TPTP Problem File: SLH0401^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Prefix_Free_Code_Combinators/0000_Prefix_Free_Code_Combinators/prob_00520_017594__11960162_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1483 ( 747 unt; 205 typ; 0 def)
% Number of atoms : 3184 (2113 equ; 0 cnn)
% Maximal formula atoms : 15 ( 2 avg)
% Number of connectives : 11848 ( 407 ~; 81 |; 353 &;9836 @)
% ( 0 <=>;1171 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 6 avg)
% Number of types : 17 ( 16 usr)
% Number of type conns : 838 ( 838 >; 0 *; 0 +; 0 <<)
% Number of symbols : 192 ( 189 usr; 14 con; 0-4 aty)
% Number of variables : 3768 ( 274 ^;3234 !; 260 ?;3768 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:58:50.545
%------------------------------------------------------------------------------
% Could-be-implicit typings (16)
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
list_list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_list_list_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__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
option_list_o: $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__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__List__Olist_I_Eo_J,type,
list_o: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (189)
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Nat__Onat,type,
bNF_Gr1872714664788909425ft_nat: set_list_nat > nat > set_list_nat ).
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_001t__List__Olist_It__Nat__Onat_J,type,
bNF_Gr3053708287304744325st_nat: set_list_list_nat > list_list_nat > set_list_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Nat__Onat,type,
bNF_Gr6352880689984616693cc_nat: set_list_nat > list_nat > set_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001tf__a,type,
bNF_Greatest_Succ_a: set_list_a > list_a > set_a ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
finite_card_list_nat: set_list_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
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__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
groups4561878855575611511st_nat: list_nat > nat ).
thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
if_list_nat: $o > list_nat > list_nat > list_nat ).
thf(sy_c_If_001t__List__Olist_Itf__a_J,type,
if_list_a: $o > list_a > list_a > list_a ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Nat__Onat,type,
semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_List_Oappend_001t__List__Olist_It__Nat__Onat_J,type,
append_list_nat: list_list_nat > list_list_nat > list_list_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__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_Obind_001t__Nat__Onat_001t__Nat__Onat,type,
bind_nat_nat: list_nat > ( nat > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001t__Nat__Onat_001tf__a,type,
bind_nat_a: list_nat > ( nat > list_a ) > list_a ).
thf(sy_c_List_Obind_001tf__a_001t__Nat__Onat,type,
bind_a_nat: list_a > ( a > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001tf__a_001tf__a,type,
bind_a_a: list_a > ( a > list_a ) > list_a ).
thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
butlast_nat: list_nat > list_nat ).
thf(sy_c_List_Obutlast_001tf__a,type,
butlast_a: list_a > list_a ).
thf(sy_c_List_Oconcat_001t__List__Olist_It__Nat__Onat_J,type,
concat_list_nat: list_list_list_nat > list_list_nat ).
thf(sy_c_List_Oconcat_001t__List__Olist_Itf__a_J,type,
concat_list_a: list_list_list_a > list_list_a ).
thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
concat_nat: list_list_nat > list_nat ).
thf(sy_c_List_Oconcat_001tf__a,type,
concat_a: list_list_a > list_a ).
thf(sy_c_List_Odistinct__adj_001t__Nat__Onat,type,
distinct_adj_nat: list_nat > $o ).
thf(sy_c_List_Odistinct__adj_001tf__a,type,
distinct_adj_a: list_a > $o ).
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_Ofilter_001t__List__Olist_It__Nat__Onat_J,type,
filter_list_nat: ( list_nat > $o ) > list_list_nat > list_list_nat ).
thf(sy_c_List_Ofilter_001t__List__Olist_Itf__a_J,type,
filter_list_a: ( list_a > $o ) > list_list_a > list_list_a ).
thf(sy_c_List_Ofilter_001t__Nat__Onat,type,
filter_nat: ( nat > $o ) > list_nat > list_nat ).
thf(sy_c_List_Ofilter_001tf__a,type,
filter_a: ( a > $o ) > list_a > list_a ).
thf(sy_c_List_Ofoldr_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
foldr_6871341030409798377st_nat: ( list_nat > list_nat > list_nat ) > list_list_nat > list_nat > list_nat ).
thf(sy_c_List_Ofoldr_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
foldr_list_nat_nat: ( list_nat > nat > nat ) > list_list_nat > nat > nat ).
thf(sy_c_List_Ofoldr_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
foldr_list_a_list_a: ( list_a > list_a > list_a ) > list_list_a > list_a > list_a ).
thf(sy_c_List_Ofoldr_001t__List__Olist_Itf__a_J_001t__Nat__Onat,type,
foldr_list_a_nat: ( list_a > nat > nat ) > list_list_a > nat > nat ).
thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: nat > list_nat > nat ).
thf(sy_c_List_Ogen__length_001tf__a,type,
gen_length_a: nat > list_a > nat ).
thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oinsert_001tf__a,type,
insert_a: a > list_a > list_a ).
thf(sy_c_List_Olast_001t__Nat__Onat,type,
last_nat: list_nat > nat ).
thf(sy_c_List_Olast_001tf__a,type,
last_a: list_a > a ).
thf(sy_c_List_Olist_OCons_001_Eo,type,
cons_o: $o > list_o > list_o ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
cons_list_nat: list_nat > list_list_nat > list_list_nat ).
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__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__Nat__Onat_J,type,
nil_list_nat: list_list_nat ).
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_Ocase__list_001_Eo_001t__Nat__Onat,type,
case_list_o_nat: $o > ( nat > list_nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist_Ocase__list_001_Eo_001tf__a,type,
case_list_o_a: $o > ( a > list_a > $o ) > list_a > $o ).
thf(sy_c_List_Olist_Ocase__list_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__Nat__Onat,type,
case_l3331202209248957608at_nat: list_list_nat > ( nat > list_nat > list_list_nat ) > list_nat > list_list_nat ).
thf(sy_c_List_Olist_Ocase__list_001t__List__Olist_It__List__Olist_Itf__a_J_J_001tf__a,type,
case_l8408404631611421914st_a_a: list_list_a > ( a > list_a > list_list_a ) > list_a > list_list_a ).
thf(sy_c_List_Olist_Ocase__list_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
case_l2340614614379431832at_nat: list_nat > ( nat > list_nat > list_nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Ocase__list_001t__List__Olist_Itf__a_J_001tf__a,type,
case_list_list_a_a: list_a > ( a > list_a > list_a ) > list_a > list_a ).
thf(sy_c_List_Olist_Ohd_001t__List__Olist_It__Nat__Onat_J,type,
hd_list_nat: list_list_nat > list_nat ).
thf(sy_c_List_Olist_Ohd_001t__List__Olist_Itf__a_J,type,
hd_list_a: list_list_a > list_a ).
thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
hd_nat: list_nat > nat ).
thf(sy_c_List_Olist_Ohd_001tf__a,type,
hd_a: list_a > a ).
thf(sy_c_List_Olist_Olist__all_001t__Nat__Onat,type,
list_all_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist_Olist__all_001tf__a,type,
list_all_a: ( a > $o ) > list_a > $o ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
map_li960784813134754710st_nat: ( list_nat > list_list_nat ) > list_list_nat > list_list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
map_li7225945977422193158st_nat: ( list_nat > list_nat ) > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
map_list_nat_nat: ( list_nat > nat ) > list_list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_Itf__a_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
map_li5729356230488778442list_a: ( list_a > list_list_a ) > list_list_a > list_list_list_a ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
map_list_a_list_a: ( list_a > list_a ) > list_list_a > list_list_a ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_Itf__a_J_001tf__a,type,
map_list_a_a: ( list_a > a ) > list_list_a > list_a ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
map_na6205611841492582150st_nat: ( nat > list_list_nat ) > list_nat > list_list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
map_nat_list_nat: ( nat > list_nat ) > list_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_Itf__a_J,type,
map_nat_list_a: ( nat > list_a ) > list_nat > list_list_a ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001tf__a,type,
map_nat_a: ( nat > a ) > list_nat > list_a ).
thf(sy_c_List_Olist_Omap_001tf__a_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
map_a_list_list_a: ( a > list_list_a ) > list_a > list_list_list_a ).
thf(sy_c_List_Olist_Omap_001tf__a_001t__List__Olist_Itf__a_J,type,
map_a_list_a: ( a > list_a ) > list_a > list_list_a ).
thf(sy_c_List_Olist_Omap_001tf__a_001t__Nat__Onat,type,
map_a_nat: ( a > nat ) > list_a > list_nat ).
thf(sy_c_List_Olist_Omap_001tf__a_001tf__a,type,
map_a_a: ( a > a ) > list_a > list_a ).
thf(sy_c_List_Olist_Orec__list_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
rec_li7516600145284979816at_nat: list_nat > ( nat > list_nat > list_nat > list_nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_Itf__a_J,type,
set_list_a2: list_list_a > set_list_a ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Olist_Osize__list_001t__Nat__Onat,type,
size_list_nat: ( nat > nat ) > list_nat > nat ).
thf(sy_c_List_Olist_Osize__list_001tf__a,type,
size_list_a: ( a > nat ) > list_a > nat ).
thf(sy_c_List_Olist_Otl_001t__List__Olist_It__Nat__Onat_J,type,
tl_list_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_Otl_001t__List__Olist_Itf__a_J,type,
tl_list_a: list_list_a > list_list_a ).
thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
tl_nat: list_nat > list_nat ).
thf(sy_c_List_Olist_Otl_001tf__a,type,
tl_a: list_a > list_a ).
thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
list_ex1_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex1_001tf__a,type,
list_ex1_a: ( a > $o ) > list_a > $o ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(sy_c_List_Olist__update_001tf__a,type,
list_update_a: list_a > nat > a > list_a ).
thf(sy_c_List_Omap__tailrec_001t__Nat__Onat_001t__Nat__Onat,type,
map_tailrec_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Omaps_001t__Nat__Onat_001t__Nat__Onat,type,
maps_nat_nat: ( nat > list_nat ) > list_nat > list_nat ).
thf(sy_c_List_Omaps_001t__Nat__Onat_001tf__a,type,
maps_nat_a: ( nat > list_a ) > list_nat > list_a ).
thf(sy_c_List_Omaps_001tf__a_001t__Nat__Onat,type,
maps_a_nat: ( a > list_nat ) > list_a > list_nat ).
thf(sy_c_List_Omaps_001tf__a_001tf__a,type,
maps_a_a: ( a > list_a ) > list_a > list_a ).
thf(sy_c_List_On__lists_001t__Nat__Onat,type,
n_lists_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_On__lists_001tf__a,type,
n_lists_a: nat > list_a > list_list_a ).
thf(sy_c_List_Onth_001_Eo,type,
nth_o: list_o > nat > $o ).
thf(sy_c_List_Onth_001t__List__Olist_It__Nat__Onat_J,type,
nth_list_nat: list_list_nat > nat > list_nat ).
thf(sy_c_List_Onth_001t__List__Olist_Itf__a_J,type,
nth_list_a: list_list_a > nat > list_a ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_List_Onths_001t__Nat__Onat,type,
nths_nat: list_nat > set_nat > list_nat ).
thf(sy_c_List_Onths_001tf__a,type,
nths_a: list_a > set_nat > list_a ).
thf(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
product_lists_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Oproduct__lists_001tf__a,type,
product_lists_a: list_list_a > list_list_a ).
thf(sy_c_List_Oreplicate_001t__List__Olist_It__Nat__Onat_J,type,
replicate_list_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_Oreplicate_001t__List__Olist_Itf__a_J,type,
replicate_list_a: nat > list_a > list_list_a ).
thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
replicate_nat: nat > nat > list_nat ).
thf(sy_c_List_Oreplicate_001tf__a,type,
replicate_a: nat > a > list_a ).
thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
rotate1_nat: list_nat > list_nat ).
thf(sy_c_List_Orotate1_001tf__a,type,
rotate1_a: list_a > list_a ).
thf(sy_c_List_Oshuffles_001t__Nat__Onat,type,
shuffles_nat: list_nat > list_nat > set_list_nat ).
thf(sy_c_List_Oshuffles_001tf__a,type,
shuffles_a: list_a > list_a > set_list_a ).
thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
subseqs_nat: list_nat > list_list_nat ).
thf(sy_c_List_Osubseqs_001tf__a,type,
subseqs_a: list_a > list_list_a ).
thf(sy_c_List_Osuccessively_001t__Nat__Onat,type,
successively_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(sy_c_List_Osuccessively_001tf__a,type,
successively_a: ( a > a > $o ) > list_a > $o ).
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_List_Otranspose_001t__Nat__Onat,type,
transpose_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Otranspose_001tf__a,type,
transpose_a: list_list_a > list_list_a ).
thf(sy_c_List_Otranspose__rel_001t__Nat__Onat,type,
transpose_rel_nat: list_list_nat > list_list_nat > $o ).
thf(sy_c_List_Otranspose__rel_001tf__a,type,
transpose_rel_a: list_list_a > list_list_a > $o ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > list_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
case_nat_o: $o > ( nat > $o ) > nat > $o ).
thf(sy_c_Nat_Onat_Ocase__nat_001t__List__Olist_It__Nat__Onat_J,type,
case_nat_list_nat: list_nat > ( nat > list_nat ) > nat > list_nat ).
thf(sy_c_Nat_Onat_Ocase__nat_001t__List__Olist_Itf__a_J,type,
case_nat_list_a: list_a > ( nat > list_a ) > nat > list_a ).
thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
case_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Nat_Onat_Ocase__nat_001tf__a,type,
case_nat_a: a > ( nat > a ) > nat > a ).
thf(sy_c_Nat_Onat_Opred,type,
pred: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
size_s3023201423986296836st_nat: list_list_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__Nat__Onat_J,type,
size_size_list_nat: list_nat > 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__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__List__Olist_It__Nat__Onat_J,type,
ord_less_list_nat: list_nat > list_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_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__List__Olist_It__Nat__Onat_J,type,
ord_less_eq_list_nat: list_nat > list_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_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
ord_max_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
thf(sy_c_Prefix__Free__Code__Combinators_OLf_092_060_094sub_062e_001tf__a,type,
prefix4097710381326367690Lf_e_a: ( a > option_list_o ) > nat > list_a > option_list_o ).
thf(sy_c_Prefix__Free__Code__Combinators_Ois__encoding_001t__List__Olist_Itf__a_J,type,
prefix5220018966750911590list_a: ( list_a > option_list_o ) > $o ).
thf(sy_c_Prefix__Free__Code__Combinators_Ois__encoding_001tf__a,type,
prefix7485107378405021920ding_a: ( a > option_list_o ) > $o ).
thf(sy_c_Prefix__Free__Code__Combinators_Oopt__comp_001_Eo,type,
prefix454693708527911765comp_o: option_list_o > option_list_o > $o ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
modulo_modulo_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Sublist_Oprefix_001t__Nat__Onat,type,
prefix_nat: list_nat > list_nat > $o ).
thf(sy_c_Sublist_Oprefix_001tf__a,type,
prefix_a: list_a > list_a > $o ).
thf(sy_c_Sublist_Oprefixes_001t__Nat__Onat,type,
prefixes_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Oprefixes_001tf__a,type,
prefixes_a: list_a > list_list_a ).
thf(sy_c_Sublist_Osublists_001t__Nat__Onat,type,
sublists_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Osublists_001tf__a,type,
sublists_a: list_a > list_list_a ).
thf(sy_c_Sublist_Osuffixes_001t__Nat__Onat,type,
suffixes_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Osuffixes_001tf__a,type,
suffixes_a: list_a > list_list_a ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
accp_list_list_nat: ( list_list_nat > list_list_nat > $o ) > list_list_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
accp_list_list_a: ( list_list_a > list_list_a > $o ) > list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
member_list_list_nat: list_list_nat > set_list_list_nat > $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__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_e,type,
e: a > option_list_o ).
thf(sy_v_na____,type,
na: nat ).
thf(sy_v_thesis____,type,
thesis: $o ).
thf(sy_v_x1____,type,
x1: list_a ).
thf(sy_v_x2____,type,
x2: a ).
thf(sy_v_x____,type,
x: list_a ).
thf(sy_v_y____,type,
y: list_a ).
% Relevant facts (1269)
thf(fact_0_c,axiom,
( ( size_size_list_a @ y )
= ( suc @ na ) ) ).
% c
thf(fact_1_x__def_I2_J,axiom,
( ( size_size_list_a @ x1 )
= na ) ).
% x_def(2)
thf(fact_2__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x1_Ax2_O_A_092_060lbrakk_062x_A_061_Ax1_A_064_A_091x2_093_059_Alength_Ax1_A_061_An_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [X1: list_a] :
( ? [X2: a] :
( x
= ( append_a @ X1 @ ( cons_a @ X2 @ nil_a ) ) )
=> ( ( size_size_list_a @ X1 )
!= na ) ) ).
% \<open>\<And>thesis. (\<And>x1 x2. \<lbrakk>x = x1 @ [x2]; length x1 = n\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_3_x__def_I1_J,axiom,
( x
= ( append_a @ x1 @ ( cons_a @ x2 @ nil_a ) ) ) ).
% x_def(1)
thf(fact_4_append1__eq__conv,axiom,
! [Xs: list_a,X: a,Ys: list_a,Y: a] :
( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_5_append1__eq__conv,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat,Y: nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) )
= ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_6_b,axiom,
( ( size_size_list_a @ x )
= ( suc @ na ) ) ).
% b
thf(fact_7_append__eq__append__conv,axiom,
! [Xs: list_a,Ys: list_a,Us: list_a,Vs: list_a] :
( ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
| ( ( size_size_list_a @ Us )
= ( size_size_list_a @ Vs ) ) )
=> ( ( ( append_a @ Xs @ Us )
= ( append_a @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_8_append__eq__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
| ( ( size_size_list_nat @ Us )
= ( size_size_list_nat @ Vs ) ) )
=> ( ( ( append_nat @ Xs @ Us )
= ( append_nat @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_9_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_10_append_Oright__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ A @ nil_nat )
= A ) ).
% append.right_neutral
thf(fact_11_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_12_append__Nil2,axiom,
! [Xs: list_nat] :
( ( append_nat @ Xs @ nil_nat )
= Xs ) ).
% append_Nil2
thf(fact_13_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_14_append__self__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_nat ) ) ).
% append_self_conv
thf(fact_15_self__append__conv,axiom,
! [Y: list_a,Ys: list_a] :
( ( Y
= ( append_a @ Y @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_16_self__append__conv,axiom,
! [Y: list_nat,Ys: list_nat] :
( ( Y
= ( append_nat @ Y @ Ys ) )
= ( Ys = nil_nat ) ) ).
% self_append_conv
thf(fact_17_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_18_append__self__conv2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_nat ) ) ).
% append_self_conv2
thf(fact_19_self__append__conv2,axiom,
! [Y: list_a,Xs: list_a] :
( ( Y
= ( append_a @ Xs @ Y ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_20_self__append__conv2,axiom,
! [Y: list_nat,Xs: list_nat] :
( ( Y
= ( append_nat @ Xs @ Y ) )
= ( Xs = nil_nat ) ) ).
% self_append_conv2
thf(fact_21_Nil__is__append__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( nil_a
= ( append_a @ Xs @ Ys ) )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_22_Nil__is__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( nil_nat
= ( append_nat @ Xs @ Ys ) )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_23_append__is__Nil__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= nil_a )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_24_append__is__Nil__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= nil_nat )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_25_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_26_list_Oinject,axiom,
! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X22 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_27_same__append__eq,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_28_same__append__eq,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_29_append__same__eq,axiom,
! [Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( append_a @ Ys @ Xs )
= ( append_a @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_30_append__same__eq,axiom,
! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( append_nat @ Ys @ Xs )
= ( append_nat @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_31_append__assoc,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( append_a @ ( append_a @ Xs @ Ys ) @ Zs )
= ( append_a @ Xs @ ( append_a @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_32_append__assoc,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( append_nat @ ( append_nat @ Xs @ Ys ) @ Zs )
= ( append_nat @ Xs @ ( append_nat @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_33_append_Oassoc,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( append_a @ ( append_a @ A @ B ) @ C )
= ( append_a @ A @ ( append_a @ B @ C ) ) ) ).
% append.assoc
thf(fact_34_append_Oassoc,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( append_nat @ ( append_nat @ A @ B ) @ C )
= ( append_nat @ A @ ( append_nat @ B @ C ) ) ) ).
% append.assoc
thf(fact_35_not__Cons__self2,axiom,
! [X: a,Xs: list_a] :
( ( cons_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_36_not__Cons__self2,axiom,
! [X: nat,Xs: list_nat] :
( ( cons_nat @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_37_neq__if__length__neq,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
!= ( size_size_list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_38_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_39_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_40_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_41_append__eq__append__conv2,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ts: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Ts ) )
= ( ? [Us2: list_a] :
( ( ( Xs
= ( append_a @ Zs @ Us2 ) )
& ( ( append_a @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_a @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_a @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_42_append__eq__append__conv2,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ts: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Zs @ Ts ) )
= ( ? [Us2: list_nat] :
( ( ( Xs
= ( append_nat @ Zs @ Us2 ) )
& ( ( append_nat @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_nat @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_nat @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_43_append__eq__appendI,axiom,
! [Xs: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us: list_a] :
( ( ( append_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_a @ Xs1 @ Us ) )
=> ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_44_append__eq__appendI,axiom,
! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us: list_nat] :
( ( ( append_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_nat @ Xs1 @ Us ) )
=> ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_45_length__Suc__conv,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y2: a,Ys2: list_a] :
( ( Xs
= ( cons_a @ Y2 @ Ys2 ) )
& ( ( size_size_list_a @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_46_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y2: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ Y2 @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_47_Suc__length__conv,axiom,
! [N: nat,Xs: list_a] :
( ( ( suc @ N )
= ( size_size_list_a @ Xs ) )
= ( ? [Y2: a,Ys2: list_a] :
( ( Xs
= ( cons_a @ Y2 @ Ys2 ) )
& ( ( size_size_list_a @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_48_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y2: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ Y2 @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_49_list__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X3: a] : ( P @ ( cons_a @ X3 @ nil_a ) )
=> ( ! [X3: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_50_list__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] : ( P @ ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X3: nat,Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_51_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a] : ( P @ ( cons_a @ X3 @ Xs2 ) @ nil_a )
=> ( ! [Y3: a,Ys3: list_a] : ( P @ nil_a @ ( cons_a @ Y3 @ Ys3 ) )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys3: list_a] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_52_list__induct2_H,axiom,
! [P: list_a > list_nat > $o,Xs: list_a,Ys: list_nat] :
( ( P @ nil_a @ nil_nat )
=> ( ! [X3: a,Xs2: list_a] : ( P @ ( cons_a @ X3 @ Xs2 ) @ nil_nat )
=> ( ! [Y3: nat,Ys3: list_nat] : ( P @ nil_a @ ( cons_nat @ Y3 @ Ys3 ) )
=> ( ! [X3: a,Xs2: list_a,Y3: nat,Ys3: list_nat] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_53_list__induct2_H,axiom,
! [P: list_nat > list_a > $o,Xs: list_nat,Ys: list_a] :
( ( P @ nil_nat @ nil_a )
=> ( ! [X3: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X3 @ Xs2 ) @ nil_a )
=> ( ! [Y3: a,Ys3: list_a] : ( P @ nil_nat @ ( cons_a @ Y3 @ Ys3 ) )
=> ( ! [X3: nat,Xs2: list_nat,Y3: a,Ys3: list_a] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_54_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X3 @ Xs2 ) @ nil_nat )
=> ( ! [Y3: nat,Ys3: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y3 @ Ys3 ) )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys3: list_nat] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_55_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y2: a,Ys2: list_a] :
( Xs
= ( cons_a @ Y2 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_56_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y2: nat,Ys2: list_nat] :
( Xs
= ( cons_nat @ Y2 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_57_remdups__adj_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X3: a] :
( X
!= ( cons_a @ X3 @ nil_a ) )
=> ~ ! [X3: a,Y3: a,Xs2: list_a] :
( X
!= ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_58_remdups__adj_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ( ! [X3: nat] :
( X
!= ( cons_nat @ X3 @ nil_nat ) )
=> ~ ! [X3: nat,Y3: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_59_transpose_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X3: a,Xs2: list_a,Xss: list_list_a] :
( X
!= ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_60_transpose_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X3: nat,Xs2: list_nat,Xss: list_list_nat] :
( X
!= ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_61_min__list_Ocases,axiom,
! [X: list_nat] :
( ! [X3: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X3 @ Xs2 ) )
=> ( X = nil_nat ) ) ).
% min_list.cases
thf(fact_62_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_63_list_Oexhaust,axiom,
! [Y: list_nat] :
( ( Y != nil_nat )
=> ~ ! [X212: nat,X222: list_nat] :
( Y
!= ( cons_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_64_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_65_mem__Collect__eq,axiom,
! [A: list_nat,P: list_nat > $o] :
( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_66_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_67_Collect__mem__eq,axiom,
! [A2: set_list_nat] :
( ( collect_list_nat
@ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_68_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_69_Collect__cong,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X3: list_nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_list_nat @ P )
= ( collect_list_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_70_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_71_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_72_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_73_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_74_Cons__eq__appendI,axiom,
! [X: a,Xs1: list_a,Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_75_Cons__eq__appendI,axiom,
! [X: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_nat @ Xs1 @ Zs ) )
=> ( ( cons_nat @ X @ Xs )
= ( append_nat @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_76_append__Cons,axiom,
! [X: a,Xs: list_a,Ys: list_a] :
( ( append_a @ ( cons_a @ X @ Xs ) @ Ys )
= ( cons_a @ X @ ( append_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_77_append__Cons,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat] :
( ( append_nat @ ( cons_nat @ X @ Xs ) @ Ys )
= ( cons_nat @ X @ ( append_nat @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_78_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_79_eq__Nil__appendI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append_nat @ nil_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_80_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_81_append_Oleft__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ nil_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_82_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_83_append__Nil,axiom,
! [Ys: list_nat] :
( ( append_nat @ nil_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_84_length__Suc__conv__rev,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y2: a,Ys2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ Y2 @ nil_a ) ) )
& ( ( size_size_list_a @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_85_length__Suc__conv__rev,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y2: nat,Ys2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ Y2 @ nil_nat ) ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_86_list__induct4,axiom,
! [Xs: 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 @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys3: list_a,Z: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_87_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_nat,P: list_a > list_a > list_a > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_a @ nil_nat )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys3: list_a,Z: a,Zs2: list_a,W: nat,Ws2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_88_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_nat,Ws: list_a,P: list_a > list_a > list_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_nat @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys3: list_a,Z: nat,Zs2: list_nat,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_89_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_nat,Ws: list_nat,P: list_a > list_a > list_nat > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_nat @ nil_nat )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys3: list_a,Z: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_90_list__induct4,axiom,
! [Xs: list_a,Ys: list_nat,Zs: list_a,Ws: list_a,P: list_a > list_nat > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_nat @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y3: nat,Ys3: list_nat,Z: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys3 ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_91_list__induct4,axiom,
! [Xs: list_a,Ys: list_nat,Zs: list_a,Ws: list_nat,P: list_a > list_nat > list_a > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_a @ nil_nat @ nil_a @ nil_nat )
=> ( ! [X3: a,Xs2: list_a,Y3: nat,Ys3: list_nat,Z: a,Zs2: list_a,W: nat,Ws2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys3 ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_92_list__induct4,axiom,
! [Xs: list_a,Ys: list_nat,Zs: list_nat,Ws: list_a,P: list_a > list_nat > list_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_nat @ nil_nat @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y3: nat,Ys3: list_nat,Z: nat,Zs2: list_nat,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys3 ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_93_list__induct4,axiom,
! [Xs: list_a,Ys: list_nat,Zs: list_nat,Ws: list_nat,P: list_a > list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_a @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X3: a,Xs2: list_a,Y3: nat,Ys3: list_nat,Z: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys3 ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_94_list__induct4,axiom,
! [Xs: list_nat,Ys: list_a,Zs: list_a,Ws: list_a,P: list_nat > list_a > list_a > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_nat @ nil_a @ nil_a @ nil_a )
=> ( ! [X3: nat,Xs2: list_nat,Y3: a,Ys3: list_a,Z: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_95_list__induct4,axiom,
! [Xs: list_nat,Ys: list_a,Zs: list_a,Ws: list_nat,P: list_nat > list_a > list_a > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_a @ nil_a @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: a,Ys3: list_a,Z: a,Zs2: list_a,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_96_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,P: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys3: list_a,Z: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_97_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_nat,P: list_a > list_a > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_a @ nil_a @ nil_nat )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys3: list_a,Z: nat,Zs2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_98_list__induct3,axiom,
! [Xs: list_a,Ys: list_nat,Zs: list_a,P: list_a > list_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_nat @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y3: nat,Ys3: list_nat,Z: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys3 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_99_list__induct3,axiom,
! [Xs: list_a,Ys: list_nat,Zs: list_nat,P: list_a > list_nat > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_a @ nil_nat @ nil_nat )
=> ( ! [X3: a,Xs2: list_a,Y3: nat,Ys3: list_nat,Z: nat,Zs2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys3 ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_100_list__induct3,axiom,
! [Xs: list_nat,Ys: list_a,Zs: list_a,P: list_nat > list_a > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_nat @ nil_a @ nil_a )
=> ( ! [X3: nat,Xs2: list_nat,Y3: a,Ys3: list_a,Z: a,Zs2: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_101_list__induct3,axiom,
! [Xs: list_nat,Ys: list_a,Zs: list_nat,P: list_nat > list_a > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_a @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: a,Ys3: list_a,Z: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_102_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_a,P: list_nat > list_nat > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_a )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys3: list_nat,Z: a,Zs2: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys3 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_103_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,P: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys3: list_nat,Z: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( ( size_size_list_nat @ Ys3 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys3 ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_104_list__induct2,axiom,
! [Xs: list_a,Ys: list_a,P: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys3: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_105_list__induct2,axiom,
! [Xs: list_a,Ys: list_nat,P: list_a > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_a @ nil_nat )
=> ( ! [X3: a,Xs2: list_a,Y3: nat,Ys3: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_106_list__induct2,axiom,
! [Xs: list_nat,Ys: list_a,P: list_nat > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_nat @ nil_a )
=> ( ! [X3: nat,Xs2: list_nat,Y3: a,Ys3: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_107_list__induct2,axiom,
! [Xs: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_108_rev__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X3: a] : ( P @ ( cons_a @ X3 @ nil_a ) )
=> ( ! [X3: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X3 @ nil_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_109_rev__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] : ( P @ ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X3: nat,Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X3 @ nil_nat ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_110_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X: a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X @ Xs ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X @ Xs ) ) )
| ? [Ys4: list_a] :
( ( Ys
= ( cons_a @ X @ Ys4 ) )
& ( ( append_a @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_111_append__eq__Cons__conv,axiom,
! [Ys: list_nat,Zs: list_nat,X: nat,Xs: list_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( cons_nat @ X @ Xs ) )
= ( ( ( Ys = nil_nat )
& ( Zs
= ( cons_nat @ X @ Xs ) ) )
| ? [Ys4: list_nat] :
( ( Ys
= ( cons_nat @ X @ Ys4 ) )
& ( ( append_nat @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_112_Cons__eq__append__conv,axiom,
! [X: a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X @ Xs )
= Zs ) )
| ? [Ys4: list_a] :
( ( ( cons_a @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_a @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_113_Cons__eq__append__conv,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( append_nat @ Ys @ Zs ) )
= ( ( ( Ys = nil_nat )
& ( ( cons_nat @ X @ Xs )
= Zs ) )
| ? [Ys4: list_nat] :
( ( ( cons_nat @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_nat @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_114_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys3: list_a,Y3: a] :
( Xs
!= ( append_a @ Ys3 @ ( cons_a @ Y3 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_115_rev__exhaust,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ~ ! [Ys3: list_nat,Y3: nat] :
( Xs
!= ( append_nat @ Ys3 @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) ).
% rev_exhaust
thf(fact_116_rev__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ( P @ nil_a )
=> ( ! [X3: a,Xs2: list_a] :
( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X3 @ nil_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_117_rev__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ( P @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat] :
( ( P @ Xs2 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X3 @ nil_nat ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_118_same__length__different,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != Ys )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ? [Pre: list_a,X3: a,Xs3: list_a,Y3: a,Ys5: list_a] :
( ( X3 != Y3 )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X3 @ nil_a ) @ Xs3 ) ) )
& ( Ys
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y3 @ nil_a ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_119_same__length__different,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != Ys )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ? [Pre: list_nat,X3: nat,Xs3: list_nat,Y3: nat,Ys5: list_nat] :
( ( X3 != Y3 )
& ( Xs
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X3 @ nil_nat ) @ Xs3 ) ) )
& ( Ys
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y3 @ nil_nat ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_120_a,axiom,
prefix454693708527911765comp_o @ ( prefix4097710381326367690Lf_e_a @ e @ ( suc @ na ) @ x ) @ ( prefix4097710381326367690Lf_e_a @ e @ ( suc @ na ) @ y ) ).
