309874995902050716st_a_b @ G2 @ U4 @ P3 @ V5 )
& ( ( set_b2 @ P3 )
= ( pre_ar3460806382551299165t_unit @ G2 ) )
& ( ( set_list_a2 @ ( arc_pr6350002437206376376st_a_b @ G2 @ U4 @ P3 ) )
= ( pre_ve1830060048215441954t_unit @ G2 ) ) ) ) ) ).
% pre_digraph.euler_trail_def
thf(fact_1121_pre__digraph_Oeuler__trail__def,axiom,
( pre_euler_trail_a_b
= ( ^ [G2: pre_pr7278220950009878019t_unit,U4: a,P3: list_b,V5: a] :
( ( arc_pre_trail_a_b @ G2 @ U4 @ P3 @ V5 )
& ( ( set_b2 @ P3 )
= ( pre_ar1395965042833527383t_unit @ G2 ) )
& ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ G2 @ U4 @ P3 ) )
= ( pre_ve642382030648772252t_unit @ G2 ) ) ) ) ) ).
% pre_digraph.euler_trail_def
thf(fact_1122_pre__digraph_Ocycle__def,axiom,
( arc_pre_cycle_a_a
= ( ^ [G2: pre_pr3327329314391289540t_unit,P3: list_a] :
? [U4: a] :
( ( arc_pre_awalk_a_a @ G2 @ U4 @ P3 @ U4 )
& ( distinct_a @ ( tl_a @ ( arc_pr7493981781705774525ts_a_a @ G2 @ U4 @ P3 ) ) )
& ( P3 != nil_a ) ) ) ) ).
% pre_digraph.cycle_def
thf(fact_1123_pre__digraph_Ocycle__def,axiom,
( arc_pre_cycle_b_a
= ( ^ [G2: pre_pr3994228789931197893t_unit,P3: list_a] :
? [U4: b] :
( ( arc_pre_awalk_b_a @ G2 @ U4 @ P3 @ U4 )
& ( distinct_b @ ( tl_b @ ( arc_pr4706526199733098492ts_b_a @ G2 @ U4 @ P3 ) ) )
& ( P3 != nil_a ) ) ) ) ).
% pre_digraph.cycle_def
thf(fact_1124_pre__digraph_Ocycle__def,axiom,
( arc_pre_cycle_b_b
= ( ^ [G2: pre_pr7945120425549786372t_unit,P3: list_b] :
? [U4: b] :
( ( arc_pre_awalk_b_b @ G2 @ U4 @ P3 @ U4 )
& ( distinct_b @ ( tl_b @ ( arc_pr4706526199733098493ts_b_b @ G2 @ U4 @ P3 ) ) )
& ( P3 != nil_b ) ) ) ) ).
% pre_digraph.cycle_def
thf(fact_1125_pre__digraph_Ocycle__def,axiom,
( arc_pr6335352977596618620st_a_b
= ( ^ [G2: pre_pr2882871181989701257t_unit,P3: list_b] :
? [U4: list_a] :
( ( arc_pr6214585750886380800st_a_b @ G2 @ U4 @ P3 @ U4 )
& ( distinct_list_a @ ( tl_list_a @ ( arc_pr6350002437206376376st_a_b @ G2 @ U4 @ P3 ) ) )
& ( P3 != nil_b ) ) ) ) ).
% pre_digraph.cycle_def
thf(fact_1126_pre__digraph_Ocycle__def,axiom,
( arc_pre_cycle_a_b
= ( ^ [G2: pre_pr7278220950009878019t_unit,P3: list_b] :
? [U4: a] :
( ( arc_pre_awalk_a_b @ G2 @ U4 @ P3 @ U4 )
& ( distinct_a @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ G2 @ U4 @ P3 ) ) )
& ( P3 != nil_b ) ) ) ) ).
% pre_digraph.cycle_def
thf(fact_1127_awalkI,axiom,
! [U: a,P: list_b,V: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P ) @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( arc_pre_cas_a_b @ t @ U @ P @ V )
=> ( arc_pre_awalk_a_b @ t @ U @ P @ V ) ) ) ) ).
% awalkI
thf(fact_1128_cas_Osimps_I1_J,axiom,
! [U: a,V: a] :
( ( arc_pre_cas_a_b @ t @ U @ nil_b @ V )
= ( U = V ) ) ).
% cas.simps(1)
thf(fact_1129_cas__ends,axiom,
! [U: a,P: list_b,V: a,U3: a,V2: a] :
( ( arc_pre_cas_a_b @ t @ U @ P @ V )
=> ( ( arc_pre_cas_a_b @ t @ U3 @ P @ V2 )
=> ( ( ( P != nil_b )
& ( U = U3 )
& ( V = V2 ) )
| ( ( P = nil_b )
& ( U = V )
& ( U3 = V2 ) ) ) ) ) ).
% cas_ends
thf(fact_1130_awhd__if__cas,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_cas_a_b @ t @ U @ P @ V )
=> ( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) )
= U ) ) ).
% awhd_if_cas
thf(fact_1131_cas__simp,axiom,
! [Es2: list_b,U: a,V: a] :
( ( Es2 != nil_b )
=> ( ( arc_pre_cas_a_b @ t @ U @ Es2 @ V )
= ( ( ( pre_ta4931606617599662728t_unit @ t @ ( hd_b @ Es2 ) )
= U )
& ( arc_pre_cas_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ ( hd_b @ Es2 ) ) @ ( tl_b @ Es2 ) @ V ) ) ) ) ).
% cas_simp
thf(fact_1132_tail__and__head__eq__impl__cas,axiom,
! [X2: a,P: list_b,Y2: a,G3: pre_pr7278220950009878019t_unit] :
( ( arc_pre_cas_a_b @ t @ X2 @ P @ Y2 )
=> ( ! [X: b] :
( ( member_b @ X @ ( set_b2 @ P ) )
=> ( ( pre_ta4931606617599662728t_unit @ t @ X )
= ( pre_ta4931606617599662728t_unit @ G3 @ X ) ) )
=> ( ! [X: b] :
( ( member_b @ X @ ( set_b2 @ P ) )
=> ( ( pre_he5236287464308401016t_unit @ t @ X )
= ( pre_he5236287464308401016t_unit @ G3 @ X ) ) )
=> ( arc_pre_cas_a_b @ G3 @ X2 @ P @ Y2 ) ) ) ) ).
% tail_and_head_eq_impl_cas
thf(fact_1133_cas_Osimps_I2_J,axiom,
! [U: a,E: b,Es2: list_b,V: a] :
( ( arc_pre_cas_a_b @ t @ U @ ( cons_b @ E @ Es2 ) @ V )
= ( ( ( pre_ta4931606617599662728t_unit @ t @ E )
= U )
& ( arc_pre_cas_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E ) @ Es2 @ V ) ) ) ).
% cas.simps(2)
thf(fact_1134_to__list__tree__awalk,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
= ( arc_pr6214585750886380800st_a_b @ ( direct3773525127397338803ee_a_b @ t ) @ ( cons_a @ U @ nil_a ) @ P @ ( cons_a @ V @ nil_a ) ) ) ).
% to_list_tree_awalk
thf(fact_1135_cas_Oelims_I3_J,axiom,
! [X2: a,Xa3: list_b,Xb: a] :
( ~ ( arc_pre_cas_a_b @ t @ X2 @ Xa3 @ Xb )
=> ( ( ( Xa3 = nil_b )
=> ( X2 = Xb ) )
=> ~ ! [E2: b,Es: list_b] :
( ( Xa3
= ( cons_b @ E2 @ Es ) )
=> ( ( ( pre_ta4931606617599662728t_unit @ t @ E2 )
= X2 )
& ( arc_pre_cas_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E2 ) @ Es @ Xb ) ) ) ) ) ).
% cas.elims(3)
thf(fact_1136_cas_Oelims_I2_J,axiom,
! [X2: a,Xa3: list_b,Xb: a] :
( ( arc_pre_cas_a_b @ t @ X2 @ Xa3 @ Xb )
=> ( ( ( Xa3 = nil_b )
=> ( X2 != Xb ) )
=> ~ ! [E2: b,Es: list_b] :
( ( Xa3
= ( cons_b @ E2 @ Es ) )
=> ~ ( ( ( pre_ta4931606617599662728t_unit @ t @ E2 )
= X2 )
& ( arc_pre_cas_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E2 ) @ Es @ Xb ) ) ) ) ) ).
