TPTP Problem File: SLH0024^1.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/0014_Set_Multiset_Extras/prob_00145_005727__27877974_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1445 ( 673 unt; 175 typ; 0 def)
% Number of atoms : 3066 (1839 equ; 0 cnn)
% Maximal formula atoms : 10 ( 2 avg)
% Number of connectives : 10701 ( 423 ~; 69 |; 298 &;8896 @)
% ( 0 <=>;1015 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 6 avg)
% Number of types : 26 ( 25 usr)
% Number of type conns : 396 ( 396 >; 0 *; 0 +; 0 <<)
% Number of symbols : 153 ( 150 usr; 19 con; 0-3 aty)
% Number of variables : 3373 ( 68 ^;3073 !; 232 ?;3373 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 15:42:23.304
%------------------------------------------------------------------------------
% Could-be-implicit typings (25)
thf(ty_n_t__Multiset__Omultiset_It__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J_J,type,
multis8895438461125693264iset_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Multiset__Omultiset_It__List__Olist_Itf__a_J_J_J,type,
multis272298859511629456list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J_J,type,
set_mu2969860852453053802iset_a: $tType ).
thf(ty_n_t__Set__Oset_It__Multiset__Omultiset_It__List__Olist_Itf__a_J_J_J,type,
set_multiset_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Multiset__Omultiset_Itf__a_J_J_J,type,
set_list_multiset_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Multiset__Omultiset_It__Nat__Onat_J_J,type,
multis1201202736280713200et_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
list_list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
set_list_list_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J,type,
multiset_multiset_a: $tType ).
thf(ty_n_t__Set__Oset_It__Multiset__Omultiset_It__Nat__Onat_J_J,type,
set_multiset_nat: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__List__Olist_Itf__a_J_J,type,
multiset_list_a: $tType ).
thf(ty_n_t__List__Olist_It__Multiset__Omultiset_Itf__a_J_J,type,
list_multiset_a: $tType ).
thf(ty_n_t__Set__Oset_It__Multiset__Omultiset_Itf__a_J_J,type,
set_multiset_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__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Nat__Onat_J,type,
multiset_nat: $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__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (150)
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001tf__a,type,
bNF_Greatest_Shift_a: set_list_a > a > set_list_a ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001tf__a,type,
bNF_Greatest_Succ_a: set_list_a > list_a > set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__List__Olist_Itf__a_J_J,type,
minus_7431248565939055793list_a: multiset_list_a > multiset_list_a > multiset_list_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J,type,
minus_3395427628221709681iset_a: multiset_multiset_a > multiset_multiset_a > multiset_multiset_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_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_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__List__Olist_Itf__a_J_J,type,
plus_p690419498615200257list_a: multiset_list_a > multiset_list_a > multiset_list_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J,type,
plus_p6738641960240532161iset_a: multiset_multiset_a > multiset_multiset_a > multiset_multiset_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_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_Ozero__class_Ozero_001t__Multiset__Omultiset_It__List__Olist_Itf__a_J_J,type,
zero_z4454100511807792257list_a: multiset_list_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J,type,
zero_z7799948378220188993iset_a: multiset_multiset_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_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_Groups__List_Omonoid__add_Osum__list_001t__Multiset__Omultiset_Itf__a_J,type,
groups778166972783649551iset_a: ( multiset_a > multiset_a > multiset_a ) > multiset_a > list_multiset_a > multiset_a ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_List_Oappend_001t__List__Olist_Itf__a_J,type,
append_list_a: list_list_a > list_list_a > list_list_a ).
thf(sy_c_List_Oappend_001t__Multiset__Omultiset_Itf__a_J,type,
append_multiset_a: list_multiset_a > list_multiset_a > list_multiset_a ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oappend_001tf__a,type,
append_a: list_a > list_a > list_a ).
thf(sy_c_List_Obutlast_001tf__a,type,
butlast_a: list_a > list_a ).
thf(sy_c_List_Oconcat_001tf__a,type,
concat_a: list_list_a > list_a ).
thf(sy_c_List_Odistinct__adj_001tf__a,type,
distinct_adj_a: list_a > $o ).
thf(sy_c_List_Odrop_001t__List__Olist_Itf__a_J,type,
drop_list_a: nat > list_list_a > list_list_a ).
thf(sy_c_List_Odrop_001t__Multiset__Omultiset_Itf__a_J,type,
drop_multiset_a: nat > list_multiset_a > list_multiset_a ).
thf(sy_c_List_Odrop_001t__Nat__Onat,type,
drop_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Odrop_001tf__a,type,
drop_a: nat > list_a > list_a ).
thf(sy_c_List_Olast_001tf__a,type,
last_a: list_a > a ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
cons_list_list_a: list_list_a > list_list_list_a > list_list_list_a ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__a_J,type,
cons_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Olist_OCons_001t__Multiset__Omultiset_Itf__a_J,type,
cons_multiset_a: multiset_a > list_multiset_a > list_multiset_a ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
nil_list_list_a: list_list_list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__a_J,type,
nil_list_a: list_list_a ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Ohd_001tf__a,type,
hd_a: list_a > a ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_Itf__a_J,type,
set_list_a2: list_list_a > set_list_a ).
thf(sy_c_List_Olist_Oset_001t__Multiset__Omultiset_Itf__a_J,type,
set_multiset_a2: list_multiset_a > set_multiset_a ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Olist_Osize__list_001tf__a,type,
size_list_a: ( a > nat ) > list_a > nat ).
thf(sy_c_List_Olist_Otl_001tf__a,type,
tl_a: list_a > list_a ).
thf(sy_c_List_Olist__update_001t__Multiset__Omultiset_Itf__a_J,type,
list_u788720842672076722iset_a: list_multiset_a > nat > multiset_a > list_multiset_a ).
thf(sy_c_List_Olist__update_001tf__a,type,
list_update_a: list_a > nat > a > list_a ).
thf(sy_c_List_On__lists_001tf__a,type,
n_lists_a: nat > list_a > list_list_a ).
thf(sy_c_List_Onth_001t__List__Olist_Itf__a_J,type,
nth_list_a: list_list_a > nat > list_a ).
thf(sy_c_List_Onth_001t__Multiset__Omultiset_Itf__a_J,type,
nth_multiset_a: list_multiset_a > nat > multiset_a ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_List_Oproduct__lists_001tf__a,type,
product_lists_a: list_list_a > list_list_a ).
thf(sy_c_List_Oremove1_001t__List__Olist_Itf__a_J,type,
remove1_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Oremove1_001t__Multiset__Omultiset_Itf__a_J,type,
remove1_multiset_a: multiset_a > list_multiset_a > list_multiset_a ).
thf(sy_c_List_Oremove1_001t__Nat__Onat,type,
remove1_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oremove1_001tf__a,type,
remove1_a: a > list_a > list_a ).
thf(sy_c_List_Orotate1_001tf__a,type,
rotate1_a: list_a > list_a ).
thf(sy_c_List_Otake_001t__List__Olist_Itf__a_J,type,
take_list_a: nat > list_list_a > list_list_a ).
thf(sy_c_List_Otake_001t__Multiset__Omultiset_Itf__a_J,type,
take_multiset_a: nat > list_multiset_a > list_multiset_a ).
thf(sy_c_List_Otake_001t__Nat__Onat,type,
take_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Otake_001tf__a,type,
take_a: nat > list_a > list_a ).
thf(sy_c_Multiset_Oadd__mset_001t__List__Olist_Itf__a_J,type,
add_mset_list_a: list_a > multiset_list_a > multiset_list_a ).
thf(sy_c_Multiset_Oadd__mset_001t__Multiset__Omultiset_Itf__a_J,type,
add_mset_multiset_a: multiset_a > multiset_multiset_a > multiset_multiset_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_001tf__a,type,
add_mset_a: a > multiset_a > multiset_a ).
thf(sy_c_Multiset_Ocomm__monoid__add_Osum__mset_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
comm_m5787568287065167983et_nat: ( multiset_nat > multiset_nat > multiset_nat ) > multiset_nat > multis1201202736280713200et_nat > multiset_nat ).
thf(sy_c_Multiset_Ocomm__monoid__add_Osum__mset_001t__Multiset__Omultiset_Itf__a_J,type,
comm_m1977238983017651125iset_a: ( multiset_a > multiset_a > multiset_a ) > multiset_a > multiset_multiset_a > multiset_a ).
thf(sy_c_Multiset_Ofold__mset_001t__Multiset__Omultiset_It__List__Olist_Itf__a_J_J_001t__Multiset__Omultiset_It__List__Olist_Itf__a_J_J,type,
fold_m7881050568294030859list_a: ( multiset_list_a > multiset_list_a > multiset_list_a ) > multiset_list_a > multis272298859511629456list_a > multiset_list_a ).
thf(sy_c_Multiset_Ofold__mset_001t__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J_001t__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J,type,
fold_m7435756534263755787iset_a: ( multiset_multiset_a > multiset_multiset_a > multiset_multiset_a ) > multiset_multiset_a > multis8895438461125693264iset_a > multiset_multiset_a ).
thf(sy_c_Multiset_Ofold__mset_001t__Multiset__Omultiset_It__Nat__Onat_J_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
fold_m1829410296857755981et_nat: ( multiset_nat > multiset_nat > multiset_nat ) > multiset_nat > multis1201202736280713200et_nat > multiset_nat ).
thf(sy_c_Multiset_Ofold__mset_001t__Multiset__Omultiset_Itf__a_J_001t__Multiset__Omultiset_Itf__a_J,type,
fold_m6601649825673331723iset_a: ( multiset_a > multiset_a > multiset_a ) > multiset_a > multiset_multiset_a > multiset_a ).
thf(sy_c_Multiset_Ofold__mset_001t__Nat__Onat_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
fold_m2600682269844132093et_nat: ( nat > multiset_nat > multiset_nat ) > multiset_nat > multiset_nat > multiset_nat ).
thf(sy_c_Multiset_Ofold__mset_001tf__a_001t__Multiset__Omultiset_Itf__a_J,type,
fold_m7320414754419674833iset_a: ( a > multiset_a > multiset_a ) > multiset_a > multiset_a > multiset_a ).
thf(sy_c_Multiset_Ois__empty_001t__Nat__Onat,type,
is_empty_nat: multiset_nat > $o ).
thf(sy_c_Multiset_Ois__empty_001tf__a,type,
is_empty_a: multiset_a > $o ).
thf(sy_c_Multiset_Olinorder__class_Osorted__list__of__multiset_001t__Nat__Onat,type,
linord3047872887403683810et_nat: multiset_nat > list_nat ).
thf(sy_c_Multiset_Omset_001t__List__Olist_Itf__a_J,type,
mset_list_a: list_list_a > multiset_list_a ).
thf(sy_c_Multiset_Omset_001t__Multiset__Omultiset_Itf__a_J,type,
mset_multiset_a: list_multiset_a > multiset_multiset_a ).
thf(sy_c_Multiset_Omset_001t__Nat__Onat,type,
mset_nat: list_nat > multiset_nat ).
thf(sy_c_Multiset_Omset_001tf__a,type,
mset_a: list_a > multiset_a ).
thf(sy_c_Multiset_Omultiset_Ocount_001t__Nat__Onat,type,
count_nat: multiset_nat > nat > nat ).
thf(sy_c_Multiset_Omultiset_Ocount_001tf__a,type,
count_a: multiset_a > a > nat ).
thf(sy_c_Multiset_Oset__mset_001t__List__Olist_Itf__a_J,type,
set_mset_list_a: multiset_list_a > set_list_a ).
thf(sy_c_Multiset_Oset__mset_001t__Multiset__Omultiset_It__List__Olist_Itf__a_J_J,type,
set_ms6457238410686372999list_a: multis272298859511629456list_a > set_multiset_list_a ).
thf(sy_c_Multiset_Oset__mset_001t__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J,type,
set_ms6500351741012541767iset_a: multis8895438461125693264iset_a > set_mu2969860852453053802iset_a ).
thf(sy_c_Multiset_Oset__mset_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
set_ms4188662328148412963et_nat: multis1201202736280713200et_nat > set_multiset_nat ).
thf(sy_c_Multiset_Oset__mset_001t__Multiset__Omultiset_Itf__a_J,type,
set_mset_multiset_a: multiset_multiset_a > set_multiset_a ).
thf(sy_c_Multiset_Oset__mset_001t__Nat__Onat,type,
set_mset_nat: multiset_nat > set_nat ).
thf(sy_c_Multiset_Oset__mset_001tf__a,type,
set_mset_a: multiset_a > set_a ).
thf(sy_c_Multiset_Osize__multiset_001t__Nat__Onat,type,
size_multiset_nat: ( nat > nat ) > multiset_nat > nat ).
thf(sy_c_Multiset_Osize__multiset_001tf__a,type,
size_multiset_a: ( a > nat ) > multiset_a > nat ).
thf(sy_c_Multiset_Osubseteq__mset_001t__Nat__Onat,type,
subseteq_mset_nat: multiset_nat > multiset_nat > $o ).
thf(sy_c_Multiset_Osubseteq__mset_001tf__a,type,
subseteq_mset_a: multiset_a > multiset_a > $o ).
thf(sy_c_Multiset_Owcount_001t__Nat__Onat,type,
wcount_nat: ( nat > nat ) > multiset_nat > nat > nat ).
thf(sy_c_Multiset_Owcount_001tf__a,type,
wcount_a: ( a > nat ) > multiset_a > a > nat ).
thf(sy_c_Multiset__More_Olist__of__mset_001t__List__Olist_Itf__a_J,type,
multis5009215944268684175list_a: multiset_list_a > list_list_a ).
thf(sy_c_Multiset__More_Olist__of__mset_001t__Nat__Onat,type,
multis105632648212199813et_nat: multiset_nat > list_nat ).
thf(sy_c_Multiset__More_Olist__of__mset_001tf__a,type,
multis4723169673647964297mset_a: multiset_a > list_a ).
thf(sy_c_Multiset__More_Oremdups__mset_001t__List__Olist_Itf__a_J,type,
multis807522658164556234list_a: multiset_list_a > multiset_list_a ).
thf(sy_c_Multiset__More_Oremdups__mset_001t__Multiset__Omultiset_Itf__a_J,type,
multis2302138969243003530iset_a: multiset_multiset_a > multiset_multiset_a ).
thf(sy_c_Multiset__More_Oremdups__mset_001t__Nat__Onat,type,
multis3892936081960800522et_nat: multiset_nat > multiset_nat ).
thf(sy_c_Multiset__More_Oremdups__mset_001tf__a,type,
multis1648347201088684100mset_a: multiset_a > multiset_a ).
thf(sy_c_Multiset__Order_Omultp_092_060_094sub_062D_092_060_094sub_062M_001tf__a,type,
multiset_multp_D_M_a: ( a > a > $o ) > multiset_a > multiset_a > $o ).
thf(sy_c_Multiset__Order_Opreorder__class_Oless__multiset_092_060_094sub_062D_092_060_094sub_062M_001t__Nat__Onat,type,
multis2595361328647931344_M_nat: multiset_nat > multiset_nat > $o ).
thf(sy_c_Multiset__Permutations_Opermutations__of__list__impl_001tf__a,type,
multis7466489000348688569impl_a: list_a > list_list_a ).
thf(sy_c_Multiset__Permutations_Opermutations__of__list__impl__aux_001tf__a,type,
multis7962179442503870237_aux_a: list_a > list_a > list_list_a ).
thf(sy_c_Multiset__Permutations_Opermutations__of__multiset_001t__List__Olist_Itf__a_J,type,
multis7786709813750966868list_a: multiset_list_a > set_list_list_a ).
thf(sy_c_Multiset__Permutations_Opermutations__of__multiset_001t__Multiset__Omultiset_Itf__a_J,type,
multis8172983762386353940iset_a: multiset_multiset_a > set_list_multiset_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_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_Itf__a_J_J,type,
size_s349497388124573686list_a: list_list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Multiset__Omultiset_Itf__a_J_J,type,
size_s3003218658903709366iset_a: list_multiset_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_Itf__a_J,type,
size_size_multiset_a: multiset_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
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__Nat__Onat,type,
ord_less_nat: nat > nat > $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__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $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_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001t__Multiset__Omultiset_Itf__a_J,type,
collect_multiset_a: ( multiset_a > $o ) > set_multiset_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_Sublist_Oprefix_001tf__a,type,
prefix_a: list_a > list_a > $o ).
thf(sy_c_Sublist_Oprefixes_001tf__a,type,
prefixes_a: list_a > list_list_a ).
thf(sy_c_Sublist_Osuffixes_001tf__a,type,
suffixes_a: list_a > list_list_a ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
member_list_list_a: list_list_a > set_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Multiset__Omultiset_Itf__a_J_J,type,
member1058549947519091315iset_a: list_multiset_a > set_list_multiset_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Multiset__Omultiset_It__List__Olist_Itf__a_J_J,type,
member391257453028489331list_a: multiset_list_a > set_multiset_list_a > $o ).
thf(sy_c_member_001t__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J,type,
member7618379257985549619iset_a: multiset_multiset_a > set_mu2969860852453053802iset_a > $o ).
thf(sy_c_member_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
member_multiset_nat: multiset_nat > set_multiset_nat > $o ).
thf(sy_c_member_001t__Multiset__Omultiset_Itf__a_J,type,
member_multiset_a: multiset_a > set_multiset_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_j1,type,
j1: nat ).
thf(sy_v_thesis,type,
thesis: $o ).
thf(sy_v_x,type,
x: a ).
thf(sy_v_xs,type,
xs: list_a ).
thf(sy_v_y,type,
y: a ).
thf(sy_v_ys____,type,
ys: list_a ).
thf(sy_v_zs____,type,
zs: list_a ).
% Relevant facts (1266)
thf(fact_0_False,axiom,
~ ( member_a @ y @ ( set_mset_a @ ( mset_a @ ys ) ) ) ).
% False
thf(fact_1_True,axiom,
x = y ).
% True
thf(fact_2_yinor,axiom,
( ( member_a @ y @ ( set_mset_a @ ( mset_a @ ys ) ) )
| ( member_a @ y @ ( set_mset_a @ ( mset_a @ zs ) ) ) ) ).
% yinor
thf(fact_3__092_060open_062y_A_092_060in_062_D_Amset_Ays_A_L_Amset_Azs_092_060close_062,axiom,
member_a @ y @ ( set_mset_a @ ( plus_plus_multiset_a @ ( mset_a @ ys ) @ ( mset_a @ zs ) ) ) ).
% \<open>y \<in># mset ys + mset zs\<close>
thf(fact_4_assms_I1_J,axiom,
member_a @ x @ ( set_mset_a @ ( mset_a @ xs ) ) ).
% assms(1)
thf(fact_5_ex__mset,axiom,
! [X: multiset_a] :
? [Xs: list_a] :
( ( mset_a @ Xs )
= X ) ).
% ex_mset
thf(fact_6_list__of__mset__exi,axiom,
! [M: multiset_a] :
? [L: list_a] :
( M
= ( mset_a @ L ) ) ).
% list_of_mset_exi
thf(fact_7_mset__list__of__mset,axiom,
! [M: multiset_a] :
( ( mset_a @ ( multis4723169673647964297mset_a @ M ) )
= M ) ).
% mset_list_of_mset
thf(fact_8_set__mset__mset,axiom,
! [Xs2: list_list_a] :
( ( set_mset_list_a @ ( mset_list_a @ Xs2 ) )
= ( set_list_a2 @ Xs2 ) ) ).
% set_mset_mset
thf(fact_9_set__mset__mset,axiom,
! [Xs2: list_multiset_a] :
( ( set_mset_multiset_a @ ( mset_multiset_a @ Xs2 ) )
= ( set_multiset_a2 @ Xs2 ) ) ).
% set_mset_mset
thf(fact_10_set__mset__mset,axiom,
! [Xs2: list_nat] :
( ( set_mset_nat @ ( mset_nat @ Xs2 ) )
= ( set_nat2 @ Xs2 ) ) ).
% set_mset_mset
thf(fact_11_set__mset__mset,axiom,
! [Xs2: list_a] :
( ( set_mset_a @ ( mset_a @ Xs2 ) )
= ( set_a2 @ Xs2 ) ) ).
% set_mset_mset
thf(fact_12_in__multiset__in__set,axiom,
! [X2: list_a,Xs2: list_list_a] :
( ( member_list_a @ X2 @ ( set_mset_list_a @ ( mset_list_a @ Xs2 ) ) )
= ( member_list_a @ X2 @ ( set_list_a2 @ Xs2 ) ) ) ).
% in_multiset_in_set
thf(fact_13_in__multiset__in__set,axiom,
! [X2: multiset_a,Xs2: list_multiset_a] :
( ( member_multiset_a @ X2 @ ( set_mset_multiset_a @ ( mset_multiset_a @ Xs2 ) ) )
= ( member_multiset_a @ X2 @ ( set_multiset_a2 @ Xs2 ) ) ) ).
% in_multiset_in_set
thf(fact_14_in__multiset__in__set,axiom,
! [X2: nat,Xs2: list_nat] :
( ( member_nat @ X2 @ ( set_mset_nat @ ( mset_nat @ Xs2 ) ) )
= ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) ) ) ).
% in_multiset_in_set
thf(fact_15_in__multiset__in__set,axiom,
! [X2: a,Xs2: list_a] :
( ( member_a @ X2 @ ( set_mset_a @ ( mset_a @ Xs2 ) ) )
= ( member_a @ X2 @ ( set_a2 @ Xs2 ) ) ) ).
% in_multiset_in_set
thf(fact_16_assms_I2_J,axiom,
member_a @ y @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ ( mset_a @ xs ) @ ( add_mset_a @ x @ zero_zero_multiset_a ) ) ) ).
% assms(2)
thf(fact_17_set__mset__remdups__mset,axiom,
! [A: multiset_multiset_a] :
( ( set_mset_multiset_a @ ( multis2302138969243003530iset_a @ A ) )
= ( set_mset_multiset_a @ A ) ) ).
% set_mset_remdups_mset
thf(fact_18_set__mset__remdups__mset,axiom,
! [A: multiset_nat] :
( ( set_mset_nat @ ( multis3892936081960800522et_nat @ A ) )
= ( set_mset_nat @ A ) ) ).
% set_mset_remdups_mset
thf(fact_19_set__mset__remdups__mset,axiom,
! [A: multiset_a] :
( ( set_mset_a @ ( multis1648347201088684100mset_a @ A ) )
= ( set_mset_a @ A ) ) ).
% set_mset_remdups_mset
thf(fact_20__092_060open_062mset_Axs_A_061_Amset_Ays_A_L_Amset_A_Ix_A_D_Azs_J_092_060close_062,axiom,
( ( mset_a @ xs )
= ( plus_plus_multiset_a @ ( mset_a @ ys ) @ ( mset_a @ ( cons_a @ x @ zs ) ) ) ) ).
% \<open>mset xs = mset ys + mset (x # zs)\<close>
thf(fact_21_zseq,axiom,
( zs
= ( drop_a @ ( suc @ j1 ) @ xs ) ) ).
% zseq
thf(fact_22_add__mset__add__mset__same__iff,axiom,
! [A2: nat,A: multiset_nat,B: multiset_nat] :
( ( ( add_mset_nat @ A2 @ A )
= ( add_mset_nat @ A2 @ B ) )
= ( A = B ) ) ).
% add_mset_add_mset_same_iff
thf(fact_23_add__mset__add__mset__same__iff,axiom,
! [A2: a,A: multiset_a,B: multiset_a] :
( ( ( add_mset_a @ A2 @ A )
= ( add_mset_a @ A2 @ B ) )
= ( A = B ) ) ).
% add_mset_add_mset_same_iff
thf(fact_24_multi__self__add__other__not__self,axiom,
! [M2: multiset_nat,X2: nat] :
( M2
!= ( add_mset_nat @ X2 @ M2 ) ) ).
% multi_self_add_other_not_self
thf(fact_25_multi__self__add__other__not__self,axiom,
! [M2: multiset_a,X2: a] :
( M2
!= ( add_mset_a @ X2 @ M2 ) ) ).
% multi_self_add_other_not_self
thf(fact_26_single__eq__single,axiom,
! [A2: nat,B2: nat] :
( ( ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat )
= ( add_mset_nat @ B2 @ zero_z7348594199698428585et_nat ) )
= ( A2 = B2 ) ) ).
% single_eq_single
thf(fact_27_single__eq__single,axiom,
! [A2: a,B2: a] :
( ( ( add_mset_a @ A2 @ zero_zero_multiset_a )
= ( add_mset_a @ B2 @ zero_zero_multiset_a ) )
= ( A2 = B2 ) ) ).
% single_eq_single
thf(fact_28_add__mset__eq__single,axiom,
! [B2: nat,M2: multiset_nat,A2: nat] :
( ( ( add_mset_nat @ B2 @ M2 )
= ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) )
= ( ( B2 = A2 )
& ( M2 = zero_z7348594199698428585et_nat ) ) ) ).
% add_mset_eq_single
thf(fact_29_add__mset__eq__single,axiom,
! [B2: a,M2: multiset_a,A2: a] :
( ( ( add_mset_a @ B2 @ M2 )
= ( add_mset_a @ A2 @ zero_zero_multiset_a ) )
= ( ( B2 = A2 )
& ( M2 = zero_zero_multiset_a ) ) ) ).
% add_mset_eq_single
thf(fact_30_single__eq__add__mset,axiom,
! [A2: nat,B2: nat,M2: multiset_nat] :
( ( ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat )
= ( add_mset_nat @ B2 @ M2 ) )
= ( ( B2 = A2 )
& ( M2 = zero_z7348594199698428585et_nat ) ) ) ).
% single_eq_add_mset
thf(fact_31_single__eq__add__mset,axiom,
! [A2: a,B2: a,M2: multiset_a] :
( ( ( add_mset_a @ A2 @ zero_zero_multiset_a )
= ( add_mset_a @ B2 @ M2 ) )
= ( ( B2 = A2 )
& ( M2 = zero_zero_multiset_a ) ) ) ).
% single_eq_add_mset
thf(fact_32_add__mset__eq__singleton__iff,axiom,
! [X2: nat,M2: multiset_nat,Y: nat] :
( ( ( add_mset_nat @ X2 @ M2 )
= ( add_mset_nat @ Y @ zero_z7348594199698428585et_nat ) )
= ( ( M2 = zero_z7348594199698428585et_nat )
& ( X2 = Y ) ) ) ).
% add_mset_eq_singleton_iff
thf(fact_33_add__mset__eq__singleton__iff,axiom,
! [X2: a,M2: multiset_a,Y: a] :
( ( ( add_mset_a @ X2 @ M2 )
= ( add_mset_a @ Y @ zero_zero_multiset_a ) )
= ( ( M2 = zero_zero_multiset_a )
& ( X2 = Y ) ) ) ).
% add_mset_eq_singleton_iff
thf(fact_34_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [X2: multiset_nat,Y: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ X2 @ Y )
= zero_z7348594199698428585et_nat )
= ( ( X2 = zero_z7348594199698428585et_nat )
& ( Y = zero_z7348594199698428585et_nat ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_35_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [X2: multiset_a,Y: multiset_a] :
( ( ( plus_plus_multiset_a @ X2 @ Y )
= zero_zero_multiset_a )
= ( ( X2 = zero_zero_multiset_a )
& ( Y = zero_zero_multiset_a ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_36_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [X2: multiset_nat,Y: multiset_nat] :
( ( zero_z7348594199698428585et_nat
= ( plus_p6334493942879108393et_nat @ X2 @ Y ) )
= ( ( X2 = zero_z7348594199698428585et_nat )
& ( Y = zero_z7348594199698428585et_nat ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_37_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [X2: multiset_a,Y: multiset_a] :
( ( zero_zero_multiset_a
= ( plus_plus_multiset_a @ X2 @ Y ) )
= ( ( X2 = zero_zero_multiset_a )
& ( Y = zero_zero_multiset_a ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_38_empty__eq__union,axiom,
! [M2: multiset_nat,N: multiset_nat] :
( ( zero_z7348594199698428585et_nat
= ( plus_p6334493942879108393et_nat @ M2 @ N ) )
= ( ( M2 = zero_z7348594199698428585et_nat )
& ( N = zero_z7348594199698428585et_nat ) ) ) ).
% empty_eq_union
thf(fact_39_empty__eq__union,axiom,
! [M2: multiset_a,N: multiset_a] :
( ( zero_zero_multiset_a
= ( plus_plus_multiset_a @ M2 @ N ) )
= ( ( M2 = zero_zero_multiset_a )
& ( N = zero_zero_multiset_a ) ) ) ).
% empty_eq_union
thf(fact_40_union__eq__empty,axiom,
! [M2: multiset_nat,N: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ M2 @ N )
= zero_z7348594199698428585et_nat )
= ( ( M2 = zero_z7348594199698428585et_nat )
& ( N = zero_z7348594199698428585et_nat ) ) ) ).
% union_eq_empty
thf(fact_41_union__eq__empty,axiom,
! [M2: multiset_a,N: multiset_a] :
( ( ( plus_plus_multiset_a @ M2 @ N )
= zero_zero_multiset_a )
= ( ( M2 = zero_zero_multiset_a )
& ( N = zero_zero_multiset_a ) ) ) ).
% union_eq_empty
thf(fact_42_union__mset__add__mset__left,axiom,
! [A2: nat,A: multiset_nat,B: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ ( add_mset_nat @ A2 @ A ) @ B )
= ( add_mset_nat @ A2 @ ( plus_p6334493942879108393et_nat @ A @ B ) ) ) ).
% union_mset_add_mset_left
thf(fact_43_union__mset__add__mset__left,axiom,
! [A2: a,A: multiset_a,B: multiset_a] :
( ( plus_plus_multiset_a @ ( add_mset_a @ A2 @ A ) @ B )
= ( add_mset_a @ A2 @ ( plus_plus_multiset_a @ A @ B ) ) ) ).
% union_mset_add_mset_left
thf(fact_44_union__mset__add__mset__right,axiom,
! [A: multiset_nat,A2: nat,B: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ A @ ( add_mset_nat @ A2 @ B ) )
= ( add_mset_nat @ A2 @ ( plus_p6334493942879108393et_nat @ A @ B ) ) ) ).
% union_mset_add_mset_right
thf(fact_45_union__mset__add__mset__right,axiom,
! [A: multiset_a,A2: a,B: multiset_a] :
( ( plus_plus_multiset_a @ A @ ( add_mset_a @ A2 @ B ) )
= ( add_mset_a @ A2 @ ( plus_plus_multiset_a @ A @ B ) ) ) ).
% union_mset_add_mset_right
thf(fact_46_diff__diff__add__mset,axiom,
! [M2: multiset_nat,N: multiset_nat,P: multiset_nat] :
( ( minus_8522176038001411705et_nat @ ( minus_8522176038001411705et_nat @ M2 @ N ) @ P )
= ( minus_8522176038001411705et_nat @ M2 @ ( plus_p6334493942879108393et_nat @ N @ P ) ) ) ).
% diff_diff_add_mset
thf(fact_47_diff__diff__add__mset,axiom,
! [M2: multiset_a,N: multiset_a,P: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ M2 @ N ) @ P )
= ( minus_3765977307040488491iset_a @ M2 @ ( plus_plus_multiset_a @ N @ P ) ) ) ).
% diff_diff_add_mset
thf(fact_48_remdups__mset__empty,axiom,
( ( multis3892936081960800522et_nat @ zero_z7348594199698428585et_nat )
= zero_z7348594199698428585et_nat ) ).
% remdups_mset_empty
thf(fact_49_remdups__mset__empty,axiom,
( ( multis1648347201088684100mset_a @ zero_zero_multiset_a )
= zero_zero_multiset_a ) ).
% remdups_mset_empty
thf(fact_50_remdups__mset__eq__empty,axiom,
! [D: multiset_nat] :
( ( ( multis3892936081960800522et_nat @ D )
= zero_z7348594199698428585et_nat )
= ( D = zero_z7348594199698428585et_nat ) ) ).
% remdups_mset_eq_empty
thf(fact_51_remdups__mset__eq__empty,axiom,
! [D: multiset_a] :
( ( ( multis1648347201088684100mset_a @ D )
= zero_zero_multiset_a )
= ( D = zero_zero_multiset_a ) ) ).
% remdups_mset_eq_empty
thf(fact_52_add__mset__remove__trivial,axiom,
! [X2: nat,M2: multiset_nat] :
( ( minus_8522176038001411705et_nat @ ( add_mset_nat @ X2 @ M2 ) @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) )
= M2 ) ).
% add_mset_remove_trivial
thf(fact_53_add__mset__remove__trivial,axiom,
! [X2: a,M2: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( add_mset_a @ X2 @ M2 ) @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) )
= M2 ) ).
% add_mset_remove_trivial
thf(fact_54_remove1__single__empty__iff,axiom,
! [L2: nat,L3: nat] :
( ( ( minus_8522176038001411705et_nat @ ( add_mset_nat @ L2 @ zero_z7348594199698428585et_nat ) @ ( add_mset_nat @ L3 @ zero_z7348594199698428585et_nat ) )
= zero_z7348594199698428585et_nat )
= ( L3 = L2 ) ) ).
% remove1_single_empty_iff
thf(fact_55_remove1__single__empty__iff,axiom,
! [L2: a,L3: a] :
( ( ( minus_3765977307040488491iset_a @ ( add_mset_a @ L2 @ zero_zero_multiset_a ) @ ( add_mset_a @ L3 @ zero_zero_multiset_a ) )
= zero_zero_multiset_a )
= ( L3 = L2 ) ) ).
% remove1_single_empty_iff
thf(fact_56_diff__add__mset__swap,axiom,
! [B2: list_a,A: multiset_list_a,M2: multiset_list_a] :
( ~ ( member_list_a @ B2 @ ( set_mset_list_a @ A ) )
=> ( ( minus_7431248565939055793list_a @ ( add_mset_list_a @ B2 @ M2 ) @ A )
= ( add_mset_list_a @ B2 @ ( minus_7431248565939055793list_a @ M2 @ A ) ) ) ) ).
% diff_add_mset_swap
thf(fact_57_diff__add__mset__swap,axiom,
! [B2: multiset_a,A: multiset_multiset_a,M2: multiset_multiset_a] :
( ~ ( member_multiset_a @ B2 @ ( set_mset_multiset_a @ A ) )
=> ( ( minus_3395427628221709681iset_a @ ( add_mset_multiset_a @ B2 @ M2 ) @ A )
= ( add_mset_multiset_a @ B2 @ ( minus_3395427628221709681iset_a @ M2 @ A ) ) ) ) ).
% diff_add_mset_swap
thf(fact_58_diff__add__mset__swap,axiom,
! [B2: nat,A: multiset_nat,M2: multiset_nat] :
( ~ ( member_nat @ B2 @ ( set_mset_nat @ A ) )
=> ( ( minus_8522176038001411705et_nat @ ( add_mset_nat @ B2 @ M2 ) @ A )
= ( add_mset_nat @ B2 @ ( minus_8522176038001411705et_nat @ M2 @ A ) ) ) ) ).
% diff_add_mset_swap
thf(fact_59_diff__add__mset__swap,axiom,
! [B2: a,A: multiset_a,M2: multiset_a] :
( ~ ( member_a @ B2 @ ( set_mset_a @ A ) )
=> ( ( minus_3765977307040488491iset_a @ ( add_mset_a @ B2 @ M2 ) @ A )
= ( add_mset_a @ B2 @ ( minus_3765977307040488491iset_a @ M2 @ A ) ) ) ) ).
% diff_add_mset_swap
thf(fact_60_remove__diff__multiset,axiom,
! [X13: list_a,A: multiset_list_a,B: multiset_list_a] :
( ~ ( member_list_a @ X13 @ ( set_mset_list_a @ A ) )
=> ( ( minus_7431248565939055793list_a @ A @ ( add_mset_list_a @ X13 @ B ) )
= ( minus_7431248565939055793list_a @ A @ B ) ) ) ).
% remove_diff_multiset
thf(fact_61_remove__diff__multiset,axiom,
! [X13: multiset_a,A: multiset_multiset_a,B: multiset_multiset_a] :
( ~ ( member_multiset_a @ X13 @ ( set_mset_multiset_a @ A ) )
=> ( ( minus_3395427628221709681iset_a @ A @ ( add_mset_multiset_a @ X13 @ B ) )
= ( minus_3395427628221709681iset_a @ A @ B ) ) ) ).
% remove_diff_multiset
thf(fact_62_remove__diff__multiset,axiom,
! [X13: nat,A: multiset_nat,B: multiset_nat] :
( ~ ( member_nat @ X13 @ ( set_mset_nat @ A ) )
=> ( ( minus_8522176038001411705et_nat @ A @ ( add_mset_nat @ X13 @ B ) )
= ( minus_8522176038001411705et_nat @ A @ B ) ) ) ).
% remove_diff_multiset
thf(fact_63_remove__diff__multiset,axiom,
! [X13: a,A: multiset_a,B: multiset_a] :
( ~ ( member_a @ X13 @ ( set_mset_a @ A ) )
=> ( ( minus_3765977307040488491iset_a @ A @ ( add_mset_a @ X13 @ B ) )
= ( minus_3765977307040488491iset_a @ A @ B ) ) ) ).
% remove_diff_multiset
thf(fact_64_remdups__mset__singleton,axiom,
! [A2: nat] :
( ( multis3892936081960800522et_nat @ ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) )
= ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) ) ).
% remdups_mset_singleton
thf(fact_65_remdups__mset__singleton,axiom,
! [A2: a] :
( ( multis1648347201088684100mset_a @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) )
= ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) ).
% remdups_mset_singleton
thf(fact_66_remdups__mset__singleton__sum,axiom,
! [A2: list_a,A: multiset_list_a] :
( ( ( member_list_a @ A2 @ ( set_mset_list_a @ A ) )
=> ( ( multis807522658164556234list_a @ ( add_mset_list_a @ A2 @ A ) )
= ( multis807522658164556234list_a @ A ) ) )
& ( ~ ( member_list_a @ A2 @ ( set_mset_list_a @ A ) )
=> ( ( multis807522658164556234list_a @ ( add_mset_list_a @ A2 @ A ) )
= ( add_mset_list_a @ A2 @ ( multis807522658164556234list_a @ A ) ) ) ) ) ).
% remdups_mset_singleton_sum
thf(fact_67_remdups__mset__singleton__sum,axiom,
! [A2: multiset_a,A: multiset_multiset_a] :
( ( ( member_multiset_a @ A2 @ ( set_mset_multiset_a @ A ) )
=> ( ( multis2302138969243003530iset_a @ ( add_mset_multiset_a @ A2 @ A ) )
= ( multis2302138969243003530iset_a @ A ) ) )
& ( ~ ( member_multiset_a @ A2 @ ( set_mset_multiset_a @ A ) )
=> ( ( multis2302138969243003530iset_a @ ( add_mset_multiset_a @ A2 @ A ) )
= ( add_mset_multiset_a @ A2 @ ( multis2302138969243003530iset_a @ A ) ) ) ) ) ).
% remdups_mset_singleton_sum
thf(fact_68_remdups__mset__singleton__sum,axiom,
! [A2: nat,A: multiset_nat] :
( ( ( member_nat @ A2 @ ( set_mset_nat @ A ) )
=> ( ( multis3892936081960800522et_nat @ ( add_mset_nat @ A2 @ A ) )
= ( multis3892936081960800522et_nat @ A ) ) )
& ( ~ ( member_nat @ A2 @ ( set_mset_nat @ A ) )
=> ( ( multis3892936081960800522et_nat @ ( add_mset_nat @ A2 @ A ) )
= ( add_mset_nat @ A2 @ ( multis3892936081960800522et_nat @ A ) ) ) ) ) ).
% remdups_mset_singleton_sum
thf(fact_69_remdups__mset__singleton__sum,axiom,
! [A2: a,A: multiset_a] :
( ( ( member_a @ A2 @ ( set_mset_a @ A ) )
=> ( ( multis1648347201088684100mset_a @ ( add_mset_a @ A2 @ A ) )
= ( multis1648347201088684100mset_a @ A ) ) )
& ( ~ ( member_a @ A2 @ ( set_mset_a @ A ) )
=> ( ( multis1648347201088684100mset_a @ ( add_mset_a @ A2 @ A ) )
= ( add_mset_a @ A2 @ ( multis1648347201088684100mset_a @ A ) ) ) ) ) ).
% remdups_mset_singleton_sum
thf(fact_70_set__sorted__list__of__multiset,axiom,
! [M2: multiset_nat] :
( ( set_nat2 @ ( linord3047872887403683810et_nat @ M2 ) )
= ( set_mset_nat @ M2 ) ) ).
% set_sorted_list_of_multiset
thf(fact_71_insert__DiffM,axiom,
! [X2: list_a,M2: multiset_list_a] :
( ( member_list_a @ X2 @ ( set_mset_list_a @ M2 ) )
=> ( ( add_mset_list_a @ X2 @ ( minus_7431248565939055793list_a @ M2 @ ( add_mset_list_a @ X2 @ zero_z4454100511807792257list_a ) ) )
= M2 ) ) ).
% insert_DiffM
thf(fact_72_insert__DiffM,axiom,
! [X2: multiset_a,M2: multiset_multiset_a] :
( ( member_multiset_a @ X2 @ ( set_mset_multiset_a @ M2 ) )
=> ( ( add_mset_multiset_a @ X2 @ ( minus_3395427628221709681iset_a @ M2 @ ( add_mset_multiset_a @ X2 @ zero_z7799948378220188993iset_a ) ) )
= M2 ) ) ).
% insert_DiffM
thf(fact_73_insert__DiffM,axiom,
! [X2: nat,M2: multiset_nat] :
( ( member_nat @ X2 @ ( set_mset_nat @ M2 ) )
=> ( ( add_mset_nat @ X2 @ ( minus_8522176038001411705et_nat @ M2 @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) ) )
= M2 ) ) ).
% insert_DiffM
thf(fact_74_insert__DiffM,axiom,
! [X2: a,M2: multiset_a] :
( ( member_a @ X2 @ ( set_mset_a @ M2 ) )
=> ( ( add_mset_a @ X2 @ ( minus_3765977307040488491iset_a @ M2 @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) ) )
= M2 ) ) ).
% insert_DiffM
thf(fact_75_diff__union__swap2,axiom,
! [Y: list_a,M2: multiset_list_a,X2: list_a] :
( ( member_list_a @ Y @ ( set_mset_list_a @ M2 ) )
=> ( ( minus_7431248565939055793list_a @ ( add_mset_list_a @ X2 @ M2 ) @ ( add_mset_list_a @ Y @ zero_z4454100511807792257list_a ) )
= ( add_mset_list_a @ X2 @ ( minus_7431248565939055793list_a @ M2 @ ( add_mset_list_a @ Y @ zero_z4454100511807792257list_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_76_diff__union__swap2,axiom,
! [Y: multiset_a,M2: multiset_multiset_a,X2: multiset_a] :
( ( member_multiset_a @ Y @ ( set_mset_multiset_a @ M2 ) )
=> ( ( minus_3395427628221709681iset_a @ ( add_mset_multiset_a @ X2 @ M2 ) @ ( add_mset_multiset_a @ Y @ zero_z7799948378220188993iset_a ) )
= ( add_mset_multiset_a @ X2 @ ( minus_3395427628221709681iset_a @ M2 @ ( add_mset_multiset_a @ Y @ zero_z7799948378220188993iset_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_77_diff__union__swap2,axiom,
! [Y: nat,M2: multiset_nat,X2: nat] :
( ( member_nat @ Y @ ( set_mset_nat @ M2 ) )
=> ( ( minus_8522176038001411705et_nat @ ( add_mset_nat @ X2 @ M2 ) @ ( add_mset_nat @ Y @ zero_z7348594199698428585et_nat ) )
= ( add_mset_nat @ X2 @ ( minus_8522176038001411705et_nat @ M2 @ ( add_mset_nat @ Y @ zero_z7348594199698428585et_nat ) ) ) ) ) ).
% diff_union_swap2
thf(fact_78_diff__union__swap2,axiom,
! [Y: a,M2: multiset_a,X2: a] :
( ( member_a @ Y @ ( set_mset_a @ M2 ) )
=> ( ( minus_3765977307040488491iset_a @ ( add_mset_a @ X2 @ M2 ) @ ( add_mset_a @ Y @ zero_zero_multiset_a ) )
= ( add_mset_a @ X2 @ ( minus_3765977307040488491iset_a @ M2 @ ( add_mset_a @ Y @ zero_zero_multiset_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_79_diff__single__trivial,axiom,
! [X2: list_a,M2: multiset_list_a] :
( ~ ( member_list_a @ X2 @ ( set_mset_list_a @ M2 ) )
=> ( ( minus_7431248565939055793list_a @ M2 @ ( add_mset_list_a @ X2 @ zero_z4454100511807792257list_a ) )
= M2 ) ) ).
% diff_single_trivial
thf(fact_80_diff__single__trivial,axiom,
! [X2: multiset_a,M2: multiset_multiset_a] :
( ~ ( member_multiset_a @ X2 @ ( set_mset_multiset_a @ M2 ) )
=> ( ( minus_3395427628221709681iset_a @ M2 @ ( add_mset_multiset_a @ X2 @ zero_z7799948378220188993iset_a ) )
= M2 ) ) ).
% diff_single_trivial
thf(fact_81_diff__single__trivial,axiom,
! [X2: nat,M2: multiset_nat] :
( ~ ( member_nat @ X2 @ ( set_mset_nat @ M2 ) )
=> ( ( minus_8522176038001411705et_nat @ M2 @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) )
= M2 ) ) ).
% diff_single_trivial
thf(fact_82_diff__single__trivial,axiom,
! [X2: a,M2: multiset_a] :
( ~ ( member_a @ X2 @ ( set_mset_a @ M2 ) )
=> ( ( minus_3765977307040488491iset_a @ M2 @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) )
= M2 ) ) ).
% diff_single_trivial
thf(fact_83__092_060open_062remove1__mset_Ax_A_Imset_Axs_J_A_061_Amset_Ays_A_L_Amset_Azs_092_060close_062,axiom,
( ( minus_3765977307040488491iset_a @ ( mset_a @ xs ) @ ( add_mset_a @ x @ zero_zero_multiset_a ) )
= ( plus_plus_multiset_a @ ( mset_a @ ys ) @ ( mset_a @ zs ) ) ) ).
% \<open>remove1_mset x (mset xs) = mset ys + mset zs\<close>
thf(fact_84_assms_I3_J,axiom,
( ( nth_a @ xs @ j1 )
= x ) ).
% assms(3)
thf(fact_85_xseq,axiom,
( xs
= ( append_a @ ys @ ( cons_a @ x @ zs ) ) ) ).
% xseq
thf(fact_86_yseq,axiom,
( ys
= ( take_a @ j1 @ xs ) ) ).
% yseq
thf(fact_87_mset_Osimps_I2_J,axiom,
! [A2: list_a,X2: list_list_a] :
( ( mset_list_a @ ( cons_list_a @ A2 @ X2 ) )
= ( add_mset_list_a @ A2 @ ( mset_list_a @ X2 ) ) ) ).