% a
thf(fact_121_length__append__singleton,axiom,
! [Xs: list_a,X: a] :
( ( size_size_list_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_append_singleton
thf(fact_122_length__append__singleton,axiom,
! [Xs: list_nat,X: nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_append_singleton
thf(fact_123_length__Cons,axiom,
! [X: a,Xs: list_a] :
( ( size_size_list_a @ ( cons_a @ X @ Xs ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_Cons
thf(fact_124_length__Cons,axiom,
! [X: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_Cons
thf(fact_125_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_126_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_127_bind__simps_I2_J,axiom,
! [X: a,Xs: list_a,F: a > list_a] :
( ( bind_a_a @ ( cons_a @ X @ Xs ) @ F )
= ( append_a @ ( F @ X ) @ ( bind_a_a @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_128_bind__simps_I2_J,axiom,
! [X: a,Xs: list_a,F: a > list_nat] :
( ( bind_a_nat @ ( cons_a @ X @ Xs ) @ F )
= ( append_nat @ ( F @ X ) @ ( bind_a_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_129_bind__simps_I2_J,axiom,
! [X: nat,Xs: list_nat,F: nat > list_a] :
( ( bind_nat_a @ ( cons_nat @ X @ Xs ) @ F )
= ( append_a @ ( F @ X ) @ ( bind_nat_a @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_130_bind__simps_I2_J,axiom,
! [X: nat,Xs: list_nat,F: nat > list_nat] :
( ( bind_nat_nat @ ( cons_nat @ X @ Xs ) @ F )
= ( append_nat @ ( F @ X ) @ ( bind_nat_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_131_gen__length__code_I2_J,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( gen_length_a @ N @ ( cons_a @ X @ Xs ) )
= ( gen_length_a @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_132_gen__length__code_I2_J,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( gen_length_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( gen_length_nat @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_133_maps__simps_I1_J,axiom,
! [F: a > list_a,X: a,Xs: list_a] :
( ( maps_a_a @ F @ ( cons_a @ X @ Xs ) )
= ( append_a @ ( F @ X ) @ ( maps_a_a @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_134_maps__simps_I1_J,axiom,
! [F: a > list_nat,X: a,Xs: list_a] :
( ( maps_a_nat @ F @ ( cons_a @ X @ Xs ) )
= ( append_nat @ ( F @ X ) @ ( maps_a_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_135_maps__simps_I1_J,axiom,
! [F: nat > list_a,X: nat,Xs: list_nat] :
( ( maps_nat_a @ F @ ( cons_nat @ X @ Xs ) )
= ( append_a @ ( F @ X ) @ ( maps_nat_a @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_136_maps__simps_I1_J,axiom,
! [F: nat > list_nat,X: nat,Xs: list_nat] :
( ( maps_nat_nat @ F @ ( cons_nat @ X @ Xs ) )
= ( append_nat @ ( F @ X ) @ ( maps_nat_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_137_insert__Nil,axiom,
! [X: a] :
( ( insert_a @ X @ nil_a )
= ( cons_a @ X @ nil_a ) ) ).
% insert_Nil
thf(fact_138_insert__Nil,axiom,
! [X: nat] :
( ( insert_nat @ X @ nil_nat )
= ( cons_nat @ X @ nil_nat ) ) ).
% insert_Nil
thf(fact_139_list__ex1__simps_I1_J,axiom,
! [P: a > $o] :
~ ( list_ex1_a @ P @ nil_a ) ).
% list_ex1_simps(1)
thf(fact_140_list__ex1__simps_I1_J,axiom,
! [P: nat > $o] :
~ ( list_ex1_nat @ P @ nil_nat ) ).
% list_ex1_simps(1)
thf(fact_141_assms,axiom,
prefix7485107378405021920ding_a @ e ).
% assms
thf(fact_142_bind__simps_I1_J,axiom,
! [F: a > list_a] :
( ( bind_a_a @ nil_a @ F )
= nil_a ) ).
% bind_simps(1)
thf(fact_143_bind__simps_I1_J,axiom,
! [F: a > list_nat] :
( ( bind_a_nat @ nil_a @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_144_bind__simps_I1_J,axiom,
! [F: nat > list_a] :
( ( bind_nat_a @ nil_nat @ F )
= nil_a ) ).
% bind_simps(1)
thf(fact_145_bind__simps_I1_J,axiom,
! [F: nat > list_nat] :
( ( bind_nat_nat @ nil_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_146_opt__comp__sym,axiom,
( prefix454693708527911765comp_o
= ( ^ [X4: option_list_o,Y2: option_list_o] : ( prefix454693708527911765comp_o @ Y2 @ X4 ) ) ) ).
% opt_comp_sym
thf(fact_147_maps__simps_I2_J,axiom,
! [F: a > list_a] :
( ( maps_a_a @ F @ nil_a )
= nil_a ) ).
% maps_simps(2)
thf(fact_148_maps__simps_I2_J,axiom,
! [F: a > list_nat] :
( ( maps_a_nat @ F @ nil_a )
= nil_nat ) ).
% maps_simps(2)
thf(fact_149_maps__simps_I2_J,axiom,
! [F: nat > list_a] :
( ( maps_nat_a @ F @ nil_nat )
= nil_a ) ).
% maps_simps(2)
thf(fact_150_maps__simps_I2_J,axiom,
! [F: nat > list_nat] :
( ( maps_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% maps_simps(2)
thf(fact_151_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_a @ N @ nil_a )
= N ) ).
% gen_length_code(1)
thf(fact_152_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_nat @ N @ nil_nat )
= N ) ).
% gen_length_code(1)
thf(fact_153_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_154_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_155_size__neq__size__imp__neq,axiom,
! [X: list_a,Y: list_a] :
( ( ( size_size_list_a @ X )
!= ( size_size_list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_156_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X )
!= ( size_size_list_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_157_prefixes__snoc,axiom,
! [Xs: list_a,X: a] :
( ( prefixes_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= ( append_list_a @ ( prefixes_a @ Xs ) @ ( cons_list_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) @ nil_list_a ) ) ) ).
% prefixes_snoc
thf(fact_158_prefixes__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( prefixes_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= ( append_list_nat @ ( prefixes_nat @ Xs ) @ ( cons_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) @ nil_list_nat ) ) ) ).
% prefixes_snoc
thf(fact_159_SuccD,axiom,
! [K: a,Kl: set_list_a,Kl2: list_a] :
( ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl @ Kl2 ) )
=> ( member_list_a @ ( append_a @ Kl2 @ ( cons_a @ K @ nil_a ) ) @ Kl ) ) ).
% SuccD
thf(fact_160_SuccD,axiom,
! [K: nat,Kl: set_list_nat,Kl2: list_nat] :
( ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl @ Kl2 ) )
=> ( member_list_nat @ ( append_nat @ Kl2 @ ( cons_nat @ K @ nil_nat ) ) @ Kl ) ) ).
% SuccD
thf(fact_161_SuccI,axiom,
! [Kl2: list_a,K: a,Kl: set_list_a] :
( ( member_list_a @ ( append_a @ Kl2 @ ( cons_a @ K @ nil_a ) ) @ Kl )
=> ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_162_SuccI,axiom,
! [Kl2: list_nat,K: nat,Kl: set_list_nat] :
( ( member_list_nat @ ( append_nat @ Kl2 @ ( cons_nat @ K @ nil_nat ) ) @ Kl )
=> ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_163_prefixes__eq__snoc,axiom,
! [Ys: list_a,Xs: list_list_a,X: list_a] :
( ( ( prefixes_a @ Ys )
= ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) ) )
= ( ( ( ( Ys = nil_a )
& ( Xs = nil_list_a ) )
| ? [Z2: a,Zs3: list_a] :
( ( Ys
= ( append_a @ Zs3 @ ( cons_a @ Z2 @ nil_a ) ) )
& ( Xs
= ( prefixes_a @ Zs3 ) ) ) )
& ( X = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_164_prefixes__eq__snoc,axiom,
! [Ys: list_nat,Xs: list_list_nat,X: list_nat] :
( ( ( prefixes_nat @ Ys )
= ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= ( ( ( ( Ys = nil_nat )
& ( Xs = nil_list_nat ) )
| ? [Z2: nat,Zs3: list_nat] :
( ( Ys
= ( append_nat @ Zs3 @ ( cons_nat @ Z2 @ nil_nat ) ) )
& ( Xs
= ( prefixes_nat @ Zs3 ) ) ) )
& ( X = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_165_sublists_Osimps_I1_J,axiom,
( ( sublists_a @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% sublists.simps(1)
thf(fact_166_sublists_Osimps_I1_J,axiom,
( ( sublists_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% sublists.simps(1)
thf(fact_167_product__lists_Osimps_I1_J,axiom,
( ( product_lists_a @ nil_list_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% product_lists.simps(1)
thf(fact_168_product__lists_Osimps_I1_J,axiom,
( ( product_lists_nat @ nil_list_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% product_lists.simps(1)
thf(fact_169_suffixes__eq__snoc,axiom,
! [Ys: list_a,Xs: list_list_a,X: list_a] :
( ( ( suffixes_a @ Ys )
= ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) ) )
= ( ( ( ( Ys = nil_a )
& ( Xs = nil_list_a ) )
| ? [Z2: a,Zs3: list_a] :
( ( Ys
= ( cons_a @ Z2 @ Zs3 ) )
& ( Xs
= ( suffixes_a @ Zs3 ) ) ) )
& ( X = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_170_suffixes__eq__snoc,axiom,
! [Ys: list_nat,Xs: list_list_nat,X: list_nat] :
( ( ( suffixes_nat @ Ys )
= ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= ( ( ( ( Ys = nil_nat )
& ( Xs = nil_list_nat ) )
| ? [Z2: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z2 @ Zs3 ) )
& ( Xs
= ( suffixes_nat @ Zs3 ) ) ) )
& ( X = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_171_subseqs_Osimps_I1_J,axiom,
( ( subseqs_a @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% subseqs.simps(1)
thf(fact_172_subseqs_Osimps_I1_J,axiom,
( ( subseqs_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_173_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,Xs5: list_a,Xss22: list_list_a] :
( ( Xss2
= ( append_list_a @ Xss1 @ ( cons_list_a @ ( append_a @ Xs4 @ Xs5 ) @ Xss22 ) ) )
& ( Ys
= ( append_a @ ( concat_a @ Xss1 ) @ Xs4 ) )
& ( Zs
= ( append_a @ Xs5 @ ( concat_a @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_174_concat__eq__append__conv,axiom,
! [Xss2: list_list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( concat_nat @ Xss2 )
= ( append_nat @ Ys @ Zs ) )
= ( ( ( Xss2 = nil_list_nat )
=> ( ( Ys = nil_nat )
& ( Zs = nil_nat ) ) )
& ( ( Xss2 != nil_list_nat )
=> ? [Xss1: list_list_nat,Xs4: list_nat,Xs5: list_nat,Xss22: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss1 @ ( cons_list_nat @ ( append_nat @ Xs4 @ Xs5 ) @ Xss22 ) ) )
& ( Ys
= ( append_nat @ ( concat_nat @ Xss1 ) @ Xs4 ) )
& ( Zs
= ( append_nat @ Xs5 @ ( concat_nat @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_175_length__suffixes,axiom,
! [Xs: list_a] :
( ( size_s349497388124573686list_a @ ( suffixes_a @ Xs ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_suffixes
thf(fact_176_length__suffixes,axiom,
! [Xs: list_nat] :
( ( size_s3023201423986296836st_nat @ ( suffixes_nat @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_suffixes
thf(fact_177_concat__append,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( concat_a @ ( append_list_a @ Xs @ Ys ) )
= ( append_a @ ( concat_a @ Xs ) @ ( concat_a @ Ys ) ) ) ).
% concat_append
thf(fact_178_concat__append,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( concat_nat @ ( append_list_nat @ Xs @ Ys ) )
= ( append_nat @ ( concat_nat @ Xs ) @ ( concat_nat @ Ys ) ) ) ).
% concat_append
thf(fact_179_concat_Osimps_I2_J,axiom,
! [X: list_a,Xs: list_list_a] :
( ( concat_a @ ( cons_list_a @ X @ Xs ) )
= ( append_a @ X @ ( concat_a @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_180_concat_Osimps_I2_J,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( concat_nat @ ( cons_list_nat @ X @ Xs ) )
= ( append_nat @ X @ ( concat_nat @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_181_concat_Osimps_I1_J,axiom,
( ( concat_a @ nil_list_a )
= nil_a ) ).
% concat.simps(1)
thf(fact_182_concat_Osimps_I1_J,axiom,
( ( concat_nat @ nil_list_nat )
= nil_nat ) ).
% concat.simps(1)
thf(fact_183_is__encodingD,axiom,
! [E: a > option_list_o,X: a,Y: a] :
( ( prefix7485107378405021920ding_a @ E )
=> ( ( prefix454693708527911765comp_o @ ( E @ X ) @ ( E @ Y ) )
=> ( X = Y ) ) ) ).
% is_encodingD
thf(fact_184_is__encodingD,axiom,
! [E: list_a > option_list_o,X: list_a,Y: list_a] :
( ( prefix5220018966750911590list_a @ E )
=> ( ( prefix454693708527911765comp_o @ ( E @ X ) @ ( E @ Y ) )
=> ( X = Y ) ) ) ).
% is_encodingD
thf(fact_185_is__encodingI__2,axiom,
! [E: a > option_list_o] :
( ! [X3: a,Y3: a] :
( ( prefix454693708527911765comp_o @ ( E @ X3 ) @ ( E @ Y3 ) )
=> ( X3 = Y3 ) )
=> ( prefix7485107378405021920ding_a @ E ) ) ).
% is_encodingI_2
thf(fact_186_is__encodingI__2,axiom,
! [E: list_a > option_list_o] :
( ! [X3: list_a,Y3: list_a] :
( ( prefix454693708527911765comp_o @ ( E @ X3 ) @ ( E @ Y3 ) )
=> ( X3 = Y3 ) )
=> ( prefix5220018966750911590list_a @ E ) ) ).
% is_encodingI_2
thf(fact_187_concat__eq__appendD,axiom,
! [Xss2: list_list_a,Ys: list_a,Zs: list_a] :
( ( ( concat_a @ Xss2 )
= ( append_a @ Ys @ Zs ) )
=> ( ( Xss2 != nil_list_a )
=> ? [Xss12: list_list_a,Xs2: list_a,Xs3: list_a,Xss23: list_list_a] :
( ( Xss2
= ( append_list_a @ Xss12 @ ( cons_list_a @ ( append_a @ Xs2 @ Xs3 ) @ Xss23 ) ) )
& ( Ys
= ( append_a @ ( concat_a @ Xss12 ) @ Xs2 ) )
& ( Zs
= ( append_a @ Xs3 @ ( concat_a @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_188_concat__eq__appendD,axiom,
! [Xss2: list_list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( concat_nat @ Xss2 )
= ( append_nat @ Ys @ Zs ) )
=> ( ( Xss2 != nil_list_nat )
=> ? [Xss12: list_list_nat,Xs2: list_nat,Xs3: list_nat,Xss23: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss12 @ ( cons_list_nat @ ( append_nat @ Xs2 @ Xs3 ) @ Xss23 ) ) )
& ( Ys
= ( append_nat @ ( concat_nat @ Xss12 ) @ Xs2 ) )
& ( Zs
= ( append_nat @ Xs3 @ ( concat_nat @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_189_prefixes_Osimps_I1_J,axiom,
( ( prefixes_a @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% prefixes.simps(1)
thf(fact_190_prefixes_Osimps_I1_J,axiom,
( ( prefixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% prefixes.simps(1)
thf(fact_191_suffixes_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( suffixes_a @ ( cons_a @ X @ Xs ) )
= ( append_list_a @ ( suffixes_a @ Xs ) @ ( cons_list_a @ ( cons_a @ X @ Xs ) @ nil_list_a ) ) ) ).
% suffixes.simps(2)
thf(fact_192_suffixes_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( suffixes_nat @ ( cons_nat @ X @ Xs ) )
= ( append_list_nat @ ( suffixes_nat @ Xs ) @ ( cons_list_nat @ ( cons_nat @ X @ Xs ) @ nil_list_nat ) ) ) ).
% suffixes.simps(2)
thf(fact_193_suffixes_Osimps_I1_J,axiom,
( ( suffixes_a @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% suffixes.simps(1)
thf(fact_194_suffixes_Osimps_I1_J,axiom,
( ( suffixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% suffixes.simps(1)
thf(fact_195_empty__Shift,axiom,
! [Kl: set_list_a,K: a] :
( ( member_list_a @ nil_a @ Kl )
=> ( ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl @ nil_a ) )
=> ( member_list_a @ nil_a @ ( bNF_Greatest_Shift_a @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_196_empty__Shift,axiom,
! [Kl: set_list_nat,K: nat] :
( ( member_list_nat @ nil_nat @ Kl )
=> ( ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl @ nil_nat ) )
=> ( member_list_nat @ nil_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_197_Succ__Shift,axiom,
! [Kl: set_list_a,K: a,Kl2: list_a] :
( ( bNF_Greatest_Succ_a @ ( bNF_Greatest_Shift_a @ Kl @ K ) @ Kl2 )
= ( bNF_Greatest_Succ_a @ Kl @ ( cons_a @ K @ Kl2 ) ) ) ).
% Succ_Shift
thf(fact_198_Succ__Shift,axiom,
! [Kl: set_list_nat,K: nat,Kl2: list_nat] :
( ( bNF_Gr6352880689984616693cc_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl @ K ) @ Kl2 )
= ( bNF_Gr6352880689984616693cc_nat @ Kl @ ( cons_nat @ K @ Kl2 ) ) ) ).
% Succ_Shift
thf(fact_199_butlast__snoc,axiom,
! [Xs: list_a,X: a] :
( ( butlast_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_200_butlast__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( butlast_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_201_sublists_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( sublists_a @ ( cons_a @ X @ Xs ) )
= ( append_list_a @ ( sublists_a @ Xs ) @ ( map_list_a_list_a @ ( cons_a @ X ) @ ( prefixes_a @ Xs ) ) ) ) ).
% sublists.simps(2)
thf(fact_202_sublists_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( sublists_nat @ ( cons_nat @ X @ Xs ) )
= ( append_list_nat @ ( sublists_nat @ Xs ) @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X ) @ ( prefixes_nat @ Xs ) ) ) ) ).
% sublists.simps(2)
thf(fact_203_last__snoc,axiom,
! [Xs: list_a,X: a] :
( ( last_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= X ) ).
% last_snoc
thf(fact_204_last__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( last_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= X ) ).
% last_snoc
thf(fact_205_rotate1_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( rotate1_a @ ( cons_a @ X @ Xs ) )
= ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) ) ).
% rotate1.simps(2)
thf(fact_206_rotate1_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( rotate1_nat @ ( cons_nat @ X @ Xs ) )
= ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) ) ).
% rotate1.simps(2)
thf(fact_207_concat__conv__foldr,axiom,
( concat_a
= ( ^ [Xss3: list_list_a] : ( foldr_list_a_list_a @ append_a @ Xss3 @ nil_a ) ) ) ).
% concat_conv_foldr
thf(fact_208_concat__conv__foldr,axiom,
( concat_nat
= ( ^ [Xss3: list_list_nat] : ( foldr_6871341030409798377st_nat @ append_nat @ Xss3 @ nil_nat ) ) ) ).
% concat_conv_foldr
thf(fact_209_prefixes_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( prefixes_a @ ( cons_a @ X @ Xs ) )
= ( cons_list_a @ nil_a @ ( map_list_a_list_a @ ( cons_a @ X ) @ ( prefixes_a @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_210_prefixes_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( prefixes_nat @ ( cons_nat @ X @ Xs ) )
= ( cons_list_nat @ nil_nat @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X ) @ ( prefixes_nat @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_211_Suc,axiom,
prefix5220018966750911590list_a @ ( prefix4097710381326367690Lf_e_a @ e @ na ) ).
% Suc
thf(fact_212_list__update__length,axiom,
! [Xs: list_a,X: a,Ys: list_a,Y: a] :
( ( list_update_a @ ( append_a @ Xs @ ( cons_a @ X @ Ys ) ) @ ( size_size_list_a @ Xs ) @ Y )
= ( append_a @ Xs @ ( cons_a @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_213_list__update__length,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat,Y: nat] :
( ( list_update_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) ) @ ( size_size_list_nat @ Xs ) @ Y )
= ( append_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_214_map__ident,axiom,
( ( map_nat_nat
@ ^ [X4: nat] : X4 )
= ( ^ [Xs4: list_nat] : Xs4 ) ) ).
% map_ident
thf(fact_215_list_Omap__disc__iff,axiom,
! [F: a > a,A: list_a] :
( ( ( map_a_a @ F @ A )
= nil_a )
= ( A = nil_a ) ) ).
% list.map_disc_iff
thf(fact_216_list_Omap__disc__iff,axiom,
! [F: nat > a,A: list_nat] :
( ( ( map_nat_a @ F @ A )
= nil_a )
= ( A = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_217_list_Omap__disc__iff,axiom,
! [F: a > nat,A: list_a] :
( ( ( map_a_nat @ F @ A )
= nil_nat )
= ( A = nil_a ) ) ).
% list.map_disc_iff
thf(fact_218_list_Omap__disc__iff,axiom,
! [F: nat > nat,A: list_nat] :
( ( ( map_nat_nat @ F @ A )
= nil_nat )
= ( A = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_219_Nil__is__map__conv,axiom,
! [F: a > a,Xs: list_a] :
( ( nil_a
= ( map_a_a @ F @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_map_conv
thf(fact_220_Nil__is__map__conv,axiom,
! [F: nat > a,Xs: list_nat] :
( ( nil_a
= ( map_nat_a @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_221_Nil__is__map__conv,axiom,
! [F: a > nat,Xs: list_a] :
( ( nil_nat
= ( map_a_nat @ F @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_map_conv
thf(fact_222_Nil__is__map__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( nil_nat
= ( map_nat_nat @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_223_map__is__Nil__conv,axiom,
! [F: a > a,Xs: list_a] :
( ( ( map_a_a @ F @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% map_is_Nil_conv
thf(fact_224_map__is__Nil__conv,axiom,
! [F: nat > a,Xs: list_nat] :
( ( ( map_nat_a @ F @ Xs )
= nil_a )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_225_map__is__Nil__conv,axiom,
! [F: a > nat,Xs: list_a] :
( ( ( map_a_nat @ F @ Xs )
= nil_nat )
= ( Xs = nil_a ) ) ).
% map_is_Nil_conv
thf(fact_226_map__is__Nil__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_227_length__map,axiom,
! [F: a > a,Xs: list_a] :
( ( size_size_list_a @ ( map_a_a @ F @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_map
thf(fact_228_length__map,axiom,
! [F: nat > a,Xs: list_nat] :
( ( size_size_list_a @ ( map_nat_a @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_229_length__map,axiom,
! [F: a > nat,Xs: list_a] :
( ( size_size_list_nat @ ( map_a_nat @ F @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_map
thf(fact_230_length__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_231_map__append,axiom,
! [F: a > a,Xs: list_a,Ys: list_a] :
( ( map_a_a @ F @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( map_a_a @ F @ Xs ) @ ( map_a_a @ F @ Ys ) ) ) ).
% map_append
thf(fact_232_map__append,axiom,
! [F: a > nat,Xs: list_a,Ys: list_a] :
( ( map_a_nat @ F @ ( append_a @ Xs @ Ys ) )
= ( append_nat @ ( map_a_nat @ F @ Xs ) @ ( map_a_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_233_map__append,axiom,
! [F: nat > a,Xs: list_nat,Ys: list_nat] :
( ( map_nat_a @ F @ ( append_nat @ Xs @ Ys ) )
= ( append_a @ ( map_nat_a @ F @ Xs ) @ ( map_nat_a @ F @ Ys ) ) ) ).
% map_append
thf(fact_234_map__append,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( map_nat_nat @ F @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_235_list__update__nonempty,axiom,
! [Xs: list_a,K: nat,X: a] :
( ( ( list_update_a @ Xs @ K @ X )
= nil_a )
= ( Xs = nil_a ) ) ).
% list_update_nonempty
thf(fact_236_list__update__nonempty,axiom,
! [Xs: list_nat,K: nat,X: nat] :
( ( ( list_update_nat @ Xs @ K @ X )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% list_update_nonempty
thf(fact_237_length__list__update,axiom,
! [Xs: list_a,I: nat,X: a] :
( ( size_size_list_a @ ( list_update_a @ Xs @ I @ X ) )
= ( size_size_list_a @ Xs ) ) ).
% length_list_update
thf(fact_238_length__list__update,axiom,
! [Xs: list_nat,I: nat,X: nat] :
( ( size_size_list_nat @ ( list_update_nat @ Xs @ I @ X ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_list_update
thf(fact_239_rotate1__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rotate1_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_240_rotate1__is__Nil__conv,axiom,
! [Xs: list_nat] :
( ( ( rotate1_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% rotate1_is_Nil_conv
thf(fact_241_length__rotate1,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( rotate1_a @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_rotate1
thf(fact_242_length__rotate1,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate1
thf(fact_243_last__appendL,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Xs ) ) ) ).
% last_appendL
thf(fact_244_last__appendL,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( Ys = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
= ( last_nat @ Xs ) ) ) ).
% last_appendL
thf(fact_245_last__appendR,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Ys ) ) ) ).
% last_appendR
thf(fact_246_last__appendR,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( Ys != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
= ( last_nat @ Ys ) ) ) ).
% last_appendR
thf(fact_247_append__butlast__last__id,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( append_a @ ( butlast_a @ Xs ) @ ( cons_a @ ( last_a @ Xs ) @ nil_a ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_248_append__butlast__last__id,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( append_nat @ ( butlast_nat @ Xs ) @ ( cons_nat @ ( last_nat @ Xs ) @ nil_nat ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_249_concat__map__singleton,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( concat_nat
@ ( map_nat_list_nat
@ ^ [X4: nat] : ( cons_nat @ ( F @ X4 ) @ nil_nat )
@ Xs ) )
= ( map_nat_nat @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_250_suffixes__snoc,axiom,
! [Xs: list_a,X: a] :
( ( suffixes_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= ( cons_list_a @ nil_a
@ ( map_list_a_list_a
@ ^ [Ys2: list_a] : ( append_a @ Ys2 @ ( cons_a @ X @ nil_a ) )
@ ( suffixes_a @ Xs ) ) ) ) ).
% suffixes_snoc
thf(fact_251_suffixes__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( suffixes_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= ( cons_list_nat @ nil_nat
@ ( map_li7225945977422193158st_nat
@ ^ [Ys2: list_nat] : ( append_nat @ Ys2 @ ( cons_nat @ X @ nil_nat ) )
@ ( suffixes_nat @ Xs ) ) ) ) ).
% suffixes_snoc
thf(fact_252_map__concat,axiom,
! [F: nat > nat,Xs: list_list_nat] :
( ( map_nat_nat @ F @ ( concat_nat @ Xs ) )
= ( concat_nat @ ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_253_list_Omap__ident,axiom,
! [T: list_nat] :
( ( map_nat_nat
@ ^ [X4: nat] : X4
@ T )
= T ) ).
% list.map_ident
thf(fact_254_map__update,axiom,
! [F: nat > nat,Xs: list_nat,K: nat,Y: nat] :
( ( map_nat_nat @ F @ ( list_update_nat @ Xs @ K @ Y ) )
= ( list_update_nat @ ( map_nat_nat @ F @ Xs ) @ K @ ( F @ Y ) ) ) ).
% map_update
thf(fact_255_map__butlast,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( map_nat_nat @ F @ ( butlast_nat @ Xs ) )
= ( butlast_nat @ ( map_nat_nat @ F @ Xs ) ) ) ).
% map_butlast
thf(fact_256_rotate1__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( rotate1_nat @ ( map_nat_nat @ F @ Xs ) )
= ( map_nat_nat @ F @ ( rotate1_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_257_last__map,axiom,
! [Xs: list_a,F: a > nat] :
( ( Xs != nil_a )
=> ( ( last_nat @ ( map_a_nat @ F @ Xs ) )
= ( F @ ( last_a @ Xs ) ) ) ) ).
% last_map
thf(fact_258_last__map,axiom,
! [Xs: list_nat,F: nat > nat] :
( ( Xs != nil_nat )
=> ( ( last_nat @ ( map_nat_nat @ F @ Xs ) )
= ( F @ ( last_nat @ Xs ) ) ) ) ).
% last_map
thf(fact_259_Shift__def,axiom,
( bNF_Greatest_Shift_a
= ( ^ [Kl3: set_list_a,K2: a] :
( collect_list_a
@ ^ [Kl4: list_a] : ( member_list_a @ ( cons_a @ K2 @ Kl4 ) @ Kl3 ) ) ) ) ).
% Shift_def
thf(fact_260_Shift__def,axiom,
( bNF_Gr1872714664788909425ft_nat
= ( ^ [Kl3: set_list_nat,K2: nat] :
( collect_list_nat
@ ^ [Kl4: list_nat] : ( member_list_nat @ ( cons_nat @ K2 @ Kl4 ) @ Kl3 ) ) ) ) ).
% Shift_def
thf(fact_261_subseqs_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( subseqs_a @ ( cons_a @ X @ Xs ) )
= ( append_list_a @ ( map_list_a_list_a @ ( cons_a @ X ) @ ( subseqs_a @ Xs ) ) @ ( subseqs_a @ Xs ) ) ) ).
% subseqs.simps(2)
thf(fact_262_subseqs_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( subseqs_nat @ ( cons_nat @ X @ Xs ) )
= ( append_list_nat @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X ) @ ( subseqs_nat @ Xs ) ) @ ( subseqs_nat @ Xs ) ) ) ).
% subseqs.simps(2)
thf(fact_263_list_Osimps_I9_J,axiom,
! [F: a > a,X21: a,X22: list_a] :
( ( map_a_a @ F @ ( cons_a @ X21 @ X22 ) )
= ( cons_a @ ( F @ X21 ) @ ( map_a_a @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_264_list_Osimps_I9_J,axiom,
! [F: a > nat,X21: a,X22: list_a] :
( ( map_a_nat @ F @ ( cons_a @ X21 @ X22 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_a_nat @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_265_list_Osimps_I9_J,axiom,
! [F: nat > a,X21: nat,X22: list_nat] :
( ( map_nat_a @ F @ ( cons_nat @ X21 @ X22 ) )
= ( cons_a @ ( F @ X21 ) @ ( map_nat_a @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_266_list_Osimps_I9_J,axiom,
! [F: nat > nat,X21: nat,X22: list_nat] :
( ( map_nat_nat @ F @ ( cons_nat @ X21 @ X22 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_nat_nat @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_267_Cons__eq__map__D,axiom,
! [X: a,Xs: list_a,F: a > a,Ys: list_a] :
( ( ( cons_a @ X @ Xs )
= ( map_a_a @ F @ Ys ) )
=> ? [Z: a,Zs2: list_a] :
( ( Ys
= ( cons_a @ Z @ Zs2 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_a_a @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_268_Cons__eq__map__D,axiom,
! [X: a,Xs: list_a,F: nat > a,Ys: list_nat] :
( ( ( cons_a @ X @ Xs )
= ( map_nat_a @ F @ Ys ) )
=> ? [Z: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z @ Zs2 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_nat_a @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_269_Cons__eq__map__D,axiom,
! [X: nat,Xs: list_nat,F: a > nat,Ys: list_a] :
( ( ( cons_nat @ X @ Xs )
= ( map_a_nat @ F @ Ys ) )
=> ? [Z: a,Zs2: list_a] :
( ( Ys
= ( cons_a @ Z @ Zs2 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_a_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_270_Cons__eq__map__D,axiom,
! [X: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( map_nat_nat @ F @ Ys ) )
=> ? [Z: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z @ Zs2 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_nat_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_271_map__eq__Cons__D,axiom,
! [F: a > a,Xs: list_a,Y: a,Ys: list_a] :
( ( ( map_a_a @ F @ Xs )
= ( cons_a @ Y @ Ys ) )
=> ? [Z: a,Zs2: list_a] :
( ( Xs
= ( cons_a @ Z @ Zs2 ) )
& ( ( F @ Z )
= Y )
& ( ( map_a_a @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_272_map__eq__Cons__D,axiom,
! [F: nat > a,Xs: list_nat,Y: a,Ys: list_a] :
( ( ( map_nat_a @ F @ Xs )
= ( cons_a @ Y @ Ys ) )
=> ? [Z: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z @ Zs2 ) )
& ( ( F @ Z )
= Y )
& ( ( map_nat_a @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_273_map__eq__Cons__D,axiom,
! [F: a > nat,Xs: list_a,Y: nat,Ys: list_nat] :
( ( ( map_a_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
=> ? [Z: a,Zs2: list_a] :
( ( Xs
= ( cons_a @ Z @ Zs2 ) )
& ( ( F @ Z )
= Y )
& ( ( map_a_nat @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_274_map__eq__Cons__D,axiom,
! [F: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
=> ? [Z: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z @ Zs2 ) )
& ( ( F @ Z )
= Y )
& ( ( map_nat_nat @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_275_Cons__eq__map__conv,axiom,
! [X: a,Xs: list_a,F: a > a,Ys: list_a] :
( ( ( cons_a @ X @ Xs )
= ( map_a_a @ F @ Ys ) )
= ( ? [Z2: a,Zs3: list_a] :
( ( Ys
= ( cons_a @ Z2 @ Zs3 ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_a_a @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_276_Cons__eq__map__conv,axiom,
! [X: a,Xs: list_a,F: nat > a,Ys: list_nat] :
( ( ( cons_a @ X @ Xs )
= ( map_nat_a @ F @ Ys ) )
= ( ? [Z2: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z2 @ Zs3 ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_nat_a @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_277_Cons__eq__map__conv,axiom,
! [X: nat,Xs: list_nat,F: a > nat,Ys: list_a] :
( ( ( cons_nat @ X @ Xs )
= ( map_a_nat @ F @ Ys ) )
= ( ? [Z2: a,Zs3: list_a] :
( ( Ys
= ( cons_a @ Z2 @ Zs3 ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_a_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_278_Cons__eq__map__conv,axiom,
! [X: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( map_nat_nat @ F @ Ys ) )
= ( ? [Z2: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z2 @ Zs3 ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_279_map__eq__Cons__conv,axiom,
! [F: a > a,Xs: list_a,Y: a,Ys: list_a] :
( ( ( map_a_a @ F @ Xs )
= ( cons_a @ Y @ Ys ) )
= ( ? [Z2: a,Zs3: list_a] :
( ( Xs
= ( cons_a @ Z2 @ Zs3 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_a_a @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_280_map__eq__Cons__conv,axiom,
! [F: nat > a,Xs: list_nat,Y: a,Ys: list_a] :
( ( ( map_nat_a @ F @ Xs )
= ( cons_a @ Y @ Ys ) )
= ( ? [Z2: nat,Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Z2 @ Zs3 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_nat_a @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_281_map__eq__Cons__conv,axiom,
! [F: a > nat,Xs: list_a,Y: nat,Ys: list_nat] :
( ( ( map_a_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
= ( ? [Z2: a,Zs3: list_a] :
( ( Xs
= ( cons_a @ Z2 @ Zs3 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_a_nat @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_282_map__eq__Cons__conv,axiom,
! [F: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
= ( ? [Z2: nat,Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Z2 @ Zs3 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_nat_nat @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_283_list_Osimps_I8_J,axiom,
! [F: a > a] :
( ( map_a_a @ F @ nil_a )
= nil_a ) ).