% cas.elims(2)
thf(fact_1137_cas_Oelims_I1_J,axiom,
! [X2: a,Xa3: list_b,Xb: a,Y2: $o] :
( ( ( arc_pre_cas_a_b @ t @ X2 @ Xa3 @ Xb )
= Y2 )
=> ( ( ( Xa3 = nil_b )
=> ( Y2
= ( X2 != Xb ) ) )
=> ~ ! [E2: b,Es: list_b] :
( ( Xa3
= ( cons_b @ E2 @ Es ) )
=> ( Y2
= ( ~ ( ( ( pre_ta4931606617599662728t_unit @ t @ E2 )
= X2 )
& ( arc_pre_cas_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E2 ) @ Es @ Xb ) ) ) ) ) ) ) ).
% cas.elims(1)
thf(fact_1138_cas__induce,axiom,
! [U: a,P: list_b,V: a,S3: set_a] :
( ( arc_pre_cas_a_b @ t @ U @ P @ V )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) @ S3 )
=> ( arc_pre_cas_a_b @ ( digrap7873285959652527175ph_a_b @ t @ S3 ) @ U @ P @ V ) ) ) ).
% cas_induce
thf(fact_1139_awalk__decomp__verts,axiom,
! [U: a,P: list_b,V: a,Xs: list_a,Y2: a,Ys: list_a] :
( ( arc_pre_cas_a_b @ t @ U @ P @ V )
=> ( ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ P )
= ( append_a @ Xs @ ( cons_a @ Y2 @ Ys ) ) )
=> ~ ! [Q3: list_b] :
( ( arc_pre_cas_a_b @ t @ U @ Q3 @ Y2 )
=> ! [R: list_b] :
( ( arc_pre_cas_a_b @ t @ Y2 @ R @ V )
=> ( ( P
= ( append_b @ Q3 @ R ) )
=> ( ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ Q3 )
= ( append_a @ Xs @ ( cons_a @ Y2 @ nil_a ) ) )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ Y2 @ R )
!= ( cons_a @ Y2 @ Ys ) ) ) ) ) ) ) ) ).
% awalk_decomp_verts
thf(fact_1140_awalk__def,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
= ( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) )
& ( ord_less_eq_set_b @ ( set_b2 @ P ) @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( arc_pre_cas_a_b @ t @ U @ P @ V ) ) ) ).
% awalk_def
thf(fact_1141_cas__append__if,axiom,
! [X2: a,Ps2: list_b,U: a,P: b,V: a] :
( ( arc_pre_cas_a_b @ t @ X2 @ Ps2 @ U )
=> ( ( ( pre_ta4931606617599662728t_unit @ t @ P )
= U )
=> ( ( ( pre_he5236287464308401016t_unit @ t @ P )
= V )
=> ( arc_pre_cas_a_b @ t @ X2 @ ( append_b @ Ps2 @ ( cons_b @ P @ nil_b ) ) @ V ) ) ) ) ).
% cas_append_if
thf(fact_1142_pre__digraph_Ocas_Osimps_I2_J,axiom,
! [G: pre_pr2882871181989701257t_unit,U: list_a,E: b,Es2: list_b,V: list_a] :
( ( arc_pre_cas_list_a_b @ G @ U @ ( cons_b @ E @ Es2 ) @ V )
= ( ( ( pre_ta8437681634429857806t_unit @ G @ E )
= U )
& ( arc_pre_cas_list_a_b @ G @ ( pre_he1293792728851071230t_unit @ G @ E ) @ Es2 @ V ) ) ) ).
% pre_digraph.cas.simps(2)
thf(fact_1143_pre__digraph_Ocas_Osimps_I2_J,axiom,
! [G: pre_pr7278220950009878019t_unit,U: a,E: b,Es2: list_b,V: a] :
( ( arc_pre_cas_a_b @ G @ U @ ( cons_b @ E @ Es2 ) @ V )
= ( ( ( pre_ta4931606617599662728t_unit @ G @ E )
= U )
& ( arc_pre_cas_a_b @ G @ ( pre_he5236287464308401016t_unit @ G @ E ) @ Es2 @ V ) ) ) ).
% pre_digraph.cas.simps(2)
thf(fact_1144_pre__digraph_Ocas_Osimps_I1_J,axiom,
! [G: pre_pr7278220950009878019t_unit,U: a,V: a] :
( ( arc_pre_cas_a_b @ G @ U @ nil_b @ V )
= ( U = V ) ) ).
% pre_digraph.cas.simps(1)
thf(fact_1145_pre__digraph_Ocas_Osimps_I1_J,axiom,
! [G: pre_pr2882871181989701257t_unit,U: list_a,V: list_a] :
( ( arc_pre_cas_list_a_b @ G @ U @ nil_b @ V )
= ( U = V ) ) ).
% pre_digraph.cas.simps(1)
thf(fact_1146_pre__digraph_Ocas_Oelims_I1_J,axiom,
! [G: pre_pr2882871181989701257t_unit,X2: list_a,Xa3: list_b,Xb: list_a,Y2: $o] :
( ( ( arc_pre_cas_list_a_b @ G @ X2 @ Xa3 @ Xb )
= Y2 )
=> ( ( ( Xa3 = nil_b )
=> ( Y2
= ( X2 != Xb ) ) )
=> ~ ! [E2: b,Es: list_b] :
( ( Xa3
= ( cons_b @ E2 @ Es ) )
=> ( Y2
= ( ~ ( ( ( pre_ta8437681634429857806t_unit @ G @ E2 )
= X2 )
& ( arc_pre_cas_list_a_b @ G @ ( pre_he1293792728851071230t_unit @ G @ E2 ) @ Es @ Xb ) ) ) ) ) ) ) ).
% pre_digraph.cas.elims(1)
thf(fact_1147_pre__digraph_Ocas_Oelims_I1_J,axiom,
! [G: pre_pr7278220950009878019t_unit,X2: a,Xa3: list_b,Xb: a,Y2: $o] :
( ( ( arc_pre_cas_a_b @ G @ X2 @ Xa3 @ Xb )
= Y2 )
=> ( ( ( Xa3 = nil_b )
=> ( Y2
= ( X2 != Xb ) ) )
=> ~ ! [E2: b,Es: list_b] :
( ( Xa3
= ( cons_b @ E2 @ Es ) )
=> ( Y2
= ( ~ ( ( ( pre_ta4931606617599662728t_unit @ G @ E2 )
= X2 )
& ( arc_pre_cas_a_b @ G @ ( pre_he5236287464308401016t_unit @ G @ E2 ) @ Es @ Xb ) ) ) ) ) ) ) ).
% pre_digraph.cas.elims(1)
thf(fact_1148_pre__digraph_Ocas_Oelims_I2_J,axiom,
! [G: pre_pr2882871181989701257t_unit,X2: list_a,Xa3: list_b,Xb: list_a] :
( ( arc_pre_cas_list_a_b @ G @ X2 @ Xa3 @ Xb )
=> ( ( ( Xa3 = nil_b )
=> ( X2 != Xb ) )
=> ~ ! [E2: b,Es: list_b] :
( ( Xa3
= ( cons_b @ E2 @ Es ) )
=> ~ ( ( ( pre_ta8437681634429857806t_unit @ G @ E2 )
= X2 )
& ( arc_pre_cas_list_a_b @ G @ ( pre_he1293792728851071230t_unit @ G @ E2 ) @ Es @ Xb ) ) ) ) ) ).
% pre_digraph.cas.elims(2)
thf(fact_1149_pre__digraph_Ocas_Oelims_I2_J,axiom,
! [G: pre_pr7278220950009878019t_unit,X2: a,Xa3: list_b,Xb: a] :
( ( arc_pre_cas_a_b @ G @ X2 @ Xa3 @ Xb )
=> ( ( ( Xa3 = nil_b )
=> ( X2 != Xb ) )
=> ~ ! [E2: b,Es: list_b] :
( ( Xa3
= ( cons_b @ E2 @ Es ) )
=> ~ ( ( ( pre_ta4931606617599662728t_unit @ G @ E2 )
= X2 )
& ( arc_pre_cas_a_b @ G @ ( pre_he5236287464308401016t_unit @ G @ E2 ) @ Es @ Xb ) ) ) ) ) ).