% mset.simps(2)
thf(fact_88_mset_Osimps_I2_J,axiom,
! [A2: nat,X2: list_nat] :
( ( mset_nat @ ( cons_nat @ A2 @ X2 ) )
= ( add_mset_nat @ A2 @ ( mset_nat @ X2 ) ) ) ).
% mset.simps(2)
thf(fact_89_mset_Osimps_I2_J,axiom,
! [A2: a,X2: list_a] :
( ( mset_a @ ( cons_a @ A2 @ X2 ) )
= ( add_mset_a @ A2 @ ( mset_a @ X2 ) ) ) ).
% mset.simps(2)
thf(fact_90_mem__Collect__eq,axiom,
! [A2: list_a,P: list_a > $o] :
( ( member_list_a @ A2 @ ( collect_list_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_91_mem__Collect__eq,axiom,
! [A2: multiset_a,P: multiset_a > $o] :
( ( member_multiset_a @ A2 @ ( collect_multiset_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_92_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_93_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_94_Collect__mem__eq,axiom,
! [A: set_list_a] :
( ( collect_list_a
@ ^ [X3: list_a] : ( member_list_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_95_Collect__mem__eq,axiom,
! [A: set_multiset_a] :
( ( collect_multiset_a
@ ^ [X3: multiset_a] : ( member_multiset_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_96_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_97_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_98_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_99_empty__neutral_I2_J,axiom,
! [X2: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ X2 @ zero_z7348594199698428585et_nat )
= X2 ) ).
% empty_neutral(2)
thf(fact_100_empty__neutral_I2_J,axiom,
! [X2: multiset_a] :
( ( plus_plus_multiset_a @ X2 @ zero_zero_multiset_a )
= X2 ) ).
% empty_neutral(2)
thf(fact_101_empty__neutral_I1_J,axiom,
! [X2: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ X2 )
= X2 ) ).
% empty_neutral(1)
thf(fact_102_empty__neutral_I1_J,axiom,
! [X2: multiset_a] :
( ( plus_plus_multiset_a @ zero_zero_multiset_a @ X2 )
= X2 ) ).
% empty_neutral(1)
thf(fact_103_Multiset_Odiff__add,axiom,
! [M2: multiset_nat,N: multiset_nat,Q: multiset_nat] :
( ( minus_8522176038001411705et_nat @ M2 @ ( plus_p6334493942879108393et_nat @ N @ Q ) )
= ( minus_8522176038001411705et_nat @ ( minus_8522176038001411705et_nat @ M2 @ N ) @ Q ) ) ).
% Multiset.diff_add
thf(fact_104_Multiset_Odiff__add,axiom,
! [M2: multiset_a,N: multiset_a,Q: multiset_a] :
( ( minus_3765977307040488491iset_a @ M2 @ ( plus_plus_multiset_a @ N @ Q ) )
= ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ M2 @ N ) @ Q ) ) ).
% Multiset.diff_add
thf(fact_105_diff__empty,axiom,
! [M2: multiset_nat] :
( ( ( minus_8522176038001411705et_nat @ M2 @ zero_z7348594199698428585et_nat )
= M2 )
& ( ( minus_8522176038001411705et_nat @ zero_z7348594199698428585et_nat @ M2 )
= zero_z7348594199698428585et_nat ) ) ).
% diff_empty
thf(fact_106_diff__empty,axiom,
! [M2: multiset_a] :
( ( ( minus_3765977307040488491iset_a @ M2 @ zero_zero_multiset_a )
= M2 )
& ( ( minus_3765977307040488491iset_a @ zero_zero_multiset_a @ M2 )
= zero_zero_multiset_a ) ) ).
% diff_empty
thf(fact_107_Multiset_Odiff__cancel,axiom,
! [A: multiset_nat] :
( ( minus_8522176038001411705et_nat @ A @ A )
= zero_z7348594199698428585et_nat ) ).
% Multiset.diff_cancel
thf(fact_108_Multiset_Odiff__cancel,axiom,
! [A: multiset_a] :
( ( minus_3765977307040488491iset_a @ A @ A )
= zero_zero_multiset_a ) ).
% Multiset.diff_cancel
thf(fact_109_union__assoc,axiom,
! [M2: multiset_nat,N: multiset_nat,K: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ ( plus_p6334493942879108393et_nat @ M2 @ N ) @ K )
= ( plus_p6334493942879108393et_nat @ M2 @ ( plus_p6334493942879108393et_nat @ N @ K ) ) ) ).
% union_assoc
thf(fact_110_union__assoc,axiom,
! [M2: multiset_a,N: multiset_a,K: multiset_a] :
( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ M2 @ N ) @ K )
= ( plus_plus_multiset_a @ M2 @ ( plus_plus_multiset_a @ N @ K ) ) ) ).
% union_assoc
thf(fact_111_union__lcomm,axiom,
! [M2: multiset_nat,N: multiset_nat,K: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ M2 @ ( plus_p6334493942879108393et_nat @ N @ K ) )
= ( plus_p6334493942879108393et_nat @ N @ ( plus_p6334493942879108393et_nat @ M2 @ K ) ) ) ).
% union_lcomm
thf(fact_112_union__lcomm,axiom,
! [M2: multiset_a,N: multiset_a,K: multiset_a] :
( ( plus_plus_multiset_a @ M2 @ ( plus_plus_multiset_a @ N @ K ) )
= ( plus_plus_multiset_a @ N @ ( plus_plus_multiset_a @ M2 @ K ) ) ) ).
% union_lcomm
thf(fact_113_insert__DiffM2,axiom,
! [X2: list_a,M2: multiset_list_a] :
( ( member_list_a @ X2 @ ( set_mset_list_a @ M2 ) )
=> ( ( plus_p690419498615200257list_a @ ( minus_7431248565939055793list_a @ M2 @ ( add_mset_list_a @ X2 @ zero_z4454100511807792257list_a ) ) @ ( add_mset_list_a @ X2 @ zero_z4454100511807792257list_a ) )
= M2 ) ) ).
% insert_DiffM2
thf(fact_114_insert__DiffM2,axiom,
! [X2: multiset_a,M2: multiset_multiset_a] :
( ( member_multiset_a @ X2 @ ( set_mset_multiset_a @ M2 ) )
=> ( ( plus_p6738641960240532161iset_a @ ( minus_3395427628221709681iset_a @ M2 @ ( add_mset_multiset_a @ X2 @ zero_z7799948378220188993iset_a ) ) @ ( add_mset_multiset_a @ X2 @ zero_z7799948378220188993iset_a ) )
= M2 ) ) ).
% insert_DiffM2
thf(fact_115_insert__DiffM2,axiom,
! [X2: nat,M2: multiset_nat] :
( ( member_nat @ X2 @ ( set_mset_nat @ M2 ) )
=> ( ( plus_p6334493942879108393et_nat @ ( minus_8522176038001411705et_nat @ M2 @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) ) @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) )
= M2 ) ) ).
% insert_DiffM2
thf(fact_116_insert__DiffM2,axiom,
! [X2: a,M2: multiset_a] :
( ( member_a @ X2 @ ( set_mset_a @ M2 ) )
=> ( ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ M2 @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) ) @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) )
= M2 ) ) ).
% insert_DiffM2
thf(fact_117_union__commute,axiom,
( plus_p6334493942879108393et_nat
= ( ^ [M3: multiset_nat,N2: multiset_nat] : ( plus_p6334493942879108393et_nat @ N2 @ M3 ) ) ) ).
% union_commute
thf(fact_118_union__commute,axiom,
( plus_plus_multiset_a
= ( ^ [M3: multiset_a,N2: multiset_a] : ( plus_plus_multiset_a @ N2 @ M3 ) ) ) ).
% union_commute
thf(fact_119_add__eq__conv__ex,axiom,
! [A2: nat,M2: multiset_nat,B2: nat,N: multiset_nat] :
( ( ( add_mset_nat @ A2 @ M2 )
= ( add_mset_nat @ B2 @ N ) )
= ( ( ( M2 = N )
& ( A2 = B2 ) )
| ? [K2: multiset_nat] :
( ( M2
= ( add_mset_nat @ B2 @ K2 ) )
& ( N
= ( add_mset_nat @ A2 @ K2 ) ) ) ) ) ).
% add_eq_conv_ex
thf(fact_120_add__eq__conv__ex,axiom,
! [A2: a,M2: multiset_a,B2: a,N: multiset_a] :
( ( ( add_mset_a @ A2 @ M2 )
= ( add_mset_a @ B2 @ N ) )
= ( ( ( M2 = N )
& ( A2 = B2 ) )
| ? [K2: multiset_a] :
( ( M2
= ( add_mset_a @ B2 @ K2 ) )
& ( N
= ( add_mset_a @ A2 @ K2 ) ) ) ) ) ).
% add_eq_conv_ex
thf(fact_121_multiset__cases,axiom,
! [M2: multiset_nat] :
( ( M2 != zero_z7348594199698428585et_nat )
=> ~ ! [X4: nat,N3: multiset_nat] :
( M2
!= ( add_mset_nat @ X4 @ N3 ) ) ) ).
% multiset_cases
thf(fact_122_multiset__cases,axiom,
! [M2: multiset_a] :
( ( M2 != zero_zero_multiset_a )
=> ~ ! [X4: a,N3: multiset_a] :
( M2
!= ( add_mset_a @ X4 @ N3 ) ) ) ).
% multiset_cases
thf(fact_123_diff__union__swap,axiom,
! [A2: nat,B2: nat,M2: multiset_nat] :
( ( A2 != B2 )
=> ( ( add_mset_nat @ B2 @ ( minus_8522176038001411705et_nat @ M2 @ ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) ) )
= ( minus_8522176038001411705et_nat @ ( add_mset_nat @ B2 @ M2 ) @ ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) ) ) ) ).
% diff_union_swap
thf(fact_124_diff__union__swap,axiom,
! [A2: a,B2: a,M2: multiset_a] :
( ( A2 != B2 )
=> ( ( add_mset_a @ B2 @ ( minus_3765977307040488491iset_a @ M2 @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) )
= ( minus_3765977307040488491iset_a @ ( add_mset_a @ B2 @ M2 ) @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) ) ) ).
% diff_union_swap
thf(fact_125_multiset__induct,axiom,
! [P: multiset_nat > $o,M2: multiset_nat] :
( ( P @ zero_z7348594199698428585et_nat )
=> ( ! [X4: nat,M4: multiset_nat] :
( ( P @ M4 )
=> ( P @ ( add_mset_nat @ X4 @ M4 ) ) )
=> ( P @ M2 ) ) ) ).
% multiset_induct
thf(fact_126_multiset__induct,axiom,
! [P: multiset_a > $o,M2: multiset_a] :
( ( P @ zero_zero_multiset_a )
=> ( ! [X4: a,M4: multiset_a] :
( ( P @ M4 )
=> ( P @ ( add_mset_a @ X4 @ M4 ) ) )
=> ( P @ M2 ) ) ) ).
% multiset_induct
thf(fact_127_single__is__union,axiom,
! [A2: nat,M2: multiset_nat,N: multiset_nat] :
( ( ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat )
= ( plus_p6334493942879108393et_nat @ M2 @ N ) )
= ( ( ( ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat )
= M2 )
& ( N = zero_z7348594199698428585et_nat ) )
| ( ( M2 = zero_z7348594199698428585et_nat )
& ( ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat )
= N ) ) ) ) ).
% single_is_union
thf(fact_128_single__is__union,axiom,
! [A2: a,M2: multiset_a,N: multiset_a] :
( ( ( add_mset_a @ A2 @ zero_zero_multiset_a )
= ( plus_plus_multiset_a @ M2 @ N ) )
= ( ( ( ( add_mset_a @ A2 @ zero_zero_multiset_a )
= M2 )
& ( N = zero_zero_multiset_a ) )
| ( ( M2 = zero_zero_multiset_a )
& ( ( add_mset_a @ A2 @ zero_zero_multiset_a )
= N ) ) ) ) ).
% single_is_union
thf(fact_129_union__is__single,axiom,
! [M2: multiset_nat,N: multiset_nat,A2: nat] :
( ( ( plus_p6334493942879108393et_nat @ M2 @ N )
= ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) )
= ( ( ( M2
= ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) )
& ( N = zero_z7348594199698428585et_nat ) )
| ( ( M2 = zero_z7348594199698428585et_nat )
& ( N
= ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) ) ) ) ) ).
% union_is_single
thf(fact_130_union__is__single,axiom,
! [M2: multiset_a,N: multiset_a,A2: a] :
( ( ( plus_plus_multiset_a @ M2 @ N )
= ( add_mset_a @ A2 @ zero_zero_multiset_a ) )
= ( ( ( M2
= ( add_mset_a @ A2 @ zero_zero_multiset_a ) )
& ( N = zero_zero_multiset_a ) )
| ( ( M2 = zero_zero_multiset_a )
& ( N
= ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) ) ) ) ).
% union_is_single
thf(fact_131_add__eq__conv__diff,axiom,
! [A2: nat,M2: multiset_nat,B2: nat,N: multiset_nat] :
( ( ( add_mset_nat @ A2 @ M2 )
= ( add_mset_nat @ B2 @ N ) )
= ( ( ( M2 = N )
& ( A2 = B2 ) )
| ( ( M2
= ( add_mset_nat @ B2 @ ( minus_8522176038001411705et_nat @ N @ ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) ) ) )
& ( N
= ( add_mset_nat @ A2 @ ( minus_8522176038001411705et_nat @ M2 @ ( add_mset_nat @ B2 @ zero_z7348594199698428585et_nat ) ) ) ) ) ) ) ).
% add_eq_conv_diff
thf(fact_132_add__eq__conv__diff,axiom,
! [A2: a,M2: multiset_a,B2: a,N: multiset_a] :
( ( ( add_mset_a @ A2 @ M2 )
= ( add_mset_a @ B2 @ N ) )
= ( ( ( M2 = N )
& ( A2 = B2 ) )
| ( ( M2
= ( add_mset_a @ B2 @ ( minus_3765977307040488491iset_a @ N @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) ) )
& ( N
= ( add_mset_a @ A2 @ ( minus_3765977307040488491iset_a @ M2 @ ( add_mset_a @ B2 @ zero_zero_multiset_a ) ) ) ) ) ) ) ).
% add_eq_conv_diff
thf(fact_133_add__mset__commute,axiom,
! [X2: nat,Y: nat,M2: multiset_nat] :
( ( add_mset_nat @ X2 @ ( add_mset_nat @ Y @ M2 ) )
= ( add_mset_nat @ Y @ ( add_mset_nat @ X2 @ M2 ) ) ) ).
% add_mset_commute
thf(fact_134_add__mset__commute,axiom,
! [X2: a,Y: a,M2: multiset_a] :
( ( add_mset_a @ X2 @ ( add_mset_a @ Y @ M2 ) )
= ( add_mset_a @ Y @ ( add_mset_a @ X2 @ M2 ) ) ) ).
% add_mset_commute
thf(fact_135_multiset__induct2,axiom,
! [P: multiset_a > multiset_nat > $o,M2: multiset_a,N: multiset_nat] :
( ( P @ zero_zero_multiset_a @ zero_z7348594199698428585et_nat )
=> ( ! [A3: a,M4: multiset_a,N3: multiset_nat] :
( ( P @ M4 @ N3 )
=> ( P @ ( add_mset_a @ A3 @ M4 ) @ N3 ) )
=> ( ! [A3: nat,M4: multiset_a,N3: multiset_nat] :
( ( P @ M4 @ N3 )
=> ( P @ M4 @ ( add_mset_nat @ A3 @ N3 ) ) )
=> ( P @ M2 @ N ) ) ) ) ).
% multiset_induct2
thf(fact_136_multiset__induct2,axiom,
! [P: multiset_nat > multiset_a > $o,M2: multiset_nat,N: multiset_a] :
( ( P @ zero_z7348594199698428585et_nat @ zero_zero_multiset_a )
=> ( ! [A3: nat,M4: multiset_nat,N3: multiset_a] :
( ( P @ M4 @ N3 )
=> ( P @ ( add_mset_nat @ A3 @ M4 ) @ N3 ) )
=> ( ! [A3: a,M4: multiset_nat,N3: multiset_a] :
( ( P @ M4 @ N3 )
=> ( P @ M4 @ ( add_mset_a @ A3 @ N3 ) ) )
=> ( P @ M2 @ N ) ) ) ) ).
% multiset_induct2
thf(fact_137_multiset__induct2,axiom,
! [P: multiset_nat > multiset_nat > $o,M2: multiset_nat,N: multiset_nat] :
( ( P @ zero_z7348594199698428585et_nat @ zero_z7348594199698428585et_nat )
=> ( ! [A3: nat,M4: multiset_nat,N3: multiset_nat] :
( ( P @ M4 @ N3 )
=> ( P @ ( add_mset_nat @ A3 @ M4 ) @ N3 ) )
=> ( ! [A3: nat,M4: multiset_nat,N3: multiset_nat] :
( ( P @ M4 @ N3 )
=> ( P @ M4 @ ( add_mset_nat @ A3 @ N3 ) ) )
=> ( P @ M2 @ N ) ) ) ) ).
% multiset_induct2
thf(fact_138_multiset__induct2,axiom,
! [P: multiset_a > multiset_a > $o,M2: multiset_a,N: multiset_a] :
( ( P @ zero_zero_multiset_a @ zero_zero_multiset_a )
=> ( ! [A3: a,M4: multiset_a,N3: multiset_a] :
( ( P @ M4 @ N3 )
=> ( P @ ( add_mset_a @ A3 @ M4 ) @ N3 ) )
=> ( ! [A3: a,M4: multiset_a,N3: multiset_a] :
( ( P @ M4 @ N3 )
=> ( P @ M4 @ ( add_mset_a @ A3 @ N3 ) ) )
=> ( P @ M2 @ N ) ) ) ) ).
% multiset_induct2
thf(fact_139_multi__member__last,axiom,
! [X2: list_a] : ( member_list_a @ X2 @ ( set_mset_list_a @ ( add_mset_list_a @ X2 @ zero_z4454100511807792257list_a ) ) ) ).
% multi_member_last
thf(fact_140_multi__member__last,axiom,
! [X2: multiset_a] : ( member_multiset_a @ X2 @ ( set_mset_multiset_a @ ( add_mset_multiset_a @ X2 @ zero_z7799948378220188993iset_a ) ) ) ).
% multi_member_last
thf(fact_141_multi__member__last,axiom,
! [X2: nat] : ( member_nat @ X2 @ ( set_mset_nat @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) ) ) ).
% multi_member_last
thf(fact_142_multi__member__last,axiom,
! [X2: a] : ( member_a @ X2 @ ( set_mset_a @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) ) ) ).
% multi_member_last
thf(fact_143_multi__member__skip,axiom,
! [X2: list_a,XS: multiset_list_a,Y: list_a] :
( ( member_list_a @ X2 @ ( set_mset_list_a @ XS ) )
=> ( member_list_a @ X2 @ ( set_mset_list_a @ ( plus_p690419498615200257list_a @ ( add_mset_list_a @ Y @ zero_z4454100511807792257list_a ) @ XS ) ) ) ) ).
% multi_member_skip
thf(fact_144_multi__member__skip,axiom,
! [X2: multiset_a,XS: multiset_multiset_a,Y: multiset_a] :
( ( member_multiset_a @ X2 @ ( set_mset_multiset_a @ XS ) )
=> ( member_multiset_a @ X2 @ ( set_mset_multiset_a @ ( plus_p6738641960240532161iset_a @ ( add_mset_multiset_a @ Y @ zero_z7799948378220188993iset_a ) @ XS ) ) ) ) ).
% multi_member_skip
thf(fact_145_multi__member__skip,axiom,
! [X2: nat,XS: multiset_nat,Y: nat] :
( ( member_nat @ X2 @ ( set_mset_nat @ XS ) )
=> ( member_nat @ X2 @ ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ ( add_mset_nat @ Y @ zero_z7348594199698428585et_nat ) @ XS ) ) ) ) ).
% multi_member_skip
thf(fact_146_multi__member__skip,axiom,
! [X2: a,XS: multiset_a,Y: a] :
( ( member_a @ X2 @ ( set_mset_a @ XS ) )
=> ( member_a @ X2 @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ Y @ zero_zero_multiset_a ) @ XS ) ) ) ) ).
% multi_member_skip
thf(fact_147_multi__member__this,axiom,
! [X2: list_a,XS: multiset_list_a] : ( member_list_a @ X2 @ ( set_mset_list_a @ ( plus_p690419498615200257list_a @ ( add_mset_list_a @ X2 @ zero_z4454100511807792257list_a ) @ XS ) ) ) ).
% multi_member_this
thf(fact_148_multi__member__this,axiom,
! [X2: multiset_a,XS: multiset_multiset_a] : ( member_multiset_a @ X2 @ ( set_mset_multiset_a @ ( plus_p6738641960240532161iset_a @ ( add_mset_multiset_a @ X2 @ zero_z7799948378220188993iset_a ) @ XS ) ) ) ).
% multi_member_this
thf(fact_149_multi__member__this,axiom,
! [X2: nat,XS: multiset_nat] : ( member_nat @ X2 @ ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) @ XS ) ) ) ).
% multi_member_this
thf(fact_150_multi__member__this,axiom,
! [X2: a,XS: multiset_a] : ( member_a @ X2 @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) @ XS ) ) ) ).
% multi_member_this
thf(fact_151_union__left__cancel,axiom,
! [K: multiset_nat,M2: multiset_nat,N: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ K @ M2 )
= ( plus_p6334493942879108393et_nat @ K @ N ) )
= ( M2 = N ) ) ).
% union_left_cancel
thf(fact_152_union__left__cancel,axiom,
! [K: multiset_a,M2: multiset_a,N: multiset_a] :
( ( ( plus_plus_multiset_a @ K @ M2 )
= ( plus_plus_multiset_a @ K @ N ) )
= ( M2 = N ) ) ).
% union_left_cancel
thf(fact_153_Multiset_Odiff__right__commute,axiom,
! [M2: multiset_nat,N: multiset_nat,Q: multiset_nat] :
( ( minus_8522176038001411705et_nat @ ( minus_8522176038001411705et_nat @ M2 @ N ) @ Q )
= ( minus_8522176038001411705et_nat @ ( minus_8522176038001411705et_nat @ M2 @ Q ) @ N ) ) ).
% Multiset.diff_right_commute
thf(fact_154_Multiset_Odiff__right__commute,axiom,
! [M2: multiset_a,N: multiset_a,Q: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ M2 @ N ) @ Q )
= ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ M2 @ Q ) @ N ) ) ).
% Multiset.diff_right_commute
thf(fact_155_diff__union__cancelL,axiom,
! [N: multiset_nat,M2: multiset_nat] :
( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ N @ M2 ) @ N )
= M2 ) ).
% diff_union_cancelL
thf(fact_156_diff__union__cancelL,axiom,
! [N: multiset_a,M2: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ N @ M2 ) @ N )
= M2 ) ).
% diff_union_cancelL
thf(fact_157_diff__union__cancelR,axiom,
! [M2: multiset_nat,N: multiset_nat] :
( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ M2 @ N ) @ N )
= M2 ) ).
% diff_union_cancelR
thf(fact_158_diff__union__cancelR,axiom,
! [M2: multiset_a,N: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ M2 @ N ) @ N )
= M2 ) ).
% diff_union_cancelR
thf(fact_159_empty__not__add__mset,axiom,
! [A2: nat,A: multiset_nat] :
( zero_z7348594199698428585et_nat
!= ( add_mset_nat @ A2 @ A ) ) ).
% empty_not_add_mset
thf(fact_160_empty__not__add__mset,axiom,
! [A2: a,A: multiset_a] :
( zero_zero_multiset_a
!= ( add_mset_a @ A2 @ A ) ) ).
% empty_not_add_mset
thf(fact_161_union__right__cancel,axiom,
! [M2: multiset_nat,K: multiset_nat,N: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ M2 @ K )
= ( plus_p6334493942879108393et_nat @ N @ K ) )
= ( M2 = N ) ) ).
% union_right_cancel
thf(fact_162_union__right__cancel,axiom,
! [M2: multiset_a,K: multiset_a,N: multiset_a] :
( ( ( plus_plus_multiset_a @ M2 @ K )
= ( plus_plus_multiset_a @ N @ K ) )
= ( M2 = N ) ) ).
% union_right_cancel
thf(fact_163_add__mset__add__single,axiom,
( add_mset_nat
= ( ^ [A4: nat,A5: multiset_nat] : ( plus_p6334493942879108393et_nat @ A5 @ ( add_mset_nat @ A4 @ zero_z7348594199698428585et_nat ) ) ) ) ).
% add_mset_add_single
thf(fact_164_add__mset__add__single,axiom,
( add_mset_a
= ( ^ [A4: a,A5: multiset_a] : ( plus_plus_multiset_a @ A5 @ ( add_mset_a @ A4 @ zero_zero_multiset_a ) ) ) ) ).
% add_mset_add_single
thf(fact_165_diff__single__eq__union,axiom,
! [X2: list_a,M2: multiset_list_a,N: multiset_list_a] :
( ( member_list_a @ X2 @ ( set_mset_list_a @ M2 ) )
=> ( ( ( minus_7431248565939055793list_a @ M2 @ ( add_mset_list_a @ X2 @ zero_z4454100511807792257list_a ) )
= N )
= ( M2
= ( add_mset_list_a @ X2 @ N ) ) ) ) ).
% diff_single_eq_union
thf(fact_166_diff__single__eq__union,axiom,
! [X2: multiset_a,M2: multiset_multiset_a,N: multiset_multiset_a] :
( ( member_multiset_a @ X2 @ ( set_mset_multiset_a @ M2 ) )
=> ( ( ( minus_3395427628221709681iset_a @ M2 @ ( add_mset_multiset_a @ X2 @ zero_z7799948378220188993iset_a ) )
= N )
= ( M2
= ( add_mset_multiset_a @ X2 @ N ) ) ) ) ).
% diff_single_eq_union
thf(fact_167_diff__single__eq__union,axiom,
! [X2: nat,M2: multiset_nat,N: multiset_nat] :
( ( member_nat @ X2 @ ( set_mset_nat @ M2 ) )
=> ( ( ( minus_8522176038001411705et_nat @ M2 @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) )
= N )
= ( M2
= ( add_mset_nat @ X2 @ N ) ) ) ) ).
% diff_single_eq_union
thf(fact_168_diff__single__eq__union,axiom,
! [X2: a,M2: multiset_a,N: multiset_a] :
( ( member_a @ X2 @ ( set_mset_a @ M2 ) )
=> ( ( ( minus_3765977307040488491iset_a @ M2 @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) )
= N )
= ( M2
= ( add_mset_a @ X2 @ N ) ) ) ) ).
% diff_single_eq_union
thf(fact_169_multi__nonempty__split,axiom,
! [M2: multiset_nat] :
( ( M2 != zero_z7348594199698428585et_nat )
=> ? [A6: multiset_nat,A3: nat] :
( M2
= ( add_mset_nat @ A3 @ A6 ) ) ) ).
% multi_nonempty_split
thf(fact_170_multi__nonempty__split,axiom,
! [M2: multiset_a] :
( ( M2 != zero_zero_multiset_a )
=> ? [A6: multiset_a,A3: a] :
( M2
= ( add_mset_a @ A3 @ A6 ) ) ) ).
% multi_nonempty_split
thf(fact_171_union__single__eq__diff,axiom,
! [X2: nat,M2: multiset_nat,N: multiset_nat] :
( ( ( add_mset_nat @ X2 @ M2 )
= N )
=> ( M2
= ( minus_8522176038001411705et_nat @ N @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) ) ) ) ).
% union_single_eq_diff
thf(fact_172_union__single__eq__diff,axiom,
! [X2: a,M2: multiset_a,N: multiset_a] :
( ( ( add_mset_a @ X2 @ M2 )
= N )
=> ( M2
= ( minus_3765977307040488491iset_a @ N @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) ) ) ) ).
% union_single_eq_diff
thf(fact_173_multi__drop__mem__not__eq,axiom,
! [C: list_a,B: multiset_list_a] :
( ( member_list_a @ C @ ( set_mset_list_a @ B ) )
=> ( ( minus_7431248565939055793list_a @ B @ ( add_mset_list_a @ C @ zero_z4454100511807792257list_a ) )
!= B ) ) ).
% multi_drop_mem_not_eq
thf(fact_174_multi__drop__mem__not__eq,axiom,
! [C: multiset_a,B: multiset_multiset_a] :
( ( member_multiset_a @ C @ ( set_mset_multiset_a @ B ) )
=> ( ( minus_3395427628221709681iset_a @ B @ ( add_mset_multiset_a @ C @ zero_z7799948378220188993iset_a ) )
!= B ) ) ).
% multi_drop_mem_not_eq
thf(fact_175_multi__drop__mem__not__eq,axiom,
! [C: nat,B: multiset_nat] :
( ( member_nat @ C @ ( set_mset_nat @ B ) )
=> ( ( minus_8522176038001411705et_nat @ B @ ( add_mset_nat @ C @ zero_z7348594199698428585et_nat ) )
!= B ) ) ).
% multi_drop_mem_not_eq
thf(fact_176_multi__drop__mem__not__eq,axiom,
! [C: a,B: multiset_a] :
( ( member_a @ C @ ( set_mset_a @ B ) )
=> ( ( minus_3765977307040488491iset_a @ B @ ( add_mset_a @ C @ zero_zero_multiset_a ) )
!= B ) ) ).
% multi_drop_mem_not_eq
thf(fact_177_remove1__mset__eqE,axiom,
! [X1: multiset_list_a,L3: list_a,M2: multiset_list_a] :
( ( ( minus_7431248565939055793list_a @ X1 @ ( add_mset_list_a @ L3 @ zero_z4454100511807792257list_a ) )
= M2 )
=> ( ( ( member_list_a @ L3 @ ( set_mset_list_a @ X1 ) )
=> ( X1
!= ( plus_p690419498615200257list_a @ M2 @ ( add_mset_list_a @ L3 @ zero_z4454100511807792257list_a ) ) ) )
=> ~ ( ~ ( member_list_a @ L3 @ ( set_mset_list_a @ X1 ) )
=> ( X1 != M2 ) ) ) ) ).
% remove1_mset_eqE
thf(fact_178_remove1__mset__eqE,axiom,
! [X1: multiset_multiset_a,L3: multiset_a,M2: multiset_multiset_a] :
( ( ( minus_3395427628221709681iset_a @ X1 @ ( add_mset_multiset_a @ L3 @ zero_z7799948378220188993iset_a ) )
= M2 )
=> ( ( ( member_multiset_a @ L3 @ ( set_mset_multiset_a @ X1 ) )
=> ( X1
!= ( plus_p6738641960240532161iset_a @ M2 @ ( add_mset_multiset_a @ L3 @ zero_z7799948378220188993iset_a ) ) ) )
=> ~ ( ~ ( member_multiset_a @ L3 @ ( set_mset_multiset_a @ X1 ) )
=> ( X1 != M2 ) ) ) ) ).
% remove1_mset_eqE
thf(fact_179_remove1__mset__eqE,axiom,
! [X1: multiset_nat,L3: nat,M2: multiset_nat] :
( ( ( minus_8522176038001411705et_nat @ X1 @ ( add_mset_nat @ L3 @ zero_z7348594199698428585et_nat ) )
= M2 )
=> ( ( ( member_nat @ L3 @ ( set_mset_nat @ X1 ) )
=> ( X1
!= ( plus_p6334493942879108393et_nat @ M2 @ ( add_mset_nat @ L3 @ zero_z7348594199698428585et_nat ) ) ) )
=> ~ ( ~ ( member_nat @ L3 @ ( set_mset_nat @ X1 ) )
=> ( X1 != M2 ) ) ) ) ).
% remove1_mset_eqE
thf(fact_180_remove1__mset__eqE,axiom,
! [X1: multiset_a,L3: a,M2: multiset_a] :
( ( ( minus_3765977307040488491iset_a @ X1 @ ( add_mset_a @ L3 @ zero_zero_multiset_a ) )
= M2 )
=> ( ( ( member_a @ L3 @ ( set_mset_a @ X1 ) )
=> ( X1
!= ( plus_plus_multiset_a @ M2 @ ( add_mset_a @ L3 @ zero_zero_multiset_a ) ) ) )
=> ~ ( ~ ( member_a @ L3 @ ( set_mset_a @ X1 ) )
=> ( X1 != M2 ) ) ) ) ).
% remove1_mset_eqE
thf(fact_181_diff__union__single__conv,axiom,
! [A2: list_a,J: multiset_list_a,I: multiset_list_a] :
( ( member_list_a @ A2 @ ( set_mset_list_a @ J ) )
=> ( ( minus_7431248565939055793list_a @ ( plus_p690419498615200257list_a @ I @ J ) @ ( add_mset_list_a @ A2 @ zero_z4454100511807792257list_a ) )
= ( plus_p690419498615200257list_a @ I @ ( minus_7431248565939055793list_a @ J @ ( add_mset_list_a @ A2 @ zero_z4454100511807792257list_a ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_182_diff__union__single__conv,axiom,
! [A2: multiset_a,J: multiset_multiset_a,I: multiset_multiset_a] :
( ( member_multiset_a @ A2 @ ( set_mset_multiset_a @ J ) )
=> ( ( minus_3395427628221709681iset_a @ ( plus_p6738641960240532161iset_a @ I @ J ) @ ( add_mset_multiset_a @ A2 @ zero_z7799948378220188993iset_a ) )
= ( plus_p6738641960240532161iset_a @ I @ ( minus_3395427628221709681iset_a @ J @ ( add_mset_multiset_a @ A2 @ zero_z7799948378220188993iset_a ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_183_diff__union__single__conv,axiom,
! [A2: nat,J: multiset_nat,I: multiset_nat] :
( ( member_nat @ A2 @ ( set_mset_nat @ J ) )
=> ( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ I @ J ) @ ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) )
= ( plus_p6334493942879108393et_nat @ I @ ( minus_8522176038001411705et_nat @ J @ ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_184_diff__union__single__conv,axiom,
! [A2: a,J: multiset_a,I: multiset_a] :
( ( member_a @ A2 @ ( set_mset_a @ J ) )
=> ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ I @ J ) @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) )
= ( plus_plus_multiset_a @ I @ ( minus_3765977307040488491iset_a @ J @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_185_add__mset__diff__bothsides,axiom,
! [A2: nat,M2: multiset_nat,A: multiset_nat] :
( ( minus_8522176038001411705et_nat @ ( add_mset_nat @ A2 @ M2 ) @ ( add_mset_nat @ A2 @ A ) )
= ( minus_8522176038001411705et_nat @ M2 @ A ) ) ).
% add_mset_diff_bothsides
thf(fact_186_add__mset__diff__bothsides,axiom,
! [A2: a,M2: multiset_a,A: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( add_mset_a @ A2 @ M2 ) @ ( add_mset_a @ A2 @ A ) )
= ( minus_3765977307040488491iset_a @ M2 @ A ) ) ).
% add_mset_diff_bothsides
thf(fact_187_in__remove1__mset__neq,axiom,
! [A2: list_a,B2: list_a,C2: multiset_list_a] :
( ( A2 != B2 )
=> ( ( member_list_a @ A2 @ ( set_mset_list_a @ ( minus_7431248565939055793list_a @ C2 @ ( add_mset_list_a @ B2 @ zero_z4454100511807792257list_a ) ) ) )
= ( member_list_a @ A2 @ ( set_mset_list_a @ C2 ) ) ) ) ).
% in_remove1_mset_neq
thf(fact_188_in__remove1__mset__neq,axiom,
! [A2: multiset_a,B2: multiset_a,C2: multiset_multiset_a] :
( ( A2 != B2 )
=> ( ( member_multiset_a @ A2 @ ( set_mset_multiset_a @ ( minus_3395427628221709681iset_a @ C2 @ ( add_mset_multiset_a @ B2 @ zero_z7799948378220188993iset_a ) ) ) )
= ( member_multiset_a @ A2 @ ( set_mset_multiset_a @ C2 ) ) ) ) ).
% in_remove1_mset_neq
thf(fact_189_in__remove1__mset__neq,axiom,
! [A2: nat,B2: nat,C2: multiset_nat] :
( ( A2 != B2 )
=> ( ( member_nat @ A2 @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ C2 @ ( add_mset_nat @ B2 @ zero_z7348594199698428585et_nat ) ) ) )
= ( member_nat @ A2 @ ( set_mset_nat @ C2 ) ) ) ) ).
% in_remove1_mset_neq
thf(fact_190_in__remove1__mset__neq,axiom,
! [A2: a,B2: a,C2: multiset_a] :
( ( A2 != B2 )
=> ( ( member_a @ A2 @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ C2 @ ( add_mset_a @ B2 @ zero_zero_multiset_a ) ) ) )
= ( member_a @ A2 @ ( set_mset_a @ C2 ) ) ) ) ).
% in_remove1_mset_neq
thf(fact_191_multi__union__self__other__eq,axiom,
! [A: multiset_nat,X: multiset_nat,Y2: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ A @ X )
= ( plus_p6334493942879108393et_nat @ A @ Y2 ) )
=> ( X = Y2 ) ) ).
% multi_union_self_other_eq
thf(fact_192_multi__union__self__other__eq,axiom,
! [A: multiset_a,X: multiset_a,Y2: multiset_a] :
( ( ( plus_plus_multiset_a @ A @ X )
= ( plus_plus_multiset_a @ A @ Y2 ) )
=> ( X = Y2 ) ) ).
% multi_union_self_other_eq
thf(fact_193_add__mset__eq__add__mset,axiom,
! [A2: list_a,M2: multiset_list_a,B2: list_a,M5: multiset_list_a] :
( ( ( add_mset_list_a @ A2 @ M2 )
= ( add_mset_list_a @ B2 @ M5 ) )
= ( ( ( A2 = B2 )
& ( M2 = M5 ) )
| ( ( A2 != B2 )
& ( member_list_a @ B2 @ ( set_mset_list_a @ M2 ) )
& ( ( add_mset_list_a @ A2 @ ( minus_7431248565939055793list_a @ M2 @ ( add_mset_list_a @ B2 @ zero_z4454100511807792257list_a ) ) )
= M5 ) ) ) ) ).
% add_mset_eq_add_mset
thf(fact_194_add__mset__eq__add__mset,axiom,
! [A2: multiset_a,M2: multiset_multiset_a,B2: multiset_a,M5: multiset_multiset_a] :
( ( ( add_mset_multiset_a @ A2 @ M2 )
= ( add_mset_multiset_a @ B2 @ M5 ) )
= ( ( ( A2 = B2 )
& ( M2 = M5 ) )
| ( ( A2 != B2 )
& ( member_multiset_a @ B2 @ ( set_mset_multiset_a @ M2 ) )
& ( ( add_mset_multiset_a @ A2 @ ( minus_3395427628221709681iset_a @ M2 @ ( add_mset_multiset_a @ B2 @ zero_z7799948378220188993iset_a ) ) )
= M5 ) ) ) ) ).
% add_mset_eq_add_mset
thf(fact_195_add__mset__eq__add__mset,axiom,
! [A2: nat,M2: multiset_nat,B2: nat,M5: multiset_nat] :
( ( ( add_mset_nat @ A2 @ M2 )
= ( add_mset_nat @ B2 @ M5 ) )
= ( ( ( A2 = B2 )
& ( M2 = M5 ) )
| ( ( A2 != B2 )
& ( member_nat @ B2 @ ( set_mset_nat @ M2 ) )
& ( ( add_mset_nat @ A2 @ ( minus_8522176038001411705et_nat @ M2 @ ( add_mset_nat @ B2 @ zero_z7348594199698428585et_nat ) ) )
= M5 ) ) ) ) ).
% add_mset_eq_add_mset
thf(fact_196_add__mset__eq__add__mset,axiom,
! [A2: a,M2: multiset_a,B2: a,M5: multiset_a] :
( ( ( add_mset_a @ A2 @ M2 )
= ( add_mset_a @ B2 @ M5 ) )
= ( ( ( A2 = B2 )
& ( M2 = M5 ) )
| ( ( A2 != B2 )
& ( member_a @ B2 @ ( set_mset_a @ M2 ) )
& ( ( add_mset_a @ A2 @ ( minus_3765977307040488491iset_a @ M2 @ ( add_mset_a @ B2 @ zero_zero_multiset_a ) ) )
= M5 ) ) ) ) ).
% add_mset_eq_add_mset
thf(fact_197_add__mset__remove__trivial__If,axiom,
! [A2: list_a,N: multiset_list_a] :
( ( ( member_list_a @ A2 @ ( set_mset_list_a @ N ) )
=> ( ( add_mset_list_a @ A2 @ ( minus_7431248565939055793list_a @ N @ ( add_mset_list_a @ A2 @ zero_z4454100511807792257list_a ) ) )
= N ) )
& ( ~ ( member_list_a @ A2 @ ( set_mset_list_a @ N ) )
=> ( ( add_mset_list_a @ A2 @ ( minus_7431248565939055793list_a @ N @ ( add_mset_list_a @ A2 @ zero_z4454100511807792257list_a ) ) )
= ( add_mset_list_a @ A2 @ N ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_198_add__mset__remove__trivial__If,axiom,
! [A2: multiset_a,N: multiset_multiset_a] :
( ( ( member_multiset_a @ A2 @ ( set_mset_multiset_a @ N ) )
=> ( ( add_mset_multiset_a @ A2 @ ( minus_3395427628221709681iset_a @ N @ ( add_mset_multiset_a @ A2 @ zero_z7799948378220188993iset_a ) ) )
= N ) )
& ( ~ ( member_multiset_a @ A2 @ ( set_mset_multiset_a @ N ) )
=> ( ( add_mset_multiset_a @ A2 @ ( minus_3395427628221709681iset_a @ N @ ( add_mset_multiset_a @ A2 @ zero_z7799948378220188993iset_a ) ) )
= ( add_mset_multiset_a @ A2 @ N ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_199_add__mset__remove__trivial__If,axiom,
! [A2: nat,N: multiset_nat] :
( ( ( member_nat @ A2 @ ( set_mset_nat @ N ) )
=> ( ( add_mset_nat @ A2 @ ( minus_8522176038001411705et_nat @ N @ ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) ) )
= N ) )
& ( ~ ( member_nat @ A2 @ ( set_mset_nat @ N ) )
=> ( ( add_mset_nat @ A2 @ ( minus_8522176038001411705et_nat @ N @ ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) ) )
= ( add_mset_nat @ A2 @ N ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_200_add__mset__remove__trivial__If,axiom,
! [A2: a,N: multiset_a] :
( ( ( member_a @ A2 @ ( set_mset_a @ N ) )
=> ( ( add_mset_a @ A2 @ ( minus_3765977307040488491iset_a @ N @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) )
= N ) )
& ( ~ ( member_a @ A2 @ ( set_mset_a @ N ) )
=> ( ( add_mset_a @ A2 @ ( minus_3765977307040488491iset_a @ N @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) )
= ( add_mset_a @ A2 @ N ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_201_add__mset__remove__trivial__eq,axiom,
! [N: multiset_list_a,A2: list_a] :
( ( N
= ( add_mset_list_a @ A2 @ ( minus_7431248565939055793list_a @ N @ ( add_mset_list_a @ A2 @ zero_z4454100511807792257list_a ) ) ) )
= ( member_list_a @ A2 @ ( set_mset_list_a @ N ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_202_add__mset__remove__trivial__eq,axiom,
! [N: multiset_multiset_a,A2: multiset_a] :
( ( N
= ( add_mset_multiset_a @ A2 @ ( minus_3395427628221709681iset_a @ N @ ( add_mset_multiset_a @ A2 @ zero_z7799948378220188993iset_a ) ) ) )
= ( member_multiset_a @ A2 @ ( set_mset_multiset_a @ N ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_203_add__mset__remove__trivial__eq,axiom,
! [N: multiset_nat,A2: nat] :
( ( N
= ( add_mset_nat @ A2 @ ( minus_8522176038001411705et_nat @ N @ ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) ) ) )
= ( member_nat @ A2 @ ( set_mset_nat @ N ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_204_add__mset__remove__trivial__eq,axiom,
! [N: multiset_a,A2: a] :
( ( N
= ( add_mset_a @ A2 @ ( minus_3765977307040488491iset_a @ N @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) ) )
= ( member_a @ A2 @ ( set_mset_a @ N ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_205_multiset__add__sub__el__shuffle,axiom,
! [C: list_a,B: multiset_list_a,B2: list_a] :
( ( member_list_a @ C @ ( set_mset_list_a @ B ) )
=> ( ( B2 != C )
=> ( ( add_mset_list_a @ B2 @ ( minus_7431248565939055793list_a @ B @ ( add_mset_list_a @ C @ zero_z4454100511807792257list_a ) ) )
= ( minus_7431248565939055793list_a @ ( add_mset_list_a @ B2 @ B ) @ ( add_mset_list_a @ C @ zero_z4454100511807792257list_a ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_206_multiset__add__sub__el__shuffle,axiom,
! [C: multiset_a,B: multiset_multiset_a,B2: multiset_a] :
( ( member_multiset_a @ C @ ( set_mset_multiset_a @ B ) )
=> ( ( B2 != C )
=> ( ( add_mset_multiset_a @ B2 @ ( minus_3395427628221709681iset_a @ B @ ( add_mset_multiset_a @ C @ zero_z7799948378220188993iset_a ) ) )
= ( minus_3395427628221709681iset_a @ ( add_mset_multiset_a @ B2 @ B ) @ ( add_mset_multiset_a @ C @ zero_z7799948378220188993iset_a ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_207_multiset__add__sub__el__shuffle,axiom,
! [C: nat,B: multiset_nat,B2: nat] :
( ( member_nat @ C @ ( set_mset_nat @ B ) )
=> ( ( B2 != C )
=> ( ( add_mset_nat @ B2 @ ( minus_8522176038001411705et_nat @ B @ ( add_mset_nat @ C @ zero_z7348594199698428585et_nat ) ) )
= ( minus_8522176038001411705et_nat @ ( add_mset_nat @ B2 @ B ) @ ( add_mset_nat @ C @ zero_z7348594199698428585et_nat ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_208_multiset__add__sub__el__shuffle,axiom,
! [C: a,B: multiset_a,B2: a] :
( ( member_a @ C @ ( set_mset_a @ B ) )
=> ( ( B2 != C )
=> ( ( add_mset_a @ B2 @ ( minus_3765977307040488491iset_a @ B @ ( add_mset_a @ C @ zero_zero_multiset_a ) ) )
= ( minus_3765977307040488491iset_a @ ( add_mset_a @ B2 @ B ) @ ( add_mset_a @ C @ zero_zero_multiset_a ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_209_more__than__one__mset__mset__diff,axiom,
! [A2: list_a,M2: multiset_list_a] :
( ( member_list_a @ A2 @ ( set_mset_list_a @ ( minus_7431248565939055793list_a @ M2 @ ( add_mset_list_a @ A2 @ zero_z4454100511807792257list_a ) ) ) )
=> ( ( set_mset_list_a @ ( minus_7431248565939055793list_a @ M2 @ ( add_mset_list_a @ A2 @ zero_z4454100511807792257list_a ) ) )
= ( set_mset_list_a @ M2 ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_210_more__than__one__mset__mset__diff,axiom,
! [A2: multiset_a,M2: multiset_multiset_a] :
( ( member_multiset_a @ A2 @ ( set_mset_multiset_a @ ( minus_3395427628221709681iset_a @ M2 @ ( add_mset_multiset_a @ A2 @ zero_z7799948378220188993iset_a ) ) ) )
=> ( ( set_mset_multiset_a @ ( minus_3395427628221709681iset_a @ M2 @ ( add_mset_multiset_a @ A2 @ zero_z7799948378220188993iset_a ) ) )
= ( set_mset_multiset_a @ M2 ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_211_more__than__one__mset__mset__diff,axiom,
! [A2: nat,M2: multiset_nat] :
( ( member_nat @ A2 @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M2 @ ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) ) ) )
=> ( ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M2 @ ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) ) )
= ( set_mset_nat @ M2 ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_212_more__than__one__mset__mset__diff,axiom,
! [A2: a,M2: multiset_a] :
( ( member_a @ A2 @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M2 @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) ) )
=> ( ( set_mset_a @ ( minus_3765977307040488491iset_a @ M2 @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) )
= ( set_mset_a @ M2 ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_213_add__mset__eq__add__mset__ne,axiom,
! [A2: list_a,B2: list_a,A: multiset_list_a,B: multiset_list_a] :
( ( A2 != B2 )
=> ( ( ( add_mset_list_a @ A2 @ A )
= ( add_mset_list_a @ B2 @ B ) )
= ( ( member_list_a @ A2 @ ( set_mset_list_a @ B ) )
& ( member_list_a @ B2 @ ( set_mset_list_a @ A ) )
& ( A
= ( add_mset_list_a @ B2 @ ( minus_7431248565939055793list_a @ B @ ( add_mset_list_a @ A2 @ zero_z4454100511807792257list_a ) ) ) ) ) ) ) ).
