TPTP Problem File: SLH0130^1.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_00532_018139__11997810_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1504 ( 772 unt; 225 typ; 0 def)
% Number of atoms : 3121 (2180 equ; 0 cnn)
% Maximal formula atoms : 18 ( 2 avg)
% Number of connectives : 11790 ( 416 ~; 87 |; 313 &;9839 @)
% ( 0 <=>;1135 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 5 avg)
% Number of types : 22 ( 21 usr)
% Number of type conns : 835 ( 835 >; 0 *; 0 +; 0 <<)
% Number of symbols : 207 ( 204 usr; 15 con; 0-3 aty)
% Number of variables : 3597 ( 255 ^;3163 !; 179 ?;3597 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 10:02:51.398
%------------------------------------------------------------------------------
% Could-be-implicit typings (21)
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
list_P6011104703257516679at_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
list_P3592885314253461005_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J_J,type,
list_P2851791750731487283_nat_a: $tType ).
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__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
list_P1396940483166286381od_a_a: $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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $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__String__Ochar,type,
char: $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 (204)
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__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__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_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_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
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_OdropWhile_001t__Nat__Onat,type,
dropWhile_nat: ( nat > $o ) > list_nat > list_nat ).
thf(sy_c_List_OdropWhile_001tf__a,type,
dropWhile_a: ( a > $o ) > list_a > list_a ).
thf(sy_c_List_Odrop_001t__Nat__Onat,type,
drop_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Odrop_001tf__a,type,
drop_a: nat > list_a > list_a ).
thf(sy_c_List_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_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_001t__Nat__Onat_001t__Nat__Onat,type,
product_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Oproduct_001t__Nat__Onat_001tf__a,type,
product_nat_a: list_nat > list_a > list_P2851791750731487283_nat_a ).
thf(sy_c_List_Oproduct_001tf__a_001t__Nat__Onat,type,
product_a_nat: list_a > list_nat > list_P3592885314253461005_a_nat ).
thf(sy_c_List_Oproduct_001tf__a_001tf__a,type,
product_a_a: list_a > list_a > list_P1396940483166286381od_a_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_Oremdups__adj_001t__Nat__Onat,type,
remdups_adj_nat: list_nat > list_nat ).
thf(sy_c_List_Oremdups__adj_001tf__a,type,
remdups_adj_a: list_a > list_a ).
thf(sy_c_List_Oremdups__adj__rel_001t__Nat__Onat,type,
remdups_adj_rel_nat: list_nat > list_nat > $o ).
thf(sy_c_List_Oremdups__adj__rel_001tf__a,type,
remdups_adj_rel_a: list_a > list_a > $o ).
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_Orotate_001t__Nat__Onat,type,
rotate_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Orotate_001tf__a,type,
rotate_a: nat > 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_OtakeWhile_001t__Nat__Onat,type,
takeWhile_nat: ( nat > $o ) > list_nat > list_nat ).
thf(sy_c_List_OtakeWhile_001tf__a,type,
takeWhile_a: ( a > $o ) > list_a > list_a ).
thf(sy_c_List_Otake_001t__Nat__Onat,type,
take_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Otake_001tf__a,type,
take_a: nat > list_a > list_a ).
thf(sy_c_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_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J_J,type,
size_s243904063682394823_nat_a: list_P2851791750731487283_nat_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
size_s984997627204368545_a_nat: list_P3592885314253461005_a_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
size_s3885678630836030617od_a_a: list_P1396940483166286381od_a_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
nat_prod_decode_aux: nat > nat > product_prod_nat_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__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
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__append,type,
prefix5314359684614007693append: option_list_o > option_list_o > option_list_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_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
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_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_Itf__a_J,type,
accp_list_a: ( list_a > list_a > $o ) > list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__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_x1____,type,
x1: list_a ).
thf(sy_v_x2____,type,
x2: a ).
thf(sy_v_x____,type,
x: list_a ).
thf(sy_v_y1____,type,
y1: list_a ).
thf(sy_v_y2____,type,
y2: a ).
thf(sy_v_y____,type,
y: list_a ).
% Relevant facts (1269)
thf(fact_0_assms,axiom,
prefix7485107378405021920ding_a @ e ).
% assms
thf(fact_1__092_060open_062opt__comp_A_Ie_Ax2_J_A_Ie_Ay2_J_092_060close_062,axiom,
prefix454693708527911765comp_o @ ( e @ x2 ) @ ( e @ y2 ) ).
% \<open>opt_comp (e x2) (e y2)\<close>
thf(fact_2_opt__comp__sym,axiom,
( prefix454693708527911765comp_o
= ( ^ [X: option_list_o,Y: option_list_o] : ( prefix454693708527911765comp_o @ Y @ X ) ) ) ).
% opt_comp_sym
thf(fact_3_d,axiom,
prefix454693708527911765comp_o @ ( prefix5314359684614007693append @ ( prefix4097710381326367690Lf_e_a @ e @ na @ x1 ) @ ( e @ x2 ) ) @ ( prefix5314359684614007693append @ ( prefix4097710381326367690Lf_e_a @ e @ na @ y1 ) @ ( e @ y2 ) ) ).
% d
thf(fact_4_x__def_I1_J,axiom,
( x
= ( append_a @ x1 @ ( cons_a @ x2 @ nil_a ) ) ) ).
% x_def(1)
thf(fact_5_y__def_I1_J,axiom,
( y
= ( append_a @ y1 @ ( cons_a @ y2 @ nil_a ) ) ) ).
% y_def(1)
thf(fact_6_e,axiom,
x1 = y1 ).
% e
thf(fact_7__092_060open_062_092_060And_062y_Ax_O_Aopt__comp_A_Ie_Ax_J_A_Ie_Ay_J_A_092_060Longrightarrow_062_Ax_A_061_Ay_092_060close_062,axiom,
! [X2: a,Y2: a] :
( ( prefix454693708527911765comp_o @ ( e @ X2 ) @ ( e @ Y2 ) )
=> ( X2 = Y2 ) ) ).
% \<open>\<And>y x. opt_comp (e x) (e y) \<Longrightarrow> x = y\<close>
thf(fact_8__092_060open_062opt__comp_A_ILf_092_060_094sub_062e_Ae_An_Ax1_J_A_ILf_092_060_094sub_062e_Ae_An_Ay1_J_092_060close_062,axiom,
prefix454693708527911765comp_o @ ( prefix4097710381326367690Lf_e_a @ e @ na @ x1 ) @ ( prefix4097710381326367690Lf_e_a @ e @ na @ y1 ) ).
% \<open>opt_comp (Lf\<^sub>e e n x1) (Lf\<^sub>e e n y1)\<close>
thf(fact_9_x__def_I2_J,axiom,
( ( size_size_list_a @ x1 )
= na ) ).
% x_def(2)
thf(fact_10_y__def_I2_J,axiom,
( ( size_size_list_a @ y1 )
= na ) ).
% y_def(2)
thf(fact_11_a,axiom,
prefix454693708527911765comp_o @ ( prefix4097710381326367690Lf_e_a @ e @ ( suc @ na ) @ x ) @ ( prefix4097710381326367690Lf_e_a @ e @ ( suc @ na ) @ y ) ).
% a
thf(fact_12_is__encodingD,axiom,
! [E: a > option_list_o,X2: a,Y2: a] :
( ( prefix7485107378405021920ding_a @ E )
=> ( ( prefix454693708527911765comp_o @ ( E @ X2 ) @ ( E @ Y2 ) )
=> ( X2 = Y2 ) ) ) ).
% is_encodingD
thf(fact_13_is__encodingD,axiom,
! [E: list_a > option_list_o,X2: list_a,Y2: list_a] :
( ( prefix5220018966750911590list_a @ E )
=> ( ( prefix454693708527911765comp_o @ ( E @ X2 ) @ ( E @ Y2 ) )
=> ( X2 = Y2 ) ) ) ).
% is_encodingD
thf(fact_14_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_15_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_16_opt__comp__append,axiom,
! [X2: option_list_o,Y2: option_list_o,Z: option_list_o] :
( ( prefix454693708527911765comp_o @ ( prefix5314359684614007693append @ X2 @ Y2 ) @ Z )
=> ( prefix454693708527911765comp_o @ X2 @ Z ) ) ).
% opt_comp_append
thf(fact_17_opt__comp__append__2,axiom,
! [X2: option_list_o,Y2: option_list_o,Z: option_list_o] :
( ( prefix454693708527911765comp_o @ X2 @ ( prefix5314359684614007693append @ Y2 @ Z ) )
=> ( prefix454693708527911765comp_o @ X2 @ Y2 ) ) ).
% opt_comp_append_2
thf(fact_18_opt__comp__append__3,axiom,
! [X2: option_list_o,Y2: option_list_o,Z: option_list_o] :
( ( prefix454693708527911765comp_o @ ( prefix5314359684614007693append @ X2 @ Y2 ) @ ( prefix5314359684614007693append @ X2 @ Z ) )
=> ( prefix454693708527911765comp_o @ Y2 @ Z ) ) ).
% opt_comp_append_3
thf(fact_19_append1__eq__conv,axiom,
! [Xs: list_a,X2: a,Ys: list_a,Y2: a] :
( ( ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y2 @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_20_append1__eq__conv,axiom,
! [Xs: list_nat,X2: nat,Ys: list_nat,Y2: nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) )
= ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) )
= ( ( Xs = Ys )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_21__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] :
( ? [X22: a] :
( x
= ( append_a @ X1 @ ( cons_a @ X22 @ 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_22__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062y1_Ay2_O_A_092_060lbrakk_062y_A_061_Ay1_A_064_A_091y2_093_059_Alength_Ay1_A_061_An_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Y1: list_a] :
( ? [Y22: a] :
( y
= ( append_a @ Y1 @ ( cons_a @ Y22 @ nil_a ) ) )
=> ( ( size_size_list_a @ Y1 )
!= na ) ) ).
% \<open>\<And>thesis. (\<And>y1 y2. \<lbrakk>y = y1 @ [y2]; length y1 = n\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_23_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_24_append_Oright__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ A @ nil_nat )
= A ) ).
% append.right_neutral
thf(fact_25_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_26_append__Nil2,axiom,
! [Xs: list_nat] :
( ( append_nat @ Xs @ nil_nat )
= Xs ) ).
% append_Nil2
thf(fact_27_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_28_append__self__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_nat ) ) ).
% append_self_conv
thf(fact_29_self__append__conv,axiom,
! [Y2: list_a,Ys: list_a] :
( ( Y2
= ( append_a @ Y2 @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_30_self__append__conv,axiom,
! [Y2: list_nat,Ys: list_nat] :
( ( Y2
= ( append_nat @ Y2 @ Ys ) )
= ( Ys = nil_nat ) ) ).
% self_append_conv
thf(fact_31_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_32_append__self__conv2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_nat ) ) ).
% append_self_conv2
thf(fact_33_self__append__conv2,axiom,
! [Y2: list_a,Xs: list_a] :
( ( Y2
= ( append_a @ Xs @ Y2 ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_34_self__append__conv2,axiom,
! [Y2: list_nat,Xs: list_nat] :
( ( Y2
= ( append_nat @ Xs @ Y2 ) )
= ( Xs = nil_nat ) ) ).
% self_append_conv2
thf(fact_35_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_36_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_37_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_38_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_39_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_40_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_41_list_Oinject,axiom,
! [X21: a,X222: list_a,Y21: a,Y222: list_a] :
( ( ( cons_a @ X21 @ X222 )
= ( cons_a @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_42_list_Oinject,axiom,
! [X21: nat,X222: list_nat,Y21: nat,Y222: list_nat] :
( ( ( cons_nat @ X21 @ X222 )
= ( cons_nat @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_43_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_44_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_45_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_46_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_47_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_48_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_49_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_50_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_51_c,axiom,
( ( size_size_list_a @ y )
= ( suc @ na ) ) ).
% c
thf(fact_52_b,axiom,
( ( size_size_list_a @ x )
= ( suc @ na ) ) ).
% b
thf(fact_53_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_54_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_55_Suc__length__conv,axiom,
! [N: nat,Xs: list_a] :
( ( ( suc @ N )
= ( size_size_list_a @ Xs ) )
= ( ? [Y: a,Ys2: list_a] :
( ( Xs
= ( cons_a @ Y @ Ys2 ) )
& ( ( size_size_list_a @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_56_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ Y @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_57_length__Suc__conv,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y: a,Ys2: list_a] :
( ( Xs
= ( cons_a @ Y @ Ys2 ) )
& ( ( size_size_list_a @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_58_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ Y @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_59_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_60_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_61_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_62_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_63_length__Suc__conv__rev,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y: a,Ys2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ Y @ nil_a ) ) )
& ( ( size_size_list_a @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_64_length__Suc__conv__rev,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y: nat,Ys2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ Y @ nil_nat ) ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_65_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,Z2: 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 @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_66_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,Z2: 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 @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_67_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,Z2: 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 @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_68_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,Z2: 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 @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_69_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,Z2: 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 @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_70_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,Z2: 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 @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_71_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,Z2: 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 @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_72_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,Z2: 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 @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_73_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,Z2: 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 @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_74_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,Z2: 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 @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_75_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,Z2: 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 @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_76_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,Z2: 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 @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_77_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,Z2: 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 @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_78_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,Z2: 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 @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_79_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,Z2: 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 @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_80_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,Z2: 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 @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_81_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,Z2: 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 @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_82_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,Z2: 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 @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_83_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_84_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_85_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_86_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_87_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_88_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_89_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_90_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,Ys4: 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 ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_91_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,Ys4: 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 ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_92_not__Cons__self2,axiom,
! [X2: a,Xs: list_a] :
( ( cons_a @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_93_not__Cons__self2,axiom,
! [X2: nat,Xs: list_nat] :
( ( cons_nat @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_94_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_95_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_96_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_97_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_98_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_99_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_100_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_101_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_102_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_103_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_104_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y: a,Ys2: list_a] :
( Xs
= ( cons_a @ Y @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_105_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y: nat,Ys2: list_nat] :
( Xs
= ( cons_nat @ Y @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_106_remdups__adj_Ocases,axiom,
! [X2: list_a] :
( ( X2 != nil_a )
=> ( ! [X3: a] :
( X2
!= ( cons_a @ X3 @ nil_a ) )
=> ~ ! [X3: a,Y3: a,Xs2: list_a] :
( X2
!= ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_107_remdups__adj_Ocases,axiom,
! [X2: list_nat] :
( ( X2 != nil_nat )
=> ( ! [X3: nat] :
( X2
!= ( cons_nat @ X3 @ nil_nat ) )
=> ~ ! [X3: nat,Y3: nat,Xs2: list_nat] :
( X2
!= ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_108_transpose_Ocases,axiom,
! [X2: list_list_a] :
( ( X2 != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X2
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X3: a,Xs2: list_a,Xss: list_list_a] :
( X2
!= ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_109_transpose_Ocases,axiom,
! [X2: list_list_nat] :
( ( X2 != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X3: nat,Xs2: list_nat,Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_110_min__list_Ocases,axiom,
! [X2: list_nat] :
( ! [X3: nat,Xs2: list_nat] :
( X2
!= ( cons_nat @ X3 @ Xs2 ) )
=> ( X2 = nil_nat ) ) ).
% min_list.cases
thf(fact_111_list_Oexhaust,axiom,
! [Y2: list_a] :
( ( Y2 != nil_a )
=> ~ ! [X212: a,X223: list_a] :
( Y2
!= ( cons_a @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_112_list_Oexhaust,axiom,
! [Y2: list_nat] :
( ( Y2 != nil_nat )
=> ~ ! [X212: nat,X223: list_nat] :
( Y2
!= ( cons_nat @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_113_list_OdiscI,axiom,
! [List: list_a,X21: a,X222: list_a] :
( ( List
= ( cons_a @ X21 @ X222 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_114_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X222: list_nat] :
( ( List
= ( cons_nat @ X21 @ X222 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_115_list_Odistinct_I1_J,axiom,
! [X21: a,X222: list_a] :
( nil_a
!= ( cons_a @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_116_list_Odistinct_I1_J,axiom,
! [X21: nat,X222: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_117_Cons__eq__appendI,axiom,
! [X2: a,Xs1: list_a,Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( cons_a @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X2 @ Xs )
= ( append_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_118_Cons__eq__appendI,axiom,
! [X2: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_nat @ Xs1 @ Zs ) )
=> ( ( cons_nat @ X2 @ Xs )
= ( append_nat @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_119_append__Cons,axiom,
! [X2: a,Xs: list_a,Ys: list_a] :
( ( append_a @ ( cons_a @ X2 @ Xs ) @ Ys )
= ( cons_a @ X2 @ ( append_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_120_append__Cons,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat] :
( ( append_nat @ ( cons_nat @ X2 @ Xs ) @ Ys )
= ( cons_nat @ X2 @ ( append_nat @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_121_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_122_eq__Nil__appendI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append_nat @ nil_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_123_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_124_append_Oleft__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ nil_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_125_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_126_append__Nil,axiom,
! [Ys: list_nat] :
( ( append_nat @ nil_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_127_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_128_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_129_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X2: a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X2 @ Xs ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X2 @ Xs ) ) )
| ? [Ys5: list_a] :
( ( Ys
= ( cons_a @ X2 @ Ys5 ) )
& ( ( append_a @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_130_append__eq__Cons__conv,axiom,
! [Ys: list_nat,Zs: list_nat,X2: nat,Xs: list_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( cons_nat @ X2 @ Xs ) )
= ( ( ( Ys = nil_nat )
& ( Zs
= ( cons_nat @ X2 @ Xs ) ) )
| ? [Ys5: list_nat] :
( ( Ys
= ( cons_nat @ X2 @ Ys5 ) )
& ( ( append_nat @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_131_Cons__eq__append__conv,axiom,
! [X2: a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X2 @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X2 @ Xs )
= Zs ) )
| ? [Ys5: list_a] :
( ( ( cons_a @ X2 @ Ys5 )
= Ys )
& ( Xs
= ( append_a @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_132_Cons__eq__append__conv,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( append_nat @ Ys @ Zs ) )
= ( ( ( Ys = nil_nat )
& ( ( cons_nat @ X2 @ Xs )
= Zs ) )
| ? [Ys5: list_nat] :
( ( ( cons_nat @ X2 @ Ys5 )
= Ys )
& ( Xs
= ( append_nat @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_133_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_134_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_135_length__append__singleton,axiom,
! [Xs: list_a,X2: a] :
( ( size_size_list_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_append_singleton
thf(fact_136_length__append__singleton,axiom,
! [Xs: list_nat,X2: nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_append_singleton
thf(fact_137_length__Cons,axiom,
! [X2: a,Xs: list_a] :
( ( size_size_list_a @ ( cons_a @ X2 @ Xs ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_Cons
thf(fact_138_length__Cons,axiom,
! [X2: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X2 @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_Cons
thf(fact_139_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_140_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_141_bind__simps_I2_J,axiom,
! [X2: a,Xs: list_a,F: a > list_a] :
( ( bind_a_a @ ( cons_a @ X2 @ Xs ) @ F )
= ( append_a @ ( F @ X2 ) @ ( bind_a_a @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_142_bind__simps_I2_J,axiom,
! [X2: a,Xs: list_a,F: a > list_nat] :
( ( bind_a_nat @ ( cons_a @ X2 @ Xs ) @ F )
= ( append_nat @ ( F @ X2 ) @ ( bind_a_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_143_bind__simps_I2_J,axiom,
! [X2: nat,Xs: list_nat,F: nat > list_a] :
( ( bind_nat_a @ ( cons_nat @ X2 @ Xs ) @ F )
= ( append_a @ ( F @ X2 ) @ ( bind_nat_a @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_144_bind__simps_I2_J,axiom,
! [X2: nat,Xs: list_nat,F: nat > list_nat] :
( ( bind_nat_nat @ ( cons_nat @ X2 @ Xs ) @ F )
= ( append_nat @ ( F @ X2 ) @ ( bind_nat_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_145_Suc,axiom,
prefix5220018966750911590list_a @ ( prefix4097710381326367690Lf_e_a @ e @ na ) ).
% Suc
thf(fact_146_gen__length__code_I2_J,axiom,
! [N: nat,X2: a,Xs: list_a] :
( ( gen_length_a @ N @ ( cons_a @ X2 @ Xs ) )
= ( gen_length_a @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_147_gen__length__code_I2_J,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( gen_length_nat @ N @ ( cons_nat @ X2 @ Xs ) )
= ( gen_length_nat @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_148_maps__simps_I1_J,axiom,
! [F: a > list_a,X2: a,Xs: list_a] :
( ( maps_a_a @ F @ ( cons_a @ X2 @ Xs ) )
= ( append_a @ ( F @ X2 ) @ ( maps_a_a @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_149_maps__simps_I1_J,axiom,
! [F: a > list_nat,X2: a,Xs: list_a] :
( ( maps_a_nat @ F @ ( cons_a @ X2 @ Xs ) )
= ( append_nat @ ( F @ X2 ) @ ( maps_a_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_150_maps__simps_I1_J,axiom,
! [F: nat > list_a,X2: nat,Xs: list_nat] :
( ( maps_nat_a @ F @ ( cons_nat @ X2 @ Xs ) )
= ( append_a @ ( F @ X2 ) @ ( maps_nat_a @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_151_maps__simps_I1_J,axiom,
! [F: nat > list_nat,X2: nat,Xs: list_nat] :
( ( maps_nat_nat @ F @ ( cons_nat @ X2 @ Xs ) )
= ( append_nat @ ( F @ X2 ) @ ( maps_nat_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_152_insert__Nil,axiom,
! [X2: a] :
( ( insert_a @ X2 @ nil_a )
= ( cons_a @ X2 @ nil_a ) ) ).
% insert_Nil
thf(fact_153_insert__Nil,axiom,
! [X2: nat] :
( ( insert_nat @ X2 @ nil_nat )
= ( cons_nat @ X2 @ nil_nat ) ) ).
% insert_Nil
thf(fact_154_bind__simps_I1_J,axiom,
! [F: a > list_a] :
( ( bind_a_a @ nil_a @ F )
= nil_a ) ).
% bind_simps(1)
thf(fact_155_bind__simps_I1_J,axiom,
! [F: a > list_nat] :
( ( bind_a_nat @ nil_a @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_156_bind__simps_I1_J,axiom,
! [F: nat > list_a] :
( ( bind_nat_a @ nil_nat @ F )
= nil_a ) ).
% bind_simps(1)
thf(fact_157_bind__simps_I1_J,axiom,
! [F: nat > list_nat] :
( ( bind_nat_nat @ nil_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_158_maps__simps_I2_J,axiom,
! [F: a > list_a] :
( ( maps_a_a @ F @ nil_a )
= nil_a ) ).
% maps_simps(2)
thf(fact_159_maps__simps_I2_J,axiom,
! [F: a > list_nat] :
( ( maps_a_nat @ F @ nil_a )
= nil_nat ) ).
% maps_simps(2)
thf(fact_160_maps__simps_I2_J,axiom,
! [F: nat > list_a] :
( ( maps_nat_a @ F @ nil_nat )
= nil_a ) ).
% maps_simps(2)
thf(fact_161_maps__simps_I2_J,axiom,
! [F: nat > list_nat] :
( ( maps_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% maps_simps(2)
thf(fact_162_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_a @ N @ nil_a )
= N ) ).
% gen_length_code(1)
thf(fact_163_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_nat @ N @ nil_nat )
= N ) ).
% gen_length_code(1)
thf(fact_164_Suc__inject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
=> ( X2 = Y2 ) ) ).
% Suc_inject
thf(fact_165_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_166_size__neq__size__imp__neq,axiom,
! [X2: list_a,Y2: list_a] :
( ( ( size_size_list_a @ X2 )
!= ( size_size_list_a @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_167_size__neq__size__imp__neq,axiom,
! [X2: char,Y2: char] :
( ( ( size_size_char @ X2 )
!= ( size_size_char @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_168_size__neq__size__imp__neq,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( ( size_size_list_nat @ X2 )
!= ( size_size_list_nat @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_169_prefixes__snoc,axiom,
! [Xs: list_a,X2: a] :
( ( prefixes_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) )
= ( append_list_a @ ( prefixes_a @ Xs ) @ ( cons_list_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) @ nil_list_a ) ) ) ).
% prefixes_snoc
thf(fact_170_prefixes__snoc,axiom,
! [Xs: list_nat,X2: nat] :
( ( prefixes_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= ( append_list_nat @ ( prefixes_nat @ Xs ) @ ( cons_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) @ nil_list_nat ) ) ) ).
% prefixes_snoc
thf(fact_171_Succ__def,axiom,
( bNF_Greatest_Succ_a
= ( ^ [Kl: set_list_a,Kl2: list_a] :
( collect_a
@ ^ [K: a] : ( member_list_a @ ( append_a @ Kl2 @ ( cons_a @ K @ nil_a ) ) @ Kl ) ) ) ) ).
% Succ_def
thf(fact_172_Succ__def,axiom,
( bNF_Gr6352880689984616693cc_nat
= ( ^ [Kl: set_list_nat,Kl2: list_nat] :
( collect_nat
@ ^ [K: nat] : ( member_list_nat @ ( append_nat @ Kl2 @ ( cons_nat @ K @ nil_nat ) ) @ Kl ) ) ) ) ).
% Succ_def
thf(fact_173_SuccD,axiom,
! [K2: a,Kl3: set_list_a,Kl4: list_a] :
( ( member_a @ K2 @ ( bNF_Greatest_Succ_a @ Kl3 @ Kl4 ) )
=> ( member_list_a @ ( append_a @ Kl4 @ ( cons_a @ K2 @ nil_a ) ) @ Kl3 ) ) ).
% SuccD
thf(fact_174_SuccD,axiom,
! [K2: nat,Kl3: set_list_nat,Kl4: list_nat] :
( ( member_nat @ K2 @ ( bNF_Gr6352880689984616693cc_nat @ Kl3 @ Kl4 ) )
=> ( member_list_nat @ ( append_nat @ Kl4 @ ( cons_nat @ K2 @ nil_nat ) ) @ Kl3 ) ) ).
% SuccD
thf(fact_175_SuccI,axiom,
! [Kl4: list_a,K2: a,Kl3: set_list_a] :
( ( member_list_a @ ( append_a @ Kl4 @ ( cons_a @ K2 @ nil_a ) ) @ Kl3 )
=> ( member_a @ K2 @ ( bNF_Greatest_Succ_a @ Kl3 @ Kl4 ) ) ) ).
% SuccI
thf(fact_176_SuccI,axiom,
! [Kl4: list_nat,K2: nat,Kl3: set_list_nat] :
( ( member_list_nat @ ( append_nat @ Kl4 @ ( cons_nat @ K2 @ nil_nat ) ) @ Kl3 )
=> ( member_nat @ K2 @ ( bNF_Gr6352880689984616693cc_nat @ Kl3 @ Kl4 ) ) ) ).