% pre_digraph.cas.elims(2)
thf(fact_1150_pre__digraph_Ocas_Oelims_I3_J,axiom,
! [G: pre_pr2882871181989701257t_unit,X2: list_a,Xa3: list_b,Xb: list_a] :
( ~ ( arc_pre_cas_list_a_b @ G @ X2 @ Xa3 @ Xb )
=> ( ( ( Xa3 = nil_b )
=> ( X2 = Xb ) )
=> ~ ! [E2: b,Es: list_b] :
( ( Xa3
= ( cons_b @ E2 @ Es ) )
=> ( ( ( pre_ta8437681634429857806t_unit @ G @ E2 )
= X2 )
& ( arc_pre_cas_list_a_b @ G @ ( pre_he1293792728851071230t_unit @ G @ E2 ) @ Es @ Xb ) ) ) ) ) ).
% pre_digraph.cas.elims(3)
thf(fact_1151_pre__digraph_Ocas_Oelims_I3_J,axiom,
! [G: pre_pr7278220950009878019t_unit,X2: a,Xa3: list_b,Xb: a] :
( ~ ( arc_pre_cas_a_b @ G @ X2 @ Xa3 @ Xb )
=> ( ( ( Xa3 = nil_b )
=> ( X2 = Xb ) )
=> ~ ! [E2: b,Es: list_b] :
( ( Xa3
= ( cons_b @ E2 @ Es ) )
=> ( ( ( pre_ta4931606617599662728t_unit @ G @ E2 )
= X2 )
& ( arc_pre_cas_a_b @ G @ ( pre_he5236287464308401016t_unit @ G @ E2 ) @ Es @ Xb ) ) ) ) ) ).
% pre_digraph.cas.elims(3)
thf(fact_1152_pre__digraph_Oawalk_Ocong,axiom,
arc_pre_awalk_a_b = arc_pre_awalk_a_b ).
% pre_digraph.awalk.cong
thf(fact_1153_pre__digraph_Oawalk_Ocong,axiom,
arc_pr6214585750886380800st_a_b = arc_pr6214585750886380800st_a_b ).
% pre_digraph.awalk.cong
thf(fact_1154_pre__digraph_Oawalk__def,axiom,
( arc_pr6214585750886380799st_a_a
= ( ^ [G2: pre_pr8155351583225888586t_unit,U4: list_a,P3: list_a,V5: list_a] :
( ( member_list_a @ U4 @ ( pre_ve7102540449451629283t_unit @ G2 ) )
& ( ord_less_eq_set_a @ ( set_a2 @ P3 ) @ ( pre_ar8733286783787486494t_unit @ G2 ) )
& ( arc_pre_cas_list_a_a @ G2 @ U4 @ P3 @ V5 ) ) ) ) ).
% pre_digraph.awalk_def
thf(fact_1155_pre__digraph_Oawalk__def,axiom,
( arc_pr441381926571271589et_a_a
= ( ^ [G2: pre_pr3647964229410195492t_unit,U4: set_a,P3: list_a,V5: set_a] :
( ( member_set_a @ U4 @ ( pre_ve2608818176351713469t_unit @ G2 ) )
& ( ord_less_eq_set_a @ ( set_a2 @ P3 ) @ ( pre_ar4979499625094109304t_unit @ G2 ) )
& ( arc_pre_cas_set_a_a @ G2 @ U4 @ P3 @ V5 ) ) ) ) ).
% pre_digraph.awalk_def
thf(fact_1156_pre__digraph_Oawalk__def,axiom,
( arc_pre_awalk_nat_a
= ( ^ [G2: pre_pr4235297827024194814t_unit,U4: nat,P3: list_a,V5: nat] :
( ( member_nat @ U4 @ ( pre_ve1231858382851255055t_unit @ G2 ) )
& ( ord_less_eq_set_a @ ( set_a2 @ P3 ) @ ( pre_ar4501138274228975252t_unit @ G2 ) )
& ( arc_pre_cas_nat_a @ G2 @ U4 @ P3 @ V5 ) ) ) ) ).
% pre_digraph.awalk_def
thf(fact_1157_pre__digraph_Oawalk__def,axiom,
( arc_pre_awalk_b_a
= ( ^ [G2: pre_pr3994228789931197893t_unit,U4: b,P3: list_a,V5: b] :
( ( member_b @ U4 @ ( pre_ve2160112157898065374t_unit @ G2 ) )
& ( ord_less_eq_set_a @ ( set_a2 @ P3 ) @ ( pre_ar2913695170082820505t_unit @ G2 ) )
& ( arc_pre_cas_b_a @ G2 @ U4 @ P3 @ V5 ) ) ) ) ).
% pre_digraph.awalk_def
thf(fact_1158_pre__digraph_Oawalk__def,axiom,
( arc_pre_awalk_a_a
= ( ^ [G2: pre_pr3327329314391289540t_unit,U4: a,P3: list_a,V5: a] :
( ( member_a @ U4 @ ( pre_ve5914862431884959581t_unit @ G2 ) )
& ( ord_less_eq_set_a @ ( set_a2 @ P3 ) @ ( pre_ar6668445444069714712t_unit @ G2 ) )
& ( arc_pre_cas_a_a @ G2 @ U4 @ P3 @ V5 ) ) ) ) ).
% pre_digraph.awalk_def
thf(fact_1159_pre__digraph_Oawalk__def,axiom,
( arc_pr441381926571271590et_a_b
= ( ^ [G2: pre_pr7598855865028783971t_unit,U4: set_a,P3: list_b,V5: set_a] :
( ( member_set_a @ U4 @ ( pre_ve6559709811970301948t_unit @ G2 ) )
& ( ord_less_eq_set_b @ ( set_b2 @ P3 ) @ ( pre_ar8930391260712697783t_unit @ G2 ) )
& ( arc_pre_cas_set_a_b @ G2 @ U4 @ P3 @ V5 ) ) ) ) ).
% pre_digraph.awalk_def
thf(fact_1160_pre__digraph_Oawalk__def,axiom,
( arc_pre_awalk_nat_b
= ( ^ [G2: pre_pr8186189462642783293t_unit,U4: nat,P3: list_b,V5: nat] :
( ( member_nat @ U4 @ ( pre_ve5182750018469843534t_unit @ G2 ) )
& ( ord_less_eq_set_b @ ( set_b2 @ P3 ) @ ( pre_ar8452029909847563731t_unit @ G2 ) )
& ( arc_pre_cas_nat_b @ G2 @ U4 @ P3 @ V5 ) ) ) ) ).
% pre_digraph.awalk_def
thf(fact_1161_pre__digraph_Oawalk__def,axiom,
( arc_pre_awalk_b_b
= ( ^ [G2: pre_pr7945120425549786372t_unit,U4: b,P3: list_b,V5: b] :
( ( member_b @ U4 @ ( pre_ve6111003793516653853t_unit @ G2 ) )
& ( ord_less_eq_set_b @ ( set_b2 @ P3 ) @ ( pre_ar6864586805701408984t_unit @ G2 ) )
& ( arc_pre_cas_b_b @ G2 @ U4 @ P3 @ V5 ) ) ) ) ).
% pre_digraph.awalk_def
thf(fact_1162_pre__digraph_Oawalk__def,axiom,
( arc_pre_awalk_a_b
= ( ^ [G2: pre_pr7278220950009878019t_unit,U4: a,P3: list_b,V5: a] :
( ( member_a @ U4 @ ( pre_ve642382030648772252t_unit @ G2 ) )
& ( ord_less_eq_set_b @ ( set_b2 @ P3 ) @ ( pre_ar1395965042833527383t_unit @ G2 ) )
& ( arc_pre_cas_a_b @ G2 @ U4 @ P3 @ V5 ) ) ) ) ).