% add_mset_eq_add_mset_ne
thf(fact_214_add__mset__eq__add__mset__ne,axiom,
! [A2: multiset_a,B2: multiset_a,A: multiset_multiset_a,B: multiset_multiset_a] :
( ( A2 != B2 )
=> ( ( ( add_mset_multiset_a @ A2 @ A )
= ( add_mset_multiset_a @ B2 @ B ) )
= ( ( member_multiset_a @ A2 @ ( set_mset_multiset_a @ B ) )
& ( member_multiset_a @ B2 @ ( set_mset_multiset_a @ A ) )
& ( A
= ( add_mset_multiset_a @ B2 @ ( minus_3395427628221709681iset_a @ B @ ( add_mset_multiset_a @ A2 @ zero_z7799948378220188993iset_a ) ) ) ) ) ) ) ).
% add_mset_eq_add_mset_ne
thf(fact_215_add__mset__eq__add__mset__ne,axiom,
! [A2: nat,B2: nat,A: multiset_nat,B: multiset_nat] :
( ( A2 != B2 )
=> ( ( ( add_mset_nat @ A2 @ A )
= ( add_mset_nat @ B2 @ B ) )
= ( ( member_nat @ A2 @ ( set_mset_nat @ B ) )
& ( member_nat @ B2 @ ( set_mset_nat @ A ) )
& ( A
= ( add_mset_nat @ B2 @ ( minus_8522176038001411705et_nat @ B @ ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) ) ) ) ) ) ) ).
% add_mset_eq_add_mset_ne
thf(fact_216_add__mset__eq__add__mset__ne,axiom,
! [A2: a,B2: a,A: multiset_a,B: multiset_a] :
( ( A2 != B2 )
=> ( ( ( add_mset_a @ A2 @ A )
= ( add_mset_a @ B2 @ B ) )
= ( ( member_a @ A2 @ ( set_mset_a @ B ) )
& ( member_a @ B2 @ ( set_mset_a @ A ) )
& ( A
= ( add_mset_a @ B2 @ ( minus_3765977307040488491iset_a @ B @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) ) ) ) ) ) ).
% add_mset_eq_add_mset_ne
thf(fact_217_remove1__mset__add__mset__If,axiom,
! [L3: nat,L2: nat,C2: multiset_nat] :
( ( ( L3 = L2 )
=> ( ( minus_8522176038001411705et_nat @ ( add_mset_nat @ L2 @ C2 ) @ ( add_mset_nat @ L3 @ zero_z7348594199698428585et_nat ) )
= C2 ) )
& ( ( L3 != L2 )
=> ( ( minus_8522176038001411705et_nat @ ( add_mset_nat @ L2 @ C2 ) @ ( add_mset_nat @ L3 @ zero_z7348594199698428585et_nat ) )
= ( plus_p6334493942879108393et_nat @ ( minus_8522176038001411705et_nat @ C2 @ ( add_mset_nat @ L3 @ zero_z7348594199698428585et_nat ) ) @ ( add_mset_nat @ L2 @ zero_z7348594199698428585et_nat ) ) ) ) ) ).
% remove1_mset_add_mset_If
thf(fact_218_remove1__mset__add__mset__If,axiom,
! [L3: a,L2: a,C2: multiset_a] :
( ( ( L3 = L2 )
=> ( ( minus_3765977307040488491iset_a @ ( add_mset_a @ L2 @ C2 ) @ ( add_mset_a @ L3 @ zero_zero_multiset_a ) )
= C2 ) )
& ( ( L3 != L2 )
=> ( ( minus_3765977307040488491iset_a @ ( add_mset_a @ L2 @ C2 ) @ ( add_mset_a @ L3 @ zero_zero_multiset_a ) )
= ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ C2 @ ( add_mset_a @ L3 @ zero_zero_multiset_a ) ) @ ( add_mset_a @ L2 @ zero_zero_multiset_a ) ) ) ) ) ).
% remove1_mset_add_mset_If
thf(fact_219_id__remove__1__mset__iff__notin,axiom,
! [M2: multiset_list_a,A2: list_a] :
( ( M2
= ( minus_7431248565939055793list_a @ M2 @ ( add_mset_list_a @ A2 @ zero_z4454100511807792257list_a ) ) )
= ( ~ ( member_list_a @ A2 @ ( set_mset_list_a @ M2 ) ) ) ) ).
% id_remove_1_mset_iff_notin
thf(fact_220_id__remove__1__mset__iff__notin,axiom,
! [M2: multiset_multiset_a,A2: multiset_a] :
( ( M2
= ( minus_3395427628221709681iset_a @ M2 @ ( add_mset_multiset_a @ A2 @ zero_z7799948378220188993iset_a ) ) )
= ( ~ ( member_multiset_a @ A2 @ ( set_mset_multiset_a @ M2 ) ) ) ) ).
% id_remove_1_mset_iff_notin
thf(fact_221_id__remove__1__mset__iff__notin,axiom,
! [M2: multiset_nat,A2: nat] :
( ( M2
= ( minus_8522176038001411705et_nat @ M2 @ ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) ) )
= ( ~ ( member_nat @ A2 @ ( set_mset_nat @ M2 ) ) ) ) ).
% id_remove_1_mset_iff_notin
thf(fact_222_id__remove__1__mset__iff__notin,axiom,
! [M2: multiset_a,A2: a] :
( ( M2
= ( minus_3765977307040488491iset_a @ M2 @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) )
= ( ~ ( member_a @ A2 @ ( set_mset_a @ M2 ) ) ) ) ).
% id_remove_1_mset_iff_notin
thf(fact_223_remove__1__mset__id__iff__notin,axiom,
! [M2: multiset_list_a,A2: list_a] :
( ( ( minus_7431248565939055793list_a @ M2 @ ( add_mset_list_a @ A2 @ zero_z4454100511807792257list_a ) )
= M2 )
= ( ~ ( member_list_a @ A2 @ ( set_mset_list_a @ M2 ) ) ) ) ).
% remove_1_mset_id_iff_notin
thf(fact_224_remove__1__mset__id__iff__notin,axiom,
! [M2: multiset_multiset_a,A2: multiset_a] :
( ( ( minus_3395427628221709681iset_a @ M2 @ ( add_mset_multiset_a @ A2 @ zero_z7799948378220188993iset_a ) )
= M2 )
= ( ~ ( member_multiset_a @ A2 @ ( set_mset_multiset_a @ M2 ) ) ) ) ).
% remove_1_mset_id_iff_notin
thf(fact_225_remove__1__mset__id__iff__notin,axiom,
! [M2: multiset_nat,A2: nat] :
( ( ( minus_8522176038001411705et_nat @ M2 @ ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) )
= M2 )
= ( ~ ( member_nat @ A2 @ ( set_mset_nat @ M2 ) ) ) ) ).
% remove_1_mset_id_iff_notin
thf(fact_226_remove__1__mset__id__iff__notin,axiom,
! [M2: multiset_a,A2: a] :
( ( ( minus_3765977307040488491iset_a @ M2 @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) )
= M2 )
= ( ~ ( member_a @ A2 @ ( set_mset_a @ M2 ) ) ) ) ).
% remove_1_mset_id_iff_notin
thf(fact_227_add__mset__remove__trivial__iff,axiom,
! [N: multiset_list_a,A2: list_a,B2: list_a] :
( ( N
= ( add_mset_list_a @ A2 @ ( minus_7431248565939055793list_a @ N @ ( add_mset_list_a @ B2 @ zero_z4454100511807792257list_a ) ) ) )
= ( ( member_list_a @ A2 @ ( set_mset_list_a @ N ) )
& ( A2 = B2 ) ) ) ).
% add_mset_remove_trivial_iff
thf(fact_228_add__mset__remove__trivial__iff,axiom,
! [N: multiset_multiset_a,A2: multiset_a,B2: multiset_a] :
( ( N
= ( add_mset_multiset_a @ A2 @ ( minus_3395427628221709681iset_a @ N @ ( add_mset_multiset_a @ B2 @ zero_z7799948378220188993iset_a ) ) ) )
= ( ( member_multiset_a @ A2 @ ( set_mset_multiset_a @ N ) )
& ( A2 = B2 ) ) ) ).
% add_mset_remove_trivial_iff
thf(fact_229_add__mset__remove__trivial__iff,axiom,
! [N: multiset_nat,A2: nat,B2: nat] :
( ( N
= ( add_mset_nat @ A2 @ ( minus_8522176038001411705et_nat @ N @ ( add_mset_nat @ B2 @ zero_z7348594199698428585et_nat ) ) ) )
= ( ( member_nat @ A2 @ ( set_mset_nat @ N ) )
& ( A2 = B2 ) ) ) ).
% add_mset_remove_trivial_iff
thf(fact_230_add__mset__remove__trivial__iff,axiom,
! [N: multiset_a,A2: a,B2: a] :
( ( N
= ( add_mset_a @ A2 @ ( minus_3765977307040488491iset_a @ N @ ( add_mset_a @ B2 @ zero_zero_multiset_a ) ) ) )
= ( ( member_a @ A2 @ ( set_mset_a @ N ) )
& ( A2 = B2 ) ) ) ).
% add_mset_remove_trivial_iff
thf(fact_231_notin__add__mset__remdups__mset,axiom,
! [A2: list_a,A: multiset_list_a] :
( ~ ( member_list_a @ A2 @ ( set_mset_list_a @ A ) )
=> ( ( add_mset_list_a @ A2 @ ( multis807522658164556234list_a @ A ) )
= ( multis807522658164556234list_a @ ( add_mset_list_a @ A2 @ A ) ) ) ) ).
% notin_add_mset_remdups_mset
thf(fact_232_notin__add__mset__remdups__mset,axiom,
! [A2: multiset_a,A: multiset_multiset_a] :
( ~ ( member_multiset_a @ A2 @ ( set_mset_multiset_a @ A ) )
=> ( ( add_mset_multiset_a @ A2 @ ( multis2302138969243003530iset_a @ A ) )
= ( multis2302138969243003530iset_a @ ( add_mset_multiset_a @ A2 @ A ) ) ) ) ).
% notin_add_mset_remdups_mset
thf(fact_233_notin__add__mset__remdups__mset,axiom,
! [A2: nat,A: multiset_nat] :
( ~ ( member_nat @ A2 @ ( set_mset_nat @ A ) )
=> ( ( add_mset_nat @ A2 @ ( multis3892936081960800522et_nat @ A ) )
= ( multis3892936081960800522et_nat @ ( add_mset_nat @ A2 @ A ) ) ) ) ).
% notin_add_mset_remdups_mset
thf(fact_234_notin__add__mset__remdups__mset,axiom,
! [A2: a,A: multiset_a] :
( ~ ( member_a @ A2 @ ( set_mset_a @ A ) )
=> ( ( add_mset_a @ A2 @ ( multis1648347201088684100mset_a @ A ) )
= ( multis1648347201088684100mset_a @ ( add_mset_a @ A2 @ A ) ) ) ) ).
% notin_add_mset_remdups_mset
thf(fact_235_trivial__add__mset__remove__iff,axiom,
! [A2: list_a,N: multiset_list_a,B2: list_a] :
( ( ( add_mset_list_a @ A2 @ ( minus_7431248565939055793list_a @ N @ ( add_mset_list_a @ B2 @ zero_z4454100511807792257list_a ) ) )
= N )
= ( ( member_list_a @ A2 @ ( set_mset_list_a @ N ) )
& ( A2 = B2 ) ) ) ).
% trivial_add_mset_remove_iff
thf(fact_236_trivial__add__mset__remove__iff,axiom,
! [A2: multiset_a,N: multiset_multiset_a,B2: multiset_a] :
( ( ( add_mset_multiset_a @ A2 @ ( minus_3395427628221709681iset_a @ N @ ( add_mset_multiset_a @ B2 @ zero_z7799948378220188993iset_a ) ) )
= N )
= ( ( member_multiset_a @ A2 @ ( set_mset_multiset_a @ N ) )
& ( A2 = B2 ) ) ) ).
% trivial_add_mset_remove_iff
thf(fact_237_trivial__add__mset__remove__iff,axiom,
! [A2: nat,N: multiset_nat,B2: nat] :
( ( ( add_mset_nat @ A2 @ ( minus_8522176038001411705et_nat @ N @ ( add_mset_nat @ B2 @ zero_z7348594199698428585et_nat ) ) )
= N )
= ( ( member_nat @ A2 @ ( set_mset_nat @ N ) )
& ( A2 = B2 ) ) ) ).
% trivial_add_mset_remove_iff
thf(fact_238_trivial__add__mset__remove__iff,axiom,
! [A2: a,N: multiset_a,B2: a] :
( ( ( add_mset_a @ A2 @ ( minus_3765977307040488491iset_a @ N @ ( add_mset_a @ B2 @ zero_zero_multiset_a ) ) )
= N )
= ( ( member_a @ A2 @ ( set_mset_a @ N ) )
& ( A2 = B2 ) ) ) ).
% trivial_add_mset_remove_iff
thf(fact_239_union__single__eq__member,axiom,
! [X2: list_a,M2: multiset_list_a,N: multiset_list_a] :
( ( ( add_mset_list_a @ X2 @ M2 )
= N )
=> ( member_list_a @ X2 @ ( set_mset_list_a @ N ) ) ) ).
% union_single_eq_member
thf(fact_240_union__single__eq__member,axiom,
! [X2: multiset_a,M2: multiset_multiset_a,N: multiset_multiset_a] :
( ( ( add_mset_multiset_a @ X2 @ M2 )
= N )
=> ( member_multiset_a @ X2 @ ( set_mset_multiset_a @ N ) ) ) ).
% union_single_eq_member
thf(fact_241_union__single__eq__member,axiom,
! [X2: nat,M2: multiset_nat,N: multiset_nat] :
( ( ( add_mset_nat @ X2 @ M2 )
= N )
=> ( member_nat @ X2 @ ( set_mset_nat @ N ) ) ) ).
% union_single_eq_member
thf(fact_242_union__single__eq__member,axiom,
! [X2: a,M2: multiset_a,N: multiset_a] :
( ( ( add_mset_a @ X2 @ M2 )
= N )
=> ( member_a @ X2 @ ( set_mset_a @ N ) ) ) ).
% union_single_eq_member
thf(fact_243_insert__noteq__member,axiom,
! [B2: list_a,B: multiset_list_a,C: list_a,C2: multiset_list_a] :
( ( ( add_mset_list_a @ B2 @ B )
= ( add_mset_list_a @ C @ C2 ) )
=> ( ( B2 != C )
=> ( member_list_a @ C @ ( set_mset_list_a @ B ) ) ) ) ).
% insert_noteq_member
thf(fact_244_insert__noteq__member,axiom,
! [B2: multiset_a,B: multiset_multiset_a,C: multiset_a,C2: multiset_multiset_a] :
( ( ( add_mset_multiset_a @ B2 @ B )
= ( add_mset_multiset_a @ C @ C2 ) )
=> ( ( B2 != C )
=> ( member_multiset_a @ C @ ( set_mset_multiset_a @ B ) ) ) ) ).
% insert_noteq_member
thf(fact_245_insert__noteq__member,axiom,
! [B2: nat,B: multiset_nat,C: nat,C2: multiset_nat] :
( ( ( add_mset_nat @ B2 @ B )
= ( add_mset_nat @ C @ C2 ) )
=> ( ( B2 != C )
=> ( member_nat @ C @ ( set_mset_nat @ B ) ) ) ) ).
% insert_noteq_member
thf(fact_246_insert__noteq__member,axiom,
! [B2: a,B: multiset_a,C: a,C2: multiset_a] :
( ( ( add_mset_a @ B2 @ B )
= ( add_mset_a @ C @ C2 ) )
=> ( ( B2 != C )
=> ( member_a @ C @ ( set_mset_a @ B ) ) ) ) ).
% insert_noteq_member
thf(fact_247_multi__member__split,axiom,
! [X2: list_a,M2: multiset_list_a] :
( ( member_list_a @ X2 @ ( set_mset_list_a @ M2 ) )
=> ? [A6: multiset_list_a] :
( M2
= ( add_mset_list_a @ X2 @ A6 ) ) ) ).
% multi_member_split
thf(fact_248_multi__member__split,axiom,
! [X2: multiset_a,M2: multiset_multiset_a] :
( ( member_multiset_a @ X2 @ ( set_mset_multiset_a @ M2 ) )
=> ? [A6: multiset_multiset_a] :
( M2
= ( add_mset_multiset_a @ X2 @ A6 ) ) ) ).
% multi_member_split
thf(fact_249_multi__member__split,axiom,
! [X2: nat,M2: multiset_nat] :
( ( member_nat @ X2 @ ( set_mset_nat @ M2 ) )
=> ? [A6: multiset_nat] :
( M2
= ( add_mset_nat @ X2 @ A6 ) ) ) ).
% multi_member_split
thf(fact_250_multi__member__split,axiom,
! [X2: a,M2: multiset_a] :
( ( member_a @ X2 @ ( set_mset_a @ M2 ) )
=> ? [A6: multiset_a] :
( M2
= ( add_mset_a @ X2 @ A6 ) ) ) ).
% multi_member_split
thf(fact_251_mset__add,axiom,
! [A2: list_a,A: multiset_list_a] :
( ( member_list_a @ A2 @ ( set_mset_list_a @ A ) )
=> ~ ! [B3: multiset_list_a] :
( A
!= ( add_mset_list_a @ A2 @ B3 ) ) ) ).
% mset_add
thf(fact_252_mset__add,axiom,
! [A2: multiset_a,A: multiset_multiset_a] :
( ( member_multiset_a @ A2 @ ( set_mset_multiset_a @ A ) )
=> ~ ! [B3: multiset_multiset_a] :
( A
!= ( add_mset_multiset_a @ A2 @ B3 ) ) ) ).
% mset_add
thf(fact_253_mset__add,axiom,
! [A2: nat,A: multiset_nat] :
( ( member_nat @ A2 @ ( set_mset_nat @ A ) )
=> ~ ! [B3: multiset_nat] :
( A
!= ( add_mset_nat @ A2 @ B3 ) ) ) ).
% mset_add
thf(fact_254_mset__add,axiom,
! [A2: a,A: multiset_a] :
( ( member_a @ A2 @ ( set_mset_a @ A ) )
=> ~ ! [B3: multiset_a] :
( A
!= ( add_mset_a @ A2 @ B3 ) ) ) ).
% mset_add
thf(fact_255_in__diffD,axiom,
! [A2: list_a,M2: multiset_list_a,N: multiset_list_a] :
( ( member_list_a @ A2 @ ( set_mset_list_a @ ( minus_7431248565939055793list_a @ M2 @ N ) ) )
=> ( member_list_a @ A2 @ ( set_mset_list_a @ M2 ) ) ) ).
% in_diffD
thf(fact_256_in__diffD,axiom,
! [A2: multiset_a,M2: multiset_multiset_a,N: multiset_multiset_a] :
( ( member_multiset_a @ A2 @ ( set_mset_multiset_a @ ( minus_3395427628221709681iset_a @ M2 @ N ) ) )
=> ( member_multiset_a @ A2 @ ( set_mset_multiset_a @ M2 ) ) ) ).
% in_diffD
thf(fact_257_in__diffD,axiom,
! [A2: nat,M2: multiset_nat,N: multiset_nat] :
( ( member_nat @ A2 @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M2 @ N ) ) )
=> ( member_nat @ A2 @ ( set_mset_nat @ M2 ) ) ) ).
% in_diffD
thf(fact_258_in__diffD,axiom,
! [A2: a,M2: multiset_a,N: multiset_a] :
( ( member_a @ A2 @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M2 @ N ) ) )
=> ( member_a @ A2 @ ( set_mset_a @ M2 ) ) ) ).
% in_diffD
thf(fact_259_union__iff,axiom,
! [A2: list_a,A: multiset_list_a,B: multiset_list_a] :
( ( member_list_a @ A2 @ ( set_mset_list_a @ ( plus_p690419498615200257list_a @ A @ B ) ) )
= ( ( member_list_a @ A2 @ ( set_mset_list_a @ A ) )
| ( member_list_a @ A2 @ ( set_mset_list_a @ B ) ) ) ) ).
% union_iff
thf(fact_260_union__iff,axiom,
! [A2: multiset_a,A: multiset_multiset_a,B: multiset_multiset_a] :
( ( member_multiset_a @ A2 @ ( set_mset_multiset_a @ ( plus_p6738641960240532161iset_a @ A @ B ) ) )
= ( ( member_multiset_a @ A2 @ ( set_mset_multiset_a @ A ) )
| ( member_multiset_a @ A2 @ ( set_mset_multiset_a @ B ) ) ) ) ).
% union_iff
thf(fact_261_union__iff,axiom,
! [A2: nat,A: multiset_nat,B: multiset_nat] :
( ( member_nat @ A2 @ ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ A @ B ) ) )
= ( ( member_nat @ A2 @ ( set_mset_nat @ A ) )
| ( member_nat @ A2 @ ( set_mset_nat @ B ) ) ) ) ).
% union_iff
thf(fact_262_union__iff,axiom,
! [A2: a,A: multiset_a,B: multiset_a] :
( ( member_a @ A2 @ ( set_mset_a @ ( plus_plus_multiset_a @ A @ B ) ) )
= ( ( member_a @ A2 @ ( set_mset_a @ A ) )
| ( member_a @ A2 @ ( set_mset_a @ B ) ) ) ) ).
% union_iff
thf(fact_263_mset__eq__setD,axiom,
! [Xs2: list_list_a,Ys: list_list_a] :
( ( ( mset_list_a @ Xs2 )
= ( mset_list_a @ Ys ) )
=> ( ( set_list_a2 @ Xs2 )
= ( set_list_a2 @ Ys ) ) ) ).
% mset_eq_setD
thf(fact_264_mset__eq__setD,axiom,
! [Xs2: list_multiset_a,Ys: list_multiset_a] :
( ( ( mset_multiset_a @ Xs2 )
= ( mset_multiset_a @ Ys ) )
=> ( ( set_multiset_a2 @ Xs2 )
= ( set_multiset_a2 @ Ys ) ) ) ).
% mset_eq_setD
thf(fact_265_mset__eq__setD,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( mset_a @ Xs2 )
= ( mset_a @ Ys ) )
=> ( ( set_a2 @ Xs2 )
= ( set_a2 @ Ys ) ) ) ).
% mset_eq_setD
thf(fact_266_multiset__nonemptyE,axiom,
! [A: multiset_list_a] :
( ( A != zero_z4454100511807792257list_a )
=> ~ ! [X4: list_a] :
~ ( member_list_a @ X4 @ ( set_mset_list_a @ A ) ) ) ).
% multiset_nonemptyE
thf(fact_267_multiset__nonemptyE,axiom,
! [A: multiset_multiset_a] :
( ( A != zero_z7799948378220188993iset_a )
=> ~ ! [X4: multiset_a] :
~ ( member_multiset_a @ X4 @ ( set_mset_multiset_a @ A ) ) ) ).
% multiset_nonemptyE
thf(fact_268_multiset__nonemptyE,axiom,
! [A: multiset_nat] :
( ( A != zero_z7348594199698428585et_nat )
=> ~ ! [X4: nat] :
~ ( member_nat @ X4 @ ( set_mset_nat @ A ) ) ) ).
% multiset_nonemptyE
thf(fact_269_multiset__nonemptyE,axiom,
! [A: multiset_a] :
( ( A != zero_zero_multiset_a )
=> ~ ! [X4: a] :
~ ( member_a @ X4 @ ( set_mset_a @ A ) ) ) ).
% multiset_nonemptyE
thf(fact_270_drop__Suc__Cons,axiom,
! [N4: nat,X2: list_a,Xs2: list_list_a] :
( ( drop_list_a @ ( suc @ N4 ) @ ( cons_list_a @ X2 @ Xs2 ) )
= ( drop_list_a @ N4 @ Xs2 ) ) ).
% drop_Suc_Cons
thf(fact_271_drop__Suc__Cons,axiom,
! [N4: nat,X2: a,Xs2: list_a] :
( ( drop_a @ ( suc @ N4 ) @ ( cons_a @ X2 @ Xs2 ) )
= ( drop_a @ N4 @ Xs2 ) ) ).
% drop_Suc_Cons
thf(fact_272_diff__add__zero,axiom,
! [A2: multiset_nat,B2: multiset_nat] :
( ( minus_8522176038001411705et_nat @ A2 @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) )
= zero_z7348594199698428585et_nat ) ).
% diff_add_zero
thf(fact_273_diff__add__zero,axiom,
! [A2: multiset_a,B2: multiset_a] :
( ( minus_3765977307040488491iset_a @ A2 @ ( plus_plus_multiset_a @ A2 @ B2 ) )
= zero_zero_multiset_a ) ).
% diff_add_zero
thf(fact_274_diff__add__zero,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_275_add__diff__cancel__left,axiom,
! [C: multiset_nat,A2: multiset_nat,B2: multiset_nat] :
( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ C @ A2 ) @ ( plus_p6334493942879108393et_nat @ C @ B2 ) )
= ( minus_8522176038001411705et_nat @ A2 @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_276_add__diff__cancel__left,axiom,
! [C: multiset_a,A2: multiset_a,B2: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ C @ A2 ) @ ( plus_plus_multiset_a @ C @ B2 ) )
= ( minus_3765977307040488491iset_a @ A2 @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_277_add__diff__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( minus_minus_nat @ A2 @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_278_add__diff__cancel__left_H,axiom,
! [A2: multiset_nat,B2: multiset_nat] :
( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) @ A2 )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_279_add__diff__cancel__left_H,axiom,
! [A2: multiset_a,B2: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A2 @ B2 ) @ A2 )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_280_add__diff__cancel__left_H,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ A2 )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_281_add__diff__cancel__right,axiom,
! [A2: multiset_nat,C: multiset_nat,B2: multiset_nat] :
( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ C ) @ ( plus_p6334493942879108393et_nat @ B2 @ C ) )
= ( minus_8522176038001411705et_nat @ A2 @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_282_add__diff__cancel__right,axiom,
! [A2: multiset_a,C: multiset_a,B2: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A2 @ C ) @ ( plus_plus_multiset_a @ B2 @ C ) )
= ( minus_3765977307040488491iset_a @ A2 @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_283_add__diff__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( minus_minus_nat @ A2 @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_284_add__diff__cancel__right_H,axiom,
! [A2: multiset_nat,B2: multiset_nat] :
( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) @ B2 )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_285_add__diff__cancel__right_H,axiom,
! [A2: multiset_a,B2: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A2 @ B2 ) @ B2 )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_286_add__diff__cancel__right_H,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_287_zero__diff,axiom,
! [A2: multiset_nat] :
( ( minus_8522176038001411705et_nat @ zero_z7348594199698428585et_nat @ A2 )
= zero_z7348594199698428585et_nat ) ).
% zero_diff
thf(fact_288_zero__diff,axiom,
! [A2: multiset_a] :
( ( minus_3765977307040488491iset_a @ zero_zero_multiset_a @ A2 )
= zero_zero_multiset_a ) ).
% zero_diff
thf(fact_289_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_290_diff__zero,axiom,
! [A2: multiset_nat] :
( ( minus_8522176038001411705et_nat @ A2 @ zero_z7348594199698428585et_nat )
= A2 ) ).
% diff_zero
thf(fact_291_diff__zero,axiom,
! [A2: multiset_a] :
( ( minus_3765977307040488491iset_a @ A2 @ zero_zero_multiset_a )
= A2 ) ).
% diff_zero
thf(fact_292_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_293_add__right__cancel,axiom,
! [B2: multiset_nat,A2: multiset_nat,C: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ B2 @ A2 )
= ( plus_p6334493942879108393et_nat @ C @ A2 ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_294_add__right__cancel,axiom,
! [B2: multiset_a,A2: multiset_a,C: multiset_a] :
( ( ( plus_plus_multiset_a @ B2 @ A2 )
= ( plus_plus_multiset_a @ C @ A2 ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_295_add__right__cancel,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_296_add__left__cancel,axiom,
! [A2: multiset_nat,B2: multiset_nat,C: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ A2 @ B2 )
= ( plus_p6334493942879108393et_nat @ A2 @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_297_add__left__cancel,axiom,
! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
( ( ( plus_plus_multiset_a @ A2 @ B2 )
= ( plus_plus_multiset_a @ A2 @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_298_add__left__cancel,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ A2 @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_299_list_Oinject,axiom,
! [X21: list_a,X22: list_list_a,Y21: list_a,Y22: list_list_a] :
( ( ( cons_list_a @ X21 @ X22 )
= ( cons_list_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_300_list_Oinject,axiom,
! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
( ( ( cons_a @ X21 @ X22 )
= ( cons_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_301_same__append__eq,axiom,
! [Xs2: list_list_a,Ys: list_list_a,Zs: list_list_a] :
( ( ( append_list_a @ Xs2 @ Ys )
= ( append_list_a @ Xs2 @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_302_same__append__eq,axiom,
! [Xs2: list_a,Ys: list_a,Zs: list_a] :
( ( ( append_a @ Xs2 @ Ys )
= ( append_a @ Xs2 @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_303_append__same__eq,axiom,
! [Ys: list_list_a,Xs2: list_list_a,Zs: list_list_a] :
( ( ( append_list_a @ Ys @ Xs2 )
= ( append_list_a @ Zs @ Xs2 ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_304_append__same__eq,axiom,
! [Ys: list_a,Xs2: list_a,Zs: list_a] :
( ( ( append_a @ Ys @ Xs2 )
= ( append_a @ Zs @ Xs2 ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_305_append__assoc,axiom,
! [Xs2: list_list_a,Ys: list_list_a,Zs: list_list_a] :
( ( append_list_a @ ( append_list_a @ Xs2 @ Ys ) @ Zs )
= ( append_list_a @ Xs2 @ ( append_list_a @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_306_append__assoc,axiom,
! [Xs2: list_a,Ys: list_a,Zs: list_a] :
( ( append_a @ ( append_a @ Xs2 @ Ys ) @ Zs )
= ( append_a @ Xs2 @ ( append_a @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_307_append_Oassoc,axiom,
! [A2: list_list_a,B2: list_list_a,C: list_list_a] :
( ( append_list_a @ ( append_list_a @ A2 @ B2 ) @ C )
= ( append_list_a @ A2 @ ( append_list_a @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_308_append_Oassoc,axiom,
! [A2: list_a,B2: list_a,C: list_a] :
( ( append_a @ ( append_a @ A2 @ B2 ) @ C )
= ( append_a @ A2 @ ( append_a @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_309_drop__drop,axiom,
! [N4: nat,M: nat,Xs2: list_a] :
( ( drop_a @ N4 @ ( drop_a @ M @ Xs2 ) )
= ( drop_a @ ( plus_plus_nat @ N4 @ M ) @ Xs2 ) ) ).
% drop_drop
thf(fact_310_drop0,axiom,
( ( drop_a @ zero_zero_nat )
= ( ^ [X3: list_a] : X3 ) ) ).
% drop0
thf(fact_311_add__0,axiom,
! [A2: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_312_add__0,axiom,
! [A2: multiset_a] :
( ( plus_plus_multiset_a @ zero_zero_multiset_a @ A2 )
= A2 ) ).
% add_0
thf(fact_313_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_314_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y ) )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_315_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_316_add__cancel__right__right,axiom,
! [A2: multiset_nat,B2: multiset_nat] :
( ( A2
= ( plus_p6334493942879108393et_nat @ A2 @ B2 ) )
= ( B2 = zero_z7348594199698428585et_nat ) ) ).
% add_cancel_right_right
thf(fact_317_add__cancel__right__right,axiom,
! [A2: multiset_a,B2: multiset_a] :
( ( A2
= ( plus_plus_multiset_a @ A2 @ B2 ) )
= ( B2 = zero_zero_multiset_a ) ) ).
% add_cancel_right_right
thf(fact_318_add__cancel__right__right,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_319_add__cancel__right__left,axiom,
! [A2: multiset_nat,B2: multiset_nat] :
( ( A2
= ( plus_p6334493942879108393et_nat @ B2 @ A2 ) )
= ( B2 = zero_z7348594199698428585et_nat ) ) ).
% add_cancel_right_left
thf(fact_320_add__cancel__right__left,axiom,
! [A2: multiset_a,B2: multiset_a] :
( ( A2
= ( plus_plus_multiset_a @ B2 @ A2 ) )
= ( B2 = zero_zero_multiset_a ) ) ).
% add_cancel_right_left
thf(fact_321_add__cancel__right__left,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ A2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_322_add__cancel__left__right,axiom,
! [A2: multiset_nat,B2: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ A2 @ B2 )
= A2 )
= ( B2 = zero_z7348594199698428585et_nat ) ) ).
% add_cancel_left_right
thf(fact_323_add__cancel__left__right,axiom,
! [A2: multiset_a,B2: multiset_a] :
( ( ( plus_plus_multiset_a @ A2 @ B2 )
= A2 )
= ( B2 = zero_zero_multiset_a ) ) ).
% add_cancel_left_right
thf(fact_324_add__cancel__left__right,axiom,
! [A2: nat,B2: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= A2 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_325_add__cancel__left__left,axiom,
! [B2: multiset_nat,A2: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ B2 @ A2 )
= A2 )
= ( B2 = zero_z7348594199698428585et_nat ) ) ).
% add_cancel_left_left
thf(fact_326_add__cancel__left__left,axiom,
! [B2: multiset_a,A2: multiset_a] :
( ( ( plus_plus_multiset_a @ B2 @ A2 )
= A2 )
= ( B2 = zero_zero_multiset_a ) ) ).
% add_cancel_left_left
thf(fact_327_add__cancel__left__left,axiom,
! [B2: nat,A2: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= A2 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_328_add_Oright__neutral,axiom,
! [A2: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ A2 @ zero_z7348594199698428585et_nat )
= A2 ) ).
% add.right_neutral
thf(fact_329_add_Oright__neutral,axiom,
! [A2: multiset_a] :
( ( plus_plus_multiset_a @ A2 @ zero_zero_multiset_a )
= A2 ) ).
% add.right_neutral
thf(fact_330_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_331_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: multiset_nat] :
( ( minus_8522176038001411705et_nat @ A2 @ A2 )
= zero_z7348594199698428585et_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_332_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: multiset_a] :
( ( minus_3765977307040488491iset_a @ A2 @ A2 )
= zero_zero_multiset_a ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_333_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_334_nth__Cons__0,axiom,
! [X2: multiset_a,Xs2: list_multiset_a] :
( ( nth_multiset_a @ ( cons_multiset_a @ X2 @ Xs2 ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_335_nth__Cons__0,axiom,
! [X2: list_a,Xs2: list_list_a] :
( ( nth_list_a @ ( cons_list_a @ X2 @ Xs2 ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_336_nth__Cons__0,axiom,
! [X2: a,Xs2: list_a] :
( ( nth_a @ ( cons_a @ X2 @ Xs2 ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_337_nth__Cons__Suc,axiom,
! [X2: multiset_a,Xs2: list_multiset_a,N4: nat] :
( ( nth_multiset_a @ ( cons_multiset_a @ X2 @ Xs2 ) @ ( suc @ N4 ) )
= ( nth_multiset_a @ Xs2 @ N4 ) ) ).
% nth_Cons_Suc
thf(fact_338_nth__Cons__Suc,axiom,
! [X2: list_a,Xs2: list_list_a,N4: nat] :
( ( nth_list_a @ ( cons_list_a @ X2 @ Xs2 ) @ ( suc @ N4 ) )
= ( nth_list_a @ Xs2 @ N4 ) ) ).
% nth_Cons_Suc
thf(fact_339_nth__Cons__Suc,axiom,
! [X2: a,Xs2: list_a,N4: nat] :
( ( nth_a @ ( cons_a @ X2 @ Xs2 ) @ ( suc @ N4 ) )
= ( nth_a @ Xs2 @ N4 ) ) ).
% nth_Cons_Suc
thf(fact_340_take__Suc__Cons,axiom,
! [N4: nat,X2: list_a,Xs2: list_list_a] :
( ( take_list_a @ ( suc @ N4 ) @ ( cons_list_a @ X2 @ Xs2 ) )
= ( cons_list_a @ X2 @ ( take_list_a @ N4 @ Xs2 ) ) ) ).
% take_Suc_Cons
thf(fact_341_take__Suc__Cons,axiom,
! [N4: nat,X2: a,Xs2: list_a] :
( ( take_a @ ( suc @ N4 ) @ ( cons_a @ X2 @ Xs2 ) )
= ( cons_a @ X2 @ ( take_a @ N4 @ Xs2 ) ) ) ).
% take_Suc_Cons
thf(fact_342_mset__append,axiom,
! [Xs2: list_list_a,Ys: list_list_a] :
( ( mset_list_a @ ( append_list_a @ Xs2 @ Ys ) )
= ( plus_p690419498615200257list_a @ ( mset_list_a @ Xs2 ) @ ( mset_list_a @ Ys ) ) ) ).
% mset_append
thf(fact_343_mset__append,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( mset_nat @ ( append_nat @ Xs2 @ Ys ) )
= ( plus_p6334493942879108393et_nat @ ( mset_nat @ Xs2 ) @ ( mset_nat @ Ys ) ) ) ).
% mset_append
thf(fact_344_mset__append,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( mset_a @ ( append_a @ Xs2 @ Ys ) )
= ( plus_plus_multiset_a @ ( mset_a @ Xs2 ) @ ( mset_a @ Ys ) ) ) ).
% mset_append
thf(fact_345_append__take__drop__id,axiom,
! [N4: nat,Xs2: list_list_a] :
( ( append_list_a @ ( take_list_a @ N4 @ Xs2 ) @ ( drop_list_a @ N4 @ Xs2 ) )
= Xs2 ) ).
% append_take_drop_id
thf(fact_346_append__take__drop__id,axiom,
! [N4: nat,Xs2: list_a] :
( ( append_a @ ( take_a @ N4 @ Xs2 ) @ ( drop_a @ N4 @ Xs2 ) )
= Xs2 ) ).
% append_take_drop_id
thf(fact_347__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062ys_Azs_O_A_092_060lbrakk_062xs_A_061_Ays_A_064_Ax_A_D_Azs_059_Ays_A_061_Atake_Aj1_Axs_059_Azs_A_061_Adrop_A_ISuc_Aj1_J_Axs_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Ys2: list_a,Zs2: list_a] :
( ( xs
= ( append_a @ Ys2 @ ( cons_a @ x @ Zs2 ) ) )
=> ( ( Ys2
= ( take_a @ j1 @ xs ) )
=> ( Zs2
!= ( drop_a @ ( suc @ j1 ) @ xs ) ) ) ) ).
% \<open>\<And>thesis. (\<And>ys zs. \<lbrakk>xs = ys @ x # zs; ys = take j1 xs; zs = drop (Suc j1) xs\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_348_take__add,axiom,
! [I2: nat,J2: nat,Xs2: list_list_a] :
( ( take_list_a @ ( plus_plus_nat @ I2 @ J2 ) @ Xs2 )
= ( append_list_a @ ( take_list_a @ I2 @ Xs2 ) @ ( take_list_a @ J2 @ ( drop_list_a @ I2 @ Xs2 ) ) ) ) ).
% take_add
thf(fact_349_take__add,axiom,
! [I2: nat,J2: nat,Xs2: list_a] :
( ( take_a @ ( plus_plus_nat @ I2 @ J2 ) @ Xs2 )
= ( append_a @ ( take_a @ I2 @ Xs2 ) @ ( take_a @ J2 @ ( drop_a @ I2 @ Xs2 ) ) ) ) ).
% take_add
thf(fact_350_drop__take,axiom,
! [N4: nat,M: nat,Xs2: list_a] :
( ( drop_a @ N4 @ ( take_a @ M @ Xs2 ) )
= ( take_a @ ( minus_minus_nat @ M @ N4 ) @ ( drop_a @ N4 @ Xs2 ) ) ) ).
% drop_take
thf(fact_351_take__drop,axiom,
! [N4: nat,M: nat,Xs2: list_a] :
( ( take_a @ N4 @ ( drop_a @ M @ Xs2 ) )
= ( drop_a @ M @ ( take_a @ ( plus_plus_nat @ N4 @ M ) @ Xs2 ) ) ) ).
% take_drop
thf(fact_352_append__eq__append__conv2,axiom,
! [Xs2: list_list_a,Ys: list_list_a,Zs: list_list_a,Ts: list_list_a] :
( ( ( append_list_a @ Xs2 @ Ys )
= ( append_list_a @ Zs @ Ts ) )
= ( ? [Us: list_list_a] :
( ( ( Xs2
= ( append_list_a @ Zs @ Us ) )
& ( ( append_list_a @ Us @ Ys )
= Ts ) )
| ( ( ( append_list_a @ Xs2 @ Us )
= Zs )
& ( Ys
= ( append_list_a @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_353_append__eq__append__conv2,axiom,
! [Xs2: list_a,Ys: list_a,Zs: list_a,Ts: list_a] :
( ( ( append_a @ Xs2 @ Ys )
= ( append_a @ Zs @ Ts ) )
= ( ? [Us: list_a] :
( ( ( Xs2
= ( append_a @ Zs @ Us ) )
& ( ( append_a @ Us @ Ys )
= Ts ) )
| ( ( ( append_a @ Xs2 @ Us )
= Zs )
& ( Ys
= ( append_a @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_354_append__eq__appendI,axiom,
! [Xs2: list_list_a,Xs1: list_list_a,Zs: list_list_a,Ys: list_list_a,Us2: list_list_a] :
( ( ( append_list_a @ Xs2 @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_list_a @ Xs1 @ Us2 ) )
=> ( ( append_list_a @ Xs2 @ Ys )
= ( append_list_a @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_355_append__eq__appendI,axiom,
! [Xs2: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us2: list_a] :
( ( ( append_a @ Xs2 @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_a @ Xs1 @ Us2 ) )
=> ( ( append_a @ Xs2 @ Ys )
= ( append_a @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_356_take__equalityI,axiom,
! [Xs2: list_a,Ys: list_a] :
( ! [I3: nat] :
( ( take_a @ I3 @ Xs2 )
= ( take_a @ I3 @ Ys ) )
=> ( Xs2 = Ys ) ) ).
% take_equalityI
thf(fact_357_in__set__takeD,axiom,
! [X2: nat,N4: nat,Xs2: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ ( take_nat @ N4 @ Xs2 ) ) )
=> ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) ) ) ).
% in_set_takeD
thf(fact_358_in__set__takeD,axiom,
! [X2: list_a,N4: nat,Xs2: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ ( take_list_a @ N4 @ Xs2 ) ) )
=> ( member_list_a @ X2 @ ( set_list_a2 @ Xs2 ) ) ) ).
% in_set_takeD
thf(fact_359_in__set__takeD,axiom,
! [X2: multiset_a,N4: nat,Xs2: list_multiset_a] :
( ( member_multiset_a @ X2 @ ( set_multiset_a2 @ ( take_multiset_a @ N4 @ Xs2 ) ) )
=> ( member_multiset_a @ X2 @ ( set_multiset_a2 @ Xs2 ) ) ) ).
% in_set_takeD
thf(fact_360_in__set__takeD,axiom,
! [X2: a,N4: nat,Xs2: list_a] :
( ( member_a @ X2 @ ( set_a2 @ ( take_a @ N4 @ Xs2 ) ) )
=> ( member_a @ X2 @ ( set_a2 @ Xs2 ) ) ) ).
% in_set_takeD
thf(fact_361_Cons__eq__appendI,axiom,
! [X2: list_a,Xs1: list_list_a,Ys: list_list_a,Xs2: list_list_a,Zs: list_list_a] :
( ( ( cons_list_a @ X2 @ Xs1 )
= Ys )
=> ( ( Xs2
= ( append_list_a @ Xs1 @ Zs ) )
=> ( ( cons_list_a @ X2 @ Xs2 )
= ( append_list_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_362_Cons__eq__appendI,axiom,
! [X2: a,Xs1: list_a,Ys: list_a,Xs2: list_a,Zs: list_a] :
( ( ( cons_a @ X2 @ Xs1 )
= Ys )
=> ( ( Xs2
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X2 @ Xs2 )
= ( append_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_363_append__Cons,axiom,
! [X2: list_a,Xs2: list_list_a,Ys: list_list_a] :
( ( append_list_a @ ( cons_list_a @ X2 @ Xs2 ) @ Ys )
= ( cons_list_a @ X2 @ ( append_list_a @ Xs2 @ Ys ) ) ) ).
% append_Cons
thf(fact_364_append__Cons,axiom,
! [X2: a,Xs2: list_a,Ys: list_a] :
( ( append_a @ ( cons_a @ X2 @ Xs2 ) @ Ys )
= ( cons_a @ X2 @ ( append_a @ Xs2 @ Ys ) ) ) ).
% append_Cons
thf(fact_365_drop__0,axiom,
! [Xs2: list_a] :
( ( drop_a @ zero_zero_nat @ Xs2 )
= Xs2 ) ).