% SuccI
thf(fact_177_prefixes__eq__snoc,axiom,
! [Ys: list_a,Xs: list_list_a,X2: list_a] :
( ( ( prefixes_a @ Ys )
= ( append_list_a @ Xs @ ( cons_list_a @ X2 @ nil_list_a ) ) )
= ( ( ( ( Ys = nil_a )
& ( Xs = nil_list_a ) )
| ? [Z3: a,Zs3: list_a] :
( ( Ys
= ( append_a @ Zs3 @ ( cons_a @ Z3 @ nil_a ) ) )
& ( Xs
= ( prefixes_a @ Zs3 ) ) ) )
& ( X2 = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_178_prefixes__eq__snoc,axiom,
! [Ys: list_nat,Xs: list_list_nat,X2: list_nat] :
( ( ( prefixes_nat @ Ys )
= ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) )
= ( ( ( ( Ys = nil_nat )
& ( Xs = nil_list_nat ) )
| ? [Z3: nat,Zs3: list_nat] :
( ( Ys
= ( append_nat @ Zs3 @ ( cons_nat @ Z3 @ nil_nat ) ) )
& ( Xs
= ( prefixes_nat @ Zs3 ) ) ) )
& ( X2 = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_179_sublists_Osimps_I1_J,axiom,
( ( sublists_a @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% sublists.simps(1)
thf(fact_180_sublists_Osimps_I1_J,axiom,
( ( sublists_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% sublists.simps(1)
thf(fact_181_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_182_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_183_suffixes__eq__snoc,axiom,
! [Ys: list_a,Xs: list_list_a,X2: list_a] :
( ( ( suffixes_a @ Ys )
= ( append_list_a @ Xs @ ( cons_list_a @ X2 @ nil_list_a ) ) )
= ( ( ( ( Ys = nil_a )
& ( Xs = nil_list_a ) )
| ? [Z3: a,Zs3: list_a] :
( ( Ys
= ( cons_a @ Z3 @ Zs3 ) )
& ( Xs
= ( suffixes_a @ Zs3 ) ) ) )
& ( X2 = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_184_suffixes__eq__snoc,axiom,
! [Ys: list_nat,Xs: list_list_nat,X2: list_nat] :
( ( ( suffixes_nat @ Ys )
= ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) )
= ( ( ( ( Ys = nil_nat )
& ( Xs = nil_list_nat ) )
| ? [Z3: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z3 @ Zs3 ) )
& ( Xs
= ( suffixes_nat @ Zs3 ) ) ) )
& ( X2 = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_185_subseqs_Osimps_I1_J,axiom,
( ( subseqs_a @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% subseqs.simps(1)
thf(fact_186_subseqs_Osimps_I1_J,axiom,
( ( subseqs_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_187_length__suffixes,axiom,
! [Xs: list_a] :
( ( size_s349497388124573686list_a @ ( suffixes_a @ Xs ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_suffixes
thf(fact_188_length__suffixes,axiom,
! [Xs: list_nat] :
( ( size_s3023201423986296836st_nat @ ( suffixes_nat @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_suffixes
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,
! [X2: a,Xs: list_a] :
( ( suffixes_a @ ( cons_a @ X2 @ Xs ) )
= ( append_list_a @ ( suffixes_a @ Xs ) @ ( cons_list_a @ ( cons_a @ X2 @ Xs ) @ nil_list_a ) ) ) ).
% suffixes.simps(2)
thf(fact_192_suffixes_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( suffixes_nat @ ( cons_nat @ X2 @ Xs ) )
= ( append_list_nat @ ( suffixes_nat @ Xs ) @ ( cons_list_nat @ ( cons_nat @ X2 @ 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,
! [Kl3: set_list_a,K2: a] :
( ( member_list_a @ nil_a @ Kl3 )
=> ( ( member_a @ K2 @ ( bNF_Greatest_Succ_a @ Kl3 @ nil_a ) )
=> ( member_list_a @ nil_a @ ( bNF_Greatest_Shift_a @ Kl3 @ K2 ) ) ) ) ).
% empty_Shift
thf(fact_196_empty__Shift,axiom,
! [Kl3: set_list_nat,K2: nat] :
( ( member_list_nat @ nil_nat @ Kl3 )
=> ( ( member_nat @ K2 @ ( bNF_Gr6352880689984616693cc_nat @ Kl3 @ nil_nat ) )
=> ( member_list_nat @ nil_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl3 @ K2 ) ) ) ) ).
% empty_Shift
thf(fact_197_Succ__Shift,axiom,
! [Kl3: set_list_a,K2: a,Kl4: list_a] :
( ( bNF_Greatest_Succ_a @ ( bNF_Greatest_Shift_a @ Kl3 @ K2 ) @ Kl4 )
= ( bNF_Greatest_Succ_a @ Kl3 @ ( cons_a @ K2 @ Kl4 ) ) ) ).
% Succ_Shift
thf(fact_198_Succ__Shift,axiom,
! [Kl3: set_list_nat,K2: nat,Kl4: list_nat] :
( ( bNF_Gr6352880689984616693cc_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl3 @ K2 ) @ Kl4 )
= ( bNF_Gr6352880689984616693cc_nat @ Kl3 @ ( cons_nat @ K2 @ Kl4 ) ) ) ).
% Succ_Shift
thf(fact_199_suffixes__snoc,axiom,
! [Xs: list_a,X2: a] :
( ( suffixes_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) )
= ( cons_list_a @ nil_a
@ ( map_list_a_list_a
@ ^ [Ys2: list_a] : ( append_a @ Ys2 @ ( cons_a @ X2 @ nil_a ) )
@ ( suffixes_a @ Xs ) ) ) ) ).
% suffixes_snoc
thf(fact_200_suffixes__snoc,axiom,
! [Xs: list_nat,X2: nat] :
( ( suffixes_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= ( cons_list_nat @ nil_nat
@ ( map_li7225945977422193158st_nat
@ ^ [Ys2: list_nat] : ( append_nat @ Ys2 @ ( cons_nat @ X2 @ nil_nat ) )
@ ( suffixes_nat @ Xs ) ) ) ) ).
% suffixes_snoc
thf(fact_201_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_202_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_203_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_204_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_205_list__ex1__simps_I1_J,axiom,
! [P: a > $o] :
~ ( list_ex1_a @ P @ nil_a ) ).
% list_ex1_simps(1)
thf(fact_206_list__ex1__simps_I1_J,axiom,
! [P: nat > $o] :
~ ( list_ex1_nat @ P @ nil_nat ) ).
% list_ex1_simps(1)
thf(fact_207_butlast__snoc,axiom,
! [Xs: list_a,X2: a] :
( ( butlast_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_208_butlast__snoc,axiom,
! [Xs: list_nat,X2: nat] :
( ( butlast_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_209_sublists_Osimps_I2_J,axiom,
! [X2: a,Xs: list_a] :
( ( sublists_a @ ( cons_a @ X2 @ Xs ) )
= ( append_list_a @ ( sublists_a @ Xs ) @ ( map_list_a_list_a @ ( cons_a @ X2 ) @ ( prefixes_a @ Xs ) ) ) ) ).
% sublists.simps(2)
thf(fact_210_sublists_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( sublists_nat @ ( cons_nat @ X2 @ Xs ) )
= ( append_list_nat @ ( sublists_nat @ Xs ) @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X2 ) @ ( prefixes_nat @ Xs ) ) ) ) ).
% sublists.simps(2)
thf(fact_211_last__snoc,axiom,
! [Xs: list_a,X2: a] :
( ( last_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) )
= X2 ) ).
% last_snoc
thf(fact_212_last__snoc,axiom,
! [Xs: list_nat,X2: nat] :
( ( last_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= X2 ) ).
% last_snoc
thf(fact_213_map__ident,axiom,
( ( map_nat_nat
@ ^ [X: nat] : X )
= ( ^ [Xs4: list_nat] : Xs4 ) ) ).
% map_ident
thf(fact_214_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_215_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_216_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_217_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_218_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_219_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_220_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_221_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_222_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_223_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_224_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_225_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_226_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_227_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_228_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_229_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_230_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_231_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_232_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_233_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_234_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_235_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_236_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_237_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_238_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_239_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_240_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_241_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_242_concat__map__singleton,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( concat_nat
@ ( map_nat_list_nat
@ ^ [X: nat] : ( cons_nat @ ( F @ X ) @ nil_nat )
@ Xs ) )
= ( map_nat_nat @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_243_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_244_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_245_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_246_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_247_list_Omap__ident,axiom,
! [T: list_nat] :
( ( map_nat_nat
@ ^ [X: nat] : X
@ T )
= T ) ).
% list.map_ident
thf(fact_248_list_Osimps_I9_J,axiom,
! [F: a > a,X21: a,X222: list_a] :
( ( map_a_a @ F @ ( cons_a @ X21 @ X222 ) )
= ( cons_a @ ( F @ X21 ) @ ( map_a_a @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_249_list_Osimps_I9_J,axiom,
! [F: a > nat,X21: a,X222: list_a] :
( ( map_a_nat @ F @ ( cons_a @ X21 @ X222 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_a_nat @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_250_list_Osimps_I9_J,axiom,
! [F: nat > a,X21: nat,X222: list_nat] :
( ( map_nat_a @ F @ ( cons_nat @ X21 @ X222 ) )
= ( cons_a @ ( F @ X21 ) @ ( map_nat_a @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_251_list_Osimps_I9_J,axiom,
! [F: nat > nat,X21: nat,X222: list_nat] :
( ( map_nat_nat @ F @ ( cons_nat @ X21 @ X222 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_nat_nat @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_252_Cons__eq__map__D,axiom,
! [X2: a,Xs: list_a,F: a > a,Ys: list_a] :
( ( ( cons_a @ X2 @ Xs )
= ( map_a_a @ F @ Ys ) )
=> ? [Z2: a,Zs2: list_a] :
( ( Ys
= ( cons_a @ Z2 @ Zs2 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_a_a @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_253_Cons__eq__map__D,axiom,
! [X2: a,Xs: list_a,F: nat > a,Ys: list_nat] :
( ( ( cons_a @ X2 @ Xs )
= ( map_nat_a @ F @ Ys ) )
=> ? [Z2: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z2 @ Zs2 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_nat_a @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_254_Cons__eq__map__D,axiom,
! [X2: nat,Xs: list_nat,F: a > nat,Ys: list_a] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_a_nat @ F @ Ys ) )
=> ? [Z2: a,Zs2: list_a] :
( ( Ys
= ( cons_a @ Z2 @ Zs2 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_a_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_255_Cons__eq__map__D,axiom,
! [X2: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_nat_nat @ F @ Ys ) )
=> ? [Z2: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z2 @ Zs2 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_256_map__eq__Cons__D,axiom,
! [F: a > a,Xs: list_a,Y2: a,Ys: list_a] :
( ( ( map_a_a @ F @ Xs )
= ( cons_a @ Y2 @ Ys ) )
=> ? [Z2: a,Zs2: list_a] :
( ( Xs
= ( cons_a @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y2 )
& ( ( map_a_a @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_257_map__eq__Cons__D,axiom,
! [F: nat > a,Xs: list_nat,Y2: a,Ys: list_a] :
( ( ( map_nat_a @ F @ Xs )
= ( cons_a @ Y2 @ Ys ) )
=> ? [Z2: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y2 )
& ( ( map_nat_a @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_258_map__eq__Cons__D,axiom,
! [F: a > nat,Xs: list_a,Y2: nat,Ys: list_nat] :
( ( ( map_a_nat @ F @ Xs )
= ( cons_nat @ Y2 @ Ys ) )
=> ? [Z2: a,Zs2: list_a] :
( ( Xs
= ( cons_a @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y2 )
& ( ( map_a_nat @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_259_map__eq__Cons__D,axiom,
! [F: nat > nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y2 @ Ys ) )
=> ? [Z2: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y2 )
& ( ( map_nat_nat @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_260_Cons__eq__map__conv,axiom,
! [X2: a,Xs: list_a,F: a > a,Ys: list_a] :
( ( ( cons_a @ X2 @ Xs )
= ( map_a_a @ F @ Ys ) )
= ( ? [Z3: a,Zs3: list_a] :
( ( Ys
= ( cons_a @ Z3 @ Zs3 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_a_a @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_261_Cons__eq__map__conv,axiom,
! [X2: a,Xs: list_a,F: nat > a,Ys: list_nat] :
( ( ( cons_a @ X2 @ Xs )
= ( map_nat_a @ F @ Ys ) )
= ( ? [Z3: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z3 @ Zs3 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_nat_a @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_262_Cons__eq__map__conv,axiom,
! [X2: nat,Xs: list_nat,F: a > nat,Ys: list_a] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_a_nat @ F @ Ys ) )
= ( ? [Z3: a,Zs3: list_a] :
( ( Ys
= ( cons_a @ Z3 @ Zs3 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_a_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_263_Cons__eq__map__conv,axiom,
! [X2: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_nat_nat @ F @ Ys ) )
= ( ? [Z3: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z3 @ Zs3 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_264_map__eq__Cons__conv,axiom,
! [F: a > a,Xs: list_a,Y2: a,Ys: list_a] :
( ( ( map_a_a @ F @ Xs )
= ( cons_a @ Y2 @ Ys ) )
= ( ? [Z3: a,Zs3: list_a] :
( ( Xs
= ( cons_a @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y2 )
& ( ( map_a_a @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_265_map__eq__Cons__conv,axiom,
! [F: nat > a,Xs: list_nat,Y2: a,Ys: list_a] :
( ( ( map_nat_a @ F @ Xs )
= ( cons_a @ Y2 @ Ys ) )
= ( ? [Z3: nat,Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y2 )
& ( ( map_nat_a @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_266_map__eq__Cons__conv,axiom,
! [F: a > nat,Xs: list_a,Y2: nat,Ys: list_nat] :
( ( ( map_a_nat @ F @ Xs )
= ( cons_nat @ Y2 @ Ys ) )
= ( ? [Z3: a,Zs3: list_a] :
( ( Xs
= ( cons_a @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y2 )
& ( ( map_a_nat @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_267_map__eq__Cons__conv,axiom,
! [F: nat > nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y2 @ Ys ) )
= ( ? [Z3: nat,Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y2 )
& ( ( map_nat_nat @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_268_list_Osimps_I8_J,axiom,
! [F: a > a] :
( ( map_a_a @ F @ nil_a )
= nil_a ) ).
% list.simps(8)
thf(fact_269_list_Osimps_I8_J,axiom,
! [F: a > nat] :
( ( map_a_nat @ F @ nil_a )
= nil_nat ) ).
% list.simps(8)
thf(fact_270_list_Osimps_I8_J,axiom,
! [F: nat > a] :
( ( map_nat_a @ F @ nil_nat )
= nil_a ) ).
% list.simps(8)
thf(fact_271_list_Osimps_I8_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_272_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_273_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_274_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_275_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_276_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_277_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_278_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_279_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_280_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_281_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_282_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_283_butlast_Osimps_I1_J,axiom,
( ( butlast_a @ nil_a )
= nil_a ) ).
% butlast.simps(1)
thf(fact_284_butlast_Osimps_I1_J,axiom,
( ( butlast_nat @ nil_nat )
= nil_nat ) ).
% butlast.simps(1)
thf(fact_285_snoc__eq__iff__butlast,axiom,
! [Xs: list_a,X2: a,Ys: list_a] :
( ( ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) )
= Ys )
= ( ( Ys != nil_a )
& ( ( butlast_a @ Ys )
= Xs )
& ( ( last_a @ Ys )
= X2 ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_286_snoc__eq__iff__butlast,axiom,
! [Xs: list_nat,X2: nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) )
= Ys )
= ( ( Ys != nil_nat )
& ( ( butlast_nat @ Ys )
= Xs )
& ( ( last_nat @ Ys )
= X2 ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_287_concat_Osimps_I2_J,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( concat_a @ ( cons_list_a @ X2 @ Xs ) )
= ( append_a @ X2 @ ( concat_a @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_288_concat_Osimps_I2_J,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( concat_nat @ ( cons_list_nat @ X2 @ Xs ) )
= ( append_nat @ X2 @ ( concat_nat @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_289_last_Osimps,axiom,
! [Xs: list_a,X2: a] :
( ( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X2 @ Xs ) )
= X2 ) )
& ( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X2 @ Xs ) )
= ( last_a @ Xs ) ) ) ) ).
% last.simps
thf(fact_290_last_Osimps,axiom,
! [Xs: list_nat,X2: nat] :
( ( ( Xs = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
= X2 ) )
& ( ( Xs != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
= ( last_nat @ Xs ) ) ) ) ).
% last.simps
thf(fact_291_last__ConsL,axiom,
! [Xs: list_a,X2: a] :
( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X2 @ Xs ) )
= X2 ) ) ).
% last_ConsL
thf(fact_292_last__ConsL,axiom,
! [Xs: list_nat,X2: nat] :
( ( Xs = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
= X2 ) ) ).
% last_ConsL
thf(fact_293_last__ConsR,axiom,
! [Xs: list_a,X2: a] :
( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X2 @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_294_last__ConsR,axiom,
! [Xs: list_nat,X2: nat] :
( ( Xs != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
= ( last_nat @ Xs ) ) ) ).
% last_ConsR
thf(fact_295_concat_Osimps_I1_J,axiom,
( ( concat_a @ nil_list_a )
= nil_a ) ).
% concat.simps(1)
thf(fact_296_concat_Osimps_I1_J,axiom,
( ( concat_nat @ nil_list_nat )
= nil_nat ) ).
% concat.simps(1)
thf(fact_297_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_298_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_299_longest__common__suffix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ss: list_a,Xs3: list_a,Ys4: list_a] :
( ( Xs
= ( append_a @ Xs3 @ Ss ) )
& ( Ys
= ( append_a @ Ys4 @ Ss ) )
& ( ( Xs3 = nil_a )
| ( Ys4 = nil_a )
| ( ( last_a @ Xs3 )
!= ( last_a @ Ys4 ) ) ) ) ).
% longest_common_suffix
thf(fact_300_longest__common__suffix,axiom,
! [Xs: list_nat,Ys: list_nat] :
? [Ss: list_nat,Xs3: list_nat,Ys4: list_nat] :
( ( Xs
= ( append_nat @ Xs3 @ Ss ) )
& ( Ys
= ( append_nat @ Ys4 @ Ss ) )
& ( ( Xs3 = nil_nat )
| ( Ys4 = nil_nat )
| ( ( last_nat @ Xs3 )
!= ( last_nat @ Ys4 ) ) ) ) ).
% longest_common_suffix
thf(fact_301_butlast_Osimps_I2_J,axiom,
! [Xs: list_a,X2: a] :
( ( ( Xs = nil_a )
=> ( ( butlast_a @ ( cons_a @ X2 @ Xs ) )
= nil_a ) )
& ( ( Xs != nil_a )
=> ( ( butlast_a @ ( cons_a @ X2 @ Xs ) )
= ( cons_a @ X2 @ ( butlast_a @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_302_butlast_Osimps_I2_J,axiom,
! [Xs: list_nat,X2: nat] :
( ( ( Xs = nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X2 @ Xs ) )
= nil_nat ) )
& ( ( Xs != nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ X2 @ ( butlast_nat @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_303_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_304_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_305_ShiftD,axiom,
! [Kl4: list_a,Kl3: set_list_a,K2: a] :
( ( member_list_a @ Kl4 @ ( bNF_Greatest_Shift_a @ Kl3 @ K2 ) )
=> ( member_list_a @ ( cons_a @ K2 @ Kl4 ) @ Kl3 ) ) ).
% ShiftD
thf(fact_306_ShiftD,axiom,
! [Kl4: list_nat,Kl3: set_list_nat,K2: nat] :
( ( member_list_nat @ Kl4 @ ( bNF_Gr1872714664788909425ft_nat @ Kl3 @ K2 ) )
=> ( member_list_nat @ ( cons_nat @ K2 @ Kl4 ) @ Kl3 ) ) ).
% ShiftD
thf(fact_307_subseqs_Osimps_I2_J,axiom,
! [X2: a,Xs: list_a] :
( ( subseqs_a @ ( cons_a @ X2 @ Xs ) )
= ( append_list_a @ ( map_list_a_list_a @ ( cons_a @ X2 ) @ ( subseqs_a @ Xs ) ) @ ( subseqs_a @ Xs ) ) ) ).
% subseqs.simps(2)
thf(fact_308_subseqs_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( subseqs_nat @ ( cons_nat @ X2 @ Xs ) )
= ( append_list_nat @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X2 ) @ ( subseqs_nat @ Xs ) ) @ ( subseqs_nat @ Xs ) ) ) ).
% subseqs.simps(2)
thf(fact_309_prefixes_Osimps_I2_J,axiom,
! [X2: a,Xs: list_a] :
( ( prefixes_a @ ( cons_a @ X2 @ Xs ) )
= ( cons_list_a @ nil_a @ ( map_list_a_list_a @ ( cons_a @ X2 ) @ ( prefixes_a @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_310_prefixes_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( prefixes_nat @ ( cons_nat @ X2 @ Xs ) )
= ( cons_list_nat @ nil_nat @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X2 ) @ ( prefixes_nat @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_311_Shift__def,axiom,
( bNF_Greatest_Shift_a
= ( ^ [Kl: set_list_a,K: a] :
( collect_list_a
@ ^ [Kl2: list_a] : ( member_list_a @ ( cons_a @ K @ Kl2 ) @ Kl ) ) ) ) ).
% Shift_def
thf(fact_312_Shift__def,axiom,
( bNF_Gr1872714664788909425ft_nat
= ( ^ [Kl: set_list_nat,K: nat] :
( collect_list_nat
@ ^ [Kl2: list_nat] : ( member_list_nat @ ( cons_nat @ K @ Kl2 ) @ Kl ) ) ) ) ).
% Shift_def
thf(fact_313_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
@ ^ [Y: a] : ( cons_a @ Y @ Ys2 )
@ Xs )
@ ( n_lists_a @ N @ Xs ) ) ) ) ).
% n_lists.simps(2)
thf(fact_314_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
@ ^ [Y: nat] : ( cons_nat @ Y @ Ys2 )
@ Xs )
@ ( n_lists_nat @ N @ Xs ) ) ) ) ).
% n_lists.simps(2)
thf(fact_315_map__rec,axiom,
( map_nat_nat
= ( ^ [F2: nat > nat] :
( rec_li7516600145284979816at_nat @ nil_nat
@ ^ [X: nat,Uu: list_nat] : ( cons_nat @ ( F2 @ X ) ) ) ) ) ).
% map_rec
thf(fact_316_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_317_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_318_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_319_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_320_map__eq__map__tailrec,axiom,
map_nat_nat = map_tailrec_nat_nat ).
% map_eq_map_tailrec
thf(fact_321_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
@ ^ [X: a] : ( map_list_a_list_a @ ( cons_a @ X ) @ ( product_lists_a @ Xss2 ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_322_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
@ ^ [X: nat] : ( map_li7225945977422193158st_nat @ ( cons_nat @ X ) @ ( product_lists_nat @ Xss2 ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_323_list__ex1__simps_I2_J,axiom,
! [P: a > $o,X2: a,Xs: list_a] :
( ( list_ex1_a @ P @ ( cons_a @ X2 @ Xs ) )
= ( ( ( P @ X2 )
=> ( list_all_a
@ ^ [Y: a] :
( ~ ( P @ Y )
| ( X2 = Y ) )
@ Xs ) )
& ( ~ ( P @ X2 )
=> ( list_ex1_a @ P @ Xs ) ) ) ) ).
% list_ex1_simps(2)
thf(fact_324_list__ex1__simps_I2_J,axiom,
! [P: nat > $o,X2: nat,Xs: list_nat] :
( ( list_ex1_nat @ P @ ( cons_nat @ X2 @ Xs ) )
= ( ( ( P @ X2 )
=> ( list_all_nat
@ ^ [Y: nat] :
( ~ ( P @ Y )
| ( X2 = Y ) )
@ Xs ) )
& ( ~ ( P @ X2 )
=> ( list_ex1_nat @ P @ Xs ) ) ) ) ).
% list_ex1_simps(2)
thf(fact_325_rotate1_Osimps_I2_J,axiom,
! [X2: a,Xs: list_a] :
( ( rotate1_a @ ( cons_a @ X2 @ Xs ) )
= ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) ) ).
% rotate1.simps(2)
thf(fact_326_rotate1_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( rotate1_nat @ ( cons_nat @ X2 @ Xs ) )
= ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) ) ).
% rotate1.simps(2)
thf(fact_327_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_328_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_329_list__all__simps_I1_J,axiom,
! [P: a > $o,X2: a,Xs: list_a] :
( ( list_all_a @ P @ ( cons_a @ X2 @ Xs ) )
= ( ( P @ X2 )
& ( list_all_a @ P @ Xs ) ) ) ).
% list_all_simps(1)
thf(fact_330_list__all__simps_I1_J,axiom,
! [P: nat > $o,X2: nat,Xs: list_nat] :
( ( list_all_nat @ P @ ( cons_nat @ X2 @ Xs ) )
= ( ( P @ X2 )
& ( list_all_nat @ P @ Xs ) ) ) ).
% list_all_simps(1)
thf(fact_331_list__all__simps_I2_J,axiom,
! [P: a > $o] : ( list_all_a @ P @ nil_a ) ).
% list_all_simps(2)
thf(fact_332_list__all__simps_I2_J,axiom,
! [P: nat > $o] : ( list_all_nat @ P @ nil_nat ) ).
% list_all_simps(2)
thf(fact_333_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_334_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_335_rotate1__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rotate1_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_336_rotate1__is__Nil__conv,axiom,
! [Xs: list_nat] :
( ( ( rotate1_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% rotate1_is_Nil_conv
thf(fact_337_length__rotate1,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( rotate1_a @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_rotate1
thf(fact_338_length__rotate1,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate1
thf(fact_339_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_340_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_341_tl__def,axiom,
( tl_a
= ( case_list_list_a_a @ nil_a
@ ^ [X213: a,X224: list_a] : X224 ) ) ).
% tl_def
thf(fact_342_tl__def,axiom,
( tl_nat
= ( case_l2340614614379431832at_nat @ nil_nat
@ ^ [X213: nat,X224: list_nat] : X224 ) ) ).
% tl_def
thf(fact_343_tl__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( case_list_list_a_a @ ( tl_a @ Ys )
@ ^ [Z3: a,Zs3: list_a] : ( append_a @ Zs3 @ Ys )
@ Xs ) ) ).
% tl_append
thf(fact_344_tl__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( tl_nat @ ( append_nat @ Xs @ Ys ) )
= ( case_l2340614614379431832at_nat @ ( tl_nat @ Ys )
@ ^ [Z3: nat,Zs3: list_nat] : ( append_nat @ Zs3 @ Ys )
@ Xs ) ) ).
% tl_append
thf(fact_345_list_Osel_I3_J,axiom,
! [X21: a,X222: list_a] :
( ( tl_a @ ( cons_a @ X21 @ X222 ) )
= X222 ) ).
