TPTP Problem File: SLH0025^1.p
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%------------------------------------------------------------------------------
% 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 : Fishers_Inequality/0034_Incidence_Matrices/prob_00874_037675__28016798_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1423 ( 541 unt; 151 typ; 0 def)
% Number of atoms : 3444 (1339 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 11276 ( 393 ~; 47 |; 255 &;9185 @)
% ( 0 <=>;1396 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Number of types : 25 ( 24 usr)
% Number of type conns : 271 ( 271 >; 0 *; 0 +; 0 <<)
% Number of symbols : 130 ( 127 usr; 17 con; 0-3 aty)
% Number of variables : 3179 ( 102 ^;2970 !; 107 ?;3179 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 15:48:12.687
%------------------------------------------------------------------------------
% Could-be-implicit typings (24)
thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J_J,type,
multis7946784228617641354_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Set__Oset_Itf__a_J_J_J_J,type,
set_list_list_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J_J,type,
set_set_list_set_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
multiset_list_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J_J,type,
set_multiset_set_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
multiset_set_set_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
list_list_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
set_list_set_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_It__Nat__Onat_J_J,type,
multiset_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_set_set_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
multiset_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
list_set_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Nat__Onat_J,type,
multiset_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_Itf__a_J,type,
multiset_a: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (127)
thf(sy_c_Design__Basics_Oincidence__system_Odesign__support_001tf__a,type,
design5397942185814921632port_a: multiset_set_a > set_set_a ).
thf(sy_c_Design__Basics_Oincidence__system_Osys__block__sizes_001tf__a,type,
design1769254222028858111izes_a: multiset_set_a > set_nat ).
thf(sy_c_Design__Basics_Opoint__replication__number_001tf__a,type,
design6637022207325878697mber_a: multiset_set_a > a > nat ).
thf(sy_c_Design__Basics_Opoints__index_001tf__a,type,
design254580327166089565ndex_a: multiset_set_a > set_a > nat ).
thf(sy_c_Design__Operations_Oincidence__system_Oadd__block_001tf__a,type,
design4001997691126659652lock_a: multiset_set_a > set_a > multiset_set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Oadd__point__to__blocks_001tf__a,type,
design2935547469388721088ocks_a: multiset_set_a > a > set_set_a > multiset_set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Odel__block_001tf__a,type,
design1146539425385464078lock_a: multiset_set_a > set_a > multiset_set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Odel__point__blocks_001tf__a,type,
design6411949732824333445ocks_a: multiset_set_a > a > multiset_set_a ).
thf(sy_c_Design__Operations_Oincidence__system_Ostr__del__point__blocks_001tf__a,type,
design5657747894866638574ocks_a: multiset_set_a > a > multiset_set_a ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
finite5064046906505528594_set_a: set_list_list_set_a > nat ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
finite_card_list_nat: set_list_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
finite4179508071619380492_set_a: set_list_set_a > nat ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_Itf__a_J,type,
finite_card_list_a: set_list_a > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_Itf__a_J,type,
finite_card_set_a: set_set_a > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
finite1971793804006318733_set_a: set_list_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
finite2815193924343055693_set_a: set_multiset_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
minus_2572999347059163665_set_a: multiset_list_set_a > multiset_list_set_a > multiset_list_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
minus_8522176038001411705et_nat: multiset_nat > multiset_nat > multiset_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
minus_706656509937749387_set_a: multiset_set_a > multiset_set_a > multiset_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_Itf__a_J,type,
minus_3765977307040488491iset_a: multiset_a > multiset_a > multiset_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__Set__Oset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
minus_5993239077472792235_set_a: set_list_set_a > set_list_set_a > set_list_set_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_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
plus_p4509188130224566113_set_a: multiset_list_set_a > multiset_list_set_a > multiset_list_set_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
plus_p6334493942879108393et_nat: multiset_nat > multiset_nat > multiset_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
plus_p2331992037799027419_set_a: multiset_set_a > multiset_set_a > multiset_set_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_Itf__a_J,type,
plus_plus_multiset_a: multiset_a > multiset_a > multiset_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
zero_z8272816460787710433_set_a: multiset_list_set_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
zero_z7348594199698428585et_nat: multiset_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
zero_z5079479921072680283_set_a: multiset_set_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_Itf__a_J,type,
zero_zero_multiset_a: multiset_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_List_Olist_ONil_001t__Set__Oset_Itf__a_J,type,
nil_set_a: list_set_a ).
thf(sy_c_List_Onth_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
nth_list_set_a: list_list_set_a > nat > list_set_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_Multiset_Oadd__mset_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
add_mset_list_set_a: list_set_a > multiset_list_set_a > multiset_list_set_a ).
thf(sy_c_Multiset_Oadd__mset_001t__Nat__Onat,type,
add_mset_nat: nat > multiset_nat > multiset_nat ).
thf(sy_c_Multiset_Oadd__mset_001t__Set__Oset_Itf__a_J,type,
add_mset_set_a: set_a > multiset_set_a > multiset_set_a ).
thf(sy_c_Multiset_Oadd__mset_001tf__a,type,
add_mset_a: a > multiset_a > multiset_a ).
thf(sy_c_Multiset_Omset_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
mset_list_set_a: list_list_set_a > multiset_list_set_a ).
thf(sy_c_Multiset_Omset_001t__Nat__Onat,type,
mset_nat: list_nat > multiset_nat ).
thf(sy_c_Multiset_Omset_001t__Set__Oset_Itf__a_J,type,
mset_set_a: list_set_a > multiset_set_a ).
thf(sy_c_Multiset_Omset_001tf__a,type,
mset_a: list_a > multiset_a ).
thf(sy_c_Multiset_Omultiset_Ocount_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
count_list_set_a: multiset_list_set_a > list_set_a > nat ).
thf(sy_c_Multiset_Omultiset_Ocount_001t__Nat__Onat,type,
count_nat: multiset_nat > nat > nat ).
thf(sy_c_Multiset_Omultiset_Ocount_001t__Set__Oset_Itf__a_J,type,
count_set_a: multiset_set_a > set_a > nat ).
thf(sy_c_Multiset_Omultiset_Ocount_001tf__a,type,
count_a: multiset_a > a > nat ).
thf(sy_c_Multiset_Orepeat__mset_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
repeat2273734958817108158_set_a: nat > multiset_list_set_a > multiset_list_set_a ).
thf(sy_c_Multiset_Orepeat__mset_001t__Nat__Onat,type,
repeat_mset_nat: nat > multiset_nat > multiset_nat ).
thf(sy_c_Multiset_Orepeat__mset_001t__Set__Oset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
repeat6861153585859484830_set_a: nat > multis7946784228617641354_set_a > multis7946784228617641354_set_a ).
thf(sy_c_Multiset_Orepeat__mset_001t__Set__Oset_It__Nat__Onat_J,type,
repeat_mset_set_nat: nat > multiset_set_nat > multiset_set_nat ).
thf(sy_c_Multiset_Orepeat__mset_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
repeat3222187171979612824_set_a: nat > multiset_set_set_a > multiset_set_set_a ).
thf(sy_c_Multiset_Orepeat__mset_001t__Set__Oset_Itf__a_J,type,
repeat_mset_set_a: nat > multiset_set_a > multiset_set_a ).
thf(sy_c_Multiset_Orepeat__mset_001tf__a,type,
repeat_mset_a: nat > multiset_a > multiset_a ).
thf(sy_c_Multiset_Oset__mset_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
set_mset_list_set_a: multiset_list_set_a > set_list_set_a ).
thf(sy_c_Multiset_Oset__mset_001t__Nat__Onat,type,
set_mset_nat: multiset_nat > set_nat ).
thf(sy_c_Multiset_Oset__mset_001t__Set__Oset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
set_ms531227285539831169_set_a: multis7946784228617641354_set_a > set_set_list_set_a ).
thf(sy_c_Multiset_Oset__mset_001t__Set__Oset_It__Nat__Onat_J,type,
set_mset_set_nat: multiset_set_nat > set_set_nat ).
thf(sy_c_Multiset_Oset__mset_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_mset_set_set_a: multiset_set_set_a > set_set_set_a ).
thf(sy_c_Multiset_Oset__mset_001t__Set__Oset_Itf__a_J,type,
set_mset_set_a: multiset_set_a > set_set_a ).
thf(sy_c_Multiset_Oset__mset_001tf__a,type,
set_mset_a: multiset_a > set_a ).
thf(sy_c_Multiset_Osubseteq__mset_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
subset925826154347786955_set_a: multiset_list_set_a > multiset_list_set_a > $o ).
thf(sy_c_Multiset_Osubseteq__mset_001t__Nat__Onat,type,
subseteq_mset_nat: multiset_nat > multiset_nat > $o ).
thf(sy_c_Multiset_Osubseteq__mset_001t__Set__Oset_Itf__a_J,type,
subseteq_mset_set_a: multiset_set_a > multiset_set_a > $o ).
thf(sy_c_Multiset_Osubseteq__mset_001tf__a,type,
subseteq_mset_a: multiset_a > multiset_a > $o ).
thf(sy_c_Multiset__More_Olist__of__mset_001t__Set__Oset_Itf__a_J,type,
multis2300860884436073961_set_a: multiset_set_a > list_set_a ).
thf(sy_c_Multiset__Permutations_Opermutations__of__multiset_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
multis802782155138758580_set_a: multiset_list_set_a > set_list_list_set_a ).
thf(sy_c_Multiset__Permutations_Opermutations__of__multiset_001t__Nat__Onat,type,
multis6201468865946971392et_nat: multiset_nat > set_list_nat ).
thf(sy_c_Multiset__Permutations_Opermutations__of__multiset_001t__Set__Oset_Itf__a_J,type,
multis5469701301851823918_set_a: multiset_set_a > set_list_set_a ).
thf(sy_c_Multiset__Permutations_Opermutations__of__multiset_001tf__a,type,
multis5886240593633752526iset_a: multiset_a > set_list_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
size_s3202270027169943894_set_a: list_list_set_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__Multiset__Omultiset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
size_s2437250918661492246_set_a: multiset_list_set_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
size_s5917832649809541300et_nat: multiset_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
size_s6566526139600085008_set_a: multiset_set_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_Itf__a_J,type,
size_size_multiset_a: multiset_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Nat__Onat,type,
size_size_nat: nat > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
bot_bo4397488018069675312_set_a: set_list_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J_J,type,
bot_bo9088538438451294192_set_a: set_multiset_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
ord_le5777773500796000884et_nat: multiset_nat > multiset_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
ord_le5765082015083327056_set_a: multiset_set_a > multiset_set_a > $o ).
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__Set__Oset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
ord_le2831269692516795760_set_a: set_list_set_a > set_list_set_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__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_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
ord_le6602235886369790592et_nat: multiset_nat > multiset_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
ord_le7905258569527593284_set_a: multiset_set_a > multiset_set_a > $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__Set__Oset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
ord_le864617614081865828_set_a: set_list_set_a > set_list_set_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__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_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
collect_list_set_a: ( list_set_a > $o ) > set_list_set_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_member_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
member_list_set_a: list_set_a > set_list_set_a > $o ).
thf(sy_c_member_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
member2747690772047059533_set_a: multiset_set_a > set_multiset_set_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
member844995724677003117_set_a: set_list_set_a > set_set_list_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $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_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v__092_060B_062s,type,
b_s: list_set_a ).
thf(sy_v_i1,type,
i1: nat ).
thf(sy_v_i2,type,
i2: nat ).
% Relevant facts (1268)
thf(fact_0_assms_I3_J,axiom,
i1 != i2 ).
% assms(3)
thf(fact_1_False,axiom,
( ( nth_set_a @ b_s @ i1 )
!= ( nth_set_a @ b_s @ i2 ) ) ).
% False
thf(fact_2_design__support__def,axiom,
( ( design5397942185814921632port_a @ ( mset_set_a @ b_s ) )
= ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ).
% design_support_def
thf(fact_3_add__point__existing__blocks,axiom,
! [Bs: set_set_a,P: a] :
( ! [Bl: set_a] :
( ( member_set_a @ Bl @ Bs )
=> ( member_a @ P @ Bl ) )
=> ( ( design2935547469388721088ocks_a @ ( mset_set_a @ b_s ) @ P @ Bs )
= ( mset_set_a @ b_s ) ) ) ).
% add_point_existing_blocks
thf(fact_4_delete__point__p__not__in__bl__blocks,axiom,
! [P: a] :
( ! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ~ ( member_a @ P @ Bl ) )
=> ( ( design6411949732824333445ocks_a @ ( mset_set_a @ b_s ) @ P )
= ( mset_set_a @ b_s ) ) ) ).
% delete_point_p_not_in_bl_blocks
thf(fact_5_delete__point__strong__block__in,axiom,
! [P: a,Bl2: set_a] :
( ~ ( member_a @ P @ Bl2 )
=> ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P ) ) ) ) ) ).
% delete_point_strong_block_in
thf(fact_6_delete__point__strong__block__in__iff,axiom,
! [Bl2: set_a,P: a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P ) ) )
= ( ~ ( member_a @ P @ Bl2 ) ) ) ) ).
% delete_point_strong_block_in_iff
thf(fact_7_delete__point__strong__block__in__orig,axiom,
! [Bl2: set_a,P: a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P ) ) )
=> ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) ).
% delete_point_strong_block_in_orig
thf(fact_8_delete__point__strong__block__not__in,axiom,
! [P: a,Bl2: set_a] :
( ( member_a @ P @ Bl2 )
=> ~ ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P ) ) ) ) ).
% delete_point_strong_block_not_in
thf(fact_9_delete__invalid__block__eq,axiom,
! [B: set_a] :
( ~ ( member_set_a @ B @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ B )
= ( mset_set_a @ b_s ) ) ) ).
% delete_invalid_block_eq
thf(fact_10_block__set__nempty__imp__block__ex,axiom,
( ( ( mset_set_a @ b_s )
!= zero_z5079479921072680283_set_a )
=> ? [Bl: set_a] : ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) ).
% block_set_nempty_imp_block_ex
thf(fact_11_blocks__indexing,axiom,
member_list_set_a @ b_s @ ( multis5469701301851823918_set_a @ ( mset_set_a @ b_s ) ) ).
% blocks_indexing
thf(fact_12_finite__blocks,axiom,
! [B: set_a] :
( ( member_set_a @ B @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( finite_finite_a @ B ) ) ).
% finite_blocks
thf(fact_13_multiple__block__in__original,axiom,
! [B: set_a,N: nat] :
( ( member_set_a @ B @ ( set_mset_set_a @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) ) )
=> ( member_set_a @ B @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) ).
% multiple_block_in_original
thf(fact_14_assms_I2_J,axiom,
ord_less_nat @ i2 @ ( size_size_list_set_a @ b_s ) ).
% assms(2)
thf(fact_15_assms_I1_J,axiom,
ord_less_nat @ i1 @ ( size_size_list_set_a @ b_s ) ).
% assms(1)
thf(fact_16_del__invalid__add__block__eq,axiom,
! [Bl2: set_a] :
( ~ ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( design4001997691126659652lock_a @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ Bl2 ) @ Bl2 )
= ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ Bl2 ) ) ) ).
% del_invalid_add_block_eq
thf(fact_17_del__add__block__inv,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( design4001997691126659652lock_a @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ Bl2 ) @ Bl2 )
= ( mset_set_a @ b_s ) ) ) ).
% del_add_block_inv
thf(fact_18_multiple__block__in,axiom,
! [N: nat,B: set_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( member_set_a @ B @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( member_set_a @ B @ ( set_mset_set_a @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) ) ) ) ) ).
% multiple_block_in
thf(fact_19_repeat__mset__empty,axiom,
! [N: nat] :
( ( repeat_mset_set_a @ N @ zero_z5079479921072680283_set_a )
= zero_z5079479921072680283_set_a ) ).
% repeat_mset_empty
thf(fact_20_in__mset__conv__nth,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_mset_a @ ( mset_a @ Xs ) ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ I )
= X ) ) ) ) ).
% in_mset_conv_nth
thf(fact_21_in__mset__conv__nth,axiom,
! [X: list_set_a,Xs: list_list_set_a] :
( ( member_list_set_a @ X @ ( set_mset_list_set_a @ ( mset_list_set_a @ Xs ) ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_s3202270027169943894_set_a @ Xs ) )
& ( ( nth_list_set_a @ Xs @ I )
= X ) ) ) ) ).
% in_mset_conv_nth
thf(fact_22_in__mset__conv__nth,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_mset_nat @ ( mset_nat @ Xs ) ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I )
= X ) ) ) ) ).
% in_mset_conv_nth
thf(fact_23_in__mset__conv__nth,axiom,
! [X: set_a,Xs: list_set_a] :
( ( member_set_a @ X @ ( set_mset_set_a @ ( mset_set_a @ Xs ) ) )
= ( ? [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_set_a @ Xs ) )
& ( ( nth_set_a @ Xs @ I )
= X ) ) ) ) ).
% in_mset_conv_nth
thf(fact_24_nth__mem__mset,axiom,
! [I2: nat,Ls: list_a] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ls ) )
=> ( member_a @ ( nth_a @ Ls @ I2 ) @ ( set_mset_a @ ( mset_a @ Ls ) ) ) ) ).
% nth_mem_mset
thf(fact_25_nth__mem__mset,axiom,
! [I2: nat,Ls: list_list_set_a] :
( ( ord_less_nat @ I2 @ ( size_s3202270027169943894_set_a @ Ls ) )
=> ( member_list_set_a @ ( nth_list_set_a @ Ls @ I2 ) @ ( set_mset_list_set_a @ ( mset_list_set_a @ Ls ) ) ) ) ).
% nth_mem_mset
thf(fact_26_nth__mem__mset,axiom,
! [I2: nat,Ls: list_nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ls ) )
=> ( member_nat @ ( nth_nat @ Ls @ I2 ) @ ( set_mset_nat @ ( mset_nat @ Ls ) ) ) ) ).
% nth_mem_mset
thf(fact_27_nth__mem__mset,axiom,
! [I2: nat,Ls: list_set_a] :
( ( ord_less_nat @ I2 @ ( size_size_list_set_a @ Ls ) )
=> ( member_set_a @ ( nth_set_a @ Ls @ I2 ) @ ( set_mset_set_a @ ( mset_set_a @ Ls ) ) ) ) ).
% nth_mem_mset
thf(fact_28_finite__set__mset,axiom,
! [M: multiset_a] : ( finite_finite_a @ ( set_mset_a @ M ) ) ).
% finite_set_mset
thf(fact_29_finite__set__mset,axiom,
! [M: multiset_nat] : ( finite_finite_nat @ ( set_mset_nat @ M ) ) ).
% finite_set_mset
thf(fact_30_finite__set__mset,axiom,
! [M: multiset_set_a] : ( finite_finite_set_a @ ( set_mset_set_a @ M ) ) ).
% finite_set_mset
thf(fact_31_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_32_delete__point__blocks__sub,axiom,
! [B: set_a,P: a] :
( ( member_set_a @ B @ ( set_mset_set_a @ ( design6411949732824333445ocks_a @ ( mset_set_a @ b_s ) @ P ) ) )
=> ~ ! [Bl: set_a] :
~ ( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
& ( ord_less_eq_set_a @ B @ Bl ) ) ) ).
% delete_point_blocks_sub
thf(fact_33_valid__blocks__index__obtains,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ~ ! [J: nat] :
~ ( ( ( nth_set_a @ b_s @ J )
= Bl2 )
& ( ord_less_nat @ J @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ) ) ).
% valid_blocks_index_obtains
thf(fact_34_valid__blocks__index__cons,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ? [J: nat] :
( ( ( nth_set_a @ b_s @ J )
= Bl2 )
& ( ord_less_nat @ J @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ) ) ).
% valid_blocks_index_cons
thf(fact_35_valid__blocks__index,axiom,
! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ( member_set_a @ ( nth_set_a @ b_s @ J2 ) @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) ).
% valid_blocks_index
thf(fact_36_finite__design__support,axiom,
finite_finite_set_a @ ( design5397942185814921632port_a @ ( mset_set_a @ b_s ) ) ).
% finite_design_support
thf(fact_37_del__point__block__count,axiom,
! [P: a] :
( ( size_s6566526139600085008_set_a @ ( design6411949732824333445ocks_a @ ( mset_set_a @ b_s ) @ P ) )
= ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ).
% del_point_block_count
thf(fact_38_blocks__list__length,axiom,
( ( size_size_list_set_a @ b_s )
= ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ).
% blocks_list_length
thf(fact_39_del__block__b_I2_J,axiom,
! [Bl2: set_a] :
( ~ ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( size_s6566526139600085008_set_a @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ Bl2 ) )
= ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ) ).
% del_block_b(2)
thf(fact_40_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_41_size__empty,axiom,
( ( size_s6566526139600085008_set_a @ zero_z5079479921072680283_set_a )
= zero_zero_nat ) ).
% size_empty
thf(fact_42_size__eq__0__iff__empty,axiom,
! [M: multiset_set_a] :
( ( ( size_s6566526139600085008_set_a @ M )
= zero_zero_nat )
= ( M = zero_z5079479921072680283_set_a ) ) ).
% size_eq_0_iff_empty
thf(fact_43_size__mset,axiom,
! [Xs: list_set_a] :
( ( size_s6566526139600085008_set_a @ ( mset_set_a @ Xs ) )
= ( size_size_list_set_a @ Xs ) ) ).
% size_mset
thf(fact_44_repeat__mset__0,axiom,
! [M: multiset_set_a] :
( ( repeat_mset_set_a @ zero_zero_nat @ M )
= zero_z5079479921072680283_set_a ) ).
% repeat_mset_0
thf(fact_45_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_46_nonempty__has__size,axiom,
! [S: multiset_set_a] :
( ( S != zero_z5079479921072680283_set_a )
= ( ord_less_nat @ zero_zero_nat @ ( size_s6566526139600085008_set_a @ S ) ) ) ).
% nonempty_has_size
thf(fact_47_repeat__mset__cancel1,axiom,
! [A: nat,A2: multiset_set_a,B2: multiset_set_a] :
( ( ( repeat_mset_set_a @ A @ A2 )
= ( repeat_mset_set_a @ A @ B2 ) )
= ( ( A2 = B2 )
| ( A = zero_zero_nat ) ) ) ).
% repeat_mset_cancel1
thf(fact_48_repeat__mset__eq__empty__iff,axiom,
! [N: nat,A2: multiset_set_a] :
( ( ( repeat_mset_set_a @ N @ A2 )
= zero_z5079479921072680283_set_a )
= ( ( N = zero_zero_nat )
| ( A2 = zero_z5079479921072680283_set_a ) ) ) ).
% repeat_mset_eq_empty_iff
thf(fact_49_zero__reorient,axiom,
! [X: multiset_set_a] :
( ( zero_z5079479921072680283_set_a = X )
= ( X = zero_z5079479921072680283_set_a ) ) ).
% zero_reorient
thf(fact_50_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_51_ex__mset,axiom,
! [X2: multiset_set_a] :
? [Xs2: list_set_a] :
( ( mset_set_a @ Xs2 )
= X2 ) ).
% ex_mset
thf(fact_52_list__of__mset__exi,axiom,
! [M2: multiset_set_a] :
? [L: list_set_a] :
( M2
= ( mset_set_a @ L ) ) ).
% list_of_mset_exi
thf(fact_53_mem__Collect__eq,axiom,
! [A: set_a,P2: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_54_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_55_mem__Collect__eq,axiom,
! [A: list_set_a,P2: list_set_a > $o] :
( ( member_list_set_a @ A @ ( collect_list_set_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_56_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_57_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_58_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_59_Collect__mem__eq,axiom,
! [A2: set_list_set_a] :
( ( collect_list_set_a
@ ^ [X3: list_set_a] : ( member_list_set_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_60_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_61_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_62_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_63_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_64_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_65_multiset__nonemptyE,axiom,
! [A2: multiset_a] :
( ( A2 != zero_zero_multiset_a )
=> ~ ! [X4: a] :
~ ( member_a @ X4 @ ( set_mset_a @ A2 ) ) ) ).
% multiset_nonemptyE
thf(fact_66_multiset__nonemptyE,axiom,
! [A2: multiset_list_set_a] :
( ( A2 != zero_z8272816460787710433_set_a )
=> ~ ! [X4: list_set_a] :
~ ( member_list_set_a @ X4 @ ( set_mset_list_set_a @ A2 ) ) ) ).
% multiset_nonemptyE
thf(fact_67_multiset__nonemptyE,axiom,
! [A2: multiset_nat] :
( ( A2 != zero_z7348594199698428585et_nat )
=> ~ ! [X4: nat] :
~ ( member_nat @ X4 @ ( set_mset_nat @ A2 ) ) ) ).
% multiset_nonemptyE
thf(fact_68_multiset__nonemptyE,axiom,
! [A2: multiset_set_a] :
( ( A2 != zero_z5079479921072680283_set_a )
=> ~ ! [X4: set_a] :
~ ( member_set_a @ X4 @ ( set_mset_set_a @ A2 ) ) ) ).
% multiset_nonemptyE
thf(fact_69_mset__eq__length,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( mset_set_a @ Xs )
= ( mset_set_a @ Ys ) )
=> ( ( size_size_list_set_a @ Xs )
= ( size_size_list_set_a @ Ys ) ) ) ).
% mset_eq_length
thf(fact_70_repeat__mset__cancel2,axiom,
! [A: nat,A2: multiset_set_a,B: nat] :
( ( ( repeat_mset_set_a @ A @ A2 )
= ( repeat_mset_set_a @ B @ A2 ) )
= ( ( A = B )
| ( A2 = zero_z5079479921072680283_set_a ) ) ) ).
% repeat_mset_cancel2
thf(fact_71_multiple__block__sizes__same,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) )
= ( design1769254222028858111izes_a @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) ) ) ) ).
% multiple_block_sizes_same
thf(fact_72_multiple__blocks__gt,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) @ ( size_s6566526139600085008_set_a @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) ) ) ) ).
% multiple_blocks_gt
thf(fact_73_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_74_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_75_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_76_multiple__blocks__sub,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( subseteq_mset_set_a @ ( mset_set_a @ b_s ) @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) ) ) ).
% multiple_blocks_sub
thf(fact_77_finite__block__sizes,axiom,
finite_finite_nat @ ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) ) ).
% finite_block_sizes
thf(fact_78_repeat__mset__not__empty,axiom,
! [N: nat,A2: multiset_set_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( A2 != zero_z5079479921072680283_set_a )
=> ( ( repeat_mset_set_a @ N @ A2 )
!= zero_z5079479921072680283_set_a ) ) ) ).
% repeat_mset_not_empty
thf(fact_79_elem__in__original__in__repeat,axiom,
! [N: nat,A: a,A2: multiset_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( member_a @ A @ ( set_mset_a @ A2 ) )
=> ( member_a @ A @ ( set_mset_a @ ( repeat_mset_a @ N @ A2 ) ) ) ) ) ).
% elem_in_original_in_repeat
thf(fact_80_elem__in__original__in__repeat,axiom,
! [N: nat,A: list_set_a,A2: multiset_list_set_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( member_list_set_a @ A @ ( set_mset_list_set_a @ A2 ) )
=> ( member_list_set_a @ A @ ( set_mset_list_set_a @ ( repeat2273734958817108158_set_a @ N @ A2 ) ) ) ) ) ).
% elem_in_original_in_repeat
thf(fact_81_elem__in__original__in__repeat,axiom,
! [N: nat,A: nat,A2: multiset_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( member_nat @ A @ ( set_mset_nat @ A2 ) )
=> ( member_nat @ A @ ( set_mset_nat @ ( repeat_mset_nat @ N @ A2 ) ) ) ) ) ).
% elem_in_original_in_repeat
thf(fact_82_elem__in__original__in__repeat,axiom,
! [N: nat,A: set_a,A2: multiset_set_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( member_set_a @ A @ ( set_mset_set_a @ A2 ) )
=> ( member_set_a @ A @ ( set_mset_set_a @ ( repeat_mset_set_a @ N @ A2 ) ) ) ) ) ).
% elem_in_original_in_repeat
thf(fact_83_multiple__1__same,axiom,
( ( repeat_mset_set_a @ one_one_nat @ ( mset_set_a @ b_s ) )
= ( mset_set_a @ b_s ) ) ).
% multiple_1_same
thf(fact_84_blocks__list__empty__iff,axiom,
( ( b_s = nil_set_a )
= ( ( mset_set_a @ b_s )
= zero_z5079479921072680283_set_a ) ) ).
% blocks_list_empty_iff
thf(fact_85_subset__mset_Odual__order_Orefl,axiom,
! [A: multiset_set_a] : ( subseteq_mset_set_a @ A @ A ) ).
% subset_mset.dual_order.refl
thf(fact_86_subset__mset_Oorder__refl,axiom,
! [X: multiset_set_a] : ( subseteq_mset_set_a @ X @ X ) ).
% subset_mset.order_refl
thf(fact_87_delete__block__subset,axiom,
! [B: set_a] : ( subseteq_mset_set_a @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ B ) @ ( mset_set_a @ b_s ) ) ).
% delete_block_subset
thf(fact_88_delete__point__strong__block__subset,axiom,
! [P: a] : ( subseteq_mset_set_a @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P ) @ ( mset_set_a @ b_s ) ) ).
% delete_point_strong_block_subset
thf(fact_89_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_90_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_91_subset__mset_Ole__zero__eq,axiom,
! [N: multiset_set_a] :
( ( subseteq_mset_set_a @ N @ zero_z5079479921072680283_set_a )
= ( N = zero_z5079479921072680283_set_a ) ) ).
% subset_mset.le_zero_eq
thf(fact_92_subset__mset_Oextremum__unique,axiom,
! [A: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ zero_z5079479921072680283_set_a )
= ( A = zero_z5079479921072680283_set_a ) ) ).
% subset_mset.extremum_unique
thf(fact_93_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_94_mset__zero__iff,axiom,
! [X: list_set_a] :
( ( ( mset_set_a @ X )
= zero_z5079479921072680283_set_a )
= ( X = nil_set_a ) ) ).
% mset_zero_iff
thf(fact_95_mset__zero__iff__right,axiom,
! [X: list_set_a] :
( ( zero_z5079479921072680283_set_a
= ( mset_set_a @ X ) )
= ( X = nil_set_a ) ) ).
% mset_zero_iff_right
thf(fact_96_subset__mset_Ofinite__has__maximal2,axiom,
! [A2: set_multiset_set_a,A: multiset_set_a] :
( ( finite2815193924343055693_set_a @ A2 )
=> ( ( member2747690772047059533_set_a @ A @ A2 )
=> ? [X4: multiset_set_a] :
( ( member2747690772047059533_set_a @ X4 @ A2 )
& ( subseteq_mset_set_a @ A @ X4 )
& ! [Xa: multiset_set_a] :
( ( member2747690772047059533_set_a @ Xa @ A2 )
=> ( ( subseteq_mset_set_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% subset_mset.finite_has_maximal2
thf(fact_97_subset__mset_Ofinite__has__minimal2,axiom,
! [A2: set_multiset_set_a,A: multiset_set_a] :
( ( finite2815193924343055693_set_a @ A2 )
=> ( ( member2747690772047059533_set_a @ A @ A2 )
=> ? [X4: multiset_set_a] :
( ( member2747690772047059533_set_a @ X4 @ A2 )
& ( subseteq_mset_set_a @ X4 @ A )
& ! [Xa: multiset_set_a] :
( ( member2747690772047059533_set_a @ Xa @ A2 )
=> ( ( subseteq_mset_set_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% subset_mset.finite_has_minimal2
thf(fact_98_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B: nat] :
( ( P2 @ K )
=> ( ! [Y: nat] :
( ( P2 @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ? [X4: nat] :
( ( P2 @ X4 )
& ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_99_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_100_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_101_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_102_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_103_le__trans,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_104_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_105_subset__mset_Odual__order_Oantisym,axiom,
! [B: multiset_set_a,A: multiset_set_a] :
( ( subseteq_mset_set_a @ B @ A )
=> ( ( subseteq_mset_set_a @ A @ B )
=> ( A = B ) ) ) ).
% subset_mset.dual_order.antisym
thf(fact_106_subset__mset_Odual__order_Oeq__iff,axiom,
( ( ^ [Y3: multiset_set_a,Z: multiset_set_a] : ( Y3 = Z ) )
= ( ^ [A3: multiset_set_a,B3: multiset_set_a] :
( ( subseteq_mset_set_a @ B3 @ A3 )
& ( subseteq_mset_set_a @ A3 @ B3 ) ) ) ) ).
% subset_mset.dual_order.eq_iff
thf(fact_107_subset__mset_Odual__order_Otrans,axiom,
! [B: multiset_set_a,A: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ B @ A )
=> ( ( subseteq_mset_set_a @ C @ B )
=> ( subseteq_mset_set_a @ C @ A ) ) ) ).
% subset_mset.dual_order.trans
thf(fact_108_subset__mset_Oord__le__eq__trans,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B )
=> ( ( B = C )
=> ( subseteq_mset_set_a @ A @ C ) ) ) ).
% subset_mset.ord_le_eq_trans
thf(fact_109_subset__mset_Oord__eq__le__trans,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( A = B )
=> ( ( subseteq_mset_set_a @ B @ C )
=> ( subseteq_mset_set_a @ A @ C ) ) ) ).
% subset_mset.ord_eq_le_trans
thf(fact_110_subset__mset_Oorder__antisym,axiom,
! [X: multiset_set_a,Y4: multiset_set_a] :
( ( subseteq_mset_set_a @ X @ Y4 )
=> ( ( subseteq_mset_set_a @ Y4 @ X )
=> ( X = Y4 ) ) ) ).
% subset_mset.order_antisym
thf(fact_111_subset__mset_Oorder__eq__iff,axiom,
( ( ^ [Y3: multiset_set_a,Z: multiset_set_a] : ( Y3 = Z ) )
= ( ^ [X3: multiset_set_a,Y5: multiset_set_a] :
( ( subseteq_mset_set_a @ X3 @ Y5 )
& ( subseteq_mset_set_a @ Y5 @ X3 ) ) ) ) ).
% subset_mset.order_eq_iff
thf(fact_112_subset__mset_Oantisym__conv,axiom,
! [Y4: multiset_set_a,X: multiset_set_a] :
( ( subseteq_mset_set_a @ Y4 @ X )
=> ( ( subseteq_mset_set_a @ X @ Y4 )
= ( X = Y4 ) ) ) ).
% subset_mset.antisym_conv
thf(fact_113_subset__mset_Oorder__trans,axiom,
! [X: multiset_set_a,Y4: multiset_set_a,Z2: multiset_set_a] :
( ( subseteq_mset_set_a @ X @ Y4 )
=> ( ( subseteq_mset_set_a @ Y4 @ Z2 )
=> ( subseteq_mset_set_a @ X @ Z2 ) ) ) ).
% subset_mset.order_trans
thf(fact_114_subset__mset_Oeq__refl,axiom,
! [X: multiset_set_a,Y4: multiset_set_a] :
( ( X = Y4 )
=> ( subseteq_mset_set_a @ X @ Y4 ) ) ).