% list.simps(8)
thf(fact_284_list_Osimps_I8_J,axiom,
! [F: a > nat] :
( ( map_a_nat @ F @ nil_a )
= nil_nat ) ).
% list.simps(8)
thf(fact_285_list_Osimps_I8_J,axiom,
! [F: nat > a] :
( ( map_nat_a @ F @ nil_nat )
= nil_a ) ).
% list.simps(8)
thf(fact_286_list_Osimps_I8_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_287_map__eq__imp__length__eq,axiom,
! [F: a > nat,Xs: list_a,G: nat > nat,Ys: list_nat] :
( ( ( map_a_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_288_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: a > nat,Ys: list_a] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_a_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_289_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_290_append__eq__map__conv,axiom,
! [Ys: list_a,Zs: list_a,F: a > a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( map_a_a @ F @ Xs ) )
= ( ? [Us2: list_a,Vs2: list_a] :
( ( Xs
= ( append_a @ Us2 @ Vs2 ) )
& ( Ys
= ( map_a_a @ F @ Us2 ) )
& ( Zs
= ( map_a_a @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_291_append__eq__map__conv,axiom,
! [Ys: list_a,Zs: list_a,F: nat > a,Xs: list_nat] :
( ( ( append_a @ Ys @ Zs )
= ( map_nat_a @ F @ Xs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys
= ( map_nat_a @ F @ Us2 ) )
& ( Zs
= ( map_nat_a @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_292_append__eq__map__conv,axiom,
! [Ys: list_nat,Zs: list_nat,F: a > nat,Xs: list_a] :
( ( ( append_nat @ Ys @ Zs )
= ( map_a_nat @ F @ Xs ) )
= ( ? [Us2: list_a,Vs2: list_a] :
( ( Xs
= ( append_a @ Us2 @ Vs2 ) )
& ( Ys
= ( map_a_nat @ F @ Us2 ) )
& ( Zs
= ( map_a_nat @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_293_append__eq__map__conv,axiom,
! [Ys: list_nat,Zs: list_nat,F: nat > nat,Xs: list_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( map_nat_nat @ F @ Xs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_294_map__eq__append__conv,axiom,
! [F: a > a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( map_a_a @ F @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ? [Us2: list_a,Vs2: list_a] :
( ( Xs
= ( append_a @ Us2 @ Vs2 ) )
& ( Ys
= ( map_a_a @ F @ Us2 ) )
& ( Zs
= ( map_a_a @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_295_map__eq__append__conv,axiom,
! [F: nat > a,Xs: list_nat,Ys: list_a,Zs: list_a] :
( ( ( map_nat_a @ F @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys
= ( map_nat_a @ F @ Us2 ) )
& ( Zs
= ( map_nat_a @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_296_map__eq__append__conv,axiom,
! [F: a > nat,Xs: list_a,Ys: list_nat,Zs: list_nat] :
( ( ( map_a_nat @ F @ Xs )
= ( append_nat @ Ys @ Zs ) )
= ( ? [Us2: list_a,Vs2: list_a] :
( ( Xs
= ( append_a @ Us2 @ Vs2 ) )
& ( Ys
= ( map_a_nat @ F @ Us2 ) )
& ( Zs
= ( map_a_nat @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_297_map__eq__append__conv,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( append_nat @ Ys @ Zs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_298_list__update__code_I1_J,axiom,
! [I: nat,Y: a] :
( ( list_update_a @ nil_a @ I @ Y )
= nil_a ) ).
% list_update_code(1)
thf(fact_299_list__update__code_I1_J,axiom,
! [I: nat,Y: nat] :
( ( list_update_nat @ nil_nat @ I @ Y )
= nil_nat ) ).
% list_update_code(1)
thf(fact_300_list__update_Osimps_I1_J,axiom,
! [I: nat,V: a] :
( ( list_update_a @ nil_a @ I @ V )
= nil_a ) ).
% list_update.simps(1)
thf(fact_301_list__update_Osimps_I1_J,axiom,
! [I: nat,V: nat] :
( ( list_update_nat @ nil_nat @ I @ V )
= nil_nat ) ).
% list_update.simps(1)
thf(fact_302_butlast_Osimps_I1_J,axiom,
( ( butlast_a @ nil_a )
= nil_a ) ).
% butlast.simps(1)
thf(fact_303_butlast_Osimps_I1_J,axiom,
( ( butlast_nat @ nil_nat )
= nil_nat ) ).
% butlast.simps(1)
thf(fact_304_rotate1_Osimps_I1_J,axiom,
( ( rotate1_a @ nil_a )
= nil_a ) ).
% rotate1.simps(1)
thf(fact_305_rotate1_Osimps_I1_J,axiom,
( ( rotate1_nat @ nil_nat )
= nil_nat ) ).
% rotate1.simps(1)
thf(fact_306_snoc__eq__iff__butlast,axiom,
! [Xs: list_a,X: a,Ys: list_a] :
( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
= Ys )
= ( ( Ys != nil_a )
& ( ( butlast_a @ Ys )
= Xs )
& ( ( last_a @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_307_snoc__eq__iff__butlast,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) )
= Ys )
= ( ( Ys != nil_nat )
& ( ( butlast_nat @ Ys )
= Xs )
& ( ( last_nat @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_308_Succ__def,axiom,
( bNF_Gr3053708287304744325st_nat
= ( ^ [Kl3: set_list_list_nat,Kl4: list_list_nat] :
( collect_list_nat
@ ^ [K2: list_nat] : ( member_list_list_nat @ ( append_list_nat @ Kl4 @ ( cons_list_nat @ K2 @ nil_list_nat ) ) @ Kl3 ) ) ) ) ).
% Succ_def
thf(fact_309_Succ__def,axiom,
( bNF_Greatest_Succ_a
= ( ^ [Kl3: set_list_a,Kl4: list_a] :
( collect_a
@ ^ [K2: a] : ( member_list_a @ ( append_a @ Kl4 @ ( cons_a @ K2 @ nil_a ) ) @ Kl3 ) ) ) ) ).
% Succ_def
thf(fact_310_Succ__def,axiom,
( bNF_Gr6352880689984616693cc_nat
= ( ^ [Kl3: set_list_nat,Kl4: list_nat] :
( collect_nat
@ ^ [K2: nat] : ( member_list_nat @ ( append_nat @ Kl4 @ ( cons_nat @ K2 @ nil_nat ) ) @ Kl3 ) ) ) ) ).
% Succ_def
thf(fact_311_list__update__code_I3_J,axiom,
! [X: a,Xs: list_a,I: nat,Y: a] :
( ( list_update_a @ ( cons_a @ X @ Xs ) @ ( suc @ I ) @ Y )
= ( cons_a @ X @ ( list_update_a @ Xs @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_312_list__update__code_I3_J,axiom,
! [X: nat,Xs: list_nat,I: nat,Y: nat] :
( ( list_update_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ I ) @ Y )
= ( cons_nat @ X @ ( list_update_nat @ Xs @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_313_last_Osimps,axiom,
! [Xs: list_a,X: a] :
( ( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= ( last_a @ Xs ) ) ) ) ).
% last.simps
thf(fact_314_last_Osimps,axiom,
! [Xs: list_nat,X: nat] :
( ( ( Xs = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs ) )
= ( last_nat @ Xs ) ) ) ) ).
% last.simps
thf(fact_315_last__ConsL,axiom,
! [Xs: list_a,X: a] :
( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_316_last__ConsL,axiom,
! [Xs: list_nat,X: nat] :
( ( Xs = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_317_last__ConsR,axiom,
! [Xs: list_a,X: a] :
( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_318_last__ConsR,axiom,
! [Xs: list_nat,X: nat] :
( ( Xs != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs ) )
= ( last_nat @ Xs ) ) ) ).
% last_ConsR
thf(fact_319_last__append,axiom,
! [Ys: list_a,Xs: list_a] :
( ( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Xs ) ) )
& ( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Ys ) ) ) ) ).
% last_append
thf(fact_320_last__append,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( ( Ys = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
= ( last_nat @ Xs ) ) )
& ( ( Ys != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
= ( last_nat @ Ys ) ) ) ) ).
% last_append
thf(fact_321_longest__common__suffix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ss: list_a,Xs3: list_a,Ys5: list_a] :
( ( Xs
= ( append_a @ Xs3 @ Ss ) )
& ( Ys
= ( append_a @ Ys5 @ Ss ) )
& ( ( Xs3 = nil_a )
| ( Ys5 = nil_a )
| ( ( last_a @ Xs3 )
!= ( last_a @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_322_longest__common__suffix,axiom,
! [Xs: list_nat,Ys: list_nat] :
? [Ss: list_nat,Xs3: list_nat,Ys5: list_nat] :
( ( Xs
= ( append_nat @ Xs3 @ Ss ) )
& ( Ys
= ( append_nat @ Ys5 @ Ss ) )
& ( ( Xs3 = nil_nat )
| ( Ys5 = nil_nat )
| ( ( last_nat @ Xs3 )
!= ( last_nat @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_323_butlast_Osimps_I2_J,axiom,
! [Xs: list_a,X: a] :
( ( ( Xs = nil_a )
=> ( ( butlast_a @ ( cons_a @ X @ Xs ) )
= nil_a ) )
& ( ( Xs != nil_a )
=> ( ( butlast_a @ ( cons_a @ X @ Xs ) )
= ( cons_a @ X @ ( butlast_a @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_324_butlast_Osimps_I2_J,axiom,
! [Xs: list_nat,X: nat] :
( ( ( Xs = nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X @ Xs ) )
= nil_nat ) )
& ( ( Xs != nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( butlast_nat @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_325_butlast__append,axiom,
! [Ys: list_a,Xs: list_a] :
( ( ( Ys = nil_a )
=> ( ( butlast_a @ ( append_a @ Xs @ Ys ) )
= ( butlast_a @ Xs ) ) )
& ( ( Ys != nil_a )
=> ( ( butlast_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ Xs @ ( butlast_a @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_326_butlast__append,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( ( Ys = nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs @ Ys ) )
= ( butlast_nat @ Xs ) ) )
& ( ( Ys != nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ Xs @ ( butlast_nat @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_327_ShiftD,axiom,
! [Kl2: list_a,Kl: set_list_a,K: a] :
( ( member_list_a @ Kl2 @ ( bNF_Greatest_Shift_a @ Kl @ K ) )
=> ( member_list_a @ ( cons_a @ K @ Kl2 ) @ Kl ) ) ).
% ShiftD
thf(fact_328_ShiftD,axiom,
! [Kl2: list_nat,Kl: set_list_nat,K: nat] :
( ( member_list_nat @ Kl2 @ ( bNF_Gr1872714664788909425ft_nat @ Kl @ K ) )
=> ( member_list_nat @ ( cons_nat @ K @ Kl2 ) @ Kl ) ) ).
% ShiftD
thf(fact_329_n__lists_Osimps_I2_J,axiom,
! [N: nat,Xs: list_a] :
( ( n_lists_a @ ( suc @ N ) @ Xs )
= ( concat_list_a
@ ( map_li5729356230488778442list_a
@ ^ [Ys2: list_a] :
( map_a_list_a
@ ^ [Y2: a] : ( cons_a @ Y2 @ Ys2 )
@ Xs )
@ ( n_lists_a @ N @ Xs ) ) ) ) ).
% n_lists.simps(2)
thf(fact_330_n__lists_Osimps_I2_J,axiom,
! [N: nat,Xs: list_nat] :
( ( n_lists_nat @ ( suc @ N ) @ Xs )
= ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ^ [Ys2: list_nat] :
( map_nat_list_nat
@ ^ [Y2: nat] : ( cons_nat @ Y2 @ Ys2 )
@ Xs )
@ ( n_lists_nat @ N @ Xs ) ) ) ) ).
% n_lists.simps(2)
thf(fact_331_map__rec,axiom,
( map_nat_nat
= ( ^ [F2: nat > nat] :
( rec_li7516600145284979816at_nat @ nil_nat
@ ^ [X4: nat,Uu: list_nat] : ( cons_nat @ ( F2 @ X4 ) ) ) ) ) ).
% map_rec
thf(fact_332_suffixes__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( suffixes_a @ ( append_a @ Xs @ Ys ) )
= ( append_list_a @ ( suffixes_a @ Ys )
@ ( map_list_a_list_a
@ ^ [Xs5: list_a] : ( append_a @ Xs5 @ Ys )
@ ( tl_list_a @ ( suffixes_a @ Xs ) ) ) ) ) ).
% suffixes_append
thf(fact_333_suffixes__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( suffixes_nat @ ( append_nat @ Xs @ Ys ) )
= ( append_list_nat @ ( suffixes_nat @ Ys )
@ ( map_li7225945977422193158st_nat
@ ^ [Xs5: list_nat] : ( append_nat @ Xs5 @ Ys )
@ ( tl_list_nat @ ( suffixes_nat @ Xs ) ) ) ) ) ).
% suffixes_append
thf(fact_334_prefixes__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( prefixes_a @ ( append_a @ Xs @ Ys ) )
= ( append_list_a @ ( prefixes_a @ Xs ) @ ( map_list_a_list_a @ ( append_a @ Xs ) @ ( tl_list_a @ ( prefixes_a @ Ys ) ) ) ) ) ).
% prefixes_append
thf(fact_335_prefixes__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( prefixes_nat @ ( append_nat @ Xs @ Ys ) )
= ( append_list_nat @ ( prefixes_nat @ Xs ) @ ( map_li7225945977422193158st_nat @ ( append_nat @ Xs ) @ ( tl_list_nat @ ( prefixes_nat @ Ys ) ) ) ) ) ).
% prefixes_append
thf(fact_336_map__eq__map__tailrec,axiom,
map_nat_nat = map_tailrec_nat_nat ).
% map_eq_map_tailrec
thf(fact_337_product__lists_Osimps_I2_J,axiom,
! [Xs: list_a,Xss2: list_list_a] :
( ( product_lists_a @ ( cons_list_a @ Xs @ Xss2 ) )
= ( concat_list_a
@ ( map_a_list_list_a
@ ^ [X4: a] : ( map_list_a_list_a @ ( cons_a @ X4 ) @ ( product_lists_a @ Xss2 ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_338_product__lists_Osimps_I2_J,axiom,
! [Xs: list_nat,Xss2: list_list_nat] :
( ( product_lists_nat @ ( cons_list_nat @ Xs @ Xss2 ) )
= ( concat_list_nat
@ ( map_na6205611841492582150st_nat
@ ^ [X4: nat] : ( map_li7225945977422193158st_nat @ ( cons_nat @ X4 ) @ ( product_lists_nat @ Xss2 ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_339_list__ex1__simps_I2_J,axiom,
! [P: a > $o,X: a,Xs: list_a] :
( ( list_ex1_a @ P @ ( cons_a @ X @ Xs ) )
= ( ( ( P @ X )
=> ( list_all_a
@ ^ [Y2: a] :
( ~ ( P @ Y2 )
| ( X = Y2 ) )
@ Xs ) )
& ( ~ ( P @ X )
=> ( list_ex1_a @ P @ Xs ) ) ) ) ).
% list_ex1_simps(2)
thf(fact_340_list__ex1__simps_I2_J,axiom,
! [P: nat > $o,X: nat,Xs: list_nat] :
( ( list_ex1_nat @ P @ ( cons_nat @ X @ Xs ) )
= ( ( ( P @ X )
=> ( list_all_nat
@ ^ [Y2: nat] :
( ~ ( P @ Y2 )
| ( X = Y2 ) )
@ Xs ) )
& ( ~ ( P @ X )
=> ( list_ex1_nat @ P @ Xs ) ) ) ) ).
% list_ex1_simps(2)
thf(fact_341_list__update_Osimps_I2_J,axiom,
! [X: a,Xs: list_a,I: nat,V: a] :
( ( list_update_a @ ( cons_a @ X @ Xs ) @ I @ V )
= ( case_nat_list_a @ ( cons_a @ V @ Xs )
@ ^ [J: nat] : ( cons_a @ X @ ( list_update_a @ Xs @ J @ V ) )
@ I ) ) ).
% list_update.simps(2)
thf(fact_342_list__update_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat,I: nat,V: nat] :
( ( list_update_nat @ ( cons_nat @ X @ Xs ) @ I @ V )
= ( case_nat_list_nat @ ( cons_nat @ V @ Xs )
@ ^ [J: nat] : ( cons_nat @ X @ ( list_update_nat @ Xs @ J @ V ) )
@ I ) ) ).
% list_update.simps(2)
thf(fact_343_list_Opred__inject_I2_J,axiom,
! [P: a > $o,A: a,Aa: list_a] :
( ( list_all_a @ P @ ( cons_a @ A @ Aa ) )
= ( ( P @ A )
& ( list_all_a @ P @ Aa ) ) ) ).
% list.pred_inject(2)
thf(fact_344_list_Opred__inject_I2_J,axiom,
! [P: nat > $o,A: nat,Aa: list_nat] :
( ( list_all_nat @ P @ ( cons_nat @ A @ Aa ) )
= ( ( P @ A )
& ( list_all_nat @ P @ Aa ) ) ) ).
% list.pred_inject(2)
thf(fact_345_list__all__simps_I1_J,axiom,
! [P: a > $o,X: a,Xs: list_a] :
( ( list_all_a @ P @ ( cons_a @ X @ Xs ) )
= ( ( P @ X )
& ( list_all_a @ P @ Xs ) ) ) ).
% list_all_simps(1)
thf(fact_346_list__all__simps_I1_J,axiom,
! [P: nat > $o,X: nat,Xs: list_nat] :
( ( list_all_nat @ P @ ( cons_nat @ X @ Xs ) )
= ( ( P @ X )
& ( list_all_nat @ P @ Xs ) ) ) ).
% list_all_simps(1)
thf(fact_347_list__all__simps_I2_J,axiom,
! [P: a > $o] : ( list_all_a @ P @ nil_a ) ).
% list_all_simps(2)
thf(fact_348_list__all__simps_I2_J,axiom,
! [P: nat > $o] : ( list_all_nat @ P @ nil_nat ) ).
% list_all_simps(2)
thf(fact_349_list__all__append,axiom,
! [P: a > $o,Xs: list_a,Ys: list_a] :
( ( list_all_a @ P @ ( append_a @ Xs @ Ys ) )
= ( ( list_all_a @ P @ Xs )
& ( list_all_a @ P @ Ys ) ) ) ).
% list_all_append
thf(fact_350_list__all__append,axiom,
! [P: nat > $o,Xs: list_nat,Ys: list_nat] :
( ( list_all_nat @ P @ ( append_nat @ Xs @ Ys ) )
= ( ( list_all_nat @ P @ Xs )
& ( list_all_nat @ P @ Ys ) ) ) ).
% list_all_append
thf(fact_351_tl__append2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_352_tl__append2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != nil_nat )
=> ( ( tl_nat @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( tl_nat @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_353_tl__def,axiom,
( tl_a
= ( case_list_list_a_a @ nil_a
@ ^ [X213: a,X223: list_a] : X223 ) ) ).
% tl_def
thf(fact_354_tl__def,axiom,
( tl_nat
= ( case_l2340614614379431832at_nat @ nil_nat
@ ^ [X213: nat,X223: list_nat] : X223 ) ) ).
% tl_def
thf(fact_355_tl__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( case_list_list_a_a @ ( tl_a @ Ys )
@ ^ [Z2: a,Zs3: list_a] : ( append_a @ Zs3 @ Ys )
@ Xs ) ) ).
% tl_append
thf(fact_356_tl__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( tl_nat @ ( append_nat @ Xs @ Ys ) )
= ( case_l2340614614379431832at_nat @ ( tl_nat @ Ys )
@ ^ [Z2: nat,Zs3: list_nat] : ( append_nat @ Zs3 @ Ys )
@ Xs ) ) ).
% tl_append
thf(fact_357_nat_Ocase__distrib,axiom,
! [H: $o > $o,F1: $o,F22: nat > $o,Nat: nat] :
( ( H @ ( case_nat_o @ F1 @ F22 @ Nat ) )
= ( case_nat_o @ ( H @ F1 )
@ ^ [X4: nat] : ( H @ ( F22 @ X4 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_358_nat_Ocase__distrib,axiom,
! [H: $o > nat,F1: $o,F22: nat > $o,Nat: nat] :
( ( H @ ( case_nat_o @ F1 @ F22 @ Nat ) )
= ( case_nat_nat @ ( H @ F1 )
@ ^ [X4: nat] : ( H @ ( F22 @ X4 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_359_nat_Ocase__distrib,axiom,
! [H: nat > $o,F1: nat,F22: nat > nat,Nat: nat] :
( ( H @ ( case_nat_nat @ F1 @ F22 @ Nat ) )
= ( case_nat_o @ ( H @ F1 )
@ ^ [X4: nat] : ( H @ ( F22 @ X4 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_360_nat_Ocase__distrib,axiom,
! [H: nat > nat,F1: nat,F22: nat > nat,Nat: nat] :
( ( H @ ( case_nat_nat @ F1 @ F22 @ Nat ) )
= ( case_nat_nat @ ( H @ F1 )
@ ^ [X4: nat] : ( H @ ( F22 @ X4 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_361_list_Osel_I3_J,axiom,
! [X21: a,X22: list_a] :
( ( tl_a @ ( cons_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_362_list_Osel_I3_J,axiom,
! [X21: nat,X22: list_nat] :
( ( tl_nat @ ( cons_nat @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_363_list_Osel_I2_J,axiom,
( ( tl_a @ nil_a )
= nil_a ) ).
% list.sel(2)
thf(fact_364_list_Osel_I2_J,axiom,
( ( tl_nat @ nil_nat )
= nil_nat ) ).
% list.sel(2)
thf(fact_365_map__tl,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( map_nat_nat @ F @ ( tl_nat @ Xs ) )
= ( tl_nat @ ( map_nat_nat @ F @ Xs ) ) ) ).
% map_tl
thf(fact_366_list_Opred__inject_I1_J,axiom,
! [P: a > $o] : ( list_all_a @ P @ nil_a ) ).
% list.pred_inject(1)
thf(fact_367_list_Opred__inject_I1_J,axiom,
! [P: nat > $o] : ( list_all_nat @ P @ nil_nat ) ).
% list.pred_inject(1)
thf(fact_368_butlast__tl,axiom,
! [Xs: list_nat] :
( ( butlast_nat @ ( tl_nat @ Xs ) )
= ( tl_nat @ ( butlast_nat @ Xs ) ) ) ).
% butlast_tl
thf(fact_369_list_Omap__cong__pred,axiom,
! [X: list_nat,Ya: list_nat,F: nat > nat,G: nat > nat] :
( ( X = Ya )
=> ( ( list_all_nat
@ ^ [Z2: nat] :
( ( F @ Z2 )
= ( G @ Z2 ) )
@ Ya )
=> ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ G @ Ya ) ) ) ) ).
% list.map_cong_pred
thf(fact_370_old_Onat_Osimps_I5_J,axiom,
! [F1: $o,F22: nat > $o,X23: nat] :
( ( case_nat_o @ F1 @ F22 @ ( suc @ X23 ) )
= ( F22 @ X23 ) ) ).
% old.nat.simps(5)
thf(fact_371_old_Onat_Osimps_I5_J,axiom,
! [F1: nat,F22: nat > nat,X23: nat] :
( ( case_nat_nat @ F1 @ F22 @ ( suc @ X23 ) )
= ( F22 @ X23 ) ) ).
% old.nat.simps(5)
thf(fact_372_tl__Nil,axiom,
! [Xs: list_a] :
( ( ( tl_a @ Xs )
= nil_a )
= ( ( Xs = nil_a )
| ? [X4: a] :
( Xs
= ( cons_a @ X4 @ nil_a ) ) ) ) ).
% tl_Nil
thf(fact_373_tl__Nil,axiom,
! [Xs: list_nat] :
( ( ( tl_nat @ Xs )
= nil_nat )
= ( ( Xs = nil_nat )
| ? [X4: nat] :
( Xs
= ( cons_nat @ X4 @ nil_nat ) ) ) ) ).
% tl_Nil
thf(fact_374_Nil__tl,axiom,
! [Xs: list_a] :
( ( nil_a
= ( tl_a @ Xs ) )
= ( ( Xs = nil_a )
| ? [X4: a] :
( Xs
= ( cons_a @ X4 @ nil_a ) ) ) ) ).
% Nil_tl
thf(fact_375_Nil__tl,axiom,
! [Xs: list_nat] :
( ( nil_nat
= ( tl_nat @ Xs ) )
= ( ( Xs = nil_nat )
| ? [X4: nat] :
( Xs
= ( cons_nat @ X4 @ nil_nat ) ) ) ) ).
% Nil_tl
thf(fact_376_tl__append__if,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( tl_a @ Ys ) ) )
& ( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_377_tl__append__if,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( Xs = nil_nat )
=> ( ( tl_nat @ ( append_nat @ Xs @ Ys ) )
= ( tl_nat @ Ys ) ) )
& ( ( Xs != nil_nat )
=> ( ( tl_nat @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( tl_nat @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_378_list_Omap__sel_I2_J,axiom,
! [A: list_a,F: a > nat] :
( ( A != nil_a )
=> ( ( tl_nat @ ( map_a_nat @ F @ A ) )
= ( map_a_nat @ F @ ( tl_a @ A ) ) ) ) ).
% list.map_sel(2)
thf(fact_379_list_Omap__sel_I2_J,axiom,
! [A: list_nat,F: nat > nat] :
( ( A != nil_nat )
=> ( ( tl_nat @ ( map_nat_nat @ F @ A ) )
= ( map_nat_nat @ F @ ( tl_nat @ A ) ) ) ) ).
% list.map_sel(2)
thf(fact_380_last__tl,axiom,
! [Xs: list_a] :
( ( ( Xs = nil_a )
| ( ( tl_a @ Xs )
!= nil_a ) )
=> ( ( last_a @ ( tl_a @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_tl
thf(fact_381_last__tl,axiom,
! [Xs: list_nat] :
( ( ( Xs = nil_nat )
| ( ( tl_nat @ Xs )
!= nil_nat ) )
=> ( ( last_nat @ ( tl_nat @ Xs ) )
= ( last_nat @ Xs ) ) ) ).
% last_tl
thf(fact_382_transpose_Osimps_I3_J,axiom,
! [X: a,Xs: list_a,Xss2: list_list_a] :
( ( transpose_a @ ( cons_list_a @ ( cons_a @ X @ Xs ) @ Xss2 ) )
= ( cons_list_a
@ ( cons_a @ X
@ ( concat_a
@ ( map_list_a_list_a
@ ( case_list_list_a_a @ nil_a
@ ^ [H2: a,T2: list_a] : ( cons_a @ H2 @ nil_a ) )
@ Xss2 ) ) )
@ ( transpose_a
@ ( cons_list_a @ Xs
@ ( concat_list_a
@ ( map_li5729356230488778442list_a
@ ( case_l8408404631611421914st_a_a @ nil_list_a
@ ^ [H2: a,T2: list_a] : ( cons_list_a @ T2 @ nil_list_a ) )
@ Xss2 ) ) ) ) ) ) ).
% transpose.simps(3)
thf(fact_383_transpose_Osimps_I3_J,axiom,
! [X: nat,Xs: list_nat,Xss2: list_list_nat] :
( ( transpose_nat @ ( cons_list_nat @ ( cons_nat @ X @ Xs ) @ Xss2 ) )
= ( cons_list_nat
@ ( cons_nat @ X
@ ( concat_nat
@ ( map_li7225945977422193158st_nat
@ ( case_l2340614614379431832at_nat @ nil_nat
@ ^ [H2: nat,T2: list_nat] : ( cons_nat @ H2 @ nil_nat ) )
@ Xss2 ) ) )
@ ( transpose_nat
@ ( cons_list_nat @ Xs
@ ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ( case_l3331202209248957608at_nat @ nil_list_nat
@ ^ [H2: nat,T2: list_nat] : ( cons_list_nat @ T2 @ nil_list_nat ) )
@ Xss2 ) ) ) ) ) ) ).
% transpose.simps(3)
thf(fact_384_transpose_Oelims,axiom,
! [X: list_list_a,Y: list_list_a] :
( ( ( transpose_a @ X )
= Y )
=> ( ( ( X = nil_list_a )
=> ( Y != nil_list_a ) )
=> ( ! [Xss: list_list_a] :
( ( X
= ( cons_list_a @ nil_a @ Xss ) )
=> ( Y
!= ( transpose_a @ Xss ) ) )
=> ~ ! [X3: a,Xs2: list_a,Xss: list_list_a] :
( ( X
= ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ Xss ) )
=> ( Y
!= ( cons_list_a
@ ( cons_a @ X3
@ ( concat_a
@ ( map_list_a_list_a
@ ( case_list_list_a_a @ nil_a
@ ^ [H2: a,T2: list_a] : ( cons_a @ H2 @ nil_a ) )
@ Xss ) ) )
@ ( transpose_a
@ ( cons_list_a @ Xs2
@ ( concat_list_a
@ ( map_li5729356230488778442list_a
@ ( case_l8408404631611421914st_a_a @ nil_list_a
@ ^ [H2: a,T2: list_a] : ( cons_list_a @ T2 @ nil_list_a ) )
@ Xss ) ) ) ) ) ) ) ) ) ) ).
% transpose.elims
thf(fact_385_transpose_Oelims,axiom,
! [X: list_list_nat,Y: list_list_nat] :
( ( ( transpose_nat @ X )
= Y )
=> ( ( ( X = nil_list_nat )
=> ( Y != nil_list_nat ) )
=> ( ! [Xss: list_list_nat] :
( ( X
= ( cons_list_nat @ nil_nat @ Xss ) )
=> ( Y
!= ( transpose_nat @ Xss ) ) )
=> ~ ! [X3: nat,Xs2: list_nat,Xss: list_list_nat] :
( ( X
= ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ Xss ) )
=> ( Y
!= ( cons_list_nat
@ ( cons_nat @ X3
@ ( concat_nat
@ ( map_li7225945977422193158st_nat
@ ( case_l2340614614379431832at_nat @ nil_nat
@ ^ [H2: nat,T2: list_nat] : ( cons_nat @ H2 @ nil_nat ) )
@ Xss ) ) )
@ ( transpose_nat
@ ( cons_list_nat @ Xs2
@ ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ( case_l3331202209248957608at_nat @ nil_list_nat
@ ^ [H2: nat,T2: list_nat] : ( cons_list_nat @ T2 @ nil_list_nat ) )
@ Xss ) ) ) ) ) ) ) ) ) ) ).
% transpose.elims
thf(fact_386_transpose__aux__filter__tail,axiom,
! [Xss2: list_list_a] :
( ( concat_list_a
@ ( map_li5729356230488778442list_a
@ ( case_l8408404631611421914st_a_a @ nil_list_a
@ ^ [H2: a,T2: list_a] : ( cons_list_a @ T2 @ nil_list_a ) )
@ Xss2 ) )
= ( map_list_a_list_a @ tl_a
@ ( filter_list_a
@ ^ [Ys2: list_a] : ( Ys2 != nil_a )
@ Xss2 ) ) ) ).
% transpose_aux_filter_tail
thf(fact_387_transpose__aux__filter__tail,axiom,
! [Xss2: list_list_nat] :
( ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ( case_l3331202209248957608at_nat @ nil_list_nat
@ ^ [H2: nat,T2: list_nat] : ( cons_list_nat @ T2 @ nil_list_nat ) )
@ Xss2 ) )
= ( map_li7225945977422193158st_nat @ tl_nat
@ ( filter_list_nat
@ ^ [Ys2: list_nat] : ( Ys2 != nil_nat )
@ Xss2 ) ) ) ).
% transpose_aux_filter_tail
thf(fact_388_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_a @ N @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_a @ N @ nil_a )
= nil_list_a ) ) ) ).
% n_lists_Nil
thf(fact_389_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= nil_list_nat ) ) ) ).
% n_lists_Nil
thf(fact_390_transpose_Opinduct,axiom,
! [A0: list_list_a,P: list_list_a > $o] :
( ( accp_list_list_a @ transpose_rel_a @ A0 )
=> ( ( ( accp_list_list_a @ transpose_rel_a @ nil_list_a )
=> ( P @ nil_list_a ) )
=> ( ! [Xss: list_list_a] :
( ( accp_list_list_a @ transpose_rel_a @ ( cons_list_a @ nil_a @ Xss ) )
=> ( ( P @ Xss )
=> ( P @ ( cons_list_a @ nil_a @ Xss ) ) ) )
=> ( ! [X3: a,Xs2: list_a,Xss: list_list_a] :
( ( accp_list_list_a @ transpose_rel_a @ ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ Xss ) )
=> ( ( P
@ ( cons_list_a @ Xs2
@ ( concat_list_a
@ ( map_li5729356230488778442list_a
@ ( case_l8408404631611421914st_a_a @ nil_list_a
@ ^ [H2: a,T2: list_a] : ( cons_list_a @ T2 @ nil_list_a ) )
@ Xss ) ) ) )
=> ( P @ ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ Xss ) ) ) )
=> ( P @ A0 ) ) ) ) ) ).