% list.sel(3)
thf(fact_346_list_Osel_I3_J,axiom,
! [X21: nat,X222: list_nat] :
( ( tl_nat @ ( cons_nat @ X21 @ X222 ) )
= X222 ) ).
% list.sel(3)
thf(fact_347_list_Osel_I2_J,axiom,
( ( tl_a @ nil_a )
= nil_a ) ).
% list.sel(2)
thf(fact_348_list_Osel_I2_J,axiom,
( ( tl_nat @ nil_nat )
= nil_nat ) ).
% list.sel(2)
thf(fact_349_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_350_list_Opred__inject_I1_J,axiom,
! [P: a > $o] : ( list_all_a @ P @ nil_a ) ).
% list.pred_inject(1)
thf(fact_351_list_Opred__inject_I1_J,axiom,
! [P: nat > $o] : ( list_all_nat @ P @ nil_nat ) ).
% list.pred_inject(1)
thf(fact_352_butlast__tl,axiom,
! [Xs: list_nat] :
( ( butlast_nat @ ( tl_nat @ Xs ) )
= ( tl_nat @ ( butlast_nat @ Xs ) ) ) ).
% butlast_tl
thf(fact_353_rotate1_Osimps_I1_J,axiom,
( ( rotate1_a @ nil_a )
= nil_a ) ).
% rotate1.simps(1)
thf(fact_354_rotate1_Osimps_I1_J,axiom,
( ( rotate1_nat @ nil_nat )
= nil_nat ) ).
% rotate1.simps(1)
thf(fact_355_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_356_list_Omap__cong__pred,axiom,
! [X2: list_nat,Ya: list_nat,F: nat > nat,G: nat > nat] :
( ( X2 = Ya )
=> ( ( list_all_nat
@ ^ [Z3: nat] :
( ( F @ Z3 )
= ( G @ Z3 ) )
@ Ya )
=> ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ G @ Ya ) ) ) ) ).
% list.map_cong_pred
thf(fact_357_tl__Nil,axiom,
! [Xs: list_a] :
( ( ( tl_a @ Xs )
= nil_a )
= ( ( Xs = nil_a )
| ? [X: a] :
( Xs
= ( cons_a @ X @ nil_a ) ) ) ) ).
% tl_Nil
thf(fact_358_tl__Nil,axiom,
! [Xs: list_nat] :
( ( ( tl_nat @ Xs )
= nil_nat )
= ( ( Xs = nil_nat )
| ? [X: nat] :
( Xs
= ( cons_nat @ X @ nil_nat ) ) ) ) ).
% tl_Nil
thf(fact_359_Nil__tl,axiom,
! [Xs: list_a] :
( ( nil_a
= ( tl_a @ Xs ) )
= ( ( Xs = nil_a )
| ? [X: a] :
( Xs
= ( cons_a @ X @ nil_a ) ) ) ) ).
% Nil_tl
thf(fact_360_Nil__tl,axiom,
! [Xs: list_nat] :
( ( nil_nat
= ( tl_nat @ Xs ) )
= ( ( Xs = nil_nat )
| ? [X: nat] :
( Xs
= ( cons_nat @ X @ nil_nat ) ) ) ) ).
% Nil_tl
thf(fact_361_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_362_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_363_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_364_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_365_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_366_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_367_transpose_Oelims,axiom,
! [X2: list_list_a,Y2: list_list_a] :
( ( ( transpose_a @ X2 )
= Y2 )
=> ( ( ( X2 = nil_list_a )
=> ( Y2 != nil_list_a ) )
=> ( ! [Xss: list_list_a] :
( ( X2
= ( cons_list_a @ nil_a @ Xss ) )
=> ( Y2
!= ( transpose_a @ Xss ) ) )
=> ~ ! [X3: a,Xs2: list_a,Xss: list_list_a] :
( ( X2
= ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ Xss ) )
=> ( Y2
!= ( cons_list_a
@ ( cons_a @ X3
@ ( concat_a
@ ( map_list_a_list_a
@ ( case_list_list_a_a @ nil_a
@ ^ [H: a,T2: list_a] : ( cons_a @ H @ nil_a ) )
@ Xss ) ) )
@ ( transpose_a
@ ( cons_list_a @ Xs2
@ ( concat_list_a
@ ( map_li5729356230488778442list_a
@ ( case_l8408404631611421914st_a_a @ nil_list_a
@ ^ [H: a,T2: list_a] : ( cons_list_a @ T2 @ nil_list_a ) )
@ Xss ) ) ) ) ) ) ) ) ) ) ).
% transpose.elims
thf(fact_368_transpose_Oelims,axiom,
! [X2: list_list_nat,Y2: list_list_nat] :
( ( ( transpose_nat @ X2 )
= Y2 )
=> ( ( ( X2 = nil_list_nat )
=> ( Y2 != nil_list_nat ) )
=> ( ! [Xss: list_list_nat] :
( ( X2
= ( cons_list_nat @ nil_nat @ Xss ) )
=> ( Y2
!= ( transpose_nat @ Xss ) ) )
=> ~ ! [X3: nat,Xs2: list_nat,Xss: list_list_nat] :
( ( X2
= ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ Xss ) )
=> ( Y2
!= ( cons_list_nat
@ ( cons_nat @ X3
@ ( concat_nat
@ ( map_li7225945977422193158st_nat
@ ( case_l2340614614379431832at_nat @ nil_nat
@ ^ [H: nat,T2: list_nat] : ( cons_nat @ H @ nil_nat ) )
@ Xss ) ) )
@ ( transpose_nat
@ ( cons_list_nat @ Xs2
@ ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ( case_l3331202209248957608at_nat @ nil_list_nat
@ ^ [H: nat,T2: list_nat] : ( cons_list_nat @ T2 @ nil_list_nat ) )
@ Xss ) ) ) ) ) ) ) ) ) ) ).
% transpose.elims
thf(fact_369_transpose_Osimps_I3_J,axiom,
! [X2: a,Xs: list_a,Xss2: list_list_a] :
( ( transpose_a @ ( cons_list_a @ ( cons_a @ X2 @ Xs ) @ Xss2 ) )
= ( cons_list_a
@ ( cons_a @ X2
@ ( concat_a
@ ( map_list_a_list_a
@ ( case_list_list_a_a @ nil_a
@ ^ [H: a,T2: list_a] : ( cons_a @ H @ nil_a ) )
@ Xss2 ) ) )
@ ( transpose_a
@ ( cons_list_a @ Xs
@ ( concat_list_a
@ ( map_li5729356230488778442list_a
@ ( case_l8408404631611421914st_a_a @ nil_list_a
@ ^ [H: a,T2: list_a] : ( cons_list_a @ T2 @ nil_list_a ) )
@ Xss2 ) ) ) ) ) ) ).
% transpose.simps(3)
thf(fact_370_transpose_Osimps_I3_J,axiom,
! [X2: nat,Xs: list_nat,Xss2: list_list_nat] :
( ( transpose_nat @ ( cons_list_nat @ ( cons_nat @ X2 @ Xs ) @ Xss2 ) )
= ( cons_list_nat
@ ( cons_nat @ X2
@ ( concat_nat
@ ( map_li7225945977422193158st_nat
@ ( case_l2340614614379431832at_nat @ nil_nat
@ ^ [H: nat,T2: list_nat] : ( cons_nat @ H @ nil_nat ) )
@ Xss2 ) ) )
@ ( transpose_nat
@ ( cons_list_nat @ Xs
@ ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ( case_l3331202209248957608at_nat @ nil_list_nat
@ ^ [H: nat,T2: list_nat] : ( cons_list_nat @ T2 @ nil_list_nat ) )
@ Xss2 ) ) ) ) ) ) ).
% transpose.simps(3)
thf(fact_371_transpose__aux__filter__tail,axiom,
! [Xss2: list_list_a] :
( ( concat_list_a
@ ( map_li5729356230488778442list_a
@ ( case_l8408404631611421914st_a_a @ nil_list_a
@ ^ [H: 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_372_transpose__aux__filter__tail,axiom,
! [Xss2: list_list_nat] :
( ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ( case_l3331202209248957608at_nat @ nil_list_nat
@ ^ [H: 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_373_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_374_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_375_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
@ ^ [H: 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_376_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
@ ^ [H: 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_377_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_378_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_379_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_380_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_381_transpose_Opelims,axiom,
! [X2: list_list_a,Y2: list_list_a] :
( ( ( transpose_a @ X2 )
= Y2 )
=> ( ( accp_list_list_a @ transpose_rel_a @ X2 )
=> ( ( ( X2 = nil_list_a )
=> ( ( Y2 = nil_list_a )
=> ~ ( accp_list_list_a @ transpose_rel_a @ nil_list_a ) ) )
=> ( ! [Xss: list_list_a] :
( ( X2
= ( cons_list_a @ nil_a @ Xss ) )
=> ( ( Y2
= ( 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] :
( ( X2
= ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ Xss ) )
=> ( ( Y2
= ( cons_list_a
@ ( cons_a @ X3
@ ( concat_a
@ ( map_list_a_list_a
@ ( case_list_list_a_a @ nil_a
@ ^ [H: a,T2: list_a] : ( cons_a @ H @ nil_a ) )
@ Xss ) ) )
@ ( transpose_a
@ ( cons_list_a @ Xs2
@ ( concat_list_a
@ ( map_li5729356230488778442list_a
@ ( case_l8408404631611421914st_a_a @ nil_list_a
@ ^ [H: 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_382_transpose_Opelims,axiom,
! [X2: list_list_nat,Y2: list_list_nat] :
( ( ( transpose_nat @ X2 )
= Y2 )
=> ( ( accp_list_list_nat @ transpose_rel_nat @ X2 )
=> ( ( ( X2 = nil_list_nat )
=> ( ( Y2 = nil_list_nat )
=> ~ ( accp_list_list_nat @ transpose_rel_nat @ nil_list_nat ) ) )
=> ( ! [Xss: list_list_nat] :
( ( X2
= ( cons_list_nat @ nil_nat @ Xss ) )
=> ( ( Y2
= ( 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] :
( ( X2
= ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ Xss ) )
=> ( ( Y2
= ( cons_list_nat
@ ( cons_nat @ X3
@ ( concat_nat
@ ( map_li7225945977422193158st_nat
@ ( case_l2340614614379431832at_nat @ nil_nat
@ ^ [H: nat,T2: list_nat] : ( cons_nat @ H @ nil_nat ) )
@ Xss ) ) )
@ ( transpose_nat
@ ( cons_list_nat @ Xs2
@ ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ( case_l3331202209248957608at_nat @ nil_list_nat
@ ^ [H: 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_383_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_384_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_385_hd__prefixes,axiom,
! [Xs: list_a] :
( ( hd_list_a @ ( prefixes_a @ Xs ) )
= nil_a ) ).
% hd_prefixes
thf(fact_386_hd__prefixes,axiom,
! [Xs: list_nat] :
( ( hd_list_nat @ ( prefixes_nat @ Xs ) )
= nil_nat ) ).
% hd_prefixes
thf(fact_387_hd__suffixes,axiom,
! [Xs: list_a] :
( ( hd_list_a @ ( suffixes_a @ Xs ) )
= nil_a ) ).
% hd_suffixes
thf(fact_388_hd__suffixes,axiom,
! [Xs: list_nat] :
( ( hd_list_nat @ ( suffixes_nat @ Xs ) )
= nil_nat ) ).
% hd_suffixes
thf(fact_389_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_390_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_391_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_392_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_393_list_Ocollapse,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_394_list_Ocollapse,axiom,
! [List: list_nat] :
( ( List != nil_nat )
=> ( ( cons_nat @ ( hd_nat @ List ) @ ( tl_nat @ List ) )
= List ) ) ).
% list.collapse
thf(fact_395_hd__Cons__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs ) @ ( tl_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_396_hd__Cons__tl,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( cons_nat @ ( hd_nat @ Xs ) @ ( tl_nat @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_397_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_398_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_399_filter_Osimps_I2_J,axiom,
! [P: a > $o,X2: a,Xs: list_a] :
( ( ( P @ X2 )
=> ( ( filter_a @ P @ ( cons_a @ X2 @ Xs ) )
= ( cons_a @ X2 @ ( filter_a @ P @ Xs ) ) ) )
& ( ~ ( P @ X2 )
=> ( ( filter_a @ P @ ( cons_a @ X2 @ Xs ) )
= ( filter_a @ P @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_400_filter_Osimps_I2_J,axiom,
! [P: nat > $o,X2: nat,Xs: list_nat] :
( ( ( P @ X2 )
=> ( ( filter_nat @ P @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ X2 @ ( filter_nat @ P @ Xs ) ) ) )
& ( ~ ( P @ X2 )
=> ( ( filter_nat @ P @ ( cons_nat @ X2 @ Xs ) )
= ( filter_nat @ P @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_401_filter_Osimps_I1_J,axiom,
! [P: a > $o] :
( ( filter_a @ P @ nil_a )
= nil_a ) ).
% filter.simps(1)
thf(fact_402_filter_Osimps_I1_J,axiom,
! [P: nat > $o] :
( ( filter_nat @ P @ nil_nat )
= nil_nat ) ).
% filter.simps(1)
thf(fact_403_list_Osel_I1_J,axiom,
! [X21: a,X222: list_a] :
( ( hd_a @ ( cons_a @ X21 @ X222 ) )
= X21 ) ).
% list.sel(1)
thf(fact_404_list_Osel_I1_J,axiom,
! [X21: nat,X222: list_nat] :
( ( hd_nat @ ( cons_nat @ X21 @ X222 ) )
= X21 ) ).
% list.sel(1)
thf(fact_405_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_406_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_407_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_408_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_409_old_Onat_Oexhaust,axiom,
! [Y2: nat] :
( ( Y2 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y2
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_410_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_411_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_412_zero__induct,axiom,
! [P: nat > $o,K2: nat] :
( ( P @ K2 )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_413_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_414_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_415_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_416_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% not0_implies_Suc
thf(fact_417_encode__unary__nat_Ocases,axiom,
! [X2: nat] :
( ! [L: nat] :
( X2
!= ( suc @ L ) )
=> ( X2 = zero_zero_nat ) ) ).
% encode_unary_nat.cases
thf(fact_418_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_419_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_420_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_421_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_422_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_423_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_424_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_425_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_426_longest__common__prefix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ps: list_a,Xs3: list_a,Ys4: list_a] :
( ( Xs
= ( append_a @ Ps @ Xs3 ) )
& ( Ys
= ( append_a @ Ps @ Ys4 ) )
& ( ( Xs3 = nil_a )
| ( Ys4 = nil_a )
| ( ( hd_a @ Xs3 )
!= ( hd_a @ Ys4 ) ) ) ) ).
% longest_common_prefix
thf(fact_427_longest__common__prefix,axiom,
! [Xs: list_nat,Ys: list_nat] :
? [Ps: list_nat,Xs3: list_nat,Ys4: list_nat] :
( ( Xs
= ( append_nat @ Ps @ Xs3 ) )
& ( Ys
= ( append_nat @ Ps @ Ys4 ) )
& ( ( Xs3 = nil_nat )
| ( Ys4 = nil_nat )
| ( ( hd_nat @ Xs3 )
!= ( hd_nat @ Ys4 ) ) ) ) ).
% longest_common_prefix
thf(fact_428_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_429_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_430_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_431_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_432_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_433_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_434_hd__Nil__eq__last,axiom,
( ( hd_a @ nil_a )
= ( last_a @ nil_a ) ) ).
% hd_Nil_eq_last
thf(fact_435_hd__Nil__eq__last,axiom,
( ( hd_nat @ nil_nat )
= ( last_nat @ nil_nat ) ) ).
% hd_Nil_eq_last
thf(fact_436_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_437_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_438_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_439_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_440_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_441_length__code,axiom,
( size_size_list_a
= ( gen_length_a @ zero_zero_nat ) ) ).
% length_code
thf(fact_442_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_443_list_Oexhaust__sel,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( List
= ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_444_list_Oexhaust__sel,axiom,
! [List: list_nat] :
( ( List != nil_nat )
=> ( List
= ( cons_nat @ ( hd_nat @ List ) @ ( tl_nat @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_445_transpose__aux__filter__head,axiom,
! [Xss2: list_list_a] :
( ( concat_a
@ ( map_list_a_list_a
@ ( case_list_list_a_a @ nil_a
@ ^ [H: a,T2: list_a] : ( cons_a @ H @ 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_446_transpose__aux__filter__head,axiom,
! [Xss2: list_list_nat] :
( ( concat_nat
@ ( map_li7225945977422193158st_nat
@ ( case_l2340614614379431832at_nat @ nil_nat
@ ^ [H: nat,T2: list_nat] : ( cons_nat @ H @ 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_447_transpose_Opsimps_I3_J,axiom,
! [X2: a,Xs: list_a,Xss2: list_list_a] :
( ( accp_list_list_a @ transpose_rel_a @ ( cons_list_a @ ( cons_a @ X2 @ Xs ) @ Xss2 ) )
=> ( ( transpose_a @ ( cons_list_a @ ( cons_a @ X2 @ Xs ) @ Xss2 ) )
= ( cons_list_a
@ ( cons_a @ X2
@ ( concat_a
@ ( map_list_a_list_a
@ ( case_list_list_a_a @ nil_a
@ ^ [H: a,T2: list_a] : ( cons_a @ H @ nil_a ) )
@ Xss2 ) ) )
@ ( transpose_a
@ ( cons_list_a @ Xs
@ ( concat_list_a
@ ( map_li5729356230488778442list_a
@ ( case_l8408404631611421914st_a_a @ nil_list_a
@ ^ [H: a,T2: list_a] : ( cons_list_a @ T2 @ nil_list_a ) )
@ Xss2 ) ) ) ) ) ) ) ).
% transpose.psimps(3)
thf(fact_448_transpose_Opsimps_I3_J,axiom,
! [X2: nat,Xs: list_nat,Xss2: list_list_nat] :
( ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ ( cons_nat @ X2 @ Xs ) @ Xss2 ) )
=> ( ( transpose_nat @ ( cons_list_nat @ ( cons_nat @ X2 @ Xs ) @ Xss2 ) )
= ( cons_list_nat
@ ( cons_nat @ X2
@ ( concat_nat
@ ( map_li7225945977422193158st_nat
@ ( case_l2340614614379431832at_nat @ nil_nat
@ ^ [H: nat,T2: list_nat] : ( cons_nat @ H @ nil_nat ) )
@ Xss2 ) ) )
@ ( transpose_nat
@ ( cons_list_nat @ Xs
@ ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ( case_l3331202209248957608at_nat @ nil_list_nat
@ ^ [H: nat,T2: list_nat] : ( cons_list_nat @ T2 @ nil_list_nat ) )
@ Xss2 ) ) ) ) ) ) ) ).
% transpose.psimps(3)
thf(fact_449_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_450_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_451_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C2: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_452_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_453_list__decode_Ocases,axiom,
! [X2: nat] :
( ( X2 != zero_zero_nat )
=> ~ ! [N2: nat] :
( X2
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_454_nths__Cons,axiom,
! [X2: a,L2: list_a,A2: set_nat] :
( ( nths_a @ ( cons_a @ X2 @ L2 ) @ A2 )
= ( append_a @ ( if_list_a @ ( member_nat @ zero_zero_nat @ A2 ) @ ( cons_a @ X2 @ nil_a ) @ nil_a )
@ ( nths_a @ L2
@ ( collect_nat
@ ^ [J: nat] : ( member_nat @ ( suc @ J ) @ A2 ) ) ) ) ) ).
% nths_Cons
thf(fact_455_nths__Cons,axiom,
! [X2: nat,L2: list_nat,A2: set_nat] :
( ( nths_nat @ ( cons_nat @ X2 @ L2 ) @ A2 )
= ( append_nat @ ( if_list_nat @ ( member_nat @ zero_zero_nat @ A2 ) @ ( cons_nat @ X2 @ nil_nat ) @ nil_nat )
@ ( nths_nat @ L2
@ ( collect_nat
@ ^ [J: nat] : ( member_nat @ ( suc @ J ) @ A2 ) ) ) ) ) ).
% nths_Cons
thf(fact_456_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_457_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_458_nths__nil,axiom,
! [A2: set_nat] :
( ( nths_a @ nil_a @ A2 )
= nil_a ) ).
% nths_nil
thf(fact_459_nths__nil,axiom,
! [A2: set_nat] :
( ( nths_nat @ nil_nat @ A2 )
= nil_nat ) ).
% nths_nil
thf(fact_460_distinct__adj__Cons__Cons,axiom,
! [X2: a,Y2: a,Xs: list_a] :
( ( distinct_adj_a @ ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs ) ) )
= ( ( X2 != Y2 )
& ( distinct_adj_a @ ( cons_a @ Y2 @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_461_distinct__adj__Cons__Cons,axiom,
! [X2: nat,Y2: nat,Xs: list_nat] :
( ( distinct_adj_nat @ ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Xs ) ) )
= ( ( X2 != Y2 )
& ( distinct_adj_nat @ ( cons_nat @ Y2 @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_462_nths__singleton,axiom,
! [A2: set_nat,X2: a] :
( ( ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_a @ ( cons_a @ X2 @ nil_a ) @ A2 )
= ( cons_a @ X2 @ nil_a ) ) )
& ( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_a @ ( cons_a @ X2 @ nil_a ) @ A2 )
= nil_a ) ) ) ).
% nths_singleton
thf(fact_463_nths__singleton,axiom,
! [A2: set_nat,X2: nat] :
( ( ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_nat @ ( cons_nat @ X2 @ nil_nat ) @ A2 )
= ( cons_nat @ X2 @ nil_nat ) ) )
& ( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_nat @ ( cons_nat @ X2 @ nil_nat ) @ A2 )
= nil_nat ) ) ) ).
% nths_singleton
thf(fact_464_nths__map,axiom,
! [F: nat > nat,Xs: list_nat,I: set_nat] :
( ( nths_nat @ ( map_nat_nat @ F @ Xs ) @ I )
= ( map_nat_nat @ F @ ( nths_nat @ Xs @ I ) ) ) ).
% nths_map
thf(fact_465_distinct__adj__ConsD,axiom,
! [X2: a,Xs: list_a] :
( ( distinct_adj_a @ ( cons_a @ X2 @ Xs ) )
=> ( distinct_adj_a @ Xs ) ) ).
% distinct_adj_ConsD
thf(fact_466_distinct__adj__ConsD,axiom,
! [X2: nat,Xs: list_nat] :
( ( distinct_adj_nat @ ( cons_nat @ X2 @ Xs ) )
=> ( distinct_adj_nat @ Xs ) ) ).
% distinct_adj_ConsD
thf(fact_467_distinct__adj__Nil,axiom,
distinct_adj_a @ nil_a ).
% distinct_adj_Nil
thf(fact_468_distinct__adj__Nil,axiom,
distinct_adj_nat @ nil_nat ).
% distinct_adj_Nil
thf(fact_469_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_470_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_471_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_472_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_473_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_474_distinct__adj__singleton,axiom,
! [X2: a] : ( distinct_adj_a @ ( cons_a @ X2 @ nil_a ) ) ).
% distinct_adj_singleton
thf(fact_475_distinct__adj__singleton,axiom,
! [X2: nat] : ( distinct_adj_nat @ ( cons_nat @ X2 @ nil_nat ) ) ).
% distinct_adj_singleton
thf(fact_476_distinct__adj__Cons,axiom,
! [X2: a,Xs: list_a] :
( ( distinct_adj_a @ ( cons_a @ X2 @ Xs ) )
= ( ( Xs = nil_a )
| ( ( X2
!= ( hd_a @ Xs ) )
& ( distinct_adj_a @ Xs ) ) ) ) ).
% distinct_adj_Cons
thf(fact_477_distinct__adj__Cons,axiom,
! [X2: nat,Xs: list_nat] :
( ( distinct_adj_nat @ ( cons_nat @ X2 @ Xs ) )
= ( ( Xs = nil_nat )
| ( ( X2
!= ( hd_nat @ Xs ) )
& ( distinct_adj_nat @ Xs ) ) ) ) ).
% distinct_adj_Cons
thf(fact_478_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_479_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_480_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_481_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_482_Cons__in__shuffles__iff,axiom,
! [Z: a,Zs: list_a,Xs: list_a,Ys: list_a] :
( ( member_list_a @ ( cons_a @ Z @ Zs ) @ ( shuffles_a @ Xs @ Ys ) )
= ( ( ( Xs != nil_a )
& ( ( hd_a @ Xs )
= Z )
& ( member_list_a @ Zs @ ( shuffles_a @ ( tl_a @ Xs ) @ Ys ) ) )
| ( ( Ys != nil_a )
& ( ( hd_a @ Ys )
= Z )
& ( member_list_a @ Zs @ ( shuffles_a @ Xs @ ( tl_a @ Ys ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_483_Cons__in__shuffles__iff,axiom,
! [Z: nat,Zs: list_nat,Xs: list_nat,Ys: list_nat] :
( ( member_list_nat @ ( cons_nat @ Z @ Zs ) @ ( shuffles_nat @ Xs @ Ys ) )
= ( ( ( Xs != nil_nat )
& ( ( hd_nat @ Xs )
= Z )
& ( member_list_nat @ Zs @ ( shuffles_nat @ ( tl_nat @ Xs ) @ Ys ) ) )
| ( ( Ys != nil_nat )
& ( ( hd_nat @ Ys )
= Z )
& ( member_list_nat @ Zs @ ( shuffles_nat @ Xs @ ( tl_nat @ Ys ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_484_remdups__adj__append_H,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
| ( Ys = nil_a )
| ( ( last_a @ Xs )
!= ( hd_a @ Ys ) ) )
=> ( ( remdups_adj_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( remdups_adj_a @ Xs ) @ ( remdups_adj_a @ Ys ) ) ) ) ).
% remdups_adj_append'
thf(fact_485_remdups__adj__append_H,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( Xs = nil_nat )
| ( Ys = nil_nat )
| ( ( last_nat @ Xs )
!= ( hd_nat @ Ys ) ) )
=> ( ( remdups_adj_nat @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( remdups_adj_nat @ Xs ) @ ( remdups_adj_nat @ Ys ) ) ) ) ).
% remdups_adj_append'
thf(fact_486_remdups__adj__append,axiom,
! [Xs_1: list_a,X2: a,Xs_2: list_a] :
( ( remdups_adj_a @ ( append_a @ Xs_1 @ ( cons_a @ X2 @ Xs_2 ) ) )
= ( append_a @ ( remdups_adj_a @ ( append_a @ Xs_1 @ ( cons_a @ X2 @ nil_a ) ) ) @ ( tl_a @ ( remdups_adj_a @ ( cons_a @ X2 @ Xs_2 ) ) ) ) ) ).
% remdups_adj_append
thf(fact_487_remdups__adj__append,axiom,
! [Xs_1: list_nat,X2: nat,Xs_2: list_nat] :
( ( remdups_adj_nat @ ( append_nat @ Xs_1 @ ( cons_nat @ X2 @ Xs_2 ) ) )
= ( append_nat @ ( remdups_adj_nat @ ( append_nat @ Xs_1 @ ( cons_nat @ X2 @ nil_nat ) ) ) @ ( tl_nat @ ( remdups_adj_nat @ ( cons_nat @ X2 @ Xs_2 ) ) ) ) ) ).