% drop_0
thf(fact_366_split__list__first__prop__iff,axiom,
! [Xs2: list_multiset_a,P: multiset_a > $o] :
( ( ? [X3: multiset_a] :
( ( member_multiset_a @ X3 @ ( set_multiset_a2 @ Xs2 ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_multiset_a,X3: multiset_a] :
( ? [Zs3: list_multiset_a] :
( Xs2
= ( append_multiset_a @ Ys3 @ ( cons_multiset_a @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y3: multiset_a] :
( ( member_multiset_a @ Y3 @ ( set_multiset_a2 @ Ys3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_367_split__list__first__prop__iff,axiom,
! [Xs2: list_list_a,P: list_a > $o] :
( ( ? [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs2 ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_list_a,X3: list_a] :
( ? [Zs3: list_list_a] :
( Xs2
= ( append_list_a @ Ys3 @ ( cons_list_a @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y3: list_a] :
( ( member_list_a @ Y3 @ ( set_list_a2 @ Ys3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_368_split__list__first__prop__iff,axiom,
! [Xs2: list_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs2 ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_a,X3: a] :
( ? [Zs3: list_a] :
( Xs2
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Ys3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_369_split__list__last__prop__iff,axiom,
! [Xs2: list_multiset_a,P: multiset_a > $o] :
( ( ? [X3: multiset_a] :
( ( member_multiset_a @ X3 @ ( set_multiset_a2 @ Xs2 ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_multiset_a,X3: multiset_a,Zs3: list_multiset_a] :
( ( Xs2
= ( append_multiset_a @ Ys3 @ ( cons_multiset_a @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y3: multiset_a] :
( ( member_multiset_a @ Y3 @ ( set_multiset_a2 @ Zs3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_370_split__list__last__prop__iff,axiom,
! [Xs2: list_list_a,P: list_a > $o] :
( ( ? [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs2 ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_list_a,X3: list_a,Zs3: list_list_a] :
( ( Xs2
= ( append_list_a @ Ys3 @ ( cons_list_a @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y3: list_a] :
( ( member_list_a @ Y3 @ ( set_list_a2 @ Zs3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_371_split__list__last__prop__iff,axiom,
! [Xs2: list_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs2 ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_a,X3: a,Zs3: list_a] :
( ( Xs2
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Zs3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_372_in__set__conv__decomp__first,axiom,
! [X2: nat,Xs2: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( ( Xs2
= ( append_nat @ Ys3 @ ( cons_nat @ X2 @ Zs3 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_373_in__set__conv__decomp__first,axiom,
! [X2: multiset_a,Xs2: list_multiset_a] :
( ( member_multiset_a @ X2 @ ( set_multiset_a2 @ Xs2 ) )
= ( ? [Ys3: list_multiset_a,Zs3: list_multiset_a] :
( ( Xs2
= ( append_multiset_a @ Ys3 @ ( cons_multiset_a @ X2 @ Zs3 ) ) )
& ~ ( member_multiset_a @ X2 @ ( set_multiset_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_374_in__set__conv__decomp__first,axiom,
! [X2: list_a,Xs2: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs2 ) )
= ( ? [Ys3: list_list_a,Zs3: list_list_a] :
( ( Xs2
= ( append_list_a @ Ys3 @ ( cons_list_a @ X2 @ Zs3 ) ) )
& ~ ( member_list_a @ X2 @ ( set_list_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_375_in__set__conv__decomp__first,axiom,
! [X2: a,Xs2: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs2 ) )
= ( ? [Ys3: list_a,Zs3: list_a] :
( ( Xs2
= ( append_a @ Ys3 @ ( cons_a @ X2 @ Zs3 ) ) )
& ~ ( member_a @ X2 @ ( set_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_376_in__set__conv__decomp__last,axiom,
! [X2: nat,Xs2: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( ( Xs2
= ( append_nat @ Ys3 @ ( cons_nat @ X2 @ Zs3 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_377_in__set__conv__decomp__last,axiom,
! [X2: multiset_a,Xs2: list_multiset_a] :
( ( member_multiset_a @ X2 @ ( set_multiset_a2 @ Xs2 ) )
= ( ? [Ys3: list_multiset_a,Zs3: list_multiset_a] :
( ( Xs2
= ( append_multiset_a @ Ys3 @ ( cons_multiset_a @ X2 @ Zs3 ) ) )
& ~ ( member_multiset_a @ X2 @ ( set_multiset_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_378_in__set__conv__decomp__last,axiom,
! [X2: list_a,Xs2: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs2 ) )
= ( ? [Ys3: list_list_a,Zs3: list_list_a] :
( ( Xs2
= ( append_list_a @ Ys3 @ ( cons_list_a @ X2 @ Zs3 ) ) )
& ~ ( member_list_a @ X2 @ ( set_list_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_379_in__set__conv__decomp__last,axiom,
! [X2: a,Xs2: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs2 ) )
= ( ? [Ys3: list_a,Zs3: list_a] :
( ( Xs2
= ( append_a @ Ys3 @ ( cons_a @ X2 @ Zs3 ) ) )
& ~ ( member_a @ X2 @ ( set_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_380_split__list__first__propE,axiom,
! [Xs2: list_multiset_a,P: multiset_a > $o] :
( ? [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_multiset_a2 @ Xs2 ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_multiset_a,X4: multiset_a] :
( ? [Zs2: list_multiset_a] :
( Xs2
= ( append_multiset_a @ Ys2 @ ( cons_multiset_a @ X4 @ Zs2 ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa: multiset_a] :
( ( member_multiset_a @ Xa @ ( set_multiset_a2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_381_split__list__first__propE,axiom,
! [Xs2: list_list_a,P: list_a > $o] :
( ? [X5: list_a] :
( ( member_list_a @ X5 @ ( set_list_a2 @ Xs2 ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_list_a,X4: list_a] :
( ? [Zs2: list_list_a] :
( Xs2
= ( append_list_a @ Ys2 @ ( cons_list_a @ X4 @ Zs2 ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa: list_a] :
( ( member_list_a @ Xa @ ( set_list_a2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_382_split__list__first__propE,axiom,
! [Xs2: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs2 ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_a,X4: a] :
( ? [Zs2: list_a] :
( Xs2
= ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs2 ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_383_split__list__last__propE,axiom,
! [Xs2: list_multiset_a,P: multiset_a > $o] :
( ? [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_multiset_a2 @ Xs2 ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_multiset_a,X4: multiset_a,Zs2: list_multiset_a] :
( ( Xs2
= ( append_multiset_a @ Ys2 @ ( cons_multiset_a @ X4 @ Zs2 ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa: multiset_a] :
( ( member_multiset_a @ Xa @ ( set_multiset_a2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_384_split__list__last__propE,axiom,
! [Xs2: list_list_a,P: list_a > $o] :
( ? [X5: list_a] :
( ( member_list_a @ X5 @ ( set_list_a2 @ Xs2 ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_list_a,X4: list_a,Zs2: list_list_a] :
( ( Xs2
= ( append_list_a @ Ys2 @ ( cons_list_a @ X4 @ Zs2 ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa: list_a] :
( ( member_list_a @ Xa @ ( set_list_a2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_385_split__list__last__propE,axiom,
! [Xs2: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs2 ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_a,X4: a,Zs2: list_a] :
( ( Xs2
= ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs2 ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_386_split__list__first__prop,axiom,
! [Xs2: list_multiset_a,P: multiset_a > $o] :
( ? [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_multiset_a2 @ Xs2 ) )
& ( P @ X5 ) )
=> ? [Ys2: list_multiset_a,X4: multiset_a] :
( ? [Zs2: list_multiset_a] :
( Xs2
= ( append_multiset_a @ Ys2 @ ( cons_multiset_a @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Xa: multiset_a] :
( ( member_multiset_a @ Xa @ ( set_multiset_a2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_387_split__list__first__prop,axiom,
! [Xs2: list_list_a,P: list_a > $o] :
( ? [X5: list_a] :
( ( member_list_a @ X5 @ ( set_list_a2 @ Xs2 ) )
& ( P @ X5 ) )
=> ? [Ys2: list_list_a,X4: list_a] :
( ? [Zs2: list_list_a] :
( Xs2
= ( append_list_a @ Ys2 @ ( cons_list_a @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Xa: list_a] :
( ( member_list_a @ Xa @ ( set_list_a2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_388_split__list__first__prop,axiom,
! [Xs2: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs2 ) )
& ( P @ X5 ) )
=> ? [Ys2: list_a,X4: a] :
( ? [Zs2: list_a] :
( Xs2
= ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_389_split__list__last__prop,axiom,
! [Xs2: list_multiset_a,P: multiset_a > $o] :
( ? [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_multiset_a2 @ Xs2 ) )
& ( P @ X5 ) )
=> ? [Ys2: list_multiset_a,X4: multiset_a,Zs2: list_multiset_a] :
( ( Xs2
= ( append_multiset_a @ Ys2 @ ( cons_multiset_a @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Xa: multiset_a] :
( ( member_multiset_a @ Xa @ ( set_multiset_a2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_390_split__list__last__prop,axiom,
! [Xs2: list_list_a,P: list_a > $o] :
( ? [X5: list_a] :
( ( member_list_a @ X5 @ ( set_list_a2 @ Xs2 ) )
& ( P @ X5 ) )
=> ? [Ys2: list_list_a,X4: list_a,Zs2: list_list_a] :
( ( Xs2
= ( append_list_a @ Ys2 @ ( cons_list_a @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Xa: list_a] :
( ( member_list_a @ Xa @ ( set_list_a2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_391_split__list__last__prop,axiom,
! [Xs2: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs2 ) )
& ( P @ X5 ) )
=> ? [Ys2: list_a,X4: a,Zs2: list_a] :
( ( Xs2
= ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_392_in__set__conv__decomp,axiom,
! [X2: nat,Xs2: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( Xs2
= ( append_nat @ Ys3 @ ( cons_nat @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_393_in__set__conv__decomp,axiom,
! [X2: multiset_a,Xs2: list_multiset_a] :
( ( member_multiset_a @ X2 @ ( set_multiset_a2 @ Xs2 ) )
= ( ? [Ys3: list_multiset_a,Zs3: list_multiset_a] :
( Xs2
= ( append_multiset_a @ Ys3 @ ( cons_multiset_a @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_394_in__set__conv__decomp,axiom,
! [X2: list_a,Xs2: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs2 ) )
= ( ? [Ys3: list_list_a,Zs3: list_list_a] :
( Xs2
= ( append_list_a @ Ys3 @ ( cons_list_a @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_395_in__set__conv__decomp,axiom,
! [X2: a,Xs2: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs2 ) )
= ( ? [Ys3: list_a,Zs3: list_a] :
( Xs2
= ( append_a @ Ys3 @ ( cons_a @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_396_append__Cons__eq__iff,axiom,
! [X2: nat,Xs2: list_nat,Ys: list_nat,Xs3: list_nat,Ys4: list_nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ~ ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
=> ( ( ( append_nat @ Xs2 @ ( cons_nat @ X2 @ Ys ) )
= ( append_nat @ Xs3 @ ( cons_nat @ X2 @ Ys4 ) ) )
= ( ( Xs2 = Xs3 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_397_append__Cons__eq__iff,axiom,
! [X2: multiset_a,Xs2: list_multiset_a,Ys: list_multiset_a,Xs3: list_multiset_a,Ys4: list_multiset_a] :
( ~ ( member_multiset_a @ X2 @ ( set_multiset_a2 @ Xs2 ) )
=> ( ~ ( member_multiset_a @ X2 @ ( set_multiset_a2 @ Ys ) )
=> ( ( ( append_multiset_a @ Xs2 @ ( cons_multiset_a @ X2 @ Ys ) )
= ( append_multiset_a @ Xs3 @ ( cons_multiset_a @ X2 @ Ys4 ) ) )
= ( ( Xs2 = Xs3 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_398_append__Cons__eq__iff,axiom,
! [X2: list_a,Xs2: list_list_a,Ys: list_list_a,Xs3: list_list_a,Ys4: list_list_a] :
( ~ ( member_list_a @ X2 @ ( set_list_a2 @ Xs2 ) )
=> ( ~ ( member_list_a @ X2 @ ( set_list_a2 @ Ys ) )
=> ( ( ( append_list_a @ Xs2 @ ( cons_list_a @ X2 @ Ys ) )
= ( append_list_a @ Xs3 @ ( cons_list_a @ X2 @ Ys4 ) ) )
= ( ( Xs2 = Xs3 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_399_append__Cons__eq__iff,axiom,
! [X2: a,Xs2: list_a,Ys: list_a,Xs3: list_a,Ys4: list_a] :
( ~ ( member_a @ X2 @ ( set_a2 @ Xs2 ) )
=> ( ~ ( member_a @ X2 @ ( set_a2 @ Ys ) )
=> ( ( ( append_a @ Xs2 @ ( cons_a @ X2 @ Ys ) )
= ( append_a @ Xs3 @ ( cons_a @ X2 @ Ys4 ) ) )
= ( ( Xs2 = Xs3 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_400_split__list__propE,axiom,
! [Xs2: list_multiset_a,P: multiset_a > $o] :
( ? [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_multiset_a2 @ Xs2 ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_multiset_a,X4: multiset_a] :
( ? [Zs2: list_multiset_a] :
( Xs2
= ( append_multiset_a @ Ys2 @ ( cons_multiset_a @ X4 @ Zs2 ) ) )
=> ~ ( P @ X4 ) ) ) ).
% split_list_propE
thf(fact_401_split__list__propE,axiom,
! [Xs2: list_list_a,P: list_a > $o] :
( ? [X5: list_a] :
( ( member_list_a @ X5 @ ( set_list_a2 @ Xs2 ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_list_a,X4: list_a] :
( ? [Zs2: list_list_a] :
( Xs2
= ( append_list_a @ Ys2 @ ( cons_list_a @ X4 @ Zs2 ) ) )
=> ~ ( P @ X4 ) ) ) ).
% split_list_propE
thf(fact_402_split__list__propE,axiom,
! [Xs2: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs2 ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_a,X4: a] :
( ? [Zs2: list_a] :
( Xs2
= ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs2 ) ) )
=> ~ ( P @ X4 ) ) ) ).
% split_list_propE
thf(fact_403_split__list__first,axiom,
! [X2: nat,Xs2: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ? [Ys2: list_nat,Zs2: list_nat] :
( ( Xs2
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs2 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_404_split__list__first,axiom,
! [X2: multiset_a,Xs2: list_multiset_a] :
( ( member_multiset_a @ X2 @ ( set_multiset_a2 @ Xs2 ) )
=> ? [Ys2: list_multiset_a,Zs2: list_multiset_a] :
( ( Xs2
= ( append_multiset_a @ Ys2 @ ( cons_multiset_a @ X2 @ Zs2 ) ) )
& ~ ( member_multiset_a @ X2 @ ( set_multiset_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_405_split__list__first,axiom,
! [X2: list_a,Xs2: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs2 ) )
=> ? [Ys2: list_list_a,Zs2: list_list_a] :
( ( Xs2
= ( append_list_a @ Ys2 @ ( cons_list_a @ X2 @ Zs2 ) ) )
& ~ ( member_list_a @ X2 @ ( set_list_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_406_split__list__first,axiom,
! [X2: a,Xs2: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs2 ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( ( Xs2
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
& ~ ( member_a @ X2 @ ( set_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_407_split__list__prop,axiom,
! [Xs2: list_multiset_a,P: multiset_a > $o] :
( ? [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_multiset_a2 @ Xs2 ) )
& ( P @ X5 ) )
=> ? [Ys2: list_multiset_a,X4: multiset_a] :
( ? [Zs2: list_multiset_a] :
( Xs2
= ( append_multiset_a @ Ys2 @ ( cons_multiset_a @ X4 @ Zs2 ) ) )
& ( P @ X4 ) ) ) ).
% split_list_prop
thf(fact_408_split__list__prop,axiom,
! [Xs2: list_list_a,P: list_a > $o] :
( ? [X5: list_a] :
( ( member_list_a @ X5 @ ( set_list_a2 @ Xs2 ) )
& ( P @ X5 ) )
=> ? [Ys2: list_list_a,X4: list_a] :
( ? [Zs2: list_list_a] :
( Xs2
= ( append_list_a @ Ys2 @ ( cons_list_a @ X4 @ Zs2 ) ) )
& ( P @ X4 ) ) ) ).
% split_list_prop
thf(fact_409_split__list__prop,axiom,
! [Xs2: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs2 ) )
& ( P @ X5 ) )
=> ? [Ys2: list_a,X4: a] :
( ? [Zs2: list_a] :
( Xs2
= ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs2 ) ) )
& ( P @ X4 ) ) ) ).
% split_list_prop
thf(fact_410_split__list__last,axiom,
! [X2: nat,Xs2: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ? [Ys2: list_nat,Zs2: list_nat] :
( ( Xs2
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs2 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_411_split__list__last,axiom,
! [X2: multiset_a,Xs2: list_multiset_a] :
( ( member_multiset_a @ X2 @ ( set_multiset_a2 @ Xs2 ) )
=> ? [Ys2: list_multiset_a,Zs2: list_multiset_a] :
( ( Xs2
= ( append_multiset_a @ Ys2 @ ( cons_multiset_a @ X2 @ Zs2 ) ) )
& ~ ( member_multiset_a @ X2 @ ( set_multiset_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_412_split__list__last,axiom,
! [X2: list_a,Xs2: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs2 ) )
=> ? [Ys2: list_list_a,Zs2: list_list_a] :
( ( Xs2
= ( append_list_a @ Ys2 @ ( cons_list_a @ X2 @ Zs2 ) ) )
& ~ ( member_list_a @ X2 @ ( set_list_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_413_split__list__last,axiom,
! [X2: a,Xs2: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs2 ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( ( Xs2
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
& ~ ( member_a @ X2 @ ( set_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_414_split__list,axiom,
! [X2: nat,Xs2: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ? [Ys2: list_nat,Zs2: list_nat] :
( Xs2
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_415_split__list,axiom,
! [X2: multiset_a,Xs2: list_multiset_a] :
( ( member_multiset_a @ X2 @ ( set_multiset_a2 @ Xs2 ) )
=> ? [Ys2: list_multiset_a,Zs2: list_multiset_a] :
( Xs2
= ( append_multiset_a @ Ys2 @ ( cons_multiset_a @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_416_split__list,axiom,
! [X2: list_a,Xs2: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs2 ) )
=> ? [Ys2: list_list_a,Zs2: list_list_a] :
( Xs2
= ( append_list_a @ Ys2 @ ( cons_list_a @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_417_split__list,axiom,
! [X2: a,Xs2: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs2 ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( Xs2
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_418_nth__via__drop,axiom,
! [N4: nat,Xs2: list_multiset_a,Y: multiset_a,Ys: list_multiset_a] :
( ( ( drop_multiset_a @ N4 @ Xs2 )
= ( cons_multiset_a @ Y @ Ys ) )
=> ( ( nth_multiset_a @ Xs2 @ N4 )
= Y ) ) ).
% nth_via_drop
thf(fact_419_nth__via__drop,axiom,
! [N4: nat,Xs2: list_list_a,Y: list_a,Ys: list_list_a] :
( ( ( drop_list_a @ N4 @ Xs2 )
= ( cons_list_a @ Y @ Ys ) )
=> ( ( nth_list_a @ Xs2 @ N4 )
= Y ) ) ).
% nth_via_drop
thf(fact_420_nth__via__drop,axiom,
! [N4: nat,Xs2: list_a,Y: a,Ys: list_a] :
( ( ( drop_a @ N4 @ Xs2 )
= ( cons_a @ Y @ Ys ) )
=> ( ( nth_a @ Xs2 @ N4 )
= Y ) ) ).
% nth_via_drop
thf(fact_421_zero__reorient,axiom,
! [X2: multiset_nat] :
( ( zero_z7348594199698428585et_nat = X2 )
= ( X2 = zero_z7348594199698428585et_nat ) ) ).
% zero_reorient
thf(fact_422_zero__reorient,axiom,
! [X2: multiset_a] :
( ( zero_zero_multiset_a = X2 )
= ( X2 = zero_zero_multiset_a ) ) ).
% zero_reorient
thf(fact_423_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_424_add__right__imp__eq,axiom,
! [B2: multiset_nat,A2: multiset_nat,C: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ B2 @ A2 )
= ( plus_p6334493942879108393et_nat @ C @ A2 ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_425_add__right__imp__eq,axiom,
! [B2: multiset_a,A2: multiset_a,C: multiset_a] :
( ( ( plus_plus_multiset_a @ B2 @ A2 )
= ( plus_plus_multiset_a @ C @ A2 ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_426_add__right__imp__eq,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_427_add__left__imp__eq,axiom,
! [A2: multiset_nat,B2: multiset_nat,C: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ A2 @ B2 )
= ( plus_p6334493942879108393et_nat @ A2 @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_428_add__left__imp__eq,axiom,
! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
( ( ( plus_plus_multiset_a @ A2 @ B2 )
= ( plus_plus_multiset_a @ A2 @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_429_add__left__imp__eq,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ A2 @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_430_add_Oleft__commute,axiom,
! [B2: multiset_nat,A2: multiset_nat,C: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ B2 @ ( plus_p6334493942879108393et_nat @ A2 @ C ) )
= ( plus_p6334493942879108393et_nat @ A2 @ ( plus_p6334493942879108393et_nat @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_431_add_Oleft__commute,axiom,
! [B2: multiset_a,A2: multiset_a,C: multiset_a] :
( ( plus_plus_multiset_a @ B2 @ ( plus_plus_multiset_a @ A2 @ C ) )
= ( plus_plus_multiset_a @ A2 @ ( plus_plus_multiset_a @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_432_add_Oleft__commute,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_433_add_Ocommute,axiom,
( plus_p6334493942879108393et_nat
= ( ^ [A4: multiset_nat,B4: multiset_nat] : ( plus_p6334493942879108393et_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_434_add_Ocommute,axiom,
( plus_plus_multiset_a
= ( ^ [A4: multiset_a,B4: multiset_a] : ( plus_plus_multiset_a @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_435_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_436_add_Oassoc,axiom,
! [A2: multiset_nat,B2: multiset_nat,C: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) @ C )
= ( plus_p6334493942879108393et_nat @ A2 @ ( plus_p6334493942879108393et_nat @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_437_add_Oassoc,axiom,
! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ A2 @ B2 ) @ C )
= ( plus_plus_multiset_a @ A2 @ ( plus_plus_multiset_a @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_438_add_Oassoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_439_group__cancel_Oadd2,axiom,
! [B: multiset_nat,K3: multiset_nat,B2: multiset_nat,A2: multiset_nat] :
( ( B
= ( plus_p6334493942879108393et_nat @ K3 @ B2 ) )
=> ( ( plus_p6334493942879108393et_nat @ A2 @ B )
= ( plus_p6334493942879108393et_nat @ K3 @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_440_group__cancel_Oadd2,axiom,
! [B: multiset_a,K3: multiset_a,B2: multiset_a,A2: multiset_a] :
( ( B
= ( plus_plus_multiset_a @ K3 @ B2 ) )
=> ( ( plus_plus_multiset_a @ A2 @ B )
= ( plus_plus_multiset_a @ K3 @ ( plus_plus_multiset_a @ A2 @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_441_group__cancel_Oadd2,axiom,
! [B: nat,K3: nat,B2: nat,A2: nat] :
( ( B
= ( plus_plus_nat @ K3 @ B2 ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K3 @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_442_group__cancel_Oadd1,axiom,
! [A: multiset_nat,K3: multiset_nat,A2: multiset_nat,B2: multiset_nat] :
( ( A
= ( plus_p6334493942879108393et_nat @ K3 @ A2 ) )
=> ( ( plus_p6334493942879108393et_nat @ A @ B2 )
= ( plus_p6334493942879108393et_nat @ K3 @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_443_group__cancel_Oadd1,axiom,
! [A: multiset_a,K3: multiset_a,A2: multiset_a,B2: multiset_a] :
( ( A
= ( plus_plus_multiset_a @ K3 @ A2 ) )
=> ( ( plus_plus_multiset_a @ A @ B2 )
= ( plus_plus_multiset_a @ K3 @ ( plus_plus_multiset_a @ A2 @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_444_group__cancel_Oadd1,axiom,
! [A: nat,K3: nat,A2: nat,B2: nat] :
( ( A
= ( plus_plus_nat @ K3 @ A2 ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K3 @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_445_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: multiset_nat,J2: multiset_nat,K3: multiset_nat,L4: multiset_nat] :
( ( ( I2 = J2 )
& ( K3 = L4 ) )
=> ( ( plus_p6334493942879108393et_nat @ I2 @ K3 )
= ( plus_p6334493942879108393et_nat @ J2 @ L4 ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_446_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J2: nat,K3: nat,L4: nat] :
( ( ( I2 = J2 )
& ( K3 = L4 ) )
=> ( ( plus_plus_nat @ I2 @ K3 )
= ( plus_plus_nat @ J2 @ L4 ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_447_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: multiset_nat,B2: multiset_nat,C: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) @ C )
= ( plus_p6334493942879108393et_nat @ A2 @ ( plus_p6334493942879108393et_nat @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_448_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ A2 @ B2 ) @ C )
= ( plus_plus_multiset_a @ A2 @ ( plus_plus_multiset_a @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_449_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_450_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: multiset_nat,C: multiset_nat,B2: multiset_nat] :
( ( minus_8522176038001411705et_nat @ ( minus_8522176038001411705et_nat @ A2 @ C ) @ B2 )
= ( minus_8522176038001411705et_nat @ ( minus_8522176038001411705et_nat @ A2 @ B2 ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_451_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: multiset_a,C: multiset_a,B2: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ A2 @ C ) @ B2 )
= ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ A2 @ B2 ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_452_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B2 )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_453_not__Cons__self2,axiom,
! [X2: list_a,Xs2: list_list_a] :
( ( cons_list_a @ X2 @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_454_not__Cons__self2,axiom,
! [X2: a,Xs2: list_a] :
( ( cons_a @ X2 @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_455_add_Ocomm__neutral,axiom,
! [A2: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ A2 @ zero_z7348594199698428585et_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_456_add_Ocomm__neutral,axiom,
! [A2: multiset_a] :
( ( plus_plus_multiset_a @ A2 @ zero_zero_multiset_a )
= A2 ) ).
% add.comm_neutral
thf(fact_457_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_458_comm__monoid__add__class_Oadd__0,axiom,
! [A2: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_459_comm__monoid__add__class_Oadd__0,axiom,
! [A2: multiset_a] :
( ( plus_plus_multiset_a @ zero_zero_multiset_a @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_460_comm__monoid__add__class_Oadd__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_461_diff__diff__eq,axiom,
! [A2: multiset_nat,B2: multiset_nat,C: multiset_nat] :
( ( minus_8522176038001411705et_nat @ ( minus_8522176038001411705et_nat @ A2 @ B2 ) @ C )
= ( minus_8522176038001411705et_nat @ A2 @ ( plus_p6334493942879108393et_nat @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_462_diff__diff__eq,axiom,
! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ A2 @ B2 ) @ C )
= ( minus_3765977307040488491iset_a @ A2 @ ( plus_plus_multiset_a @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_463_diff__diff__eq,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C )
= ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_464_add__implies__diff,axiom,
! [C: multiset_nat,B2: multiset_nat,A2: multiset_nat] :
( ( ( plus_p6334493942879108393et_nat @ C @ B2 )
= A2 )
=> ( C
= ( minus_8522176038001411705et_nat @ A2 @ B2 ) ) ) ).
% add_implies_diff
thf(fact_465_add__implies__diff,axiom,
! [C: multiset_a,B2: multiset_a,A2: multiset_a] :
( ( ( plus_plus_multiset_a @ C @ B2 )
= A2 )
=> ( C
= ( minus_3765977307040488491iset_a @ A2 @ B2 ) ) ) ).
% add_implies_diff
thf(fact_466_add__implies__diff,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ( plus_plus_nat @ C @ B2 )
= A2 )
=> ( C
= ( minus_minus_nat @ A2 @ B2 ) ) ) ).
% add_implies_diff
thf(fact_467_set__ConsD,axiom,
! [Y: nat,X2: nat,Xs2: list_nat] :
( ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X2 @ Xs2 ) ) )
=> ( ( Y = X2 )
| ( member_nat @ Y @ ( set_nat2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_468_set__ConsD,axiom,
! [Y: multiset_a,X2: multiset_a,Xs2: list_multiset_a] :
( ( member_multiset_a @ Y @ ( set_multiset_a2 @ ( cons_multiset_a @ X2 @ Xs2 ) ) )
=> ( ( Y = X2 )
| ( member_multiset_a @ Y @ ( set_multiset_a2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_469_set__ConsD,axiom,
! [Y: list_a,X2: list_a,Xs2: list_list_a] :
( ( member_list_a @ Y @ ( set_list_a2 @ ( cons_list_a @ X2 @ Xs2 ) ) )
=> ( ( Y = X2 )
| ( member_list_a @ Y @ ( set_list_a2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_470_set__ConsD,axiom,
! [Y: a,X2: a,Xs2: list_a] :
( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X2 @ Xs2 ) ) )
=> ( ( Y = X2 )
| ( member_a @ Y @ ( set_a2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_471_list_Oset__cases,axiom,
! [E: nat,A2: list_nat] :
( ( member_nat @ E @ ( set_nat2 @ A2 ) )
=> ( ! [Z2: list_nat] :
( A2
!= ( cons_nat @ E @ Z2 ) )
=> ~ ! [Z1: nat,Z2: list_nat] :
( ( A2
= ( cons_nat @ Z1 @ Z2 ) )
=> ~ ( member_nat @ E @ ( set_nat2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_472_list_Oset__cases,axiom,
! [E: multiset_a,A2: list_multiset_a] :
( ( member_multiset_a @ E @ ( set_multiset_a2 @ A2 ) )
=> ( ! [Z2: list_multiset_a] :
( A2
!= ( cons_multiset_a @ E @ Z2 ) )
=> ~ ! [Z1: multiset_a,Z2: list_multiset_a] :
( ( A2
= ( cons_multiset_a @ Z1 @ Z2 ) )
=> ~ ( member_multiset_a @ E @ ( set_multiset_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_473_list_Oset__cases,axiom,
! [E: list_a,A2: list_list_a] :
( ( member_list_a @ E @ ( set_list_a2 @ A2 ) )
=> ( ! [Z2: list_list_a] :
( A2
!= ( cons_list_a @ E @ Z2 ) )
=> ~ ! [Z1: list_a,Z2: list_list_a] :
( ( A2
= ( cons_list_a @ Z1 @ Z2 ) )
=> ~ ( member_list_a @ E @ ( set_list_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_474_list_Oset__cases,axiom,
! [E: a,A2: list_a] :
( ( member_a @ E @ ( set_a2 @ A2 ) )
=> ( ! [Z2: list_a] :
( A2
!= ( cons_a @ E @ Z2 ) )
=> ~ ! [Z1: a,Z2: list_a] :
( ( A2
= ( cons_a @ Z1 @ Z2 ) )
=> ~ ( member_a @ E @ ( set_a2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_475_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_476_list_Oset__intros_I1_J,axiom,
! [X21: multiset_a,X22: list_multiset_a] : ( member_multiset_a @ X21 @ ( set_multiset_a2 @ ( cons_multiset_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_477_list_Oset__intros_I1_J,axiom,
! [X21: list_a,X22: list_list_a] : ( member_list_a @ X21 @ ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_478_list_Oset__intros_I1_J,axiom,
! [X21: a,X22: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_479_list_Oset__intros_I2_J,axiom,
! [Y: nat,X22: list_nat,X21: nat] :
( ( member_nat @ Y @ ( set_nat2 @ X22 ) )
=> ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_480_list_Oset__intros_I2_J,axiom,
! [Y: multiset_a,X22: list_multiset_a,X21: multiset_a] :
( ( member_multiset_a @ Y @ ( set_multiset_a2 @ X22 ) )
=> ( member_multiset_a @ Y @ ( set_multiset_a2 @ ( cons_multiset_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_481_list_Oset__intros_I2_J,axiom,
! [Y: list_a,X22: list_list_a,X21: list_a] :
( ( member_list_a @ Y @ ( set_list_a2 @ X22 ) )
=> ( member_list_a @ Y @ ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_482_list_Oset__intros_I2_J,axiom,
! [Y: a,X22: list_a,X21: a] :
( ( member_a @ Y @ ( set_a2 @ X22 ) )
=> ( member_a @ Y @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_483_in__set__dropD,axiom,
! [X2: nat,N4: nat,Xs2: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ ( drop_nat @ N4 @ Xs2 ) ) )
=> ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) ) ) ).
% in_set_dropD
thf(fact_484_in__set__dropD,axiom,
! [X2: list_a,N4: nat,Xs2: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ ( drop_list_a @ N4 @ Xs2 ) ) )
=> ( member_list_a @ X2 @ ( set_list_a2 @ Xs2 ) ) ) ).
% in_set_dropD
thf(fact_485_in__set__dropD,axiom,
! [X2: multiset_a,N4: nat,Xs2: list_multiset_a] :
( ( member_multiset_a @ X2 @ ( set_multiset_a2 @ ( drop_multiset_a @ N4 @ Xs2 ) ) )
=> ( member_multiset_a @ X2 @ ( set_multiset_a2 @ Xs2 ) ) ) ).
% in_set_dropD
thf(fact_486_in__set__dropD,axiom,
! [X2: a,N4: nat,Xs2: list_a] :
( ( member_a @ X2 @ ( set_a2 @ ( drop_a @ N4 @ Xs2 ) ) )
=> ( member_a @ X2 @ ( set_a2 @ Xs2 ) ) ) ).
% in_set_dropD
thf(fact_487_multiset__remove__induct,axiom,
! [P: multiset_list_a > $o,A: multiset_list_a] :
( ( P @ zero_z4454100511807792257list_a )
=> ( ! [A6: multiset_list_a] :
( ( A6 != zero_z4454100511807792257list_a )
=> ( ! [X5: list_a] :
( ( member_list_a @ X5 @ ( set_mset_list_a @ A6 ) )
=> ( P @ ( minus_7431248565939055793list_a @ A6 @ ( add_mset_list_a @ X5 @ zero_z4454100511807792257list_a ) ) ) )
=> ( P @ A6 ) ) )
=> ( P @ A ) ) ) ).
% multiset_remove_induct
thf(fact_488_multiset__remove__induct,axiom,
! [P: multiset_multiset_a > $o,A: multiset_multiset_a] :
( ( P @ zero_z7799948378220188993iset_a )
=> ( ! [A6: multiset_multiset_a] :
( ( A6 != zero_z7799948378220188993iset_a )
=> ( ! [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ A6 ) )
=> ( P @ ( minus_3395427628221709681iset_a @ A6 @ ( add_mset_multiset_a @ X5 @ zero_z7799948378220188993iset_a ) ) ) )
=> ( P @ A6 ) ) )
=> ( P @ A ) ) ) ).
% multiset_remove_induct
thf(fact_489_multiset__remove__induct,axiom,
! [P: multiset_nat > $o,A: multiset_nat] :
( ( P @ zero_z7348594199698428585et_nat )
=> ( ! [A6: multiset_nat] :
( ( A6 != zero_z7348594199698428585et_nat )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_mset_nat @ A6 ) )
=> ( P @ ( minus_8522176038001411705et_nat @ A6 @ ( add_mset_nat @ X5 @ zero_z7348594199698428585et_nat ) ) ) )
=> ( P @ A6 ) ) )
=> ( P @ A ) ) ) ).
% multiset_remove_induct
thf(fact_490_multiset__remove__induct,axiom,
! [P: multiset_a > $o,A: multiset_a] :
( ( P @ zero_zero_multiset_a )
=> ( ! [A6: multiset_a] :
( ( A6 != zero_zero_multiset_a )
=> ( ! [X5: a] :
( ( member_a @ X5 @ ( set_mset_a @ A6 ) )
=> ( P @ ( minus_3765977307040488491iset_a @ A6 @ ( add_mset_a @ X5 @ zero_zero_multiset_a ) ) ) )
=> ( P @ A6 ) ) )
=> ( P @ A ) ) ) ).
% multiset_remove_induct
thf(fact_491_Suc__diff__diff,axiom,
! [M: nat,N4: nat,K3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N4 ) @ ( suc @ K3 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N4 ) @ K3 ) ) ).
% Suc_diff_diff
thf(fact_492_diff__Suc__Suc,axiom,
! [M: nat,N4: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N4 ) )
= ( minus_minus_nat @ M @ N4 ) ) ).
% diff_Suc_Suc
thf(fact_493_add__Suc__right,axiom,
! [M: nat,N4: nat] :
( ( plus_plus_nat @ M @ ( suc @ N4 ) )
= ( suc @ ( plus_plus_nat @ M @ N4 ) ) ) ).
% add_Suc_right
thf(fact_494_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_495_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_496_add__is__0,axiom,
! [M: nat,N4: nat] :
( ( ( plus_plus_nat @ M @ N4 )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N4 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_497_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_498_diff__0__eq__0,axiom,
! [N4: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N4 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_499_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_500_diff__diff__left,axiom,
! [I2: nat,J2: nat,K3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K3 )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J2 @ K3 ) ) ) ).
% diff_diff_left
thf(fact_501_plus__nat_Oadd__0,axiom,
! [N4: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N4 )
= N4 ) ).
% plus_nat.add_0
thf(fact_502_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_503_diff__add__0,axiom,
! [N4: nat,M: nat] :
( ( minus_minus_nat @ N4 @ ( plus_plus_nat @ N4 @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_504_Nat_Odiff__cancel,axiom,
! [K3: nat,M: nat,N4: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K3 @ M ) @ ( plus_plus_nat @ K3 @ N4 ) )
= ( minus_minus_nat @ M @ N4 ) ) ).
% Nat.diff_cancel
thf(fact_505_diff__cancel2,axiom,
! [M: nat,K3: nat,N4: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K3 ) @ ( plus_plus_nat @ N4 @ K3 ) )
= ( minus_minus_nat @ M @ N4 ) ) ).
% diff_cancel2
thf(fact_506_diff__commute,axiom,
! [I2: nat,J2: nat,K3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K3 )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K3 ) @ J2 ) ) ).
% diff_commute
thf(fact_507_add__eq__self__zero,axiom,
! [M: nat,N4: nat] :
( ( ( plus_plus_nat @ M @ N4 )
= M )
=> ( N4 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_508_diff__add__inverse,axiom,
! [N4: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N4 @ M ) @ N4 )
= M ) ).
% diff_add_inverse
thf(fact_509_diffs0__imp__equal,axiom,
! [M: nat,N4: nat] :
( ( ( minus_minus_nat @ M @ N4 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N4 @ M )
= zero_zero_nat )
=> ( M = N4 ) ) ) ).
% diffs0_imp_equal
thf(fact_510_diff__add__inverse2,axiom,
! [M: nat,N4: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N4 ) @ N4 )
= M ) ).
% diff_add_inverse2
thf(fact_511_Suc__inject,axiom,
! [X2: nat,Y: nat] :
( ( ( suc @ X2 )
= ( suc @ Y ) )
=> ( X2 = Y ) ) ).
% Suc_inject
thf(fact_512_n__not__Suc__n,axiom,
! [N4: nat] :
( N4
!= ( suc @ N4 ) ) ).
% n_not_Suc_n
thf(fact_513_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_514_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_515_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_516_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_517_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_518_nat__induct,axiom,
! [P: nat > $o,N4: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N5: nat] :
( ( P @ N5 )
=> ( P @ ( suc @ N5 ) ) )
=> ( P @ N4 ) ) ) ).
% nat_induct
thf(fact_519_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N4: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X4: nat,Y4: nat] :
( ( P @ X4 @ Y4 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N4 ) ) ) ) ).
% diff_induct
thf(fact_520_zero__induct,axiom,
! [P: nat > $o,K3: nat] :
( ( P @ K3 )
=> ( ! [N5: nat] :
( ( P @ ( suc @ N5 ) )
=> ( P @ N5 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_521_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_522_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_523_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_524_not0__implies__Suc,axiom,
! [N4: nat] :
( ( N4 != zero_zero_nat )
=> ? [M6: nat] :
( N4
= ( suc @ M6 ) ) ) ).
% not0_implies_Suc
thf(fact_525_nat__arith_Osuc1,axiom,
! [A: nat,K3: nat,A2: nat] :
( ( A
= ( plus_plus_nat @ K3 @ A2 ) )
=> ( ( suc @ A )
= ( plus_plus_nat @ K3 @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_526_add__Suc,axiom,
! [M: nat,N4: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N4 )
= ( suc @ ( plus_plus_nat @ M @ N4 ) ) ) ).
% add_Suc
thf(fact_527_add__Suc__shift,axiom,
! [M: nat,N4: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N4 )
= ( plus_plus_nat @ M @ ( suc @ N4 ) ) ) ).
% add_Suc_shift
thf(fact_528_zero__induct__lemma,axiom,
! [P: nat > $o,K3: nat,I2: nat] :
( ( P @ K3 )
=> ( ! [N5: nat] :
( ( P @ ( suc @ N5 ) )
=> ( P @ N5 ) )
=> ( P @ ( minus_minus_nat @ K3 @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_529_one__is__add,axiom,
! [M: nat,N4: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N4 ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N4 = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N4
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_530_add__is__1,axiom,
! [M: nat,N4: nat] :
( ( ( plus_plus_nat @ M @ N4 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N4 = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N4
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_531_subset__mset_Osum__mset__0__iff,axiom,
! [M2: multis1201202736280713200et_nat] :
( ( ( comm_m5787568287065167983et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ M2 )
= zero_z7348594199698428585et_nat )
= ( ! [X3: multiset_nat] :
( ( member_multiset_nat @ X3 @ ( set_ms4188662328148412963et_nat @ M2 ) )
=> ( X3 = zero_z7348594199698428585et_nat ) ) ) ) ).
% subset_mset.sum_mset_0_iff
thf(fact_532_subset__mset_Osum__mset__0__iff,axiom,
! [M2: multiset_multiset_a] :
( ( ( comm_m1977238983017651125iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ M2 )
= zero_zero_multiset_a )
= ( ! [X3: multiset_a] :
( ( member_multiset_a @ X3 @ ( set_mset_multiset_a @ M2 ) )
=> ( X3 = zero_zero_multiset_a ) ) ) ) ).
% subset_mset.sum_mset_0_iff
thf(fact_533_Multiset_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A5: multiset_nat] : ( A5 = zero_z7348594199698428585et_nat ) ) ) ).
% Multiset.is_empty_def
thf(fact_534_Multiset_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A5: multiset_a] : ( A5 = zero_zero_multiset_a ) ) ) ).
% Multiset.is_empty_def
thf(fact_535_wcount__add__mset,axiom,
! [F: nat > nat,X2: nat,M2: multiset_nat,A2: nat] :
( ( wcount_nat @ F @ ( add_mset_nat @ X2 @ M2 ) @ A2 )
= ( plus_plus_nat @ ( if_nat @ ( X2 = A2 ) @ ( suc @ ( F @ A2 ) ) @ zero_zero_nat ) @ ( wcount_nat @ F @ M2 @ A2 ) ) ) ).
% wcount_add_mset
thf(fact_536_wcount__add__mset,axiom,
! [F: a > nat,X2: a,M2: multiset_a,A2: a] :
( ( wcount_a @ F @ ( add_mset_a @ X2 @ M2 ) @ A2 )
= ( plus_plus_nat @ ( if_nat @ ( X2 = A2 ) @ ( suc @ ( F @ A2 ) ) @ zero_zero_nat ) @ ( wcount_a @ F @ M2 @ A2 ) ) ) ).
% wcount_add_mset
thf(fact_537_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A3: nat,B5: nat] :
( ( P @ A3 @ B5 )
= ( P @ B5 @ A3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
=> ( ! [A3: nat,B5: nat] :
( ( P @ A3 @ B5 )
=> ( P @ A3 @ ( plus_plus_nat @ A3 @ B5 ) ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_538_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N5: nat] :
( ~ ( P @ N5 )
& ( P @ ( suc @ N5 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_539_add__0__iff,axiom,
! [B2: nat,A2: nat] :
( ( B2
= ( plus_plus_nat @ B2 @ A2 ) )
= ( A2 = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_540_comm__monoid__add_Osum__mset_Ocong,axiom,
comm_m1977238983017651125iset_a = comm_m1977238983017651125iset_a ).
% comm_monoid_add.sum_mset.cong
thf(fact_541_wcount__union,axiom,
! [F: nat > nat,M2: multiset_nat,N: multiset_nat,A2: nat] :
( ( wcount_nat @ F @ ( plus_p6334493942879108393et_nat @ M2 @ N ) @ A2 )
= ( plus_plus_nat @ ( wcount_nat @ F @ M2 @ A2 ) @ ( wcount_nat @ F @ N @ A2 ) ) ) ).
% wcount_union
thf(fact_542_wcount__union,axiom,
! [F: a > nat,M2: multiset_a,N: multiset_a,A2: a] :
( ( wcount_a @ F @ ( plus_plus_multiset_a @ M2 @ N ) @ A2 )
= ( plus_plus_nat @ ( wcount_a @ F @ M2 @ A2 ) @ ( wcount_a @ F @ N @ A2 ) ) ) ).
% wcount_union
thf(fact_543_verit__sum__simplify,axiom,
! [A2: multiset_nat] :
( ( plus_p6334493942879108393et_nat @ A2 @ zero_z7348594199698428585et_nat )
= A2 ) ).
% verit_sum_simplify
thf(fact_544_verit__sum__simplify,axiom,
! [A2: multiset_a] :
( ( plus_plus_multiset_a @ A2 @ zero_zero_multiset_a )
= A2 ) ).
% verit_sum_simplify
thf(fact_545_verit__sum__simplify,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% verit_sum_simplify
thf(fact_546_size__multiset__add__mset,axiom,
! [F: nat > nat,A2: nat,M2: multiset_nat] :
( ( size_multiset_nat @ F @ ( add_mset_nat @ A2 @ M2 ) )
= ( plus_plus_nat @ ( suc @ ( F @ A2 ) ) @ ( size_multiset_nat @ F @ M2 ) ) ) ).
% size_multiset_add_mset
thf(fact_547_size__multiset__add__mset,axiom,
! [F: a > nat,A2: a,M2: multiset_a] :
( ( size_multiset_a @ F @ ( add_mset_a @ A2 @ M2 ) )
= ( plus_plus_nat @ ( suc @ ( F @ A2 ) ) @ ( size_multiset_a @ F @ M2 ) ) ) ).
% size_multiset_add_mset
thf(fact_548_permutations__of__multiset__Cons__iff,axiom,
! [X2: list_a,Xs2: list_list_a,A: multiset_list_a] :
( ( member_list_list_a @ ( cons_list_a @ X2 @ Xs2 ) @ ( multis7786709813750966868list_a @ A ) )
= ( ( member_list_a @ X2 @ ( set_mset_list_a @ A ) )
& ( member_list_list_a @ Xs2 @ ( multis7786709813750966868list_a @ ( minus_7431248565939055793list_a @ A @ ( add_mset_list_a @ X2 @ zero_z4454100511807792257list_a ) ) ) ) ) ) ).
% permutations_of_multiset_Cons_iff
thf(fact_549_permutations__of__multiset__Cons__iff,axiom,
! [X2: multiset_a,Xs2: list_multiset_a,A: multiset_multiset_a] :
( ( member1058549947519091315iset_a @ ( cons_multiset_a @ X2 @ Xs2 ) @ ( multis8172983762386353940iset_a @ A ) )
= ( ( member_multiset_a @ X2 @ ( set_mset_multiset_a @ A ) )
& ( member1058549947519091315iset_a @ Xs2 @ ( multis8172983762386353940iset_a @ ( minus_3395427628221709681iset_a @ A @ ( add_mset_multiset_a @ X2 @ zero_z7799948378220188993iset_a ) ) ) ) ) ) ).