% transpose.pinduct
thf(fact_391_transpose_Opinduct,axiom,
! [A0: list_list_nat,P: list_list_nat > $o] :
( ( accp_list_list_nat @ transpose_rel_nat @ A0 )
=> ( ( ( accp_list_list_nat @ transpose_rel_nat @ nil_list_nat )
=> ( P @ nil_list_nat ) )
=> ( ! [Xss: list_list_nat] :
( ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ nil_nat @ Xss ) )
=> ( ( P @ Xss )
=> ( P @ ( cons_list_nat @ nil_nat @ Xss ) ) ) )
=> ( ! [X3: nat,Xs2: list_nat,Xss: list_list_nat] :
( ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ Xss ) )
=> ( ( P
@ ( cons_list_nat @ Xs2
@ ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ( case_l3331202209248957608at_nat @ nil_list_nat
@ ^ [H2: nat,T2: list_nat] : ( cons_list_nat @ T2 @ nil_list_nat ) )
@ Xss ) ) ) )
=> ( P @ ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ Xss ) ) ) )
=> ( P @ A0 ) ) ) ) ) ).
% transpose.pinduct
thf(fact_392_rotate1__hd__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( rotate1_a @ Xs )
= ( append_a @ ( tl_a @ Xs ) @ ( cons_a @ ( hd_a @ Xs ) @ nil_a ) ) ) ) ).
% rotate1_hd_tl
thf(fact_393_rotate1__hd__tl,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( rotate1_nat @ Xs )
= ( append_nat @ ( tl_nat @ Xs ) @ ( cons_nat @ ( hd_nat @ Xs ) @ nil_nat ) ) ) ) ).
% rotate1_hd_tl
thf(fact_394_n__lists_Osimps_I1_J,axiom,
! [Xs: list_a] :
( ( n_lists_a @ zero_zero_nat @ Xs )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% n_lists.simps(1)
thf(fact_395_n__lists_Osimps_I1_J,axiom,
! [Xs: list_nat] :
( ( n_lists_nat @ zero_zero_nat @ Xs )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% n_lists.simps(1)
thf(fact_396_transpose_Opelims,axiom,
! [X: list_list_a,Y: list_list_a] :
( ( ( transpose_a @ X )
= Y )
=> ( ( accp_list_list_a @ transpose_rel_a @ X )
=> ( ( ( X = nil_list_a )
=> ( ( Y = nil_list_a )
=> ~ ( accp_list_list_a @ transpose_rel_a @ nil_list_a ) ) )
=> ( ! [Xss: list_list_a] :
( ( X
= ( cons_list_a @ nil_a @ Xss ) )
=> ( ( Y
= ( transpose_a @ Xss ) )
=> ~ ( accp_list_list_a @ transpose_rel_a @ ( cons_list_a @ nil_a @ Xss ) ) ) )
=> ~ ! [X3: a,Xs2: list_a,Xss: list_list_a] :
( ( X
= ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ Xss ) )
=> ( ( Y
= ( cons_list_a
@ ( cons_a @ X3
@ ( concat_a
@ ( map_list_a_list_a
@ ( case_list_list_a_a @ nil_a
@ ^ [H2: a,T2: list_a] : ( cons_a @ H2 @ nil_a ) )
@ Xss ) ) )
@ ( transpose_a
@ ( cons_list_a @ Xs2
@ ( concat_list_a
@ ( map_li5729356230488778442list_a
@ ( case_l8408404631611421914st_a_a @ nil_list_a
@ ^ [H2: a,T2: list_a] : ( cons_list_a @ T2 @ nil_list_a ) )
@ Xss ) ) ) ) ) )
=> ~ ( accp_list_list_a @ transpose_rel_a @ ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ Xss ) ) ) ) ) ) ) ) ).
% transpose.pelims
thf(fact_397_transpose_Opelims,axiom,
! [X: list_list_nat,Y: list_list_nat] :
( ( ( transpose_nat @ X )
= Y )
=> ( ( accp_list_list_nat @ transpose_rel_nat @ X )
=> ( ( ( X = nil_list_nat )
=> ( ( Y = nil_list_nat )
=> ~ ( accp_list_list_nat @ transpose_rel_nat @ nil_list_nat ) ) )
=> ( ! [Xss: list_list_nat] :
( ( X
= ( cons_list_nat @ nil_nat @ Xss ) )
=> ( ( Y
= ( transpose_nat @ Xss ) )
=> ~ ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ nil_nat @ Xss ) ) ) )
=> ~ ! [X3: nat,Xs2: list_nat,Xss: list_list_nat] :
( ( X
= ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ Xss ) )
=> ( ( Y
= ( cons_list_nat
@ ( cons_nat @ X3
@ ( concat_nat
@ ( map_li7225945977422193158st_nat
@ ( case_l2340614614379431832at_nat @ nil_nat
@ ^ [H2: nat,T2: list_nat] : ( cons_nat @ H2 @ nil_nat ) )
@ Xss ) ) )
@ ( transpose_nat
@ ( cons_list_nat @ Xs2
@ ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ( case_l3331202209248957608at_nat @ nil_list_nat
@ ^ [H2: nat,T2: list_nat] : ( cons_list_nat @ T2 @ nil_list_nat ) )
@ Xss ) ) ) ) ) )
=> ~ ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ Xss ) ) ) ) ) ) ) ) ).
% transpose.pelims
thf(fact_398_filter__append,axiom,
! [P: a > $o,Xs: list_a,Ys: list_a] :
( ( filter_a @ P @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( filter_a @ P @ Xs ) @ ( filter_a @ P @ Ys ) ) ) ).
% filter_append
thf(fact_399_filter__append,axiom,
! [P: nat > $o,Xs: list_nat,Ys: list_nat] :
( ( filter_nat @ P @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( filter_nat @ P @ Xs ) @ ( filter_nat @ P @ Ys ) ) ) ).
% filter_append
thf(fact_400_hd__prefixes,axiom,
! [Xs: list_a] :
( ( hd_list_a @ ( prefixes_a @ Xs ) )
= nil_a ) ).
% hd_prefixes
thf(fact_401_hd__prefixes,axiom,
! [Xs: list_nat] :
( ( hd_list_nat @ ( prefixes_nat @ Xs ) )
= nil_nat ) ).
% hd_prefixes
thf(fact_402_hd__suffixes,axiom,
! [Xs: list_a] :
( ( hd_list_a @ ( suffixes_a @ Xs ) )
= nil_a ) ).
% hd_suffixes
thf(fact_403_hd__suffixes,axiom,
! [Xs: list_nat] :
( ( hd_list_nat @ ( suffixes_nat @ Xs ) )
= nil_nat ) ).
% hd_suffixes
thf(fact_404_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_405_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_406_hd__append2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ).
% hd_append2
thf(fact_407_hd__append2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs @ Ys ) )
= ( hd_nat @ Xs ) ) ) ).
% hd_append2
thf(fact_408_list_Ocollapse,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_409_list_Ocollapse,axiom,
! [List: list_nat] :
( ( List != nil_nat )
=> ( ( cons_nat @ ( hd_nat @ List ) @ ( tl_nat @ List ) )
= List ) ) ).
% list.collapse
thf(fact_410_hd__Cons__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs ) @ ( tl_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_411_hd__Cons__tl,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( cons_nat @ ( hd_nat @ Xs ) @ ( tl_nat @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_412_nat_Odisc__eq__case_I2_J,axiom,
! [Nat: nat] :
( ( Nat != zero_zero_nat )
= ( case_nat_o @ $false
@ ^ [Uu: nat] : $true
@ Nat ) ) ).
% nat.disc_eq_case(2)
thf(fact_413_nat_Odisc__eq__case_I1_J,axiom,
! [Nat: nat] :
( ( Nat = zero_zero_nat )
= ( case_nat_o @ $true
@ ^ [Uu: nat] : $false
@ Nat ) ) ).
% nat.disc_eq_case(1)
thf(fact_414_transpose_Opsimps_I2_J,axiom,
! [Xss2: list_list_a] :
( ( accp_list_list_a @ transpose_rel_a @ ( cons_list_a @ nil_a @ Xss2 ) )
=> ( ( transpose_a @ ( cons_list_a @ nil_a @ Xss2 ) )
= ( transpose_a @ Xss2 ) ) ) ).
% transpose.psimps(2)
thf(fact_415_transpose_Opsimps_I2_J,axiom,
! [Xss2: list_list_nat] :
( ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ nil_nat @ Xss2 ) )
=> ( ( transpose_nat @ ( cons_list_nat @ nil_nat @ Xss2 ) )
= ( transpose_nat @ Xss2 ) ) ) ).
% transpose.psimps(2)
thf(fact_416_filter_Osimps_I2_J,axiom,
! [P: a > $o,X: a,Xs: list_a] :
( ( ( P @ X )
=> ( ( filter_a @ P @ ( cons_a @ X @ Xs ) )
= ( cons_a @ X @ ( filter_a @ P @ Xs ) ) ) )
& ( ~ ( P @ X )
=> ( ( filter_a @ P @ ( cons_a @ X @ Xs ) )
= ( filter_a @ P @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_417_filter_Osimps_I2_J,axiom,
! [P: nat > $o,X: nat,Xs: list_nat] :
( ( ( P @ X )
=> ( ( filter_nat @ P @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( filter_nat @ P @ Xs ) ) ) )
& ( ~ ( P @ X )
=> ( ( filter_nat @ P @ ( cons_nat @ X @ Xs ) )
= ( filter_nat @ P @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_418_filter_Osimps_I1_J,axiom,
! [P: a > $o] :
( ( filter_a @ P @ nil_a )
= nil_a ) ).
% filter.simps(1)
thf(fact_419_filter_Osimps_I1_J,axiom,
! [P: nat > $o] :
( ( filter_nat @ P @ nil_nat )
= nil_nat ) ).
% filter.simps(1)
thf(fact_420_list_Osel_I1_J,axiom,
! [X21: a,X22: list_a] :
( ( hd_a @ ( cons_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_421_list_Osel_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( ( hd_nat @ ( cons_nat @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_422_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M: nat] :
( N
= ( suc @ M ) ) ) ).
% not0_implies_Suc
thf(fact_423_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_424_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_425_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_426_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_427_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_428_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_429_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_430_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_431_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_432_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_433_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_434_encode__unary__nat_Ocases,axiom,
! [X: nat] :
( ! [L: nat] :
( X
!= ( suc @ L ) )
=> ( X = zero_zero_nat ) ) ).
% encode_unary_nat.cases
thf(fact_435_old_Onat_Osimps_I4_J,axiom,
! [F1: $o,F22: nat > $o] :
( ( case_nat_o @ F1 @ F22 @ zero_zero_nat )
= F1 ) ).
% old.nat.simps(4)
thf(fact_436_old_Onat_Osimps_I4_J,axiom,
! [F1: nat,F22: nat > nat] :
( ( case_nat_nat @ F1 @ F22 @ zero_zero_nat )
= F1 ) ).
% old.nat.simps(4)
thf(fact_437_hd__concat,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( ( hd_list_a @ Xs )
!= nil_a )
=> ( ( hd_a @ ( concat_a @ Xs ) )
= ( hd_a @ ( hd_list_a @ Xs ) ) ) ) ) ).
% hd_concat
thf(fact_438_hd__concat,axiom,
! [Xs: list_list_nat] :
( ( Xs != nil_list_nat )
=> ( ( ( hd_list_nat @ Xs )
!= nil_nat )
=> ( ( hd_nat @ ( concat_nat @ Xs ) )
= ( hd_nat @ ( hd_list_nat @ Xs ) ) ) ) ) ).
% hd_concat
thf(fact_439_list_Odisc__eq__case_I1_J,axiom,
! [List: list_a] :
( ( List = nil_a )
= ( case_list_o_a @ $true
@ ^ [Uu: a,Uv: list_a] : $false
@ List ) ) ).
% list.disc_eq_case(1)
thf(fact_440_list_Odisc__eq__case_I1_J,axiom,
! [List: list_nat] :
( ( List = nil_nat )
= ( case_list_o_nat @ $true
@ ^ [Uu: nat,Uv: list_nat] : $false
@ List ) ) ).
% list.disc_eq_case(1)
thf(fact_441_list_Odisc__eq__case_I2_J,axiom,
! [List: list_a] :
( ( List != nil_a )
= ( case_list_o_a @ $false
@ ^ [Uu: a,Uv: list_a] : $true
@ List ) ) ).
% list.disc_eq_case(2)
thf(fact_442_list_Odisc__eq__case_I2_J,axiom,
! [List: list_nat] :
( ( List != nil_nat )
= ( case_list_o_nat @ $false
@ ^ [Uu: nat,Uv: list_nat] : $true
@ List ) ) ).
% list.disc_eq_case(2)
thf(fact_443_hd__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Ys ) ) )
& ( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ) ).
% hd_append
thf(fact_444_hd__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( Xs = nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs @ Ys ) )
= ( hd_nat @ Ys ) ) )
& ( ( Xs != nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs @ Ys ) )
= ( hd_nat @ Xs ) ) ) ) ).
% hd_append
thf(fact_445_longest__common__prefix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ps: list_a,Xs3: list_a,Ys5: list_a] :
( ( Xs
= ( append_a @ Ps @ Xs3 ) )
& ( Ys
= ( append_a @ Ps @ Ys5 ) )
& ( ( Xs3 = nil_a )
| ( Ys5 = nil_a )
| ( ( hd_a @ Xs3 )
!= ( hd_a @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_446_longest__common__prefix,axiom,
! [Xs: list_nat,Ys: list_nat] :
? [Ps: list_nat,Xs3: list_nat,Ys5: list_nat] :
( ( Xs
= ( append_nat @ Ps @ Xs3 ) )
& ( Ys
= ( append_nat @ Ps @ Ys5 ) )
& ( ( Xs3 = nil_nat )
| ( Ys5 = nil_nat )
| ( ( hd_nat @ Xs3 )
!= ( hd_nat @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_447_list_Omap__sel_I1_J,axiom,
! [A: list_a,F: a > nat] :
( ( A != nil_a )
=> ( ( hd_nat @ ( map_a_nat @ F @ A ) )
= ( F @ ( hd_a @ A ) ) ) ) ).
% list.map_sel(1)
thf(fact_448_list_Omap__sel_I1_J,axiom,
! [A: list_nat,F: nat > nat] :
( ( A != nil_nat )
=> ( ( hd_nat @ ( map_nat_nat @ F @ A ) )
= ( F @ ( hd_nat @ A ) ) ) ) ).
% list.map_sel(1)
thf(fact_449_hd__map,axiom,
! [Xs: list_a,F: a > nat] :
( ( Xs != nil_a )
=> ( ( hd_nat @ ( map_a_nat @ F @ Xs ) )
= ( F @ ( hd_a @ Xs ) ) ) ) ).
% hd_map
thf(fact_450_hd__map,axiom,
! [Xs: list_nat,F: nat > nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ ( map_nat_nat @ F @ Xs ) )
= ( F @ ( hd_nat @ Xs ) ) ) ) ).
% hd_map
thf(fact_451_list_Oexpand,axiom,
! [List: list_a,List2: list_a] :
( ( ( List = nil_a )
= ( List2 = nil_a ) )
=> ( ( ( List != nil_a )
=> ( ( List2 != nil_a )
=> ( ( ( hd_a @ List )
= ( hd_a @ List2 ) )
& ( ( tl_a @ List )
= ( tl_a @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_452_list_Oexpand,axiom,
! [List: list_nat,List2: list_nat] :
( ( ( List = nil_nat )
= ( List2 = nil_nat ) )
=> ( ( ( List != nil_nat )
=> ( ( List2 != nil_nat )
=> ( ( ( hd_nat @ List )
= ( hd_nat @ List2 ) )
& ( ( tl_nat @ List )
= ( tl_nat @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_453_hd__Nil__eq__last,axiom,
( ( hd_a @ nil_a )
= ( last_a @ nil_a ) ) ).
% hd_Nil_eq_last
thf(fact_454_hd__Nil__eq__last,axiom,
( ( hd_nat @ nil_nat )
= ( last_nat @ nil_nat ) ) ).
% hd_Nil_eq_last
thf(fact_455_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_456_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_457_list__update__code_I2_J,axiom,
! [X: a,Xs: list_a,Y: a] :
( ( list_update_a @ ( cons_a @ X @ Xs ) @ zero_zero_nat @ Y )
= ( cons_a @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_458_list__update__code_I2_J,axiom,
! [X: nat,Xs: list_nat,Y: nat] :
( ( list_update_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat @ Y )
= ( cons_nat @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_459_transpose_Osimps_I2_J,axiom,
! [Xss2: list_list_a] :
( ( transpose_a @ ( cons_list_a @ nil_a @ Xss2 ) )
= ( transpose_a @ Xss2 ) ) ).
% transpose.simps(2)
thf(fact_460_transpose_Osimps_I2_J,axiom,
! [Xss2: list_list_nat] :
( ( transpose_nat @ ( cons_list_nat @ nil_nat @ Xss2 ) )
= ( transpose_nat @ Xss2 ) ) ).
% transpose.simps(2)
thf(fact_461_transpose__map__map,axiom,
! [F: nat > nat,Xs: list_list_nat] :
( ( transpose_nat @ ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ Xs ) )
= ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ ( transpose_nat @ Xs ) ) ) ).
% transpose_map_map
thf(fact_462_length__code,axiom,
( size_size_list_a
= ( gen_length_a @ zero_zero_nat ) ) ).
% length_code
thf(fact_463_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_464_list_Oexhaust__sel,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( List
= ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_465_list_Oexhaust__sel,axiom,
! [List: list_nat] :
( ( List != nil_nat )
=> ( List
= ( cons_nat @ ( hd_nat @ List ) @ ( tl_nat @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_466_transpose__aux__filter__head,axiom,
! [Xss2: list_list_a] :
( ( concat_a
@ ( map_list_a_list_a
@ ( case_list_list_a_a @ nil_a
@ ^ [H2: a,T2: list_a] : ( cons_a @ H2 @ nil_a ) )
@ Xss2 ) )
= ( map_list_a_a @ hd_a
@ ( filter_list_a
@ ^ [Ys2: list_a] : ( Ys2 != nil_a )
@ Xss2 ) ) ) ).
% transpose_aux_filter_head
thf(fact_467_transpose__aux__filter__head,axiom,
! [Xss2: list_list_nat] :
( ( concat_nat
@ ( map_li7225945977422193158st_nat
@ ( case_l2340614614379431832at_nat @ nil_nat
@ ^ [H2: nat,T2: list_nat] : ( cons_nat @ H2 @ nil_nat ) )
@ Xss2 ) )
= ( map_list_nat_nat @ hd_nat
@ ( filter_list_nat
@ ^ [Ys2: list_nat] : ( Ys2 != nil_nat )
@ Xss2 ) ) ) ).
% transpose_aux_filter_head
thf(fact_468_transpose_Opsimps_I3_J,axiom,
! [X: a,Xs: list_a,Xss2: list_list_a] :
( ( accp_list_list_a @ transpose_rel_a @ ( cons_list_a @ ( cons_a @ X @ Xs ) @ Xss2 ) )
=> ( ( transpose_a @ ( cons_list_a @ ( cons_a @ X @ Xs ) @ Xss2 ) )
= ( cons_list_a
@ ( cons_a @ X
@ ( concat_a
@ ( map_list_a_list_a
@ ( case_list_list_a_a @ nil_a
@ ^ [H2: a,T2: list_a] : ( cons_a @ H2 @ nil_a ) )
@ Xss2 ) ) )
@ ( transpose_a
@ ( cons_list_a @ Xs
@ ( concat_list_a
@ ( map_li5729356230488778442list_a
@ ( case_l8408404631611421914st_a_a @ nil_list_a
@ ^ [H2: a,T2: list_a] : ( cons_list_a @ T2 @ nil_list_a ) )
@ Xss2 ) ) ) ) ) ) ) ).
% transpose.psimps(3)
thf(fact_469_transpose_Opsimps_I3_J,axiom,
! [X: nat,Xs: list_nat,Xss2: list_list_nat] :
( ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ ( cons_nat @ X @ Xs ) @ Xss2 ) )
=> ( ( transpose_nat @ ( cons_list_nat @ ( cons_nat @ X @ Xs ) @ Xss2 ) )
= ( cons_list_nat
@ ( cons_nat @ X
@ ( concat_nat
@ ( map_li7225945977422193158st_nat
@ ( case_l2340614614379431832at_nat @ nil_nat
@ ^ [H2: nat,T2: list_nat] : ( cons_nat @ H2 @ nil_nat ) )
@ Xss2 ) ) )
@ ( transpose_nat
@ ( cons_list_nat @ Xs
@ ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ( case_l3331202209248957608at_nat @ nil_list_nat
@ ^ [H2: nat,T2: list_nat] : ( cons_list_nat @ T2 @ nil_list_nat ) )
@ Xss2 ) ) ) ) ) ) ) ).
% transpose.psimps(3)
thf(fact_470_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_471_Nitpick_Osize__list__simp_I2_J,axiom,
( size_size_list_nat
= ( ^ [Xs4: list_nat] : ( if_nat @ ( Xs4 = nil_nat ) @ zero_zero_nat @ ( suc @ ( size_size_list_nat @ ( tl_nat @ Xs4 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_472_nat_Osplit__sels_I1_J,axiom,
! [P: $o > $o,F1: $o,F22: nat > $o,Nat: nat] :
( ( P @ ( case_nat_o @ F1 @ F22 @ Nat ) )
= ( ( ( Nat = zero_zero_nat )
=> ( P @ F1 ) )
& ( ( Nat
= ( suc @ ( pred @ Nat ) ) )
=> ( P @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ).
% nat.split_sels(1)
thf(fact_473_nat_Osplit__sels_I1_J,axiom,
! [P: nat > $o,F1: nat,F22: nat > nat,Nat: nat] :
( ( P @ ( case_nat_nat @ F1 @ F22 @ Nat ) )
= ( ( ( Nat = zero_zero_nat )
=> ( P @ F1 ) )
& ( ( Nat
= ( suc @ ( pred @ Nat ) ) )
=> ( P @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ).
% nat.split_sels(1)
thf(fact_474_nat_Osplit__sels_I2_J,axiom,
! [P: $o > $o,F1: $o,F22: nat > $o,Nat: nat] :
( ( P @ ( case_nat_o @ F1 @ F22 @ Nat ) )
= ( ~ ( ( ( Nat = zero_zero_nat )
& ~ ( P @ F1 ) )
| ( ( Nat
= ( suc @ ( pred @ Nat ) ) )
& ~ ( P @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ) ).
% nat.split_sels(2)
thf(fact_475_nat_Osplit__sels_I2_J,axiom,
! [P: nat > $o,F1: nat,F22: nat > nat,Nat: nat] :
( ( P @ ( case_nat_nat @ F1 @ F22 @ Nat ) )
= ( ~ ( ( ( Nat = zero_zero_nat )
& ~ ( P @ F1 ) )
| ( ( Nat
= ( suc @ ( pred @ Nat ) ) )
& ~ ( P @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ) ).
% nat.split_sels(2)
thf(fact_476_transpose__max__length,axiom,
! [Xs: list_list_a] :
( ( foldr_list_a_nat
@ ^ [Xs4: list_a] : ( ord_max_nat @ ( size_size_list_a @ Xs4 ) )
@ ( transpose_a @ Xs )
@ zero_zero_nat )
= ( size_s349497388124573686list_a
@ ( filter_list_a
@ ^ [X4: list_a] : ( X4 != nil_a )
@ Xs ) ) ) ).
% transpose_max_length
thf(fact_477_transpose__max__length,axiom,
! [Xs: list_list_nat] :
( ( foldr_list_nat_nat
@ ^ [Xs4: list_nat] : ( ord_max_nat @ ( size_size_list_nat @ Xs4 ) )
@ ( transpose_nat @ Xs )
@ zero_zero_nat )
= ( size_s3023201423986296836st_nat
@ ( filter_list_nat
@ ^ [X4: list_nat] : ( X4 != nil_nat )
@ Xs ) ) ) ).
% transpose_max_length
thf(fact_478_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N2: nat] :
( X
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_479_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N2: nat] :
( ~ ( P @ N2 )
& ( P @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_480_max__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_max_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( suc @ ( ord_max_nat @ M2 @ N ) ) ) ).
% max_Suc_Suc
thf(fact_481_max__0R,axiom,
! [N: nat] :
( ( ord_max_nat @ N @ zero_zero_nat )
= N ) ).
% max_0R
thf(fact_482_max__0L,axiom,
! [N: nat] :
( ( ord_max_nat @ zero_zero_nat @ N )
= N ) ).
% max_0L
thf(fact_483_max__nat_Oright__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ zero_zero_nat )
= A ) ).
% max_nat.right_neutral
thf(fact_484_max__nat_Oneutr__eq__iff,axiom,
! [A: nat,B: nat] :
( ( zero_zero_nat
= ( ord_max_nat @ A @ B ) )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_485_max__nat_Oleft__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ zero_zero_nat @ A )
= A ) ).
% max_nat.left_neutral
thf(fact_486_max__nat_Oeq__neutr__iff,axiom,
! [A: nat,B: nat] :
( ( ( ord_max_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_487_max__Suc1,axiom,
! [N: nat,M2: nat] :
( ( ord_max_nat @ ( suc @ N ) @ M2 )
= ( case_nat_nat @ ( suc @ N )
@ ^ [M3: nat] : ( suc @ ( ord_max_nat @ N @ M3 ) )
@ M2 ) ) ).
% max_Suc1
thf(fact_488_max__Suc2,axiom,
! [M2: nat,N: nat] :
( ( ord_max_nat @ M2 @ ( suc @ N ) )
= ( case_nat_nat @ ( suc @ N )
@ ^ [M3: nat] : ( suc @ ( ord_max_nat @ M3 @ N ) )
@ M2 ) ) ).
% max_Suc2
thf(fact_489_pred__def,axiom,
( pred
= ( case_nat_nat @ zero_zero_nat
@ ^ [X24: nat] : X24 ) ) ).
% pred_def
thf(fact_490_length__transpose,axiom,
! [Xs: list_list_a] :
( ( size_s349497388124573686list_a @ ( transpose_a @ Xs ) )
= ( foldr_list_a_nat
@ ^ [Xs4: list_a] : ( ord_max_nat @ ( size_size_list_a @ Xs4 ) )
@ Xs
@ zero_zero_nat ) ) ).
% length_transpose
thf(fact_491_length__transpose,axiom,
! [Xs: list_list_nat] :
( ( size_s3023201423986296836st_nat @ ( transpose_nat @ Xs ) )
= ( foldr_list_nat_nat
@ ^ [Xs4: list_nat] : ( ord_max_nat @ ( size_size_list_nat @ Xs4 ) )
@ Xs
@ zero_zero_nat ) ) ).
% length_transpose
thf(fact_492_transpose__aux__max,axiom,
! [Xs: list_a,Xss2: list_list_a] :
( ( ord_max_nat @ ( suc @ ( size_size_list_a @ Xs ) )
@ ( foldr_list_a_nat
@ ^ [Xs4: list_a] : ( ord_max_nat @ ( size_size_list_a @ Xs4 ) )
@ Xss2
@ zero_zero_nat ) )
= ( suc
@ ( ord_max_nat @ ( size_size_list_a @ Xs )
@ ( foldr_list_a_nat
@ ^ [X4: list_a] : ( ord_max_nat @ ( minus_minus_nat @ ( size_size_list_a @ X4 ) @ ( suc @ zero_zero_nat ) ) )
@ ( filter_list_a
@ ^ [Ys2: list_a] : ( Ys2 != nil_a )
@ Xss2 )
@ zero_zero_nat ) ) ) ) ).
% transpose_aux_max
thf(fact_493_transpose__aux__max,axiom,
! [Xs: list_a,Xss2: list_list_nat] :
( ( ord_max_nat @ ( suc @ ( size_size_list_a @ Xs ) )
@ ( foldr_list_nat_nat
@ ^ [Xs4: list_nat] : ( ord_max_nat @ ( size_size_list_nat @ Xs4 ) )
@ Xss2
@ zero_zero_nat ) )
= ( suc
@ ( ord_max_nat @ ( size_size_list_a @ Xs )
@ ( foldr_list_nat_nat
@ ^ [X4: list_nat] : ( ord_max_nat @ ( minus_minus_nat @ ( size_size_list_nat @ X4 ) @ ( suc @ zero_zero_nat ) ) )
@ ( filter_list_nat
@ ^ [Ys2: list_nat] : ( Ys2 != nil_nat )
@ Xss2 )
@ zero_zero_nat ) ) ) ) ).
% transpose_aux_max
thf(fact_494_transpose__aux__max,axiom,
! [Xs: list_nat,Xss2: list_list_a] :
( ( ord_max_nat @ ( suc @ ( size_size_list_nat @ Xs ) )
@ ( foldr_list_a_nat
@ ^ [Xs4: list_a] : ( ord_max_nat @ ( size_size_list_a @ Xs4 ) )
@ Xss2
@ zero_zero_nat ) )
= ( suc
@ ( ord_max_nat @ ( size_size_list_nat @ Xs )
@ ( foldr_list_a_nat
@ ^ [X4: list_a] : ( ord_max_nat @ ( minus_minus_nat @ ( size_size_list_a @ X4 ) @ ( suc @ zero_zero_nat ) ) )
@ ( filter_list_a
@ ^ [Ys2: list_a] : ( Ys2 != nil_a )
@ Xss2 )
@ zero_zero_nat ) ) ) ) ).
% transpose_aux_max
thf(fact_495_transpose__aux__max,axiom,
! [Xs: list_nat,Xss2: list_list_nat] :
( ( ord_max_nat @ ( suc @ ( size_size_list_nat @ Xs ) )
@ ( foldr_list_nat_nat
@ ^ [Xs4: list_nat] : ( ord_max_nat @ ( size_size_list_nat @ Xs4 ) )
@ Xss2
@ zero_zero_nat ) )
= ( suc
@ ( ord_max_nat @ ( size_size_list_nat @ Xs )
@ ( foldr_list_nat_nat
@ ^ [X4: list_nat] : ( ord_max_nat @ ( minus_minus_nat @ ( size_size_list_nat @ X4 ) @ ( suc @ zero_zero_nat ) ) )
@ ( filter_list_nat
@ ^ [Ys2: list_nat] : ( Ys2 != nil_nat )
@ Xss2 )
@ zero_zero_nat ) ) ) ) ).
% transpose_aux_max
thf(fact_496_nths__Cons,axiom,
! [X: a,L2: list_a,A2: set_nat] :
( ( nths_a @ ( cons_a @ X @ L2 ) @ A2 )
= ( append_a @ ( if_list_a @ ( member_nat @ zero_zero_nat @ A2 ) @ ( cons_a @ X @ nil_a ) @ nil_a )
@ ( nths_a @ L2
@ ( collect_nat
@ ^ [J: nat] : ( member_nat @ ( suc @ J ) @ A2 ) ) ) ) ) ).
% nths_Cons
thf(fact_497_nths__Cons,axiom,
! [X: nat,L2: list_nat,A2: set_nat] :
( ( nths_nat @ ( cons_nat @ X @ L2 ) @ A2 )
= ( append_nat @ ( if_list_nat @ ( member_nat @ zero_zero_nat @ A2 ) @ ( cons_nat @ X @ nil_nat ) @ nil_nat )
@ ( nths_nat @ L2
@ ( collect_nat
@ ^ [J: nat] : ( member_nat @ ( suc @ J ) @ A2 ) ) ) ) ) ).
% nths_Cons
thf(fact_498_distinct__adj__append__iff,axiom,
! [Xs: list_a,Ys: list_a] :
( ( distinct_adj_a @ ( append_a @ Xs @ Ys ) )
= ( ( distinct_adj_a @ Xs )
& ( distinct_adj_a @ Ys )
& ( ( Xs = nil_a )
| ( Ys = nil_a )
| ( ( last_a @ Xs )
!= ( hd_a @ Ys ) ) ) ) ) ).
% distinct_adj_append_iff
thf(fact_499_distinct__adj__append__iff,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( distinct_adj_nat @ ( append_nat @ Xs @ Ys ) )
= ( ( distinct_adj_nat @ Xs )
& ( distinct_adj_nat @ Ys )
& ( ( Xs = nil_nat )
| ( Ys = nil_nat )
| ( ( last_nat @ Xs )
!= ( hd_nat @ Ys ) ) ) ) ) ).
% distinct_adj_append_iff
thf(fact_500_take__Suc,axiom,
! [Xs: list_a,N: nat] :
( ( Xs != nil_a )
=> ( ( take_a @ ( suc @ N ) @ Xs )
= ( cons_a @ ( hd_a @ Xs ) @ ( take_a @ N @ ( tl_a @ Xs ) ) ) ) ) ).
% take_Suc
thf(fact_501_take__Suc,axiom,
! [Xs: list_nat,N: nat] :
( ( Xs != nil_nat )
=> ( ( take_nat @ ( suc @ N ) @ Xs )
= ( cons_nat @ ( hd_nat @ Xs ) @ ( take_nat @ N @ ( tl_nat @ Xs ) ) ) ) ) ).
% take_Suc
thf(fact_502_successively__append__iff,axiom,
! [P: a > a > $o,Xs: list_a,Ys: list_a] :
( ( successively_a @ P @ ( append_a @ Xs @ Ys ) )
= ( ( successively_a @ P @ Xs )
& ( successively_a @ P @ Ys )
& ( ( Xs = nil_a )
| ( Ys = nil_a )
| ( P @ ( last_a @ Xs ) @ ( hd_a @ Ys ) ) ) ) ) ).
% successively_append_iff
thf(fact_503_successively__append__iff,axiom,
! [P: nat > nat > $o,Xs: list_nat,Ys: list_nat] :
( ( successively_nat @ P @ ( append_nat @ Xs @ Ys ) )
= ( ( successively_nat @ P @ Xs )
& ( successively_nat @ P @ Ys )
& ( ( Xs = nil_nat )
| ( Ys = nil_nat )
| ( P @ ( last_nat @ Xs ) @ ( hd_nat @ Ys ) ) ) ) ) ).