% remdups_adj_append
thf(fact_488_list_Osize_I4_J,axiom,
! [X21: a,X222: list_a] :
( ( size_size_list_a @ ( cons_a @ X21 @ X222 ) )
= ( plus_plus_nat @ ( size_size_list_a @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_489_list_Osize_I4_J,axiom,
! [X21: nat,X222: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X21 @ X222 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_490_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_491_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_492_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_493_remdups__adj__Nil__iff,axiom,
! [Xs: list_a] :
( ( ( remdups_adj_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% remdups_adj_Nil_iff
thf(fact_494_remdups__adj__Nil__iff,axiom,
! [Xs: list_nat] :
( ( ( remdups_adj_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% remdups_adj_Nil_iff
thf(fact_495_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_496_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_497_hd__remdups__adj,axiom,
! [Xs: list_nat] :
( ( hd_nat @ ( remdups_adj_nat @ Xs ) )
= ( hd_nat @ Xs ) ) ).
% hd_remdups_adj
thf(fact_498_last__remdups__adj,axiom,
! [Xs: list_nat] :
( ( last_nat @ ( remdups_adj_nat @ Xs ) )
= ( last_nat @ Xs ) ) ).
% last_remdups_adj
thf(fact_499_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_500_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_501_take__Suc__Cons,axiom,
! [N: nat,X2: a,Xs: list_a] :
( ( take_a @ ( suc @ N ) @ ( cons_a @ X2 @ Xs ) )
= ( cons_a @ X2 @ ( take_a @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_502_take__Suc__Cons,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( take_nat @ ( suc @ N ) @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ X2 @ ( take_nat @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_503_take0,axiom,
( ( take_a @ zero_zero_nat )
= ( ^ [Xs4: list_a] : nil_a ) ) ).
% take0
thf(fact_504_take0,axiom,
( ( take_nat @ zero_zero_nat )
= ( ^ [Xs4: list_nat] : nil_nat ) ) ).
% take0
thf(fact_505_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_506_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_507_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_508_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_509_remdups__adj__Cons__alt,axiom,
! [X2: a,Xs: list_a] :
( ( cons_a @ X2 @ ( tl_a @ ( remdups_adj_a @ ( cons_a @ X2 @ Xs ) ) ) )
= ( remdups_adj_a @ ( cons_a @ X2 @ Xs ) ) ) ).
% remdups_adj_Cons_alt
thf(fact_510_remdups__adj__Cons__alt,axiom,
! [X2: nat,Xs: list_nat] :
( ( cons_nat @ X2 @ ( tl_nat @ ( remdups_adj_nat @ ( cons_nat @ X2 @ Xs ) ) ) )
= ( remdups_adj_nat @ ( cons_nat @ X2 @ Xs ) ) ) ).
% remdups_adj_Cons_alt
thf(fact_511_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_512_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_513_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_514_take__Nil,axiom,
! [N: nat] :
( ( take_a @ N @ nil_a )
= nil_a ) ).
% take_Nil
thf(fact_515_take__Nil,axiom,
! [N: nat] :
( ( take_nat @ N @ nil_nat )
= nil_nat ) ).
% take_Nil
thf(fact_516_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_517_remdups__adj_Osimps_I3_J,axiom,
! [X2: a,Y2: a,Xs: list_a] :
( ( ( X2 = Y2 )
=> ( ( remdups_adj_a @ ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs ) ) )
= ( remdups_adj_a @ ( cons_a @ X2 @ Xs ) ) ) )
& ( ( X2 != Y2 )
=> ( ( remdups_adj_a @ ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs ) ) )
= ( cons_a @ X2 @ ( remdups_adj_a @ ( cons_a @ Y2 @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_518_remdups__adj_Osimps_I3_J,axiom,
! [X2: nat,Y2: nat,Xs: list_nat] :
( ( ( X2 = Y2 )
=> ( ( remdups_adj_nat @ ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Xs ) ) )
= ( remdups_adj_nat @ ( cons_nat @ X2 @ Xs ) ) ) )
& ( ( X2 != Y2 )
=> ( ( remdups_adj_nat @ ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Xs ) ) )
= ( cons_nat @ X2 @ ( remdups_adj_nat @ ( cons_nat @ Y2 @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_519_remdups__adj_Osimps_I1_J,axiom,
( ( remdups_adj_a @ nil_a )
= nil_a ) ).
% remdups_adj.simps(1)
thf(fact_520_remdups__adj_Osimps_I1_J,axiom,
( ( remdups_adj_nat @ nil_nat )
= nil_nat ) ).
% remdups_adj.simps(1)
thf(fact_521_nat__arith_Osuc1,axiom,
! [A2: nat,K2: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K2 @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_522_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_523_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_524_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_525_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_526_Cons__in__shuffles__leftI,axiom,
! [Zs: list_a,Xs: list_a,Ys: list_a,Z: a] :
( ( member_list_a @ Zs @ ( shuffles_a @ Xs @ Ys ) )
=> ( member_list_a @ ( cons_a @ Z @ Zs ) @ ( shuffles_a @ ( cons_a @ Z @ Xs ) @ Ys ) ) ) ).
% Cons_in_shuffles_leftI
thf(fact_527_Cons__in__shuffles__leftI,axiom,
! [Zs: list_nat,Xs: list_nat,Ys: list_nat,Z: nat] :
( ( member_list_nat @ Zs @ ( shuffles_nat @ Xs @ Ys ) )
=> ( member_list_nat @ ( cons_nat @ Z @ Zs ) @ ( shuffles_nat @ ( cons_nat @ Z @ Xs ) @ Ys ) ) ) ).
% Cons_in_shuffles_leftI
thf(fact_528_Cons__in__shuffles__rightI,axiom,
! [Zs: list_a,Xs: list_a,Ys: list_a,Z: a] :
( ( member_list_a @ Zs @ ( shuffles_a @ Xs @ Ys ) )
=> ( member_list_a @ ( cons_a @ Z @ Zs ) @ ( shuffles_a @ Xs @ ( cons_a @ Z @ Ys ) ) ) ) ).
% Cons_in_shuffles_rightI
thf(fact_529_Cons__in__shuffles__rightI,axiom,
! [Zs: list_nat,Xs: list_nat,Ys: list_nat,Z: nat] :
( ( member_list_nat @ Zs @ ( shuffles_nat @ Xs @ Ys ) )
=> ( member_list_nat @ ( cons_nat @ Z @ Zs ) @ ( shuffles_nat @ Xs @ ( cons_nat @ Z @ Ys ) ) ) ) ).
% Cons_in_shuffles_rightI
thf(fact_530_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_531_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_532_successively_Oelims_I3_J,axiom,
! [X2: a > a > $o,Xa: list_a] :
( ~ ( successively_a @ X2 @ Xa )
=> ~ ! [X3: a,Y3: a,Xs2: list_a] :
( ( Xa
= ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs2 ) ) )
=> ( ( X2 @ X3 @ Y3 )
& ( successively_a @ X2 @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ).
% successively.elims(3)
thf(fact_533_successively_Oelims_I3_J,axiom,
! [X2: nat > nat > $o,Xa: list_nat] :
( ~ ( successively_nat @ X2 @ Xa )
=> ~ ! [X3: nat,Y3: nat,Xs2: list_nat] :
( ( Xa
= ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs2 ) ) )
=> ( ( X2 @ X3 @ Y3 )
& ( successively_nat @ X2 @ ( cons_nat @ Y3 @ Xs2 ) ) ) ) ) ).
% successively.elims(3)
thf(fact_534_successively_Osimps_I3_J,axiom,
! [P: a > a > $o,X2: a,Y2: a,Xs: list_a] :
( ( successively_a @ P @ ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs ) ) )
= ( ( P @ X2 @ Y2 )
& ( successively_a @ P @ ( cons_a @ Y2 @ Xs ) ) ) ) ).
% successively.simps(3)
thf(fact_535_successively_Osimps_I3_J,axiom,
! [P: nat > nat > $o,X2: nat,Y2: nat,Xs: list_nat] :
( ( successively_nat @ P @ ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Xs ) ) )
= ( ( P @ X2 @ Y2 )
& ( successively_nat @ P @ ( cons_nat @ Y2 @ Xs ) ) ) ) ).
% successively.simps(3)
thf(fact_536_successively_Osimps_I1_J,axiom,
! [P: a > a > $o] : ( successively_a @ P @ nil_a ) ).
% successively.simps(1)
thf(fact_537_successively_Osimps_I1_J,axiom,
! [P: nat > nat > $o] : ( successively_nat @ P @ nil_nat ) ).
% successively.simps(1)
thf(fact_538_successively__map,axiom,
! [P: nat > nat > $o,F: nat > nat,Xs: list_nat] :
( ( successively_nat @ P @ ( map_nat_nat @ F @ Xs ) )
= ( successively_nat
@ ^ [X: nat,Y: nat] : ( P @ ( F @ X ) @ ( F @ Y ) )
@ Xs ) ) ).
% successively_map
thf(fact_539_take__0,axiom,
! [Xs: list_a] :
( ( take_a @ zero_zero_nat @ Xs )
= nil_a ) ).
% take_0
thf(fact_540_take__0,axiom,
! [Xs: list_nat] :
( ( take_nat @ zero_zero_nat @ Xs )
= nil_nat ) ).
% take_0
thf(fact_541_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_542_remdups__adj_Osimps_I2_J,axiom,
! [X2: a] :
( ( remdups_adj_a @ ( cons_a @ X2 @ nil_a ) )
= ( cons_a @ X2 @ nil_a ) ) ).
% remdups_adj.simps(2)
thf(fact_543_remdups__adj_Osimps_I2_J,axiom,
! [X2: nat] :
( ( remdups_adj_nat @ ( cons_nat @ X2 @ nil_nat ) )
= ( cons_nat @ X2 @ nil_nat ) ) ).
% remdups_adj.simps(2)
thf(fact_544_remdups__adj_Oelims,axiom,
! [X2: list_a,Y2: list_a] :
( ( ( remdups_adj_a @ X2 )
= Y2 )
=> ( ( ( X2 = nil_a )
=> ( Y2 != nil_a ) )
=> ( ! [X3: a] :
( ( X2
= ( cons_a @ X3 @ nil_a ) )
=> ( Y2
!= ( cons_a @ X3 @ nil_a ) ) )
=> ~ ! [X3: a,Y3: a,Xs2: list_a] :
( ( X2
= ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs2 ) ) )
=> ~ ( ( ( X3 = Y3 )
=> ( Y2
= ( remdups_adj_a @ ( cons_a @ X3 @ Xs2 ) ) ) )
& ( ( X3 != Y3 )
=> ( Y2
= ( cons_a @ X3 @ ( remdups_adj_a @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_545_remdups__adj_Oelims,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( ( remdups_adj_nat @ X2 )
= Y2 )
=> ( ( ( X2 = nil_nat )
=> ( Y2 != nil_nat ) )
=> ( ! [X3: nat] :
( ( X2
= ( cons_nat @ X3 @ nil_nat ) )
=> ( Y2
!= ( cons_nat @ X3 @ nil_nat ) ) )
=> ~ ! [X3: nat,Y3: nat,Xs2: list_nat] :
( ( X2
= ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs2 ) ) )
=> ~ ( ( ( X3 = Y3 )
=> ( Y2
= ( remdups_adj_nat @ ( cons_nat @ X3 @ Xs2 ) ) ) )
& ( ( X3 != Y3 )
=> ( Y2
= ( cons_nat @ X3 @ ( remdups_adj_nat @ ( cons_nat @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_546_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_547_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_548_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 ) )
=> ! [Z2: a,Zs4: list_a] :
( ( Zs
= ( cons_a @ Z2 @ Zs4 ) )
=> ( ( X3 = Z2 )
=> ~ ( member_list_a @ Zs4 @ ( shuffles_a @ Xs3 @ Ys ) ) ) ) )
=> ~ ! [Y3: a,Ys4: list_a] :
( ( Ys
= ( cons_a @ Y3 @ Ys4 ) )
=> ! [Z2: a,Zs4: list_a] :
( ( Zs
= ( cons_a @ Z2 @ Zs4 ) )
=> ( ( Y3 = Z2 )
=> ~ ( member_list_a @ Zs4 @ ( shuffles_a @ Xs @ Ys4 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_549_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 ) )
=> ! [Z2: nat,Zs4: list_nat] :
( ( Zs
= ( cons_nat @ Z2 @ Zs4 ) )
=> ( ( X3 = Z2 )
=> ~ ( member_list_nat @ Zs4 @ ( shuffles_nat @ Xs3 @ Ys ) ) ) ) )
=> ~ ! [Y3: nat,Ys4: list_nat] :
( ( Ys
= ( cons_nat @ Y3 @ Ys4 ) )
=> ! [Z2: nat,Zs4: list_nat] :
( ( Zs
= ( cons_nat @ Z2 @ Zs4 ) )
=> ( ( Y3 = Z2 )
=> ~ ( member_list_nat @ Zs4 @ ( shuffles_nat @ Xs @ Ys4 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_550_successively_Oelims_I2_J,axiom,
! [X2: a > a > $o,Xa: list_a] :
( ( successively_a @ X2 @ 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 ) ) )
=> ~ ( ( X2 @ X3 @ Y3 )
& ( successively_a @ X2 @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ) ) ).
% successively.elims(2)
thf(fact_551_successively_Oelims_I2_J,axiom,
! [X2: nat > nat > $o,Xa: list_nat] :
( ( successively_nat @ X2 @ 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 ) ) )
=> ~ ( ( X2 @ X3 @ Y3 )
& ( successively_nat @ X2 @ ( cons_nat @ Y3 @ Xs2 ) ) ) ) ) ) ) ).
% successively.elims(2)
thf(fact_552_successively_Oelims_I1_J,axiom,
! [X2: a > a > $o,Xa: list_a,Y2: $o] :
( ( ( successively_a @ X2 @ Xa )
= Y2 )
=> ( ( ( Xa = nil_a )
=> ~ Y2 )
=> ( ( ? [X3: a] :
( Xa
= ( cons_a @ X3 @ nil_a ) )
=> ~ Y2 )
=> ~ ! [X3: a,Y3: a,Xs2: list_a] :
( ( Xa
= ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs2 ) ) )
=> ( Y2
= ( ~ ( ( X2 @ X3 @ Y3 )
& ( successively_a @ X2 @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% successively.elims(1)
thf(fact_553_successively_Oelims_I1_J,axiom,
! [X2: nat > nat > $o,Xa: list_nat,Y2: $o] :
( ( ( successively_nat @ X2 @ Xa )
= Y2 )
=> ( ( ( Xa = nil_nat )
=> ~ Y2 )
=> ( ( ? [X3: nat] :
( Xa
= ( cons_nat @ X3 @ nil_nat ) )
=> ~ Y2 )
=> ~ ! [X3: nat,Y3: nat,Xs2: list_nat] :
( ( Xa
= ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs2 ) ) )
=> ( Y2
= ( ~ ( ( X2 @ X3 @ Y3 )
& ( successively_nat @ X2 @ ( cons_nat @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% successively.elims(1)
thf(fact_554_successively_Osimps_I2_J,axiom,
! [P: a > a > $o,X2: a] : ( successively_a @ P @ ( cons_a @ X2 @ nil_a ) ) ).
% successively.simps(2)
thf(fact_555_successively_Osimps_I2_J,axiom,
! [P: nat > nat > $o,X2: nat] : ( successively_nat @ P @ ( cons_nat @ X2 @ nil_nat ) ) ).
% successively.simps(2)
thf(fact_556_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
@ ^ [X: a] :
~ ( P @ X )
@ Xs ) ) )
= ( size_size_list_a @ Xs ) ) ).
% sum_length_filter_compl
thf(fact_557_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
@ ^ [X: nat] :
~ ( P @ X )
@ Xs ) ) )
= ( size_size_list_nat @ Xs ) ) ).
% sum_length_filter_compl
thf(fact_558_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_559_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_560_remdups__adj__append__two,axiom,
! [Xs: list_a,X2: a,Y2: a] :
( ( remdups_adj_a @ ( append_a @ Xs @ ( cons_a @ X2 @ ( cons_a @ Y2 @ nil_a ) ) ) )
= ( append_a @ ( remdups_adj_a @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) ) @ ( if_list_a @ ( X2 = Y2 ) @ nil_a @ ( cons_a @ Y2 @ nil_a ) ) ) ) ).
% remdups_adj_append_two
thf(fact_561_remdups__adj__append__two,axiom,
! [Xs: list_nat,X2: nat,Y2: nat] :
( ( remdups_adj_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ ( cons_nat @ Y2 @ nil_nat ) ) ) )
= ( append_nat @ ( remdups_adj_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) ) @ ( if_list_nat @ ( X2 = Y2 ) @ nil_nat @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) ).
% remdups_adj_append_two
thf(fact_562_successively__Cons,axiom,
! [P: a > a > $o,X2: a,Xs: list_a] :
( ( successively_a @ P @ ( cons_a @ X2 @ Xs ) )
= ( ( Xs = nil_a )
| ( ( P @ X2 @ ( hd_a @ Xs ) )
& ( successively_a @ P @ Xs ) ) ) ) ).
% successively_Cons
thf(fact_563_successively__Cons,axiom,
! [P: nat > nat > $o,X2: nat,Xs: list_nat] :
( ( successively_nat @ P @ ( cons_nat @ X2 @ Xs ) )
= ( ( Xs = nil_nat )
| ( ( P @ X2 @ ( hd_nat @ Xs ) )
& ( successively_nat @ P @ Xs ) ) ) ) ).
% successively_Cons
thf(fact_564_remdups__adj__Cons,axiom,
! [X2: a,Xs: list_a] :
( ( remdups_adj_a @ ( cons_a @ X2 @ Xs ) )
= ( case_list_list_a_a @ ( cons_a @ X2 @ nil_a )
@ ^ [Y: a,Xs4: list_a] : ( if_list_a @ ( X2 = Y ) @ ( cons_a @ Y @ Xs4 ) @ ( cons_a @ X2 @ ( cons_a @ Y @ Xs4 ) ) )
@ ( remdups_adj_a @ Xs ) ) ) ).
% remdups_adj_Cons
thf(fact_565_remdups__adj__Cons,axiom,
! [X2: nat,Xs: list_nat] :
( ( remdups_adj_nat @ ( cons_nat @ X2 @ Xs ) )
= ( case_l2340614614379431832at_nat @ ( cons_nat @ X2 @ nil_nat )
@ ^ [Y: nat,Xs4: list_nat] : ( if_list_nat @ ( X2 = Y ) @ ( cons_nat @ Y @ Xs4 ) @ ( cons_nat @ X2 @ ( cons_nat @ Y @ Xs4 ) ) )
@ ( remdups_adj_nat @ Xs ) ) ) ).
% remdups_adj_Cons
thf(fact_566_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_567_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_568_remdups__adj_Opelims,axiom,
! [X2: list_a,Y2: list_a] :
( ( ( remdups_adj_a @ X2 )
= Y2 )
=> ( ( accp_list_a @ remdups_adj_rel_a @ X2 )
=> ( ( ( X2 = nil_a )
=> ( ( Y2 = nil_a )
=> ~ ( accp_list_a @ remdups_adj_rel_a @ nil_a ) ) )
=> ( ! [X3: a] :
( ( X2
= ( cons_a @ X3 @ nil_a ) )
=> ( ( Y2
= ( cons_a @ X3 @ nil_a ) )
=> ~ ( accp_list_a @ remdups_adj_rel_a @ ( cons_a @ X3 @ nil_a ) ) ) )
=> ~ ! [X3: a,Y3: a,Xs2: list_a] :
( ( X2
= ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs2 ) ) )
=> ( ( ( ( X3 = Y3 )
=> ( Y2
= ( remdups_adj_a @ ( cons_a @ X3 @ Xs2 ) ) ) )
& ( ( X3 != Y3 )
=> ( Y2
= ( cons_a @ X3 @ ( remdups_adj_a @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) )
=> ~ ( accp_list_a @ remdups_adj_rel_a @ ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% remdups_adj.pelims
thf(fact_569_remdups__adj_Opelims,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( ( remdups_adj_nat @ X2 )
= Y2 )
=> ( ( accp_list_nat @ remdups_adj_rel_nat @ X2 )
=> ( ( ( X2 = nil_nat )
=> ( ( Y2 = nil_nat )
=> ~ ( accp_list_nat @ remdups_adj_rel_nat @ nil_nat ) ) )
=> ( ! [X3: nat] :
( ( X2
= ( cons_nat @ X3 @ nil_nat ) )
=> ( ( Y2
= ( cons_nat @ X3 @ nil_nat ) )
=> ~ ( accp_list_nat @ remdups_adj_rel_nat @ ( cons_nat @ X3 @ nil_nat ) ) ) )
=> ~ ! [X3: nat,Y3: nat,Xs2: list_nat] :
( ( X2
= ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs2 ) ) )
=> ( ( ( ( X3 = Y3 )
=> ( Y2
= ( remdups_adj_nat @ ( cons_nat @ X3 @ Xs2 ) ) ) )
& ( ( X3 != Y3 )
=> ( Y2
= ( cons_nat @ X3 @ ( remdups_adj_nat @ ( cons_nat @ Y3 @ Xs2 ) ) ) ) ) )
=> ~ ( accp_list_nat @ remdups_adj_rel_nat @ ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% remdups_adj.pelims
thf(fact_570_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_571_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_572_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_573_list_Osize__gen_I2_J,axiom,
! [X2: a > nat,X21: a,X222: list_a] :
( ( size_list_a @ X2 @ ( cons_a @ X21 @ X222 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( X2 @ X21 ) @ ( size_list_a @ X2 @ X222 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size_gen(2)
thf(fact_574_list_Osize__gen_I2_J,axiom,
! [X2: nat > nat,X21: nat,X222: list_nat] :
( ( size_list_nat @ X2 @ ( cons_nat @ X21 @ X222 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( X2 @ X21 ) @ ( size_list_nat @ X2 @ X222 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size_gen(2)
thf(fact_575_remdups__adj__singleton__iff,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ ( remdups_adj_a @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_a )
& ( Xs
= ( replicate_a @ ( size_size_list_a @ Xs ) @ ( hd_a @ Xs ) ) ) ) ) ).
% remdups_adj_singleton_iff
thf(fact_576_remdups__adj__singleton__iff,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_nat )
& ( Xs
= ( replicate_nat @ ( size_size_list_nat @ Xs ) @ ( hd_nat @ Xs ) ) ) ) ) ).
% remdups_adj_singleton_iff
thf(fact_577_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_578_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_579_concat__replicate__trivial,axiom,
! [I3: nat] :
( ( concat_a @ ( replicate_list_a @ I3 @ nil_a ) )
= nil_a ) ).
% concat_replicate_trivial
thf(fact_580_concat__replicate__trivial,axiom,
! [I3: nat] :
( ( concat_nat @ ( replicate_list_nat @ I3 @ nil_nat ) )
= nil_nat ) ).
% concat_replicate_trivial
thf(fact_581_length__replicate,axiom,
! [N: nat,X2: a] :
( ( size_size_list_a @ ( replicate_a @ N @ X2 ) )
= N ) ).
% length_replicate
thf(fact_582_length__replicate,axiom,
! [N: nat,X2: nat] :
( ( size_size_list_nat @ ( replicate_nat @ N @ X2 ) )
= N ) ).
% length_replicate
thf(fact_583_map__replicate,axiom,
! [F: nat > nat,N: nat,X2: nat] :
( ( map_nat_nat @ F @ ( replicate_nat @ N @ X2 ) )
= ( replicate_nat @ N @ ( F @ X2 ) ) ) ).
% map_replicate
thf(fact_584_replicate__empty,axiom,
! [N: nat,X2: a] :
( ( ( replicate_a @ N @ X2 )
= nil_a )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_585_replicate__empty,axiom,
! [N: nat,X2: nat] :
( ( ( replicate_nat @ N @ X2 )
= nil_nat )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_586_empty__replicate,axiom,
! [N: nat,X2: a] :
( ( nil_a
= ( replicate_a @ N @ X2 ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_587_empty__replicate,axiom,
! [N: nat,X2: nat] :
( ( nil_nat
= ( replicate_nat @ N @ X2 ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_588_hd__replicate,axiom,
! [N: nat,X2: nat] :
( ( N != zero_zero_nat )
=> ( ( hd_nat @ ( replicate_nat @ N @ X2 ) )
= X2 ) ) ).
% hd_replicate
thf(fact_589_last__replicate,axiom,
! [N: nat,X2: nat] :
( ( N != zero_zero_nat )
=> ( ( last_nat @ ( replicate_nat @ N @ X2 ) )
= X2 ) ) ).
% last_replicate
thf(fact_590_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_591_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_592_append__replicate__commute,axiom,
! [N: nat,X2: a,K2: nat] :
( ( append_a @ ( replicate_a @ N @ X2 ) @ ( replicate_a @ K2 @ X2 ) )
= ( append_a @ ( replicate_a @ K2 @ X2 ) @ ( replicate_a @ N @ X2 ) ) ) ).
% append_replicate_commute
thf(fact_593_append__replicate__commute,axiom,
! [N: nat,X2: nat,K2: nat] :
( ( append_nat @ ( replicate_nat @ N @ X2 ) @ ( replicate_nat @ K2 @ X2 ) )
= ( append_nat @ ( replicate_nat @ K2 @ X2 ) @ ( replicate_nat @ N @ X2 ) ) ) ).
% append_replicate_commute
thf(fact_594_replicate__Suc,axiom,
! [N: nat,X2: a] :
( ( replicate_a @ ( suc @ N ) @ X2 )
= ( cons_a @ X2 @ ( replicate_a @ N @ X2 ) ) ) ).
% replicate_Suc
thf(fact_595_replicate__Suc,axiom,
! [N: nat,X2: nat] :
( ( replicate_nat @ ( suc @ N ) @ X2 )
= ( cons_nat @ X2 @ ( replicate_nat @ N @ X2 ) ) ) ).
% replicate_Suc
thf(fact_596_replicate__0,axiom,
! [X2: a] :
( ( replicate_a @ zero_zero_nat @ X2 )
= nil_a ) ).
% replicate_0
thf(fact_597_replicate__0,axiom,
! [X2: nat] :
( ( replicate_nat @ zero_zero_nat @ X2 )
= nil_nat ) ).
% replicate_0
thf(fact_598_replicate__app__Cons__same,axiom,
! [N: nat,X2: a,Xs: list_a] :
( ( append_a @ ( replicate_a @ N @ X2 ) @ ( cons_a @ X2 @ Xs ) )
= ( cons_a @ X2 @ ( append_a @ ( replicate_a @ N @ X2 ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_599_replicate__app__Cons__same,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( append_nat @ ( replicate_nat @ N @ X2 ) @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ X2 @ ( append_nat @ ( replicate_nat @ N @ X2 ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_600_replicate__add,axiom,
! [N: nat,M: nat,X2: a] :
( ( replicate_a @ ( plus_plus_nat @ N @ M ) @ X2 )
= ( append_a @ ( replicate_a @ N @ X2 ) @ ( replicate_a @ M @ X2 ) ) ) ).