% pre_digraph.awalk_def
thf(fact_1163_pre__digraph_Oawalk__def,axiom,
( arc_pr6214585750886380800st_a_b
= ( ^ [G2: pre_pr2882871181989701257t_unit,U4: list_a,P3: list_b,V5: list_a] :
( ( member_list_a @ U4 @ ( pre_ve1830060048215441954t_unit @ G2 ) )
& ( ord_less_eq_set_b @ ( set_b2 @ P3 ) @ ( pre_ar3460806382551299165t_unit @ G2 ) )
& ( arc_pre_cas_list_a_b @ G2 @ U4 @ P3 @ V5 ) ) ) ) ).
% pre_digraph.awalk_def
thf(fact_1164_pre__digraph_Ocas__simp,axiom,
! [Es2: list_b,G: pre_pr2882871181989701257t_unit,U: list_a,V: list_a] :
( ( Es2 != nil_b )
=> ( ( arc_pre_cas_list_a_b @ G @ U @ Es2 @ V )
= ( ( ( pre_ta8437681634429857806t_unit @ G @ ( hd_b @ Es2 ) )
= U )
& ( arc_pre_cas_list_a_b @ G @ ( pre_he1293792728851071230t_unit @ G @ ( hd_b @ Es2 ) ) @ ( tl_b @ Es2 ) @ V ) ) ) ) ).
% pre_digraph.cas_simp
thf(fact_1165_pre__digraph_Ocas__simp,axiom,
! [Es2: list_b,G: pre_pr7278220950009878019t_unit,U: a,V: a] :
( ( Es2 != nil_b )
=> ( ( arc_pre_cas_a_b @ G @ U @ Es2 @ V )
= ( ( ( pre_ta4931606617599662728t_unit @ G @ ( hd_b @ Es2 ) )
= U )
& ( arc_pre_cas_a_b @ G @ ( pre_he5236287464308401016t_unit @ G @ ( hd_b @ Es2 ) ) @ ( tl_b @ Es2 ) @ V ) ) ) ) ).
% pre_digraph.cas_simp
thf(fact_1166_pre__digraph_Ocycle_Ocong,axiom,
arc_pre_cycle_a_b = arc_pre_cycle_a_b ).
% pre_digraph.cycle.cong
thf(fact_1167_wf__digraph_Oclosed__w_Ocong,axiom,
arc_wf_closed_w_a_b = arc_wf_closed_w_a_b ).
% wf_digraph.closed_w.cong
thf(fact_1168_pre__digraph_Oawalk__verts__ne__eq,axiom,
! [P: list_b,G: pre_pr7278220950009878019t_unit,U: a,V: a] :
( ( P != nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ G @ U @ P )
= ( arc_pr7493981781705774526ts_a_b @ G @ V @ P ) ) ) ).
% pre_digraph.awalk_verts_ne_eq
thf(fact_1169_pre__digraph_Oawalk__verts__non__Nil,axiom,
! [G: pre_pr7278220950009878019t_unit,U: a,P: list_b] :
( ( arc_pr7493981781705774526ts_a_b @ G @ U @ P )
!= nil_a ) ).
% pre_digraph.awalk_verts_non_Nil
thf(fact_1170_pre__digraph_Oawalk__verts_Osimps_I1_J,axiom,
! [G: pre_pr3327329314391289540t_unit,U: a] :
( ( arc_pr7493981781705774525ts_a_a @ G @ U @ nil_a )
= ( cons_a @ U @ nil_a ) ) ).
% pre_digraph.awalk_verts.simps(1)
thf(fact_1171_pre__digraph_Oawalk__verts_Osimps_I1_J,axiom,
! [G: pre_pr3994228789931197893t_unit,U: b] :
( ( arc_pr4706526199733098492ts_b_a @ G @ U @ nil_a )
= ( cons_b @ U @ nil_b ) ) ).
% pre_digraph.awalk_verts.simps(1)
thf(fact_1172_pre__digraph_Oawalk__verts_Osimps_I1_J,axiom,
! [G: pre_pr7945120425549786372t_unit,U: b] :
( ( arc_pr4706526199733098493ts_b_b @ G @ U @ nil_b )
= ( cons_b @ U @ nil_b ) ) ).
% pre_digraph.awalk_verts.simps(1)
thf(fact_1173_pre__digraph_Oawalk__verts_Osimps_I1_J,axiom,
! [G: pre_pr7278220950009878019t_unit,U: a] :
( ( arc_pr7493981781705774526ts_a_b @ G @ U @ nil_b )
= ( cons_a @ U @ nil_a ) ) ).
% pre_digraph.awalk_verts.simps(1)
thf(fact_1174_pre__digraph_Otrail__def,axiom,
( arc_pr7309874995902050716st_a_b
= ( ^ [G2: pre_pr2882871181989701257t_unit,U4: list_a,P3: list_b,V5: list_a] :
( ( arc_pr6214585750886380800st_a_b @ G2 @ U4 @ P3 @ V5 )
& ( distinct_b @ P3 ) ) ) ) ).
% pre_digraph.trail_def
thf(fact_1175_pre__digraph_Otrail__def,axiom,
( arc_pre_trail_a_b
= ( ^ [G2: pre_pr7278220950009878019t_unit,U4: a,P3: list_b,V5: a] :
( ( arc_pre_awalk_a_b @ G2 @ U4 @ P3 @ V5 )
& ( distinct_b @ P3 ) ) ) ) ).
% pre_digraph.trail_def
thf(fact_1176_pre__digraph_Ocas_Ocases,axiom,
! [X2: produc8766925488660474953list_b] :
( ! [U5: list_b,V3: list_b] :
( X2
!= ( produc305491333965050169list_b @ U5 @ ( produc1564554178308465111list_b @ nil_b @ V3 ) ) )
=> ~ ! [U5: list_b,E2: b,Es: list_b,V3: list_b] :
( X2
!= ( produc305491333965050169list_b @ U5 @ ( produc1564554178308465111list_b @ ( cons_b @ E2 @ Es ) @ V3 ) ) ) ) ).
% pre_digraph.cas.cases
thf(fact_1177_pre__digraph_Ocas_Ocases,axiom,
! [X2: produc1553995403754578250list_a] :
( ! [U5: list_a,V3: list_a] :
( X2
!= ( produc1910438886824740410list_a @ U5 @ ( produc6837034575241423639list_a @ nil_a @ V3 ) ) )
=> ~ ! [U5: list_a,E2: a,Es: list_a,V3: list_a] :
( X2
!= ( produc1910438886824740410list_a @ U5 @ ( produc6837034575241423639list_a @ ( cons_a @ E2 @ Es ) @ V3 ) ) ) ) ).
% pre_digraph.cas.cases
thf(fact_1178_pre__digraph_Ocas_Ocases,axiom,
! [X2: produc7945266988514096265st_b_a] :
( ! [U5: a,V3: a] :
( X2
!= ( produc7119031474978700025st_b_a @ U5 @ ( produc4145578316043568848st_b_a @ nil_b @ V3 ) ) )
=> ~ ! [U5: a,E2: b,Es: list_b,V3: a] :
( X2
!= ( produc7119031474978700025st_b_a @ U5 @ ( produc4145578316043568848st_b_a @ ( cons_b @ E2 @ Es ) @ V3 ) ) ) ) ).
% pre_digraph.cas.cases
thf(fact_1179_pre__digraph_Oawalk__verts_Osimps_I2_J,axiom,
! [G: pre_pr3327329314391289540t_unit,U: a,E: a,Es2: list_a] :
( ( arc_pr7493981781705774525ts_a_a @ G @ U @ ( cons_a @ E @ Es2 ) )
= ( cons_a @ ( pre_ta980714981981074249t_unit @ G @ E ) @ ( arc_pr7493981781705774525ts_a_a @ G @ ( pre_he1285395828689812537t_unit @ G @ E ) @ Es2 ) ) ) ).
% pre_digraph.awalk_verts.simps(2)
thf(fact_1180_pre__digraph_Oawalk__verts_Osimps_I2_J,axiom,
! [G: pre_pr3994228789931197893t_unit,U: b,E: a,Es2: list_a] :
( ( arc_pr4706526199733098492ts_b_a @ G @ U @ ( cons_a @ E @ Es2 ) )
= ( cons_b @ ( pre_ta6449336744848955850t_unit @ G @ E ) @ ( arc_pr4706526199733098492ts_b_a @ G @ ( pre_he6754017591557694138t_unit @ G @ E ) @ Es2 ) ) ) ).