% subset_mset.eq_refl
thf(fact_115_subset__mset_Oantisym,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B )
=> ( ( subseteq_mset_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_mset.antisym
thf(fact_116_subset__mset_Oeq__iff,axiom,
( ( ^ [Y3: multiset_set_a,Z: multiset_set_a] : ( Y3 = Z ) )
= ( ^ [A3: multiset_set_a,B3: multiset_set_a] :
( ( subseteq_mset_set_a @ A3 @ B3 )
& ( subseteq_mset_set_a @ B3 @ A3 ) ) ) ) ).
% subset_mset.eq_iff
thf(fact_117_subset__mset_Otrans,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B )
=> ( ( subseteq_mset_set_a @ B @ C )
=> ( subseteq_mset_set_a @ A @ C ) ) ) ).
% subset_mset.trans
thf(fact_118_size__mset__mono,axiom,
! [A2: multiset_set_a,B2: multiset_set_a] :
( ( subseteq_mset_set_a @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( size_s6566526139600085008_set_a @ A2 ) @ ( size_s6566526139600085008_set_a @ B2 ) ) ) ).
% size_mset_mono
thf(fact_119_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
& ( M3 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_120_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_121_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_122_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_123_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_124_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J2: nat] :
( ! [I3: nat,J: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_125_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_126_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_127_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_128_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_129_mset__subset__eqD,axiom,
! [A2: multiset_a,B2: multiset_a,X: a] :
( ( subseteq_mset_a @ A2 @ B2 )
=> ( ( member_a @ X @ ( set_mset_a @ A2 ) )
=> ( member_a @ X @ ( set_mset_a @ B2 ) ) ) ) ).
% mset_subset_eqD
thf(fact_130_mset__subset__eqD,axiom,
! [A2: multiset_list_set_a,B2: multiset_list_set_a,X: list_set_a] :
( ( subset925826154347786955_set_a @ A2 @ B2 )
=> ( ( member_list_set_a @ X @ ( set_mset_list_set_a @ A2 ) )
=> ( member_list_set_a @ X @ ( set_mset_list_set_a @ B2 ) ) ) ) ).
% mset_subset_eqD
thf(fact_131_mset__subset__eqD,axiom,
! [A2: multiset_nat,B2: multiset_nat,X: nat] :
( ( subseteq_mset_nat @ A2 @ B2 )
=> ( ( member_nat @ X @ ( set_mset_nat @ A2 ) )
=> ( member_nat @ X @ ( set_mset_nat @ B2 ) ) ) ) ).
% mset_subset_eqD
thf(fact_132_mset__subset__eqD,axiom,
! [A2: multiset_set_a,B2: multiset_set_a,X: set_a] :
( ( subseteq_mset_set_a @ A2 @ B2 )
=> ( ( member_set_a @ X @ ( set_mset_set_a @ A2 ) )
=> ( member_set_a @ X @ ( set_mset_set_a @ B2 ) ) ) ) ).
% mset_subset_eqD
thf(fact_133_subset__mset_Ozero__le,axiom,
! [X: multiset_set_a] : ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ X ) ).
% subset_mset.zero_le
thf(fact_134_subset__mset_Obot__least,axiom,
! [A: multiset_set_a] : ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ A ) ).
% subset_mset.bot_least
thf(fact_135_subset__mset_Oextremum__uniqueI,axiom,
! [A: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ zero_z5079479921072680283_set_a )
=> ( A = zero_z5079479921072680283_set_a ) ) ).
% subset_mset.extremum_uniqueI
thf(fact_136_empty__le,axiom,
! [A2: multiset_set_a] : ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ A2 ) ).
% empty_le
thf(fact_137_subseteq__mset__size__eql,axiom,
! [X2: multiset_set_a,Y6: multiset_set_a] :
( ( subseteq_mset_set_a @ X2 @ Y6 )
=> ( ( ( size_s6566526139600085008_set_a @ Y6 )
= ( size_s6566526139600085008_set_a @ X2 ) )
=> ( X2 = Y6 ) ) ) ).
% subseteq_mset_size_eql
thf(fact_138_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P2 @ I4 ) )
& ( P2 @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_139_set__mset__mono,axiom,
! [A2: multiset_set_a,B2: multiset_set_a] :
( ( subseteq_mset_set_a @ A2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( set_mset_set_a @ A2 ) @ ( set_mset_set_a @ B2 ) ) ) ).
% set_mset_mono
thf(fact_140_set__mset__mono,axiom,
! [A2: multiset_a,B2: multiset_a] :
( ( subseteq_mset_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ ( set_mset_a @ A2 ) @ ( set_mset_a @ B2 ) ) ) ).
% set_mset_mono
thf(fact_141_mset_Osimps_I1_J,axiom,
( ( mset_set_a @ nil_set_a )
= zero_z5079479921072680283_set_a ) ).
% mset.simps(1)
thf(fact_142_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_143_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_144_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_145_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_146_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_147_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( P2 @ M4 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_148_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P2 @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P2 @ M4 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_149_linorder__neqE__nat,axiom,
! [X: nat,Y4: nat] :
( ( X != Y4 )
=> ( ~ ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ Y4 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_150_size__neq__size__imp__neq,axiom,
! [X: list_set_a,Y4: list_set_a] :
( ( ( size_size_list_set_a @ X )
!= ( size_size_list_set_a @ Y4 ) )
=> ( X != Y4 ) ) ).
% size_neq_size_imp_neq
thf(fact_151_size__neq__size__imp__neq,axiom,
! [X: multiset_set_a,Y4: multiset_set_a] :
( ( ( size_s6566526139600085008_set_a @ X )
!= ( size_s6566526139600085008_set_a @ Y4 ) )
=> ( X != Y4 ) ) ).
% size_neq_size_imp_neq
thf(fact_152_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_153_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_154_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_155_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_156_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_157_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_158_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P2 @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P2 @ M4 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_159_elem__in__repeat__in__original,axiom,
! [A: a,N: nat,A2: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( repeat_mset_a @ N @ A2 ) ) )
=> ( member_a @ A @ ( set_mset_a @ A2 ) ) ) ).
% elem_in_repeat_in_original
thf(fact_160_elem__in__repeat__in__original,axiom,
! [A: list_set_a,N: nat,A2: multiset_list_set_a] :
( ( member_list_set_a @ A @ ( set_mset_list_set_a @ ( repeat2273734958817108158_set_a @ N @ A2 ) ) )
=> ( member_list_set_a @ A @ ( set_mset_list_set_a @ A2 ) ) ) ).
% elem_in_repeat_in_original
thf(fact_161_elem__in__repeat__in__original,axiom,
! [A: nat,N: nat,A2: multiset_nat] :
( ( member_nat @ A @ ( set_mset_nat @ ( repeat_mset_nat @ N @ A2 ) ) )
=> ( member_nat @ A @ ( set_mset_nat @ A2 ) ) ) ).
% elem_in_repeat_in_original
thf(fact_162_elem__in__repeat__in__original,axiom,
! [A: set_a,N: nat,A2: multiset_set_a] :
( ( member_set_a @ A @ ( set_mset_set_a @ ( repeat_mset_set_a @ N @ A2 ) ) )
=> ( member_set_a @ A @ ( set_mset_set_a @ A2 ) ) ) ).
% elem_in_repeat_in_original
thf(fact_163_repeat__mset__subset__in,axiom,
! [A2: multiset_set_set_a,B2: set_set_a,X2: set_set_a,N: nat,X: set_a] :
( ! [A4: set_set_a] :
( ( member_set_set_a @ A4 @ ( set_mset_set_set_a @ A2 ) )
=> ( ord_le3724670747650509150_set_a @ A4 @ B2 ) )
=> ( ( member_set_set_a @ X2 @ ( set_mset_set_set_a @ ( repeat3222187171979612824_set_a @ N @ A2 ) ) )
=> ( ( member_set_a @ X @ X2 )
=> ( member_set_a @ X @ B2 ) ) ) ) ).
% repeat_mset_subset_in
thf(fact_164_repeat__mset__subset__in,axiom,
! [A2: multis7946784228617641354_set_a,B2: set_list_set_a,X2: set_list_set_a,N: nat,X: list_set_a] :
( ! [A4: set_list_set_a] :
( ( member844995724677003117_set_a @ A4 @ ( set_ms531227285539831169_set_a @ A2 ) )
=> ( ord_le864617614081865828_set_a @ A4 @ B2 ) )
=> ( ( member844995724677003117_set_a @ X2 @ ( set_ms531227285539831169_set_a @ ( repeat6861153585859484830_set_a @ N @ A2 ) ) )
=> ( ( member_list_set_a @ X @ X2 )
=> ( member_list_set_a @ X @ B2 ) ) ) ) ).
% repeat_mset_subset_in
thf(fact_165_repeat__mset__subset__in,axiom,
! [A2: multiset_set_nat,B2: set_nat,X2: set_nat,N: nat,X: nat] :
( ! [A4: set_nat] :
( ( member_set_nat @ A4 @ ( set_mset_set_nat @ A2 ) )
=> ( ord_less_eq_set_nat @ A4 @ B2 ) )
=> ( ( member_set_nat @ X2 @ ( set_mset_set_nat @ ( repeat_mset_set_nat @ N @ A2 ) ) )
=> ( ( member_nat @ X @ X2 )
=> ( member_nat @ X @ B2 ) ) ) ) ).
% repeat_mset_subset_in
thf(fact_166_repeat__mset__subset__in,axiom,
! [A2: multiset_set_a,B2: set_a,X2: set_a,N: nat,X: a] :
( ! [A4: set_a] :
( ( member_set_a @ A4 @ ( set_mset_set_a @ A2 ) )
=> ( ord_less_eq_set_a @ A4 @ B2 ) )
=> ( ( member_set_a @ X2 @ ( set_mset_set_a @ ( repeat_mset_set_a @ N @ A2 ) ) )
=> ( ( member_a @ X @ X2 )
=> ( member_a @ X @ B2 ) ) ) ) ).
% repeat_mset_subset_in
thf(fact_167_length__greater__0__conv,axiom,
! [Xs: list_set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_set_a @ Xs ) )
= ( Xs != nil_set_a ) ) ).
% length_greater_0_conv
thf(fact_168_length__0__conv,axiom,
! [Xs: list_set_a] :
( ( ( size_size_list_set_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_set_a ) ) ).
% length_0_conv
thf(fact_169_del__block__b_I1_J,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( size_s6566526139600085008_set_a @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ Bl2 ) )
= ( minus_minus_nat @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) @ one_one_nat ) ) ) ).
% del_block_b(1)
thf(fact_170_block__sizes__non__empty,axiom,
( ( ( mset_set_a @ b_s )
!= zero_z5079479921072680283_set_a )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) ) ) ) ) ).
% block_sizes_non_empty
thf(fact_171_block__sizes__non__empty__set,axiom,
( ( ( mset_set_a @ b_s )
!= zero_z5079479921072680283_set_a )
=> ( ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) )
!= bot_bot_set_nat ) ) ).
% block_sizes_non_empty_set
thf(fact_172_block__original__count__le,axiom,
! [N: nat,B: set_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( count_set_a @ ( mset_set_a @ b_s ) @ B ) @ ( count_set_a @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) @ B ) ) ) ).
% block_original_count_le
thf(fact_173_sys__block__sizes__obtain__bl,axiom,
! [X: nat] :
( ( member_nat @ X @ ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) ) )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
& ( ( finite_card_a @ X4 )
= X ) ) ) ).
% sys_block_sizes_obtain_bl
thf(fact_174_sys__block__sizes__in,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( member_nat @ ( finite_card_a @ Bl2 ) @ ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) ) ) ) ).
% sys_block_sizes_in
thf(fact_175_length__finite__permutations__of__multiset,axiom,
! [Xs: list_set_a,A2: multiset_set_a] :
( ( member_list_set_a @ Xs @ ( multis5469701301851823918_set_a @ A2 ) )
=> ( ( size_size_list_set_a @ Xs )
= ( size_s6566526139600085008_set_a @ A2 ) ) ) ).
% length_finite_permutations_of_multiset
thf(fact_176_elem__permutation__of__mset__empty__iff,axiom,
! [Xs: list_set_a,A2: multiset_set_a] :
( ( member_list_set_a @ Xs @ ( multis5469701301851823918_set_a @ A2 ) )
=> ( ( Xs = nil_set_a )
= ( A2 = zero_z5079479921072680283_set_a ) ) ) ).
% elem_permutation_of_mset_empty_iff
thf(fact_177_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_178_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: multiset_set_a] :
( ( minus_706656509937749387_set_a @ A @ A )
= zero_z5079479921072680283_set_a ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_179_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_180_diff__zero,axiom,
! [A: multiset_set_a] :
( ( minus_706656509937749387_set_a @ A @ zero_z5079479921072680283_set_a )
= A ) ).
% diff_zero
thf(fact_181_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_182_zero__diff,axiom,
! [A: multiset_set_a] :
( ( minus_706656509937749387_set_a @ zero_z5079479921072680283_set_a @ A )
= zero_z5079479921072680283_set_a ) ).
% zero_diff
thf(fact_183_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_184_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_185_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_186_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_187_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_188_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_189_set__mset__eq__empty__iff,axiom,
! [M: multiset_set_a] :
( ( ( set_mset_set_a @ M )
= bot_bot_set_set_a )
= ( M = zero_z5079479921072680283_set_a ) ) ).
% set_mset_eq_empty_iff
thf(fact_190_set__mset__eq__empty__iff,axiom,
! [M: multiset_nat] :
( ( ( set_mset_nat @ M )
= bot_bot_set_nat )
= ( M = zero_z7348594199698428585et_nat ) ) ).
% set_mset_eq_empty_iff
thf(fact_191_set__mset__empty,axiom,
( ( set_mset_set_a @ zero_z5079479921072680283_set_a )
= bot_bot_set_set_a ) ).
% set_mset_empty
thf(fact_192_set__mset__empty,axiom,
( ( set_mset_nat @ zero_z7348594199698428585et_nat )
= bot_bot_set_nat ) ).
% set_mset_empty
thf(fact_193_count__empty,axiom,
! [A: set_a] :
( ( count_set_a @ zero_z5079479921072680283_set_a @ A )
= zero_zero_nat ) ).
% count_empty
thf(fact_194_count__greater__zero__iff,axiom,
! [M: multiset_a,X: a] :
( ( ord_less_nat @ zero_zero_nat @ ( count_a @ M @ X ) )
= ( member_a @ X @ ( set_mset_a @ M ) ) ) ).
% count_greater_zero_iff
thf(fact_195_count__greater__zero__iff,axiom,
! [M: multiset_list_set_a,X: list_set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( count_list_set_a @ M @ X ) )
= ( member_list_set_a @ X @ ( set_mset_list_set_a @ M ) ) ) ).
% count_greater_zero_iff
thf(fact_196_count__greater__zero__iff,axiom,
! [M: multiset_nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( count_nat @ M @ X ) )
= ( member_nat @ X @ ( set_mset_nat @ M ) ) ) ).
% count_greater_zero_iff
thf(fact_197_count__greater__zero__iff,axiom,
! [M: multiset_set_a,X: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( count_set_a @ M @ X ) )
= ( member_set_a @ X @ ( set_mset_set_a @ M ) ) ) ).
% count_greater_zero_iff
thf(fact_198_count__greater__eq__one__iff,axiom,
! [M: multiset_a,X: a] :
( ( ord_less_eq_nat @ one_one_nat @ ( count_a @ M @ X ) )
= ( member_a @ X @ ( set_mset_a @ M ) ) ) ).
% count_greater_eq_one_iff
thf(fact_199_count__greater__eq__one__iff,axiom,
! [M: multiset_list_set_a,X: list_set_a] :
( ( ord_less_eq_nat @ one_one_nat @ ( count_list_set_a @ M @ X ) )
= ( member_list_set_a @ X @ ( set_mset_list_set_a @ M ) ) ) ).
% count_greater_eq_one_iff
thf(fact_200_count__greater__eq__one__iff,axiom,
! [M: multiset_nat,X: nat] :
( ( ord_less_eq_nat @ one_one_nat @ ( count_nat @ M @ X ) )
= ( member_nat @ X @ ( set_mset_nat @ M ) ) ) ).
% count_greater_eq_one_iff
thf(fact_201_count__greater__eq__one__iff,axiom,
! [M: multiset_set_a,X: set_a] :
( ( ord_less_eq_nat @ one_one_nat @ ( count_set_a @ M @ X ) )
= ( member_set_a @ X @ ( set_mset_set_a @ M ) ) ) ).
% count_greater_eq_one_iff
thf(fact_202_diff__commute,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J2 ) ) ).
% diff_commute
thf(fact_203_count__inject,axiom,
! [X: multiset_set_a,Y4: multiset_set_a] :
( ( ( count_set_a @ X )
= ( count_set_a @ Y4 ) )
= ( X = Y4 ) ) ).
% count_inject
thf(fact_204_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_205_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: multiset_set_a,C: multiset_set_a,B: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ A @ C ) @ B )
= ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_206_multiset__eqI,axiom,
! [A2: multiset_set_a,B2: multiset_set_a] :
( ! [X4: set_a] :
( ( count_set_a @ A2 @ X4 )
= ( count_set_a @ B2 @ X4 ) )
=> ( A2 = B2 ) ) ).
% multiset_eqI
thf(fact_207_multiset__eq__iff,axiom,
( ( ^ [Y3: multiset_set_a,Z: multiset_set_a] : ( Y3 = Z ) )
= ( ^ [M5: multiset_set_a,N4: multiset_set_a] :
! [A3: set_a] :
( ( count_set_a @ M5 @ A3 )
= ( count_set_a @ N4 @ A3 ) ) ) ) ).
% multiset_eq_iff
thf(fact_208_less__imp__diff__less,axiom,
! [J2: nat,K: nat,N: nat] :
( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_209_diff__less__mono2,axiom,
! [M2: nat,N: nat,L2: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L2 )
=> ( ord_less_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_210_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_211_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_212_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_213_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_214_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_215_diff__le__mono,axiom,
! [M2: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L2 ) @ ( minus_minus_nat @ N @ L2 ) ) ) ).
% diff_le_mono
thf(fact_216_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_217_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_218_diff__le__mono2,axiom,
! [M2: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_219_count__inI,axiom,
! [M: multiset_a,X: a] :
( ( ( count_a @ M @ X )
!= zero_zero_nat )
=> ( member_a @ X @ ( set_mset_a @ M ) ) ) ).
% count_inI
thf(fact_220_count__inI,axiom,
! [M: multiset_list_set_a,X: list_set_a] :
( ( ( count_list_set_a @ M @ X )
!= zero_zero_nat )
=> ( member_list_set_a @ X @ ( set_mset_list_set_a @ M ) ) ) ).
% count_inI
thf(fact_221_count__inI,axiom,
! [M: multiset_nat,X: nat] :
( ( ( count_nat @ M @ X )
!= zero_zero_nat )
=> ( member_nat @ X @ ( set_mset_nat @ M ) ) ) ).
% count_inI
thf(fact_222_count__inI,axiom,
! [M: multiset_set_a,X: set_a] :
( ( ( count_set_a @ M @ X )
!= zero_zero_nat )
=> ( member_set_a @ X @ ( set_mset_set_a @ M ) ) ) ).
% count_inI
thf(fact_223_count__eq__zero__iff,axiom,
! [M: multiset_a,X: a] :
( ( ( count_a @ M @ X )
= zero_zero_nat )
= ( ~ ( member_a @ X @ ( set_mset_a @ M ) ) ) ) ).
% count_eq_zero_iff
thf(fact_224_count__eq__zero__iff,axiom,
! [M: multiset_list_set_a,X: list_set_a] :
( ( ( count_list_set_a @ M @ X )
= zero_zero_nat )
= ( ~ ( member_list_set_a @ X @ ( set_mset_list_set_a @ M ) ) ) ) ).
% count_eq_zero_iff
thf(fact_225_count__eq__zero__iff,axiom,
! [M: multiset_nat,X: nat] :
( ( ( count_nat @ M @ X )
= zero_zero_nat )
= ( ~ ( member_nat @ X @ ( set_mset_nat @ M ) ) ) ) ).
% count_eq_zero_iff
thf(fact_226_count__eq__zero__iff,axiom,
! [M: multiset_set_a,X: set_a] :
( ( ( count_set_a @ M @ X )
= zero_zero_nat )
= ( ~ ( member_set_a @ X @ ( set_mset_set_a @ M ) ) ) ) ).
% count_eq_zero_iff
thf(fact_227_zero__multiset_Orep__eq,axiom,
( ( count_set_a @ zero_z5079479921072680283_set_a )
= ( ^ [A3: set_a] : zero_zero_nat ) ) ).
% zero_multiset.rep_eq
thf(fact_228_mset__subset__eqI,axiom,
! [A2: multiset_set_a,B2: multiset_set_a] :
( ! [A4: set_a] : ( ord_less_eq_nat @ ( count_set_a @ A2 @ A4 ) @ ( count_set_a @ B2 @ A4 ) )
=> ( subseteq_mset_set_a @ A2 @ B2 ) ) ).
% mset_subset_eqI
thf(fact_229_subseteq__mset__def,axiom,
( subseteq_mset_set_a
= ( ^ [A5: multiset_set_a,B4: multiset_set_a] :
! [A3: set_a] : ( ord_less_eq_nat @ ( count_set_a @ A5 @ A3 ) @ ( count_set_a @ B4 @ A3 ) ) ) ) ).
% subseteq_mset_def
thf(fact_230_mset__subset__eq__count,axiom,
! [A2: multiset_set_a,B2: multiset_set_a,A: set_a] :
( ( subseteq_mset_set_a @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( count_set_a @ A2 @ A ) @ ( count_set_a @ B2 @ A ) ) ) ).
% mset_subset_eq_count
thf(fact_231_set__count__size__min,axiom,
! [N: nat,A2: multiset_set_a,A: set_a] :
( ( ord_less_eq_nat @ N @ ( count_set_a @ A2 @ A ) )
=> ( ord_less_eq_nat @ N @ ( size_s6566526139600085008_set_a @ A2 ) ) ) ).
% set_count_size_min
thf(fact_232_elem__exists__non__empty__set,axiom,
! [A2: set_set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_a @ A2 ) )
=> ~ ! [X4: set_a] :
~ ( member_set_a @ X4 @ A2 ) ) ).
% elem_exists_non_empty_set
thf(fact_233_elem__exists__non__empty__set,axiom,
! [A2: set_list_set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite4179508071619380492_set_a @ A2 ) )
=> ~ ! [X4: list_set_a] :
~ ( member_list_set_a @ X4 @ A2 ) ) ).
% elem_exists_non_empty_set
thf(fact_234_elem__exists__non__empty__set,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
=> ~ ! [X4: nat] :
~ ( member_nat @ X4 @ A2 ) ) ).
% elem_exists_non_empty_set
thf(fact_235_elem__exists__non__empty__set,axiom,
! [A2: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A2 ) )
=> ~ ! [X4: a] :
~ ( member_a @ X4 @ A2 ) ) ).
% elem_exists_non_empty_set
thf(fact_236_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_237_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_238_less__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_239_mset__nempty__set__nempty,axiom,
! [A2: multiset_set_a] :
( ( A2 != zero_z5079479921072680283_set_a )
= ( ( set_mset_set_a @ A2 )
!= bot_bot_set_set_a ) ) ).
% mset_nempty_set_nempty
thf(fact_240_mset__nempty__set__nempty,axiom,
! [A2: multiset_nat] :
( ( A2 != zero_z7348594199698428585et_nat )
= ( ( set_mset_nat @ A2 )
!= bot_bot_set_nat ) ) ).
% mset_nempty_set_nempty
thf(fact_241_card__subset__not__gt__card,axiom,
! [A2: set_set_a,Ps: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( ord_less_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ Ps ) )
=> ~ ( ord_le3724670747650509150_set_a @ Ps @ A2 ) ) ) ).
% card_subset_not_gt_card
thf(fact_242_card__subset__not__gt__card,axiom,
! [A2: set_nat,Ps: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ Ps ) )
=> ~ ( ord_less_eq_set_nat @ Ps @ A2 ) ) ) ).
% card_subset_not_gt_card
thf(fact_243_card__subset__not__gt__card,axiom,
! [A2: set_a,Ps: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ Ps ) )
=> ~ ( ord_less_eq_set_a @ Ps @ A2 ) ) ) ).
% card_subset_not_gt_card
thf(fact_244_neq__if__length__neq,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( size_size_list_set_a @ Xs )
!= ( size_size_list_set_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_245_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_set_a] :
( ( size_size_list_set_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_246_mset__size__ne0__set__card,axiom,
! [A2: multiset_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s5917832649809541300et_nat @ A2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( set_mset_nat @ A2 ) ) ) ) ).
% mset_size_ne0_set_card
thf(fact_247_mset__size__ne0__set__card,axiom,
! [A2: multiset_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_multiset_a @ A2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ ( set_mset_a @ A2 ) ) ) ) ).
% mset_size_ne0_set_card
thf(fact_248_mset__size__ne0__set__card,axiom,
! [A2: multiset_set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s6566526139600085008_set_a @ A2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_a @ ( set_mset_set_a @ A2 ) ) ) ) ).
% mset_size_ne0_set_card
thf(fact_249_finite__permutations__of__multiset,axiom,
! [A2: multiset_set_a] : ( finite1971793804006318733_set_a @ ( multis5469701301851823918_set_a @ A2 ) ) ).
% finite_permutations_of_multiset
thf(fact_250_finite__maxlen,axiom,
! [M: set_list_set_a] :
( ( finite1971793804006318733_set_a @ M )
=> ? [N3: nat] :
! [X5: list_set_a] :
( ( member_list_set_a @ X5 @ M )
=> ( ord_less_nat @ ( size_size_list_set_a @ X5 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_251_length__induct,axiom,
! [P2: list_set_a > $o,Xs: list_set_a] :
( ! [Xs2: list_set_a] :
( ! [Ys2: list_set_a] :
( ( ord_less_nat @ ( size_size_list_set_a @ Ys2 ) @ ( size_size_list_set_a @ Xs2 ) )
=> ( P2 @ Ys2 ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_252_permutations__of__multisetD,axiom,
! [Xs: list_set_a,A2: multiset_set_a] :
( ( member_list_set_a @ Xs @ ( multis5469701301851823918_set_a @ A2 ) )
=> ( ( mset_set_a @ Xs )
= A2 ) ) ).
% permutations_of_multisetD
thf(fact_253_permutations__of__multisetI,axiom,
! [Xs: list_set_a,A2: multiset_set_a] :
( ( ( mset_set_a @ Xs )
= A2 )
=> ( member_list_set_a @ Xs @ ( multis5469701301851823918_set_a @ A2 ) ) ) ).
% permutations_of_multisetI
thf(fact_254_list_Osize_I3_J,axiom,
( ( size_size_list_set_a @ nil_set_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_255_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_set_a,Z: list_set_a] : ( Y3 = Z ) )
= ( ^ [Xs3: list_set_a,Ys3: list_set_a] :
( ( ( size_size_list_set_a @ Xs3 )
= ( size_size_list_set_a @ Ys3 ) )
& ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_set_a @ Xs3 ) )
=> ( ( nth_set_a @ Xs3 @ I )
= ( nth_set_a @ Ys3 @ I ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_256_Skolem__list__nth,axiom,
! [K: nat,P2: nat > set_a > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ? [X6: set_a] : ( P2 @ I @ X6 ) ) )
= ( ? [Xs3: list_set_a] :
( ( ( size_size_list_set_a @ Xs3 )
= K )
& ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ( P2 @ I @ ( nth_set_a @ Xs3 @ I ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_257_nth__equalityI,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( size_size_list_set_a @ Xs )
= ( size_size_list_set_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_set_a @ Xs ) )
=> ( ( nth_set_a @ Xs @ I3 )
= ( nth_set_a @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_258_card__0__eq,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ( finite_card_a @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_a ) ) ) ).
% card_0_eq
thf(fact_259_card__0__eq,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( ( finite_card_set_a @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_set_a ) ) ) ).
% card_0_eq
thf(fact_260_card__0__eq,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_261_card_Oinfinite,axiom,
! [A2: set_a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_card_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_262_card_Oinfinite,axiom,
! [A2: set_set_a] :
( ~ ( finite_finite_set_a @ A2 )
=> ( ( finite_card_set_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_263_card_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_card_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_264_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_265_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_266_card__gt__0__iff,axiom,
! [A2: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A2 ) )
= ( ( A2 != bot_bot_set_a )
& ( finite_finite_a @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_267_card__gt__0__iff,axiom,
! [A2: set_set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_a @ A2 ) )
= ( ( A2 != bot_bot_set_set_a )
& ( finite_finite_set_a @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_268_card__gt__0__iff,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
= ( ( A2 != bot_bot_set_nat )
& ( finite_finite_nat @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_269_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_270_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_271_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_272_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_273_card__mono,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B2 ) ) ) ) ).
% card_mono
thf(fact_274_card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_275_card__mono,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ).
% card_mono
thf(fact_276_card__seteq,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_a @ B2 ) @ ( finite_card_set_a @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_277_card__seteq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_278_card__seteq,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B2 ) @ ( finite_card_a @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_279_Diff__empty,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_280_empty__Diff,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_281_Diff__cancel,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ A2 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_282_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_283_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_284_empty__iff,axiom,
! [C: list_set_a] :
~ ( member_list_set_a @ C @ bot_bo4397488018069675312_set_a ) ).
% empty_iff
thf(fact_285_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_286_all__not__in__conv,axiom,
! [A2: set_set_a] :
( ( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_287_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_288_all__not__in__conv,axiom,
! [A2: set_list_set_a] :
( ( ! [X3: list_set_a] :
~ ( member_list_set_a @ X3 @ A2 ) )
= ( A2 = bot_bo4397488018069675312_set_a ) ) ).
% all_not_in_conv
thf(fact_289_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_290_Collect__empty__eq,axiom,
! [P2: nat > $o] :
( ( ( collect_nat @ P2 )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_291_empty__Collect__eq,axiom,
! [P2: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P2 ) )
= ( ! [X3: nat] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_292_subsetI,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ( member_set_a @ X4 @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_293_subsetI,axiom,
! [A2: set_list_set_a,B2: set_list_set_a] :
( ! [X4: list_set_a] :
( ( member_list_set_a @ X4 @ A2 )
=> ( member_list_set_a @ X4 @ B2 ) )
=> ( ord_le864617614081865828_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_294_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat @ X4 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_295_subsetI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( member_a @ X4 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_296_subset__antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_297_psubsetI,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_298_finite__Diff,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_299_finite__Diff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_300_finite__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_301_finite__Diff2,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( finite_finite_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_302_finite__Diff2,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
= ( finite_finite_set_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_303_finite__Diff2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_304_Diff__eq__empty__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( minus_minus_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_305_Diff__eq__empty__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( minus_minus_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_306_count__diff,axiom,
! [M: multiset_set_a,N5: multiset_set_a,A: set_a] :
( ( count_set_a @ ( minus_706656509937749387_set_a @ M @ N5 ) @ A )
= ( minus_minus_nat @ ( count_set_a @ M @ A ) @ ( count_set_a @ N5 @ A ) ) ) ).
% count_diff
thf(fact_307_double__diff,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ( minus_minus_set_a @ B2 @ ( minus_minus_set_a @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_308_Diff__subset,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_309_Diff__mono,axiom,
! [A2: set_a,C2: set_a,D: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ D @ B2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( minus_minus_set_a @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_310_Diff__infinite__finite,axiom,
! [T2: set_a,S: set_a] :
( ( finite_finite_a @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_311_Diff__infinite__finite,axiom,
! [T2: set_set_a,S: set_set_a] :
( ( finite_finite_set_a @ T2 )
=> ( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_312_Diff__infinite__finite,axiom,
! [T2: set_nat,S: set_nat] :
( ( finite_finite_nat @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_313_in__diffD,axiom,
! [A: a,M: multiset_a,N5: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M @ N5 ) ) )
=> ( member_a @ A @ ( set_mset_a @ M ) ) ) ).
% in_diffD
thf(fact_314_in__diffD,axiom,
! [A: list_set_a,M: multiset_list_set_a,N5: multiset_list_set_a] :
( ( member_list_set_a @ A @ ( set_mset_list_set_a @ ( minus_2572999347059163665_set_a @ M @ N5 ) ) )
=> ( member_list_set_a @ A @ ( set_mset_list_set_a @ M ) ) ) ).
% in_diffD
thf(fact_315_in__diffD,axiom,
! [A: nat,M: multiset_nat,N5: multiset_nat] :
( ( member_nat @ A @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M @ N5 ) ) )
=> ( member_nat @ A @ ( set_mset_nat @ M ) ) ) ).
% in_diffD
thf(fact_316_in__diffD,axiom,
! [A: set_a,M: multiset_set_a,N5: multiset_set_a] :
( ( member_set_a @ A @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ M @ N5 ) ) )
=> ( member_set_a @ A @ ( set_mset_set_a @ M ) ) ) ).
% in_diffD
thf(fact_317_diff__subset__eq__self,axiom,
! [M: multiset_set_a,N5: multiset_set_a] : ( subseteq_mset_set_a @ ( minus_706656509937749387_set_a @ M @ N5 ) @ M ) ).
% diff_subset_eq_self
thf(fact_318_diff__empty,axiom,
! [M: multiset_set_a] :
( ( ( minus_706656509937749387_set_a @ M @ zero_z5079479921072680283_set_a )
= M )
& ( ( minus_706656509937749387_set_a @ zero_z5079479921072680283_set_a @ M )
= zero_z5079479921072680283_set_a ) ) ).
% diff_empty
thf(fact_319_Multiset_Odiff__cancel,axiom,
! [A2: multiset_set_a] :
( ( minus_706656509937749387_set_a @ A2 @ A2 )
= zero_z5079479921072680283_set_a ) ).
% Multiset.diff_cancel
thf(fact_320_permutations__of__multiset__not__empty,axiom,
! [A2: multiset_set_a] :
( ( multis5469701301851823918_set_a @ A2 )
!= bot_bo4397488018069675312_set_a ) ).
% permutations_of_multiset_not_empty
thf(fact_321_card__less__sym__Diff,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_322_card__less__sym__Diff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( finite_finite_set_a @ B2 )
=> ( ( ord_less_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B2 ) )
=> ( ord_less_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_323_card__less__sym__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_324_card__le__sym__Diff,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_325_card__le__sym__Diff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( finite_finite_set_a @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_326_card__le__sym__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_327_set__diff__non__empty__not__subset,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C2 ) )
=> ( ( C2 != bot_bot_set_nat )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( B2 != bot_bot_set_nat )
=> ~ ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ) ) ).
% set_diff_non_empty_not_subset
thf(fact_328_set__diff__non__empty__not__subset,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ C2 ) )
=> ( ( C2 != bot_bot_set_a )
=> ( ( A2 != bot_bot_set_a )
=> ( ( B2 != bot_bot_set_a )
=> ~ ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ) ) ).
% set_diff_non_empty_not_subset
thf(fact_329_Diff__eq__empty__iff__mset,axiom,
! [A2: multiset_set_a,B2: multiset_set_a] :
( ( ( minus_706656509937749387_set_a @ A2 @ B2 )
= zero_z5079479921072680283_set_a )
= ( subseteq_mset_set_a @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff_mset
thf(fact_330_minus__multiset_Orep__eq,axiom,
! [X: multiset_set_a,Xa2: multiset_set_a] :
( ( count_set_a @ ( minus_706656509937749387_set_a @ X @ Xa2 ) )
= ( ^ [A3: set_a] : ( minus_minus_nat @ ( count_set_a @ X @ A3 ) @ ( count_set_a @ Xa2 @ A3 ) ) ) ) ).