% successively_append_iff
thf(fact_504_nths__singleton,axiom,
! [A2: set_nat,X: a] :
( ( ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_a @ ( cons_a @ X @ nil_a ) @ A2 )
= ( cons_a @ X @ nil_a ) ) )
& ( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_a @ ( cons_a @ X @ nil_a ) @ A2 )
= nil_a ) ) ) ).
% nths_singleton
thf(fact_505_nths__singleton,axiom,
! [A2: set_nat,X: nat] :
( ( ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_nat @ ( cons_nat @ X @ nil_nat ) @ A2 )
= ( cons_nat @ X @ nil_nat ) ) )
& ( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_nat @ ( cons_nat @ X @ nil_nat ) @ A2 )
= nil_nat ) ) ) ).
% nths_singleton
thf(fact_506_Suc__diff__diff,axiom,
! [M2: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_507_diff__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_Suc_Suc
thf(fact_508_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_509_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_510_nths__nil,axiom,
! [A2: set_nat] :
( ( nths_a @ nil_a @ A2 )
= nil_a ) ).
% nths_nil
thf(fact_511_nths__nil,axiom,
! [A2: set_nat] :
( ( nths_nat @ nil_nat @ A2 )
= nil_nat ) ).
% nths_nil
thf(fact_512_distinct__adj__Cons__Cons,axiom,
! [X: a,Y: a,Xs: list_a] :
( ( distinct_adj_a @ ( cons_a @ X @ ( cons_a @ Y @ Xs ) ) )
= ( ( X != Y )
& ( distinct_adj_a @ ( cons_a @ Y @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_513_distinct__adj__Cons__Cons,axiom,
! [X: nat,Y: nat,Xs: list_nat] :
( ( distinct_adj_nat @ ( cons_nat @ X @ ( cons_nat @ Y @ Xs ) ) )
= ( ( X != Y )
& ( distinct_adj_nat @ ( cons_nat @ Y @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_514_take__Suc__Cons,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( take_a @ ( suc @ N ) @ ( cons_a @ X @ Xs ) )
= ( cons_a @ X @ ( take_a @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_515_take__Suc__Cons,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( take_nat @ ( suc @ N ) @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( take_nat @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_516_take0,axiom,
( ( take_a @ zero_zero_nat )
= ( ^ [Xs4: list_a] : nil_a ) ) ).
% take0
thf(fact_517_take0,axiom,
( ( take_nat @ zero_zero_nat )
= ( ^ [Xs4: list_nat] : nil_nat ) ) ).
% take0
thf(fact_518_take__eq__Nil,axiom,
! [N: nat,Xs: list_a] :
( ( ( take_a @ N @ Xs )
= nil_a )
= ( ( N = zero_zero_nat )
| ( Xs = nil_a ) ) ) ).
% take_eq_Nil
thf(fact_519_take__eq__Nil,axiom,
! [N: nat,Xs: list_nat] :
( ( ( take_nat @ N @ Xs )
= nil_nat )
= ( ( N = zero_zero_nat )
| ( Xs = nil_nat ) ) ) ).
% take_eq_Nil
thf(fact_520_take__eq__Nil2,axiom,
! [N: nat,Xs: list_a] :
( ( nil_a
= ( take_a @ N @ Xs ) )
= ( ( N = zero_zero_nat )
| ( Xs = nil_a ) ) ) ).
% take_eq_Nil2
thf(fact_521_take__eq__Nil2,axiom,
! [N: nat,Xs: list_nat] :
( ( nil_nat
= ( take_nat @ N @ Xs ) )
= ( ( N = zero_zero_nat )
| ( Xs = nil_nat ) ) ) ).
% take_eq_Nil2
thf(fact_522_take__append,axiom,
! [N: nat,Xs: list_a,Ys: list_a] :
( ( take_a @ N @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( take_a @ N @ Xs ) @ ( take_a @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_523_take__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( take_nat @ N @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( take_nat @ N @ Xs ) @ ( take_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_524_take__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ! [I2: nat] :
( ( take_nat @ I2 @ Xs )
= ( take_nat @ I2 @ Ys ) )
=> ( Xs = Ys ) ) ).
% take_equalityI
thf(fact_525_diff__commute,axiom,
! [I: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J2 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J2 ) ) ).
% diff_commute
thf(fact_526_take__Nil,axiom,
! [N: nat] :
( ( take_a @ N @ nil_a )
= nil_a ) ).
% take_Nil
thf(fact_527_take__Nil,axiom,
! [N: nat] :
( ( take_nat @ N @ nil_nat )
= nil_nat ) ).
% take_Nil
thf(fact_528_take__map,axiom,
! [N: nat,F: nat > nat,Xs: list_nat] :
( ( take_nat @ N @ ( map_nat_nat @ F @ Xs ) )
= ( map_nat_nat @ F @ ( take_nat @ N @ Xs ) ) ) ).
% take_map
thf(fact_529_take__update__swap,axiom,
! [M2: nat,Xs: list_nat,N: nat,X: nat] :
( ( take_nat @ M2 @ ( list_update_nat @ Xs @ N @ X ) )
= ( list_update_nat @ ( take_nat @ M2 @ Xs ) @ N @ X ) ) ).
% take_update_swap
thf(fact_530_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_531_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_532_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_533_successively_Oelims_I3_J,axiom,
! [X: a > a > $o,Xa: list_a] :
( ~ ( successively_a @ X @ Xa )
=> ~ ! [X3: a,Y3: a,Xs2: list_a] :
( ( Xa
= ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs2 ) ) )
=> ( ( X @ X3 @ Y3 )
& ( successively_a @ X @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ).
% successively.elims(3)
thf(fact_534_successively_Oelims_I3_J,axiom,
! [X: nat > nat > $o,Xa: list_nat] :
( ~ ( successively_nat @ X @ Xa )
=> ~ ! [X3: nat,Y3: nat,Xs2: list_nat] :
( ( Xa
= ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs2 ) ) )
=> ( ( X @ X3 @ Y3 )
& ( successively_nat @ X @ ( cons_nat @ Y3 @ Xs2 ) ) ) ) ) ).
% successively.elims(3)
thf(fact_535_successively_Osimps_I3_J,axiom,
! [P: a > a > $o,X: a,Y: a,Xs: list_a] :
( ( successively_a @ P @ ( cons_a @ X @ ( cons_a @ Y @ Xs ) ) )
= ( ( P @ X @ Y )
& ( successively_a @ P @ ( cons_a @ Y @ Xs ) ) ) ) ).
% successively.simps(3)
thf(fact_536_successively_Osimps_I3_J,axiom,
! [P: nat > nat > $o,X: nat,Y: nat,Xs: list_nat] :
( ( successively_nat @ P @ ( cons_nat @ X @ ( cons_nat @ Y @ Xs ) ) )
= ( ( P @ X @ Y )
& ( successively_nat @ P @ ( cons_nat @ Y @ Xs ) ) ) ) ).
% successively.simps(3)
thf(fact_537_successively_Osimps_I1_J,axiom,
! [P: a > a > $o] : ( successively_a @ P @ nil_a ) ).
% successively.simps(1)
thf(fact_538_successively_Osimps_I1_J,axiom,
! [P: nat > nat > $o] : ( successively_nat @ P @ nil_nat ) ).
% successively.simps(1)
thf(fact_539_nths__map,axiom,
! [F: nat > nat,Xs: list_nat,I3: set_nat] :
( ( nths_nat @ ( map_nat_nat @ F @ Xs ) @ I3 )
= ( map_nat_nat @ F @ ( nths_nat @ Xs @ I3 ) ) ) ).
% nths_map
thf(fact_540_distinct__adj__ConsD,axiom,
! [X: a,Xs: list_a] :
( ( distinct_adj_a @ ( cons_a @ X @ Xs ) )
=> ( distinct_adj_a @ Xs ) ) ).
% distinct_adj_ConsD
thf(fact_541_distinct__adj__ConsD,axiom,
! [X: nat,Xs: list_nat] :
( ( distinct_adj_nat @ ( cons_nat @ X @ Xs ) )
=> ( distinct_adj_nat @ Xs ) ) ).
% distinct_adj_ConsD
thf(fact_542_distinct__adj__Nil,axiom,
distinct_adj_a @ nil_a ).
% distinct_adj_Nil
thf(fact_543_distinct__adj__Nil,axiom,
distinct_adj_nat @ nil_nat ).
% distinct_adj_Nil
thf(fact_544_distinct__adj__appendD1,axiom,
! [Xs: list_a,Ys: list_a] :
( ( distinct_adj_a @ ( append_a @ Xs @ Ys ) )
=> ( distinct_adj_a @ Xs ) ) ).
% distinct_adj_appendD1
thf(fact_545_distinct__adj__appendD1,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( distinct_adj_nat @ ( append_nat @ Xs @ Ys ) )
=> ( distinct_adj_nat @ Xs ) ) ).
% distinct_adj_appendD1
thf(fact_546_distinct__adj__appendD2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( distinct_adj_a @ ( append_a @ Xs @ Ys ) )
=> ( distinct_adj_a @ Ys ) ) ).
% distinct_adj_appendD2
thf(fact_547_distinct__adj__appendD2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( distinct_adj_nat @ ( append_nat @ Xs @ Ys ) )
=> ( distinct_adj_nat @ Ys ) ) ).
% distinct_adj_appendD2
thf(fact_548_distinct__adj__mapD,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( distinct_adj_nat @ ( map_nat_nat @ F @ Xs ) )
=> ( distinct_adj_nat @ Xs ) ) ).
% distinct_adj_mapD
thf(fact_549_successively__map,axiom,
! [P: nat > nat > $o,F: nat > nat,Xs: list_nat] :
( ( successively_nat @ P @ ( map_nat_nat @ F @ Xs ) )
= ( successively_nat
@ ^ [X4: nat,Y2: nat] : ( P @ ( F @ X4 ) @ ( F @ Y2 ) )
@ Xs ) ) ).
% successively_map
thf(fact_550_take__0,axiom,
! [Xs: list_a] :
( ( take_a @ zero_zero_nat @ Xs )
= nil_a ) ).
% take_0
thf(fact_551_take__0,axiom,
! [Xs: list_nat] :
( ( take_nat @ zero_zero_nat @ Xs )
= nil_nat ) ).
% take_0
thf(fact_552_take__tl,axiom,
! [N: nat,Xs: list_nat] :
( ( take_nat @ N @ ( tl_nat @ Xs ) )
= ( tl_nat @ ( take_nat @ ( suc @ N ) @ Xs ) ) ) ).
% take_tl
thf(fact_553_successively_Oelims_I2_J,axiom,
! [X: a > a > $o,Xa: list_a] :
( ( successively_a @ X @ Xa )
=> ( ( Xa != nil_a )
=> ( ! [X3: a] :
( Xa
!= ( cons_a @ X3 @ nil_a ) )
=> ~ ! [X3: a,Y3: a,Xs2: list_a] :
( ( Xa
= ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs2 ) ) )
=> ~ ( ( X @ X3 @ Y3 )
& ( successively_a @ X @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ) ) ).
% successively.elims(2)
thf(fact_554_successively_Oelims_I2_J,axiom,
! [X: nat > nat > $o,Xa: list_nat] :
( ( successively_nat @ X @ Xa )
=> ( ( Xa != nil_nat )
=> ( ! [X3: nat] :
( Xa
!= ( cons_nat @ X3 @ nil_nat ) )
=> ~ ! [X3: nat,Y3: nat,Xs2: list_nat] :
( ( Xa
= ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs2 ) ) )
=> ~ ( ( X @ X3 @ Y3 )
& ( successively_nat @ X @ ( cons_nat @ Y3 @ Xs2 ) ) ) ) ) ) ) ).
% successively.elims(2)
thf(fact_555_successively_Oelims_I1_J,axiom,
! [X: a > a > $o,Xa: list_a,Y: $o] :
( ( ( successively_a @ X @ Xa )
= Y )
=> ( ( ( Xa = nil_a )
=> ~ Y )
=> ( ( ? [X3: a] :
( Xa
= ( cons_a @ X3 @ nil_a ) )
=> ~ Y )
=> ~ ! [X3: a,Y3: a,Xs2: list_a] :
( ( Xa
= ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs2 ) ) )
=> ( Y
= ( ~ ( ( X @ X3 @ Y3 )
& ( successively_a @ X @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% successively.elims(1)
thf(fact_556_successively_Oelims_I1_J,axiom,
! [X: nat > nat > $o,Xa: list_nat,Y: $o] :
( ( ( successively_nat @ X @ Xa )
= Y )
=> ( ( ( Xa = nil_nat )
=> ~ Y )
=> ( ( ? [X3: nat] :
( Xa
= ( cons_nat @ X3 @ nil_nat ) )
=> ~ Y )
=> ~ ! [X3: nat,Y3: nat,Xs2: list_nat] :
( ( Xa
= ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs2 ) ) )
=> ( Y
= ( ~ ( ( X @ X3 @ Y3 )
& ( successively_nat @ X @ ( cons_nat @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% successively.elims(1)
thf(fact_557_successively_Osimps_I2_J,axiom,
! [P: a > a > $o,X: a] : ( successively_a @ P @ ( cons_a @ X @ nil_a ) ) ).
% successively.simps(2)
thf(fact_558_successively_Osimps_I2_J,axiom,
! [P: nat > nat > $o,X: nat] : ( successively_nat @ P @ ( cons_nat @ X @ nil_nat ) ) ).
% successively.simps(2)
thf(fact_559_diff__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N ) )
= ( case_nat_nat @ zero_zero_nat
@ ^ [K2: nat] : K2
@ ( minus_minus_nat @ M2 @ N ) ) ) ).
% diff_Suc
thf(fact_560_distinct__adj__singleton,axiom,
! [X: a] : ( distinct_adj_a @ ( cons_a @ X @ nil_a ) ) ).
% distinct_adj_singleton
thf(fact_561_distinct__adj__singleton,axiom,
! [X: nat] : ( distinct_adj_nat @ ( cons_nat @ X @ nil_nat ) ) ).
% distinct_adj_singleton
thf(fact_562_successively__Cons,axiom,
! [P: a > a > $o,X: a,Xs: list_a] :
( ( successively_a @ P @ ( cons_a @ X @ Xs ) )
= ( ( Xs = nil_a )
| ( ( P @ X @ ( hd_a @ Xs ) )
& ( successively_a @ P @ Xs ) ) ) ) ).
% successively_Cons
thf(fact_563_successively__Cons,axiom,
! [P: nat > nat > $o,X: nat,Xs: list_nat] :
( ( successively_nat @ P @ ( cons_nat @ X @ Xs ) )
= ( ( Xs = nil_nat )
| ( ( P @ X @ ( hd_nat @ Xs ) )
& ( successively_nat @ P @ Xs ) ) ) ) ).
% successively_Cons
thf(fact_564_take__Cons,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( take_a @ N @ ( cons_a @ X @ Xs ) )
= ( case_nat_list_a @ nil_a
@ ^ [M4: nat] : ( cons_a @ X @ ( take_a @ M4 @ Xs ) )
@ N ) ) ).
% take_Cons
thf(fact_565_take__Cons,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( take_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( case_nat_list_nat @ nil_nat
@ ^ [M4: nat] : ( cons_nat @ X @ ( take_nat @ M4 @ Xs ) )
@ N ) ) ).
% take_Cons
thf(fact_566_distinct__adj__Cons,axiom,
! [X: a,Xs: list_a] :
( ( distinct_adj_a @ ( cons_a @ X @ Xs ) )
= ( ( Xs = nil_a )
| ( ( X
!= ( hd_a @ Xs ) )
& ( distinct_adj_a @ Xs ) ) ) ) ).
% distinct_adj_Cons
thf(fact_567_distinct__adj__Cons,axiom,
! [X: nat,Xs: list_nat] :
( ( distinct_adj_nat @ ( cons_nat @ X @ Xs ) )
= ( ( Xs = nil_nat )
| ( ( X
!= ( hd_nat @ Xs ) )
& ( distinct_adj_nat @ Xs ) ) ) ) ).
% distinct_adj_Cons
thf(fact_568_last__list__update,axiom,
! [Xs: list_a,K: nat,X: a] :
( ( Xs != nil_a )
=> ( ( ( K
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
=> ( ( last_a @ ( list_update_a @ Xs @ K @ X ) )
= X ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
=> ( ( last_a @ ( list_update_a @ Xs @ K @ X ) )
= ( last_a @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_569_last__list__update,axiom,
! [Xs: list_nat,K: nat,X: nat] :
( ( Xs != nil_nat )
=> ( ( ( K
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( last_nat @ ( list_update_nat @ Xs @ K @ X ) )
= X ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( last_nat @ ( list_update_nat @ Xs @ K @ X ) )
= ( last_nat @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_570_take__Cons_H,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( ( N = zero_zero_nat )
=> ( ( take_a @ N @ ( cons_a @ X @ Xs ) )
= nil_a ) )
& ( ( N != zero_zero_nat )
=> ( ( take_a @ N @ ( cons_a @ X @ Xs ) )
= ( cons_a @ X @ ( take_a @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_571_take__Cons_H,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( take_nat @ N @ ( cons_nat @ X @ Xs ) )
= nil_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( take_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( take_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_572_butlast__list__update,axiom,
! [K: nat,Xs: list_a,X: a] :
( ( ( K
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
=> ( ( butlast_a @ ( list_update_a @ Xs @ K @ X ) )
= ( butlast_a @ Xs ) ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
=> ( ( butlast_a @ ( list_update_a @ Xs @ K @ X ) )
= ( list_update_a @ ( butlast_a @ Xs ) @ K @ X ) ) ) ) ).
% butlast_list_update
thf(fact_573_butlast__list__update,axiom,
! [K: nat,Xs: list_nat,X: nat] :
( ( ( K
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( butlast_nat @ ( list_update_nat @ Xs @ K @ X ) )
= ( butlast_nat @ Xs ) ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( butlast_nat @ ( list_update_nat @ Xs @ K @ X ) )
= ( list_update_nat @ ( butlast_nat @ Xs ) @ K @ X ) ) ) ) ).
% butlast_list_update
thf(fact_574_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_575_butlast__conv__take,axiom,
( butlast_nat
= ( ^ [Xs4: list_nat] : ( take_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs4 ) @ one_one_nat ) @ Xs4 ) ) ) ).
% butlast_conv_take
thf(fact_576_Cons__in__shuffles__iff,axiom,
! [Z3: a,Zs: list_a,Xs: list_a,Ys: list_a] :
( ( member_list_a @ ( cons_a @ Z3 @ Zs ) @ ( shuffles_a @ Xs @ Ys ) )
= ( ( ( Xs != nil_a )
& ( ( hd_a @ Xs )
= Z3 )
& ( member_list_a @ Zs @ ( shuffles_a @ ( tl_a @ Xs ) @ Ys ) ) )
| ( ( Ys != nil_a )
& ( ( hd_a @ Ys )
= Z3 )
& ( member_list_a @ Zs @ ( shuffles_a @ Xs @ ( tl_a @ Ys ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_577_Cons__in__shuffles__iff,axiom,
! [Z3: nat,Zs: list_nat,Xs: list_nat,Ys: list_nat] :
( ( member_list_nat @ ( cons_nat @ Z3 @ Zs ) @ ( shuffles_nat @ Xs @ Ys ) )
= ( ( ( Xs != nil_nat )
& ( ( hd_nat @ Xs )
= Z3 )
& ( member_list_nat @ Zs @ ( shuffles_nat @ ( tl_nat @ Xs ) @ Ys ) ) )
| ( ( Ys != nil_nat )
& ( ( hd_nat @ Ys )
= Z3 )
& ( member_list_nat @ Zs @ ( shuffles_nat @ Xs @ ( tl_nat @ Ys ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_578_Nil__in__shuffles,axiom,
! [Xs: list_a,Ys: list_a] :
( ( member_list_a @ nil_a @ ( shuffles_a @ Xs @ Ys ) )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_in_shuffles
thf(fact_579_Nil__in__shuffles,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( member_list_nat @ nil_nat @ ( shuffles_nat @ Xs @ Ys ) )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% Nil_in_shuffles
thf(fact_580_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_581_length__tl,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( tl_a @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) ).
% length_tl
thf(fact_582_length__tl,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( tl_nat @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ).
% length_tl
thf(fact_583_length__butlast,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( butlast_a @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) ).
% length_butlast
thf(fact_584_length__butlast,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( butlast_nat @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ).
% length_butlast
thf(fact_585_Cons__in__shuffles__rightI,axiom,
! [Zs: list_a,Xs: list_a,Ys: list_a,Z3: a] :
( ( member_list_a @ Zs @ ( shuffles_a @ Xs @ Ys ) )
=> ( member_list_a @ ( cons_a @ Z3 @ Zs ) @ ( shuffles_a @ Xs @ ( cons_a @ Z3 @ Ys ) ) ) ) ).
% Cons_in_shuffles_rightI
thf(fact_586_Cons__in__shuffles__rightI,axiom,
! [Zs: list_nat,Xs: list_nat,Ys: list_nat,Z3: nat] :
( ( member_list_nat @ Zs @ ( shuffles_nat @ Xs @ Ys ) )
=> ( member_list_nat @ ( cons_nat @ Z3 @ Zs ) @ ( shuffles_nat @ Xs @ ( cons_nat @ Z3 @ Ys ) ) ) ) ).
% Cons_in_shuffles_rightI
thf(fact_587_Cons__in__shuffles__leftI,axiom,
! [Zs: list_a,Xs: list_a,Ys: list_a,Z3: a] :
( ( member_list_a @ Zs @ ( shuffles_a @ Xs @ Ys ) )
=> ( member_list_a @ ( cons_a @ Z3 @ Zs ) @ ( shuffles_a @ ( cons_a @ Z3 @ Xs ) @ Ys ) ) ) ).
% Cons_in_shuffles_leftI
thf(fact_588_Cons__in__shuffles__leftI,axiom,
! [Zs: list_nat,Xs: list_nat,Ys: list_nat,Z3: nat] :
( ( member_list_nat @ Zs @ ( shuffles_nat @ Xs @ Ys ) )
=> ( member_list_nat @ ( cons_nat @ Z3 @ Zs ) @ ( shuffles_nat @ ( cons_nat @ Z3 @ Xs ) @ Ys ) ) ) ).
% Cons_in_shuffles_leftI
thf(fact_589_Nil__in__shufflesI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = nil_a )
=> ( ( Ys = nil_a )
=> ( member_list_a @ nil_a @ ( shuffles_a @ Xs @ Ys ) ) ) ) ).
% Nil_in_shufflesI
thf(fact_590_Nil__in__shufflesI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs = nil_nat )
=> ( ( Ys = nil_nat )
=> ( member_list_nat @ nil_nat @ ( shuffles_nat @ Xs @ Ys ) ) ) ) ).
% Nil_in_shufflesI
thf(fact_591_shufflesE,axiom,
! [Zs: list_a,Xs: list_a,Ys: list_a] :
( ( member_list_a @ Zs @ ( shuffles_a @ Xs @ Ys ) )
=> ( ( ( Zs = Xs )
=> ( Ys != nil_a ) )
=> ( ( ( Zs = Ys )
=> ( Xs != nil_a ) )
=> ( ! [X3: a,Xs3: list_a] :
( ( Xs
= ( cons_a @ X3 @ Xs3 ) )
=> ! [Z: a,Zs4: list_a] :
( ( Zs
= ( cons_a @ Z @ Zs4 ) )
=> ( ( X3 = Z )
=> ~ ( member_list_a @ Zs4 @ ( shuffles_a @ Xs3 @ Ys ) ) ) ) )
=> ~ ! [Y3: a,Ys5: list_a] :
( ( Ys
= ( cons_a @ Y3 @ Ys5 ) )
=> ! [Z: a,Zs4: list_a] :
( ( Zs
= ( cons_a @ Z @ Zs4 ) )
=> ( ( Y3 = Z )
=> ~ ( member_list_a @ Zs4 @ ( shuffles_a @ Xs @ Ys5 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_592_shufflesE,axiom,
! [Zs: list_nat,Xs: list_nat,Ys: list_nat] :
( ( member_list_nat @ Zs @ ( shuffles_nat @ Xs @ Ys ) )
=> ( ( ( Zs = Xs )
=> ( Ys != nil_nat ) )
=> ( ( ( Zs = Ys )
=> ( Xs != nil_nat ) )
=> ( ! [X3: nat,Xs3: list_nat] :
( ( Xs
= ( cons_nat @ X3 @ Xs3 ) )
=> ! [Z: nat,Zs4: list_nat] :
( ( Zs
= ( cons_nat @ Z @ Zs4 ) )
=> ( ( X3 = Z )
=> ~ ( member_list_nat @ Zs4 @ ( shuffles_nat @ Xs3 @ Ys ) ) ) ) )
=> ~ ! [Y3: nat,Ys5: list_nat] :
( ( Ys
= ( cons_nat @ Y3 @ Ys5 ) )
=> ! [Z: nat,Zs4: list_nat] :
( ( Zs
= ( cons_nat @ Z @ Zs4 ) )
=> ( ( Y3 = Z )
=> ~ ( member_list_nat @ Zs4 @ ( shuffles_nat @ Xs @ Ys5 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_593_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_594_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_595_Nitpick_Ocase__nat__unfold,axiom,
( case_nat_o
= ( ^ [X4: $o,F2: nat > $o,N3: nat] :
( ( ( N3 = zero_zero_nat )
=> X4 )
& ( ( N3 != zero_zero_nat )
=> ( F2 @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% Nitpick.case_nat_unfold
thf(fact_596_Nitpick_Ocase__nat__unfold,axiom,
( case_nat_nat
= ( ^ [X4: nat,F2: nat > nat,N3: nat] : ( if_nat @ ( N3 = zero_zero_nat ) @ X4 @ ( F2 @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ).
% Nitpick.case_nat_unfold
thf(fact_597_tl__take,axiom,
! [N: nat,Xs: list_nat] :
( ( tl_nat @ ( take_nat @ N @ Xs ) )
= ( take_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( tl_nat @ Xs ) ) ) ).
% tl_take
thf(fact_598_length__prefixes,axiom,
! [Xs: list_a] :
( ( size_s349497388124573686list_a @ ( prefixes_a @ Xs ) )
= ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) ).
% length_prefixes
thf(fact_599_length__prefixes,axiom,
! [Xs: list_nat] :
( ( size_s3023201423986296836st_nat @ ( prefixes_nat @ Xs ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ).
% length_prefixes
thf(fact_600_butlast__take,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_eq_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( butlast_a @ ( take_a @ N @ Xs ) )
= ( take_a @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ).
% butlast_take
thf(fact_601_butlast__take,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( butlast_nat @ ( take_nat @ N @ Xs ) )
= ( take_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ).
% butlast_take
thf(fact_602_take__Cons__numeral,axiom,
! [V: num,X: a,Xs: list_a] :
( ( take_a @ ( numeral_numeral_nat @ V ) @ ( cons_a @ X @ Xs ) )
= ( cons_a @ X @ ( take_a @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) @ Xs ) ) ) ).
% take_Cons_numeral
thf(fact_603_take__Cons__numeral,axiom,
! [V: num,X: nat,Xs: list_nat] :
( ( take_nat @ ( numeral_numeral_nat @ V ) @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( take_nat @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) @ Xs ) ) ) ).
% take_Cons_numeral
thf(fact_604_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_605_last__conv__nth,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( last_a @ Xs )
= ( nth_a @ Xs @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_606_last__conv__nth,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( last_nat @ Xs )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_607_list__update__append,axiom,
! [N: nat,Xs: list_a,Ys: list_a,X: a] :
( ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ ( append_a @ Xs @ Ys ) @ N @ X )
= ( append_a @ ( list_update_a @ Xs @ N @ X ) @ Ys ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ ( append_a @ Xs @ Ys ) @ N @ X )
= ( append_a @ Xs @ ( list_update_a @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_608_list__update__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat,X: nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N @ X )
= ( append_nat @ ( list_update_nat @ Xs @ N @ X ) @ Ys ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N @ X )
= ( append_nat @ Xs @ ( list_update_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_609_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_610_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_611_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_612_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_613_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_614_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_615_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_616_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_617_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_618_add__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc_right
thf(fact_619_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_620_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_621_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_622_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_623_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_624_diff__diff__left,axiom,
! [I: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J2 ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% diff_diff_left
thf(fact_625_list__update__id,axiom,
! [Xs: list_nat,I: nat] :
( ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_626_nth__list__update__neq,axiom,
! [I: nat,J2: nat,Xs: list_nat,X: nat] :
( ( I != J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J2 )
= ( nth_nat @ Xs @ J2 ) ) ) ).
% nth_list_update_neq
thf(fact_627_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_628_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_629_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_630_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_631_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_632_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_633_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_634_Nat_Odiff__diff__right,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_635_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_636_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J2 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_637_nth__Cons__Suc,axiom,
! [X: a,Xs: list_a,N: nat] :
( ( nth_a @ ( cons_a @ X @ Xs ) @ ( suc @ N ) )
= ( nth_a @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_638_nth__Cons__Suc,axiom,
! [X: nat,Xs: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ N ) )
= ( nth_nat @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_639_nth__Cons__0,axiom,
! [X: a,Xs: list_a] :
( ( nth_a @ ( cons_a @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_640_nth__Cons__0,axiom,
! [X: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_641_length__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( size_size_list_a @ ( append_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) ) ) ).
% length_append
thf(fact_642_length__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_append
thf(fact_643_take__all,axiom,
! [Xs: list_a,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
=> ( ( take_a @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_644_take__all,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
=> ( ( take_nat @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_645_take__all__iff,axiom,
! [N: nat,Xs: list_a] :
( ( ( take_a @ N @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_646_take__all__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ( take_nat @ N @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_647_list__update__beyond,axiom,
! [Xs: list_a,I: nat,X: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ I )
=> ( ( list_update_a @ Xs @ I @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_648_list__update__beyond,axiom,
! [Xs: list_nat,I: nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
=> ( ( list_update_nat @ Xs @ I @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_649_nth__take,axiom,
! [I: nat,N: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_nat @ ( take_nat @ N @ Xs ) @ I )
= ( nth_nat @ Xs @ I ) ) ) ).
% nth_take
thf(fact_650_take__update__cancel,axiom,
! [N: nat,M2: nat,Xs: list_nat,Y: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( take_nat @ N @ ( list_update_nat @ Xs @ M2 @ Y ) )
= ( take_nat @ N @ Xs ) ) ) ).
% take_update_cancel
thf(fact_651_length__greater__0__conv,axiom,
! [Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) )
= ( Xs != nil_a ) ) ).
% length_greater_0_conv
thf(fact_652_length__greater__0__conv,axiom,
! [Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
= ( Xs != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_653_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_654_diff__Suc__diff__eq2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_655_diff__Suc__diff__eq1,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J2 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_656_nth__append__length,axiom,
! [Xs: list_a,X: a,Ys: list_a] :
( ( nth_a @ ( append_a @ Xs @ ( cons_a @ X @ Ys ) ) @ ( size_size_list_a @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_657_nth__append__length,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat] :
( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) ) @ ( size_size_list_nat @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_658_nth__map,axiom,
! [N: nat,Xs: list_a,F: a > nat] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_nat @ ( map_a_nat @ F @ Xs ) @ N )
= ( F @ ( nth_a @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_659_nth__map,axiom,
! [N: nat,Xs: list_nat,F: nat > nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ Xs ) @ N )
= ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_660_nth__append__length__plus,axiom,
! [Xs: list_a,Ys: list_a,N: nat] :
( ( nth_a @ ( append_a @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ N ) )
= ( nth_a @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_661_nth__append__length__plus,axiom,
! [Xs: list_nat,Ys: list_nat,N: nat] :
( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ N ) )
= ( nth_nat @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_662_nth__list__update__eq,axiom,
! [I: nat,Xs: list_a,X: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( list_update_a @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_663_nth__list__update__eq,axiom,
! [I: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_664_hd__take,axiom,
! [J2: nat,Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ J2 )
=> ( ( hd_nat @ ( take_nat @ J2 @ Xs ) )
= ( hd_nat @ Xs ) ) ) ).
% hd_take
thf(fact_665_rotate1__length01,axiom,
! [Xs: list_a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ one_one_nat )
=> ( ( rotate1_a @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_666_rotate1__length01,axiom,
! [Xs: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( ( rotate1_nat @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_667_nth__Cons__numeral,axiom,
! [X: a,Xs: list_a,V: num] :
( ( nth_a @ ( cons_a @ X @ Xs ) @ ( numeral_numeral_nat @ V ) )
= ( nth_a @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_668_nth__Cons__numeral,axiom,
! [X: nat,Xs: list_nat,V: num] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( numeral_numeral_nat @ V ) )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_669_nth__Cons__pos,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_a @ ( cons_a @ X @ Xs ) @ N )
= ( nth_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_670_nth__Cons__pos,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_671_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_672_less__imp__add__positive,axiom,
! [I: nat,J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_673_map__equality__iff,axiom,
! [F: a > nat,Xs: list_a,G: nat > nat,Ys: list_nat] :
( ( ( map_a_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
= ( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_a @ Xs @ I5 ) )
= ( G @ ( nth_nat @ Ys @ I5 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_674_map__equality__iff,axiom,
! [F: nat > nat,Xs: list_nat,G: a > nat,Ys: list_a] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_a_nat @ G @ Ys ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_a @ Ys ) )
=> ( ( F @ ( nth_nat @ Xs @ I5 ) )
= ( G @ ( nth_a @ Ys @ I5 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_675_map__equality__iff,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_nat @ Xs @ I5 ) )
= ( G @ ( nth_nat @ Ys @ I5 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_676_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_677_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D: nat] :
( ( A
= ( plus_plus_nat @ B @ D ) )
=> ( P @ D ) ) ) ) ).
% nat_diff_split
thf(fact_678_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D: nat] :
( ( A
= ( plus_plus_nat @ B @ D ) )
& ~ ( P @ D ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_679_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M2: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_680_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_681_less__imp__Suc__add,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_682_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M4: nat,N3: nat] :
? [K2: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M4 @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_683_less__add__Suc2,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).