% replicate_add
thf(fact_601_replicate__add,axiom,
! [N: nat,M: nat,X2: nat] :
( ( replicate_nat @ ( plus_plus_nat @ N @ M ) @ X2 )
= ( append_nat @ ( replicate_nat @ N @ X2 ) @ ( replicate_nat @ M @ X2 ) ) ) ).
% replicate_add
thf(fact_602_filter__replicate,axiom,
! [P: a > $o,X2: a,N: nat] :
( ( ( P @ X2 )
=> ( ( filter_a @ P @ ( replicate_a @ N @ X2 ) )
= ( replicate_a @ N @ X2 ) ) )
& ( ~ ( P @ X2 )
=> ( ( filter_a @ P @ ( replicate_a @ N @ X2 ) )
= nil_a ) ) ) ).
% filter_replicate
thf(fact_603_filter__replicate,axiom,
! [P: nat > $o,X2: nat,N: nat] :
( ( ( P @ X2 )
=> ( ( filter_nat @ P @ ( replicate_nat @ N @ X2 ) )
= ( replicate_nat @ N @ X2 ) ) )
& ( ~ ( P @ X2 )
=> ( ( filter_nat @ P @ ( replicate_nat @ N @ X2 ) )
= nil_nat ) ) ) ).
% filter_replicate
thf(fact_604_comm__append__are__replicate,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Ys @ Xs ) )
=> ? [M2: nat,N2: nat,Zs2: list_a] :
( ( ( concat_a @ ( replicate_list_a @ M2 @ Zs2 ) )
= Xs )
& ( ( concat_a @ ( replicate_list_a @ N2 @ Zs2 ) )
= Ys ) ) ) ).
% comm_append_are_replicate
thf(fact_605_comm__append__are__replicate,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Ys @ Xs ) )
=> ? [M2: nat,N2: nat,Zs2: list_nat] :
( ( ( concat_nat @ ( replicate_list_nat @ M2 @ Zs2 ) )
= Xs )
& ( ( concat_nat @ ( replicate_list_nat @ N2 @ Zs2 ) )
= Ys ) ) ) ).
% comm_append_are_replicate
thf(fact_606_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_607_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_608_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_609_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_610_map__replicate__const,axiom,
! [K2: nat,Lst: list_nat] :
( ( map_nat_nat
@ ^ [X: nat] : K2
@ Lst )
= ( replicate_nat @ ( size_size_list_nat @ Lst ) @ K2 ) ) ).
% map_replicate_const
thf(fact_611_replicate__length__filter,axiom,
! [X2: a,Xs: list_a] :
( ( replicate_a
@ ( size_size_list_a
@ ( filter_a
@ ( ^ [Y4: a,Z4: a] : ( Y4 = Z4 )
@ X2 )
@ Xs ) )
@ X2 )
= ( filter_a
@ ( ^ [Y4: a,Z4: a] : ( Y4 = Z4 )
@ X2 )
@ Xs ) ) ).
% replicate_length_filter
thf(fact_612_replicate__length__filter,axiom,
! [X2: nat,Xs: list_nat] :
( ( replicate_nat
@ ( size_size_list_nat
@ ( filter_nat
@ ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 )
@ X2 )
@ Xs ) )
@ X2 )
= ( filter_nat
@ ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 )
@ X2 )
@ Xs ) ) ).
% replicate_length_filter
thf(fact_613_list_Osize__gen_I1_J,axiom,
! [X2: a > nat] :
( ( size_list_a @ X2 @ nil_a )
= zero_zero_nat ) ).
% list.size_gen(1)
thf(fact_614_list_Osize__gen_I1_J,axiom,
! [X2: nat > nat] :
( ( size_list_nat @ X2 @ nil_nat )
= zero_zero_nat ) ).
% list.size_gen(1)
thf(fact_615_replicate__append__same,axiom,
! [I3: nat,X2: a] :
( ( append_a @ ( replicate_a @ I3 @ X2 ) @ ( cons_a @ X2 @ nil_a ) )
= ( cons_a @ X2 @ ( replicate_a @ I3 @ X2 ) ) ) ).
% replicate_append_same
thf(fact_616_replicate__append__same,axiom,
! [I3: nat,X2: nat] :
( ( append_nat @ ( replicate_nat @ I3 @ X2 ) @ ( cons_nat @ X2 @ nil_nat ) )
= ( cons_nat @ X2 @ ( replicate_nat @ I3 @ X2 ) ) ) ).
% replicate_append_same
thf(fact_617_remdups__adj__replicate,axiom,
! [N: nat,X2: a] :
( ( ( N = zero_zero_nat )
=> ( ( remdups_adj_a @ ( replicate_a @ N @ X2 ) )
= nil_a ) )
& ( ( N != zero_zero_nat )
=> ( ( remdups_adj_a @ ( replicate_a @ N @ X2 ) )
= ( cons_a @ X2 @ nil_a ) ) ) ) ).
% remdups_adj_replicate
thf(fact_618_remdups__adj__replicate,axiom,
! [N: nat,X2: nat] :
( ( ( N = zero_zero_nat )
=> ( ( remdups_adj_nat @ ( replicate_nat @ N @ X2 ) )
= nil_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( remdups_adj_nat @ ( replicate_nat @ N @ X2 ) )
= ( cons_nat @ X2 @ nil_nat ) ) ) ) ).
% remdups_adj_replicate
thf(fact_619_remdups__adj__singleton,axiom,
! [Xs: list_a,X2: a] :
( ( ( remdups_adj_a @ Xs )
= ( cons_a @ X2 @ nil_a ) )
=> ( Xs
= ( replicate_a @ ( size_size_list_a @ Xs ) @ X2 ) ) ) ).
% remdups_adj_singleton
thf(fact_620_remdups__adj__singleton,axiom,
! [Xs: list_nat,X2: nat] :
( ( ( remdups_adj_nat @ Xs )
= ( cons_nat @ X2 @ nil_nat ) )
=> ( Xs
= ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X2 ) ) ) ).
% remdups_adj_singleton
thf(fact_621_remdups__adj__length__ge1,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_a @ ( remdups_adj_a @ Xs ) ) ) ) ).
% remdups_adj_length_ge1
thf(fact_622_remdups__adj__length__ge1,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) ) ) ) ).
% remdups_adj_length_ge1
thf(fact_623_take__Cons_H,axiom,
! [N: nat,X2: a,Xs: list_a] :
( ( ( N = zero_zero_nat )
=> ( ( take_a @ N @ ( cons_a @ X2 @ Xs ) )
= nil_a ) )
& ( ( N != zero_zero_nat )
=> ( ( take_a @ N @ ( cons_a @ X2 @ Xs ) )
= ( cons_a @ X2 @ ( take_a @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_624_take__Cons_H,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( take_nat @ N @ ( cons_nat @ X2 @ Xs ) )
= nil_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( take_nat @ N @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ X2 @ ( take_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_625_take__Cons,axiom,
! [N: nat,X2: a,Xs: list_a] :
( ( take_a @ N @ ( cons_a @ X2 @ Xs ) )
= ( case_nat_list_a @ nil_a
@ ^ [M3: nat] : ( cons_a @ X2 @ ( take_a @ M3 @ Xs ) )
@ N ) ) ).
% take_Cons
thf(fact_626_take__Cons,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( take_nat @ N @ ( cons_nat @ X2 @ Xs ) )
= ( case_nat_list_nat @ nil_nat
@ ^ [M3: nat] : ( cons_nat @ X2 @ ( take_nat @ M3 @ Xs ) )
@ N ) ) ).
% take_Cons
thf(fact_627_tl__remdups__adj,axiom,
! [Ys: list_a] :
( ( Ys != nil_a )
=> ( ( tl_a @ ( remdups_adj_a @ Ys ) )
= ( remdups_adj_a
@ ( dropWhile_a
@ ^ [X: a] :
( X
= ( hd_a @ Ys ) )
@ ( tl_a @ Ys ) ) ) ) ) ).
% tl_remdups_adj
thf(fact_628_tl__remdups__adj,axiom,
! [Ys: list_nat] :
( ( Ys != nil_nat )
=> ( ( tl_nat @ ( remdups_adj_nat @ Ys ) )
= ( remdups_adj_nat
@ ( dropWhile_nat
@ ^ [X: nat] :
( X
= ( hd_nat @ Ys ) )
@ ( tl_nat @ Ys ) ) ) ) ) ).
% tl_remdups_adj
thf(fact_629_remdups__adj__append_H_H,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( remdups_adj_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( remdups_adj_a @ Xs )
@ ( remdups_adj_a
@ ( dropWhile_a
@ ^ [Y: a] :
( Y
= ( last_a @ Xs ) )
@ Ys ) ) ) ) ) ).
% remdups_adj_append''
thf(fact_630_remdups__adj__append_H_H,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != nil_nat )
=> ( ( remdups_adj_nat @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( remdups_adj_nat @ Xs )
@ ( remdups_adj_nat
@ ( dropWhile_nat
@ ^ [Y: nat] :
( Y
= ( last_nat @ Xs ) )
@ Ys ) ) ) ) ) ).
% remdups_adj_append''
thf(fact_631_remdups__adj__append__dropWhile,axiom,
! [Xs: list_a,Y2: a,Ys: list_a] :
( ( remdups_adj_a @ ( append_a @ Xs @ ( cons_a @ Y2 @ Ys ) ) )
= ( append_a @ ( remdups_adj_a @ ( append_a @ Xs @ ( cons_a @ Y2 @ nil_a ) ) )
@ ( remdups_adj_a
@ ( dropWhile_a
@ ^ [X: a] : ( X = Y2 )
@ Ys ) ) ) ) ).
% remdups_adj_append_dropWhile
thf(fact_632_remdups__adj__append__dropWhile,axiom,
! [Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( remdups_adj_nat @ ( append_nat @ Xs @ ( cons_nat @ Y2 @ Ys ) ) )
= ( append_nat @ ( remdups_adj_nat @ ( append_nat @ Xs @ ( cons_nat @ Y2 @ nil_nat ) ) )
@ ( remdups_adj_nat
@ ( dropWhile_nat
@ ^ [X: nat] : ( X = Y2 )
@ Ys ) ) ) ) ).
% remdups_adj_append_dropWhile
thf(fact_633_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_634_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_635_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_636_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_637_Suc__diff__diff,axiom,
! [M: nat,N: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_638_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_639_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_640_nat__add__left__cancel__le,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_641_diff__diff__cancel,axiom,
! [I3: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I3 ) )
= I3 ) ) ).
% diff_diff_cancel
thf(fact_642_diff__diff__left,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J2 ) @ K2 )
= ( minus_minus_nat @ I3 @ ( plus_plus_nat @ J2 @ K2 ) ) ) ).
% diff_diff_left
thf(fact_643_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_644_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_645_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J2 @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J2 ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_646_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K2 ) @ I3 )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I3 ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_647_Nat_Odiff__diff__right,axiom,
! [K2: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( minus_minus_nat @ I3 @ ( minus_minus_nat @ J2 @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K2 ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_648_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_649_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_650_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_651_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_652_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_653_dropWhile__replicate,axiom,
! [P: a > $o,X2: a,N: nat] :
( ( ( P @ X2 )
=> ( ( dropWhile_a @ P @ ( replicate_a @ N @ X2 ) )
= nil_a ) )
& ( ~ ( P @ X2 )
=> ( ( dropWhile_a @ P @ ( replicate_a @ N @ X2 ) )
= ( replicate_a @ N @ X2 ) ) ) ) ).
% dropWhile_replicate
thf(fact_654_dropWhile__replicate,axiom,
! [P: nat > $o,X2: nat,N: nat] :
( ( ( P @ X2 )
=> ( ( dropWhile_nat @ P @ ( replicate_nat @ N @ X2 ) )
= nil_nat ) )
& ( ~ ( P @ X2 )
=> ( ( dropWhile_nat @ P @ ( replicate_nat @ N @ X2 ) )
= ( replicate_nat @ N @ X2 ) ) ) ) ).
% dropWhile_replicate
thf(fact_655_diff__Suc__diff__eq1,axiom,
! [K2: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( minus_minus_nat @ I3 @ ( suc @ ( minus_minus_nat @ J2 @ K2 ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( suc @ J2 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_656_diff__Suc__diff__eq2,axiom,
! [K2: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K2 ) ) @ I3 )
= ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K2 @ I3 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_657_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_658_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_659_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_660_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_661_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_662_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_663_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_664_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_665_tl__replicate,axiom,
! [N: nat,X2: nat] :
( ( tl_nat @ ( replicate_nat @ N @ X2 ) )
= ( replicate_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ X2 ) ) ).
% tl_replicate
thf(fact_666_nat_Ocase__distrib,axiom,
! [H2: nat > nat,F1: nat,F22: nat > nat,Nat: nat] :
( ( H2 @ ( case_nat_nat @ F1 @ F22 @ Nat ) )
= ( case_nat_nat @ ( H2 @ F1 )
@ ^ [X: nat] : ( H2 @ ( F22 @ X ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_667_nat_Ocase__distrib,axiom,
! [H2: nat > $o,F1: nat,F22: nat > nat,Nat: nat] :
( ( H2 @ ( case_nat_nat @ F1 @ F22 @ Nat ) )
= ( case_nat_o @ ( H2 @ F1 )
@ ^ [X: nat] : ( H2 @ ( F22 @ X ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_668_nat_Ocase__distrib,axiom,
! [H2: $o > nat,F1: $o,F22: nat > $o,Nat: nat] :
( ( H2 @ ( case_nat_o @ F1 @ F22 @ Nat ) )
= ( case_nat_nat @ ( H2 @ F1 )
@ ^ [X: nat] : ( H2 @ ( F22 @ X ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_669_nat_Ocase__distrib,axiom,
! [H2: $o > $o,F1: $o,F22: nat > $o,Nat: nat] :
( ( H2 @ ( case_nat_o @ F1 @ F22 @ Nat ) )
= ( case_nat_o @ ( H2 @ F1 )
@ ^ [X: nat] : ( H2 @ ( F22 @ X ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_670_length__dropWhile__le,axiom,
! [P: a > $o,Xs: list_a] : ( ord_less_eq_nat @ ( size_size_list_a @ ( dropWhile_a @ P @ Xs ) ) @ ( size_size_list_a @ Xs ) ) ).
% length_dropWhile_le
thf(fact_671_length__dropWhile__le,axiom,
! [P: nat > $o,Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( dropWhile_nat @ P @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% length_dropWhile_le
thf(fact_672_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_673_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_674_diff__le__mono2,axiom,
! [M: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ).
% diff_le_mono2
thf(fact_675_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_676_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_677_diff__le__mono,axiom,
! [M: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L2 ) @ ( minus_minus_nat @ N @ L2 ) ) ) ).
% diff_le_mono
thf(fact_678_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_679_diff__commute,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J2 ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I3 @ K2 ) @ J2 ) ) ).
% diff_commute
thf(fact_680_le__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_681_eq__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ( minus_minus_nat @ M @ K2 )
= ( minus_minus_nat @ N @ K2 ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_682_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_683_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_684_le__trans,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K2 )
=> ( ord_less_eq_nat @ I3 @ K2 ) ) ) ).
% le_trans
thf(fact_685_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_686_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_687_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_688_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_689_le__diff__conv,axiom,
! [J2: nat,K2: nat,I3: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K2 ) @ I3 )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I3 @ K2 ) ) ) ).
% le_diff_conv
thf(fact_690_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ J2 @ K2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_691_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J2 ) @ K2 )
= ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J2 @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_692_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I3 ) @ K2 )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K2 ) @ I3 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_693_Nat_Ole__imp__diff__is__add,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I3 )
= K2 )
= ( J2
= ( plus_plus_nat @ K2 @ I3 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_694_dropWhile_Osimps_I2_J,axiom,
! [P: a > $o,X2: a,Xs: list_a] :
( ( ( P @ X2 )
=> ( ( dropWhile_a @ P @ ( cons_a @ X2 @ Xs ) )
= ( dropWhile_a @ P @ Xs ) ) )
& ( ~ ( P @ X2 )
=> ( ( dropWhile_a @ P @ ( cons_a @ X2 @ Xs ) )
= ( cons_a @ X2 @ Xs ) ) ) ) ).
% dropWhile.simps(2)
thf(fact_695_dropWhile_Osimps_I2_J,axiom,
! [P: nat > $o,X2: nat,Xs: list_nat] :
( ( ( P @ X2 )
=> ( ( dropWhile_nat @ P @ ( cons_nat @ X2 @ Xs ) )
= ( dropWhile_nat @ P @ Xs ) ) )
& ( ~ ( P @ X2 )
=> ( ( dropWhile_nat @ P @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ X2 @ Xs ) ) ) ) ).
% dropWhile.simps(2)
thf(fact_696_dropWhile_Osimps_I1_J,axiom,
! [P: a > $o] :
( ( dropWhile_a @ P @ nil_a )
= nil_a ) ).
% dropWhile.simps(1)
thf(fact_697_dropWhile_Osimps_I1_J,axiom,
! [P: nat > $o] :
( ( dropWhile_nat @ P @ nil_nat )
= nil_nat ) ).
% dropWhile.simps(1)
thf(fact_698_zero__induct__lemma,axiom,
! [P: nat > $o,K2: nat,I3: nat] :
( ( P @ K2 )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K2 @ I3 ) ) ) ) ).
% zero_induct_lemma
thf(fact_699_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_700_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_701_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_702_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_703_diff__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K2 ) @ ( plus_plus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_704_Nat_Odiff__cancel,axiom,
! [K2: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_705_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_706_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_707_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_708_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
=> ? [M2: nat] :
( M4
= ( suc @ M2 ) ) ) ).
% Suc_le_D
thf(fact_709_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_710_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_711_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_712_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_713_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_714_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z2: nat] :
( ( R @ X3 @ Y3 )
=> ( ( R @ Y3 @ Z2 )
=> ( R @ X3 @ Z2 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_715_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_716_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_717_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_718_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_719_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
? [K: nat] :
( N3
= ( plus_plus_nat @ M3 @ K ) ) ) ) ).
% nat_le_iff_add
thf(fact_720_trans__le__add2,axiom,
! [I3: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_le_add2
thf(fact_721_trans__le__add1,axiom,
! [I3: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_le_add1
thf(fact_722_add__le__mono1,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J2 @ K2 ) ) ) ).
% add_le_mono1
thf(fact_723_add__le__mono,axiom,
! [I3: nat,J2: nat,K2: nat,L2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ K2 @ L2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ) ).
% add_le_mono
thf(fact_724_le__Suc__ex,axiom,
! [K2: nat,L2: nat] :
( ( ord_less_eq_nat @ K2 @ L2 )
=> ? [N2: nat] :
( L2
= ( plus_plus_nat @ K2 @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_725_add__leD2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ K2 @ N ) ) ).
% add_leD2
thf(fact_726_add__leD1,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_727_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_728_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_729_add__leE,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K2 @ N ) ) ) ).
% add_leE
thf(fact_730_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_731_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_732_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_733_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_734_dropWhile__append3,axiom,
! [P: a > $o,Y2: a,Xs: list_a,Ys: list_a] :
( ~ ( P @ Y2 )
=> ( ( dropWhile_a @ P @ ( append_a @ Xs @ ( cons_a @ Y2 @ Ys ) ) )
= ( append_a @ ( dropWhile_a @ P @ Xs ) @ ( cons_a @ Y2 @ Ys ) ) ) ) ).
% dropWhile_append3
thf(fact_735_dropWhile__append3,axiom,
! [P: nat > $o,Y2: nat,Xs: list_nat,Ys: list_nat] :
( ~ ( P @ Y2 )
=> ( ( dropWhile_nat @ P @ ( append_nat @ Xs @ ( cons_nat @ Y2 @ Ys ) ) )
= ( append_nat @ ( dropWhile_nat @ P @ Xs ) @ ( cons_nat @ Y2 @ Ys ) ) ) ) ).
% dropWhile_append3
thf(fact_736_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_737_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_738_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_739_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_740_impossible__Cons,axiom,
! [Xs: list_a,Ys: list_a,X2: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
=> ( Xs
!= ( cons_a @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_741_impossible__Cons,axiom,
! [Xs: list_nat,Ys: list_nat,X2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
=> ( Xs
!= ( cons_nat @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_742_hd__dropWhile,axiom,
! [P: a > $o,Xs: list_a] :
( ( ( dropWhile_a @ P @ Xs )
!= nil_a )
=> ~ ( P @ ( hd_a @ ( dropWhile_a @ P @ Xs ) ) ) ) ).
% hd_dropWhile
thf(fact_743_hd__dropWhile,axiom,
! [P: nat > $o,Xs: list_nat] :
( ( ( dropWhile_nat @ P @ Xs )
!= nil_nat )
=> ~ ( P @ ( hd_nat @ ( dropWhile_nat @ P @ Xs ) ) ) ) ).
% hd_dropWhile
thf(fact_744_dropWhile__eq__self__iff,axiom,
! [P: a > $o,Xs: list_a] :
( ( ( dropWhile_a @ P @ Xs )
= Xs )
= ( ( Xs = nil_a )
| ~ ( P @ ( hd_a @ Xs ) ) ) ) ).
% dropWhile_eq_self_iff
thf(fact_745_dropWhile__eq__self__iff,axiom,
! [P: nat > $o,Xs: list_nat] :
( ( ( dropWhile_nat @ P @ Xs )
= Xs )
= ( ( Xs = nil_nat )
| ~ ( P @ ( hd_nat @ Xs ) ) ) ) ).
% dropWhile_eq_self_iff
thf(fact_746_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_747_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_748_remdups__adj__length,axiom,
! [Xs: list_a] : ( ord_less_eq_nat @ ( size_size_list_a @ ( remdups_adj_a @ Xs ) ) @ ( size_size_list_a @ Xs ) ) ).
% remdups_adj_length
thf(fact_749_remdups__adj__length,axiom,
! [Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% remdups_adj_length
thf(fact_750_remdups__adj__Cons_H,axiom,
! [X2: a,Xs: list_a] :
( ( remdups_adj_a @ ( cons_a @ X2 @ Xs ) )
= ( cons_a @ X2
@ ( remdups_adj_a
@ ( dropWhile_a
@ ^ [Y: a] : ( Y = X2 )
@ Xs ) ) ) ) ).
% remdups_adj_Cons'
thf(fact_751_remdups__adj__Cons_H,axiom,
! [X2: nat,Xs: list_nat] :
( ( remdups_adj_nat @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ X2
@ ( remdups_adj_nat
@ ( dropWhile_nat
@ ^ [Y: nat] : ( Y = X2 )
@ Xs ) ) ) ) ).
% remdups_adj_Cons'
thf(fact_752_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_a @ Xs ) )
= ( ? [X: a,Ys2: list_a] :
( ( Xs
= ( cons_a @ X @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_a @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_753_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ X @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_754_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_755_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_756_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_757_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_758_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_759_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_760_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_761_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_762_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_763_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_764_Cons__replicate__eq,axiom,
! [X2: a,Xs: list_a,N: nat,Y2: a] :
( ( ( cons_a @ X2 @ Xs )
= ( replicate_a @ N @ Y2 ) )
= ( ( X2 = Y2 )
& ( ord_less_nat @ zero_zero_nat @ N )
& ( Xs
= ( replicate_a @ ( minus_minus_nat @ N @ one_one_nat ) @ X2 ) ) ) ) ).
% Cons_replicate_eq
thf(fact_765_Cons__replicate__eq,axiom,
! [X2: nat,Xs: list_nat,N: nat,Y2: nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( replicate_nat @ N @ Y2 ) )
= ( ( X2 = Y2 )
& ( ord_less_nat @ zero_zero_nat @ N )
& ( Xs
= ( replicate_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ X2 ) ) ) ) ).
% Cons_replicate_eq
thf(fact_766_last__list__update,axiom,
! [Xs: list_a,K2: nat,X2: a] :
( ( Xs != nil_a )
=> ( ( ( K2
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
=> ( ( last_a @ ( list_update_a @ Xs @ K2 @ X2 ) )
= X2 ) )
& ( ( K2
!= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
=> ( ( last_a @ ( list_update_a @ Xs @ K2 @ X2 ) )
= ( last_a @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_767_last__list__update,axiom,
! [Xs: list_nat,K2: nat,X2: nat] :
( ( Xs != nil_nat )
=> ( ( ( K2
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( last_nat @ ( list_update_nat @ Xs @ K2 @ X2 ) )
= X2 ) )
& ( ( K2
!= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( last_nat @ ( list_update_nat @ Xs @ K2 @ X2 ) )
= ( last_nat @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_768_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_769_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_770_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_771_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_772_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_773_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_774_nat__add__left__cancel__less,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_775_list__update__nonempty,axiom,
! [Xs: list_a,K2: nat,X2: a] :
( ( ( list_update_a @ Xs @ K2 @ X2 )
= nil_a )
= ( Xs = nil_a ) ) ).
% list_update_nonempty
thf(fact_776_list__update__nonempty,axiom,
! [Xs: list_nat,K2: nat,X2: nat] :
( ( ( list_update_nat @ Xs @ K2 @ X2 )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% list_update_nonempty
thf(fact_777_length__list__update,axiom,
! [Xs: list_a,I3: nat,X2: a] :
( ( size_size_list_a @ ( list_update_a @ Xs @ I3 @ X2 ) )
= ( size_size_list_a @ Xs ) ) ).
% length_list_update
thf(fact_778_length__list__update,axiom,
! [Xs: list_nat,I3: nat,X2: nat] :
( ( size_size_list_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_list_update
thf(fact_779_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_780_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_781_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_782_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_783_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_784_list__update__beyond,axiom,
! [Xs: list_a,I3: nat,X2: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ I3 )
=> ( ( list_update_a @ Xs @ I3 @ X2 )
= Xs ) ) ).
% list_update_beyond
thf(fact_785_list__update__beyond,axiom,
! [Xs: list_nat,I3: nat,X2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I3 )
=> ( ( list_update_nat @ Xs @ I3 @ X2 )
= Xs ) ) ).
% list_update_beyond
thf(fact_786_take__update__cancel,axiom,
! [N: nat,M: nat,Xs: list_nat,Y2: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( take_nat @ N @ ( list_update_nat @ Xs @ M @ Y2 ) )
= ( take_nat @ N @ Xs ) ) ) ).
% take_update_cancel
thf(fact_787_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_788_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_789_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_790_list__update__length,axiom,
! [Xs: list_a,X2: a,Ys: list_a,Y2: a] :
( ( list_update_a @ ( append_a @ Xs @ ( cons_a @ X2 @ Ys ) ) @ ( size_size_list_a @ Xs ) @ Y2 )
= ( append_a @ Xs @ ( cons_a @ Y2 @ Ys ) ) ) ).