% pre_digraph.awalk_verts.simps(2)
thf(fact_1181_pre__digraph_Oawalk__verts_Osimps_I2_J,axiom,
! [G: pre_pr7945120425549786372t_unit,U: b,E: b,Es2: list_b] :
( ( arc_pr4706526199733098493ts_b_b @ G @ U @ ( cons_b @ E @ Es2 ) )
= ( cons_b @ ( pre_ta1176856343612768521t_unit @ G @ E ) @ ( arc_pr4706526199733098493ts_b_b @ G @ ( pre_he1481537190321506809t_unit @ G @ E ) @ Es2 ) ) ) ).
% pre_digraph.awalk_verts.simps(2)
thf(fact_1182_pre__digraph_Oawalk__verts_Osimps_I2_J,axiom,
! [G: pre_pr7278220950009878019t_unit,U: a,E: b,Es2: list_b] :
( ( arc_pr7493981781705774526ts_a_b @ G @ U @ ( cons_b @ E @ Es2 ) )
= ( cons_a @ ( pre_ta4931606617599662728t_unit @ G @ E ) @ ( arc_pr7493981781705774526ts_a_b @ G @ ( pre_he5236287464308401016t_unit @ G @ E ) @ Es2 ) ) ) ).
% pre_digraph.awalk_verts.simps(2)
thf(fact_1183_awalk__conv,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
= ( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) @ ( pre_ve642382030648772252t_unit @ t ) )
& ( ord_less_eq_set_b @ ( set_b2 @ P ) @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) )
= U )
& ( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) )
= V )
& ( arc_pre_cas_a_b @ t @ U @ P @ V ) ) ) ).
% awalk_conv
thf(fact_1184_awalkE_H,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ~ ( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P ) @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( arc_pre_cas_a_b @ t @ U @ P @ V )
=> ( ( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) )
= U )
=> ( ( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) )
= V )
=> ( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) )
=> ~ ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ) ) ) ) ) ).
% awalkE'
thf(fact_1185_awalkE,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ~ ( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P ) @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( arc_pre_cas_a_b @ t @ U @ P @ V )
=> ( ( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) )
= U )
=> ( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) )
!= V ) ) ) ) ) ) ).
% awalkE
thf(fact_1186_awlast__append,axiom,
! [U: a,P: list_b,Q: list_b] :
( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ ( append_b @ P @ Q ) ) )
= ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) @ Q ) ) ) ).
% awlast_append
thf(fact_1187_awlast__if__cas,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_cas_a_b @ t @ U @ P @ V )
=> ( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) )
= V ) ) ).
% awlast_if_cas
thf(fact_1188_to__list__tree__cas,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_cas_a_b @ t @ U @ P @ V )
= ( arc_pre_cas_list_a_b @ ( direct3773525127397338803ee_a_b @ t ) @ ( cons_a @ U @ nil_a ) @ P @ ( cons_a @ V @ nil_a ) ) ) ).
% to_list_tree_cas
thf(fact_1189_reachable__vwalk__conv,axiom,
! [U: a,V: a] :
( ( reachable_a_b @ t @ U @ V )
= ( ? [P3: list_a] :
( ( vertex_vwalk_a_b @ P3 @ t )
& ( ( hd_a @ P3 )
= U )
& ( ( last_a @ P3 )
= V ) ) ) ) ).
% reachable_vwalk_conv
thf(fact_1190_reachable__vpath__conv,axiom,
! [U: a,V: a] :
( ( reachable_a_b @ t @ U @ V )
= ( ? [P3: list_a] :
( ( vertex_vpath_a_b @ P3 @ t )
& ( ( hd_a @ P3 )
= U )
& ( ( last_a @ P3 )
= V ) ) ) ) ).
% reachable_vpath_conv
thf(fact_1191_awlast__in__verts,axiom,
! [U: a,P: list_b] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P ) @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( member_a @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% awlast_in_verts
thf(fact_1192_awalk__verts__append,axiom,
! [U: a,P: list_b,Q: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ ( append_b @ P @ Q ) @ V )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ ( append_b @ P @ Q ) )
= ( append_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) @ Q ) ) ) ) ) ).
% awalk_verts_append
thf(fact_1193_awalk__verts__append__cas,axiom,
! [U: a,P: list_b,Q: list_b,V: a] :
( ( arc_pre_cas_a_b @ t @ U @ ( append_b @ P @ Q ) @ V )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ ( append_b @ P @ Q ) )
= ( append_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) @ Q ) ) ) ) ) ).
% awalk_verts_append_cas
thf(fact_1194_last__appendL,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Xs ) ) ) ).
% last_appendL
thf(fact_1195_last__appendL,axiom,
! [Ys: list_b,Xs: list_b] :
( ( Ys = nil_b )
=> ( ( last_b @ ( append_b @ Xs @ Ys ) )
= ( last_b @ Xs ) ) ) ).
% last_appendL
thf(fact_1196_last__appendR,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Ys ) ) ) ).
% last_appendR
thf(fact_1197_last__appendR,axiom,
! [Ys: list_b,Xs: list_b] :
( ( Ys != nil_b )
=> ( ( last_b @ ( append_b @ Xs @ Ys ) )
= ( last_b @ Ys ) ) ) ).
% last_appendR
thf(fact_1198_last__snoc,axiom,
! [Xs: list_a,X2: a] :
( ( last_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) )
= X2 ) ).
% last_snoc
thf(fact_1199_last__snoc,axiom,
! [Xs: list_b,X2: b] :
( ( last_b @ ( append_b @ Xs @ ( cons_b @ X2 @ nil_b ) ) )
= X2 ) ).
% last_snoc
thf(fact_1200_awalk__append__iff,axiom,
! [U: a,P: list_b,Q: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ ( append_b @ P @ Q ) @ V )
= ( ( arc_pre_awalk_a_b @ t @ U @ P @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) )
& ( arc_pre_awalk_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) @ Q @ V ) ) ) ).
% awalk_append_iff
thf(fact_1201_cas__append__iff,axiom,
! [U: a,P: list_b,Q: list_b,V: a] :
( ( arc_pre_cas_a_b @ t @ U @ ( append_b @ P @ Q ) @ V )
= ( ( arc_pre_cas_a_b @ t @ U @ P @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) )
& ( arc_pre_cas_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) @ Q @ V ) ) ) ).
% cas_append_iff
thf(fact_1202_awalk__verts__conv_H,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_cas_a_b @ t @ U @ P @ V )
=> ( ( ( P = nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ P )
= ( cons_a @ U @ nil_a ) ) )
& ( ( P != nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ P )
= ( cons_a @ ( pre_ta4931606617599662728t_unit @ t @ ( hd_b @ P ) ) @ ( map_b_a @ ( pre_he5236287464308401016t_unit @ t ) @ P ) ) ) ) ) ) ).
% awalk_verts_conv'
thf(fact_1203_awalk__verts__append2,axiom,
! [U: a,P: list_b,Q: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ ( append_b @ P @ Q ) @ V )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ ( append_b @ P @ Q ) )
= ( append_a @ ( butlast_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) @ ( arc_pr7493981781705774526ts_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) @ Q ) ) ) ) ).
% awalk_verts_append2
thf(fact_1204_inner__verts__def,axiom,
! [P: list_b] :
( ( pre_inner_verts_a_b @ t @ P )
= ( tl_a @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ t ) @ P ) ) ) ).
% inner_verts_def
thf(fact_1205_inner__verts__conv,axiom,
! [P: list_b,U: a] :
( ( pre_inner_verts_a_b @ t @ P )
= ( butlast_a @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) ) ) ).
% inner_verts_conv
thf(fact_1206_awalk__verts__conv,axiom,
! [P: list_b,U: a] :
( ( ( P = nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ P )
= ( cons_a @ U @ nil_a ) ) )
& ( ( P != nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ P )
= ( append_a @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ t ) @ P ) @ ( cons_a @ ( pre_he5236287464308401016t_unit @ t @ ( last_b @ P ) ) @ nil_a ) ) ) ) ) ).