% minus_multiset.rep_eq
thf(fact_331_left__diff__repeat__mset__distrib_H,axiom,
! [I2: nat,J2: nat,U: multiset_set_a] :
( ( repeat_mset_set_a @ ( minus_minus_nat @ I2 @ J2 ) @ U )
= ( minus_706656509937749387_set_a @ ( repeat_mset_set_a @ I2 @ U ) @ ( repeat_mset_set_a @ J2 @ U ) ) ) ).
% left_diff_repeat_mset_distrib'
thf(fact_332_subset__mset_Ofinite__has__maximal,axiom,
! [A2: set_multiset_set_a] :
( ( finite2815193924343055693_set_a @ A2 )
=> ( ( A2 != bot_bo9088538438451294192_set_a )
=> ? [X4: multiset_set_a] :
( ( member2747690772047059533_set_a @ X4 @ A2 )
& ! [Xa: multiset_set_a] :
( ( member2747690772047059533_set_a @ Xa @ A2 )
=> ( ( subseteq_mset_set_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% subset_mset.finite_has_maximal
thf(fact_333_subset__mset_Ofinite__has__minimal,axiom,
! [A2: set_multiset_set_a] :
( ( finite2815193924343055693_set_a @ A2 )
=> ( ( A2 != bot_bo9088538438451294192_set_a )
=> ? [X4: multiset_set_a] :
( ( member2747690772047059533_set_a @ X4 @ A2 )
& ! [Xa: multiset_set_a] :
( ( member2747690772047059533_set_a @ Xa @ A2 )
=> ( ( subseteq_mset_set_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% subset_mset.finite_has_minimal
thf(fact_334_card__Diff__subset,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_335_card__Diff__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_336_card__Diff__subset,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_337_diff__card__le__card__Diff,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_338_diff__card__le__card__Diff,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B2 ) ) @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_339_diff__card__le__card__Diff,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_340_in__diff__count,axiom,
! [A: a,M: multiset_a,N5: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M @ N5 ) ) )
= ( ord_less_nat @ ( count_a @ N5 @ A ) @ ( count_a @ M @ A ) ) ) ).
% in_diff_count
thf(fact_341_in__diff__count,axiom,
! [A: list_set_a,M: multiset_list_set_a,N5: multiset_list_set_a] :
( ( member_list_set_a @ A @ ( set_mset_list_set_a @ ( minus_2572999347059163665_set_a @ M @ N5 ) ) )
= ( ord_less_nat @ ( count_list_set_a @ N5 @ A ) @ ( count_list_set_a @ M @ A ) ) ) ).
% in_diff_count
thf(fact_342_in__diff__count,axiom,
! [A: nat,M: multiset_nat,N5: multiset_nat] :
( ( member_nat @ A @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M @ N5 ) ) )
= ( ord_less_nat @ ( count_nat @ N5 @ A ) @ ( count_nat @ M @ A ) ) ) ).
% in_diff_count
thf(fact_343_in__diff__count,axiom,
! [A: set_a,M: multiset_set_a,N5: multiset_set_a] :
( ( member_set_a @ A @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ M @ N5 ) ) )
= ( ord_less_nat @ ( count_set_a @ N5 @ A ) @ ( count_set_a @ M @ A ) ) ) ).
% in_diff_count
thf(fact_344_diff__size__le__size__Diff,axiom,
! [M: multiset_set_a,M6: multiset_set_a] : ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s6566526139600085008_set_a @ M ) @ ( size_s6566526139600085008_set_a @ M6 ) ) @ ( size_s6566526139600085008_set_a @ ( minus_706656509937749387_set_a @ M @ M6 ) ) ) ).
% diff_size_le_size_Diff
thf(fact_345_size__Diff__submset,axiom,
! [M: multiset_set_a,M6: multiset_set_a] :
( ( subseteq_mset_set_a @ M @ M6 )
=> ( ( size_s6566526139600085008_set_a @ ( minus_706656509937749387_set_a @ M6 @ M ) )
= ( minus_minus_nat @ ( size_s6566526139600085008_set_a @ M6 ) @ ( size_s6566526139600085008_set_a @ M ) ) ) ) ).
% size_Diff_submset
thf(fact_346_not__psubset__empty,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_347_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_348_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_349_emptyE,axiom,
! [A: list_set_a] :
~ ( member_list_set_a @ A @ bot_bo4397488018069675312_set_a ) ).
% emptyE
thf(fact_350_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_351_equals0D,axiom,
! [A2: set_set_a,A: set_a] :
( ( A2 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_352_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_353_equals0D,axiom,
! [A2: set_list_set_a,A: list_set_a] :
( ( A2 = bot_bo4397488018069675312_set_a )
=> ~ ( member_list_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_354_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_355_equals0I,axiom,
! [A2: set_set_a] :
( ! [Y: set_a] :
~ ( member_set_a @ Y @ A2 )
=> ( A2 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_356_equals0I,axiom,
! [A2: set_a] :
( ! [Y: a] :
~ ( member_a @ Y @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_357_equals0I,axiom,
! [A2: set_list_set_a] :
( ! [Y: list_set_a] :
~ ( member_list_set_a @ Y @ A2 )
=> ( A2 = bot_bo4397488018069675312_set_a ) ) ).
% equals0I
thf(fact_358_equals0I,axiom,
! [A2: set_nat] :
( ! [Y: nat] :
~ ( member_nat @ Y @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_359_ex__in__conv,axiom,
! [A2: set_set_a] :
( ( ? [X3: set_a] : ( member_set_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_360_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_361_ex__in__conv,axiom,
! [A2: set_list_set_a] :
( ( ? [X3: list_set_a] : ( member_list_set_a @ X3 @ A2 ) )
= ( A2 != bot_bo4397488018069675312_set_a ) ) ).
% ex_in_conv
thf(fact_362_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_363_in__mono,axiom,
! [A2: set_set_a,B2: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( member_set_a @ X @ A2 )
=> ( member_set_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_364_in__mono,axiom,
! [A2: set_list_set_a,B2: set_list_set_a,X: list_set_a] :
( ( ord_le864617614081865828_set_a @ A2 @ B2 )
=> ( ( member_list_set_a @ X @ A2 )
=> ( member_list_set_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_365_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_366_in__mono,axiom,
! [A2: set_a,B2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_367_subsetD,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_368_subsetD,axiom,
! [A2: set_list_set_a,B2: set_list_set_a,C: list_set_a] :
( ( ord_le864617614081865828_set_a @ A2 @ B2 )
=> ( ( member_list_set_a @ C @ A2 )
=> ( member_list_set_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_369_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_370_subsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_371_equalityE,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_372_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_373_subset__eq,axiom,
( ord_le864617614081865828_set_a
= ( ^ [A5: set_list_set_a,B4: set_list_set_a] :
! [X3: list_set_a] :
( ( member_list_set_a @ X3 @ A5 )
=> ( member_list_set_a @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_374_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_375_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_376_equalityD1,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_377_equalityD2,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_378_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B4: set_set_a] :
! [T3: set_a] :
( ( member_set_a @ T3 @ A5 )
=> ( member_set_a @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_379_subset__iff,axiom,
( ord_le864617614081865828_set_a
= ( ^ [A5: set_list_set_a,B4: set_list_set_a] :
! [T3: list_set_a] :
( ( member_list_set_a @ T3 @ A5 )
=> ( member_list_set_a @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_380_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A5 )
=> ( member_nat @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_381_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B4: set_a] :
! [T3: a] :
( ( member_a @ T3 @ A5 )
=> ( member_a @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_382_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_383_Collect__mono,axiom,
! [P2: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P2 @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_384_subset__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_385_set__eq__subset,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = 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_386_Collect__mono__iff,axiom,
! [P2: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P2 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_387_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_388_subset__psubset__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C2 )
=> ( ord_less_set_a @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_389_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_390_psubset__subset__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_set_a @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_391_psubset__imp__subset,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_392_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_393_psubsetE,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_394_finite__psubset__induct,axiom,
! [A2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ! [B5: set_a] :
( ( ord_less_set_a @ B5 @ A6 )
=> ( P2 @ B5 ) )
=> ( P2 @ A6 ) ) )
=> ( P2 @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_395_finite__psubset__induct,axiom,
! [A2: set_set_a,P2: set_set_a > $o] :
( ( finite_finite_set_a @ A2 )
=> ( ! [A6: set_set_a] :
( ( finite_finite_set_a @ A6 )
=> ( ! [B5: set_set_a] :
( ( ord_less_set_set_a @ B5 @ A6 )
=> ( P2 @ B5 ) )
=> ( P2 @ A6 ) ) )
=> ( P2 @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_396_finite__psubset__induct,axiom,
! [A2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [B5: set_nat] :
( ( ord_less_set_nat @ B5 @ A6 )
=> ( P2 @ B5 ) )
=> ( P2 @ A6 ) ) )
=> ( P2 @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_397_set__card__diff__ge__zero,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( A2 != B2 )
=> ( ( ( finite_card_a @ A2 )
= ( finite_card_a @ B2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ) ) ) ).
% set_card_diff_ge_zero
thf(fact_398_set__card__diff__ge__zero,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( finite_finite_set_a @ B2 )
=> ( ( A2 != B2 )
=> ( ( ( finite_card_set_a @ A2 )
= ( finite_card_set_a @ B2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ) ) ) ) ).
% set_card_diff_ge_zero
thf(fact_399_set__card__diff__ge__zero,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( A2 != B2 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ) ) ).
% set_card_diff_ge_zero
thf(fact_400_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_401_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_402_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_403_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_404_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
& ( ord_less_eq_set_a @ A @ X4 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_405_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_nat @ A @ X4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_406_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
& ( ord_less_eq_set_a @ X4 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_407_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_nat @ X4 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_408_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_409_finite_OemptyI,axiom,
finite_finite_set_a @ bot_bot_set_set_a ).
% finite.emptyI
thf(fact_410_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_411_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_412_infinite__imp__nonempty,axiom,
! [S: set_set_a] :
( ~ ( finite_finite_set_a @ S )
=> ( S != bot_bot_set_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_413_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_414_finite__subset,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( finite_finite_set_a @ B2 )
=> ( finite_finite_set_a @ A2 ) ) ) ).
% finite_subset
thf(fact_415_finite__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_416_finite__subset,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_417_infinite__super,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ S @ T2 )
=> ( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ T2 ) ) ) ).
% infinite_super
thf(fact_418_infinite__super,axiom,
! [S: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_419_infinite__super,axiom,
! [S: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_420_rev__finite__subset,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( finite_finite_set_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_421_rev__finite__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_422_rev__finite__subset,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_423_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_424_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_425_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_426_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_427_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_428_infinite__arbitrarily__large,axiom,
! [A2: set_set_a,N: nat] :
( ~ ( finite_finite_set_a @ A2 )
=> ? [B6: set_set_a] :
( ( finite_finite_set_a @ B6 )
& ( ( finite_card_set_a @ B6 )
= N )
& ( ord_le3724670747650509150_set_a @ B6 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_429_infinite__arbitrarily__large,axiom,
! [A2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ( finite_card_nat @ B6 )
= N )
& ( ord_less_eq_set_nat @ B6 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_430_infinite__arbitrarily__large,axiom,
! [A2: set_a,N: nat] :
( ~ ( finite_finite_a @ A2 )
=> ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ( finite_card_a @ B6 )
= N )
& ( ord_less_eq_set_a @ B6 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_431_card__subset__eq,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ( finite_card_set_a @ A2 )
= ( finite_card_set_a @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_432_card__subset__eq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_433_card__subset__eq,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( finite_card_a @ A2 )
= ( finite_card_a @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_434_psubset__card__mono,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_set_a @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_435_psubset__card__mono,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_436_psubset__card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_set_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_437_card__psubset,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B2 ) )
=> ( ord_less_set_set_a @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_438_card__psubset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_439_card__psubset,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_440_card__eq__0__iff,axiom,
! [A2: set_a] :
( ( ( finite_card_a @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_a )
| ~ ( finite_finite_a @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_441_card__eq__0__iff,axiom,
! [A2: set_set_a] :
( ( ( finite_card_set_a @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_set_a )
| ~ ( finite_finite_set_a @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_442_card__eq__0__iff,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_443_card__ge__0__finite,axiom,
! [A2: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A2 ) )
=> ( finite_finite_a @ A2 ) ) ).
% card_ge_0_finite
thf(fact_444_card__ge__0__finite,axiom,
! [A2: set_set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_a @ A2 ) )
=> ( finite_finite_set_a @ A2 ) ) ).
% card_ge_0_finite
thf(fact_445_card__ge__0__finite,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
=> ( finite_finite_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_446_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_set_a,C2: nat] :
( ! [G: set_set_a] :
( ( ord_le3724670747650509150_set_a @ G @ F2 )
=> ( ( finite_finite_set_a @ G )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ G ) @ C2 ) ) )
=> ( ( finite_finite_set_a @ F2 )
& ( ord_less_eq_nat @ ( finite_card_set_a @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_447_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C2: nat] :
( ! [G: set_nat] :
( ( ord_less_eq_set_nat @ G @ F2 )
=> ( ( finite_finite_nat @ G )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C2 ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_448_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_a,C2: nat] :
( ! [G: set_a] :
( ( ord_less_eq_set_a @ G @ F2 )
=> ( ( finite_finite_a @ G )
=> ( ord_less_eq_nat @ ( finite_card_a @ G ) @ C2 ) ) )
=> ( ( finite_finite_a @ F2 )
& ( ord_less_eq_nat @ ( finite_card_a @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_449_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_set_a @ S ) )
=> ~ ! [T4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ T4 @ S )
=> ( ( ( finite_card_set_a @ T4 )
= N )
=> ~ ( finite_finite_set_a @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_450_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
=> ~ ! [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ S )
=> ( ( ( finite_card_nat @ T4 )
= N )
=> ~ ( finite_finite_nat @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_451_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_a @ S ) )
=> ~ ! [T4: set_a] :
( ( ord_less_eq_set_a @ T4 @ S )
=> ( ( ( finite_card_a @ T4 )
= N )
=> ~ ( finite_finite_a @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_452_exists__subset__between,axiom,
! [A2: set_set_a,N: nat,C2: set_set_a] :
( ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_set_a @ C2 ) )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ C2 )
=> ( ( finite_finite_set_a @ C2 )
=> ? [B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B6 )
& ( ord_le3724670747650509150_set_a @ B6 @ C2 )
& ( ( finite_card_set_a @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_453_exists__subset__between,axiom,
! [A2: set_nat,N: nat,C2: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C2 ) )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( finite_finite_nat @ C2 )
=> ? [B6: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B6 )
& ( ord_less_eq_set_nat @ B6 @ C2 )
& ( ( finite_card_nat @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_454_exists__subset__between,axiom,
! [A2: set_a,N: nat,C2: set_a] :
( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_a @ C2 ) )
=> ( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( finite_finite_a @ C2 )
=> ? [B6: set_a] :
( ( ord_less_eq_set_a @ A2 @ B6 )
& ( ord_less_eq_set_a @ B6 @ C2 )
& ( ( finite_card_a @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_455_Ramsey,axiom,
! [Z3: set_set_a,R: nat,F: set_set_a > nat,S2: nat] :
( ~ ( finite_finite_set_a @ Z3 )
=> ( ! [X7: set_set_a] :
( ( ( ord_le3724670747650509150_set_a @ X7 @ Z3 )
& ( finite_finite_set_a @ X7 )
& ( ( finite_card_set_a @ X7 )
= R ) )
=> ( ord_less_nat @ ( F @ X7 ) @ S2 ) )
=> ? [Y7: set_set_a,T5: nat] :
( ( ord_le3724670747650509150_set_a @ Y7 @ Z3 )
& ~ ( finite_finite_set_a @ Y7 )
& ( ord_less_nat @ T5 @ S2 )
& ! [X8: set_set_a] :
( ( ( ord_le3724670747650509150_set_a @ X8 @ Y7 )
& ( finite_finite_set_a @ X8 )
& ( ( finite_card_set_a @ X8 )
= R ) )
=> ( ( F @ X8 )
= T5 ) ) ) ) ) ).
% Ramsey
thf(fact_456_Ramsey,axiom,
! [Z3: set_nat,R: nat,F: set_nat > nat,S2: nat] :
( ~ ( finite_finite_nat @ Z3 )
=> ( ! [X7: set_nat] :
( ( ( ord_less_eq_set_nat @ X7 @ Z3 )
& ( finite_finite_nat @ X7 )
& ( ( finite_card_nat @ X7 )
= R ) )
=> ( ord_less_nat @ ( F @ X7 ) @ S2 ) )
=> ? [Y7: set_nat,T5: nat] :
( ( ord_less_eq_set_nat @ Y7 @ Z3 )
& ~ ( finite_finite_nat @ Y7 )
& ( ord_less_nat @ T5 @ S2 )
& ! [X8: set_nat] :
( ( ( ord_less_eq_set_nat @ X8 @ Y7 )
& ( finite_finite_nat @ X8 )
& ( ( finite_card_nat @ X8 )
= R ) )
=> ( ( F @ X8 )
= T5 ) ) ) ) ) ).
% Ramsey
thf(fact_457_Ramsey,axiom,
! [Z3: set_a,R: nat,F: set_a > nat,S2: nat] :
( ~ ( finite_finite_a @ Z3 )
=> ( ! [X7: set_a] :
( ( ( ord_less_eq_set_a @ X7 @ Z3 )
& ( finite_finite_a @ X7 )
& ( ( finite_card_a @ X7 )
= R ) )
=> ( ord_less_nat @ ( F @ X7 ) @ S2 ) )
=> ? [Y7: set_a,T5: nat] :
( ( ord_less_eq_set_a @ Y7 @ Z3 )
& ~ ( finite_finite_a @ Y7 )
& ( ord_less_nat @ T5 @ S2 )
& ! [X8: set_a] :
( ( ( ord_less_eq_set_a @ X8 @ Y7 )
& ( finite_finite_a @ X8 )
& ( ( finite_card_a @ X8 )
= R ) )
=> ( ( F @ X8 )
= T5 ) ) ) ) ) ).
% Ramsey
thf(fact_458_del__block__def,axiom,
! [B: set_a] :
( ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ B )
= ( minus_706656509937749387_set_a @ ( mset_set_a @ b_s ) @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) ) ) ).
% del_block_def
thf(fact_459_add__block__index__not__in,axiom,
! [Ps: set_a,B: set_a] :
( ~ ( ord_less_eq_set_a @ Ps @ B )
=> ( ( design254580327166089565ndex_a @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ B ) @ Ps )
= ( design254580327166089565ndex_a @ ( mset_set_a @ b_s ) @ Ps ) ) ) ).
% add_block_index_not_in
thf(fact_460_length__list__of__mset,axiom,
! [A2: multiset_set_a] :
( ( size_size_list_set_a @ ( multis2300860884436073961_set_a @ A2 ) )
= ( size_s6566526139600085008_set_a @ A2 ) ) ).
% length_list_of_mset
thf(fact_461_Ex__list__of__length__P,axiom,
! [N: nat,P2: set_a > nat > $o] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ? [X5: set_a] : ( P2 @ X5 @ I3 ) )
=> ? [Xs2: list_set_a] :
( ( ( size_size_list_set_a @ Xs2 )
= N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P2 @ ( nth_set_a @ Xs2 @ I4 ) @ I4 ) ) ) ) ).
% Ex_list_of_length_P
thf(fact_462_list__of__mset__empty,axiom,
! [M2: multiset_set_a] :
( ( ( multis2300860884436073961_set_a @ M2 )
= nil_set_a )
= ( M2 = zero_z5079479921072680283_set_a ) ) ).
% list_of_mset_empty
thf(fact_463_card__le__if__inj__on__rel,axiom,
! [B2: set_a,A2: set_nat,R: nat > a > $o] :
( ( finite_finite_a @ B2 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ? [B7: a] :
( ( member_a @ B7 @ B2 )
& ( R @ A4 @ B7 ) ) )
=> ( ! [A1: nat,A22: nat,B8: a] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_a @ B8 @ B2 )
=> ( ( R @ A1 @ B8 )
=> ( ( R @ A22 @ B8 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_464_card__le__if__inj__on__rel,axiom,
! [B2: set_a,A2: set_a,R: a > a > $o] :
( ( finite_finite_a @ B2 )
=> ( ! [A4: a] :
( ( member_a @ A4 @ A2 )
=> ? [B7: a] :
( ( member_a @ B7 @ B2 )
& ( R @ A4 @ B7 ) ) )
=> ( ! [A1: a,A22: a,B8: a] :
( ( member_a @ A1 @ A2 )
=> ( ( member_a @ A22 @ A2 )
=> ( ( member_a @ B8 @ B2 )
=> ( ( R @ A1 @ B8 )
=> ( ( R @ A22 @ B8 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_465_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ? [B7: nat] :
( ( member_nat @ B7 @ B2 )
& ( R @ A4 @ B7 ) ) )
=> ( ! [A1: nat,A22: nat,B8: nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_nat @ B8 @ B2 )
=> ( ( R @ A1 @ B8 )
=> ( ( R @ A22 @ B8 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_466_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_a,R: a > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A4: a] :
( ( member_a @ A4 @ A2 )
=> ? [B7: nat] :
( ( member_nat @ B7 @ B2 )
& ( R @ A4 @ B7 ) ) )
=> ( ! [A1: a,A22: a,B8: nat] :
( ( member_a @ A1 @ A2 )
=> ( ( member_a @ A22 @ A2 )
=> ( ( member_nat @ B8 @ B2 )
=> ( ( R @ A1 @ B8 )
=> ( ( R @ A22 @ B8 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_467_card__le__if__inj__on__rel,axiom,
! [B2: set_a,A2: set_set_a,R: set_a > a > $o] :
( ( finite_finite_a @ B2 )
=> ( ! [A4: set_a] :
( ( member_set_a @ A4 @ A2 )
=> ? [B7: a] :
( ( member_a @ B7 @ B2 )
& ( R @ A4 @ B7 ) ) )
=> ( ! [A1: set_a,A22: set_a,B8: a] :
( ( member_set_a @ A1 @ A2 )
=> ( ( member_set_a @ A22 @ A2 )
=> ( ( member_a @ B8 @ B2 )
=> ( ( R @ A1 @ B8 )
=> ( ( R @ A22 @ B8 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_468_card__le__if__inj__on__rel,axiom,
! [B2: set_set_a,A2: set_nat,R: nat > set_a > $o] :
( ( finite_finite_set_a @ B2 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ? [B7: set_a] :
( ( member_set_a @ B7 @ B2 )
& ( R @ A4 @ B7 ) ) )
=> ( ! [A1: nat,A22: nat,B8: set_a] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_set_a @ B8 @ B2 )
=> ( ( R @ A1 @ B8 )
=> ( ( R @ A22 @ B8 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_set_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_469_card__le__if__inj__on__rel,axiom,
! [B2: set_set_a,A2: set_a,R: a > set_a > $o] :
( ( finite_finite_set_a @ B2 )
=> ( ! [A4: a] :
( ( member_a @ A4 @ A2 )
=> ? [B7: set_a] :
( ( member_set_a @ B7 @ B2 )
& ( R @ A4 @ B7 ) ) )
=> ( ! [A1: a,A22: a,B8: set_a] :
( ( member_a @ A1 @ A2 )
=> ( ( member_a @ A22 @ A2 )
=> ( ( member_set_a @ B8 @ B2 )
=> ( ( R @ A1 @ B8 )
=> ( ( R @ A22 @ B8 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_set_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_470_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_set_a,R: set_a > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A4: set_a] :
( ( member_set_a @ A4 @ A2 )
=> ? [B7: nat] :
( ( member_nat @ B7 @ B2 )
& ( R @ A4 @ B7 ) ) )
=> ( ! [A1: set_a,A22: set_a,B8: nat] :
( ( member_set_a @ A1 @ A2 )
=> ( ( member_set_a @ A22 @ A2 )
=> ( ( member_nat @ B8 @ B2 )
=> ( ( R @ A1 @ B8 )
=> ( ( R @ A22 @ B8 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_471_card__le__if__inj__on__rel,axiom,
! [B2: set_list_set_a,A2: set_nat,R: nat > list_set_a > $o] :
( ( finite1971793804006318733_set_a @ B2 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ? [B7: list_set_a] :
( ( member_list_set_a @ B7 @ B2 )
& ( R @ A4 @ B7 ) ) )
=> ( ! [A1: nat,A22: nat,B8: list_set_a] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_list_set_a @ B8 @ B2 )
=> ( ( R @ A1 @ B8 )
=> ( ( R @ A22 @ B8 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite4179508071619380492_set_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_472_card__le__if__inj__on__rel,axiom,
! [B2: set_list_set_a,A2: set_a,R: a > list_set_a > $o] :
( ( finite1971793804006318733_set_a @ B2 )
=> ( ! [A4: a] :
( ( member_a @ A4 @ A2 )
=> ? [B7: list_set_a] :
( ( member_list_set_a @ B7 @ B2 )
& ( R @ A4 @ B7 ) ) )
=> ( ! [A1: a,A22: a,B8: list_set_a] :
( ( member_a @ A1 @ A2 )
=> ( ( member_a @ A22 @ A2 )
=> ( ( member_list_set_a @ B8 @ B2 )
=> ( ( R @ A1 @ B8 )
=> ( ( R @ A22 @ B8 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite4179508071619380492_set_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_473_Diff__iff,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
= ( ( member_set_a @ C @ A2 )
& ~ ( member_set_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_474_Diff__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
& ~ ( member_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_475_Diff__iff,axiom,
! [C: list_set_a,A2: set_list_set_a,B2: set_list_set_a] :
( ( member_list_set_a @ C @ ( minus_5993239077472792235_set_a @ A2 @ B2 ) )
= ( ( member_list_set_a @ C @ A2 )
& ~ ( member_list_set_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_476_Diff__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
& ~ ( member_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_477_DiffI,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ A2 )
=> ( ~ ( member_set_a @ C @ B2 )
=> ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_478_DiffI,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_479_DiffI,axiom,
! [C: list_set_a,A2: set_list_set_a,B2: set_list_set_a] :
( ( member_list_set_a @ C @ A2 )
=> ( ~ ( member_list_set_a @ C @ B2 )
=> ( member_list_set_a @ C @ ( minus_5993239077472792235_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_480_DiffI,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_481_add__mset__add__mset__same__iff,axiom,
! [A: set_a,A2: multiset_set_a,B2: multiset_set_a] :
( ( ( add_mset_set_a @ A @ A2 )
= ( add_mset_set_a @ A @ B2 ) )
= ( A2 = B2 ) ) ).
% add_mset_add_mset_same_iff
thf(fact_482_multi__self__add__other__not__self,axiom,
! [M: multiset_set_a,X: set_a] :
( M
!= ( add_mset_set_a @ X @ M ) ) ).
% multi_self_add_other_not_self
thf(fact_483_add__block__alt,axiom,
! [B: set_a] :
( ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ B )
= ( add_mset_set_a @ B @ ( mset_set_a @ b_s ) ) ) ).
% add_block_alt
thf(fact_484_add__mset__eq__singleton__iff,axiom,
! [X: set_a,M: multiset_set_a,Y4: set_a] :
( ( ( add_mset_set_a @ X @ M )
= ( add_mset_set_a @ Y4 @ zero_z5079479921072680283_set_a ) )
= ( ( M = zero_z5079479921072680283_set_a )
& ( X = Y4 ) ) ) ).
% add_mset_eq_singleton_iff
thf(fact_485_single__eq__add__mset,axiom,
! [A: set_a,B: set_a,M: multiset_set_a] :
( ( ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a )
= ( add_mset_set_a @ B @ M ) )
= ( ( B = A )
& ( M = zero_z5079479921072680283_set_a ) ) ) ).
% single_eq_add_mset
thf(fact_486_add__mset__eq__single,axiom,
! [B: set_a,M: multiset_set_a,A: set_a] :
( ( ( add_mset_set_a @ B @ M )
= ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) )
= ( ( B = A )
& ( M = zero_z5079479921072680283_set_a ) ) ) ).
% add_mset_eq_single
thf(fact_487_single__eq__single,axiom,
! [A: set_a,B: set_a] :
( ( ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a )
= ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) )
= ( A = B ) ) ).
% single_eq_single
thf(fact_488_mset__list__of__mset,axiom,
! [M2: multiset_set_a] :
( ( mset_set_a @ ( multis2300860884436073961_set_a @ M2 ) )
= M2 ) ).
% mset_list_of_mset
thf(fact_489_add__mset__subseteq__single__iff,axiom,
! [A: set_a,M: multiset_set_a,B: set_a] :
( ( subseteq_mset_set_a @ ( add_mset_set_a @ A @ M ) @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) )
= ( ( M = zero_z5079479921072680283_set_a )
& ( A = B ) ) ) ).
% add_mset_subseteq_single_iff
thf(fact_490_diff__add__mset__swap,axiom,
! [B: a,A2: multiset_a,M: multiset_a] :
( ~ ( member_a @ B @ ( set_mset_a @ A2 ) )
=> ( ( minus_3765977307040488491iset_a @ ( add_mset_a @ B @ M ) @ A2 )
= ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ M @ A2 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_491_diff__add__mset__swap,axiom,
! [B: list_set_a,A2: multiset_list_set_a,M: multiset_list_set_a] :
( ~ ( member_list_set_a @ B @ ( set_mset_list_set_a @ A2 ) )
=> ( ( minus_2572999347059163665_set_a @ ( add_mset_list_set_a @ B @ M ) @ A2 )
= ( add_mset_list_set_a @ B @ ( minus_2572999347059163665_set_a @ M @ A2 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_492_diff__add__mset__swap,axiom,
! [B: nat,A2: multiset_nat,M: multiset_nat] :
( ~ ( member_nat @ B @ ( set_mset_nat @ A2 ) )
=> ( ( minus_8522176038001411705et_nat @ ( add_mset_nat @ B @ M ) @ A2 )
= ( add_mset_nat @ B @ ( minus_8522176038001411705et_nat @ M @ A2 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_493_diff__add__mset__swap,axiom,
! [B: set_a,A2: multiset_set_a,M: multiset_set_a] :
( ~ ( member_set_a @ B @ ( set_mset_set_a @ A2 ) )
=> ( ( minus_706656509937749387_set_a @ ( add_mset_set_a @ B @ M ) @ A2 )
= ( add_mset_set_a @ B @ ( minus_706656509937749387_set_a @ M @ A2 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_494_remove__diff__multiset,axiom,
! [X13: a,A2: multiset_a,B2: multiset_a] :
( ~ ( member_a @ X13 @ ( set_mset_a @ A2 ) )
=> ( ( minus_3765977307040488491iset_a @ A2 @ ( add_mset_a @ X13 @ B2 ) )
= ( minus_3765977307040488491iset_a @ A2 @ B2 ) ) ) ).
% remove_diff_multiset
thf(fact_495_remove__diff__multiset,axiom,
! [X13: list_set_a,A2: multiset_list_set_a,B2: multiset_list_set_a] :
( ~ ( member_list_set_a @ X13 @ ( set_mset_list_set_a @ A2 ) )
=> ( ( minus_2572999347059163665_set_a @ A2 @ ( add_mset_list_set_a @ X13 @ B2 ) )
= ( minus_2572999347059163665_set_a @ A2 @ B2 ) ) ) ).
% remove_diff_multiset
thf(fact_496_remove__diff__multiset,axiom,
! [X13: nat,A2: multiset_nat,B2: multiset_nat] :
( ~ ( member_nat @ X13 @ ( set_mset_nat @ A2 ) )
=> ( ( minus_8522176038001411705et_nat @ A2 @ ( add_mset_nat @ X13 @ B2 ) )
= ( minus_8522176038001411705et_nat @ A2 @ B2 ) ) ) ).
% remove_diff_multiset
thf(fact_497_remove__diff__multiset,axiom,
! [X13: set_a,A2: multiset_set_a,B2: multiset_set_a] :
( ~ ( member_set_a @ X13 @ ( set_mset_set_a @ A2 ) )
=> ( ( minus_706656509937749387_set_a @ A2 @ ( add_mset_set_a @ X13 @ B2 ) )
= ( minus_706656509937749387_set_a @ A2 @ B2 ) ) ) ).
% remove_diff_multiset
thf(fact_498_remove1__single__empty__iff,axiom,
! [L3: set_a,L4: set_a] :
( ( ( minus_706656509937749387_set_a @ ( add_mset_set_a @ L3 @ zero_z5079479921072680283_set_a ) @ ( add_mset_set_a @ L4 @ zero_z5079479921072680283_set_a ) )
= zero_z5079479921072680283_set_a )
= ( L4 = L3 ) ) ).
% remove1_single_empty_iff
thf(fact_499_add__mset__remove__trivial,axiom,
! [X: set_a,M: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( add_mset_set_a @ X @ M ) @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) )
= M ) ).
% add_mset_remove_trivial
thf(fact_500_single__subset__iff,axiom,
! [A: a,M: multiset_a] :
( ( subseteq_mset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) @ M )
= ( member_a @ A @ ( set_mset_a @ M ) ) ) ).
% single_subset_iff
thf(fact_501_single__subset__iff,axiom,
! [A: list_set_a,M: multiset_list_set_a] :
( ( subset925826154347786955_set_a @ ( add_mset_list_set_a @ A @ zero_z8272816460787710433_set_a ) @ M )
= ( member_list_set_a @ A @ ( set_mset_list_set_a @ M ) ) ) ).
% single_subset_iff
thf(fact_502_single__subset__iff,axiom,
! [A: nat,M: multiset_nat] :
( ( subseteq_mset_nat @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) @ M )
= ( member_nat @ A @ ( set_mset_nat @ M ) ) ) ).
% single_subset_iff
thf(fact_503_single__subset__iff,axiom,
! [A: set_a,M: multiset_set_a] :
( ( subseteq_mset_set_a @ ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) @ M )
= ( member_set_a @ A @ ( set_mset_set_a @ M ) ) ) ).
% single_subset_iff
thf(fact_504_diff__single__trivial,axiom,
! [X: a,M: multiset_a] :
( ~ ( member_a @ X @ ( set_mset_a @ M ) )
=> ( ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
= M ) ) ).
% diff_single_trivial
thf(fact_505_diff__single__trivial,axiom,
! [X: list_set_a,M: multiset_list_set_a] :
( ~ ( member_list_set_a @ X @ ( set_mset_list_set_a @ M ) )
=> ( ( minus_2572999347059163665_set_a @ M @ ( add_mset_list_set_a @ X @ zero_z8272816460787710433_set_a ) )
= M ) ) ).
% diff_single_trivial
thf(fact_506_diff__single__trivial,axiom,
! [X: nat,M: multiset_nat] :
( ~ ( member_nat @ X @ ( set_mset_nat @ M ) )
=> ( ( minus_8522176038001411705et_nat @ M @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) )
= M ) ) ).
% diff_single_trivial
thf(fact_507_diff__single__trivial,axiom,
! [X: set_a,M: multiset_set_a] :
( ~ ( member_set_a @ X @ ( set_mset_set_a @ M ) )
=> ( ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) )
= M ) ) ).