% list_update_length
thf(fact_791_list__update__length,axiom,
! [Xs: list_nat,X2: nat,Ys: list_nat,Y2: nat] :
( ( list_update_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys ) ) @ ( size_size_list_nat @ Xs ) @ Y2 )
= ( append_nat @ Xs @ ( cons_nat @ Y2 @ Ys ) ) ) ).
% list_update_length
thf(fact_792_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_793_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_794_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_795_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_796_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_797_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_798_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_799_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_800_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_801_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_802_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_803_linorder__neqE__nat,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_804_list__update__append1,axiom,
! [I3: nat,Xs: list_a,Ys: list_a,X2: a] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ ( append_a @ Xs @ Ys ) @ I3 @ X2 )
= ( append_a @ ( list_update_a @ Xs @ I3 @ X2 ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_805_list__update__append1,axiom,
! [I3: nat,Xs: list_nat,Ys: list_nat,X2: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ I3 @ X2 )
= ( append_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_806_list__update_Osimps_I1_J,axiom,
! [I3: nat,V: a] :
( ( list_update_a @ nil_a @ I3 @ V )
= nil_a ) ).
% list_update.simps(1)
thf(fact_807_list__update_Osimps_I1_J,axiom,
! [I3: nat,V: nat] :
( ( list_update_nat @ nil_nat @ I3 @ V )
= nil_nat ) ).
% list_update.simps(1)
thf(fact_808_list__update__code_I1_J,axiom,
! [I3: nat,Y2: a] :
( ( list_update_a @ nil_a @ I3 @ Y2 )
= nil_a ) ).
% list_update_code(1)
thf(fact_809_list__update__code_I1_J,axiom,
! [I3: nat,Y2: nat] :
( ( list_update_nat @ nil_nat @ I3 @ Y2 )
= nil_nat ) ).
% list_update_code(1)
thf(fact_810_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_811_strict__inc__induct,axiom,
! [I3: nat,J2: nat,P: nat > $o] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ! [I2: nat] :
( ( J2
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I3 ) ) ) ) ).
% strict_inc_induct
thf(fact_812_less__Suc__induct,axiom,
! [I3: nat,J2: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I3 @ 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 @ I3 @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_813_less__trans__Suc,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ord_less_nat @ ( suc @ I3 ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_814_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_815_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_816_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M6: nat] :
( ( M
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_817_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_818_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_819_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_820_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_821_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_822_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_823_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_824_Suc__lessE,axiom,
! [I3: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ K2 )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( K2
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_825_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_826_Nat_OlessE,axiom,
! [I3: nat,K2: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ( ( K2
!= ( suc @ I3 ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( K2
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_827_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_828_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_829_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_830_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_831_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_832_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_833_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_834_map__update,axiom,
! [F: nat > nat,Xs: list_nat,K2: nat,Y2: nat] :
( ( map_nat_nat @ F @ ( list_update_nat @ Xs @ K2 @ Y2 ) )
= ( list_update_nat @ ( map_nat_nat @ F @ Xs ) @ K2 @ ( F @ Y2 ) ) ) ).
% map_update
thf(fact_835_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_836_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_837_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_838_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_839_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_840_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J2: nat] :
( ! [I2: nat,J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_841_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_842_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_843_add__lessD1,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I3 @ J2 ) @ K2 )
=> ( ord_less_nat @ I3 @ K2 ) ) ).
% add_lessD1
thf(fact_844_add__less__mono,axiom,
! [I3: nat,J2: nat,K2: nat,L2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ K2 @ L2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ) ).
% add_less_mono
thf(fact_845_not__add__less1,axiom,
! [I3: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I3 @ J2 ) @ I3 ) ).
% not_add_less1
thf(fact_846_not__add__less2,axiom,
! [J2: nat,I3: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I3 ) @ I3 ) ).
% not_add_less2
thf(fact_847_add__less__mono1,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J2 @ K2 ) ) ) ).
% add_less_mono1
thf(fact_848_trans__less__add1,axiom,
! [I3: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_less_add1
thf(fact_849_trans__less__add2,axiom,
! [I3: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_less_add2
thf(fact_850_less__add__eq__less,axiom,
! [K2: nat,L2: nat,M: nat,N: nat] :
( ( ord_less_nat @ K2 @ L2 )
=> ( ( ( plus_plus_nat @ M @ L2 )
= ( plus_plus_nat @ K2 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_851_diff__less__mono2,axiom,
! [M: nat,N: nat,L2: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L2 )
=> ( ord_less_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_852_less__imp__diff__less,axiom,
! [J2: nat,K2: nat,N: nat] :
( ( ord_less_nat @ J2 @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_853_take__update__swap,axiom,
! [M: nat,Xs: list_nat,N: nat,X2: nat] :
( ( take_nat @ M @ ( list_update_nat @ Xs @ N @ X2 ) )
= ( list_update_nat @ ( take_nat @ M @ Xs ) @ N @ X2 ) ) ).
% take_update_swap
thf(fact_854_pred__def,axiom,
( pred
= ( case_nat_nat @ zero_zero_nat
@ ^ [X24: nat] : X24 ) ) ).
% pred_def
thf(fact_855_list__update__append,axiom,
! [N: nat,Xs: list_a,Ys: list_a,X2: a] :
( ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ ( append_a @ Xs @ Ys ) @ N @ X2 )
= ( append_a @ ( list_update_a @ Xs @ N @ X2 ) @ Ys ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ ( append_a @ Xs @ Ys ) @ N @ X2 )
= ( append_a @ Xs @ ( list_update_a @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) @ X2 ) ) ) ) ) ).
% list_update_append
thf(fact_856_list__update__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat,X2: nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N @ X2 )
= ( append_nat @ ( list_update_nat @ Xs @ N @ X2 ) @ Ys ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N @ X2 )
= ( append_nat @ Xs @ ( list_update_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ X2 ) ) ) ) ) ).
% list_update_append
thf(fact_857_list__update__code_I3_J,axiom,
! [X2: a,Xs: list_a,I3: nat,Y2: a] :
( ( list_update_a @ ( cons_a @ X2 @ Xs ) @ ( suc @ I3 ) @ Y2 )
= ( cons_a @ X2 @ ( list_update_a @ Xs @ I3 @ Y2 ) ) ) ).
% list_update_code(3)
thf(fact_858_list__update__code_I3_J,axiom,
! [X2: nat,Xs: list_nat,I3: nat,Y2: nat] :
( ( list_update_nat @ ( cons_nat @ X2 @ Xs ) @ ( suc @ I3 ) @ Y2 )
= ( cons_nat @ X2 @ ( list_update_nat @ Xs @ I3 @ Y2 ) ) ) ).
% list_update_code(3)
thf(fact_859_list__update__code_I2_J,axiom,
! [X2: a,Xs: list_a,Y2: a] :
( ( list_update_a @ ( cons_a @ X2 @ Xs ) @ zero_zero_nat @ Y2 )
= ( cons_a @ Y2 @ Xs ) ) ).
% list_update_code(2)
thf(fact_860_list__update__code_I2_J,axiom,
! [X2: nat,Xs: list_nat,Y2: nat] :
( ( list_update_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat @ Y2 )
= ( cons_nat @ Y2 @ Xs ) ) ).
% list_update_code(2)
thf(fact_861_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_862_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_863_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J: nat] :
( ( M
= ( suc @ J ) )
& ( ord_less_nat @ J @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_864_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% gr0_implies_Suc
thf(fact_865_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_866_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_867_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_868_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_869_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_870_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_871_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_872_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_873_inc__induct,axiom,
! [I3: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( P @ J2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ N2 @ J2 )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% inc_induct
thf(fact_874_dec__induct,axiom,
! [I3: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( P @ I3 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ N2 @ J2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J2 ) ) ) ) ).
% dec_induct
thf(fact_875_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_876_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_877_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K3 )
=> ~ ( P @ I5 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_878_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_879_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
? [K: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M3 @ K ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_880_less__add__Suc2,axiom,
! [I3: nat,M: nat] : ( ord_less_nat @ I3 @ ( suc @ ( plus_plus_nat @ M @ I3 ) ) ) ).
% less_add_Suc2
thf(fact_881_less__add__Suc1,axiom,
! [I3: nat,M: nat] : ( ord_less_nat @ I3 @ ( suc @ ( plus_plus_nat @ I3 @ M ) ) ) ).
% less_add_Suc1
thf(fact_882_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% less_natE
thf(fact_883_less__imp__add__positive,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I3 @ K3 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_884_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_885_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_886_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_887_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K2: nat] :
( ! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( F @ M2 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K2 ) @ ( F @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_888_less__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_889_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_890_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_891_less__diff__conv,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ ( minus_minus_nat @ J2 @ K2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ J2 ) ) ).
% less_diff_conv
thf(fact_892_nths__all,axiom,
! [Xs: list_a,I: set_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( member_nat @ I2 @ I ) )
=> ( ( nths_a @ Xs @ I )
= Xs ) ) ).
% nths_all
thf(fact_893_nths__all,axiom,
! [Xs: list_nat,I: set_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ I2 @ I ) )
=> ( ( nths_nat @ Xs @ I )
= Xs ) ) ).
% nths_all
thf(fact_894_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K3 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_895_diff__Suc__less,axiom,
! [N: nat,I3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_896_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_897_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_898_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_899_less__diff__conv2,axiom,
! [K2: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K2 ) @ I3 )
= ( ord_less_nat @ J2 @ ( plus_plus_nat @ I3 @ K2 ) ) ) ) ).
% less_diff_conv2
thf(fact_900_list__update_Osimps_I2_J,axiom,
! [X2: a,Xs: list_a,I3: nat,V: a] :
( ( list_update_a @ ( cons_a @ X2 @ Xs ) @ I3 @ V )
= ( case_nat_list_a @ ( cons_a @ V @ Xs )
@ ^ [J: nat] : ( cons_a @ X2 @ ( list_update_a @ Xs @ J @ V ) )
@ I3 ) ) ).
% list_update.simps(2)
thf(fact_901_list__update_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat,I3: nat,V: nat] :
( ( list_update_nat @ ( cons_nat @ X2 @ Xs ) @ I3 @ V )
= ( case_nat_list_nat @ ( cons_nat @ V @ Xs )
@ ^ [J: nat] : ( cons_nat @ X2 @ ( list_update_nat @ Xs @ J @ V ) )
@ I3 ) ) ).
% list_update.simps(2)
thf(fact_902_less__eq__nat_Osimps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
% less_eq_nat.simps(2)
thf(fact_903_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_904_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_905_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_906_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_907_butlast__list__update,axiom,
! [K2: nat,Xs: list_a,X2: a] :
( ( ( K2
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
=> ( ( butlast_a @ ( list_update_a @ Xs @ K2 @ X2 ) )
= ( butlast_a @ Xs ) ) )
& ( ( K2
!= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
=> ( ( butlast_a @ ( list_update_a @ Xs @ K2 @ X2 ) )
= ( list_update_a @ ( butlast_a @ Xs ) @ K2 @ X2 ) ) ) ) ).
% butlast_list_update
thf(fact_908_butlast__list__update,axiom,
! [K2: nat,Xs: list_nat,X2: nat] :
( ( ( K2
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( butlast_nat @ ( list_update_nat @ Xs @ K2 @ X2 ) )
= ( butlast_nat @ Xs ) ) )
& ( ( K2
!= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( butlast_nat @ ( list_update_nat @ Xs @ K2 @ X2 ) )
= ( list_update_nat @ ( butlast_nat @ Xs ) @ K2 @ X2 ) ) ) ) ).
% butlast_list_update
thf(fact_909_diff__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( case_nat_nat @ zero_zero_nat
@ ^ [K: nat] : K
@ ( minus_minus_nat @ M @ N ) ) ) ).
% diff_Suc
thf(fact_910_Cons__le__Cons,axiom,
! [A: nat,X2: list_nat,B: nat,Y2: list_nat] :
( ( ord_less_eq_list_nat @ ( cons_nat @ A @ X2 ) @ ( cons_nat @ B @ Y2 ) )
= ( ( ord_less_nat @ A @ B )
| ( ( A = B )
& ( ord_less_eq_list_nat @ X2 @ Y2 ) ) ) ) ).
% Cons_le_Cons
thf(fact_911_Cons__less__Cons,axiom,
! [A: nat,X2: list_nat,B: nat,Y2: list_nat] :
( ( ord_less_list_nat @ ( cons_nat @ A @ X2 ) @ ( cons_nat @ B @ Y2 ) )
= ( ( ord_less_nat @ A @ B )
| ( ( A = B )
& ( ord_less_list_nat @ X2 @ Y2 ) ) ) ) ).
% Cons_less_Cons
thf(fact_912_le__Nil,axiom,
! [X2: list_nat] :
( ( ord_less_eq_list_nat @ X2 @ nil_nat )
= ( X2 = nil_nat ) ) ).
% le_Nil
thf(fact_913_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_914_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_915_drop0,axiom,
( ( drop_nat @ zero_zero_nat )
= ( ^ [X: list_nat] : X ) ) ).
% drop0
thf(fact_916_drop__drop,axiom,
! [N: nat,M: nat,Xs: list_nat] :
( ( drop_nat @ N @ ( drop_nat @ M @ Xs ) )
= ( drop_nat @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ).
% drop_drop
thf(fact_917_drop__Suc__Cons,axiom,
! [N: nat,X2: a,Xs: list_a] :
( ( drop_a @ ( suc @ N ) @ ( cons_a @ X2 @ Xs ) )
= ( drop_a @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_918_drop__Suc__Cons,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( drop_nat @ ( suc @ N ) @ ( cons_nat @ X2 @ Xs ) )
= ( drop_nat @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_919_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_920_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_921_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_922_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_923_drop__update__cancel,axiom,
! [N: nat,M: nat,Xs: list_nat,X2: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( drop_nat @ M @ ( list_update_nat @ Xs @ N @ X2 ) )
= ( drop_nat @ M @ Xs ) ) ) ).
% drop_update_cancel
thf(fact_924_drop__replicate,axiom,
! [I3: nat,K2: nat,X2: nat] :
( ( drop_nat @ I3 @ ( replicate_nat @ K2 @ X2 ) )
= ( replicate_nat @ ( minus_minus_nat @ K2 @ I3 ) @ X2 ) ) ).
% drop_replicate
thf(fact_925_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_926_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_927_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_928_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_929_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_930_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_931_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_932_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_933_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_934_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_935_less__list__code_I1_J,axiom,
! [Xs: list_nat] :
~ ( ord_less_list_nat @ Xs @ nil_nat ) ).
% less_list_code(1)
thf(fact_936_not__less__Nil,axiom,
! [X2: list_nat] :
~ ( ord_less_list_nat @ X2 @ nil_nat ) ).
% not_less_Nil
thf(fact_937_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_938_tl__drop,axiom,
! [N: nat,Xs: list_nat] :
( ( tl_nat @ ( drop_nat @ N @ Xs ) )
= ( drop_nat @ N @ ( tl_nat @ Xs ) ) ) ).
% tl_drop
thf(fact_939_drop__butlast,axiom,
! [N: nat,Xs: list_nat] :
( ( drop_nat @ N @ ( butlast_nat @ Xs ) )
= ( butlast_nat @ ( drop_nat @ N @ Xs ) ) ) ).
% drop_butlast
thf(fact_940_Nil__less__Cons,axiom,
! [A: nat,X2: list_nat] : ( ord_less_list_nat @ nil_nat @ ( cons_nat @ A @ X2 ) ) ).
% Nil_less_Cons
thf(fact_941_less__list__code_I2_J,axiom,
! [X2: nat,Xs: list_nat] : ( ord_less_list_nat @ nil_nat @ ( cons_nat @ X2 @ Xs ) ) ).
% less_list_code(2)
thf(fact_942_drop__Nil,axiom,
! [N: nat] :
( ( drop_a @ N @ nil_a )
= nil_a ) ).
% drop_Nil
thf(fact_943_drop__Nil,axiom,
! [N: nat] :
( ( drop_nat @ N @ nil_nat )
= nil_nat ) ).
% drop_Nil
thf(fact_944_drop__0,axiom,
! [Xs: list_nat] :
( ( drop_nat @ zero_zero_nat @ Xs )
= Xs ) ).
% drop_0
thf(fact_945_take__drop,axiom,
! [N: nat,M: nat,Xs: list_nat] :
( ( take_nat @ N @ ( drop_nat @ M @ Xs ) )
= ( drop_nat @ M @ ( take_nat @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ) ).
% take_drop
thf(fact_946_drop__take,axiom,
! [N: nat,M: nat,Xs: list_nat] :
( ( drop_nat @ N @ ( take_nat @ M @ Xs ) )
= ( take_nat @ ( minus_minus_nat @ M @ N ) @ ( drop_nat @ N @ Xs ) ) ) ).
% drop_take
thf(fact_947_drop__Suc,axiom,
! [N: nat,Xs: list_nat] :
( ( drop_nat @ ( suc @ N ) @ Xs )
= ( drop_nat @ N @ ( tl_nat @ Xs ) ) ) ).
% drop_Suc
thf(fact_948_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_949_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_950_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_951_take__add,axiom,
! [I3: nat,J2: nat,Xs: list_a] :
( ( take_a @ ( plus_plus_nat @ I3 @ J2 ) @ Xs )
= ( append_a @ ( take_a @ I3 @ Xs ) @ ( take_a @ J2 @ ( drop_a @ I3 @ Xs ) ) ) ) ).
% take_add
thf(fact_952_take__add,axiom,
! [I3: nat,J2: nat,Xs: list_nat] :
( ( take_nat @ ( plus_plus_nat @ I3 @ J2 ) @ Xs )
= ( append_nat @ ( take_nat @ I3 @ Xs ) @ ( take_nat @ J2 @ ( drop_nat @ I3 @ Xs ) ) ) ) ).
% take_add
thf(fact_953_drop__update__swap,axiom,
! [M: nat,N: nat,Xs: list_nat,X2: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( drop_nat @ M @ ( list_update_nat @ Xs @ N @ X2 ) )
= ( list_update_nat @ ( drop_nat @ M @ Xs ) @ ( minus_minus_nat @ N @ M ) @ X2 ) ) ) ).
% drop_update_swap
thf(fact_954_drop__Cons,axiom,
! [N: nat,X2: a,Xs: list_a] :
( ( drop_a @ N @ ( cons_a @ X2 @ Xs ) )
= ( case_nat_list_a @ ( cons_a @ X2 @ Xs )
@ ^ [M3: nat] : ( drop_a @ M3 @ Xs )
@ N ) ) ).
% drop_Cons
thf(fact_955_drop__Cons,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( drop_nat @ N @ ( cons_nat @ X2 @ Xs ) )
= ( case_nat_list_nat @ ( cons_nat @ X2 @ Xs )
@ ^ [M3: nat] : ( drop_nat @ M3 @ Xs )
@ N ) ) ).
% drop_Cons
thf(fact_956_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_957_Nil__le__Cons,axiom,
! [X2: list_nat] : ( ord_less_eq_list_nat @ nil_nat @ X2 ) ).
% Nil_le_Cons
thf(fact_958_drop__Cons_H,axiom,
! [N: nat,X2: a,Xs: list_a] :
( ( ( N = zero_zero_nat )
=> ( ( drop_a @ N @ ( cons_a @ X2 @ Xs ) )
= ( cons_a @ X2 @ Xs ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_a @ N @ ( cons_a @ X2 @ Xs ) )
= ( drop_a @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_959_drop__Cons_H,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( drop_nat @ N @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ X2 @ Xs ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_nat @ N @ ( cons_nat @ X2 @ Xs ) )
= ( drop_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_960_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_961_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_962_less__list__code_I3_J,axiom,
! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( ord_less_list_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) )
= ( ( ord_less_nat @ X2 @ Y2 )
| ( ( X2 = Y2 )
& ( ord_less_list_nat @ Xs @ Ys ) ) ) ) ).
% less_list_code(3)
thf(fact_963_less__eq__list__code_I3_J,axiom,
! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( ord_less_eq_list_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) )
= ( ( ord_less_nat @ X2 @ Y2 )
| ( ( X2 = Y2 )
& ( ord_less_eq_list_nat @ Xs @ Ys ) ) ) ) ).
% less_eq_list_code(3)
thf(fact_964_less__eq__list__code_I1_J,axiom,
! [X2: nat,Xs: list_nat] :
~ ( ord_less_eq_list_nat @ ( cons_nat @ X2 @ Xs ) @ nil_nat ) ).
% less_eq_list_code(1)
thf(fact_965_upd__conv__take__nth__drop,axiom,
! [I3: nat,Xs: list_a,A: a] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ Xs @ I3 @ A )
= ( append_a @ ( take_a @ I3 @ Xs ) @ ( cons_a @ A @ ( drop_a @ ( suc @ I3 ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_966_upd__conv__take__nth__drop,axiom,
! [I3: nat,Xs: list_nat,A: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ Xs @ I3 @ A )
= ( append_nat @ ( take_nat @ I3 @ Xs ) @ ( cons_nat @ A @ ( drop_nat @ ( suc @ I3 ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_967_id__take__nth__drop,axiom,
! [I3: nat,Xs: list_a] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( Xs
= ( append_a @ ( take_a @ I3 @ Xs ) @ ( cons_a @ ( nth_a @ Xs @ I3 ) @ ( drop_a @ ( suc @ I3 ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_968_id__take__nth__drop,axiom,
! [I3: nat,Xs: list_nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( Xs
= ( append_nat @ ( take_nat @ I3 @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I3 ) @ ( drop_nat @ ( suc @ I3 ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_969_take__Suc__conv__app__nth,axiom,
! [I3: nat,Xs: list_a] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( ( take_a @ ( suc @ I3 ) @ Xs )
= ( append_a @ ( take_a @ I3 @ Xs ) @ ( cons_a @ ( nth_a @ Xs @ I3 ) @ nil_a ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_970_take__Suc__conv__app__nth,axiom,
! [I3: nat,Xs: list_nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( take_nat @ ( suc @ I3 ) @ Xs )
= ( append_nat @ ( take_nat @ I3 @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I3 ) @ nil_nat ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_971_nth__Cons__pos,axiom,
! [N: nat,X2: a,Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_a @ ( cons_a @ X2 @ Xs ) @ N )
= ( nth_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_972_nth__Cons__pos,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_973_nth__list__update__neq,axiom,
! [I3: nat,J2: nat,Xs: list_nat,X2: nat] :
( ( I3 != J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ J2 )
= ( nth_nat @ Xs @ J2 ) ) ) ).
% nth_list_update_neq
thf(fact_974_list__update__id,axiom,
! [Xs: list_nat,I3: nat] :
( ( list_update_nat @ Xs @ I3 @ ( nth_nat @ Xs @ I3 ) )
= Xs ) ).
% list_update_id
thf(fact_975_nth__Cons__Suc,axiom,
! [X2: a,Xs: list_a,N: nat] :
( ( nth_a @ ( cons_a @ X2 @ Xs ) @ ( suc @ N ) )
= ( nth_a @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_976_nth__Cons__Suc,axiom,
! [X2: nat,Xs: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ ( suc @ N ) )
= ( nth_nat @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_977_nth__Cons__0,axiom,
! [X2: a,Xs: list_a] :
( ( nth_a @ ( cons_a @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_978_nth__Cons__0,axiom,
! [X2: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_979_nth__take,axiom,
! [I3: nat,N: nat,Xs: list_nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ( nth_nat @ ( take_nat @ N @ Xs ) @ I3 )
= ( nth_nat @ Xs @ I3 ) ) ) ).
% nth_take
thf(fact_980_nth__replicate,axiom,
! [I3: nat,N: nat,X2: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ( nth_nat @ ( replicate_nat @ N @ X2 ) @ I3 )
= X2 ) ) ).
% nth_replicate
thf(fact_981_nth__append__length,axiom,
! [Xs: list_a,X2: a,Ys: list_a] :
( ( nth_a @ ( append_a @ Xs @ ( cons_a @ X2 @ Ys ) ) @ ( size_size_list_a @ Xs ) )
= X2 ) ).
% nth_append_length
thf(fact_982_nth__append__length,axiom,
! [Xs: list_nat,X2: nat,Ys: list_nat] :
( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys ) ) @ ( size_size_list_nat @ Xs ) )
= X2 ) ).
% nth_append_length
thf(fact_983_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_984_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_985_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_986_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_987_nth__list__update__eq,axiom,
! [I3: nat,Xs: list_a,X2: a] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( list_update_a @ Xs @ I3 @ X2 ) @ I3 )
= X2 ) ) ).
% nth_list_update_eq
thf(fact_988_nth__list__update__eq,axiom,
! [I3: nat,Xs: list_nat,X2: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ I3 )
= X2 ) ) ).
% nth_list_update_eq
thf(fact_989_nth__drop,axiom,
! [N: nat,Xs: list_a,I3: nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( drop_a @ N @ Xs ) @ I3 )
= ( nth_a @ Xs @ ( plus_plus_nat @ N @ I3 ) ) ) ) ).
% nth_drop
thf(fact_990_nth__drop,axiom,
! [N: nat,Xs: list_nat,I3: nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( drop_nat @ N @ Xs ) @ I3 )
= ( nth_nat @ Xs @ ( plus_plus_nat @ N @ I3 ) ) ) ) ).
% nth_drop
thf(fact_991_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_a,Z4: list_a] : ( Y4 = Z4 ) )
= ( ^ [Xs4: list_a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs4 )
= ( size_size_list_a @ Ys2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs4 ) )
=> ( ( nth_a @ Xs4 @ I4 )
= ( nth_a @ Ys2 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_992_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_nat,Z4: list_nat] : ( Y4 = Z4 ) )
= ( ^ [Xs4: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs4 )
= ( size_size_list_nat @ Ys2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs4 ) )
=> ( ( nth_nat @ Xs4 @ I4 )
= ( nth_nat @ Ys2 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_993_Skolem__list__nth,axiom,
! [K2: nat,P: nat > a > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ? [X4: a] : ( P @ I4 @ X4 ) ) )
= ( ? [Xs4: list_a] :
( ( ( size_size_list_a @ Xs4 )
= K2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ( P @ I4 @ ( nth_a @ Xs4 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_994_Skolem__list__nth,axiom,
! [K2: nat,P: nat > nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ? [X4: nat] : ( P @ I4 @ X4 ) ) )
= ( ? [Xs4: list_nat] :
( ( ( size_size_list_nat @ Xs4 )
= K2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ( P @ I4 @ ( nth_nat @ Xs4 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_995_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_996_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_997_nth__via__drop,axiom,
! [N: nat,Xs: list_a,Y2: a,Ys: list_a] :
( ( ( drop_a @ N @ Xs )
= ( cons_a @ Y2 @ Ys ) )
=> ( ( nth_a @ Xs @ N )
= Y2 ) ) ).
% nth_via_drop
thf(fact_998_nth__via__drop,axiom,
! [N: nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( ( drop_nat @ N @ Xs )
= ( cons_nat @ Y2 @ Ys ) )
=> ( ( nth_nat @ Xs @ N )
= Y2 ) ) ).