% less_add_Suc2
thf(fact_684_less__add__Suc1,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_685_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M2 @ Q2 ) ) ) ) ).
% less_natE
thf(fact_686_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_687_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_688_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_689_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_690_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_691_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_692_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_693_inc__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( P @ J2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J2 )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_694_dec__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J2 ) ) ) ) ).
% dec_induct
thf(fact_695_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_696_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_697_add__leE,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_698_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_699_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_700_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_701_add__leD1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_702_add__leD2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_703_le__trans,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_704_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_705_le__Suc__ex,axiom,
! [K: nat,L2: nat] :
( ( ord_less_eq_nat @ K @ L2 )
=> ? [N2: nat] :
( L2
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_706_add__lessD1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J2 ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_707_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_708_add__le__mono,axiom,
! [I: nat,J2: nat,K: nat,L2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ K @ L2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ) ).
% add_le_mono
thf(fact_709_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
& ( M4 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_710_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_711_add__le__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_le_mono1
thf(fact_712_add__less__mono,axiom,
! [I: nat,J2: nat,K: nat,L2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ K @ L2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ) ).
% add_less_mono
thf(fact_713_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_714_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_715_not__add__less1,axiom,
! [I: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J2 ) @ I ) ).
% not_add_less1
thf(fact_716_not__add__less2,axiom,
! [J2: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I ) @ I ) ).
% not_add_less2
thf(fact_717_trans__le__add1,axiom,
! [I: nat,J2: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).
% trans_le_add1
thf(fact_718_trans__le__add2,axiom,
! [I: nat,J2: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).
% trans_le_add2
thf(fact_719_add__less__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_less_mono1
thf(fact_720_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_721_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_722_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N3: nat] :
? [K2: nat] :
( N3
= ( plus_plus_nat @ M4 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_723_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_724_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_725_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
=> ( P @ M5 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_726_trans__less__add1,axiom,
! [I: nat,J2: nat,M2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).
% trans_less_add1
thf(fact_727_trans__less__add2,axiom,
! [I: nat,J2: nat,M2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).
% trans_less_add2
thf(fact_728_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
& ~ ( P @ M5 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_729_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
| ( M4 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_730_less__add__eq__less,axiom,
! [K: nat,L2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K @ L2 )
=> ( ( ( plus_plus_nat @ M2 @ L2 )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_731_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_732_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_733_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ ( F @ M ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_734_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_735_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_736_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J2: nat] :
( ! [I2: nat,J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_737_nth__equalityI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ Xs @ I2 )
= ( nth_a @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_738_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_739_Skolem__list__nth,axiom,
! [K: nat,P: nat > a > $o] :
( ( ! [I5: nat] :
( ( ord_less_nat @ I5 @ K )
=> ? [X5: a] : ( P @ I5 @ X5 ) ) )
= ( ? [Xs4: list_a] :
( ( ( size_size_list_a @ Xs4 )
= K )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K )
=> ( P @ I5 @ ( nth_a @ Xs4 @ I5 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_740_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I5: nat] :
( ( ord_less_nat @ I5 @ K )
=> ? [X5: nat] : ( P @ I5 @ X5 ) ) )
= ( ? [Xs4: list_nat] :
( ( ( size_size_list_nat @ Xs4 )
= K )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K )
=> ( P @ I5 @ ( nth_nat @ Xs4 @ I5 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_741_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_a,Z4: list_a] : ( Y5 = Z4 ) )
= ( ^ [Xs4: list_a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs4 )
= ( size_size_list_a @ Ys2 ) )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_a @ Xs4 ) )
=> ( ( nth_a @ Xs4 @ I5 )
= ( nth_a @ Ys2 @ I5 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_742_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_nat,Z4: list_nat] : ( Y5 = Z4 ) )
= ( ^ [Xs4: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs4 )
= ( size_size_list_nat @ Ys2 ) )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs4 ) )
=> ( ( nth_nat @ Xs4 @ I5 )
= ( nth_nat @ Ys2 @ I5 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_743_nth__take__lemma,axiom,
! [K: nat,Xs: list_a,Ys: list_a] :
( ( ord_less_eq_nat @ K @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_size_list_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( nth_a @ Xs @ I2 )
= ( nth_a @ Ys @ I2 ) ) )
=> ( ( take_a @ K @ Xs )
= ( take_a @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_744_nth__take__lemma,axiom,
! [K: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys @ I2 ) ) )
=> ( ( take_nat @ K @ Xs )
= ( take_nat @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_745_add__diff__inverse__nat,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less_nat @ M2 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_746_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I )
= K )
= ( J2
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_747_less__diff__conv2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K ) @ I )
= ( ord_less_nat @ J2 @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_748_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_749_less__diff__conv,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ).
% less_diff_conv
thf(fact_750_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_751_Nat_Odiff__add__assoc,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J2 ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J2 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_752_less__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_753_Nat_Ole__diff__conv2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_754_le__diff__conv,axiom,
! [J2: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K ) @ I )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_755_list__update__same__conv,axiom,
! [I: nat,Xs: list_a,X: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ( list_update_a @ Xs @ I @ X )
= Xs )
= ( ( nth_a @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_756_list__update__same__conv,axiom,
! [I: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ( list_update_nat @ Xs @ I @ X )
= Xs )
= ( ( nth_nat @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_757_nth__list__update,axiom,
! [I: nat,Xs: list_a,J2: nat,X: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ( I = J2 )
=> ( ( nth_a @ ( list_update_a @ Xs @ I @ X ) @ J2 )
= X ) )
& ( ( I != J2 )
=> ( ( nth_a @ ( list_update_a @ Xs @ I @ X ) @ J2 )
= ( nth_a @ Xs @ J2 ) ) ) ) ) ).
% nth_list_update
thf(fact_758_nth__list__update,axiom,
! [I: nat,Xs: list_nat,J2: nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ( I = J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J2 )
= X ) )
& ( ( I != J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J2 )
= ( nth_nat @ Xs @ J2 ) ) ) ) ) ).
% nth_list_update
thf(fact_759_nth__butlast,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ ( butlast_a @ Xs ) ) )
=> ( ( nth_a @ ( butlast_a @ Xs ) @ N )
= ( nth_a @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_760_nth__butlast,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ ( butlast_nat @ Xs ) ) )
=> ( ( nth_nat @ ( butlast_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_761_list__all__length,axiom,
( list_all_a
= ( ^ [P2: a > $o,Xs4: list_a] :
! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs4 ) )
=> ( P2 @ ( nth_a @ Xs4 @ N3 ) ) ) ) ) ).
% list_all_length
thf(fact_762_list__all__length,axiom,
( list_all_nat
= ( ^ [P2: nat > $o,Xs4: list_nat] :
! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs4 ) )
=> ( P2 @ ( nth_nat @ Xs4 @ N3 ) ) ) ) ) ).
% list_all_length
thf(fact_763_add__Suc__shift,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( plus_plus_nat @ M2 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_764_add__Suc,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc
thf(fact_765_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_766_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_767_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_768_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_769_strict__inc__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J2 )
=> ( ! [I2: nat] :
( ( J2
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_770_less__Suc__induct,axiom,
! [I: nat,J2: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J2 )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J3: nat,K3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ( ord_less_nat @ J3 @ K3 )
=> ( ( P @ I2 @ J3 )
=> ( ( P @ J3 @ K3 )
=> ( P @ I2 @ K3 ) ) ) ) )
=> ( P @ I @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_771_less__trans__Suc,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_772_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_773_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_774_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M3: nat] :
( ( M2
= ( suc @ M3 ) )
& ( ord_less_nat @ N @ M3 ) ) ) ) ).
% Suc_less_eq2
thf(fact_775_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( suc @ N ) )
=> ( P @ I5 ) ) )
= ( ( P @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ N )
=> ( P @ I5 ) ) ) ) ).
% All_less_Suc
thf(fact_776_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_777_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_778_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I5: nat] :
( ( ord_less_nat @ I5 @ ( suc @ N ) )
& ( P @ I5 ) ) )
= ( ( P @ N )
| ? [I5: nat] :
( ( ord_less_nat @ I5 @ N )
& ( P @ I5 ) ) ) ) ).
% Ex_less_Suc
thf(fact_779_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_780_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_781_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_782_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_783_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_784_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_785_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_786_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_787_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_788_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_789_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_790_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_791_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_792_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z: nat] :
( ( R @ X3 @ Y3 )
=> ( ( R @ Y3 @ Z )
=> ( R @ X3 @ Z ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_793_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P @ M2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_794_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N2 )
=> ( P @ M5 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_795_not__less__eq__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_796_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_797_le__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M2 @ N )
| ( M2
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_798_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M: nat] :
( M6
= ( suc @ M ) ) ) ).
% Suc_le_D
thf(fact_799_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_800_le__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( M2
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_801_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_802_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_803_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_804_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_805_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_806_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% Nat.diff_cancel
thf(fact_807_diff__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_cancel2
thf(fact_808_diff__add__inverse,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
= M2 ) ).
% diff_add_inverse
thf(fact_809_diff__add__inverse2,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
= M2 ) ).
% diff_add_inverse2
thf(fact_810_length__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ! [Xs2: list_a] :
( ! [Ys6: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys6 ) @ ( size_size_list_a @ Xs2 ) )
=> ( P @ Ys6 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_811_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys6: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys6 ) @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ Ys6 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_812_diff__less__mono2,axiom,
! [M2: nat,N: nat,L2: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L2 )
=> ( ord_less_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_813_less__imp__diff__less,axiom,
! [J2: nat,K: nat,N: nat] :
( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_814_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_815_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_816_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_817_diff__le__mono,axiom,
! [M2: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L2 ) @ ( minus_minus_nat @ N @ L2 ) ) ) ).
% diff_le_mono
thf(fact_818_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_819_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_820_diff__le__mono2,axiom,
! [M2: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_821_nat__add__max__left,axiom,
! [M2: nat,N: nat,Q3: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ M2 @ N ) @ Q3 )
= ( ord_max_nat @ ( plus_plus_nat @ M2 @ Q3 ) @ ( plus_plus_nat @ N @ Q3 ) ) ) ).
% nat_add_max_left
thf(fact_822_nat__add__max__right,axiom,
! [M2: nat,N: nat,Q3: nat] :
( ( plus_plus_nat @ M2 @ ( ord_max_nat @ N @ Q3 ) )
= ( ord_max_nat @ ( plus_plus_nat @ M2 @ N ) @ ( plus_plus_nat @ M2 @ Q3 ) ) ) ).
% nat_add_max_right
thf(fact_823_nth__append,axiom,
! [N: nat,Xs: list_a,Ys: list_a] :
( ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
= ( nth_a @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
= ( nth_a @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_824_nth__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
= ( nth_nat @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
= ( nth_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_825_nth__tl,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ ( tl_a @ Xs ) ) )
=> ( ( nth_a @ ( tl_a @ Xs ) @ N )
= ( nth_a @ Xs @ ( suc @ N ) ) ) ) ).
% nth_tl
thf(fact_826_nth__tl,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ ( tl_nat @ Xs ) ) )
=> ( ( nth_nat @ ( tl_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ ( suc @ N ) ) ) ) ).
% nth_tl
thf(fact_827_filter__eq__nths,axiom,
( filter_a
= ( ^ [P2: a > $o,Xs4: list_a] :
( nths_a @ Xs4
@ ( collect_nat
@ ^ [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_a @ Xs4 ) )
& ( P2 @ ( nth_a @ Xs4 @ I5 ) ) ) ) ) ) ) ).
% filter_eq_nths
thf(fact_828_filter__eq__nths,axiom,
( filter_nat
= ( ^ [P2: nat > $o,Xs4: list_nat] :
( nths_nat @ Xs4
@ ( collect_nat
@ ^ [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs4 ) )
& ( P2 @ ( nth_nat @ Xs4 @ I5 ) ) ) ) ) ) ) ).
% filter_eq_nths
thf(fact_829_nth__non__equal__first__eq,axiom,
! [X: a,Y: a,Xs: list_a,N: nat] :
( ( X != Y )
=> ( ( ( nth_a @ ( cons_a @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_830_nth__non__equal__first__eq,axiom,
! [X: nat,Y: nat,Xs: list_nat,N: nat] :
( ( X != Y )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_831_nth__transpose,axiom,
! [I: nat,Xs: list_list_a] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ ( transpose_a @ Xs ) ) )
=> ( ( nth_list_a @ ( transpose_a @ Xs ) @ I )
= ( map_list_a_a
@ ^ [Xs4: list_a] : ( nth_a @ Xs4 @ I )
@ ( filter_list_a
@ ^ [Ys2: list_a] : ( ord_less_nat @ I @ ( size_size_list_a @ Ys2 ) )
@ Xs ) ) ) ) ).
% nth_transpose
thf(fact_832_nth__transpose,axiom,
! [I: nat,Xs: list_list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ ( transpose_nat @ Xs ) ) )
=> ( ( nth_list_nat @ ( transpose_nat @ Xs ) @ I )
= ( map_list_nat_nat
@ ^ [Xs4: list_nat] : ( nth_nat @ Xs4 @ I )
@ ( filter_list_nat
@ ^ [Ys2: list_nat] : ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
@ Xs ) ) ) ) ).
% nth_transpose
thf(fact_833_one__is__add,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_834_add__is__1,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_835_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J: nat] :
( ( M2
= ( suc @ J ) )
& ( ord_less_nat @ J @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_836_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M: nat] :
( N
= ( suc @ M ) ) ) ).
% gr0_implies_Suc
thf(fact_837_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( suc @ N ) )
=> ( P @ I5 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ N )
=> ( P @ ( suc @ I5 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_838_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_839_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I5: nat] :
( ( ord_less_nat @ I5 @ ( suc @ N ) )
& ( P @ I5 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I5: nat] :
( ( ord_less_nat @ I5 @ N )
& ( P @ ( suc @ I5 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_840_diff__add__0,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_841_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_842_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_843_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_844_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_845_Suc__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_846_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_847_impossible__Cons,axiom,
! [Xs: list_a,Ys: list_a,X: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
=> ( Xs
!= ( cons_a @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_848_impossible__Cons,axiom,
! [Xs: list_nat,Ys: list_nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
=> ( Xs
!= ( cons_nat @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_849_Suc__diff__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_850_length__filter__le,axiom,
! [P: a > $o,Xs: list_a] : ( ord_less_eq_nat @ ( size_size_list_a @ ( filter_a @ P @ Xs ) ) @ ( size_size_list_a @ Xs ) ) ).
% length_filter_le
thf(fact_851_length__filter__le,axiom,
! [P: nat > $o,Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( filter_nat @ P @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% length_filter_le
thf(fact_852_length__shuffles,axiom,
! [Zs: list_a,Xs: list_a,Ys: list_a] :
( ( member_list_a @ Zs @ ( shuffles_a @ Xs @ Ys ) )
=> ( ( size_size_list_a @ Zs )
= ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) ) ) ) ).
% length_shuffles
thf(fact_853_length__shuffles,axiom,
! [Zs: list_nat,Xs: list_nat,Ys: list_nat] :
( ( member_list_nat @ Zs @ ( shuffles_nat @ Xs @ Ys ) )
=> ( ( size_size_list_nat @ Zs )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ) ).
% length_shuffles
thf(fact_854_nat__minus__add__max,axiom,
! [N: nat,M2: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ N @ M2 ) @ M2 )
= ( ord_max_nat @ N @ M2 ) ) ).
% nat_minus_add_max
thf(fact_855_nths__all,axiom,
! [Xs: list_a,I3: set_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( member_nat @ I2 @ I3 ) )
=> ( ( nths_a @ Xs @ I3 )
= Xs ) ) ).
% nths_all
thf(fact_856_nths__all,axiom,
! [Xs: list_nat,I3: set_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ I2 @ I3 ) )
=> ( ( nths_nat @ Xs @ I3 )
= Xs ) ) ).
% nths_all
thf(fact_857_nth__Cons,axiom,
! [X: a,Xs: list_a,N: nat] :
( ( nth_a @ ( cons_a @ X @ Xs ) @ N )
= ( case_nat_a @ X @ ( nth_a @ Xs ) @ N ) ) ).
% nth_Cons
thf(fact_858_nth__Cons,axiom,
! [X: $o,Xs: list_o,N: nat] :
( ( nth_o @ ( cons_o @ X @ Xs ) @ N )
= ( case_nat_o @ X @ ( nth_o @ Xs ) @ N ) ) ).
% nth_Cons
thf(fact_859_nth__Cons,axiom,
! [X: nat,Xs: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= ( case_nat_nat @ X @ ( nth_nat @ Xs ) @ N ) ) ).
% nth_Cons
thf(fact_860_sum__length__filter__compl,axiom,
! [P: a > $o,Xs: list_a] :
( ( plus_plus_nat @ ( size_size_list_a @ ( filter_a @ P @ Xs ) )
@ ( size_size_list_a
@ ( filter_a
@ ^ [X4: a] :
~ ( P @ X4 )
@ Xs ) ) )
= ( size_size_list_a @ Xs ) ) ).
% sum_length_filter_compl
thf(fact_861_sum__length__filter__compl,axiom,
! [P: nat > $o,Xs: list_nat] :
( ( plus_plus_nat @ ( size_size_list_nat @ ( filter_nat @ P @ Xs ) )
@ ( size_size_list_nat
@ ( filter_nat
@ ^ [X4: nat] :
~ ( P @ X4 )
@ Xs ) ) )
= ( size_size_list_nat @ Xs ) ) ).
% sum_length_filter_compl
thf(fact_862_gen__length__def,axiom,
( gen_length_a
= ( ^ [N3: nat,Xs4: list_a] : ( plus_plus_nat @ N3 @ ( size_size_list_a @ Xs4 ) ) ) ) ).
% gen_length_def
thf(fact_863_gen__length__def,axiom,
( gen_length_nat
= ( ^ [N3: nat,Xs4: list_nat] : ( plus_plus_nat @ N3 @ ( size_size_list_nat @ Xs4 ) ) ) ) ).
% gen_length_def
thf(fact_864_take__Suc__conv__app__nth,axiom,
! [I: nat,Xs: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( take_a @ ( suc @ I ) @ Xs )
= ( append_a @ ( take_a @ I @ Xs ) @ ( cons_a @ ( nth_a @ Xs @ I ) @ nil_a ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_865_take__Suc__conv__app__nth,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( take_nat @ ( suc @ I ) @ Xs )
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I ) @ nil_nat ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_866_hd__conv__nth,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ Xs )
= ( nth_a @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_867_hd__conv__nth,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ Xs )
= ( nth_nat @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_868_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_869_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_a @ Xs ) )
= ( ? [X4: a,Ys2: list_a] :
( ( Xs
= ( cons_a @ X4 @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_a @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_870_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X4: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ X4 @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_871_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_872_less__eq__nat_Osimps_I2_J,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( case_nat_o @ $false @ ( ord_less_eq_nat @ M2 ) @ N ) ) ).
% less_eq_nat.simps(2)
thf(fact_873_list__update__append1,axiom,
! [I: nat,Xs: list_a,Ys: list_a,X: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ ( append_a @ Xs @ Ys ) @ I @ X )
= ( append_a @ ( list_update_a @ Xs @ I @ X ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_874_list__update__append1,axiom,
! [I: nat,Xs: list_nat,Ys: list_nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ I @ X )
= ( append_nat @ ( list_update_nat @ Xs @ I @ X ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_875_take__butlast,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( take_a @ N @ ( butlast_a @ Xs ) )
= ( take_a @ N @ Xs ) ) ) ).
% take_butlast
thf(fact_876_take__butlast,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( take_nat @ N @ ( butlast_nat @ Xs ) )
= ( take_nat @ N @ Xs ) ) ) ).
% take_butlast
thf(fact_877_nths__append,axiom,
! [L2: list_a,L3: list_a,A2: set_nat] :
( ( nths_a @ ( append_a @ L2 @ L3 ) @ A2 )
= ( append_a @ ( nths_a @ L2 @ A2 )
@ ( nths_a @ L3
@ ( collect_nat
@ ^ [J: nat] : ( member_nat @ ( plus_plus_nat @ J @ ( size_size_list_a @ L2 ) ) @ A2 ) ) ) ) ) ).
% nths_append
thf(fact_878_nths__append,axiom,
! [L2: list_nat,L3: list_nat,A2: set_nat] :
( ( nths_nat @ ( append_nat @ L2 @ L3 ) @ A2 )
= ( append_nat @ ( nths_nat @ L2 @ A2 )
@ ( nths_nat @ L3
@ ( collect_nat
@ ^ [J: nat] : ( member_nat @ ( plus_plus_nat @ J @ ( size_size_list_nat @ L2 ) ) @ A2 ) ) ) ) ) ).
% nths_append
thf(fact_879_nth__Cons_H,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( ( N = zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X @ Xs ) @ N )
= ( nth_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_880_nth__Cons_H,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_881_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_882_list_Osize_I4_J,axiom,
! [X21: nat,X22: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_883_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M4: nat,N3: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_884_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_885_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_886_Cons__less__Cons,axiom,
! [A: nat,X: list_nat,B: nat,Y: list_nat] :
( ( ord_less_list_nat @ ( cons_nat @ A @ X ) @ ( cons_nat @ B @ Y ) )
= ( ( ord_less_nat @ A @ B )
| ( ( A = B )
& ( ord_less_list_nat @ X @ Y ) ) ) ) ).
% Cons_less_Cons
thf(fact_887_Nitpick_Osize__list__simp_I1_J,axiom,
( size_list_a
= ( ^ [F2: a > nat,Xs4: list_a] : ( if_nat @ ( Xs4 = nil_a ) @ zero_zero_nat @ ( suc @ ( plus_plus_nat @ ( F2 @ ( hd_a @ Xs4 ) ) @ ( size_list_a @ F2 @ ( tl_a @ Xs4 ) ) ) ) ) ) ) ).
% Nitpick.size_list_simp(1)
thf(fact_888_Nitpick_Osize__list__simp_I1_J,axiom,
( size_list_nat
= ( ^ [F2: nat > nat,Xs4: list_nat] : ( if_nat @ ( Xs4 = nil_nat ) @ zero_zero_nat @ ( suc @ ( plus_plus_nat @ ( F2 @ ( hd_nat @ Xs4 ) ) @ ( size_list_nat @ F2 @ ( tl_nat @ Xs4 ) ) ) ) ) ) ) ).
% Nitpick.size_list_simp(1)
thf(fact_889_max__def__raw,axiom,
( ord_max_nat
= ( ^ [A3: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).
% max_def_raw
thf(fact_890_take__hd__drop,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( append_a @ ( take_a @ N @ Xs ) @ ( cons_a @ ( hd_a @ ( drop_a @ N @ Xs ) ) @ nil_a ) )
= ( take_a @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_891_take__hd__drop,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( append_nat @ ( take_nat @ N @ Xs ) @ ( cons_nat @ ( hd_nat @ ( drop_nat @ N @ Xs ) ) @ nil_nat ) )
= ( take_nat @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_892_transpose__rectangle,axiom,
! [Xs: list_list_a,N: nat] :
( ( ( Xs = nil_list_a )
=> ( N = zero_zero_nat ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( size_size_list_a @ ( nth_list_a @ Xs @ I2 ) )
= N ) )
=> ( ( transpose_a @ Xs )
= ( map_nat_list_a
@ ^ [I5: nat] :
( map_nat_a
@ ^ [J: nat] : ( nth_a @ ( nth_list_a @ Xs @ J ) @ I5 )
@ ( upt @ zero_zero_nat @ ( size_s349497388124573686list_a @ Xs ) ) )
@ ( upt @ zero_zero_nat @ N ) ) ) ) ) ).
% transpose_rectangle
thf(fact_893_transpose__rectangle,axiom,
! [Xs: list_list_nat,N: nat] :
( ( ( Xs = nil_list_nat )
=> ( N = zero_zero_nat ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I2 ) )
= N ) )
=> ( ( transpose_nat @ Xs )
= ( map_nat_list_nat
@ ^ [I5: nat] :
( map_nat_nat
@ ^ [J: nat] : ( nth_nat @ ( nth_list_nat @ Xs @ J ) @ I5 )
@ ( upt @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs ) ) )
@ ( upt @ zero_zero_nat @ N ) ) ) ) ) ).
% transpose_rectangle
thf(fact_894_le__Nil,axiom,
! [X: list_nat] :
( ( ord_less_eq_list_nat @ X @ nil_nat )
= ( X = nil_nat ) ) ).
% le_Nil
thf(fact_895_Cons__le__Cons,axiom,
! [A: nat,X: list_nat,B: nat,Y: list_nat] :
( ( ord_less_eq_list_nat @ ( cons_nat @ A @ X ) @ ( cons_nat @ B @ Y ) )
= ( ( ord_less_nat @ A @ B )
| ( ( A = B )
& ( ord_less_eq_list_nat @ X @ Y ) ) ) ) ).
% Cons_le_Cons
thf(fact_896_drop0,axiom,
( ( drop_nat @ zero_zero_nat )
= ( ^ [X4: list_nat] : X4 ) ) ).
% drop0
thf(fact_897_tl__upt,axiom,
! [M2: nat,N: nat] :
( ( tl_nat @ ( upt @ M2 @ N ) )
= ( upt @ ( suc @ M2 ) @ N ) ) ).
% tl_upt
thf(fact_898_hd__upt,axiom,
! [I: nat,J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( hd_nat @ ( upt @ I @ J2 ) )
= I ) ) ).
% hd_upt
thf(fact_899_upt__conv__Nil,axiom,
! [J2: nat,I: nat] :
( ( ord_less_eq_nat @ J2 @ I )
=> ( ( upt @ I @ J2 )
= nil_nat ) ) ).
% upt_conv_Nil
thf(fact_900_drop__drop,axiom,
! [N: nat,M2: nat,Xs: list_nat] :
( ( drop_nat @ N @ ( drop_nat @ M2 @ Xs ) )
= ( drop_nat @ ( plus_plus_nat @ N @ M2 ) @ Xs ) ) ).
% drop_drop
thf(fact_901_drop__upt,axiom,
! [M2: nat,I: nat,J2: nat] :
( ( drop_nat @ M2 @ ( upt @ I @ J2 ) )
= ( upt @ ( plus_plus_nat @ I @ M2 ) @ J2 ) ) ).
% drop_upt
thf(fact_902_length__upt,axiom,
! [I: nat,J2: nat] :
( ( size_size_list_nat @ ( upt @ I @ J2 ) )
= ( minus_minus_nat @ J2 @ I ) ) ).
% length_upt
thf(fact_903_drop__Suc__Cons,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( drop_a @ ( suc @ N ) @ ( cons_a @ X @ Xs ) )
= ( drop_a @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_904_drop__Suc__Cons,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( drop_nat @ ( suc @ N ) @ ( cons_nat @ X @ Xs ) )
= ( drop_nat @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_905_upt__eq__Nil__conv,axiom,
! [I: nat,J2: nat] :
( ( ( upt @ I @ J2 )
= nil_nat )
= ( ( J2 = zero_zero_nat )
| ( ord_less_eq_nat @ J2 @ I ) ) ) ).
% upt_eq_Nil_conv
thf(fact_906_nth__upt,axiom,
! [I: nat,K: nat,J2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J2 )
=> ( ( nth_nat @ ( upt @ I @ J2 ) @ K )
= ( plus_plus_nat @ I @ K ) ) ) ).
% nth_upt
thf(fact_907_take__upt,axiom,
! [I: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M2 ) @ N )
=> ( ( take_nat @ M2 @ ( upt @ I @ N ) )
= ( upt @ I @ ( plus_plus_nat @ I @ M2 ) ) ) ) ).
% take_upt
thf(fact_908_length__drop,axiom,
! [N: nat,Xs: list_a] :
( ( size_size_list_a @ ( drop_a @ N @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% length_drop
thf(fact_909_length__drop,axiom,
! [N: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( drop_nat @ N @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% length_drop
thf(fact_910_append__take__drop__id,axiom,
! [N: nat,Xs: list_a] :
( ( append_a @ ( take_a @ N @ Xs ) @ ( drop_a @ N @ Xs ) )
= Xs ) ).
% append_take_drop_id
thf(fact_911_append__take__drop__id,axiom,
! [N: nat,Xs: list_nat] :
( ( append_nat @ ( take_nat @ N @ Xs ) @ ( drop_nat @ N @ Xs ) )
= Xs ) ).
% append_take_drop_id
thf(fact_912_drop__update__cancel,axiom,
! [N: nat,M2: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( ( drop_nat @ M2 @ ( list_update_nat @ Xs @ N @ X ) )
= ( drop_nat @ M2 @ Xs ) ) ) ).
% drop_update_cancel
thf(fact_913_size__list__append,axiom,
! [F: a > nat,Xs: list_a,Ys: list_a] :
( ( size_list_a @ F @ ( append_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_list_a @ F @ Xs ) @ ( size_list_a @ F @ Ys ) ) ) ).
% size_list_append
thf(fact_914_size__list__append,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( size_list_nat @ F @ ( append_nat @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_list_nat @ F @ Xs ) @ ( size_list_nat @ F @ Ys ) ) ) ).
% size_list_append
thf(fact_915_upt__rec__numeral,axiom,
! [M2: num,N: num] :
( ( ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= ( cons_nat @ ( numeral_numeral_nat @ M2 ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
& ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= nil_nat ) ) ) ).
% upt_rec_numeral
thf(fact_916_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_a] :
( ( nil_a
= ( drop_a @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_917_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_nat] :
( ( nil_nat
= ( drop_nat @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_918_drop__eq__Nil,axiom,
! [N: nat,Xs: list_a] :
( ( ( drop_a @ N @ Xs )
= nil_a )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_919_drop__eq__Nil,axiom,
! [N: nat,Xs: list_nat] :
( ( ( drop_nat @ N @ Xs )
= nil_nat )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_920_drop__all,axiom,
! [Xs: list_a,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
=> ( ( drop_a @ N @ Xs )
= nil_a ) ) ).
% drop_all
thf(fact_921_drop__all,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
=> ( ( drop_nat @ N @ Xs )
= nil_nat ) ) ).
% drop_all
thf(fact_922_drop__append,axiom,
! [N: nat,Xs: list_a,Ys: list_a] :
( ( drop_a @ N @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( drop_a @ N @ Xs ) @ ( drop_a @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_923_drop__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( drop_nat @ N @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( drop_nat @ N @ Xs ) @ ( drop_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_924_last__upt,axiom,
! [I: nat,J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( last_nat @ ( upt @ I @ J2 ) )
= ( minus_minus_nat @ J2 @ one_one_nat ) ) ) ).
% last_upt
thf(fact_925_last__drop,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( last_a @ ( drop_a @ N @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_drop
thf(fact_926_last__drop,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( last_nat @ ( drop_nat @ N @ Xs ) )
= ( last_nat @ Xs ) ) ) ).
% last_drop
thf(fact_927_drop__Cons__numeral,axiom,
! [V: num,X: a,Xs: list_a] :
( ( drop_a @ ( numeral_numeral_nat @ V ) @ ( cons_a @ X @ Xs ) )
= ( drop_a @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) @ Xs ) ) ).
% drop_Cons_numeral
thf(fact_928_drop__Cons__numeral,axiom,
! [V: num,X: nat,Xs: list_nat] :
( ( drop_nat @ ( numeral_numeral_nat @ V ) @ ( cons_nat @ X @ Xs ) )
= ( drop_nat @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) @ Xs ) ) ).
% drop_Cons_numeral
thf(fact_929_nth__drop,axiom,
! [N: nat,Xs: list_a,I: nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( drop_a @ N @ Xs ) @ I )
= ( nth_a @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_930_nth__drop,axiom,
! [N: nat,Xs: list_nat,I: nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( drop_nat @ N @ Xs ) @ I )
= ( nth_nat @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_931_not__less__Nil,axiom,
! [X: list_nat] :
~ ( ord_less_list_nat @ X @ nil_nat ) ).
% not_less_Nil
thf(fact_932_less__list__code_I1_J,axiom,
! [Xs: list_nat] :
~ ( ord_less_list_nat @ Xs @ nil_nat ) ).
% less_list_code(1)
thf(fact_933_Nil__le__Cons,axiom,
! [X: list_nat] : ( ord_less_eq_list_nat @ nil_nat @ X ) ).
% Nil_le_Cons
thf(fact_934_less__eq__list__code_I2_J,axiom,
! [Xs: list_nat] : ( ord_less_eq_list_nat @ nil_nat @ Xs ) ).
% less_eq_list_code(2)
thf(fact_935_drop__0,axiom,
! [Xs: list_nat] :
( ( drop_nat @ zero_zero_nat @ Xs )
= Xs ) ).
% drop_0
thf(fact_936_drop__Nil,axiom,
! [N: nat] :
( ( drop_a @ N @ nil_a )
= nil_a ) ).
% drop_Nil
thf(fact_937_drop__Nil,axiom,
! [N: nat] :
( ( drop_nat @ N @ nil_nat )
= nil_nat ) ).
% drop_Nil
thf(fact_938_map__Suc__upt,axiom,
! [M2: nat,N: nat] :
( ( map_nat_nat @ suc @ ( upt @ M2 @ N ) )
= ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_939_upt__conv__Cons__Cons,axiom,
! [M2: nat,N: nat,Ns: list_nat,Q3: nat] :
( ( ( cons_nat @ M2 @ ( cons_nat @ N @ Ns ) )
= ( upt @ M2 @ Q3 ) )
= ( ( cons_nat @ N @ Ns )
= ( upt @ ( suc @ M2 ) @ Q3 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_940_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ zero_zero_nat )
= nil_nat ) ).
% upt_0
thf(fact_941_drop__map,axiom,
! [N: nat,F: nat > nat,Xs: list_nat] :
( ( drop_nat @ N @ ( map_nat_nat @ F @ Xs ) )
= ( map_nat_nat @ F @ ( drop_nat @ N @ Xs ) ) ) ).