% diff_single_trivial
thf(fact_508_diff__union__swap2,axiom,
! [Y4: a,M: multiset_a,X: a] :
( ( member_a @ Y4 @ ( set_mset_a @ M ) )
=> ( ( minus_3765977307040488491iset_a @ ( add_mset_a @ X @ M ) @ ( add_mset_a @ Y4 @ zero_zero_multiset_a ) )
= ( add_mset_a @ X @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ Y4 @ zero_zero_multiset_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_509_diff__union__swap2,axiom,
! [Y4: list_set_a,M: multiset_list_set_a,X: list_set_a] :
( ( member_list_set_a @ Y4 @ ( set_mset_list_set_a @ M ) )
=> ( ( minus_2572999347059163665_set_a @ ( add_mset_list_set_a @ X @ M ) @ ( add_mset_list_set_a @ Y4 @ zero_z8272816460787710433_set_a ) )
= ( add_mset_list_set_a @ X @ ( minus_2572999347059163665_set_a @ M @ ( add_mset_list_set_a @ Y4 @ zero_z8272816460787710433_set_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_510_diff__union__swap2,axiom,
! [Y4: nat,M: multiset_nat,X: nat] :
( ( member_nat @ Y4 @ ( set_mset_nat @ M ) )
=> ( ( minus_8522176038001411705et_nat @ ( add_mset_nat @ X @ M ) @ ( add_mset_nat @ Y4 @ zero_z7348594199698428585et_nat ) )
= ( add_mset_nat @ X @ ( minus_8522176038001411705et_nat @ M @ ( add_mset_nat @ Y4 @ zero_z7348594199698428585et_nat ) ) ) ) ) ).
% diff_union_swap2
thf(fact_511_diff__union__swap2,axiom,
! [Y4: set_a,M: multiset_set_a,X: set_a] :
( ( member_set_a @ Y4 @ ( set_mset_set_a @ M ) )
=> ( ( minus_706656509937749387_set_a @ ( add_mset_set_a @ X @ M ) @ ( add_mset_set_a @ Y4 @ zero_z5079479921072680283_set_a ) )
= ( add_mset_set_a @ X @ ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ Y4 @ zero_z5079479921072680283_set_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_512_insert__DiffM,axiom,
! [X: a,M: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M ) )
=> ( ( add_mset_a @ X @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) )
= M ) ) ).
% insert_DiffM
thf(fact_513_insert__DiffM,axiom,
! [X: list_set_a,M: multiset_list_set_a] :
( ( member_list_set_a @ X @ ( set_mset_list_set_a @ M ) )
=> ( ( add_mset_list_set_a @ X @ ( minus_2572999347059163665_set_a @ M @ ( add_mset_list_set_a @ X @ zero_z8272816460787710433_set_a ) ) )
= M ) ) ).
% insert_DiffM
thf(fact_514_insert__DiffM,axiom,
! [X: nat,M: multiset_nat] :
( ( member_nat @ X @ ( set_mset_nat @ M ) )
=> ( ( add_mset_nat @ X @ ( minus_8522176038001411705et_nat @ M @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) )
= M ) ) ).
% insert_DiffM
thf(fact_515_insert__DiffM,axiom,
! [X: set_a,M: multiset_set_a] :
( ( member_set_a @ X @ ( set_mset_set_a @ M ) )
=> ( ( add_mset_set_a @ X @ ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) )
= M ) ) ).
% insert_DiffM
thf(fact_516_DiffD2,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
=> ~ ( member_set_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_517_DiffD2,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( member_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_518_DiffD2,axiom,
! [C: list_set_a,A2: set_list_set_a,B2: set_list_set_a] :
( ( member_list_set_a @ C @ ( minus_5993239077472792235_set_a @ A2 @ B2 ) )
=> ~ ( member_list_set_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_519_DiffD2,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( member_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_520_DiffD1,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
=> ( member_set_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_521_DiffD1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_522_DiffD1,axiom,
! [C: list_set_a,A2: set_list_set_a,B2: set_list_set_a] :
( ( member_list_set_a @ C @ ( minus_5993239077472792235_set_a @ A2 @ B2 ) )
=> ( member_list_set_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_523_DiffD1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_524_DiffE,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
=> ~ ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_525_DiffE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_526_DiffE,axiom,
! [C: list_set_a,A2: set_list_set_a,B2: set_list_set_a] :
( ( member_list_set_a @ C @ ( minus_5993239077472792235_set_a @ A2 @ B2 ) )
=> ~ ( ( member_list_set_a @ C @ A2 )
=> ( member_list_set_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_527_DiffE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_528_Multiset_Odiff__right__commute,axiom,
! [M: multiset_set_a,N5: multiset_set_a,Q: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ M @ N5 ) @ Q )
= ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ M @ Q ) @ N5 ) ) ).
% Multiset.diff_right_commute
thf(fact_529_add__mset__diff__bothsides,axiom,
! [A: set_a,M: multiset_set_a,A2: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( add_mset_set_a @ A @ M ) @ ( add_mset_set_a @ A @ A2 ) )
= ( minus_706656509937749387_set_a @ M @ A2 ) ) ).
% add_mset_diff_bothsides
thf(fact_530_psubset__imp__ex__mem,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ? [B8: set_a] : ( member_set_a @ B8 @ ( minus_5736297505244876581_set_a @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_531_psubset__imp__ex__mem,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ? [B8: a] : ( member_a @ B8 @ ( minus_minus_set_a @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_532_psubset__imp__ex__mem,axiom,
! [A2: set_list_set_a,B2: set_list_set_a] :
( ( ord_le2831269692516795760_set_a @ A2 @ B2 )
=> ? [B8: list_set_a] : ( member_list_set_a @ B8 @ ( minus_5993239077472792235_set_a @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_533_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ? [B8: nat] : ( member_nat @ B8 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_534_psubsetD,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_535_psubsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_536_psubsetD,axiom,
! [A2: set_list_set_a,B2: set_list_set_a,C: list_set_a] :
( ( ord_le2831269692516795760_set_a @ A2 @ B2 )
=> ( ( member_list_set_a @ C @ A2 )
=> ( member_list_set_a @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_537_psubsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_538_add__eq__conv__ex,axiom,
! [A: set_a,M: multiset_set_a,B: set_a,N5: multiset_set_a] :
( ( ( add_mset_set_a @ A @ M )
= ( add_mset_set_a @ B @ N5 ) )
= ( ( ( M = N5 )
& ( A = B ) )
| ? [K3: multiset_set_a] :
( ( M
= ( add_mset_set_a @ B @ K3 ) )
& ( N5
= ( add_mset_set_a @ A @ K3 ) ) ) ) ) ).
% add_eq_conv_ex
thf(fact_539_add__mset__commute,axiom,
! [X: set_a,Y4: set_a,M: multiset_set_a] :
( ( add_mset_set_a @ X @ ( add_mset_set_a @ Y4 @ M ) )
= ( add_mset_set_a @ Y4 @ ( add_mset_set_a @ X @ M ) ) ) ).
% add_mset_commute
thf(fact_540_mset__add,axiom,
! [A: a,A2: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ A2 ) )
=> ~ ! [B6: multiset_a] :
( A2
!= ( add_mset_a @ A @ B6 ) ) ) ).
% mset_add
thf(fact_541_mset__add,axiom,
! [A: list_set_a,A2: multiset_list_set_a] :
( ( member_list_set_a @ A @ ( set_mset_list_set_a @ A2 ) )
=> ~ ! [B6: multiset_list_set_a] :
( A2
!= ( add_mset_list_set_a @ A @ B6 ) ) ) ).
% mset_add
thf(fact_542_mset__add,axiom,
! [A: nat,A2: multiset_nat] :
( ( member_nat @ A @ ( set_mset_nat @ A2 ) )
=> ~ ! [B6: multiset_nat] :
( A2
!= ( add_mset_nat @ A @ B6 ) ) ) ).
% mset_add
thf(fact_543_mset__add,axiom,
! [A: set_a,A2: multiset_set_a] :
( ( member_set_a @ A @ ( set_mset_set_a @ A2 ) )
=> ~ ! [B6: multiset_set_a] :
( A2
!= ( add_mset_set_a @ A @ B6 ) ) ) ).
% mset_add
thf(fact_544_multi__member__split,axiom,
! [X: a,M: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M ) )
=> ? [A6: multiset_a] :
( M
= ( add_mset_a @ X @ A6 ) ) ) ).
% multi_member_split
thf(fact_545_multi__member__split,axiom,
! [X: list_set_a,M: multiset_list_set_a] :
( ( member_list_set_a @ X @ ( set_mset_list_set_a @ M ) )
=> ? [A6: multiset_list_set_a] :
( M
= ( add_mset_list_set_a @ X @ A6 ) ) ) ).
% multi_member_split
thf(fact_546_multi__member__split,axiom,
! [X: nat,M: multiset_nat] :
( ( member_nat @ X @ ( set_mset_nat @ M ) )
=> ? [A6: multiset_nat] :
( M
= ( add_mset_nat @ X @ A6 ) ) ) ).
% multi_member_split
thf(fact_547_multi__member__split,axiom,
! [X: set_a,M: multiset_set_a] :
( ( member_set_a @ X @ ( set_mset_set_a @ M ) )
=> ? [A6: multiset_set_a] :
( M
= ( add_mset_set_a @ X @ A6 ) ) ) ).
% multi_member_split
thf(fact_548_insert__noteq__member,axiom,
! [B: a,B2: multiset_a,C: a,C2: multiset_a] :
( ( ( add_mset_a @ B @ B2 )
= ( add_mset_a @ C @ C2 ) )
=> ( ( B != C )
=> ( member_a @ C @ ( set_mset_a @ B2 ) ) ) ) ).
% insert_noteq_member
thf(fact_549_insert__noteq__member,axiom,
! [B: list_set_a,B2: multiset_list_set_a,C: list_set_a,C2: multiset_list_set_a] :
( ( ( add_mset_list_set_a @ B @ B2 )
= ( add_mset_list_set_a @ C @ C2 ) )
=> ( ( B != C )
=> ( member_list_set_a @ C @ ( set_mset_list_set_a @ B2 ) ) ) ) ).
% insert_noteq_member
thf(fact_550_insert__noteq__member,axiom,
! [B: nat,B2: multiset_nat,C: nat,C2: multiset_nat] :
( ( ( add_mset_nat @ B @ B2 )
= ( add_mset_nat @ C @ C2 ) )
=> ( ( B != C )
=> ( member_nat @ C @ ( set_mset_nat @ B2 ) ) ) ) ).
% insert_noteq_member
thf(fact_551_insert__noteq__member,axiom,
! [B: set_a,B2: multiset_set_a,C: set_a,C2: multiset_set_a] :
( ( ( add_mset_set_a @ B @ B2 )
= ( add_mset_set_a @ C @ C2 ) )
=> ( ( B != C )
=> ( member_set_a @ C @ ( set_mset_set_a @ B2 ) ) ) ) ).
% insert_noteq_member
thf(fact_552_union__single__eq__member,axiom,
! [X: a,M: multiset_a,N5: multiset_a] :
( ( ( add_mset_a @ X @ M )
= N5 )
=> ( member_a @ X @ ( set_mset_a @ N5 ) ) ) ).
% union_single_eq_member
thf(fact_553_union__single__eq__member,axiom,
! [X: list_set_a,M: multiset_list_set_a,N5: multiset_list_set_a] :
( ( ( add_mset_list_set_a @ X @ M )
= N5 )
=> ( member_list_set_a @ X @ ( set_mset_list_set_a @ N5 ) ) ) ).
% union_single_eq_member
thf(fact_554_union__single__eq__member,axiom,
! [X: nat,M: multiset_nat,N5: multiset_nat] :
( ( ( add_mset_nat @ X @ M )
= N5 )
=> ( member_nat @ X @ ( set_mset_nat @ N5 ) ) ) ).
% union_single_eq_member
thf(fact_555_union__single__eq__member,axiom,
! [X: set_a,M: multiset_set_a,N5: multiset_set_a] :
( ( ( add_mset_set_a @ X @ M )
= N5 )
=> ( member_set_a @ X @ ( set_mset_set_a @ N5 ) ) ) ).
% union_single_eq_member
thf(fact_556_mset__subset__eq__add__mset__cancel,axiom,
! [A: set_a,A2: multiset_set_a,B2: multiset_set_a] :
( ( subseteq_mset_set_a @ ( add_mset_set_a @ A @ A2 ) @ ( add_mset_set_a @ A @ B2 ) )
= ( subseteq_mset_set_a @ A2 @ B2 ) ) ).
% mset_subset_eq_add_mset_cancel
thf(fact_557_subset__mset__imp__subset__add__mset,axiom,
! [A2: multiset_set_a,B2: multiset_set_a,X: set_a] :
( ( subseteq_mset_set_a @ A2 @ B2 )
=> ( subseteq_mset_set_a @ A2 @ ( add_mset_set_a @ X @ B2 ) ) ) ).
% subset_mset_imp_subset_add_mset
thf(fact_558_multi__nonempty__split,axiom,
! [M: multiset_set_a] :
( ( M != zero_z5079479921072680283_set_a )
=> ? [A6: multiset_set_a,A4: set_a] :
( M
= ( add_mset_set_a @ A4 @ A6 ) ) ) ).
% multi_nonempty_split
thf(fact_559_empty__not__add__mset,axiom,
! [A: set_a,A2: multiset_set_a] :
( zero_z5079479921072680283_set_a
!= ( add_mset_set_a @ A @ A2 ) ) ).
% empty_not_add_mset
thf(fact_560_multiset__induct2,axiom,
! [P2: multiset_set_a > multiset_set_a > $o,M: multiset_set_a,N5: multiset_set_a] :
( ( P2 @ zero_z5079479921072680283_set_a @ zero_z5079479921072680283_set_a )
=> ( ! [A4: set_a,M7: multiset_set_a,N6: multiset_set_a] :
( ( P2 @ M7 @ N6 )
=> ( P2 @ ( add_mset_set_a @ A4 @ M7 ) @ N6 ) )
=> ( ! [A4: set_a,M7: multiset_set_a,N6: multiset_set_a] :
( ( P2 @ M7 @ N6 )
=> ( P2 @ M7 @ ( add_mset_set_a @ A4 @ N6 ) ) )
=> ( P2 @ M @ N5 ) ) ) ) ).
% multiset_induct2
thf(fact_561_multiset__induct,axiom,
! [P2: multiset_set_a > $o,M: multiset_set_a] :
( ( P2 @ zero_z5079479921072680283_set_a )
=> ( ! [X4: set_a,M7: multiset_set_a] :
( ( P2 @ M7 )
=> ( P2 @ ( add_mset_set_a @ X4 @ M7 ) ) )
=> ( P2 @ M ) ) ) ).
% multiset_induct
thf(fact_562_multiset__cases,axiom,
! [M: multiset_set_a] :
( ( M != zero_z5079479921072680283_set_a )
=> ~ ! [X4: set_a,N6: multiset_set_a] :
( M
!= ( add_mset_set_a @ X4 @ N6 ) ) ) ).
% multiset_cases
thf(fact_563_add__mset__less__imp__less__remove1__mset,axiom,
! [X: set_a,M: multiset_set_a,N5: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ ( add_mset_set_a @ X @ M ) @ N5 )
=> ( ord_le5765082015083327056_set_a @ M @ ( minus_706656509937749387_set_a @ N5 @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) ) ) ).
% add_mset_less_imp_less_remove1_mset
thf(fact_564_union__single__eq__diff,axiom,
! [X: set_a,M: multiset_set_a,N5: multiset_set_a] :
( ( ( add_mset_set_a @ X @ M )
= N5 )
=> ( M
= ( minus_706656509937749387_set_a @ N5 @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) ) ) ).
% union_single_eq_diff
thf(fact_565_add__eq__conv__diff,axiom,
! [A: set_a,M: multiset_set_a,B: set_a,N5: multiset_set_a] :
( ( ( add_mset_set_a @ A @ M )
= ( add_mset_set_a @ B @ N5 ) )
= ( ( ( M = N5 )
& ( A = B ) )
| ( ( M
= ( add_mset_set_a @ B @ ( minus_706656509937749387_set_a @ N5 @ ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) ) ) )
& ( N5
= ( add_mset_set_a @ A @ ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) ) ) ) ) ) ) ).
% add_eq_conv_diff
thf(fact_566_diff__union__swap,axiom,
! [A: set_a,B: set_a,M: multiset_set_a] :
( ( A != B )
=> ( ( add_mset_set_a @ B @ ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) ) )
= ( minus_706656509937749387_set_a @ ( add_mset_set_a @ B @ M ) @ ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) ) ) ) ).
% diff_union_swap
thf(fact_567_less__multiset__doubletons,axiom,
! [Y4: set_a,T: set_a,S2: set_a,X: set_a] :
( ( ( ord_less_set_a @ Y4 @ T )
| ( ord_less_set_a @ Y4 @ S2 ) )
=> ( ( ( ord_less_set_a @ X @ T )
| ( ord_less_set_a @ X @ S2 ) )
=> ( ord_le5765082015083327056_set_a @ ( add_mset_set_a @ Y4 @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) @ ( add_mset_set_a @ T @ ( add_mset_set_a @ S2 @ zero_z5079479921072680283_set_a ) ) ) ) ) ).
% less_multiset_doubletons
thf(fact_568_less__multiset__doubletons,axiom,
! [Y4: nat,T: nat,S2: nat,X: nat] :
( ( ( ord_less_nat @ Y4 @ T )
| ( ord_less_nat @ Y4 @ S2 ) )
=> ( ( ( ord_less_nat @ X @ T )
| ( ord_less_nat @ X @ S2 ) )
=> ( ord_le5777773500796000884et_nat @ ( add_mset_nat @ Y4 @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) @ ( add_mset_nat @ T @ ( add_mset_nat @ S2 @ zero_z7348594199698428585et_nat ) ) ) ) ) ).
% less_multiset_doubletons
thf(fact_569_subset__add__mset__notin__subset__mset,axiom,
! [A2: multiset_a,B: a,B2: multiset_a] :
( ( subseteq_mset_a @ A2 @ ( add_mset_a @ B @ B2 ) )
=> ( ~ ( member_a @ B @ ( set_mset_a @ A2 ) )
=> ( subseteq_mset_a @ A2 @ B2 ) ) ) ).
% subset_add_mset_notin_subset_mset
thf(fact_570_subset__add__mset__notin__subset__mset,axiom,
! [A2: multiset_list_set_a,B: list_set_a,B2: multiset_list_set_a] :
( ( subset925826154347786955_set_a @ A2 @ ( add_mset_list_set_a @ B @ B2 ) )
=> ( ~ ( member_list_set_a @ B @ ( set_mset_list_set_a @ A2 ) )
=> ( subset925826154347786955_set_a @ A2 @ B2 ) ) ) ).
% subset_add_mset_notin_subset_mset
thf(fact_571_subset__add__mset__notin__subset__mset,axiom,
! [A2: multiset_nat,B: nat,B2: multiset_nat] :
( ( subseteq_mset_nat @ A2 @ ( add_mset_nat @ B @ B2 ) )
=> ( ~ ( member_nat @ B @ ( set_mset_nat @ A2 ) )
=> ( subseteq_mset_nat @ A2 @ B2 ) ) ) ).
% subset_add_mset_notin_subset_mset
thf(fact_572_subset__add__mset__notin__subset__mset,axiom,
! [A2: multiset_set_a,B: set_a,B2: multiset_set_a] :
( ( subseteq_mset_set_a @ A2 @ ( add_mset_set_a @ B @ B2 ) )
=> ( ~ ( member_set_a @ B @ ( set_mset_set_a @ A2 ) )
=> ( subseteq_mset_set_a @ A2 @ B2 ) ) ) ).
% subset_add_mset_notin_subset_mset
thf(fact_573_multi__member__last,axiom,
! [X: a] : ( member_a @ X @ ( set_mset_a @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) ).
% multi_member_last
thf(fact_574_multi__member__last,axiom,
! [X: list_set_a] : ( member_list_set_a @ X @ ( set_mset_list_set_a @ ( add_mset_list_set_a @ X @ zero_z8272816460787710433_set_a ) ) ) ).
% multi_member_last
thf(fact_575_multi__member__last,axiom,
! [X: nat] : ( member_nat @ X @ ( set_mset_nat @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) ) ).
% multi_member_last
thf(fact_576_multi__member__last,axiom,
! [X: set_a] : ( member_set_a @ X @ ( set_mset_set_a @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) ) ).
% multi_member_last
thf(fact_577_multiset__remove__induct,axiom,
! [P2: multiset_a > $o,A2: multiset_a] :
( ( P2 @ zero_zero_multiset_a )
=> ( ! [A6: multiset_a] :
( ( A6 != zero_zero_multiset_a )
=> ( ! [X5: a] :
( ( member_a @ X5 @ ( set_mset_a @ A6 ) )
=> ( P2 @ ( minus_3765977307040488491iset_a @ A6 @ ( add_mset_a @ X5 @ zero_zero_multiset_a ) ) ) )
=> ( P2 @ A6 ) ) )
=> ( P2 @ A2 ) ) ) ).
% multiset_remove_induct
thf(fact_578_multiset__remove__induct,axiom,
! [P2: multiset_list_set_a > $o,A2: multiset_list_set_a] :
( ( P2 @ zero_z8272816460787710433_set_a )
=> ( ! [A6: multiset_list_set_a] :
( ( A6 != zero_z8272816460787710433_set_a )
=> ( ! [X5: list_set_a] :
( ( member_list_set_a @ X5 @ ( set_mset_list_set_a @ A6 ) )
=> ( P2 @ ( minus_2572999347059163665_set_a @ A6 @ ( add_mset_list_set_a @ X5 @ zero_z8272816460787710433_set_a ) ) ) )
=> ( P2 @ A6 ) ) )
=> ( P2 @ A2 ) ) ) ).
% multiset_remove_induct
thf(fact_579_multiset__remove__induct,axiom,
! [P2: multiset_nat > $o,A2: multiset_nat] :
( ( P2 @ zero_z7348594199698428585et_nat )
=> ( ! [A6: multiset_nat] :
( ( A6 != zero_z7348594199698428585et_nat )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_mset_nat @ A6 ) )
=> ( P2 @ ( minus_8522176038001411705et_nat @ A6 @ ( add_mset_nat @ X5 @ zero_z7348594199698428585et_nat ) ) ) )
=> ( P2 @ A6 ) ) )
=> ( P2 @ A2 ) ) ) ).
% multiset_remove_induct
thf(fact_580_multiset__remove__induct,axiom,
! [P2: multiset_set_a > $o,A2: multiset_set_a] :
( ( P2 @ zero_z5079479921072680283_set_a )
=> ( ! [A6: multiset_set_a] :
( ( A6 != zero_z5079479921072680283_set_a )
=> ( ! [X5: set_a] :
( ( member_set_a @ X5 @ ( set_mset_set_a @ A6 ) )
=> ( P2 @ ( minus_706656509937749387_set_a @ A6 @ ( add_mset_set_a @ X5 @ zero_z5079479921072680283_set_a ) ) ) )
=> ( P2 @ A6 ) ) )
=> ( P2 @ A2 ) ) ) ).
% multiset_remove_induct
thf(fact_581_trivial__add__mset__remove__iff,axiom,
! [A: a,N5: multiset_a,B: a] :
( ( ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ N5 @ ( add_mset_a @ B @ zero_zero_multiset_a ) ) )
= N5 )
= ( ( member_a @ A @ ( set_mset_a @ N5 ) )
& ( A = B ) ) ) ).
% trivial_add_mset_remove_iff
thf(fact_582_trivial__add__mset__remove__iff,axiom,
! [A: list_set_a,N5: multiset_list_set_a,B: list_set_a] :
( ( ( add_mset_list_set_a @ A @ ( minus_2572999347059163665_set_a @ N5 @ ( add_mset_list_set_a @ B @ zero_z8272816460787710433_set_a ) ) )
= N5 )
= ( ( member_list_set_a @ A @ ( set_mset_list_set_a @ N5 ) )
& ( A = B ) ) ) ).
% trivial_add_mset_remove_iff
thf(fact_583_trivial__add__mset__remove__iff,axiom,
! [A: nat,N5: multiset_nat,B: nat] :
( ( ( add_mset_nat @ A @ ( minus_8522176038001411705et_nat @ N5 @ ( add_mset_nat @ B @ zero_z7348594199698428585et_nat ) ) )
= N5 )
= ( ( member_nat @ A @ ( set_mset_nat @ N5 ) )
& ( A = B ) ) ) ).
% trivial_add_mset_remove_iff
thf(fact_584_trivial__add__mset__remove__iff,axiom,
! [A: set_a,N5: multiset_set_a,B: set_a] :
( ( ( add_mset_set_a @ A @ ( minus_706656509937749387_set_a @ N5 @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) ) )
= N5 )
= ( ( member_set_a @ A @ ( set_mset_set_a @ N5 ) )
& ( A = B ) ) ) ).
% trivial_add_mset_remove_iff
thf(fact_585_add__mset__remove__trivial__iff,axiom,
! [N5: multiset_a,A: a,B: a] :
( ( N5
= ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ N5 @ ( add_mset_a @ B @ zero_zero_multiset_a ) ) ) )
= ( ( member_a @ A @ ( set_mset_a @ N5 ) )
& ( A = B ) ) ) ).
% add_mset_remove_trivial_iff
thf(fact_586_add__mset__remove__trivial__iff,axiom,
! [N5: multiset_list_set_a,A: list_set_a,B: list_set_a] :
( ( N5
= ( add_mset_list_set_a @ A @ ( minus_2572999347059163665_set_a @ N5 @ ( add_mset_list_set_a @ B @ zero_z8272816460787710433_set_a ) ) ) )
= ( ( member_list_set_a @ A @ ( set_mset_list_set_a @ N5 ) )
& ( A = B ) ) ) ).
% add_mset_remove_trivial_iff
thf(fact_587_add__mset__remove__trivial__iff,axiom,
! [N5: multiset_nat,A: nat,B: nat] :
( ( N5
= ( add_mset_nat @ A @ ( minus_8522176038001411705et_nat @ N5 @ ( add_mset_nat @ B @ zero_z7348594199698428585et_nat ) ) ) )
= ( ( member_nat @ A @ ( set_mset_nat @ N5 ) )
& ( A = B ) ) ) ).
% add_mset_remove_trivial_iff
thf(fact_588_add__mset__remove__trivial__iff,axiom,
! [N5: multiset_set_a,A: set_a,B: set_a] :
( ( N5
= ( add_mset_set_a @ A @ ( minus_706656509937749387_set_a @ N5 @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) ) ) )
= ( ( member_set_a @ A @ ( set_mset_set_a @ N5 ) )
& ( A = B ) ) ) ).
% add_mset_remove_trivial_iff
thf(fact_589_remove__1__mset__id__iff__notin,axiom,
! [M: multiset_a,A: a] :
( ( ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ A @ zero_zero_multiset_a ) )
= M )
= ( ~ ( member_a @ A @ ( set_mset_a @ M ) ) ) ) ).
% remove_1_mset_id_iff_notin
thf(fact_590_remove__1__mset__id__iff__notin,axiom,
! [M: multiset_list_set_a,A: list_set_a] :
( ( ( minus_2572999347059163665_set_a @ M @ ( add_mset_list_set_a @ A @ zero_z8272816460787710433_set_a ) )
= M )
= ( ~ ( member_list_set_a @ A @ ( set_mset_list_set_a @ M ) ) ) ) ).
% remove_1_mset_id_iff_notin
thf(fact_591_remove__1__mset__id__iff__notin,axiom,
! [M: multiset_nat,A: nat] :
( ( ( minus_8522176038001411705et_nat @ M @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) )
= M )
= ( ~ ( member_nat @ A @ ( set_mset_nat @ M ) ) ) ) ).
% remove_1_mset_id_iff_notin
thf(fact_592_remove__1__mset__id__iff__notin,axiom,
! [M: multiset_set_a,A: set_a] :
( ( ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) )
= M )
= ( ~ ( member_set_a @ A @ ( set_mset_set_a @ M ) ) ) ) ).
% remove_1_mset_id_iff_notin
thf(fact_593_id__remove__1__mset__iff__notin,axiom,
! [M: multiset_a,A: a] :
( ( M
= ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= ( ~ ( member_a @ A @ ( set_mset_a @ M ) ) ) ) ).
% id_remove_1_mset_iff_notin
thf(fact_594_id__remove__1__mset__iff__notin,axiom,
! [M: multiset_list_set_a,A: list_set_a] :
( ( M
= ( minus_2572999347059163665_set_a @ M @ ( add_mset_list_set_a @ A @ zero_z8272816460787710433_set_a ) ) )
= ( ~ ( member_list_set_a @ A @ ( set_mset_list_set_a @ M ) ) ) ) ).
% id_remove_1_mset_iff_notin
thf(fact_595_id__remove__1__mset__iff__notin,axiom,
! [M: multiset_nat,A: nat] :
( ( M
= ( minus_8522176038001411705et_nat @ M @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) )
= ( ~ ( member_nat @ A @ ( set_mset_nat @ M ) ) ) ) ).
% id_remove_1_mset_iff_notin
thf(fact_596_id__remove__1__mset__iff__notin,axiom,
! [M: multiset_set_a,A: set_a] :
( ( M
= ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) ) )
= ( ~ ( member_set_a @ A @ ( set_mset_set_a @ M ) ) ) ) ).
% id_remove_1_mset_iff_notin
thf(fact_597_add__mset__eq__add__mset__ne,axiom,
! [A: a,B: a,A2: multiset_a,B2: multiset_a] :
( ( A != B )
=> ( ( ( add_mset_a @ A @ A2 )
= ( add_mset_a @ B @ B2 ) )
= ( ( member_a @ A @ ( set_mset_a @ B2 ) )
& ( member_a @ B @ ( set_mset_a @ A2 ) )
& ( A2
= ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ B2 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ) ) ) ) ).
% add_mset_eq_add_mset_ne
thf(fact_598_add__mset__eq__add__mset__ne,axiom,
! [A: list_set_a,B: list_set_a,A2: multiset_list_set_a,B2: multiset_list_set_a] :
( ( A != B )
=> ( ( ( add_mset_list_set_a @ A @ A2 )
= ( add_mset_list_set_a @ B @ B2 ) )
= ( ( member_list_set_a @ A @ ( set_mset_list_set_a @ B2 ) )
& ( member_list_set_a @ B @ ( set_mset_list_set_a @ A2 ) )
& ( A2
= ( add_mset_list_set_a @ B @ ( minus_2572999347059163665_set_a @ B2 @ ( add_mset_list_set_a @ A @ zero_z8272816460787710433_set_a ) ) ) ) ) ) ) ).
% add_mset_eq_add_mset_ne
thf(fact_599_add__mset__eq__add__mset__ne,axiom,
! [A: nat,B: nat,A2: multiset_nat,B2: multiset_nat] :
( ( A != B )
=> ( ( ( add_mset_nat @ A @ A2 )
= ( add_mset_nat @ B @ B2 ) )
= ( ( member_nat @ A @ ( set_mset_nat @ B2 ) )
& ( member_nat @ B @ ( set_mset_nat @ A2 ) )
& ( A2
= ( add_mset_nat @ B @ ( minus_8522176038001411705et_nat @ B2 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) ) ) ) ) ) ).
% add_mset_eq_add_mset_ne
thf(fact_600_add__mset__eq__add__mset__ne,axiom,
! [A: set_a,B: set_a,A2: multiset_set_a,B2: multiset_set_a] :
( ( A != B )
=> ( ( ( add_mset_set_a @ A @ A2 )
= ( add_mset_set_a @ B @ B2 ) )
= ( ( member_set_a @ A @ ( set_mset_set_a @ B2 ) )
& ( member_set_a @ B @ ( set_mset_set_a @ A2 ) )
& ( A2
= ( add_mset_set_a @ B @ ( minus_706656509937749387_set_a @ B2 @ ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) ) ) ) ) ) ) ).