% nth_via_drop
thf(fact_999_nth__Cons,axiom,
! [X2: a,Xs: list_a,N: nat] :
( ( nth_a @ ( cons_a @ X2 @ Xs ) @ N )
= ( case_nat_a @ X2 @ ( nth_a @ Xs ) @ N ) ) ).
% nth_Cons
thf(fact_1000_nth__Cons,axiom,
! [X2: nat,Xs: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
= ( case_nat_nat @ X2 @ ( nth_nat @ Xs ) @ N ) ) ).
% nth_Cons
thf(fact_1001_nth__Cons,axiom,
! [X2: $o,Xs: list_o,N: nat] :
( ( nth_o @ ( cons_o @ X2 @ Xs ) @ N )
= ( case_nat_o @ X2 @ ( nth_o @ Xs ) @ N ) ) ).
% nth_Cons
thf(fact_1002_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 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_a @ Xs @ I4 ) )
= ( G @ ( nth_nat @ Ys @ I4 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_1003_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 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Ys ) )
=> ( ( F @ ( nth_nat @ Xs @ I4 ) )
= ( G @ ( nth_a @ Ys @ I4 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_1004_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 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_nat @ Xs @ I4 ) )
= ( G @ ( nth_nat @ Ys @ I4 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_1005_hd__conv__nth,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ Xs )
= ( nth_a @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_1006_hd__conv__nth,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ Xs )
= ( nth_nat @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_1007_nth__list__update,axiom,
! [I3: nat,Xs: list_a,J2: nat,X2: a] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( ( ( I3 = J2 )
=> ( ( nth_a @ ( list_update_a @ Xs @ I3 @ X2 ) @ J2 )
= X2 ) )
& ( ( I3 != J2 )
=> ( ( nth_a @ ( list_update_a @ Xs @ I3 @ X2 ) @ J2 )
= ( nth_a @ Xs @ J2 ) ) ) ) ) ).
% nth_list_update
thf(fact_1008_nth__list__update,axiom,
! [I3: nat,Xs: list_nat,J2: nat,X2: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( I3 = J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ J2 )
= X2 ) )
& ( ( I3 != J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ J2 )
= ( nth_nat @ Xs @ J2 ) ) ) ) ) ).
% nth_list_update
thf(fact_1009_list__update__same__conv,axiom,
! [I3: nat,Xs: list_a,X2: a] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( ( ( list_update_a @ Xs @ I3 @ X2 )
= Xs )
= ( ( nth_a @ Xs @ I3 )
= X2 ) ) ) ).
% list_update_same_conv
thf(fact_1010_list__update__same__conv,axiom,
! [I3: nat,Xs: list_nat,X2: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( list_update_nat @ Xs @ I3 @ X2 )
= Xs )
= ( ( nth_nat @ Xs @ I3 )
= X2 ) ) ) ).
% list_update_same_conv
thf(fact_1011_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_1012_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_1013_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_1014_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_1015_nth__transpose,axiom,
! [I3: nat,Xs: list_list_a] :
( ( ord_less_nat @ I3 @ ( size_s349497388124573686list_a @ ( transpose_a @ Xs ) ) )
=> ( ( nth_list_a @ ( transpose_a @ Xs ) @ I3 )
= ( map_list_a_a
@ ^ [Xs4: list_a] : ( nth_a @ Xs4 @ I3 )
@ ( filter_list_a
@ ^ [Ys2: list_a] : ( ord_less_nat @ I3 @ ( size_size_list_a @ Ys2 ) )
@ Xs ) ) ) ) ).
% nth_transpose
thf(fact_1016_nth__transpose,axiom,
! [I3: nat,Xs: list_list_nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ ( transpose_nat @ Xs ) ) )
=> ( ( nth_list_nat @ ( transpose_nat @ Xs ) @ I3 )
= ( map_list_nat_nat
@ ^ [Xs4: list_nat] : ( nth_nat @ Xs4 @ I3 )
@ ( filter_list_nat
@ ^ [Ys2: list_nat] : ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys2 ) )
@ Xs ) ) ) ) ).
% nth_transpose
thf(fact_1017_nth__Cons_H,axiom,
! [N: nat,X2: a,Xs: list_a] :
( ( ( N = zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X2 @ Xs ) @ N )
= X2 ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X2 @ Xs ) @ N )
= ( nth_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_1018_nth__Cons_H,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
= X2 ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_1019_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_1020_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_1021_nth__take__lemma,axiom,
! [K2: nat,Xs: list_a,Ys: list_a] :
( ( ord_less_eq_nat @ K2 @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_eq_nat @ K2 @ ( size_size_list_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ( ( nth_a @ Xs @ I2 )
= ( nth_a @ Ys @ I2 ) ) )
=> ( ( take_a @ K2 @ Xs )
= ( take_a @ K2 @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_1022_nth__take__lemma,axiom,
! [K2: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_eq_nat @ K2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ K2 @ ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys @ I2 ) ) )
=> ( ( take_nat @ K2 @ Xs )
= ( take_nat @ K2 @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_1023_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_1024_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_1025_remdups__adj__adjacent,axiom,
! [I3: nat,Xs: list_a] :
( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_a @ ( remdups_adj_a @ Xs ) ) )
=> ( ( nth_a @ ( remdups_adj_a @ Xs ) @ I3 )
!= ( nth_a @ ( remdups_adj_a @ Xs ) @ ( suc @ I3 ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_1026_remdups__adj__adjacent,axiom,
! [I3: nat,Xs: list_nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) ) )
=> ( ( nth_nat @ ( remdups_adj_nat @ Xs ) @ I3 )
!= ( nth_nat @ ( remdups_adj_nat @ Xs ) @ ( suc @ I3 ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_1027_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_1028_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_1029_filter__eq__nths,axiom,
( filter_a
= ( ^ [P2: a > $o,Xs4: list_a] :
( nths_a @ Xs4
@ ( collect_nat
@ ^ [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs4 ) )
& ( P2 @ ( nth_a @ Xs4 @ I4 ) ) ) ) ) ) ) ).
% filter_eq_nths
thf(fact_1030_filter__eq__nths,axiom,
( filter_nat
= ( ^ [P2: nat > $o,Xs4: list_nat] :
( nths_nat @ Xs4
@ ( collect_nat
@ ^ [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs4 ) )
& ( P2 @ ( nth_nat @ Xs4 @ I4 ) ) ) ) ) ) ) ).
% filter_eq_nths
thf(fact_1031_nth__non__equal__first__eq,axiom,
! [X2: a,Y2: a,Xs: list_a,N: nat] :
( ( X2 != Y2 )
=> ( ( ( nth_a @ ( cons_a @ X2 @ Xs ) @ N )
= Y2 )
= ( ( ( nth_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_1032_nth__non__equal__first__eq,axiom,
! [X2: nat,Y2: nat,Xs: list_nat,N: nat] :
( ( X2 != Y2 )
=> ( ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
= Y2 )
= ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_1033_Cons__nth__drop__Suc,axiom,
! [I3: nat,Xs: list_a] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( ( cons_a @ ( nth_a @ Xs @ I3 ) @ ( drop_a @ ( suc @ I3 ) @ Xs ) )
= ( drop_a @ I3 @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_1034_Cons__nth__drop__Suc,axiom,
! [I3: nat,Xs: list_nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( cons_nat @ ( nth_nat @ Xs @ I3 ) @ ( drop_nat @ ( suc @ I3 ) @ Xs ) )
= ( drop_nat @ I3 @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_1035_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_1036_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_1037_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
@ ^ [I4: nat] :
( map_nat_a
@ ^ [J: nat] : ( nth_a @ ( nth_list_a @ Xs @ J ) @ I4 )
@ ( upt @ zero_zero_nat @ ( size_s349497388124573686list_a @ Xs ) ) )
@ ( upt @ zero_zero_nat @ N ) ) ) ) ) ).
% transpose_rectangle
thf(fact_1038_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
@ ^ [I4: nat] :
( map_nat_nat
@ ^ [J: nat] : ( nth_nat @ ( nth_list_nat @ Xs @ J ) @ I4 )
@ ( upt @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs ) ) )
@ ( upt @ zero_zero_nat @ N ) ) ) ) ) ).
% transpose_rectangle
thf(fact_1039_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_1040_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_1041_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A3 )
@ ^ [X: nat] : ( member_nat @ X @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_1042_Cons__prefix__Cons,axiom,
! [X2: a,Xs: list_a,Y2: a,Ys: list_a] :
( ( prefix_a @ ( cons_a @ X2 @ Xs ) @ ( cons_a @ Y2 @ Ys ) )
= ( ( X2 = Y2 )
& ( prefix_a @ Xs @ Ys ) ) ) ).
% Cons_prefix_Cons
thf(fact_1043_Cons__prefix__Cons,axiom,
! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( prefix_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) )
= ( ( X2 = Y2 )
& ( prefix_nat @ Xs @ Ys ) ) ) ).
% Cons_prefix_Cons
thf(fact_1044_prefix__Nil,axiom,
! [Xs: list_a] :
( ( prefix_a @ Xs @ nil_a )
= ( Xs = nil_a ) ) ).
% prefix_Nil
thf(fact_1045_prefix__Nil,axiom,
! [Xs: list_nat] :
( ( prefix_nat @ Xs @ nil_nat )
= ( Xs = nil_nat ) ) ).
% prefix_Nil
thf(fact_1046_prefix__bot_Oextremum__unique,axiom,
! [A: list_a] :
( ( prefix_a @ A @ nil_a )
= ( A = nil_a ) ) ).
% prefix_bot.extremum_unique
thf(fact_1047_prefix__bot_Oextremum__unique,axiom,
! [A: list_nat] :
( ( prefix_nat @ A @ nil_nat )
= ( A = nil_nat ) ) ).
% prefix_bot.extremum_unique
thf(fact_1048_prefix__code_I1_J,axiom,
! [Xs: list_a] : ( prefix_a @ nil_a @ Xs ) ).
% prefix_code(1)
thf(fact_1049_prefix__code_I1_J,axiom,
! [Xs: list_nat] : ( prefix_nat @ nil_nat @ Xs ) ).
% prefix_code(1)
thf(fact_1050_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_1051_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_1052_tl__upt,axiom,
! [M: nat,N: nat] :
( ( tl_nat @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ N ) ) ).
% tl_upt
thf(fact_1053_hd__upt,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( hd_nat @ ( upt @ I3 @ J2 ) )
= I3 ) ) ).
% hd_upt
thf(fact_1054_upt__conv__Nil,axiom,
! [J2: nat,I3: nat] :
( ( ord_less_eq_nat @ J2 @ I3 )
=> ( ( upt @ I3 @ J2 )
= nil_nat ) ) ).
% upt_conv_Nil
thf(fact_1055_drop__upt,axiom,
! [M: nat,I3: nat,J2: nat] :
( ( drop_nat @ M @ ( upt @ I3 @ J2 ) )
= ( upt @ ( plus_plus_nat @ I3 @ M ) @ J2 ) ) ).
% drop_upt
thf(fact_1056_length__upt,axiom,
! [I3: nat,J2: nat] :
( ( size_size_list_nat @ ( upt @ I3 @ J2 ) )
= ( minus_minus_nat @ J2 @ I3 ) ) ).
% length_upt
thf(fact_1057_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_1058_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_1059_upt__eq__Nil__conv,axiom,
! [I3: nat,J2: nat] :
( ( ( upt @ I3 @ J2 )
= nil_nat )
= ( ( J2 = zero_zero_nat )
| ( ord_less_eq_nat @ J2 @ I3 ) ) ) ).
% upt_eq_Nil_conv
thf(fact_1060_nth__upt,axiom,
! [I3: nat,K2: nat,J2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ J2 )
=> ( ( nth_nat @ ( upt @ I3 @ J2 ) @ K2 )
= ( plus_plus_nat @ I3 @ K2 ) ) ) ).
% nth_upt
thf(fact_1061_take__upt,axiom,
! [I3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ M ) @ N )
=> ( ( take_nat @ M @ ( upt @ I3 @ N ) )
= ( upt @ I3 @ ( plus_plus_nat @ I3 @ M ) ) ) ) ).
% take_upt
thf(fact_1062_prefix__snoc,axiom,
! [Xs: list_a,Ys: list_a,Y2: a] :
( ( prefix_a @ Xs @ ( append_a @ Ys @ ( cons_a @ Y2 @ nil_a ) ) )
= ( ( Xs
= ( append_a @ Ys @ ( cons_a @ Y2 @ nil_a ) ) )
| ( prefix_a @ Xs @ Ys ) ) ) ).
% prefix_snoc
thf(fact_1063_prefix__snoc,axiom,
! [Xs: list_nat,Ys: list_nat,Y2: nat] :
( ( prefix_nat @ Xs @ ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) )
= ( ( Xs
= ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) )
| ( prefix_nat @ Xs @ Ys ) ) ) ).
% prefix_snoc
thf(fact_1064_last__upt,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( last_nat @ ( upt @ I3 @ J2 ) )
= ( minus_minus_nat @ J2 @ one_one_nat ) ) ) ).
% last_upt
thf(fact_1065_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_1066_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_1067_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_1068_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_1069_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_1070_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_1071_prefix__def,axiom,
( prefix_a
= ( ^ [Xs4: list_a,Ys2: list_a] :
? [Zs3: list_a] :
( Ys2
= ( append_a @ Xs4 @ Zs3 ) ) ) ) ).
% prefix_def
thf(fact_1072_prefix__def,axiom,
( prefix_nat
= ( ^ [Xs4: list_nat,Ys2: list_nat] :
? [Zs3: list_nat] :
( Ys2
= ( append_nat @ Xs4 @ Zs3 ) ) ) ) ).
% prefix_def
thf(fact_1073_prefixI,axiom,
! [Ys: list_a,Xs: list_a,Zs: list_a] :
( ( Ys
= ( append_a @ Xs @ Zs ) )
=> ( prefix_a @ Xs @ Ys ) ) ).
% prefixI
thf(fact_1074_prefixI,axiom,
! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( Ys
= ( append_nat @ Xs @ Zs ) )
=> ( prefix_nat @ Xs @ Ys ) ) ).
% prefixI
thf(fact_1075_prefixE,axiom,
! [Xs: list_a,Ys: list_a] :
( ( prefix_a @ Xs @ Ys )
=> ~ ! [Zs2: list_a] :
( Ys
!= ( append_a @ Xs @ Zs2 ) ) ) ).
% prefixE
thf(fact_1076_prefixE,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( prefix_nat @ Xs @ Ys )
=> ~ ! [Zs2: list_nat] :
( Ys
!= ( append_nat @ Xs @ Zs2 ) ) ) ).
% prefixE
thf(fact_1077_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_1078_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_1079_take__is__prefix,axiom,
! [N: nat,Xs: list_nat] : ( prefix_nat @ ( take_nat @ N @ Xs ) @ Xs ) ).
% take_is_prefix
thf(fact_1080_Nil__prefix,axiom,
! [Xs: list_a] : ( prefix_a @ nil_a @ Xs ) ).
% Nil_prefix
thf(fact_1081_Nil__prefix,axiom,
! [Xs: list_nat] : ( prefix_nat @ nil_nat @ Xs ) ).
% Nil_prefix
thf(fact_1082_prefix__bot_Oextremum__uniqueI,axiom,
! [A: list_a] :
( ( prefix_a @ A @ nil_a )
=> ( A = nil_a ) ) ).
% prefix_bot.extremum_uniqueI
thf(fact_1083_prefix__bot_Oextremum__uniqueI,axiom,
! [A: list_nat] :
( ( prefix_nat @ A @ nil_nat )
=> ( A = nil_nat ) ) ).
% prefix_bot.extremum_uniqueI
thf(fact_1084_prefix__bot_Obot__least,axiom,
! [A: list_a] : ( prefix_a @ nil_a @ A ) ).
% prefix_bot.bot_least
thf(fact_1085_prefix__bot_Obot__least,axiom,
! [A: list_nat] : ( prefix_nat @ nil_nat @ A ) ).
% prefix_bot.bot_least
thf(fact_1086_upt__0,axiom,
! [I3: nat] :
( ( upt @ I3 @ zero_zero_nat )
= nil_nat ) ).
% upt_0
thf(fact_1087_map__Suc__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_1088_upt__conv__Cons__Cons,axiom,
! [M: nat,N: nat,Ns: list_nat,Q3: nat] :
( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
= ( upt @ M @ Q3 ) )
= ( ( cons_nat @ N @ Ns )
= ( upt @ ( suc @ M ) @ Q3 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_1089_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_1090_subset__CollectI,axiom,
! [B3: set_nat,A2: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B3 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ B3 )
& ( Q @ X ) ) )
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1091_subset__Collect__iff,axiom,
! [B3: set_nat,A2: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( ord_less_eq_set_nat @ B3
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ X ) ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ B3 )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1092_upt__rec,axiom,
( upt
= ( ^ [I4: nat,J: nat] : ( if_list_nat @ ( ord_less_nat @ I4 @ J ) @ ( cons_nat @ I4 @ ( upt @ ( suc @ I4 ) @ J ) ) @ nil_nat ) ) ) ).
% upt_rec
thf(fact_1093_upt__conv__Cons,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( upt @ I3 @ J2 )
= ( cons_nat @ I3 @ ( upt @ ( suc @ I3 ) @ J2 ) ) ) ) ).
% upt_conv_Cons
thf(fact_1094_upt__Suc__append,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( upt @ I3 @ ( suc @ J2 ) )
= ( append_nat @ ( upt @ I3 @ J2 ) @ ( cons_nat @ J2 @ nil_nat ) ) ) ) ).
% upt_Suc_append
thf(fact_1095_upt__Suc,axiom,
! [I3: nat,J2: nat] :
( ( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( upt @ I3 @ ( suc @ J2 ) )
= ( append_nat @ ( upt @ I3 @ J2 ) @ ( cons_nat @ J2 @ nil_nat ) ) ) )
& ( ~ ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( upt @ I3 @ ( suc @ J2 ) )
= nil_nat ) ) ) ).
% upt_Suc
thf(fact_1096_prefix__code_I2_J,axiom,
! [X2: a,Xs: list_a] :
~ ( prefix_a @ ( cons_a @ X2 @ Xs ) @ nil_a ) ).
% prefix_code(2)
thf(fact_1097_prefix__code_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
~ ( prefix_nat @ ( cons_nat @ X2 @ Xs ) @ nil_nat ) ).
% prefix_code(2)
thf(fact_1098_prefix__Cons,axiom,
! [Xs: list_a,Y2: a,Ys: list_a] :
( ( prefix_a @ Xs @ ( cons_a @ Y2 @ Ys ) )
= ( ( Xs = nil_a )
| ? [Zs3: list_a] :
( ( Xs
= ( cons_a @ Y2 @ Zs3 ) )
& ( prefix_a @ Zs3 @ Ys ) ) ) ) ).
% prefix_Cons
thf(fact_1099_prefix__Cons,axiom,
! [Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( prefix_nat @ Xs @ ( cons_nat @ Y2 @ Ys ) )
= ( ( Xs = nil_nat )
| ? [Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Y2 @ Zs3 ) )
& ( prefix_nat @ Zs3 @ Ys ) ) ) ) ).
% prefix_Cons
thf(fact_1100_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_1101_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_1102_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_1103_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_1104_upt__add__eq__append,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( upt @ I3 @ ( plus_plus_nat @ J2 @ K2 ) )
= ( append_nat @ ( upt @ I3 @ J2 ) @ ( upt @ J2 @ ( plus_plus_nat @ J2 @ K2 ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_1105_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_1106_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_1107_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_1108_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_1109_map__add__upt,axiom,
! [N: nat,M: nat] :
( ( map_nat_nat
@ ^ [I4: nat] : ( plus_plus_nat @ I4 @ N )
@ ( upt @ zero_zero_nat @ M ) )
= ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).
% map_add_upt
thf(fact_1110_map__replicate__trivial,axiom,
! [X2: nat,I3: nat] :
( ( map_nat_nat
@ ^ [I4: nat] : X2
@ ( upt @ zero_zero_nat @ I3 ) )
= ( replicate_nat @ I3 @ X2 ) ) ).
% map_replicate_trivial
thf(fact_1111_upt__eq__Cons__conv,axiom,
! [I3: nat,J2: nat,X2: nat,Xs: list_nat] :
( ( ( upt @ I3 @ J2 )
= ( cons_nat @ X2 @ Xs ) )
= ( ( ord_less_nat @ I3 @ J2 )
& ( I3 = X2 )
& ( ( upt @ ( plus_plus_nat @ I3 @ one_one_nat ) @ J2 )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_1112_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
@ ^ [I4: nat] : ( F @ ( suc @ I4 ) )
@ ( upt @ zero_zero_nat @ N ) ) ) ) ).
% map_upt_Suc
thf(fact_1113_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
@ ^ [I4: nat] : ( F @ ( suc @ I4 ) )
@ ( upt @ zero_zero_nat @ N ) ) ) ) ).
% map_upt_Suc
thf(fact_1114_map__decr__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat
@ ^ [N3: nat] : ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) )
@ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
= ( upt @ M @ N ) ) ).
% map_decr_upt
thf(fact_1115_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_1116_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_1117_nth__map__upt,axiom,
! [I3: nat,N: nat,M: nat,F: nat > nat] :
( ( ord_less_nat @ I3 @ ( minus_minus_nat @ N @ M ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ ( upt @ M @ N ) ) @ I3 )
= ( F @ ( plus_plus_nat @ M @ I3 ) ) ) ) ).
% nth_map_upt
thf(fact_1118_map__upt__eqI,axiom,
! [Xs: list_a,N: nat,M: nat,F: nat > a] :
( ( ( size_size_list_a @ Xs )
= ( minus_minus_nat @ N @ M ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ Xs @ I2 )
= ( F @ ( plus_plus_nat @ M @ I2 ) ) ) )
=> ( ( map_nat_a @ F @ ( upt @ M @ N ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_1119_map__upt__eqI,axiom,
! [Xs: list_nat,N: nat,M: nat,F: nat > nat] :
( ( ( size_size_list_nat @ Xs )
= ( minus_minus_nat @ N @ M ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I2 )
= ( F @ ( plus_plus_nat @ M @ I2 ) ) ) )
=> ( ( map_nat_nat @ F @ ( upt @ M @ N ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_1120_pred__subset__eq,axiom,
! [R: set_nat,S2: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ R )
@ ^ [X: nat] : ( member_nat @ X @ S2 ) )
= ( ord_less_eq_set_nat @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_1121_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
@ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X )
@ zero_zero_nat ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_1122_drop__Cons__numeral,axiom,
! [V: num,X2: a,Xs: list_a] :
( ( drop_a @ ( numeral_numeral_nat @ V ) @ ( cons_a @ X2 @ Xs ) )
= ( drop_a @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) @ Xs ) ) ).
% drop_Cons_numeral
thf(fact_1123_drop__Cons__numeral,axiom,
! [V: num,X2: nat,Xs: list_nat] :
( ( drop_nat @ ( numeral_numeral_nat @ V ) @ ( cons_nat @ X2 @ Xs ) )
= ( drop_nat @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) @ Xs ) ) ).
% drop_Cons_numeral
thf(fact_1124_nth__Cons__numeral,axiom,
! [X2: a,Xs: list_a,V: num] :
( ( nth_a @ ( cons_a @ X2 @ Xs ) @ ( numeral_numeral_nat @ V ) )
= ( nth_a @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_1125_nth__Cons__numeral,axiom,
! [X2: nat,Xs: list_nat,V: num] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ ( numeral_numeral_nat @ V ) )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_1126_take__Cons__numeral,axiom,
! [V: num,X2: a,Xs: list_a] :
( ( take_a @ ( numeral_numeral_nat @ V ) @ ( cons_a @ X2 @ Xs ) )
= ( cons_a @ X2 @ ( take_a @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) @ Xs ) ) ) ).
% take_Cons_numeral
thf(fact_1127_take__Cons__numeral,axiom,
! [V: num,X2: nat,Xs: list_nat] :
( ( take_nat @ ( numeral_numeral_nat @ V ) @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ X2 @ ( take_nat @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) @ Xs ) ) ) ).
% take_Cons_numeral
thf(fact_1128_upt__rec__numeral,axiom,
! [M: num,N: num] :
( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
& ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= nil_nat ) ) ) ).
% upt_rec_numeral
thf(fact_1129_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_1130_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_1131_dropWhile__nth,axiom,
! [J2: nat,P: a > $o,Xs: list_a] :
( ( ord_less_nat @ J2 @ ( size_size_list_a @ ( dropWhile_a @ P @ Xs ) ) )
=> ( ( nth_a @ ( dropWhile_a @ P @ Xs ) @ J2 )
= ( nth_a @ Xs @ ( plus_plus_nat @ J2 @ ( size_size_list_a @ ( takeWhile_a @ P @ Xs ) ) ) ) ) ) ).
% dropWhile_nth
thf(fact_1132_dropWhile__nth,axiom,
! [J2: nat,P: nat > $o,Xs: list_nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_nat @ ( dropWhile_nat @ P @ Xs ) ) )
=> ( ( nth_nat @ ( dropWhile_nat @ P @ Xs ) @ J2 )
= ( nth_nat @ Xs @ ( plus_plus_nat @ J2 @ ( size_size_list_nat @ ( takeWhile_nat @ P @ Xs ) ) ) ) ) ) ).
% dropWhile_nth
thf(fact_1133_takeWhile__replicate,axiom,
! [P: a > $o,X2: a,N: nat] :
( ( ( P @ X2 )
=> ( ( takeWhile_a @ P @ ( replicate_a @ N @ X2 ) )
= ( replicate_a @ N @ X2 ) ) )
& ( ~ ( P @ X2 )
=> ( ( takeWhile_a @ P @ ( replicate_a @ N @ X2 ) )
= nil_a ) ) ) ).
% takeWhile_replicate
thf(fact_1134_takeWhile__replicate,axiom,
! [P: nat > $o,X2: nat,N: nat] :
( ( ( P @ X2 )
=> ( ( takeWhile_nat @ P @ ( replicate_nat @ N @ X2 ) )
= ( replicate_nat @ N @ X2 ) ) )
& ( ~ ( P @ X2 )
=> ( ( takeWhile_nat @ P @ ( replicate_nat @ N @ X2 ) )
= nil_nat ) ) ) ).
% takeWhile_replicate
thf(fact_1135_takeWhile__dropWhile__id,axiom,
! [P: a > $o,Xs: list_a] :
( ( append_a @ ( takeWhile_a @ P @ Xs ) @ ( dropWhile_a @ P @ Xs ) )
= Xs ) ).
% takeWhile_dropWhile_id
thf(fact_1136_takeWhile__dropWhile__id,axiom,
! [P: nat > $o,Xs: list_nat] :
( ( append_nat @ ( takeWhile_nat @ P @ Xs ) @ ( dropWhile_nat @ P @ Xs ) )
= Xs ) ).