% awalk_verts_conv
thf(fact_1207_set__awalk__verts__not__Nil__cas,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_cas_a_b @ t @ U @ P @ V )
=> ( ( P != nil_b )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) )
= ( sup_sup_set_a @ ( set_a2 @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ t ) @ P ) ) @ ( set_a2 @ ( map_b_a @ ( pre_he5236287464308401016t_unit @ t ) @ P ) ) ) ) ) ) ).
% set_awalk_verts_not_Nil_cas
thf(fact_1208_set__awalk__verts__not__Nil,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( ( P != nil_b )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) )
= ( sup_sup_set_a @ ( set_a2 @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ t ) @ P ) ) @ ( set_a2 @ ( map_b_a @ ( pre_he5236287464308401016t_unit @ t ) @ P ) ) ) ) ) ) ).
% set_awalk_verts_not_Nil
thf(fact_1209_set__awalk__verts__append,axiom,
! [U: a,P: list_b,V: a,Q: list_b,W: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( ( arc_pre_awalk_a_b @ t @ V @ Q @ W )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ ( append_b @ P @ Q ) ) )
= ( sup_sup_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ V @ Q ) ) ) ) ) ) ).
% set_awalk_verts_append
thf(fact_1210_set__awalk__verts__append__cas,axiom,
! [U: a,P: list_b,V: a,Q: list_b,W: a] :
( ( arc_pre_cas_a_b @ t @ U @ P @ V )
=> ( ( arc_pre_cas_a_b @ t @ V @ Q @ W )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ ( append_b @ P @ Q ) ) )
= ( sup_sup_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ V @ Q ) ) ) ) ) ) ).
% set_awalk_verts_append_cas
thf(fact_1211_awlast__of__awalk,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( ( nOMATCH_a @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) @ V )
=> ( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) )
= V ) ) ) ).
% awlast_of_awalk
thf(fact_1212_connected__minimal,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ~ ( reachable_a_b @ ( pre_del_arc_a_b @ t @ E ) @ ( pre_ta4931606617599662728t_unit @ t @ E ) @ ( pre_he5236287464308401016t_unit @ t @ E ) ) ) ).
% connected_minimal
thf(fact_1213_del__arc__commute,axiom,
! [B: b,A: b] :
( ( pre_del_arc_a_b @ ( pre_del_arc_a_b @ t @ B ) @ A )
= ( pre_del_arc_a_b @ ( pre_del_arc_a_b @ t @ A ) @ B ) ) ).
% del_arc_commute
thf(fact_1214_del__arc__in,axiom,
! [A: b] :
( ~ ( member_b @ A @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( pre_del_arc_a_b @ t @ A )
= t ) ) ).
% del_arc_in
thf(fact_1215_del__del__arc__collapse,axiom,
! [A: b] :
( ( pre_del_arc_a_b @ ( pre_del_arc_a_b @ t @ A ) @ A )
= ( pre_del_arc_a_b @ t @ A ) ) ).
% del_del_arc_collapse
thf(fact_1216_verts__del__arc,axiom,
! [A: b] :
( ( pre_ve642382030648772252t_unit @ ( pre_del_arc_a_b @ t @ A ) )
= ( pre_ve642382030648772252t_unit @ t ) ) ).
% verts_del_arc
thf(fact_1217_head__del__arc,axiom,
! [A: b] :
( ( pre_he5236287464308401016t_unit @ ( pre_del_arc_a_b @ t @ A ) )
= ( pre_he5236287464308401016t_unit @ t ) ) ).
% head_del_arc
thf(fact_1218_tail__del__arc,axiom,
! [A: b] :
( ( pre_ta4931606617599662728t_unit @ ( pre_del_arc_a_b @ t @ A ) )
= ( pre_ta4931606617599662728t_unit @ t ) ) ).
% tail_del_arc
thf(fact_1219_apath__Cons__iff,axiom,
! [U: a,E: b,Es2: list_b,W: a] :
( ( arc_pre_apath_a_b @ t @ U @ ( cons_b @ E @ Es2 ) @ W )
= ( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( ( pre_ta4931606617599662728t_unit @ t @ E )
= U )
& ( arc_pre_apath_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E ) @ Es2 @ W )
& ~ ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ E ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E ) @ Es2 ) ) ) ) ) ).
% apath_Cons_iff
thf(fact_1220_apath__if__awalk,axiom,
! [R2: a,P: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ R2 @ P @ V )
=> ( arc_pre_apath_a_b @ t @ R2 @ P @ V ) ) ).
% apath_if_awalk
thf(fact_1221_awalkI__apath,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_apath_a_b @ t @ U @ P @ V )
=> ( arc_pre_awalk_a_b @ t @ U @ P @ V ) ) ).
% awalkI_apath
thf(fact_1222_reachable__apath,axiom,
! [U: a,V: a] :
( ( reachable_a_b @ t @ U @ V )
= ( ? [P3: list_b] : ( arc_pre_apath_a_b @ t @ U @ P3 @ V ) ) ) ).
% reachable_apath
thf(fact_1223_apath__ends,axiom,
! [U: a,P: list_b,V: a,U3: a,V2: a] :
( ( arc_pre_apath_a_b @ t @ U @ P @ V )
=> ( ( arc_pre_apath_a_b @ t @ U3 @ P @ V2 )
=> ( ( ( P != nil_b )
& ( U != V )
& ( U = U3 )
& ( V = V2 ) )
| ( ( P = nil_b )
& ( U = V )
& ( U3 = V2 ) ) ) ) ) ).
% apath_ends
thf(fact_1224_apath__nonempty__ends,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_apath_a_b @ t @ U @ P @ V )
=> ( ( P != nil_b )
=> ( U != V ) ) ) ).
% apath_nonempty_ends
thf(fact_1225_hd__in__awalk__verts_I2_J,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_apath_a_b @ t @ U @ P @ V )
=> ( member_a @ U @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) ) ) ).
% hd_in_awalk_verts(2)
thf(fact_1226_apath__Nil__iff,axiom,
! [U: a,V: a] :
( ( arc_pre_apath_a_b @ t @ U @ nil_b @ V )
= ( ( U = V )
& ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% apath_Nil_iff
thf(fact_1227_apath__awalk__to__apath,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( arc_pre_apath_a_b @ t @ U @ ( arc_wf446166946845163101th_a_b @ t @ P ) @ V ) ) ).
% apath_awalk_to_apath
thf(fact_1228_unique__apath__verts__in__awalk,axiom,
! [X2: a,U: a,P1: list_b,V: a,P22: list_b] :
( ( member_a @ X2 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P1 ) ) )
=> ( ( arc_pre_apath_a_b @ t @ U @ P1 @ V )
=> ( ( arc_pre_awalk_a_b @ t @ U @ P22 @ V )
=> ( ? [X4: list_b] :
( ( arc_pre_apath_a_b @ t @ U @ X4 @ V )
& ! [Y3: list_b] :
( ( arc_pre_apath_a_b @ t @ U @ Y3 @ V )
=> ( Y3 = X4 ) ) )
=> ( member_a @ X2 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P22 ) ) ) ) ) ) ) ).
% unique_apath_verts_in_awalk
thf(fact_1229_apath__def,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_apath_a_b @ t @ U @ P @ V )
= ( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
& ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) ) ) ).
% apath_def
thf(fact_1230_no__loops__in__apath,axiom,
! [U: a,P: list_b,V: a,A: b] :
( ( arc_pre_apath_a_b @ t @ U @ P @ V )
=> ( ( member_b @ A @ ( set_b2 @ P ) )
=> ( ( pre_ta4931606617599662728t_unit @ t @ A )
!= ( pre_he5236287464308401016t_unit @ t @ A ) ) ) ) ).
% no_loops_in_apath
thf(fact_1231_unique__apath__verts__sub__awalk,axiom,
! [U: a,P: list_b,V: a,Q: list_b] :
( ( arc_pre_apath_a_b @ t @ U @ P @ V )
=> ( ( arc_pre_awalk_a_b @ t @ U @ Q @ V )
=> ( ? [X4: list_b] :
( ( arc_pre_apath_a_b @ t @ U @ X4 @ V )
& ! [Y3: list_b] :
( ( arc_pre_apath_a_b @ t @ U @ Y3 @ V )
=> ( Y3 = X4 ) ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ Q ) ) ) ) ) ) ).