% add_mset_eq_add_mset_ne
thf(fact_601_more__than__one__mset__mset__diff,axiom,
! [A: a,M: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) )
=> ( ( set_mset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= ( set_mset_a @ M ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_602_more__than__one__mset__mset__diff,axiom,
! [A: list_set_a,M: multiset_list_set_a] :
( ( member_list_set_a @ A @ ( set_mset_list_set_a @ ( minus_2572999347059163665_set_a @ M @ ( add_mset_list_set_a @ A @ zero_z8272816460787710433_set_a ) ) ) )
=> ( ( set_mset_list_set_a @ ( minus_2572999347059163665_set_a @ M @ ( add_mset_list_set_a @ A @ zero_z8272816460787710433_set_a ) ) )
= ( set_mset_list_set_a @ M ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_603_more__than__one__mset__mset__diff,axiom,
! [A: nat,M: multiset_nat] :
( ( member_nat @ A @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) ) )
=> ( ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) )
= ( set_mset_nat @ M ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_604_more__than__one__mset__mset__diff,axiom,
! [A: set_a,M: multiset_set_a] :
( ( member_set_a @ A @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) ) ) )
=> ( ( set_mset_set_a @ ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) ) )
= ( set_mset_set_a @ M ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_605_multiset__add__sub__el__shuffle,axiom,
! [C: a,B2: multiset_a,B: a] :
( ( member_a @ C @ ( set_mset_a @ B2 ) )
=> ( ( B != C )
=> ( ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ B2 @ ( add_mset_a @ C @ zero_zero_multiset_a ) ) )
= ( minus_3765977307040488491iset_a @ ( add_mset_a @ B @ B2 ) @ ( add_mset_a @ C @ zero_zero_multiset_a ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_606_multiset__add__sub__el__shuffle,axiom,
! [C: list_set_a,B2: multiset_list_set_a,B: list_set_a] :
( ( member_list_set_a @ C @ ( set_mset_list_set_a @ B2 ) )
=> ( ( B != C )
=> ( ( add_mset_list_set_a @ B @ ( minus_2572999347059163665_set_a @ B2 @ ( add_mset_list_set_a @ C @ zero_z8272816460787710433_set_a ) ) )
= ( minus_2572999347059163665_set_a @ ( add_mset_list_set_a @ B @ B2 ) @ ( add_mset_list_set_a @ C @ zero_z8272816460787710433_set_a ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_607_multiset__add__sub__el__shuffle,axiom,
! [C: nat,B2: multiset_nat,B: nat] :
( ( member_nat @ C @ ( set_mset_nat @ B2 ) )
=> ( ( B != C )
=> ( ( add_mset_nat @ B @ ( minus_8522176038001411705et_nat @ B2 @ ( add_mset_nat @ C @ zero_z7348594199698428585et_nat ) ) )
= ( minus_8522176038001411705et_nat @ ( add_mset_nat @ B @ B2 ) @ ( add_mset_nat @ C @ zero_z7348594199698428585et_nat ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_608_multiset__add__sub__el__shuffle,axiom,
! [C: set_a,B2: multiset_set_a,B: set_a] :
( ( member_set_a @ C @ ( set_mset_set_a @ B2 ) )
=> ( ( B != C )
=> ( ( add_mset_set_a @ B @ ( minus_706656509937749387_set_a @ B2 @ ( add_mset_set_a @ C @ zero_z5079479921072680283_set_a ) ) )
= ( minus_706656509937749387_set_a @ ( add_mset_set_a @ B @ B2 ) @ ( add_mset_set_a @ C @ zero_z5079479921072680283_set_a ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_609_add__mset__remove__trivial__eq,axiom,
! [N5: multiset_a,A: a] :
( ( N5
= ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ N5 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) )
= ( member_a @ A @ ( set_mset_a @ N5 ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_610_add__mset__remove__trivial__eq,axiom,
! [N5: multiset_list_set_a,A: list_set_a] :
( ( N5
= ( add_mset_list_set_a @ A @ ( minus_2572999347059163665_set_a @ N5 @ ( add_mset_list_set_a @ A @ zero_z8272816460787710433_set_a ) ) ) )
= ( member_list_set_a @ A @ ( set_mset_list_set_a @ N5 ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_611_add__mset__remove__trivial__eq,axiom,
! [N5: multiset_nat,A: nat] :
( ( N5
= ( add_mset_nat @ A @ ( minus_8522176038001411705et_nat @ N5 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) ) )
= ( member_nat @ A @ ( set_mset_nat @ N5 ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_612_add__mset__remove__trivial__eq,axiom,
! [N5: multiset_set_a,A: set_a] :
( ( N5
= ( add_mset_set_a @ A @ ( minus_706656509937749387_set_a @ N5 @ ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) ) ) )
= ( member_set_a @ A @ ( set_mset_set_a @ N5 ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_613_add__mset__remove__trivial__If,axiom,
! [A: a,N5: multiset_a] :
( ( ( member_a @ A @ ( set_mset_a @ N5 ) )
=> ( ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ N5 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= N5 ) )
& ( ~ ( member_a @ A @ ( set_mset_a @ N5 ) )
=> ( ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ N5 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= ( add_mset_a @ A @ N5 ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_614_add__mset__remove__trivial__If,axiom,
! [A: list_set_a,N5: multiset_list_set_a] :
( ( ( member_list_set_a @ A @ ( set_mset_list_set_a @ N5 ) )
=> ( ( add_mset_list_set_a @ A @ ( minus_2572999347059163665_set_a @ N5 @ ( add_mset_list_set_a @ A @ zero_z8272816460787710433_set_a ) ) )
= N5 ) )
& ( ~ ( member_list_set_a @ A @ ( set_mset_list_set_a @ N5 ) )
=> ( ( add_mset_list_set_a @ A @ ( minus_2572999347059163665_set_a @ N5 @ ( add_mset_list_set_a @ A @ zero_z8272816460787710433_set_a ) ) )
= ( add_mset_list_set_a @ A @ N5 ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_615_add__mset__remove__trivial__If,axiom,
! [A: nat,N5: multiset_nat] :
( ( ( member_nat @ A @ ( set_mset_nat @ N5 ) )
=> ( ( add_mset_nat @ A @ ( minus_8522176038001411705et_nat @ N5 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) )
= N5 ) )
& ( ~ ( member_nat @ A @ ( set_mset_nat @ N5 ) )
=> ( ( add_mset_nat @ A @ ( minus_8522176038001411705et_nat @ N5 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) )
= ( add_mset_nat @ A @ N5 ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_616_add__mset__remove__trivial__If,axiom,
! [A: set_a,N5: multiset_set_a] :
( ( ( member_set_a @ A @ ( set_mset_set_a @ N5 ) )
=> ( ( add_mset_set_a @ A @ ( minus_706656509937749387_set_a @ N5 @ ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) ) )
= N5 ) )
& ( ~ ( member_set_a @ A @ ( set_mset_set_a @ N5 ) )
=> ( ( add_mset_set_a @ A @ ( minus_706656509937749387_set_a @ N5 @ ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) ) )
= ( add_mset_set_a @ A @ N5 ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_617_add__mset__eq__add__mset,axiom,
! [A: a,M: multiset_a,B: a,M6: multiset_a] :
( ( ( add_mset_a @ A @ M )
= ( add_mset_a @ B @ M6 ) )
= ( ( ( A = B )
& ( M = M6 ) )
| ( ( A != B )
& ( member_a @ B @ ( set_mset_a @ M ) )
& ( ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ B @ zero_zero_multiset_a ) ) )
= M6 ) ) ) ) ).
% add_mset_eq_add_mset
thf(fact_618_add__mset__eq__add__mset,axiom,
! [A: list_set_a,M: multiset_list_set_a,B: list_set_a,M6: multiset_list_set_a] :
( ( ( add_mset_list_set_a @ A @ M )
= ( add_mset_list_set_a @ B @ M6 ) )
= ( ( ( A = B )
& ( M = M6 ) )
| ( ( A != B )
& ( member_list_set_a @ B @ ( set_mset_list_set_a @ M ) )
& ( ( add_mset_list_set_a @ A @ ( minus_2572999347059163665_set_a @ M @ ( add_mset_list_set_a @ B @ zero_z8272816460787710433_set_a ) ) )
= M6 ) ) ) ) ).
% add_mset_eq_add_mset
thf(fact_619_add__mset__eq__add__mset,axiom,
! [A: nat,M: multiset_nat,B: nat,M6: multiset_nat] :
( ( ( add_mset_nat @ A @ M )
= ( add_mset_nat @ B @ M6 ) )
= ( ( ( A = B )
& ( M = M6 ) )
| ( ( A != B )
& ( member_nat @ B @ ( set_mset_nat @ M ) )
& ( ( add_mset_nat @ A @ ( minus_8522176038001411705et_nat @ M @ ( add_mset_nat @ B @ zero_z7348594199698428585et_nat ) ) )
= M6 ) ) ) ) ).
% add_mset_eq_add_mset
thf(fact_620_add__mset__eq__add__mset,axiom,
! [A: set_a,M: multiset_set_a,B: set_a,M6: multiset_set_a] :
( ( ( add_mset_set_a @ A @ M )
= ( add_mset_set_a @ B @ M6 ) )
= ( ( ( A = B )
& ( M = M6 ) )
| ( ( A != B )
& ( member_set_a @ B @ ( set_mset_set_a @ M ) )
& ( ( add_mset_set_a @ A @ ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) ) )
= M6 ) ) ) ) ).
% add_mset_eq_add_mset
thf(fact_621_in__remove1__mset__neq,axiom,
! [A: a,B: a,C2: multiset_a] :
( ( A != B )
=> ( ( member_a @ A @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ C2 @ ( add_mset_a @ B @ zero_zero_multiset_a ) ) ) )
= ( member_a @ A @ ( set_mset_a @ C2 ) ) ) ) ).
% in_remove1_mset_neq
thf(fact_622_in__remove1__mset__neq,axiom,
! [A: list_set_a,B: list_set_a,C2: multiset_list_set_a] :
( ( A != B )
=> ( ( member_list_set_a @ A @ ( set_mset_list_set_a @ ( minus_2572999347059163665_set_a @ C2 @ ( add_mset_list_set_a @ B @ zero_z8272816460787710433_set_a ) ) ) )
= ( member_list_set_a @ A @ ( set_mset_list_set_a @ C2 ) ) ) ) ).
% in_remove1_mset_neq
thf(fact_623_in__remove1__mset__neq,axiom,
! [A: nat,B: nat,C2: multiset_nat] :
( ( A != B )
=> ( ( member_nat @ A @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ C2 @ ( add_mset_nat @ B @ zero_z7348594199698428585et_nat ) ) ) )
= ( member_nat @ A @ ( set_mset_nat @ C2 ) ) ) ) ).
% in_remove1_mset_neq
thf(fact_624_in__remove1__mset__neq,axiom,
! [A: set_a,B: set_a,C2: multiset_set_a] :
( ( A != B )
=> ( ( member_set_a @ A @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ C2 @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) ) ) )
= ( member_set_a @ A @ ( set_mset_set_a @ C2 ) ) ) ) ).
% in_remove1_mset_neq
thf(fact_625_multi__drop__mem__not__eq,axiom,
! [C: a,B2: multiset_a] :
( ( member_a @ C @ ( set_mset_a @ B2 ) )
=> ( ( minus_3765977307040488491iset_a @ B2 @ ( add_mset_a @ C @ zero_zero_multiset_a ) )
!= B2 ) ) ).
% multi_drop_mem_not_eq
thf(fact_626_multi__drop__mem__not__eq,axiom,
! [C: list_set_a,B2: multiset_list_set_a] :
( ( member_list_set_a @ C @ ( set_mset_list_set_a @ B2 ) )
=> ( ( minus_2572999347059163665_set_a @ B2 @ ( add_mset_list_set_a @ C @ zero_z8272816460787710433_set_a ) )
!= B2 ) ) ).
% multi_drop_mem_not_eq
thf(fact_627_multi__drop__mem__not__eq,axiom,
! [C: nat,B2: multiset_nat] :
( ( member_nat @ C @ ( set_mset_nat @ B2 ) )
=> ( ( minus_8522176038001411705et_nat @ B2 @ ( add_mset_nat @ C @ zero_z7348594199698428585et_nat ) )
!= B2 ) ) ).
% multi_drop_mem_not_eq
thf(fact_628_multi__drop__mem__not__eq,axiom,
! [C: set_a,B2: multiset_set_a] :
( ( member_set_a @ C @ ( set_mset_set_a @ B2 ) )
=> ( ( minus_706656509937749387_set_a @ B2 @ ( add_mset_set_a @ C @ zero_z5079479921072680283_set_a ) )
!= B2 ) ) ).
% multi_drop_mem_not_eq
thf(fact_629_diff__single__eq__union,axiom,
! [X: a,M: multiset_a,N5: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M ) )
=> ( ( ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
= N5 )
= ( M
= ( add_mset_a @ X @ N5 ) ) ) ) ).
% diff_single_eq_union
thf(fact_630_diff__single__eq__union,axiom,
! [X: list_set_a,M: multiset_list_set_a,N5: multiset_list_set_a] :
( ( member_list_set_a @ X @ ( set_mset_list_set_a @ M ) )
=> ( ( ( minus_2572999347059163665_set_a @ M @ ( add_mset_list_set_a @ X @ zero_z8272816460787710433_set_a ) )
= N5 )
= ( M
= ( add_mset_list_set_a @ X @ N5 ) ) ) ) ).
% diff_single_eq_union
thf(fact_631_diff__single__eq__union,axiom,
! [X: nat,M: multiset_nat,N5: multiset_nat] :
( ( member_nat @ X @ ( set_mset_nat @ M ) )
=> ( ( ( minus_8522176038001411705et_nat @ M @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) )
= N5 )
= ( M
= ( add_mset_nat @ X @ N5 ) ) ) ) ).
% diff_single_eq_union
thf(fact_632_diff__single__eq__union,axiom,
! [X: set_a,M: multiset_set_a,N5: multiset_set_a] :
( ( member_set_a @ X @ ( set_mset_set_a @ M ) )
=> ( ( ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) )
= N5 )
= ( M
= ( add_mset_set_a @ X @ N5 ) ) ) ) ).
% diff_single_eq_union
thf(fact_633_multiset__induct__min,axiom,
! [P2: multiset_nat > $o,M: multiset_nat] :
( ( P2 @ zero_z7348594199698428585et_nat )
=> ( ! [X4: nat,M7: multiset_nat] :
( ( P2 @ M7 )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_mset_nat @ M7 ) )
=> ( ord_less_eq_nat @ X4 @ Xa ) )
=> ( P2 @ ( add_mset_nat @ X4 @ M7 ) ) ) )
=> ( P2 @ M ) ) ) ).
% multiset_induct_min
thf(fact_634_multiset__induct__max,axiom,
! [P2: multiset_nat > $o,M: multiset_nat] :
( ( P2 @ zero_z7348594199698428585et_nat )
=> ( ! [X4: nat,M7: multiset_nat] :
( ( P2 @ M7 )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_mset_nat @ M7 ) )
=> ( ord_less_eq_nat @ Xa @ X4 ) )
=> ( P2 @ ( add_mset_nat @ X4 @ M7 ) ) ) )
=> ( P2 @ M ) ) ) ).
% multiset_induct_max
thf(fact_635_mset__subset__eq__single,axiom,
! [A: a,B2: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ B2 ) )
=> ( subseteq_mset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) @ B2 ) ) ).
% mset_subset_eq_single
thf(fact_636_mset__subset__eq__single,axiom,
! [A: list_set_a,B2: multiset_list_set_a] :
( ( member_list_set_a @ A @ ( set_mset_list_set_a @ B2 ) )
=> ( subset925826154347786955_set_a @ ( add_mset_list_set_a @ A @ zero_z8272816460787710433_set_a ) @ B2 ) ) ).
% mset_subset_eq_single
thf(fact_637_mset__subset__eq__single,axiom,
! [A: nat,B2: multiset_nat] :
( ( member_nat @ A @ ( set_mset_nat @ B2 ) )
=> ( subseteq_mset_nat @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) @ B2 ) ) ).
% mset_subset_eq_single
thf(fact_638_mset__subset__eq__single,axiom,
! [A: set_a,B2: multiset_set_a] :
( ( member_set_a @ A @ ( set_mset_set_a @ B2 ) )
=> ( subseteq_mset_set_a @ ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) @ B2 ) ) ).
% mset_subset_eq_single
thf(fact_639_multi__subset__induct,axiom,
! [F2: multiset_a,A2: multiset_a,P2: multiset_a > $o] :
( ( subseteq_mset_a @ F2 @ A2 )
=> ( ( P2 @ zero_zero_multiset_a )
=> ( ! [A4: a,F3: multiset_a] :
( ( member_a @ A4 @ ( set_mset_a @ A2 ) )
=> ( ( P2 @ F3 )
=> ( P2 @ ( add_mset_a @ A4 @ F3 ) ) ) )
=> ( P2 @ F2 ) ) ) ) ).
% multi_subset_induct
thf(fact_640_multi__subset__induct,axiom,
! [F2: multiset_list_set_a,A2: multiset_list_set_a,P2: multiset_list_set_a > $o] :
( ( subset925826154347786955_set_a @ F2 @ A2 )
=> ( ( P2 @ zero_z8272816460787710433_set_a )
=> ( ! [A4: list_set_a,F3: multiset_list_set_a] :
( ( member_list_set_a @ A4 @ ( set_mset_list_set_a @ A2 ) )
=> ( ( P2 @ F3 )
=> ( P2 @ ( add_mset_list_set_a @ A4 @ F3 ) ) ) )
=> ( P2 @ F2 ) ) ) ) ).
% multi_subset_induct
thf(fact_641_multi__subset__induct,axiom,
! [F2: multiset_nat,A2: multiset_nat,P2: multiset_nat > $o] :
( ( subseteq_mset_nat @ F2 @ A2 )
=> ( ( P2 @ zero_z7348594199698428585et_nat )
=> ( ! [A4: nat,F3: multiset_nat] :
( ( member_nat @ A4 @ ( set_mset_nat @ A2 ) )
=> ( ( P2 @ F3 )
=> ( P2 @ ( add_mset_nat @ A4 @ F3 ) ) ) )
=> ( P2 @ F2 ) ) ) ) ).
% multi_subset_induct
thf(fact_642_multi__subset__induct,axiom,
! [F2: multiset_set_a,A2: multiset_set_a,P2: multiset_set_a > $o] :
( ( subseteq_mset_set_a @ F2 @ A2 )
=> ( ( P2 @ zero_z5079479921072680283_set_a )
=> ( ! [A4: set_a,F3: multiset_set_a] :
( ( member_set_a @ A4 @ ( set_mset_set_a @ A2 ) )
=> ( ( P2 @ F3 )
=> ( P2 @ ( add_mset_set_a @ A4 @ F3 ) ) ) )
=> ( P2 @ F2 ) ) ) ) ).
% multi_subset_induct
thf(fact_643_insert__subset__eq__iff,axiom,
! [A: a,A2: multiset_a,B2: multiset_a] :
( ( subseteq_mset_a @ ( add_mset_a @ A @ A2 ) @ B2 )
= ( ( member_a @ A @ ( set_mset_a @ B2 ) )
& ( subseteq_mset_a @ A2 @ ( minus_3765977307040488491iset_a @ B2 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ) ) ).
% insert_subset_eq_iff
thf(fact_644_insert__subset__eq__iff,axiom,
! [A: list_set_a,A2: multiset_list_set_a,B2: multiset_list_set_a] :
( ( subset925826154347786955_set_a @ ( add_mset_list_set_a @ A @ A2 ) @ B2 )
= ( ( member_list_set_a @ A @ ( set_mset_list_set_a @ B2 ) )
& ( subset925826154347786955_set_a @ A2 @ ( minus_2572999347059163665_set_a @ B2 @ ( add_mset_list_set_a @ A @ zero_z8272816460787710433_set_a ) ) ) ) ) ).
% insert_subset_eq_iff
thf(fact_645_insert__subset__eq__iff,axiom,
! [A: nat,A2: multiset_nat,B2: multiset_nat] :
( ( subseteq_mset_nat @ ( add_mset_nat @ A @ A2 ) @ B2 )
= ( ( member_nat @ A @ ( set_mset_nat @ B2 ) )
& ( subseteq_mset_nat @ A2 @ ( minus_8522176038001411705et_nat @ B2 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) ) ) ) ).
% insert_subset_eq_iff
thf(fact_646_insert__subset__eq__iff,axiom,
! [A: set_a,A2: multiset_set_a,B2: multiset_set_a] :
( ( subseteq_mset_set_a @ ( add_mset_set_a @ A @ A2 ) @ B2 )
= ( ( member_set_a @ A @ ( set_mset_set_a @ B2 ) )
& ( subseteq_mset_set_a @ A2 @ ( minus_706656509937749387_set_a @ B2 @ ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) ) ) ) ) ).
% insert_subset_eq_iff
thf(fact_647_size__Diff1__le,axiom,
! [M: multiset_set_a,X: set_a] : ( ord_less_eq_nat @ ( size_s6566526139600085008_set_a @ ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) ) @ ( size_s6566526139600085008_set_a @ M ) ) ).
% size_Diff1_le
thf(fact_648_size__single,axiom,
! [B: set_a] :
( ( size_s6566526139600085008_set_a @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) )
= one_one_nat ) ).
% size_single
thf(fact_649_size__1__singleton__mset,axiom,
! [M: multiset_set_a] :
( ( ( size_s6566526139600085008_set_a @ M )
= one_one_nat )
=> ? [A4: set_a] :
( M
= ( add_mset_set_a @ A4 @ zero_z5079479921072680283_set_a ) ) ) ).
% size_1_singleton_mset
thf(fact_650_bounded__Max__nat,axiom,
! [P2: nat > $o,X: nat,M: nat] :
( ( P2 @ X )
=> ( ! [X4: nat] :
( ( P2 @ X4 )
=> ( ord_less_eq_nat @ X4 @ M ) )
=> ~ ! [M8: nat] :
( ( P2 @ M8 )
=> ~ ! [X5: nat] :
( ( P2 @ X5 )
=> ( ord_less_eq_nat @ X5 @ M8 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_651_size__mset__remove1__mset__le__iff,axiom,
! [M: multiset_a,X: a] :
( ( ord_less_nat @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) @ ( size_size_multiset_a @ M ) )
= ( member_a @ X @ ( set_mset_a @ M ) ) ) ).
% size_mset_remove1_mset_le_iff
thf(fact_652_size__mset__remove1__mset__le__iff,axiom,
! [M: multiset_list_set_a,X: list_set_a] :
( ( ord_less_nat @ ( size_s2437250918661492246_set_a @ ( minus_2572999347059163665_set_a @ M @ ( add_mset_list_set_a @ X @ zero_z8272816460787710433_set_a ) ) ) @ ( size_s2437250918661492246_set_a @ M ) )
= ( member_list_set_a @ X @ ( set_mset_list_set_a @ M ) ) ) ).
% size_mset_remove1_mset_le_iff
thf(fact_653_size__mset__remove1__mset__le__iff,axiom,
! [M: multiset_nat,X: nat] :
( ( ord_less_nat @ ( size_s5917832649809541300et_nat @ ( minus_8522176038001411705et_nat @ M @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) ) @ ( size_s5917832649809541300et_nat @ M ) )
= ( member_nat @ X @ ( set_mset_nat @ M ) ) ) ).
% size_mset_remove1_mset_le_iff
thf(fact_654_size__mset__remove1__mset__le__iff,axiom,
! [M: multiset_set_a,X: set_a] :
( ( ord_less_nat @ ( size_s6566526139600085008_set_a @ ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) ) @ ( size_s6566526139600085008_set_a @ M ) )
= ( member_set_a @ X @ ( set_mset_set_a @ M ) ) ) ).
% size_mset_remove1_mset_le_iff
thf(fact_655_size__Diff2__less,axiom,
! [X: a,M: multiset_a,Y4: a] :
( ( member_a @ X @ ( set_mset_a @ M ) )
=> ( ( member_a @ Y4 @ ( set_mset_a @ M ) )
=> ( ord_less_nat @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) @ ( add_mset_a @ Y4 @ zero_zero_multiset_a ) ) ) @ ( size_size_multiset_a @ M ) ) ) ) ).
% size_Diff2_less
thf(fact_656_size__Diff2__less,axiom,
! [X: list_set_a,M: multiset_list_set_a,Y4: list_set_a] :
( ( member_list_set_a @ X @ ( set_mset_list_set_a @ M ) )
=> ( ( member_list_set_a @ Y4 @ ( set_mset_list_set_a @ M ) )
=> ( ord_less_nat @ ( size_s2437250918661492246_set_a @ ( minus_2572999347059163665_set_a @ ( minus_2572999347059163665_set_a @ M @ ( add_mset_list_set_a @ X @ zero_z8272816460787710433_set_a ) ) @ ( add_mset_list_set_a @ Y4 @ zero_z8272816460787710433_set_a ) ) ) @ ( size_s2437250918661492246_set_a @ M ) ) ) ) ).
% size_Diff2_less
thf(fact_657_size__Diff2__less,axiom,
! [X: nat,M: multiset_nat,Y4: nat] :
( ( member_nat @ X @ ( set_mset_nat @ M ) )
=> ( ( member_nat @ Y4 @ ( set_mset_nat @ M ) )
=> ( ord_less_nat @ ( size_s5917832649809541300et_nat @ ( minus_8522176038001411705et_nat @ ( minus_8522176038001411705et_nat @ M @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) @ ( add_mset_nat @ Y4 @ zero_z7348594199698428585et_nat ) ) ) @ ( size_s5917832649809541300et_nat @ M ) ) ) ) ).
% size_Diff2_less
thf(fact_658_size__Diff2__less,axiom,
! [X: set_a,M: multiset_set_a,Y4: set_a] :
( ( member_set_a @ X @ ( set_mset_set_a @ M ) )
=> ( ( member_set_a @ Y4 @ ( set_mset_set_a @ M ) )
=> ( ord_less_nat @ ( size_s6566526139600085008_set_a @ ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) @ ( add_mset_set_a @ Y4 @ zero_z5079479921072680283_set_a ) ) ) @ ( size_s6566526139600085008_set_a @ M ) ) ) ) ).
% size_Diff2_less
thf(fact_659_size__Diff1__less,axiom,
! [X: a,M: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M ) )
=> ( ord_less_nat @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) @ ( size_size_multiset_a @ M ) ) ) ).
% size_Diff1_less
thf(fact_660_size__Diff1__less,axiom,
! [X: list_set_a,M: multiset_list_set_a] :
( ( member_list_set_a @ X @ ( set_mset_list_set_a @ M ) )
=> ( ord_less_nat @ ( size_s2437250918661492246_set_a @ ( minus_2572999347059163665_set_a @ M @ ( add_mset_list_set_a @ X @ zero_z8272816460787710433_set_a ) ) ) @ ( size_s2437250918661492246_set_a @ M ) ) ) ).
% size_Diff1_less
thf(fact_661_size__Diff1__less,axiom,
! [X: nat,M: multiset_nat] :
( ( member_nat @ X @ ( set_mset_nat @ M ) )
=> ( ord_less_nat @ ( size_s5917832649809541300et_nat @ ( minus_8522176038001411705et_nat @ M @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) ) @ ( size_s5917832649809541300et_nat @ M ) ) ) ).
% size_Diff1_less
thf(fact_662_size__Diff1__less,axiom,
! [X: set_a,M: multiset_set_a] :
( ( member_set_a @ X @ ( set_mset_set_a @ M ) )
=> ( ord_less_nat @ ( size_s6566526139600085008_set_a @ ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) ) @ ( size_s6566526139600085008_set_a @ M ) ) ) ).
% size_Diff1_less
thf(fact_663_count__single,axiom,
! [B: set_a,A: set_a] :
( ( ( B = A )
=> ( ( count_set_a @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) @ A )
= one_one_nat ) )
& ( ( B != A )
=> ( ( count_set_a @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) @ A )
= zero_zero_nat ) ) ) ).
% count_single
thf(fact_664_obtain__two__items__mset,axiom,
! [A2: multiset_a] :
( ( ord_less_nat @ one_one_nat @ ( size_size_multiset_a @ A2 ) )
=> ~ ! [X4: a] :
( ( member_a @ X4 @ ( set_mset_a @ A2 ) )
=> ! [Y: a] :
~ ( member_a @ Y @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ A2 @ ( add_mset_a @ X4 @ zero_zero_multiset_a ) ) ) ) ) ) ).
% obtain_two_items_mset
thf(fact_665_obtain__two__items__mset,axiom,
! [A2: multiset_list_set_a] :
( ( ord_less_nat @ one_one_nat @ ( size_s2437250918661492246_set_a @ A2 ) )
=> ~ ! [X4: list_set_a] :
( ( member_list_set_a @ X4 @ ( set_mset_list_set_a @ A2 ) )
=> ! [Y: list_set_a] :
~ ( member_list_set_a @ Y @ ( set_mset_list_set_a @ ( minus_2572999347059163665_set_a @ A2 @ ( add_mset_list_set_a @ X4 @ zero_z8272816460787710433_set_a ) ) ) ) ) ) ).
% obtain_two_items_mset
thf(fact_666_obtain__two__items__mset,axiom,
! [A2: multiset_nat] :
( ( ord_less_nat @ one_one_nat @ ( size_s5917832649809541300et_nat @ A2 ) )
=> ~ ! [X4: nat] :
( ( member_nat @ X4 @ ( set_mset_nat @ A2 ) )
=> ! [Y: nat] :
~ ( member_nat @ Y @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ A2 @ ( add_mset_nat @ X4 @ zero_z7348594199698428585et_nat ) ) ) ) ) ) ).
% obtain_two_items_mset
thf(fact_667_obtain__two__items__mset,axiom,
! [A2: multiset_set_a] :
( ( ord_less_nat @ one_one_nat @ ( size_s6566526139600085008_set_a @ A2 ) )
=> ~ ! [X4: set_a] :
( ( member_set_a @ X4 @ ( set_mset_set_a @ A2 ) )
=> ! [Y: set_a] :
~ ( member_set_a @ Y @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ A2 @ ( add_mset_set_a @ X4 @ zero_z5079479921072680283_set_a ) ) ) ) ) ) ).
% obtain_two_items_mset
thf(fact_668_size__Diff__singleton,axiom,
! [X: a,M: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M ) )
=> ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) )
= ( minus_minus_nat @ ( size_size_multiset_a @ M ) @ one_one_nat ) ) ) ).
% size_Diff_singleton
thf(fact_669_size__Diff__singleton,axiom,
! [X: list_set_a,M: multiset_list_set_a] :
( ( member_list_set_a @ X @ ( set_mset_list_set_a @ M ) )
=> ( ( size_s2437250918661492246_set_a @ ( minus_2572999347059163665_set_a @ M @ ( add_mset_list_set_a @ X @ zero_z8272816460787710433_set_a ) ) )
= ( minus_minus_nat @ ( size_s2437250918661492246_set_a @ M ) @ one_one_nat ) ) ) ).
% size_Diff_singleton
thf(fact_670_size__Diff__singleton,axiom,
! [X: nat,M: multiset_nat] :
( ( member_nat @ X @ ( set_mset_nat @ M ) )
=> ( ( size_s5917832649809541300et_nat @ ( minus_8522176038001411705et_nat @ M @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) )
= ( minus_minus_nat @ ( size_s5917832649809541300et_nat @ M ) @ one_one_nat ) ) ) ).
% size_Diff_singleton
thf(fact_671_size__Diff__singleton,axiom,
! [X: set_a,M: multiset_set_a] :
( ( member_set_a @ X @ ( set_mset_set_a @ M ) )
=> ( ( size_s6566526139600085008_set_a @ ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) )
= ( minus_minus_nat @ ( size_s6566526139600085008_set_a @ M ) @ one_one_nat ) ) ) ).
% size_Diff_singleton
thf(fact_672_size__Diff__singleton__if,axiom,
! [X: a,A2: multiset_a] :
( ( ( member_a @ X @ ( set_mset_a @ A2 ) )
=> ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ A2 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) )
= ( minus_minus_nat @ ( size_size_multiset_a @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_a @ X @ ( set_mset_a @ A2 ) )
=> ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ A2 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) )
= ( size_size_multiset_a @ A2 ) ) ) ) ).
% size_Diff_singleton_if
thf(fact_673_size__Diff__singleton__if,axiom,
! [X: list_set_a,A2: multiset_list_set_a] :
( ( ( member_list_set_a @ X @ ( set_mset_list_set_a @ A2 ) )
=> ( ( size_s2437250918661492246_set_a @ ( minus_2572999347059163665_set_a @ A2 @ ( add_mset_list_set_a @ X @ zero_z8272816460787710433_set_a ) ) )
= ( minus_minus_nat @ ( size_s2437250918661492246_set_a @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_list_set_a @ X @ ( set_mset_list_set_a @ A2 ) )
=> ( ( size_s2437250918661492246_set_a @ ( minus_2572999347059163665_set_a @ A2 @ ( add_mset_list_set_a @ X @ zero_z8272816460787710433_set_a ) ) )
= ( size_s2437250918661492246_set_a @ A2 ) ) ) ) ).
% size_Diff_singleton_if
thf(fact_674_size__Diff__singleton__if,axiom,
! [X: nat,A2: multiset_nat] :
( ( ( member_nat @ X @ ( set_mset_nat @ A2 ) )
=> ( ( size_s5917832649809541300et_nat @ ( minus_8522176038001411705et_nat @ A2 @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) )
= ( minus_minus_nat @ ( size_s5917832649809541300et_nat @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_nat @ X @ ( set_mset_nat @ A2 ) )
=> ( ( size_s5917832649809541300et_nat @ ( minus_8522176038001411705et_nat @ A2 @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) )
= ( size_s5917832649809541300et_nat @ A2 ) ) ) ) ).
% size_Diff_singleton_if
thf(fact_675_size__Diff__singleton__if,axiom,
! [X: set_a,A2: multiset_set_a] :
( ( ( member_set_a @ X @ ( set_mset_set_a @ A2 ) )
=> ( ( size_s6566526139600085008_set_a @ ( minus_706656509937749387_set_a @ A2 @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) )
= ( minus_minus_nat @ ( size_s6566526139600085008_set_a @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_set_a @ X @ ( set_mset_set_a @ A2 ) )
=> ( ( size_s6566526139600085008_set_a @ ( minus_706656509937749387_set_a @ A2 @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) )
= ( size_s6566526139600085008_set_a @ A2 ) ) ) ) ).
% size_Diff_singleton_if
thf(fact_676_index__remove1__mset__ne,axiom,
! [X: a,Xs: list_a,Y4: a,J1: nat] :
( ( member_a @ X @ ( set_mset_a @ ( mset_a @ Xs ) ) )
=> ( ( member_a @ Y4 @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ ( mset_a @ Xs ) @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) )
=> ( ( ( nth_a @ Xs @ J1 )
= X )
=> ( ( ord_less_nat @ J1 @ ( size_size_list_a @ Xs ) )
=> ~ ! [J22: nat] :
( ( ( nth_a @ Xs @ J22 )
= Y4 )
=> ( ( ord_less_nat @ J22 @ ( size_size_list_a @ Xs ) )
=> ( J1 = J22 ) ) ) ) ) ) ) ).
% index_remove1_mset_ne
thf(fact_677_index__remove1__mset__ne,axiom,
! [X: list_set_a,Xs: list_list_set_a,Y4: list_set_a,J1: nat] :
( ( member_list_set_a @ X @ ( set_mset_list_set_a @ ( mset_list_set_a @ Xs ) ) )
=> ( ( member_list_set_a @ Y4 @ ( set_mset_list_set_a @ ( minus_2572999347059163665_set_a @ ( mset_list_set_a @ Xs ) @ ( add_mset_list_set_a @ X @ zero_z8272816460787710433_set_a ) ) ) )
=> ( ( ( nth_list_set_a @ Xs @ J1 )
= X )
=> ( ( ord_less_nat @ J1 @ ( size_s3202270027169943894_set_a @ Xs ) )
=> ~ ! [J22: nat] :
( ( ( nth_list_set_a @ Xs @ J22 )
= Y4 )
=> ( ( ord_less_nat @ J22 @ ( size_s3202270027169943894_set_a @ Xs ) )
=> ( J1 = J22 ) ) ) ) ) ) ) ).
% index_remove1_mset_ne
thf(fact_678_index__remove1__mset__ne,axiom,
! [X: nat,Xs: list_nat,Y4: nat,J1: nat] :
( ( member_nat @ X @ ( set_mset_nat @ ( mset_nat @ Xs ) ) )
=> ( ( member_nat @ Y4 @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ ( mset_nat @ Xs ) @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) ) )
=> ( ( ( nth_nat @ Xs @ J1 )
= X )
=> ( ( ord_less_nat @ J1 @ ( size_size_list_nat @ Xs ) )
=> ~ ! [J22: nat] :
( ( ( nth_nat @ Xs @ J22 )
= Y4 )
=> ( ( ord_less_nat @ J22 @ ( size_size_list_nat @ Xs ) )
=> ( J1 = J22 ) ) ) ) ) ) ) ).
% index_remove1_mset_ne
thf(fact_679_index__remove1__mset__ne,axiom,
! [X: set_a,Xs: list_set_a,Y4: set_a,J1: nat] :
( ( member_set_a @ X @ ( set_mset_set_a @ ( mset_set_a @ Xs ) ) )
=> ( ( member_set_a @ Y4 @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ ( mset_set_a @ Xs ) @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) ) )
=> ( ( ( nth_set_a @ Xs @ J1 )
= X )
=> ( ( ord_less_nat @ J1 @ ( size_size_list_set_a @ Xs ) )
=> ~ ! [J22: nat] :
( ( ( nth_set_a @ Xs @ J22 )
= Y4 )
=> ( ( ord_less_nat @ J22 @ ( size_size_list_set_a @ Xs ) )
=> ( J1 = J22 ) ) ) ) ) ) ) ).
% index_remove1_mset_ne
thf(fact_680_size__remove1__mset__If,axiom,
! [M: multiset_a,X: a] :
( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) )
= ( minus_minus_nat @ ( size_size_multiset_a @ M ) @ ( if_nat @ ( member_a @ X @ ( set_mset_a @ M ) ) @ one_one_nat @ zero_zero_nat ) ) ) ).
% size_remove1_mset_If
thf(fact_681_size__remove1__mset__If,axiom,
! [M: multiset_list_set_a,X: list_set_a] :
( ( size_s2437250918661492246_set_a @ ( minus_2572999347059163665_set_a @ M @ ( add_mset_list_set_a @ X @ zero_z8272816460787710433_set_a ) ) )
= ( minus_minus_nat @ ( size_s2437250918661492246_set_a @ M ) @ ( if_nat @ ( member_list_set_a @ X @ ( set_mset_list_set_a @ M ) ) @ one_one_nat @ zero_zero_nat ) ) ) ).