% takeWhile_dropWhile_id
thf(fact_1137_takeWhile_Osimps_I2_J,axiom,
! [P: a > $o,X2: a,Xs: list_a] :
( ( ( P @ X2 )
=> ( ( takeWhile_a @ P @ ( cons_a @ X2 @ Xs ) )
= ( cons_a @ X2 @ ( takeWhile_a @ P @ Xs ) ) ) )
& ( ~ ( P @ X2 )
=> ( ( takeWhile_a @ P @ ( cons_a @ X2 @ Xs ) )
= nil_a ) ) ) ).
% takeWhile.simps(2)
thf(fact_1138_takeWhile_Osimps_I2_J,axiom,
! [P: nat > $o,X2: nat,Xs: list_nat] :
( ( ( P @ X2 )
=> ( ( takeWhile_nat @ P @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ X2 @ ( takeWhile_nat @ P @ Xs ) ) ) )
& ( ~ ( P @ X2 )
=> ( ( takeWhile_nat @ P @ ( cons_nat @ X2 @ Xs ) )
= nil_nat ) ) ) ).
% takeWhile.simps(2)
thf(fact_1139_takeWhile__tail,axiom,
! [P: a > $o,X2: a,Xs: list_a,L2: list_a] :
( ~ ( P @ X2 )
=> ( ( takeWhile_a @ P @ ( append_a @ Xs @ ( cons_a @ X2 @ L2 ) ) )
= ( takeWhile_a @ P @ Xs ) ) ) ).
% takeWhile_tail
thf(fact_1140_takeWhile__tail,axiom,
! [P: nat > $o,X2: nat,Xs: list_nat,L2: list_nat] :
( ~ ( P @ X2 )
=> ( ( takeWhile_nat @ P @ ( append_nat @ Xs @ ( cons_nat @ X2 @ L2 ) ) )
= ( takeWhile_nat @ P @ Xs ) ) ) ).
% takeWhile_tail
thf(fact_1141_length__takeWhile__le,axiom,
! [P: a > $o,Xs: list_a] : ( ord_less_eq_nat @ ( size_size_list_a @ ( takeWhile_a @ P @ Xs ) ) @ ( size_size_list_a @ Xs ) ) ).
% length_takeWhile_le
thf(fact_1142_length__takeWhile__le,axiom,
! [P: nat > $o,Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( takeWhile_nat @ P @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% length_takeWhile_le
thf(fact_1143_takeWhile_Osimps_I1_J,axiom,
! [P: a > $o] :
( ( takeWhile_a @ P @ nil_a )
= nil_a ) ).
% takeWhile.simps(1)
thf(fact_1144_takeWhile_Osimps_I1_J,axiom,
! [P: nat > $o] :
( ( takeWhile_nat @ P @ nil_nat )
= nil_nat ) ).
% takeWhile.simps(1)
thf(fact_1145_takeWhile__eq__take,axiom,
( takeWhile_a
= ( ^ [P2: a > $o,Xs4: list_a] : ( take_a @ ( size_size_list_a @ ( takeWhile_a @ P2 @ Xs4 ) ) @ Xs4 ) ) ) ).
% takeWhile_eq_take
thf(fact_1146_takeWhile__eq__take,axiom,
( takeWhile_nat
= ( ^ [P2: nat > $o,Xs4: list_nat] : ( take_nat @ ( size_size_list_nat @ ( takeWhile_nat @ P2 @ Xs4 ) ) @ Xs4 ) ) ) ).
% takeWhile_eq_take
thf(fact_1147_takeWhile__eq__Nil__iff,axiom,
! [P: a > $o,Xs: list_a] :
( ( ( takeWhile_a @ P @ Xs )
= nil_a )
= ( ( Xs = nil_a )
| ~ ( P @ ( hd_a @ Xs ) ) ) ) ).
% takeWhile_eq_Nil_iff
thf(fact_1148_takeWhile__eq__Nil__iff,axiom,
! [P: nat > $o,Xs: list_nat] :
( ( ( takeWhile_nat @ P @ Xs )
= nil_nat )
= ( ( Xs = nil_nat )
| ~ ( P @ ( hd_nat @ Xs ) ) ) ) ).
% takeWhile_eq_Nil_iff
thf(fact_1149_nth__length__takeWhile,axiom,
! [P: a > $o,Xs: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ ( takeWhile_a @ P @ Xs ) ) @ ( size_size_list_a @ Xs ) )
=> ~ ( P @ ( nth_a @ Xs @ ( size_size_list_a @ ( takeWhile_a @ P @ Xs ) ) ) ) ) ).
% nth_length_takeWhile
thf(fact_1150_nth__length__takeWhile,axiom,
! [P: nat > $o,Xs: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ ( takeWhile_nat @ P @ Xs ) ) @ ( size_size_list_nat @ Xs ) )
=> ~ ( P @ ( nth_nat @ Xs @ ( size_size_list_nat @ ( takeWhile_nat @ P @ Xs ) ) ) ) ) ).
% nth_length_takeWhile
thf(fact_1151_takeWhile__nth,axiom,
! [J2: nat,P: a > $o,Xs: list_a] :
( ( ord_less_nat @ J2 @ ( size_size_list_a @ ( takeWhile_a @ P @ Xs ) ) )
=> ( ( nth_a @ ( takeWhile_a @ P @ Xs ) @ J2 )
= ( nth_a @ Xs @ J2 ) ) ) ).
% takeWhile_nth
thf(fact_1152_takeWhile__nth,axiom,
! [J2: nat,P: nat > $o,Xs: list_nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_nat @ ( takeWhile_nat @ P @ Xs ) ) )
=> ( ( nth_nat @ ( takeWhile_nat @ P @ Xs ) @ J2 )
= ( nth_nat @ Xs @ J2 ) ) ) ).
% takeWhile_nth
thf(fact_1153_dropWhile__eq__Cons__conv,axiom,
! [P: a > $o,Xs: list_a,Y2: a,Ys: list_a] :
( ( ( dropWhile_a @ P @ Xs )
= ( cons_a @ Y2 @ Ys ) )
= ( ( Xs
= ( append_a @ ( takeWhile_a @ P @ Xs ) @ ( cons_a @ Y2 @ Ys ) ) )
& ~ ( P @ Y2 ) ) ) ).
% dropWhile_eq_Cons_conv
thf(fact_1154_dropWhile__eq__Cons__conv,axiom,
! [P: nat > $o,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( ( dropWhile_nat @ P @ Xs )
= ( cons_nat @ Y2 @ Ys ) )
= ( ( Xs
= ( append_nat @ ( takeWhile_nat @ P @ Xs ) @ ( cons_nat @ Y2 @ Ys ) ) )
& ~ ( P @ Y2 ) ) ) ).
% dropWhile_eq_Cons_conv
thf(fact_1155_dropWhile__eq__drop,axiom,
( dropWhile_a
= ( ^ [P2: a > $o,Xs4: list_a] : ( drop_a @ ( size_size_list_a @ ( takeWhile_a @ P2 @ Xs4 ) ) @ Xs4 ) ) ) ).
% dropWhile_eq_drop
thf(fact_1156_dropWhile__eq__drop,axiom,
( dropWhile_nat
= ( ^ [P2: nat > $o,Xs4: list_nat] : ( drop_nat @ ( size_size_list_nat @ ( takeWhile_nat @ P2 @ Xs4 ) ) @ Xs4 ) ) ) ).
% dropWhile_eq_drop
thf(fact_1157_length__takeWhile__less__P__nth,axiom,
! [J2: nat,P: a > $o,Xs: list_a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( P @ ( nth_a @ Xs @ I2 ) ) )
=> ( ( ord_less_eq_nat @ J2 @ ( size_size_list_a @ Xs ) )
=> ( ord_less_eq_nat @ J2 @ ( size_size_list_a @ ( takeWhile_a @ P @ Xs ) ) ) ) ) ).
% length_takeWhile_less_P_nth
thf(fact_1158_length__takeWhile__less__P__nth,axiom,
! [J2: nat,P: nat > $o,Xs: list_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( P @ ( nth_nat @ Xs @ I2 ) ) )
=> ( ( ord_less_eq_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ J2 @ ( size_size_list_nat @ ( takeWhile_nat @ P @ Xs ) ) ) ) ) ).
% length_takeWhile_less_P_nth
thf(fact_1159_takeWhile__eq__take__P__nth,axiom,
! [N: nat,Xs: list_a,P: a > $o] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( P @ ( nth_a @ Xs @ I2 ) ) ) )
=> ( ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ~ ( P @ ( nth_a @ Xs @ N ) ) )
=> ( ( takeWhile_a @ P @ Xs )
= ( take_a @ N @ Xs ) ) ) ) ).
% takeWhile_eq_take_P_nth
thf(fact_1160_takeWhile__eq__take__P__nth,axiom,
! [N: nat,Xs: list_nat,P: nat > $o] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I2 ) ) ) )
=> ( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ~ ( P @ ( nth_nat @ Xs @ N ) ) )
=> ( ( takeWhile_nat @ P @ Xs )
= ( take_nat @ N @ Xs ) ) ) ) ).
% takeWhile_eq_take_P_nth
thf(fact_1161_mod__by__Suc__0,axiom,
! [M: nat] :
( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% mod_by_Suc_0
thf(fact_1162_mod__Suc__le__divisor,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).
% mod_Suc_le_divisor
thf(fact_1163_mod__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% mod_Suc_eq
thf(fact_1164_mod__Suc__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% mod_Suc_Suc_eq
thf(fact_1165_mod__Suc,axiom,
! [M: nat,N: nat] :
( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
= N )
=> ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
= zero_zero_nat ) )
& ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
!= N )
=> ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
= ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).
% mod_Suc
thf(fact_1166_mod__induct,axiom,
! [P: nat > $o,N: nat,P3: nat,M: nat] :
( ( P @ N )
=> ( ( ord_less_nat @ N @ P3 )
=> ( ( ord_less_nat @ M @ P3 )
=> ( ! [N2: nat] :
( ( ord_less_nat @ N2 @ P3 )
=> ( ( P @ N2 )
=> ( P @ ( modulo_modulo_nat @ ( suc @ N2 ) @ P3 ) ) ) )
=> ( P @ M ) ) ) ) ) ).
% mod_induct
thf(fact_1167_rotate__drop__take,axiom,
( rotate_a
= ( ^ [N3: nat,Xs4: list_a] : ( append_a @ ( drop_a @ ( modulo_modulo_nat @ N3 @ ( size_size_list_a @ Xs4 ) ) @ Xs4 ) @ ( take_a @ ( modulo_modulo_nat @ N3 @ ( size_size_list_a @ Xs4 ) ) @ Xs4 ) ) ) ) ).
% rotate_drop_take
thf(fact_1168_rotate__drop__take,axiom,
( rotate_nat
= ( ^ [N3: nat,Xs4: list_nat] : ( append_nat @ ( drop_nat @ ( modulo_modulo_nat @ N3 @ ( size_size_list_nat @ Xs4 ) ) @ Xs4 ) @ ( take_nat @ ( modulo_modulo_nat @ N3 @ ( size_size_list_nat @ Xs4 ) ) @ Xs4 ) ) ) ) ).
% rotate_drop_take
thf(fact_1169_hd__rotate__conv__nth,axiom,
! [Xs: list_a,N: nat] :
( ( Xs != nil_a )
=> ( ( hd_a @ ( rotate_a @ N @ Xs ) )
= ( nth_a @ Xs @ ( modulo_modulo_nat @ N @ ( size_size_list_a @ Xs ) ) ) ) ) ).
% hd_rotate_conv_nth
thf(fact_1170_hd__rotate__conv__nth,axiom,
! [Xs: list_nat,N: nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ ( rotate_nat @ N @ Xs ) )
= ( nth_nat @ Xs @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% hd_rotate_conv_nth
thf(fact_1171_rotate__is__Nil__conv,axiom,
! [N: nat,Xs: list_a] :
( ( ( rotate_a @ N @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rotate_is_Nil_conv
thf(fact_1172_rotate__is__Nil__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( rotate_nat @ N @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% rotate_is_Nil_conv
thf(fact_1173_length__rotate,axiom,
! [N: nat,Xs: list_a] :
( ( size_size_list_a @ ( rotate_a @ N @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_rotate
thf(fact_1174_length__rotate,axiom,
! [N: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( rotate_nat @ N @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate
thf(fact_1175_rotate__length01,axiom,
! [Xs: list_a,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ one_one_nat )
=> ( ( rotate_a @ N @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_1176_rotate__length01,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( ( rotate_nat @ N @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_1177_rotate__id,axiom,
! [N: nat,Xs: list_a] :
( ( ( modulo_modulo_nat @ N @ ( size_size_list_a @ Xs ) )
= zero_zero_nat )
=> ( ( rotate_a @ N @ Xs )
= Xs ) ) ).
% rotate_id
thf(fact_1178_rotate__id,axiom,
! [N: nat,Xs: list_nat] :
( ( ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Xs ) )
= zero_zero_nat )
=> ( ( rotate_nat @ N @ Xs )
= Xs ) ) ).
% rotate_id
thf(fact_1179_rotate__map,axiom,
! [N: nat,F: nat > nat,Xs: list_nat] :
( ( rotate_nat @ N @ ( map_nat_nat @ F @ Xs ) )
= ( map_nat_nat @ F @ ( rotate_nat @ N @ Xs ) ) ) ).
% rotate_map
thf(fact_1180_rotate__append,axiom,
! [L2: list_a,Q3: list_a] :
( ( rotate_a @ ( size_size_list_a @ L2 ) @ ( append_a @ L2 @ Q3 ) )
= ( append_a @ Q3 @ L2 ) ) ).
% rotate_append
thf(fact_1181_rotate__append,axiom,
! [L2: list_nat,Q3: list_nat] :
( ( rotate_nat @ ( size_size_list_nat @ L2 ) @ ( append_nat @ L2 @ Q3 ) )
= ( append_nat @ Q3 @ L2 ) ) ).
% rotate_append
thf(fact_1182_rotate__conv__mod,axiom,
( rotate_a
= ( ^ [N3: nat,Xs4: list_a] : ( rotate_a @ ( modulo_modulo_nat @ N3 @ ( size_size_list_a @ Xs4 ) ) @ Xs4 ) ) ) ).
% rotate_conv_mod
thf(fact_1183_rotate__conv__mod,axiom,
( rotate_nat
= ( ^ [N3: nat,Xs4: list_nat] : ( rotate_nat @ ( modulo_modulo_nat @ N3 @ ( size_size_list_nat @ Xs4 ) ) @ Xs4 ) ) ) ).
% rotate_conv_mod
thf(fact_1184_nth__rotate,axiom,
! [N: nat,Xs: list_a,M: nat] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( rotate_a @ M @ Xs ) @ N )
= ( nth_a @ Xs @ ( modulo_modulo_nat @ ( plus_plus_nat @ M @ N ) @ ( size_size_list_a @ Xs ) ) ) ) ) ).
% nth_rotate
thf(fact_1185_nth__rotate,axiom,
! [N: nat,Xs: list_nat,M: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rotate_nat @ M @ Xs ) @ N )
= ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( plus_plus_nat @ M @ N ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% nth_rotate
thf(fact_1186_Suc__times__mod__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
= one_one_nat ) ) ).
% Suc_times_mod_eq
thf(fact_1187_Suc__times__numeral__mod__eq,axiom,
! [K2: num,N: nat] :
( ( ( numeral_numeral_nat @ K2 )
!= one_one_nat )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K2 ) @ N ) ) @ ( numeral_numeral_nat @ K2 ) )
= one_one_nat ) ) ).
% Suc_times_numeral_mod_eq
thf(fact_1188_mult__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ( times_times_nat @ M @ K2 )
= ( times_times_nat @ N @ K2 ) )
= ( ( M = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1189_mult__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K2 @ M )
= ( times_times_nat @ K2 @ N ) )
= ( ( M = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1190_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1191_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1192_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1193_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1194_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1195_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1196_mult__less__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1197_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1198_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_1199_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1200_mult__le__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1201_Suc__mod__mult__self4,axiom,
! [N: nat,K2: nat,M: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K2 ) @ M ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self4
thf(fact_1202_Suc__mod__mult__self3,axiom,
! [K2: nat,N: nat,M: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K2 @ N ) @ M ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self3
thf(fact_1203_Suc__mod__mult__self2,axiom,
! [M: nat,N: nat,K2: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K2 ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self2
thf(fact_1204_Suc__mod__mult__self1,axiom,
! [M: nat,K2: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K2 @ N ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self1
thf(fact_1205_mult__le__mono2,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K2 @ I3 ) @ ( times_times_nat @ K2 @ J2 ) ) ) ).
% mult_le_mono2
thf(fact_1206_mult__le__mono1,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K2 ) @ ( times_times_nat @ J2 @ K2 ) ) ) ).
% mult_le_mono1
thf(fact_1207_mult__le__mono,axiom,
! [I3: nat,J2: nat,K2: nat,L2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ K2 @ L2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K2 ) @ ( times_times_nat @ J2 @ L2 ) ) ) ) ).
% mult_le_mono
thf(fact_1208_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1209_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1210_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1211_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1212_diff__mult__distrib2,axiom,
! [K2: nat,M: nat,N: nat] :
( ( times_times_nat @ K2 @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1213_diff__mult__distrib,axiom,
! [M: nat,N: nat,K2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K2 )
= ( minus_minus_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).
% diff_mult_distrib
thf(fact_1214_add__mult__distrib2,axiom,
! [K2: nat,M: nat,N: nat] :
( ( times_times_nat @ K2 @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1215_add__mult__distrib,axiom,
! [M: nat,N: nat,K2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K2 )
= ( plus_plus_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).
% add_mult_distrib
thf(fact_1216_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1217_Suc__mult__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K2 ) @ M )
= ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_1218_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1219_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_1220_Suc__mult__le__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K2 ) @ M ) @ ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1221_mult__less__mono2,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ K2 @ I3 ) @ ( times_times_nat @ K2 @ J2 ) ) ) ) ).
% mult_less_mono2
thf(fact_1222_mult__less__mono1,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ I3 @ K2 ) @ ( times_times_nat @ J2 @ K2 ) ) ) ) ).
% mult_less_mono1
thf(fact_1223_Suc__mult__less__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K2 ) @ M ) @ ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1224_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_1225_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1226_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1227_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1228_lambda__one,axiom,
( ( ^ [X: nat] : X )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_1229_lambda__zero,axiom,
( ( ^ [H: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_1230_length__product,axiom,
! [Xs: list_a,Ys: list_a] :
( ( size_s3885678630836030617od_a_a @ ( product_a_a @ Xs @ Ys ) )
= ( times_times_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) ) ) ).
% length_product
thf(fact_1231_length__product,axiom,
! [Xs: list_a,Ys: list_nat] :
( ( size_s984997627204368545_a_nat @ ( product_a_nat @ Xs @ Ys ) )
= ( times_times_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_product
thf(fact_1232_length__product,axiom,
! [Xs: list_nat,Ys: list_a] :
( ( size_s243904063682394823_nat_a @ ( product_nat_a @ Xs @ Ys ) )
= ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_a @ Ys ) ) ) ).
% length_product
thf(fact_1233_length__product,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( size_s5460976970255530739at_nat @ ( product_nat_nat @ Xs @ Ys ) )
= ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_product
thf(fact_1234_set__swap,axiom,
! [I3: nat,Xs: list_a,J2: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_a @ Xs ) )
=> ( ( set_a2 @ ( list_update_a @ ( list_update_a @ Xs @ I3 @ ( nth_a @ Xs @ J2 ) ) @ J2 @ ( nth_a @ Xs @ I3 ) ) )
= ( set_a2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_1235_set__swap,axiom,
! [I3: nat,Xs: list_nat,J2: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I3 @ ( nth_nat @ Xs @ J2 ) ) @ J2 @ ( nth_nat @ Xs @ I3 ) ) )
= ( set_nat2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_1236_map__eq__conv,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Xs ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_1237_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_a] :
( ( ( concat_a @ Xss2 )
= nil_a )
= ( ! [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xss2 ) )
=> ( X = nil_a ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_1238_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_nat] :
( ( ( concat_nat @ Xss2 )
= nil_nat )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xss2 ) )
=> ( X = nil_nat ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_1239_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_a] :
( ( nil_a
= ( concat_a @ Xss2 ) )
= ( ! [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xss2 ) )
=> ( X = nil_a ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_1240_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_nat] :
( ( nil_nat
= ( concat_nat @ Xss2 ) )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xss2 ) )
=> ( X = nil_nat ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_1241_in__set__insert,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_1242_set__filter,axiom,
! [P: nat > $o,Xs: list_nat] :
( ( set_nat2 @ ( filter_nat @ P @ Xs ) )
= ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
& ( P @ X ) ) ) ) ).
% set_filter
thf(fact_1243_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_1244_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_1245_in__set__replicate,axiom,
! [X2: nat,N: nat,Y2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ ( replicate_nat @ N @ Y2 ) ) )
= ( ( X2 = Y2 )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_1246_dropWhile__eq__Nil__conv,axiom,
! [P: a > $o,Xs: list_a] :
( ( ( dropWhile_a @ P @ Xs )
= nil_a )
= ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( P @ X ) ) ) ) ).
% dropWhile_eq_Nil_conv
thf(fact_1247_dropWhile__eq__Nil__conv,axiom,
! [P: nat > $o,Xs: list_nat] :
( ( ( dropWhile_nat @ P @ Xs )
= nil_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( P @ X ) ) ) ) ).
% dropWhile_eq_Nil_conv
thf(fact_1248_takeWhile__append2,axiom,
! [Xs: list_a,P: a > $o,Ys: list_a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( P @ X3 ) )
=> ( ( takeWhile_a @ P @ ( append_a @ Xs @ Ys ) )
= ( append_a @ Xs @ ( takeWhile_a @ P @ Ys ) ) ) ) ).
% takeWhile_append2
thf(fact_1249_takeWhile__append2,axiom,
! [Xs: list_nat,P: nat > $o,Ys: list_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( P @ X3 ) )
=> ( ( takeWhile_nat @ P @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ Xs @ ( takeWhile_nat @ P @ Ys ) ) ) ) ).
% takeWhile_append2
thf(fact_1250_takeWhile__append1,axiom,
! [X2: a,Xs: list_a,P: a > $o,Ys: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ~ ( P @ X2 )
=> ( ( takeWhile_a @ P @ ( append_a @ Xs @ Ys ) )
= ( takeWhile_a @ P @ Xs ) ) ) ) ).
% takeWhile_append1
thf(fact_1251_takeWhile__append1,axiom,
! [X2: nat,Xs: list_nat,P: nat > $o,Ys: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ~ ( P @ X2 )
=> ( ( takeWhile_nat @ P @ ( append_nat @ Xs @ Ys ) )
= ( takeWhile_nat @ P @ Xs ) ) ) ) ).
% takeWhile_append1
thf(fact_1252_dropWhile__append1,axiom,
! [X2: a,Xs: list_a,P: a > $o,Ys: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ~ ( P @ X2 )
=> ( ( dropWhile_a @ P @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( dropWhile_a @ P @ Xs ) @ Ys ) ) ) ) ).
% dropWhile_append1
thf(fact_1253_dropWhile__append1,axiom,
! [X2: nat,Xs: list_nat,P: nat > $o,Ys: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ~ ( P @ X2 )
=> ( ( dropWhile_nat @ P @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( dropWhile_nat @ P @ Xs ) @ Ys ) ) ) ) ).
% dropWhile_append1
thf(fact_1254_dropWhile__append2,axiom,
! [Xs: list_nat,P: nat > $o,Ys: list_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( P @ X3 ) )
=> ( ( dropWhile_nat @ P @ ( append_nat @ Xs @ Ys ) )
= ( dropWhile_nat @ P @ Ys ) ) ) ).
% dropWhile_append2
thf(fact_1255_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_eq_nat @ I4 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_1256_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_1257_card__less__Suc2,axiom,
! [M7: set_nat,I3: nat] :
( ~ ( member_nat @ zero_zero_nat @ M7 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat @ ( suc @ K ) @ M7 )
& ( ord_less_nat @ K @ I3 ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat @ K @ M7 )
& ( ord_less_nat @ K @ ( suc @ I3 ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_1258_card__less__Suc,axiom,
! [M7: set_nat,I3: nat] :
( ( member_nat @ zero_zero_nat @ M7 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat @ ( suc @ K ) @ M7 )
& ( ord_less_nat @ K @ I3 ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat @ K @ M7 )
& ( ord_less_nat @ K @ ( suc @ I3 ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_1259_card__less,axiom,
! [M7: set_nat,I3: nat] :
( ( member_nat @ zero_zero_nat @ M7 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat @ K @ M7 )
& ( ord_less_nat @ K @ ( suc @ I3 ) ) ) ) )
!= zero_zero_nat ) ) ).
% card_less
thf(fact_1260_split__div_H,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
& ( P @ zero_zero_nat ) )
| ? [Q4: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M )
& ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
& ( P @ Q4 ) ) ) ) ).
% split_div'
thf(fact_1261_le__div__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ M @ N )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% le_div_geq
thf(fact_1262_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_1263_div__Suc,axiom,
! [M: nat,N: nat] :
( ( ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
= zero_zero_nat )
=> ( ( divide_divide_nat @ ( suc @ M ) @ N )
= ( suc @ ( divide_divide_nat @ M @ N ) ) ) )
& ( ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
!= zero_zero_nat )
=> ( ( divide_divide_nat @ ( suc @ M ) @ N )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% div_Suc
thf(fact_1264_Suc__div__le__mono,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_1265_div__if,axiom,
( divide_divide_nat
= ( ^ [M3: nat,N3: nat] :
( if_nat
@ ( ( ord_less_nat @ M3 @ N3 )
| ( N3 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M3 @ N3 ) @ N3 ) ) ) ) ) ).
% div_if
thf(fact_1266_div__nat__eqI,axiom,
! [N: nat,Q3: nat,M: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q3 ) @ M )
=> ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q3 ) ) )
=> ( ( divide_divide_nat @ M @ N )
= Q3 ) ) ) ).
% div_nat_eqI
thf(fact_1267_encode__bounded__nat_Ocases,axiom,
! [X2: product_prod_nat_nat] :
( ! [L: nat,N2: nat] :
( X2
!= ( product_Pair_nat_nat @ ( suc @ L ) @ N2 ) )
=> ~ ! [Uu2: nat] :
( X2
!= ( product_Pair_nat_nat @ zero_zero_nat @ Uu2 ) ) ) ).
% encode_bounded_nat.cases
thf(fact_1268_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K: nat,M3: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M3 @ K ) @ ( product_Pair_nat_nat @ M3 @ ( minus_minus_nat @ K @ M3 ) ) @ ( nat_prod_decode_aux @ ( suc @ K ) @ ( minus_minus_nat @ M3 @ ( suc @ K ) ) ) ) ) ) ).
% prod_decode_aux.simps
% Helper facts (9)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X2: list_a,Y2: list_a] :
( ( if_list_a @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X2: list_a,Y2: list_a] :
( ( if_list_a @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( if_list_nat @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( if_list_nat @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $true @ X2 @ Y2 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
x2 = y2 ).
%------------------------------------------------------------------------------