% permutations_of_multiset_Cons_iff
thf(fact_550_permutations__of__multiset__Cons__iff,axiom,
! [X2: nat,Xs2: list_nat,A: multiset_nat] :
( ( member_list_nat @ ( cons_nat @ X2 @ Xs2 ) @ ( multis6201468865946971392et_nat @ A ) )
= ( ( member_nat @ X2 @ ( set_mset_nat @ A ) )
& ( member_list_nat @ Xs2 @ ( multis6201468865946971392et_nat @ ( minus_8522176038001411705et_nat @ A @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) ) ) ) ) ) ).
% permutations_of_multiset_Cons_iff
thf(fact_551_permutations__of__multiset__Cons__iff,axiom,
! [X2: a,Xs2: list_a,A: multiset_a] :
( ( member_list_a @ ( cons_a @ X2 @ Xs2 ) @ ( multis5886240593633752526iset_a @ A ) )
= ( ( member_a @ X2 @ ( set_mset_a @ A ) )
& ( member_list_a @ Xs2 @ ( multis5886240593633752526iset_a @ ( minus_3765977307040488491iset_a @ A @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) ) ) ) ) ) ).
% permutations_of_multiset_Cons_iff
thf(fact_552_in__mset__fold__plus__iff,axiom,
! [X2: list_a,M2: multiset_list_a,NN: multis272298859511629456list_a] :
( ( member_list_a @ X2 @ ( set_mset_list_a @ ( fold_m7881050568294030859list_a @ plus_p690419498615200257list_a @ M2 @ NN ) ) )
= ( ( member_list_a @ X2 @ ( set_mset_list_a @ M2 ) )
| ? [N2: multiset_list_a] :
( ( member391257453028489331list_a @ N2 @ ( set_ms6457238410686372999list_a @ NN ) )
& ( member_list_a @ X2 @ ( set_mset_list_a @ N2 ) ) ) ) ) ).
% in_mset_fold_plus_iff
thf(fact_553_in__mset__fold__plus__iff,axiom,
! [X2: multiset_a,M2: multiset_multiset_a,NN: multis8895438461125693264iset_a] :
( ( member_multiset_a @ X2 @ ( set_mset_multiset_a @ ( fold_m7435756534263755787iset_a @ plus_p6738641960240532161iset_a @ M2 @ NN ) ) )
= ( ( member_multiset_a @ X2 @ ( set_mset_multiset_a @ M2 ) )
| ? [N2: multiset_multiset_a] :
( ( member7618379257985549619iset_a @ N2 @ ( set_ms6500351741012541767iset_a @ NN ) )
& ( member_multiset_a @ X2 @ ( set_mset_multiset_a @ N2 ) ) ) ) ) ).
% in_mset_fold_plus_iff
thf(fact_554_in__mset__fold__plus__iff,axiom,
! [X2: nat,M2: multiset_nat,NN: multis1201202736280713200et_nat] :
( ( member_nat @ X2 @ ( set_mset_nat @ ( fold_m1829410296857755981et_nat @ plus_p6334493942879108393et_nat @ M2 @ NN ) ) )
= ( ( member_nat @ X2 @ ( set_mset_nat @ M2 ) )
| ? [N2: multiset_nat] :
( ( member_multiset_nat @ N2 @ ( set_ms4188662328148412963et_nat @ NN ) )
& ( member_nat @ X2 @ ( set_mset_nat @ N2 ) ) ) ) ) ).
% in_mset_fold_plus_iff
thf(fact_555_in__mset__fold__plus__iff,axiom,
! [X2: a,M2: multiset_a,NN: multiset_multiset_a] :
( ( member_a @ X2 @ ( set_mset_a @ ( fold_m6601649825673331723iset_a @ plus_plus_multiset_a @ M2 @ NN ) ) )
= ( ( member_a @ X2 @ ( set_mset_a @ M2 ) )
| ? [N2: multiset_a] :
( ( member_multiset_a @ N2 @ ( set_mset_multiset_a @ NN ) )
& ( member_a @ X2 @ ( set_mset_a @ N2 ) ) ) ) ) ).
% in_mset_fold_plus_iff
thf(fact_556_sorted__list__of__multiset__singleton,axiom,
! [X2: nat] :
( ( linord3047872887403683810et_nat @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) )
= ( cons_nat @ X2 @ nil_nat ) ) ).
% sorted_list_of_multiset_singleton
thf(fact_557_mset__remove1,axiom,
! [A2: nat,Xs2: list_nat] :
( ( mset_nat @ ( remove1_nat @ A2 @ Xs2 ) )
= ( minus_8522176038001411705et_nat @ ( mset_nat @ Xs2 ) @ ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) ) ) ).
% mset_remove1
thf(fact_558_mset__remove1,axiom,
! [A2: a,Xs2: list_a] :
( ( mset_a @ ( remove1_a @ A2 @ Xs2 ) )
= ( minus_3765977307040488491iset_a @ ( mset_a @ Xs2 ) @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) ) ).
% mset_remove1
thf(fact_559_mset__single__iff__right,axiom,
! [X2: list_a,Xs2: list_list_a] :
( ( ( add_mset_list_a @ X2 @ zero_z4454100511807792257list_a )
= ( mset_list_a @ Xs2 ) )
= ( Xs2
= ( cons_list_a @ X2 @ nil_list_a ) ) ) ).
% mset_single_iff_right
thf(fact_560_mset__single__iff__right,axiom,
! [X2: nat,Xs2: list_nat] :
( ( ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat )
= ( mset_nat @ Xs2 ) )
= ( Xs2
= ( cons_nat @ X2 @ nil_nat ) ) ) ).
% mset_single_iff_right
thf(fact_561_mset__single__iff__right,axiom,
! [X2: a,Xs2: list_a] :
( ( ( add_mset_a @ X2 @ zero_zero_multiset_a )
= ( mset_a @ Xs2 ) )
= ( Xs2
= ( cons_a @ X2 @ nil_a ) ) ) ).
% mset_single_iff_right
thf(fact_562_append_Oright__neutral,axiom,
! [A2: list_list_a] :
( ( append_list_a @ A2 @ nil_list_a )
= A2 ) ).
% append.right_neutral
thf(fact_563_append_Oright__neutral,axiom,
! [A2: list_a] :
( ( append_a @ A2 @ nil_a )
= A2 ) ).
% append.right_neutral
thf(fact_564_append__Nil2,axiom,
! [Xs2: list_list_a] :
( ( append_list_a @ Xs2 @ nil_list_a )
= Xs2 ) ).
% append_Nil2
thf(fact_565_append__Nil2,axiom,
! [Xs2: list_a] :
( ( append_a @ Xs2 @ nil_a )
= Xs2 ) ).
% append_Nil2
thf(fact_566_append__self__conv,axiom,
! [Xs2: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs2 @ Ys )
= Xs2 )
= ( Ys = nil_list_a ) ) ).
% append_self_conv
thf(fact_567_append__self__conv,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( append_a @ Xs2 @ Ys )
= Xs2 )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_568_self__append__conv,axiom,
! [Y: list_list_a,Ys: list_list_a] :
( ( Y
= ( append_list_a @ Y @ Ys ) )
= ( Ys = nil_list_a ) ) ).
% self_append_conv
thf(fact_569_self__append__conv,axiom,
! [Y: list_a,Ys: list_a] :
( ( Y
= ( append_a @ Y @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_570_append__self__conv2,axiom,
! [Xs2: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs2 @ Ys )
= Ys )
= ( Xs2 = nil_list_a ) ) ).
% append_self_conv2
thf(fact_571_append__self__conv2,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( append_a @ Xs2 @ Ys )
= Ys )
= ( Xs2 = nil_a ) ) ).
% append_self_conv2
thf(fact_572_self__append__conv2,axiom,
! [Y: list_list_a,Xs2: list_list_a] :
( ( Y
= ( append_list_a @ Xs2 @ Y ) )
= ( Xs2 = nil_list_a ) ) ).
% self_append_conv2
thf(fact_573_self__append__conv2,axiom,
! [Y: list_a,Xs2: list_a] :
( ( Y
= ( append_a @ Xs2 @ Y ) )
= ( Xs2 = nil_a ) ) ).
% self_append_conv2
thf(fact_574_Nil__is__append__conv,axiom,
! [Xs2: list_list_a,Ys: list_list_a] :
( ( nil_list_a
= ( append_list_a @ Xs2 @ Ys ) )
= ( ( Xs2 = nil_list_a )
& ( Ys = nil_list_a ) ) ) ).
% Nil_is_append_conv
thf(fact_575_Nil__is__append__conv,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( nil_a
= ( append_a @ Xs2 @ Ys ) )
= ( ( Xs2 = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_576_append__is__Nil__conv,axiom,
! [Xs2: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs2 @ Ys )
= nil_list_a )
= ( ( Xs2 = nil_list_a )
& ( Ys = nil_list_a ) ) ) ).
% append_is_Nil_conv
thf(fact_577_append__is__Nil__conv,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( append_a @ Xs2 @ Ys )
= nil_a )
= ( ( Xs2 = nil_a )
& ( Ys = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_578_in__set__remove1,axiom,
! [A2: nat,B2: nat,Xs2: list_nat] :
( ( A2 != B2 )
=> ( ( member_nat @ A2 @ ( set_nat2 @ ( remove1_nat @ B2 @ Xs2 ) ) )
= ( member_nat @ A2 @ ( set_nat2 @ Xs2 ) ) ) ) ).
% in_set_remove1
thf(fact_579_in__set__remove1,axiom,
! [A2: list_a,B2: list_a,Xs2: list_list_a] :
( ( A2 != B2 )
=> ( ( member_list_a @ A2 @ ( set_list_a2 @ ( remove1_list_a @ B2 @ Xs2 ) ) )
= ( member_list_a @ A2 @ ( set_list_a2 @ Xs2 ) ) ) ) ).
% in_set_remove1
thf(fact_580_in__set__remove1,axiom,
! [A2: multiset_a,B2: multiset_a,Xs2: list_multiset_a] :
( ( A2 != B2 )
=> ( ( member_multiset_a @ A2 @ ( set_multiset_a2 @ ( remove1_multiset_a @ B2 @ Xs2 ) ) )
= ( member_multiset_a @ A2 @ ( set_multiset_a2 @ Xs2 ) ) ) ) ).
% in_set_remove1
thf(fact_581_in__set__remove1,axiom,
! [A2: a,B2: a,Xs2: list_a] :
( ( A2 != B2 )
=> ( ( member_a @ A2 @ ( set_a2 @ ( remove1_a @ B2 @ Xs2 ) ) )
= ( member_a @ A2 @ ( set_a2 @ Xs2 ) ) ) ) ).
% in_set_remove1
thf(fact_582_fold__mset__empty,axiom,
! [F: multiset_a > multiset_a > multiset_a,S: multiset_a] :
( ( fold_m6601649825673331723iset_a @ F @ S @ zero_z7799948378220188993iset_a )
= S ) ).
% fold_mset_empty
thf(fact_583_fold__mset__empty,axiom,
! [F: a > multiset_a > multiset_a,S: multiset_a] :
( ( fold_m7320414754419674833iset_a @ F @ S @ zero_zero_multiset_a )
= S ) ).
% fold_mset_empty
thf(fact_584_append1__eq__conv,axiom,
! [Xs2: list_list_a,X2: list_a,Ys: list_list_a,Y: list_a] :
( ( ( append_list_a @ Xs2 @ ( cons_list_a @ X2 @ nil_list_a ) )
= ( append_list_a @ Ys @ ( cons_list_a @ Y @ nil_list_a ) ) )
= ( ( Xs2 = Ys )
& ( X2 = Y ) ) ) ).
% append1_eq_conv
thf(fact_585_append1__eq__conv,axiom,
! [Xs2: list_a,X2: a,Ys: list_a,Y: a] :
( ( ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
= ( ( Xs2 = Ys )
& ( X2 = Y ) ) ) ).
% append1_eq_conv
thf(fact_586_take__eq__Nil2,axiom,
! [N4: nat,Xs2: list_list_a] :
( ( nil_list_a
= ( take_list_a @ N4 @ Xs2 ) )
= ( ( N4 = zero_zero_nat )
| ( Xs2 = nil_list_a ) ) ) ).
% take_eq_Nil2
thf(fact_587_take__eq__Nil2,axiom,
! [N4: nat,Xs2: list_a] :
( ( nil_a
= ( take_a @ N4 @ Xs2 ) )
= ( ( N4 = zero_zero_nat )
| ( Xs2 = nil_a ) ) ) ).
% take_eq_Nil2
thf(fact_588_take__eq__Nil,axiom,
! [N4: nat,Xs2: list_list_a] :
( ( ( take_list_a @ N4 @ Xs2 )
= nil_list_a )
= ( ( N4 = zero_zero_nat )
| ( Xs2 = nil_list_a ) ) ) ).
% take_eq_Nil
thf(fact_589_take__eq__Nil,axiom,
! [N4: nat,Xs2: list_a] :
( ( ( take_a @ N4 @ Xs2 )
= nil_a )
= ( ( N4 = zero_zero_nat )
| ( Xs2 = nil_a ) ) ) ).
% take_eq_Nil
thf(fact_590_take0,axiom,
( ( take_list_a @ zero_zero_nat )
= ( ^ [Xs4: list_list_a] : nil_list_a ) ) ).
% take0
thf(fact_591_take0,axiom,
( ( take_a @ zero_zero_nat )
= ( ^ [Xs4: list_a] : nil_a ) ) ).
% take0
thf(fact_592_mset__zero__iff,axiom,
! [X2: list_list_a] :
( ( ( mset_list_a @ X2 )
= zero_z4454100511807792257list_a )
= ( X2 = nil_list_a ) ) ).
% mset_zero_iff
thf(fact_593_mset__zero__iff,axiom,
! [X2: list_nat] :
( ( ( mset_nat @ X2 )
= zero_z7348594199698428585et_nat )
= ( X2 = nil_nat ) ) ).
% mset_zero_iff
thf(fact_594_mset__zero__iff,axiom,
! [X2: list_a] :
( ( ( mset_a @ X2 )
= zero_zero_multiset_a )
= ( X2 = nil_a ) ) ).
% mset_zero_iff
thf(fact_595_mset__zero__iff__right,axiom,
! [X2: list_list_a] :
( ( zero_z4454100511807792257list_a
= ( mset_list_a @ X2 ) )
= ( X2 = nil_list_a ) ) ).
% mset_zero_iff_right
thf(fact_596_mset__zero__iff__right,axiom,
! [X2: list_nat] :
( ( zero_z7348594199698428585et_nat
= ( mset_nat @ X2 ) )
= ( X2 = nil_nat ) ) ).
% mset_zero_iff_right
thf(fact_597_mset__zero__iff__right,axiom,
! [X2: list_a] :
( ( zero_zero_multiset_a
= ( mset_a @ X2 ) )
= ( X2 = nil_a ) ) ).
% mset_zero_iff_right
thf(fact_598_size__multiset__empty,axiom,
! [F: nat > nat] :
( ( size_multiset_nat @ F @ zero_z7348594199698428585et_nat )
= zero_zero_nat ) ).
% size_multiset_empty
thf(fact_599_size__multiset__empty,axiom,
! [F: a > nat] :
( ( size_multiset_a @ F @ zero_zero_multiset_a )
= zero_zero_nat ) ).
% size_multiset_empty
thf(fact_600_size__multiset__eq__0__iff__empty,axiom,
! [F: nat > nat,M2: multiset_nat] :
( ( ( size_multiset_nat @ F @ M2 )
= zero_zero_nat )
= ( M2 = zero_z7348594199698428585et_nat ) ) ).
% size_multiset_eq_0_iff_empty
thf(fact_601_size__multiset__eq__0__iff__empty,axiom,
! [F: a > nat,M2: multiset_a] :
( ( ( size_multiset_a @ F @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_multiset_a ) ) ).
% size_multiset_eq_0_iff_empty
thf(fact_602_sorted__list__of__multiset__empty,axiom,
( ( linord3047872887403683810et_nat @ zero_z7348594199698428585et_nat )
= nil_nat ) ).
% sorted_list_of_multiset_empty
thf(fact_603_sorted__list__of__multiset__eq__Nil,axiom,
! [M2: multiset_nat] :
( ( ( linord3047872887403683810et_nat @ M2 )
= nil_nat )
= ( M2 = zero_z7348594199698428585et_nat ) ) ).
% sorted_list_of_multiset_eq_Nil
thf(fact_604_size__multiset__union,axiom,
! [F: nat > nat,M2: multiset_nat,N: multiset_nat] :
( ( size_multiset_nat @ F @ ( plus_p6334493942879108393et_nat @ M2 @ N ) )
= ( plus_plus_nat @ ( size_multiset_nat @ F @ M2 ) @ ( size_multiset_nat @ F @ N ) ) ) ).
% size_multiset_union
thf(fact_605_size__multiset__union,axiom,
! [F: a > nat,M2: multiset_a,N: multiset_a] :
( ( size_multiset_a @ F @ ( plus_plus_multiset_a @ M2 @ N ) )
= ( plus_plus_nat @ ( size_multiset_a @ F @ M2 ) @ ( size_multiset_a @ F @ N ) ) ) ).
% size_multiset_union
thf(fact_606_list__of__mset__empty,axiom,
! [M: multiset_list_a] :
( ( ( multis5009215944268684175list_a @ M )
= nil_list_a )
= ( M = zero_z4454100511807792257list_a ) ) ).
% list_of_mset_empty
thf(fact_607_list__of__mset__empty,axiom,
! [M: multiset_nat] :
( ( ( multis105632648212199813et_nat @ M )
= nil_nat )
= ( M = zero_z7348594199698428585et_nat ) ) ).
% list_of_mset_empty
thf(fact_608_list__of__mset__empty,axiom,
! [M: multiset_a] :
( ( ( multis4723169673647964297mset_a @ M )
= nil_a )
= ( M = zero_zero_multiset_a ) ) ).
% list_of_mset_empty
thf(fact_609_mset__single__iff,axiom,
! [Xs2: list_list_a,X2: list_a] :
( ( ( mset_list_a @ Xs2 )
= ( add_mset_list_a @ X2 @ zero_z4454100511807792257list_a ) )
= ( Xs2
= ( cons_list_a @ X2 @ nil_list_a ) ) ) ).
% mset_single_iff
thf(fact_610_mset__single__iff,axiom,
! [Xs2: list_nat,X2: nat] :
( ( ( mset_nat @ Xs2 )
= ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) )
= ( Xs2
= ( cons_nat @ X2 @ nil_nat ) ) ) ).
% mset_single_iff
thf(fact_611_mset__single__iff,axiom,
! [Xs2: list_a,X2: a] :
( ( ( mset_a @ Xs2 )
= ( add_mset_a @ X2 @ zero_zero_multiset_a ) )
= ( Xs2
= ( cons_a @ X2 @ nil_a ) ) ) ).
% mset_single_iff
thf(fact_612_transpose_Ocases,axiom,
! [X2: list_list_list_a] :
( ( X2 != nil_list_list_a )
=> ( ! [Xss: list_list_list_a] :
( X2
!= ( cons_list_list_a @ nil_list_a @ Xss ) )
=> ~ ! [X4: list_a,Xs: list_list_a,Xss: list_list_list_a] :
( X2
!= ( cons_list_list_a @ ( cons_list_a @ X4 @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_613_transpose_Ocases,axiom,
! [X2: list_list_a] :
( ( X2 != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X2
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X4: a,Xs: list_a,Xss: list_list_a] :
( X2
!= ( cons_list_a @ ( cons_a @ X4 @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_614_remove1__commute,axiom,
! [X2: a,Y: a,Zs: list_a] :
( ( remove1_a @ X2 @ ( remove1_a @ Y @ Zs ) )
= ( remove1_a @ Y @ ( remove1_a @ X2 @ Zs ) ) ) ).
% remove1_commute
thf(fact_615_remove1_Osimps_I1_J,axiom,
! [X2: a] :
( ( remove1_a @ X2 @ nil_a )
= nil_a ) ).
% remove1.simps(1)
thf(fact_616_remove1_Osimps_I1_J,axiom,
! [X2: list_a] :
( ( remove1_list_a @ X2 @ nil_list_a )
= nil_list_a ) ).
% remove1.simps(1)
thf(fact_617_elem__permutation__of__mset__empty__iff,axiom,
! [Xs2: list_list_a,A: multiset_list_a] :
( ( member_list_list_a @ Xs2 @ ( multis7786709813750966868list_a @ A ) )
=> ( ( Xs2 = nil_list_a )
= ( A = zero_z4454100511807792257list_a ) ) ) ).
% elem_permutation_of_mset_empty_iff
thf(fact_618_elem__permutation__of__mset__empty__iff,axiom,
! [Xs2: list_nat,A: multiset_nat] :
( ( member_list_nat @ Xs2 @ ( multis6201468865946971392et_nat @ A ) )
=> ( ( Xs2 = nil_nat )
= ( A = zero_z7348594199698428585et_nat ) ) ) ).
% elem_permutation_of_mset_empty_iff
thf(fact_619_elem__permutation__of__mset__empty__iff,axiom,
! [Xs2: list_a,A: multiset_a] :
( ( member_list_a @ Xs2 @ ( multis5886240593633752526iset_a @ A ) )
=> ( ( Xs2 = nil_a )
= ( A = zero_zero_multiset_a ) ) ) ).
% elem_permutation_of_mset_empty_iff
thf(fact_620_list_Odistinct_I1_J,axiom,
! [X21: list_a,X22: list_list_a] :
( nil_list_a
!= ( cons_list_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_621_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_622_list_OdiscI,axiom,
! [List: list_list_a,X21: list_a,X22: list_list_a] :
( ( List
= ( cons_list_a @ X21 @ X22 ) )
=> ( List != nil_list_a ) ) ).
% list.discI
thf(fact_623_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_624_list_Oexhaust,axiom,
! [Y: list_list_a] :
( ( Y != nil_list_a )
=> ~ ! [X212: list_a,X222: list_list_a] :
( Y
!= ( cons_list_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_625_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_626_remdups__adj_Ocases,axiom,
! [X2: list_list_a] :
( ( X2 != nil_list_a )
=> ( ! [X4: list_a] :
( X2
!= ( cons_list_a @ X4 @ nil_list_a ) )
=> ~ ! [X4: list_a,Y4: list_a,Xs: list_list_a] :
( X2
!= ( cons_list_a @ X4 @ ( cons_list_a @ Y4 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_627_remdups__adj_Ocases,axiom,
! [X2: list_a] :
( ( X2 != nil_a )
=> ( ! [X4: a] :
( X2
!= ( cons_a @ X4 @ nil_a ) )
=> ~ ! [X4: a,Y4: a,Xs: list_a] :
( X2
!= ( cons_a @ X4 @ ( cons_a @ Y4 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_628_neq__Nil__conv,axiom,
! [Xs2: list_list_a] :
( ( Xs2 != nil_list_a )
= ( ? [Y3: list_a,Ys3: list_list_a] :
( Xs2
= ( cons_list_a @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_629_neq__Nil__conv,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
= ( ? [Y3: a,Ys3: list_a] :
( Xs2
= ( cons_a @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_630_list__induct2_H,axiom,
! [P: list_a > list_list_a > $o,Xs2: list_a,Ys: list_list_a] :
( ( P @ nil_a @ nil_list_a )
=> ( ! [X4: a,Xs: list_a] : ( P @ ( cons_a @ X4 @ Xs ) @ nil_list_a )
=> ( ! [Y4: list_a,Ys2: list_list_a] : ( P @ nil_a @ ( cons_list_a @ Y4 @ Ys2 ) )
=> ( ! [X4: a,Xs: list_a,Y4: list_a,Ys2: list_list_a] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_a @ X4 @ Xs ) @ ( cons_list_a @ Y4 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_631_list__induct2_H,axiom,
! [P: list_list_a > list_a > $o,Xs2: list_list_a,Ys: list_a] :
( ( P @ nil_list_a @ nil_a )
=> ( ! [X4: list_a,Xs: list_list_a] : ( P @ ( cons_list_a @ X4 @ Xs ) @ nil_a )
=> ( ! [Y4: a,Ys2: list_a] : ( P @ nil_list_a @ ( cons_a @ Y4 @ Ys2 ) )
=> ( ! [X4: list_a,Xs: list_list_a,Y4: a,Ys2: list_a] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_list_a @ X4 @ Xs ) @ ( cons_a @ Y4 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_632_list__induct2_H,axiom,
! [P: list_list_a > list_list_a > $o,Xs2: list_list_a,Ys: list_list_a] :
( ( P @ nil_list_a @ nil_list_a )
=> ( ! [X4: list_a,Xs: list_list_a] : ( P @ ( cons_list_a @ X4 @ Xs ) @ nil_list_a )
=> ( ! [Y4: list_a,Ys2: list_list_a] : ( P @ nil_list_a @ ( cons_list_a @ Y4 @ Ys2 ) )
=> ( ! [X4: list_a,Xs: list_list_a,Y4: list_a,Ys2: list_list_a] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_list_a @ X4 @ Xs ) @ ( cons_list_a @ Y4 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_633_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs2: list_a,Ys: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X4: a,Xs: list_a] : ( P @ ( cons_a @ X4 @ Xs ) @ nil_a )
=> ( ! [Y4: a,Ys2: list_a] : ( P @ nil_a @ ( cons_a @ Y4 @ Ys2 ) )
=> ( ! [X4: a,Xs: list_a,Y4: a,Ys2: list_a] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_a @ X4 @ Xs ) @ ( cons_a @ Y4 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_634_list__nonempty__induct,axiom,
! [Xs2: list_list_a,P: list_list_a > $o] :
( ( Xs2 != nil_list_a )
=> ( ! [X4: list_a] : ( P @ ( cons_list_a @ X4 @ nil_list_a ) )
=> ( ! [X4: list_a,Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( P @ Xs )
=> ( P @ ( cons_list_a @ X4 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_635_list__nonempty__induct,axiom,
! [Xs2: list_a,P: list_a > $o] :
( ( Xs2 != nil_a )
=> ( ! [X4: a] : ( P @ ( cons_a @ X4 @ nil_a ) )
=> ( ! [X4: a,Xs: list_a] :
( ( Xs != nil_a )
=> ( ( P @ Xs )
=> ( P @ ( cons_a @ X4 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_636_remove1_Osimps_I2_J,axiom,
! [X2: list_a,Y: list_a,Xs2: list_list_a] :
( ( ( X2 = Y )
=> ( ( remove1_list_a @ X2 @ ( cons_list_a @ Y @ Xs2 ) )
= Xs2 ) )
& ( ( X2 != Y )
=> ( ( remove1_list_a @ X2 @ ( cons_list_a @ Y @ Xs2 ) )
= ( cons_list_a @ Y @ ( remove1_list_a @ X2 @ Xs2 ) ) ) ) ) ).
% remove1.simps(2)
thf(fact_637_remove1_Osimps_I2_J,axiom,
! [X2: a,Y: a,Xs2: list_a] :
( ( ( X2 = Y )
=> ( ( remove1_a @ X2 @ ( cons_a @ Y @ Xs2 ) )
= Xs2 ) )
& ( ( X2 != Y )
=> ( ( remove1_a @ X2 @ ( cons_a @ Y @ Xs2 ) )
= ( cons_a @ Y @ ( remove1_a @ X2 @ Xs2 ) ) ) ) ) ).
% remove1.simps(2)
thf(fact_638_append__Nil,axiom,
! [Ys: list_list_a] :
( ( append_list_a @ nil_list_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_639_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_640_append_Oleft__neutral,axiom,
! [A2: list_list_a] :
( ( append_list_a @ nil_list_a @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_641_append_Oleft__neutral,axiom,
! [A2: list_a] :
( ( append_a @ nil_a @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_642_eq__Nil__appendI,axiom,
! [Xs2: list_list_a,Ys: list_list_a] :
( ( Xs2 = Ys )
=> ( Xs2
= ( append_list_a @ nil_list_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_643_eq__Nil__appendI,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( Xs2 = Ys )
=> ( Xs2
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_644_remove1__idem,axiom,
! [X2: nat,Xs2: list_nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ( remove1_nat @ X2 @ Xs2 )
= Xs2 ) ) ).
% remove1_idem
thf(fact_645_remove1__idem,axiom,
! [X2: list_a,Xs2: list_list_a] :
( ~ ( member_list_a @ X2 @ ( set_list_a2 @ Xs2 ) )
=> ( ( remove1_list_a @ X2 @ Xs2 )
= Xs2 ) ) ).
% remove1_idem
thf(fact_646_remove1__idem,axiom,
! [X2: multiset_a,Xs2: list_multiset_a] :
( ~ ( member_multiset_a @ X2 @ ( set_multiset_a2 @ Xs2 ) )
=> ( ( remove1_multiset_a @ X2 @ Xs2 )
= Xs2 ) ) ).
% remove1_idem
thf(fact_647_remove1__idem,axiom,
! [X2: a,Xs2: list_a] :
( ~ ( member_a @ X2 @ ( set_a2 @ Xs2 ) )
=> ( ( remove1_a @ X2 @ Xs2 )
= Xs2 ) ) ).
% remove1_idem
thf(fact_648_notin__set__remove1,axiom,
! [X2: nat,Xs2: list_nat,Y: nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ~ ( member_nat @ X2 @ ( set_nat2 @ ( remove1_nat @ Y @ Xs2 ) ) ) ) ).
% notin_set_remove1
thf(fact_649_notin__set__remove1,axiom,
! [X2: list_a,Xs2: list_list_a,Y: list_a] :
( ~ ( member_list_a @ X2 @ ( set_list_a2 @ Xs2 ) )
=> ~ ( member_list_a @ X2 @ ( set_list_a2 @ ( remove1_list_a @ Y @ Xs2 ) ) ) ) ).
% notin_set_remove1
thf(fact_650_notin__set__remove1,axiom,
! [X2: multiset_a,Xs2: list_multiset_a,Y: multiset_a] :
( ~ ( member_multiset_a @ X2 @ ( set_multiset_a2 @ Xs2 ) )
=> ~ ( member_multiset_a @ X2 @ ( set_multiset_a2 @ ( remove1_multiset_a @ Y @ Xs2 ) ) ) ) ).
% notin_set_remove1
thf(fact_651_notin__set__remove1,axiom,
! [X2: a,Xs2: list_a,Y: a] :
( ~ ( member_a @ X2 @ ( set_a2 @ Xs2 ) )
=> ~ ( member_a @ X2 @ ( set_a2 @ ( remove1_a @ Y @ Xs2 ) ) ) ) ).
% notin_set_remove1
thf(fact_652_take__Nil,axiom,
! [N4: nat] :
( ( take_list_a @ N4 @ nil_list_a )
= nil_list_a ) ).
% take_Nil
thf(fact_653_take__Nil,axiom,
! [N4: nat] :
( ( take_a @ N4 @ nil_a )
= nil_a ) ).
% take_Nil
thf(fact_654_drop__Nil,axiom,
! [N4: nat] :
( ( drop_list_a @ N4 @ nil_list_a )
= nil_list_a ) ).
% drop_Nil
thf(fact_655_drop__Nil,axiom,
! [N4: nat] :
( ( drop_a @ N4 @ nil_a )
= nil_a ) ).
% drop_Nil
thf(fact_656_permutations__of__multisetI,axiom,
! [Xs2: list_a,A: multiset_a] :
( ( ( mset_a @ Xs2 )
= A )
=> ( member_list_a @ Xs2 @ ( multis5886240593633752526iset_a @ A ) ) ) ).
% permutations_of_multisetI
thf(fact_657_permutations__of__multisetD,axiom,
! [Xs2: list_a,A: multiset_a] :
( ( member_list_a @ Xs2 @ ( multis5886240593633752526iset_a @ A ) )
=> ( ( mset_a @ Xs2 )
= A ) ) ).
% permutations_of_multisetD
thf(fact_658_rev__nonempty__induct,axiom,
! [Xs2: list_list_a,P: list_list_a > $o] :
( ( Xs2 != nil_list_a )
=> ( ! [X4: list_a] : ( P @ ( cons_list_a @ X4 @ nil_list_a ) )
=> ( ! [X4: list_a,Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( P @ Xs )
=> ( P @ ( append_list_a @ Xs @ ( cons_list_a @ X4 @ nil_list_a ) ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_659_rev__nonempty__induct,axiom,
! [Xs2: list_a,P: list_a > $o] :
( ( Xs2 != nil_a )
=> ( ! [X4: a] : ( P @ ( cons_a @ X4 @ nil_a ) )
=> ( ! [X4: a,Xs: list_a] :
( ( Xs != nil_a )
=> ( ( P @ Xs )
=> ( P @ ( append_a @ Xs @ ( cons_a @ X4 @ nil_a ) ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_660_append__eq__Cons__conv,axiom,
! [Ys: list_list_a,Zs: list_list_a,X2: list_a,Xs2: list_list_a] :
( ( ( append_list_a @ Ys @ Zs )
= ( cons_list_a @ X2 @ Xs2 ) )
= ( ( ( Ys = nil_list_a )
& ( Zs
= ( cons_list_a @ X2 @ Xs2 ) ) )
| ? [Ys5: list_list_a] :
( ( Ys
= ( cons_list_a @ X2 @ Ys5 ) )
& ( ( append_list_a @ Ys5 @ Zs )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_661_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X2: a,Xs2: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X2 @ Xs2 ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X2 @ Xs2 ) ) )
| ? [Ys5: list_a] :
( ( Ys
= ( cons_a @ X2 @ Ys5 ) )
& ( ( append_a @ Ys5 @ Zs )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_662_Cons__eq__append__conv,axiom,
! [X2: list_a,Xs2: list_list_a,Ys: list_list_a,Zs: list_list_a] :
( ( ( cons_list_a @ X2 @ Xs2 )
= ( append_list_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_list_a )
& ( ( cons_list_a @ X2 @ Xs2 )
= Zs ) )
| ? [Ys5: list_list_a] :
( ( ( cons_list_a @ X2 @ Ys5 )
= Ys )
& ( Xs2
= ( append_list_a @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_663_Cons__eq__append__conv,axiom,
! [X2: a,Xs2: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X2 @ Xs2 )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X2 @ Xs2 )
= Zs ) )
| ? [Ys5: list_a] :
( ( ( cons_a @ X2 @ Ys5 )
= Ys )
& ( Xs2
= ( append_a @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_664_rev__exhaust,axiom,
! [Xs2: list_list_a] :
( ( Xs2 != nil_list_a )
=> ~ ! [Ys2: list_list_a,Y4: list_a] :
( Xs2
!= ( append_list_a @ Ys2 @ ( cons_list_a @ Y4 @ nil_list_a ) ) ) ) ).
% rev_exhaust
thf(fact_665_rev__exhaust,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
=> ~ ! [Ys2: list_a,Y4: a] :
( Xs2
!= ( append_a @ Ys2 @ ( cons_a @ Y4 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_666_rev__induct,axiom,
! [P: list_list_a > $o,Xs2: list_list_a] :
( ( P @ nil_list_a )
=> ( ! [X4: list_a,Xs: list_list_a] :
( ( P @ Xs )
=> ( P @ ( append_list_a @ Xs @ ( cons_list_a @ X4 @ nil_list_a ) ) ) )
=> ( P @ Xs2 ) ) ) ).
% rev_induct
thf(fact_667_rev__induct,axiom,
! [P: list_a > $o,Xs2: list_a] :
( ( P @ nil_a )
=> ( ! [X4: a,Xs: list_a] :
( ( P @ Xs )
=> ( P @ ( append_a @ Xs @ ( cons_a @ X4 @ nil_a ) ) ) )
=> ( P @ Xs2 ) ) ) ).
% rev_induct
thf(fact_668_remove1__append,axiom,
! [X2: nat,Xs2: list_nat,Ys: list_nat] :
( ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ( remove1_nat @ X2 @ ( append_nat @ Xs2 @ Ys ) )
= ( append_nat @ ( remove1_nat @ X2 @ Xs2 ) @ Ys ) ) )
& ( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ( remove1_nat @ X2 @ ( append_nat @ Xs2 @ Ys ) )
= ( append_nat @ Xs2 @ ( remove1_nat @ X2 @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_669_remove1__append,axiom,
! [X2: list_a,Xs2: list_list_a,Ys: list_list_a] :
( ( ( member_list_a @ X2 @ ( set_list_a2 @ Xs2 ) )
=> ( ( remove1_list_a @ X2 @ ( append_list_a @ Xs2 @ Ys ) )
= ( append_list_a @ ( remove1_list_a @ X2 @ Xs2 ) @ Ys ) ) )
& ( ~ ( member_list_a @ X2 @ ( set_list_a2 @ Xs2 ) )
=> ( ( remove1_list_a @ X2 @ ( append_list_a @ Xs2 @ Ys ) )
= ( append_list_a @ Xs2 @ ( remove1_list_a @ X2 @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_670_remove1__append,axiom,
! [X2: multiset_a,Xs2: list_multiset_a,Ys: list_multiset_a] :
( ( ( member_multiset_a @ X2 @ ( set_multiset_a2 @ Xs2 ) )
=> ( ( remove1_multiset_a @ X2 @ ( append_multiset_a @ Xs2 @ Ys ) )
= ( append_multiset_a @ ( remove1_multiset_a @ X2 @ Xs2 ) @ Ys ) ) )
& ( ~ ( member_multiset_a @ X2 @ ( set_multiset_a2 @ Xs2 ) )
=> ( ( remove1_multiset_a @ X2 @ ( append_multiset_a @ Xs2 @ Ys ) )
= ( append_multiset_a @ Xs2 @ ( remove1_multiset_a @ X2 @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_671_remove1__append,axiom,
! [X2: a,Xs2: list_a,Ys: list_a] :
( ( ( member_a @ X2 @ ( set_a2 @ Xs2 ) )
=> ( ( remove1_a @ X2 @ ( append_a @ Xs2 @ Ys ) )
= ( append_a @ ( remove1_a @ X2 @ Xs2 ) @ Ys ) ) )
& ( ~ ( member_a @ X2 @ ( set_a2 @ Xs2 ) )
=> ( ( remove1_a @ X2 @ ( append_a @ Xs2 @ Ys ) )
= ( append_a @ Xs2 @ ( remove1_a @ X2 @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_672_take__0,axiom,
! [Xs2: list_list_a] :
( ( take_list_a @ zero_zero_nat @ Xs2 )
= nil_list_a ) ).
% take_0
thf(fact_673_take__0,axiom,
! [Xs2: list_a] :
( ( take_a @ zero_zero_nat @ Xs2 )
= nil_a ) ).
% take_0
thf(fact_674_mset_Osimps_I1_J,axiom,
( ( mset_list_a @ nil_list_a )
= zero_z4454100511807792257list_a ) ).
% mset.simps(1)
thf(fact_675_mset_Osimps_I1_J,axiom,
( ( mset_nat @ nil_nat )
= zero_z7348594199698428585et_nat ) ).
% mset.simps(1)
thf(fact_676_mset_Osimps_I1_J,axiom,
( ( mset_a @ nil_a )
= zero_zero_multiset_a ) ).
% mset.simps(1)
thf(fact_677_union__fold__mset__add__mset,axiom,
( plus_p6334493942879108393et_nat
= ( fold_m2600682269844132093et_nat @ add_mset_nat ) ) ).
% union_fold_mset_add_mset
thf(fact_678_union__fold__mset__add__mset,axiom,
( plus_plus_multiset_a
= ( fold_m7320414754419674833iset_a @ add_mset_a ) ) ).
% union_fold_mset_add_mset
thf(fact_679_remove1__split,axiom,
! [A2: list_a,Xs2: list_list_a,Ys: list_list_a] :
( ( member_list_a @ A2 @ ( set_list_a2 @ Xs2 ) )
=> ( ( ( remove1_list_a @ A2 @ Xs2 )
= Ys )
= ( ? [Ls: list_list_a,Rs: list_list_a] :
( ( Xs2
= ( append_list_a @ Ls @ ( cons_list_a @ A2 @ Rs ) ) )
& ~ ( member_list_a @ A2 @ ( set_list_a2 @ Ls ) )
& ( Ys
= ( append_list_a @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_680_remove1__split,axiom,
! [A2: a,Xs2: list_a,Ys: list_a] :
( ( member_a @ A2 @ ( set_a2 @ Xs2 ) )
=> ( ( ( remove1_a @ A2 @ Xs2 )
= Ys )
= ( ? [Ls: list_a,Rs: list_a] :
( ( Xs2
= ( append_a @ Ls @ ( cons_a @ A2 @ Rs ) ) )
& ~ ( member_a @ A2 @ ( set_a2 @ Ls ) )
& ( Ys
= ( append_a @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_681_size__multiset__single,axiom,
! [F: a > nat,B2: a] :
( ( size_multiset_a @ F @ ( add_mset_a @ B2 @ zero_zero_multiset_a ) )
= ( suc @ ( F @ B2 ) ) ) ).
% size_multiset_single
thf(fact_682_the__elem__set,axiom,
! [X2: a] :
( ( the_elem_a @ ( set_a2 @ ( cons_a @ X2 @ nil_a ) ) )
= X2 ) ).
% the_elem_set
thf(fact_683_set__permutations__of__list__impl__aux,axiom,
! [Xs2: list_a] :
( ( set_list_a2 @ ( multis7962179442503870237_aux_a @ nil_a @ Xs2 ) )
= ( multis5886240593633752526iset_a @ ( mset_a @ Xs2 ) ) ) ).
% set_permutations_of_list_impl_aux
thf(fact_684_take__Cons_H,axiom,
! [N4: nat,X2: a,Xs2: list_a] :
( ( ( N4 = zero_zero_nat )
=> ( ( take_a @ N4 @ ( cons_a @ X2 @ Xs2 ) )
= nil_a ) )
& ( ( N4 != zero_zero_nat )
=> ( ( take_a @ N4 @ ( cons_a @ X2 @ Xs2 ) )
= ( cons_a @ X2 @ ( take_a @ ( minus_minus_nat @ N4 @ one_one_nat ) @ Xs2 ) ) ) ) ) ).
% take_Cons'
thf(fact_685_assms_I4_J,axiom,
ord_less_nat @ j1 @ ( size_size_list_a @ xs ) ).
% assms(4)
thf(fact_686_that,axiom,
! [J22: nat] :
( ( ( nth_a @ xs @ J22 )
= y )
=> ( ( ord_less_nat @ J22 @ ( size_size_list_a @ xs ) )
=> ( ( j1 != J22 )
=> thesis ) ) ) ).
% that
thf(fact_687_not__gr__zero,axiom,
! [N4: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N4 ) )
= ( N4 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_688_add__less__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_689_add__less__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_690_less__nat__zero__code,axiom,
! [N4: nat] :
~ ( ord_less_nat @ N4 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_691_neq0__conv,axiom,
! [N4: nat] :
( ( N4 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N4 ) ) ).
% neq0_conv
thf(fact_692_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_693_Suc__less__eq,axiom,
! [M: nat,N4: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N4 ) )
= ( ord_less_nat @ M @ N4 ) ) ).
% Suc_less_eq
thf(fact_694_Suc__mono,axiom,
! [M: nat,N4: nat] :
( ( ord_less_nat @ M @ N4 )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N4 ) ) ) ).
% Suc_mono
thf(fact_695_lessI,axiom,
! [N4: nat] : ( ord_less_nat @ N4 @ ( suc @ N4 ) ) ).
% lessI
thf(fact_696_nat__add__left__cancel__less,axiom,
! [K3: nat,M: nat,N4: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K3 @ M ) @ ( plus_plus_nat @ K3 @ N4 ) )
= ( ord_less_nat @ M @ N4 ) ) ).
% nat_add_left_cancel_less
thf(fact_697_append__eq__append__conv,axiom,
! [Xs2: list_a,Ys: list_a,Us2: list_a,Vs: list_a] :
( ( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
| ( ( size_size_list_a @ Us2 )
= ( size_size_list_a @ Vs ) ) )
=> ( ( ( append_a @ Xs2 @ Us2 )
= ( append_a @ Ys @ Vs ) )
= ( ( Xs2 = Ys )
& ( Us2 = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_698_less__add__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_699_less__add__same__cancel1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_700_add__less__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_701_add__less__same__cancel1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_702_less__Suc0,axiom,
! [N4: nat] :
( ( ord_less_nat @ N4 @ ( suc @ zero_zero_nat ) )
= ( N4 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_703_zero__less__Suc,axiom,
! [N4: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N4 ) ) ).
% zero_less_Suc
thf(fact_704_length__0__conv,axiom,
! [Xs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_a ) ) ).
% length_0_conv
thf(fact_705_add__gr__0,axiom,
! [M: nat,N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N4 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% add_gr_0
thf(fact_706_zero__less__diff,axiom,
! [N4: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N4 @ M ) )
= ( ord_less_nat @ M @ N4 ) ) ).
% zero_less_diff
thf(fact_707_less__one,axiom,
! [N4: nat] :
( ( ord_less_nat @ N4 @ one_one_nat )
= ( N4 = zero_zero_nat ) ) ).
% less_one
thf(fact_708_length__append,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( size_size_list_a @ ( append_a @ Xs2 @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_a @ Xs2 ) @ ( size_size_list_a @ Ys ) ) ) ).
% length_append
thf(fact_709_diff__Suc__1,axiom,
! [N4: nat] :
( ( minus_minus_nat @ ( suc @ N4 ) @ one_one_nat )
= N4 ) ).
% diff_Suc_1
thf(fact_710_length__drop,axiom,
! [N4: nat,Xs2: list_a] :
( ( size_size_list_a @ ( drop_a @ N4 @ Xs2 ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs2 ) @ N4 ) ) ).
% length_drop
thf(fact_711_nth__take,axiom,
! [I2: nat,N4: nat,Xs2: list_a] :
( ( ord_less_nat @ I2 @ N4 )
=> ( ( nth_a @ ( take_a @ N4 @ Xs2 ) @ I2 )
= ( nth_a @ Xs2 @ I2 ) ) ) ).
% nth_take
thf(fact_712_length__greater__0__conv,axiom,
! [Xs2: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs2 ) )
= ( Xs2 != nil_a ) ) ).
% length_greater_0_conv
thf(fact_713_Suc__pred,axiom,
! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( suc @ ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) ) )
= N4 ) ) ).
% Suc_pred
thf(fact_714_nth__append__length,axiom,
! [Xs2: list_a,X2: a,Ys: list_a] :
( ( nth_a @ ( append_a @ Xs2 @ ( cons_a @ X2 @ Ys ) ) @ ( size_size_list_a @ Xs2 ) )
= X2 ) ).
% nth_append_length
thf(fact_715_nth__append__length__plus,axiom,
! [Xs2: list_a,Ys: list_a,N4: nat] :
( ( nth_a @ ( append_a @ Xs2 @ Ys ) @ ( plus_plus_nat @ ( size_size_list_a @ Xs2 ) @ N4 ) )
= ( nth_a @ Ys @ N4 ) ) ).
% nth_append_length_plus
thf(fact_716_take__append,axiom,
! [N4: nat,Xs2: list_a,Ys: list_a] :
( ( take_a @ N4 @ ( append_a @ Xs2 @ Ys ) )
= ( append_a @ ( take_a @ N4 @ Xs2 ) @ ( take_a @ ( minus_minus_nat @ N4 @ ( size_size_list_a @ Xs2 ) ) @ Ys ) ) ) ).