% size_remove1_mset_If
thf(fact_682_size__remove1__mset__If,axiom,
! [M: multiset_nat,X: nat] :
( ( size_s5917832649809541300et_nat @ ( minus_8522176038001411705et_nat @ M @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) )
= ( minus_minus_nat @ ( size_s5917832649809541300et_nat @ M ) @ ( if_nat @ ( member_nat @ X @ ( set_mset_nat @ M ) ) @ one_one_nat @ zero_zero_nat ) ) ) ).
% size_remove1_mset_If
thf(fact_683_size__remove1__mset__If,axiom,
! [M: multiset_set_a,X: set_a] :
( ( size_s6566526139600085008_set_a @ ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) )
= ( minus_minus_nat @ ( size_s6566526139600085008_set_a @ M ) @ ( if_nat @ ( member_set_a @ X @ ( set_mset_set_a @ M ) ) @ one_one_nat @ zero_zero_nat ) ) ) ).
% size_remove1_mset_If
thf(fact_684_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M3: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N4 )
=> ( ord_less_nat @ X3 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_685_bounded__nat__set__is__finite,axiom,
! [N5: set_nat,N: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ N5 )
=> ( ord_less_nat @ X4 @ N ) )
=> ( finite_finite_nat @ N5 ) ) ).
% bounded_nat_set_is_finite
thf(fact_686_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M3: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N4 )
=> ( ord_less_eq_nat @ X3 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_687_mset__lt__single__right__iff,axiom,
! [M: multiset_nat,Y4: nat] :
( ( ord_le5777773500796000884et_nat @ M @ ( add_mset_nat @ Y4 @ zero_z7348594199698428585et_nat ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_mset_nat @ M ) )
=> ( ord_less_nat @ X3 @ Y4 ) ) ) ) ).
% mset_lt_single_right_iff
thf(fact_688_mset__le__single__right__iff,axiom,
! [M: multiset_nat,Y4: nat] :
( ( ord_le6602235886369790592et_nat @ M @ ( add_mset_nat @ Y4 @ zero_z7348594199698428585et_nat ) )
= ( ( M
= ( add_mset_nat @ Y4 @ zero_z7348594199698428585et_nat ) )
| ! [X3: nat] :
( ( member_nat @ X3 @ ( set_mset_nat @ M ) )
=> ( ord_less_nat @ X3 @ Y4 ) ) ) ) ).
% mset_le_single_right_iff
thf(fact_689_points__index__empty,axiom,
! [Ps: set_a] :
( ( design254580327166089565ndex_a @ zero_z5079479921072680283_set_a @ Ps )
= zero_zero_nat ) ).
% points_index_empty
thf(fact_690_points__index__one__not__unique__block,axiom,
! [B2: multiset_set_a,Ps: set_a,Bl2: set_a,Bl3: set_a] :
( ( ( design254580327166089565ndex_a @ B2 @ Ps )
= one_one_nat )
=> ( ( ord_less_eq_set_a @ Ps @ Bl2 )
=> ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ B2 ) )
=> ( ( member_set_a @ Bl3 @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ B2 @ ( add_mset_set_a @ Bl2 @ zero_z5079479921072680283_set_a ) ) ) )
=> ~ ( ord_less_eq_set_a @ Ps @ Bl3 ) ) ) ) ) ).
% points_index_one_not_unique_block
thf(fact_691_mset__lt__single__iff,axiom,
! [X: set_a,Y4: set_a] :
( ( ord_le5765082015083327056_set_a @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) @ ( add_mset_set_a @ Y4 @ zero_z5079479921072680283_set_a ) )
= ( ord_less_set_a @ X @ Y4 ) ) ).
% mset_lt_single_iff
thf(fact_692_mset__lt__single__iff,axiom,
! [X: nat,Y4: nat] :
( ( ord_le5777773500796000884et_nat @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) @ ( add_mset_nat @ Y4 @ zero_z7348594199698428585et_nat ) )
= ( ord_less_nat @ X @ Y4 ) ) ).
% mset_lt_single_iff
thf(fact_693_mset__le__single__iff,axiom,
! [X: set_a,Y4: set_a] :
( ( ord_le7905258569527593284_set_a @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) @ ( add_mset_set_a @ Y4 @ zero_z5079479921072680283_set_a ) )
= ( ord_less_eq_set_a @ X @ Y4 ) ) ).
% mset_le_single_iff
thf(fact_694_mset__le__single__iff,axiom,
! [X: nat,Y4: nat] :
( ( ord_le6602235886369790592et_nat @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) @ ( add_mset_nat @ Y4 @ zero_z7348594199698428585et_nat ) )
= ( ord_less_eq_nat @ X @ Y4 ) ) ).
% mset_le_single_iff
thf(fact_695_subset__eq__imp__le__multiset,axiom,
! [M: multiset_set_a,N5: multiset_set_a] :
( ( subseteq_mset_set_a @ M @ N5 )
=> ( ord_le7905258569527593284_set_a @ M @ N5 ) ) ).
% subset_eq_imp_le_multiset
thf(fact_696_less__eq__multiset__empty__left,axiom,
! [M: multiset_set_a] : ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ M ) ).
% less_eq_multiset_empty_left
thf(fact_697_less__eq__multiset__empty__right,axiom,
! [M: multiset_set_a] :
( ( M != zero_z5079479921072680283_set_a )
=> ~ ( ord_le7905258569527593284_set_a @ M @ zero_z5079479921072680283_set_a ) ) ).
% less_eq_multiset_empty_right
thf(fact_698_le__multiset__empty__left,axiom,
! [M: multiset_set_a] :
( ( M != zero_z5079479921072680283_set_a )
=> ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ M ) ) ).
% le_multiset_empty_left
thf(fact_699_le__multiset__empty__right,axiom,
! [M: multiset_set_a] :
~ ( ord_le5765082015083327056_set_a @ M @ zero_z5079479921072680283_set_a ) ).
% le_multiset_empty_right
thf(fact_700_incidence__system_Osys__block__sizes_Ocong,axiom,
design1769254222028858111izes_a = design1769254222028858111izes_a ).
% incidence_system.sys_block_sizes.cong
thf(fact_701_incidence__system_Odesign__support_Ocong,axiom,
design5397942185814921632port_a = design5397942185814921632port_a ).
% incidence_system.design_support.cong
thf(fact_702_ex__gt__imp__less__multiset,axiom,
! [N5: multiset_set_a,M: multiset_set_a] :
( ? [Y2: set_a] :
( ( member_set_a @ Y2 @ ( set_mset_set_a @ N5 ) )
& ! [X4: set_a] :
( ( member_set_a @ X4 @ ( set_mset_set_a @ M ) )
=> ( ord_less_set_a @ X4 @ Y2 ) ) )
=> ( ord_le5765082015083327056_set_a @ M @ N5 ) ) ).
% ex_gt_imp_less_multiset
thf(fact_703_ex__gt__imp__less__multiset,axiom,
! [N5: multiset_nat,M: multiset_nat] :
( ? [Y2: nat] :
( ( member_nat @ Y2 @ ( set_mset_nat @ N5 ) )
& ! [X4: nat] :
( ( member_nat @ X4 @ ( set_mset_nat @ M ) )
=> ( ord_less_nat @ X4 @ Y2 ) ) )
=> ( ord_le5777773500796000884et_nat @ M @ N5 ) ) ).
% ex_gt_imp_less_multiset
thf(fact_704_lt__imp__ex__count__lt,axiom,
! [M: multiset_set_a,N5: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ M @ N5 )
=> ? [Y: set_a] : ( ord_less_nat @ ( count_set_a @ M @ Y ) @ ( count_set_a @ N5 @ Y ) ) ) ).
% lt_imp_ex_count_lt
thf(fact_705_less__eq__multiset_092_060_094sub_062H_092_060_094sub_062O,axiom,
( ord_le7905258569527593284_set_a
= ( ^ [M5: multiset_set_a,N4: multiset_set_a] :
! [Y5: set_a] :
( ( ord_less_nat @ ( count_set_a @ N4 @ Y5 ) @ ( count_set_a @ M5 @ Y5 ) )
=> ? [X3: set_a] :
( ( ord_less_set_a @ Y5 @ X3 )
& ( ord_less_nat @ ( count_set_a @ M5 @ X3 ) @ ( count_set_a @ N4 @ X3 ) ) ) ) ) ) ).
% less_eq_multiset\<^sub>H\<^sub>O
thf(fact_706_less__eq__multiset_092_060_094sub_062H_092_060_094sub_062O,axiom,
( ord_le6602235886369790592et_nat
= ( ^ [M5: multiset_nat,N4: multiset_nat] :
! [Y5: nat] :
( ( ord_less_nat @ ( count_nat @ N4 @ Y5 ) @ ( count_nat @ M5 @ Y5 ) )
=> ? [X3: nat] :
( ( ord_less_nat @ Y5 @ X3 )
& ( ord_less_nat @ ( count_nat @ M5 @ X3 ) @ ( count_nat @ N4 @ X3 ) ) ) ) ) ) ).
% less_eq_multiset\<^sub>H\<^sub>O
thf(fact_707_less__multiset_092_060_094sub_062H_092_060_094sub_062O,axiom,
( ord_le5765082015083327056_set_a
= ( ^ [M5: multiset_set_a,N4: multiset_set_a] :
( ( M5 != N4 )
& ! [Y5: set_a] :
( ( ord_less_nat @ ( count_set_a @ N4 @ Y5 ) @ ( count_set_a @ M5 @ Y5 ) )
=> ? [X3: set_a] :
( ( ord_less_set_a @ Y5 @ X3 )
& ( ord_less_nat @ ( count_set_a @ M5 @ X3 ) @ ( count_set_a @ N4 @ X3 ) ) ) ) ) ) ) ).
% less_multiset\<^sub>H\<^sub>O
thf(fact_708_less__multiset_092_060_094sub_062H_092_060_094sub_062O,axiom,
( ord_le5777773500796000884et_nat
= ( ^ [M5: multiset_nat,N4: multiset_nat] :
( ( M5 != N4 )
& ! [Y5: nat] :
( ( ord_less_nat @ ( count_nat @ N4 @ Y5 ) @ ( count_nat @ M5 @ Y5 ) )
=> ? [X3: nat] :
( ( ord_less_nat @ Y5 @ X3 )
& ( ord_less_nat @ ( count_nat @ M5 @ X3 ) @ ( count_nat @ N4 @ X3 ) ) ) ) ) ) ) ).
% less_multiset\<^sub>H\<^sub>O
thf(fact_709_points__index__0__right__imp,axiom,
! [B2: multiset_set_a,Ps: set_a] :
( ! [B8: set_a] :
( ( member_set_a @ B8 @ ( set_mset_set_a @ B2 ) )
=> ~ ( ord_less_eq_set_a @ Ps @ B8 ) )
=> ( ( design254580327166089565ndex_a @ B2 @ Ps )
= zero_zero_nat ) ) ).
% points_index_0_right_imp
thf(fact_710_points__index__0__left__imp,axiom,
! [B2: multiset_set_a,Ps: set_a,B: set_a] :
( ( ( design254580327166089565ndex_a @ B2 @ Ps )
= zero_zero_nat )
=> ( ( member_set_a @ B @ ( set_mset_set_a @ B2 ) )
=> ~ ( ord_less_eq_set_a @ Ps @ B ) ) ) ).
% points_index_0_left_imp
thf(fact_711_points__index__0__iff,axiom,
! [B2: multiset_set_a,Ps: set_a] :
( ( ( design254580327166089565ndex_a @ B2 @ Ps )
= zero_zero_nat )
= ( ! [B3: set_a] :
( ( member_set_a @ B3 @ ( set_mset_set_a @ B2 ) )
=> ~ ( ord_less_eq_set_a @ Ps @ B3 ) ) ) ) ).
% points_index_0_iff
thf(fact_712_points__index__one__unique,axiom,
! [B2: multiset_set_a,Ps: set_a,Bl2: set_a,Bl3: set_a] :
( ( ( design254580327166089565ndex_a @ B2 @ Ps )
= one_one_nat )
=> ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ B2 ) )
=> ( ( ord_less_eq_set_a @ Ps @ Bl2 )
=> ( ( member_set_a @ Bl3 @ ( set_mset_set_a @ B2 ) )
=> ( ( ord_less_eq_set_a @ Ps @ Bl3 )
=> ( Bl2 = Bl3 ) ) ) ) ) ) ).
% points_index_one_unique
thf(fact_713_points__index__one__unique__block,axiom,
! [B2: multiset_set_a,Ps: set_a] :
( ( ( design254580327166089565ndex_a @ B2 @ Ps )
= one_one_nat )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ ( set_mset_set_a @ B2 ) )
& ( ord_less_eq_set_a @ Ps @ X4 )
& ! [Y2: set_a] :
( ( ( member_set_a @ Y2 @ ( set_mset_set_a @ B2 ) )
& ( ord_less_eq_set_a @ Ps @ Y2 ) )
=> ( Y2 = X4 ) ) ) ) ).
% points_index_one_unique_block
thf(fact_714_points__index__gt0__impl__existance,axiom,
! [B2: multiset_set_a,Ps: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( design254580327166089565ndex_a @ B2 @ Ps ) )
=> ? [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ B2 ) )
& ( ord_less_eq_set_a @ Ps @ Bl ) ) ) ).
% points_index_gt0_impl_existance
thf(fact_715_points__index__singleton__zero,axiom,
! [Ps: set_a,B: set_a] :
( ~ ( ord_less_eq_set_a @ Ps @ B )
=> ( ( design254580327166089565ndex_a @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) @ Ps )
= zero_zero_nat ) ) ).
% points_index_singleton_zero
thf(fact_716_points__index__singleton,axiom,
! [B: set_a,Ps: set_a] :
( ( ( design254580327166089565ndex_a @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) @ Ps )
= one_one_nat )
= ( ord_less_eq_set_a @ Ps @ B ) ) ).
% points_index_singleton
thf(fact_717_add__block__index__in,axiom,
! [Ps: set_a,B: set_a] :
( ( ord_less_eq_set_a @ Ps @ B )
=> ( ( design254580327166089565ndex_a @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ B ) @ Ps )
= ( plus_plus_nat @ ( design254580327166089565ndex_a @ ( mset_set_a @ b_s ) @ Ps ) @ one_one_nat ) ) ) ).
% add_block_index_in
thf(fact_718_add__block__def,axiom,
! [B: set_a] :
( ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ B )
= ( plus_p2331992037799027419_set_a @ ( mset_set_a @ b_s ) @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) ) ) ).
% add_block_def
thf(fact_719_multiple__point__index,axiom,
! [N: nat,Ps: set_a] :
( ( design254580327166089565ndex_a @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) @ Ps )
= ( times_times_nat @ ( design254580327166089565ndex_a @ ( mset_set_a @ b_s ) @ Ps ) @ N ) ) ).
% multiple_point_index
thf(fact_720_mbs_Ominimal,axiom,
! [X: nat,A2: set_nat,P2: nat > $o] :
( ( member_nat @ X @ A2 )
=> ( ( P2 @ X )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_nat @ ( size_size_nat @ X4 ) @ ( size_size_nat @ X ) )
& ( P2 @ X4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_nat @ ( size_size_nat @ Xa ) @ ( size_size_nat @ X4 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ) ).
% mbs.minimal
thf(fact_721_mbs_Ominimal,axiom,
! [X: list_set_a,A2: set_list_set_a,P2: list_set_a > $o] :
( ( member_list_set_a @ X @ A2 )
=> ( ( P2 @ X )
=> ? [X4: list_set_a] :
( ( member_list_set_a @ X4 @ A2 )
& ( ord_less_eq_nat @ ( size_size_list_set_a @ X4 ) @ ( size_size_list_set_a @ X ) )
& ( P2 @ X4 )
& ! [Xa: list_set_a] :
( ( member_list_set_a @ Xa @ A2 )
=> ( ( ord_less_nat @ ( size_size_list_set_a @ Xa ) @ ( size_size_list_set_a @ X4 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ) ).
% mbs.minimal
thf(fact_722_mbs_Ominimal,axiom,
! [X: multiset_set_a,A2: set_multiset_set_a,P2: multiset_set_a > $o] :
( ( member2747690772047059533_set_a @ X @ A2 )
=> ( ( P2 @ X )
=> ? [X4: multiset_set_a] :
( ( member2747690772047059533_set_a @ X4 @ A2 )
& ( ord_less_eq_nat @ ( size_s6566526139600085008_set_a @ X4 ) @ ( size_s6566526139600085008_set_a @ X ) )
& ( P2 @ X4 )
& ! [Xa: multiset_set_a] :
( ( member2747690772047059533_set_a @ Xa @ A2 )
=> ( ( ord_less_nat @ ( size_s6566526139600085008_set_a @ Xa ) @ ( size_s6566526139600085008_set_a @ X4 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ) ).
% mbs.minimal
thf(fact_723_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_724_add__left__cancel,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ A @ B )
= ( plus_p2331992037799027419_set_a @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_725_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_726_add__right__cancel,axiom,
! [B: multiset_set_a,A: multiset_set_a,C: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ B @ A )
= ( plus_p2331992037799027419_set_a @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_727_add__0,axiom,
! [A: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ zero_z5079479921072680283_set_a @ A )
= A ) ).
% add_0
thf(fact_728_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_729_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y4: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y4 ) )
= ( ( X = zero_zero_nat )
& ( Y4 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_730_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y4: nat] :
( ( ( plus_plus_nat @ X @ Y4 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y4 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_731_add__cancel__right__right,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( A
= ( plus_p2331992037799027419_set_a @ A @ B ) )
= ( B = zero_z5079479921072680283_set_a ) ) ).
% add_cancel_right_right
thf(fact_732_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_733_add__cancel__right__left,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( A
= ( plus_p2331992037799027419_set_a @ B @ A ) )
= ( B = zero_z5079479921072680283_set_a ) ) ).
% add_cancel_right_left
thf(fact_734_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_735_add__cancel__left__right,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ A @ B )
= A )
= ( B = zero_z5079479921072680283_set_a ) ) ).
% add_cancel_left_right
thf(fact_736_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_737_add__cancel__left__left,axiom,
! [B: multiset_set_a,A: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ B @ A )
= A )
= ( B = zero_z5079479921072680283_set_a ) ) ).
% add_cancel_left_left
thf(fact_738_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_739_add_Oright__neutral,axiom,
! [A: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ A @ zero_z5079479921072680283_set_a )
= A ) ).
% add.right_neutral
thf(fact_740_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_741_add__le__cancel__right,axiom,
! [A: multiset_set_a,C: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ ( plus_p2331992037799027419_set_a @ B @ C ) )
= ( ord_le7905258569527593284_set_a @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_742_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_743_add__le__cancel__left,axiom,
! [C: multiset_set_a,A: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ C @ A ) @ ( plus_p2331992037799027419_set_a @ C @ B ) )
= ( ord_le7905258569527593284_set_a @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_744_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_745_add__less__cancel__right,axiom,
! [A: multiset_set_a,C: multiset_set_a,B: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ ( plus_p2331992037799027419_set_a @ B @ C ) )
= ( ord_le5765082015083327056_set_a @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_746_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_747_add__less__cancel__left,axiom,
! [C: multiset_set_a,A: multiset_set_a,B: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ C @ A ) @ ( plus_p2331992037799027419_set_a @ C @ B ) )
= ( ord_le5765082015083327056_set_a @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_748_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_749_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_750_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_751_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_752_add__diff__cancel__left,axiom,
! [C: multiset_set_a,A: multiset_set_a,B: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ C @ A ) @ ( plus_p2331992037799027419_set_a @ C @ B ) )
= ( minus_706656509937749387_set_a @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_753_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_754_add__diff__cancel__left_H,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_755_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_756_add__diff__cancel__right,axiom,
! [A: multiset_set_a,C: multiset_set_a,B: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ ( plus_p2331992037799027419_set_a @ B @ C ) )
= ( minus_706656509937749387_set_a @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_757_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_758_add__diff__cancel__right_H,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_759_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_760_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_761_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_762_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_763_mult__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_764_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_765_mult__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_766_mult__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M2 @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_767_diff__diff__left,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% diff_diff_left
thf(fact_768_subset__mset_Oadd__le__cancel__left,axiom,
! [C: multiset_set_a,A: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ C @ A ) @ ( plus_p2331992037799027419_set_a @ C @ B ) )
= ( subseteq_mset_set_a @ A @ B ) ) ).
% subset_mset.add_le_cancel_left
thf(fact_769_subset__mset_Oadd__le__cancel__right,axiom,
! [A: multiset_set_a,C: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ ( plus_p2331992037799027419_set_a @ B @ C ) )
= ( subseteq_mset_set_a @ A @ B ) ) ).
% subset_mset.add_le_cancel_right
thf(fact_770_mset__subset__eq__mono__add__left__cancel,axiom,
! [C2: multiset_set_a,A2: multiset_set_a,B2: multiset_set_a] :
( ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ C2 @ A2 ) @ ( plus_p2331992037799027419_set_a @ C2 @ B2 ) )
= ( subseteq_mset_set_a @ A2 @ B2 ) ) ).
% mset_subset_eq_mono_add_left_cancel
thf(fact_771_mset__subset__eq__mono__add__right__cancel,axiom,
! [A2: multiset_set_a,C2: multiset_set_a,B2: multiset_set_a] :
( ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ C2 ) @ ( plus_p2331992037799027419_set_a @ B2 @ C2 ) )
= ( subseteq_mset_set_a @ A2 @ B2 ) ) ).
% mset_subset_eq_mono_add_right_cancel
thf(fact_772_nat__mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_773_nat__1__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N ) )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_774_union__eq__empty,axiom,
! [M: multiset_set_a,N5: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ M @ N5 )
= zero_z5079479921072680283_set_a )
= ( ( M = zero_z5079479921072680283_set_a )
& ( N5 = zero_z5079479921072680283_set_a ) ) ) ).
% union_eq_empty
thf(fact_775_empty__eq__union,axiom,
! [M: multiset_set_a,N5: multiset_set_a] :
( ( zero_z5079479921072680283_set_a
= ( plus_p2331992037799027419_set_a @ M @ N5 ) )
= ( ( M = zero_z5079479921072680283_set_a )
& ( N5 = zero_z5079479921072680283_set_a ) ) ) ).
% empty_eq_union
thf(fact_776_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [X: multiset_set_a,Y4: multiset_set_a] :
( ( zero_z5079479921072680283_set_a
= ( plus_p2331992037799027419_set_a @ X @ Y4 ) )
= ( ( X = zero_z5079479921072680283_set_a )
& ( Y4 = zero_z5079479921072680283_set_a ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_777_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [X: multiset_set_a,Y4: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ X @ Y4 )
= zero_z5079479921072680283_set_a )
= ( ( X = zero_z5079479921072680283_set_a )
& ( Y4 = zero_z5079479921072680283_set_a ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_778_union__mset__add__mset__left,axiom,
! [A: set_a,A2: multiset_set_a,B2: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ ( add_mset_set_a @ A @ A2 ) @ B2 )
= ( add_mset_set_a @ A @ ( plus_p2331992037799027419_set_a @ A2 @ B2 ) ) ) ).
% union_mset_add_mset_left
thf(fact_779_union__mset__add__mset__right,axiom,
! [A2: multiset_set_a,A: set_a,B2: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ A2 @ ( add_mset_set_a @ A @ B2 ) )
= ( add_mset_set_a @ A @ ( plus_p2331992037799027419_set_a @ A2 @ B2 ) ) ) ).
% union_mset_add_mset_right
thf(fact_780_diff__diff__add__mset,axiom,
! [M: multiset_set_a,N5: multiset_set_a,P2: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ M @ N5 ) @ P2 )
= ( minus_706656509937749387_set_a @ M @ ( plus_p2331992037799027419_set_a @ N5 @ P2 ) ) ) ).
% diff_diff_add_mset
thf(fact_781_repeat__mset__right,axiom,
! [A: nat,B: nat,A2: multiset_set_a] :
( ( repeat_mset_set_a @ A @ ( repeat_mset_set_a @ B @ A2 ) )
= ( repeat_mset_set_a @ ( times_times_nat @ A @ B ) @ A2 ) ) ).
% repeat_mset_right
thf(fact_782_repeat__mset__distrib2,axiom,
! [N: nat,A2: multiset_set_a,B2: multiset_set_a] :
( ( repeat_mset_set_a @ N @ ( plus_p2331992037799027419_set_a @ A2 @ B2 ) )
= ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ N @ A2 ) @ ( repeat_mset_set_a @ N @ B2 ) ) ) ).
% repeat_mset_distrib2
thf(fact_783_add__le__same__cancel1,axiom,
! [B: multiset_set_a,A: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ B @ A ) @ B )
= ( ord_le7905258569527593284_set_a @ A @ zero_z5079479921072680283_set_a ) ) ).
% add_le_same_cancel1
thf(fact_784_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_785_add__le__same__cancel2,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ B )
= ( ord_le7905258569527593284_set_a @ A @ zero_z5079479921072680283_set_a ) ) ).
% add_le_same_cancel2
thf(fact_786_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_787_le__add__same__cancel1,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ A @ ( plus_p2331992037799027419_set_a @ A @ B ) )
= ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ B ) ) ).
% le_add_same_cancel1
thf(fact_788_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_789_le__add__same__cancel2,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ A @ ( plus_p2331992037799027419_set_a @ B @ A ) )
= ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ B ) ) ).
% le_add_same_cancel2
thf(fact_790_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_791_less__add__same__cancel2,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ A @ ( plus_p2331992037799027419_set_a @ B @ A ) )
= ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ B ) ) ).
% less_add_same_cancel2
thf(fact_792_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_793_less__add__same__cancel1,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ A @ ( plus_p2331992037799027419_set_a @ A @ B ) )
= ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ B ) ) ).
% less_add_same_cancel1
thf(fact_794_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_795_add__less__same__cancel2,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ B )
= ( ord_le5765082015083327056_set_a @ A @ zero_z5079479921072680283_set_a ) ) ).
% add_less_same_cancel2
thf(fact_796_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_797_add__less__same__cancel1,axiom,
! [B: multiset_set_a,A: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ B @ A ) @ B )
= ( ord_le5765082015083327056_set_a @ A @ zero_z5079479921072680283_set_a ) ) ).
% add_less_same_cancel1
thf(fact_798_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_799_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_800_diff__add__zero,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( minus_706656509937749387_set_a @ A @ ( plus_p2331992037799027419_set_a @ A @ B ) )
= zero_z5079479921072680283_set_a ) ).
% diff_add_zero
thf(fact_801_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_802_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_803_nat__0__less__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_804_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_805_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I2 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_806_Nat_Odiff__diff__right,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_807_subset__mset_Oadd__le__same__cancel1,axiom,
! [B: multiset_set_a,A: multiset_set_a] :
( ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ B @ A ) @ B )
= ( subseteq_mset_set_a @ A @ zero_z5079479921072680283_set_a ) ) ).
% subset_mset.add_le_same_cancel1
thf(fact_808_subset__mset_Oadd__le__same__cancel2,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ B )
= ( subseteq_mset_set_a @ A @ zero_z5079479921072680283_set_a ) ) ).
% subset_mset.add_le_same_cancel2
thf(fact_809_subset__mset_Ole__add__same__cancel1,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ ( plus_p2331992037799027419_set_a @ A @ B ) )
= ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ B ) ) ).
% subset_mset.le_add_same_cancel1
thf(fact_810_subset__mset_Ole__add__same__cancel2,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ ( plus_p2331992037799027419_set_a @ B @ A ) )
= ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ B ) ) ).
% subset_mset.le_add_same_cancel2
thf(fact_811_count__union,axiom,
! [M: multiset_set_a,N5: multiset_set_a,A: set_a] :
( ( count_set_a @ ( plus_p2331992037799027419_set_a @ M @ N5 ) @ A )
= ( plus_plus_nat @ ( count_set_a @ M @ A ) @ ( count_set_a @ N5 @ A ) ) ) ).
% count_union
thf(fact_812_size__union,axiom,
! [M: multiset_set_a,N5: multiset_set_a] :
( ( size_s6566526139600085008_set_a @ ( plus_p2331992037799027419_set_a @ M @ N5 ) )
= ( plus_plus_nat @ ( size_s6566526139600085008_set_a @ M ) @ ( size_s6566526139600085008_set_a @ N5 ) ) ) ).
% size_union
thf(fact_813_mset__subset__eq__multiset__union__diff__commute,axiom,
! [B2: multiset_set_a,A2: multiset_set_a,C2: multiset_set_a] :
( ( subseteq_mset_set_a @ B2 @ A2 )
=> ( ( plus_p2331992037799027419_set_a @ ( minus_706656509937749387_set_a @ A2 @ B2 ) @ C2 )
= ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ C2 ) @ B2 ) ) ) ).
% mset_subset_eq_multiset_union_diff_commute
thf(fact_814_subset__mset_Oadd__diff__assoc2,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B )
=> ( ( plus_p2331992037799027419_set_a @ ( minus_706656509937749387_set_a @ B @ A ) @ C )
= ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ B @ C ) @ A ) ) ) ).
% subset_mset.add_diff_assoc2
thf(fact_815_subset__mset_Oadd__diff__assoc,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B )
=> ( ( plus_p2331992037799027419_set_a @ C @ ( minus_706656509937749387_set_a @ B @ A ) )
= ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ C @ B ) @ A ) ) ) ).
% subset_mset.add_diff_assoc
thf(fact_816_count__repeat__mset,axiom,
! [I2: nat,A2: multiset_set_a,A: set_a] :
( ( count_set_a @ ( repeat_mset_set_a @ I2 @ A2 ) @ A )
= ( times_times_nat @ I2 @ ( count_set_a @ A2 @ A ) ) ) ).
% count_repeat_mset
thf(fact_817_repeat__mset__size,axiom,
! [N: nat,A2: multiset_set_a] :
( ( size_s6566526139600085008_set_a @ ( repeat_mset_set_a @ N @ A2 ) )
= ( times_times_nat @ N @ ( size_s6566526139600085008_set_a @ A2 ) ) ) ).
% repeat_mset_size
thf(fact_818_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_819_multiple__blocks__num,axiom,
! [N: nat] :
( ( size_s6566526139600085008_set_a @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) )
= ( times_times_nat @ N @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ) ).
% multiple_blocks_num
thf(fact_820_multiple__block__multiplicity,axiom,
! [N: nat,Bl2: set_a] :
( ( count_set_a @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) @ Bl2 )
= ( times_times_nat @ ( count_set_a @ ( mset_set_a @ b_s ) @ Bl2 ) @ N ) ) ).
% multiple_block_multiplicity
thf(fact_821_comm__monoid__add__class_Oadd__0,axiom,
! [A: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ zero_z5079479921072680283_set_a @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_822_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_823_add_Ocomm__neutral,axiom,
! [A: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ A @ zero_z5079479921072680283_set_a )
= A ) ).
% add.comm_neutral
thf(fact_824_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_825_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: multiset_set_a,J2: multiset_set_a,K: multiset_set_a,L2: multiset_set_a] :
( ( ( ord_le7905258569527593284_set_a @ I2 @ J2 )
& ( K = L2 ) )
=> ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ I2 @ K ) @ ( plus_p2331992037799027419_set_a @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_826_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: nat,J2: nat,K: nat,L2: nat] :
( ( ( ord_less_eq_nat @ I2 @ J2 )
& ( K = L2 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_827_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: multiset_set_a,J2: multiset_set_a,K: multiset_set_a,L2: multiset_set_a] :
( ( ( I2 = J2 )
& ( ord_le7905258569527593284_set_a @ K @ L2 ) )
=> ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ I2 @ K ) @ ( plus_p2331992037799027419_set_a @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_828_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: nat,J2: nat,K: nat,L2: nat] :
( ( ( I2 = J2 )
& ( ord_less_eq_nat @ K @ L2 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_829_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: multiset_set_a,J2: multiset_set_a,K: multiset_set_a,L2: multiset_set_a] :
( ( ( ord_le7905258569527593284_set_a @ I2 @ J2 )
& ( ord_le7905258569527593284_set_a @ K @ L2 ) )
=> ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ I2 @ K ) @ ( plus_p2331992037799027419_set_a @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_830_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: nat,J2: nat,K: nat,L2: nat] :
( ( ( ord_less_eq_nat @ I2 @ J2 )
& ( ord_less_eq_nat @ K @ L2 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_831_add__mono,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a,D2: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ A @ B )
=> ( ( ord_le7905258569527593284_set_a @ C @ D2 )
=> ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ ( plus_p2331992037799027419_set_a @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_832_add__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_833_add__left__mono,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ A @ B )
=> ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ C @ A ) @ ( plus_p2331992037799027419_set_a @ C @ B ) ) ) ).
% add_left_mono
thf(fact_834_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_835_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_836_add__right__mono,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ A @ B )
=> ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ ( plus_p2331992037799027419_set_a @ B @ C ) ) ) ).
% add_right_mono
thf(fact_837_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_838_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
? [C4: nat] :
( B3
= ( plus_plus_nat @ A3 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_839_add__le__imp__le__left,axiom,
! [C: multiset_set_a,A: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ C @ A ) @ ( plus_p2331992037799027419_set_a @ C @ B ) )
=> ( ord_le7905258569527593284_set_a @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_840_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_841_add__le__imp__le__right,axiom,
! [A: multiset_set_a,C: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ ( plus_p2331992037799027419_set_a @ B @ C ) )
=> ( ord_le7905258569527593284_set_a @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_842_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_843_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: multiset_set_a,J2: multiset_set_a,K: multiset_set_a,L2: multiset_set_a] :
( ( ( ord_le5765082015083327056_set_a @ I2 @ J2 )
& ( ord_le5765082015083327056_set_a @ K @ L2 ) )
=> ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ I2 @ K ) @ ( plus_p2331992037799027419_set_a @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_844_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: nat,J2: nat,K: nat,L2: nat] :
( ( ( ord_less_nat @ I2 @ J2 )
& ( ord_less_nat @ K @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_845_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: multiset_set_a,J2: multiset_set_a,K: multiset_set_a,L2: multiset_set_a] :
( ( ( I2 = J2 )
& ( ord_le5765082015083327056_set_a @ K @ L2 ) )
=> ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ I2 @ K ) @ ( plus_p2331992037799027419_set_a @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_846_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: nat,J2: nat,K: nat,L2: nat] :
( ( ( I2 = J2 )
& ( ord_less_nat @ K @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_847_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: multiset_set_a,J2: multiset_set_a,K: multiset_set_a,L2: multiset_set_a] :
( ( ( ord_le5765082015083327056_set_a @ I2 @ J2 )
& ( K = L2 ) )
=> ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ I2 @ K ) @ ( plus_p2331992037799027419_set_a @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_848_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: nat,J2: nat,K: nat,L2: nat] :
( ( ( ord_less_nat @ I2 @ J2 )
& ( K = L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_849_add__strict__mono,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a,D2: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ A @ B )
=> ( ( ord_le5765082015083327056_set_a @ C @ D2 )
=> ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ ( plus_p2331992037799027419_set_a @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_850_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_851_add__strict__left__mono,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ A @ B )
=> ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ C @ A ) @ ( plus_p2331992037799027419_set_a @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_852_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_853_add__strict__right__mono,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ A @ B )
=> ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ ( plus_p2331992037799027419_set_a @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_854_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_855_add__less__imp__less__left,axiom,
! [C: multiset_set_a,A: multiset_set_a,B: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ C @ A ) @ ( plus_p2331992037799027419_set_a @ C @ B ) )
=> ( ord_le5765082015083327056_set_a @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_856_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_857_add__less__imp__less__right,axiom,
! [A: multiset_set_a,C: multiset_set_a,B: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ ( plus_p2331992037799027419_set_a @ B @ C ) )
=> ( ord_le5765082015083327056_set_a @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_858_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_859_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_860_diff__diff__eq,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ A @ B ) @ C )
= ( minus_706656509937749387_set_a @ A @ ( plus_p2331992037799027419_set_a @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_861_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_862_add__implies__diff,axiom,
! [C: multiset_set_a,B: multiset_set_a,A: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ C @ B )
= A )
=> ( C
= ( minus_706656509937749387_set_a @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_863_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_864_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_865_add__lessD1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_866_add__less__mono,axiom,
! [I2: nat,J2: nat,K: nat,L2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ K @ L2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ) ).