% unique_apath_verts_sub_awalk
thf(fact_1232_apath__decomp__disjoint,axiom,
! [U: a,P: list_b,V: a,Q: list_b,R2: list_b,X2: a] :
( ( arc_pre_apath_a_b @ t @ U @ P @ V )
=> ( ( P
= ( append_b @ Q @ R2 ) )
=> ( ( member_a @ X2 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ Q ) ) )
=> ~ ( member_a @ X2 @ ( set_a2 @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ Q ) ) @ R2 ) ) ) ) ) ) ) ).
% apath_decomp_disjoint
thf(fact_1233_bidirected__digraphI,axiom,
! [Arev: b > b] :
( ! [A4: b] :
( ~ ( member_b @ A4 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( Arev @ A4 )
= A4 ) )
=> ( ! [A4: b] :
( ( member_b @ A4 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( Arev @ A4 )
!= A4 ) )
=> ( ! [A4: b] :
( ( member_b @ A4 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( Arev @ ( Arev @ A4 ) )
= A4 ) )
=> ( ! [A4: b] :
( ( member_b @ A4 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( pre_ta4931606617599662728t_unit @ t @ ( Arev @ A4 ) )
= ( pre_he5236287464308401016t_unit @ t @ A4 ) ) )
=> ( bidire6463457107099887885ph_a_b @ t @ Arev ) ) ) ) ) ).
% bidirected_digraphI
thf(fact_1234_awalk__to__path__no__neg__cyc__cost,axiom,
! [U: a,P: list_b,V: a,F: b > real] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( ~ ? [W2: a,C2: list_b] :
( ( arc_pre_awalk_a_b @ t @ W2 @ C2 @ W2 )
& ( member_a @ W2 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) )
& ( ord_less_real @ ( weight7472181610322534790cost_b @ F @ C2 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( weight7472181610322534790cost_b @ F @ ( arc_wf446166946845163101th_a_b @ t @ P ) ) @ ( weight7472181610322534790cost_b @ F @ P ) ) ) ) ).
% awalk_to_path_no_neg_cyc_cost
thf(fact_1235_verts__reachable__connected,axiom,
( ( ( pre_ve642382030648772252t_unit @ t )
!= bot_bot_set_a )
=> ( ! [X: a] :
( ( member_a @ X @ ( pre_ve642382030648772252t_unit @ t ) )
=> ! [Xa: a] :
( ( member_a @ Xa @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( reachable_a_b @ t @ X @ Xa ) ) )
=> ( digrap8783888973171253482ed_a_b @ t ) ) ) ).
% verts_reachable_connected
thf(fact_1236_non__empty,axiom,
( ( pre_ve642382030648772252t_unit @ t )
!= bot_bot_set_a ) ).
% non_empty
thf(fact_1237_merging__empty,axiom,
( ( graph_2957805489637798020ts_a_b @ t )
= bot_bot_set_a ) ).
% merging_empty
thf(fact_1238_in__sccs__verts__conv__reachable,axiom,
! [S3: set_a] :
( ( member_set_a @ S3 @ ( digrap2871191568752656621ts_a_b @ t ) )
= ( ( S3 != bot_bot_set_a )
& ! [X3: a] :
( ( member_a @ X3 @ S3 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ S3 )
=> ( reachable_a_b @ t @ X3 @ Y4 ) ) )
& ! [X3: a] :
( ( member_a @ X3 @ S3 )
=> ! [V5: a] :
( ~ ( member_a @ V5 @ S3 )
=> ( ~ ( reachable_a_b @ t @ X3 @ V5 )
| ~ ( reachable_a_b @ t @ V5 @ X3 ) ) ) ) ) ) ).
% in_sccs_verts_conv_reachable
thf(fact_1239_scc__of__empty__conv,axiom,
! [U: a] :
( ( ( digrap2937667069914300949of_a_b @ t @ U )
= bot_bot_set_a )
= ( ~ ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% scc_of_empty_conv
thf(fact_1240_is__chain_H__def,axiom,
( ( graph_8150681439568091980in_a_b @ t )
= ( ( graph_2957805489637798020ts_a_b @ t )
= bot_bot_set_a ) ) ).
% is_chain'_def
thf(fact_1241_pos__cost__pos__awalk__cost,axiom,
! [U: a,P: list_b,V: a,C: b > real] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( ! [E2: b] :
( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( C @ E2 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( weight7472181610322534790cost_b @ C @ P ) ) ) ) ).
% pos_cost_pos_awalk_cost
thf(fact_1242_awalk__cost__append,axiom,
! [F: b > real,Xs: list_b,Ys: list_b] :
( ( weight7472181610322534790cost_b @ F @ ( append_b @ Xs @ Ys ) )
= ( plus_plus_real @ ( weight7472181610322534790cost_b @ F @ Xs ) @ ( weight7472181610322534790cost_b @ F @ Ys ) ) ) ).
% awalk_cost_append
thf(fact_1243_awalk__cost__Cons,axiom,
! [F: b > real,X2: b,Xs: list_b] :
( ( weight7472181610322534790cost_b @ F @ ( cons_b @ X2 @ Xs ) )
= ( plus_plus_real @ ( F @ X2 ) @ ( weight7472181610322534790cost_b @ F @ Xs ) ) ) ).
% awalk_cost_Cons
thf(fact_1244_awalk__cost__Nil,axiom,
! [F: b > real] :
( ( weight7472181610322534790cost_b @ F @ nil_b )
= zero_zero_real ) ).
% awalk_cost_Nil
thf(fact_1245_set__awalk__verts__cas,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_cas_a_b @ t @ U @ P @ V )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) )
= ( sup_sup_set_a @ ( sup_sup_set_a @ ( insert_a @ U @ bot_bot_set_a ) @ ( set_a2 @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ t ) @ P ) ) ) @ ( set_a2 @ ( map_b_a @ ( pre_he5236287464308401016t_unit @ t ) @ P ) ) ) ) ) ).
% set_awalk_verts_cas
thf(fact_1246_set__awalk__verts,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P @ V )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) )
= ( sup_sup_set_a @ ( sup_sup_set_a @ ( insert_a @ U @ bot_bot_set_a ) @ ( set_a2 @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ t ) @ P ) ) ) @ ( set_a2 @ ( map_b_a @ ( pre_he5236287464308401016t_unit @ t ) @ P ) ) ) ) ) ).
% set_awalk_verts
thf(fact_1247_verts__add__vert,axiom,
! [U: a] :
( ( pre_ve642382030648772252t_unit @ ( pre_add_vert_a_b @ t @ U ) )
= ( insert_a @ U @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% verts_add_vert
thf(fact_1248_connected__arcs__empty,axiom,
( ( digrap8783888973171253482ed_a_b @ t )
=> ( ( ( pre_ar1395965042833527383t_unit @ t )
= bot_bot_set_b )
=> ( ( ( pre_ve642382030648772252t_unit @ t )
!= bot_bot_set_a )
=> ~ ! [V3: a] :
( ( pre_ve642382030648772252t_unit @ t )
!= ( insert_a @ V3 @ bot_bot_set_a ) ) ) ) ) ).
% connected_arcs_empty
thf(fact_1249_verts__del__vert,axiom,
! [U: a] :
( ( pre_ve642382030648772252t_unit @ ( pre_del_vert_a_b @ t @ U ) )
= ( minus_minus_set_a @ ( pre_ve642382030648772252t_unit @ t ) @ ( insert_a @ U @ bot_bot_set_a ) ) ) ).
% verts_del_vert
thf(fact_1250_set__inner__verts,axiom,
! [U: a,P: list_b,V: a] :
( ( arc_pre_apath_a_b @ t @ U @ P @ V )
=> ( ( set_a2 @ ( pre_inner_verts_a_b @ t @ P ) )
= ( minus_minus_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) @ ( insert_a @ U @ ( insert_a @ V @ bot_bot_set_a ) ) ) ) ) ).