% drop_map
thf(fact_942_tl__drop,axiom,
! [N: nat,Xs: list_nat] :
( ( tl_nat @ ( drop_nat @ N @ Xs ) )
= ( drop_nat @ N @ ( tl_nat @ Xs ) ) ) ).
% tl_drop
thf(fact_943_drop__butlast,axiom,
! [N: nat,Xs: list_nat] :
( ( drop_nat @ N @ ( butlast_nat @ Xs ) )
= ( butlast_nat @ ( drop_nat @ N @ Xs ) ) ) ).
% drop_butlast
thf(fact_944_less__eq__list__code_I3_J,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ord_less_eq_list_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( ( ord_less_nat @ X @ Y )
| ( ( X = Y )
& ( ord_less_eq_list_nat @ Xs @ Ys ) ) ) ) ).
% less_eq_list_code(3)
thf(fact_945_Nil__less__Cons,axiom,
! [A: nat,X: list_nat] : ( ord_less_list_nat @ nil_nat @ ( cons_nat @ A @ X ) ) ).
% Nil_less_Cons
thf(fact_946_less__list__code_I2_J,axiom,
! [X: nat,Xs: list_nat] : ( ord_less_list_nat @ nil_nat @ ( cons_nat @ X @ Xs ) ) ).
% less_list_code(2)
thf(fact_947_less__eq__list__code_I1_J,axiom,
! [X: nat,Xs: list_nat] :
~ ( ord_less_eq_list_nat @ ( cons_nat @ X @ Xs ) @ nil_nat ) ).
% less_eq_list_code(1)
thf(fact_948_upt__conv__Cons,axiom,
! [I: nat,J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( upt @ I @ J2 )
= ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J2 ) ) ) ) ).
% upt_conv_Cons
thf(fact_949_upt__rec,axiom,
( upt
= ( ^ [I5: nat,J: nat] : ( if_list_nat @ ( ord_less_nat @ I5 @ J ) @ ( cons_nat @ I5 @ ( upt @ ( suc @ I5 ) @ J ) ) @ nil_nat ) ) ) ).
% upt_rec
thf(fact_950_upt__Suc,axiom,
! [I: nat,J2: nat] :
( ( ( ord_less_eq_nat @ I @ J2 )
=> ( ( upt @ I @ ( suc @ J2 ) )
= ( append_nat @ ( upt @ I @ J2 ) @ ( cons_nat @ J2 @ nil_nat ) ) ) )
& ( ~ ( ord_less_eq_nat @ I @ J2 )
=> ( ( upt @ I @ ( suc @ J2 ) )
= nil_nat ) ) ) ).
% upt_Suc
thf(fact_951_upt__Suc__append,axiom,
! [I: nat,J2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( upt @ I @ ( suc @ J2 ) )
= ( append_nat @ ( upt @ I @ J2 ) @ ( cons_nat @ J2 @ nil_nat ) ) ) ) ).
% upt_Suc_append
thf(fact_952_upt__add__eq__append,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( upt @ I @ ( plus_plus_nat @ J2 @ K ) )
= ( append_nat @ ( upt @ I @ J2 ) @ ( upt @ J2 @ ( plus_plus_nat @ J2 @ K ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_953_nth__via__drop,axiom,
! [N: nat,Xs: list_a,Y: a,Ys: list_a] :
( ( ( drop_a @ N @ Xs )
= ( cons_a @ Y @ Ys ) )
=> ( ( nth_a @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_954_nth__via__drop,axiom,
! [N: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( drop_nat @ N @ Xs )
= ( cons_nat @ Y @ Ys ) )
=> ( ( nth_nat @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_955_take__drop,axiom,
! [N: nat,M2: nat,Xs: list_nat] :
( ( take_nat @ N @ ( drop_nat @ M2 @ Xs ) )
= ( drop_nat @ M2 @ ( take_nat @ ( plus_plus_nat @ N @ M2 ) @ Xs ) ) ) ).
% take_drop
thf(fact_956_drop__take,axiom,
! [N: nat,M2: nat,Xs: list_nat] :
( ( drop_nat @ N @ ( take_nat @ M2 @ Xs ) )
= ( take_nat @ ( minus_minus_nat @ M2 @ N ) @ ( drop_nat @ N @ Xs ) ) ) ).
% drop_take
thf(fact_957_drop__Suc,axiom,
! [N: nat,Xs: list_nat] :
( ( drop_nat @ ( suc @ N ) @ Xs )
= ( drop_nat @ N @ ( tl_nat @ Xs ) ) ) ).
% drop_Suc
thf(fact_958_map__add__upt,axiom,
! [N: nat,M2: nat] :
( ( map_nat_nat
@ ^ [I5: nat] : ( plus_plus_nat @ I5 @ N )
@ ( upt @ zero_zero_nat @ M2 ) )
= ( upt @ N @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% map_add_upt
thf(fact_959_drop__eq__nths,axiom,
( drop_nat
= ( ^ [N3: nat,Xs4: list_nat] : ( nths_nat @ Xs4 @ ( collect_nat @ ( ord_less_eq_nat @ N3 ) ) ) ) ) ).
% drop_eq_nths
thf(fact_960_list_Osize__gen_I1_J,axiom,
! [X: a > nat] :
( ( size_list_a @ X @ nil_a )
= zero_zero_nat ) ).
% list.size_gen(1)
thf(fact_961_list_Osize__gen_I1_J,axiom,
! [X: nat > nat] :
( ( size_list_nat @ X @ nil_nat )
= zero_zero_nat ) ).
% list.size_gen(1)
thf(fact_962_upt__eq__Cons__conv,axiom,
! [I: nat,J2: nat,X: nat,Xs: list_nat] :
( ( ( upt @ I @ J2 )
= ( cons_nat @ X @ Xs ) )
= ( ( ord_less_nat @ I @ J2 )
& ( I = X )
& ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J2 )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_963_append__eq__conv__conj,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( append_a @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take_a @ ( size_size_list_a @ Xs ) @ Zs ) )
& ( Ys
= ( drop_a @ ( size_size_list_a @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_964_append__eq__conv__conj,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take_nat @ ( size_size_list_nat @ Xs ) @ Zs ) )
& ( Ys
= ( drop_nat @ ( size_size_list_nat @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_965_take__add,axiom,
! [I: nat,J2: nat,Xs: list_a] :
( ( take_a @ ( plus_plus_nat @ I @ J2 ) @ Xs )
= ( append_a @ ( take_a @ I @ Xs ) @ ( take_a @ J2 @ ( drop_a @ I @ Xs ) ) ) ) ).
% take_add
thf(fact_966_take__add,axiom,
! [I: nat,J2: nat,Xs: list_nat] :
( ( take_nat @ ( plus_plus_nat @ I @ J2 ) @ Xs )
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( take_nat @ J2 @ ( drop_nat @ I @ Xs ) ) ) ) ).
% take_add
thf(fact_967_drop__update__swap,axiom,
! [M2: nat,N: nat,Xs: list_nat,X: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( drop_nat @ M2 @ ( list_update_nat @ Xs @ N @ X ) )
= ( list_update_nat @ ( drop_nat @ M2 @ Xs ) @ ( minus_minus_nat @ N @ M2 ) @ X ) ) ) ).
% drop_update_swap
thf(fact_968_drop__Cons,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( drop_a @ N @ ( cons_a @ X @ Xs ) )
= ( case_nat_list_a @ ( cons_a @ X @ Xs )
@ ^ [M4: nat] : ( drop_a @ M4 @ Xs )
@ N ) ) ).
% drop_Cons
thf(fact_969_drop__Cons,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( drop_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( case_nat_list_nat @ ( cons_nat @ X @ Xs )
@ ^ [M4: nat] : ( drop_nat @ M4 @ Xs )
@ N ) ) ).
% drop_Cons
thf(fact_970_map__upt__Suc,axiom,
! [F: nat > a,N: nat] :
( ( map_nat_a @ F @ ( upt @ zero_zero_nat @ ( suc @ N ) ) )
= ( cons_a @ ( F @ zero_zero_nat )
@ ( map_nat_a
@ ^ [I5: nat] : ( F @ ( suc @ I5 ) )
@ ( upt @ zero_zero_nat @ N ) ) ) ) ).
% map_upt_Suc
thf(fact_971_map__upt__Suc,axiom,
! [F: nat > nat,N: nat] :
( ( map_nat_nat @ F @ ( upt @ zero_zero_nat @ ( suc @ N ) ) )
= ( cons_nat @ ( F @ zero_zero_nat )
@ ( map_nat_nat
@ ^ [I5: nat] : ( F @ ( suc @ I5 ) )
@ ( upt @ zero_zero_nat @ N ) ) ) ) ).
% map_upt_Suc
thf(fact_972_map__decr__upt,axiom,
! [M2: nat,N: nat] :
( ( map_nat_nat
@ ^ [N3: nat] : ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) )
@ ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( upt @ M2 @ N ) ) ).
% map_decr_upt
thf(fact_973_map__nth,axiom,
! [Xs: list_a] :
( ( map_nat_a @ ( nth_a @ Xs ) @ ( upt @ zero_zero_nat @ ( size_size_list_a @ Xs ) ) )
= Xs ) ).
% map_nth
thf(fact_974_map__nth,axiom,
! [Xs: list_nat] :
( ( map_nat_nat @ ( nth_nat @ Xs ) @ ( upt @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) )
= Xs ) ).
% map_nth
thf(fact_975_drop__Cons_H,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( ( N = zero_zero_nat )
=> ( ( drop_a @ N @ ( cons_a @ X @ Xs ) )
= ( cons_a @ X @ Xs ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_a @ N @ ( cons_a @ X @ Xs ) )
= ( drop_a @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_976_drop__Cons_H,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( drop_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ Xs ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( drop_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_977_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_978_append__eq__append__conv__if,axiom,
! [Xs_1: list_nat,Xs_2: list_nat,Ys_1: list_nat,Ys_2: list_nat] :
( ( ( append_nat @ Xs_1 @ Xs_2 )
= ( append_nat @ Ys_1 @ Ys_2 ) )
= ( ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs_1 ) @ ( size_size_list_nat @ Ys_1 ) )
=> ( ( Xs_1
= ( take_nat @ ( size_size_list_nat @ Xs_1 ) @ Ys_1 ) )
& ( Xs_2
= ( append_nat @ ( drop_nat @ ( size_size_list_nat @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( size_size_list_nat @ Xs_1 ) @ ( size_size_list_nat @ Ys_1 ) )
=> ( ( ( take_nat @ ( size_size_list_nat @ Ys_1 ) @ Xs_1 )
= Ys_1 )
& ( ( append_nat @ ( drop_nat @ ( size_size_list_nat @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
= Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
thf(fact_979_nth__map__upt,axiom,
! [I: nat,N: nat,M2: nat,F: nat > nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ N @ M2 ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ ( upt @ M2 @ N ) ) @ I )
= ( F @ ( plus_plus_nat @ M2 @ I ) ) ) ) ).
% nth_map_upt
thf(fact_980_hd__drop__conv__nth,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( hd_a @ ( drop_a @ N @ Xs ) )
= ( nth_a @ Xs @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_981_hd__drop__conv__nth,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( hd_nat @ ( drop_nat @ N @ Xs ) )
= ( nth_nat @ Xs @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_982_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( cons_a @ ( nth_a @ Xs @ I ) @ ( drop_a @ ( suc @ I ) @ Xs ) )
= ( drop_a @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_983_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( cons_nat @ ( nth_nat @ Xs @ I ) @ ( drop_nat @ ( suc @ I ) @ Xs ) )
= ( drop_nat @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_984_map__upt__eqI,axiom,
! [Xs: list_a,N: nat,M2: nat,F: nat > a] :
( ( ( size_size_list_a @ Xs )
= ( minus_minus_nat @ N @ M2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ Xs @ I2 )
= ( F @ ( plus_plus_nat @ M2 @ I2 ) ) ) )
=> ( ( map_nat_a @ F @ ( upt @ M2 @ N ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_985_map__upt__eqI,axiom,
! [Xs: list_nat,N: nat,M2: nat,F: nat > nat] :
( ( ( size_size_list_nat @ Xs )
= ( minus_minus_nat @ N @ M2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I2 )
= ( F @ ( plus_plus_nat @ M2 @ I2 ) ) ) )
=> ( ( map_nat_nat @ F @ ( upt @ M2 @ N ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_986_list_Osize__gen_I2_J,axiom,
! [X: a > nat,X21: a,X22: list_a] :
( ( size_list_a @ X @ ( cons_a @ X21 @ X22 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( X @ X21 ) @ ( size_list_a @ X @ X22 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size_gen(2)
thf(fact_987_list_Osize__gen_I2_J,axiom,
! [X: nat > nat,X21: nat,X22: list_nat] :
( ( size_list_nat @ X @ ( cons_nat @ X21 @ X22 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( X @ X21 ) @ ( size_list_nat @ X @ X22 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size_gen(2)
thf(fact_988_less__list__code_I3_J,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ord_less_list_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( ( ord_less_nat @ X @ Y )
| ( ( X = Y )
& ( ord_less_list_nat @ Xs @ Ys ) ) ) ) ).
% less_list_code(3)
thf(fact_989_id__take__nth__drop,axiom,
! [I: nat,Xs: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( Xs
= ( append_a @ ( take_a @ I @ Xs ) @ ( cons_a @ ( nth_a @ Xs @ I ) @ ( drop_a @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_990_id__take__nth__drop,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( Xs
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I ) @ ( drop_nat @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_991_upd__conv__take__nth__drop,axiom,
! [I: nat,Xs: list_a,A: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ Xs @ I @ A )
= ( append_a @ ( take_a @ I @ Xs ) @ ( cons_a @ A @ ( drop_a @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_992_upd__conv__take__nth__drop,axiom,
! [I: nat,Xs: list_nat,A: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ Xs @ I @ A )
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( cons_nat @ A @ ( drop_nat @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_993_max__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
@ ^ [X4: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X4 )
@ ^ [X4: nat,Y2: nat] : ( ord_less_nat @ Y2 @ X4 ) ) ).
% max_nat.semilattice_neutr_order_axioms
thf(fact_994_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_995_append__one__prefix,axiom,
! [Xs: list_a,Ys: list_a] :
( ( prefix_a @ Xs @ Ys )
=> ( ( ord_less_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
=> ( prefix_a @ ( append_a @ Xs @ ( cons_a @ ( nth_a @ Ys @ ( size_size_list_a @ Xs ) ) @ nil_a ) ) @ Ys ) ) ) ).
% append_one_prefix
thf(fact_996_append__one__prefix,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( prefix_nat @ Xs @ Ys )
=> ( ( ord_less_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
=> ( prefix_nat @ ( append_nat @ Xs @ ( cons_nat @ ( nth_nat @ Ys @ ( size_size_list_nat @ Xs ) ) @ nil_nat ) ) @ Ys ) ) ) ).
% append_one_prefix
thf(fact_997_Cons__prefix__Cons,axiom,
! [X: a,Xs: list_a,Y: a,Ys: list_a] :
( ( prefix_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) )
= ( ( X = Y )
& ( prefix_a @ Xs @ Ys ) ) ) ).
% Cons_prefix_Cons
thf(fact_998_Cons__prefix__Cons,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( prefix_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( ( X = Y )
& ( prefix_nat @ Xs @ Ys ) ) ) ).
% Cons_prefix_Cons
thf(fact_999_prefix__Nil,axiom,
! [Xs: list_a] :
( ( prefix_a @ Xs @ nil_a )
= ( Xs = nil_a ) ) ).
% prefix_Nil
thf(fact_1000_prefix__Nil,axiom,
! [Xs: list_nat] :
( ( prefix_nat @ Xs @ nil_nat )
= ( Xs = nil_nat ) ) ).
% prefix_Nil
thf(fact_1001_prefix__bot_Oextremum__unique,axiom,
! [A: list_a] :
( ( prefix_a @ A @ nil_a )
= ( A = nil_a ) ) ).
% prefix_bot.extremum_unique
thf(fact_1002_prefix__bot_Oextremum__unique,axiom,
! [A: list_nat] :
( ( prefix_nat @ A @ nil_nat )
= ( A = nil_nat ) ) ).
% prefix_bot.extremum_unique
thf(fact_1003_prefix__code_I1_J,axiom,
! [Xs: list_a] : ( prefix_a @ nil_a @ Xs ) ).
% prefix_code(1)
thf(fact_1004_prefix__code_I1_J,axiom,
! [Xs: list_nat] : ( prefix_nat @ nil_nat @ Xs ) ).
% prefix_code(1)
thf(fact_1005_same__prefix__prefix,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( prefix_a @ ( append_a @ Xs @ Ys ) @ ( append_a @ Xs @ Zs ) )
= ( prefix_a @ Ys @ Zs ) ) ).
% same_prefix_prefix
thf(fact_1006_same__prefix__prefix,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( prefix_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) )
= ( prefix_nat @ Ys @ Zs ) ) ).
% same_prefix_prefix
thf(fact_1007_same__prefix__nil,axiom,
! [Xs: list_a,Ys: list_a] :
( ( prefix_a @ ( append_a @ Xs @ Ys ) @ Xs )
= ( Ys = nil_a ) ) ).
% same_prefix_nil
thf(fact_1008_same__prefix__nil,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( prefix_nat @ ( append_nat @ Xs @ Ys ) @ Xs )
= ( Ys = nil_nat ) ) ).
% same_prefix_nil
thf(fact_1009_prefix__snoc,axiom,
! [Xs: list_a,Ys: list_a,Y: a] :
( ( prefix_a @ Xs @ ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
= ( ( Xs
= ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
| ( prefix_a @ Xs @ Ys ) ) ) ).
% prefix_snoc
thf(fact_1010_prefix__snoc,axiom,
! [Xs: list_nat,Ys: list_nat,Y: nat] :
( ( prefix_nat @ Xs @ ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) )
= ( ( Xs
= ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) )
| ( prefix_nat @ Xs @ Ys ) ) ) ).
% prefix_snoc
thf(fact_1011_Nil__prefix,axiom,
! [Xs: list_a] : ( prefix_a @ nil_a @ Xs ) ).
% Nil_prefix
thf(fact_1012_Nil__prefix,axiom,
! [Xs: list_nat] : ( prefix_nat @ nil_nat @ Xs ) ).
% Nil_prefix
thf(fact_1013_prefix__bot_Oextremum__uniqueI,axiom,
! [A: list_a] :
( ( prefix_a @ A @ nil_a )
=> ( A = nil_a ) ) ).
% prefix_bot.extremum_uniqueI
thf(fact_1014_prefix__bot_Oextremum__uniqueI,axiom,
! [A: list_nat] :
( ( prefix_nat @ A @ nil_nat )
=> ( A = nil_nat ) ) ).
% prefix_bot.extremum_uniqueI
thf(fact_1015_prefix__bot_Obot__least,axiom,
! [A: list_a] : ( prefix_a @ nil_a @ A ) ).
% prefix_bot.bot_least
thf(fact_1016_prefix__bot_Obot__least,axiom,
! [A: list_nat] : ( prefix_nat @ nil_nat @ A ) ).
% prefix_bot.bot_least
thf(fact_1017_append__prefixD,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( prefix_a @ ( append_a @ Xs @ Ys ) @ Zs )
=> ( prefix_a @ Xs @ Zs ) ) ).
% append_prefixD
thf(fact_1018_append__prefixD,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( prefix_nat @ ( append_nat @ Xs @ Ys ) @ Zs )
=> ( prefix_nat @ Xs @ Zs ) ) ).
% append_prefixD
thf(fact_1019_prefix__prefix,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( prefix_a @ Xs @ Ys )
=> ( prefix_a @ Xs @ ( append_a @ Ys @ Zs ) ) ) ).
% prefix_prefix
thf(fact_1020_prefix__prefix,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( prefix_nat @ Xs @ Ys )
=> ( prefix_nat @ Xs @ ( append_nat @ Ys @ Zs ) ) ) ).
% prefix_prefix
thf(fact_1021_prefix__append,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( prefix_a @ Xs @ ( append_a @ Ys @ Zs ) )
= ( ( prefix_a @ Xs @ Ys )
| ? [Us2: list_a] :
( ( Xs
= ( append_a @ Ys @ Us2 ) )
& ( prefix_a @ Us2 @ Zs ) ) ) ) ).
% prefix_append
thf(fact_1022_prefix__append,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( prefix_nat @ Xs @ ( append_nat @ Ys @ Zs ) )
= ( ( prefix_nat @ Xs @ Ys )
| ? [Us2: list_nat] :
( ( Xs
= ( append_nat @ Ys @ Us2 ) )
& ( prefix_nat @ Us2 @ Zs ) ) ) ) ).
% prefix_append
thf(fact_1023_prefix__def,axiom,
( prefix_a
= ( ^ [Xs4: list_a,Ys2: list_a] :
? [Zs3: list_a] :
( Ys2
= ( append_a @ Xs4 @ Zs3 ) ) ) ) ).
% prefix_def
thf(fact_1024_prefix__def,axiom,
( prefix_nat
= ( ^ [Xs4: list_nat,Ys2: list_nat] :
? [Zs3: list_nat] :
( Ys2
= ( append_nat @ Xs4 @ Zs3 ) ) ) ) ).
% prefix_def
thf(fact_1025_prefixI,axiom,
! [Ys: list_a,Xs: list_a,Zs: list_a] :
( ( Ys
= ( append_a @ Xs @ Zs ) )
=> ( prefix_a @ Xs @ Ys ) ) ).
% prefixI
thf(fact_1026_prefixI,axiom,
! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( Ys
= ( append_nat @ Xs @ Zs ) )
=> ( prefix_nat @ Xs @ Ys ) ) ).
% prefixI
thf(fact_1027_prefixE,axiom,
! [Xs: list_a,Ys: list_a] :
( ( prefix_a @ Xs @ Ys )
=> ~ ! [Zs2: list_a] :
( Ys
!= ( append_a @ Xs @ Zs2 ) ) ) ).
% prefixE
thf(fact_1028_prefixE,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( prefix_nat @ Xs @ Ys )
=> ~ ! [Zs2: list_nat] :
( Ys
!= ( append_nat @ Xs @ Zs2 ) ) ) ).
% prefixE
thf(fact_1029_prefix__map__rightE,axiom,
! [Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( prefix_nat @ Xs @ ( map_nat_nat @ F @ Ys ) )
=> ? [Xs3: list_nat] :
( ( prefix_nat @ Xs3 @ Ys )
& ( Xs
= ( map_nat_nat @ F @ Xs3 ) ) ) ) ).
% prefix_map_rightE
thf(fact_1030_map__mono__prefix,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > nat] :
( ( prefix_nat @ Xs @ Ys )
=> ( prefix_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ F @ Ys ) ) ) ).
% map_mono_prefix
thf(fact_1031_take__is__prefix,axiom,
! [N: nat,Xs: list_nat] : ( prefix_nat @ ( take_nat @ N @ Xs ) @ Xs ) ).
% take_is_prefix
thf(fact_1032_prefix__code_I2_J,axiom,
! [X: a,Xs: list_a] :
~ ( prefix_a @ ( cons_a @ X @ Xs ) @ nil_a ) ).
% prefix_code(2)
thf(fact_1033_prefix__code_I2_J,axiom,
! [X: nat,Xs: list_nat] :
~ ( prefix_nat @ ( cons_nat @ X @ Xs ) @ nil_nat ) ).
% prefix_code(2)
thf(fact_1034_prefix__Cons,axiom,
! [Xs: list_a,Y: a,Ys: list_a] :
( ( prefix_a @ Xs @ ( cons_a @ Y @ Ys ) )
= ( ( Xs = nil_a )
| ? [Zs3: list_a] :
( ( Xs
= ( cons_a @ Y @ Zs3 ) )
& ( prefix_a @ Zs3 @ Ys ) ) ) ) ).
% prefix_Cons
thf(fact_1035_prefix__Cons,axiom,
! [Xs: list_nat,Y: nat,Ys: list_nat] :
( ( prefix_nat @ Xs @ ( cons_nat @ Y @ Ys ) )
= ( ( Xs = nil_nat )
| ? [Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Y @ Zs3 ) )
& ( prefix_nat @ Zs3 @ Ys ) ) ) ) ).
% prefix_Cons
thf(fact_1036_not__prefix__cases,axiom,
! [Ps2: list_a,Ls: list_a] :
( ~ ( prefix_a @ Ps2 @ Ls )
=> ( ( ( Ps2 != nil_a )
=> ( Ls != nil_a ) )
=> ( ! [A4: a,As: list_a] :
( ( Ps2
= ( cons_a @ A4 @ As ) )
=> ! [X3: a,Xs2: list_a] :
( ( Ls
= ( cons_a @ X3 @ Xs2 ) )
=> ( ( X3 = A4 )
=> ( prefix_a @ As @ Xs2 ) ) ) )
=> ~ ! [A4: a] :
( ? [As: list_a] :
( Ps2
= ( cons_a @ A4 @ As ) )
=> ! [X3: a] :
( ? [Xs2: list_a] :
( Ls
= ( cons_a @ X3 @ Xs2 ) )
=> ( X3 = A4 ) ) ) ) ) ) ).
% not_prefix_cases
thf(fact_1037_not__prefix__cases,axiom,
! [Ps2: list_nat,Ls: list_nat] :
( ~ ( prefix_nat @ Ps2 @ Ls )
=> ( ( ( Ps2 != nil_nat )
=> ( Ls != nil_nat ) )
=> ( ! [A4: nat,As: list_nat] :
( ( Ps2
= ( cons_nat @ A4 @ As ) )
=> ! [X3: nat,Xs2: list_nat] :
( ( Ls
= ( cons_nat @ X3 @ Xs2 ) )
=> ( ( X3 = A4 )
=> ( prefix_nat @ As @ Xs2 ) ) ) )
=> ~ ! [A4: nat] :
( ? [As: list_nat] :
( Ps2
= ( cons_nat @ A4 @ As ) )
=> ! [X3: nat] :
( ? [Xs2: list_nat] :
( Ls
= ( cons_nat @ X3 @ Xs2 ) )
=> ( X3 = A4 ) ) ) ) ) ) ).
% not_prefix_cases
thf(fact_1038_not__prefix__induct,axiom,
! [Ps2: list_a,Ls: list_a,P: list_a > list_a > $o] :
( ~ ( prefix_a @ Ps2 @ Ls )
=> ( ! [X3: a,Xs2: list_a] : ( P @ ( cons_a @ X3 @ Xs2 ) @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys3: list_a] :
( ( X3 != Y3 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) ) )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys3: list_a] :
( ( X3 = Y3 )
=> ( ~ ( prefix_a @ Xs2 @ Ys3 )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys3 ) ) ) ) )
=> ( P @ Ps2 @ Ls ) ) ) ) ) ).
% not_prefix_induct
thf(fact_1039_not__prefix__induct,axiom,
! [Ps2: list_nat,Ls: list_nat,P: list_nat > list_nat > $o] :
( ~ ( prefix_nat @ Ps2 @ Ls )
=> ( ! [X3: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X3 @ Xs2 ) @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys3: list_nat] :
( ( X3 != Y3 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys3 ) ) )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys3: list_nat] :
( ( X3 = Y3 )
=> ( ~ ( prefix_nat @ Xs2 @ Ys3 )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys3 ) ) ) ) )
=> ( P @ Ps2 @ Ls ) ) ) ) ) ).
% not_prefix_induct
thf(fact_1040_prefix__length__le,axiom,
! [Xs: list_a,Ys: list_a] :
( ( prefix_a @ Xs @ Ys )
=> ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) ) ) ).
% prefix_length_le
thf(fact_1041_prefix__length__le,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( prefix_nat @ Xs @ Ys )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% prefix_length_le
thf(fact_1042_prefix__length__prefix,axiom,
! [Ps2: list_a,Xs: list_a,Qs: list_a] :
( ( prefix_a @ Ps2 @ Xs )
=> ( ( prefix_a @ Qs @ Xs )
=> ( ( ord_less_eq_nat @ ( size_size_list_a @ Ps2 ) @ ( size_size_list_a @ Qs ) )
=> ( prefix_a @ Ps2 @ Qs ) ) ) ) ).
% prefix_length_prefix
thf(fact_1043_prefix__length__prefix,axiom,
! [Ps2: list_nat,Xs: list_nat,Qs: list_nat] :
( ( prefix_nat @ Ps2 @ Xs )
=> ( ( prefix_nat @ Qs @ Xs )
=> ( ( ord_less_eq_nat @ ( size_size_list_nat @ Ps2 ) @ ( size_size_list_nat @ Qs ) )
=> ( prefix_nat @ Ps2 @ Qs ) ) ) ) ).
% prefix_length_prefix
thf(fact_1044_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B3: set_nat] :
( ord_less_eq_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ A5 )
@ ^ [X4: nat] : ( member_nat @ X4 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_1045_pred__subset__eq,axiom,
! [R: set_nat,S2: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ R )
@ ^ [X4: nat] : ( member_nat @ X4 @ S2 ) )
= ( ord_less_eq_set_nat @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_1046_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X4: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X4 )
@ ^ [X4: nat,Y2: nat] : ( ord_less_nat @ Y2 @ X4 )
@ zero_zero_nat ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_1047_subset__Collect__iff,axiom,
! [B4: set_nat,A2: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( ( ord_less_eq_set_nat @ B4
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ X4 ) ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ B4 )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1048_subset__Collect__iff,axiom,
! [B4: set_list_nat,A2: set_list_nat,P: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B4 @ A2 )
=> ( ( ord_le6045566169113846134st_nat @ B4
@ ( collect_list_nat
@ ^ [X4: list_nat] :
( ( member_list_nat @ X4 @ A2 )
& ( P @ X4 ) ) ) )
= ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ B4 )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1049_subset__CollectI,axiom,
! [B4: set_nat,A2: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ B4 )
& ( Q @ X4 ) ) )
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1050_subset__CollectI,axiom,
! [B4: set_list_nat,A2: set_list_nat,Q: list_nat > $o,P: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B4 @ A2 )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ B4 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X4: list_nat] :
( ( member_list_nat @ X4 @ B4 )
& ( Q @ X4 ) ) )
@ ( collect_list_nat
@ ^ [X4: list_nat] :
( ( member_list_nat @ X4 @ A2 )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1051_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ X4 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_1052_Collect__subset,axiom,
! [A2: set_list_nat,P: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X4: list_nat] :
( ( member_list_nat @ X4 @ A2 )
& ( P @ X4 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_1053_nth__rotate1,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( rotate1_a @ Xs ) @ N )
= ( nth_a @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_a @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_1054_nth__rotate1,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rotate1_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_1055_set__swap,axiom,
! [I: nat,Xs: list_a,J2: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_a @ Xs ) )
=> ( ( set_a2 @ ( list_update_a @ ( list_update_a @ Xs @ I @ ( nth_a @ Xs @ J2 ) ) @ J2 @ ( nth_a @ Xs @ I ) ) )
= ( set_a2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_1056_set__swap,axiom,
! [I: nat,Xs: list_nat,J2: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J2 ) ) @ J2 @ ( nth_nat @ Xs @ I ) ) )
= ( set_nat2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_1057_comm__append__is__replicate,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( Ys != nil_a )
=> ( ( ( append_a @ Xs @ Ys )
= ( append_a @ Ys @ Xs ) )
=> ? [N2: nat,Zs2: list_a] :
( ( ord_less_nat @ one_one_nat @ N2 )
& ( ( concat_a @ ( replicate_list_a @ N2 @ Zs2 ) )
= ( append_a @ Xs @ Ys ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_1058_comm__append__is__replicate,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != nil_nat )
=> ( ( Ys != nil_nat )
=> ( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Ys @ Xs ) )
=> ? [N2: nat,Zs2: list_nat] :
( ( ord_less_nat @ one_one_nat @ N2 )
& ( ( concat_nat @ ( replicate_list_nat @ N2 @ Zs2 ) )
= ( append_nat @ Xs @ Ys ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_1059_map__eq__conv,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Xs ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) ) ) ) ).
% map_eq_conv
thf(fact_1060_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_a] :
( ( ( concat_a @ Xss2 )
= nil_a )
= ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Xss2 ) )
=> ( X4 = nil_a ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_1061_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_nat] :
( ( ( concat_nat @ Xss2 )
= nil_nat )
= ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xss2 ) )
=> ( X4 = nil_nat ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_1062_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_a] :
( ( nil_a
= ( concat_a @ Xss2 ) )
= ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Xss2 ) )
=> ( X4 = nil_a ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_1063_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_nat] :
( ( nil_nat
= ( concat_nat @ Xss2 ) )
= ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xss2 ) )
=> ( X4 = nil_nat ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_1064_length__replicate,axiom,
! [N: nat,X: a] :
( ( size_size_list_a @ ( replicate_a @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_1065_length__replicate,axiom,
! [N: nat,X: nat] :
( ( size_size_list_nat @ ( replicate_nat @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_1066_map__replicate,axiom,
! [F: nat > nat,N: nat,X: nat] :
( ( map_nat_nat @ F @ ( replicate_nat @ N @ X ) )
= ( replicate_nat @ N @ ( F @ X ) ) ) ).
% map_replicate
thf(fact_1067_in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_1068_set__filter,axiom,
! [P: nat > $o,Xs: list_nat] :
( ( set_nat2 @ ( filter_nat @ P @ Xs ) )
= ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
& ( P @ X4 ) ) ) ) ).
% set_filter
thf(fact_1069_set__filter,axiom,
! [P: list_nat > $o,Xs: list_list_nat] :
( ( set_list_nat2 @ ( filter_list_nat @ P @ Xs ) )
= ( collect_list_nat
@ ^ [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs ) )
& ( P @ X4 ) ) ) ) ).
% set_filter
thf(fact_1070_filter__False,axiom,
! [Xs: list_a,P: a > $o] :
( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ~ ( P @ X3 ) )
=> ( ( filter_a @ P @ Xs )
= nil_a ) ) ).