% take_append
thf(fact_717_drop__append,axiom,
! [N4: nat,Xs2: list_a,Ys: list_a] :
( ( drop_a @ N4 @ ( append_a @ Xs2 @ Ys ) )
= ( append_a @ ( drop_a @ N4 @ Xs2 ) @ ( drop_a @ ( minus_minus_nat @ N4 @ ( size_size_list_a @ Xs2 ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_718_Suc__diff__1,axiom,
! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( suc @ ( minus_minus_nat @ N4 @ one_one_nat ) )
= N4 ) ) ).
% Suc_diff_1
thf(fact_719_nth__Cons__pos,axiom,
! [N4: nat,X2: a,Xs2: list_a] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( nth_a @ ( cons_a @ X2 @ Xs2 ) @ N4 )
= ( nth_a @ Xs2 @ ( minus_minus_nat @ N4 @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_720_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_721_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_a,Z: list_a] : ( Y5 = Z ) )
= ( ^ [Xs4: list_a,Ys3: list_a] :
( ( ( size_size_list_a @ Xs4 )
= ( size_size_list_a @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs4 ) )
=> ( ( nth_a @ Xs4 @ I4 )
= ( nth_a @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_722_Skolem__list__nth,axiom,
! [K3: nat,P: nat > a > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ? [X6: a] : ( P @ I4 @ X6 ) ) )
= ( ? [Xs4: list_a] :
( ( ( size_size_list_a @ Xs4 )
= K3 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ( P @ I4 @ ( nth_a @ Xs4 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_723_nth__equalityI,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs2 ) )
=> ( ( nth_a @ Xs2 @ I3 )
= ( nth_a @ Ys @ I3 ) ) )
=> ( Xs2 = Ys ) ) ) ).
% nth_equalityI
thf(fact_724_nat__neq__iff,axiom,
! [M: nat,N4: nat] :
( ( M != N4 )
= ( ( ord_less_nat @ M @ N4 )
| ( ord_less_nat @ N4 @ M ) ) ) ).
% nat_neq_iff
thf(fact_725_less__not__refl,axiom,
! [N4: nat] :
~ ( ord_less_nat @ N4 @ N4 ) ).
% less_not_refl
thf(fact_726_less__not__refl2,axiom,
! [N4: nat,M: nat] :
( ( ord_less_nat @ N4 @ M )
=> ( M != N4 ) ) ).
% less_not_refl2
thf(fact_727_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_728_less__irrefl__nat,axiom,
! [N4: nat] :
~ ( ord_less_nat @ N4 @ N4 ) ).
% less_irrefl_nat
thf(fact_729_nat__less__induct,axiom,
! [P: nat > $o,N4: nat] :
( ! [N5: nat] :
( ! [M7: nat] :
( ( ord_less_nat @ M7 @ N5 )
=> ( P @ M7 ) )
=> ( P @ N5 ) )
=> ( P @ N4 ) ) ).
% nat_less_induct
thf(fact_730_infinite__descent,axiom,
! [P: nat > $o,N4: nat] :
( ! [N5: nat] :
( ~ ( P @ N5 )
=> ? [M7: nat] :
( ( ord_less_nat @ M7 @ N5 )
& ~ ( P @ M7 ) ) )
=> ( P @ N4 ) ) ).
% infinite_descent
thf(fact_731_linorder__neqE__nat,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_732_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N4: nat,M: nat] :
( ! [N5: nat] : ( ord_less_nat @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
=> ( ( ord_less_nat @ ( F @ N4 ) @ ( F @ M ) )
= ( ord_less_nat @ N4 @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_733_lift__Suc__mono__less,axiom,
! [F: nat > nat,N4: nat,N6: nat] :
( ! [N5: nat] : ( ord_less_nat @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
=> ( ( ord_less_nat @ N4 @ N6 )
=> ( ord_less_nat @ ( F @ N4 ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_734_length__induct,axiom,
! [P: list_a > $o,Xs2: list_a] :
( ! [Xs: list_a] :
( ! [Ys6: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys6 ) @ ( size_size_list_a @ Xs ) )
=> ( P @ Ys6 ) )
=> ( P @ Xs ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_735_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_736_Ex__list__of__length,axiom,
! [N4: nat] :
? [Xs: list_a] :
( ( size_size_list_a @ Xs )
= N4 ) ).
% Ex_list_of_length
thf(fact_737_neq__if__length__neq,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs2 )
!= ( size_size_list_a @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_738_length__n__lists__elem,axiom,
! [Ys: list_a,N4: nat,Xs2: list_a] :
( ( member_list_a @ Ys @ ( set_list_a2 @ ( n_lists_a @ N4 @ Xs2 ) ) )
=> ( ( size_size_list_a @ Ys )
= N4 ) ) ).
% length_n_lists_elem
thf(fact_739_size__neq__size__imp__neq,axiom,
! [X2: list_a,Y: list_a] :
( ( ( size_size_list_a @ X2 )
!= ( size_size_list_a @ Y ) )
=> ( X2 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_740_size__neq__size__imp__neq,axiom,
! [X2: char,Y: char] :
( ( ( size_size_char @ X2 )
!= ( size_size_char @ Y ) )
=> ( X2 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_741_length__pos__if__in__set,axiom,
! [X2: a,Xs2: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_742_nth__mem,axiom,
! [N4: nat,Xs2: list_a] :
( ( ord_less_nat @ N4 @ ( size_size_list_a @ Xs2 ) )
=> ( member_a @ ( nth_a @ Xs2 @ N4 ) @ ( set_a2 @ Xs2 ) ) ) ).
% nth_mem
thf(fact_743_list__ball__nth,axiom,
! [N4: nat,Xs2: list_a,P: a > $o] :
( ( ord_less_nat @ N4 @ ( size_size_list_a @ Xs2 ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs2 ) )
=> ( P @ X4 ) )
=> ( P @ ( nth_a @ Xs2 @ N4 ) ) ) ) ).
% list_ball_nth
thf(fact_744_in__set__conv__nth,axiom,
! [X2: a,Xs2: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs2 ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs2 ) )
& ( ( nth_a @ Xs2 @ I4 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_745_all__nth__imp__all__set,axiom,
! [Xs2: list_a,P: a > $o,X2: a] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs2 ) )
=> ( P @ ( nth_a @ Xs2 @ I3 ) ) )
=> ( ( member_a @ X2 @ ( set_a2 @ Xs2 ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_746_all__set__conv__all__nth,axiom,
! [Xs2: list_a,P: a > $o] :
( ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs2 ) )
=> ( P @ X3 ) ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs2 ) )
=> ( P @ ( nth_a @ Xs2 @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_747_less__multiset__doubletons,axiom,
! [Y: nat,T: nat,S: nat,X2: nat] :
( ( ( ord_less_nat @ Y @ T )
| ( ord_less_nat @ Y @ S ) )
=> ( ( ( ord_less_nat @ X2 @ T )
| ( ord_less_nat @ X2 @ S ) )
=> ( ord_le5777773500796000884et_nat @ ( add_mset_nat @ Y @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) ) @ ( add_mset_nat @ T @ ( add_mset_nat @ S @ zero_z7348594199698428585et_nat ) ) ) ) ) ).
% less_multiset_doubletons
thf(fact_748_nat__induct__non__zero,axiom,
! [N4: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( P @ one_one_nat )
=> ( ! [N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
=> ( ( P @ N5 )
=> ( P @ ( suc @ N5 ) ) ) )
=> ( P @ N4 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_749_zero__less__iff__neq__zero,axiom,
! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
= ( N4 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_750_gr__implies__not__zero,axiom,
! [M: nat,N4: nat] :
( ( ord_less_nat @ M @ N4 )
=> ( N4 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_751_not__less__zero,axiom,
! [N4: nat] :
~ ( ord_less_nat @ N4 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_752_gr__zeroI,axiom,
! [N4: nat] :
( ( N4 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N4 ) ) ).
% gr_zeroI
thf(fact_753_add__less__imp__less__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_754_add__less__imp__less__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_755_add__strict__right__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_756_add__strict__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_757_add__strict__mono,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_758_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: nat,J2: nat,K3: nat,L4: nat] :
( ( ( ord_less_nat @ I2 @ J2 )
& ( K3 = L4 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J2 @ L4 ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_759_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: nat,J2: nat,K3: nat,L4: nat] :
( ( ( I2 = J2 )
& ( ord_less_nat @ K3 @ L4 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J2 @ L4 ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_760_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: nat,J2: nat,K3: nat,L4: nat] :
( ( ( ord_less_nat @ I2 @ J2 )
& ( ord_less_nat @ K3 @ L4 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J2 @ L4 ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_761_infinite__descent0,axiom,
! [P: nat > $o,N4: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
=> ( ~ ( P @ N5 )
=> ? [M7: nat] :
( ( ord_less_nat @ M7 @ N5 )
& ~ ( P @ M7 ) ) ) )
=> ( P @ N4 ) ) ) ).
% infinite_descent0
thf(fact_762_gr__implies__not0,axiom,
! [M: nat,N4: nat] :
( ( ord_less_nat @ M @ N4 )
=> ( N4 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_763_less__zeroE,axiom,
! [N4: nat] :
~ ( ord_less_nat @ N4 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_764_not__less0,axiom,
! [N4: nat] :
~ ( ord_less_nat @ N4 @ zero_zero_nat ) ).
% not_less0
thf(fact_765_not__gr0,axiom,
! [N4: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N4 ) )
= ( N4 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_766_gr0I,axiom,
! [N4: nat] :
( ( N4 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N4 ) ) ).
% gr0I
thf(fact_767_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_768_not__less__less__Suc__eq,axiom,
! [N4: nat,M: nat] :
( ~ ( ord_less_nat @ N4 @ M )
=> ( ( ord_less_nat @ N4 @ ( suc @ M ) )
= ( N4 = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_769_strict__inc__induct,axiom,
! [I2: nat,J2: nat,P: nat > $o] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ! [I3: nat] :
( ( J2
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_770_less__Suc__induct,axiom,
! [I2: nat,J2: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J3: nat,K4: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ K4 )
=> ( ( P @ I3 @ J3 )
=> ( ( P @ J3 @ K4 )
=> ( P @ I3 @ K4 ) ) ) ) )
=> ( P @ I2 @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_771_less__trans__Suc,axiom,
! [I2: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K3 )
=> ( ord_less_nat @ ( suc @ I2 ) @ K3 ) ) ) ).
% less_trans_Suc
thf(fact_772_Suc__less__SucD,axiom,
! [M: nat,N4: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N4 ) )
=> ( ord_less_nat @ M @ N4 ) ) ).
% Suc_less_SucD
thf(fact_773_less__antisym,axiom,
! [N4: nat,M: nat] :
( ~ ( ord_less_nat @ N4 @ M )
=> ( ( ord_less_nat @ N4 @ ( suc @ M ) )
=> ( M = N4 ) ) ) ).
% less_antisym
thf(fact_774_Suc__less__eq2,axiom,
! [N4: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N4 ) @ M )
= ( ? [M8: nat] :
( ( M
= ( suc @ M8 ) )
& ( ord_less_nat @ N4 @ M8 ) ) ) ) ).
% Suc_less_eq2
thf(fact_775_All__less__Suc,axiom,
! [N4: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N4 ) )
=> ( P @ I4 ) ) )
= ( ( P @ N4 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N4 )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_776_not__less__eq,axiom,
! [M: nat,N4: nat] :
( ( ~ ( ord_less_nat @ M @ N4 ) )
= ( ord_less_nat @ N4 @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_777_less__Suc__eq,axiom,
! [M: nat,N4: nat] :
( ( ord_less_nat @ M @ ( suc @ N4 ) )
= ( ( ord_less_nat @ M @ N4 )
| ( M = N4 ) ) ) ).
% less_Suc_eq
thf(fact_778_Ex__less__Suc,axiom,
! [N4: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N4 ) )
& ( P @ I4 ) ) )
= ( ( P @ N4 )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N4 )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_779_less__SucI,axiom,
! [M: nat,N4: nat] :
( ( ord_less_nat @ M @ N4 )
=> ( ord_less_nat @ M @ ( suc @ N4 ) ) ) ).
% less_SucI
thf(fact_780_less__SucE,axiom,
! [M: nat,N4: nat] :
( ( ord_less_nat @ M @ ( suc @ N4 ) )
=> ( ~ ( ord_less_nat @ M @ N4 )
=> ( M = N4 ) ) ) ).
% less_SucE
thf(fact_781_Suc__lessI,axiom,
! [M: nat,N4: nat] :
( ( ord_less_nat @ M @ N4 )
=> ( ( ( suc @ M )
!= N4 )
=> ( ord_less_nat @ ( suc @ M ) @ N4 ) ) ) ).
% Suc_lessI
thf(fact_782_Suc__lessE,axiom,
! [I2: nat,K3: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K3 )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( K3
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_783_Suc__lessD,axiom,
! [M: nat,N4: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N4 )
=> ( ord_less_nat @ M @ N4 ) ) ).
% Suc_lessD
thf(fact_784_Nat_OlessE,axiom,
! [I2: nat,K3: nat] :
( ( ord_less_nat @ I2 @ K3 )
=> ( ( K3
!= ( suc @ I2 ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( K3
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_785_mset__eq__length,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( mset_a @ Xs2 )
= ( mset_a @ Ys ) )
=> ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) ) ) ).
% mset_eq_length
thf(fact_786_less__add__eq__less,axiom,
! [K3: nat,L4: nat,M: nat,N4: nat] :
( ( ord_less_nat @ K3 @ L4 )
=> ( ( ( plus_plus_nat @ M @ L4 )
= ( plus_plus_nat @ K3 @ N4 ) )
=> ( ord_less_nat @ M @ N4 ) ) ) ).
% less_add_eq_less
thf(fact_787_trans__less__add2,axiom,
! [I2: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_less_add2
thf(fact_788_trans__less__add1,axiom,
! [I2: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_less_add1
thf(fact_789_add__less__mono1,axiom,
! [I2: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J2 @ K3 ) ) ) ).
% add_less_mono1
thf(fact_790_not__add__less2,axiom,
! [J2: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_791_not__add__less1,axiom,
! [I2: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J2 ) @ I2 ) ).
% not_add_less1
thf(fact_792_add__less__mono,axiom,
! [I2: nat,J2: nat,K3: nat,L4: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ K3 @ L4 )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J2 @ L4 ) ) ) ) ).
% add_less_mono
thf(fact_793_add__lessD1,axiom,
! [I2: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K3 )
=> ( ord_less_nat @ I2 @ K3 ) ) ).
% add_lessD1
thf(fact_794_less__imp__diff__less,axiom,
! [J2: nat,K3: nat,N4: nat] :
( ( ord_less_nat @ J2 @ K3 )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N4 ) @ K3 ) ) ).
% less_imp_diff_less
thf(fact_795_diff__less__mono2,axiom,
! [M: nat,N4: nat,L4: nat] :
( ( ord_less_nat @ M @ N4 )
=> ( ( ord_less_nat @ M @ L4 )
=> ( ord_less_nat @ ( minus_minus_nat @ L4 @ N4 ) @ ( minus_minus_nat @ L4 @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_796_nth__append,axiom,
! [N4: nat,Xs2: list_a,Ys: list_a] :
( ( ( ord_less_nat @ N4 @ ( size_size_list_a @ Xs2 ) )
=> ( ( nth_a @ ( append_a @ Xs2 @ Ys ) @ N4 )
= ( nth_a @ Xs2 @ N4 ) ) )
& ( ~ ( ord_less_nat @ N4 @ ( size_size_list_a @ Xs2 ) )
=> ( ( nth_a @ ( append_a @ Xs2 @ Ys ) @ N4 )
= ( nth_a @ Ys @ ( minus_minus_nat @ N4 @ ( size_size_list_a @ Xs2 ) ) ) ) ) ) ).
% nth_append
thf(fact_797_Suc__diff__eq__diff__pred,axiom,
! [N4: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N4 )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N4 @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_798_Suc__pred_H,axiom,
! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( N4
= ( suc @ ( minus_minus_nat @ N4 @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_799_nth__mem__mset,axiom,
! [I2: nat,Ls2: list_a] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ls2 ) )
=> ( member_a @ ( nth_a @ Ls2 @ I2 ) @ ( set_mset_a @ ( mset_a @ Ls2 ) ) ) ) ).
% nth_mem_mset
thf(fact_800_in__mset__conv__nth,axiom,
! [X2: a,Xs2: list_a] :
( ( member_a @ X2 @ ( set_mset_a @ ( mset_a @ Xs2 ) ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs2 ) )
& ( ( nth_a @ Xs2 @ I4 )
= X2 ) ) ) ) ).
% in_mset_conv_nth
thf(fact_801_length__remove1,axiom,
! [X2: a,Xs2: list_a] :
( ( ( member_a @ X2 @ ( set_a2 @ Xs2 ) )
=> ( ( size_size_list_a @ ( remove1_a @ X2 @ Xs2 ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs2 ) @ one_one_nat ) ) )
& ( ~ ( member_a @ X2 @ ( set_a2 @ Xs2 ) )
=> ( ( size_size_list_a @ ( remove1_a @ X2 @ Xs2 ) )
= ( size_size_list_a @ Xs2 ) ) ) ) ).
% length_remove1
thf(fact_802_Cons__nth__drop__Suc,axiom,
! [I2: nat,Xs2: list_a] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs2 ) )
=> ( ( cons_a @ ( nth_a @ Xs2 @ I2 ) @ ( drop_a @ ( suc @ I2 ) @ Xs2 ) )
= ( drop_a @ I2 @ Xs2 ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_803_nth__non__equal__first__eq,axiom,
! [X2: a,Y: a,Xs2: list_a,N4: nat] :
( ( X2 != Y )
=> ( ( ( nth_a @ ( cons_a @ X2 @ Xs2 ) @ N4 )
= Y )
= ( ( ( nth_a @ Xs2 @ ( minus_minus_nat @ N4 @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_804_length__Suc__conv,axiom,
! [Xs2: list_a,N4: nat] :
( ( ( size_size_list_a @ Xs2 )
= ( suc @ N4 ) )
= ( ? [Y3: a,Ys3: list_a] :
( ( Xs2
= ( cons_a @ Y3 @ Ys3 ) )
& ( ( size_size_list_a @ Ys3 )
= N4 ) ) ) ) ).
% length_Suc_conv
thf(fact_805_Suc__length__conv,axiom,
! [N4: nat,Xs2: list_a] :
( ( ( suc @ N4 )
= ( size_size_list_a @ Xs2 ) )
= ( ? [Y3: a,Ys3: list_a] :
( ( Xs2
= ( cons_a @ Y3 @ Ys3 ) )
& ( ( size_size_list_a @ Ys3 )
= N4 ) ) ) ) ).
% Suc_length_conv
thf(fact_806_list__induct4,axiom,
! [Xs2: list_a,Ys: list_a,Zs: list_a,Ws: list_a,P: list_a > list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X4: a,Xs: list_a,Y4: a,Ys2: list_a,Z3: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X4 @ Xs ) @ ( cons_a @ Y4 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_807_list__induct3,axiom,
! [Xs2: list_a,Ys: list_a,Zs: list_a,P: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_a @ nil_a )
=> ( ! [X4: a,Xs: list_a,Y4: a,Ys2: list_a,Z3: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs @ Ys2 @ Zs2 )
=> ( P @ ( cons_a @ X4 @ Xs ) @ ( cons_a @ Y4 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_808_list__induct2,axiom,
! [Xs2: list_a,Ys: list_a,P: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_a @ nil_a )
=> ( ! [X4: a,Xs: list_a,Y4: a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_a @ X4 @ Xs ) @ ( cons_a @ Y4 @ Ys2 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_809_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_810_pos__add__strict,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_811_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ! [C3: nat] :
( ( B2
= ( plus_plus_nat @ A2 @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_812_add__pos__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_813_add__neg__neg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_814_Ex__less__Suc2,axiom,
! [N4: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N4 ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N4 )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_815_gr0__conv__Suc,axiom,
! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
= ( ? [M9: nat] :
( N4
= ( suc @ M9 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_816_All__less__Suc2,axiom,
! [N4: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N4 ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N4 )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_817_gr0__implies__Suc,axiom,
! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ? [M6: nat] :
( N4
= ( suc @ M6 ) ) ) ).
% gr0_implies_Suc
thf(fact_818_less__Suc__eq__0__disj,axiom,
! [M: nat,N4: nat] :
( ( ord_less_nat @ M @ ( suc @ N4 ) )
= ( ( M = zero_zero_nat )
| ? [J4: nat] :
( ( M
= ( suc @ J4 ) )
& ( ord_less_nat @ J4 @ N4 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_819_less__imp__add__positive,axiom,
! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ? [K4: nat] :
( ( ord_less_nat @ zero_zero_nat @ K4 )
& ( ( plus_plus_nat @ I2 @ K4 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_820_less__natE,axiom,
! [M: nat,N4: nat] :
( ( ord_less_nat @ M @ N4 )
=> ~ ! [Q2: nat] :
( N4
!= ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% less_natE
thf(fact_821_less__add__Suc1,axiom,
! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M ) ) ) ).
% less_add_Suc1
thf(fact_822_less__add__Suc2,axiom,
! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_823_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M9: nat,N7: nat] :
? [K5: nat] :
( N7
= ( suc @ ( plus_plus_nat @ M9 @ K5 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_824_less__imp__Suc__add,axiom,
! [M: nat,N4: nat] :
( ( ord_less_nat @ M @ N4 )
=> ? [K4: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M @ K4 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_825_diff__less,axiom,
! [N4: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N4 ) @ M ) ) ) ).
% diff_less
thf(fact_826_Suc__diff__Suc,axiom,
! [N4: nat,M: nat] :
( ( ord_less_nat @ N4 @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N4 ) ) )
= ( minus_minus_nat @ M @ N4 ) ) ) ).
% Suc_diff_Suc
thf(fact_827_diff__less__Suc,axiom,
! [M: nat,N4: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N4 ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_828_add__diff__inverse__nat,axiom,
! [M: nat,N4: nat] :
( ~ ( ord_less_nat @ M @ N4 )
=> ( ( plus_plus_nat @ N4 @ ( minus_minus_nat @ M @ N4 ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_829_less__diff__conv,axiom,
! [I2: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J2 @ K3 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ J2 ) ) ).
% less_diff_conv
thf(fact_830_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_831_Suc__eq__plus1,axiom,
( suc
= ( ^ [N7: nat] : ( plus_plus_nat @ N7 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_832_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_833_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_834_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N4: nat] :
( ( minus_minus_nat @ M @ ( suc @ N4 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N4 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_835_take__Suc__conv__app__nth,axiom,
! [I2: nat,Xs2: list_a] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs2 ) )
=> ( ( take_a @ ( suc @ I2 ) @ Xs2 )
= ( append_a @ ( take_a @ I2 @ Xs2 ) @ ( cons_a @ ( nth_a @ Xs2 @ I2 ) @ nil_a ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_836_same__length__different,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( Xs2 != Ys )
=> ( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys ) )
=> ? [Pre: list_a,X4: a,Xs5: list_a,Y4: a,Ys7: list_a] :
( ( X4 != Y4 )
& ( Xs2
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X4 @ nil_a ) @ Xs5 ) ) )
& ( Ys
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y4 @ nil_a ) @ Ys7 ) ) ) ) ) ) ).
% same_length_different
thf(fact_837_id__take__nth__drop,axiom,
! [I2: nat,Xs2: list_a] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs2 ) )
=> ( Xs2
= ( append_a @ ( take_a @ I2 @ Xs2 ) @ ( cons_a @ ( nth_a @ Xs2 @ I2 ) @ ( drop_a @ ( suc @ I2 ) @ Xs2 ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_838_diff__Suc__less,axiom,
! [N4: nat,I2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ord_less_nat @ ( minus_minus_nat @ N4 @ ( suc @ I2 ) ) @ N4 ) ) ).
% diff_Suc_less
thf(fact_839_nat__diff__split__asm,axiom,
! [P: nat > $o,A2: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B2 ) )
= ( ~ ( ( ( ord_less_nat @ A2 @ B2 )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_840_nat__diff__split,axiom,
! [P: nat > $o,A2: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B2 ) )
= ( ( ( ord_less_nat @ A2 @ B2 )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_841_append__eq__conv__conj,axiom,
! [Xs2: list_a,Ys: list_a,Zs: list_a] :
( ( ( append_a @ Xs2 @ Ys )
= Zs )
= ( ( Xs2
= ( take_a @ ( size_size_list_a @ Xs2 ) @ Zs ) )
& ( Ys
= ( drop_a @ ( size_size_list_a @ Xs2 ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_842_list_Osize_I4_J,axiom,
! [X21: a,X22: list_a] :
( ( size_size_list_a @ ( cons_a @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_a @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_843_length__Suc__conv__rev,axiom,
! [Xs2: list_a,N4: nat] :
( ( ( size_size_list_a @ Xs2 )
= ( suc @ N4 ) )
= ( ? [Y3: a,Ys3: list_a] :
( ( Xs2
= ( append_a @ Ys3 @ ( cons_a @ Y3 @ nil_a ) ) )
& ( ( size_size_list_a @ Ys3 )
= N4 ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_844_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M9: nat,N7: nat] : ( if_nat @ ( M9 = zero_zero_nat ) @ N7 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M9 @ one_one_nat ) @ N7 ) ) ) ) ) ).
% add_eq_if
thf(fact_845_nth__Cons_H,axiom,
! [N4: nat,X2: a,Xs2: list_a] :
( ( ( N4 = zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X2 @ Xs2 ) @ N4 )
= X2 ) )
& ( ( N4 != zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X2 @ Xs2 ) @ N4 )
= ( nth_a @ Xs2 @ ( minus_minus_nat @ N4 @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_846_drop__Cons_H,axiom,
! [N4: nat,X2: a,Xs2: list_a] :
( ( ( N4 = zero_zero_nat )
=> ( ( drop_a @ N4 @ ( cons_a @ X2 @ Xs2 ) )
= ( cons_a @ X2 @ Xs2 ) ) )
& ( ( N4 != zero_zero_nat )
=> ( ( drop_a @ N4 @ ( cons_a @ X2 @ Xs2 ) )
= ( drop_a @ ( minus_minus_nat @ N4 @ one_one_nat ) @ Xs2 ) ) ) ) ).
% drop_Cons'
thf(fact_847_mset__lt__single__right__iff,axiom,
! [M2: multiset_nat,Y: nat] :
( ( ord_le5777773500796000884et_nat @ M2 @ ( add_mset_nat @ Y @ zero_z7348594199698428585et_nat ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_mset_nat @ M2 ) )
=> ( ord_less_nat @ X3 @ Y ) ) ) ) ).
% mset_lt_single_right_iff
thf(fact_848_mset__lt__single__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_le5777773500796000884et_nat @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) @ ( add_mset_nat @ Y @ zero_z7348594199698428585et_nat ) )
= ( ord_less_nat @ X2 @ Y ) ) ).
% mset_lt_single_iff
thf(fact_849_length__append__singleton,axiom,
! [Xs2: list_a,X2: a] :
( ( size_size_list_a @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) )
= ( suc @ ( size_size_list_a @ Xs2 ) ) ) ).
% length_append_singleton
thf(fact_850_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_851_size__empty,axiom,
( ( size_size_multiset_a @ zero_zero_multiset_a )
= zero_zero_nat ) ).
% size_empty
thf(fact_852_size__eq__0__iff__empty,axiom,
! [M2: multiset_a] :
( ( ( size_size_multiset_a @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_multiset_a ) ) ).
% size_eq_0_iff_empty
thf(fact_853_size__add__mset,axiom,
! [A2: a,A: multiset_a] :
( ( size_size_multiset_a @ ( add_mset_a @ A2 @ A ) )
= ( suc @ ( size_size_multiset_a @ A ) ) ) ).
% size_add_mset
thf(fact_854_size__union,axiom,
! [M2: multiset_a,N: multiset_a] :
( ( size_size_multiset_a @ ( plus_plus_multiset_a @ M2 @ N ) )
= ( plus_plus_nat @ ( size_size_multiset_a @ M2 ) @ ( size_size_multiset_a @ N ) ) ) ).
% size_union
thf(fact_855_size__mset,axiom,
! [Xs2: list_a] :
( ( size_size_multiset_a @ ( mset_a @ Xs2 ) )
= ( size_size_list_a @ Xs2 ) ) ).
% size_mset
thf(fact_856_length__list__of__mset,axiom,
! [A: multiset_a] :
( ( size_size_list_a @ ( multis4723169673647964297mset_a @ A ) )
= ( size_size_multiset_a @ A ) ) ).
% length_list_of_mset
thf(fact_857_size__eq__Suc__imp__elem,axiom,
! [M2: multiset_a,N4: nat] :
( ( ( size_size_multiset_a @ M2 )
= ( suc @ N4 ) )
=> ? [A3: a] : ( member_a @ A3 @ ( set_mset_a @ M2 ) ) ) ).
% size_eq_Suc_imp_elem
thf(fact_858_size__eq__Suc__imp__eq__union,axiom,
! [M2: multiset_a,N4: nat] :
( ( ( size_size_multiset_a @ M2 )
= ( suc @ N4 ) )
=> ? [A3: a,N3: multiset_a] :
( M2
= ( add_mset_a @ A3 @ N3 ) ) ) ).
% size_eq_Suc_imp_eq_union
thf(fact_859_length__finite__permutations__of__multiset,axiom,
! [Xs2: list_a,A: multiset_a] :
( ( member_list_a @ Xs2 @ ( multis5886240593633752526iset_a @ A ) )
=> ( ( size_size_list_a @ Xs2 )
= ( size_size_multiset_a @ A ) ) ) ).
% length_finite_permutations_of_multiset
thf(fact_860_nonempty__has__size,axiom,
! [S2: multiset_a] :
( ( S2 != zero_zero_multiset_a )
= ( ord_less_nat @ zero_zero_nat @ ( size_size_multiset_a @ S2 ) ) ) ).
% nonempty_has_size
thf(fact_861_size__single,axiom,
! [B2: a] :
( ( size_size_multiset_a @ ( add_mset_a @ B2 @ zero_zero_multiset_a ) )
= one_one_nat ) ).
% size_single
thf(fact_862_size__1__singleton__mset,axiom,
! [M2: multiset_a] :
( ( ( size_size_multiset_a @ M2 )
= one_one_nat )
=> ? [A3: a] :
( M2
= ( add_mset_a @ A3 @ zero_zero_multiset_a ) ) ) ).
% size_1_singleton_mset
thf(fact_863_size__mset__SucE,axiom,
! [A: multiset_a,N4: nat] :
( ( ( size_size_multiset_a @ A )
= ( suc @ N4 ) )
=> ~ ! [A3: a,B3: multiset_a] :
( ( A
= ( plus_plus_multiset_a @ ( add_mset_a @ A3 @ zero_zero_multiset_a ) @ B3 ) )
=> ( ( size_size_multiset_a @ B3 )
!= N4 ) ) ) ).
% size_mset_SucE
thf(fact_864_size__Suc__Diff1,axiom,
! [X2: a,M2: multiset_a] :
( ( member_a @ X2 @ ( set_mset_a @ M2 ) )
=> ( ( suc @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M2 @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) ) ) )
= ( size_size_multiset_a @ M2 ) ) ) ).
% size_Suc_Diff1
thf(fact_865_size__Diff1__less,axiom,
! [X2: a,M2: multiset_a] :
( ( member_a @ X2 @ ( set_mset_a @ M2 ) )
=> ( ord_less_nat @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M2 @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) ) ) @ ( size_size_multiset_a @ M2 ) ) ) ).
% size_Diff1_less
thf(fact_866_size__Diff2__less,axiom,
! [X2: a,M2: multiset_a,Y: a] :
( ( member_a @ X2 @ ( set_mset_a @ M2 ) )
=> ( ( member_a @ Y @ ( set_mset_a @ M2 ) )
=> ( ord_less_nat @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ M2 @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) ) @ ( add_mset_a @ Y @ zero_zero_multiset_a ) ) ) @ ( size_size_multiset_a @ M2 ) ) ) ) ).
% size_Diff2_less
thf(fact_867_size__mset__remove1__mset__le__iff,axiom,
! [M2: multiset_a,X2: a] :
( ( ord_less_nat @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M2 @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) ) ) @ ( size_size_multiset_a @ M2 ) )
= ( member_a @ X2 @ ( set_mset_a @ M2 ) ) ) ).
% size_mset_remove1_mset_le_iff
thf(fact_868_obtain__two__items__mset,axiom,
! [A: multiset_a] :
( ( ord_less_nat @ one_one_nat @ ( size_size_multiset_a @ A ) )
=> ~ ! [X4: a] :
( ( member_a @ X4 @ ( set_mset_a @ A ) )
=> ! [Y4: a] :
~ ( member_a @ Y4 @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ A @ ( add_mset_a @ X4 @ zero_zero_multiset_a ) ) ) ) ) ) ).
% obtain_two_items_mset
thf(fact_869_size__Diff__singleton,axiom,
! [X2: a,M2: multiset_a] :
( ( member_a @ X2 @ ( set_mset_a @ M2 ) )
=> ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M2 @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) ) )
= ( minus_minus_nat @ ( size_size_multiset_a @ M2 ) @ one_one_nat ) ) ) ).
% size_Diff_singleton
thf(fact_870_size__Diff__singleton__if,axiom,
! [X2: a,A: multiset_a] :
( ( ( member_a @ X2 @ ( set_mset_a @ A ) )
=> ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ A @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) ) )
= ( minus_minus_nat @ ( size_size_multiset_a @ A ) @ one_one_nat ) ) )
& ( ~ ( member_a @ X2 @ ( set_mset_a @ A ) )
=> ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ A @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) ) )
= ( size_size_multiset_a @ A ) ) ) ) ).
% size_Diff_singleton_if
thf(fact_871_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_872_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_873_size__remove1__mset__If,axiom,
! [M2: multiset_a,X2: a] :
( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M2 @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) ) )
= ( minus_minus_nat @ ( size_size_multiset_a @ M2 ) @ ( if_nat @ ( member_a @ X2 @ ( set_mset_a @ M2 ) ) @ one_one_nat @ zero_zero_nat ) ) ) ).
% size_remove1_mset_If
thf(fact_874_ex__gt__imp__less__multiset,axiom,
! [N: multiset_nat,M2: multiset_nat] :
( ? [Y6: nat] :
( ( member_nat @ Y6 @ ( set_mset_nat @ N ) )
& ! [X4: nat] :
( ( member_nat @ X4 @ ( set_mset_nat @ M2 ) )
=> ( ord_less_nat @ X4 @ Y6 ) ) )
=> ( ord_le5777773500796000884et_nat @ M2 @ N ) ) ).
% ex_gt_imp_less_multiset
thf(fact_875_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_876_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_877_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_878_add__mono1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( plus_plus_nat @ B2 @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_879_less__add__one,axiom,
! [A2: nat] : ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ).
% less_add_one
thf(fact_880_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A2: nat,B2: nat] :
( ~ ( ord_less_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A2 @ B2 ) )
= A2 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_881_length__Cons,axiom,
! [X2: a,Xs2: list_a] :
( ( size_size_list_a @ ( cons_a @ X2 @ Xs2 ) )
= ( suc @ ( size_size_list_a @ Xs2 ) ) ) ).
% length_Cons
thf(fact_882_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C4: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_883_crm,axiom,
ord_less_eq_nat @ one_one_nat @ ( count_a @ ( minus_3765977307040488491iset_a @ ( mset_a @ xs ) @ ( add_mset_a @ x @ zero_zero_multiset_a ) ) @ x ) ).
% crm
thf(fact_884_take__hd__drop,axiom,
! [N4: nat,Xs2: list_a] :
( ( ord_less_nat @ N4 @ ( size_size_list_a @ Xs2 ) )
=> ( ( append_a @ ( take_a @ N4 @ Xs2 ) @ ( cons_a @ ( hd_a @ ( drop_a @ N4 @ Xs2 ) ) @ nil_a ) )
= ( take_a @ ( suc @ N4 ) @ Xs2 ) ) ) ).
% take_hd_drop
thf(fact_885_prefixes__snoc,axiom,
! [Xs2: list_a,X2: a] :
( ( prefixes_a @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) )
= ( append_list_a @ ( prefixes_a @ Xs2 ) @ ( cons_list_a @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) @ nil_list_a ) ) ) ).
% prefixes_snoc
thf(fact_886_le__zero__eq,axiom,
! [N4: nat] :
( ( ord_less_eq_nat @ N4 @ zero_zero_nat )
= ( N4 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_887_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_888_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_889_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_890_le0,axiom,
! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N4 ) ).
% le0
thf(fact_891_Suc__le__mono,axiom,
! [N4: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N4 ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N4 @ M ) ) ).
% Suc_le_mono
thf(fact_892_nat__add__left__cancel__le,axiom,
! [K3: nat,M: nat,N4: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K3 @ M ) @ ( plus_plus_nat @ K3 @ N4 ) )
= ( ord_less_eq_nat @ M @ N4 ) ) ).
% nat_add_left_cancel_le
thf(fact_893_diff__diff__cancel,axiom,
! [I2: nat,N4: nat] :
( ( ord_less_eq_nat @ I2 @ N4 )
=> ( ( minus_minus_nat @ N4 @ ( minus_minus_nat @ N4 @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_894_le__add__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_895_le__add__same__cancel1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_896_add__le__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_897_add__le__same__cancel1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_898_le__add__diff__inverse2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_899_le__add__diff__inverse,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A2 @ B2 ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_900_mset__le__single__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_le6602235886369790592et_nat @ ( add_mset_nat @ X2 @ zero_z7348594199698428585et_nat ) @ ( add_mset_nat @ Y @ zero_z7348594199698428585et_nat ) )
= ( ord_less_eq_nat @ X2 @ Y ) ) ).
% mset_le_single_iff
thf(fact_901_diff__is__0__eq_H,axiom,
! [M: nat,N4: nat] :
( ( ord_less_eq_nat @ M @ N4 )
=> ( ( minus_minus_nat @ M @ N4 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_902_diff__is__0__eq,axiom,
! [M: nat,N4: nat] :
( ( ( minus_minus_nat @ M @ N4 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N4 ) ) ).
% diff_is_0_eq
thf(fact_903_Nat_Odiff__diff__right,axiom,
! [K3: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K3 @ J2 )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J2 @ K3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K3 ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_904_Nat_Oadd__diff__assoc2,axiom,
! [K3: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K3 @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K3 ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I2 ) @ K3 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_905_Nat_Oadd__diff__assoc,axiom,
! [K3: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K3 @ J2 )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J2 @ K3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K3 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_906_count__empty,axiom,
! [A2: a] :
( ( count_a @ zero_zero_multiset_a @ A2 )
= zero_zero_nat ) ).
% count_empty
thf(fact_907_count__add__mset,axiom,
! [B2: a,A2: a,A: multiset_a] :
( ( ( B2 = A2 )
=> ( ( count_a @ ( add_mset_a @ B2 @ A ) @ A2 )
= ( suc @ ( count_a @ A @ A2 ) ) ) )
& ( ( B2 != A2 )
=> ( ( count_a @ ( add_mset_a @ B2 @ A ) @ A2 )
= ( count_a @ A @ A2 ) ) ) ) ).
% count_add_mset
thf(fact_908_take__all,axiom,
! [Xs2: list_a,N4: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs2 ) @ N4 )
=> ( ( take_a @ N4 @ Xs2 )
= Xs2 ) ) ).
% take_all
thf(fact_909_take__all__iff,axiom,
! [N4: nat,Xs2: list_a] :
( ( ( take_a @ N4 @ Xs2 )
= Xs2 )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs2 ) @ N4 ) ) ).
% take_all_iff
thf(fact_910_count__union,axiom,
! [M2: multiset_a,N: multiset_a,A2: a] :
( ( count_a @ ( plus_plus_multiset_a @ M2 @ N ) @ A2 )
= ( plus_plus_nat @ ( count_a @ M2 @ A2 ) @ ( count_a @ N @ A2 ) ) ) ).
% count_union
thf(fact_911_count__diff,axiom,
! [M2: multiset_a,N: multiset_a,A2: a] :
( ( count_a @ ( minus_3765977307040488491iset_a @ M2 @ N ) @ A2 )
= ( minus_minus_nat @ ( count_a @ M2 @ A2 ) @ ( count_a @ N @ A2 ) ) ) ).
% count_diff
thf(fact_912_hd__append2,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( Xs2 != nil_a )
=> ( ( hd_a @ ( append_a @ Xs2 @ Ys ) )
= ( hd_a @ Xs2 ) ) ) ).
% hd_append2
thf(fact_913_diff__Suc__diff__eq1,axiom,
! [K3: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K3 @ J2 )
=> ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J2 @ K3 ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( suc @ J2 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_914_diff__Suc__diff__eq2,axiom,
! [K3: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K3 @ J2 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K3 ) ) @ I2 )
= ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K3 @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_915_count__greater__zero__iff,axiom,
! [M2: multiset_a,X2: a] :
( ( ord_less_nat @ zero_zero_nat @ ( count_a @ M2 @ X2 ) )
= ( member_a @ X2 @ ( set_mset_a @ M2 ) ) ) ).
% count_greater_zero_iff
thf(fact_916_drop__eq__Nil2,axiom,
! [N4: nat,Xs2: list_a] :
( ( nil_a
= ( drop_a @ N4 @ Xs2 ) )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs2 ) @ N4 ) ) ).
% drop_eq_Nil2
thf(fact_917_drop__eq__Nil,axiom,
! [N4: nat,Xs2: list_a] :
( ( ( drop_a @ N4 @ Xs2 )
= nil_a )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs2 ) @ N4 ) ) ).
% drop_eq_Nil
thf(fact_918_drop__all,axiom,
! [Xs2: list_a,N4: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs2 ) @ N4 )
=> ( ( drop_a @ N4 @ Xs2 )
= nil_a ) ) ).
% drop_all
thf(fact_919_count__greater__eq__one__iff,axiom,
! [M2: multiset_a,X2: a] :
( ( ord_less_eq_nat @ one_one_nat @ ( count_a @ M2 @ X2 ) )
= ( member_a @ X2 @ ( set_mset_a @ M2 ) ) ) ).
% count_greater_eq_one_iff
thf(fact_920_count__mset__0__iff,axiom,
! [Xs2: list_a,X2: a] :
( ( ( count_a @ ( mset_a @ Xs2 ) @ X2 )
= zero_zero_nat )
= ( ~ ( member_a @ X2 @ ( set_a2 @ Xs2 ) ) ) ) ).
% count_mset_0_iff
thf(fact_921_hd__take,axiom,
! [J2: nat,Xs2: list_a] :
( ( ord_less_nat @ zero_zero_nat @ J2 )
=> ( ( hd_a @ ( take_a @ J2 @ Xs2 ) )
= ( hd_a @ Xs2 ) ) ) ).
% hd_take
thf(fact_922_length__prefixes,axiom,
! [Xs2: list_a] :
( ( size_s349497388124573686list_a @ ( prefixes_a @ Xs2 ) )
= ( plus_plus_nat @ ( size_size_list_a @ Xs2 ) @ one_one_nat ) ) ).
% length_prefixes
thf(fact_923_count__greater__eq__Suc__zero__iff,axiom,
! [M2: multiset_a,X2: a] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( count_a @ M2 @ X2 ) )
= ( member_a @ X2 @ ( set_mset_a @ M2 ) ) ) ).
% count_greater_eq_Suc_zero_iff
thf(fact_924_nth__drop,axiom,
! [N4: nat,Xs2: list_a,I2: nat] :
( ( ord_less_eq_nat @ N4 @ ( size_size_list_a @ Xs2 ) )
=> ( ( nth_a @ ( drop_a @ N4 @ Xs2 ) @ I2 )
= ( nth_a @ Xs2 @ ( plus_plus_nat @ N4 @ I2 ) ) ) ) ).
% nth_drop
thf(fact_925_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_926_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_927_list_Osel_I1_J,axiom,
! [X21: a,X22: list_a] :
( ( hd_a @ ( cons_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_928_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A7: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A7 ) )
= ( ord_less_nat @ A7 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_929_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: nat,J2: nat,K3: nat,L4: nat] :
( ( ( ord_less_eq_nat @ I2 @ J2 )
& ( K3 = L4 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J2 @ L4 ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_930_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: nat,J2: nat,K3: nat,L4: nat] :
( ( ( I2 = J2 )
& ( ord_less_eq_nat @ K3 @ L4 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J2 @ L4 ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_931_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: nat,J2: nat,K3: nat,L4: nat] :
( ( ( ord_less_eq_nat @ I2 @ J2 )
& ( ord_less_eq_nat @ K3 @ L4 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J2 @ L4 ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_932_add__mono,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_mono
thf(fact_933_add__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_934_less__eqE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ! [C3: nat] :
( B2
!= ( plus_plus_nat @ A2 @ C3 ) ) ) ).
% less_eqE
thf(fact_935_add__right__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_936_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
? [C4: nat] :
( B4
= ( plus_plus_nat @ A4 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_937_add__le__imp__le__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_938_add__le__imp__le__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_939_less__eq__nat_Osimps_I1_J,axiom,
! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N4 ) ).
% less_eq_nat.simps(1)
thf(fact_940_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_941_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_942_le__0__eq,axiom,
! [N4: nat] :
( ( ord_less_eq_nat @ N4 @ zero_zero_nat )
= ( N4 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_943_Suc__leD,axiom,
! [M: nat,N4: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N4 )
=> ( ord_less_eq_nat @ M @ N4 ) ) ).
% Suc_leD
thf(fact_944_le__SucE,axiom,
! [M: nat,N4: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N4 ) )
=> ( ~ ( ord_less_eq_nat @ M @ N4 )
=> ( M
= ( suc @ N4 ) ) ) ) ).
% le_SucE
thf(fact_945_le__SucI,axiom,
! [M: nat,N4: nat] :
( ( ord_less_eq_nat @ M @ N4 )
=> ( ord_less_eq_nat @ M @ ( suc @ N4 ) ) ) ).
% le_SucI
thf(fact_946_Suc__le__D,axiom,
! [N4: nat,M10: nat] :
( ( ord_less_eq_nat @ ( suc @ N4 ) @ M10 )
=> ? [M6: nat] :
( M10
= ( suc @ M6 ) ) ) ).
% Suc_le_D
thf(fact_947_le__Suc__eq,axiom,
! [M: nat,N4: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N4 ) )
= ( ( ord_less_eq_nat @ M @ N4 )
| ( M
= ( suc @ N4 ) ) ) ) ).
% le_Suc_eq
thf(fact_948_Suc__n__not__le__n,axiom,
! [N4: nat] :
~ ( ord_less_eq_nat @ ( suc @ N4 ) @ N4 ) ).
% Suc_n_not_le_n
thf(fact_949_not__less__eq__eq,axiom,
! [M: nat,N4: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N4 ) )
= ( ord_less_eq_nat @ ( suc @ N4 ) @ M ) ) ).
% not_less_eq_eq
thf(fact_950_full__nat__induct,axiom,
! [P: nat > $o,N4: nat] :
( ! [N5: nat] :
( ! [M7: nat] :
( ( ord_less_eq_nat @ ( suc @ M7 ) @ N5 )
=> ( P @ M7 ) )
=> ( P @ N5 ) )
=> ( P @ N4 ) ) ).