% add_less_mono
thf(fact_867_not__add__less1,axiom,
! [I2: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J2 ) @ I2 ) ).
% not_add_less1
thf(fact_868_not__add__less2,axiom,
! [J2: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_869_add__less__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_less_mono1
thf(fact_870_trans__less__add1,axiom,
! [I2: nat,J2: nat,M2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).
% trans_less_add1
thf(fact_871_trans__less__add2,axiom,
! [I2: nat,J2: nat,M2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).
% trans_less_add2
thf(fact_872_less__add__eq__less,axiom,
! [K: nat,L2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K @ L2 )
=> ( ( ( plus_plus_nat @ M2 @ L2 )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_873_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_874_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_875_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_876_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
? [K4: nat] :
( N2
= ( plus_plus_nat @ M3 @ K4 ) ) ) ) ).
% nat_le_iff_add
thf(fact_877_trans__le__add2,axiom,
! [I2: nat,J2: nat,M2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).
% trans_le_add2
thf(fact_878_trans__le__add1,axiom,
! [I2: nat,J2: nat,M2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).
% trans_le_add1
thf(fact_879_add__le__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_le_mono1
thf(fact_880_add__le__mono,axiom,
! [I2: nat,J2: nat,K: nat,L2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ K @ L2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ) ).
% add_le_mono
thf(fact_881_le__Suc__ex,axiom,
! [K: nat,L2: nat] :
( ( ord_less_eq_nat @ K @ L2 )
=> ? [N3: nat] :
( L2
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_882_add__leD2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_883_add__leD1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_884_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_885_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_886_add__leE,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_887_union__iff,axiom,
! [A: a,A2: multiset_a,B2: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( plus_plus_multiset_a @ A2 @ B2 ) ) )
= ( ( member_a @ A @ ( set_mset_a @ A2 ) )
| ( member_a @ A @ ( set_mset_a @ B2 ) ) ) ) ).
% union_iff
thf(fact_888_union__iff,axiom,
! [A: list_set_a,A2: multiset_list_set_a,B2: multiset_list_set_a] :
( ( member_list_set_a @ A @ ( set_mset_list_set_a @ ( plus_p4509188130224566113_set_a @ A2 @ B2 ) ) )
= ( ( member_list_set_a @ A @ ( set_mset_list_set_a @ A2 ) )
| ( member_list_set_a @ A @ ( set_mset_list_set_a @ B2 ) ) ) ) ).
% union_iff
thf(fact_889_union__iff,axiom,
! [A: nat,A2: multiset_nat,B2: multiset_nat] :
( ( member_nat @ A @ ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) ) )
= ( ( member_nat @ A @ ( set_mset_nat @ A2 ) )
| ( member_nat @ A @ ( set_mset_nat @ B2 ) ) ) ) ).
% union_iff
thf(fact_890_union__iff,axiom,
! [A: set_a,A2: multiset_set_a,B2: multiset_set_a] :
( ( member_set_a @ A @ ( set_mset_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ B2 ) ) )
= ( ( member_set_a @ A @ ( set_mset_set_a @ A2 ) )
| ( member_set_a @ A @ ( set_mset_set_a @ B2 ) ) ) ) ).
% union_iff
thf(fact_891_mset__subset__eq__exists__conv,axiom,
( subseteq_mset_set_a
= ( ^ [A5: multiset_set_a,B4: multiset_set_a] :
? [C5: multiset_set_a] :
( B4
= ( plus_p2331992037799027419_set_a @ A5 @ C5 ) ) ) ) ).
% mset_subset_eq_exists_conv
thf(fact_892_mset__subset__eq__add__right,axiom,
! [B2: multiset_set_a,A2: multiset_set_a] : ( subseteq_mset_set_a @ B2 @ ( plus_p2331992037799027419_set_a @ A2 @ B2 ) ) ).
% mset_subset_eq_add_right
thf(fact_893_mset__subset__eq__mono__add,axiom,
! [A2: multiset_set_a,B2: multiset_set_a,C2: multiset_set_a,D: multiset_set_a] :
( ( subseteq_mset_set_a @ A2 @ B2 )
=> ( ( subseteq_mset_set_a @ C2 @ D )
=> ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ C2 ) @ ( plus_p2331992037799027419_set_a @ B2 @ D ) ) ) ) ).
% mset_subset_eq_mono_add
thf(fact_894_mset__subset__eq__add__left,axiom,
! [A2: multiset_set_a,B2: multiset_set_a] : ( subseteq_mset_set_a @ A2 @ ( plus_p2331992037799027419_set_a @ A2 @ B2 ) ) ).
% mset_subset_eq_add_left
thf(fact_895_subset__mset_Oadd__le__imp__le__right,axiom,
! [A: multiset_set_a,C: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ ( plus_p2331992037799027419_set_a @ B @ C ) )
=> ( subseteq_mset_set_a @ A @ B ) ) ).
% subset_mset.add_le_imp_le_right
thf(fact_896_subset__mset_Oadd__le__imp__le__left,axiom,
! [C: multiset_set_a,A: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ C @ A ) @ ( plus_p2331992037799027419_set_a @ C @ B ) )
=> ( subseteq_mset_set_a @ A @ B ) ) ).
% subset_mset.add_le_imp_le_left
thf(fact_897_subset__mset_Oadd__right__mono,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B )
=> ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ ( plus_p2331992037799027419_set_a @ B @ C ) ) ) ).
% subset_mset.add_right_mono
thf(fact_898_subset__mset_Oadd__left__mono,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B )
=> ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ C @ A ) @ ( plus_p2331992037799027419_set_a @ C @ B ) ) ) ).
% subset_mset.add_left_mono
thf(fact_899_subset__mset_Ole__iff__add,axiom,
( subseteq_mset_set_a
= ( ^ [A3: multiset_set_a,B3: multiset_set_a] :
? [C4: multiset_set_a] :
( B3
= ( plus_p2331992037799027419_set_a @ A3 @ C4 ) ) ) ) ).
% subset_mset.le_iff_add
thf(fact_900_subset__mset_Oless__eqE,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B )
=> ~ ! [C3: multiset_set_a] :
( B
!= ( plus_p2331992037799027419_set_a @ A @ C3 ) ) ) ).
% subset_mset.less_eqE
thf(fact_901_subset__mset_Oadd__mono,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a,D2: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B )
=> ( ( subseteq_mset_set_a @ C @ D2 )
=> ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ ( plus_p2331992037799027419_set_a @ B @ D2 ) ) ) ) ).
% subset_mset.add_mono
thf(fact_902_mult__le__mono2,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J2 ) ) ) ).
% mult_le_mono2
thf(fact_903_mult__le__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ).
% mult_le_mono1
thf(fact_904_mult__le__mono,axiom,
! [I2: nat,J2: nat,K: nat,L2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ K @ L2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J2 @ L2 ) ) ) ) ).
% mult_le_mono
thf(fact_905_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_906_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_907_empty__neutral_I2_J,axiom,
! [X: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ X @ zero_z5079479921072680283_set_a )
= X ) ).
% empty_neutral(2)
thf(fact_908_empty__neutral_I1_J,axiom,
! [X: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ zero_z5079479921072680283_set_a @ X )
= X ) ).
% empty_neutral(1)
thf(fact_909_plus__multiset_Orep__eq,axiom,
! [X: multiset_set_a,Xa2: multiset_set_a] :
( ( count_set_a @ ( plus_p2331992037799027419_set_a @ X @ Xa2 ) )
= ( ^ [A3: set_a] : ( plus_plus_nat @ ( count_set_a @ X @ A3 ) @ ( count_set_a @ Xa2 @ A3 ) ) ) ) ).
% plus_multiset.rep_eq
thf(fact_910_add__mult__distrib,axiom,
! [M2: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M2 @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_911_add__mult__distrib2,axiom,
! [K: nat,M2: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_912_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_913_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ C )
= ( plus_p2331992037799027419_set_a @ A @ ( plus_p2331992037799027419_set_a @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_914_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_915_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J2: nat,K: nat,L2: nat] :
( ( ( I2 = J2 )
& ( K = L2 ) )
=> ( ( plus_plus_nat @ I2 @ K )
= ( plus_plus_nat @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_916_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: multiset_set_a,J2: multiset_set_a,K: multiset_set_a,L2: multiset_set_a] :
( ( ( I2 = J2 )
& ( K = L2 ) )
=> ( ( plus_p2331992037799027419_set_a @ I2 @ K )
= ( plus_p2331992037799027419_set_a @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_917_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_918_group__cancel_Oadd1,axiom,
! [A2: multiset_set_a,K: multiset_set_a,A: multiset_set_a,B: multiset_set_a] :
( ( A2
= ( plus_p2331992037799027419_set_a @ K @ A ) )
=> ( ( plus_p2331992037799027419_set_a @ A2 @ B )
= ( plus_p2331992037799027419_set_a @ K @ ( plus_p2331992037799027419_set_a @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_919_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_920_group__cancel_Oadd2,axiom,
! [B2: multiset_set_a,K: multiset_set_a,B: multiset_set_a,A: multiset_set_a] :
( ( B2
= ( plus_p2331992037799027419_set_a @ K @ B ) )
=> ( ( plus_p2331992037799027419_set_a @ A @ B2 )
= ( plus_p2331992037799027419_set_a @ K @ ( plus_p2331992037799027419_set_a @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_921_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_922_add_Oassoc,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ C )
= ( plus_p2331992037799027419_set_a @ A @ ( plus_p2331992037799027419_set_a @ B @ C ) ) ) ).
% add.assoc
thf(fact_923_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_924_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_925_add_Ocommute,axiom,
( plus_p2331992037799027419_set_a
= ( ^ [A3: multiset_set_a,B3: multiset_set_a] : ( plus_p2331992037799027419_set_a @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_926_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_927_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_928_add_Oleft__commute,axiom,
! [B: multiset_set_a,A: multiset_set_a,C: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ B @ ( plus_p2331992037799027419_set_a @ A @ C ) )
= ( plus_p2331992037799027419_set_a @ A @ ( plus_p2331992037799027419_set_a @ B @ C ) ) ) ).
% add.left_commute
thf(fact_929_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_930_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_931_add__left__imp__eq,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ A @ B )
= ( plus_p2331992037799027419_set_a @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_932_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_933_add__right__imp__eq,axiom,
! [B: multiset_set_a,A: multiset_set_a,C: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ B @ A )
= ( plus_p2331992037799027419_set_a @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_934_union__assoc,axiom,
! [M: multiset_set_a,N5: multiset_set_a,K5: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ ( plus_p2331992037799027419_set_a @ M @ N5 ) @ K5 )
= ( plus_p2331992037799027419_set_a @ M @ ( plus_p2331992037799027419_set_a @ N5 @ K5 ) ) ) ).
% union_assoc
thf(fact_935_union__lcomm,axiom,
! [M: multiset_set_a,N5: multiset_set_a,K5: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ M @ ( plus_p2331992037799027419_set_a @ N5 @ K5 ) )
= ( plus_p2331992037799027419_set_a @ N5 @ ( plus_p2331992037799027419_set_a @ M @ K5 ) ) ) ).
% union_lcomm
thf(fact_936_union__commute,axiom,
( plus_p2331992037799027419_set_a
= ( ^ [M5: multiset_set_a,N4: multiset_set_a] : ( plus_p2331992037799027419_set_a @ N4 @ M5 ) ) ) ).
% union_commute
thf(fact_937_union__left__cancel,axiom,
! [K5: multiset_set_a,M: multiset_set_a,N5: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ K5 @ M )
= ( plus_p2331992037799027419_set_a @ K5 @ N5 ) )
= ( M = N5 ) ) ).
% union_left_cancel
thf(fact_938_union__right__cancel,axiom,
! [M: multiset_set_a,K5: multiset_set_a,N5: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ M @ K5 )
= ( plus_p2331992037799027419_set_a @ N5 @ K5 ) )
= ( M = N5 ) ) ).
% union_right_cancel
thf(fact_939_multi__union__self__other__eq,axiom,
! [A2: multiset_set_a,X2: multiset_set_a,Y6: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ A2 @ X2 )
= ( plus_p2331992037799027419_set_a @ A2 @ Y6 ) )
=> ( X2 = Y6 ) ) ).
% multi_union_self_other_eq
thf(fact_940_left__add__mult__distrib__mset,axiom,
! [I2: nat,U: multiset_set_a,J2: nat,K: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ I2 @ U ) @ ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ J2 @ U ) @ K ) )
= ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ ( plus_plus_nat @ I2 @ J2 ) @ U ) @ K ) ) ).
% left_add_mult_distrib_mset
thf(fact_941_repeat__mset__distrib,axiom,
! [M2: nat,N: nat,A2: multiset_set_a] :
( ( repeat_mset_set_a @ ( plus_plus_nat @ M2 @ N ) @ A2 )
= ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ M2 @ A2 ) @ ( repeat_mset_set_a @ N @ A2 ) ) ) ).
% repeat_mset_distrib
thf(fact_942_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% Nat.diff_cancel
thf(fact_943_diff__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_cancel2
thf(fact_944_diff__add__inverse,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
= M2 ) ).
% diff_add_inverse
thf(fact_945_diff__add__inverse2,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
= M2 ) ).
% diff_add_inverse2
thf(fact_946_diff__mult__distrib,axiom,
! [M2: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M2 @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_947_diff__mult__distrib2,axiom,
! [K: nat,M2: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_948_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_949_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_950_Multiset_Odiff__add,axiom,
! [M: multiset_set_a,N5: multiset_set_a,Q: multiset_set_a] :
( ( minus_706656509937749387_set_a @ M @ ( plus_p2331992037799027419_set_a @ N5 @ Q ) )
= ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ M @ N5 ) @ Q ) ) ).
% Multiset.diff_add
thf(fact_951_diff__union__cancelL,axiom,
! [N5: multiset_set_a,M: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ N5 @ M ) @ N5 )
= M ) ).
% diff_union_cancelL
thf(fact_952_diff__union__cancelR,axiom,
! [M: multiset_set_a,N5: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ M @ N5 ) @ N5 )
= M ) ).
% diff_union_cancelR
thf(fact_953_union__le__mono1,axiom,
! [B2: multiset_set_a,D: multiset_set_a,C2: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ B2 @ D )
=> ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ B2 @ C2 ) @ ( plus_p2331992037799027419_set_a @ D @ C2 ) ) ) ).
% union_le_mono1
thf(fact_954_union__le__mono2,axiom,
! [B2: multiset_set_a,D: multiset_set_a,C2: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ B2 @ D )
=> ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ C2 @ B2 ) @ ( plus_p2331992037799027419_set_a @ C2 @ D ) ) ) ).
% union_le_mono2
thf(fact_955_union__less__mono,axiom,
! [A2: multiset_set_a,C2: multiset_set_a,B2: multiset_set_a,D: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ A2 @ C2 )
=> ( ( ord_le5765082015083327056_set_a @ B2 @ D )
=> ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ B2 ) @ ( plus_p2331992037799027419_set_a @ C2 @ D ) ) ) ) ).
% union_less_mono
thf(fact_956_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_957_card__permutations__of__multiset__insert__aux,axiom,
! [A2: multiset_set_a,X: set_a] :
( ( times_times_nat @ ( finite4179508071619380492_set_a @ ( multis5469701301851823918_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) ) ) @ ( plus_plus_nat @ ( count_set_a @ A2 @ X ) @ one_one_nat ) )
= ( times_times_nat @ ( plus_plus_nat @ ( size_s6566526139600085008_set_a @ A2 ) @ one_one_nat ) @ ( finite4179508071619380492_set_a @ ( multis5469701301851823918_set_a @ A2 ) ) ) ) ).
% card_permutations_of_multiset_insert_aux
thf(fact_958_add__decreasing,axiom,
! [A: multiset_set_a,C: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ A @ zero_z5079479921072680283_set_a )
=> ( ( ord_le7905258569527593284_set_a @ C @ B )
=> ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_959_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_960_add__increasing,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ A )
=> ( ( ord_le7905258569527593284_set_a @ B @ C )
=> ( ord_le7905258569527593284_set_a @ B @ ( plus_p2331992037799027419_set_a @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_961_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_962_add__decreasing2,axiom,
! [C: multiset_set_a,A: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ C @ zero_z5079479921072680283_set_a )
=> ( ( ord_le7905258569527593284_set_a @ A @ B )
=> ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_963_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_964_add__increasing2,axiom,
! [C: multiset_set_a,B: multiset_set_a,A: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ C )
=> ( ( ord_le7905258569527593284_set_a @ B @ A )
=> ( ord_le7905258569527593284_set_a @ B @ ( plus_p2331992037799027419_set_a @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_965_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_966_add__nonneg__nonneg,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ A )
=> ( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ B )
=> ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_967_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_968_add__nonpos__nonpos,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ A @ zero_z5079479921072680283_set_a )
=> ( ( ord_le7905258569527593284_set_a @ B @ zero_z5079479921072680283_set_a )
=> ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ zero_z5079479921072680283_set_a ) ) ) ).
% add_nonpos_nonpos
thf(fact_969_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_970_add__nonneg__eq__0__iff,axiom,
! [X: multiset_set_a,Y4: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ X )
=> ( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ Y4 )
=> ( ( ( plus_p2331992037799027419_set_a @ X @ Y4 )
= zero_z5079479921072680283_set_a )
= ( ( X = zero_z5079479921072680283_set_a )
& ( Y4 = zero_z5079479921072680283_set_a ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_971_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y4 )
=> ( ( ( plus_plus_nat @ X @ Y4 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y4 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_972_add__nonpos__eq__0__iff,axiom,
! [X: multiset_set_a,Y4: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ X @ zero_z5079479921072680283_set_a )
=> ( ( ord_le7905258569527593284_set_a @ Y4 @ zero_z5079479921072680283_set_a )
=> ( ( ( plus_p2331992037799027419_set_a @ X @ Y4 )
= zero_z5079479921072680283_set_a )
= ( ( X = zero_z5079479921072680283_set_a )
& ( Y4 = zero_z5079479921072680283_set_a ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_973_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y4 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y4 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y4 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_974_pos__add__strict,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ A )
=> ( ( ord_le5765082015083327056_set_a @ B @ C )
=> ( ord_le5765082015083327056_set_a @ B @ ( plus_p2331992037799027419_set_a @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_975_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_976_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_977_add__pos__pos,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ A )
=> ( ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ B )
=> ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_978_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_979_add__neg__neg,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ A @ zero_z5079479921072680283_set_a )
=> ( ( ord_le5765082015083327056_set_a @ B @ zero_z5079479921072680283_set_a )
=> ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ zero_z5079479921072680283_set_a ) ) ) ).
% add_neg_neg
thf(fact_980_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_981_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: multiset_set_a,J2: multiset_set_a,K: multiset_set_a,L2: multiset_set_a] :
( ( ( ord_le7905258569527593284_set_a @ I2 @ J2 )
& ( ord_le5765082015083327056_set_a @ K @ L2 ) )
=> ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ I2 @ K ) @ ( plus_p2331992037799027419_set_a @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_982_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: nat,J2: nat,K: nat,L2: nat] :
( ( ( ord_less_eq_nat @ I2 @ J2 )
& ( ord_less_nat @ K @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_983_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: multiset_set_a,J2: multiset_set_a,K: multiset_set_a,L2: multiset_set_a] :
( ( ( ord_le5765082015083327056_set_a @ I2 @ J2 )
& ( ord_le7905258569527593284_set_a @ K @ L2 ) )
=> ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ I2 @ K ) @ ( plus_p2331992037799027419_set_a @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_984_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: nat,J2: nat,K: nat,L2: nat] :
( ( ( ord_less_nat @ I2 @ J2 )
& ( ord_less_eq_nat @ K @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_985_add__le__less__mono,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a,D2: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ A @ B )
=> ( ( ord_le5765082015083327056_set_a @ C @ D2 )
=> ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ ( plus_p2331992037799027419_set_a @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_986_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_987_add__less__le__mono,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a,D2: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ A @ B )
=> ( ( ord_le7905258569527593284_set_a @ C @ D2 )
=> ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ ( plus_p2331992037799027419_set_a @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_988_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_989_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_990_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_991_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_992_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_993_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_994_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_995_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_996_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_997_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_998_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_999_less__imp__add__positive,axiom,
! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I2 @ K2 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_1000_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M8: nat,N3: nat] :
( ( ord_less_nat @ M8 @ N3 )
=> ( ord_less_nat @ ( F @ M8 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1001_mult__less__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1002_mult__less__mono2,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J2 ) ) ) ) ).
% mult_less_mono2
thf(fact_1003_add__diff__inverse__nat,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less_nat @ M2 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1004_less__diff__conv,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 ) ) ).
% less_diff_conv
thf(fact_1005_diff__add__0,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1006_subset__mset_Oadd__decreasing,axiom,
! [A: multiset_set_a,C: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ zero_z5079479921072680283_set_a )
=> ( ( subseteq_mset_set_a @ C @ B )
=> ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ B ) ) ) ).
% subset_mset.add_decreasing
thf(fact_1007_subset__mset_Oadd__increasing,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ A )
=> ( ( subseteq_mset_set_a @ B @ C )
=> ( subseteq_mset_set_a @ B @ ( plus_p2331992037799027419_set_a @ A @ C ) ) ) ) ).
% subset_mset.add_increasing
thf(fact_1008_subset__mset_Oadd__decreasing2,axiom,
! [C: multiset_set_a,A: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ C @ zero_z5079479921072680283_set_a )
=> ( ( subseteq_mset_set_a @ A @ B )
=> ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ B ) ) ) ).
% subset_mset.add_decreasing2
thf(fact_1009_subset__mset_Oadd__increasing2,axiom,
! [C: multiset_set_a,B: multiset_set_a,A: multiset_set_a] :
( ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ C )
=> ( ( subseteq_mset_set_a @ B @ A )
=> ( subseteq_mset_set_a @ B @ ( plus_p2331992037799027419_set_a @ A @ C ) ) ) ) ).
% subset_mset.add_increasing2
thf(fact_1010_subset__mset_Oadd__nonneg__nonneg,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ A )
=> ( ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ B )
=> ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) ) ) ) ).
% subset_mset.add_nonneg_nonneg
thf(fact_1011_subset__mset_Oadd__nonpos__nonpos,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ zero_z5079479921072680283_set_a )
=> ( ( subseteq_mset_set_a @ B @ zero_z5079479921072680283_set_a )
=> ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ zero_z5079479921072680283_set_a ) ) ) ).
% subset_mset.add_nonpos_nonpos
thf(fact_1012_subset__mset_Oadd__nonneg__eq__0__iff,axiom,
! [X: multiset_set_a,Y4: multiset_set_a] :
( ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ X )
=> ( ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ Y4 )
=> ( ( ( plus_p2331992037799027419_set_a @ X @ Y4 )
= zero_z5079479921072680283_set_a )
= ( ( X = zero_z5079479921072680283_set_a )
& ( Y4 = zero_z5079479921072680283_set_a ) ) ) ) ) ).
% subset_mset.add_nonneg_eq_0_iff
thf(fact_1013_subset__mset_Oadd__nonpos__eq__0__iff,axiom,
! [X: multiset_set_a,Y4: multiset_set_a] :
( ( subseteq_mset_set_a @ X @ zero_z5079479921072680283_set_a )
=> ( ( subseteq_mset_set_a @ Y4 @ zero_z5079479921072680283_set_a )
=> ( ( ( plus_p2331992037799027419_set_a @ X @ Y4 )
= zero_z5079479921072680283_set_a )
= ( ( X = zero_z5079479921072680283_set_a )
& ( Y4 = zero_z5079479921072680283_set_a ) ) ) ) ) ).
% subset_mset.add_nonpos_eq_0_iff
thf(fact_1014_le__diff__conv,axiom,
! [J2: nat,K: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% le_diff_conv
thf(fact_1015_Nat_Ole__diff__conv2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1016_Nat_Odiff__add__assoc,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1017_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I2 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1018_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I2 )
= K )
= ( J2
= ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1019_mult__eq__self__implies__10,axiom,
! [M2: nat,N: nat] :
( ( M2
= ( times_times_nat @ M2 @ N ) )
=> ( ( N = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1020_add__mset__add__single,axiom,
( add_mset_set_a
= ( ^ [A3: set_a,A5: multiset_set_a] : ( plus_p2331992037799027419_set_a @ A5 @ ( add_mset_set_a @ A3 @ zero_z5079479921072680283_set_a ) ) ) ) ).
% add_mset_add_single
thf(fact_1021_union__is__single,axiom,
! [M: multiset_set_a,N5: multiset_set_a,A: set_a] :
( ( ( plus_p2331992037799027419_set_a @ M @ N5 )
= ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) )
= ( ( ( M
= ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) )
& ( N5 = zero_z5079479921072680283_set_a ) )
| ( ( M = zero_z5079479921072680283_set_a )
& ( N5
= ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) ) ) ) ) ).
% union_is_single
thf(fact_1022_single__is__union,axiom,
! [A: set_a,M: multiset_set_a,N5: multiset_set_a] :
( ( ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a )
= ( plus_p2331992037799027419_set_a @ M @ N5 ) )
= ( ( ( ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a )
= M )
& ( N5 = zero_z5079479921072680283_set_a ) )
| ( ( M = zero_z5079479921072680283_set_a )
& ( ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a )
= N5 ) ) ) ) ).
% single_is_union
thf(fact_1023_multiset__diff__union__assoc,axiom,
! [C2: multiset_set_a,B2: multiset_set_a,A2: multiset_set_a] :
( ( subseteq_mset_set_a @ C2 @ B2 )
=> ( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ B2 ) @ C2 )
= ( plus_p2331992037799027419_set_a @ A2 @ ( minus_706656509937749387_set_a @ B2 @ C2 ) ) ) ) ).
% multiset_diff_union_assoc
thf(fact_1024_subset__eq__diff__conv,axiom,
! [A2: multiset_set_a,C2: multiset_set_a,B2: multiset_set_a] :
( ( subseteq_mset_set_a @ ( minus_706656509937749387_set_a @ A2 @ C2 ) @ B2 )
= ( subseteq_mset_set_a @ A2 @ ( plus_p2331992037799027419_set_a @ B2 @ C2 ) ) ) ).
% subset_eq_diff_conv
thf(fact_1025_subset__mset_Ole__imp__diff__is__add,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B )
=> ( ( subseteq_mset_set_a @ A @ B )
=> ( ( ( minus_706656509937749387_set_a @ B @ A )
= C )
= ( B
= ( plus_p2331992037799027419_set_a @ C @ A ) ) ) ) ) ).
% subset_mset.le_imp_diff_is_add
thf(fact_1026_subset__mset_Oadd__diff__inverse,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B )
=> ( ( plus_p2331992037799027419_set_a @ A @ ( minus_706656509937749387_set_a @ B @ A ) )
= B ) ) ).
% subset_mset.add_diff_inverse
thf(fact_1027_subset__mset_Odiff__diff__right,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B )
=> ( ( minus_706656509937749387_set_a @ C @ ( minus_706656509937749387_set_a @ B @ A ) )
= ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ C @ A ) @ B ) ) ) ).
% subset_mset.diff_diff_right
thf(fact_1028_subset__mset_Odiff__add__assoc2,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B )
=> ( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ B @ C ) @ A )
= ( plus_p2331992037799027419_set_a @ ( minus_706656509937749387_set_a @ B @ A ) @ C ) ) ) ).
% subset_mset.diff_add_assoc2
thf(fact_1029_subset__mset_Odiff__add__assoc,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B )
=> ( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ C @ B ) @ A )
= ( plus_p2331992037799027419_set_a @ C @ ( minus_706656509937749387_set_a @ B @ A ) ) ) ) ).
% subset_mset.diff_add_assoc
thf(fact_1030_subset__mset_Ole__diff__conv2,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B )
=> ( ( subseteq_mset_set_a @ C @ ( minus_706656509937749387_set_a @ B @ A ) )
= ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ C @ A ) @ B ) ) ) ).
% subset_mset.le_diff_conv2
thf(fact_1031_subset__mset_Ole__add__diff,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B )
=> ( subseteq_mset_set_a @ C @ ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ B @ C ) @ A ) ) ) ).
% subset_mset.le_add_diff
thf(fact_1032_subset__mset_Odiff__add,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B )
=> ( ( plus_p2331992037799027419_set_a @ ( minus_706656509937749387_set_a @ B @ A ) @ A )
= B ) ) ).
% subset_mset.diff_add
thf(fact_1033_point__index__diff,axiom,
! [B22: multiset_set_a] :
( design254580327166089565ndex_a
= ( ^ [B1: multiset_set_a,Ps2: set_a] : ( minus_minus_nat @ ( design254580327166089565ndex_a @ ( plus_p2331992037799027419_set_a @ B1 @ B22 ) @ Ps2 ) @ ( design254580327166089565ndex_a @ B22 @ Ps2 ) ) ) ) ).
% point_index_diff
thf(fact_1034_repeat__mset_Orep__eq,axiom,
! [X: nat,Xa2: multiset_set_a] :
( ( count_set_a @ ( repeat_mset_set_a @ X @ Xa2 ) )
= ( ^ [A3: set_a] : ( times_times_nat @ X @ ( count_set_a @ Xa2 @ A3 ) ) ) ) ).
% repeat_mset.rep_eq
thf(fact_1035_le__multiset__plus__left__nonempty,axiom,
! [M: multiset_set_a,N5: multiset_set_a] :
( ( M != zero_z5079479921072680283_set_a )
=> ( ord_le5765082015083327056_set_a @ N5 @ ( plus_p2331992037799027419_set_a @ M @ N5 ) ) ) ).
% le_multiset_plus_left_nonempty
thf(fact_1036_le__multiset__plus__right__nonempty,axiom,
! [N5: multiset_set_a,M: multiset_set_a] :
( ( N5 != zero_z5079479921072680283_set_a )
=> ( ord_le5765082015083327056_set_a @ M @ ( plus_p2331992037799027419_set_a @ M @ N5 ) ) ) ).
% le_multiset_plus_right_nonempty
thf(fact_1037_add__strict__increasing2,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ A )
=> ( ( ord_le5765082015083327056_set_a @ B @ C )
=> ( ord_le5765082015083327056_set_a @ B @ ( plus_p2331992037799027419_set_a @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1038_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_1039_add__strict__increasing,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ A )
=> ( ( ord_le7905258569527593284_set_a @ B @ C )
=> ( ord_le5765082015083327056_set_a @ B @ ( plus_p2331992037799027419_set_a @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1040_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_1041_add__pos__nonneg,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ A )
=> ( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ B )
=> ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1042_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_1043_add__nonpos__neg,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ A @ zero_z5079479921072680283_set_a )
=> ( ( ord_le5765082015083327056_set_a @ B @ zero_z5079479921072680283_set_a )
=> ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ zero_z5079479921072680283_set_a ) ) ) ).
% add_nonpos_neg
thf(fact_1044_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_1045_add__nonneg__pos,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ A )
=> ( ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ B )
=> ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1046_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_1047_add__neg__nonpos,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ A @ zero_z5079479921072680283_set_a )
=> ( ( ord_le7905258569527593284_set_a @ B @ zero_z5079479921072680283_set_a )
=> ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ zero_z5079479921072680283_set_a ) ) ) ).
% add_neg_nonpos
thf(fact_1048_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_1049_nat__diff__split,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P2 @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P2 @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1050_nat__diff__split__asm,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P2 @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P2 @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1051_less__diff__conv2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 )
= ( ord_less_nat @ J2 @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1052_multi__member__this,axiom,
! [X: a,XS: multiset_a] : ( member_a @ X @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ X @ zero_zero_multiset_a ) @ XS ) ) ) ).
% multi_member_this
thf(fact_1053_multi__member__this,axiom,
! [X: list_set_a,XS: multiset_list_set_a] : ( member_list_set_a @ X @ ( set_mset_list_set_a @ ( plus_p4509188130224566113_set_a @ ( add_mset_list_set_a @ X @ zero_z8272816460787710433_set_a ) @ XS ) ) ) ).
% multi_member_this
thf(fact_1054_multi__member__this,axiom,
! [X: nat,XS: multiset_nat] : ( member_nat @ X @ ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) @ XS ) ) ) ).
% multi_member_this
thf(fact_1055_multi__member__this,axiom,
! [X: set_a,XS: multiset_set_a] : ( member_set_a @ X @ ( set_mset_set_a @ ( plus_p2331992037799027419_set_a @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) @ XS ) ) ) ).
% multi_member_this
thf(fact_1056_multi__member__skip,axiom,
! [X: a,XS: multiset_a,Y4: a] :
( ( member_a @ X @ ( set_mset_a @ XS ) )
=> ( member_a @ X @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ Y4 @ zero_zero_multiset_a ) @ XS ) ) ) ) ).
% multi_member_skip
thf(fact_1057_multi__member__skip,axiom,
! [X: list_set_a,XS: multiset_list_set_a,Y4: list_set_a] :
( ( member_list_set_a @ X @ ( set_mset_list_set_a @ XS ) )
=> ( member_list_set_a @ X @ ( set_mset_list_set_a @ ( plus_p4509188130224566113_set_a @ ( add_mset_list_set_a @ Y4 @ zero_z8272816460787710433_set_a ) @ XS ) ) ) ) ).
% multi_member_skip
thf(fact_1058_multi__member__skip,axiom,
! [X: nat,XS: multiset_nat,Y4: nat] :
( ( member_nat @ X @ ( set_mset_nat @ XS ) )
=> ( member_nat @ X @ ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ ( add_mset_nat @ Y4 @ zero_z7348594199698428585et_nat ) @ XS ) ) ) ) ).
% multi_member_skip
thf(fact_1059_multi__member__skip,axiom,
! [X: set_a,XS: multiset_set_a,Y4: set_a] :
( ( member_set_a @ X @ ( set_mset_set_a @ XS ) )
=> ( member_set_a @ X @ ( set_mset_set_a @ ( plus_p2331992037799027419_set_a @ ( add_mset_set_a @ Y4 @ zero_z5079479921072680283_set_a ) @ XS ) ) ) ) ).