% set_inner_verts
thf(fact_1251_verts__add__arc__conv,axiom,
! [A: b] :
( ( pre_ve642382030648772252t_unit @ ( pre_add_arc_a_b @ t @ A ) )
= ( sup_sup_set_a @ ( pre_ve642382030648772252t_unit @ t ) @ ( insert_a @ ( pre_ta4931606617599662728t_unit @ t @ A ) @ ( insert_a @ ( pre_he5236287464308401016t_unit @ t @ A ) @ bot_bot_set_a ) ) ) ) ).
% verts_add_arc_conv
thf(fact_1252_apath__append__iff,axiom,
! [U: a,P: list_b,Q: list_b,V: a] :
( ( arc_pre_apath_a_b @ t @ U @ ( append_b @ P @ Q ) @ V )
= ( ( arc_pre_apath_a_b @ t @ U @ P @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) )
& ( arc_pre_apath_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) @ Q @ V )
& ( ( inf_inf_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) @ ( set_a2 @ ( tl_a @ ( arc_pr7493981781705774526ts_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P ) ) @ Q ) ) ) )
= bot_bot_set_a ) ) ) ).
% apath_append_iff
thf(fact_1253_add__arc__commute,axiom,
! [B: b,A: b] :
( ( pre_add_arc_a_b @ ( pre_add_arc_a_b @ t @ B ) @ A )
= ( pre_add_arc_a_b @ ( pre_add_arc_a_b @ t @ A ) @ B ) ) ).
% add_arc_commute
thf(fact_1254_add__arc__in,axiom,
! [A: b] :
( ( member_b @ A @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( pre_add_arc_a_b @ t @ A )
= t ) ) ).
% add_arc_in
thf(fact_1255_sccs__verts__disjoint,axiom,
! [S3: set_a,T3: set_a] :
( ( member_set_a @ S3 @ ( digrap2871191568752656621ts_a_b @ t ) )
=> ( ( member_set_a @ T3 @ ( digrap2871191568752656621ts_a_b @ t ) )
=> ( ( S3 != T3 )
=> ( ( inf_inf_set_a @ S3 @ T3 )
= bot_bot_set_a ) ) ) ) ).
% sccs_verts_disjoint
thf(fact_1256_to__list__tree__disjoint__verts,axiom,
! [U: list_a,V: list_a] :
( ( member_list_a @ U @ ( pre_ve1830060048215441954t_unit @ ( direct3773525127397338803ee_a_b @ t ) ) )
=> ( ( member_list_a @ V @ ( pre_ve1830060048215441954t_unit @ ( direct3773525127397338803ee_a_b @ t ) ) )
=> ( ( U != V )
=> ( ( inf_inf_set_a @ ( set_a2 @ U ) @ ( set_a2 @ V ) )
= bot_bot_set_a ) ) ) ) ).
% to_list_tree_disjoint_verts
thf(fact_1257_arcs__del__leaf,axiom,
! [E: b,V: a] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( ( pre_he5236287464308401016t_unit @ t @ E )
= V )
=> ( ( shorte1213025427933718126af_a_b @ t @ V )
=> ( ( pre_ar1395965042833527383t_unit @ ( pre_del_vert_a_b @ t @ V ) )
= ( minus_minus_set_b @ ( pre_ar1395965042833527383t_unit @ t ) @ ( insert_b @ E @ bot_bot_set_b ) ) ) ) ) ) ).
% arcs_del_leaf
thf(fact_1258_forward__app_H,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ S1 )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ S2 )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ S1 ) @ ( set_a2 @ S2 ) )
= bot_bot_set_a )
=> ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ S1 ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ S2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Xa2 ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ S1 @ S2 ) ) ) ) ) ) ).
% forward_app'
thf(fact_1259_forward__arc__to__head,axiom,
! [Ys: list_a,Xs: list_a,X2: a,Y2: a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ Ys )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) )
= bot_bot_set_a )
=> ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ( member_a @ Y2 @ ( set_a2 @ Ys ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ ( arcs_ends_a_b @ t ) )
=> ( Y2
= ( hd_a @ Ys ) ) ) ) ) ) ) ).
% forward_arc_to_head
thf(fact_1260_before__def,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S2 )
= ( ( iKKBZ_4622586873178280511rm_a_b @ t @ S1 )
& ( iKKBZ_4622586873178280511rm_a_b @ t @ S2 )
& ( ( inf_inf_set_a @ ( set_a2 @ S1 ) @ ( set_a2 @ S2 ) )
= bot_bot_set_a )
& ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ S1 ) )
& ? [Y4: a] :
( ( member_a @ Y4 @ ( set_a2 @ S2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ).
% before_def
thf(fact_1261_reachable1__append__old__if__arcU,axiom,
! [Xs: list_a,Ys: list_a,U2: list_a,Z2: a,Y2: a] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Xa2 ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ U2 ) @ ( set_a2 @ Xs ) )
= bot_bot_set_a )
=> ( ( member_a @ Z2 @ ( set_a2 @ U2 ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs )
=> ( ( member_a @ Y2 @ ( set_a2 @ ( append_a @ Xs @ Ys ) ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z2 @ Y2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z2 @ X ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ) ) ).
% reachable1_append_old_if_arcU
thf(fact_1262_hd__reachable1__from__outside,axiom,
! [X2: a,Y2: a,Ys: list_a,Xs: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ Ys )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) )
= bot_bot_set_a )
=> ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ( member_a @ Y2 @ ( set_a2 @ Ys ) )
=> ( ? [X: a] : ( member_a @ X @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ ( hd_a @ Ys ) ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ) ).
% hd_reachable1_from_outside
thf(fact_1263_no__back__before__aux,axiom,
! [Xs: list_a,Ys: list_a] :
( ( iKKBZ_4622586873178280511rm_a_b @ t @ Xs )
=> ( ( iKKBZ_4622586873178280511rm_a_b @ t @ Ys )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) )
= bot_bot_set_a )
=> ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Xa2 ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ( iKKBZ_3684931046464919648ck_a_b @ t @ ( append_a @ Xs @ Ys ) ) ) ) ) ) ).
% no_back_before_aux
thf(fact_1264_add__add__arc__collapse,axiom,
! [A: b] :
( ( pre_add_arc_a_b @ ( pre_add_arc_a_b @ t @ A ) @ A )
= ( pre_add_arc_a_b @ t @ A ) ) ).
% add_add_arc_collapse
thf(fact_1265_arcs__add__arc,axiom,
! [A: b] :
( ( pre_ar1395965042833527383t_unit @ ( pre_add_arc_a_b @ t @ A ) )
= ( insert_b @ A @ ( pre_ar1395965042833527383t_unit @ t ) ) ) ).
% arcs_add_arc
thf(fact_1266_head__add__arc,axiom,
! [A: b] :
( ( pre_he5236287464308401016t_unit @ ( pre_add_arc_a_b @ t @ A ) )
= ( pre_he5236287464308401016t_unit @ t ) ) ).
% head_add_arc
thf(fact_1267_tail__add__arc,axiom,
! [A: b] :
( ( pre_ta4931606617599662728t_unit @ ( pre_add_arc_a_b @ t @ A ) )
= ( pre_ta4931606617599662728t_unit @ t ) ) ).
% tail_add_arc
thf(fact_1268_add__del__arc__collapse,axiom,
! [A: b] :
( ( pre_add_arc_a_b @ ( pre_del_arc_a_b @ t @ A ) @ A )
= ( pre_add_arc_a_b @ t @ A ) ) ).
% add_del_arc_collapse
thf(fact_1269_verts__add__arc,axiom,
! [A: b] :
( ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ A ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( member_a @ ( pre_he5236287464308401016t_unit @ t @ A ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( pre_ve642382030648772252t_unit @ ( pre_add_arc_a_b @ t @ A ) )
= ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% verts_add_arc
% Conjectures (1)
thf(conj_0,conjecture,
? [J4: nat] :
( ( ord_less_nat @ J4 @ i )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ ( append_a @ as @ ( append_a @ u @ ( append_a @ v @ ( append_a @ bs @ cs ) ) ) ) @ J4 ) @ ( nth_a @ ( append_a @ as @ ( append_a @ u @ ( append_a @ v @ ( append_a @ bs @ cs ) ) ) ) @ i ) ) @ ( arcs_ends_a_b @ t ) ) ) ).
%------------------------------------------------------------------------------