% full_nat_induct
thf(fact_951_nat__induct__at__least,axiom,
! [M: nat,N4: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N4 )
=> ( ( P @ M )
=> ( ! [N5: nat] :
( ( ord_less_eq_nat @ M @ N5 )
=> ( ( P @ N5 )
=> ( P @ ( suc @ N5 ) ) ) )
=> ( P @ N4 ) ) ) ) ).
% nat_induct_at_least
thf(fact_952_transitive__stepwise__le,axiom,
! [M: nat,N4: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N4 )
=> ( ! [X4: nat] : ( R @ X4 @ X4 )
=> ( ! [X4: nat,Y4: nat,Z3: nat] :
( ( R @ X4 @ Y4 )
=> ( ( R @ Y4 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [N5: nat] : ( R @ N5 @ ( suc @ N5 ) )
=> ( R @ M @ N4 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_953_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J2: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_954_le__neq__implies__less,axiom,
! [M: nat,N4: nat] :
( ( ord_less_eq_nat @ M @ N4 )
=> ( ( M != N4 )
=> ( ord_less_nat @ M @ N4 ) ) ) ).
% le_neq_implies_less
thf(fact_955_less__or__eq__imp__le,axiom,
! [M: nat,N4: nat] :
( ( ( ord_less_nat @ M @ N4 )
| ( M = N4 ) )
=> ( ord_less_eq_nat @ M @ N4 ) ) ).
% less_or_eq_imp_le
thf(fact_956_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M9: nat,N7: nat] :
( ( ord_less_nat @ M9 @ N7 )
| ( M9 = N7 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_957_less__imp__le__nat,axiom,
! [M: nat,N4: nat] :
( ( ord_less_nat @ M @ N4 )
=> ( ord_less_eq_nat @ M @ N4 ) ) ).
% less_imp_le_nat
thf(fact_958_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M9: nat,N7: nat] :
( ( ord_less_eq_nat @ M9 @ N7 )
& ( M9 != N7 ) ) ) ) ).
% nat_less_le
thf(fact_959_add__leE,axiom,
! [M: nat,K3: nat,N4: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K3 ) @ N4 )
=> ~ ( ( ord_less_eq_nat @ M @ N4 )
=> ~ ( ord_less_eq_nat @ K3 @ N4 ) ) ) ).
% add_leE
thf(fact_960_le__add1,axiom,
! [N4: nat,M: nat] : ( ord_less_eq_nat @ N4 @ ( plus_plus_nat @ N4 @ M ) ) ).
% le_add1
thf(fact_961_le__add2,axiom,
! [N4: nat,M: nat] : ( ord_less_eq_nat @ N4 @ ( plus_plus_nat @ M @ N4 ) ) ).
% le_add2
thf(fact_962_add__leD1,axiom,
! [M: nat,K3: nat,N4: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K3 ) @ N4 )
=> ( ord_less_eq_nat @ M @ N4 ) ) ).
% add_leD1
thf(fact_963_add__leD2,axiom,
! [M: nat,K3: nat,N4: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K3 ) @ N4 )
=> ( ord_less_eq_nat @ K3 @ N4 ) ) ).
% add_leD2
thf(fact_964_le__Suc__ex,axiom,
! [K3: nat,L4: nat] :
( ( ord_less_eq_nat @ K3 @ L4 )
=> ? [N5: nat] :
( L4
= ( plus_plus_nat @ K3 @ N5 ) ) ) ).
% le_Suc_ex
thf(fact_965_add__le__mono,axiom,
! [I2: nat,J2: nat,K3: nat,L4: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ K3 @ L4 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J2 @ L4 ) ) ) ) ).
% add_le_mono
thf(fact_966_add__le__mono1,axiom,
! [I2: nat,J2: nat,K3: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J2 @ K3 ) ) ) ).
% add_le_mono1
thf(fact_967_trans__le__add1,axiom,
! [I2: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_le_add1
thf(fact_968_trans__le__add2,axiom,
! [I2: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_le_add2
thf(fact_969_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M9: nat,N7: nat] :
? [K5: nat] :
( N7
= ( plus_plus_nat @ M9 @ K5 ) ) ) ) ).
% nat_le_iff_add
thf(fact_970_eq__diff__iff,axiom,
! [K3: nat,M: nat,N4: nat] :
( ( ord_less_eq_nat @ K3 @ M )
=> ( ( ord_less_eq_nat @ K3 @ N4 )
=> ( ( ( minus_minus_nat @ M @ K3 )
= ( minus_minus_nat @ N4 @ K3 ) )
= ( M = N4 ) ) ) ) ).
% eq_diff_iff
thf(fact_971_le__diff__iff,axiom,
! [K3: nat,M: nat,N4: nat] :
( ( ord_less_eq_nat @ K3 @ M )
=> ( ( ord_less_eq_nat @ K3 @ N4 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K3 ) @ ( minus_minus_nat @ N4 @ K3 ) )
= ( ord_less_eq_nat @ M @ N4 ) ) ) ) ).
% le_diff_iff
thf(fact_972_Nat_Odiff__diff__eq,axiom,
! [K3: nat,M: nat,N4: nat] :
( ( ord_less_eq_nat @ K3 @ M )
=> ( ( ord_less_eq_nat @ K3 @ N4 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K3 ) @ ( minus_minus_nat @ N4 @ K3 ) )
= ( minus_minus_nat @ M @ N4 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_973_diff__le__mono,axiom,
! [M: nat,N4: nat,L4: nat] :
( ( ord_less_eq_nat @ M @ N4 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L4 ) @ ( minus_minus_nat @ N4 @ L4 ) ) ) ).
% diff_le_mono
thf(fact_974_diff__le__self,axiom,
! [M: nat,N4: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N4 ) @ M ) ).
% diff_le_self
thf(fact_975_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_976_diff__le__mono2,axiom,
! [M: nat,N4: nat,L4: nat] :
( ( ord_less_eq_nat @ M @ N4 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L4 @ N4 ) @ ( minus_minus_nat @ L4 @ M ) ) ) ).
% diff_le_mono2
thf(fact_977_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_978_verit__la__disequality,axiom,
! [A2: nat,B2: nat] :
( ( A2 = B2 )
| ~ ( ord_less_eq_nat @ A2 @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_979_le__refl,axiom,
! [N4: nat] : ( ord_less_eq_nat @ N4 @ N4 ) ).
% le_refl
thf(fact_980_le__trans,axiom,
! [I2: nat,J2: nat,K3: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K3 )
=> ( ord_less_eq_nat @ I2 @ K3 ) ) ) ).
% le_trans
thf(fact_981_eq__imp__le,axiom,
! [M: nat,N4: nat] :
( ( M = N4 )
=> ( ord_less_eq_nat @ M @ N4 ) ) ).
% eq_imp_le
thf(fact_982_le__antisym,axiom,
! [M: nat,N4: nat] :
( ( ord_less_eq_nat @ M @ N4 )
=> ( ( ord_less_eq_nat @ N4 @ M )
=> ( M = N4 ) ) ) ).
% le_antisym
thf(fact_983_nat__le__linear,axiom,
! [M: nat,N4: nat] :
( ( ord_less_eq_nat @ M @ N4 )
| ( ord_less_eq_nat @ N4 @ M ) ) ).
% nat_le_linear
thf(fact_984_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K3: nat,B2: nat] :
( ( P @ K3 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B2 ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_985_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N4: nat,N6: nat] :
( ! [N5: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N5 ) ) @ ( F @ N5 ) )
=> ( ( ord_less_eq_nat @ N4 @ N6 )
=> ( ord_less_eq_nat @ ( F @ N6 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_986_lift__Suc__mono__le,axiom,
! [F: nat > nat,N4: nat,N6: nat] :
( ! [N5: nat] : ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
=> ( ( ord_less_eq_nat @ N4 @ N6 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_987_count__inject,axiom,
! [X2: multiset_a,Y: multiset_a] :
( ( ( count_a @ X2 )
= ( count_a @ Y ) )
= ( X2 = Y ) ) ).
% count_inject
thf(fact_988_multiset__eqI,axiom,
! [A: multiset_a,B: multiset_a] :
( ! [X4: a] :
( ( count_a @ A @ X4 )
= ( count_a @ B @ X4 ) )
=> ( A = B ) ) ).
% multiset_eqI
thf(fact_989_multiset__eq__iff,axiom,
( ( ^ [Y5: multiset_a,Z: multiset_a] : ( Y5 = Z ) )
= ( ^ [M3: multiset_a,N2: multiset_a] :
! [A4: a] :
( ( count_a @ M3 @ A4 )
= ( count_a @ N2 @ A4 ) ) ) ) ).
% multiset_eq_iff
thf(fact_990_zero__multiset_Orep__eq,axiom,
( ( count_a @ zero_zero_multiset_a )
= ( ^ [A4: a] : zero_zero_nat ) ) ).
% zero_multiset.rep_eq
thf(fact_991_count__eq__zero__iff,axiom,
! [M2: multiset_a,X2: a] :
( ( ( count_a @ M2 @ X2 )
= zero_zero_nat )
= ( ~ ( member_a @ X2 @ ( set_mset_a @ M2 ) ) ) ) ).
% count_eq_zero_iff
thf(fact_992_count__inI,axiom,
! [M2: multiset_a,X2: a] :
( ( ( count_a @ M2 @ X2 )
!= zero_zero_nat )
=> ( member_a @ X2 @ ( set_mset_a @ M2 ) ) ) ).
% count_inI
thf(fact_993_in__countE,axiom,
! [X2: a,M2: multiset_a] :
( ( member_a @ X2 @ ( set_mset_a @ M2 ) )
=> ~ ! [N5: nat] :
( ( count_a @ M2 @ X2 )
!= ( suc @ N5 ) ) ) ).
% in_countE
thf(fact_994_add__mset_Orep__eq,axiom,
! [X2: a,Xa2: multiset_a] :
( ( count_a @ ( add_mset_a @ X2 @ Xa2 ) )
= ( ^ [B4: a] : ( if_nat @ ( B4 = X2 ) @ ( suc @ ( count_a @ Xa2 @ B4 ) ) @ ( count_a @ Xa2 @ B4 ) ) ) ) ).
% add_mset.rep_eq
thf(fact_995_ex__gt__count__imp__le__multiset,axiom,
! [M2: multiset_nat,N: multiset_nat,X2: nat] :
( ! [Y4: nat] :
( ( member_nat @ Y4 @ ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ M2 @ N ) ) )
=> ( ord_less_eq_nat @ Y4 @ X2 ) )
=> ( ( ord_less_nat @ ( count_nat @ M2 @ X2 ) @ ( count_nat @ N @ X2 ) )
=> ( ord_le5777773500796000884et_nat @ M2 @ N ) ) ) ).
% ex_gt_count_imp_le_multiset
thf(fact_996_plus__multiset_Orep__eq,axiom,
! [X2: multiset_a,Xa2: multiset_a] :
( ( count_a @ ( plus_plus_multiset_a @ X2 @ Xa2 ) )
= ( ^ [A4: a] : ( plus_plus_nat @ ( count_a @ X2 @ A4 ) @ ( count_a @ Xa2 @ A4 ) ) ) ) ).
% plus_multiset.rep_eq
thf(fact_997_minus__multiset_Orep__eq,axiom,
! [X2: multiset_a,Xa2: multiset_a] :
( ( count_a @ ( minus_3765977307040488491iset_a @ X2 @ Xa2 ) )
= ( ^ [A4: a] : ( minus_minus_nat @ ( count_a @ X2 @ A4 ) @ ( count_a @ Xa2 @ A4 ) ) ) ) ).
% minus_multiset.rep_eq
thf(fact_998_add__nonpos__eq__0__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_999_add__nonneg__eq__0__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1000_add__nonpos__nonpos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1001_add__nonneg__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1002_add__increasing2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_1003_add__decreasing2,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_1004_add__increasing,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_1005_add__decreasing,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_1006_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_1007_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_1008_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1009_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: nat,J2: nat,K3: nat,L4: nat] :
( ( ( ord_less_eq_nat @ I2 @ J2 )
& ( ord_less_nat @ K3 @ L4 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J2 @ L4 ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1010_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: nat,J2: nat,K3: nat,L4: nat] :
( ( ( ord_less_nat @ I2 @ J2 )
& ( ord_less_eq_nat @ K3 @ L4 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J2 @ L4 ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1011_add__le__less__mono,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_1012_add__less__le__mono,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_1013_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( minus_minus_nat @ B2 @ A2 )
= C )
= ( B2
= ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1014_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B2 @ A2 ) )
= B2 ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1015_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1016_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 )
= ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1017_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1018_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A2 )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1019_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1020_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1021_le__add__diff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 ) ) ) ).
% le_add_diff
thf(fact_1022_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ A2 )
= B2 ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_1023_add__le__add__imp__diff__le,axiom,
! [I2: nat,K3: nat,N4: nat,J2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K3 ) @ N4 )
=> ( ( ord_less_eq_nat @ N4 @ ( plus_plus_nat @ J2 @ K3 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K3 ) @ N4 )
=> ( ( ord_less_eq_nat @ N4 @ ( plus_plus_nat @ J2 @ K3 ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N4 @ K3 ) @ J2 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1024_add__le__imp__le__diff,axiom,
! [I2: nat,K3: nat,N4: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K3 ) @ N4 )
=> ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N4 @ K3 ) ) ) ).
% add_le_imp_le_diff
thf(fact_1025_hd__in__set,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( member_a @ ( hd_a @ Xs2 ) @ ( set_a2 @ Xs2 ) ) ) ).
% hd_in_set
thf(fact_1026_list_Oset__sel_I1_J,axiom,
! [A2: list_a] :
( ( A2 != nil_a )
=> ( member_a @ ( hd_a @ A2 ) @ ( set_a2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_1027_longest__common__prefix,axiom,
! [Xs2: list_a,Ys: list_a] :
? [Ps: list_a,Xs5: list_a,Ys7: list_a] :
( ( Xs2
= ( append_a @ Ps @ Xs5 ) )
& ( Ys
= ( append_a @ Ps @ Ys7 ) )
& ( ( Xs5 = nil_a )
| ( Ys7 = nil_a )
| ( ( hd_a @ Xs5 )
!= ( hd_a @ Ys7 ) ) ) ) ).
% longest_common_prefix
thf(fact_1028_hd__append,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( Xs2 = nil_a )
=> ( ( hd_a @ ( append_a @ Xs2 @ Ys ) )
= ( hd_a @ Ys ) ) )
& ( ( Xs2 != nil_a )
=> ( ( hd_a @ ( append_a @ Xs2 @ Ys ) )
= ( hd_a @ Xs2 ) ) ) ) ).
% hd_append
thf(fact_1029_ex__least__nat__le,axiom,
! [P: nat > $o,N4: nat] :
( ( P @ N4 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N4 )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K4 )
=> ~ ( P @ I5 ) )
& ( P @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1030_le__imp__less__Suc,axiom,
! [M: nat,N4: nat] :
( ( ord_less_eq_nat @ M @ N4 )
=> ( ord_less_nat @ M @ ( suc @ N4 ) ) ) ).
% le_imp_less_Suc
thf(fact_1031_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N7: nat] : ( ord_less_eq_nat @ ( suc @ N7 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1032_less__Suc__eq__le,axiom,
! [M: nat,N4: nat] :
( ( ord_less_nat @ M @ ( suc @ N4 ) )
= ( ord_less_eq_nat @ M @ N4 ) ) ).
% less_Suc_eq_le
thf(fact_1033_le__less__Suc__eq,axiom,
! [M: nat,N4: nat] :
( ( ord_less_eq_nat @ M @ N4 )
=> ( ( ord_less_nat @ N4 @ ( suc @ M ) )
= ( N4 = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1034_Suc__le__lessD,axiom,
! [M: nat,N4: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N4 )
=> ( ord_less_nat @ M @ N4 ) ) ).
% Suc_le_lessD
thf(fact_1035_inc__induct,axiom,
! [I2: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( P @ J2 )
=> ( ! [N5: nat] :
( ( ord_less_eq_nat @ I2 @ N5 )
=> ( ( ord_less_nat @ N5 @ J2 )
=> ( ( P @ ( suc @ N5 ) )
=> ( P @ N5 ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_1036_dec__induct,axiom,
! [I2: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( P @ I2 )
=> ( ! [N5: nat] :
( ( ord_less_eq_nat @ I2 @ N5 )
=> ( ( ord_less_nat @ N5 @ J2 )
=> ( ( P @ N5 )
=> ( P @ ( suc @ N5 ) ) ) ) )
=> ( P @ J2 ) ) ) ) ).
% dec_induct
thf(fact_1037_Suc__le__eq,axiom,
! [M: nat,N4: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N4 )
= ( ord_less_nat @ M @ N4 ) ) ).
% Suc_le_eq
thf(fact_1038_Suc__leI,axiom,
! [M: nat,N4: nat] :
( ( ord_less_nat @ M @ N4 )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N4 ) ) ).
% Suc_leI
thf(fact_1039_impossible__Cons,axiom,
! [Xs2: list_a,Ys: list_a,X2: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs2 ) @ ( size_size_list_a @ Ys ) )
=> ( Xs2
!= ( cons_a @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_1040_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K3: nat] :
( ! [M6: nat,N5: nat] :
( ( ord_less_nat @ M6 @ N5 )
=> ( ord_less_nat @ ( F @ M6 ) @ ( F @ N5 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K3 ) @ ( F @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1041_Suc__diff__le,axiom,
! [N4: nat,M: nat] :
( ( ord_less_eq_nat @ N4 @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N4 )
= ( suc @ ( minus_minus_nat @ M @ N4 ) ) ) ) ).
% Suc_diff_le
thf(fact_1042_less__diff__iff,axiom,
! [K3: nat,M: nat,N4: nat] :
( ( ord_less_eq_nat @ K3 @ M )
=> ( ( ord_less_eq_nat @ K3 @ N4 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K3 ) @ ( minus_minus_nat @ N4 @ K3 ) )
= ( ord_less_nat @ M @ N4 ) ) ) ) ).
% less_diff_iff
thf(fact_1043_diff__less__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1044_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J2: nat,K3: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I2 )
= K3 )
= ( J2
= ( plus_plus_nat @ K3 @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1045_Nat_Odiff__add__assoc2,axiom,
! [K3: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K3 @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I2 ) @ K3 )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K3 ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1046_Nat_Odiff__add__assoc,axiom,
! [K3: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K3 @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K3 )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J2 @ K3 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1047_Nat_Ole__diff__conv2,axiom,
! [K3: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K3 @ J2 )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J2 @ K3 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K3 ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1048_le__diff__conv,axiom,
! [J2: nat,K3: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K3 ) @ I2 )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I2 @ K3 ) ) ) ).
% le_diff_conv
thf(fact_1049_diff__size__le__size__Diff,axiom,
! [M2: multiset_a,M5: multiset_a] : ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_multiset_a @ M2 ) @ ( size_size_multiset_a @ M5 ) ) @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M2 @ M5 ) ) ) ).
% diff_size_le_size_Diff
thf(fact_1050_minus__remove1__mset__if,axiom,
! [B2: a,B: multiset_a,A: multiset_a] :
( ( ( ( member_a @ B2 @ ( set_mset_a @ B ) )
& ( member_a @ B2 @ ( set_mset_a @ A ) )
& ( ord_less_eq_nat @ ( count_a @ B @ B2 ) @ ( count_a @ A @ B2 ) ) )
=> ( ( minus_3765977307040488491iset_a @ A @ ( minus_3765977307040488491iset_a @ B @ ( add_mset_a @ B2 @ zero_zero_multiset_a ) ) )
= ( plus_plus_multiset_a @ ( add_mset_a @ B2 @ zero_zero_multiset_a ) @ ( minus_3765977307040488491iset_a @ A @ B ) ) ) )
& ( ~ ( ( member_a @ B2 @ ( set_mset_a @ B ) )
& ( member_a @ B2 @ ( set_mset_a @ A ) )
& ( ord_less_eq_nat @ ( count_a @ B @ B2 ) @ ( count_a @ A @ B2 ) ) )
=> ( ( minus_3765977307040488491iset_a @ A @ ( minus_3765977307040488491iset_a @ B @ ( add_mset_a @ B2 @ zero_zero_multiset_a ) ) )
= ( minus_3765977307040488491iset_a @ A @ B ) ) ) ) ).
% minus_remove1_mset_if
thf(fact_1051_in__diff__count,axiom,
! [A2: a,M2: multiset_a,N: multiset_a] :
( ( member_a @ A2 @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M2 @ N ) ) )
= ( ord_less_nat @ ( count_a @ N @ A2 ) @ ( count_a @ M2 @ A2 ) ) ) ).
% in_diff_count
thf(fact_1052_count__in__diffI,axiom,
! [N: multiset_a,X2: a,M2: multiset_a] :
( ! [N5: nat] :
( ( count_a @ N @ X2 )
!= ( plus_plus_nat @ N5 @ ( count_a @ M2 @ X2 ) ) )
=> ( member_a @ X2 @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M2 @ N ) ) ) ) ).
% count_in_diffI
thf(fact_1053_add__strict__increasing2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1054_add__strict__increasing,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1055_add__pos__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_1056_add__nonpos__neg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1057_add__nonneg__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_1058_add__neg__nonpos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1059_hd__conv__nth,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( hd_a @ Xs2 )
= ( nth_a @ Xs2 @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_1060_ex__least__nat__less,axiom,
! [P: nat > $o,N4: nat] :
( ( P @ N4 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_nat @ K4 @ N4 )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K4 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K4 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1061_Suc__le__length__iff,axiom,
! [N4: nat,Xs2: list_a] :
( ( ord_less_eq_nat @ ( suc @ N4 ) @ ( size_size_list_a @ Xs2 ) )
= ( ? [X3: a,Ys3: list_a] :
( ( Xs2
= ( cons_a @ X3 @ Ys3 ) )
& ( ord_less_eq_nat @ N4 @ ( size_size_list_a @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1062_count__remdups__mset__eq__1,axiom,
! [A2: a,A: multiset_a] :
( ( member_a @ A2 @ ( set_mset_a @ ( multis1648347201088684100mset_a @ A ) ) )
= ( ( count_a @ ( multis1648347201088684100mset_a @ A ) @ A2 )
= one_one_nat ) ) ).
% count_remdups_mset_eq_1
thf(fact_1063_less__diff__conv2,axiom,
! [K3: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K3 @ J2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K3 ) @ I2 )
= ( ord_less_nat @ J2 @ ( plus_plus_nat @ I2 @ K3 ) ) ) ) ).
% less_diff_conv2
thf(fact_1064_multiset__induct__min,axiom,
! [P: multiset_nat > $o,M2: multiset_nat] :
( ( P @ zero_z7348594199698428585et_nat )
=> ( ! [X4: nat,M4: multiset_nat] :
( ( P @ M4 )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_mset_nat @ M4 ) )
=> ( ord_less_eq_nat @ X4 @ Xa ) )
=> ( P @ ( add_mset_nat @ X4 @ M4 ) ) ) )
=> ( P @ M2 ) ) ) ).
% multiset_induct_min
thf(fact_1065_multiset__induct__max,axiom,
! [P: multiset_nat > $o,M2: multiset_nat] :
( ( P @ zero_z7348594199698428585et_nat )
=> ( ! [X4: nat,M4: multiset_nat] :
( ( P @ M4 )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_mset_nat @ M4 ) )
=> ( ord_less_eq_nat @ Xa @ X4 ) )
=> ( P @ ( add_mset_nat @ X4 @ M4 ) ) ) )
=> ( P @ M2 ) ) ) ).
% multiset_induct_max
thf(fact_1066_size__Diff1__le,axiom,
! [M2: multiset_a,X2: a] : ( ord_less_eq_nat @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M2 @ ( add_mset_a @ X2 @ zero_zero_multiset_a ) ) ) @ ( size_size_multiset_a @ M2 ) ) ).
% size_Diff1_le
thf(fact_1067_count__mset__gt__0,axiom,
! [X2: a,Xs2: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( count_a @ ( mset_a @ Xs2 ) @ X2 ) ) ) ).
% count_mset_gt_0
thf(fact_1068_count__single,axiom,
! [B2: a,A2: a] :
( ( ( B2 = A2 )
=> ( ( count_a @ ( add_mset_a @ B2 @ zero_zero_multiset_a ) @ A2 )
= one_one_nat ) )
& ( ( B2 != A2 )
=> ( ( count_a @ ( add_mset_a @ B2 @ zero_zero_multiset_a ) @ A2 )
= zero_zero_nat ) ) ) ).
% count_single
thf(fact_1069_in__diff__countE,axiom,
! [X2: a,M2: multiset_a,N: multiset_a] :
( ( member_a @ X2 @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M2 @ N ) ) )
=> ~ ! [N5: nat] :
( ( count_a @ M2 @ X2 )
!= ( plus_plus_nat @ ( suc @ N5 ) @ ( count_a @ N @ X2 ) ) ) ) ).
% in_diff_countE
thf(fact_1070_count__remdups__mset__If,axiom,
! [A2: a,A: multiset_a] :
( ( ( member_a @ A2 @ ( set_mset_a @ A ) )
=> ( ( count_a @ ( multis1648347201088684100mset_a @ A ) @ A2 )
= one_one_nat ) )
& ( ~ ( member_a @ A2 @ ( set_mset_a @ A ) )
=> ( ( count_a @ ( multis1648347201088684100mset_a @ A ) @ A2 )
= zero_zero_nat ) ) ) ).
% count_remdups_mset_If
thf(fact_1071_hd__drop__conv__nth,axiom,
! [N4: nat,Xs2: list_a] :
( ( ord_less_nat @ N4 @ ( size_size_list_a @ Xs2 ) )
=> ( ( hd_a @ ( drop_a @ N4 @ Xs2 ) )
= ( nth_a @ Xs2 @ N4 ) ) ) ).
% hd_drop_conv_nth
thf(fact_1072_nth__take__lemma,axiom,
! [K3: nat,Xs2: list_a,Ys: list_a] :
( ( ord_less_eq_nat @ K3 @ ( size_size_list_a @ Xs2 ) )
=> ( ( ord_less_eq_nat @ K3 @ ( size_size_list_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K3 )
=> ( ( nth_a @ Xs2 @ I3 )
= ( nth_a @ Ys @ I3 ) ) )
=> ( ( take_a @ K3 @ Xs2 )
= ( take_a @ K3 @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_1073_append__eq__append__conv__if,axiom,
! [Xs_1: list_a,Xs_2: list_a,Ys_1: list_a,Ys_2: list_a] :
( ( ( append_a @ Xs_1 @ Xs_2 )
= ( append_a @ Ys_1 @ Ys_2 ) )
= ( ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs_1 ) @ ( size_size_list_a @ Ys_1 ) )
=> ( ( Xs_1
= ( take_a @ ( size_size_list_a @ Xs_1 ) @ Ys_1 ) )
& ( Xs_2
= ( append_a @ ( drop_a @ ( size_size_list_a @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( size_size_list_a @ Xs_1 ) @ ( size_size_list_a @ Ys_1 ) )
=> ( ( ( take_a @ ( size_size_list_a @ Ys_1 ) @ Xs_1 )
= Ys_1 )
& ( ( append_a @ ( drop_a @ ( size_size_list_a @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
= Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
thf(fact_1074_nth__equal__first__eq,axiom,
! [X2: a,Xs2: list_a,N4: nat] :
( ~ ( member_a @ X2 @ ( set_a2 @ Xs2 ) )
=> ( ( ord_less_eq_nat @ N4 @ ( size_size_list_a @ Xs2 ) )
=> ( ( ( nth_a @ ( cons_a @ X2 @ Xs2 ) @ N4 )
= X2 )
= ( N4 = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1075_prefixes__eq__snoc,axiom,
! [Ys: list_a,Xs2: list_list_a,X2: list_a] :
( ( ( prefixes_a @ Ys )
= ( append_list_a @ Xs2 @ ( cons_list_a @ X2 @ nil_list_a ) ) )
= ( ( ( ( Ys = nil_a )
& ( Xs2 = nil_list_a ) )
| ? [Z4: a,Zs3: list_a] :
( ( Ys
= ( append_a @ Zs3 @ ( cons_a @ Z4 @ nil_a ) ) )
& ( Xs2
= ( prefixes_a @ Zs3 ) ) ) )
& ( X2 = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_1076_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C4: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_1077_mset__le__single__right__iff,axiom,
! [M2: multiset_nat,Y: nat] :
( ( ord_le6602235886369790592et_nat @ M2 @ ( add_mset_nat @ Y @ zero_z7348594199698428585et_nat ) )
= ( ( M2
= ( add_mset_nat @ Y @ zero_z7348594199698428585et_nat ) )
| ! [X3: nat] :
( ( member_nat @ X3 @ ( set_mset_nat @ M2 ) )
=> ( ord_less_nat @ X3 @ Y ) ) ) ) ).
% mset_le_single_right_iff
thf(fact_1078_subset__code_I1_J,axiom,
! [Xs2: list_a,B: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ B )
= ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs2 ) )
=> ( member_a @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_1079_set__subset__Cons,axiom,
! [Xs2: list_a,X2: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ ( cons_a @ X2 @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_1080_set__take__subset,axiom,
! [N4: nat,Xs2: list_a] : ( ord_less_eq_set_a @ ( set_a2 @ ( take_a @ N4 @ Xs2 ) ) @ ( set_a2 @ Xs2 ) ) ).
% set_take_subset
thf(fact_1081_set__drop__subset,axiom,
! [N4: nat,Xs2: list_a] : ( ord_less_eq_set_a @ ( set_a2 @ ( drop_a @ N4 @ Xs2 ) ) @ ( set_a2 @ Xs2 ) ) ).
% set_drop_subset
thf(fact_1082_set__remove1__subset,axiom,
! [X2: a,Xs2: list_a] : ( ord_less_eq_set_a @ ( set_a2 @ ( remove1_a @ X2 @ Xs2 ) ) @ ( set_a2 @ Xs2 ) ) ).
% set_remove1_subset
thf(fact_1083_set__take__subset__set__take,axiom,
! [M: nat,N4: nat,Xs2: list_a] :
( ( ord_less_eq_nat @ M @ N4 )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( take_a @ M @ Xs2 ) ) @ ( set_a2 @ ( take_a @ N4 @ Xs2 ) ) ) ) ).
% set_take_subset_set_take
thf(fact_1084_set__drop__subset__set__drop,axiom,
! [N4: nat,M: nat,Xs2: list_a] :
( ( ord_less_eq_nat @ N4 @ M )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( drop_a @ M @ Xs2 ) ) @ ( set_a2 @ ( drop_a @ N4 @ Xs2 ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_1085_mset__update,axiom,
! [I2: nat,Ls2: list_a,V: a] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ls2 ) )
=> ( ( mset_a @ ( list_update_a @ Ls2 @ I2 @ V ) )
= ( add_mset_a @ V @ ( minus_3765977307040488491iset_a @ ( mset_a @ Ls2 ) @ ( add_mset_a @ ( nth_a @ Ls2 @ I2 ) @ zero_zero_multiset_a ) ) ) ) ) ).
% mset_update
thf(fact_1086_upd__conv__take__nth__drop,axiom,
! [I2: nat,Xs2: list_a,A2: a] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs2 ) )
=> ( ( list_update_a @ Xs2 @ I2 @ A2 )
= ( append_a @ ( take_a @ I2 @ Xs2 ) @ ( cons_a @ A2 @ ( drop_a @ ( suc @ I2 ) @ Xs2 ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_1087_length__list__update,axiom,
! [Xs2: list_a,I2: nat,X2: a] :
( ( size_size_list_a @ ( list_update_a @ Xs2 @ I2 @ X2 ) )
= ( size_size_list_a @ Xs2 ) ) ).
% length_list_update
thf(fact_1088_list__update__id,axiom,
! [Xs2: list_a,I2: nat] :
( ( list_update_a @ Xs2 @ I2 @ ( nth_a @ Xs2 @ I2 ) )
= Xs2 ) ).
% list_update_id
thf(fact_1089_nth__list__update__neq,axiom,
! [I2: nat,J2: nat,Xs2: list_a,X2: a] :
( ( I2 != J2 )
=> ( ( nth_a @ ( list_update_a @ Xs2 @ I2 @ X2 ) @ J2 )
= ( nth_a @ Xs2 @ J2 ) ) ) ).
% nth_list_update_neq
thf(fact_1090_list__update__beyond,axiom,
! [Xs2: list_a,I2: nat,X2: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs2 ) @ I2 )
=> ( ( list_update_a @ Xs2 @ I2 @ X2 )
= Xs2 ) ) ).
% list_update_beyond
thf(fact_1091_take__update__cancel,axiom,
! [N4: nat,M: nat,Xs2: list_a,Y: a] :
( ( ord_less_eq_nat @ N4 @ M )
=> ( ( take_a @ N4 @ ( list_update_a @ Xs2 @ M @ Y ) )
= ( take_a @ N4 @ Xs2 ) ) ) ).
% take_update_cancel
thf(fact_1092_drop__update__cancel,axiom,
! [N4: nat,M: nat,Xs2: list_a,X2: a] :
( ( ord_less_nat @ N4 @ M )
=> ( ( drop_a @ M @ ( list_update_a @ Xs2 @ N4 @ X2 ) )
= ( drop_a @ M @ Xs2 ) ) ) ).
% drop_update_cancel
thf(fact_1093_list__update__length,axiom,
! [Xs2: list_a,X2: a,Ys: list_a,Y: a] :
( ( list_update_a @ ( append_a @ Xs2 @ ( cons_a @ X2 @ Ys ) ) @ ( size_size_list_a @ Xs2 ) @ Y )
= ( append_a @ Xs2 @ ( cons_a @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_1094_nth__list__update__eq,axiom,
! [I2: nat,Xs2: list_a,X2: a] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs2 ) )
=> ( ( nth_a @ ( list_update_a @ Xs2 @ I2 @ X2 ) @ I2 )
= X2 ) ) ).
% nth_list_update_eq
thf(fact_1095_set__swap,axiom,
! [I2: nat,Xs2: list_a,J2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs2 ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_a @ Xs2 ) )
=> ( ( set_a2 @ ( list_update_a @ ( list_update_a @ Xs2 @ I2 @ ( nth_a @ Xs2 @ J2 ) ) @ J2 @ ( nth_a @ Xs2 @ I2 ) ) )
= ( set_a2 @ Xs2 ) ) ) ) ).
% set_swap
thf(fact_1096_set__update__subsetI,axiom,
! [Xs2: list_a,A: set_a,X2: a,I2: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ A )
=> ( ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( list_update_a @ Xs2 @ I2 @ X2 ) ) @ A ) ) ) ).
% set_update_subsetI
thf(fact_1097_list__update__code_I2_J,axiom,
! [X2: a,Xs2: list_a,Y: a] :
( ( list_update_a @ ( cons_a @ X2 @ Xs2 ) @ zero_zero_nat @ Y )
= ( cons_a @ Y @ Xs2 ) ) ).
% list_update_code(2)
thf(fact_1098_list__update__code_I3_J,axiom,
! [X2: a,Xs2: list_a,I2: nat,Y: a] :
( ( list_update_a @ ( cons_a @ X2 @ Xs2 ) @ ( suc @ I2 ) @ Y )
= ( cons_a @ X2 @ ( list_update_a @ Xs2 @ I2 @ Y ) ) ) ).
% list_update_code(3)
thf(fact_1099_take__update__swap,axiom,
! [M: nat,Xs2: list_a,N4: nat,X2: a] :
( ( take_a @ M @ ( list_update_a @ Xs2 @ N4 @ X2 ) )
= ( list_update_a @ ( take_a @ M @ Xs2 ) @ N4 @ X2 ) ) ).
% take_update_swap
thf(fact_1100_set__update__memI,axiom,
! [N4: nat,Xs2: list_a,X2: a] :
( ( ord_less_nat @ N4 @ ( size_size_list_a @ Xs2 ) )
=> ( member_a @ X2 @ ( set_a2 @ ( list_update_a @ Xs2 @ N4 @ X2 ) ) ) ) ).
% set_update_memI
thf(fact_1101_list__update__append1,axiom,
! [I2: nat,Xs2: list_a,Ys: list_a,X2: a] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs2 ) )
=> ( ( list_update_a @ ( append_a @ Xs2 @ Ys ) @ I2 @ X2 )
= ( append_a @ ( list_update_a @ Xs2 @ I2 @ X2 ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_1102_list__update__same__conv,axiom,
! [I2: nat,Xs2: list_a,X2: a] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs2 ) )
=> ( ( ( list_update_a @ Xs2 @ I2 @ X2 )
= Xs2 )
= ( ( nth_a @ Xs2 @ I2 )
= X2 ) ) ) ).
% list_update_same_conv
thf(fact_1103_nth__list__update,axiom,
! [I2: nat,Xs2: list_a,J2: nat,X2: a] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs2 ) )
=> ( ( ( I2 = J2 )
=> ( ( nth_a @ ( list_update_a @ Xs2 @ I2 @ X2 ) @ J2 )
= X2 ) )
& ( ( I2 != J2 )
=> ( ( nth_a @ ( list_update_a @ Xs2 @ I2 @ X2 ) @ J2 )
= ( nth_a @ Xs2 @ J2 ) ) ) ) ) ).
% nth_list_update
thf(fact_1104_drop__update__swap,axiom,
! [M: nat,N4: nat,Xs2: list_a,X2: a] :
( ( ord_less_eq_nat @ M @ N4 )
=> ( ( drop_a @ M @ ( list_update_a @ Xs2 @ N4 @ X2 ) )
= ( list_update_a @ ( drop_a @ M @ Xs2 ) @ ( minus_minus_nat @ N4 @ M ) @ X2 ) ) ) ).
% drop_update_swap
thf(fact_1105_list__update__append,axiom,
! [N4: nat,Xs2: list_a,Ys: list_a,X2: a] :
( ( ( ord_less_nat @ N4 @ ( size_size_list_a @ Xs2 ) )
=> ( ( list_update_a @ ( append_a @ Xs2 @ Ys ) @ N4 @ X2 )
= ( append_a @ ( list_update_a @ Xs2 @ N4 @ X2 ) @ Ys ) ) )
& ( ~ ( ord_less_nat @ N4 @ ( size_size_list_a @ Xs2 ) )
=> ( ( list_update_a @ ( append_a @ Xs2 @ Ys ) @ N4 @ X2 )
= ( append_a @ Xs2 @ ( list_update_a @ Ys @ ( minus_minus_nat @ N4 @ ( size_size_list_a @ Xs2 ) ) @ X2 ) ) ) ) ) ).
% list_update_append
thf(fact_1106_mset__swap,axiom,
! [I2: nat,Ls2: list_a,J2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ls2 ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_a @ Ls2 ) )
=> ( ( mset_a @ ( list_update_a @ ( list_update_a @ Ls2 @ J2 @ ( nth_a @ Ls2 @ I2 ) ) @ I2 @ ( nth_a @ Ls2 @ J2 ) ) )
= ( mset_a @ Ls2 ) ) ) ) ).
% mset_swap
thf(fact_1107_subset__mset_Osum__list__update,axiom,
! [K3: nat,Xs2: list_multiset_a,X2: multiset_a] :
( ( ord_less_nat @ K3 @ ( size_s3003218658903709366iset_a @ Xs2 ) )
=> ( ( groups778166972783649551iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ ( list_u788720842672076722iset_a @ Xs2 @ K3 @ X2 ) )
= ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ ( groups778166972783649551iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ Xs2 ) @ X2 ) @ ( nth_multiset_a @ Xs2 @ K3 ) ) ) ) ).
% subset_mset.sum_list_update
thf(fact_1108_subset__mset_Osum__list__eq__0__iff,axiom,
! [Ns: list_multiset_a] :
( ( ( groups778166972783649551iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ Ns )
= zero_zero_multiset_a )
= ( ! [X3: multiset_a] :
( ( member_multiset_a @ X3 @ ( set_multiset_a2 @ Ns ) )
=> ( X3 = zero_zero_multiset_a ) ) ) ) ).
% subset_mset.sum_list_eq_0_iff
thf(fact_1109_set__permutations__of__list__impl,axiom,
! [Xs2: list_a] :
( ( set_list_a2 @ ( multis7466489000348688569impl_a @ Xs2 ) )
= ( multis5886240593633752526iset_a @ ( mset_a @ Xs2 ) ) ) ).
% set_permutations_of_list_impl
thf(fact_1110_subset__mset_Oelem__le__sum__list,axiom,
! [K3: nat,Ns: list_multiset_a] :
( ( ord_less_nat @ K3 @ ( size_s3003218658903709366iset_a @ Ns ) )
=> ( subseteq_mset_a @ ( nth_multiset_a @ Ns @ K3 ) @ ( groups778166972783649551iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ Ns ) ) ) ).
% subset_mset.elem_le_sum_list
thf(fact_1111_SuccI,axiom,
! [Kl: list_a,K3: a,Kl2: set_list_a] :
( ( member_list_a @ ( append_a @ Kl @ ( cons_a @ K3 @ nil_a ) ) @ Kl2 )
=> ( member_a @ K3 @ ( bNF_Greatest_Succ_a @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_1112_subset__mset_Ole__zero__eq,axiom,
! [N4: multiset_a] :
( ( subseteq_mset_a @ N4 @ zero_zero_multiset_a )
= ( N4 = zero_zero_multiset_a ) ) ).
% subset_mset.le_zero_eq
thf(fact_1113_subset__mset_Oextremum__unique,axiom,
! [A2: multiset_a] :
( ( subseteq_mset_a @ A2 @ zero_zero_multiset_a )
= ( A2 = zero_zero_multiset_a ) ) ).
% subset_mset.extremum_unique
thf(fact_1114_subset__mset_Oadd__le__cancel__left,axiom,
! [C: multiset_a,A2: multiset_a,B2: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C @ A2 ) @ ( plus_plus_multiset_a @ C @ B2 ) )
= ( subseteq_mset_a @ A2 @ B2 ) ) ).
% subset_mset.add_le_cancel_left
thf(fact_1115_subset__mset_Oadd__le__cancel__right,axiom,
! [A2: multiset_a,C: multiset_a,B2: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C ) @ ( plus_plus_multiset_a @ B2 @ C ) )
= ( subseteq_mset_a @ A2 @ B2 ) ) ).
% subset_mset.add_le_cancel_right
thf(fact_1116_mset__subset__eq__mono__add__left__cancel,axiom,
! [C2: multiset_a,A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C2 @ A ) @ ( plus_plus_multiset_a @ C2 @ B ) )
= ( subseteq_mset_a @ A @ B ) ) ).
% mset_subset_eq_mono_add_left_cancel
thf(fact_1117_mset__subset__eq__mono__add__right__cancel,axiom,
! [A: multiset_a,C2: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C2 ) @ ( plus_plus_multiset_a @ B @ C2 ) )
= ( subseteq_mset_a @ A @ B ) ) ).
% mset_subset_eq_mono_add_right_cancel
thf(fact_1118_add__mset__subseteq__single__iff,axiom,
! [A2: a,M2: multiset_a,B2: a] :
( ( subseteq_mset_a @ ( add_mset_a @ A2 @ M2 ) @ ( add_mset_a @ B2 @ zero_zero_multiset_a ) )
= ( ( M2 = zero_zero_multiset_a )
& ( A2 = B2 ) ) ) ).
% add_mset_subseteq_single_iff
thf(fact_1119_subset__mset_Ole__add__same__cancel2,axiom,
! [A2: multiset_a,B2: multiset_a] :
( ( subseteq_mset_a @ A2 @ ( plus_plus_multiset_a @ B2 @ A2 ) )
= ( subseteq_mset_a @ zero_zero_multiset_a @ B2 ) ) ).
% subset_mset.le_add_same_cancel2
thf(fact_1120_subset__mset_Ole__add__same__cancel1,axiom,
! [A2: multiset_a,B2: multiset_a] :
( ( subseteq_mset_a @ A2 @ ( plus_plus_multiset_a @ A2 @ B2 ) )
= ( subseteq_mset_a @ zero_zero_multiset_a @ B2 ) ) ).
% subset_mset.le_add_same_cancel1
thf(fact_1121_subset__mset_Oadd__le__same__cancel2,axiom,
! [A2: multiset_a,B2: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ B2 ) @ B2 )
= ( subseteq_mset_a @ A2 @ zero_zero_multiset_a ) ) ).
% subset_mset.add_le_same_cancel2
thf(fact_1122_subset__mset_Oadd__le__same__cancel1,axiom,
! [B2: multiset_a,A2: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ B2 @ A2 ) @ B2 )
= ( subseteq_mset_a @ A2 @ zero_zero_multiset_a ) ) ).
% subset_mset.add_le_same_cancel1
thf(fact_1123_mset__subset__eq__multiset__union__diff__commute,axiom,
! [B: multiset_a,A: multiset_a,C2: multiset_a] :
( ( subseteq_mset_a @ B @ A )
=> ( ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ A @ B ) @ C2 )
= ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A @ C2 ) @ B ) ) ) ).
% mset_subset_eq_multiset_union_diff_commute
thf(fact_1124_subset__mset_Oadd__diff__assoc2,axiom,
! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A2 @ B2 )
=> ( ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ B2 @ A2 ) @ C )
= ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ B2 @ C ) @ A2 ) ) ) ).
% subset_mset.add_diff_assoc2
thf(fact_1125_subset__mset_Oadd__diff__assoc,axiom,
! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A2 @ B2 )
=> ( ( plus_plus_multiset_a @ C @ ( minus_3765977307040488491iset_a @ B2 @ A2 ) )
= ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ C @ B2 ) @ A2 ) ) ) ).
% subset_mset.add_diff_assoc
thf(fact_1126_single__subset__iff,axiom,
! [A2: a,M2: multiset_a] :
( ( subseteq_mset_a @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) @ M2 )
= ( member_a @ A2 @ ( set_mset_a @ M2 ) ) ) ).
% single_subset_iff
thf(fact_1127_subset__mset_Oadd__decreasing,axiom,
! [A2: multiset_a,C: multiset_a,B2: multiset_a] :
( ( subseteq_mset_a @ A2 @ zero_zero_multiset_a )
=> ( ( subseteq_mset_a @ C @ B2 )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C ) @ B2 ) ) ) ).
% subset_mset.add_decreasing
thf(fact_1128_subset__mset_Oadd__increasing,axiom,
! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ zero_zero_multiset_a @ A2 )
=> ( ( subseteq_mset_a @ B2 @ C )
=> ( subseteq_mset_a @ B2 @ ( plus_plus_multiset_a @ A2 @ C ) ) ) ) ).
% subset_mset.add_increasing
thf(fact_1129_subset__mset_Oadd__decreasing2,axiom,
! [C: multiset_a,A2: multiset_a,B2: multiset_a] :
( ( subseteq_mset_a @ C @ zero_zero_multiset_a )
=> ( ( subseteq_mset_a @ A2 @ B2 )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C ) @ B2 ) ) ) ).
% subset_mset.add_decreasing2
thf(fact_1130_subset__mset_Oadd__increasing2,axiom,
! [C: multiset_a,B2: multiset_a,A2: multiset_a] :
( ( subseteq_mset_a @ zero_zero_multiset_a @ C )
=> ( ( subseteq_mset_a @ B2 @ A2 )
=> ( subseteq_mset_a @ B2 @ ( plus_plus_multiset_a @ A2 @ C ) ) ) ) ).