% filter_False
thf(fact_1071_filter__False,axiom,
! [Xs: list_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ~ ( P @ X3 ) )
=> ( ( filter_nat @ P @ Xs )
= nil_nat ) ) ).
% filter_False
thf(fact_1072_empty__replicate,axiom,
! [N: nat,X: a] :
( ( nil_a
= ( replicate_a @ N @ X ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_1073_empty__replicate,axiom,
! [N: nat,X: nat] :
( ( nil_nat
= ( replicate_nat @ N @ X ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_1074_replicate__empty,axiom,
! [N: nat,X: a] :
( ( ( replicate_a @ N @ X )
= nil_a )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_1075_replicate__empty,axiom,
! [N: nat,X: nat] :
( ( ( replicate_nat @ N @ X )
= nil_nat )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_1076_in__set__replicate,axiom,
! [X: nat,N: nat,Y: nat] :
( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_1077_nth__replicate,axiom,
! [I: nat,N: nat,X: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_nat @ ( replicate_nat @ N @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_1078_hd__replicate,axiom,
! [N: nat,X: nat] :
( ( N != zero_zero_nat )
=> ( ( hd_nat @ ( replicate_nat @ N @ X ) )
= X ) ) ).
% hd_replicate
thf(fact_1079_drop__replicate,axiom,
! [I: nat,K: nat,X: nat] :
( ( drop_nat @ I @ ( replicate_nat @ K @ X ) )
= ( replicate_nat @ ( minus_minus_nat @ K @ I ) @ X ) ) ).
% drop_replicate
thf(fact_1080_last__replicate,axiom,
! [N: nat,X: nat] :
( ( N != zero_zero_nat )
=> ( ( last_nat @ ( replicate_nat @ N @ X ) )
= X ) ) ).
% last_replicate
thf(fact_1081_not__in__set__insert,axiom,
! [X: a,Xs: list_a] :
( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ( insert_a @ X @ Xs )
= ( cons_a @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_1082_not__in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= ( cons_nat @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_1083_concat__replicate__trivial,axiom,
! [I: nat] :
( ( concat_a @ ( replicate_list_a @ I @ nil_a ) )
= nil_a ) ).
% concat_replicate_trivial
thf(fact_1084_concat__replicate__trivial,axiom,
! [I: nat] :
( ( concat_nat @ ( replicate_list_nat @ I @ nil_nat ) )
= nil_nat ) ).
% concat_replicate_trivial
thf(fact_1085_tl__replicate,axiom,
! [N: nat,X: nat] :
( ( tl_nat @ ( replicate_nat @ N @ X ) )
= ( replicate_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ X ) ) ).
% tl_replicate
thf(fact_1086_set__n__lists,axiom,
! [N: nat,Xs: list_a] :
( ( set_list_a2 @ ( n_lists_a @ N @ Xs ) )
= ( collect_list_a
@ ^ [Ys2: list_a] :
( ( ( size_size_list_a @ Ys2 )
= N )
& ( ord_less_eq_set_a @ ( set_a2 @ Ys2 ) @ ( set_a2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_1087_set__n__lists,axiom,
! [N: nat,Xs: list_nat] :
( ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) )
= ( collect_list_nat
@ ^ [Ys2: list_nat] :
( ( ( size_size_list_nat @ Ys2 )
= N )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Ys2 ) @ ( set_nat2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_1088_subset__code_I1_J,axiom,
! [Xs: list_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B4 )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X4 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_1089_set__subset__Cons,axiom,
! [Xs: list_a,X: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1090_set__subset__Cons,axiom,
! [Xs: list_nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1091_replicate__Suc,axiom,
! [N: nat,X: a] :
( ( replicate_a @ ( suc @ N ) @ X )
= ( cons_a @ X @ ( replicate_a @ N @ X ) ) ) ).
% replicate_Suc
thf(fact_1092_replicate__Suc,axiom,
! [N: nat,X: nat] :
( ( replicate_nat @ ( suc @ N ) @ X )
= ( cons_nat @ X @ ( replicate_nat @ N @ X ) ) ) ).
% replicate_Suc
thf(fact_1093_replicate__0,axiom,
! [X: a] :
( ( replicate_a @ zero_zero_nat @ X )
= nil_a ) ).
% replicate_0
thf(fact_1094_replicate__0,axiom,
! [X: nat] :
( ( replicate_nat @ zero_zero_nat @ X )
= nil_nat ) ).
% replicate_0
thf(fact_1095_replicate__app__Cons__same,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( append_a @ ( replicate_a @ N @ X ) @ ( cons_a @ X @ Xs ) )
= ( cons_a @ X @ ( append_a @ ( replicate_a @ N @ X ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_1096_replicate__app__Cons__same,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( append_nat @ ( replicate_nat @ N @ X ) @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( append_nat @ ( replicate_nat @ N @ X ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_1097_split__list__first__prop__iff,axiom,
! [Xs: list_a,P: a > $o] :
( ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys2: list_a,X4: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Y2: a] :
( ( member_a @ Y2 @ ( set_a2 @ Ys2 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_1098_split__list__first__prop__iff,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys2: list_nat,X4: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Y2: nat] :
( ( member_nat @ Y2 @ ( set_nat2 @ Ys2 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_1099_split__list__last__prop__iff,axiom,
! [Xs: list_a,P: a > $o] :
( ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys2: list_a,X4: a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Y2: a] :
( ( member_a @ Y2 @ ( set_a2 @ Zs3 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_1100_split__list__last__prop__iff,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys2: list_nat,X4: nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Y2: nat] :
( ( member_nat @ Y2 @ ( set_nat2 @ Zs3 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_1101_in__set__conv__decomp__first,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys2: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs3 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1102_in__set__conv__decomp__first,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1103_in__set__conv__decomp__last,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys2: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs3 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1104_in__set__conv__decomp__last,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1105_split__list__first__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys3: list_a,X3: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_1106_split__list__first__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys3: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_nat2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_1107_split__list__last__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys3: list_a,X3: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_1108_split__list__last__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys3: list_nat,X3: nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_nat2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_1109_split__list__first__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys3: list_a,X3: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_1110_split__list__first__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys3: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_nat2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_1111_split__list__last__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys3: list_a,X3: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_1112_split__list__last__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys3: list_nat,X3: nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_nat2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_1113_in__set__conv__decomp,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys2: list_a,Zs3: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1114_in__set__conv__decomp,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs3: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1115_append__Cons__eq__iff,axiom,
! [X: a,Xs: list_a,Ys: list_a,Xs6: list_a,Ys7: list_a] :
( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ~ ( member_a @ X @ ( set_a2 @ Ys ) )
=> ( ( ( append_a @ Xs @ ( cons_a @ X @ Ys ) )
= ( append_a @ Xs6 @ ( cons_a @ X @ Ys7 ) ) )
= ( ( Xs = Xs6 )
& ( Ys = Ys7 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1116_append__Cons__eq__iff,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat,Xs6: list_nat,Ys7: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ~ ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ( ( ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) )
= ( append_nat @ Xs6 @ ( cons_nat @ X @ Ys7 ) ) )
= ( ( Xs = Xs6 )
& ( Ys = Ys7 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1117_split__list__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys3: list_a,X3: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs2 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_1118_split__list__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys3: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs2 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_1119_split__list__first,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys3: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_1120_split__list__first,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys3: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_1121_split__list__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys3: list_a,X3: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs2 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_1122_split__list__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys3: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs2 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_1123_split__list__last,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys3: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1124_split__list__last,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys3: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1125_split__list,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys3: list_a,Zs2: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1126_split__list,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys3: list_nat,Zs2: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1127_replicate__add,axiom,
! [N: nat,M2: nat,X: a] :
( ( replicate_a @ ( plus_plus_nat @ N @ M2 ) @ X )
= ( append_a @ ( replicate_a @ N @ X ) @ ( replicate_a @ M2 @ X ) ) ) ).
% replicate_add
thf(fact_1128_replicate__add,axiom,
! [N: nat,M2: nat,X: nat] :
( ( replicate_nat @ ( plus_plus_nat @ N @ M2 ) @ X )
= ( append_nat @ ( replicate_nat @ N @ X ) @ ( replicate_nat @ M2 @ X ) ) ) ).
% replicate_add
thf(fact_1129_filter__replicate,axiom,
! [P: a > $o,X: a,N: nat] :
( ( ( P @ X )
=> ( ( filter_a @ P @ ( replicate_a @ N @ X ) )
= ( replicate_a @ N @ X ) ) )
& ( ~ ( P @ X )
=> ( ( filter_a @ P @ ( replicate_a @ N @ X ) )
= nil_a ) ) ) ).
% filter_replicate
thf(fact_1130_filter__replicate,axiom,
! [P: nat > $o,X: nat,N: nat] :
( ( ( P @ X )
=> ( ( filter_nat @ P @ ( replicate_nat @ N @ X ) )
= ( replicate_nat @ N @ X ) ) )
& ( ~ ( P @ X )
=> ( ( filter_nat @ P @ ( replicate_nat @ N @ X ) )
= nil_nat ) ) ) ).
% filter_replicate
thf(fact_1131_filter__empty__conv,axiom,
! [P: a > $o,Xs: list_a] :
( ( ( filter_a @ P @ Xs )
= nil_a )
= ( ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
=> ~ ( P @ X4 ) ) ) ) ).
% filter_empty_conv
thf(fact_1132_filter__empty__conv,axiom,
! [P: nat > $o,Xs: list_nat] :
( ( ( filter_nat @ P @ Xs )
= nil_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ~ ( P @ X4 ) ) ) ) ).
% filter_empty_conv
thf(fact_1133_empty__filter__conv,axiom,
! [P: a > $o,Xs: list_a] :
( ( nil_a
= ( filter_a @ P @ Xs ) )
= ( ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
=> ~ ( P @ X4 ) ) ) ) ).
% empty_filter_conv
thf(fact_1134_empty__filter__conv,axiom,
! [P: nat > $o,Xs: list_nat] :
( ( nil_nat
= ( filter_nat @ P @ Xs ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ~ ( P @ X4 ) ) ) ) ).
% empty_filter_conv
thf(fact_1135_filter__cong,axiom,
! [Xs: list_nat,Ys: list_nat,P: nat > $o,Q: nat > $o] :
( ( Xs = Ys )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( filter_nat @ P @ Xs )
= ( filter_nat @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_1136_successively__mono,axiom,
! [P: nat > nat > $o,Xs: list_nat,Q: nat > nat > $o] :
( ( successively_nat @ P @ Xs )
=> ( ! [X3: nat,Y3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( member_nat @ Y3 @ ( set_nat2 @ Xs ) )
=> ( ( P @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) ) ) )
=> ( successively_nat @ Q @ Xs ) ) ) ).
% successively_mono
thf(fact_1137_successively__cong,axiom,
! [Xs: list_nat,P: nat > nat > $o,Q: nat > nat > $o,Ys: list_nat] :
( ! [X3: nat,Y3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( member_nat @ Y3 @ ( set_nat2 @ Xs ) )
=> ( ( P @ X3 @ Y3 )
= ( Q @ X3 @ Y3 ) ) ) )
=> ( ( Xs = Ys )
=> ( ( successively_nat @ P @ Xs )
= ( successively_nat @ Q @ Ys ) ) ) ) ).
% successively_cong
thf(fact_1138_in__set__butlastD,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Xs ) ) )
=> ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_1139_list__all__cong,axiom,
! [X: list_nat,Ya: list_nat,P: nat > $o,Pa: nat > $o] :
( ( X = Ya )
=> ( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ Ya ) )
=> ( ( P @ Z )
= ( Pa @ Z ) ) )
=> ( ( list_all_nat @ P @ X )
= ( list_all_nat @ Pa @ Ya ) ) ) ) ).
% list_all_cong
thf(fact_1140_list_Opred__mono__strong,axiom,
! [P: nat > $o,X: list_nat,Pa: nat > $o] :
( ( list_all_nat @ P @ X )
=> ( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ X ) )
=> ( ( P @ Z )
=> ( Pa @ Z ) ) )
=> ( list_all_nat @ Pa @ X ) ) ) ).
% list.pred_mono_strong
thf(fact_1141_notin__set__nthsI,axiom,
! [X: nat,Xs: list_nat,I3: set_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ~ ( member_nat @ X @ ( set_nat2 @ ( nths_nat @ Xs @ I3 ) ) ) ) ).
% notin_set_nthsI
thf(fact_1142_in__set__nthsD,axiom,
! [X: nat,Xs: list_nat,I3: set_nat] :
( ( member_nat @ X @ ( set_nat2 @ ( nths_nat @ Xs @ I3 ) ) )
=> ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_1143_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P2: nat > $o,Xs4: list_nat] :
? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs4 ) )
& ( P2 @ X4 )
& ! [Y2: nat] :
( ( ( member_nat @ Y2 @ ( set_nat2 @ Xs4 ) )
& ( P2 @ Y2 ) )
=> ( Y2 = X4 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_1144_replicate__length__same,axiom,
! [Xs: list_a,X: a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( X3 = X ) )
=> ( ( replicate_a @ ( size_size_list_a @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_1145_replicate__length__same,axiom,
! [Xs: list_nat,X: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( X3 = X ) )
=> ( ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_1146_replicate__eqI,axiom,
! [Xs: list_a,N: nat,X: a] :
( ( ( size_size_list_a @ Xs )
= N )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Xs ) )
=> ( Y3 = X ) )
=> ( Xs
= ( replicate_a @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_1147_replicate__eqI,axiom,
! [Xs: list_nat,N: nat,X: nat] :
( ( ( size_size_list_nat @ Xs )
= N )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ Xs ) )
=> ( Y3 = X ) )
=> ( Xs
= ( replicate_nat @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_1148_in__set__takeD,axiom,
! [X: nat,N: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) )
=> ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_1149_list_Omap__cong,axiom,
! [X: list_nat,Ya: list_nat,F: nat > nat,G: nat > nat] :
( ( X = Ya )
=> ( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_1150_list_Omap__cong0,axiom,
! [X: list_nat,F: nat > nat,G: nat > nat] :
( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ X ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_1151_list_Oinj__map__strong,axiom,
! [X: list_nat,Xa: list_nat,F: nat > nat,Fa: nat > nat] :
( ! [Z: nat,Za: nat] :
( ( member_nat @ Z @ ( set_nat2 @ X ) )
=> ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_1152_list_Omap__ident__strong,axiom,
! [T: list_nat,F: nat > nat] :
( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_nat_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_1153_map__ext,axiom,
! [Xs: list_nat,F: nat > nat,G: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_1154_map__idI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_nat_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_1155_map__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > nat,G: nat > nat] :
( ( Xs = Ys )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_1156_ex__map__conv,axiom,
! [Ys: list_nat,F: nat > nat] :
( ( ? [Xs4: list_nat] :
( Ys
= ( map_nat_nat @ F @ Xs4 ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
=> ? [Y2: nat] :
( X4
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_1157_append__replicate__commute,axiom,
! [N: nat,X: a,K: nat] :
( ( append_a @ ( replicate_a @ N @ X ) @ ( replicate_a @ K @ X ) )
= ( append_a @ ( replicate_a @ K @ X ) @ ( replicate_a @ N @ X ) ) ) ).
% append_replicate_commute
thf(fact_1158_append__replicate__commute,axiom,
! [N: nat,X: nat,K: nat] :
( ( append_nat @ ( replicate_nat @ N @ X ) @ ( replicate_nat @ K @ X ) )
= ( append_nat @ ( replicate_nat @ K @ X ) @ ( replicate_nat @ N @ X ) ) ) ).
% append_replicate_commute
thf(fact_1159_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_1160_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_1161_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_1162_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_1163_list_Oset__cases,axiom,
! [E: a,A: list_a] :
( ( member_a @ E @ ( set_a2 @ A ) )
=> ( ! [Z22: list_a] :
( A
!= ( cons_a @ E @ Z22 ) )
=> ~ ! [Z1: a,Z22: list_a] :
( ( A
= ( cons_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1164_list_Oset__cases,axiom,
! [E: nat,A: list_nat] :
( ( member_nat @ E @ ( set_nat2 @ A ) )
=> ( ! [Z22: list_nat] :
( A
!= ( cons_nat @ E @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1165_set__ConsD,axiom,
! [Y: a,X: a,Xs: list_a] :
( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_a @ Y @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1166_set__ConsD,axiom,
! [Y: nat,X: nat,Xs: list_nat] :
( ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_nat @ Y @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1167_in__set__dropD,axiom,
! [X: nat,N: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ ( drop_nat @ N @ Xs ) ) )
=> ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_1168_set__take__subset,axiom,
! [N: nat,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).
% set_take_subset
thf(fact_1169_set__drop__subset,axiom,
! [N: nat,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( drop_nat @ N @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).
% set_drop_subset
thf(fact_1170_comm__append__are__replicate,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Ys @ Xs ) )
=> ? [M: nat,N2: nat,Zs2: list_a] :
( ( ( concat_a @ ( replicate_list_a @ M @ Zs2 ) )
= Xs )
& ( ( concat_a @ ( replicate_list_a @ N2 @ Zs2 ) )
= Ys ) ) ) ).
% comm_append_are_replicate
thf(fact_1171_comm__append__are__replicate,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Ys @ Xs ) )
=> ? [M: nat,N2: nat,Zs2: list_nat] :
( ( ( concat_nat @ ( replicate_list_nat @ M @ Zs2 ) )
= Xs )
& ( ( concat_nat @ ( replicate_list_nat @ N2 @ Zs2 ) )
= Ys ) ) ) ).
% comm_append_are_replicate
thf(fact_1172_set__update__subsetI,axiom,
! [Xs: list_nat,A2: set_nat,X: nat,I: nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_1173_list_Oset__sel_I1_J,axiom,
! [A: list_a] :
( ( A != nil_a )
=> ( member_a @ ( hd_a @ A ) @ ( set_a2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_1174_list_Oset__sel_I1_J,axiom,
! [A: list_nat] :
( ( A != nil_nat )
=> ( member_nat @ ( hd_nat @ A ) @ ( set_nat2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_1175_hd__in__set,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( member_a @ ( hd_a @ Xs ) @ ( set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1176_hd__in__set,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( member_nat @ ( hd_nat @ Xs ) @ ( set_nat2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1177_list_Oset__sel_I2_J,axiom,
! [A: list_a,X: a] :
( ( A != nil_a )
=> ( ( member_a @ X @ ( set_a2 @ ( tl_a @ A ) ) )
=> ( member_a @ X @ ( set_a2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_1178_list_Oset__sel_I2_J,axiom,
! [A: list_nat,X: nat] :
( ( A != nil_nat )
=> ( ( member_nat @ X @ ( set_nat2 @ ( tl_nat @ A ) ) )
=> ( member_nat @ X @ ( set_nat2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_1179_last__in__set,axiom,
! [As2: list_a] :
( ( As2 != nil_a )
=> ( member_a @ ( last_a @ As2 ) @ ( set_a2 @ As2 ) ) ) ).
% last_in_set
thf(fact_1180_last__in__set,axiom,
! [As2: list_nat] :
( ( As2 != nil_nat )
=> ( member_nat @ ( last_nat @ As2 ) @ ( set_nat2 @ As2 ) ) ) ).
% last_in_set
thf(fact_1181_in__set__butlast__appendI,axiom,
! [X: a,Xs: list_a,Ys: list_a] :
( ( ( member_a @ X @ ( set_a2 @ ( butlast_a @ Xs ) ) )
| ( member_a @ X @ ( set_a2 @ ( butlast_a @ Ys ) ) ) )
=> ( member_a @ X @ ( set_a2 @ ( butlast_a @ ( append_a @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_1182_in__set__butlast__appendI,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat] :
( ( ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Xs ) ) )
| ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Ys ) ) ) )
=> ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ ( append_nat @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_1183_size__list__estimation,axiom,
! [X: nat,Xs: list_nat,Y: nat,F: nat > nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_nat @ Y @ ( F @ X ) )
=> ( ord_less_nat @ Y @ ( size_list_nat @ F @ Xs ) ) ) ) ).
% size_list_estimation
thf(fact_1184_size__list__estimation_H,axiom,
! [X: nat,Xs: list_nat,Y: nat,F: nat > nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ Y @ ( F @ X ) )
=> ( ord_less_eq_nat @ Y @ ( size_list_nat @ F @ Xs ) ) ) ) ).
% size_list_estimation'
thf(fact_1185_size__list__pointwise,axiom,
! [Xs: list_nat,F: nat > nat,G: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_nat @ ( size_list_nat @ F @ Xs ) @ ( size_list_nat @ G @ Xs ) ) ) ).
% size_list_pointwise
thf(fact_1186_Cons__in__subseqsD,axiom,
! [Y: a,Ys: list_a,Xs: list_a] :
( ( member_list_a @ ( cons_a @ Y @ Ys ) @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) )
=> ( member_list_a @ Ys @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_1187_Cons__in__subseqsD,axiom,
! [Y: nat,Ys: list_nat,Xs: list_nat] :
( ( member_list_nat @ ( cons_nat @ Y @ Ys ) @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
=> ( member_list_nat @ Ys @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_1188_List_Oinsert__def,axiom,
( insert_a
= ( ^ [X4: a,Xs4: list_a] : ( if_list_a @ ( member_a @ X4 @ ( set_a2 @ Xs4 ) ) @ Xs4 @ ( cons_a @ X4 @ Xs4 ) ) ) ) ).
% List.insert_def
thf(fact_1189_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X4: nat,Xs4: list_nat] : ( if_list_nat @ ( member_nat @ X4 @ ( set_nat2 @ Xs4 ) ) @ Xs4 @ ( cons_nat @ X4 @ Xs4 ) ) ) ) ).
% List.insert_def
thf(fact_1190_map__replicate__const,axiom,
! [K: nat,Lst: list_nat] :
( ( map_nat_nat
@ ^ [X4: nat] : K
@ Lst )
= ( replicate_nat @ ( size_size_list_nat @ Lst ) @ K ) ) ).
% map_replicate_const
thf(fact_1191_replicate__length__filter,axiom,
! [X: a,Xs: list_a] :
( ( replicate_a
@ ( size_size_list_a
@ ( filter_a
@ ( ^ [Y5: a,Z4: a] : ( Y5 = Z4 )
@ X )
@ Xs ) )
@ X )
= ( filter_a
@ ( ^ [Y5: a,Z4: a] : ( Y5 = Z4 )
@ X )
@ Xs ) ) ).
% replicate_length_filter
thf(fact_1192_replicate__length__filter,axiom,
! [X: nat,Xs: list_nat] :
( ( replicate_nat
@ ( size_size_list_nat
@ ( filter_nat
@ ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
@ X )
@ Xs ) )
@ X )
= ( filter_nat
@ ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
@ X )
@ Xs ) ) ).
% replicate_length_filter
thf(fact_1193_length__n__lists__elem,axiom,
! [Ys: list_a,N: nat,Xs: list_a] :
( ( member_list_a @ Ys @ ( set_list_a2 @ ( n_lists_a @ N @ Xs ) ) )
=> ( ( size_size_list_a @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_1194_length__n__lists__elem,axiom,
! [Ys: list_nat,N: nat,Xs: list_nat] :
( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) ) )
=> ( ( size_size_list_nat @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_1195_set__prefixes__eq,axiom,
! [Xs: list_nat] :
( ( set_list_nat2 @ ( prefixes_nat @ Xs ) )
= ( collect_list_nat
@ ^ [Ys2: list_nat] : ( prefix_nat @ Ys2 @ Xs ) ) ) ).
% set_prefixes_eq
thf(fact_1196_replicate__append__same,axiom,
! [I: nat,X: a] :
( ( append_a @ ( replicate_a @ I @ X ) @ ( cons_a @ X @ nil_a ) )
= ( cons_a @ X @ ( replicate_a @ I @ X ) ) ) ).
% replicate_append_same
thf(fact_1197_replicate__append__same,axiom,
! [I: nat,X: nat] :
( ( append_nat @ ( replicate_nat @ I @ X ) @ ( cons_nat @ X @ nil_nat ) )
= ( cons_nat @ X @ ( replicate_nat @ I @ X ) ) ) ).
% replicate_append_same
thf(fact_1198_length__pos__if__in__set,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1199_length__pos__if__in__set,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1200_nth__mem,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( member_a @ ( nth_a @ Xs @ N ) @ ( set_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1201_nth__mem,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1202_list__ball__nth,axiom,
! [N: nat,Xs: list_a,P: a > $o] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_a @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1203_list__ball__nth,axiom,
! [N: nat,Xs: list_nat,P: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1204_in__set__conv__nth,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ I5 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1205_in__set__conv__nth,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I5 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1206_all__nth__imp__all__set,axiom,
! [Xs: list_a,P: a > $o,X: a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( P @ ( nth_a @ Xs @ I2 ) ) )
=> ( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1207_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P: nat > $o,X: nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I2 ) ) )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1208_all__set__conv__all__nth,axiom,
! [Xs: list_a,P: a > $o] :
( ( ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
=> ( P @ X4 ) ) )
= ( ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_a @ Xs ) )
=> ( P @ ( nth_a @ Xs @ I5 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1209_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( P @ X4 ) ) )
= ( ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I5 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1210_set__take__subset__set__take,axiom,
! [M2: nat,N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ M2 @ Xs ) ) @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_1211_Cons__eq__filterD,axiom,
! [X: a,Xs: list_a,P: a > $o,Ys: list_a] :
( ( ( cons_a @ X @ Xs )
= ( filter_a @ P @ Ys ) )
=> ? [Us3: list_a,Vs3: list_a] :
( ( Ys
= ( append_a @ Us3 @ ( cons_a @ X @ Vs3 ) ) )
& ! [X6: a] :
( ( member_a @ X6 @ ( set_a2 @ Us3 ) )
=> ~ ( P @ X6 ) )
& ( P @ X )
& ( Xs
= ( filter_a @ P @ Vs3 ) ) ) ) ).
% Cons_eq_filterD
thf(fact_1212_Cons__eq__filterD,axiom,
! [X: nat,Xs: list_nat,P: nat > $o,Ys: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( filter_nat @ P @ Ys ) )
=> ? [Us3: list_nat,Vs3: list_nat] :
( ( Ys
= ( append_nat @ Us3 @ ( cons_nat @ X @ Vs3 ) ) )
& ! [X6: nat] :
( ( member_nat @ X6 @ ( set_nat2 @ Us3 ) )
=> ~ ( P @ X6 ) )
& ( P @ X )
& ( Xs
= ( filter_nat @ P @ Vs3 ) ) ) ) ).
% Cons_eq_filterD
thf(fact_1213_mod__by__Suc__0,axiom,
! [M2: nat] :
( ( modulo_modulo_nat @ M2 @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% mod_by_Suc_0
thf(fact_1214_mod__Suc__le__divisor,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ ( suc @ N ) ) @ N ) ).
% mod_Suc_le_divisor
thf(fact_1215_mod__induct,axiom,
! [P: nat > $o,N: nat,P3: nat,M2: nat] :
( ( P @ N )
=> ( ( ord_less_nat @ N @ P3 )
=> ( ( ord_less_nat @ M2 @ P3 )
=> ( ! [N2: nat] :
( ( ord_less_nat @ N2 @ P3 )
=> ( ( P @ N2 )
=> ( P @ ( modulo_modulo_nat @ ( suc @ N2 ) @ P3 ) ) ) )
=> ( P @ M2 ) ) ) ) ) ).
% mod_induct
thf(fact_1216_mod__Suc__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M2 @ N ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ ( suc @ M2 ) ) @ N ) ) ).
% mod_Suc_Suc_eq
thf(fact_1217_mod__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M2 @ N ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M2 ) @ N ) ) ).
% mod_Suc_eq
thf(fact_1218_mod__Suc,axiom,
! [M2: nat,N: nat] :
( ( ( ( suc @ ( modulo_modulo_nat @ M2 @ N ) )
= N )
=> ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N )
= zero_zero_nat ) )
& ( ( ( suc @ ( modulo_modulo_nat @ M2 @ N ) )
!= N )
=> ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( modulo_modulo_nat @ M2 @ N ) ) ) ) ) ).
% mod_Suc
thf(fact_1219_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I5: nat] : ( ord_less_eq_nat @ I5 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_1220_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I5: nat] : ( ord_less_nat @ I5 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_1221_card__less__Suc2,axiom,
! [M7: set_nat,I: nat] :
( ~ ( member_nat @ zero_zero_nat @ M7 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( member_nat @ ( suc @ K2 ) @ M7 )
& ( ord_less_nat @ K2 @ I ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( member_nat @ K2 @ M7 )
& ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_1222_card__less__Suc,axiom,
! [M7: set_nat,I: nat] :
( ( member_nat @ zero_zero_nat @ M7 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( member_nat @ ( suc @ K2 ) @ M7 )
& ( ord_less_nat @ K2 @ I ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( member_nat @ K2 @ M7 )
& ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_1223_card__less,axiom,
! [M7: set_nat,I: nat] :
( ( member_nat @ zero_zero_nat @ M7 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( member_nat @ K2 @ M7 )
& ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) )
!= zero_zero_nat ) ) ).
% card_less
thf(fact_1224_Suc__times__mod__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M2 @ N ) ) @ M2 )
= one_one_nat ) ) ).
% Suc_times_mod_eq
thf(fact_1225_mult__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M2 @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1226_mult__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1227_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1228_mult__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1229_nat__mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1230_nat__1__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N ) )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1231_mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1232_one__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M2 @ N ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1233_nat__0__less__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1234_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1235_mult__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( times_times_nat @ M2 @ ( suc @ N ) )
= ( plus_plus_nat @ M2 @ ( times_times_nat @ M2 @ N ) ) ) ).
% mult_Suc_right
thf(fact_1236_one__le__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M2 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1237_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1238_Suc__mod__mult__self4,axiom,
! [N: nat,K: nat,M2: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M2 ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_mod_mult_self4
thf(fact_1239_Suc__mod__mult__self3,axiom,
! [K: nat,N: nat,M2: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M2 ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_mod_mult_self3
thf(fact_1240_Suc__mod__mult__self2,axiom,
! [M2: nat,N: nat,K: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M2 @ ( times_times_nat @ N @ K ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_mod_mult_self2
thf(fact_1241_Suc__mod__mult__self1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M2 @ ( times_times_nat @ K @ N ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_mod_mult_self1
thf(fact_1242_Suc__times__numeral__mod__eq,axiom,
! [K: num,N: nat] :
( ( ( numeral_numeral_nat @ K )
!= one_one_nat )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N ) ) @ ( numeral_numeral_nat @ K ) )
= one_one_nat ) ) ).
% Suc_times_numeral_mod_eq
thf(fact_1243_mult__eq__self__implies__10,axiom,
! [M2: nat,N: nat] :
( ( M2
= ( times_times_nat @ M2 @ N ) )
=> ( ( N = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1244_mult__le__mono2,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J2 ) ) ) ).
% mult_le_mono2
thf(fact_1245_mult__le__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ).
% mult_le_mono1
thf(fact_1246_mult__le__mono,axiom,
! [I: nat,J2: nat,K: nat,L2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ K @ L2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J2 @ L2 ) ) ) ) ).
% mult_le_mono
thf(fact_1247_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_1248_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_1249_Suc__mult__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M2 )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M2 = N ) ) ).
% Suc_mult_cancel1
thf(fact_1250_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1251_add__mult__distrib2,axiom,
! [K: nat,M2: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1252_add__mult__distrib,axiom,
! [M2: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M2 @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1253_diff__mult__distrib2,axiom,
! [K: nat,M2: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1254_diff__mult__distrib,axiom,
! [M2: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M2 @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1255_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1256_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1257_nat__mult__max__right,axiom,
! [M2: nat,N: nat,Q3: nat] :
( ( times_times_nat @ M2 @ ( ord_max_nat @ N @ Q3 ) )
= ( ord_max_nat @ ( times_times_nat @ M2 @ N ) @ ( times_times_nat @ M2 @ Q3 ) ) ) ).
% nat_mult_max_right
thf(fact_1258_nat__mult__max__left,axiom,
! [M2: nat,N: nat,Q3: nat] :
( ( times_times_nat @ ( ord_max_nat @ M2 @ N ) @ Q3 )
= ( ord_max_nat @ ( times_times_nat @ M2 @ Q3 ) @ ( times_times_nat @ N @ Q3 ) ) ) ).
% nat_mult_max_left
thf(fact_1259_mult__Suc,axiom,
! [M2: nat,N: nat] :
( ( times_times_nat @ ( suc @ M2 ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M2 @ N ) ) ) ).
% mult_Suc
thf(fact_1260_Suc__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M2 ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1261_mult__less__mono2,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J2 ) ) ) ) ).
% mult_less_mono2
thf(fact_1262_mult__less__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1263_Suc__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M2 ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1264_one__less__mult,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% one_less_mult
thf(fact_1265_n__less__m__mult__n,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1266_n__less__n__mult__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M2 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1267_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M4: nat,N3: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1268_card__length__sum__list__rec,axiom,
! [M2: nat,N5: nat] :
( ( ord_less_eq_nat @ one_one_nat @ M2 )
=> ( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L4: list_nat] :
( ( ( size_size_list_nat @ L4 )
= M2 )
& ( ( groups4561878855575611511st_nat @ L4 )
= N5 ) ) ) )
= ( plus_plus_nat
@ ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L4: list_nat] :
( ( ( size_size_list_nat @ L4 )
= ( minus_minus_nat @ M2 @ one_one_nat ) )
& ( ( groups4561878855575611511st_nat @ L4 )
= N5 ) ) ) )
@ ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L4: list_nat] :
( ( ( size_size_list_nat @ L4 )
= M2 )
& ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L4 ) @ one_one_nat )
= N5 ) ) ) ) ) ) ) ).
% card_length_sum_list_rec
% Helper facts (7)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
! [Y1: list_a,Y24: a] :
( ( y
= ( append_a @ Y1 @ ( cons_a @ Y24 @ nil_a ) ) )
=> ( ( ( size_size_list_a @ Y1 )
= na )
=> thesis ) ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------