% multi_member_skip
thf(fact_1060_remove1__mset__add__mset__If,axiom,
! [L4: set_a,L3: set_a,C2: multiset_set_a] :
( ( ( L4 = L3 )
=> ( ( minus_706656509937749387_set_a @ ( add_mset_set_a @ L3 @ C2 ) @ ( add_mset_set_a @ L4 @ zero_z5079479921072680283_set_a ) )
= C2 ) )
& ( ( L4 != L3 )
=> ( ( minus_706656509937749387_set_a @ ( add_mset_set_a @ L3 @ C2 ) @ ( add_mset_set_a @ L4 @ zero_z5079479921072680283_set_a ) )
= ( plus_p2331992037799027419_set_a @ ( minus_706656509937749387_set_a @ C2 @ ( add_mset_set_a @ L4 @ zero_z5079479921072680283_set_a ) ) @ ( add_mset_set_a @ L3 @ zero_z5079479921072680283_set_a ) ) ) ) ) ).
% remove1_mset_add_mset_If
thf(fact_1061_count__in__diffI,axiom,
! [N5: multiset_a,X: a,M: multiset_a] :
( ! [N3: nat] :
( ( count_a @ N5 @ X )
!= ( plus_plus_nat @ N3 @ ( count_a @ M @ X ) ) )
=> ( member_a @ X @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M @ N5 ) ) ) ) ).
% count_in_diffI
thf(fact_1062_count__in__diffI,axiom,
! [N5: multiset_list_set_a,X: list_set_a,M: multiset_list_set_a] :
( ! [N3: nat] :
( ( count_list_set_a @ N5 @ X )
!= ( plus_plus_nat @ N3 @ ( count_list_set_a @ M @ X ) ) )
=> ( member_list_set_a @ X @ ( set_mset_list_set_a @ ( minus_2572999347059163665_set_a @ M @ N5 ) ) ) ) ).
% count_in_diffI
thf(fact_1063_count__in__diffI,axiom,
! [N5: multiset_nat,X: nat,M: multiset_nat] :
( ! [N3: nat] :
( ( count_nat @ N5 @ X )
!= ( plus_plus_nat @ N3 @ ( count_nat @ M @ X ) ) )
=> ( member_nat @ X @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M @ N5 ) ) ) ) ).
% count_in_diffI
thf(fact_1064_count__in__diffI,axiom,
! [N5: multiset_set_a,X: set_a,M: multiset_set_a] :
( ! [N3: nat] :
( ( count_set_a @ N5 @ X )
!= ( plus_plus_nat @ N3 @ ( count_set_a @ M @ X ) ) )
=> ( member_set_a @ X @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ M @ N5 ) ) ) ) ).
% count_in_diffI
thf(fact_1065_union__le__diff__plus,axiom,
! [P2: multiset_set_a,M: multiset_set_a,N5: multiset_set_a] :
( ( subseteq_mset_set_a @ P2 @ M )
=> ( ( ord_le5765082015083327056_set_a @ N5 @ P2 )
=> ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ ( minus_706656509937749387_set_a @ M @ P2 ) @ N5 ) @ M ) ) ) ).
% union_le_diff_plus
thf(fact_1066_insert__DiffM2,axiom,
! [X: a,M: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M ) )
=> ( ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
= M ) ) ).
% insert_DiffM2
thf(fact_1067_insert__DiffM2,axiom,
! [X: list_set_a,M: multiset_list_set_a] :
( ( member_list_set_a @ X @ ( set_mset_list_set_a @ M ) )
=> ( ( plus_p4509188130224566113_set_a @ ( minus_2572999347059163665_set_a @ M @ ( add_mset_list_set_a @ X @ zero_z8272816460787710433_set_a ) ) @ ( add_mset_list_set_a @ X @ zero_z8272816460787710433_set_a ) )
= M ) ) ).
% insert_DiffM2
thf(fact_1068_insert__DiffM2,axiom,
! [X: nat,M: multiset_nat] :
( ( member_nat @ X @ ( set_mset_nat @ M ) )
=> ( ( plus_p6334493942879108393et_nat @ ( minus_8522176038001411705et_nat @ M @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) )
= M ) ) ).
% insert_DiffM2
thf(fact_1069_insert__DiffM2,axiom,
! [X: set_a,M: multiset_set_a] :
( ( member_set_a @ X @ ( set_mset_set_a @ M ) )
=> ( ( plus_p2331992037799027419_set_a @ ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) )
= M ) ) ).
% insert_DiffM2
thf(fact_1070_remove1__mset__eqE,axiom,
! [X1: multiset_a,L4: a,M: multiset_a] :
( ( ( minus_3765977307040488491iset_a @ X1 @ ( add_mset_a @ L4 @ zero_zero_multiset_a ) )
= M )
=> ( ( ( member_a @ L4 @ ( set_mset_a @ X1 ) )
=> ( X1
!= ( plus_plus_multiset_a @ M @ ( add_mset_a @ L4 @ zero_zero_multiset_a ) ) ) )
=> ~ ( ~ ( member_a @ L4 @ ( set_mset_a @ X1 ) )
=> ( X1 != M ) ) ) ) ).
% remove1_mset_eqE
thf(fact_1071_remove1__mset__eqE,axiom,
! [X1: multiset_list_set_a,L4: list_set_a,M: multiset_list_set_a] :
( ( ( minus_2572999347059163665_set_a @ X1 @ ( add_mset_list_set_a @ L4 @ zero_z8272816460787710433_set_a ) )
= M )
=> ( ( ( member_list_set_a @ L4 @ ( set_mset_list_set_a @ X1 ) )
=> ( X1
!= ( plus_p4509188130224566113_set_a @ M @ ( add_mset_list_set_a @ L4 @ zero_z8272816460787710433_set_a ) ) ) )
=> ~ ( ~ ( member_list_set_a @ L4 @ ( set_mset_list_set_a @ X1 ) )
=> ( X1 != M ) ) ) ) ).
% remove1_mset_eqE
thf(fact_1072_remove1__mset__eqE,axiom,
! [X1: multiset_nat,L4: nat,M: multiset_nat] :
( ( ( minus_8522176038001411705et_nat @ X1 @ ( add_mset_nat @ L4 @ zero_z7348594199698428585et_nat ) )
= M )
=> ( ( ( member_nat @ L4 @ ( set_mset_nat @ X1 ) )
=> ( X1
!= ( plus_p6334493942879108393et_nat @ M @ ( add_mset_nat @ L4 @ zero_z7348594199698428585et_nat ) ) ) )
=> ~ ( ~ ( member_nat @ L4 @ ( set_mset_nat @ X1 ) )
=> ( X1 != M ) ) ) ) ).
% remove1_mset_eqE
thf(fact_1073_remove1__mset__eqE,axiom,
! [X1: multiset_set_a,L4: set_a,M: multiset_set_a] :
( ( ( minus_706656509937749387_set_a @ X1 @ ( add_mset_set_a @ L4 @ zero_z5079479921072680283_set_a ) )
= M )
=> ( ( ( member_set_a @ L4 @ ( set_mset_set_a @ X1 ) )
=> ( X1
!= ( plus_p2331992037799027419_set_a @ M @ ( add_mset_set_a @ L4 @ zero_z5079479921072680283_set_a ) ) ) )
=> ~ ( ~ ( member_set_a @ L4 @ ( set_mset_set_a @ X1 ) )
=> ( X1 != M ) ) ) ) ).
% remove1_mset_eqE
thf(fact_1074_diff__union__single__conv,axiom,
! [A: a,J3: multiset_a,I5: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ J3 ) )
=> ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ I5 @ J3 ) @ ( add_mset_a @ A @ zero_zero_multiset_a ) )
= ( plus_plus_multiset_a @ I5 @ ( minus_3765977307040488491iset_a @ J3 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_1075_diff__union__single__conv,axiom,
! [A: list_set_a,J3: multiset_list_set_a,I5: multiset_list_set_a] :
( ( member_list_set_a @ A @ ( set_mset_list_set_a @ J3 ) )
=> ( ( minus_2572999347059163665_set_a @ ( plus_p4509188130224566113_set_a @ I5 @ J3 ) @ ( add_mset_list_set_a @ A @ zero_z8272816460787710433_set_a ) )
= ( plus_p4509188130224566113_set_a @ I5 @ ( minus_2572999347059163665_set_a @ J3 @ ( add_mset_list_set_a @ A @ zero_z8272816460787710433_set_a ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_1076_diff__union__single__conv,axiom,
! [A: nat,J3: multiset_nat,I5: multiset_nat] :
( ( member_nat @ A @ ( set_mset_nat @ J3 ) )
=> ( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ I5 @ J3 ) @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) )
= ( plus_p6334493942879108393et_nat @ I5 @ ( minus_8522176038001411705et_nat @ J3 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_1077_diff__union__single__conv,axiom,
! [A: set_a,J3: multiset_set_a,I5: multiset_set_a] :
( ( member_set_a @ A @ ( set_mset_set_a @ J3 ) )
=> ( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ I5 @ J3 ) @ ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) )
= ( plus_p2331992037799027419_set_a @ I5 @ ( minus_706656509937749387_set_a @ J3 @ ( add_mset_set_a @ A @ zero_z5079479921072680283_set_a ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_1078_mset__subseteq__add__iff2,axiom,
! [I2: nat,J2: nat,U: multiset_set_a,M2: multiset_set_a,N: multiset_set_a] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ I2 @ U ) @ M2 ) @ ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ J2 @ U ) @ N ) )
= ( subseteq_mset_set_a @ M2 @ ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ ( minus_minus_nat @ J2 @ I2 ) @ U ) @ N ) ) ) ) ).
% mset_subseteq_add_iff2
thf(fact_1079_mset__subseteq__add__iff1,axiom,
! [J2: nat,I2: nat,U: multiset_set_a,M2: multiset_set_a,N: multiset_set_a] :
( ( ord_less_eq_nat @ J2 @ I2 )
=> ( ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ I2 @ U ) @ M2 ) @ ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ J2 @ U ) @ N ) )
= ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ ( minus_minus_nat @ I2 @ J2 ) @ U ) @ M2 ) @ N ) ) ) ).
% mset_subseteq_add_iff1
thf(fact_1080_size__multiset__set__mset__const__count,axiom,
! [A2: multiset_list_set_a,Ca: nat,Ca2: nat] :
( ( ( finite4179508071619380492_set_a @ ( set_mset_list_set_a @ A2 ) )
= Ca )
=> ( ! [P3: list_set_a] :
( ( member_list_set_a @ P3 @ ( set_mset_list_set_a @ A2 ) )
=> ( ( count_list_set_a @ A2 @ P3 )
= Ca2 ) )
=> ( ( size_s2437250918661492246_set_a @ A2 )
= ( times_times_nat @ Ca @ Ca2 ) ) ) ) ).
% size_multiset_set_mset_const_count
thf(fact_1081_size__multiset__set__mset__const__count,axiom,
! [A2: multiset_nat,Ca: nat,Ca2: nat] :
( ( ( finite_card_nat @ ( set_mset_nat @ A2 ) )
= Ca )
=> ( ! [P3: nat] :
( ( member_nat @ P3 @ ( set_mset_nat @ A2 ) )
=> ( ( count_nat @ A2 @ P3 )
= Ca2 ) )
=> ( ( size_s5917832649809541300et_nat @ A2 )
= ( times_times_nat @ Ca @ Ca2 ) ) ) ) ).
% size_multiset_set_mset_const_count
thf(fact_1082_size__multiset__set__mset__const__count,axiom,
! [A2: multiset_a,Ca: nat,Ca2: nat] :
( ( ( finite_card_a @ ( set_mset_a @ A2 ) )
= Ca )
=> ( ! [P3: a] :
( ( member_a @ P3 @ ( set_mset_a @ A2 ) )
=> ( ( count_a @ A2 @ P3 )
= Ca2 ) )
=> ( ( size_size_multiset_a @ A2 )
= ( times_times_nat @ Ca @ Ca2 ) ) ) ) ).
% size_multiset_set_mset_const_count
thf(fact_1083_size__multiset__set__mset__const__count,axiom,
! [A2: multiset_set_a,Ca: nat,Ca2: nat] :
( ( ( finite_card_set_a @ ( set_mset_set_a @ A2 ) )
= Ca )
=> ( ! [P3: set_a] :
( ( member_set_a @ P3 @ ( set_mset_set_a @ A2 ) )
=> ( ( count_set_a @ A2 @ P3 )
= Ca2 ) )
=> ( ( size_s6566526139600085008_set_a @ A2 )
= ( times_times_nat @ Ca @ Ca2 ) ) ) ) ).
% size_multiset_set_mset_const_count
thf(fact_1084_mset__le__add__iff2,axiom,
! [I2: nat,J2: nat,U: multiset_set_a,M2: multiset_set_a,N: multiset_set_a] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ I2 @ U ) @ M2 ) @ ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ J2 @ U ) @ N ) )
= ( ord_le7905258569527593284_set_a @ M2 @ ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ ( minus_minus_nat @ J2 @ I2 ) @ U ) @ N ) ) ) ) ).
% mset_le_add_iff2
thf(fact_1085_mset__le__add__iff1,axiom,
! [J2: nat,I2: nat,U: multiset_set_a,M2: multiset_set_a,N: multiset_set_a] :
( ( ord_less_eq_nat @ J2 @ I2 )
=> ( ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ I2 @ U ) @ M2 ) @ ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ J2 @ U ) @ N ) )
= ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ ( minus_minus_nat @ I2 @ J2 ) @ U ) @ M2 ) @ N ) ) ) ).
% mset_le_add_iff1
thf(fact_1086_ex__gt__count__imp__le__multiset,axiom,
! [M: multiset_set_a,N5: multiset_set_a,X: set_a] :
( ! [Y: set_a] :
( ( member_set_a @ Y @ ( set_mset_set_a @ ( plus_p2331992037799027419_set_a @ M @ N5 ) ) )
=> ( ord_less_eq_set_a @ Y @ X ) )
=> ( ( ord_less_nat @ ( count_set_a @ M @ X ) @ ( count_set_a @ N5 @ X ) )
=> ( ord_le5765082015083327056_set_a @ M @ N5 ) ) ) ).
% ex_gt_count_imp_le_multiset
thf(fact_1087_ex__gt__count__imp__le__multiset,axiom,
! [M: multiset_nat,N5: multiset_nat,X: nat] :
( ! [Y: nat] :
( ( member_nat @ Y @ ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ M @ N5 ) ) )
=> ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ord_less_nat @ ( count_nat @ M @ X ) @ ( count_nat @ N5 @ X ) )
=> ( ord_le5777773500796000884et_nat @ M @ N5 ) ) ) ).
% ex_gt_count_imp_le_multiset
thf(fact_1088_minus__remove1__mset__if,axiom,
! [B: a,B2: multiset_a,A2: multiset_a] :
( ( ( ( member_a @ B @ ( set_mset_a @ B2 ) )
& ( member_a @ B @ ( set_mset_a @ A2 ) )
& ( ord_less_eq_nat @ ( count_a @ B2 @ B ) @ ( count_a @ A2 @ B ) ) )
=> ( ( minus_3765977307040488491iset_a @ A2 @ ( minus_3765977307040488491iset_a @ B2 @ ( add_mset_a @ B @ zero_zero_multiset_a ) ) )
= ( plus_plus_multiset_a @ ( add_mset_a @ B @ zero_zero_multiset_a ) @ ( minus_3765977307040488491iset_a @ A2 @ B2 ) ) ) )
& ( ~ ( ( member_a @ B @ ( set_mset_a @ B2 ) )
& ( member_a @ B @ ( set_mset_a @ A2 ) )
& ( ord_less_eq_nat @ ( count_a @ B2 @ B ) @ ( count_a @ A2 @ B ) ) )
=> ( ( minus_3765977307040488491iset_a @ A2 @ ( minus_3765977307040488491iset_a @ B2 @ ( add_mset_a @ B @ zero_zero_multiset_a ) ) )
= ( minus_3765977307040488491iset_a @ A2 @ B2 ) ) ) ) ).
% minus_remove1_mset_if
thf(fact_1089_minus__remove1__mset__if,axiom,
! [B: list_set_a,B2: multiset_list_set_a,A2: multiset_list_set_a] :
( ( ( ( member_list_set_a @ B @ ( set_mset_list_set_a @ B2 ) )
& ( member_list_set_a @ B @ ( set_mset_list_set_a @ A2 ) )
& ( ord_less_eq_nat @ ( count_list_set_a @ B2 @ B ) @ ( count_list_set_a @ A2 @ B ) ) )
=> ( ( minus_2572999347059163665_set_a @ A2 @ ( minus_2572999347059163665_set_a @ B2 @ ( add_mset_list_set_a @ B @ zero_z8272816460787710433_set_a ) ) )
= ( plus_p4509188130224566113_set_a @ ( add_mset_list_set_a @ B @ zero_z8272816460787710433_set_a ) @ ( minus_2572999347059163665_set_a @ A2 @ B2 ) ) ) )
& ( ~ ( ( member_list_set_a @ B @ ( set_mset_list_set_a @ B2 ) )
& ( member_list_set_a @ B @ ( set_mset_list_set_a @ A2 ) )
& ( ord_less_eq_nat @ ( count_list_set_a @ B2 @ B ) @ ( count_list_set_a @ A2 @ B ) ) )
=> ( ( minus_2572999347059163665_set_a @ A2 @ ( minus_2572999347059163665_set_a @ B2 @ ( add_mset_list_set_a @ B @ zero_z8272816460787710433_set_a ) ) )
= ( minus_2572999347059163665_set_a @ A2 @ B2 ) ) ) ) ).
% minus_remove1_mset_if
thf(fact_1090_minus__remove1__mset__if,axiom,
! [B: nat,B2: multiset_nat,A2: multiset_nat] :
( ( ( ( member_nat @ B @ ( set_mset_nat @ B2 ) )
& ( member_nat @ B @ ( set_mset_nat @ A2 ) )
& ( ord_less_eq_nat @ ( count_nat @ B2 @ B ) @ ( count_nat @ A2 @ B ) ) )
=> ( ( minus_8522176038001411705et_nat @ A2 @ ( minus_8522176038001411705et_nat @ B2 @ ( add_mset_nat @ B @ zero_z7348594199698428585et_nat ) ) )
= ( plus_p6334493942879108393et_nat @ ( add_mset_nat @ B @ zero_z7348594199698428585et_nat ) @ ( minus_8522176038001411705et_nat @ A2 @ B2 ) ) ) )
& ( ~ ( ( member_nat @ B @ ( set_mset_nat @ B2 ) )
& ( member_nat @ B @ ( set_mset_nat @ A2 ) )
& ( ord_less_eq_nat @ ( count_nat @ B2 @ B ) @ ( count_nat @ A2 @ B ) ) )
=> ( ( minus_8522176038001411705et_nat @ A2 @ ( minus_8522176038001411705et_nat @ B2 @ ( add_mset_nat @ B @ zero_z7348594199698428585et_nat ) ) )
= ( minus_8522176038001411705et_nat @ A2 @ B2 ) ) ) ) ).
% minus_remove1_mset_if
thf(fact_1091_minus__remove1__mset__if,axiom,
! [B: set_a,B2: multiset_set_a,A2: multiset_set_a] :
( ( ( ( member_set_a @ B @ ( set_mset_set_a @ B2 ) )
& ( member_set_a @ B @ ( set_mset_set_a @ A2 ) )
& ( ord_less_eq_nat @ ( count_set_a @ B2 @ B ) @ ( count_set_a @ A2 @ B ) ) )
=> ( ( minus_706656509937749387_set_a @ A2 @ ( minus_706656509937749387_set_a @ B2 @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) ) )
= ( plus_p2331992037799027419_set_a @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) @ ( minus_706656509937749387_set_a @ A2 @ B2 ) ) ) )
& ( ~ ( ( member_set_a @ B @ ( set_mset_set_a @ B2 ) )
& ( member_set_a @ B @ ( set_mset_set_a @ A2 ) )
& ( ord_less_eq_nat @ ( count_set_a @ B2 @ B ) @ ( count_set_a @ A2 @ B ) ) )
=> ( ( minus_706656509937749387_set_a @ A2 @ ( minus_706656509937749387_set_a @ B2 @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) ) )
= ( minus_706656509937749387_set_a @ A2 @ B2 ) ) ) ) ).
% minus_remove1_mset_if
thf(fact_1092_less__multiset_092_060_094sub_062D_092_060_094sub_062M,axiom,
( ord_le5765082015083327056_set_a
= ( ^ [M5: multiset_set_a,N4: multiset_set_a] :
? [X6: multiset_set_a,Y8: multiset_set_a] :
( ( X6 != zero_z5079479921072680283_set_a )
& ( subseteq_mset_set_a @ X6 @ N4 )
& ( M5
= ( plus_p2331992037799027419_set_a @ ( minus_706656509937749387_set_a @ N4 @ X6 ) @ Y8 ) )
& ! [K4: set_a] :
( ( member_set_a @ K4 @ ( set_mset_set_a @ Y8 ) )
=> ? [A3: set_a] :
( ( member_set_a @ A3 @ ( set_mset_set_a @ X6 ) )
& ( ord_less_set_a @ K4 @ A3 ) ) ) ) ) ) ).
% less_multiset\<^sub>D\<^sub>M
thf(fact_1093_less__multiset_092_060_094sub_062D_092_060_094sub_062M,axiom,
( ord_le5777773500796000884et_nat
= ( ^ [M5: multiset_nat,N4: multiset_nat] :
? [X6: multiset_nat,Y8: multiset_nat] :
( ( X6 != zero_z7348594199698428585et_nat )
& ( subseteq_mset_nat @ X6 @ N4 )
& ( M5
= ( plus_p6334493942879108393et_nat @ ( minus_8522176038001411705et_nat @ N4 @ X6 ) @ Y8 ) )
& ! [K4: nat] :
( ( member_nat @ K4 @ ( set_mset_nat @ Y8 ) )
=> ? [A3: nat] :
( ( member_nat @ A3 @ ( set_mset_nat @ X6 ) )
& ( ord_less_nat @ K4 @ A3 ) ) ) ) ) ) ).
% less_multiset\<^sub>D\<^sub>M
thf(fact_1094_mbs_Oless__not__eq,axiom,
! [X: nat,A2: set_nat,Y4: nat] :
( ( member_nat @ X @ A2 )
=> ( ( ord_less_nat @ ( size_size_nat @ X ) @ ( size_size_nat @ Y4 ) )
=> ( X != Y4 ) ) ) ).
% mbs.less_not_eq
thf(fact_1095_mbs_Oless__not__eq,axiom,
! [X: list_set_a,A2: set_list_set_a,Y4: list_set_a] :
( ( member_list_set_a @ X @ A2 )
=> ( ( ord_less_nat @ ( size_size_list_set_a @ X ) @ ( size_size_list_set_a @ Y4 ) )
=> ( X != Y4 ) ) ) ).
% mbs.less_not_eq
thf(fact_1096_mbs_Oless__not__eq,axiom,
! [X: multiset_set_a,A2: set_multiset_set_a,Y4: multiset_set_a] :
( ( member2747690772047059533_set_a @ X @ A2 )
=> ( ( ord_less_nat @ ( size_s6566526139600085008_set_a @ X ) @ ( size_s6566526139600085008_set_a @ Y4 ) )
=> ( X != Y4 ) ) ) ).
% mbs.less_not_eq
thf(fact_1097_card__permutations__of__multiset__remove__aux,axiom,
! [X: a,A2: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ A2 ) )
=> ( ( times_times_nat @ ( finite_card_list_a @ ( multis5886240593633752526iset_a @ A2 ) ) @ ( count_a @ A2 @ X ) )
= ( times_times_nat @ ( size_size_multiset_a @ A2 ) @ ( finite_card_list_a @ ( multis5886240593633752526iset_a @ ( minus_3765977307040488491iset_a @ A2 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) ) ) ) ) ).
% card_permutations_of_multiset_remove_aux
thf(fact_1098_card__permutations__of__multiset__remove__aux,axiom,
! [X: list_set_a,A2: multiset_list_set_a] :
( ( member_list_set_a @ X @ ( set_mset_list_set_a @ A2 ) )
=> ( ( times_times_nat @ ( finite5064046906505528594_set_a @ ( multis802782155138758580_set_a @ A2 ) ) @ ( count_list_set_a @ A2 @ X ) )
= ( times_times_nat @ ( size_s2437250918661492246_set_a @ A2 ) @ ( finite5064046906505528594_set_a @ ( multis802782155138758580_set_a @ ( minus_2572999347059163665_set_a @ A2 @ ( add_mset_list_set_a @ X @ zero_z8272816460787710433_set_a ) ) ) ) ) ) ) ).
% card_permutations_of_multiset_remove_aux
thf(fact_1099_card__permutations__of__multiset__remove__aux,axiom,
! [X: nat,A2: multiset_nat] :
( ( member_nat @ X @ ( set_mset_nat @ A2 ) )
=> ( ( times_times_nat @ ( finite_card_list_nat @ ( multis6201468865946971392et_nat @ A2 ) ) @ ( count_nat @ A2 @ X ) )
= ( times_times_nat @ ( size_s5917832649809541300et_nat @ A2 ) @ ( finite_card_list_nat @ ( multis6201468865946971392et_nat @ ( minus_8522176038001411705et_nat @ A2 @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) ) ) ) ) ) ).
% card_permutations_of_multiset_remove_aux
thf(fact_1100_card__permutations__of__multiset__remove__aux,axiom,
! [X: set_a,A2: multiset_set_a] :
( ( member_set_a @ X @ ( set_mset_set_a @ A2 ) )
=> ( ( times_times_nat @ ( finite4179508071619380492_set_a @ ( multis5469701301851823918_set_a @ A2 ) ) @ ( count_set_a @ A2 @ X ) )
= ( times_times_nat @ ( size_s6566526139600085008_set_a @ A2 ) @ ( finite4179508071619380492_set_a @ ( multis5469701301851823918_set_a @ ( minus_706656509937749387_set_a @ A2 @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) ) ) ) ) ) ).
% card_permutations_of_multiset_remove_aux
thf(fact_1101_nat__mult__le__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1102_nat__mult__less__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1103_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_1104_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_1105_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_1106_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_1107_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_1108_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1109_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1110_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_1111_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_1112_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_1113_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_1114_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_1115_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_1116_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1117_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1118_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M2 = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1119_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1120_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1121_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1122_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1123_mult__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 @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1124_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_1125_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_1126_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_1127_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1128_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_1129_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1130_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_1131_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_1132_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_1133_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_1134_linordered__semiring__strict__class_Omult__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 @ ( times_times_nat @ A @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_1135_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_1136_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_1137_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_1138_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_1139_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_1140_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1141_zero__less__one__class_Ozero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_less_one
thf(fact_1142_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1143_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_1144_add__le__add__imp__diff__le,axiom,
! [I2: nat,K: nat,N: nat,J2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J2 @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J2 @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J2 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1145_less__1__mult,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M2 )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% less_1_mult
thf(fact_1146_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_1147_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_1148_nat__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1149_nat__mult__eq__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( M2 = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1150_nat__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1151_nat__diff__add__eq2,axiom,
! [I2: nat,J2: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( minus_minus_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1152_nat__diff__add__eq1,axiom,
! [J2: nat,I2: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ J2 @ I2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J2 ) @ U ) @ M2 ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1153_nat__le__add__iff2,axiom,
! [I2: nat,J2: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1154_nat__le__add__iff1,axiom,
! [J2: nat,I2: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ J2 @ I2 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J2 ) @ U ) @ M2 ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1155_nat__eq__add__iff2,axiom,
! [I2: nat,J2: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 )
= ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( M2
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1156_nat__eq__add__iff1,axiom,
! [J2: nat,I2: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ J2 @ I2 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 )
= ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J2 ) @ U ) @ M2 )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1157_nat__less__add__iff1,axiom,
! [J2: nat,I2: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ J2 @ I2 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J2 ) @ U ) @ M2 ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1158_nat__less__add__iff2,axiom,
! [I2: nat,J2: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( ord_less_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1159_multiple__unfold__1,axiom,
! [N: nat] :
( ( repeat_mset_set_a @ ( suc @ N ) @ ( mset_set_a @ b_s ) )
= ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) @ ( mset_set_a @ b_s ) ) ) ).
% multiple_unfold_1
thf(fact_1160_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1161_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1162_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1163_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1164_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_1165_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_1166_add__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc_right
thf(fact_1167_Suc__diff__diff,axiom,
! [M2: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1168_diff__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_Suc_Suc
thf(fact_1169_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1170_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1171_mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1172_one__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M2 @ N ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1173_mult__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( times_times_nat @ M2 @ ( suc @ N ) )
= ( plus_plus_nat @ M2 @ ( times_times_nat @ M2 @ N ) ) ) ).
% mult_Suc_right
thf(fact_1174_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1175_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1176_one__le__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M2 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1177_diff__Suc__diff__eq1,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ ( suc @ J2 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1178_diff__Suc__diff__eq2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) @ I2 )
= ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1179_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1180_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1181_add__Suc,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc
thf(fact_1182_add__Suc__shift,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( plus_plus_nat @ M2 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1183_Suc__mult__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M2 )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M2 = N ) ) ).
% Suc_mult_cancel1
thf(fact_1184_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J4: nat] :
( ( M2
= ( suc @ J4 ) )
& ( ord_less_nat @ J4 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1185_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M8: nat] :
( N
= ( suc @ M8 ) ) ) ).
% gr0_implies_Suc
thf(fact_1186_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1187_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_1188_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_1189_dec__induct,axiom,
! [I2: nat,J2: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( P2 @ I2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I2 @ N3 )
=> ( ( ord_less_nat @ N3 @ J2 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) ) )
=> ( P2 @ J2 ) ) ) ) ).
% dec_induct
thf(fact_1190_inc__induct,axiom,
! [I2: nat,J2: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( P2 @ J2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I2 @ N3 )
=> ( ( ord_less_nat @ N3 @ J2 )
=> ( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% inc_induct
thf(fact_1191_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_1192_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_1193_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_1194_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1195_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1196_less__imp__Suc__add,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1197_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M3: nat,N2: nat] :
? [K4: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M3 @ K4 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1198_less__add__Suc2,axiom,
! [I2: nat,M2: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M2 @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_1199_less__add__Suc1,axiom,
! [I2: nat,M2: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1200_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M2 @ Q2 ) ) ) ) ).
% less_natE
thf(fact_1201_add__is__1,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1202_one__is__add,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1203_Suc__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1204_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_1205_Suc__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M2 ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1206_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1207_Suc__diff__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1208_Suc__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M2 ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1209_mult__Suc,axiom,
! [M2: nat,N: nat] :
( ( times_times_nat @ ( suc @ M2 ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M2 @ N ) ) ) ).
% mult_Suc
thf(fact_1210_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1211_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1212_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1213_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1214_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( K
!= ( suc @ J ) ) ) ) ) ).
% Nat.lessE
thf(fact_1215_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_1216_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( K
!= ( suc @ J ) ) ) ) ).
% Suc_lessE
thf(fact_1217_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1218_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_1219_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1220_Ex__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P2 @ I ) ) )
= ( ( P2 @ N )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P2 @ I ) ) ) ) ).
% Ex_less_Suc
thf(fact_1221_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_1222_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_1223_All__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P2 @ I ) ) )
= ( ( P2 @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P2 @ I ) ) ) ) ).
% All_less_Suc
thf(fact_1224_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M9: nat] :
( ( M2
= ( suc @ M9 ) )
& ( ord_less_nat @ N @ M9 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1225_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_1226_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_1227_less__trans__Suc,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1228_less__Suc__induct,axiom,
! [I2: nat,J2: nat,P2: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ! [I3: nat] : ( P2 @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ( P2 @ I3 @ J )
=> ( ( P2 @ J @ K2 )
=> ( P2 @ I3 @ K2 ) ) ) ) )
=> ( P2 @ I2 @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_1229_strict__inc__induct,axiom,
! [I2: nat,J2: nat,P2: nat > $o] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ! [I3: nat] :
( ( J2
= ( suc @ I3 ) )
=> ( P2 @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( P2 @ ( suc @ I3 ) )
=> ( P2 @ I3 ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_1230_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1231_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M8: nat] :
( N
= ( suc @ M8 ) ) ) ).
% not0_implies_Suc
thf(fact_1232_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_1233_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_1234_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1235_zero__induct,axiom,
! [P2: nat > $o,K: nat] :
( ( P2 @ K )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1236_diff__induct,axiom,
! [P2: nat > nat > $o,M2: nat,N: nat] :
( ! [X4: nat] : ( P2 @ X4 @ zero_zero_nat )
=> ( ! [Y: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y ) )
=> ( ! [X4: nat,Y: nat] :
( ( P2 @ X4 @ Y )
=> ( P2 @ ( suc @ X4 ) @ ( suc @ Y ) ) )
=> ( P2 @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_1237_nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) )
=> ( P2 @ N ) ) ) ).
% nat_induct
thf(fact_1238_old_Onat_Oexhaust,axiom,
! [Y4: nat] :
( ( Y4 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y4
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1239_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1240_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1241_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1242_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1243_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ! [X4: nat] : ( R2 @ X4 @ X4 )
=> ( ! [X4: nat,Y: nat,Z4: nat] :
( ( R2 @ X4 @ Y )
=> ( ( R2 @ Y @ Z4 )
=> ( R2 @ X4 @ Z4 ) ) )
=> ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
=> ( R2 @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1244_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P2 @ M2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1245_full__nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N3 )
=> ( P2 @ M4 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% full_nat_induct
thf(fact_1246_not__less__eq__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_1247_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1248_le__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M2 @ N )
| ( M2
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1249_Suc__le__D,axiom,
! [N: nat,M10: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M10 )
=> ? [M8: nat] :
( M10
= ( suc @ M8 ) ) ) ).
% Suc_le_D
thf(fact_1250_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1251_le__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( M2
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1252_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_1253_zero__induct__lemma,axiom,
! [P2: nat > $o,K: nat,I2: nat] :
( ( P2 @ K )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_1254_Suc__inject,axiom,
! [X: nat,Y4: nat] :
( ( ( suc @ X )
= ( suc @ Y4 ) )
=> ( X = Y4 ) ) ).
% Suc_inject
thf(fact_1255_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1256_ex__least__nat__less,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P2 @ I4 ) )
& ( P2 @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1257_diff__Suc__less,axiom,
! [N: nat,I2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1258_one__less__mult,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% one_less_mult
thf(fact_1259_n__less__m__mult__n,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1260_n__less__n__mult__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M2 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1261_nat__induct__non__zero,axiom,
! [N: nat,P2: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P2 @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1262_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1263_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1264_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_1265_add__block__rep__number__in,axiom,
! [X: a,B: set_a] :
( ( member_a @ X @ B )
=> ( ( design6637022207325878697mber_a @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ B ) @ X )
= ( plus_plus_nat @ ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X ) @ one_one_nat ) ) ) ).
% add_block_rep_number_in
thf(fact_1266_add__block__rep__number__not__in,axiom,
! [X: a,B: set_a] :
( ~ ( member_a @ X @ B )
=> ( ( design6637022207325878697mber_a @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ B ) @ X )
= ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X ) ) ) ).
% add_block_rep_number_not_in
thf(fact_1267_multiple__point__rep__num,axiom,
! [N: nat,X: a] :
( ( design6637022207325878697mber_a @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) @ X )
= ( times_times_nat @ ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X ) @ N ) ) ).
% multiple_point_rep_num
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y4: nat] :
( ( if_nat @ $false @ X @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y4: nat] :
( ( if_nat @ $true @ X @ Y4 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
member_set_a @ ( nth_set_a @ b_s @ i2 ) @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ).
%------------------------------------------------------------------------------