% subset_mset.add_increasing2
thf(fact_1131_subset__mset_Oadd__nonneg__nonneg,axiom,
! [A2: multiset_a,B2: multiset_a] :
( ( subseteq_mset_a @ zero_zero_multiset_a @ A2 )
=> ( ( subseteq_mset_a @ zero_zero_multiset_a @ B2 )
=> ( subseteq_mset_a @ zero_zero_multiset_a @ ( plus_plus_multiset_a @ A2 @ B2 ) ) ) ) ).
% subset_mset.add_nonneg_nonneg
thf(fact_1132_subset__mset_Oadd__nonpos__nonpos,axiom,
! [A2: multiset_a,B2: multiset_a] :
( ( subseteq_mset_a @ A2 @ zero_zero_multiset_a )
=> ( ( subseteq_mset_a @ B2 @ zero_zero_multiset_a )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ B2 ) @ zero_zero_multiset_a ) ) ) ).
% subset_mset.add_nonpos_nonpos
thf(fact_1133_subset__mset_Oadd__nonneg__eq__0__iff,axiom,
! [X2: multiset_a,Y: multiset_a] :
( ( subseteq_mset_a @ zero_zero_multiset_a @ X2 )
=> ( ( subseteq_mset_a @ zero_zero_multiset_a @ Y )
=> ( ( ( plus_plus_multiset_a @ X2 @ Y )
= zero_zero_multiset_a )
= ( ( X2 = zero_zero_multiset_a )
& ( Y = zero_zero_multiset_a ) ) ) ) ) ).
% subset_mset.add_nonneg_eq_0_iff
thf(fact_1134_subset__mset_Oadd__nonpos__eq__0__iff,axiom,
! [X2: multiset_a,Y: multiset_a] :
( ( subseteq_mset_a @ X2 @ zero_zero_multiset_a )
=> ( ( subseteq_mset_a @ Y @ zero_zero_multiset_a )
=> ( ( ( plus_plus_multiset_a @ X2 @ Y )
= zero_zero_multiset_a )
= ( ( X2 = zero_zero_multiset_a )
& ( Y = zero_zero_multiset_a ) ) ) ) ) ).
% subset_mset.add_nonpos_eq_0_iff
thf(fact_1135_subset__add__mset__notin__subset__mset,axiom,
! [A: multiset_a,B2: a,B: multiset_a] :
( ( subseteq_mset_a @ A @ ( add_mset_a @ B2 @ B ) )
=> ( ~ ( member_a @ B2 @ ( set_mset_a @ A ) )
=> ( subseteq_mset_a @ A @ B ) ) ) ).
% subset_add_mset_notin_subset_mset
thf(fact_1136_Diff__eq__empty__iff__mset,axiom,
! [A: multiset_a,B: multiset_a] :
( ( ( minus_3765977307040488491iset_a @ A @ B )
= zero_zero_multiset_a )
= ( subseteq_mset_a @ A @ B ) ) ).
% Diff_eq_empty_iff_mset
thf(fact_1137_subset__mset_Odiff__add,axiom,
! [A2: multiset_a,B2: multiset_a] :
( ( subseteq_mset_a @ A2 @ B2 )
=> ( ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ B2 @ A2 ) @ A2 )
= B2 ) ) ).
% subset_mset.diff_add
thf(fact_1138_subset__mset_Ole__add__diff,axiom,
! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A2 @ B2 )
=> ( subseteq_mset_a @ C @ ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ B2 @ C ) @ A2 ) ) ) ).
% subset_mset.le_add_diff
thf(fact_1139_subset__mset_Ole__diff__conv2,axiom,
! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A2 @ B2 )
=> ( ( subseteq_mset_a @ C @ ( minus_3765977307040488491iset_a @ B2 @ A2 ) )
= ( subseteq_mset_a @ ( plus_plus_multiset_a @ C @ A2 ) @ B2 ) ) ) ).
% subset_mset.le_diff_conv2
thf(fact_1140_subset__mset_Odiff__add__assoc,axiom,
! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A2 @ B2 )
=> ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ C @ B2 ) @ A2 )
= ( plus_plus_multiset_a @ C @ ( minus_3765977307040488491iset_a @ B2 @ A2 ) ) ) ) ).
% subset_mset.diff_add_assoc
thf(fact_1141_subset__mset_Odiff__add__assoc2,axiom,
! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A2 @ B2 )
=> ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ B2 @ C ) @ A2 )
= ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ B2 @ A2 ) @ C ) ) ) ).
% subset_mset.diff_add_assoc2
thf(fact_1142_subset__mset_Odiff__diff__right,axiom,
! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A2 @ B2 )
=> ( ( minus_3765977307040488491iset_a @ C @ ( minus_3765977307040488491iset_a @ B2 @ A2 ) )
= ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ C @ A2 ) @ B2 ) ) ) ).
% subset_mset.diff_diff_right
thf(fact_1143_subset__mset_Oadd__diff__inverse,axiom,
! [A2: multiset_a,B2: multiset_a] :
( ( subseteq_mset_a @ A2 @ B2 )
=> ( ( plus_plus_multiset_a @ A2 @ ( minus_3765977307040488491iset_a @ B2 @ A2 ) )
= B2 ) ) ).
% subset_mset.add_diff_inverse
thf(fact_1144_subset__mset_Ole__imp__diff__is__add,axiom,
! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A2 @ B2 )
=> ( ( subseteq_mset_a @ A2 @ B2 )
=> ( ( ( minus_3765977307040488491iset_a @ B2 @ A2 )
= C )
= ( B2
= ( plus_plus_multiset_a @ C @ A2 ) ) ) ) ) ).
% subset_mset.le_imp_diff_is_add
thf(fact_1145_subset__eq__diff__conv,axiom,
! [A: multiset_a,C2: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( minus_3765977307040488491iset_a @ A @ C2 ) @ B )
= ( subseteq_mset_a @ A @ ( plus_plus_multiset_a @ B @ C2 ) ) ) ).
% subset_eq_diff_conv
thf(fact_1146_multiset__diff__union__assoc,axiom,
! [C2: multiset_a,B: multiset_a,A: multiset_a] :
( ( subseteq_mset_a @ C2 @ B )
=> ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A @ B ) @ C2 )
= ( plus_plus_multiset_a @ A @ ( minus_3765977307040488491iset_a @ B @ C2 ) ) ) ) ).
% multiset_diff_union_assoc
thf(fact_1147_mset__take__subseteq,axiom,
! [N4: nat,Xs2: list_a] : ( subseteq_mset_a @ ( mset_a @ ( take_a @ N4 @ Xs2 ) ) @ ( mset_a @ Xs2 ) ) ).
% mset_take_subseteq
thf(fact_1148_subset__mset_Ozero__le,axiom,
! [X2: multiset_a] : ( subseteq_mset_a @ zero_zero_multiset_a @ X2 ) ).
% subset_mset.zero_le
thf(fact_1149_subset__mset_Obot__least,axiom,
! [A2: multiset_a] : ( subseteq_mset_a @ zero_zero_multiset_a @ A2 ) ).
% subset_mset.bot_least
thf(fact_1150_subset__mset_Oextremum__uniqueI,axiom,
! [A2: multiset_a] :
( ( subseteq_mset_a @ A2 @ zero_zero_multiset_a )
=> ( A2 = zero_zero_multiset_a ) ) ).
% subset_mset.extremum_uniqueI
thf(fact_1151_empty__le,axiom,
! [A: multiset_a] : ( subseteq_mset_a @ zero_zero_multiset_a @ A ) ).
% empty_le
thf(fact_1152_mset__subset__eqD,axiom,
! [A: multiset_a,B: multiset_a,X2: a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( member_a @ X2 @ ( set_mset_a @ A ) )
=> ( member_a @ X2 @ ( set_mset_a @ B ) ) ) ) ).
% mset_subset_eqD
thf(fact_1153_mset__subset__eq__add__mset__cancel,axiom,
! [A2: a,A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( add_mset_a @ A2 @ A ) @ ( add_mset_a @ A2 @ B ) )
= ( subseteq_mset_a @ A @ B ) ) ).
% mset_subset_eq_add_mset_cancel
thf(fact_1154_subset__mset__imp__subset__add__mset,axiom,
! [A: multiset_a,B: multiset_a,X2: a] :
( ( subseteq_mset_a @ A @ B )
=> ( subseteq_mset_a @ A @ ( add_mset_a @ X2 @ B ) ) ) ).
% subset_mset_imp_subset_add_mset
thf(fact_1155_subset__mset_Oadd__mono,axiom,
! [A2: multiset_a,B2: multiset_a,C: multiset_a,D2: multiset_a] :
( ( subseteq_mset_a @ A2 @ B2 )
=> ( ( subseteq_mset_a @ C @ D2 )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C ) @ ( plus_plus_multiset_a @ B2 @ D2 ) ) ) ) ).
% subset_mset.add_mono
thf(fact_1156_subset__mset_Oless__eqE,axiom,
! [A2: multiset_a,B2: multiset_a] :
( ( subseteq_mset_a @ A2 @ B2 )
=> ~ ! [C3: multiset_a] :
( B2
!= ( plus_plus_multiset_a @ A2 @ C3 ) ) ) ).
% subset_mset.less_eqE
thf(fact_1157_subset__mset_Ole__iff__add,axiom,
( subseteq_mset_a
= ( ^ [A4: multiset_a,B4: multiset_a] :
? [C4: multiset_a] :
( B4
= ( plus_plus_multiset_a @ A4 @ C4 ) ) ) ) ).
% subset_mset.le_iff_add
thf(fact_1158_subset__mset_Oadd__left__mono,axiom,
! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A2 @ B2 )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ C @ A2 ) @ ( plus_plus_multiset_a @ C @ B2 ) ) ) ).
% subset_mset.add_left_mono
thf(fact_1159_subset__mset_Oadd__right__mono,axiom,
! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A2 @ B2 )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C ) @ ( plus_plus_multiset_a @ B2 @ C ) ) ) ).
% subset_mset.add_right_mono
thf(fact_1160_subset__mset_Oadd__le__imp__le__left,axiom,
! [C: multiset_a,A2: multiset_a,B2: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C @ A2 ) @ ( plus_plus_multiset_a @ C @ B2 ) )
=> ( subseteq_mset_a @ A2 @ B2 ) ) ).
% subset_mset.add_le_imp_le_left
thf(fact_1161_subset__mset_Oadd__le__imp__le__right,axiom,
! [A2: multiset_a,C: multiset_a,B2: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C ) @ ( plus_plus_multiset_a @ B2 @ C ) )
=> ( subseteq_mset_a @ A2 @ B2 ) ) ).
% subset_mset.add_le_imp_le_right
thf(fact_1162_mset__subset__eq__add__left,axiom,
! [A: multiset_a,B: multiset_a] : ( subseteq_mset_a @ A @ ( plus_plus_multiset_a @ A @ B ) ) ).
% mset_subset_eq_add_left
thf(fact_1163_mset__subset__eq__mono__add,axiom,
! [A: multiset_a,B: multiset_a,C2: multiset_a,D: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( subseteq_mset_a @ C2 @ D )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C2 ) @ ( plus_plus_multiset_a @ B @ D ) ) ) ) ).
% mset_subset_eq_mono_add
thf(fact_1164_mset__subset__eq__add__right,axiom,
! [B: multiset_a,A: multiset_a] : ( subseteq_mset_a @ B @ ( plus_plus_multiset_a @ A @ B ) ) ).
% mset_subset_eq_add_right
thf(fact_1165_mset__subset__eq__exists__conv,axiom,
( subseteq_mset_a
= ( ^ [A5: multiset_a,B7: multiset_a] :
? [C5: multiset_a] :
( B7
= ( plus_plus_multiset_a @ A5 @ C5 ) ) ) ) ).
% mset_subset_eq_exists_conv
thf(fact_1166_diff__subset__eq__self,axiom,
! [M2: multiset_a,N: multiset_a] : ( subseteq_mset_a @ ( minus_3765977307040488491iset_a @ M2 @ N ) @ M2 ) ).
% diff_subset_eq_self
thf(fact_1167_mset__subset__eqI,axiom,
! [A: multiset_a,B: multiset_a] :
( ! [A3: a] : ( ord_less_eq_nat @ ( count_a @ A @ A3 ) @ ( count_a @ B @ A3 ) )
=> ( subseteq_mset_a @ A @ B ) ) ).
% mset_subset_eqI
thf(fact_1168_subseteq__mset__def,axiom,
( subseteq_mset_a
= ( ^ [A5: multiset_a,B7: multiset_a] :
! [A4: a] : ( ord_less_eq_nat @ ( count_a @ A5 @ A4 ) @ ( count_a @ B7 @ A4 ) ) ) ) ).
% subseteq_mset_def
thf(fact_1169_mset__subset__eq__count,axiom,
! [A: multiset_a,B: multiset_a,A2: a] :
( ( subseteq_mset_a @ A @ B )
=> ( ord_less_eq_nat @ ( count_a @ A @ A2 ) @ ( count_a @ B @ A2 ) ) ) ).
% mset_subset_eq_count
thf(fact_1170_set__mset__mono,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ord_less_eq_set_a @ ( set_mset_a @ A ) @ ( set_mset_a @ B ) ) ) ).
% set_mset_mono
thf(fact_1171_mset__subset__eq__single,axiom,
! [A2: a,B: multiset_a] :
( ( member_a @ A2 @ ( set_mset_a @ B ) )
=> ( subseteq_mset_a @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) @ B ) ) ).
% mset_subset_eq_single
thf(fact_1172_multi__subset__induct,axiom,
! [F2: multiset_a,A: multiset_a,P: multiset_a > $o] :
( ( subseteq_mset_a @ F2 @ A )
=> ( ( P @ zero_zero_multiset_a )
=> ( ! [A3: a,F3: multiset_a] :
( ( member_a @ A3 @ ( set_mset_a @ A ) )
=> ( ( P @ F3 )
=> ( P @ ( add_mset_a @ A3 @ F3 ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% multi_subset_induct
thf(fact_1173_size__Diff__submset,axiom,
! [M2: multiset_a,M5: multiset_a] :
( ( subseteq_mset_a @ M2 @ M5 )
=> ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M5 @ M2 ) )
= ( minus_minus_nat @ ( size_size_multiset_a @ M5 ) @ ( size_size_multiset_a @ M2 ) ) ) ) ).
% size_Diff_submset
thf(fact_1174_subset__mset_Osum__list__nonneg,axiom,
! [Xs2: list_multiset_a] :
( ! [X4: multiset_a] :
( ( member_multiset_a @ X4 @ ( set_multiset_a2 @ Xs2 ) )
=> ( subseteq_mset_a @ zero_zero_multiset_a @ X4 ) )
=> ( subseteq_mset_a @ zero_zero_multiset_a @ ( groups778166972783649551iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ Xs2 ) ) ) ).
% subset_mset.sum_list_nonneg
thf(fact_1175_subset__mset_Osum__list__nonpos,axiom,
! [Xs2: list_multiset_a] :
( ! [X4: multiset_a] :
( ( member_multiset_a @ X4 @ ( set_multiset_a2 @ Xs2 ) )
=> ( subseteq_mset_a @ X4 @ zero_zero_multiset_a ) )
=> ( subseteq_mset_a @ ( groups778166972783649551iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ Xs2 ) @ zero_zero_multiset_a ) ) ).
% subset_mset.sum_list_nonpos
thf(fact_1176_subset__mset_Omember__le__sum__list,axiom,
! [X2: multiset_a,Xs2: list_multiset_a] :
( ( member_multiset_a @ X2 @ ( set_multiset_a2 @ Xs2 ) )
=> ( subseteq_mset_a @ X2 @ ( groups778166972783649551iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ Xs2 ) ) ) ).
% subset_mset.member_le_sum_list
thf(fact_1177_subset__mset_Osum__list__nonneg__eq__0__iff,axiom,
! [Xs2: list_multiset_a] :
( ! [X4: multiset_a] :
( ( member_multiset_a @ X4 @ ( set_multiset_a2 @ Xs2 ) )
=> ( subseteq_mset_a @ zero_zero_multiset_a @ X4 ) )
=> ( ( ( groups778166972783649551iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ Xs2 )
= zero_zero_multiset_a )
= ( ! [X3: multiset_a] :
( ( member_multiset_a @ X3 @ ( set_multiset_a2 @ Xs2 ) )
=> ( X3 = zero_zero_multiset_a ) ) ) ) ) ).
% subset_mset.sum_list_nonneg_eq_0_iff
thf(fact_1178_insert__subset__eq__iff,axiom,
! [A2: a,A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( add_mset_a @ A2 @ A ) @ B )
= ( ( member_a @ A2 @ ( set_mset_a @ B ) )
& ( subseteq_mset_a @ A @ ( minus_3765977307040488491iset_a @ B @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) ) ) ) ).
% insert_subset_eq_iff
thf(fact_1179_less__multiset_092_060_094sub_062D_092_060_094sub_062M,axiom,
( ord_le5777773500796000884et_nat
= ( ^ [M3: multiset_nat,N2: multiset_nat] :
? [X6: multiset_nat,Y7: multiset_nat] :
( ( X6 != zero_z7348594199698428585et_nat )
& ( subseteq_mset_nat @ X6 @ N2 )
& ( M3
= ( plus_p6334493942879108393et_nat @ ( minus_8522176038001411705et_nat @ N2 @ X6 ) @ Y7 ) )
& ! [K5: nat] :
( ( member_nat @ K5 @ ( set_mset_nat @ Y7 ) )
=> ? [A4: nat] :
( ( member_nat @ A4 @ ( set_mset_nat @ X6 ) )
& ( ord_less_nat @ K5 @ A4 ) ) ) ) ) ) ).
% less_multiset\<^sub>D\<^sub>M
thf(fact_1180_SuccD,axiom,
! [K3: a,Kl2: set_list_a,Kl: list_a] :
( ( member_a @ K3 @ ( bNF_Greatest_Succ_a @ Kl2 @ Kl ) )
=> ( member_list_a @ ( append_a @ Kl @ ( cons_a @ K3 @ nil_a ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_1181_less__multiset_092_060_094sub_062D_092_060_094sub_062M__def,axiom,
( multis2595361328647931344_M_nat
= ( ^ [M3: multiset_nat,N2: multiset_nat] :
? [X6: multiset_nat,Y7: multiset_nat] :
( ( X6 != zero_z7348594199698428585et_nat )
& ( subseteq_mset_nat @ X6 @ N2 )
& ( M3
= ( plus_p6334493942879108393et_nat @ ( minus_8522176038001411705et_nat @ N2 @ X6 ) @ Y7 ) )
& ! [K5: nat] :
( ( member_nat @ K5 @ ( set_mset_nat @ Y7 ) )
=> ? [A4: nat] :
( ( member_nat @ A4 @ ( set_mset_nat @ X6 ) )
& ( ord_less_nat @ K5 @ A4 ) ) ) ) ) ) ).
% less_multiset\<^sub>D\<^sub>M_def
thf(fact_1182_multp_092_060_094sub_062D_092_060_094sub_062M__def,axiom,
( multiset_multp_D_M_a
= ( ^ [R2: a > a > $o,M3: multiset_a,N2: multiset_a] :
? [X6: multiset_a,Y7: multiset_a] :
( ( X6 != zero_zero_multiset_a )
& ( subseteq_mset_a @ X6 @ N2 )
& ( M3
= ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ N2 @ X6 ) @ Y7 ) )
& ! [K5: a] :
( ( member_a @ K5 @ ( set_mset_a @ Y7 ) )
=> ? [A4: a] :
( ( member_a @ A4 @ ( set_mset_a @ X6 ) )
& ( R2 @ K5 @ A4 ) ) ) ) ) ) ).
% multp\<^sub>D\<^sub>M_def
thf(fact_1183_empty__Shift,axiom,
! [Kl2: set_list_a,K3: a] :
( ( member_list_a @ nil_a @ Kl2 )
=> ( ( member_a @ K3 @ ( bNF_Greatest_Succ_a @ Kl2 @ nil_a ) )
=> ( member_list_a @ nil_a @ ( bNF_Greatest_Shift_a @ Kl2 @ K3 ) ) ) ) ).
% empty_Shift
thf(fact_1184_Succ__Shift,axiom,
! [Kl2: set_list_a,K3: a,Kl: list_a] :
( ( bNF_Greatest_Succ_a @ ( bNF_Greatest_Shift_a @ Kl2 @ K3 ) @ Kl )
= ( bNF_Greatest_Succ_a @ Kl2 @ ( cons_a @ K3 @ Kl ) ) ) ).
% Succ_Shift
thf(fact_1185_ShiftD,axiom,
! [Kl: list_a,Kl2: set_list_a,K3: a] :
( ( member_list_a @ Kl @ ( bNF_Greatest_Shift_a @ Kl2 @ K3 ) )
=> ( member_list_a @ ( cons_a @ K3 @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_1186_suffixes__eq__snoc,axiom,
! [Ys: list_a,Xs2: list_list_a,X2: list_a] :
( ( ( suffixes_a @ Ys )
= ( append_list_a @ Xs2 @ ( cons_list_a @ X2 @ nil_list_a ) ) )
= ( ( ( ( Ys = nil_a )
& ( Xs2 = nil_list_a ) )
| ? [Z4: a,Zs3: list_a] :
( ( Ys
= ( cons_a @ Z4 @ Zs3 ) )
& ( Xs2
= ( suffixes_a @ Zs3 ) ) ) )
& ( X2 = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_1187_in__set__product__lists__length,axiom,
! [Xs2: list_a,Xss2: list_list_a] :
( ( member_list_a @ Xs2 @ ( set_list_a2 @ ( product_lists_a @ Xss2 ) ) )
=> ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Xss2 ) ) ) ).
% in_set_product_lists_length
thf(fact_1188_length__suffixes,axiom,
! [Xs2: list_a] :
( ( size_s349497388124573686list_a @ ( suffixes_a @ Xs2 ) )
= ( suc @ ( size_size_list_a @ Xs2 ) ) ) ).
% length_suffixes
thf(fact_1189_suffixes_Osimps_I2_J,axiom,
! [X2: a,Xs2: list_a] :
( ( suffixes_a @ ( cons_a @ X2 @ Xs2 ) )
= ( append_list_a @ ( suffixes_a @ Xs2 ) @ ( cons_list_a @ ( cons_a @ X2 @ Xs2 ) @ nil_list_a ) ) ) ).
% suffixes.simps(2)
thf(fact_1190_mset__tl,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( mset_a @ ( tl_a @ Xs2 ) )
= ( minus_3765977307040488491iset_a @ ( mset_a @ Xs2 ) @ ( add_mset_a @ ( hd_a @ Xs2 ) @ zero_zero_multiset_a ) ) ) ) ).
% mset_tl
thf(fact_1191_tl__append2,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( Xs2 != nil_a )
=> ( ( tl_a @ ( append_a @ Xs2 @ Ys ) )
= ( append_a @ ( tl_a @ Xs2 ) @ Ys ) ) ) ).
% tl_append2
thf(fact_1192_length__tl,axiom,
! [Xs2: list_a] :
( ( size_size_list_a @ ( tl_a @ Xs2 ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs2 ) @ one_one_nat ) ) ).
% length_tl
thf(fact_1193_list_Ocollapse,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_1194_hd__Cons__tl,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs2 ) @ ( tl_a @ Xs2 ) )
= Xs2 ) ) ).
% hd_Cons_tl
thf(fact_1195_Nil__tl,axiom,
! [Xs2: list_a] :
( ( nil_a
= ( tl_a @ Xs2 ) )
= ( ( Xs2 = nil_a )
| ? [X3: a] :
( Xs2
= ( cons_a @ X3 @ nil_a ) ) ) ) ).
% Nil_tl
thf(fact_1196_tl__Nil,axiom,
! [Xs2: list_a] :
( ( ( tl_a @ Xs2 )
= nil_a )
= ( ( Xs2 = nil_a )
| ? [X3: a] :
( Xs2
= ( cons_a @ X3 @ nil_a ) ) ) ) ).
% tl_Nil
thf(fact_1197_tl__append__if,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( Xs2 = nil_a )
=> ( ( tl_a @ ( append_a @ Xs2 @ Ys ) )
= ( tl_a @ Ys ) ) )
& ( ( Xs2 != nil_a )
=> ( ( tl_a @ ( append_a @ Xs2 @ Ys ) )
= ( append_a @ ( tl_a @ Xs2 ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_1198_take__tl,axiom,
! [N4: nat,Xs2: list_a] :
( ( take_a @ N4 @ ( tl_a @ Xs2 ) )
= ( tl_a @ ( take_a @ ( suc @ N4 ) @ Xs2 ) ) ) ).
% take_tl
thf(fact_1199_drop__Suc,axiom,
! [N4: nat,Xs2: list_a] :
( ( drop_a @ ( suc @ N4 ) @ Xs2 )
= ( drop_a @ N4 @ ( tl_a @ Xs2 ) ) ) ).
% drop_Suc
thf(fact_1200_list_Osel_I3_J,axiom,
! [X21: a,X22: list_a] :
( ( tl_a @ ( cons_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_1201_tl__drop,axiom,
! [N4: nat,Xs2: list_a] :
( ( tl_a @ ( drop_a @ N4 @ Xs2 ) )
= ( drop_a @ N4 @ ( tl_a @ Xs2 ) ) ) ).
% tl_drop
thf(fact_1202_list_Oset__sel_I2_J,axiom,
! [A2: list_a,X2: a] :
( ( A2 != nil_a )
=> ( ( member_a @ X2 @ ( set_a2 @ ( tl_a @ A2 ) ) )
=> ( member_a @ X2 @ ( set_a2 @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_1203_list_Oexhaust__sel,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( List
= ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_1204_tl__take,axiom,
! [N4: nat,Xs2: list_a] :
( ( tl_a @ ( take_a @ N4 @ Xs2 ) )
= ( take_a @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( tl_a @ Xs2 ) ) ) ).
% tl_take
thf(fact_1205_nth__tl,axiom,
! [N4: nat,Xs2: list_a] :
( ( ord_less_nat @ N4 @ ( size_size_list_a @ ( tl_a @ Xs2 ) ) )
=> ( ( nth_a @ ( tl_a @ Xs2 ) @ N4 )
= ( nth_a @ Xs2 @ ( suc @ N4 ) ) ) ) ).
% nth_tl
thf(fact_1206_take__Suc,axiom,
! [Xs2: list_a,N4: nat] :
( ( Xs2 != nil_a )
=> ( ( take_a @ ( suc @ N4 ) @ Xs2 )
= ( cons_a @ ( hd_a @ Xs2 ) @ ( take_a @ N4 @ ( tl_a @ Xs2 ) ) ) ) ) ).
% take_Suc
thf(fact_1207_Nitpick_Osize__list__simp_I2_J,axiom,
( size_size_list_a
= ( ^ [Xs4: list_a] : ( if_nat @ ( Xs4 = nil_a ) @ zero_zero_nat @ ( suc @ ( size_size_list_a @ ( tl_a @ Xs4 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_1208_rotate1__hd__tl,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( rotate1_a @ Xs2 )
= ( append_a @ ( tl_a @ Xs2 ) @ ( cons_a @ ( hd_a @ Xs2 ) @ nil_a ) ) ) ) ).
% rotate1_hd_tl
thf(fact_1209_set__rotate1,axiom,
! [Xs2: list_a] :
( ( set_a2 @ ( rotate1_a @ Xs2 ) )
= ( set_a2 @ Xs2 ) ) ).
% set_rotate1
thf(fact_1210_length__rotate1,axiom,
! [Xs2: list_a] :
( ( size_size_list_a @ ( rotate1_a @ Xs2 ) )
= ( size_size_list_a @ Xs2 ) ) ).
% length_rotate1
thf(fact_1211_rotate1__length01,axiom,
! [Xs2: list_a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs2 ) @ one_one_nat )
=> ( ( rotate1_a @ Xs2 )
= Xs2 ) ) ).
% rotate1_length01
thf(fact_1212_rotate1_Osimps_I2_J,axiom,
! [X2: a,Xs2: list_a] :
( ( rotate1_a @ ( cons_a @ X2 @ Xs2 ) )
= ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) ) ).
% rotate1.simps(2)
thf(fact_1213_size__list__append,axiom,
! [F: a > nat,Xs2: list_a,Ys: list_a] :
( ( size_list_a @ F @ ( append_a @ Xs2 @ Ys ) )
= ( plus_plus_nat @ ( size_list_a @ F @ Xs2 ) @ ( size_list_a @ F @ Ys ) ) ) ).
% size_list_append
thf(fact_1214_size__list__estimation_H,axiom,
! [X2: a,Xs2: list_a,Y: nat,F: a > nat] :
( ( member_a @ X2 @ ( set_a2 @ Xs2 ) )
=> ( ( ord_less_eq_nat @ Y @ ( F @ X2 ) )
=> ( ord_less_eq_nat @ Y @ ( size_list_a @ F @ Xs2 ) ) ) ) ).
% size_list_estimation'
thf(fact_1215_size__list__pointwise,axiom,
! [Xs2: list_a,F: a > nat,G: a > nat] :
( ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs2 ) )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq_nat @ ( size_list_a @ F @ Xs2 ) @ ( size_list_a @ G @ Xs2 ) ) ) ).
% size_list_pointwise
thf(fact_1216_size__list__estimation,axiom,
! [X2: a,Xs2: list_a,Y: nat,F: a > nat] :
( ( member_a @ X2 @ ( set_a2 @ Xs2 ) )
=> ( ( ord_less_nat @ Y @ ( F @ X2 ) )
=> ( ord_less_nat @ Y @ ( size_list_a @ F @ Xs2 ) ) ) ) ).
% size_list_estimation
thf(fact_1217_butlast__take,axiom,
! [N4: nat,Xs2: list_a] :
( ( ord_less_eq_nat @ N4 @ ( size_size_list_a @ Xs2 ) )
=> ( ( butlast_a @ ( take_a @ N4 @ Xs2 ) )
= ( take_a @ ( minus_minus_nat @ N4 @ one_one_nat ) @ Xs2 ) ) ) ).
% butlast_take
thf(fact_1218_butlast__snoc,axiom,
! [Xs2: list_a,X2: a] :
( ( butlast_a @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) )
= Xs2 ) ).
% butlast_snoc
thf(fact_1219_length__butlast,axiom,
! [Xs2: list_a] :
( ( size_size_list_a @ ( butlast_a @ Xs2 ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs2 ) @ one_one_nat ) ) ).
% length_butlast
thf(fact_1220_in__set__butlast__appendI,axiom,
! [X2: a,Xs2: list_a,Ys: list_a] :
( ( ( member_a @ X2 @ ( set_a2 @ ( butlast_a @ Xs2 ) ) )
| ( member_a @ X2 @ ( set_a2 @ ( butlast_a @ Ys ) ) ) )
=> ( member_a @ X2 @ ( set_a2 @ ( butlast_a @ ( append_a @ Xs2 @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_1221_in__set__butlastD,axiom,
! [X2: a,Xs2: list_a] :
( ( member_a @ X2 @ ( set_a2 @ ( butlast_a @ Xs2 ) ) )
=> ( member_a @ X2 @ ( set_a2 @ Xs2 ) ) ) ).
% in_set_butlastD
thf(fact_1222_butlast_Osimps_I2_J,axiom,
! [Xs2: list_a,X2: a] :
( ( ( Xs2 = nil_a )
=> ( ( butlast_a @ ( cons_a @ X2 @ Xs2 ) )
= nil_a ) )
& ( ( Xs2 != nil_a )
=> ( ( butlast_a @ ( cons_a @ X2 @ Xs2 ) )
= ( cons_a @ X2 @ ( butlast_a @ Xs2 ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_1223_drop__butlast,axiom,
! [N4: nat,Xs2: list_a] :
( ( drop_a @ N4 @ ( butlast_a @ Xs2 ) )
= ( butlast_a @ ( drop_a @ N4 @ Xs2 ) ) ) ).
% drop_butlast
thf(fact_1224_butlast__append,axiom,
! [Ys: list_a,Xs2: list_a] :
( ( ( Ys = nil_a )
=> ( ( butlast_a @ ( append_a @ Xs2 @ Ys ) )
= ( butlast_a @ Xs2 ) ) )
& ( ( Ys != nil_a )
=> ( ( butlast_a @ ( append_a @ Xs2 @ Ys ) )
= ( append_a @ Xs2 @ ( butlast_a @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_1225_nth__butlast,axiom,
! [N4: nat,Xs2: list_a] :
( ( ord_less_nat @ N4 @ ( size_size_list_a @ ( butlast_a @ Xs2 ) ) )
=> ( ( nth_a @ ( butlast_a @ Xs2 ) @ N4 )
= ( nth_a @ Xs2 @ N4 ) ) ) ).
% nth_butlast
thf(fact_1226_take__butlast,axiom,
! [N4: nat,Xs2: list_a] :
( ( ord_less_nat @ N4 @ ( size_size_list_a @ Xs2 ) )
=> ( ( take_a @ N4 @ ( butlast_a @ Xs2 ) )
= ( take_a @ N4 @ Xs2 ) ) ) ).
% take_butlast
thf(fact_1227_butlast__conv__take,axiom,
( butlast_a
= ( ^ [Xs4: list_a] : ( take_a @ ( minus_minus_nat @ ( size_size_list_a @ Xs4 ) @ one_one_nat ) @ Xs4 ) ) ) ).
% butlast_conv_take
thf(fact_1228_butlast__list__update,axiom,
! [K3: nat,Xs2: list_a,X2: a] :
( ( ( K3
= ( minus_minus_nat @ ( size_size_list_a @ Xs2 ) @ one_one_nat ) )
=> ( ( butlast_a @ ( list_update_a @ Xs2 @ K3 @ X2 ) )
= ( butlast_a @ Xs2 ) ) )
& ( ( K3
!= ( minus_minus_nat @ ( size_size_list_a @ Xs2 ) @ one_one_nat ) )
=> ( ( butlast_a @ ( list_update_a @ Xs2 @ K3 @ X2 ) )
= ( list_update_a @ ( butlast_a @ Xs2 ) @ K3 @ X2 ) ) ) ) ).
% butlast_list_update
thf(fact_1229_append__one__prefix,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( prefix_a @ Xs2 @ Ys )
=> ( ( ord_less_nat @ ( size_size_list_a @ Xs2 ) @ ( size_size_list_a @ Ys ) )
=> ( prefix_a @ ( append_a @ Xs2 @ ( cons_a @ ( nth_a @ Ys @ ( size_size_list_a @ Xs2 ) ) @ nil_a ) ) @ Ys ) ) ) ).
% append_one_prefix
thf(fact_1230_Cons__prefix__Cons,axiom,
! [X2: a,Xs2: list_a,Y: a,Ys: list_a] :
( ( prefix_a @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y @ Ys ) )
= ( ( X2 = Y )
& ( prefix_a @ Xs2 @ Ys ) ) ) ).
% Cons_prefix_Cons
thf(fact_1231_same__prefix__prefix,axiom,
! [Xs2: list_a,Ys: list_a,Zs: list_a] :
( ( prefix_a @ ( append_a @ Xs2 @ Ys ) @ ( append_a @ Xs2 @ Zs ) )
= ( prefix_a @ Ys @ Zs ) ) ).
% same_prefix_prefix
thf(fact_1232_same__prefix__nil,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( prefix_a @ ( append_a @ Xs2 @ Ys ) @ Xs2 )
= ( Ys = nil_a ) ) ).
% same_prefix_nil
thf(fact_1233_prefix__snoc,axiom,
! [Xs2: list_a,Ys: list_a,Y: a] :
( ( prefix_a @ Xs2 @ ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
= ( ( Xs2
= ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
| ( prefix_a @ Xs2 @ Ys ) ) ) ).
% prefix_snoc
thf(fact_1234_prefix__code_I2_J,axiom,
! [X2: a,Xs2: list_a] :
~ ( prefix_a @ ( cons_a @ X2 @ Xs2 ) @ nil_a ) ).
% prefix_code(2)
thf(fact_1235_prefix__Cons,axiom,
! [Xs2: list_a,Y: a,Ys: list_a] :
( ( prefix_a @ Xs2 @ ( cons_a @ Y @ Ys ) )
= ( ( Xs2 = nil_a )
| ? [Zs3: list_a] :
( ( Xs2
= ( cons_a @ Y @ Zs3 ) )
& ( prefix_a @ Zs3 @ Ys ) ) ) ) ).
% prefix_Cons
thf(fact_1236_not__prefix__cases,axiom,
! [Ps2: list_a,Ls2: list_a] :
( ~ ( prefix_a @ Ps2 @ Ls2 )
=> ( ( ( Ps2 != nil_a )
=> ( Ls2 != nil_a ) )
=> ( ! [A3: a,As: list_a] :
( ( Ps2
= ( cons_a @ A3 @ As ) )
=> ! [X4: a,Xs: list_a] :
( ( Ls2
= ( cons_a @ X4 @ Xs ) )
=> ( ( X4 = A3 )
=> ( prefix_a @ As @ Xs ) ) ) )
=> ~ ! [A3: a] :
( ? [As: list_a] :
( Ps2
= ( cons_a @ A3 @ As ) )
=> ! [X4: a] :
( ? [Xs: list_a] :
( Ls2
= ( cons_a @ X4 @ Xs ) )
=> ( X4 = A3 ) ) ) ) ) ) ).
% not_prefix_cases
thf(fact_1237_not__prefix__induct,axiom,
! [Ps2: list_a,Ls2: list_a,P: list_a > list_a > $o] :
( ~ ( prefix_a @ Ps2 @ Ls2 )
=> ( ! [X4: a,Xs: list_a] : ( P @ ( cons_a @ X4 @ Xs ) @ nil_a )
=> ( ! [X4: a,Xs: list_a,Y4: a,Ys2: list_a] :
( ( X4 != Y4 )
=> ( P @ ( cons_a @ X4 @ Xs ) @ ( cons_a @ Y4 @ Ys2 ) ) )
=> ( ! [X4: a,Xs: list_a,Y4: a,Ys2: list_a] :
( ( X4 = Y4 )
=> ( ~ ( prefix_a @ Xs @ Ys2 )
=> ( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_a @ X4 @ Xs ) @ ( cons_a @ Y4 @ Ys2 ) ) ) ) )
=> ( P @ Ps2 @ Ls2 ) ) ) ) ) ).
% not_prefix_induct
thf(fact_1238_take__is__prefix,axiom,
! [N4: nat,Xs2: list_a] : ( prefix_a @ ( take_a @ N4 @ Xs2 ) @ Xs2 ) ).
% take_is_prefix
thf(fact_1239_prefixE,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( prefix_a @ Xs2 @ Ys )
=> ~ ! [Zs2: list_a] :
( Ys
!= ( append_a @ Xs2 @ Zs2 ) ) ) ).
% prefixE
thf(fact_1240_prefixI,axiom,
! [Ys: list_a,Xs2: list_a,Zs: list_a] :
( ( Ys
= ( append_a @ Xs2 @ Zs ) )
=> ( prefix_a @ Xs2 @ Ys ) ) ).
% prefixI
thf(fact_1241_prefix__def,axiom,
( prefix_a
= ( ^ [Xs4: list_a,Ys3: list_a] :
? [Zs3: list_a] :
( Ys3
= ( append_a @ Xs4 @ Zs3 ) ) ) ) ).
% prefix_def
thf(fact_1242_prefix__append,axiom,
! [Xs2: list_a,Ys: list_a,Zs: list_a] :
( ( prefix_a @ Xs2 @ ( append_a @ Ys @ Zs ) )
= ( ( prefix_a @ Xs2 @ Ys )
| ? [Us: list_a] :
( ( Xs2
= ( append_a @ Ys @ Us ) )
& ( prefix_a @ Us @ Zs ) ) ) ) ).
% prefix_append
thf(fact_1243_prefix__prefix,axiom,
! [Xs2: list_a,Ys: list_a,Zs: list_a] :
( ( prefix_a @ Xs2 @ Ys )
=> ( prefix_a @ Xs2 @ ( append_a @ Ys @ Zs ) ) ) ).
% prefix_prefix
thf(fact_1244_append__prefixD,axiom,
! [Xs2: list_a,Ys: list_a,Zs: list_a] :
( ( prefix_a @ ( append_a @ Xs2 @ Ys ) @ Zs )
=> ( prefix_a @ Xs2 @ Zs ) ) ).
% append_prefixD
thf(fact_1245_set__mono__prefix,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( prefix_a @ Xs2 @ Ys )
=> ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ Ys ) ) ) ).
% set_mono_prefix
thf(fact_1246_append__butlast__last__id,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( append_a @ ( butlast_a @ Xs2 ) @ ( cons_a @ ( last_a @ Xs2 ) @ nil_a ) )
= Xs2 ) ) ).
% append_butlast_last_id
thf(fact_1247_last__list__update,axiom,
! [Xs2: list_a,K3: nat,X2: a] :
( ( Xs2 != nil_a )
=> ( ( ( K3
= ( minus_minus_nat @ ( size_size_list_a @ Xs2 ) @ one_one_nat ) )
=> ( ( last_a @ ( list_update_a @ Xs2 @ K3 @ X2 ) )
= X2 ) )
& ( ( K3
!= ( minus_minus_nat @ ( size_size_list_a @ Xs2 ) @ one_one_nat ) )
=> ( ( last_a @ ( list_update_a @ Xs2 @ K3 @ X2 ) )
= ( last_a @ Xs2 ) ) ) ) ) ).
% last_list_update
thf(fact_1248_last__appendL,axiom,
! [Ys: list_a,Xs2: list_a] :
( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs2 @ Ys ) )
= ( last_a @ Xs2 ) ) ) ).
% last_appendL
thf(fact_1249_last__appendR,axiom,
! [Ys: list_a,Xs2: list_a] :
( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs2 @ Ys ) )
= ( last_a @ Ys ) ) ) ).
% last_appendR
thf(fact_1250_last__snoc,axiom,
! [Xs2: list_a,X2: a] :
( ( last_a @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) )
= X2 ) ).
% last_snoc
thf(fact_1251_last__drop,axiom,
! [N4: nat,Xs2: list_a] :
( ( ord_less_nat @ N4 @ ( size_size_list_a @ Xs2 ) )
=> ( ( last_a @ ( drop_a @ N4 @ Xs2 ) )
= ( last_a @ Xs2 ) ) ) ).
% last_drop
thf(fact_1252_last__in__set,axiom,
! [As2: list_a] :
( ( As2 != nil_a )
=> ( member_a @ ( last_a @ As2 ) @ ( set_a2 @ As2 ) ) ) ).
% last_in_set
thf(fact_1253_longest__common__suffix,axiom,
! [Xs2: list_a,Ys: list_a] :
? [Ss: list_a,Xs5: list_a,Ys7: list_a] :
( ( Xs2
= ( append_a @ Xs5 @ Ss ) )
& ( Ys
= ( append_a @ Ys7 @ Ss ) )
& ( ( Xs5 = nil_a )
| ( Ys7 = nil_a )
| ( ( last_a @ Xs5 )
!= ( last_a @ Ys7 ) ) ) ) ).
% longest_common_suffix
thf(fact_1254_last__append,axiom,
! [Ys: list_a,Xs2: list_a] :
( ( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs2 @ Ys ) )
= ( last_a @ Xs2 ) ) )
& ( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs2 @ Ys ) )
= ( last_a @ Ys ) ) ) ) ).
% last_append
thf(fact_1255_last__ConsR,axiom,
! [Xs2: list_a,X2: a] :
( ( Xs2 != nil_a )
=> ( ( last_a @ ( cons_a @ X2 @ Xs2 ) )
= ( last_a @ Xs2 ) ) ) ).
% last_ConsR
thf(fact_1256_last__ConsL,axiom,
! [Xs2: list_a,X2: a] :
( ( Xs2 = nil_a )
=> ( ( last_a @ ( cons_a @ X2 @ Xs2 ) )
= X2 ) ) ).
% last_ConsL
thf(fact_1257_last_Osimps,axiom,
! [Xs2: list_a,X2: a] :
( ( ( Xs2 = nil_a )
=> ( ( last_a @ ( cons_a @ X2 @ Xs2 ) )
= X2 ) )
& ( ( Xs2 != nil_a )
=> ( ( last_a @ ( cons_a @ X2 @ Xs2 ) )
= ( last_a @ Xs2 ) ) ) ) ).
% last.simps
thf(fact_1258_snoc__eq__iff__butlast,axiom,
! [Xs2: list_a,X2: a,Ys: list_a] :
( ( ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) )
= Ys )
= ( ( Ys != nil_a )
& ( ( butlast_a @ Ys )
= Xs2 )
& ( ( last_a @ Ys )
= X2 ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_1259_last__conv__nth,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( last_a @ Xs2 )
= ( nth_a @ Xs2 @ ( minus_minus_nat @ ( size_size_list_a @ Xs2 ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_1260_concat__eq__append__conv,axiom,
! [Xss2: list_list_a,Ys: list_a,Zs: list_a] :
( ( ( concat_a @ Xss2 )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Xss2 = nil_list_a )
=> ( ( Ys = nil_a )
& ( Zs = nil_a ) ) )
& ( ( Xss2 != nil_list_a )
=> ? [Xss1: list_list_a,Xs4: list_a,Xs6: list_a,Xss22: list_list_a] :
( ( Xss2
= ( append_list_a @ Xss1 @ ( cons_list_a @ ( append_a @ Xs4 @ Xs6 ) @ Xss22 ) ) )
& ( Ys
= ( append_a @ ( concat_a @ Xss1 ) @ Xs4 ) )
& ( Zs
= ( append_a @ Xs6 @ ( concat_a @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_1261_distinct__adj__append__iff,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( distinct_adj_a @ ( append_a @ Xs2 @ Ys ) )
= ( ( distinct_adj_a @ Xs2 )
& ( distinct_adj_a @ Ys )
& ( ( Xs2 = nil_a )
| ( Ys = nil_a )
| ( ( last_a @ Xs2 )
!= ( hd_a @ Ys ) ) ) ) ) ).
% distinct_adj_append_iff
thf(fact_1262_distinct__adj__Cons__Cons,axiom,
! [X2: a,Y: a,Xs2: list_a] :
( ( distinct_adj_a @ ( cons_a @ X2 @ ( cons_a @ Y @ Xs2 ) ) )
= ( ( X2 != Y )
& ( distinct_adj_a @ ( cons_a @ Y @ Xs2 ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_1263_concat__append,axiom,
! [Xs2: list_list_a,Ys: list_list_a] :
( ( concat_a @ ( append_list_a @ Xs2 @ Ys ) )
= ( append_a @ ( concat_a @ Xs2 ) @ ( concat_a @ Ys ) ) ) ).
% concat_append
thf(fact_1264_distinct__adj__singleton,axiom,
! [X2: a] : ( distinct_adj_a @ ( cons_a @ X2 @ nil_a ) ) ).
% distinct_adj_singleton
thf(fact_1265_concat_Osimps_I2_J,axiom,
! [X2: list_a,Xs2: list_list_a] :
( ( concat_a @ ( cons_list_a @ X2 @ Xs2 ) )
= ( append_a @ X2 @ ( concat_a @ Xs2 ) ) ) ).
% concat.simps(2)
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
member_a @ y @ ( set_mset_a @ ( mset_a @ zs ) ) ).
%------------------------------------------------------------------------------